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# Precision Measurements of the Cluster Red Sequence using an Error Corrected
Gaussian Mixture Model
Jiangang Hao11affiliation: Center for Particle Astrophysics, Fermi National
Accelerator Laboratory, Batavia, IL 60510 22affiliation: Department of
Physics, University of Michigan, Ann Arbor, MI 48109 , Benjamin P.
Koester33affiliation: Department of Astronomy and Astrophysics, The University
of Chicago, Chicago, IL60637 , Timothy A. Mckay 22affiliation: Department of
Physics, University of Michigan, Ann Arbor, MI 48109 44affiliation: Department
of Astronomy, University of Michigan, Ann Arbor, MI48109 , Eli S.
Rykoff55affiliation: TABASGO Fellow, Physics Department, University of
California at Santa Barbara, 2233B Broida Hall, Santa Barbara, CA 93106 ,
Eduardo Rozo66affiliation: Center for Cosmology and Astro-Particle Physics
(CCAPP), The Ohio State University, Columbus, OH 43210 , August
Evrard22affiliation: Department of Physics, University of Michigan, Ann Arbor,
MI 48109 44affiliation: Department of Astronomy, University of Michigan, Ann
Arbor, MI48109 , James Annis11affiliation: Center for Particle Astrophysics,
Fermi National Accelerator Laboratory, Batavia, IL 60510 , Matthew
Becker77affiliation: Department of Physics, The University of Chicago,
Chicago, IL 60637 , Michael Busha88affiliation: Kavli Institute for Particle
Astrophysics & Cosmology, Physics Department, and Stanford Linear Accelerator
Center, Stanford University, Stanford, CA 94305 , David Gerdes22affiliation:
Department of Physics, University of Michigan, Ann Arbor, MI 48109 , David E.
Johnston99affiliation: Department Physics & Astronomy, Northwestern
University, Evanston, IL 60208 , Erin Sheldon1010affiliation: Brookhaven
National Laboratory, Upton, New York 11973, USA , Risa H.
Wechsler88affiliation: Kavli Institute for Particle Astrophysics & Cosmology,
Physics Department, and Stanford Linear Accelerator Center, Stanford
University, Stanford, CA 94305
###### Abstract
The red sequence is an important feature of galaxy clusters and plays a
crucial role in optical cluster detection. Measurement of the slope and
scatter of the red sequence are affected both by selection of red sequence
galaxies and measurement errors. In this paper, we describe a new error
corrected Gaussian Mixture Model for red sequence galaxy identification. Using
this technique, we can remove the effects of measurement error and extract
unbiased information about the intrinsic properties of the red sequence. We
use this method to select red sequence galaxies in each of the 13,823 clusters
in the maxBCG catalog, and measure the red sequence ridgeline location and
scatter of each. These measurements provide precise constraints on the
variation of the average red galaxy populations in the observed frame with
redshift. We find that the scatter of the red sequence ridgeline increase
mildly with redshift, and that the slope decreases with redshift. We also
observe that the slope does not strongly depend on cluster richness. Using
similar methods, we show that this behavior is mirrored in a spectroscopic
sample of field galaxies, further emphasizing that ridgeline properties are
independent of environment. These precise measurements serve as an important
observational check on simulations and mock galaxy catalogs. The observed
trends in the slope and scatter of the red sequence ridgeline with redshift
are clues to possible intrinsic evolution of the cluster red-sequence itself.
Most importantly, the methods presented in this work lay the groundwork for
further improvements in optically-based cluster cosmology. The codes for ECGMM
can be accessed from: https://sites.google.com/site/jiangangecgmm/
Galaxies: clusters - Cosmology: observations - Methods: Data analysis,
Gaussian Mixture, Bootstrap
## 1 Introduction
Galaxy clusters are the largest gravitationally bound systems in our Universe,
whose masses, abundance and spatial distribution reflect the growth of
structure, composition, and expansion history of the Universe (Evrard, 1989;
Oukbir & Blanchard, 1992). The feasibility of constraining cosmological
parameters using galaxy clusters has been demonstrated by many authors
(Majumdar & Mohr, 2004; Hu, 2003; Lima & Hu, 2004, 2005) and realistic
constraints on cosmological parameters from optically selected galaxy clusters
have been implemented recently by Gladders et al. (2007) and Rozo et al.
(2009) on the RCS cluster catalog (Gladders & Yee, 2005) and maxBCG catalog
(Koester et al., 2007a, b) respectively.
The predominantly red, bright, passively evolving red sequence, or “E/S0
ridgeline” (Visvanathan & Sandage, 1977; Annis et al., 1999) found in the
cores of clusters of varied richness up to at least $z\sim 1.4$ (Bower et al.,
1992; Smail et al., 1998; van Dokkum et al., 1998; Barrientos, 1999; Blakeslee
et al., 2003; Mullis et al., 2005; Eisenhardt et al., 2005; De Lucia et al.,
2007) provide an efficient means for cluster detection, and have been an
integral part of modern cluster cosmology. The red sequence itself is
ubiquitous in the galaxy population (Renzini, 2006, e.g.), and in clusters red
sequence galaxies dominate the bright end of the cluster luminosity function
(Sandage et al., 1985; Barger et al., 1998). They are extremely tightly
clustered in color space, containing old populations of stars whose observed
color varies smoothly with redshift (e.g. Gladders & Yee, 2000). The
pervasiveness of this phenomenon in clusters enables efficient optical cluster
detection while a fortuitous color-redshift relation yields accurate
photometric redshifts (Gladders & Yee, 2000; Koester et al., 2007b). Simple
counting of photometrically identified cluster red sequence galaxies (Koester
et al., 2007a, e.g) has also been shown to be an effective proxy for cluster
mass (Becker et al., 2007; Sheldon et al., 2007; Johnston et al., 2007), with
more sophisticated applications yielding improvements in richness as a cluster
mass proxy (Rozo et al., 2008). In the era of precision measurements, the
extent to which the red sequence can be exploited for cluster cosmology
depends on how accurately its characteristics can be measured at a given
redshift.
In addition to its relevance to cluster cosmology, the red sequence plays an
important role in constraining the complex physical processes that drive
galaxy formation and evolution. At the field scale this includes measurements
of the red galaxy luminosity function (Wake et al., 2006; Faber et al., 2007),
the clustering of red galaxies in various environments (Zehavi et al., 2005;
Coil et al., 2008), and color-magnitude relations of spectroscopically and
morphologically identified early-type galaxies (Cool et al., 2006; Mei et al.,
2006; Stanford et al., 1998). The high density environments of clusters of
galaxies are dominated by red sequence galaxies; the red sequence portion of
the galaxy populations in the cores of rich clusters to at least $z\sim 1$
form the basis for various monolithic collapse scenarios (e.g. Bower et al.,
1992; Blakeslee et al., 2003; Mei et al., 2009). Faber et al. (2007) summarize
some of these results to fill out a picture of galaxy formation that includes
a mechanism for the formation of the red sequence.
In color-magnitude space, the red sequence is typically characterized by its
slope, zero point, and scatter. Many theoretical modeling have been proposed
to explain the red sequence (e.g. Arimoto & Yoshii, 1987; Kauffmann & Charlot,
1998). Various models posit that in the rest frame, the scatter in the red
sequence is driven primarily by age effects, its slope is a manifestation of
the mass-metallicity relation, and the zero point is set by combination of age
and mass-metallicity differences (e.g. Bernardi et al., 2005; De Lucia et al.,
2007; Faber et al., 2007).
Studies of the cluster red sequence have been accomplished by simply measuring
the photometric color-magnitude relation (e.g. López-Cruz et al., 2004; De
Lucia et al., 2007; Stott et al., 2009), supplemented with HST morphological
information (e.g Gladders et al., 1998) and spectroscopy at higher redshift
(Mei et al., 2006; Stott et al., 2009). Extra morphological and spectroscopic
data allow precise separation of E and S0-types from the rest of the galaxy
population, as well as refined identification of cluster members (Blakeslee et
al., 2003; Mei et al., 2009). The situation also benefits significantly from
precise color measurements afforded by deep, CCD-based imaging (e.g van Dokkum
et al., 1998). In the literature, the red sequence has been measured with
various levels of scrutiny in dozens of individual clusters.
In the past several years, researchers have turned to the considerable
resources of the the Sloan Digital Sky Survey (SDSS) and similar wide field
surveys to probe red sequence and elliptical galaxies in various environments
(Hogg et al., 2004; Bernardi et al., 2005, 2006; Cool et al., 2006). These
studies include both spectroscopically and morphologically identified red
galaxies at $z\sim 0.1$ that aim to constrain galaxy evolution scenarios for
the cosmologically-relevant luminous red galaxy (LRG) samples that extend to
$z\sim 0.6$ (e.g. Cool et al., 2006).
With the maxBCG cluster catalog, we are positioned to use the SDSS to make one
of the most statistically robust photometric measurements of the cluster red
sequence, using nearly 14,000 clusters between $0.1\leq z\leq 0.3$. In this
paper we focus on the slope and scatter of the red sequence. We clearly show
the systematic effects photometric errors have on the measurement of the
underlying slope and scatter of the red sequence, and introduce a method for
properly handling these effects. This method, based on an error-corrected
Gaussian Mixture Model (ECGMM), reliably recovers the properties of the
ridgeline by taking measurement errors into account. After presenting the
method, we describe its application to measurement of maxBCG clusters. Of
particular relevance to cluster cosmology are the observed mean, scatter, and
slope of the E/S0 ridgeline for all maxBCG clusters. These results are
presented, along with a discussion of observed trends with redshift.
## 2 Methods
### 2.1 Intrinsic properties of red sequence ridgeline
The existence of the red sequence ridgeline is evidence that cluster galaxies
are clustered in color space in addition to real space. The emission from
early-type galaxies is dominated by old stellar populations, which gives rise
to these remarkably similar galaxy colors. In addition, there is a close
mapping between galaxy color and redshift for these galaxies as a result of
the restframe 4000 Å break in their spectra. For the SDSS filter sets, the
4000 Å break is within the $g$ band as long as the redshift is below 0.35.
Therefore, the most informative ridgeline color for the maxBCG catalog is
$g-r$.
In the projected vicinity of a detected cluster, there are both cluster member
galaxies and field galaxies. Red sequence galaxies form a part of the member
population, whose colors are clustered tightly and can be approximated with a
Gaussian distribution with narrow width. On the other hand, the field
galaxies’ and blue member galaxies’ colors are not tightly clustered and can
be approximated by a Gaussian distribution with a broader width111There are
complicated situations where the distribution in color space is not simply
unimodal or bimodal, for example when two clusters are seen in projection. For
maxBCG catalog, it covers about 7,500 square degrees with about 14,000
clusters. This leads to about 2 clusters per square degree. Each cluster is
about the size of a few arcminutes across, so the chance of two or more
overlapped clusters is low. Therefore, a unimodal or bimodal distribution in
color space is a good approximation.. The problem of separating the ridgeline
from the field can be specified as following: What are the two Gaussian
components (one for the ridgeline and one for the field) that represent the
color distribution in the vicinity of a galaxy cluster? If this double
Gaussian is an adequate model for describing this color distribution, the one
dimensional Gausian Mixture Model (GMM) is well suited to the problem.
In the traditional applications of GMM (Titterington et al., 1985),
measurement errors are not considered. In our case, there are non-negligible
measurement errors associated with the galaxy colors. We are interested in
measuring the intrinsic color scatter of cluster members, absent contamination
by the increasing measurement errors of faint galaxies. Without accounting for
the increasing photometric errors, we expect that the color scatter will
increase as the measurement errors become larger. While the intrinsic color
scatter may increase as redshift increases (because the 4000 Å line break is
shifting toward $r$ band and making the $g-r$ color less discriminative.),
measurement errors may make us overestimate the increase in intrinsic scatter
with redshift. To avoid this problem, we include measurement error into our
likelihood function to remove the contamination. We will refer to this as the
error-corrected Gaussian Mixture Model and derive the corresponding
Expectation Maximization (EM) recursive relation in the following section.
It is clear that we can always improve the fit by adding more Gaussian
components, although this is clearly not good in the sense of parsimony. So,
we need to somehow decide on the number of Gaussian components by trading off
quality of fit against the number of introduced free parameters. To do so, we
use the Bayesian Information Criterion (BIC) (Schwarz, 1978; Connolly et al.,
2000) to determine how many mixtures we should use. The BIC is defined as:
$BIC=-2\log\mathcal{L}_{max}+k\log(M)$ (1)
Where k is the number of free parameters. For mixture models with different
number of components, we compare their corresponding BIC and select the model
with the smallest BIC.
### 2.2 Error-corrected Gaussian Mixture Model
In what follows, we describe how to fit a multi-component Gaussian mixture
model to a one dimensional distribution of data with both intrinsic scatter
and Gaussian measurement errors. Our method is an extension of the traditional
expectation maximization method for GMM (Dempster et al., 1977).
We assume the data are to be modeled by a mixture of $N$ Gaussians fit to the
distribution of $M$ data points. The subscript $i$ cycles through $N$ and $j$
cycles through $M$, and we use $\mu_{i}$, $\sigma_{i}$ and $w_{i}$ to denote
the location, width and weight of each Gaussian component. $y_{j}$ and
$\delta_{j}$ denote the data points and their measurement errors which are
assumed to be Gaussian. For brevity, we denote the parameters ($\mu_{i}$,
$\sigma_{i}$ and $w_{i}$) collectively by $\theta$. Then the likelihood of the
parameters given the data and measurement errors is:
$\mathcal{L}(\theta|y)=\prod_{j=1}^{M}\bigg{\\{}\sum_{i=1}^{N}\frac{w_{i}}{\sqrt{2\pi(\sigma_{i}^{2}+\delta_{j}^{2})}}\exp\bigg{[}-\frac{(y_{j}-\mu_{i})^{2}}{2(\sigma_{i}^{2}+\delta_{j}^{2})}\bigg{]}\bigg{\\}}$
(2)
The optimal parameters $\theta$ could be estimated by maximizing the above
likelihood function. The Expectation Maximization algorithm provides an
efficient way to get the maximum likelihood estimators in such a setting. To
utilize this, we need to introduce a hidden variable, $z_{j}$, which tells
which Gaussian component the data point $y_{j}$ is sampled from. In our case,
different from the standard EM prescription, we have non-negligible
measurement errors present. After some algebra, we arrive at a set of
recursive relations that lead to the maximum of the likelihood (see appendix
for details).
### 2.3 Bootstrapping to increase the robustness of ECGMM
Though the ECGMM is generally stable for estimating the parameters of the
Gaussian Components, it can fail occasionally due to very inappropriate choice
of initial parameters or some very big measurement errors of certain galaxies.
To make our measurement more robust, we introduced a bootstrap-like scheme.
Suppose we have M data points. We randomly pick one of the data points and
record it. We then repeat this process M times and get M recorded data points.
These M points form one resampling set of the original data set. Now, we apply
the ECGMM to this new data sample and measure the corresponding parameters.
After this, we start a second round, getting another resampling set with M
data points in it and measure the parameters using ECGMM again. We repeat this
process X times and have X estimates of each parameters. We throw away those
outlier estimates (estimates beyond the upper and lower inner fences222In
statistics, lower inner fence is defined by $Q_{1}-1.5IQR$ and higher inner
fence is $Q_{3}+1.5IQR$, where $Q_{1}$ and $Q_{3}$ are the first and third
quartiles respectively. The IQR is the the interquartile range, defined as
$Q_{3}-Q_{1}$.) for each parameter and use the mean of good estimates as the
value of each parameter. Using this scheme, our resulting parameter estimates
are much more robust, at a cost of a tolerable increase in computation time.
In this application, we took X to be 50.
### 2.4 Monte Carlo test of the ECGMM for our application
Before we delve into real data, we first conduct Monte Carlo tests to see
whether the ECGMM approach can reliably identify the cluster and background
Gaussian components. These tests are used to determine whether this method can
reliably recover the true parameters input in the simulation, and to see
whether the extracted parameters are generally unbiased with respect to
measurement errors.
Figure 1: Basic Monte Carlo tests of the ECGMM method for fitting mock galaxy
$g-r$ color distributions (see text). $\mu$ and $\sigma$ denote the locations
and widths of the corresponding Gaussian components, for clusters of
increasing richness. The true $\mu$ are 0 and 0.5 for BG and CL sets
respectively. The true $\sigma$ are 0.3 and 0.04 for BG and CL sets
respectively.
For this purpose, we generate two Gaussian random data sets, one representing
cluster member colors, denoted CL, and the other representing the field
galaxies/blue galaxies’ colors, denoted as BG. The CL set is generated from
$\sim N(0.5,0.04^{2})$ and BG set is generated from $\sim N(0,0.3^{2})$. To
represent clusters with different richness, we allow the normalization (also
denoted as $N_{gals}$ in the plots) of CL data set to vary as 10, 15, 20, 25,
30, 40, 50, 60 and 70 while keep the normalization of BG set as 30. All the
parameters used to generate the mock data are chosen to make the simulation as
close to the real data as possible. Then we combine CL and BG to create a mock
data set that mimics the colors of both cluster members and background
galaxies in a field. It is worth noting that these mock colors are error free
so far. Next, we will add some noise to them to mimic the measurements errors.
To do this, we first generate random numbers from a uniform distribution in
the range of [0, 0.1], which play the role of $\delta_{j}$ in Eq.2. Then, we
generate from $N(0,\delta_{j}^{2})$ and add them to the noise free data set to
produce a noise added mock color data set. In Fig.1, we plots the results from
the ECGMM fitting. The results show that for clusters with $N_{gals}\geq 10$,
the method can recover the locations ($\mu$) and widths ($\sigma$) of the
Gaussian components very well.
Figure 2: Monte Carlo test of the bias, ($E(\hat{\theta})-\theta$), of the
estimators for the location and width using GMM (bottom two panels) and ECGMM
(top two panels) as a function of richness for the cluster component of the
mock clusters (see text). The scales of the plots are chosen to be close to
the size of the true parameters in the plot to illustrate the fraction
precision.
Next we test for possible bias in the estimators. For each cluster richness
$N_{gals}$, we re-generate the data as well as errors 200 times and then apply
our methods to each to obtain estimates for the parameters. In each case, we
calculate the bias of parameters $\theta$ (the $\sigma$ and $\mu$ in our case)
defined as $E(\hat{\theta})-\theta$. In Fig.2, we plot the results from both
GMM and ECGMM for comparison. Clearly, the introduction of error correction(as
shown in the bottom two panels) is essential for removal of the bias of the
width resulting from measurement error(as shown in the top two panels).
## 3 Data
### 3.1 SDSS
Three main resources are included in this work: the SDSS galaxy catalog, the
maxBCG catalog, and a value-added SDSS spectroscopic catalog.
The maxBCG cluster sample and the galaxy catalogs used to remeasure cluster
richness in this paper are derived from the SDSS (Adelman-McCarthy et al.,
2006). The maxBCG cluster sample covers a sky area of about 7500 square
degrees. The camera design (Gunn et al., 2006) and drift-scan imaging strategy
of the SDSS enable acquisition of nearly simultaneous observations in the
$u,g,r,i,z$ filter system (Fukugita et al., 1996). Calibration (Hogg et al.,
2001; Smith et al., 2002; Tucker et al., 2006), astrometric (Pier et al.,
2003), and photometric (Lupton et al., 2001) pipelines reduce the data into
object catalogs containing a host of measured parameters for each object.
Galaxies are selected from SDSS object catalogs as described in (Sheldon et
al., 2007). In this work we use $\tt{CMODEL\\_COUNTS}$ as our total
magnitudes, and $\tt{MODEL\\_COUNTS}$ when computing colors. Bright stars,
survey edges and regions of poor seeing are masked as previously described
(Koester et al., 2007a; Sheldon et al., 2007).
The spectroscopic galaxy catalog is comprised of galaxies from the DR6 of SDSS
Value Added Galaxy Catalog. A detailed description about this catalog can be
found in Blanton et al. (2005, VAGC).
### 3.2 Cluster Sample
We obtain sky locations, redshift estimates, and initial richness values from
the maxBCG cluster catalog. Details of the selection algorithm and catalog
properties are published elsewhere (Koester et al., 2007b, a). In brief,
maxBCG selection relies on the observation that the galaxy population of rich
clusters is dominated by luminous, red galaxies clustered tightly in color
(the E/S0 ridgeline). Since these galaxies have old, passively evolving
stellar populations, their $g-r$ color closely reflects their redshift. The
brightest such red galaxy, typically located at the peak of the galaxy
density, defines the cluster center.
The maxBCG catalog is approximately volume limited in the redshift range
$0.1\leq z\leq 0.3$, with very accurate photometric redshifts ($\delta{}z\sim
0.01$). Studies of the maxBCG algorithm applied to mock SDSS catalogs indicate
that the completeness and purity are very high, above $90\%$ (Koester et al.,
2007a). The maxBCG catalog has been used to investigate the scaling of galaxy
velocity dispersion with cluster richness (Becker et al., 2007) and to derive
constraints on the power spectrum normalization, $\sigma_{8}$, from cluster
number counts (Rozo et al., 2009).
## 4 Measuring the ridgeline location and width of maxBCG clusters
We apply the above prescriptions of ECGMM to the maxBCG cluster catalog and
the galaxy catalog (Koester et al., 2007a), measuring the red sequence $g-r$
ridgeline. The procedures are as follows: for each cluster in maxBCG catalog,
we choose a scaled aperture $R_{200}^{lens}$ to ensure we are considering
equivalent regions of clusters of varied masses and therefore varied richness.
$R_{200}^{lens}$ is the critical radius, interior to which the mean mass
density of the cluster is 200 times of the critical energy density. Based on
the weak lensing analysis (Johnston et al., 2007; Hansen et al., 2007), the
scaling relation between $R_{200}^{lens}$ and the original maxBCG richness
$N_{200}$ is given by $R_{200}^{lens}=0.182(N_{200})^{0.42}$, which ranges
from 0.47 Mpc to 1.68 Mpc.
Next, we identify all SDSS galaxies inside this aperture range, fainter than
the BCG, and brighter than an i band magnitude corresponding to 0.4 L* at the
redshift of the cluster (Koester et al., 2007a). Then, we apply the ECGMM
procedure to the $g-r$ colors and corresponding measurement errors of these
galaxies. One of the resulting two Gaussian components from the ECGMM will
represent the cluster red sequence color distribution while the other
represents the background/blue galaxy color distribution. To determine which
Gaussian Component belongs to the cluster, we calculate the likelihood of the
BCG’s $g-r$ color on each Gaussian Component. The component for which the BCG
has a higher likelihood is assigned as the cluster component and the other is
declared background. By this way, each maxBCG cluster gets a new richness,
$N_{200}^{lens}$. It is worth noting that we apply the above measurements to
all maxBCG clusters whose original $N200\geq 10$. But we will only continue
our analysis on a subsample of the clusters whose new measured richness
$N_{200}^{lens}\geq 10$ and have two identified Gaussian mixture components in
order to guarantee the reliability of our measurements. After this selection,
we are left with about 7100 clusters and all our further analysis are based on
them. We need to point out that the clusters falling outside of this selection
are not necessarily bad clusters. They are just fall below the richness
threshold we imposed for quality control. In Fig.3 and Fig.4, we show the
ECGMM fitting of 9 big and 9 small clusters as described above. Their
corresponding CMRs are plotted in Fig.5 and Fig.6.
Figure 3: The ECGMM fitting to the galaxy color distribution around 9 rich
clusters. Note the fact that we corrected for measurement errors, the two
Gaussians appear to be narrower than the histogram. Figure 4: The ECGMM
fitting to the galaxy color distribution around 9 small clusters. Note that
when there are fewer galaxies histogram is no longer a good way to show the
distribution. Figure 5: The CMR around the 9 rich clusters in Fig.3. The red
diamonds and error bars are those from the selected members (red sequence) and
the blue dots and error bars are those from field galaxies. The red cross
symbol represent the BCG of that cluster. All the galaxies are within
$R^{lens}_{200}$ around the BCG, fainter than BCG but brighter than the
0.4L*(see text). The green line is a weighted least square fit (weighted by
the inverse square of color errors) to the cluster galaxies. Figure 6: The CMR
around the 9 small clusters in Fig.4. Figure 7: Tracking the ($g-r$) red
sequence zeropoint and width as a function of redshift, measured using
ordinary GMM (upper panels) and ECGMM (lower panels) respectively. We bin the
measured ridgeline color and width into redshift bin of 0.04 and then fit the
means with a straight line. After error correction, the broadening of the
observed red-sequence width with redshift is greatly suppressed, revealing the
effect of photometric errors on the observed broadening.
For comparison, we measure the red-sequence location and width using both
ordinary GMM and ECGMM. The top panel of Fig. 7 shows the evolution of the
average g-r ridgeline location and width measured using ordinary GMM. We
observe the well-known trend in the average ridgeline zeropoint, and there is
additional apparent strong evolution in the average ridgeline width, which
becomes nearly $140\%$ larger by $z=0.3$. However, from the lower two panels
which are measured using ECGMM, one can see very clearly the power of ECGMM in
constraining the intrinsic width of the ridgeline without contamination from
measurement error. The results show that the mean observed $g-r$ ridgeline
location retains the same linear dependence on redshift while the mean scatter
of the ridgeline shows a weak dependence on redshift, with the $g-r$ scatter
$\sigma(z=0.1)=0.051\pm 0.003$ and $\sigma(z=0.3)=0.079\pm 0.005$ or a
broadening by $\sim 55\%$ from $z=0.1$ to $z=0.3$. The strong dependence of
the scatter on redshift from the GMM is mostly due to the increased
measurement errors for cluster members at higher redshift.
## 5 The red sequence ridgeline slope
### 5.1 Ridgeline slope from galaxy clusters
It has been pointed out that the color-magnitude relation (CMR) of cluster
member galaxies has a negative slope (e.g Kodama & Arimoto, 1997; Gladders et
al., 1998), so that fainter member galaxes are generally bluer. The evolution
of these CMR slopes with respect to redshift and richness has been difficult
to address, largely due to the lack of a sufficiently large cluster catalog
with well measured photometry for all its galaxies. The maxBCG catalog
provides about 14,000 galaxy clusters, extending over $0.1\leq z\leq 0.3$,
which enables us to measure the slope of the CMR for clusters with good
statistics across a range in both richness and redshift.
Measurement of the slope of the CMR typically proceeds by identification of
the cluster red-sequence, followed by some iterative process of outlier
removal, and a determination of cluster “member” galaxies which are then used
to measure the slope and zeropoint of the CMR.333In Andreon (2006), a slightly
different method was introduced by directly modeling the CMR and measurement
errors into the likelihood function without separating the red sequence
galaxies first. We apply the method described in previous sections to measure
the color distribution of individual clusters and to assign the memberships
for every cluster by requiring the color difference between the member
galaxies and ridgeline within $\pm 2\sigma$ ($\sigma$ is the convolved
ridgeline width, given by the best-fit ECGMM, and the measurement errors of
individual member galaxy’s color). The richness of the cluster measured by
this way is denoted as $N_{200}^{lens}$. We choose $2\sigma$ because this is
roughly where the background component’s likelihood dominates over the cluster
component’s likelihood. Based on this identification of membership driven by
the ECGMM, we fit for the CMR of clusters galaxies with a straight line using
weighted least square fitting. The weights we used are the inverse square of
the measurement errors of $g-r$. We call the slope of the fitted line as the
slope fo the ridgeline.
Figure 8: The distributions of measured ridgeline slopes for clusters in steps
of 0.03 in redshift. $\mu$ and $\sigma$ denote the mean and width of the
distribution. The dashed line corresponds to zero. Figure 9: Tracking the
observed red-sequence slope vs redshift. The gray clouds represent the slope
measurements from individual clusters. The black solid circles and error bars
are weighted mean and the standard deviation to the weighted mean for each
redshift bin ($\Delta=0.03$). Note that the error bars in the plot are smaller
than the symbols. Figure 10: The evolution of mean ridgeline slope vs richness
at different redshift slices. The richness bins brackets are chosen as
$N_{200}^{lens}=$[10,20,30,40,60,80,161]. The light dark points are from
individual clusters. The black solid dots and error bars are weighted mean and
standard deviation of the slopes in each $N_{gals}$ bin for every redshift
slice. From the plot, we did not see strong trends of the slope evolution
w.r.t richness.
The distribution of ridgeline slopes for maxBCG clusters is shown in Fig.8 in
bins of $\Delta z=0.03$. Despite the substantial scatter in slope among
individual clusters, we can see from Fig.8 and Fig.9 that the mean slope of
the red sequence ridgelines for clusters deviates from zero for $0.1\leq z\leq
0.3$. For any bin, the error on the mean places the measurement many standard
deviations from zero.
In Fig.9, it is apparent that the observed trend of the mean ridgeline slope
with redshift is statistically significant: the slope becomes steeper by a
factor of 2.5 by $z=0.3$. In Fig.10, we plot the evolution of the slopes vs
richness in each redshift slice, which shows that the dependencies of the
ridgeline slope with respect to richness is weak, as shown elsewhere (e.g.
Hogg et al., 2004). Clearly, the observed slope of the red-sequence is not
associated with cluster richness, and is unsurprisingly a strong function of
redshift (see Discussion).
### 5.2 Ridgeline slope from spectroscopic data
The above measurement is based only on a photometric determination of red
sequence galaxies. The level to which projection plays into this selection is
as yet unknown. The true red-sequence galaxy population in some physical
volume, either in a cluster or in the field, is contaminated by dusty
foreground galaxies which can be rejected via spectroscopy, and by the
peculiar velocities of the galaxies themselves.
To address the possibility of foreground contamination, it is interesting to
see if the above results are preserved in a spectroscopic sample of galaxies.
To achieve this goal, we use galaxies with spectra from DR6 of SDSS Value
Added Galaxy Catalog (Blanton et al., 2005, VAGC) for a comparison. Due to the
selection effects of the spectroscopic data, we will choose only the galaxies
in redshift from 0.1 to 0.2 and brighter than 0.4 L* magnitude at their
respective redshifts. By extension from the photometric sample and from
previous work (Hogg et al., 2004), we know that the slope does not vary with
environment, so the field sample represented by our spectroscopy should be a
fair representation of the expected slope in clusters.
Our procedures are as follows: we first bin the galaxies into bins of size
$\Delta z=0.003$, which corresponds to velocity slices of 900km/sec. The color
distribution of the galaxies in each bin shows clear bimodality (top panel of
Fig. 11). Then, we separate the red sequence galaxies in each bin using ECGMM.
The red sequence galaxies correspond to the Gaussian component with bigger
$g-r$ value and we choose $\pm 2\sigma$ from the peak location as red galaxy
samples for each redshift slice, in a fashion similar to the one we used for
cluster galaxies. Then, in every bin, we fit the CMR of galaxies’ $g-r$ colors
and $i$-band magnitude with a straight line using weighted least square
fitting with the weights come from the inverse sequare of the color
measurement errors. The weights are the inverse square of the $g-r$
measurement errors. We record the corresponding slopes. In the bottom panel
Fig.11, we choose 6 redshift bins ($\Delta z=0.003$) to illustrate the
red/blue galaxy separation and the ridgeline slope fitting in each bin.
Finally, we fit the variation of slope with redshift with a line to look for a
trend, the results are shown in the left panel of Fig.12. As a comparison to
the cluster sample, we also plot the mean variation of the ridgeline slope
from clusters in the same redshift range [0.1, 0.2] in the right panel of
Fig.12. When the redshift range is changed, the slope of the fitted line for
the cluster sample becomes steeper as compared to Fig.9. The reason lies in
that the linear fit to the trend is only the first order approximation. But
for our purpose here, we just need to require the cluster sample and
spectroscopic sample on the same redshift range so as to compare them fairly.
Figure 11: Evaluating the ECGMM-derived red sequence slopes in SDSS
spectroscopy of field galaxies. The normalized color histograms (top panel)
for $\Delta z=0.003$ slices in spectroscopic redshift clearly show the
presence of the red and blue components in the field galaxy distribution.
ECGMM is used to separate the two components, the redder of which is to
measure the CMR (bottom panel). Figure 12: The comparison of the evolution of
the slopes of CMR for spectroscopic sample and cluster sample. Since the
spectroscopic sample is biased due to selection effects at $z\geq 0.2$, we
choose both sample in the redshift range from 0.1 to 0.2 and then bin the
slopes in redshift bin of 0.02. We then fit straight line to the mean slopes
in each bin for spectroscopic sample and cluster sample respectively.
Comparing the two panels in Fig.12 shows that the two slopes from
spectroscopic sample and cluster sample are not different in a statistically
significant way. This further confirms our previous observation that the
cluster environment will not affect the slope.
## 6 Discussion
To this point, the measurements have been presented in the observed frame.
Photometric cluster detection and the quantities derived (e.g. richness)
operate in the frame of the observer, and predictions from galaxy formation
models and mock galaxy catalogs can be evaluated in light of these precision
measurements. They have particular applicability to calibration of optical
cluster detection efforts, especially to those that rely on the properties of
the red sequence. The methodologies developed herein allow the “bootstrapping”
of optical algorithms: basic cluster finders locate the clusters, and
precision measurements (such as these) of said clusters lead to refinements in
those algorithms. The extent to which each of these agrees with previous
measurements from spectroscopy, other cluster samples, and simulations is left
for future work. However, for illustrative purposes, we list the relevant
observational considerations to be made in understanding the context of these
measurements with respect to previous work in the literature, and then
highlight a few of our more interesting results.
In general, there are five places where the comparison to previous work must
be treated with caution, which can be summarized as follows: 1) redshifting of
the galaxy spectra through the bandpasses under consideration, which imparts
trends in the observed colors, 2) selection effects imposed by the color
selection (e.g. Franzetti et al., 2007), 3) aperture effects, i.e. the
aperture used to measure the color in different banspasses (e.g. Scodeggio,
2001; Blakeslee et al., 2006), 4) projection effects. 5) actual evolution in
the red sequence.
To some level, any of the aforementioned issues may play into our results: (i)
at $z\simeq 0.1$, the CMR of photometrically-selected galaxies is noticeably
shallower than previous spectroscopic measurements of the color magnitude
relation (Hogg et al., 2004; Cool et al., 2006), (ii) the slopes are almost
independent of cluster richness; (iii) the photometric error-corrected scatter
of the red-sequence broadens mildly with redshift; (iv) the observed mean
slope of the CMR is negative and it becomes more negative as redshift
increases.
Naively, we expect that our measurement of the slope of the red-sequence,
$-0.013\pm 0.0003$ mags mag-1 at $z=0.1$, corresponds to the SDSS
spectroscopic analysis of Hogg et al. (2004), for which the slope is -0.022
mags mag-1 in ${}^{0.1}(g-r)$. In addition to the fact that the Hogg et al.
(2004) measurements are k-corrected to the $z=0.1$ rest-frame, one possible
difference comes from our definition of the red sequence: Hogg et al. (2004)
use a $2\sigma$ clipping algorithm to define the red sequence and to
iteratively reject outliers. While they split the sample by Sersic index,
sigma-clipping may be more permissive of objects near the “blue cloud” to be
included in the red-sequence, while the method presented in this paper
automatically accounts for the presence of these objects. Our slope
measurements at a given redshift may also be biased shallow, as the initial 2
$\sigma$ cut derived from the ECGMM fit does not account for the slope in the
red-sequence itself, i.e. the cut is applied in the same way regardless of
magnitude. Ideally, an iterative procedure would be employed to determine the
best-fit line for each cluster and the $2\sigma$ cut would be applied as a
function of magnitude. Unfortunately, the small number statistics for low
richness clusters do not permit this to be implemented in a robust fashion.
Insofar as richness and local density are similar indicators of environment,
the second observation (ii) that the slope is almost independent of
environment is in basic agreement with Hogg et al. (2004), who use SDSS
spectroscopy at $z\sim 0.1$ to compare galaxies with high ($n\geq 2$) Sersic
indicies in different environments characterized by their local density.
After the photometric error correction performed by ECGMM, a trend in the
scatter with redshift remains (iii), such that the scatter increases with
increasing redshift. At high redshift $z\simeq 1$, the color-magnitude
relation has been measured in a handful of clusters (Mei et al., 2009; Koester
et al., 2009; Santos et al., 2009, e.g.) with the general conclusion that the
restframe scatter in the CMR does not evolve with redshift. More locally, the
SDSS Luminous Red Galaxy (LRG) Sample has been used to measure various
redshifted frames of bright ($L\gtrsim 2.2L_{*}$) red galaxies (Cool et al.,
2006). Cool et al. (2006) find the intrinsic rest-frame scatter
${}^{0.16}(g-r)=35.4\pm 3.7$ and ${}^{0.37}(g-r)=43.5\pm 6.2$ mmags,
consistent with no evolution. However, with increased cluster sample, our
$observed$ frame measurements reveal an increase in the scatter, shown by a
statistically significant non-zero slope (the bottom right panel in Fig. 7).
Result (iv) is in qualitative agreement with the results in (Gladders et al.,
1998) who find a similar trend in the slope for a sample 44 Abell clusters at
$z\leq 0.15$ and 6 clusters at $0.2\leq z\leq 0.75$, the largest previous
study of its kind. In their study of the scatter of the CMR in LRGs, Cool et
al. (2006) report no significant trend with redshift in the rest-frame slope
of LRGs over $0.16<z<0.37$ in either the cluster or the field, but caution
that the sample is not-well suited to measuring the slope. The observed factor
of 2.5 increase in the magnitude in our measurement of the slope is likely due
to a combination of the lack of k-corrections and selection effects (e.g.
Franzetti et al., 2007) derived from color cuts that may preferentially
include a larger and larger fraction of galaxies with significant star-
formation at increasing redshifts.
A further contribution to the inflated slope may come from the choice of the
color aperture. van Dokkum et al. (1998) and Scodeggio (2001) note the
importance of the use of adaptive apertures, which place the color
measurements of large and small galaxies on the same footing. This point
motivates our choice of $\tt{MODEL\\_MAGS}$ from the SDSS, which are derived
from the best-fit convolution of the local PSF with a deVaucoleurs model in
the $r$-band. This same best-fit model is then used to compute the flux in
both the $g$ and $r$-bands.
As our results indicated, the intrinsic ridgeline slope may decrease as
redshift increase. Though there are many factors that may complicate the
interpretation of this results as we discussed in the preceding paragraphs, it
is still worth to speculate what may lead to the intrinsic evolution. In
general, galaxies are more active at higher redshift (Cowie et al., 1996), and
their color distributon spreads wider and toward bluer end. Therefore,
clusters at higher redshift are more likely to have member galaxies with wider
and bluer color distribution. As a result, when we fit the CMR with a
straightline, we tend to have more negative slopes at higher redshift.
## 7 Summary
In this paper, we have presented the ECGMM, a new purely photometric method
which characterizes the red sequence ridgeline in cluster samples with large
statistics. This provides precise measures of the mean variation of the red
sequence ridgeline location and scatter (width) with respect to redshift,
properly corrected for photometric errors. The measured slopes, scatters, and
zeropoints are directly applicable to improved cluster finding efforts and to
characterization of known galaxy clusters.
Applying the method to maxBCG clusters approximately recovers known properties
of the red sequence, namely its slope and the variation of the slope with
redshift, and the insensitivity of the slope to environment. It also suggests
that the scatter of the red-sequence increases mildly with redshift, and that
the slope of the red-sequence grows substantially by $z\simeq 0.3$, but we
caution that these observed trends may be attributable to a host of
observational effects that we have made no attempt to correct. Color selection
effects, the lack of k-corrections, and the details of the measurement of the
individual cluster CMRs require proper attention before applying these results
to models of galaxy formation. Nonetheless, these measurements can serve as an
important observational check on simulation and mock galaxy catalogs.
## Acknowledgments
JH thank Hyunsook Lee for helpful conversation about the BIC and Mixture
models. JH and TM gratefully acknowledge support from NSF grant AST 0807304
and DoE Grant DE-FG02-95ER40899. ESR would like to thank the TABASGO
foundation. ER was funded by the Center for Cosmology and Astro-Particle
Physics at The Ohio State University and by NSF grant AST 0707985. This
project was made possible by workshop support from the Michigan Center for
Theoretical Physics.
Funding for the creation and distribution of the SDSS Archive has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Aeronautics and Space Administration, the National Science
Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and
the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is
managed by the Astrophysical Research Consortium (ARC) for the Participating
Institutions. The Participating Institutions are The University of Chicago,
Fermilab, the Institute for Advanced Study, the Japan Participation Group, The
Johns Hopkins University, the Korean Scientist Group, Los Alamos National
Laboratory, the Max-Planck- Institute for Astronomy (MPIA), theMax-Planck-
Institute for Astrophysics (MPA), New Mexico State University, University of
Pittsburgh, University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington
## Appendix A The recursive relation for the error corrected Gaussian Mixture
Model
In this appendix, we show the derivation of the likelihood function Eq. 2 and
the EM recursive relations for the error corrected GMM. To begin with, we
introduce the following notations in Table.1. For brevity, we denote the
parameters ($\mu_{i}$, $\sigma_{i}$ and $w_{i}$) collectively by $\theta$ and
$(t)$ represents the $t^{th}$ iteration. $M$ represents number of data points
and $N$ represents the number of mixtures.
Notations | Meaning
---|---
$y_{1}$,$...$, $y_{j}$, $...$, $y_{m}$: | Observed colors of BCGs and member galaxies.
$\bar{y}_{1}$,$...$, $\bar{y}_{j}$, $...$, $\bar{y}_{m}$: | True colors of BCGs and member galaxies.
$z_{1}$,$...$, $z_{j}$, $...$, $z_{m}$: | Hidden variables that tell which Gaussian component the $\bar{y}_{j}$ is sampled from.
$\delta_{1}$,$...$, $\delta_{j}$,$...$, $\delta_{m}$: | Measurement errors for every $y_{j}$.
$\mu_{1}$, $...$, $\mu_{i}$, $...$, $\mu_{n}$: | Mean of each Gaussian component.
$\sigma_{1}$, $...$, $\sigma_{i}$, $...$, $\sigma_{n}$: | Width of each Gaussian component.
$w_{1}$, $...$, $w_{i}$, $...$, $w_{n}$: | Weights of corresponding Gaussian components.
Table 1: The notations used in our derivation of ECGMM algorithm
Since we assume the true color distribution can be approximated by mixture of
Gaussian distributions, we have the following probability density function for
$p(\bar{y_{j}}|\theta)$:
$p(\bar{y_{j}}|\theta)=\sum_{i=1}^{N}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\bigg{[}-\frac{(\bar{y}_{j}-\mu_{i})^{2}}{2\sigma_{i}^{2}}\bigg{]}$
(A1)
Though the true colors are not directly observable, we know that its
distribution given the observed colors and measurement errors is approximately
Gaussian:
$p(\bar{y_{j}}|y_{j})=\frac{1}{\sqrt{2\pi\delta_{j}^{2}}}\exp\bigg{[}-\frac{(\bar{y}_{j}-y_{j})^{2}}{2\delta_{j}^{2}}\bigg{]}$
(A2)
then, the likelihood function (under the flat priors for $\theta$)
$L(\theta|y_{j})=\int p(\bar{y_{j}}|\theta)p(\bar{y_{j}}|y_{j})d\bar{y_{j}}$
(A3)
After integrating over $\bar{y}_{j}$ and extending to all data points
($\prod_{j=1}^{M}$), we arrive at Eq.2. The optimal parameters could be
obtained by maximizing the above likelihood. However, if we introduce hidden
variables, $z$, that tell us which Gaussian component the $y_{j}$ is sampled
from, then the whole maximization process could be significantly simplified.
The corresponding pdf of data given $z$ and $\theta$ is
$p(y|z_{j}=i,\theta^{(t)})=\prod_{j=1}^{M}p(y_{j}|z_{j}=i,\theta_{i}^{(t)})=\prod_{j=1}^{M}\frac{1}{\sqrt{2\pi(\sigma_{i}^{(t)2}+\delta_{j}^{2})}}\exp\bigg{[}-\frac{(y_{j}-\mu_{i}^{(t)})^{2}}{2(\sigma_{i}^{(t)2}+\delta_{j}^{2})}\bigg{]}$
(A4)
We use $w_{i}$ denote the weight of each Gaussian Component in the mixture and
is given by $w_{i}=p(z_{j}=i|\theta)$. The estimation of hidden variable could
be related to Eq.A4 by Bayes’ Theorem as following:
$p(z_{j}=i|y_{j},\theta^{(t)})=\frac{p(z_{j}=i,y_{j}|\theta^{(t)})}{p(y_{j}|\theta^{(t)})}=\frac{p(y_{j}|z_{j}=i,\theta^{(t)})p(z_{j}=i|\theta^{(t)})}{\sum_{i=1}^{N}p(y_{j}|z_{j}=i,\theta^{(t)})p(z_{j}=i|\theta^{(t)})}$
(A5)
The EM algorithm iteratively update the parameters $\theta$ by maximizing the
expected log likelihood
$Q(\theta)=\sum_{i=1}^{N}\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta^{(t)})\bigg{[}-\frac{1}{2}\ln(2\pi)-\frac{1}{2}\ln(\sigma_{i}^{2}+\delta_{j}^{2})-\frac{(y_{j}-\mu_{i})^{2}}{2(\sigma_{i}^{2}+\delta_{j}^{2})}+\ln
p(z_{j}=i|\theta^{(t)})\bigg{]}$ (A6)
under the constraint $\sum_{i=1}^{N}p(z_{j}=i|\theta^{(t)})=1$. Using the
Lagrange Multiplier approach, we redefine
$\tilde{Q}(\theta)=Q(\theta)-\lambda\bigg{[}\sum_{i=1}^{N}p(z_{j}=i|\theta^{(t)})-1\bigg{]}$
(A7)
with $\lambda$ as the multiplier.
$\frac{\partial\tilde{Q}(\theta)}{\partial\mu_{i}}=\sum_{j=1}^{M}\bigg{[}p(z_{j}=i|y_{j},\theta^{(t)})\bigg{(}\frac{y_{j}-\mu_{i}}{\sigma_{i}^{2}+\delta_{j}^{2}}\bigg{)}\bigg{]}=0$
(A8)
From Eq.A8, we can arrive at the following recursive relation for $\mu$:
$\mu_{i}^{(t+1)}=\frac{\sum_{j=1}^{M}y_{j}p(z_{j}=i|y_{j},\theta_{i}^{(t)})/(1+\delta_{j}^{2}/\sigma_{i}^{(t)2})}{\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta_{i}^{(t)})/(1+\delta_{j}^{2}/\sigma_{i}^{(t)2})}$
(A9)
Similarly, we have
$\frac{\partial\tilde{Q}(\theta)}{\partial\sigma_{i}}=\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta^{(t)})\bigg{[}\frac{\sigma_{i}^{2}(1+\delta_{j}^{2}/\sigma_{i}^{2})-(y_{j}-\mu_{i})^{2}}{\sigma_{i}^{4}(1+\delta_{j}^{2}/\sigma_{i}^{2})^{2}}\bigg{]}=0$
(A10)
Note that since $\sigma_{i}$ and $\delta_{j}$ are entangled within the
summation, there would not be an simple analytic solution for $\sigma_{i}$.
However, since the algorithm is iterative in nature and the major contribution
for the update of $\sigma_{i}$ is from $(y_{j}-\mu_{i})^{2}$, we could
approximate $\sigma_{i}$ in $\delta_{j}^{2}/\sigma_{i}^{2}$ with its value in
$t^{th}$ iteration. Then we can solve for the $(t+1)^{th}$ iteration relation
for $\sigma_{i}$ as:
$\sigma_{i}^{(t+1)}=\bigg{[}\frac{\sum_{j=1}^{M}(y_{j}-\mu_{i})^{2}p(z_{j}=i|y_{j},\theta_{i}^{(t)})/(1+\delta_{j}^{2}/\sigma_{i}^{(t)2})^{2}}{\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta_{i}^{(t)})/(1+\delta_{j}^{2}/\sigma_{i}^{(t)2})}\bigg{]}^{1/2}$
(A11)
Our numerical test shows that such an approximation works fine in practice.
For $w_{i}=p(z_{j}=i|\theta)$, we have
$\frac{\partial\tilde{Q}(\theta)}{\partial
w_{i}}=\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta^{(t)})/w_{i}-\lambda=0$ (A12)
which leads to
$w_{i}=p(z_{j}=i|\theta)=\frac{1}{\lambda}\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta^{(t)})$
(A13)
Using the condition $\sum w_{i}=1$, we have $\lambda=M$. Substitute $\lambda$
back to Eq. A13, we arrive at:
$w_{i}^{(t+1)}=\frac{1}{M}\sum_{j=1}^{M}p(z_{j}=i|y_{j},\theta_{i}^{(t)})$
(A14)
In the above iteration relations Eq.A9, Eq.A11 and Eq.A14, $t$ and $t+1$
denote the round of iterations. When we ignore the measurement errors
$\delta_{j}$, the above recursive relation reduces to the standard EM
recursive relation for Gaussian Mixture Model. The above relations could be
easily generalized to multiviate case by simply substituting the data with
data matrix, mean with mean vector and variance with covariance matrix, which
we will not repeat the formula here. A C++ class that implements the above
algorithm is available upon request.
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* Wake et al. (2006) Wake, D. A., et al. 2006, MNRAS, 372, 537
* Zehavi et al. (2005) Zehavi, I., et al. 2005, ApJ, 630, 1
|
arxiv-papers
| 2009-07-24T22:40:48 |
2024-09-04T02:49:04.167272
|
{
"license": "Public Domain",
"authors": "Jiangang Hao, Benjamin P. Koester, Timothy A. Mckay, Eli S. Rykoff,\n Eduardo Rozo, August Evrard, James Annis, Matthew Becker, Michael Busha,\n David Gerdes, David E. Johnston, Erin Sheldon, Risa H. Wechsler",
"submitter": "Jiangang Hao",
"url": "https://arxiv.org/abs/0907.4383"
}
|
0907.4435
|
# Signals of the QCD Critical Point in Hydrodynamic Evolutions
Chiho Nonakaa, M. Asakawab, S. A. Bassc, B. Müllerc aDepartment of Physics,
Nagoya University, Nagoya 464-8602, Japan
bDepartment of Physics, Osaka University, Toyonaka 560-0043, Japan
cDepartment of Physics, Duke University, Durham, North Carolina 27708, USA
###### Abstract
The presence of a critical point in the QCD phase diagram can deform the
trajectories describing the evolution of the expanding fireball in the
$\mu_{B}$-$T$ phase diagram. The deformation of the hydrodynamic trajectories
will change the transverse velocity ($\beta_{T}$) dependence of the proton-
antiproton ratio when the fireball passes in the vicinity of the critical
point. An unusual $\beta_{T}$ dependence of the $\bar{p}/{p}$ ratio in a
narrow beam energy window would thus signal the presence of the critical
point.
††journal: Nuclear Physics A
## 1 Towards quantitative analyses of QCP
The existence and location of the QCD critical point (QCP) in the QCD phase
diagram have been attracting many physists’ interests in heavy ion collision
physics. However recent studies based on effective theories show many possible
location of the QCP in the QCD phase diagram. In addition, the latest finite
temperature and imaginary chemical potential lattice QCD calculation shows
that even the existence of the QCD critical point is uncertain [1]. At present
experiments and quantitative phenomenological analyses for the QCP are needed,
because it seems to be very difficult to reach a solid conclusion about the
QCP just from lattice QCD and effective theories. Here we discuss the
consequences of the QCP from point of view of the quantitative
phenomenological analyses in heavy ion collisions. Towards quantitative
analyses of the QCP we need the following three steps: a realistic dynamical
model which describes expansion of hot and dense matter after collisions, an
equation of state (EoS) with QCP and appropriate physical observables which
show signals of QCP clearly.
For a realistic dynamical model, we use a combined fully three-dimensional
macroscopic / microscopic transport approach employing relativistic
3d-hydrodynamics for the early, dense, deconfined stage of the reaction and a
microscopic non-equilibrium model for the later hadronic stage where the
equilibrium assumptions are not valid anymore [2]. Within this approach we
study the dynamics of hot, bulk QCD matter, which is being created in ultra-
relativistic heavy ion collisions at RHIC. Our approach is capable of self-
consistently calculating the freezeout of the hadronic system, while
accounting for the collective flow on the hadronization hypersurface generated
by the QGP expansion. We succeed in explaining a lot of experimental data
consistently at RHIC: $P_{T}$ spectra of various particles including
multistrangeness particles, rapidity distributions, mean $P_{T}$ as a function
of particle mass, freezeout time distribution of particles, and elliptic flow.
In this calculation, we adopt the EoS with the 1st order phase transition
without QCP which is used in most hydrodynamic models. Now we replace the EoS
in 3d hydro + UrQMD model by an EoS with QCP.
The EoS of QCP which we construct here is composed of two parts: one is a
singular part around QCP and another part is non-singular part which is
described by usual QGP phase and hadron phase [3]. For the singular part of
the EoS, we assume that QCD has the same universality class as 3d Ising model.
First we construct the EoS of 3d Ising model as a function of reduced
temperature ($r$) and external magnetic field ($h$) [4] and map it on the QCD
phase diagram, $\mu_{B}$-$T$ plane. The magnetization as a functions of $r$
and $h$ in 3d Ising model shows different behavior of phase transition between
negative $r$ (1st order) and positive $r$ (crossover). In mapping the EoS with
3d Ising model as a function of $r$ and $h$ on the $\mu_{B}$-$T$ plane,
however, there is no universality between QCD and 3d Ising model. We can not
determine the direction of $h$ axis to $r$ axis, though we can fix $r$ axis as
it is parallel to tangential line at QCP to the phase boundary. Here we choose
the direction of $h$ to be perpendicular to $r$ axis. Besides not only the
size of critical region around the QCD and location of it on the QCD phase
diagram are input parameters in our model. The EoS with the QCP has a very
interesting feature in isentropic trajectories on $\mu_{B}$-$T$ plane: the QCP
works as an attractor of isentropic trajectories on the QCD phase diagram [5,
3], which gives us a clue of finding the QCP.
## 2 Signals of QCP
Next we discuss how to find clear consequences of QCP in heavy ion collisions.
The promising signals of QCP should survive not only in expanding fireball but
also even after the freezeout process. Here we investigate two candidates of
them: fluctuations and hadron ratios. For fluctuation, naively, we have to
pick up fluctuations of conserved values such as charge, baryon number and
strangeness during whole process of collisions. Hadron ratios are fixed at
chemical freezeout temperature and hold the same value during freezeout
process and final state interactions.
Figure 2 shows behavior of static and dynamical fluctuations in 1d
hydrodynamic expansion along isentropic trajectories on the $\mu_{B}$-$T$
plane [3]. In both cases of static and dynamic fluctuations, we can see the
effect of QCP, i.e enhancement of fluctuation around QCP. For static case
fluctuation becomes maximum just at QCP. However for dynamic case maximum
value of fluctuation appears after passing in the vicinity of QCP because of
the critical slowing down and the maximum value itself is not so large as the
static case. There is possibility that fluctuations which are induced by QCP
do not become so large as a signal of QCP, if the expansion of fireball is
fast.
For hadron ratios key issue from the point of view of QCP is that a chemical
freezeout temperature depends on transverse velocity of hadrons. Figure 2
shows that $\bar{p}/p$ ratio has transverse velocity dependence in a
microscopic transport model, UrQMD in which the QCP does not exist. We find
that on isentropic trajectories the freezeout process occurs gradually [6]:
particles with higher transverse velocity are emitted at earlier time of
expansion and those with lower transverse velocity are produced at later time.
This suggests that hadron ratios may change on isentropic trajectory between a
hadronization point on the QCD phase boundary on the $\mu_{B}$-$T$ plane and
chemical freezeout point. The hadron ratio, especially $\bar{p}/p$ ratio as a
function of transverse velocity (momentum) is sensitive to behavior of
isentropic trajectories on $\mu_{B}$-$T$ plane and may show a consequence of
QCP clearly.
Figure 1: Fluctuations as a function of temperature along isentropic
trajectories.
Figure 2: $\bar{p}/p$ ratio as a function of transverse velocity which is
obtained with UrQMD.
Next we do a demonstrative calculation to show how the signal of QCP appears
through $\bar{p}/p$ ratio in heavy ion collisions. Here we focus on SPS energy
region and put the QCP which is parameter in our model and the chemical
freezeout point which is obtained in a statistical model [7] to $(\mu_{B},T)$
= (550, 159) MeV and (406, 105) MeV on the $\mu_{B}$-$T$ plane, respectively.
Figure 4 shows that hydrodynamical trajectories in the QCD phase diagram with
and without the presence of a critical point. Possible trajectories in the
plane in the absence of a critical point are shown as solid line (for a
crossover transition (CO)) or dash-dotted line (for a first-order transition
(FO)); the trajectory in the presence of a critical point is shown as dashed
line (QCP). All trajectories meet at the bulk chemical freezeout point. This
dotted line in which clear focusing effect appears stands for isentropic
trajectory of the QCP. Hadronization occurs between the phase boundary and
chemical freezeout point. And between them we can see clear differences in
three cases. In the case of QCD critical point the ratio of $\bar{p}{p}$
decreases along the line or almost the same. On the other hand, in the case of
1st order phase transition and crossover this value increases along isentropic
trajectory.
Antiproton-to-proton ratio along the trajectories is shown in Fig. 4 as a
function of the entropy density which is proportion to transverse momentum.
The curves start at the phase boundary 160 MeV and continue down to chemical
freezeout temperature (145 MeV). The location of the chemical freezeout point
deduced from experimental data is indicated by the open and solid squares.
Note that the ratio only rises for the trajectory deformed by the critical
point. In actual experimental data this evidence should appear as steeper
$\bar{p}$ spectra at high $P_{T}$.
Figure 3: Isentropic trajectories with and without QCP on the QCD phase
diagram. Arrows indicate the direction of time evolution.
Figure 4: Antiproton-to-proton ratio along the trajectories as a function of
the entropy density. Arrows indicate the direction of time evolution.
We find interesting experimental data which may suggest a signal of QCP:
$\bar{p}$ spectra obtained by NA49 [8]. They show $\bar{p}$ and $p$ spectra on
collision energies from 20 GeV to 158 GeV. Only at 40 GeV collision energy
slope of $P_{T}$ spectra of $\bar{p}$ seems to be steeper compared to other
collision energies, which would require a trajectory of the type expected in
the vicinity of the QCP (Fig. 4). The size of the statistical errors of the
measurement does not permit a firm conclusion about this anomaly, but it is
certainly compatible with the arguments presented here.
Finally we show an example of realistic calculation by 3d hydro + UrQMD model
with EoS including QCP. For initial conditions of our hybrid model we set
maximum value of energy density and baryon number density to be 2.0 GeV/fm3
and 0.15 fm-3 , respectively, which corresponds to SPS energy region. In the
following calculations we use the same initial conditions for the both cases
of EoS with and without QCP and set a switching temperature from hydrodynamic
model to UrQMD to be 150 MeV. Because of different behavior in isentropic
trajectories between EoS with and without QCP, chemical potential at the
switching temperature is 430 MeV in presence of QCP which is larger than one
in absence of QCP (250 MeV). This difference appears in hadron ratio. Figure 5
shows $P_{T}$ spectra of $\pi$, $K$, $p$ with QCP (left) and without QCP
(right). We can see the effect of different isentrpic trajectoris in hadron
ratios.
In summary, we have shown that the evolution of the $\bar{p}/p$ ratio along
isentropic curves between the phase boundary in the QCD phase diagram and the
chemical freeze-out point is strongly dependent on the presence or absence of
a critical point. When a critical point exists, the isentropic trajectory
approximately corresponding to hydrodynamical expansion is deformed, and the
$\bar{p}/p$ ratio grows during the approach to chemical freeze-out. Depending
on the actual size of the attractive region around the critical point, the
search for an anomaly in the $P_{T}$ dependence of the $\bar{p}/p$ ratio may
require small beam energy steps.
Figure 5: $P_{T}$ spectra for $\pi$ (solid circles), $K$ (solid square) and
$p$ (solid triangle) with QCP (left) and without QCP (right).
## References
* [1] P. Petreczky, these proceedings, C. Sasaki, these proceedings.
* [2] C.Nonaka, S.A.Bass, Phys. Rev. C 75 (2007) 014902.
* [3] C. Nonaka, M. Asakawa, Phys. Rev. C71(2005) 044904.
* [4] R.Guida and J.Zinn-Justin Nucl. Phys. B486(1997)626.
* [5] M. A. Stephanov, K. Rajagopal, E. V. Shuryak, Phys. Rev. Lett. 81(1998) 4816.
* [6] M. Asakawa, S. A. Bass, B. Mul̈ler and C. Nonaka, Phys. Rev. Lett. 101(2008)122302.
* [7] P. Braun-Munzinger, K. Redlich, and J. Stachel, arXiv: nucl-th/0304013.
* [8] C. Alt et al. (NA49 Collaboration), Phys. Rev. C73(2006) 044910\.
|
arxiv-papers
| 2009-07-25T17:05:19 |
2024-09-04T02:49:04.175498
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chiho Nonaka, M. Asakawa, S. A. Bass, B. Muller",
"submitter": "Chiho Nonaka",
"url": "https://arxiv.org/abs/0907.4435"
}
|
0907.4457
|
# Measurement of $\pi^{0}$ and $\gamma$ in d+Au collisions at $\sqrt{s_{NN}}=$
200 GeV by PHENIX experiment
Ondřej Chválaa for the PHENIX collaboration a University of California –
Riverside, 900 University Ave, Riverside CA 92521, USA
###### Abstract
Previous results indicated that high $p_{T}$ particle suppression in Au+Au
interactions is a final state effect, since RdA ratios were compatible with
unity, albeit within large experimental errors. It is important to test this
conclusion to higher precision since the modification of structure functions
may be involved. Recent d+Au data taken in 2008 improve the integrated
luminosity by about a factor of thirty compared to the 2003 data. A more
precise measurement of both $\pi^{0}$ and $\gamma$ at higher $p_{T}$ will shed
new light on whether the initial state in the heavy nuclei is modified.
††journal: Nuclear Physics A
## 1 Introduction
The PHENIX experiment at RHIC is designed for high rate measurement of
electromagnetic probes, specifically direct photons $\gamma$ and neutral pions
$\pi^{0}$ at mid-rapidity ($|\eta|<$ 0.35). The high event rate allows unique
access to rare probes, such as particles produced at very high transverse
momentum ($p_{T}$).
Early results for the nuclear modification factors RAA, the spectrum in Au+Au
collisions over the spectrum in p+p interactions scaled by the respective
nucleon overlap integrals, showed suppression of $\pi^{0}$ and charged hadron
production at high-$p_{T}$ ($p_{T}\gtrsim$ 6 GeV/c) [1, 2, 3]. This result,
combined with the apparent lack of suppression for high-$p_{T}$ direct
$\gamma$ production [4] has been interpreted as an indication of the formation
of a dense strongly interacting medium, the sQGP, in heavy ion collisions.
Furthermore, the preliminary analysis of the high statistics 2004 Au+Au run
showed a possible decrease of direct photon RAA above 14 GeV/c of transverse
momentum [5]. Possible explanations for this observation include the isospin
effect (the difference of partonic content between protons and neutrons), the
EMC effect (modifications of the distribution of partons in the heavy nuclei),
and the suppression of the direct photons originating from fragmenting partons
which are quenched by the medium.
RHIC measurements of particle production in d+Au collisions from 2003 show no
suppression of produced particles within large experimental error bars [6, 7],
indicating little or no modification of the initial state in gold nuclei. This
result confirmed the attribution of the large suppression observed in central
Au+Au interactions to final state effects in the sQGP medium.
The final analysis of the 2003 data set revealed some suppression of $\pi^{0}$
production at the highest $p_{T}$ in the most central d+Au interactions [8],
however the experimental uncertainties are too large to estimate cold nuclear
matter effects quantitatively [9, 10].
## 2 RHIC year 2008 data set
The RHIC run in year 2008 improved the total integrated luminosity of the d+Au
sample by a factor of $\approx$30 compared to the 2003 run: 1.65$\times$109
minimum bias (MB) events, and 3.68$\times$109 events from a high-$p_{T}$
photon (ERT) trigger were recorded. There were also 0.53$\times$109 MB
triggered and 1.17$\times$109 ERT triggered p+p collisions taken in 2008. The
uncorrected $\pi^{0}$ yields are shown in Fig. 1.
The improvement in total d+Au integrated luminosity can be appreciated from
the relative statistical errors in the $\pi^{0}$ measurement using PHENIX PbSc
EMCal. The error at $p_{T}=$ 15 GeV/c is 32 % using the 2003 sample, and is
reduced to 3.5 % in the recent high statistics run.
Figure 1: Raw mid-rapidity yields per event of $\pi^{0}$ production from the
2008 data sets: p+p minbias, d+Au minbias, and three centrality bins in d+Au
collisions using PHENIX PbSc EMCal. The spectra from ERT triggered runs (red)
are scaled to match MB yields (blue) in 6–13 GeV/c region of $p_{T}$. Only
statistical errors are shown.
## 3 Conclusions
Previous results from data taken in year 2003 indicate that the high $p_{T}$
suppression observed in Au+Au interactions is a final state effect since the
RdA ratios are consistent with unity, albeit within large experimental errors.
It is important to test this conclusion to higher precision. Recent d+Au data
taken in 2008 improve the integrated luminosity by about a factor of thirty
compared to the 2003 run. The 2008 PHENIX measurement will shed more light on
the origin of cold nuclear matter effects.
## References
* [1] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88, 022301 (2002) [arXiv:nucl-ex/0109003].
* [2] K. Adcox et al. [PHENIX Collaboration], Phys. Lett. B 561, 82 (2003) [arXiv:nucl-ex/0207009].
* [3] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072301 (2003) [arXiv:nucl-ex/0304022].
* [4] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 94, 232301 (2005) [arXiv:nucl-ex/0503003].
* [5] K. Reygers [PHENIX Collaboration], J. Phys. G 35, 104045 (2008) [arXiv:0804.4562 [nucl-ex]].
* [6] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072303 (2003) [arXiv:nucl-ex/0306021].
* [7] D. Peressounko [PHENIX Collaboration], Nucl. Phys. A 783, 577 (2007) [arXiv:hep-ex/0609037].
* [8] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 98, 172302 (2007) [arXiv:nucl-ex/0610036].
* [9] G. G. Barnafoldi, G. Fai, P. Levai, B. A. Cole and G. Papp, [arXiv:0805.3360 [hep-ph]].
* [10] B. W. Zhang and I. Vitev, arXiv:0810.3194 [nucl-th].
|
arxiv-papers
| 2009-07-26T03:52:15 |
2024-09-04T02:49:04.180063
|
{
"license": "Public Domain",
"authors": "Ondrej Chvala (for the PHENIX collaboration)",
"submitter": "Ond\\v{r}ej Chv\\'ala",
"url": "https://arxiv.org/abs/0907.4457"
}
|
0907.4468
|
# What type of distribution for packet delay in a global network should be
used in the control theory?
Andrei M. Sukhov
Samara State Aerospace University Moskovskoe sh., 34 Samara, 443086, Russia
amskh@yandex.ru Natalia Kuznetsova
Samara State Aerospace University,
Togliatti branch Voskresenskaya st., 1 Togliatti, 445000, Russia
meneger_job@mail.ru
(July 26, 2009)
###### Abstract
In this paper correspondence between experimental data for packet delay and
two theoretical types of distribution is investigated. Calculations have shown
that the exponential distribution describes the data on network delay better,
than truncated normal distribution. Precision experimental data to within
microseconds are gathered by means of the RIPE Test Box. In addition to exact
measurements the data gathered by means of the utility ping has been parsed
that has not changed the main result. As a result, the equation for an
exponential distribution, in the best way describing process of packet delay
in a TCP/IP based network is written. The search algorithm for key parameters
as for normal, and an exponential distribution is resulted.
###### category:
I.2.8 ARTIFICIAL INTELLIGENCE Problem Solving, Control Methods, and Search
###### keywords:
Control theory
###### category:
C.2.5 COMPUTER-COMMUNICATION NETWORKS Local and Wide-Area Networks
###### keywords:
Internet (e.g., TCP/IP)
###### keywords:
exponential distribution for network delay, truncated normal distribution,
expression for cumulative distribution function, RIPE Test Box, parameters for
delay distribution
††titlenote: corresponding author
## 1 Introduction
The special area of the control theory, named networked control systems in
which transfers as environment of operating signals were used computer
networks, has arisen in the late nineties of the XX-th century [16].
Originally, as the network environment of control systems local networks [8]
which differ high-speed data transfer and in the minimum percent of packet
loss were used.
Framing of criteria of stability for networked control systems in which as the
handle environment the global network Internet is used, is extremely
complicated because of random character of distribution of packet delay and
their big absolute values [10, 14, 15]. Non-use of knowledge of network
processes, including measurement methods of an available bandwidth, knowledge
of types of distribution of packet delay, methods on compensation of packet
losses is an obvious obstacle in path of development of networked control
systems.
However till now, results of the advanced network researches are not used in
the control theory, no less than algorithms on their basis are not created.
The present project assumes introduction of new network decisions in networked
control systems.
For the decision of problems of the networked control systems on the basis of
stack TCP/IP it is more convenient to use the process approach to the control
theory, based on idea of existence of some universal functions of control. The
purpose of our research is the finding of these function for network
components. In the modern theory of computer networks there were many
utilities working with a network delay, there is a progress in studying and
modelling of transmiting of packages. Our problem consists in trying to
describe process of a network delay of management packages and to show ways of
practical calculation of all parametres entering into corresponding
distribution functions [9].
By transmission of control signals through TCP/IP etwork, the separate
packages of the controlling data flow transferring the information, are
supplied non-uniformly, and the part of packages in general is lost by
transmission on a network and does not reach a target. For rise of efficiency
of control algorithms it is necessary to reduce to a minimum of packets delay
and their variation, and also percent of packet loss. Similar algorithms are
used for transmission voice and video streams, in grid systems, at control of
robust systems, in network computer games, etc.
At first it would be desirable to result the brightest research on a
distribution type for network delay. To understand, about what there is a
speech in described papers, will give definitions of notations used in them:
* •
Round-trip time (RTT) time is the time required for a packet to travel from
the testing host to a the remote computer that receives the packet and
retransmits it back to the source.
* •
The One-Way Delay (OWD) value is calculated between two synchronized points A
and B of an IP network, and it is the time in seconds that a packet spends in
travelling across the IP network from A to B.
In particular, Elteto and Molnar [6] have spent measurements of round-trip
delay in in the Ericsson Corporate Network, complex analysis of the received
data has allowed to build the supposition about distribution type for network
delay. The main finding of their research is that the round-trip delay can be
well approximated by a truncated normal distribution.
Konstantina Papagiannaki et al [11] in the research have measured and have
analyzed packet delay between two adjacent routers in the core network. On the
basis of the received measurements, they have made the supposition about the
factors influencing occurrence of delay, and very big delays which cannot be
explained in the way of batch processing in routers on algorithm FIFO have
been noticed.
Recently, fulfilling a series of operations on measurement of an available
bandwidth [12], we have installed that for a type definition of delay
distribution we should research only a variable part of delay while its most
part remains constant. This fact also has served as a starting point of our
operation.
## 2 Premises for model
In 1999 Downey [5] for the first time has detected linear dependence of the
minimum possible round trip time on the size of transferred packets. In 2004
precise experiments by Choi et al [1] proved that the minimum fixed delay
component $D^{fixed}(W)$ for a packet of size $W$ is a linear (or precisely,
an affine) function of its size,
$D^{fixed}(W)=W\sum_{i=1}^{h}1/C_{i}+\sum_{i=1}^{h}\delta_{i}$ (1)
where $C_{i}$ is each link of capacity of $h$ hops and $\delta_{i}$ is
propagation delay. To validate this assumption, they check the minimum delay
of packets of the same size for three path, and plot the minimum delay against
the packet size.
Let $D(W)$ represents the point-to-point delay of a packet. Here we refer to
it as the minimum path transit time for the given packet size $W$, denoted by
$D^{fixed}(W)=\min D(W)$. With the fixed delay component $D^{fixed}(W)$
identified, we can now substract it from the point-to-point delay of each
packet to study the variable delay component $d^{var}$. The variable delay
component of the packet, $d^{var}$, is given by
$D(W)=D^{fixed}(W)+d^{var}$ (2)
Computed minimal delay $D^{fixed}(W)$ is
$D^{fixed}(W)=D_{min}+W/C,$ (3)
where $C$ is end-to-end capacity and
$D_{min}=\lim_{W\rightarrow 0}D^{fixed}(W)$ (4)
The value $D_{min}$ is related to the distance between the sites (i.e.
propagation delay) and per-packet router processing time at each hop along the
path between the sites [2, 3]. This value represents as the minimum delay
$D_{min}$ for which the very small package can be transmitted on a network
from one point in another.
The minimal delay [12] of datagram transmission $D_{min}$ may be calculated as
$D_{min}=\frac{W_{2}D_{1}-W_{1}D_{2}}{W_{2}-W_{1}}$ (5)
This value as well as the methods of its measurement has a important
significance in applied tasks of control theory [16]. The second significant
question of networking control theory is the distribution type for variable
delay component $d^{var}$ which should be studied. To know the expression for
this parameter we may easy calculate the duration of buffer for streaming
aplication on receiving side.
## 3 Experimental search
To determine distribution type for a variable delay component $d^{var}$ we
should gather enough considerable quantity of measurements between various
hosts in the Internet, made with a precise accuracy. The basic problem of
experimental testing is the precise of delay measurements that is necessary
for accurate result. The exact metering demands micro second precision for
delay measurements; we are reaching such accuracy with help of RIPE Test Box
mechanism [7, 13]. In order to prepare the experiments three Test Boxes have
been installed in Moscow, Samara and Rostov on Don during 2006-2008 years in
framework of RFBR grant 06-07-89074. Each RIPE Test Box represents a server
under management of an FreeBSD operating system with the GPS receiver
connected to it.
Characteristic times of investigated processes (a packet delay, jitter) have
the order from 10 $ms$ to 1 $sec$, therefore is quite enough accuracy of
system hours of a RIPE Test Box for their reliable measurement. At the first
stage experiment between tt01.ripe.net (RIPE NCC at AMS-IX, Amsterdam),
tt143.ripe.net (Samara, SSAU), tt17.ripe.net (Bolonia) and tt74.ripe.net
(Melburn) have been made which included precision measurement of packet delay
with accuracy 2-12 $\mu s$. Testing results are available in telnet to
corresponding RIPE Test Box on port 9142. It is important to come and write
down simultaneously the data on both ends of the investigated channel.
On the basis of the received data set it is easy to construct a cumulative
distribution function for network delay $D$:
$F(D)=P(x\leq D)$ (6)
For initial comparison truncated normal and exponential distributions have
been chosen, expressions for which are written down.
For truncated normal distribution it is possible to select following
approximation:
$F(D)=\left\\{\begin{aligned} 0,\quad D\leq D_{min};\\\
\frac{\sqrt{2/\pi}}{\sigma}\int\limits_{D_{min}}^{D}\exp\left\\{-\frac{(x-D_{min})^{2}}{2\sigma^{2}}\right\\}dx,\\\
\quad D>D_{min}\end{aligned}\right.$ (7)
where
$\sigma=D_{av}-D_{min}$ (8)
is the difference between average network delay $D_{av}(W)=\mathbb{E}[D(W)]$
and minimum delay $D_{min}(W)$.
It is necessary to mark that all statistics was gathered by us for the fixed
size of a packets $W$. By default for RIPE Test Box it equals to 100 bytes.
Later we update a cumulative distribution function $F(D,W)$ taking into
account the packets size $W$.
The alternative type of allocation which will be checked on correspondence is
an exponential distribution, expression for which is written below.
$F(D)=\left\\{\begin{aligned} 0,\quad D\leq D_{min}\\\
1-\exp\left\\{-\lambda(D-D_{min})\right\\},\quad
D>D_{min}\end{aligned}\right.$ (9)
where
$\lambda=1/(D_{av}-D_{min})$ (10)
is reciprocal to the difference between average network delay
$D_{av}(W)=\mathbb{E}[D(W)]$ and minimum delay $D_{min}(W)$.
For check of conformity to distribution type two methods will be used:
calculation of Pearson correlation coefficient and a graphic method. We will
designate as $K_{nor}$ correlation coefficient between experimental and normal
distributions then $K_{exp}$ is correlation coefficient between experimental
and exponential distributions.
The data obtained by us is shown in Table 1, where the column host corresponds
to a direction between two RIPE Test Boxes, and the column $W$ specifies in
the size of a testing packet.
N | host | $W$ (bytes) | $K_{nor}$ | $K_{exp}$
---|---|---|---|---
1 | bolonia | | |
| tt01->tt17 | 100 | 0.76 | 0.97
2 | samara | | |
| tt01->tt143 | 100 | 0.87 | 0.98
3 | samara | | |
| tt01->tt143 | 1024 | 0.99 | 0.99
4 | melburn | | |
| tt01->tt74 | 100 | 0.66 | 0.97
Table 1: Precise measurements
Except correlation coefficients it is possible to compare and graphics
representation of cumulative distribution functions, representing all three
functions on one schedule. On the uniform graphics (see Figure 1) red color
selects an experimental curve, blue color corresponds to normal allocation. In
black colour the exponential distribution is painted.
Figure 1: Experimental (red), normal (blue) and exponential (black) cumulative
distribution function, precise testing
All experiments resulted above testify that the best type of distribution
describing packet delay in a global network, represents an exponential
distribution. Thus, as have shown our researches, the random variable of
packet delay between two network points is arranged on an exponential low with
the parameter calculated from experimental values under the Equation (10).
## 4 The elementary check
However, it is not each investigator who is engaged in the control theory, has
at the instruction the precision measuring system, similarly RIPE Test Boxes.
Therefore in this part it would be desirable to test delay distributions,
leaning against the data of well-known utilities which are not demanding the
expensive equipment.
For testing it is possible to use the utility ping as it is the most
widespread and readily available resource for connection quality check in
TCP/IP networks. Let’s mark only that this utility measures round-trip time,
instead of one way delay, correspondence between these values approximately
equally $OWD\approx RTT/2$.
We wish to be convinced that the data received ping, is exact enough that it
was possible to judge delay distribution. Using the utilityping, we have
tested connection between points AIST - New Zealand (tt47.ripe.net),
Volgatelekom - Australia (tt74.ripe.net) and SSAU-Melbourn (tt74.ripe.net). As
remote hosts were used servers of RIPE measurement system, AIST, Volgatelecom
and SSAU is local internet Service Providers from Samara region, Russia.
Processing the obtained data on the above described algorithm, we have
received the results presented in the Table 2.
N | host | $W$ (bytes) | $K_{nor}$ | $K_{exp}$
---|---|---|---|---
1 | AIST | | |
| New Zeland | 32 | 0.94 | 0.95
2 | Volgatelecom | | |
| Australia | 32 | 0.96 | 0.98
3 | SSAU | | |
| Melburn | 64 | 0.66 | 0.97
4 | Infolada | | |
| Athens | 32 | 0.98 | 0.98
Table 2: ping measurements
The evident illustration is resulted in definition of distribution type in
Figure 2.
Figure 2: Experimental (red), normal (blue) and exponential (black) cumulative
distribution function, ping testing
It should be noted that the utility ping allows finding automatically values
of variables $D_{av}$ and $D_{min}$ (see Eqns. (7) and (9)) which completely
define the distribution form, both normal and exponential types. It is enough
to give sequence from 10 packets to obtain the given values with a split-hair
accuracy, sufficient for the description of processes of the control theory.
## 5 Distribution type for delay
In real the Internet processes the size of transferred packages can vary,
therefore the cumulative distribution function should be updated. For each
size of a packet $W$ there is the minimum time $D^{fixed}(W)$ defined by the
equation (3).
Then, the final cumulative distribution function $F(D,W)$ is
$F(D,W)=\left\\{\begin{aligned} 0,\quad D\leq D_{min}+W/C\\\
1-\exp\left\\{-\lambda(D-D_{min}-W/C)\right\\},\\\ \quad
D>D_{min}+W/C\end{aligned}\right.$ (11)
where
$\lambda=1/(D_{av}-D_{min})$ (12)
is reciprocal to the difference between average network delay
$D_{av}(W)=\mathbb{E}[D(W)]$ and minimum delay $D_{min}(W)$ and $C$ is end-to-
end capacity.
It is necessary to mark that control signals are told by packets of the
different size that brings the additional contribution to a delay variation
(network jitter $j$). And, the less available bandwidth of end-to-end
connectivity, the network jitter will be stronger [4]. Therefore, the best the
controlling algorithm will form packages of the identical size. If we use the
utility ping for delay definition in it there is a special key for resizing
of a testing package ($-l$ in Windows, $-s$ in Linux).
## 6 Conclusion
In the present work for the description of process of the packet delay in a
global networks it has been chosen exponential distribution. In comparison
with truncated normal distribution it has shown the best correlation with
experimental results.
Experimental data were gathered by means of the precision RIPE measuring
system to within microseconds, and also by means of the standard utility ping.
This utility measures round-trip time to within milliseconds. During small
periods about several minutes when it is possible to consider conditions of
transmission on a TCP/IP network invariable, such approach gives correlations
from above 0.99. At change of network conditions the elementary ping testing
by a series from 10 packets will allow to change exponential distribution
parameters instantly.
In summary we would like to thank Leonid Fridmana, the professor from
University of Mexico for fruitful dialogues in which course the idea of this
article has taken shape.
## References
* [1] Choi, B.-Y., Moon, S., Zhang, Z.-L., Papagiannaki, K. and Diot, C.: Analysis of Point-To-Point Packet Delay In an Operational Network. In: Infocom 2004, Hong Kong, pp. 1797-1807 (2004)
* [2] Cottrell, L., Matthews, W. and Logg C.: Tutorial on Internet Monitoring $\&$ PingER at SLAC. http://www.slac.stanford.edu/comp/net/wan-mon/tutorial.html
* [3] Crovella, M.E. and Carter, R.L.: Dynamic Server Selection in the Internet. In: Proc. of the Third IEEE Workshop on the Architecture and Implementation of High Performance Communication Subsystems (1995)
* [4] Dovrolis C., Ramanathan P., and Moore D., Packet-Dispersion Techniques and a Capacity-Estimation Methodology, IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 12, NO. 6, DECEMBER 2004, p. 963-977
* [5] Downey A.B., Using Pathchar to estimate internet link characteristics, in Proc. ACM SICCOMM, Sept. 1999, pp. 222–223.
* [6] Elteto, T., Molnar, S., On the distribution of round-trip delays in TCP/IP networks, in The Proceedings of the Local Computer Networks (LCN 99) Conference, IEEE, 1999, pp. 172–181
* [7] Georgatos, F., Gruber, F., Karrenberg, D., Santcroos, M., Susanj, A., Uijterwaal, H. and Wilhelm R., Providing active measurements as a regular service for ISP’s. In: PAM2001
* [8] Georges J.-P., Divoux T., and Rondeau E., Confronting the performances of a switched ethernet network with industrial constraints by using the network calculus, _International Journal of Communication Systems(IJCS)_ , vol. 18, no. 9, pp. 877–903, 2005
* [9] Fridman E., Seuret A., and Richard J.-P., “Robust sampled-data stabilization of linear systems: an input delay approach,” _Automatica_ , vol. 40, pp. 1441–1446, 2004
* [10] Hespanha J.P., Naghshtabrizi P., and Xu Y., “A survey of recent results in networked control systems,” _Proceedings of the IEEE_ , vol. 95, pp. 138–162, January 2007
* [11] D. Papagiannaki, S. Moon, C. Fraleigh, P. Thiran, F. Tobagi, and C. Diot, Analysis of measured single-hop delay from an operational backbone network, in Proc. IEEE INFOCOM 2002, New York, New York, June 2002
* [12] A.M. Sukhov, T.G. Sultanov, M.V. Strizhov, A.P. Platonov, Throughput metrics and packet delay in TCP/IP networks, RIPE59 Meeting, Lisbon, 2009; arXiv:0907.3710
* [13] Ripe Test Box, http://ripe.net/projects/ttm/
* [14] Y. Tipsuwan and M.-Y. Chow, "Control methodologies in networked control systems," Control Engineering Practice, vol. 11, pp. 1099-1111, 2003
* [15] Zampieri S., “Trends in networked control systems,” _Proceedings of the 17th World Congress, The International federation of Automatic Control_ , July 2008
* [16] W. Zhang and M.S. Branicky and S.M. Phillips, Stability of Networked Control Systems, IEEE Control System Magazine, 21(1):84-99, February 2001
|
arxiv-papers
| 2009-07-26T09:39:39 |
2024-09-04T02:49:04.183952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.M. Sukhov, N. Kuznetsova",
"submitter": "Andrei Sukhov M",
"url": "https://arxiv.org/abs/0907.4468"
}
|
0907.4624
|
# Higher moments of charge fluctuations in QCD at high temperature
C. Miao for the RBC-Bielefeld collaboration
###### Abstract
We present lattice results for baryon number, strangeness and electric charge
fluctuations as well as their correlations at finite temperature and vanishing
chemical potentials, i.e. under conditions relevant for RHIC and LHC. We find
that the fluctuations change rapidly at the transition temperature $T_{c}$ and
approach the ideal quark gas limit already at approximately $1.5T_{c}$. This
indicates that quarks are the relevant degrees of freedom that carry the
quantum numbers of conserved charges at $T\geq 1.5T_{c}$. At low temperature,
qualitative features of the lattice results are well described by a hadron
resonance gas model.
††journal: Nuclear Physics A
## 1 Introduction
Fluctuations of conserved charges, like baryon number, electric charge and
strangeness, are generally considered to be sensitive indicators for the
structure of a thermal medium produced in heavy ion collisions [1]. Under
conditions met in current experiments at RHIC as well as in the upcoming heavy
ion experiments at LHC the net baryon number is small and QCD at vanishing
chemical potential provides a good approximation. In this region the
transition from the low temperature hadronic to the high temperature plasma
regime is continuous and fluctuations are not expected to lead to any singular
behavior. Indications for the existence of critical points can only show up in
higher order derivatives of the QCD partition function with respect to
temperature or chemical potentials [2]. Through the analysis of fluctuations
of conserved charges as well as their higher moments and correlations we thus
gain insight into the relevant degrees of freedom of the system under
consideration and at the same time gather information on possible nearby
singularities in the QCD phase diagram.
At vanishing baryon number ($B$), electric charge ($Q$) and strangeness ($S$)
fluctuations of these quantities can be obtained by starting from the QCD
partition function with non-zero light and strange quark chemical potentials,
$\hat{\mu}_{u,d,s}\equiv\mu_{u,d,s}/T$. The quark chemical potential can be
expressed in terms of chemical potentials for baryon number ($\mu_{B}$),
strangeness ($\mu_{S}$) and electric charge ($\mu_{Q}$),
$\mu_{u}=\frac{1}{3}\mu_{B}+\frac{2}{3}\mu_{Q}\
,\qquad\mu_{d}=\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q}\
,\qquad\mu_{s}=\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q}-\mu_{S}\ .$ (1)
Moments of charge fluctuations, $\delta N_{X}\equiv N_{X}-\left<N_{X}\right>$,
with $X=B$, $Q$ or $S$ and their correlations are then obtained from
derivatives of the logarithm of the QCD partition function, _i.e._ the
pressure, evaluated at $\mu_{B,Q,S}=0$,
$\chi_{i,j,k}^{B,Q,S}=\left.\frac{\partial^{i+j+k}p/T^{4}}{\partial\hat{\mu}_{B}^{i}\partial\hat{\mu}_{Q}^{j}\partial\hat{\mu}_{S}^{k}}\right|_{\mu=0}\
,$ (2)
with $\hat{\mu}_{X}\equiv\mu_{X}/T$. While the first derivatives, _i.e._
baryon number, electric charge and strangeness densities, vanish for
$B,Q,S=0$, their moments and correlation functions with $i+j+k$ even are non-
zero. The basic quantities we will analyze here are the quadratic, quartic and
6th order cumulant of fluctuations,
$\chi_{2}^{X}=\frac{1}{VT^{3}}\langle
N_{X}^{2}\rangle,\quad\chi_{4}^{X}=\frac{1}{VT^{3}}(\langle
N_{X}^{4}\rangle-3\langle
N_{X}^{2}\rangle^{2}),\quad\chi_{6}^{X}=\frac{1}{VT^{3}}(\langle
N_{X}^{6}\rangle-15\langle N_{X}^{4}\rangle\langle N_{X}^{2}\rangle+30\langle
N_{X}^{2}\rangle^{3}).$ (3)
Figure 1: The quadratic fluctuations of light (left) and strange (middle)
quark number versus temperature on lattices of size $24^{3}\times 6$ and
$32^{3}\times 8$. The horizontal solid lines show results for the ideal
massless quark gas; the vertical lines mark the temperature interval
$185~{}\mathrm{MeV}\leq T\leq 195~{}\mathrm{MeV}$. The ratio of s and u quark
fluctuations (right) are calculated on lattices with temporal extent
$N_{\tau}=8$. Curves in the right hand figure show results for a hadron
resonance gas including resonance up to $m_{max}=1.5\mathrm{GeV}$ (upper
branch) and $2.5\mathrm{GeV}$ (lower branch), respectively.
The gauge field configurations that have been used to evaluate the above
observables, had been generated previously in calculations of the QCD equation
of state [3, 4] and the transition temperature [5, 6]. An improved staggered
fermion action (p4-action) [7] that strongly reduces lattice cut-off effects
in bulk thermodynamics at high temperature has been used. The strange quark
mass has been tuned close to its physical value and the light quark masses
have been chosen to be one tenth of the strange quark mass. This corresponds
to a line of constant physics on which the kaon mass is close to its physical
value and the lightest pseudo-scalar mass is about $220$ MeV.
These calculations have been performed on $16^{3}\times 4$, $24^{3}\times 6$
and $32^{3}\times 8$ lattices. This allows us to judge the magnitude of
systematic effects arising from discretization errors in our improved action
calculations. The spatial volume has been chosen to be $V^{1/3}T=4$, which
insures that finite volume effects are small. In this calculations, the above
observables have been evaluated in the temperature interval $0.8\lesssim
T/T_{c}\lesssim 2.5$.
## 2 Results
Before entering a discussion of fluctuations of B, Q and S it is instructive
to look into fluctuations of the partonic degrees of freedom, the light and
strange quarks. In Fig. 1 we show the temperature dependence of quadratic
fluctuations for light quarks $\chi_{2}^{l}$ and strange quarks
$\chi_{2}^{s}$. They change rapidly in the transition region as the carriers
of the quantum numbers are heavy hadrons at low temperatures but much lighter
quarks at high temperatures. In the continuum and high temperature limit,
these fluctuations quickly approach the Stephan-Boltzmann limit, _i.e._ an
ideal massless quark gas.
The lattice cut-off effects for quadratic fluctuations are shown to be small
by comparing the measurements on $N_{\tau}=6$ and 8 lattices. Moreover, we
have used two different cut-off schemes, p4 and asqtad [8]. Both schemes are
improved up to $O(a^{2})$ for thermodynamic quantities at high temperature.
Indeed, the quark fluctuations have small deviations from SB limit at high
temperature on $N_{\tau}=8$ lattices for both schemes. The measurements using
p4 and asqtad actions are generally in good agreement, although we observe
differences in the temperature interval between $T_{c}$ and $1.5~{}T_{c}$.
We observe that the strange quark fluctuations rise slower than the light
quark fluctuations. As temperature decreases, their ratio
$\chi_{2}^{s}/\chi_{2}^{l}$ (Fig. 1) drops quickly in the transition
region.The hadron resonance gas (HRG) model calculations show a similar trend
of exponential fall at low temperature. This can be understood in the zero
temperature limit: the light quark fluctuations are sensitive to pions,
$\chi_{2}^{l}/T^{2}\sim\exp(-m_{\pi}/T)$, while the strange quark fluctuations
are sensitive to the lightest hadronic state that carry strangeness,
$\chi_{2}^{s}/T^{2}\sim\exp(-m_{K}/T)$. The HRG models can not reproduce the
lattice measurements at higher temperature, as it breaks down at $T\simeq 150$
MeV, where HRG models with different spectrum cuts start to show deviations.
Figure 2: Quadratic, quartic and 6th order cumulant of fluctuations for baryon
number, electric charge and strangeness. All quantities have been normalized
to the corresponding free quark gas values.
In Fig. 2 we show results for quadratic fluctuations $\chi_{2}^{X}$, quartic
fluctuations $\chi_{4}^{X}$ on $N_{\tau}=4$ and 6 lattices and 6th order
cumulants of fluctuations $\chi_{6}^{X}$ on $N_{\tau}=4$ lattices, where
$X=B,~{}Q$ and $S$. As can be seen, in all cases the quadratic fluctuations
rise rapidly in the transition region where the quartic fluctuations show a
maximum and 6th order cumulants change sign. This generic form is indeed
expected. In the chiral limit the 4th order cumulants will have a peak and the
6th order cumulants will diverge, e.g. the Baryon number cumulants are
expected to scale like
$\chi_{2n}^{B}\sim\left|\frac{T-T_{c}}{T_{c}}\right|^{2-n-\alpha}$ (4)
where $\alpha\simeq-0.25[-0.015]$ denoting the critical exponent of the
specific heat in 3-d, $O(4)$ [$O(2)$] spin models [9].
At temperature $T\gtrsim 1.5T_{c}$ the quadratic, quartic and 6th order
cumulants of fluctuations of $B$, $Q$ and $S$ are well described by those of
an ideal, massless quark gas. At low temperature, we compare lattice results
for the ratios of the cumulants with HRG model calculations (Fig. 3). In the
framework of a HRG model the ratio of fourth and second order cumulants of
baryon number fluctuations is independent of the actual value of hadron
masses; $(\chi_{4}^{B}/\chi_{2}^{B})_{HRG}=1$ if all hadrons are heavy on the
scale of the temperature. This is well reproduced by the lattice results.
However, in the chiral limit it is expected that the cusp in $\chi_{4}^{B}$
(Fig. 2) is expected to become more pronounced and thus more prominent also in
the ratio $\chi_{4}^{B}/\chi_{2}^{B}$.
For $\chi_{4}^{S}/\chi_{2}^{S}$ and $\chi_{4}^{Q}/\chi_{2}^{Q}$, the lattice
results also agree with HRG models qualitatively. The cumulants of electric
charge fluctuations receive contribution from the lightest hadrons, pions. In
our calculations the masses of the lightest pseudoscalar states, $220$ MeV,
are close to but not exact as the physical pion masses. Therefore, we have
analyzed the cumulant ratio $\chi_{4}^{Q}/\chi_{2}^{Q}$ in the HRG models with
the pion masses at physical value $140$ MeV, $220$ MeV and infinity. We
observe that the cumulant ratio is sensitive to the pion sector; with pion
masses that are $50\%$ percent larger than the physical value, the
contribution of the pion sector is drastically reduced. The pion sector
contribution are more important for the 6th order cumulants than lower order
cumulants. In Fig. 3 (right) we show the cumulant ratio
$\chi_{6}^{Q}/\chi_{2}^{Q}$. The lattice results agree with both HRG models
without pion sector or with $220$ MeV pseudoscalar scalars, while the HRG with
physical pions shows much larger discrepancy.
Figure 3: The ratio of fourth and second order cumulants of baryon number
(first), strangeness (second) and electric charge (third) fluctuations and the
ratio of sixth and second order cumulants of electric charge fluctuations
(fourth). In the last two cases we show curves for a HRG model calculated with
physical pion masses (upper curve), pions of mass 220 MeV (middle) and
infinitely heavy pions (lower curve).
## 3 Conclusion
We have analyzed the fluctuations of baryon number, electric charge and
strangeness in finite temperature QCD at vanishing chemical potential. We find
fluctuations and correlations of conserved charges are well described by an
ideal, massless quark gas already for temperatures of about ($1.5\sim 1.7$)
times the transition temperature. At low temperature we find that fluctuations
of conserved charges are well described by a hadron resonance gas up to
temperatures close to the transition temperature.
The current analysis has been performed with light quarks that are one tenth
of the strange quark mass. We have shown, that higher order cumulant ratios
like, for instance $\chi_{6}^{Q}/\chi_{2}^{Q}$, become quite sensitive to the
pseudoscalar Goldstone mass. In numerical calculations with staggered fermions
this also means that results become sensitive to a correct representation of
the entire Goldstone multiplet. Calculations with smaller quark masses closer
to the continuum limit will thus be needed in the future to correctly resolve
these higher order cumulants, which will give deeper insight into the range of
applicability of the resonance gas model at low temperature and the non-
perturbative features of the QGP above but close to Tc.
## References
* [1] for a review see: V. Koch, Hadronic Fluctuations and Correlations, arXiv:0810.2520.
* [2] for a review see: M. Stephanov, The phase diagram of QCD and the critical point, Acta Phys. Polon. B 35, 2939 (2004).
* [3] M. Cheng et al., Phys. Rev. D 77, 014511 (2008).
* [4] A. Bazavov et al., arXiv:0903.4379 [hep-lat]
* [5] M. Cheng et al., Phys. Rev. D 74, 054507 (2006).
* [6] HotQCD Collaboration, in preparation.
* [7] A. Peikert, B. Beinlich, A. Bicker, F. Karsch and E. Laermann, Nucl. Phys. Proc. Suppl. 63, 895 (1998).
* [8] K. Orginos and D. Toussaint, Phys. Rev. D 59, 014501 (1999); G.P. Lepage, Phys. Rev. D 59, 074502 (1999).
* [9] A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002)
|
arxiv-papers
| 2009-07-27T13:30:20 |
2024-09-04T02:49:04.192248
|
{
"license": "Public Domain",
"authors": "C. Miao (for the RBC-Bielefeld collaboration)",
"submitter": "Chuan Miao",
"url": "https://arxiv.org/abs/0907.4624"
}
|
0907.4627
|
# Highlights from PHENIX-I: Initial State and Early Times
Michael Leitcha (for the PHENIX collaboration) Los Alamos National
Laboratory, P-25 MS H-846, Los Alamos, NM, 87545, USA
###### Abstract
We will review the latest physics developments from PHENIX concentrating on
cold nuclear matter effects, the initial state for heavy-ion collisions, and
probes of the earliest stages of the hot-dense medium created in those
collisions. Recent physics results from $p+p$ and $d+Au$ collisions; and from
direct photons, quarkonia and low-mass vector mesons in A+A collisions will be
highlighted. Insights from these measurements into the characteristics of the
initial state and about the earliest times in heavy-ion collisions will be
discussed.
††journal: Nuclear Physics A
## 1 Introduction
In this overview, we will discuss selected highlights from PHENIX in the areas
relating to the initial state and early times, focusing only on those which we
believe to be the most significant new results. These will include 1) the
suppression of rapidity-separated hadron pairs in $d+Au$ collisions, 2) the
contributions of quarkonia and Drell-Yan to the non-photonic single electrons
used to detect heavy quarks, 3) the continued suppression of the $J/\psi$ in
$Cu+Cu$ collisions to high-$p_{T}$, 4) the suppression of $\Upsilon$s in
$Au+Au$ collisions, and 5) estimates of the initial temperature from direct
photons in $Au+Au$ collisions.
## 2 Cold Nuclear Matter (CNM) and Gluon Saturation
The physics that modifies hard processes in nuclei relative to those on a free
nucleon, often called cold nuclear matter (CNM) effects, includes 1)
traditional shadowing either from global fits or from coherence models, 2)
gluon saturation at small momentum fraction ($x$) which is amplified in the
nuclear environment, and 3) initial-state energy loss and multiple scattering.
For hadron pairs with a rapidity separation between the two hadrons in the
pair, where one ”triggers” on a mid-rapidity ($|\eta|<0.35$) hadron and
studies correlations with a forward-rapidity ($3.1<\eta<3.9$) hadron, there
are two pictures which attempt to describe the characteristics of the process.
QCD based pictures, such as those used by Vitev [1] that include non-leading-
twist shadowing, give suppression of the pairs compared to the mid-rapidity
trigger particle. An alternative approach which represents gluon saturation in
the color-glass-condensate (CGC) model [2] also gives suppression and gives
broadening of the angular correlation peak between the two particles in the
pair. In the CGC picture, a mono-jet mechanism becomes important, where a
single jet has its momentum balanced by multiple gluons coupling to the
saturated gluon field.
Figure 1: (Color online) Centrality dependence of $I_{dAu}$ for rapidity-
separated hadron pairs (left). Correlation width, $\Delta\phi$, vs $p_{T}$ of
the associated mid-rapidity ($|\eta|<0.35$) $\pi^{0}$ (filled points) for
$p+p$ and for different centrality $d+Au$ collisions, showing no broadening
within the substantial uncertainties of the data points (right). Also shown
are similar results for higher energy clusters (open symbols) where
$\pi^{0}$’s and photons are not resolved.
Using the new Muon Piston Calorimeters (MPC) in PHENIX we are able to study
correlations of rapidity-separated hadron (h± or $\pi^{0}$) pairs, where one
triggers on a hadron at mid-rapidity and studies correlations with hadrons at
forward rapidity in the MPC. For these studies we use the ratio $I_{dAu}$,
which is the pair efficiency relative to the mid-rapidity ”trigger” hadron for
$d+Au$ divided by that for $p+p$,
$I_{dAu}={{N^{pair}_{d+Au}[(\eta=3.5)+(\eta=0)]/N^{trig}_{d+Au}(\eta=0)}\over{N^{pair}_{p+p}[(\eta=3.5)+(\eta=0)]/N^{trig}_{p+p}(\eta=0)}}$
(1)
Preliminary results [3] for the centrality dependence (in terms of number of
collisions, $N_{coll}$) of $I_{dAu}$ in Fig. 1 (left) show increasing
suppression for more central collisions. The angular correlations of the pairs
were also studied, but showed no broadening in the relative angle $\Delta\Phi$
outside the substantial uncertainties in the present preliminary result, Fig.
1 (right).
We have also studied hadron pairs in $d+Au$ collisions where both hadrons are
at mid-rapidity, this time in terms of $J_{dAu}$ which is basically the same
as $R_{dAu}$ for a single particle, but in this case for pairs,
$J_{dAu}={{(PairYield)_{dAu}}\over{<N_{coll}>(PairYield)_{pp}}}$ (2)
These pairs exhibit a very large Cronin-like enhancement, i.e. they scale
faster than $N_{coll}$ ($J_{dAu}>1$) and both $J_{dAu}$ and the angular
correlation width decrease for larger $p_{T}$ [4].
## 3 Open Heavy Quarks
Recent studies of the contribution of quarkonia and Drell-Yan to the spectrum
of single electrons from heavy quarks have determined that for transverse
momenta above about 5 GeV/$c$ these contributions can amount to up to 16% of
the total non-photonic electron yield [5]. The contributions of $J/\psi$,
$\Upsilon$, and Drell-Yan are shown in Fig. 2 (left) and one can see that the
$J/\psi$ gives the dominant contribution. After subtracting off these
estimates of the contributions to the $p+p$ collision data, with careful
attention to their uncertainty, shown in Fig. 2 (right), the net yield of
electrons from heavy quark decay has moved from a little above, to slightly
below, the FONLL model’s upper uncertainty limit. Similar corrections, but
with larger uncertainties have been applied for $Au+Au$ collisions. However,
because both $p+p$ and $Au+Au$ are lowered by about the same amount, the
resulting nuclear depencence in $Au+Au$ collisions, $R_{AA}$, is not
significantly changed.
Figure 2: (Color online) Contributions of quarkonia and Drell-Yan to the
background for single electrons from heavy meson decays and the change
relative to the background before their inclusion (left). Single electron
spectrum from heavy quarks corrected for these contributions compared to a
FONLL calculation [6] (right).
Most open-heavy flavor meson measurements at RHIC to date are not able to
separate contributions to the single electrons from charm and beauty, while
theoretical predictions of energy loss and flow in the hot-dense medium
created in high-energy heavy ion collisions are generally quite different for
charm and beauty. Recently a new method has been employed, where one studies
the correlations of hadrons near the observed electron and exploits the fact
that the decay of a beauty meson into an electron and hadron produces a
broader correlation and lower efficiency for observing the pair than that of a
charm meson. Using this technique, the fraction of $(b\rightarrow
e)/(b+c\rightarrow e)$ has been determined [7] and is shown vs $p_{T}$ in Fig.
3 (left).
PHENIX has also measured open-heavy flavor mesons at forward rapidity via
their decay to single muons, but so far not with enough precision to define
the shape of the cross section vs rapidity. However, three different methods
in PHENIX now yield consistent cross sections in $p+p$ collisions at mid
rapidity: single electrons via cocktail subtraction, a converter method, and
with di-electrons. Using the beauty fraction determined above, a beauty cross
section of $\sigma_{b{\bar{b}}}=3.2^{+1.2}_{-1.1}(stat)^{+1.4}_{-1.3}(sys)\
{\mu}b$ has also been determined [7].
Finally, the first proof-of-princible measurement of charm pairs via electron-
muon correlations in $p+p$ collisions has been made [8] and is shown in Fig. 3
(right). The peak at $\pi$ radians in $\Delta\Phi$ is from these correlated
pairs. This method promises to provide another independent measurement of
charm in the near future, as luminosities increase and allow substantial
yields for this rare signal.
## 4 Quarkonia Production and Suppression
The simultaneous theoretical description of both the cross section and the
polarization of the $J/\psi$ in hadron production has long been a challenge. A
new analysis of the 2006 PHENIX $p+p$ data agrees well with the previous
results, has significantly higher precision, and agrees well with the Lansberg
s-channel cut color-singlet model [9]. The decay polarization of the $J/\psi$
measured by PHENIX at mid and forward rapidity is shown in Fig. 4 (left),
where the Lansberg model reproduces the small polarization falling with
$p_{T}$ at mid rapidity (red points), but predicts a larger polization than
the null polarization seen at forward rapidity (by 2-3 sigma) [10]. Improved
polarization measurements at forward rapidity in several bins in $p_{T}$ are
expected soon, and may help clarify the situation.
Figure 3: (Color online) Fraction of beauty, $(b\rightarrow e)/(b+c\rightarrow
e)$, vs single-electron transverse momentum compared to FONLL calculations [6]
(left). Early electron-muon pair charm signal for $p+p$ collisions (right).
Figure 4: (Color online) Polarization vs $p_{T}$ in the helicity frame for
$J/\psi$ production in 200 GeV $p+p$ collisions with mid-rapidity points as
red circles and forward rapidity points as blue squares (left). $R_{CP}$ vs
rapidity for $J/\psi$ production in 200 GeV $d+Au$ collisions for three
different centrality bins, with the most central collisions (0-20%) on the
bottom (right).
New results for the $J/\psi$ from the 2008 $d+Au$ run with approximately
thirty times larger integrated luminosity than that of the previous (2003)
$d+Au$ results are beginning to emerge, with the first preliminary result in
terms of $R_{CP}$,
$R_{CP}^{0-20\%}={{N_{inv}^{0-20\%}/<N_{coll}^{0-20\%}>}\over{N_{inv}^{60-88\%}/<N_{coll}^{60-88\%}>}},$
(3)
shown in Fig. 4 (right) vs rapidity for three different centrality bins [10,
11]. One sees essentially no nuclear dependence at backward rapidity, a little
at mid rapidity, and increasing suppression with centrality at forward
rapidity in the nuclear shadowing region (large rapidity corresponds to small
momentum fraction down to about $x=2\times 10^{-3}$ and is in the shadowing
region). PHENIX is working on more comprehensive results for the near future
in terms of $R_{dAu}$, the nuclear dependence relative to $p+p$ \- the much
higher statistical precision of this new data requires precision systematics
and more careful analysis.
New preliminary results for $J/\psi$ $R_{AA}$ in $Cu+Cu$ collisions show
continuing suppression up to at least 7 GeV/$c$ in $p_{T}$. In Fig. 5 (left)
this suppression is compared to several theoretical models, including the
”hot-wind” AdS/CFT inspired model [12] which is inconsistent with the data.
Eventually, due to the Cronin effect seen in $d+Au$ collisons, which causes a
change from suppression to enhancement at high-$p_{T}$, one would expect
$R_{AA}$ to return to unity at large $p_{T}$, but there is no evidence of that
yet from these results.
Figure 5: (Color online) $R_{CuCu}$ vs $p_{T}$ for mid-rapidity $J/\psi$s out
to 9 GeV/$c$ compared to several theoretical models [12, 17] (left). Invariant
mass spectrum at mid rapidity for $p+p$ collisions at high mass showing the
$\Upsilon$ family and other components of the spectrum.
Figure 6: (Color online) Di-electron pair mass at mid rapidity for 200 GeV
Au+Au collisions in the $\Upsilon$ mass region, where $e^{+}e^{-}$ pairs are
shown as black points, like-sign pairs as red points, and mixed $e^{+}e^{-}$
background pairs in green (left). The probability distribution vs $R_{AuAu}$
determined from the $Au+Au$ and $p+p$ data (right).
With the increasing luminosities provided by the RHIC machine, PHENIX is now
beginning to accumulate useful number of $\Upsilon$s for various kinds of
collisions. From the 2006 $p+p$ run, as shown in Fig. 5 (right), we now have a
preliminary cross section for dielectron events in the $\Upsilon(1S+2S+3S)$
mass region [8.5,11.5 GeV/$c^{2}$] of $BR*d\sigma/dy\
(|y|<0.35)=114^{+46}_{-45}\ pb$ . A small number of dielectron pairs from
Drell-Yan and from open beauty pairs may also contribute in that mass region,
but this contribution is estimated to be less than 15% and is included in the
systematic uncertaintiy. Using a similar signal for $Au+Au$ collisions, shown
in Fig. 6 (left), and doing a very careful statistical analysis which takes
into account the small numbers of counts in both the $Au+Au$ and $p+p$
$\Upsilon$ mass regions, we have obtained the probability distribution for
$R_{AuAu}$ in this mass region shown in Fig. 6 (right). From this an upper
limit of $R_{AuAu}<0.64$ at 90% C.L. is determined [13].
Although $\Upsilon$s have long been touted as the standard candle for the
melting of quarkonia in the Quark Gluon Plasma (QGP), i.e. that they would not
be screened up to very high temperatures, it is clear that there are a number
of simple non-QGP effects that could easily cause a suppression at or below
the upper limit determined above. These include 1) the suppression of
$\Upsilon$ states seen in fixed target experiments [14] which would give about
$0.81^{2}$ in $R_{AuAu}$, 2) the fact that only about 52% of the
$\Upsilon_{1S}$ do not come from feeddown from the higher mass (2S, 3S)
$\Upsilon$ states and B decays [15], and 3) that we do not resolve the three
$\Upsilon$ states (1S+2S+3S) and the 1S is only about 73% of the total [16].
## 5 Initial State and Temperature
Figure 7: (Color online) Direct photon spectra vs $p_{T}$ in $p+p$ and $Au+Au$
collisions at 200 GeV, with the scaled $p+p$ reference compared to the data
from various centrality $Au+Au$ data as a black dashed line. The enhancement
over the $p+p$ reference at small $p_{T}$ for $Au+Au$ collisions is fit to an
exponential (solid black lines) to extract an inverse slope and an effective
temperature (left). Estimates of the initial temperature of the hot-dense
medium from several different models for the expansion of that medium, given
the average temperature of 221 MeV determined from the central $Au+Au$ spectra
(right).
Direct photon production in nucleus-nucleus collisions, although a difficult
measurement, is a clean probe of the initial-state gluon distributions in the
colliding nuclei. The latest measurements in PHENIX show no modification
relative to $p+p$ collisions except for transverse momenta above about 12
GeV/$c$. These modifications are likely due to CNM effects (Cronin) and isopin
(neutrons vs protons) [18].
A new method has been used recently to extract the yield of photons in the
low-$p_{T}$ thermal region where the production of these photons is inferred
from the low-mass ($M_{ee}<\ $300 GeV/$c^{2}$) $1<p_{T}<\ $5 GeV/$c$
$e^{+}e^{-}$ spectrum [19]. These low-mass photons show an enhancement over
the scaled $p+p$ reference, as shown in Fig. 7 (left). If interpreted as
thermal photons from the hot-dense medium and fit to an exponential slope, an
average temperature of the medium (for central collisions) of $T_{avg}=221\pm
23\pm$18 MeV is obtained. Since this is the average over the expansion, one
can ask within various theoretical descriptions for that expansion, what the
initial temperature is. Fig. 7 (right) shows various initial temperatures vs
the formation time assumed in each theoretical picture. All models indicate an
initial temperature of at least 300 MeV, well above the predicted QGP phase
transition at 170 MeV.
## 6 Summary
We highlight here some of the recent PHENIX results that we believe to be most
interesting in areas that relate to the initial state and early times,
including:
1. 1.
Quarkonia contribute substantially to the electrons from heavy flavor for
$p_{T}>\ $5 GeV/$c$ and should be taken into account when comparing to
theoretical predictions.
2. 2.
Beauty decays give 50% or more of the single electrons for $p_{T}>\ $4
GeV/$c$, so any differences between beauty and charm for energy loss and flow
may become apparent at these $p_{T}$ values.
3. 3.
$J/\psi$ polarization measurements at mid rapidity agree with the Lansberg
color singlet model, but at forward rapidity the $p_{T}$-integrated value does
not.
4. 4.
For $Cu+Cu$ collisions, $J/\psi$s continue to be strongly supressed up to
$p_{T}\simeq\ $8 GeV/$c$.
5. 5.
Events in the ${\Upsilon}(1S+2S+3S)$ mass region at mid rapidity are
suppressed in $Au+Au$ collisions by at least 36%, but this is not unexpected
given cold nuclear matter effects and the likely strong suppression for
central $Au+Au$ collisions of the higher mass $\Upsilon$ states.
6. 6.
Direct photons measured in the thermal region for central $Au+Au$ collisions
indicate intial temperatures of at least 300 MeV, well above the expected QGP
phase transition.
## References
* [1] J. Qiu and I. Vitev, Phys.Lett. B632, 507 (2006) [hep-ph/0405068].
* [2] D. Kharzeev et al., Nucl. Phys. A 748, 727 (2005).
* [3] B. Meredith (for the PHENIX Collaboration), these proceedings.
* [4] J. Jia (for the PHENIX Collaboration), QM09 poster [arXiv:0906.3776].
* [5] A. Dion (for the PHENIX Collaboration), these proceedings.
* [6] M. Cacciari et al., Phys. Rev. Lett 95, 122001 (2005); and private communication.
* [7] A. Adare et al. (PHENIX Collaboration), [arXiv:0903.4851].
* [8] T. Engelmore (for the PHENIX Collaboration), these proceedings.
* [9] H. Haberzettl, J.P. Lansberg, Phys. Rev. Lett. 100, 032006 (2008) [arXiv:0709.3471].
* [10] C. da Silva (for the PHENIX Collaboration), these proceedings.
* [11] D. McGlinchey (for the PHENIX Collaboration), QM09 poster.
* [12] H. Liu, K. Rajogopal, U. Wiedemann, Phys. Rev. Lett. 98, 182301 (2007) [arXiv:hep-ph/0607062].
* [13] E. Atomssa (for the PHENIX Collaboration), these proceedings.
* [14] D. Alde, et al. (E772 Collaboration), Phys. Rev. Lett. 64, 2479 (1990).
* [15] S. Digal, et al. Phys. Rev. D 64, 094015 (2001).
* [16] The CDF Collaboration, Published Proceedings Les Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy, March 3-9, 1996. FERMILAB-CONF-96/110-E; and G. Moreno et al. (E605) Phys. Rev. D43, 2815 (1991).
* [17] X. Zhao, R. Rapp Phys. Lett. B664, 253 (2008) [hep-ph/07122407]; X.M. Xu, D. Kharzeev, H. Satz, X.N. Wang, Phys. Rev. C53 3051 (1996) [hep-ph/9511331]; B.K. Patra, V.J. Menon, Eur. Phys. J C44, 567 (2005) [nucl-th/0503034].
* [18] B. Zhang, I. Vitev, [nucl-th/0810.3194].
* [19] Y. Yamaguchi (for the PHENIX Collaboration), these proceedings, and [nucl-ex/0804.4168].
|
arxiv-papers
| 2009-07-27T14:58:05 |
2024-09-04T02:49:04.196885
|
{
"license": "Public Domain",
"authors": "Michael Leitch (for the PHENIX collaboration)",
"submitter": "Michael Leitch",
"url": "https://arxiv.org/abs/0907.4627"
}
|
0907.4685
|
# The uncertainty in Galactic parameters
Paul J. McMillan, and James J. Binney
Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP,
UK E-mail: p.mcmillan1@physics.ox.ac.uk
###### Abstract
We reanalyse the measurements of parallax, proper motion, and line-of-sight
velocity for 18 masers in high mass star-forming regions presented by Reid et
al. (2009). We use a likelihood analysis to investigate the distance of the
Sun from the Galactic centre, $R_{0}$, the rotational speed of the local
standard of rest, $v_{0}$, and the peculiar velocity of the Sun,
$\mathbf{v_{\odot}}$, for various models of the rotation curve, and models
which allow for a typical peculiar motion of the high mass star-forming
regions.
We find that these data are best fit by models with non-standard values for
$\mathbf{v_{\odot}}$ or a net peculiar motion of the high mass star-forming
regions. We argue that a correction to $\mathbf{v_{\odot}}$ is much more
likely, and that these data support the conclusion of Binney (2009) that
$V_{\odot}$ should be revised upwards from $5.2\,\mathrm{km\,s}^{-1}$ to
$11\,\mathrm{km\,s}^{-1}$. We find that the values of $R_{0}$ and $v_{0}$ that
we determine are heavily dependent on the model we use for the rotation curve,
with model-dependent estimates of $R_{0}$ ranging from $6.7\pm
0.5\,\mathrm{kpc}$ to $8.9\pm 0.9\,\mathrm{kpc}$, and those of $v_{0}$ ranging
from $200\pm 20\,\mathrm{km\,s}^{-1}$ to $279\pm 33\,\mathrm{km\,s}^{-1}$. We
argue that these data cannot be thought of as implying any particular values
of $R_{0}$ or $v_{0}$. However, we find that $v_{0}/R_{0}$ is better
constrained, lying in the range
$29.9-31.6\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$ for all models but one.
###### keywords:
Galaxy: fundamental parameters – methods: statistical – Galaxy: kinematics and
dynamics
## 1 Introduction
The fundamental parameters that define the Solar position and velocity within
the Galaxy remain uncertain to a remarkable degree. The major remaining
uncertainty is that in the distance from the Sun to the Galactic centre,
$R_{0}$. The most recent results from studies of stellar orbits in the
Galactic centre (Ghez et al., 2008; Gillessen et al., 2009) give values of
$R_{0}=(8.4\pm 0.4)\,\mathrm{kpc}$ and $R_{0}=(8.33\pm 0.35)\,\mathrm{kpc}$
respectively. These can be compared to the earlier estimate in the review by
Reid (1993) of $R_{0}=(8.0\pm 0.5)\,\mathrm{kpc}$.
The total velocity of the Sun about the Galactic centre is the sum of the
velocity of the local standard of rest (LSR), $v_{0}$, and the peculiar motion
of the Sun with respect to the LSR in the same direction, $V_{\odot}$. This
total velocity can be determined using the apparent proper motion of Sgr A*,
$\mu_{A*}$, since it is expected to be moving with a peculiar velocity less
than $\sim 1\,\mathrm{km\,s}^{-1}$ at the Galactic Centre. This constraint on
the peculiar motion of Sgr A* is justified by the observation that the
velocity of Sgr A* perpendicular to the plane is consistent with zero, with
uncertainties $\sim 1\,\mathrm{km\,s}^{-1}$ (Reid & Brunthaler, 2004), and
because this motion is thought to be due to stochastic forces from discrete
interactions with individual stars, so the velocity in the plane should be
similar to that perpendicular to it (Chatterjee, Hernquist, & Loeb, 2002).
Reid & Brunthaler (2004) found $\mu_{A*}=(6.379\pm 0.024)\,\mathrm{mas\
yr}^{-1}$, which corresponds to $(v_{0}+V_{\odot})/R_{0}=(30.2\pm
0.2)\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$, or a velocity about the
Galactic centre (using the Ghez et al. result for $R_{0}$) of
$v_{0}+V_{\odot}=(252\pm 11)\,\mathrm{km\,s}^{-1}$, where by far the dominant
uncertainty comes from the value of $R_{0}$. Analysis of the GD-1 stellar
stream (Koposov, Rix, & Hogg, 2009) has recently been used to suggest a
significantly lower value, $v_{0}=221^{+16}_{-20}\,\mathrm{km\,s}^{-1}$.
The velocity of the Sun with respect to the local standard of rest,
$\mathbf{v_{\odot}}$, is often assumed to be well known and constrained to
within $\sim 0.5\,\mathrm{km\,s}^{-1}$ because of the analysis of the dynamics
of nearby stars conducted by Dehnen & Binney (1998, henceforth DB98), and
again more recently, using identical techniques, by Aumer & Binney (2009).
DB98 found
$\begin{array}[]{ll}\mathbf{v_{\odot}}&\equiv(U_{\odot},V_{\odot},W_{\odot})\\\
&=(10.00\pm 0.36,\,5.25\pm 0.62,\,7.17\pm 0.38)\,\mathrm{km\,s}^{-1},\\\
\end{array}$ (1)
which has been widely accepted and used. However Binney (2009, henceforth B09)
suggests that the value for $V_{\odot}$ determined in these papers may be an
underestimate by $\sim 6\,\mathrm{km\,s}^{-1}$. This is because the analysis
by DB98 uses Stromberg’s equation, which is derived under the assumption that
the Galactic potential is axisymmetric, and extrapolates to zero velocity
dispersion. In practice, the Galactic potential is not axisymmetric, and the
smaller the velocity dispersion of a population, the more it is affected by
departures from axisymmetry. B09 uses a more global approach, ensuring that
more emphasis is placed on stellar populations with high velocity dispersions,
which one would expect to be less affected by the non-axisymmetry of the
Galactic potential.
Reid et al. (2009) brought together observations of masers seen in high mass
star-forming regions (hmsfrs) in the Milky Way. They used simple statistical
tools in an effort to determine the values of $R_{0}$ and $v_{0}$, using the
DB98 value of $\mathbf{v_{\odot}}$, initially under the assumption that the
hmsfrs were moving on circular orbits in a flat rotation curve, for which they
offer best-fitting parameters $R_{0}=(8.24\pm 0.55)\,\mathrm{kpc}$ and
$v_{0}=(265\pm 26)\,\mathrm{km\,s}^{-1}$, but with a high $\chi^{2}$ value.
In addition they considered a model in which the hmsfrs were moving with a
characteristic velocity with respect to their local circular velocity. They
found that they achieved a significantly improved fit to their data using this
model with a peculiar velocity $\sim 15\,\mathrm{km\,s}^{-1}$ in the opposite
direction to rotation. This fit yielded $R_{0}=(8.40\pm 0.36)\,\mathrm{kpc}$
and $v_{0}=(254\pm 16)\,\mathrm{km\,s}^{-1}$. They briefly considered a model
in which the value of $\mathbf{v_{\odot}}$ was allowed to vary, finding an
acceptable fit with $\mathbf{v_{\odot}}=(9,20,10)\,\mathrm{km\,s}^{-1}$.
However, believing the DB98 result to be “well determined” they did not pursue
the matter further.
In this paper we re-examine the data described in Reid et al. (2009), and
conduct a likelihood analysis for various models of the velocity distribution
of the maser sources. This enables us to exploit these data more thoroughly.
We also consider the implications of the results from B09 on the
interpretation of these data. In Section 2 we explain what the data consist
of, describe our models and our statistical technique; in Section 3 we give
the raw results and examine their significance. We discuss the implications of
these results in Section 4.
## 2 Methods
### 2.1 The data
These data, as given in Table 1 of Reid et al. (2009), consist of measurements
for 18 masers of: Galactic coordinates ($l_{i}$, $b_{i}$), which we can assume
to be exact; parallaxes $\pi_{i}$; proper motions $\mu_{x,i}$ and $\mu_{y,i}$;
line-of-sight velocities $v_{\mathrm{LSR},i}$. For each quantity Reid et al.
give an error and we assume that this error together with the measured value
of the quantity defines a Gaussian probability distribution for the true value
of the quantity. For the proper motions $\mu_{x}$ and $\mu_{y}$ we shall use
the more conventional notation $\mu_{\alpha}$ and $\mu_{\delta}$. The line-of-
sight velocity $v_{\rm LSR}$ is relative to an obsolete estimate of the LSR.
Fortunately the underlying heliocentric line-of-sight velocity $v_{r}$ can be
recovered from $v_{\rm LSR}$ without impact on the associated uncertainty
(Reid et al., 2009, Appendix).
### 2.2 Our models
The motion of both the Sun and the masers is dominated by circular motion
around the Galactic centre with velocity
$-v_{c}(R)\mathbf{e_{\phi}},$ (2)
where $v_{c}(R)>0$ and the minus sign reflects the fact that the Galaxy
rotates clockwise in our coordinate system. We explore three forms for
$v_{c}(R)$
* •
A flat rotation curve, $v_{c}(R)=v_{0}$, with $v_{0}$ being a (positive) free
parameter.
* •
A power-law rotation curve $v_{c}(R)=v_{0}(R/R_{0})^{\alpha}$, with $v_{0}$,
$R_{0}$ and $\alpha$ being free parameters.
* •
A rotation curve corresponding to that given by the Galactic potential Model I
in §2.7 of Binney & Tremaine (2008, henceforth GDII), linearly scaled to
variable values of $R_{0}$ and $v_{0}$.
In each case, we take the parameters of $v_{c}(R)$ to have uniform prior
probability distributions. This choice ensures that we determine what _these_
data tell us, without prejudice.
Figure 1: Diagram showing a pole-on view of the Galaxy illustrating the
various peculiar velocities we consider (e.g. equation 4). This therefore only
shows the in plane components. The Sun (represented by a solid circle) is
placed at $(0,8)\,\mathrm{kpc}$, and components of its velocity
$\mathbf{v_{\odot}}$ are indicated. Three points (empty squares) are plotted
to represent masers, and the directions of the components of any systematic
offset from circular velocity in each case ($\mathbf{v}_{\mathrm{SFR}}$) are
shown, as is the peculiar velocity due to any bias in the observed radial
velocity, $v_{\mathrm{m}}$. This diagram is purely illustrative and does not
show any real data.
The velocity of a maser can be expected to differ from the local circular
speed. We separate this difference into a random component and (in some cases)
two systematic components. Like Reid et al. (2009), we consider the
possibility that the velocity of hmsfrs has a systematic offset from the
circular velocity, $\mathbf{v}_{\mathrm{SFR}}$. We also consider the
possibility that the expansion of the shell within which a maser occurs will
displace the maser’s velocity from that of the star, and if we are biased
towards seeing masers on the near side of that shell, we will see a bias in
radial velocity. We represent this as an average peculiar motion
$v_{\mathrm{m}}\mathbf{e_{*}}$, where $\mathbf{e_{*}}$ is the unit vector from
the Sun to the maser. We assume that the random component of the maser
velocities has a Gaussian distribution of dispersion $\Delta_{v}$. In total we
take the probability distribution of the velocities of masers to be
$p(\mathbf{v})\,\mathrm{d}^{3}\mathbf{v}=\frac{\,\mathrm{d}^{3}\mathbf{v}}{(2\pi\Delta_{v}^{2})^{3/2}}\exp{\left(\frac{-|\mathbf{v}-\overline{\mathbf{v}}|^{2}}{2\Delta_{v}^{2}}\right)},$
(3)
where
$\overline{\mathbf{v}}=-v_{c}(R)\mathbf{e_{\phi}}-\mathbf{v}_{\mathrm{SFR}}+v_{\mathrm{m}}\mathbf{e_{*}}.$
(4)
Notice that positive values of $v_{\phi,\rm SFR}$ imply that the masing stars
lead Galactic rotation, and that motion of the masers towards us will be
reflected in negative values of $v_{\mathrm{m}}$. In most cases we fix both
$\mathbf{v}_{\mathrm{SFR}}=0$ and $v_{\mathrm{m}}=0$. The various velocities
are illustrated in Fig. 1.
We sometimes take the Sun’s motion with respect to the LSR,
$\mathbf{v_{\odot}}$, to be specified, and sometimes we fit it to the data.
For given values of $l_{i}$, $b_{i}$, the heliocentric distance $s_{i}$, and
$R_{0}$, we deduce the probability distributions in proper motion and line-of-
sight velocity given the above probability distribution of the velocity (eq.
3). Since our focus is on the Galaxy’s parameters, we marginalise over the
$s_{i}$. We do this under the assumption that the probability distribution of
the $s_{i}$ is that implied by the assumption of Gaussian errors in the
parallaxes.
### 2.3 Statistical analysis
#### 2.3.1 Likelihood function
In order to determine the best-fitting model parameters, we maximise the
likelihood function. This is
$\mathcal{L}(\mbox{\boldmath$\theta$})\propto\prod_{i}\int\,\mathrm{d}s_{i}\,p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$}),$
(5)
where $p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$})$ is the conditional
probability of the $i$th observation, given the model with parameters
$\mbox{\boldmath$\theta$}\equiv(v_{0},R_{0},\alpha,\mathbf{v_{\odot}},\Delta_{v},s_{i},\ldots).$
(6)
For each distance $s_{i}$, the model provides a multivariate Gaussian
probability distribution in $v_{r,i}$, $\mu_{\alpha,i}$ and $\mu_{\delta,i}$.
The data also provide a multivariate Gaussian distribution for the underlying
values of these variables. We obtain
$p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$})$ by integrating the product of
these two Gaussian distributions over all three variables.
The likelihood function defines the a posteriori probability distribution of
the model parameters $\theta$. We use a Metropolis algorithm (Metropolis et
al., 1953) to identify the peak in this probability distribution, and to
characterise its width around the most probable model. The Metroplis algorithm
is a Markov Chain Monte Carlo method for drawing a representative sample from
a probability distribution, such as the likelihood function. We start with
some choice for the parameters $\theta$, and calculate the associated
likelihood $\mathcal{L}(\mbox{\boldmath$\theta$})$. We then
1. 1.
choose a trial parameter set $\mbox{\boldmath$\theta$}^{\prime}$ by moving
from $\theta$ in all directions in parameter space, by an amount chosen at
random (from a Gaussian distribution);
2. 2.
determine $\mathcal{L}(\mbox{\boldmath$\theta$}^{\prime})$;
3. 3.
choose a random variable $r$ from a uniform distribution in the range [0,1];
4. 4.
if
$\mathcal{L}(\mbox{\boldmath$\theta$}^{\prime})/\mathcal{L}(\mbox{\boldmath$\theta$})>r$,
accept the trial parameter set, and set
$\mbox{\boldmath$\theta$}=\mbox{\boldmath$\theta$}^{\prime}$. Otherwise do not
accept it.
5. 5.
Return to step (i).
The first few values of $\theta$ are ignored as “burn-in”, which helps to
remove the dependence on the initial value of $\theta$. We repeat the
procedure until the chain of $\theta$ values constitutes a fair sample of the
probability distribution – we establish that the burn-in period is
sufficiently long by comparing chains that have different starting $\theta$
(e.g. Gelman & Rubin, 1992).
#### 2.3.2 Bayesian evidence
In addition to comparing models that differ only in the values taken by a
given set of parameters, we have to assess the value of adding a parameter to
a model. Adding a parameter is guaranteed to increase the maximum likelihood
achievable for given data, but is that increase statistically significant or
just the consequence of an enhanced ability to fit noise? The Bayesian
methodology for making such assessments is now well established and described
by Heavens (2009), for example.
One calculates the “evidence” for the model under different priors for
whatever parameters are either fixed a priori or varied. The evidence is the
total probability of the data after integrating over all parameters:
$p({\rm data}|{\rm
Model})=\int\,\mathrm{d}^{n}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm Model})\,p(\mbox{\boldmath$\theta$}|{\rm
Model}).$ (7)
Here $p(\mbox{\boldmath$\theta$}|{\rm Model})$ is the prior on the parameters.
For a parameter $\theta_{i}$ that is varied, we take the prior to be uniform
within a range of width $\Delta\theta_{i}$ that is large enough to encompass
any plausible value of $\theta_{i}$ (so $p(\theta_{i}|{\rm
Model})\,\mathrm{d}\theta_{i}=d\theta_{i}/\Delta\theta_{i}$ over this range).
The priors on parameters that are always varied play little role because we
are interested the ratio of the evidence when a parameter $\theta_{n}$ is
fixed to when it is varied. When $\theta_{n}$ is fixed, its prior is a delta
function at the chosen value, so the ratio of the evidences when $\theta_{n}$
is fixed to when it is varied is
${p({\rm data}|{\rm Model},\theta_{n}\,\mathrm{fixed})\over p({\rm data}|{\rm
Model})}={\int\,\mathrm{d}^{n-1}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm
Model})\over\int\,\mathrm{d}^{n}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm Model})}\Delta\theta_{n},$ (8)
where upper integral excludes $\theta_{n}$ and the lower one includes it.
By Bayes theorem the ratio of the a posteriori probabilities of the model when
$\theta_{n}$ is fixed to when it is varied is the ratio of the corresponding
evidences times the ratios of the prior on $\theta_{n}$ being fixed to that on
it being variable. We take this second factor to be unity, so the a posteriori
probabilities of the models is simply the ratio of the evidences.
The integration over the space of parameters that is required to calculate the
evidence is exceedingly costly if done by brute force. We approximate it by
assuming that in the vicinity of its peak, $p({\rm
data}|\mbox{\boldmath$\theta$},{\rm model})$ can be approximated by a Gaussian
$p\propto\exp(-\mbox{\boldmath$\theta$}^{T}\cdot
K\cdot\mbox{\boldmath$\theta$})$, where $K$ is a matrix whose eigenvectors and
eigenvalues can be estimated from the output of the Metropolis algorithm. With
this approximation, the integral over parameters becomes analytic.
The Metropolis algorithm yields a set of points in parameter space which
sample $p({\rm data}|{\rm Model})$. Principal component analysis of this
sample yields the eigenvectors and eigenvalues of $K$.
## 3 Results
Table 1: Log likelihoods for best-fitting models with flat rotation curves ($\alpha=0$), power law rotation curves ($\alpha\neq 0$) and rotation curves taken from GDII. The first 9 likelihoods are for models in which the sources are assumed to have no typical velocity – for the first 3 the value of $\mathbf{v_{\odot}}$ is that found in DB98; the next 3 are for models in which $\mathbf{v_{\odot}}$ is assumed to be that suggested by B09; and the next 3 are for models in which $\mathbf{v_{\odot}}$ is allowed to vary freely; the final 3 are for cases in which $\mathbf{v_{\odot}}$ is taken to be the DB98 value, and the mean source peculiar motion with respect to their local standard of rest, $\mathbf{v}_{\mathrm{SFR}}$, is allowed to vary freely. | $v_{0}$ | $R_{0}$ | $\alpha$ | $U_{\odot}$ | $V_{\odot}$ | $W_{\odot}$ | $\Delta_{v}$ | $v_{0}/R_{0}$ | $\log(\mathcal{L})$
---|---|---|---|---|---|---|---|---|---
| $(\mathrm{km\,s}^{-1})$ | $(\mathrm{kpc})$ | | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1})$ |
DB98 | $200\pm 20$ | $6.7\pm 0.5$ | $0$ | $10.0$ | $5.2$ | $7.2$ | $10.0\pm 1.3$ | $30.1\pm 1.7$ | $26.8$
$209\pm 26$ | $6.9\pm 0.7$ | $0.10\pm 0.16$ | $10.0$ | $5.2$ | $7.2$ | $10.0\pm 1.3$ | $30.4\pm 1.7$ | $27.1$
$218\pm 20$ | $7.1\pm 0.5$ | GDII | $10.0$ | $5.2$ | $7.2$ | $9.3\pm 1.2$ | $30.8\pm 1.6$ | $30.7$
B09 | $215\pm 19$ | $7.0\pm 0.5$ | $0$ | $10.0$ | $11.0$ | $7.2$ | $8.1\pm 1.1$ | $30.5\pm 1.5$ | $35.7$
$228\pm 24$ | $7.4\pm 0.6$ | $0.15\pm 0.13$ | $10.0$ | $11.0$ | $7.2$ | $8.0\pm 1.1$ | $30.8\pm 1.5$ | $36.5$
$235\pm 19$ | $7.6\pm 0.5$ | GDII | $10.0$ | $11.0$ | $7.2$ | $7.6\pm 1.0$ | $31.1\pm 1.5$ | $38.9$
$\mathbf{v_{\odot}}$ free | $232\pm 24$ | $7.7\pm 0.6$ | $0$ | $8.1\pm 2.9$ | $18.6\pm 2.4$ | $9.7\pm 2.0$ | $7.1\pm 1.0$ | $30.0\pm 1.8$ | $42.5$
$258\pm 32$ | $8.6\pm 0.9$ | $0.27\pm 0.13$ | $8.1\pm 2.8$ | $19.5\pm 2.5$ | $10.1\pm 2.0$ | $7.1\pm 1.0$ | $29.9\pm 1.7$ | $45.4$
$246\pm 24$ | $8.1\pm 0.6$ | GDII | $8.3\pm 2.8$ | $16.5\pm 2.4$ | $9.9\pm 2.0$ | $7.0\pm 1.0$ | $30.3\pm 1.8$ | $43.5$
| $v_{0}$ | $R_{0}$ | $\alpha$ | $v_{R,\mathrm{SFR}}$ | $v_{\phi,\mathrm{SFR}}$ | $v_{z,\mathrm{SFR}}$ | $\Delta_{v}$ | $v_{0}/R_{0}$ | $\log(\mathcal{L})$
$\mathbf{v}_{\mathrm{SFR}}$ free | $241\pm 24$ | $7.7\pm 0.6$ | $0$ | $3.3\pm 2.8$ | $-12.9\pm 2.4$ | $2.5\pm 2.0$ | $7.0\pm 1.0$ | $31.1\pm 1.7$ | $42.9$
$279\pm 33$ | $8.9\pm 0.9$ | $0.25\pm 0.12$ | $2.0\pm 2.8$ | $-14.8\pm 2.5$ | $3.0\pm 2.0$ | $6.6\pm 1.0$ | $31.5\pm 1.6$ | $45.5$
$259\pm 23$ | $8.2\pm 0.6$ | GDII | $2.5\pm 2.8$ | $-11.0\pm 2.4$ | $2.8\pm 2.0$ | $7.1\pm 1.0$ | $31.6\pm 1.7$ | $43.8$
Figure 2: Plot showing contours of the Likelihood function (marginalised over
$\Delta_{v}$) for a model with the GDII rotation curve, and the DB98
$\mathbf{v_{\odot}}$. There is clearly a strong correlation between the values
found for $v_{0}$ and $R_{0}$. Contours are drawn at likelihood differing from
the maximum likelihood $\Delta\mathcal{L}=\,0.25$, $0.5$, $1$, $2$.
Figure 3: Residual velocities left after the best-fitting model velocities
are subtracted, for models with GDII rotation curves and $\mathbf{v_{\odot}}$
taking the DB98 value (top-left); the B09 value (top-right); or taken to be a
free parameter (bottom-left). We also plot the residual velocities left after
subtracting the best fitting model velocities for a model with the B09
$\mathbf{v_{\odot}}$ value, and $v_{\mathrm{m}}$ taken to be a free parameter
(bottom-right). _Open squares_ correspond to the objects for which the
likelihood improves most significantly between the best-fitting DB98 model and
the best-fitting free Solar velocity one ($\Delta(\log\mathcal{L}_{i})>1$),
and _crosses_ to all other sources. The Sun is represented by a solid circle.
The _solid lined ellipses_ around each point are $1\,\sigma$ measurement error
ellipses, where the velocity error perpendicular to the line of sight in each
case is due to a combination of the uncertainty in the proper motion and in
the parallax. The uncertainty in the parallax also means that the residual
velocities shown here are not an ideal illustration of the difference between
the model and the data, because the peculiar motions can only be given for a
chosen position (in this case the position corresponding to the quoted
parallax). However this does provide a useful guide, especially for sources
relatively close to the Sun, which have small position uncertainties. The
_dashed ellipse_ in the bottom-right plot is drawn around the three objects
for which the likelihood improves most significantly when $v_{\mathrm{m}}$ is
taken to be a free parameter ($\Delta(\log\mathcal{L}_{i})>1$ for B09
$\mathbf{v_{\odot}}$)
We first investigate models in which the masers are at rest with respect to
their host stars (i.e., $v_{\rm m}=0$).
Table 1 gives the peak likelihoods of our models, as well as the best-fitting
parameters and the corresponding uncertainties. The best-fitting value of
$v_{0}$ varies in the range $(200-279)\,\mathrm{km\,s}^{-1}$, and that of
$R_{0}$ between $6.7$ and $8.9\,\mathrm{kpc}$. As Fig. 2 illustrates, $v_{0}$
and $R_{0}$ are strongly correlated with the result that $v_{0}/R_{0}$ is
confined to the relatively narrow range
$29.9-31.6\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$.
When $\mathbf{v_{\odot}}$ is a free parameter, the best-fitting values of
$U_{\odot}$ and $W_{\odot}$ are close to the DB98 values, and essentially
independent of the form of the rotation curve, whereas $V_{\odot}$ varies in
the range $16.5-19.5\,\mathrm{km\,s}^{-1}$. Similarly, when
$\mathbf{v_{\odot}}$ is fixed at the DB98 value and
$\mathbf{v}_{\mathrm{SFR}}$ is a free parameter, only $v_{\phi,\rm SFR}$ takes
a value that is far removed from that which one would naively expect – it
moves in the range $-11.0$ to $-14.8\,\mathrm{km\,s}^{-1}$. Thus the data
suggest either that the Sun is circulating significantly faster than the
circular speed, or that the hmsfr have an appreciable rotational lag.
The fits are almost perfectly degenerate between $\mathbf{v_{\odot}}$ and
$\mathbf{v}_{\mathrm{SFR}}$: they constrain only the difference between these
velocities. However, we shall argue in Section 4 that significantly non-zero
values of $\mathbf{v}_{\mathrm{SFR}}$ are physically implausible, so here we
focus on what can be inferred about $\mathbf{v_{\odot}}$ given
$\mathbf{v}_{\mathrm{SFR}}=0$. In view of the degeneracy between
$\mathbf{v_{\odot}}$ and $\mathbf{v}_{\mathrm{SFR}}$ we do not report results
obtained when both $\mathbf{v_{\odot}}$ and $\mathbf{v}_{\mathrm{SFR}}$ are
varied.
When $\mathbf{v}_{\mathrm{SFR}}=0$, the peak likelihood is higher when
$\mathbf{v_{\odot}}$ is fixed to the B09 value than when it is fixed to the
DB98 value. To assess the significance of this increase in likelihood, we
calculate the ratio of the evidences for the two models as described in
Section 2.3.2
$\frac{p(\mathrm{DB98}|\mathrm{data})}{p(\mathrm{B09}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}2\times
10^{-4}&\;\mbox{for }\alpha=0\\\ 1\times 10^{-4}&\;\;\alpha\mbox{ variable}\\\
4\times 10^{-4}&\;\;\mbox{GDII}.\\\ \end{array}\right.$ (9)
Thus regardless of the adopted rotation curve, the data strongly favour upward
revision of $V_{\odot}$ from $5.2\,\mathrm{km\,s}^{-1}$ to
$11\,\mathrm{km\,s}^{-1}$.
When $\mathbf{v_{\odot}}$ is a free parameter, the peak likelihood of the B09
model is surpassed at yet larger values of $V_{\odot}$. To determine whether
the increase in likelihood that occurs when $\mathbf{v_{\odot}}$ is set free
from the B09 value, we again calculate the relevant ratio of the evidences.
Since the two models now differ in whether $\mathbf{v_{\odot}}$ is fixed or
free, the priors on the components of $\mathbf{v_{\odot}}$ now become relevant
(cf eq. 8). We have adopted $\Delta U_{\odot}=\Delta V_{\odot}=\Delta
W_{\odot}=100\,\mathrm{km\,s}^{-1}$. With these values we have
$\frac{p(\mathrm{B09}|\mathrm{data})}{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}5&\;\mbox{for
}\alpha=0\\\ 0.6&\;\;\alpha\mbox{ variable}\\\ 40&\;\;\mbox{GDII}.\\\
\end{array}\right.$ (10)
Therefore under these assumptions the increase in likelihood attained on
setting $\mathbf{v_{\odot}}$ free from the B09 value is not statistically
significant. The ratio of evidences for the case when $\mathbf{v_{\odot}}$ is
set to the DB98 value or is set free is
$\frac{p(\mathrm{DB98}|\mathrm{data})}{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}8\times
10^{-4}&\;\mbox{for }\alpha=0\\\ 7\times 10^{-5}&\;\;\alpha\mbox{ variable}\\\
0.01&\;\;\mbox{GDII}.\\\ \end{array}\right.$ (11)
Therefore, even with a generous choice of $\Delta U_{\odot}$, etc., the data
reject the possibility that the Sun has the DB98 value of
$\mathbf{v_{\odot}}$. Reducing the widths $\Delta U_{\odot}$, etc., of the
priors on $\mathbf{v_{\odot}}$ would strengthen the case against the DB98
value of $\mathbf{v_{\odot}}$.
The choice of $\Delta U_{\odot}\sim 100\,\mathrm{km\,s}^{-1}$ is reasonably
generous, and it is sensible to ask what value of $\Delta U_{\odot}$, etc.,
would bring ${p(\mathrm{B09}|\mathrm{data})}/{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}$ down to unity. In the case where the GDII rotation
curve is used, it would require a reduction to a value $\Delta U_{\odot}\sim
30\,\mathrm{km\,s}^{-1}$ – this is smaller than is reasonable given that the
value for $V_{\odot}$ found when it is allowed to vary is already $\sim
15\,\mathrm{km\,s}^{-1}$ greater than the canonical DB98 value.
It is worth noting that the case for setting $\mathbf{v_{\odot}}$ free from
either the DB98 value of the B09 value is weakest when the best-motivated
rotation curve is adopted – the GDII curve.
$v_{0}(\,\mathrm{km\,s}^{-1})$ | $R_{0}(\,\mathrm{kpc})$ | $U_{\odot}$ | $V_{\odot}$ | $W_{\odot}$ | $v_{\mathrm{m}}$ | $\Delta_{v}$ | $v_{0}/R_{0}(\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1})$ | $\log(\mathcal{L})$
---|---|---|---|---|---|---|---|---
$233\pm 20$ | $7.3\pm 0.5$ | $10.0$ | $5.2$ | $7.2$ | $-8.1\pm 2.7$ | $8.0\pm 1.1$ | $31.6\pm 1.5$ | $36.2$
$247\pm 19$ | $7.8\pm 0.4$ | $10.0$ | $11.0$ | $7.2$ | $-6.2\pm 2.4$ | $6.8\pm 0.9$ | $31.4\pm 1.4$ | $43.1$
$259\pm 25$ | $8.2\pm 0.5$ | $10.7\pm 3.3$ | $15.0\pm 2.7$ | $10.0\pm 2.0$ | $-4.8\pm 2.9$ | $6.5\pm 1.0$ | $31.4\pm 1.9$ | $46.0$
$v_{0}(\,\mathrm{km\,s}^{-1})$ | $R_{0}(\,\mathrm{kpc})$ | $v_{R,\mathrm{SFR}}$ | $v_{\phi,\mathrm{SFR}}$ | $v_{z,\mathrm{SFR}}$ | $v_{\mathrm{m}}$ | $\Delta_{v}$ | $v_{0}/R_{0}(\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1})$ | $\log(\mathcal{L})$
$271\pm 25$ | $8.2\pm 0.6$ | $2.0\pm 3.1$ | $-9.7\pm 2.8$ | $-2.5\pm 2.2$ | $-5.8\pm 3.3$ | $6.4\pm 1.1$ | $32.9\pm 2.0$ | $46.0$
Table 2: Similar to Table 1. This shows the best-fitting parameters (and
corresponding log likelihoods) for models with a GDII rotation curve, and with
the value $v_{\mathrm{m}}$ allowed to vary.
The lowest values of both $v_{0}$ and $R_{0}$ are found for the models with
the DB98 value of $\mathbf{v_{\odot}}$, and $\mathbf{v}_{\mathrm{SFR}}$ set to
zero. Both $v_{0}$ and $R_{0}$ take larger values for models with the B09
value of $\mathbf{v_{\odot}}$, and larger still for models with either
$\mathbf{v_{\odot}}$ or $\mathbf{v}_{\mathrm{SFR}}$ allowed to vary. The
lowest values of $v_{0}$ and $R_{0}$ for the models in which the peculiar
velocities are allowed to vary are $232\,\mathrm{km\,s}^{-1}$ and
$7.7\,\mathrm{kpc}$ respectively.
Fig. 3 suggests a reason why the value of $\mathbf{v_{\odot}}$ or
$\mathbf{v}_{\mathrm{SFR}}$ in the model has such a large impact on the best
fitting values of $v_{0}$ and $R_{0}$. It shows the residual velocities of the
objects after the expected velocity is subtracted (ignoring the uncertainty in
parallax). The objects that provide the strongest indication that some change
in peculiar motion is required (indicated by open squares) are all relatively
close to the Sun. Consequently, changes in $R_{0}$ and $v_{0}$ have a
relatively small “lever arm” with which to bring the model closer to the data,
with rather large changes required in order to have any significant impact on
the expected values of the observables for those objects. The introduction of
a new peculiar velocity (either of the sources or the Sun) allows the model to
compensate for the significant discrepancy between the velocity of these
objects and those expected from the rotation curve, without producing a major
effect on the best-fitting values of $R_{0}$ and $v_{0}$.
### 3.1 Maser line-of-sight velocities
Finally we consider models in which we allow for a systematic difference
between the measured line-of-sight velocity of a maser and the line-of-sight
velocity of the exciting star – such a difference would arise if masing
occurred on the near side of an expanding shell around the star. We have done
this for all the rotation curves described in this paper, but the results are
sufficiently similar to one another that we only present (in Table 2) the
results for the GDII rotation curve.
We again compare the different models by calculating the ratios of their
evidences, using the prior for $v_{\mathrm{m}}$ uniform in a range of width
$\Delta v_{\mathrm{m}}=20\,\mathrm{km\,s}^{-1}$. We find
$\frac{p(v_{\mathrm{m}}=0|\mathrm{data})}{p(v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq\;\left\\{\begin{array}[]{ll}0.01&\;\mbox{for
DB98}\\\ 0.06&\;\;\mbox{B09}\\\ 0.3&\;\;\mathbf{v_{\odot}}\mathrm{\ free}\\\
0.3&\;\;\mathbf{v}_{\mathrm{SFR}}\mathrm{\ free}\\\ \end{array}\right.$ (12)
so in each case the data support adding the extra free parameter.
If we accept this extra parameter in our models, we have to reconsider the
evidence for a change in $\mathbf{v_{\odot}}$ (or
$\mathbf{v}_{\mathrm{SFR}}$). The relevant ratios of evidences are:
$\frac{p(\mathrm{DB98},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathrm{B09},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 0.001,$ (13)
$\frac{p(\mathrm{DB98},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathbf{v_{\odot}},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 0.3,$ (14)
$\frac{p(\mathrm{B09},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathbf{v_{\odot}},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 220.$ (15)
So the B09 value of $\mathbf{v_{\odot}}$ is still favoured. Moreover, when
$\mathbf{v_{\odot}}$ is taken to be free, the likelihood still peaks at an
even larger value of $V$ than that of B09.
We note that the sources providing the strongest evidence that a non-zero
value of $v_{\mathrm{m}}$ is needed are those ringed in the the bottom right
plot of Fig. 3, all associated with the Perseus spiral arm (Reid et al.,
2009). If we exclude these sources, the best fitting value of $v_{\mathrm{m}}$
is $\sim-3\,\mathrm{km\,s}^{-1}$ for any assumption about $\mathbf{v_{\odot}}$
we consider. This is approximately within the uncertainty on $v_{\mathrm{m}}$,
and if we considered only this subset of the data in equation (12), the
evidence would not support adding the extra free parameter $v_{\mathrm{m}}$
(though it should be noted that it is still a radial velocity _towards_ the
Sun). It is, therefore, possible that what we have modelled as an offset in
the radial velocities of all observations is actually primarily due to a large
peculiar velocity of the objects in the Perseus arm, directed approximately in
the direction of the Sun.
## 4 Discussion
These results, and in particular the large variation in $v_{0}$ and $R_{0}$
depending on the other model parameters, makes it impossible to constrain
tightly either $v_{0}$ or $R_{0}$ from these data – the smallest best-fitting
values of these parameters are, respectively, 40 per cent and 33 per cent
smaller than the largest best-fitting values. The choice of rotation curve has
a significant impact, and we do not investigate all possible rotation curves.
It is worth noting that our two most favoured models in Table 1 (according to
Bayesian evidence, so somewhat dependent on our choice of priors) have best-
fitting values of $v_{0}$ that differ by $30\,\mathrm{km\,s}^{-1}$ and of
$R_{0}$ that differ by $1\,\mathrm{kpc}$ from one another (the models are
those with B09 $\mathbf{v_{\odot}}$ $+$ GD rotation curve and free
$\mathbf{v_{\odot}}$ $+$ power-law rotation curve, respectively).
However, for all but one of our models the best fitting $v_{0}/R_{0}$ lie in
the narrow range $(29.8-31.5)\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$, with
typical uncertainties $\sim 1.5\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$. This
corresponds to best fitting $(v_{0}+V_{\odot})/R_{0}$ in the range
$(30.9-32.5)\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$ (again, depending on
the model), with similar uncertainties. These values are slightly larger than
the value of $(v_{0}+V_{\odot})/R_{0}=30.2\pm
0.2\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$ found from the proper motion of
Sgr A* (Reid & Brunthaler, 2004), but consistent to within (less than) twice
the uncertainties on our values.
If we allow for a bias in the maser radial velocities with respect to the Sun
by allowing the parameter $v_{\mathrm{m}}$ to take a non-zero value in
equation (4), this does improve the fit. However we have seen that this is
primarily due to a group of maser sources in the Perseus spiral arm, so the
cause may be a large peculiar motion in that part of the Galaxy. Even if the
bias in measured radial velocity is real, it does not affect the result: (i)
that in different models these data are fit best by very different values of
$v_{0}$ and $R_{0}$; (ii) that the DB98 $\mathbf{v_{\odot}}$ is rejected by
these data, and (iii) that the B09 $\mathbf{v_{\odot}}$ is favoured over
making $\mathbf{v_{\odot}}$ a free parameter.
Reid et al. (2009) focused on the idea that these data show that the hmsfrs
are orbiting the Galaxy with a velocity that lags the circular speed by
$-v_{\phi,\mathrm{SFR}}\sim 15\,\mathrm{km\,s}^{-1}$. Our analysis of similar
models shows that the data are fit better by a slightly smaller offset of
$(11-14.8)\,\mathrm{km\,s}^{-1}$ to the circular speed. We have also shown
that an equivalent fit to the data can be obtained by instead increasing to
$\sim 17\,\mathrm{km\,s}^{-1}$ the amount $V_{\odot}$ by which the Sun is
assumed to circulate faster than the circular speed. In fact, the sources that
provide the strongest statistical support for
$v_{\phi,\mathrm{SFR}}\sim-15\,\mathrm{km\,s}^{-1}$ are found near the Sun
(Fig. 3), so a change in $\mathbf{v_{\odot}}$ has a very similar effect to a
change in $\mathbf{v}_{\mathrm{SFR}}$. More maser data at different Galactic
azimuths could break this degeneracy, as well as reducing the correlation
between the values of $v_{0}$ and $R_{0}$ seen in Fig. 2.
If the proposal of Reid et al. (2009) that
$v_{\phi,\mathrm{SFR}}\simeq-15\,\mathrm{km\,s}^{-1}$ were correct, the hmsfrs
would all have to be close to apocentre. Since the lifetime of an hmsfr is
short compared to a typical epicycle period and spiral structure must play a
significant role in their formation, it is not inherently implausible that the
maser stars are all close to apocentre. However, two arguments make it
unlikely that the maser stars are on orbits as eccentric as is implied by a
$15\,\mathrm{km\,s}^{-1}$ offset from circular motion at apocentre.
First, non-axisymmetric structure in the Galaxy’s potential modulates the
tangential velocity of gas by only $\sim 7\,\mathrm{km\,s}^{-1}$ (e.g. Binney
& Merrifield, 1998, §9.2.3), so the hmsfrs would be moving with a peculiar
velocity significantly higher than the gas from which they formed. Second, by
GDII (equation 3.100), a quarter of an epicycle period later the $U$
velocities of these stars would be
$(2\Omega/\kappa)15\,\mathrm{km\,s}^{-1}\simeq 22\,\mathrm{km\,s}^{-1}$, where
$\Omega$ and $\kappa$ are the circular and radial frequencies at the star’s
location. Hence the radial velocity dispersion of a population of such stars
would be at least $\sim 22/\surd 2\simeq 15\,\mathrm{km\,s}^{-1}$. The
velocity dispersion of stars observed locally increases with age on account of
heating by irregularities in the Galaxy’s gravitational field (e.g. GDII
§10.4.1) and the velocity dispersion of the bluer stars in the Hipparcos
catalogue is $\sim 10\,\mathrm{km\,s}^{-1}$ (Aumer & Binney, 2009) rather than
$15\,\mathrm{km\,s}^{-1}$. Thus the conjecture of Reid et al. (2009) not only
requires the masing stars to be confined to apocentre but also requires them
to have more eccentric orbits than the generality of young stars. Recognising
this problem, Reid et al. suggested that the orbits of these stars became more
circular early in their lives. However, any random scattering process spreads
stars more widely in phase space and therefore increase the mean eccentricity
of the masing stars. Only a dissipative process could increase the phase-space
density of these stars by moving them to more circular orbits, and no such
process is known.
If the B09 value of $V_{\odot}$ is correct, the lag to circular motion
required to optimise the fit to the data is so small ($\sim
5\,\mathrm{km\,s}^{-1}$) that the above objections to the proposal that hmsfr
systematically lag rotation become moot. However, the formalism of Bayesian
inference says that when the B09 value of $V_{\odot}$ is accepted, there is no
convincing evidence for a systematic lag of the hmsfr.
Throughout this paper we have proceeded under the assumption that the velocity
of the LSR is the same thing as the circular velocity at $R_{0}$. It is worth
noting that this is not necessarily the case. The LSR is defined to be the
velocity of a closed orbit as it passes the current Solar position, which will
only be a circular orbit if the Galactic potential is axisymmetric. Therefore
the velocity of the LSR may be offset from the circular velocity curve.
However (much like the possibility of a systematic lag of the hmsfr) we should
note that if we accept the B09 value of $V_{\odot}$, there is no convincing
evidence for this offset.
We conclude that these data provide a compelling case for revising $V_{\odot}$
upward. Our Bayesian analysis (equation 10) supports the revision upwards of
$V_{\odot}$ to $11\,\mathrm{km\,s}^{-1}$ suggested by B09 on the basis of
modelling predominantly old disc stars with relatively large random
velocities. It should, however, be recognised that an even better fit to
_these_ data comes from somewhat larger upward revisions to $V_{\odot}\sim
16-20\,\mathrm{km\,s}^{-1}$.
## 5 Conclusions
We have reanalysed observations of hmsfrs reported in Reid et al. (2009) using
a maximum-likelihood approach to exploit fully the data, in an effort to
determine the values of $R_{0}$, the Galactocentric radius of the Sun, and
$v_{0}$, the circular velocity of the LSR. We have found that the best-fitting
values, considered separately, are strongly dependent on the Galaxy model used
to interpret the data, but that the ratio $v_{0}/R_{0}$ is consistently found
to lie in the range $29.8-31.5\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$.
We have also used these data to explore the value of the Sun’s peculiar
velocity $\mathbf{v_{\odot}}$, in light of the recent argument of B09 that the
canonical DB98 value is incorrect. We find that these data support the
conclusion that $V_{\odot}$ is significantly higher than the DB98 value. By a
small but significant margin, the data prefer models with the B09 revision of
$V_{\odot}$ to $\sim 11\,\mathrm{km\,s}^{-1}$ over models in which
$\mathbf{v_{\odot}}$ is left as a free parameter (equation 10). The best-
fitting models have $U_{\odot}$ and $W_{\odot}$ near to the DB98 values, and
the even larger value of $V_{\odot}\sim 16-20\,\mathrm{km\,s}^{-1}$.
Reid et al. (2009) suggested that hmsfrs significantly lag circular rotation.
We have investigated the possibility that the hmsfrs have a typical peculiar
velocity $\mathbf{v}_{\mathrm{SFR}}$ and find that the models only constrain
the velocity difference $\mathbf{v}_{\mathrm{SFR}}-\mathbf{v_{\odot}}$. We
have argued that models in which the hmsfrs have large peculiar velocities in
the opposite direction to Galactic rotation are neither needed nor plausible.
This work, in conjunction with B09, casts severe doubt on the accuracy of the
widely used DB98 value for $V_{\odot}$. This must be a concern for anyone
interested in the dynamics within the Milky Way because velocities are
inevitably measured with respect to the Sun. Both observations of more masers
and the development of a detailed dynamical model of the Galaxy’s spiral
structure would contribute to establishing more securely what the true value
of $V_{\odot}$ is.
## Acknowledgments
We thank John Magorrian and the other members of the Oxford dynamics group for
valuable discussions. PJM is supported by a grant from the Science and
Technology Facilities Council.
## Note added after acceptance
After this paper was accepted for publication, Rygl et al. (2009) presented
similar observations of maser sources. Incorporating these observations in our
analysis does not materially affect our conclusions.
## References
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* Metropolis et al. (1953) Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., Teller E., 1953, Journal of Chemical Physics, 21, 1087–1092
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|
arxiv-papers
| 2009-07-27T16:30:27 |
2024-09-04T02:49:04.202767
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul J. McMillan, James J. Binney",
"submitter": "Paul McMillan",
"url": "https://arxiv.org/abs/0907.4685"
}
|
0907.4766
|
# On the Dearth of Compact, Massive, Red Sequence Galaxies in the Local
Universe
Edward N Taylor1,2 Marijn Franx1 Karl Glazebrook3 Jarle Brinchmann1 Arjen van
der Wel4 Pieter G van Dokkum5 1 Sterrewacht Leiden, Leiden University, NL-2300
RA Leiden, Netherlands; ent@strw.leidenuniv.nl, 2 School of Physics, the
University of Melbourne, Parkville, 3010, Australia, 3 Centre for Astrophysics
& Supercomputing, Swinburne University of Technology, Hawthorn, 3122,
Australia 4 Max Planck Institut für Astronomie, D-69117 Heidelberg, Germany, 5
Department of Astronomy, Yale University, New Haven, CT 06520-8101
###### Abstract
Using data from the Sloan Digital Sky Survey (SDSS; data release 7), we have
conducted a search for local analogs to the extremely compact, massive,
quiescent galaxies that have been identified at $z\gtrsim 2$. We show that
incompleteness is a concern for such compact galaxies, particularly for low
redshifts ($z\lesssim 0.05$), as a result of the SDSS spectroscopic target
selection algorithm. We have identified 63 $M_{*}>10^{10.7}$ M⊙ ($\approx
5\times 10^{10}$ M⊙) red sequence galaxies at $0.066<z_{\textrm{spec}}<0.12$
which are smaller than the median size–mass relation by a factor of 2 or more.
Consistent with expectations from the virial theorem, the median offset from
the mass–velocity dispersion relation for these galaxies is 0.12 dex. We do
not, however, find any galaxies with sizes and masses comparable to those
observed at $z\sim 2.3$, implying a decrease in the comoving number density of
these galaxies (at fixed size and mass) by a factor of $\gtrsim 5000$. This
result cannot be explained by incompleteness: in the $0.066<z<0.12$ interval,
we estimate that the SDSS spectroscopic sample should typically be $\gtrsim
75$% complete for galaxies with the sizes and masses seen at high redshift,
although for the very smallest galaxies it may be as low as $\sim 20$%. In
order to confirm that the absence of such compact massive galaxies in SDSS is
not produced by spectroscopic selection effects, we have also looked for such
galaxies in the basic SDSS photometric catalog, using photometric redshifts.
While we do find signs of a bias against massive, compact galaxies, this
analysis suggests that the SDSS spectroscopic sample is missing at most a few
objects in the regime we consider. Accepting the high redshift results, it is
clear that massive galaxies must undergo significant structural evolution over
$z\lesssim 2$ in order to match the population seen in the local universe. Our
results suggest that a highly stochastic mechanism like major mergers cannot
be the primary driver of this strong size evolution.
###### Subject headings:
galaxies: evolution—galaxies: formation—galaxies: fundamental parameters
## 1\. Introduction
In the simplest possible terms, the naïve expectation from hierarchical
structure formation scenarios is that the most massive galaxies form last.
This is in contrast to the observation that the bulk of cosmic star formation
occurs in galaxies with progressively lower stellar masses at later times
(e.g. Juneau et al., 2005; Zheng et al., 2007; Damen et al., 2008); the
so–called downsizing of galaxy growth. These observations have been
accommodated within the $\Lambda$CDM framework with the introduction of a
quenching mechanism (e.g. Menci et al., 2005; Croton et al., 2006; Cattaneo et
al, 2008), which operates to shut down star formation in the most massive
galaxies; this mechanism is also required to correctly predict the absolute
and relative numbers of red galaxies (Dekel & Birnboim, 2006; Bell et al.,
2007; Faber et al., 2007). With this inclusion, models thus predict that a
significant fraction of the local massive galaxy population should have
finished their star formation relatively early in the history of the universe,
with later mergers working to build up the most massive galaxies.
There is thus a crucial distinction to be made between a galaxy’s mean stellar
age, and the time since that galaxy has assumed its present form (see, e.g.,
De Lucia et al., 2006): the most massive galaxies are expected to be both the
oldest and the youngest galaxies. They are the oldest in the sense that their
progenitors are expected to form first in the highest cosmic
overdensities—however, these stars are only assembled into their $z=0$
configuration relatively recently.
This leaves (at least) two open questions relating to the quenching of star
formation and the formation and evolution of massive galaxies: 1.) When does
star formation stop in massive galaxies, and 2.) What happens to galaxies
after they have stopped forming stars?
In connection with the first of these questions, deep spectroscopic surveys
have identified massive galaxies with little or no ongoing star formation at
$1\lesssim z\lesssim 2$ (e.g. Cimatti, 2004; Glazebrook et al., 2004;
McCarthy, 2004a; Daddi et al., 2004). At the same time, color selection
techniques like the ERO (McCarthy, 2004b, and references therein), DRG (Franx
et al., 2003), or BzK (Daddi et al., 2005) criteria have been used to identify
massive, passive galaxies at high redshifts. While these techniques are
deliberately biased towards certain kinds of galaxies and certain redshift
intervals, advances in techniques for photometric redshift estimation and
stellar population modeling have allowed the selection of mass-limited
samples, and so the construction of representative samples of the high
redshift massive galaxy population (e.g. van Dokkum et al., 2006).
By obtaining very deep rest-frame optical spectra of a photometric-redshift
selected sample of massive galaxies at $z\gtrsim 2$, Kriek et al. (2008a) made
a significant advance on previous spectroscopic and photometric studies. Of
the 36 $z_{\textrm{spec}}>2$, $M_{*}>10^{11}$ M⊙ galaxies in the Kriek et al.
(2008a) sample, 16 were shown unambiguously to have evolved stellar
populations and little or no ongoing star formation. These galaxies also seem
to form a red sequence in $(B-V)$ color, although at low significance
($3.3\sigma$; Kriek et al., 2008b). In other words, these massive galaxies
appear both to have assembled stellar populations similar to galaxies of
comparable mass in the local universe, and to have had their star formation
effectively quenched.
Using Keck laser guide-star assisted adaptive optics and Hubble Space
Telescope imaging, van Dokkum et al. (2008, hereafter vD08) measured sizes for
9 of the 16 strongly quenched galaxies from the Kriek et al. (2008a) sample.
They found (rest-frame optical) effective radii in the range 0.5—2.4 kpc; that
is, smaller than typical galaxies of the same mass in the local universe by
factors of 3—10. These galaxies have stellar mass densities, measured within
the central 1 kpc, which are 2—3 times higher than typical local galaxies of
the same mass (Bezanson et al., 2009). Cimatti et al. (2008) and Damjanov et
al. (2009, hereafter D09) have found similarly compact sizes for massive
galaxy samples drawn from $1<z<2$ spectroscopic surveys. Further, van Dokkum,
Kriek & Franx (2009) have recently measured a velocity dispersion of
$510^{+165}_{-95}$ km/s for one of the galaxies in the vD08 sample, based on a
29 hr NIR spectrum; this extremely high value is consistent with the galaxy’s
measured mass and size. (See also Cappellari et al. 2009.)
By providing rest-frame optical size measurements for a representative, mass-
limited sample of galaxies spectroscopically-confirmed to have little or no
ongoing star formation and $z\gtrsim 2$, these results confirm and consolidate
the work of Daddi et al. (2005), Trujillo et al. (2006), Trujillo et al.
(2007), Zirm et al. (2007), and Toft et al. (2007), as well as $1<z<2$ results
from, e.g., Longhetti et al. (2007) and Saracco et al. (2009), and $z\lesssim
1$ results from van der Wel et al. (2008). (See also Buitrago et al., 2008.)
The significance of these results is that while the massive and largely
quiescent galaxies at $z\gtrsim 2$ have stellar populations that are
consistent with their being more or less ‘fully formed’ early type galaxies,
they must each undergo significant structural evolution in order to develop
into galaxies like the ones seen in the local universe. Taken together, these
results thus paint a consistent picture of strong size evolution among
massive, early type and/or red sequence galaxies111There is considerable, but
not total, overlap between color–selected samples of red sequence galaxies,
and morphology–selected samples of early type galaxies. While it is common to
use these terms as if they were more or less interchangeable, it should be
remembered that they are not.—both as a population and individually—even after
their star formation has been quenched (see also Franx et al., 2008). Whatever
the mechanism for this growth in size (see, e.g., Fan et al., 2008; Hopkins et
al., 2009; Naab et al., 2009; Khochfar & Silk, 2009), the formation of
massive, passive galaxies is not monolithic.
The aim of this paper is to test the proposition that there are no galaxies in
the local universe with sizes and masses comparable to those found at
$z\gtrsim 1.5$ — this is the crux of the argument against the monolithic
formation of massive galaxies. This work is based on the latest data products
from the Sloan Digital Sky Survey (SDSS; York et al., 2000; Strauss et al.,
2002). In particular, we will focus on the possibility that such galaxies have
been overlooked in SDSS due to selection effects associated with the
construction of the spectroscopic target sample.
The structure of this paper is as follows: We describe the basic SDSS data
that we have used in §2. In §3, we define our sample of compact galaxy
candidates, and present several checks to confirm that these galaxies are
indeed unusually small for their stellar masses. Then, in §4, we consider the
importance of the SDSS spectroscopic selection for massive, compact galaxies.
In this Section, we also compare our $z\sim 0.1$ compact galaxy candidates
with the vD08 and D09 samples. In Appendix A, we provide a complementary
analysis in order to confirm our conclusion that the apparent differences
between the high- and low-redshift samples cannot be explained by selection
effects, including an estimate for the number of compact galaxies that may be
missing from the SDSS spectroscopic sample. Finally, in §5, we compare our
results to a similar studies by Trujillo et al. (2009) and Valentinuzzi et al.
(2009), and briefly examine the properties of our compact galaxies’ stellar
populations in comparison to the general $z\sim 0.1$ red sequence galaxy
population. A summary of our main results is given in §6. Throughout this
work, we assume the concordance cosmology; viz.: $\Omega_{\mathrm{m}}=0.3$,
$\Omega_{\Lambda}=0.7$, and $H_{0}=70$ km/s/Mpc.
## 2\. Basic Data and Analysis
The present work is based on Data Release 7 (DR7; Abazajian et al., 2009) of
the SDSS, accessed via the Catalog Archive
Server222http://casjobs.sdss.org/CasJobs/ (CAS; Thakar et al., 2008). In this
section, we describe the basic SDSS data that we have used, and our analysis
of it. We will search for compact galaxy candidates in the SDSS spectroscopic
catalog; to this end, we will only consider sciencePrimary objects (a flag
indicating a ‘science-grade spectrum, and weeding out multiple observations of
individual objects) with either a ‘star’ or ‘galaxy’ photometric type (ie., a
genuine astronomical source). The details of the SDSS spectroscopic sample
selection are given in Strauss et al. (2002); we will summarize the most
relevant aspects of this process in §4.1.
### 2.1. The Basic SDSS Catalog
For the basic SDSS catalog, there are two different methods for performing
photometry. The first, the ‘Petrosian’ magnitude, is derived from the
observed, azimuthally averaged (1D) light profile. The Petrosian radius is
defined as the point where the mean surface brightness in an annulus drops to
a set fraction (viz. 0.2) of the mean surface brightness within a circular
aperture of the same radius. Within SDSS, the Petrosian aperture is defined to
be twice the Petrosian radius; this aperture will contain 99 % of the total
light for a well resolved exponential disk, but may miss as much as 18 % of
the light for a de Vaucouleurs $R^{1/4}$ profile (Strauss et al., 2002;
Blanton et al., 2005).
The second photometric measure is derived from fits to the observed (2D)
distribution of light in each band, using a sector-fitting technique, in which
concentric annuli are divided into 12 30∘ sectors, as described in Appendix
A.1 of Strauss et al. (2002). These fits are done assuming either an
exponential or a de Vaucouleurs profile, convolved with a fit to the
appropriate PSF. For each profile, the structural parameters (viz. axis ratio,
position angle, and scalelength) are determined from the $r$ band image. The
more likely (in a $\chi^{2}$ sense) of the two profile fits is used to define
‘model’ magnitudes for each galaxy. For the $ugiz$ bands, these parameters are
then held fixed, and only overall normalization (ie. total flux) is fit for.
The basic catalog also provides two different measures of size, associated
with these two magnitude measurements. The Petrosian half-light radius,
$R_{50}$, is defined as the radius enclosing half the ‘total’ Petrosian flux.
The catalog also contains best fit structural parameters, including the
effective radius, from a separate set of fits to each band independently,
again for both an exponential and a de Vaucouleurs profile. Note that whereas
the Petrosian magnitude and size are derived from the observed, PSF-convolved
radial profile, the model values provide a PSF-corrected measure of the
intrinsic size.
We use model magnitudes to construct $ugriz$ SEDs for each object, since these
measurements are seeing–corrected. From DR7, the basic SDSS photometric
calibration has been refined so that the photometry is given in the AB
magnitude system without the need for any further corrections (Padmanabhan et
al., 2008). For measuring sizes, we will rely on the best-fit model effective
radius, $R_{\mathrm{e}}$, as determined from the $z$ band. We also adopt a
minimum measured size of $0\farcs 75$, corresponding to half the median PSF
FWHM for the SDSS imaging; we will plot all galaxies with observed sizes
smaller than $0\farcs 75$ as upper limits. (None of our conclusions depend on
the choice of this limit, which ultimately affects only 5 of our lower-mass
compact galaxy candidates.)
### 2.2. Derived Quantities
We have derived rest-frame photometry for each object, based on its observed
$ugriz$ SED and redshift, using the IDL utility InterRest (Taylor et al.,
2009), with a redshift grid of $\Delta z=0.001$. In order to minimize the
k-corrections and their associated errors, we determine rest-frame photometry
through the $ugriz$ filters redshifted to $z=0.1$, which we denote with a
superscript 0.1. We estimate that the systematic uncertainties are at the
level of $\lesssim 0.02$ mag. The agreement between our interpolated rest-
frame photometry and that derived using the SDSS kcorrect algorithm (Blanton &
Roweis, 2007) is very good: our derived $(u-r)$ colors are $\sim 0.02$ mag
bluer for blue galaxies, and $\sim 0.03$ mag redder for red galaxies.
Figure 1.— The mass–to–light ratios, $M_{*}/L_{i}$, of $0.066<z<0.12$ galaxies
as a function of their $(g-i)$ color. (Here, $M_{*}/L_{i}$ is understood to
relate to the $i$ band redshifted to $z=0.1$.) The greyscale shows the
(linear) data density in cells, where the data density is high. In the main
panel, the red line shows the median $M_{*}/L_{i}$ in narrow color bins; the
blue line is a linear fit to these points. In the lower panel, we have simply
subtracted away the median relation; in this panel, the error bars show the
16/84 percentiles in color bins. The simple linear relation shown provides an
acceptable description of the observed relation, to within 0.02—0.04 dex; the
global RMS offset from this relation is 0.032 dex. In order to avoid selecting
‘catastrophic failures’ in terms of stellar mass estimates, we will consider
only those galaxies that fall within 0.25 dex of the median
$M_{*}/L_{i}$–${}^{0.1}(g-i)$ relation, and with $0.4<\,^{0.1}(g-i)<1.8$, as
shown by the box in the lower panel. Figure 2.— The size–mass relation for
massive, red sequence galaxies showing the importance of the SDSS
spectroscopic selection criteria. The points show SDSS galaxies
($0.066<z<0.12$) selected to have ${}^{0.1}(u-r)>2.5$. The yellow points show
the median size in narrow bins of stellar mass; the error bars show the 16/84
percentiles in each bin. A fit to this median size–mass relation for red
sequence galaxies is consistent with the Shen et al. (2003) relation for early
type galaxies (dashed line), albeit offset by 0.05 dex. Individual galaxies
that we have visually inspected ($M_{*}>10^{10.7}$ M⊙; $\Delta\log
R_{\mathrm{e}}<-0.3$ dex) are marked with large symbols. Galaxies with $M/L$s
that differ significantly from the main color–$M/L$ relation shown in Figure 1
are marked with small blue crosses. Galaxies with obvious problems in their
photometry (especially those affected by the presence of a bright nearby star
or blended with other galaxies) are marked with a small red cross; those that
look okay are plotted as circles. Galaxies with observed sizes smaller than
$0\farcs 75$ are plotted as upper limits, assuming a size of $0\farcs 75$. The
different lines show how the principal selection limits for spectroscopic
followup translate onto the $(M_{*},R_{\mathrm{e}}$) plane for $z=0.12$, 0.10,
0.066, 0.050, and 0.35 (top to bottom): the diagonal, long-dashed lines show
the star/galaxy discriminator; the short-dashed boxes show the ‘saturation’
selection limit, and the diagonal dotted lines show the‘cross-talk’ selection
limit (see §4.1 for a detailed discussion). Galaxies lying below these lines
will not be targeted spectroscopically.
We make use of stellar mass estimates provided by the MPIA Garching
group.333Available via http://www.mpa-garching.mpg.de/SDSS/DR7 . JB has fit
the $ugriz$ photometry of all galaxies using the synthetic stellar population
library described by Gallazzi et al. (2005), based on Bruzual & Charlot (2003)
models and assuming a Chabrier (2003) stellar initial mass function (IMF) in
the range 0.1—100 M⊙. The Gallazzi et al. (2005) library contains a large
number of Monte Carlo realizations of star formation histories, parameterized
by a formation time ($1.5<t_{\mathrm{form}}/[\mathrm{Gyr}]<13.5$), an
exponential decay rate ($0<\gamma/[\mathrm{Gyr}^{-1}]<1$), and including a
number of random star formation bursts (randomly distributed between
$t_{\mathrm{form}}$ and 0, normalized such that 10 % of galaxies experience a
burst in the last 2 Gyr). In the fitting, the photometry has been corrected
for emission lines under the assumption that the global emission line
contribution is the same as in the spectroscopic fiber aperture.
The agreement between these SED-fit mass estimates and those of Kauffmann et
al. (2003a), which were derived from spectral line indices, are excellent: the
median offset is -0.01 dex, with a scatter on the order of 0.1 dex. For the
highest masses, however, the SED-fit results are slightly less robust: for
$M_{*}>10^{11}$ M⊙, the median formal error is $\lesssim 0.10$ dex, compared
to $\lesssim 0.06$ dex for the Kauffmann et al. (2003a) estimates.
In the upper panel of Figure 1, we show the stellar mass to light ratios,
$M_{*}/L_{i}$, for $0.066<z<0.12$ galaxies as a function of their
${}^{0.1}(g-i)$ color; here again, $L_{i}$ should be understood as referring
to the $i$-band filter redshifted to $z=0.1$, or ${}^{0.1}i$. Notice that, at
least for these mass estimates, $M_{*}/L$ is a very tight function of color.
In the main panel of this Figure, the red line shows the median $M_{*}/L_{i}$
in narrow color bins. Making a simple linear fit to these points, we find:
$\log(M_{*}/L_{i})=-0.82+0.83\times\,^{0.1}(g-i)~{},$ (1)
where both $M_{*}$ and $L_{i}$ are in solar units. (The absolute magnitude of
the sun in the ${}^{0.1}i$ band is 4.58.) This relation is shown in Figure 1
as the solid blue line. We present this relation as an alternative to the
popular Bell & de Jong (2001) or Bell et al. (2003) relations.
In the lower panel of Figure 1, we show the dispersion around the median
relation; in this Figure, the error bars show the 16/84 percentiles in bins of
color. Overall, the dispersion around this relation is just 0.032 dex. Note
that while the simple linear relation given above provides an acceptable
description of the ‘true’ relation, systematic offsets exist at the 0.02—0.04
dex level. The global mean and random offset from this linear relation are
0.002 dex and 0.040 dex, respectively.
Figure 3.— Illustrative examples of the galaxies we consider. Clockwise from
the top-right, we show a ‘normal’, massive early type galaxy that lies very
close to the median size–mass and velocity dispersion–mass relations, two
compact galaxy candidates where visual inspection suggested problematic size
measurements, and three of our compact galaxy candidates. Each of the three
compact galaxy candidates shown in this Figure have observed velocity
dispersions that are approximately consistent with their small measured sizes
(see §3.3). For each galaxy, we show the SDSS SkyServer thumbnail image used
for visual inspection, as well as the galaxies’ observed spectra; the boxes
show the SEDs from the photometry, scaled to match the spectroscopic flux in
the $r$ band.
In both panels, the small grey pluses show points that fall outside the
plotted range. Notice that there are a small fraction of galaxies that fall
well off the main $M_{*}/L$–color relation, some by an order of magnitude or
more. These galaxies also lie significantly off the main stellar
mass–dynamical mass relation and are very likely to represent catastrophic
failures of the stellar mass SED-fitting algorithm. This presents a problem
when it comes to looking for outliers in the mass–size plot: selecting the
most extreme objects may well include those objects with the largest errors.
For this reason, we will restrict our attention to those objects that fall
within 0.25 dex ($\approx 7.8\sigma$) of the main $M_{*}/L$–color relation,
and with $0.2<\,^{0.1}(g-i)<1.8$, as shown by the box in the lower panel of
Figure 1. This selection excludes just under 600 of the 223292 galaxies shown
in Figure 1.
## 3\. Searching for Massive, Compact, Early-Type Galaxies in the Local
Universe
### 3.1. Identifying Massive, Compact Galaxy Candidates
Table 1Properties of our Compact Galaxy Candidates RA | dec | $z$ | $(u-r)_{\mathrm{obs}}$ | ${}^{0.1}(u-r)$ | $\log M_{*}$ | $\Theta_{50,z}$ | $n$ | $\Theta_{50,z}$ | $R_{\mathrm{e}}$ | $\sigma$ | $T$ | $Z$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1)a | (2)a | (3)a | (4)a | (5) | (6)b | (7)a | (8)c | (9)c | (10) | (11)b | (12)d | (13)d |
190.16652 | 13.81563 | 0.0855 | 2.577 | 2.642 | 10.700 | 0.912 | 4.31 | 0.917 | 1.462 | 160 | … | … |
127.02722 | 55.37988 | 0.0665 | 2.828 | 2.999 | 10.701 | 1.152 | 3.26 | 1.211 | 1.468 | 191 | … | … |
225.31708 | 30.58266 | 0.0980 | 2.858 | 2.877 | 10.705 | 0.808 | 3.87 | 0.773 | 1.464 | 195 | … | … |
227.08531 | 7.25325 | 0.0764 | 2.963 | 3.113 | 10.709 | 0.929 | 5.13 | 1.188 | 1.345 | 199 | … | … |
215.41043 | 40.03233 | 0.0998 | 2.803 | 2.813 | 10.709 | 0.795 | 4.11 | 0.670 | 1.464 | 176 | 9.775 | 0.035 |
222.12988 | 26.48791 | 0.1058 | 2.549 | 2.540 | 10.712 | 0.750 | 4.65 | 0.722 | 1.453 | 155 | … | … |
118.81702 | 33.22864 | 0.0980 | 2.680 | 2.697 | 10.713 | 0.803 | 2.67 | 0.739 | 1.454 | 154 | -99 | -99 |
143.05707 | 11.70454 | 0.0811 | 2.690 | 2.776 | 10.726 | 0.750 | 5.52 | 0.830 | 1.146 | 166 | 9.255 | 0.132 |
204.66577 | 59.81854 | 0.0704 | 2.857 | 3.012 | 10.731 | 0.943 | 3.92 | 0.974 | 1.267 | 235 | 9.845 | 0.229 |
230.28553 | 24.21978 | 0.0809 | 2.922 | 3.028 | 10.733 | 0.952 | 5.90 | 1.298 | 1.453 | 153 | … | … |
Figure 2 shows the size–mass plot for a sample of massive, red-sequence
galaxies drawn from the SDSS DR7 spectroscopic sample. These galaxies have
been selected to have ${}^{0.1}(u-r)>2.5$ and $0.066<z<0.12$. These redshift
limits have been chosen to minimize the importance of selection effects and
measurement biases, which we will discuss in §4.1. For now, we note that,
mapping the $r<17.77$ spectroscopic limit onto $M_{*}(z)$, we should be highly
complete (volume limited) for $M_{*}>10^{10.7}$ M⊙ and $z<0.12$. As a very
simple check on this, we note that for this sample, the median redshift in
narrow bins of stellar mass is within the range $z=0.098$—0.102 for all
$M_{*}>10^{10.7}$ M⊙; the volumetric center of the $0.066<z<0.12$ bin is
$z=0.10$.
The yellow points in this Figure show the median size in narrow bins of
stellar mass; the error bars show the 14/86 percentiles. For comparison, the
long-dashed line shows the local size–mass relation for early-type galaxies
from Shen et al. (2003), corrected for differences in assumed IMF and
cosmology. Contrary to the findings of Valentinuzzi et al. (2009), a simple
fit to the size–mass relation for red sequence galaxies (${}^{0.1}(u-r)>2.5$)
shown in Figure 2 is consistent with the Shen et al. (2003) relation for early
type ($n>2.5$) galaxies, albeit offset in size by $0.05$ dex or, equivalently,
by $-0.09$ dex in mass. At fixed mass, the mode of the distribution is
similarly offset (see Figure 7); this does not appear to be due to large
numbers of late type galaxies in the sample.
We next select and study very compact galaxies from within the red sequence
sample shown in Figure 2. At first glance, it appears that there may be a few
galaxies that lie well below the main size–mass relation. However, it must be
remembered that by selecting the most extreme outliers, we will also be
selecting those objects with most egregious measurement errors.
For this reason, we have individually visually inspected all $M_{*}>10^{10.7}$
M⊙ galaxies with inferred sizes that are less than half the size predicted
from the Shen et al. (2003) relation; ie. $\Delta R_{\mathrm{e}}<-0.3$ dex.
For sizes smaller than the median relation, the distribution of sizes around
the Shen et al. (2003) relation is very well described by a Gaussian with
$\sigma=0.11$ dex; this $\Delta R_{\mathrm{e}}$ cut thus corresponds to
selecting those galaxies whose sizes are smaller than the mean size (at fixed
mass) at the $\gtrsim 2.7\sigma$ level. (Adopting our own fit to the size–mass
relation, this selection translates to $\Delta R_{\mathrm{e}}<-0.35$ dex; our
results are otherwise unchanged.)
We have inspected 280 such objects, and discarded those where there are
obvious reasons to distrust the size measurements. The most common reasons for
discarding galaxies were confusion with other galaxies (99 galaxies, including
19 good merger candidates, and two possible lenses), or with the extended
halos, diffraction spikes, and/or reflections of bright stars (62 galaxies). A
further 19 galaxies were clearly disk-like, 5 showed marked asymmetries, and 1
had a very strong AGN spectrum; these candidates were also discarded. We
discarded a further 3 objects with bad or missing data.
In Figure 3, we show several illustrative examples of the galaxies we are
considering. On the right-hand side of this Figure, we show a ‘normal’ early
type galaxy, with $M_{*}\approx 10^{11}$ M⊙, which falls very close to the
Shen et al. (2003) relation. Below this, we show two of the compact galaxy
candidates that we have rejected on the basis of visual inspection. On the
left-hand side of this Figure, we show three of the compact galaxy candidates
of different stellar masses that we have retained after visual inspection. For
each galaxy, we show the thumbnail image from the SDSS SkyServer444Also
accessible via CAS at http://cas.sdss.org., used for visual inspection. We
also show each galaxy’s observed spectrum and photometry; here, we have scaled
the photometry to match to the integrated $r$-band flux from the observed
spectrum.
In addition to these galaxies with suspect size measurements, we have excluded
a further 27 galaxies whose SED-fit $M_{*}/L$s are offset from the main
color–$M_{*}/L$ relation shown in Figure 1 by more than 0.25 dex. If we use
Equation 1 to derive new stellar mass estimates for these galaxies, all of
these galaxies move back into the main cloud in both Figure 2 and a stellar
mass-dynamical mass plot, with mean/median offsets of $\lesssim 0.02$ dex in
both cases.
The 190 galaxies discarded on the basis of inspection are shown in Figure 2 as
small red crosses; the small blue crosses show the 27 galaxies with discrepant
$M_{*}/L$s. As a function of $\Delta R_{\mathrm{e}}$, the fraction of
inspected sources that have been discarded goes fairly smoothly from 60 % for
$\Delta R_{\mathrm{e}}\sim-0.3$ dex to $\sim 100$ % for $\Delta
R_{\mathrm{e}}<-0.5$ dex. The discarded fraction has a similar dependence on
mass: it is $\sim 60$ % for $M_{*}\sim 10^{10.7}$ M⊙, rising to $\sim 85$ %
for $M_{*}\sim 10^{11}$ M⊙, and 100 % for $M_{*}>10^{11.4}$ M⊙.
This leaves us with a sample of 63 massive, compact, early-type and red
sequence galaxy candidates; these are are marked in Figure 2 with heavy black
circles. Of those galaxies that we have retained, 8 % (5/63) have observed
sizes smaller than $0\farcs 75$; all of these have $M_{*}<10^{11}$ M⊙. We have
provided the properties of our compact galaxy candidates in Table 1.
### 3.2. Are the Size Measurements Wrong?
We have performed a number of checks to validate the small measured sizes of
our compact galaxy candidates. The compact galaxy candidates do not have
significantly larger size measurement errors in comparison to the full sample
shown in Figure 2. For both the $r$\- and $z$-bands, our candidates are not
anomalous in a plot of Petrosian half-light radius versus model effective
radius, nor are they anomalous in a plot of $r$-band size versus $z$-band
size. For all but two of the candidates, the Petrosian and model magnitudes
agree to within 0.15 mag. The mean offset between model and Petrosian
magnitudes is -0.06 mag for the compact galaxies, compared to -0.08 mag for
the full sample shown in Figure 2. That is, the compact candidates appear to
be well described by the de Vaucouleurs model fits.
For the New York University (NYU) Value Added Galaxy Catalog (VAGC), Blanton
et al. (2005) have fit the radially-averaged light profiles of each object,
fitting for the Sérsic index as a free parameter over the range $0\leq n<6$.
In order to explore further the issue of the quality of the de Vaucouleurs
profile fits, we have gone to the NYU VAGC for DR7, and looked up the Sérsic
fit results for each of our candidates.
In Figure 4, we show the distribution of Sérsic parameters for our candidates,
as well as a comparison between the Sérsic and de Vaucouleurs sizes. First, we
note that nearly all (59/63) of our compact galaxy candidates have $n>3$;
these are not late-type (exponential) galaxies. It is therefore
unsurprising—but still reassuring—that the two size measures agree quite well:
for the median galaxy among our candidates, the de Vaucouleurs size is $\sim
10$% smaller than the Sérsic size; the RMS dispersion is 0.10 dex. For
comparison, the median quoted error for the de Vaucouleurs size measurements
is 4.6%.
Notice that about a quarter (17/63) of our candidates have $n=5.9$ in the NYU
VAGC; this is the maximum value allowed in the fits. These galaxies are
considerably more centrally-concentrated than the canonical de Vaucouleurs
$R^{1/4}$-law profile. However, the trend with increasing Sérsic index is for
the de Vaucouleurs size, $R_{\mathrm{DeV}}$, to be systematically lower than
the Sérsic size, $R_{\mathrm{Ser}}$: making a least-squares fit to the data
shown in Figure 4, we find $\log
R_{\mathrm{DeV}}/R_{\mathrm{Ser}}=-0.02-0.05~{}(n-4)$. If these galaxies do
have $n>6$, then we may well be underestimating their sizes by $\gtrsim 25$ %.
Figure 4.— Comparison between effective radii derived assuming a de
Vaucouleurs ($n=4$) profile and assuming a Sérsic profile ($0<n<6$). Whereas
the basic SDSS catalog uses a sector-fitting technique to fit either an
exponential ($n=1$) or a de Vaucouleurs ($n=4$) profile, for the NYU VAGC,
Blanton et al. (2005) have fit the radial profiles of each object assuming a
general Sérsic model ($0<n<6$). This Figure shows the ratio of these two sizes
for our compact galaxy candidates, based on the $z$-band data, as a function
of Sérsic index $n$. Almost all candidates have $n>3$—these galaxies are not
obviously exponentials. However, approximately 25 % have $n=5.9$; the maximum
value allowed in the Blanton et al. (2005) fits. For these galaxies, the
median ratio between the two size measurements is 0.88, with an RMS scatter of
0.1 dex. In the main panel, we show a least-squares fit to the data, the
dispersion around this relation is $\lesssim 0.1$ dex.
Guo et al. (2009) have recently demonstrated that as a result of biases in the
way the background sky level is estimated for the Sérsic fitting, the NYU-VAGC
sizes are systematically underestimated at the $\gtrsim 15$ % level for
$n\gtrsim 5$. This problem is progressively worse for large sizes
($\Theta_{\mathrm{e}}\gtrsim 1^{\prime\prime}$) and bright magnitudes
($r\lesssim 16$); for our compact galaxy candidates, the effect is likely to
be at the $\sim 20$ % level. But note this implies that the difference between
the de Vaucoleurs and Sérsic sizes is even greater than Figure 4 might
suggest: the sizes of the $n\gtrsim 5$ compact galaxies may be underestimated
by as much as $\gtrsim 30$ %.
As a final check, therefore, we have also re-derived Sérsic effective radii
for our compact galaxy candidates using GALFIT (Peng et al., 2002) and done a
similar comparison as for the NYU VAGC sizes. The agreement between the GALFIT
and VAGC Sérsic indices is quite good, with an rms difference in $n$ of 1.1.
Again the vast majority of objects have $n>3$. There are 19 objects that are
assigned the maximum allowed value of $n=8$, but only 9 of these have $n=5.9$
in the VAGC. Making a similar fit to the difference between the default De
Vaucouleurs and the GALFIT Sérsic effective radii, we find $\log
R_{\mathrm{DeV}}/R_{\mathrm{Ser}}=0.08-0.08~{}(n-4)$. As before, we may be
underestimating the sizes of high $n$ galaxies by 10—35 %, although this
comparison suggests that we may also be overestimating the sizes of the few
candidates with $n<4$. The median galaxy has a GALFIT Sérsic effective radius
15 % smaller than the default De Vaucouleurs value. Lastly, we note that there
is a definite mass-dependence to the agreement between the GALFIT Sérsic and
default De Vaucouleurs effective radii, such that all but one of the galaxies
for which the sizes agree to within 20 % have $M_{*}>10^{11}$ M⊙.
To summarize the results of this section, then: comparison to 1D and 2D Sérsic
fits does not suggest that the default De Vaucouleurs effective radii from the
SDSS catalog are catastrophically wrong for our compact galaxy candidates; if
anything, these comparisons suggest that we may in fact be underestimating the
sizes of these galaxies by 10—30 %.
### 3.3. A Consistency Check Based on Velocity Dispersions
Assuming that elliptical galaxies are structurally self-similar, the virial
theorem implies that $M_{*}\propto R_{\mathrm{e}}\sigma^{2}$. At fixed mass,
galaxies with small sizes should therefore have higher velocity dispersions,
with $\Delta\sigma\propto(\Delta R_{\mathrm{e}})^{-1/2}$.
In order to verify that the observed velocity dispersions of our compact
galaxy candidates are consistent with their being genuinely small, in the
lower panel of Figure 5 we plot the offset from the local size–mass relation
for early type galaxies, $\Delta\log R_{\mathrm{e}}$, against the offset from
the $M_{*}$–$\sigma$ relation, $\Delta\log\sigma$; the $M_{*}$–$\sigma$
relation itself is shown in the upper panel of the Figure. For the lower panel
of this plot, we have shifted the Shen et al. (2003) relation upwards in size
by 0.05 dex to be consistent with the present data set; our conclusions do not
depend on this decision. The greyscale and points show those $0.066<z<0.12$
galaxies with ${}^{0.1}(u-r)>2.5$ and $M_{*}>10^{10.7}$ M⊙; the red circles
indicate those galaxies that we have identified as compact.
Figure 5.— Using observed velocity dispersions to validate the measured sizes
of our compact galaxy candidates. Upper panel: the mass–velocity dispersion
relation for red sequence galaxies at $0.066<z<0.12$. The points and greyscale
show the SDSS data. The yellow plusses show the median velocity dispersion in
narrow bins of stellar mass; the solid yellow line shows a simple fit to these
points for $10.7<\log M_{*}<11.5$. The red circles highlight our compact
galaxy candidates. Lower panel: the offset from the $M_{*}$–$R_{\mathrm{e}}$
relation, plotted against the offset from the $M_{*}$–$\sigma$ relation for
$M_{*}>10^{10.7}$ M⊙ galaxies with ${}^{0.1}(u-r)>2.5$. If the offsets from
these two relations is a function of galaxy size, then we expect
$\Delta\log\sigma=-0.5\times\Delta\log R_{\mathrm{e}}$ (long dashed line). Our
compact galaxy candidates are shown as the red circles. In general, the
observed velocity dispersions support the idea that our compact galaxy
candidates are indeed relatively small; this is particularly true for those
with $\sigma>200$ km/s. There is one clear exception, marked with a cross in
both panels; this galaxy is also the most extreme outlier in Figure 4.
For our compact galaxy candidates, the median offset from the size–mass
relation is $\Delta\log R_{\mathrm{e}}=-0.38$ dex. We would therefore expect a
median offset from the $M_{*}$–$\sigma$ relation relation of
$\Delta\log\sigma=-0.5\times-0.38=0.19$ dex. The median value for
$\Delta\log\sigma$ is 0.12 dex—roughly 85 % of the expected value, and $\sim
1.5$ times greater than the intrinsic scatter in the relation. Overall, these
results are fairly consistent, although they do indicate that the sizes may be
underestimated and/or the masses may be overestimated at the level of 10–20 %.
We note that the difference between the default SDSS and the NYU VAGC size
measurements can account for at least half of this effect (see §3.2).
There is one of our compact galaxy candidates however, whose velocity
dispersion is clearly inconsistent with its being massive and compact, which
we have marked in Figure 5 with a cross; indeed, it has the lowest observed
velocity dispersions of all of our compact galaxy candidates. This galaxy is
also the biggest outlier in Figure 4. We will discuss this object in more
detail in §4.2.
We also note that the observed velocity dispersions of the most extreme
outliers from the size–mass relation ($\Delta\log R_{\mathrm{e}}\lesssim-0.4$)
are only marginally higher than for galaxies with ‘normal’ sizes. Only one of
these candidates ($\log M_{*}=10.73$) has $\Delta\log\sigma>0.18$ dex; the
median value of $\Delta\log\sigma$ is 0.03 dex. It would seem that the effects
of ‘outlier noise’ (ie. objects being pushed to the edge of the observed
distribution by measurement errors, rather than their true, intrinsic
properties) become dominant at these very extreme values of $\Delta\log
R_{\mathrm{e}}$.
With these caveats, the observed velocity dispersions generally support the
idea that the offsets from both the $M_{*}$—$R_{\mathrm{e}}$ and
$M_{*}$—$\sigma$ relations for our compact galaxy candidates can be explained
by their having small sizes for their masses/velocity dispersions.
## 4\. The Importance of Selection Effects for Compact Galaxies
### 4.1. SDSS Spectroscopic Sample Selection
In order to be targeted for SDSS spectroscopic follow-up (and so to appear in
Figure 2), galaxies have to satisfy a complicated set of selection criteria
(Strauss et al., 2002). In brief, there is a magnitude cut: objects must be
detected at $>5\sigma$ significance, and have $r_{\mathrm{P}}<17.77$. Any
objects that have been marked as blended and then segmented into smaller
objects are rejected, as are any objects that include saturated pixels, or
have been deblended from objects with saturated pixels. There are also a
series of (low) surface-brightness-dependent criteria that are not relevant
here.
The first important selection criterion for our purposes is the star/galaxy
separation criteria, since we are concerned about bright, compact galaxies
being mistakenly identified as stars. Star/galaxy separation is done on the
basis of the difference between the ‘PSF’ and ‘model’ magnitudes in the $r$
band. (Here, the PSF magnitude is derived by fitting the PSF model to each
object, in analogy to the exponential/de Vaucouleurs model fits described in
§2.1, and then aperture corrected to $7\farcs 4$.) Specifically, objects are
only selected for spectroscopic follow-up where:
$\Delta_{\mathrm{SG}}\equiv r_{\mathrm{PSF}}-r_{\mathrm{model}}\geq 0.3~{}.$
(2)
Further to this star/galaxy discriminator, in order to avoid cross talk
between spectroscopic fibers, galaxies with fiber magnitudes $g<15$, $r<15$,
and $i<14.5$ are also rejected. Lastly, all objects with $r_{\mathrm{P}}<15$
and a Petrosian radius $\Theta_{\mathrm{P}}<2^{\prime\prime}$ are rejected.
This criterion was introduced to eliminate “a small number of bright stars
that that managed to satisfy equation [2] during the commissioning phase of
the survey, when the star/galaxy separation threshold was
$\Delta_{\mathrm{SG}}=0.15$ mag, and was retained for later runs to avoid
saturating the spectroscopic CCDs (Strauss et al., 2002). Strauss et al.
(2002) also note that of the approximately 240000 $r<17.77$ objects in runs
752 and 756, none were rejected by the $r_{\mathrm{P}}<15$,
$\Theta_{\mathrm{P}}<2^{\prime\prime}$ criterion alone.
In order to model these selections, we need to relate the relevant observed
quantities (viz., the apparent Petrosian magnitude, $r_{\mathrm{P}}$, $gri$
fiber magnitudes, the apparent Petrosian size, $\Theta_{P}$, and the
star/galaxy separation parameter, $\Delta_{\mathrm{SG}}$) to intrinsic size
and stellar mass.
For a given redshift/distance, the intrinsic size can be trivially related to
the observed effective radius, $\Theta_{\mathrm{e}}$. In order to relate
$r_{\mathrm{P}}$ to $M_{*}$, we have made a simple fit to the relation between
stellar mass and absolute magnitude in the observers frame $r$ band (ie. with
no K-correction) for red sequence galaxies at $0.066<z<0.12$ with $M_{r}>-21$.
Note that this method naturally accounts for mass-dependent trends in, e.g.,
metallicity along the red sequence. The scatter around this relation is $\sim
0.06$ dex, with no obvious magnitude dependence. We have derived similar
relations for both $g_{\mathrm{P}}$ and $i_{\mathrm{P}}$.
We have also derived empirical relations for $\Theta_{\mathrm{P}}$,
$\Delta_{\mathrm{SG}}$, and the difference between the Petrosian and fiber
magnitudes, $\Delta_{\mathrm{fib}}=m_{\mathrm{P}}-m_{\mathrm{fiber}}$, as
functions of $\Theta_{\mathrm{e}}$ and $r_{\mathrm{P}}$, using the sample of
massive, red sequence galaxies shown in Figure 2. The scatter around these
relations is 0.059 dex (15 %) , 0.18 mag (18 %), and 0.11 mag (9 %)
respectively, with no obvious systematic residuals.
Figure 6.— The size–mass relation for massive, red sequence galaxies at low
and high redshifts. The points and circles are for SDSS galaxies, as in Figure
2; those galaxies that we have rejected as per §3 are not shown. The contours
show the relative volume completeness of the SDSS spectroscopic sample for
$0.066<z<0.12$, as marked. The orange points with error bars are the D09
sample of $1.2<z<2.0$ galaxies from the GDDS and MUNICS. The blue points with
error bars are the vD08 sample of $z\sim 2.3$ galaxies from MUSYC. There are
no red galaxies in the local universe with sizes and masses comparable to the
compact galaxies found at high redshift. This lack cannot be explained by
selection effects.
Note that there is a danger of circularity in this argument: any objects that
do not satisfy the selection criteria will not be present in the sample that
we are using to model the selection criteria. The crucial assumption here,
then, is that we can extrapolate the functions for
$\Theta_{\mathrm{P}}(\Theta_{\mathrm{e}},~{}r_{\mathrm{P}})$,
$\Delta_{\mathrm{SG}}(\Theta_{\mathrm{e}},~{}r_{\mathrm{P}})$, and
$\Delta_{\mathrm{fib}}(\Theta_{\mathrm{e}},~{}r_{\mathrm{P}})$ down past the
limits of the spectroscopic sample. In this regard, it is significant both
that the derived functions are smooth all the way down to the selection
limits, and that we do not see obvious cut-offs in the data associated with
these limits.
In Figure 2, we show how these selection criteria translate onto the
$(M_{*},R_{\mathrm{e}})$ plane for several example redshifts between 0.035 and
0.12. The thicker, roughly diagonal, long-dashed lines show the star/galaxy
separation criterion; the dotted lines show the ‘cross-talk’ fiber magnitude
selection; the thinner, short-dashed boxes show the effect of the ‘saturation’
selection against bright, compact objects. Note that, for example, a galaxy
with $M_{*}\gtrsim 10^{11}$ M⊙ and $R_{\mathrm{e}}\lesssim 2$ kpc would not be
selected as an SDSS spectroscopic target for $z\lesssim 0.05$.
### 4.2. Compact Galaxies at High and Low Redshifts
In Figure 6 we again show the size–mass relation for our sample of massive,
red sequence galaxies at $0.066<z<0.12$, with the exception that we have not
plotted those galaxies rejected as per §3.1. Furthermore, in contrast to
Figure 2, we have used the selection limits derived in §4.1 to estimate the
relative completeness of the SDSS spectroscopic sample across the
$0.066<z<0.12$ volume; these are shown by the contours. These completeness
estimates also include the $r_{\mathrm{P}}<17.77$ selection limit, which can
be seen to affect galaxies with $M_{*}\lesssim 10^{10.6}$ M⊙ at the distant
end of our redshift window.
For comparison, we have also overplotted the high-redshift samples of D09
(yellow points) and vD08 (, blue) (blue points). Where we have used size
measurements from the $z$-band for the SDSS galaxies, these high-redshift
studies use the NICMOS F160W filter, which corresponds to rest-frame $r$ at
$z=1.6$, moving close to $g$ by $z=2.3$. Locally, the difference between $z$\-
and $r$-band measured sizes leads to a slightly different slope to the
size–mass relation for red sequence galaxies (a slope of 0.65, rather than
0.56). The $r$\- and $z$\- band size–mass relations intersect at around
$M_{*}\sim 10^{10}$ M⊙; the mean $r$-band size at $M_{*}\sim 10^{11}$ M⊙ is 15
% larger than in the $z-$band. That is, by using $z$-band derived effective
radii, we are, if anything, underestimating the sizes of the local galaxies in
comparison to those at high redshift. Similarly, our decision to use the De
Vaucouleurs effective radii given in the basic SDSS catalog, rather than more
general Sérsic ones appears to lead to an underestimate of galaxy sizes. In
other words, adopting $r$\- or $g$-band derived sizes, or using Sérsic instead
of De Vaucouleurs effective radii, would increase the discrepancy between the
high- and low-redshift samples.
There is one of our candidates (marked with a cross) that appears to have
similar properties to one of the larger of the vD08 galaxies. This turns out
to be the galaxy whose observed velocity dispersion is inconsistent with its
being genuinely compact (§3.3); where we would predict $\Delta\log\sigma=0.24$
dex, or $\sigma=310\pm 70$ km/s, what we observe is $\Delta\log\sigma=-0.17$
dex and $\sigma=129\pm 14$ km/s. This is also the galaxy with the largest
difference between the Sérsic– and De Vaucouleurs–sizes ($\log
R_{\mathrm{Dev}}/R_{\mathrm{Ser}}=-0.34$; see §3.2). Adopting the NYU VAGC
Sérsic size measurement is not sufficient to reconcile the observed size and
mass with the velocity dispersion: the observed velocity dispersion would
still be too small by $\sim 0.2$ dex. This galaxy also sits nearly 0.25 dex
above the median color–mass-to-light relation shown in Figure 1; using the
Bell & de Jong (2001) prescription for $M_{*}/L$ as a function of $(B-V)$
leads to a stellar mass estimate that is 0.17 dex lower. Adopting both this
mass estimate and the NYU VAGC size estimate, we do find consistency between
$\Delta\log R_{\mathrm{e}}$ and $\Delta\log\sigma$. In this sense, this galaxy
is the weakest of our compact galaxy candidates—it seems to have had its size
underestimated and/or its mass overestimated.
We also stress that the observed velocity dispersions of the candidates that
lie furthest from the main size–mass relation suggest that these galaxies have
had their sizes significantly underestimated (see §3.3).
If the vD08 galaxies were placed at $0.066<z<0.12$, the SDSS spectroscopic
completeness would typically be $\gtrsim 75$ %. Note, however, that there are
two $R_{\mathrm{e}}<0.5$ kpc galaxies from the vD08 sample and one from the
D09 sample for which the SDSS completeness is just 20–40 %. The average SDSS
completeness for the vD08 galaxies placed at $0.066<z<0.12$ would be 80 %.
If the Kriek et al. (2008a)/vD08 galaxies were not to evolve in either size
or number density from $z\sim 2$ to the present day, we would expect there to
be $\sim 6500$ $M_{*}>10^{11}$ M⊙ galaxies with $\Delta\log
R_{\mathrm{e}}<-0.4$ dex at $0.066<z<0.12$, of which $\sim 5250$ should appear
in the SDSS spectroscopic sample. Instead, we have only one weak candidate.
As an interim conclusion, then, we have shown that there are no galaxies in
the local universe (at least as probed by the SDSS spectroscopic sample) that
are directly analogous to the compact galaxies found at high redshift. This
dearth of compact galaxies cannot be explained by selection effects. In
Appendix A, we confirm this conclusion by searching for compact galaxy
candidates from within the SDSS photometric sample, using photometric
redshifts.
Moreover, we stress that those galaxies which we have identified as ‘compact’
are not qualitatively similar to the compact galaxies found at higher
redshifts, which are offset from the local size–mass relation by at least
twice as much again as our local compact galaxy candidates.
### 4.3. The Number Density of Massive, Compact Galaxies
Figure 7.— The observed size distribution of massive, red galaxies at $z\sim
0.1$, $z\sim 1.6$, and $z\sim 2.3$; each panel is for a different mass range
as marked. In each panel, the solid histogram represents the SDSS
spectroscopic sample. The blue, diagonally-hatched histogram represents the
vD08 sample of nine massive, passive galaxies at $z\sim 2.3$; the yellow,
horizontally-hatched histogram represents the 10 $z\sim 1.6$ GDDS galaxies
from D09 . The arrows at the bottom of each panel indicate the positions of
the individual high redshift galaxies. The $z\sim 2.3$ galaxies have to
undergo significant structural evolution over $z\lesssim 2.3$ to match the
properties of local universe galaxies; at least part of this evolution has
already occurred by $z\sim 1.6$.
In Figure 7, we provide a more quantitative statement of our conclusion with
respect to the size evolution of massive galaxies from $z\sim 2$ to $z\sim
0.1$ by plotting the size distribution for massive, red galaxies in different
mass bins. In this figure, the filled histograms represent the main SDSS
spectroscopic sample described above. The horizontal-hatched histograms show,
for comparison, the situation at $z\sim 1.6$, based on the ten D09 galaxies
drawn from the GDDS; similarly, the diagonal-hatched histograms show the nine
$z\sim 2.3$ Kriek et al. (2008a) galaxies with sizes from vD08 .
The Kriek et al. (2008a)/vD08 sample is representative, but not complete. In
order to derive the densities plotted in Figure 7, we have scaled each of the
vD08 galaxies as follows: first, we have normalized the distribution to have
a density of $1.5\times 10^{-4}$ Mpc-3, which corresponds to the total number
density of all $2<z<3$ galaxies to the mass limit of Kriek et al. (2008a),
derived using the mass function fit given by Marchesini et al. (2008); then,
we have scaled this distribution by a factor of 16/36 to count only those
galaxies with little or no ongoing star formation from Kriek et al. (2008a)
that seem to form a red sequence (Kriek et al., 2008b). For the D09 sample,
we are able to use $1/V_{\mathrm{max}}$ scalings from Glazebrook et al.
(2004).
The location of each individual high-redshift galaxy is marked in Figure 7
with an arrow: the slightly lower blue arrows show the vD08 galaxies; the
slightly higher yellow arrows are for the D09 galaxies. Clearly, given the
small numbers, the uncertainties on these high redshift values are quite
large, but they do provide a useful order of magnitude estimate for comparison
to the local values.
The clear implication from the comparison between the $z\sim 0.1$ and $z\sim
2.3$ data in Figure 7 is that, consistent with the conclusions of vD08 , not
one of the vD08 galaxies is consistent with the properties of the $z\sim 0$
galaxy population. With the results we have now presented, we can extend this
conclusion by confirming that this discrepancy cannot be explained by
selection effects in the low redshift sample.
There are local analogs for less than half of the $z\sim 1.6$ galaxies, albeit
with considerably higher number densities. This would imply that at least some
($\lesssim 50$ %) of the $z\lesssim 2.3$ evolution has already occurred by
$z\sim 1.6$.
## 5\. Discussion
### 5.1. Compact Galaxy Properties
Figure 8.— The properties of our compact galaxy candidates in comparison to
the general population of massive, red sequence galaxies. As in other Figures,
the black points and greyscale show the data density of all galaxies with
$M_{*}>10^{10.7}$ M⊙, ${}^{0.1}(u-r)>2.5$, and $0.066<z<0.12$; the red circles
show our compact galaxy candidates. The grey boxes with error bars show the
mean and rms scatter in each quantity for quintiles of the velocity dispersion
distribution; the red boxes show the same for those of our compact galaxy
candidates with $\sigma<200$ km/s and $\sigma>200$ km/s separately. At fixed
velocity dispersion, our compact galaxies have slightly lower than average
mean ages and slightly higher metallicities—however, this result is only
significant at the $2\sigma$ level.
In Figure 8, we plot the properties of our compact galaxies in comparison to
the general massive, red galaxy population. In each panel, the circles
highlight our compact galaxies, while the points and greyscale show all
galaxies with $M_{*}>10^{10.7}$ M⊙, ${}^{0.1}(u-r)>2.5$, and $0.066<z<0.12$.
The large grey boxes with error bars show the mean and standard deviation of
each plotted property in quintiles of the velocity dispersion distribution.
Similarly, the red boxes with error bars show the mean and standard deviations
for our compact galaxy candidates in two bins, separated at $\sigma=200$ km/s;
the median for this sample.
In each of the panels of Figure 8 (from left to right), we show the equivalent
width of the H$\delta$ line (where negative values imply emission), the
luminosity weighted mean stellar metallicity, and the luminosity weighted mean
stellar age, as derived from the DR4 SDSS spectra by Gallazzi et al. (2005).
Because these estimates are available only for DR4, only around half of our
compact candidates can be plotted in these panels, and only 3/10 of those with
$M_{*}>10^{11}$ M⊙.
We have also matched our compact galaxy sample to the AGN sample described by
Kauffmann et al. (2003b), for SDSS DR4. These AGN hosts have been selected by
their [OIII]/H$\beta$ and [NII]/H$\alpha$ emission line ratios; ie. the
Baldwin, Phillips & Terlevich (1981, BPT) diagram. 34 of our 63 galaxies
appear in the DR4 catalog; of these, 11 are classified as AGN on the basis of
their emission line ratios. This is slightly higher than the AGN fraction of
the parent sample, which is in the range 20—26 % for the mass range we are
considering. Of the 11 galaxies identified as AGN hosts, five sit on or
slightly above the main $M_{*}$—$L_{\mathrm{[OIII]}}$ relation, with
$L_{\mathrm{[OIII]}}\approx 10^{6}$ L⊙, four have $L_{\mathrm{[OIII]}}\sim
10^{7-8}$ L⊙, and one is quite high luminosity, with
$L_{\mathrm{[OIII]}}=10^{8.7}$ L⊙. These 11 galaxies are marked in each panel
of Figure 8 with a small blue cross.
Kauffmann et al. (2003b) also provide revised stellar mass and velocity
dispersion measurements for these galaxies. Accounting for the presence of an
AGN does not have a major impact on these measurements: the masses and
velocity dispersions change at the level of 0.05 dex and 16 km/s,
respectively. That is, while it is possible that an optically bright point
source may bias the measured sizes of these galaxies downwards, within the
stated errors, the AGN does not significantly affect the derived values of
$M_{*}$ or $\sigma$. (It is relevant here that only one of our compact galaxy
candidates shows a significant residual point-source after subtracting off the
best-fit Sérsic profile, as produced by GALFIT; see §3.2.)
Looking now at Figure 8, it is clear that the majority of our compact galaxy
candidates have quite old stellar populations. For the $\sigma<200$ km/s bin,
the median age is 6 Gyr, although the ages do range from 2 to 10 Gyr, while
all but one of the $\sigma>200$ km/s candidates have $T>6$ Gyr. Among the
lower velocity dispersion candidates, there is a clear tendency towards
relatively high equivalent widths for H$\delta$ absorption, suggestive of a
relatively recent ($\lesssim 2$ Gyr) star formation event.
At fixed velocity dispersion, our compact galaxy candidates may have slightly
higher metallicities, and be slightly younger than average. Using bootstrap-
resampling on similar sized samples drawn randomly from the mass-limited
sample, and controlling for velocity dispersion, these results are weakly
significant at best: $1.9\sigma$ for the age, and $<1\sigma$ for metallicity.
Considering only the higher velocity dispersion candidates ($\sigma>200$
km/s), the significance of these differences become $1.9\sigma$ and
$2.2\sigma$ for the age and metallicity offsets, respectively. This weakly
significant result should be contrasted with the results of Shankar & Bernardi
(2009) and van der Wel et al. (2009), who find that, on average and at fixed
dynamical mass, early type galaxies with higher velocity dispersions (or,
equivalently, smaller sizes) have older mean stellar ages.
While the younger mean stellar ages and lower metallicities of our compact
galaxy sample are only weakly significant, both would imply a relatively late
start to star formation for these galaxies and/or their progenitors. But if
these galaxies grow in size through mergers (for example) then it is possible
that these galaxies are small not because their formation is delayed relative
to other galaxies of the same mass or velocity dispersion, but rather because
they have had fewer mergers overall, or perhaps just fewer recent mergers.
That is, it may be that, at fixed mass, these compact galaxies are in fact
older, in the sense that they have been assembled earlier, and existed in
(more or less) their present form for longer than other galaxies of the same
mass or velocity dispersion.
### 5.2. Comparison to Other Recent Works
In a similar study to this, using sizes and photometry from the NYU VAGC for
SDSS DR6, Trujillo et al. (2009) have recently reported the detection of 29
$z<0.2$ galaxies with $M_{*}>8\times 10^{10}$ M⊙ and $R_{\mathrm{e}}<1.5$ kpc.
In contrast, we find just one galaxy from our $0.066<z<0.12$ red sequence
galaxy sample that satisfy these mass and size criteria; this implies a
difference in volume densities of a factor of 5.5. Most of this difference is
explained by the fact that we have preselected our compact galaxies to be red.
Of the Trujillo et al. (2009) galaxies, only around 30 % (9/29) satisfy our
${}^{0.1}(u-r)>2.5$ criterion, bringing our number densities into agreement.
On the other hand, if we also look at ${}^{0.1}(u-r)<2.5$ galaxies, inspected
as per §3.1, we find only 7 additional candidates, only 3 of which have
$M_{*}>8\times 10^{10}$ M⊙; the most massive of these blue compact galaxy
candidates is $7.2\times 10^{10}$ M⊙.
We also note that the Trujillo et al. (2009) galaxies have considerably
smaller observed sizes than the galaxies we consider here. (Recall that we
adopt a minimum observed size of $0\farcs 75$; for galaxies with inferred
sizes smaller than this, we adopt a size of $0\farcs 75$ as an upper limit on
the true size.) The largest observed size among the Trujillo et al. (2009)
sample is $0\farcs 70$; the median is just $0\farcs 48$. Enforcing our minimum
allowed size of $0\farcs 75$, only one of the Trujillo et al. (2009) galaxies,
irrespective of color, would have $R_{\mathrm{e}}<1.5$ kpc; the median size
for the sample would become 2.1 kpc. It is also relevant here that only one of
the Trujillo et al. (2009) galaxies is at $z<0.12$; the other 28 are all found
at $0.12<z<0.20$. This suggests that the Trujillo et al. (2009) size
measurements may be biased by inadequate resolution.
The key difference between our compact galaxy sample and that of Trujillo et
al. (2009) is that they find a median velocity dispersion which is only 0.04
dex higher than their control sample, even though the mean size and mass are
offset from the Shen et al. (2003) relation by $-0.5$ dex. This discrepancy
can only be explained by either very large structural differences, or if the
Trujillo et al. (2009) sample is disproportionally affected by large
measurement errors in size and/or mass. In this context it is significant
that, the observed velocity dispersions for our candidates with $\Delta\log
R_{\mathrm{e}}<-0.4$ dex imply that their very small inferred sizes are
produced by large errors in the measured sizes (see §3.3). Only with followup
observations will we be able to determine the true nature of the Trujillo et
al. (2009) galaxies. In any case, Trujillo et al. (2009) also do not find any
galaxies directly comparable to those found at $z\gtrsim 2$.
Even more recently, Valentinuzzi et al. (2009) have described a sample of 147
compact galaxies selected from the WIde-field Nearby Galaxy-cluster Survey
(WINGS) of X-ray selected clusters at $0.04<z<0.07$. Unlike in this work,
Valentinuzzi et al. (2009) do find galaxies with properties comparable to the
3 largest vD08 galaxies; similarly, there are local WINGS analogs for 8 of
the 10 GDDS galaxies from D09 . However, this relies on their scaling the high
redshift galaxies’ masses down by 0.15 dex to account for the poor treatment
of NIR-luminous thermally pulsating asymptotic giant branch (TPAGB) stars in
the Bruzual & Charlot (2003) stellar population models. While both Kriek et
al. (2009) and Muzzin et al. (2009) show that the stellar masses for the
$z\sim 2.3$ implied by different models vary by $\sim 0.1$ dex, we have not
applied such a correction here.
We note, however, that the high- and low-redshift samples have been treated
consistently here, including the fact that all masses were derived from the
rest-frame optical. Further, we note that van der Wel et al. (2006) have shown
that stellar masses derived from the rest-frame optical and using Bruzual &
Charlot (2003) models are consistent with the dynamical masses of $z<1$
galaxies, and are unaffected by the TPAGB uncertainties.
The Valentinuzzi et al. (2009) compact galaxies sample is selected by
effective surface mass density, $\Sigma_{\mathrm{e}}=M_{*}/2\pi
R_{\mathrm{e}}^{2}>4\times 10^{9}$ M⊙ kpc-2, in the range 3—50 $\times
10^{10}$ M⊙. Our compact galaxy selection is roughly equivalent to
$\Sigma_{\mathrm{e}}\gtrsim 3.6\times 10^{9}$ M⊙ kpc-2; nearly half (28/63) of
our compact galaxy candidates satisfy the Valentinuzzi et al. (2009)
$\Sigma_{\mathrm{e}}$ criterion. For their sample, Valentinuzzi et al. (2009)
derive a number density of $1.6\times 10^{-5}$ Mpc-3; to our mass limit of
$M_{*}>10^{10.7}$ M⊙, this value becomes $1.2\times 10^{-5}$ Mpc-3. These
values are solid lower limits, as they assume that no such galaxies exist
outside of the clusters observed by WINGS. For our sample, however, the number
density of $M_{*}>10^{10.7}$ M⊙ galaxies with $\Sigma_{\mathrm{e}}>4\times
10^{9}$ M⊙ kpc-3 is just $2.85\times 10^{-7}$ Mpc-3.
That is, after correcting as best we can for the different stellar mass
limits, our number densities are inconsistent by a factor of more than 40 with
those found by Valentinuzzi et al. (2009). Again, our use of $z$-band
effective radii leads to smaller measured sizes than for bluer bands; this
discrepancy would only increase using $r$\- or $g$-band measured sizes. Either
our results are badly affected by unexplained selection effects, or there are
large discrepancies between our size and mass estimates and those of
Valentinuzzi et al. (2009).
We have considered possible spectroscopic selection effects that could bias
against bright, compact objects in §4.1, and shown these to be relevant for
$z\lesssim 0.05$. These effects may well explain why Valentinuzzi et al.
(2009) were able to match only a small fraction of their compact galaxies
(which have $0.04<z<0.07$) to objects in the (DR4) SDSS spectroscopic catalog.
We have shown, however, that our $0.065<z<0.12$ results are not strongly
affected by these kinds of selection effects (Figure 6, see also Appendix A).
Our estimated completeness is more than 60 % for all galaxies in the
Valentinuzzi et al. (2009) sample and greater than 90 % for 90 % of the
sample. The selection effects considered in §4.1 thus cannot explain the
difference in our inferred number densities.
An alternative explanation is that the Valentinuzzi et al. (2009) galaxies
only exist in rich clusters, and that SDSS suffers much higher spectroscopic
incompleteness in such dense fields because of fiber collisions. A completely
indepdendent estimate can be obtained from the Faber et al. (1989) sample: we
find that 5/319 of these galaxies have sizes smaller than the mass–size
relation by a factor of 2 or more. This fraction for clusters that is
approximately 15 times higher than what we find for all galaxies in SDSS.
While not conclusive, this does suggest that SDSS may suffer from additional
incompleteness beyond the effects we consider here. That said, we note that
several studies (e.g. Kauffmann et al., 2004; Park et al., 2007; Weinmann et
al., 2009) have found little or no evidence for an environmental dependence of
the size–mass relation within SDSS.
Thus we can find no easy explanation for the difference between the
Valentinuzzi et al. (2009) results and our own. Here again, velocity
dispersion measurements would provide an useful consistency check on the
Valentinuzzi et al. (2009) size and mass measurements.
Even despite these differences, however, we note that Valentinuzzi et al.
(2009) conclude that—barring large systematic errors in the high-redshift
measurements—at least 65 % (cf. our value of 100%) of the $z\sim 2.3$ galaxies
from vD08 and at least 20 % (cf. our value of 60 %) of the $z\sim 1.6$
galaxies from D09 have disappeared from the local universe. Accepting the
high-redshift results, these galaxies simply cannot evolve passively and
statically into the red sequence and/or early type galaxies found in the local
universe.
## 6\. Summary and Conclusions
The central question of this work has been the existence or otherwise of
massive, compact, quiescent and/or early type galaxies in the local universe,
and particularly the importance of selection effects in the SDSS spectroscopic
sample for such galaxies. We have shown that, especially for lower redshifts
($z\lesssim 0.05$), galaxies with the masses and sizes of those found at
$z\gtrsim 2$ would not be targeted for spectroscopic followup (Figure 2). The
main reason for this is not the star/galaxy separation criterion, but rather
the exclusion of bright and compact targets in order to avoid saturation and
cross-talk in the spectrograph (see §4.1).
We have therefore conducted a search for massive, compact galaxies at
$0.066<z<0.12$, where these selection effects should be less important. We
estimate that for $0.066<z<0.12$, the average completeness for galaxies like
those from vD08 and D09 would be $\gtrsim 25$ % at worst, and $\sim 80$ % on
average (Figure 6).
Starting from a sample of massive ($M*>10^{10.7}$ M⊙) red sequence
(${}^{0.1}(u-r)>2.5$) galaxies, we have selected the 280 galaxies with
inferred sizes that are a factor of 2 or more smaller than would be expected
from the Shen et al. (2003) $M_{*}$–$R_{\mathrm{e}}$ relation for early type
galaxies. In order to confirm their photometry and size measurements, we have
visually inspected all of these objects. Unsurprisingly, by selecting the most
extreme outliers, a large fraction of these objects ($\sim 70$%) appear to be
instances where the size and/or stellar mass estimates are unreliable (§3.1).
For the 63 galaxies with no obvious reason to suspect their size or stellar
mass estimates, there is good agreement between the default SDSS size
measurement (based on the 2D light distribution, using a sector-fitting
algorithm, and assuming a de Vaucouleurs profile), and those given in the NYU
VAGC (based on the azimuthally average growth curve, assuming a more general
Sérsic profile). However, particularly for galaxies with high $n$, the de
Vaucouleurs size measurement is systematically smaller than the Sérsic one, at
the level of $\lesssim 25$ % (§3.2).
In general, and as expected, our 63 compact galaxy candidates have
significantly higher than average velocity dispersions (Figure 5). While it
remains possible that the sizes of at least some of our compact galaxy
candidates may have had their sizes underestimated by $\sim 30$ %, in general,
the relatively high observed velocity dispersions support the notion that they
are indeed unusually compact given their stellar masses (§3.3).
Among our compact galaxy candidates, there are no galaxies with sizes
comparable to those found $z\sim 2.3$ by vD08 ; we find analogs for $\lesssim
50$ % of the D09 galaxies at $z\sim 1.6$ (Figures 6 and 7). This lack cannot
be explained by selection effects. To confirm this, we have also compared the
size–mass diagram, constructed using photometric redshifts, based on both the
full photometric sample and the spectroscopic sub-sample (Appendix A). While
it is conceivable that SDSS is missing a few massive, compact galaxies, there
are again no signs of galaxies comparable to those of vD08 or D09 .
It is not impossible that some systematic errors in the estimation of
$M_{*}/L$s for the high redshift galaxies (e.g., an evolving stellar IMF) mean
that their stellar masses are vastly overestimated, however it would require
an overestimate of $\gtrsim 0.7$ dex to reconcile the vD08 galaxies with the
sizes of the smallest galaxies we have identified in the SDSS catalog.
Accepting the high redshift observations at face value, then, our results
confirm that massive galaxies, both individually and as a population, must
undergo considerable structural evolution over the interval $z\lesssim 2.3$ in
order to develop into the kinds of galaxies seen locally—even after star
formation in these galaxies has effectively ended. We see some hints that a
significant amount of this evolution ($\lesssim 50$ %) may have already
occurred by $z\sim 1.6$.
The fact that each and every one of the vD08 galaxies must undergo
significant structural evolution to match the properties of present-day
galaxies implies that the mechanism that drives this growth must apply more or
less evenly to all galaxies. To see this, let us assume that some external
process drives the size evolution of these galaxies, and that even a single
event is sufficient to move an individual galaxy onto the main size–mass
relation. Then, we can assume some simple probability distribution for the
number of events, $N$, among individual galaxies. (For example, we could
assume that events occur randomly across the time interval $z<2.3$, or that
each galaxy experiences $N\pm\sqrt{N}$ events.) Now, our results suggest that
the number density of vD08 –like galaxies drops by at least a factor of 5000
since $z\sim 2.3$. In order to ensure that at most 1/5000 galaxies have $N=0$
after $z\sim 2.3$, simple probabilistic arguments imply that the average
galaxy must undergo $\gtrsim 20$ events. This would imply that a strongly
stochastic process like major mergers cannot be the primary mechanism for the
strong size evolution of massive galaxies.
Apart from their small sizes and high velocity dispersions, our compact galaxy
candidates are not obviously distinct from the general population (Figure 8).
If anything, at fixed velocity dispersion, our compact galaxies have stellar
populations that are slightly younger than average (at $\sim 1.9\sigma$
significance). Even so, the majority of these galaxies’ stellar populations
are definitely ‘old’, with luminosity-weighted mean stellar ages typically in
the range 6–10 Gyr. But if some external mechanism drives the size evolution
of these galaxies, we speculate that their small sizes may indicate that they
have assumed their present form comparatively early, and in this sense they
may actually be relatively old (see also, e.g., van der Wel et al., 2009). If
so, with better understanding of the processes that determine the sizes of
early type galaxies, and in particular the role of merging, the properties of
these galaxies could provide a means of constraining the evolution of massive
galaxies after they have completed their star formation, including their late-
time merger histories.
## Appendix A Looking for Massive Compact Galaxies in the SDSS Photometric
Sample
In this Appendix, we present a complementary analysis in which we directly
compare the spectroscopic and photometric samples, in order to test the
conclusion that the lack of massive, compact galaxies in the spectroscopic
sample cannot be explained by the selection effects.
Figure 9.— Selecting $z\lesssim 0.1$ galaxies based on color alone. Each panel
shows the observed ($u-g$)–($r-z$) color–color plot for objects in the SDSS
spectroscopic sample; the right panel simply shows the central region in
greater detail. Points are color–coded according to their spectral
classification, viz.: galaxies (grey), galaxies with $0.066<z<0.10$ (light
red), galaxies with $0.066<z<0.10$ and ${}^{0.1}(u-r)>2.5$ (dark red), quasars
(yellow), late-type stars (light blue), and ordinary stars (dark blue), . The
box shows the color selection that we use to select $z\lesssim 0.1$ galaxies.
This selection should produce a reasonably complete sample of $z\lesssim 0.1$
galaxies, with some contamination from both stars and quasars. In particular,
later-type stars and some quasars have observed SEDs that are very similar to
red sequence galaxies at $z\sim 0.1$.
### A.1. Selecting Galaxies by Color Alone
Before we can address the question of massive compact galaxies in the SDSS
photometric sample, we must first devise a means of separating stars and
galaxies without selecting on the basis of observed size or light profile. Our
method for doing so is shown in Figure 9, which plots the observed
(extinction-corrected) $ugrz$ colors of different classes of objects from the
spectroscopic sample; we show: all galaxies (grey), $0.066<z<0.12$ galaxies
(bright red), and those with ${}^{0.1}(u-r)>2.5$ (dark red), O–K stars (dark
blue), M-type or later stars (light blue), and quasars (yellow).
The black box shown in Figure 9 shows our criteria for selecting
$0.066<z<0.12$ galaxies based on their $ugrz$ colors:
$\displaystyle 0.6$ $\displaystyle<$
$\displaystyle(u-g)<2.4~{}~{}~{}~{}~{}\mathrm{and}$ $\displaystyle
0.3\times(u-g)$ $\displaystyle<$ $\displaystyle(r-z)<1.2~{}.$ (A1)
Again, we apply this selection in terms of model colors. Note how, whereas the
stellar sequence is reasonably well separated from the region of color space
occupied by galaxies for $(u-g)\lesssim 2.5$, beyond this point, the late-type
stellar sequence turns up, such that late-type stars and galaxies are blended.
In the most general terms possible, the mean galaxy redshift increases towards
redder $(u-g)$ colors. This means that our ability to distinguish red galaxies
from late-type stars on the basis of their optical SEDs is limited to
$z\lesssim 0.12$.
In the right-hand panel of Figure 9, we zoom in on this selection region. From
this panel, it is clear that a large proportion of quasars will also be
included in our color-selected ‘galaxy’ sample. Similarly, it is clear that
this color selection is not 100 % efficient in excluding stars from our
sample: more quantitatively, with this selection we are able to exclude more
than 80 % of spectroscopically identified stars that are given
$0.066<z_{\textrm{phot}}<0.12$, while retaining more than 97 % of all
$0.066<z_{\textrm{phot}}<0.12$ galaxies. Furthermore, it should be remembered
that stars are already heavily selected against for the spectroscopic sample
plotted in Figure 9; the relative number of stellar ‘contaminants’ may well be
considerably higher for the photometric sample.
### A.2. Photometric Redshifts and Stellar Mass Estimates
A major improvement in DR7 is a complete revision in how the basic (photoz)
photometric redshifts are derived (Abazajian et al., 2009). Rather than using
some combination of synthetic template spectra to reproduce the observed
colors of individual galaxies, the new photoz algorithm directly compares the
observed photometry of individual galaxies to that of galaxies that have
spectroscopic redshifts. Specifically, for each individual object, the
algorithm finds the 100 closest neighbours in $ugriz$ color space, and fits a
hyper-plane to these points, rejecting outliers; the redshift is then
determined by interpolating along this 4D surface. In comparison to the DR6
algorithm, this reduces the RMS redshift error by more than 75 %
($\left<\Delta z\right>$ = 0.025), and significantly reduces systematic errors
(Abazajian et al., 2009).
For this analysis, rather than full SED-fit stellar mass estimates assuming
the photometric redshifts, we will simply make use of the empirical relation
between ${}^{0.1}(g-i)$ color and $M_{*}/L$ (Equation 1). In this way, we are
able to recover the $z_{\textrm{spec}}$-derived, SED-fit $M_{*}/L$s of the
sample of galaxies shown in Figure 1 to 0.045 dex ($1\sigma$); including the
effects of photometric redshift errors, k-corrections, and $M_{*}/L$ errors,
the total ($1\sigma$) error in $M_{*}$ is 0.13 dex. This should be compared to
the median formal error on the original SED-fit stellar mass estimates, 0.10
dex; that is, the errors on $M_{*}$ based on photometric masses (adding these
two errors in quadrature) are only about 60 % greater than those based on
spectroscopic redshifts.
### A.3. The Size Distribution of Massive, Red Sequence Galaxies
Figure 10.— The size–mass plot for massive, red sequence galaxies at
$0.066<z<0.12$, based on the spectroscopic and the full photometric SDSS
catalogs. Each panel shows the sizes and masses of galaxies based on, from top
to bottom, the spectroscopic sample using spectroscopic redshifts, the
spectroscopic sample using photometric redshifts, and the photometric sample
using photometric redshifts; in each case, only those objects inferred to have
${}^{0.1}(u-r)>2.5$, and $0.066<z<0.10$ are shown. In panel 3, many more
objects with inferred sizes $\lesssim 0.3$ kpc can be seen; these are largely
stars misclassified (in terms of their photometric redshifts) as galaxies. For
$M_{*}\lesssim 10^{10.8}$ M⊙ and $R_{\mathrm{e}}\lesssim 10^{-0.2}$ kpc,
comparison between panels 2 and 3 suggest that there may be a few additional
galaxies in the photometric sample that do not appear in the spectroscopic
sample. Figure 11.— The observed size distribution of massive, red galaxies
at $0.066<z<0.12$; each panel is for a different mass range as marked. In each
panel, the solid histogram represents the SDSS spectroscopic sample, analyzed
using spectroscopic redshifts. The black and red histograms are the SDSS
spectroscopic and photometric samples, respectively, analyzed using
photometric redshifts. We have visually inspected all objects with inferred
$M_{*}>10^{11}$ M⊙ and $\Delta R_{\mathrm{e}}<-0.5$ dex; not one of these
objects is a plausible massive, comapct galaxy candidate. The fact that the
shape of the red histogram does not differ significantly from that of the
black histogram for $\Delta R_{\mathrm{e}}>-0.5$ dex indicates that the
spectroscopic sample is not significantly biased against compact galaxies.
In Figure 10, we show three size–mass diagrams corresponding to, from top to
bottom: (a.) the spectroscopic sample, analyzed using spectroscopic redshifts;
(b.) the spectroscopic sample, analyzed using photometric redshifts; and (c.)
the photometric sample, analyzed using photometric redshifts. In all three
cases, the only selections applied to each sample are on photometric type (to
exclude optical artifacts, etc., we require either a star or galaxy type
classification) and $ugrz$ color (to exclude stars); then, as in Figures 2 and
6, we are only showing those galaxies inferred to have $0.066<z<0.12$ and
${}^{0.1}(u-r)>2.5$. Again, objects with measured sizes smaller than $0\farcs
75$ are shown as upper limits, assuming a size of $0\farcs 75$. When comparing
these three different analyses, the difference between (a.) and (b.) shows the
effect of using spectroscopic versus photometric redshifts, and the difference
between (b.) and (c.) shows the difference between the SDSS spectroscopic and
photometric selection. That is, the comparison between (b.) and (c.) gives a
direct indication of the level of incompleteness in the spectroscopic sample.
Looking at panels (a.) and (b.), it is clear that the use of photometric
redshifts produces a considerably greater scatter in the size–mass diagram,
including a rather large number of galaxies with inferred stellar masses of
$10^{12}$ M⊙ or greater. There is a clear excess of unresolved objects with
inferred stellar masses greater than $\sim 10^{11}$ M⊙ in panel (b.) in
comparison to panel (a.) However, we already know from section 3 that there
are no objects in the spectroscopic sample with these sizes and masses—these
objects cannot be genuine compact galaxies. Of the 34 with inferred
$M_{*}>10^{11}$ M⊙, 16 of these objects are spectrally identified as being
stars, and one as a quasar at $z=0.102$. Of the 17 spectrally confirmed
galaxies, all have $|z_{\mathrm{phot}}-z_{\mathrm{spec}}|\gtrsim 0.02$. Of
these, 15 have had their redshifts, and thus stellar masses, seriously
overestimated; the other two are at $z>0.12$, and so have had their intrinsic
sizes underestimated.
Turning now to the comparison between panels (b.) and (c.), the first point to
make is that the excess of unresolved sources is even more pronounced. We have
matched all of these objects to the 2MASS point source catalog in order to
investigate their NIR colors. 90 % of these objects fall in the stellar region
of the $(J-K)$–$K$ color–magnitude plot; similarly, 80 % fall in the stellar
region of a $(g-z)$–$(J-K)$ color–color plot.
Further, we have visually inspected the 434 objects with inferred
$M_{*}>10^{11}$ M⊙ and with sizes smaller than the main
$M_{*}$–$R_{\mathrm{e}}$ relation by $0.4$ dex or more. Roughly 70 % of these
objects are obviously stars: 133 come from crowded Galactic fields covered as
part of SEGUE; 126 are double stars; 49 have clear diffraction spikes and/or
are clearly saturated. Another 12 objects have been cross-matched with the
USNO-B star catalog (within $1^{\prime\prime}$), and have measured proper
motions of 1—4′′/yr. 19 objects are the central point sources of very large
spiral galaxies; most of these are also found in the ROSAT and/or FIRST
catalogs. We also note that there are 17 very small disk or irregular galaxies
with red point sources at or very near their centers. Most of these also have
proper motion measurements from the USNO-B catalog, and several are spectrally
identified as late type stars; it seems plausible that these galaxies simply
have foreground stars coincidentally superposed very near their centers.
In short, of the 434 objects from the full photometric sample that, on the
basis of photometric redshifts, are inferred to have $M_{*}>10^{11}$ M⊙ and
$\Delta R_{\mathrm{e}}<-0.4$ dex, not one remains as a viable compact galaxy
candidate.
### A.4. Estimating the Importance of Spectroscopic Selection Effects
The conclusion from both the analyses that we have now presented is that there
are no galaxies in the local universe with sizes and masses comparable to the
compact galaxies found at higher redshifts. In Figure 11, we provide a more
quantitative statement of this conclusion, by plotting the size distribution
for massive, red galaxies in different mass bins.
In this figure, the filled histograms represent the main SDSS spectroscopic
sample, analyzed using spectroscopic redshifts, as in §3. The heavy black and
red histograms represent the spectroscopic and photometric samples,
respectively, analyzed using photometric redshifts, as in §A.3. In all cases,
objects excluded on the basis of visual inspection are not included; this
accounts for the sharp cutoffs at $\Delta\log R_{\mathrm{e}}=-0.3$ and at
$\Delta\log R_{\mathrm{e}}=-0.4$ for the filled and open histograms,
respectively. Immediately above these cutoffs, where we have not visually
inspected individual objects, but where there is likely to still be
significant contamination, these distributions should be regarded as upper
limits on the true distribution. In the upper panel, we plot those objects
with observed sizes smaller than $0\farcs 75$ separately as the light grey
filled histogram, and the thin black and red histograms.
As in §A.3, the difference between the filled and solid black histogram, both
of which are derived from the spectroscopic sample, shows the increased
scatter due to the use of photometric redshifts.
Similarly, the difference between the black and red histograms show the
difference between the spectroscopic and photometric samples, and so allow a
quantification of the bias in the spectroscopic sample. By simply tallying the
numbers of galaxies with $-0.4<\Delta\log R_{\mathrm{e}}<-0.3$, we find that
the ‘completeness’ (the ratio between the number of galaxies in the
spectroscopic sample compared to the full photometric sample) is 75 %, 68 %,
67 %, and 43 % for each of these mass bins, from lowest to highest.
In order to improve on these estimates, we have done the following. Using the
approach described above, we have assigned each object a weight according to
its $z_{\textrm{phot}}$–derived mass and size. Then, going back to the
spectroscopic sample, we use these to compute the mean weight in cells of
$z_{\textrm{spec}}$–derived mass and size. The completeness contours we derive
in this way are in good qualitative agreement with those shown in Figure 2,
although they suggest incompleteness at the 2–5 % level for mean–sized
galaxies with $M_{*}\gtrsim 10^{11}$ M⊙. Using these values to estimate the
number of $M_{*}>10^{11}$ M⊙ galaxies with $\Delta R_{\mathrm{e}}<-0.3$ dex,
this suggests that the spectroscopic sample is missing on the order of 4 such
galaxies.
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|
arxiv-papers
| 2009-07-27T21:15:00 |
2024-09-04T02:49:04.212290
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edward N Taylor, Marijn Franx, Karl Glazebrook, Jarle Brinchmann,\n Arjen van der Wel, Pieter G van Dokkum",
"submitter": "Edward N Taylor",
"url": "https://arxiv.org/abs/0907.4766"
}
|
0907.4817
|
# Photon–added squeezed thermal states: statistical properties and its
decoherence in a photon-loss channel††thanks: Work supported by the National
Natural Science Foundation of China (Nos.10775097 and 10874174).
Xue-xiang Xu1,2, Li-yun Hu1,2, and Hong-yi Fan1
1Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030,
China;
2College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang, 330022, China. Corresponding author. Tel./fax: +86 7918120370. Email
address: hlyun2008@gmail.com; hlyun2008@126.com. (L-Y Hu).
###### Abstract
Using the normally ordered Gaussian form of displaced-squeezed thermal field
characteristic of average photon number $\bar{n}$, we introduce the photon-
added squeezed thermo state (PASTS) and investigate its statistical
properties, such as Mandel’s Q-parameter, number distribution (as a Legendre
polynomial), the Wigner function. We then study its decoherence in a photon-
loss channel in term of the negativity of WF by deriving the analytical
expression of WF for PASTS. It is found that the WF with single photon-added
is always partial negative for the arbitrary values of $\bar{n}$ and the
squeezing parameter $r$.
PACS: 03.65.Yz, 42.50.Dv, 03.67.-a, 03.65.Wj
Keywords: Open quantum systems; Decoherence; Photon-added squeezed thermal
states; photon-loss channel; continuous variable systems; IWOP technique
## 1 Introduction
Nonclassicality of fields has been a topic of great interest in quantum optics
and quantum information processing [1]. Experimentally, the traditional
quantum states, such as Fock states and coherent states as well as squeezed
states, have been generated but there are some limitations in using them for
various tasks of quantum information process [2]. Alternately, it is possible
to generate and manipulate various nonclassical optical fields by quantum
superpositions and subtracting or adding photons from/to traditional quantum
states [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
On the other hand, the single mode displaced, squeezed, mixed Gaussian states
have been paid enough attention by both experimentalists and theoreticians.
Marian et. al [17, 18] investegated the superposition of a squeezed thermal
radiation and a coherent one. They examined the squeezing properties of the
field using the distribution functions of the quadratures. For Gaussian
squeezed states of light, a scheme is also presented experimentally to measure
its squeezing, purity and entanglement [19, 20]. As is well known, dissipative
quantum channels tend to deteriorate the degree of nonclassicality (i.e.,
render quantum features unobservable). Thus, it is usually necessary to
investigate the decoherence properties in dissipative channels, such as
dynamical behaviors of the partial negativity of Wigner function (WF) and how
long a nonclassical field preserves its partial negativity of WF. For
instances, the nonclassicality of single photon-added thermal states in the
thermal channel is investigated by exploring the volume of the negative part
of the WF [21]; Souza and Nemes have derived an upper limit for the mixedness
of the single bosonic mode Gaussian states [22] in a thermal channel.
In this paper, we shall introduce the photon-added squeezed thermo state
(PASTS) and investigate its statistical properties, such as Mandel’s
Q-parameter, number distribution, the Wigner function. We then study its
decoherence in a photon-loss channel in term of the negativity of WF by
deriving the analytical expression of WF for PASTS. It is found that the WF
with single photon-added is always partial negative for the arbitrary values
of $\bar{n}$ and the squeezing parameter $r$. The work is arranged as follows:
In section 2, we introduce the state PASTS and derive its normalized constant,
and photon number distribution is discussed in section 3\. Section 4 is
devoted to calculating the WF. In the last section, we explore the decoherence
of PASTS in a photon-loss channel by discussing the evolution of WF.
## 2 PASTS and its normalization
For a displaced-squeezed thermal field, the density operator is
$\rho_{s}=D(\beta)S(r)\rho_{c}S^{\dagger}(r)D^{\dagger}(\beta),$ (1)
where $D(\beta)=\exp(\beta a^{\dagger}-\beta^{\ast}a),$ and
$S(r)=\exp[$i$r(QP+PQ)/2],$ are the displacement operator and the squeezing
operator [23, 24], respectively, $\beta=(q+\mathtt{i}p)/\sqrt{2},$
$Q=\frac{a+a^{\dagger}}{\sqrt{2}},P=\frac{a-a^{\dagger}}{\sqrt{2}\mathtt{i}},\left[a,a^{\dagger}\right]=1,$
and
$\rho_{c}=(1-e^{-\frac{\hbar\omega}{kT}})e^{-\frac{\hbar\omega
a^{\dagger}a}{kT}},$
($k$ is the Boltzmann constant, $T$ denoting temperature), is qualified to be
a density operator of thermal (chaotic) field, since tr$\rho_{c}=1$. For a
coherent state
$\left|z\right\rangle=\exp\left(za^{\dagger}-z^{\ast}a\right)\left|0\right\rangle$
[25, 26], due to $a\left|z\right\rangle=z\left|z\right\rangle$, matrix
elements of any normally ordered operators
$\colon\hat{O}\left(a^{\dagger},a\right)\colon$ (the symbol
$\colon\colon$denotes normally ordering) in the coherent state is easily
obtained, i.e,
$\displaystyle\left\langle
z\right|\colon\hat{O}\left(a^{\dagger},a\right)\colon\left|z^{\prime}\right\rangle$
$\displaystyle=O\left(z^{\ast},z^{\prime}\right)\left\langle
z\right.\left|z^{\prime}\right\rangle$
$\displaystyle=O\left(z^{\ast},z^{\prime}\right)\exp\left\\{-\frac{\left|z\right|^{2}+\left|z^{\prime}\right|^{2}}{2}+z^{\ast}z^{\prime}\right\\},$
(2)
so in Ref.[27, 28] by using the Weyl ordering invariance under similarity
transformations and the technique of integration within an ordered product of
operators (IWOP) Fan et al have converted $\rho_{s}$ to its normally ordered
Gaussian form
$\rho_{s}=\frac{1}{\tau_{1}\tau_{2}}\colon\exp\left\\{-\frac{(q-Q)^{2}}{2\tau_{1}^{2}}-\frac{(p-P)^{2}}{2\tau_{2}^{2}}\right\\}\colon,$
(3)
where
$2\tau_{1}^{2}=(2\bar{n}+1)e^{2r}+1,2\tau_{2}^{2}=(2\bar{n}+1)e^{-2r}+1,$ (4)
and $\bar{n}$ is the average photon number for $\rho_{c},$ i.e.
$\bar{n}=(e^{\hbar\omega/kT}-1)^{-1}$ [29]. The form in Eq.(3) is similar to
the bivariate normal distribution in statistics, which is useful for us to
further derive the marginal distributions of $\rho_{s}$.
Theoretically, the PASTS can be obtained by repeatedly operating the photon
creation operator $a^{\dagger}$ on a displacement squeezed thermal state, so
its density operator is defined as
$\rho_{m}=N_{m}^{-1}a^{\dagger m}\rho_{s}a^{m},$ (5)
where $m$ is a non-negative integer, $N_{m}=$tr$(a^{\dagger m}\rho_{s}a^{m})$
is the normalization constant. Using Eq.(3) we known immediately the normally
ordered Gaussian form of $\rho_{m}$, i.e.,
$\rho_{m}=\frac{N_{m}^{-1}}{\tau_{1}\tau_{2}}\colon a^{\dagger
m}\exp\left\\{-\frac{(q-Q)^{2}}{2\tau_{1}^{2}}-\frac{(p-P)^{2}}{2\tau_{2}^{2}}\right\\}a^{m}\colon.$
(6)
Next we shall determine the normalization constant $N_{m}$. Using the
completness relation of coherent states
$\int\frac{\text{d}^{2}z}{\pi}\left|z\right\rangle\left\langle z\right|=1$ as
well as Eq.(2), we have
$\displaystyle\mathtt{tr}\rho_{m}$ $\displaystyle=$
$\displaystyle\mathtt{tr}(\rho_{m}\int\frac{\text{d}^{2}z}{\pi}\left|z\right\rangle\left\langle
z\right|)$ (7) $\displaystyle=$
$\displaystyle\frac{N_{m}^{-1}}{\tau_{1}\tau_{2}}\int\frac{\text{d}^{2}z}{\pi}\left\langle
z\right|\colon a^{\dagger
m}\exp\left\\{-\frac{(q-Q)^{2}}{2\tau_{1}^{2}}-\frac{(p-P)^{2}}{2\tau_{2}^{2}}\right\\}a^{m}\colon\left|z\right\rangle$
$\displaystyle=$
$\displaystyle\frac{N_{m}^{-1}}{\tau_{1}\tau_{2}}\int\frac{\text{d}^{2}z}{\pi}z^{\ast
m}z^{m}\exp\left[-A\left|z\right|^{2}+B^{\ast}z+Bz^{\ast}+Cz^{2}+Cz^{\ast
2}+D\right],$
where we have set
$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau_{1}^{2}}+\frac{1}{2\tau_{2}^{2}},B=\frac{1}{\sqrt{2}}\left(\frac{q}{\tau_{1}^{2}}+\frac{\mathtt{i}p}{\tau_{2}^{2}}\right),$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{1}{4\tau_{1}^{2}}+\frac{1}{4\tau_{2}^{2}},D=-\frac{q^{2}}{2\tau_{1}^{2}}-\frac{p^{2}}{2\tau_{2}^{2}}.$
(8)
Due to $\mathtt{tr}\rho_{m}=1,$ thus we know
$\displaystyle N_{m}$ $\displaystyle=$
$\displaystyle\frac{e^{D}}{\tau_{1}\tau_{2}}\int\frac{\text{d}^{2}z}{\pi}z^{\ast
m}z^{m}\exp\left\\{-A\left|z\right|^{2}+B^{\ast}z+Bz^{\ast}+Cz^{2}+Cz^{\ast
2}\right\\}$ (9) $\displaystyle=$
$\displaystyle\frac{(-1)^{m}e^{D}}{\tau_{1}\tau_{2}}\frac{\partial^{m}}{\partial
A^{m}}\int\frac{\text{d}^{2}z}{\pi}\exp\left\\{-A\left|z\right|^{2}+B^{\ast}z+Bz^{\ast}+Cz^{2}+Cz^{\ast
2}\right\\}.$
Further using the following integral formula [30]
$\int\frac{d^{2}z}{\pi}\exp\left\\{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}+fz^{2}+gz^{\ast
2}\right\\}=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left\\{\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right\\},$
(10)
whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0$ and$\
\mathtt{Re}\left(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g}\right)<0$, Eq.(9) can
be rewritten as follows
$N_{m}=\frac{(-1)^{m}e^{D}}{\tau_{1}\tau_{2}}\frac{\partial^{m}}{\partial
A^{m}}\left\\{\left(A^{2}-4C^{2}\right)^{-1/2}\exp\left[\frac{A\left|B\right|^{2}+CB^{\ast
2}+CB^{2}}{A^{2}-4C^{2}}\right]\right\\},$ (11)
which is the normalization constant of PASTS for photon-added number $m$. In
particular, when $\beta=0$ leading to $B=D=0,$ Eq.(11) reduces to the
following form,
$N_{m}=\frac{(-1)^{m}}{\tau_{1}\tau_{2}}\frac{\partial^{m}}{\partial
A^{m}}\left(A^{2}-4C^{2}\right)^{-1/2}.$ (12)
Especially when $m=0,1,2$ then the normalization constants are given by
$N_{0}=1$, $N_{1}=\frac{1}{2}\left(\tau_{1}^{2}+\tau_{2}^{2}\right)$ and
$N_{2}=\allowbreak\frac{1}{4}\left(3\tau_{1}^{4}+2\tau_{1}^{2}\tau_{2}^{2}+3\tau_{2}^{4}\right)\allowbreak$,
respectively.
To see clearly the photon statistical properties of the PASTS, we will examine
the Mandel’s $Q$-parameter defined as
$Q_{M}=\frac{\left\langle a^{{\dagger}2}a^{2}\right\rangle}{\left\langle
a^{{\dagger}}a\right\rangle}-\left\langle a^{{\dagger}}a\right\rangle,$ (13)
which measures the deviation of the variance of the photon number distribution
of the field state under consideration from the Poissonian distribution of the
coherent state. If $Q_{M}=0$ we say the field has Poissonian photon statistics
while for $Q>0$($Q<0$) we say that the field has super-(sub-) Poissonian
photon statistics.From Eq.(11) and $N_{m}=$tr$(a^{\dagger m}\rho_{s}a^{m})$,
we can easily calculate $\left\langle
a^{{\dagger}}a\right\rangle=\frac{N_{m+1}}{N_{m}}-1,$ and $\left\langle
a^{{\dagger}2}a^{2}\right\rangle=\frac{N_{m+2}}{N_{m}}-\frac{4N_{m+1}}{N_{m}}+2,$thus
we obtain the $Q$-parameter of the PASTS
$Q_{M}=\frac{N_{m+2}-4N_{m+1}+2N_{m}}{N_{m+1}-N_{m}}-\frac{N_{m+1}-N_{m}}{N_{m}}.$
(14)
It is well known that the negativity of the $Q_{M}$-parameter refers to sub-
Possonian statistics of the state. But a state can be nonclassical even though
$Q_{M}$ is positive. This case is true for the present state. From Fig.1, one
can clearly see that for the cases of $m=0$ (Fig.1(a)), $Q_{M}$ is always
positive; while for $m\neq 0$ (for instance $m=1$) and a given $\bar{n}$
value, $Q_{M}$ becomes positive only when the squeezing parameter $r$ is more
than a certain threshold value that increases as $m$ increases. In addition,
from Fig.1(a) and Fig.1(b) one can see that the threshold value of $r$
decreases as $\bar{n}$ increases. We emphasize that the WF has negative region
for all $r$ and $\bar{n}$ thus the PASTS is nonclassical (see next section
below).
## 3 Photon number distribution of PASTS
In this section, we study photon number distribution of PASTS optical field.
Using the un-normalized coherent state $\left|\alpha\right\rangle=\exp[\alpha
a^{{\dagger}}]\left|0\right\rangle$, leading to
$\left|n\right\rangle=\frac{1}{\sqrt{n!}}\frac{\mathtt{d}^{n}}{\mathtt{d}\alpha^{n}}\left|\alpha\right\rangle\left|{}_{\alpha=0}\right.,$
$\left(\left\langle\alpha\right.\left|\alpha^{\prime}\right\rangle=e^{\alpha^{\prime}\alpha^{\ast}}\right)$,
it is easy to see that the photon number distribution formula is given by
$\displaystyle\mathcal{P}(n)$ $\displaystyle=$
$\displaystyle\text{tr}\left(\rho_{m}\left|n\right\rangle\left\langle
n\right|\right)=\left\langle n\right|\rho_{m}\left|n\right\rangle$ (15)
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{1}\tau_{2}}\frac{1}{n!}\frac{\text{d}^{2n}}{\text{d}\alpha^{\ast
n}\text{d}\alpha^{\prime
n}}\left\langle\alpha\right|\rho_{m}\left|\alpha^{\prime}\right\rangle\left|{}_{\alpha=\alpha^{\prime}=0}\right..$
Employing the normal ordering form of $\rho_{m}$ in Eq.(2), Eq.(15) can be put
into the following form
$\mathcal{P}(n)=\frac{N_{m}^{-1}e^{D}}{n!\tau_{1}\tau_{2}}\frac{\text{d}^{2n}}{\mathtt{d}\alpha^{\ast
n}\text{d}\alpha^{\prime n}}\left\\{\alpha^{\ast m}\alpha^{\prime
m}\exp\left[B^{\ast}\alpha^{\prime}+B\alpha^{\ast}+C\alpha^{\prime
2}+C\alpha^{\ast
2}+\left(1-A\right)\alpha^{\ast}\alpha^{\prime}\right]\right\\}\left|{}_{\alpha=\alpha^{\prime}=0}\right..$
(16)
Further expanding the exponential term
$\exp\left[\left(1-A\right)\alpha^{\ast}\alpha^{\prime}\right]$ as series and
using the generating function of single-variable Hermite polynomials,
$H_{n}(x)=\frac{\partial^{n}}{\partial
t^{n}}\exp\left(2xt-t^{2}\right)\left|{}_{t=0}\right.,$ (17)
we can calculate the photon number distribution (PND) of PASTS, i.e.,
$\displaystyle\mathcal{P}(n)$
$\displaystyle=\frac{N_{m}^{-1}e^{D}}{n!\tau_{1}\tau_{2}}\sum_{l=0}^{\infty}\frac{\left(1-A\right)^{l}}{l!}\frac{\partial^{m+l}\partial^{m+l}}{\partial
B^{m+l}\partial B^{\ast m+l}}$
$\displaystyle\times\frac{\text{d}^{2n}}{\text{d}\alpha^{\prime
n}\text{d}\alpha^{\ast
n}}\exp\left\\{B^{\ast}\alpha^{\prime}+B\alpha^{\ast}+C\alpha^{\prime
2}+C\alpha^{\ast 2}\right\\}\left|{}_{\alpha=\alpha^{\prime}=0}\right.$
$\displaystyle=\frac{N_{m}^{-1}\left|C\right|^{n}e^{D}}{n!\tau_{1}\tau_{2}}\sum_{l=0}^{\infty}\frac{(1-A)^{l}}{l!}\left|\frac{\partial^{m+l}}{\partial
B^{m+l}}H_{n}\left[\mathtt{i}B/(2\sqrt{C})\right]\right|^{2}.$ (18)
After making the scale transformation and noticing the recurrence relation
$\frac{\mathtt{d}^{l}}{\mathtt{d}x^{l}}H_{n}(x)=\frac{2^{l}n!}{\left(n-l\right)!}H_{n-l}(x)$,
we can easily obtain
$\mathcal{P}(n)=\frac{N_{m}^{-1}e^{D}}{\tau_{1}\tau_{2}}\sum_{l=0}^{n-m}\frac{n!(1-A)^{l}\left|C\right|^{n-m-l}}{l!\left[\left(n-m-l\right)!\right]^{2}}\left|H_{n-m-l}\left[\mathtt{i}B/(2\sqrt{C})\right]\right|^{2},$
(19)
Especially when $\beta=0$, Eq.(19) reduces to
$\displaystyle\mathcal{P}(n)$
$\displaystyle=\frac{(-1)^{m}\sum_{j=0}^{\left[\left(n-m\right)/2\right]}\frac{n!(1-A)^{\left(n-m-2j\right)}\left|C\right|^{2j}}{\left(n-m-2j\right)!\left(j!\right)^{2}}}{\frac{\partial^{m}}{\partial
A^{m}}\left(A^{2}-4C^{2}\right)^{-1/2}}$
$\displaystyle=\frac{n!\sigma^{n-m}P_{n-m}(\frac{1-A}{\sigma})}{\left(n-m\right)!\frac{\partial^{m}}{\partial
A^{m}}\left(A^{2}-4C^{2}\right)^{-1/2}},$ (20)
where
$\sigma=\sqrt{\left(1-A\right)^{2}-4C^{2}}=\sqrt{(\tau_{1}^{2}-1)(\tau_{2}^{2}-1)}/(\tau_{1}\tau_{2}),$
(21)
and in the last step of (20) we have used the new expression of Legendre
polynomials [8]
$P_{m}(x)=x^{m}\sum_{l=0}^{[m/2]}\frac{m!(1-\frac{1}{x^{2}})^{l}}{2^{2l}(l!)^{2}\left(m-2l\right)!}.$
(22)
Eq. (19) or (20) is just the the analytical expression of the PND of PASTS. In
particular, when $m=0,$ Eq.(20) becomes (with $\beta=0$)
$\mathcal{P}(n)=\frac{\sigma^{n}}{\tau_{1}\tau_{2}}P_{n}(\frac{1-A}{\sigma}).$
(23)
Eq.(23) is just the PND of the squeezed thermo state which seems a new result.
The PNDs of PASTS for some given parameters ($\bar{n},r$) and $m$ are plotted
in Fig.2. From Fig. 2 it is found that the PND is constrained by $n\geqslant
m$. By adding photons, we have been able to move the peak from zero photons to
nonzero photons (see Fig.2 (a)-(c)). The position of peak depends on how many
photons are created and how much the state is squeezed initially. In addition,
comparing Fig.2(b) and Fig.2(d) we see that, for a given $m$, the “tail” of
PND becomes more “wide” with the increasing parameter $r$.
## 4 Wigner function of PASTS
The Wigner function (WF) [31] was first introduced by Wigner in 1932 to
calculate quantum correction to a classic distribution function of a quantum-
mechanical system. It now becomes a very popular tool to study the
nonclassical properties of quantum states. It is well known that WFs are
quasiprobability distributions because it may be negative in phase space [32].
Nevertheless, the partial negativity of the WF is indeed a good indication of
the highly nonclassical character of the state. Thus, to study the dynamical
behaviors of the partial negativity of WF and understand that a nonclassical
field preserves its partial negativity, Wigner distribution may be very
desirable for experimentally quantifying the variation of nonclassicality
[33].
The presence of negativity of the WF for an optical field is a signature of
its nonclassicality. In this section, using the normally ordered form of
PASTS, we evaluate its WF. For a single-mode system, the WF in the coherent
state representation $\left|z\right\rangle$ is given by [34]
$W(\alpha,\alpha^{\ast})=\frac{e^{2\left|\alpha\right|^{2}}}{\pi}\int\frac{d^{2}z}{\pi}\left\langle-z\right|\rho_{m}\left|z\right\rangle
e^{-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)},$ (24)
where $\alpha=\left(x+\mathtt{i}y\right)/\sqrt{2}$. Then substituting Eq. (6)
into Eq. (24) and using Eq. (2), we derive the WF of PASTS
$W(\alpha,\alpha^{\ast})=\frac{N_{m}^{-1}e^{2\left|\alpha\right|^{2}+D}}{\pi\tau_{1}\tau_{2}}\frac{\partial^{m}}{\partial
F^{m}}\left\\{\left(F^{2}-4C^{2}\right)^{-1/2}\exp\left[\frac{-F\left|E\right|^{2}+E^{\ast
2}C+E^{2}C}{F^{2}-4C^{2}}\right]\right\\},$ (25)
where we have set
$F=2-A,\text{ }E=B-2\alpha,$ (26)
and used Eq.(10). Especially when $\beta=0$, Eq.(25) reduces to
$W(\alpha,\alpha^{\ast})=\frac{(-1)^{m}e^{2\left|\alpha\right|^{2}}}{\pi}\frac{\frac{\partial^{m}}{\partial
F^{m}}\left\\{\left(F^{2}-4C^{2}\right)^{-1/2}\exp\left[\frac{-4F\left|\alpha\right|^{2}+4\alpha^{\ast
2}C+4\alpha^{2}C}{F^{2}-4C^{2}}\right]\right\\}}{\frac{\partial^{m}}{\partial
A^{m}}\left(A^{2}-4C^{2}\right)^{-1/2}},$ (27)
and further when $m=0,1$ Eq.(27) becomes to
$W_{m=0}(\alpha,\alpha^{\ast})=\frac{1}{\pi(2\bar{n}+1)}\exp\left(-\frac{e^{-2r}x^{2}+e^{2r}y^{2}}{2\bar{n}+1}\right),$
(28)
which is the WF of the squeezed thermo state, and
$W_{m=1}(\alpha,\alpha^{\ast})=\frac{\mathfrak{M}x^{2}+\mathfrak{N}y^{2}+\Upsilon}{\pi}\exp\left(-\frac{e^{-2r}x^{2}+e^{2r}y^{2}}{2\bar{n}+1}\right),$
(29)
respectively, where we have set
$\displaystyle\mathfrak{M}$
$\displaystyle=\frac{1}{\left(2\bar{n}+1\right)^{3}}\frac{\left(2\tau_{1}^{2}e^{-2r}\right)^{2}}{\tau_{1}^{2}+\tau_{2}^{2}},$
(30) $\displaystyle\mathfrak{N}$
$\displaystyle=\frac{1}{\left(2\bar{n}+1\right)^{3}}\frac{\left(2\tau_{2}^{2}e^{2r}\right)^{2}}{\tau_{1}^{2}+\tau_{2}^{2}},$
(31) $\displaystyle\Upsilon$
$\displaystyle=\frac{1}{\left(2\bar{n}+1\right)^{3}}\frac{\tau_{1}^{2}+\tau_{2}^{2}-4\tau_{1}^{2}\tau_{2}^{2}}{\tau_{1}^{2}+\tau_{2}^{2}}.$
(32)
Eq.(28) just agrees with the result of Eq.(48) in Ref. [27], whose form is
normal distribution.
From Eq.(27) one can see that the WF of the PASTS is always real, as expected.
When the factor $\mathfrak{M}x^{2}+\mathfrak{N}y^{2}+\Upsilon<0$ in Eq.(29),
the WF of the PASTS with $m=1$ has its negative distribution in phase space.
Noticing $\mathfrak{M,N}$ always positive, this indicates that the WF of the
PASTS always has the negative values under the condition $\Upsilon<0$ (i.e.,
$\left(\tau_{1}^{2}+\tau_{2}^{2}-4\tau_{1}^{2}\tau_{2}^{2}<0\right)$) at the
phase space center $q=p=0$. In fact, by substituting Eqs. (4), (30)-(32) into
$\Upsilon<0$, we find that for the arbitrary values of $\bar{n}$ and $r$, the
WF with $m=1$ is always partial negative.
Using Eq.(27), the WFs of the PASTS are dipicted in phase space for several
different values of $m,\bar{n}$ and $r$ in phase space. Fig. 3 exhibits the
WFs of the PASTS in phase space with $m=1$ for different $\bar{n}$, $r$. It is
easy to see that the WFs of single-PASTS always have the negative region. The
minimum in the negative region becomes larger with the increasing of $\bar{n}$
(see Fig3.(a) and (c)). In Fig. 4, we have presented the WFs with
$\bar{n}=0.5$, $r=0.3$ for different $m,$ which indicates that the peak
(absolute) value of WF become smaller as the increasing parameter $m$. The
partial negativity of WF indicates the nonclassical nature of the PASTS field.
## 5 Evolution of WF in a photon-loss channel
When the PASTS evolves in the amplitude decay channel, the evolution of the
density matrix can be described by the following master equation in the
interaction picture [35],
$\frac{d\rho}{dt}=\kappa\left(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a\right),$ (33)
where $\kappa$ represents the rate of decay. By using the thermal field
dynamics theory and thermal entangled state representation, the time evolution
of WF at time $t$ to be given by the following form [36], i.e.,
$\displaystyle W\left(\alpha,t\right)$
$\displaystyle=\frac{2}{\mathcal{T}}\int\frac{d^{2}z}{\pi}e^{-\frac{2}{\mathcal{T}}\left|\alpha-
ze^{-\kappa t}\right|^{2}}W\left(z,0\right),$ (34) $\displaystyle\mathcal{T}$
$\displaystyle=1-e^{-2\kappa t}.$
Eq.(34) is just the evolution formula of WF of single mode quantum state in
photon-loss channel. By observing Eq.(34), we see that when $t\rightarrow
0,\mathcal{T}\rightarrow 0,$
$\frac{2}{\pi\mathcal{T}}\exp(-\frac{2}{\mathcal{T}}\left|\alpha-ze^{-\kappa
t}\right|^{2})\rightarrow\delta\left(\alpha-z\right)\delta\left(\alpha^{\ast}-z^{\ast}\right),$
so $W\left(\alpha,t\right)\rightarrow W\left(\alpha,0\right)$ as expected.
Thus the WF at any time can be obtained by performing the integration when the
initial WF is known.
For simplicity, here we only discuss the special case $\beta=0$. Substituting
Eq.(27) into Eq.(34) and using Eq.(10), we derive the time evolution of WF for
PASTS in photon-loss channel:
$W\left(\alpha,t\right)=\frac{2(-1)^{m}}{\pi\mathcal{T}}\frac{\frac{\partial^{m}}{\partial
F^{m}}\left(\sqrt{\mathbb{N}}e^{\mathbb{R}\left|\alpha\right|^{2}+\Bbbk\alpha^{\ast
2}+\Bbbk\alpha^{2}}\right)}{\frac{\partial^{m}}{\partial
A^{m}}\left(A^{2}-4C^{2}\right)^{-1/2}},$ (35)
where we have set
$\displaystyle\mathbb{N}$
$\displaystyle=\frac{F^{2}-4C^{2}}{\left[4F+\left(\frac{2e^{-2\kappa
t}}{\mathcal{T}}-2\right)\left(F^{2}-4C^{2}\right)\right]^{2}-64C^{2}\allowbreak\allowbreak},$
$\displaystyle\mathbb{R}$ $\displaystyle=\frac{4\mathbb{N}e^{-2\kappa
t}}{\mathcal{T}^{2}}\left[4F+\left(\frac{2e^{-2\kappa
t}}{\mathcal{T}}-2\right)\left(F^{2}-4C^{2}\right)\right]-\frac{2}{\mathcal{T}},$
(36) $\displaystyle\Bbbk$ $\displaystyle=\frac{16\mathbb{N}Ce^{-2\kappa
t}}{\mathcal{T}^{2}}.$
In particular, when $m=0$ Eq.(35) becomes
$W_{m=0}\left(\alpha,t\right)=\frac{2\sqrt{\mathbb{N}\left(A^{2}-4C^{2}\right)}}{\pi\mathcal{T}}e^{\mathbb{R}\left|\alpha\right|^{2}+\Bbbk\alpha^{\ast
2}+\Bbbk\alpha^{2}}$ (37)
which is just the WF of the squeezed thermo state in photon-loss channel. This
result can also be checked by substituting Eq.(28) into Eq.(34).
When $\kappa t$ exceeds a threshold value, the WF has no chance to be negative
in the whole phase space. At long time $\kappa t\rightarrow\infty,$leading to
$\mathbb{N}\rightarrow\frac{1}{\allowbreak
4\left(F-2\right)^{2}\allowbreak-16C^{2}},$ $\mathbb{R}\rightarrow-2,$
$\Bbbk\rightarrow 0,$ the WF in Eq.(35) becomes
$W\left(\alpha,\infty\right)=\frac{e^{-2\left|\alpha\right|^{2}}}{\pi},$ (38)
which corresponds to the Gaussian state. In Fig.5, the WFs of PASTS are
depicted in phase space with $m=1$ and $r=0.3$ for several different $\kappa
t$. It is easily seen that the negative region of WF gradually disappears as
$\kappa t$ increases. This implies that the system state reduces to a Gaussian
state after a long time interaction in the channel. Thus the loss of channel
causes the absence of the partial negativity of the WF if the decay time
$\kappa t$ exceeds a threshold value.
In Figs. 6, we have also presented the time-evolution of WF for different $r$.
One can see clearly that the partial negativity of WF decreases gradually as
$r$ increases. The squeezing effect in one of the quadratures can be seen in
Fig.6. In Eq.(35), for the PASTS we have obtained the expression of the time
evolution of WF. In principle, by differentiating as shown in Eq.(35) we can
derive the WF of other PASTS ($m\geqslant 1$). But its form is too
complicated. However, we can draw the Wigner distributions of PASTS for
$m=2,3,5$ by numerical simulation, as shown in Fig.7 (a)-(c), respectively,
from which one can see that the absolute value of the negative minimum of the
WF decreases as $m$ increases.
## 6 Conclusions
Based on the normally ordered Gaussian form of displaced-squeezed thermal
field, we have introduced a kind state: the photon-added squeezed thermo state
(PASTS). Then we have investigated the statistical properties of PASTS (such
as Mandel’s Q-parameter, number distribution, the Wigner function) and its
decoherence in photon-loss channel with dissipative coefficient $\kappa$ in
term of the negativity of WF by deriving the analytical expression of WF for
PASTS. It is found that the photon number distribution is just a Legendre
polynomial and that the WF with single photon-added is always partial negative
for the arbitrary values of $\bar{n}$ and the squeezing parameter $r$. The
technique of integration within an ordered product of operators brings
convenience in our derivation.
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Figures caption:
Fig.1 Mandel’s $Q$-parameter of PASTS as a fuction of $r$ with $m=0,1,5,10,30$
(from top to bottom) for (a) $\bar{n}=0.3;$ (b) $\bar{n}=1.$
Fig.2 Photon number distributions of PASTS with $\bar{n}=1$ for (a)
$r=0.3,m=0;$ (b) $r=0.3,m=1;$(c) $r=0.3,m=5;$(d) $r=0.8,m=1.$
Fig. 3 WF of PASTS for $m=1$ (a) $\bar{n}=0,r=0.3;$ (b) $\bar{n}=0,r=0.8;$(c)
$\bar{n}=0.5,r=0.3;$(d) $\bar{n}=0.5,r=0.8.$
Fig. 4 WF of PASTS for $\bar{n}=0.5,r=0.3$ (a)$m=0;$ (b) $m=2;$(c)$m=3$ ;(d)
$m=5.$
Fig. 5 The time evolution of WF of PASTS for $m=1,r=0.3,$ and $\bar{n}=0.5$
with (a) $\kappa t=0.05;$ (b) $\kappa t=0.15;$(c) $\kappa t=0.2;$(d) $\kappa
t=0.4.$
Fig. 6 WF of PASTS for $m=1,$ $\bar{n}=0.5$ at $\kappa t=0.05,$ with (a)
$r=0.01;$ (b) $r=0.5;$(c) $r=1.$
Fig. 7 WF of PASTS for $r=0.3$, $\bar{n}=0.5$ at $\kappa t=0.05,$ with
(a)$m=2;$ (b) $m=3.$ (c) $m=5.$
Figure 1: Mandel’s $Q$-parameter of PASTS as a fuction of $r$ with
$m=0,1,5,10,30$ (from top to bottom) for (a) $\bar{n}=0.3;$ (b) $\bar{n}=1.$
Figure 2: Photon number distributions of PASTS with $\bar{n}=1$ for (a)
$r=0.3,m=0;$ (b) $r=0.3,m=1;$(c) $r=0.3,m=5;$(d) $r=0.8,m=1.$ Figure 3: WF of
PASTS for $m=1$ (a) $\bar{n}=0,r=0.3;$ (b) $\bar{n}=0,r=0.8;$(c)
$\bar{n}=0.5,r=0.3;$(d) $\bar{n}=0.5,r=0.8.$ Figure 4: WF of PASTS for
$\bar{n}=0.5,r=0.3$ (a)$m=0;$ (b) $m=2;$(c)$m=3$ ;(d) $m=5.$ Figure 5: The
time evolution of WF of PASTS for $m=1,r=0.3,$ and $\bar{n}=0.5$ with (a)
$\kappa t=0.05;$ (b) $\kappa t=0.15;$(c) $\kappa t=0.2;$(d) $\kappa t=0.4.$
Figure 6: WF of PASTS for $m=1,$ $\bar{n}=0.5$ at $\kappa t=0.05,$ with (a)
$r=0.01;$ (b) $r=0.5;$(c) $r=1.$ Figure 7: WF of PASTS for $r=0.3$,
$\bar{n}=0.5$ at $\kappa t=0.05,$ with (a)$m=2;$ (b) $m=3.$ (c) $m=5.$
|
arxiv-papers
| 2009-07-28T01:57:02 |
2024-09-04T02:49:04.223532
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-xiang Xu, Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0907.4817"
}
|
0907.4840
|
# Generators of supersymmetric polynomials in positive characteristic
A.N. Grishkov Departamento de Matematica, Universidade de Sao Paulo, Caixa
Postal 66281, 05315-970 - S o Paulo, Brazil shuragri@gmail.com , F. Marko
Penn State Hazleton, 76 University Drive, Hazleton PA 18202, USA
fxm13@psu.edu and A.N. Zubkov Omsk State Pedagogical University, Chair of
Geometry, 644099 Omsk-99, Tuhachevskogo Embankment 14, Russia
zubkov@iitam.omsk.net.ru
###### Abstract.
In [1], Kantor and Trishin described the algebra of polynomial invariants of
the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related
algebra $A_{s}$ of what they called pseudosymmetric polynomials over an
algebraically closed field $K$ of characteristic zero. The algebra $A_{s}$ was
investigated earlier by Stembridge who in [4] called the elements of $A_{s}$
supersymmetric polynomials and determined generators of $A_{s}$.
The case of positive characteristic $p$ has been recently investigated by La
Scala and Zubkov in [3]. They formulated two conjectures describing generators
of polynomial invariants of the adjoint action of the general linear
supergroup $GL(m|n)$ and generators of $A_{s}$, respectively. In the present
paper we prove both conjectures.
## Introduction and notation
Let $K$ be an algebraically closed field $K$ of positive characteristic $p\neq
2$. The following notation is related to general linear supergroup
$G=GL(m|n)$. Let $K[c_{ij}]$ be a commutative superalgebra freely generated by
elements $c_{ij}$ for $1\leq i,j\leq m+n$, where $c_{ij}$ is even if either
$1\leq i,j\leq m$ or $m+1\leq i,j\leq m+n$, and $c_{ij}$ is odd otherwise.
Denote by $C$ the generic matrix $(c_{ij})_{1\leq i,j\leq m+n}$ and write $C$
as a block matrix $\begin{pmatrix}C_{00}&C_{01}\\\
C_{10}&C_{11}\end{pmatrix}$, where entries of $C_{00}$ and $C_{11}$ are even
and entries of $C_{01}$ and $C_{10}$ are odd. The localization of $K[c_{ij}]$
by elements $det(C_{00})$ and $det(C_{11})$ is the coordinate superalgebra
$K[G]$ of the general linear supergroup $G=GL(m|n)$. The general linear
supergroup $G=GL(m|n)$ is a group functor from the category $SAlg_{K}$ of
commutative superalgebras over $K$ to the category of groups, represented by
its coordinate ring $K[G]$, that is $G(A)=Hom_{SAlg_{K}}(K[G],A)$ for $A\in
SAlg_{K}$. Here, for $g\in G(A)$ and $f\in K[G]$ we define $f(g)=g(f)$. Denote
by $Ber(C)=det(C_{00}-C_{01}C_{11}^{-1}C_{10})det(C_{11})^{-1}$ the Berezinian
element.
Let $T$ be the standard maximal torus in $G$ and $X(T)$ be a set of
characters. Let $V$ be a $G$-supermodule with weight decomposition
$V=\oplus_{\lambda\in X(T)}V_{\lambda}$, where
$\lambda=(\lambda_{1},\ldots,\lambda_{m+n})$ and each $V_{\lambda}$ splits
into a sum of its even subspace $(V_{\lambda})_{0}$ and odd subspace
$(V_{\lambda})_{1}$. The (formal) supercharacter $\chi_{sup}(V)$ of $V$ is
defined as
$\chi_{sup}(V)=\sum_{\lambda\in
X(T)}(dim(V_{\lambda})_{0}-dim(V_{\lambda})_{1})x_{1}^{\lambda_{1}}\ldots
x_{m}^{\lambda_{m}}y_{1}^{\lambda_{m+1}}\ldots y_{n}^{\lambda_{m+n}}.$
If $V$ is a $G$-supermodule with a homogeneous basis
$\\{v_{1},\ldots,v_{a},v_{a+1},$ $\ldots,v_{a+b}\\}$ such that $v_{i}$ is even
for $1\leq i\leq a$ and $v_{i}$ is odd for $a+1\leq i\leq a+b$, and the image
$\rho_{V}(v_{i})$ of a basis element $v_{i}$ under a comultiplication
$\rho_{V}$ is given as $\rho_{V}(v_{i})=\sum_{1\leq j\leq a+b}v_{j}\otimes
f_{ji}$, then the supertrace $Tr(V)$ is defined as $\sum_{1\leq i\leq
a}f_{ii}-\sum_{a+1\leq i\leq a+b}f_{ii}$.
Let $E$ be a standard $G$-supermodule given by basis elements $e_{1},\ldots,$
$e_{m}$ that are even and $e_{m+1},\ldots,e_{m+n}$ that are odd and by
comultiplication $\rho_{E}(e_{i})=\sum_{1\leq j\leq m+n}e_{j}\otimes c_{ji}$.
Denote by $\Lambda^{r}(E)$ the $r$-th exterior power of $E$ and by $C_{r}$ the
supertrace of $\Lambda^{r}(E)$.
The algebra $R$ of invariants with respect to adjoint action of $G$ is a set
of functions $f\in K[G]$ satisfying $f(g_{1}^{-1}g_{2}g_{1})=f(g_{2})$ for any
$g_{1},g_{2}\in G(A)$ and any commutative supergalgebra $A$ over $K$. The
algebra $R_{pol}$ of polynomial invariants is a subalgebra of $R$ consisting
of polynomial functions.
As in the characteristic zero case, the description of polynomial invariants
$R_{pol}$ can be reduced to that of the algebra $A_{s}$ of supersymmetric
polynomials. The algebra $A_{s}$ consists of polynomials
$f(x|y)=f(x_{1},\ldots,x_{m},y_{1},\ldots y_{n})$ that are symmetric in
variables $x_{1},\ldots x_{m}$ and $y_{1},\ldots,y_{n}$ separately and such
that $\frac{d}{dT}f(x|y)_{x_{1}=y_{1}=T}=0$. The main tool for this reduction
is the Chevalley epimorphism $\phi:K[G]\to A$, where $A=K[x_{1}^{\pm
1},\ldots,x_{m}^{\pm 1},y_{1}^{\pm 1}\ldots y_{n}^{\pm 1}]$, and
$\phi(c_{ij})=\delta_{ij}x_{i}$ for $1\leq i\leq m$ and
$\phi(c_{ij})=\delta_{ij}y_{i-m}$ for $m+1\leq i\leq m+n$. Then for any
$G$-supermodule $V$ we have $\phi(Tr(V))=\chi_{sup}(V)$. In particular, for
$0\leq r$ we have
$\phi(C_{r})=c_{r}=\sum_{0\leq i\leq
min(r,m)}(-1)^{r-i}\sigma_{i}(x_{1},\ldots,x_{m})p_{r-i}(y_{1},\ldots,y_{n}),$
where $\sigma_{i}$ is the $i$-th elementary symmetric function and $p_{j}$ is
the $j$-th complete symmetric function.
A homogeneous polynomial $f(x|y)=\sum a_{\lambda}x_{1}^{\lambda_{1}}\ldots
x_{m}^{\lambda_{m}}y_{1}^{\lambda_{m+1}}\ldots y_{n}^{\lambda_{m+n}}$ is
called $p$-balanced if $p|(\lambda_{i}+\lambda_{j})$ whenever $1\leq i\leq
m<j\leq m+n$ and $a_{\lambda}\neq 0$. Denote by $A_{s}(p)$ the subalgebra of
$A_{s}$ generated by $p$-balanced polynomials.
The following theorem is the main result of this paper.
###### Theorem 1.
(Conjecture 5.2 of [3]) The algebra $A_{s}$ is generated over $A_{s}(p)$ by
elements $c_{r}$ for $r\geq 0$.
For a matrix $M$, denote by $\sigma_{i}(M)$ the $i$-th coefficient of the
characteristic polynomial of $M$. Then all elements $C_{r}$,
$\sigma_{i}(C_{00})^{p}$, $\sigma_{j}(C_{11})^{p}$,
$\sigma_{n}(C_{11})^{p}Ber(C)^{k}\in R_{pol}$. As a consequence of previous
theorem we obtain that these are generators of algebra $R_{pol}$.
###### Theorem 2.
(Conjecture 5.1 of [3]) The algebra $R_{pol}$ is generated by elements
$C_{r},\sigma_{i}(C_{00})^{p},\sigma_{j}(C_{11})^{p},\sigma_{n}(C_{11})^{p}Ber(C)^{k},$
where $0\leq r,1\leq i\leq m,1\leq j\leq n,0<k<p$.
###### Proof.
Theorem 5.2 of [3] states that the restriction of $\phi$ on $R$ is a
monomorphism. It was also noticed there that $\phi(R_{pol})\subset A_{s}$
which is a consequence of arguments analogous to Theorem 1.1 of [1].
Proposition 5.1 of [3] states that $\sigma_{i}(x_{1},\ldots,x_{m})^{p}$ for
$1\leq i\leq m$, $\sigma_{j}(y_{1},\ldots,y_{n})^{p}$ for $1\leq j\leq n$ and
$u_{k}(x|y)=\sigma_{m}(x_{1},\ldots,x_{m})^{k}\sigma_{m}(y_{1},\ldots,y_{n})^{p-k}$
for $0<k<p$ are generators of algebra $A_{s}(p)$. Since $\phi(C_{r})=c_{r}$,
$\phi(\sigma_{i}(C_{00})^{p})=\sigma_{i}(x_{1},\ldots,x_{m})^{p}$,
$\phi(\sigma_{j}(C_{11})^{p})=\sigma_{j}(y_{1},\ldots,y_{n})^{p}$ and
$\phi(\sigma_{n}(C_{11})^{p}Ber(C)^{k})$ $=u_{k}(x|y)$, the statement follows
from Theorem 1. ∎
###### Corollary 1.
Algebra $R$ is equaled to $R_{pol}[\sigma_{m}(C_{00})^{\pm
p},\sigma_{n}(C_{11})^{\pm p}]$.
###### Proof.
If $f\in R$, then its multiple by a sufficiently large power of
$\sigma_{m}(C_{00})^{p}\sigma_{n}(C_{11})^{p}$ is a polynomial invariant. ∎
## 1\. Nice supersymmetric polynomials
In this section we will compare algebras corresponding to different values of
$m,n$ and apply the Schur functor. Therefore we adjust the notation slightly
to reflect the dependence on $m,n$. For example, we will write $R(m|n)$
instead of $R$ and $A_{s}(m|n)$ instead of $A_{s}$.
Denote by $A_{ns}(m|n)$ the algebra of ”nice” supersymmetric polynomials. It
is a subalgebra of $A_{s}(m|n)$ generated by polynomials
$c_{r}(m|n)=\sum_{0\leq i\leq
r}(-1)^{r-i}\sigma_{i}(x_{1},\ldots,x_{m})p_{r-i}(y_{1},\ldots,y_{n}),$
$\sigma_{i}(x,m)^{p}=\sigma_{i}(x_{1},\ldots,x_{m})^{p},\sigma_{j}(y,n)^{p}=\sigma_{j}(y_{1},\ldots,y_{n})^{p}$
and
$u_{k}(m|n)=\sigma_{m}(x,m)^{k}\sigma_{n}(y,n)^{p-k}$
for $1\leq i\leq m$, $1\leq j\leq n$ and $0<k<p$. Since
$\phi(C_{r}(m|n))=c_{r}(m|n)$, using Proposition 5.1 of [3] we conclude that
$A_{ns}(m|n)\subset\phi(R_{pol}(m|n))$. We will see below that this inclusion
is actually an equality. Denote by $A_{ns}(m|n,t)$ the homogeneous component
of $A_{ns}(m|n)$ of degree $t$.
For any integers $M\geq m,N\geq n$ there is a graded superalgebra morphism
$p_{e}:K[x_{1},\ldots,x_{M},y_{1},\ldots,y_{N}]\to
K[x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}]$ that maps $x_{i}\mapsto x_{i}$ for
$i\leq m$, $y_{j}\mapsto y_{j}$ for $j\leq n$ and the remaining generators
$x_{i}$, $y_{j}$ to zero. Clearly $p_{e}$ restricts to a map from $A_{s}(M|N)$
into $A_{s}(m|n)$.
###### Lemma 1.1.
The map $p_{e}$ takes $A_{ns}(M|N)$ to $A_{ns}(m|n)$.
###### Proof.
Assume $M>m$ or $N>n$. It follows from
$p_{e}(\sigma_{i}(x,M))=\sigma_{i}(x,m)$ if $i\leq m$ and
$p_{e}(\sigma_{i}(x,M))=0$ otherwise; $p_{e}(\sigma_{j}(y,N))=\sigma_{j}(y,n)$
if $j\leq n$ and $p_{e}(\sigma_{j}(y,N))=0$ otherwise;
$p_{e}(c_{r}(M,N))=c_{r}(m|n)$ if $r\leq m,n$ and $p_{e}(c_{r}(M,N))=0$
otherwise, and $p_{e}(u_{k}(M|N))$ $=0$. ∎
For the integers $M\geq m,N\geq n$ consider the Schur superalgebra $S(M|N,r)$
and its idempotent $e=\sum_{\mu}\xi_{\mu}$, where the sum runs over all
weights $\mu$ for which $\mu_{i}=0$ whenever $m<i\leq M$ or $M+n<i\leq M+N$.
Then $S(m|n,r)\simeq eS(M|N,r)e$ and there is a natural Schur functor
$S(M|N,r)-mod\to S(m|n,r)-mod$ given by $V\mapsto eV$. If $V$ is a
$S(M|N,r)$-supermodule, then $eV$ is a supersubspace of $V$ and therefore,
$eV$ has the canonical $S(m|n,r)$-supermodule structure.
###### Lemma 1.2.
The map $p_{e}$ induces an epimorphism of graded algebras
$\phi(R_{pol}(M|N))\to\phi(R_{pol}(m|n))$.
###### Proof.
Applying the Chevalley map $\phi$ to the collection of simple polynomial
$G$-supermodules $L$ and using Theorem 5.3 of [3] we obtain that the algebra
$\phi(R_{pol})$ is spanned by the supercharacters $\chi_{sup}(L)$. If
$\lambda$ is a highest weight of $L$, then $\chi_{sup}(L)$ is a homogeneous
polynomial of degree $r=|\lambda|=\sum_{1\leq i\leq m+n}\lambda_{i}$. By the
standard property of a Schur functor there is a simple $S(M|N,r)$-supermodule
$L^{\prime}$ such that $eL^{\prime}\simeq L$. Since
$p_{e}(\chi_{sup}(L^{\prime}))=\chi_{sup}(L)$, the claim follows. ∎
###### Proposition 1.1.
$\phi(R_{pol}(m|n))=A_{ns}(m|n)$.
###### Proof.
Fix a homogeneous element $f\in\phi(R_{pol}(m|n))$ of degree $r$ and choose
$M\geq m$ strictly greater than $r$. By Lemma 1.2 there is a homogeneous
polynomial $f^{\prime}\in\phi(R_{pol}(M|n))$ of degree $r$ such that
$p_{e}(f^{\prime})=f$. Using Chevalley map and applying Theorem 5.3 of [3] to
the collection of costandard polynomial modules $\nabla(\mu)$ we obtain that
$f^{\prime}$ is a linear combination of supercharacters
$\chi_{sup}(\nabla(\mu))$, where $\mu$ runs over polynomial dominant weights
with $|\mu|=r$.
The assumption $M>r$, Theorem 5.4 and Proposition 5.6 of [2] imply that for
the highest weight $\mu=(\mu_{+}|\mu_{-})$ we have $\mu_{-}=p\overline{\mu}$
and $\nabla(\mu)\simeq\nabla(\mu_{+}|0)\otimes
F(\overline{\nabla}(\overline{\mu}))$, where $F$ is the Frobenius map and
$\overline{\nabla}(\overline{\mu})$ is the costandard $GL(n)$-module with the
highest weight $\overline{\mu}$. Therefore
$\chi_{sup}(\nabla(\mu))=\chi_{sup}(\nabla(\mu_{+}|0))\chi(\overline{\nabla}(\overline{\mu}))^{p}.$
Since $\chi(\overline{\nabla}(\overline{\mu}))^{p}$ is a polynomial in
$\sigma_{j}(y,n)^{p}$, all that remains to show is that
$\chi_{sup}(\nabla(\mu_{+}|0))$ belongs to $A_{ns}(M|n)$ and use Lemma 1.1.
By Theorem 6.6 of [2] the character $\chi_{sup}(\nabla(\pi|0))$ for a
polynomial weight $\pi$ does not depend on the characteristic of the ground
field. Therefore we can temporarily assume that $charK=0$. In this case the
category $S(M|n,r)-mod$ is semisimple and its simple modules are
$\nabla(\lambda)$, where $\lambda$ runs over $(M|n)$-hook weights.
An exterior power $\Lambda^{t}(E(M|n))$ for $t\leq M$ has a unique maximal
weight $(1^{t}|0)$. Consequently, a $S(M|n,r)$-supermodule $V=$
$\Lambda^{M}(E(M|n))^{\otimes\pi_{M}}\otimes\Lambda^{M-1}(E(M|n))^{\otimes(\pi_{M-1}-\pi_{M})}\otimes\ldots\otimes\Lambda^{1}(E(M|n))^{\otimes(\pi_{1}-\pi_{2})}$
has the unique maximal weight $(\pi|0)$ and supercharacter
$\chi_{sup}(V)=c_{1}^{\pi_{1}-\pi_{2}}\ldots
c_{M-1}^{\pi_{M-1}-\pi_{M}}c_{M}^{\pi_{M}}.$
The module $V$ is a direct sum of $L(\pi|0)=\nabla(\pi|0)$ and
$L(\kappa)=\nabla(\kappa)$ with $\kappa<(\pi|0)$. Since $\kappa$ is a
polynomial weight, it implies that $\kappa=(\kappa_{+}|0)$ and
$\kappa_{+}<\pi$. Using induction on $\pi$ we derive that
$\chi_{sup}(\nabla(\pi|0))$ is a polynomial in $c_{1},\ldots,c_{M}$ hence it
belongs to $A_{ns}(M|n)$. ∎
###### Corollary 1.1.
The morphism $p_{e}$ maps $A_{ns}(M|N,t)$ onto $A_{ns}(m|n,t)$.
## 2\. Proof of Theorem 1
We will need the following crucial observation.
###### Lemma 2.1.
If $f\in A_{s}(m|n)$ is divided by $x_{m}$, then $f$ is divided by a
nonconstant element of $A_{ns}(m|n)$.
###### Proof.
We can assume $f\neq 0$ and use the symmetricity of $f$ in variables
$x_{1},\ldots,x_{m}$ and $y_{1},\ldots,y_{n}$ to write $f=x_{1}^{a}\ldots
x_{m}^{a}y_{1}^{b}\ldots y_{n}^{b}g$, where exponents $a>0,b\geq 0$ and a
polynomial $g$ such that $g|_{x_{m}=y_{n}=0}\neq 0$ are unique. Then
$f|_{x_{m}=y_{n}=T}=T^{a+b}x_{1}^{a}\ldots x_{m-1}^{a}y_{1}^{b}\ldots
y_{n-1}^{b}g|_{x_{m}=y_{n}=T}$ $=T^{a+b}x_{1}^{a}\ldots
x_{m-1}^{a}y_{1}^{b}\ldots y_{n-1}^{b}g_{0}+T^{a+b+1}x_{1}^{a}\ldots
x_{m-1}^{a}y_{1}^{b}\ldots y_{n-1}^{b}g_{1},$
where we write $g|_{x_{m}=y_{n}=T}=g_{0}+Tg_{1}$. The requirement
$g|_{x_{m}=y_{n}=0}\neq 0$ implies $g_{0}\neq 0$. Since
$\frac{d}{dT}f|_{x_{m}=y_{n}=T}=0$, this is only possible if $a+b\equiv
0\pmod{p}$. Since $a>0$, the polynomial $x_{1}^{a}\ldots
x_{m}^{a}y_{1}^{b}\ldots y_{n}^{b}$ is not constant, and is a product of
$\sigma_{m}(x,m)^{p}$, $\sigma_{n}(y,n)^{p}$ and $u_{k}(m|n)$ which belongs to
$A_{ns}(m|n)$. In fact, since $a>0$, we have that $f$ is divisible either by
$\sigma_{m}(x,m)^{p}$ or by some $u_{k}(m|n)$. ∎
Proof of Theorem 1. Using Proposition 5.1 of [3] the statement of the theorem
is equivalent to the equality $A_{s}(m|n)=A_{ns}(m|n)$.
Fix $n$ and assume that $m$ is minimal such that there exists a polynomial
$f\in A_{s}(m|n)\setminus A_{ns}(m|n)$ and choose $f$ such that it is
homogeneous and of the minimal degree. Then its reduction $f|_{x_{m}=0}\in
A_{ns}(m-1|n)$ is a nonzero polynomial
$h(c_{t}(m-1|n),\sigma_{i}(x,m-1)^{p},\sigma_{j}(y,n)^{p},u_{k}(m-1|n))$ in
elements $c_{t}(m-1|n)$, $\sigma_{i}(x,m-1)^{p}$, $\sigma_{j}(y,n)^{p}$ and
$u_{k}(m-1,n)$ where $t\geq 0$, $1\leq i\leq m-1$, $1\leq j\leq n$ and
$0<k<p$. By Corollary 1.1 there are elements $v_{k}\in A_{ns}(m|n)$ of degree
$mk+(p-k)n$ such that $v_{k}|_{x_{m}=0}=u_{k}(m-1|n)$. Since
$c_{t}(m|n)|_{x_{m}=0}=c_{t}(m-1|n)$,
$\sigma_{i}(x,m)^{p}|_{x_{m}=0}=\sigma_{i}(x,m-1)^{p}$ and
$\sigma_{j}(y,n)^{p}|_{x_{m}=0}=\sigma_{j}(y,n)^{p}$, the polynomial
$l=f-h(c_{t}(m|n),\sigma_{i}(x,m)^{p},\sigma_{j}(y,n)^{p},v_{k}(m|n))$
satisfies $l|_{x_{m}=0}=0$. Since the degree of $l$ does not exceed the degree
of $f$, $l\in A_{s}(m|n)$ and $x_{m}$ divides $l$, Lemma 2.1 implies that
$l=l_{0}l_{1}$, where $l_{0}\in A_{ns}(m|n)$ and the degree of $l_{1}$ is
strictly less than the degree of $f$. But $l_{1}\in A_{s}(m|n)\setminus
A_{ns}(m|n)$ which is a contradiction with our choice of $f$. ∎
## 3\. Elementary proof of Theorem 1
A closer look at the proof of Theorem 1 reveals that Corollary 1.1 is the only
result from Section 1 that was used in it. Actually, only the following weaker
statement was required in the proof of Theorem 1.
###### Proposition 3.1.
For each $0<k<p$ there is a polynomial $v_{k}\in A_{ns}(m|n)$ of degree
$(m-1)k+(p-k)n$ such that $v_{k}|_{x_{m}=0}=u_{k}(m-1|n)$.
In this section we give a constructive elementary proof of Proposition 3.1
that bypasses a use of the Schur functor and results about costandard modules
derived in [2].
Fix $0<k<p$ and denote $s=\lceil\frac{k}{p-k}\rceil$. Then for $i=0,\ldots
s-1$ define $k_{i}=(i+1)k-ip>0$ and $k_{p}=sp-(s+1)k\geq 0.$ The relations
$k_{i}+(p-k)=k_{i-1},\,k_{p}+k=s(p-k),\,k_{i}+k_{p}=(s-i)(p-k)$
will be used repeatedly.
A symbol $\Delta$ will denote a nondecreasing sequence $(i_{1}\leq\ldots\leq
i_{t})$ of natural numbers, where $0\leq t<s$. We denote $||\Delta||=t$ and
$|\Delta|=\sum_{j=1}^{t}i_{j}$. In particular, we allow $\Delta=(\emptyset)$
and set $||(\emptyset)||=|(\emptyset)|=0$. Denote by $Supp(\Delta)$ a maximal
increasing subsequence of $\Delta$. Further, denote by
$Sym_{M}(x_{1}^{a_{1}}\dots x_{r}^{a_{r}})$ and $Sym_{N}(y_{1}^{b_{1}}\dots
y_{r}^{b_{r}})$ respectively a homogeneous symmetric polynomial in variables
$x_{1},\ldots,x_{M}$ and $y_{1},\ldots,y_{N}$ respectively with a general
monomial term $x_{1}^{a_{1}}\dots x_{r}^{a_{r}}$ and $y_{1}^{b_{1}}\dots
y_{r}^{b_{r}}$ respectively. Denote
$\begin{array}[]{l}(\Delta,j)_{M,N}=\\\ Sym_{M}(x_{1}^{k}\dots
x_{M-t}^{k}x_{M-t+1}^{k_{i_{1}}}\dots
x_{M}^{k_{i_{t}}})Sym_{N}(y_{1}^{p-k}\dots
y_{N-j-1}^{p-k}y_{N-j}^{k_{p}})\end{array}$
for $0\leq||\Delta||=t\leq M$ and $0\leq j<N$, and $(\Delta,l)_{M,N}=0$
otherwise;
$\begin{array}[]{l}[\Delta,j]_{M,N}=\\\ Sym_{M}(x_{1}^{k}\dots
x_{M-t}^{k}x_{M-t+1}^{k_{i_{1}}}\dots
x_{M}^{k_{i_{t}}})Sym_{N}(y_{1}^{p-k}\dots y_{N-j}^{p-k})\end{array}$
for $0\leq||\Delta||=t\leq M$ and $0\leq j\leq N$, and $[\Delta,j]_{M,N}=0$
otherwise;
$\begin{array}[]{l}\\{\Delta,l,j\\}_{M,N}=\\\ Sym_{M}(x_{1}^{k}\dots
x_{M-t-1}^{k}x_{M-t}^{l(p-k)}x_{M-t+1}^{k_{i_{1}}}\dots
x_{M}^{k_{i_{t}}})Sym_{N}(y_{1}^{p-k}\dots y_{N-j}^{p-k})\end{array}$
for $0\leq||\Delta||=t<M$ and $0\leq j\leq N$ and any $l$, and
$\\{\Delta,l,j\\}_{M,N}=0$ otherwise.
For $f\in K[x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}]$ define
$\psi(f)=f|_{x_{m}=y_{n}=T}$ and for $g,h\in
K[x_{1},\ldots,x_{m-1},y_{1},\ldots,y_{n-1},T]$ write $g\equiv h$ if and only
if $\frac{d}{dT}(g-h)=0.$
For simplicity write $(\Delta,j)$, $[\Delta,j]$ and $\\{\Delta,l,j\\}$ short
for $(\Delta,j)_{m-1,n-1}$, $[\Delta,j]_{m-1,n-1}$ and
$\\{\Delta,l,j\\}_{m-1,n-1}$.
###### Lemma 3.1.
We have
$\begin{array}[]{l}\psi\\{\Delta,l,j\\}_{m,n}\equiv\\\
T^{k}\\{\Delta,l,j-1\\}+T^{(l+1)(p-k)}[\Delta,j]+T^{l(p-k)}[\Delta,j-1]+\\\
\sum_{i\in Supp(\Delta)}(T^{k_{i}}\\{\Delta\setminus
i,l,j-1\\}+T^{k_{i-1}}\\{\Delta\setminus i,l,j\\}).\end{array}$
and
$\begin{array}[]{l}\psi(\Delta,j)_{m,n}\equiv\\\
T^{k}(\Delta,j-1)+T^{s(p-k)}[\Delta,j]+\\\ \sum_{i\in
Supp(\Delta)}(T^{k_{i}}(\Delta\setminus i,j-1)+T^{k_{i-1}}(\Delta\setminus
i,j)+T^{(s-i)(p-k)}[\Delta\setminus i,j]).\end{array}$
###### Proof.
The first relation follows from
$\begin{array}[]{l}\psi(Sym_{m}(x_{1}^{k}\dots
x_{m-t-1}^{k}x_{m-t}^{l(p-k)}x_{m-t+1}^{k_{i_{1}}}\dots
x_{m}^{k_{i_{t}}}))=\\\ \delta_{t,m-1}T^{k}Sym_{m-1}(x_{1}^{k}\dots
x_{m-1-t-1}^{k}x_{m-1-t}^{l(p-k)}x_{m-1-t+1}^{k_{i_{1}}}\dots
x_{m-1}^{k_{i_{t}}})\\\ +T^{l(p-k)}Sym_{m-1}(x_{1}^{k}\dots
x_{m-t-1}^{k}x_{m-t}^{k_{i_{1}}}\dots x_{m-1}^{k_{i_{t}}})+\\\ \sum_{i\in
Supp(\Delta)}T^{k_{i}}Sym_{m-1}(x_{1}^{k}\dots
x_{m-t-1}^{k}x_{m-t}^{l(p-k)}x_{m-t+1}^{k_{i_{1}}}\dots\widehat{x_{m-t+i}^{k_{i}}}\dots
x_{m-1}^{k_{i_{t}}}),\end{array}$
$\begin{array}[]{l}\psi(Sym_{n}(y_{1}^{p-k}\dots y_{n-j}^{p-k}))=\\\
\delta_{j,n}T^{p-k}Sym_{n-1}(y_{1}^{p-k}\dots
y_{n-1-j}^{p-k})+\delta_{j,0}Sym_{n-1}(y_{1}^{p-k}\dots
y_{n-j}^{p-k})\end{array}$
and definitions of $(\Delta,j)$, $[\Delta,j]$ and $\\{\Delta,l,j\\}$.
Second relations follows from
$\begin{array}[]{l}\psi(Sym_{m}(x_{1}^{k}\dots
x_{m-t}^{k}x_{m-t+1}^{k_{i_{1}}}\dots x_{m}^{k_{i_{t}}}))=\\\
\delta_{t,m}T^{k}Sym_{m-1}(x_{1}^{k}\dots
x_{m-t-1}^{k}x_{m-t}^{k_{i_{1}}}\dots x_{m-1}^{k_{i_{t}}})\\\ +\sum_{i\in
Supp(\Delta)}T^{k_{i}}Sym_{m-1}(x_{1}^{k}\dots
x_{m-t}^{k}x_{m-t+1}^{k_{i_{1}}}\dots\widehat{x_{m-t+i}^{k_{i}}}\dots
x_{m-1}^{k_{i_{t}}}),\end{array}$
$\begin{array}[]{l}\psi(Sym_{n}(y_{1}^{p-k}\dots
y_{n-j}^{p-k}y_{n-1-j}^{k_{p}}))=\\\
\delta_{j,n-1}T^{p-k}Sym_{n-1}(y_{1}^{p-k}\dots
y_{n-1-j-1}^{p-k}y_{n-1-j}^{k_{p}})\\\ +T^{k_{p}}Sym_{n-1}(y_{1}^{p-k}\dots
y_{n-1-j}^{p-k})+\delta_{j,0}Sym_{n-1}(y_{1}^{p-k}\dots
y_{n-1-j}^{p-k}y_{n-j}^{k_{p}})\end{array}$
and definitions of $(\Delta,j)$, $[\Delta,j]$ and $\\{\Delta,l,j\\}$. ∎
Let us define
$\begin{array}[]{ll}w=&\sum_{l=1}^{s-1}\sum_{l-n\leq|\Delta|\leq
l}(-1)^{|\Delta|+s+l}(s-l)\\{\Delta,l,l-|\Delta|\\}_{m,n}\\\
&+\sum_{s-1-n<|\Delta|<s}(-1)^{|\Delta|}(\Delta,s-1-|\Delta|)_{m,n}.\end{array}$.
This expression is identical to
$\begin{array}[]{ll}&\sum_{l=1}^{s-1}\sum_{0\leq|\Delta|\leq
l}(-1)^{|\Delta|+s+l}(s-l)\\{\Delta,l,l-|\Delta|\\}_{m,n}\\\
&+\sum_{\Delta}(-1)^{|\Delta|}(\Delta,s-1-|\Delta|)_{m,n}\end{array}$
obtained by inserting additional terms that are all equal to zero. Then
$\begin{array}[]{ll}\psi(w)=&\sum_{l=1}^{s-1}\sum_{0\leq|\Delta|\leq
l}(-1)^{|\Delta|+s+l}(s-l)\psi\\{\Delta,l,l-|\Delta|\\}_{m,n}\\\
&+\sum_{\Delta}(-1)^{|\Delta|}\psi(\Delta,s-1-|\Delta|)_{m,n}\end{array}$
which by Lemma 3.1 equals
$\begin{array}[]{l}\sum_{l=1}^{s-1}\sum_{0\leq|\Delta|\leq
l}(-1)^{s+l+|\Delta|}(s-l)\Big{(}T^{k}\\{\Delta,l,l-|\Delta|-1\\}\\\
+T^{(l+1)(p-k)}[\Delta,l-|\Delta|]+T^{l(p-k)}[\Delta,l-|\Delta|-1]\\\
+\sum_{i\in Supp(\Delta)}(T^{k_{i}}\\{\Delta\setminus
i,l,l-|\Delta|-1\\}+T^{k_{i-1}}\\{\Delta\setminus i,l,l-|\Delta|\\})\Big{)}\\\
+\sum_{\Delta}(-1)^{|\Delta|}\\\
\Big{(}T^{k}(\Delta,s-|\Delta|-2)+T^{s(p-k)}[\Delta,s-1-|\Delta|]\\\
+\sum_{i\in Supp(\Delta)}(T^{k_{i}}(\Delta\setminus
i,s-|\Delta|-2)+T^{k_{i-1}}(\Delta\setminus i,s-|\Delta|-1)\\\
+T^{(s-i)(p-k)}[\Delta\setminus i,s-|\Delta|-1])\Big{)}.\end{array}$
To analyze this expression, denote by $C(a)$ the coefficient corresponding to
a term $a$ of $\psi(w)$.
Then
$C(T^{k_{i}}(\Delta,s-2-i-|\Delta|)=(-1)^{|\Delta|+i}+(-1)^{|\Delta|+i+1}=0$
for each $i=0,\ldots,s-2$ and admissible $\Delta$ and
$C(T^{k_{i}}\\{\Delta,l,l-1-|\Delta|-i\\})=(-1)^{s+l+|\Delta|+i}+(-1)^{s+l+|\Delta|+i+1}=0$
for each $i=0,\ldots,s-1$ and admissible $\Delta$.
Next,
$C(T^{s(p-k)}[\Delta,s-|\Delta|-1])=(-1)^{|\Delta|+s+s-1}(s-(s-1))+(-1)^{|\Delta|}=0$
and if $l-1-|\Delta|\neq 0$ or $\Delta\neq\emptyset$, then
$C(T^{l(p-k)}[\Delta,l-|\Delta|-1])=(-1)^{|\Delta|+s+l}(s-l)+(-1)^{|\Delta|+s+l-1}(s-l+1)+(-1)^{|\Delta_{l}|},$
where $\Delta_{l}=\Delta\cup(s-l).$ Since
$(-1)^{|\Delta_{l}|}=(-1)^{|\Delta|+s-l}$ we conclude that
$C(T^{l(p-k)}[\Delta,l-|\Delta|-1])=0$. Finally,
$C(T^{p-k}[\emptyset,0])=(-1)^{s+1}(s-1)+(-1)^{|\Delta_{1}|}=(-1)^{s+1}s.$
Therefore
$\begin{array}[]{ll}\psi(w)&=(-1)^{s+1}sT^{p-k}[\emptyset,0]\\\
&=(-1)^{s+1}sT^{p-k}Sym_{m-1}(x_{1}^{k}\dots
x_{m-1}^{k})Sym_{n-1}(y_{1}^{p-k}\dots y_{n-1}^{p-k}).\end{array}$
We can now easily prove Proposition 3.1.
Proof of Proposition 3.1. Since $s<p$ we can take
$v_{k}=\frac{(-1)^{s}}{s}w+Sym_{m}(x_{1}^{k}\dots
x_{m-1}^{k})(y_{1}^{p-k}\dots y_{n}^{p-k}).$
Then $\psi(v_{k})=0$, hence $v_{k}\in A_{s}(m|n)$ and
$v_{k}|_{x_{m}=0}=u_{k}(m-1|n)$. It is easy to check that $v_{k}$ is
homogeneous of degree $(m-1)k+(p-k)n$. ∎.
## 4\. Concluding remarks
The proof of Theorem 1 was influenced by Theorem 1 in [4]. Although our proof
uses different arguments we would like to remark that an analogue of Theorem 1
of [4] remains valid over arbitrary commutative ring $A$ of any
characteristic.
###### Proposition 4.1.
The algebra of polynomials $f(x_{1},\ldots,x_{m},y_{1},\ldots,y_{n})$ over a
commutative ring $A$, symmetric in variables $x_{1},\dots,x_{m}$ and
$y_{1},\ldots,y_{n}$ separately and such that $f|_{x_{m}=y_{n}=T}$ does not
depend on $T$ is generated by polynomials $c_{r}(x|y)$.
###### Proof.
Proof is a complete analogue of Theorem 1 of [4]. Just replace every
appearance of $\sigma_{m,n}^{(r)}$ by $c_{r}(x|y)$. ∎
Let us comment that if characteristic of $A$ is positive, then condition that
$f|_{x_{m}=y_{n}=T}$ does not depend on $T$ is stronger than
$\frac{d}{dT}f|_{x_{m}=y_{n}=T}=0$.
Proposition 3.1 of [1] states that, in the case of characteristic zero, the
algebra $A_{s}$ is infinitely generated. In the case of positive
characteristic we have the following.
###### Proposition 4.2.
Algebra $A_{s}$ is finitely generated.
###### Proof.
The algebra $A_{s}$ is contained in $B=K[\sigma_{i}(x|m),\sigma_{j}(y|n)|1\leq
i\leq m,1\leq j\leq n]$. Algebra $B$ is finitely generated over its subalgebra
$B^{\prime}=K[\sigma_{i}(x|m)^{p},\sigma_{j}(y|n)^{p}|1\leq i\leq m,1\leq
j\leq n]$ hence is a Noetherian $B^{\prime}$-module. However, $A_{s}$ contains
$B^{\prime}$ and is therefore finitely generated $B^{\prime}$-module. Since
$B^{\prime}$ is finitely generated, so is $A_{s}$. ∎
## References
* [1] I.Kantor and I.Trishin, The algebra of polynomial invariants of the adjoint representation of the Lie superalgebra $gl(m|n)$, Comm. Algebra, 25(7) 1997, 2039-2070.
* [2] R. La Scala and A.N.Zubkov, Costandard modules over Schur superalgebras in characteristic $p$, J.Algebra and its Appl., 7(2008), no 2, 147-166.
* [3] R. La Scala and A.N.Zubkov, Donkin-Koppinen filtration for general linear supergroup, submitted to Algebra and Representation Theory.
* [4] J.R.Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95 (1985), N.2, 439-444.
|
arxiv-papers
| 2009-07-28T05:54:30 |
2024-09-04T02:49:04.229233
|
{
"license": "Public Domain",
"authors": "A.N. Grishkov, F. Marko and A.N. Zubkov",
"submitter": "Alexander Zubkov Nikolaevich",
"url": "https://arxiv.org/abs/0907.4840"
}
|
0907.4865
|
11footnotetext: Weierstrass Institute for Applied Analysis and Stochastics,
Mohrenstr. 39, 10117 Berlin, Germany. belomest@wias-berlin.de
# Estimating the distribution of jumps in regular affine models: uniform rates
of convergence
Denis Belomestny1
###### Abstract
The problem of separating the jump part of a multidimensional regular affine
process from its continuous part is considered. In particular, we present an
algorithm for a nonparametric estimation of the jump distribution under the
presence of a nonzero diffusion component. An estimation methodology is
proposed which is based on the log-affine representation of the conditional
characteristic function of a regular affine process and employees a smoothed
in time version of the empirical characteristic function in order to estimate
the derivatives of the conditional characteristic function. We derive almost
sure uniform rates of convergence for the estimated Lévy density and prove
that these rates are optimal in the minimax sense. Finally, the performance of
the estimation algorithm is illustrated in the case of the Bates stochastic
volatility model.
_Keywords:_ Regular affine processes, estimation, empirical characteristic
function
## 1 Introduction
The problem of nonparametric statistical inference for jump processes or more
generally for semimartingales models has long history and goes back to the
works of Rubin and Tucker (1959) and Basawa and Brockwell (1982). The recent
revival of interest in this topic documented, for example, in Figueroa-López
(2004) and Figueroa-López (2009), is mainly related to a wide availability of
financial and economical time series data and new types of statistical issues
that have not been addressed before. For instance, there is now considerable
evidence (see, e.g. Cont and Mancini (2007)) that most financial time series
contain a continuous martingale component. This is why a number of works
appeared in recent years deal with the problem of estimating some
characteristics of jumps in the general semimartingale models with a nonzero
continuous part. Without further assumptions such kind of statistical
inference is not possible because the behavior of the jump component becomes
statistically indistinguishable from the behavior of the diffusion part as the
activity of small jumps tends to infinity. In the case of Lévy processes the
activity of small jumps can be measured by the so called Blumenthal-Getoor
index. The nearer is the Blumenthal-Getoor index to $2$, the more difficult
becomes the problem of separating the jump component from the diffusion
component and hence the problem of statistical inference on the
characteristics of jumps (see, e.g. Neumann and Reiß (2007)). Suppose that
values of a log-return process $X(t)=\log(S(t))$ on a time grid
$\pi=\\{t_{0},t_{1},\ldots,t_{N}\\}$ are observed. If $|\pi|$ is small (high-
frequency data) then a large increment $X(t_{i})-X(t_{i-1})$ indicates that a
jump occurred between time $t_{i-1}$ and $t_{i}$. Based on this insight and
the continuous-time observation analogue, inference for various
characteristics of jumps of the underlying semimartingale can be conducted.
For example, in Aït-Sahalia and Jacod (2008) the problem of statistical
inference on the degree of jump activity in the general semimartingale models
based on high-frequency data is considered. They proposed an estimation
procedure which is able to “see through” the continuous part of semimartingale
and consistently estimate the degree of small jump activity under some
restrictions on the structure of the underlying semimartingale. In fact, these
restrictions keep the highest degree of activity of small jumps away from $2$,
thus allowing for a consistent estimation of the degree of jump activity.
In this work we consider the problem of estimating the characteristics of
jumps in the class of regular affine models with a nonzero continuous part.
Affine Itô-Lévy processes are nowadays rather popular in financial and
econometric modeling. Due to their analytical tractability on the one hand and
their rather rich dynamics and implied volatility patterns on the other hand,
they are particularly useful in the context of option pricing. Many well known
models such as Heston and Bates stochastic volatility models fall into the
class of affine Itô-Lévy processes. Option pricing in these models can be
conveniently done via the Fourier method. The literature on affine processes
is rather extensive. Let us mention two seminal papers of Duffie, Pan and
Singleton (2000) and Duffie, Filipović and Schachermayer (2003), where
theoretical analysis of regular affine models has been conducted. More recent
literature includes Glasserman and Kim (2007) and Keller-Ressel and Steiner
(2008).
We propose an approach based on the log-affine representation of the
conditional characteristic function of a regular affine process. This
representation together with some transformation allows one to consistently
estimate the characteristics of the jump component under some prior bound on
the highest degree of activity of small jumps. We present uniform convergence
rates for the so constructed estimate of a transformed Lévy density which turn
out to be optimal in minimax sense. As a main technical result that may be of
independent interest we provide uniform convergence rates for a smoothed in
time version of the empirical characteristic function.
The problem of parametric estimation of the characteristics of an affine jump-
diffusion process (processes with finite intensity of jumps) $X(t)$ from a
high-frequency time series of the asset $S(t)=\exp(X(t))$ in has been recently
considered in the literature by Singleton (2000) and Bates (2005). In
Singleton (2000) the general method of moments (GMM) based on the empirical
characteristic function is employed and asymptotic properties of the estimator
are investigated. Bates (2005) proposed a filtration-based maximum likelihood
methodology for estimating the parameters and realizations of latent affine
processes. Filtration is conducted in the transform space of characteristic
functions, using a version of Bayes rule for recursively updating the joint
characteristic function of latent variables and the data conditional upon past
data. Since the characteristics of an affine process are a priori an infinite-
dimensional object, a parametric approach is always exposed to the problem of
misspecification, in particular when there is no inherent economic foundation
of the parameters and they are only used to generate different shapes of
possible jump distributions. The problem of a semi-parametric inference for
the characteristics of some special type affine processes $X(t)$ has been
studied in the literature as well. In the case of high-frequency observations
the problem of a nonparametric inference on the Lévy measure for time changed
Lévy processes, belonging sometimes to the class of affine processes, has been
recently studied in Figueroa-López (2009) where consistency was proved. In
Jongbloed, van der Meulen and van der Vaart (2005) the case of a one-
dimensional Lévy driven Ornstein-Uhlenbeck process, affine process with zero
diffusion part, is considered. The authors assume that the corresponding jump
component is self-decomposable, i.e. the degree of the jump activity is less
than $1$. They propose a cumulant $M$-estimator to estimate the so called
canonical function of the driving self-decomposable process from low-frequency
data and prove consistency of the resulting estimate. As to the special case
of Lévy processes, a semi-parametric estimation problems for Lévy models under
low-frequency data has recently been studied in Neumann and Reiß (2007). Let
us mention that in Neumann and Reiß (2007) a diffusion component is assumed to
be known. So, the above works do not encounter the problem of separating
diffusion and jump components as the activity of small jumps increases.
Furthermore, the challenge of devising nonparametric estimation methods for
the Lévy density in general regular affine models lies in the fact that the
structure of the conditional characteristic function of a regular affine
process has not such explicit form as in the case of Lévy processes and is
related to the parameters of the underlying affine process not directly but
via a Riccati equation. The last but not the least: the increments of an
affine process are not any longer independent, hence advanced tools from time
series analysis have to be used.
The paper is organized as follows. In Section 2 we recall the definition and
basic properties of regular affine processes. In Section 3 we formulate a
spectral estimation algorithm which applies as soon as an estimates of the
corresponding conditional characteristic function and its time derivatives are
available. Section 5 is devoted to a nonparametric estimation of the above
characteristic function and its derivatives. We derive uniform rates of
convergence and prove that these rates are optimal in minimax sense. Section 5
concludes the paper.
## 2 Affine processes
Let us fix a probability space $(\Omega,\mathcal{F},P)$ and an information
filtration $(\mathcal{F}_{t})_{t\geq 0}$. The process $X(t)$ is an affine
process if it is stochastically continuous, time-homogenous Markov process
with state space $\mathcal{D}\subset\mathbb{R}^{d}$, such that the
characteristic function of $X(t)$ given $X(0)$ is an affine function of the
initial state $X(0)$:
(1)
$\phi(u|s,x):=\mathbb{E}\left(\left.e^{\mathfrak{i}u^{\top}X(s)}\right|X(0)=x\right)=e^{\psi_{0}(u,s)+x^{\top}\psi_{1}(u,s)},\quad
u\in\mathbb{R}^{d}.$
The affine process $X(t)$ is called regular, if the derivatives
$\displaystyle F_{0}(u):=\left.\frac{\partial\psi_{0}(u,s)}{\partial
s}\right|_{s=0},\quad F_{1}(u):=\left.\frac{\partial\psi_{1}(u,s)}{\partial
s}\right|_{s=0}$
exist and are continuous at $u=0.$ The following theorem provides the
characterization of affine processes and is proved in Duffie, Filipović and
Schachermayer (2003).
###### Theorem 2.1.
If $(X(t))_{t\geq 0}$ is a regular affine process, then $\psi_{0}$ and
$\psi_{1}$ satisfy the generalized Riccati equations
(2) $\displaystyle\frac{\partial\psi_{0}(u,s)}{\partial s}$ $\displaystyle=$
$\displaystyle F_{0}(\psi_{1}(u,s)),\quad\psi_{0}(u,0)=0,$ (3)
$\displaystyle\frac{\partial\psi_{1}(u,s)}{\partial s}$ $\displaystyle=$
$\displaystyle F_{1}(\psi_{1}(u,s)),\quad\psi_{1}(u,0)=\mathfrak{i}u,$
where
$\displaystyle F_{0}(z)$ $\displaystyle=$
$\displaystyle(\alpha^{(0)}z,z)+(z,\beta^{(0)})-\gamma^{(0)}+\int_{\mathcal{D}\setminus\\{0\\}}\left(e^{z^{\top}u}-1-(\chi(u),z)\right)\,\nu^{(0)}(du)$
$\displaystyle F_{1,j}(z)$ $\displaystyle=$
$\displaystyle(\alpha^{(1)}_{j}z,z)+(z,\beta_{j}^{(1)})-\gamma_{j}^{(1)}+\int_{\mathcal{D}\setminus\\{0\\}}\left(e^{z^{\top}u}-1-(\chi(u),z)\right)\,\nu^{(1)}_{j}(du)$
for $j=1,\ldots,d$ and $\chi(u)=(\chi_{1}(u),\ldots,\chi_{d}(u))$ with
$\displaystyle\chi_{k}(u)=\begin{cases}0,&u_{k}=0,\\\
(1\wedge|u_{k}|)\frac{u_{k}}{|u_{k}|},&\mbox{otherwise}\end{cases}$
for $k=1,\ldots,d.$ Here
$\alpha=(\alpha^{(0)},\alpha^{(1)})\in\mathbb{R}^{d\times
d}\times\mathbb{R}^{d\times d\times d},$
$\beta=(\beta^{(0)},\beta^{(1)})\in\mathbb{R}^{d}\times\mathbb{R}^{d\times
d},$ $\gamma=(\gamma^{(0)},\gamma^{(1)})\in\mathbb{R}\times\mathbb{R}^{d}$and
$\nu=(\nu^{(0)},\nu^{(1)}_{1},\ldots,\nu^{(1)}_{d})$ is a vector of measures
on $\mathbb{R}^{d},$ satisfying
$\displaystyle\int_{\mathcal{D}\setminus\\{0\\}}\|\chi(u)\|_{2}^{2}\,\nu^{(0)}(du)<\infty,\quad\int_{\mathcal{D}\setminus\\{0\\}}\|\chi(u)\|_{2}^{2}\,\nu^{(1)}_{j}(du)<\infty,\quad
j=1,\ldots,d,$
where here and in the sequel $\|x\|_{2}:=\sqrt{x^{2}_{1}+\ldots+x^{2}_{d}}$
for any $x\in\mathbb{R}^{d}.$
Under some admissibility conditions a regular affine process $X(t)$ is a
Feller process in the domain
$\mathcal{D}=\mathbb{R}_{+}^{m}\times\mathbb{R}^{d-m}$ (see Duffie, Filipović
and Schachermayer, 2003, Section 2) with the infinitesimal generator
$\displaystyle\mathcal{A}f(x)$ $\displaystyle=$
$\displaystyle\sum_{k,l=m+1}^{d}\left(\alpha^{(0)}_{kl}+\sum_{i=1}^{m}\alpha^{(1)}_{kl,i}x_{i}\right)\frac{\partial^{2}f(x)}{\partial
x_{k}\partial
x_{l}}+\sum_{i=1}^{m}\alpha^{(1)}_{ii,i}x_{i}\frac{\partial^{2}f(x)}{\partial^{2}x_{i}}$
$\displaystyle+\sum_{k=1}^{d}\beta_{k}^{(0)}\frac{\partial f(x)}{\partial
x_{k}}+\sum_{k=1}^{m}\sum_{i=1}^{m}\beta^{(1)}_{k,i}x_{i}\frac{\partial
f(x)}{\partial
x_{k}}+\sum_{k=m+1}^{d}\sum_{i=m+1}^{d}\beta^{(1)}_{k,i}x_{i}\frac{\partial
f(x)}{\partial x_{k}}$
$\displaystyle+\int_{\mathcal{D}\setminus\\{0\\}}\left(f(x+\xi)-f(x)-\sum_{k=m+1}^{d}\chi_{k}(\xi)\nabla_{k}f(x)\right)\nu^{(0)}(d\xi)$
$\displaystyle+\sum_{i=1}^{m}x_{i}\int_{\mathcal{D}\setminus\\{0\\}}\left(f(x+\xi)-f(x)-\chi_{i}(\xi)\nabla_{i}f(x)-\sum_{k=m+1}^{d}\chi_{k}(\xi)\nabla_{k}f(x)\right)\nu^{(1)}_{i}(d\xi)$
where $0\leq m\leq d$ and $f\in C^{2}(\mathcal{D}).$ In the sequel we assume
that the above admissibility conditions hold. Moreover, we restrict our
analysis to the class of regular affine processes with
(4) $\displaystyle\nu^{(1)}_{1}\equiv\ldots\equiv\nu^{(1)}_{d}\equiv 0.$
On the one hand this assumption reduces the dimensionality of the jump
component of $X$. On the other hand the class of affine models satisfying (4)
remains rather large and includes such well known models as Heston, Bates and
Barndorff-Nielsen and Shephard stochastic volatility models.
The goal of this paper is to investigate the problem of a nonparametric
inference for the Lévy measure $\nu^{(0)}$ based on a time series of the asset
prices $S(t)=(S_{1}(t),\ldots,S_{k}(t))$ that follow exponential regular
affine model:
(5) $S_{k}(t)=S_{k}(0)e^{X_{k}(t)},\quad t\in[0,T],\quad k=1,\ldots,d,$
where $X(t)=(X_{1}(t),\ldots,X_{d}(t))$ is a regular affine process with
characteristics $\chi=(\alpha,\beta,\gamma,\nu).$
### 2.1 Regularity properties of affine processes
In the sequel we shall need to know how fast the derivatives of the
conditional characteristic function of an affine process can grow. The
following lemma provides the corresponding bounds.
###### Lemma 2.2.
Let for some natural $k>0$ the Lévy measure $\nu^{(0)}$ satisfies
(6) $\displaystyle\int_{\\{\|x\|>1\\}}\|x\|^{k}\nu^{(0)}(dx)<\infty,$
then functions $\psi_{0}(u,s)$ and $\psi_{1}(u,s)$ from the representation (1)
are in $C^{k+1}(\mathbb{R}_{+})$ as functions of $s$. Moreover, for any fixed
$x\in\mathcal{D}$ and any $u\in\mathbb{R}^{d}$ the following estimates hold
(7)
$\displaystyle\sup_{s\in[0,T]}\left|\frac{\partial^{l}\phi(u|s,x)}{\partial
s^{l}}\right|\leq C\|u\|_{2}^{2l},\quad l=0,\ldots,k+1,$
where $C$ is a positive constant depending on $T$ and $x$.
## 3 Spectral estimation
If values of the log-return process $X(t)$ on a time grid
$\pi=\\{t_{0},t_{1},\ldots,t_{N}\\}$ are observed and $|\pi|$ is small (high-
frequency data) then a large increment $X(t_{i})-X(t_{i-1})$ indicates that a
jump occurred between time $t_{i-1}$ and $t_{i}$. Considering different
functions of increments and averaging over all increments, we can estimate
various integrals w.r.t. the underlying Lévy measure and hence various
characteristics of the jump component. Another possibility to estimate the
characteristics of jumps is to make use of the spectral representation (1) by
first estimating the c.f. $\phi(u|s,x)$ and then its derivative
$\partial_{s}\log(\phi(u|s,x))$ at the point $s=0.$ Finally, the
representation of Theorem 2.1 and some transformation allow one to eliminate
the diffusion part of $X$ and to estimate the Lévy measure $\nu^{(0)}.$
Suppose that for any fixed $x\in\mathcal{D}$ two sequences of estimates
$\\{\widehat{\phi}_{N}(u|s,x)\\}$ and $\\{\widehat{\phi}_{s,N}(u|s,x)\\}$ of
the conditional characteristic function $\phi(u|s,x)$ and its derivative
$\partial_{s}\phi(u|s,x)$ respectively, where $u\in\mathbb{R}^{d}$ and
$s\in\mathcal{S}:=[0,T]$ for some $T>0$, are constructed. At this stage it is
not important how they were obtained. We assume only that the estimates
$\widehat{\phi}_{N}(u|s,x)$ and $\widehat{\phi}_{s,N}(u|s,x)$ are uniformly
consistent in $u$, i.e. for any fixed $x\in\mathcal{D}$ and $s\in\mathcal{S}$
(8)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widehat{\phi}_{N}(u|s,x)-\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle O_{a.s.}(\zeta_{0,N}),$ (9)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widehat{\phi}_{s,N}(u|s,x)-\partial_{s}\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle O_{a.s.}(\zeta_{1,N}),$
where $w$ is a non-negative weighting function on $\mathbb{R}_{+}$,
$\|u\|=\max_{i=1,\ldots,d}|u_{i}|$ for any $u\in\mathbb{R}^{d}$ and
$\zeta_{N}:=\max\\{\zeta_{0,N},\zeta_{1,N}\\}\to 0$ as $N\to\infty$. In the
next sections we discuss how to construct such consistent estimates from a
time series data under random sampling using a smoothed it time version of
empirical characteristic function.
### 3.1 Algorithm
Our aim is to consistently estimate the Lévy measure $\nu^{(0)}$ using the
estimates $\widehat{\phi}_{N}(u|s,x)$ and $\widehat{\phi}_{s,N}(u|s,x)$. Note
that in this formulation other parameters of the affine process $X(t)$ are
viewed as nuisance components which are to be filtered out during estimation.
The main idea of estimation algorithm is as follows. Denote
$\psi(u|s,x):=\psi_{0}(u,s)+x^{\top}\psi_{1}(u,s)$ and introduce function
(10)
$\displaystyle\Psi(u):=\int_{[-1,1]^{d}}\left(\partial_{s}\psi(u|0,x)-\partial_{s}\psi(u+w|0,x)\right)\,dw$
that fulfills (due to (2) and (3))
$\displaystyle\Psi(u)=\frac{2^{d}}{3}\operatorname{tr}(\alpha^{(0)})+\frac{2^{d}}{3}\sum_{j=1}^{d}x_{j}\operatorname{tr}(\alpha^{(1)}_{j})+\int_{\mathcal{D}}e^{\mathfrak{i}z^{\top}u}\rho^{(0)}(dz),$
where
(11) $\displaystyle\rho^{(0)}(dz)$ $\displaystyle:=$ $\displaystyle
2^{d}\prod_{k=1}^{d}\left(1-\frac{\sin z_{k}}{z_{k}}\right)\nu^{(0)}(dz)$
is a finite measure. Function $\Psi(u)$ satisfies, due the Riemann-Lebesgue
theorem, the following asymptotic relation
$\displaystyle\lim_{\|u\|\to\infty}\Psi(u)$ $\displaystyle=$
$\displaystyle\frac{2^{d}}{3}\operatorname{tr}(\alpha^{(0)})+\frac{2^{d}}{3}\sum_{j=1}^{d}x_{j}\operatorname{tr}(\alpha^{(1)}_{j})=:\mathcal{L}.$
So, in order to reconstruct a transformed Lévy measure $\rho^{(0)}$ and hence
$\nu^{(0)}$ one can first estimate function $\Psi(u)$ then estimate its limit
as $\|u\|\to\infty$ and finally the Fourier transform of $\rho^{(0)}$ using
(10).
###### Remark 3.1.
Transformation (10) which transforms the Lévy measure $\nu^{(0)}$ to a finite
measure is not unique. Motivated by the results of Neumann and Reiß (2007) for
the case of one dimensional Lévy processes, one can consider the derivative
$\frac{\partial\psi(u|0,x)}{\partial u^{2}_{1}\ldots\partial u^{2}_{d}}$
instead. However, the latter type of transformation would require existence of
the second order moments of the process $X(t).$ Moreover, empirical results
indicate that integration in (10) may reduce the variance of an estimate for
$\partial\psi(u|0,x).$
###### Remark 3.2.
If we are interested in reconstructing only one particular component of $\nu$
then it is enough to construct the estimates for the corresponding marginal
conditional c.f. and its derivative in time. These estimates in turn can be
constructed using only a time series of the corresponding component of $X$
(see Section 6 for numerical illustration).
In the sequel we shall assume that the measure $\rho^{(0)}$ is absolutely
continuous w.r.t. Lebesgue measure on $\mathbb{R}^{d}$ and possesses bounded
density denoted (with some abuse of notations) by $\rho^{(0)}(x)$. In fact,
this means that the index of jump activity of the process $X(t)$ introduced in
Aït-Sahalia and Jacod (2008) for general semimartingales is smaller than $1,$
i.e.
$\displaystyle\max_{k=1,\ldots,d}\inf\left\\{r\geq
0:\int_{\\{|x_{1}|>1,\ldots,|x_{k-1}|>1,\,|x_{k}|\leq
1,\,|x_{k+1}|>1,\ldots,|x_{d}|>1\\}}|x_{k}|^{r}\nu^{(0)}(dx)<\infty\right\\}<1.$
Note that according to the admissibility conditions in Duffie, Filipović and
Schachermayer (2003) it must hold
$\displaystyle\max_{k=1,\ldots,m}\inf\left\\{r\geq
0:\int_{\\{|x_{1}|>1,\ldots,|x_{k-1}|>1,\,|x_{k}|\leq
1,\,|x_{k+1}|>1,\ldots,|x_{d}|>1\\}}|x_{k}|^{r}\nu^{(0)}(dx)<\infty\right\\}<1.$
So, our assumption puts an upper bound on the degree of activity of small
jumps for components $X_{m+1}(t),\ldots,X_{d}(t)$. Namely, we consider the
case of medium activity of small jumps. The case of highly active small jumps
can be treated in our framework as well. In this case, however, we can not any
longer use the Fourier inversion formulas for densities since the density
$\rho^{(0)}$ becomes unbounded. But we can still employ the Fourier inversion
formula for distributions instead. For the sake of the brevity we shall not
pursue this possibility in this paper. Let $\varrho_{0}(v)$ and
$\varrho_{1}(v)$ be any two functions satisfying
$\varrho_{0}(v)\leq\inf_{\|u\|=v}\inf_{s\in\mathcal{S}}|\phi(u|s,x)|,$
$\varrho_{1}(v)\geq\sup_{\|u\|=v}\sup_{s\in\mathcal{S}}|\partial_{s}\psi(u|s,x)|=\sup_{\|u\|=v}\sup_{s\in\mathcal{S}}|\partial_{s}\phi(u|s,x)/\phi(u|s,x)|$
with $v\in\mathbb{R}_{+}$. Functions $\varrho_{0}(v)$ and $\varrho_{1}(v)$ can
be found if some prior bounds on the eigenvalues of the matrix $\alpha^{(0)}$
are available. The following lemma can be used to construct $\varrho_{0}(v)$
and $\varrho_{1}(v)$.
###### Lemma 3.3.
Let $X$ be a regular affine process with admissible characteristics
$\chi=(\alpha,\beta,\gamma,\nu)$ such that (4) is fulfilled, then the
following estimates hold
$\displaystyle\|u\|_{2}^{-2}|\log(\phi(u|s,x))|$ $\displaystyle\lesssim$
$\displaystyle\lambda_{\max}(\mathfrak{A})\int_{0}^{s}\lambda^{2}_{\max}(e^{t\mathfrak{B}})\,dt,\quad\|u\|_{2}\to\infty,$
$\displaystyle\|u\|_{2}^{-2}|\partial_{s}\psi(u|s,x)|$ $\displaystyle\lesssim$
$\displaystyle\lambda_{\max}(\mathfrak{A})\lambda^{2}_{\max}(e^{s\mathfrak{B}}),\quad\|u\|_{2}\to\infty,$
where $\mathfrak{A}:=(\alpha^{(0)}_{ij})_{m+1\leq i,j\leq d}$,
$\mathfrak{B}:=(\beta^{(1)}_{i,j})_{m+1\leq i,j\leq d}$ and for any matrix $a$
$\lambda_{\max}(a)$ stands for the maximal eigenvalue of $a.$
Let us now formulate our estimation procedure. It will be, for reader
convenience, separated into several steps.
##### Step 1
Construct estimates for $\psi(u|s,x)$ and $\partial_{s}\psi(u|s,x)$ as follows
$\displaystyle\widehat{\psi}_{N}(u|s,x)$ $\displaystyle=$
$\displaystyle\log(T_{\varrho_{0}}[\widehat{\phi}_{N}](u|s,x)),$
$\displaystyle\widehat{\psi}_{s,N}(u|s,x)$ $\displaystyle=$ $\displaystyle
T_{\varrho_{1}}[\widehat{\phi}_{s,N}/\widehat{\phi}_{N}](u|s,x),$
where
$T_{\varrho_{0}}[\widehat{\phi}_{N}]=\begin{cases}1,&|\widehat{\phi}_{N}|>1,\\\
\widehat{\phi}_{N},&\varrho_{0}\leq|\widehat{\phi}_{N}|\leq 1,\\\
\varrho_{0}[\widehat{\phi}_{N}/|\widehat{\phi}_{N}|],&|\widehat{\phi}_{N}|<\varrho_{0}\end{cases}$
and
$T_{\varrho_{1}}[\widehat{\phi}_{s,N}/\widehat{\phi}_{N}]=\begin{cases}\widehat{\phi}_{s,N}/\widehat{\phi}_{N},&|\widehat{\phi}_{s,N}|\leq\varrho_{1}|\widehat{\phi}_{N}|,\\\
\varrho_{1}[\widehat{\phi}_{s,N}|\widehat{\phi}_{N}|/(\phi_{N}|\widehat{\phi}_{s,N}|)],&|\widehat{\phi}_{s,N}|>\varrho_{1}|\widehat{\phi}_{N}|.\end{cases}$
##### Step 2
Define
$\displaystyle\widehat{\Psi}_{N}(u)$ $\displaystyle:=$
$\displaystyle\int_{[-1,1]^{d}}\left(\widehat{\psi}_{s,N}(u|0,x)-\widehat{\psi}_{s,N}(u+w|0,x)\right)\,dw.$
Let $\mathcal{K}(u)$ be a non-negative function supported on $[-1,1]$ that
satisfies
$\int_{-1}^{1}\mathcal{K}(u)\,du=1.$
For any $U>0$ put
$\mathcal{K}^{U}(u)=U^{-1}\mathcal{K}(uU^{-1})$
and define an estimate for the limit $\mathcal{L}$ as
$\displaystyle\mathcal{L}_{U,N}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\widehat{\Psi}_{N}(u)\,du,$
##### Step 3
Reconstruct density $\rho^{(0)}(x)$ using the Fourier inversion formula
$\displaystyle\widetilde{\rho}^{(0)}_{N}(x;U)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{-U}^{U}\ldots\int_{-U}^{U}e^{-\mathfrak{i}u^{\top}x}\left[\widehat{\Psi}_{N}(u)-\mathcal{L}_{U,N}\right]\,du.$
In the next section we present uniform convergence rates for the estimate
$\widetilde{\rho}_{N}^{(0)}(x;U)$.
## 4 Theoretical properties
First, the following lemma shows that both estimates
$\widehat{\psi}_{N}(u|s,x)$ and $\widehat{\psi}_{s,N}(u|s,x)$ are uniformly
consistent
###### Lemma 4.1.
If estimates $\widehat{\phi}_{N}(u|s,x)$ and $\widehat{\phi}_{s,N}(u|s,x)$
fulfill (8)-(9), then it holds for any fixed $s\in\mathcal{S}$ and
$x\in\mathcal{D}$
(12)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w_{0}(\|u\|)|\widehat{\psi}_{N}(u|s,x)-\psi(u|s,x)|\right]=O_{a.s.}(\zeta_{N}),$
(13)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w_{1}(\|u\|)|\widehat{\psi}_{s,N}(u|s,x)-\partial_{s}\psi(u|s,x)|\right]=O_{a.s.}(\zeta_{N}),$
where $w_{0}(\|u\|)=\varrho_{0}(\|u\|)w(\|u\|)$ and
$w_{1}(\|u\|)=\varrho_{0}(\|u\|)(2^{-1}\varrho_{1}^{-1}(\|u\|)\wedge
1)w(\|u\|)$.
Next we prove minimax upper and lower risk bounds for the estimate
$\widetilde{\rho}^{(0)}_{N}(x;U).$
### 4.1 Upper risk bounds
For any $\Lambda>0$, $1\leq\varkappa<2$ and $R>0$ let
$\mathcal{A}(\Lambda,\varkappa,R)$ stand for a class of regular affine models
with admissible characteristics $\chi=(\alpha,\beta,\gamma,\nu)$ satisfying
(4) and
1. 1.
It holds for any $s\in\mathcal{S}$
$\displaystyle\max\left\\{\lambda_{\max}(\mathfrak{A})\int_{0}^{s}\lambda^{2}_{\max}(e^{t\mathfrak{B}})\,dt,\lambda_{\max}(\mathfrak{A})\lambda^{2}_{\max}(e^{s\mathfrak{B}})\right\\}\leq\Lambda,$
where matrices $\mathfrak{A}$ and $\mathfrak{B}$ are defined in Lemma 3.3.
2. 2.
For any $u\in\mathbb{R}^{d}$
(14)
$\displaystyle\left(\prod_{k=1}^{d}u_{k}\right)^{\varkappa}\mathcal{F}[\rho^{(0)}](u)\leq
R,$
where measure $\rho^{(0)}$ is related to the Lévy measure $\nu^{(0)}$ via
(11).
Here and in the sequel $\mathcal{F}[\rho](u)$ stands for the Fourier transform
of a measure $\rho$.
###### Remark 4.2.
It can be shown that if the Lévy measure $\nu^{(0)}$ satisfies
$\displaystyle\max_{k=1,\ldots,d}\inf\left\\{r\geq
0:\int_{\\{|x_{1}|>1,\ldots,|x_{k-1}|>1,\,|x_{k}|\leq
1,\,|x_{k+1}|>1,\ldots,|x_{d}|>1\\}}|x_{k}|^{r}\nu^{(0)}(dx)<\infty\right\\}<2-\varkappa,$
i.e. the degree of jump activity of the each component of the process $X(t)$
is less than $2-\varkappa$, then inequality (14) holds with some $R>0$.
Based on the above assumption and Lemma 4.1 the following proposition on a
uniform convergence of the estimate $\widetilde{\rho}^{(0)}_{N}(x;U)$ can be
proved.
###### Proposition 4.3.
Let $X(t)$ be a regular affine process with characteristics
$\chi=(\alpha,\beta,\lambda,\nu)\in\mathcal{A}(\Lambda,\varkappa,R)$ and with
a conditional characteristic function $\phi(u|s,x)$. Suppose that
(15) $\displaystyle\int_{-1}^{1}|u|^{-\varkappa}|\mathcal{K}(u)|\,du<\infty.$
If the sequences of estimates $\\{\widehat{\phi}_{N}(u|s,x)\\}$ and
$\\{\widehat{\phi}_{s,N}(u|s,x)\\}$ for $\phi(u|s,x)$ and
$\partial_{s}\phi(u|s,x)$ respectively are available and fulfill (8)-(9), then
for any large enough $U$ the estimate $\widetilde{\rho}^{(0)}_{N}(x;U)$
satisfies
(16)
$\displaystyle\sup_{x\in\mathcal{D}}|\rho^{(0)}(x)-\widetilde{\rho}^{(0)}_{N}(x;U)|$
$\displaystyle=$ $\displaystyle
O_{a.s.}\left(\zeta_{N}\,w^{-1}(U)U^{2+d}e^{\Lambda
U^{2}}+U^{-(\varkappa-1)}\right).$
#### Discussion
As will be shown in Section 5, one can construct estimates
$\widehat{\phi}_{N}(u|s,x)$ and $\widehat{\phi}_{s,N}(u|s,x)$ such that (8)
and (9) hold with $w(v)=\min\\{1,v^{-4}\\}$ and
$\zeta_{N}=\max\\{\zeta_{0,N},\zeta_{1,N}\\}=O(N^{-r}\log^{q}N)$ for some
$r>0$ and $q>0$. As a result the rates in Theorem 4.3 are logarithmic if
$\Lambda>0$. More precisely, setting
$U_{N}:=\Lambda^{-1/2}\sqrt{r\log
N-\left(\frac{(\varkappa-1)}{2}+3+d/2+q\right)\log\log N},$
we get as $N\to\infty$
(17)
$\displaystyle\sup_{x\in\mathcal{D}}|\rho^{(0)}(x)-\widetilde{\rho}^{(0)}_{N}(x,U_{N})|$
$\displaystyle=$ $\displaystyle O_{a.s}\left((\log
N)^{-(\varkappa-1)/2}\right).$
Several observations can be made from the inspection of these rates. First,
the convergence rates are logarithmic and correspond to a severely ill-posed
problem. The reason for the severe ill-posedness is that we face a
deconvolution like problem: the law of the continuous part of $X(T)$ is
convolved (in generalized sense) with that of the jump part to give the
distribution of $X(T)$. Second, the nearer is $\varkappa$ to $1$ the more
complex becomes the estimation problem. Since the degree of jump activity is
equal to $2-\varkappa$ this means that estimating the characteristics of jumps
becomes more difficult as the degree of jump activity increases to $1$.
### 4.2 Lower risk bounds
The rates in (17) are in fact optimal in the minimax sense for the class
$\mathcal{A}(\Lambda,\varkappa,R)$.
###### Proposition 4.4.
The following minimax lower risk bounds hold
$\displaystyle\liminf_{N\to\infty}\inf_{\widetilde{\rho}^{(0)}_{N}}\sup_{\chi\in\mathcal{A}(\Lambda,\varkappa,R)}\operatorname{P}_{\chi}\left((\log
N)^{(\varkappa-1)/2}\sup_{x\in\mathcal{D}}|\rho^{(0)}(x)-\widetilde{\rho}^{(0)}_{N}(x)|>\varepsilon\right)>0,$
where $\varepsilon$ is some positive number and $\widetilde{\rho}^{(0)}_{N}$
is any estimator of $\rho^{(0)}$ based on $N$ observations.
## 5 Nonparametric estimation of $\phi$ and $\partial_{s}\phi$
In this section we present a method for estimating the conditional
characteristic function $\phi(u|s,x)$ and its derivatives in time from a time
series of $X(t)$. For our theoretical study we adopt the so called random
design observational model, i.e. we assume that for some $x\in\mathcal{D}$ a
trajectory of the process $X$ containing pairs
$\displaystyle(X(t_{n}),X(t_{n}+\delta_{n})),\quad X(t_{n})=x,\quad
n=1,\ldots,N,$
is available, where $\delta_{n},\,n=1,\ldots,N$ are i.i.d. random variables on
$[0,T]$ for some fixed $T>0$ with a common bounded density $p_{\delta}(x)$ and
$\min_{n}(t_{n+1}-t_{n})\geq 2T$. This assumption implies that the time
horizon $T+t_{N}$ of observations tends to $\infty$ as $N\to\infty,$ a
condition which is not unusual even in the literature on statistical inference
from high-frequency data (see, e.g. Figueroa-López (2009)). In Figure 1 a
typical trajectory of a one-dimensional Ornstein-Uhlenbeck process (without
jump component) is shown together with the level line $x=0.$
Figure 1: A typical path of the one-dimensional Ornstein-Uhlenbeck process
$dX(t)=-5X(t)\,dt+3.5\,dW(t)$ along with the line $x=0$.
We estimate $\phi_{N}(u|s,x)$ and $\partial_{s}\phi_{N}(u|s,x)$ by a local
linear smoothing of empirical characteristic process. Define
(18) $\displaystyle\widehat{\phi}_{N}(u|s,x)$ $\displaystyle:=$
$\displaystyle\sum_{n=1}^{N}\tau_{0,n}(s)\exp(\mathfrak{i}u^{\top}X(t_{n}+\delta_{n})),$
(19) $\displaystyle\widehat{\phi}_{s,N}(u|s,x)$ $\displaystyle:=$
$\displaystyle\sum_{n=1}^{N}\tau_{1,n}(s)\exp(\mathfrak{i}u^{\top}X(t_{n}+\delta_{n})),$
where
$\displaystyle\tau_{j,n}(s)=\frac{b_{j,n}(s)}{\sum_{k=1}^{N}b_{0,k}(s)},\quad
j=0,1,$ $\displaystyle b_{0,n}(s)$ $\displaystyle=$ $\displaystyle
K\left(\frac{\delta_{n}-s}{h}\right)(S_{N,2}-(\delta_{n}-s)S_{N,1}),$
$\displaystyle b_{1,n}(s)$ $\displaystyle=$ $\displaystyle
K\left(\frac{\delta_{n}-s}{h}\right)((\delta_{n}-s)S_{N,0}-S_{N,1})$
and
$\displaystyle
S_{N,j}=\sum_{n=1}^{N}K\left(\frac{\delta_{n}-s}{h}\right)(\delta_{n}-s)^{j},\quad
j=0,1,2.$
Here $K$ is a kernel and $h$ is a bandwidth. Denote
$\Gamma(s)=N^{-1}\begin{pmatrix}h^{-1}S_{N,0}&h^{-2}S_{N,1}\\\
h^{-2}S_{N,1}&h^{-3}S_{N,2}\end{pmatrix},\quad\bar{\Gamma}(s)=\begin{pmatrix}\mu_{0}(s)&\mu_{1}(s)\\\
\mu_{1}(s)&\mu_{2}(s)\end{pmatrix}$
with $\mu_{l}(s)=\int_{\mathbb{R}}z^{l}K(z)p_{\delta}(s+hz)\,dz.$ It easy to
show that if matrix $\Gamma$ is invertible then it holds
$(\widehat{\phi}_{N},h\widehat{\phi}_{s,N})=\Gamma^{-1}Y$, where $Y$ is a two-
dimensional vector with components
$\displaystyle
Y_{k}=\frac{1}{Nh}\sum_{n=1}^{N}\exp(\mathfrak{i}u^{\top}X(t_{n}+\delta_{n}))\left(\frac{\delta_{n}-s}{h}\right)^{k}K\left(\frac{\delta_{n}-s}{h}\right),\quad
k=0,1.$
In order to be able to prove the convergence of estimates
$\widehat{\phi}_{N}(u|s,x)$ and $\widehat{\phi}_{s,N}(u|s,x)$ we need the
following assumptions
##### Assumptions
(AX1)
The sequence $\bar{X}_{n}:=X(t_{n}+\delta_{n})-X(t_{n})$ is exponentially
strongly mixing, i.e. the mixing coefficients $\alpha_{\bar{X}}$ (see Appendix
for definition) satisfy
$\displaystyle\alpha_{\bar{X}}(n)\leq\bar{\alpha}\exp(-cn),\quad n\geq 1,$
for some $\bar{\alpha}>0$ and $c>0.$
(AX2)
The Lévy measure $\nu^{(0)}$ satisfies for some $p>2$
$\displaystyle\int_{\\{\|x\|>1\\}}\|x\|^{p}\nu^{(0)}(dx)<\infty.$
(AS)
The minimal eigenvalue of the matrix $\bar{\Gamma}(s)$ is bounded away from
below by some positive constant $\gamma_{0}$ uniformly in $s,$ i.e.
$\displaystyle\min_{s\in\mathcal{S}}\lambda_{\min}(\bar{\Gamma}(s))\geq\gamma_{0}>0.$
(AK)
Kernel $K$ is bounded and has a support on $[-1,1]$.
###### Remark 5.1.
Exponentially strongly mixing holds for a wide class of Itô-Lévy processes. In
Masuda (2007) conditions are formulated that ensure that a multidimensional
Itô-Lévy process is exponentially $\beta$-mixing (hence exponentially
$\alpha$-mixing). Suppose that $X$ is a regular affine process with
characteristics $\chi$ satisfying admissibility conditions, the condition (4)
and $\int_{|x|>1}|x|^{q}\nu^{(0)}(dx)<\infty$ for some $q>0$. Moreover, assume
that
$\beta_{k,i}^{(1)}=0,\quad\alpha^{(1)}_{kl,i}=0,\quad
i,k\in\bar{\mathcal{I}}:=\\{1,\ldots,d\\}\setminus\mathcal{I},$
where $\mathcal{I}$ is a subset of $\\{1,\ldots,d\\}$. This situation is
typical for stochastic volatility models. If the maximal eigenvalue of the
matrix $(\beta_{i,j}^{(1)})_{i,j\in\mathcal{I}}$ is negative then both
sequences $X_{\mathcal{I}}(t_{n}+\delta_{n})$ and
$X_{\bar{\mathcal{I}}}(t_{n}+\delta_{n})-X_{\bar{\mathcal{I}}}(t_{n})$ are
exponentially $\beta$-mixing and ergodic. Taking the initial distribution for
$X_{\mathcal{I}}(0)$ to be the invariant one, both sequences
$X_{\mathcal{I}}(t_{n}+\delta_{n})$ and
$X_{\bar{\mathcal{I}}}(t_{n}+\delta_{n})-X_{\bar{\mathcal{I}}}(t_{n})$ become
stationary.
###### Remark 5.2.
Assumption (AS) imposes some regularity conditions on the design
$\\{\delta_{n}\\}_{n=1}^{N}.$ In particular, it ensures that points arbitrary
close to $0$ appear in the sample with positive probability as $N\to\infty$.
This in turn allows us to consistently estimate the derivative
$\partial_{s}\phi(u|s,x)|_{s=0}.$
### 5.1 Asymptotic properties of $\widehat{\phi}$ and $\widehat{\phi}_{s}$.
In this section we investigate the asymptotic properties of estimates
$\widehat{\phi}_{N}$ and $\widehat{\phi}_{s,N}$. Consider truncated versions
$\widetilde{\phi}_{N}(u)$ and $\widetilde{\phi}_{s,N}(u)$ of estimates
$\widehat{\phi}_{N}(u)$ and $\widehat{\phi}_{s,N}(u)$ respectively which are
defined as follows. If the smallest eigenvalue of the matrix $\Gamma$ is
greater than $\gamma_{0}/2$ we set
$\widetilde{\phi}_{N}(u)=\widehat{\phi}_{N}(u)$ and
$\widetilde{\phi}_{s,N}(u)=\widehat{\phi}_{s,N}(u).$ Otherwise, we put
$\widetilde{\phi}_{N}(u)=\widetilde{\phi}_{s,N}(u)=0$. The following
proposition shows that both estimates $\widehat{\phi}_{N}$ and
$\widehat{\phi}_{s,N}$ are uniformly consistent in a weighted sup-norm.
###### Proposition 5.3.
Suppose that assumptions (AX1), (AX2), (AS) and (AK) are fulfilled and the
bandwidth $h_{N}$ satisfies $h_{N}^{-1}=o(N)$. Let $w$ be a positive Lipschitz
function on $\mathbb{R}_{+}$ such that
(20) $0<w(z)\leq\min\\{1,z^{-4}\\},\quad z\in\mathbb{R}_{+}.$
Then for any fixed $s\in\mathcal{S}$ and $x\in\mathcal{D}$
(21)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widetilde{\phi}_{N}(u|s,x)-\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle
O_{a.s.}\left(\sqrt{\frac{\log^{(1+r)}N}{Nh_{N}}}+h_{N}^{2}\right),$ (22)
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widetilde{\phi}_{s,N}(u|s,x)-\partial_{s}\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle
O_{a.s.}\left(\sqrt{\frac{\log^{(1+r)}N}{Nh_{N}^{3}}}+h_{N}\right)$
with some $r>0.$
##### Discussion
The right-hand sides of (21) and (22) are sums of two terms corresponding to
“variance” and “bias” respectively. The optimal choice of the bandwidth
parameter $h_{N}$ which minimizes (21) and (22) is given by
$\displaystyle h_{N}=\left[N^{-1}\log^{(1+r)}N\right]^{1/5}$
in which case uniform convergence rates are given by
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widetilde{\phi}_{N}(u|s,x)-\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle
O_{a.s.}\left(\left(N^{-1}\log^{(1+r)}N\right)^{2/5}\right),$
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\widetilde{\phi}_{s,N}(u|s,x)-\partial_{s}\phi(u|s,x)|\right]$
$\displaystyle=$ $\displaystyle
O_{a.s.}\left(\left(N^{-1}\log^{(1+r)}N\right)^{1/5}\right).$
###### Remark 5.4.
The condition (20) on the decay of weighting function $w$ can not be, in
general, weakened. For example, in the case of a one-dimensional Brownian
motion with volatility $\sigma^{2}$, the simplest affine process, we get
$\partial_{ss}\phi(u|s,x)|_{s=0}=\sigma^{4}u^{4}/4.$ This means that
approximation errors of local linear estimates in (18) and (19) at the point
$0$ are of order $h^{2}\sigma^{4}u^{4}/8$ and $h\sigma^{4}u^{4}/8$
respectively. So in order to be able to prove uniform consistency in $u$ we
have to assume (20).
## 6 Numerical example
Let us consider a class of stochastic volatility models of the type
(23) $\displaystyle dX(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}V(t)\,dt+\sqrt{V(t)}dW^{S}(t)+dZ_{t},$
$\displaystyle dV(t)$ $\displaystyle=$
$\displaystyle\lambda(\theta-V(t))dt+\zeta\sqrt{V(t)}dW^{V}(t),$
where $X(t)=\log(S(t))$ is the log-price process, $W^{S}$ and $W^{V}$ are two
independent Brownian motions, $\lambda,\theta,\zeta$ are positive constants
and $Z_{t}$ is a pure-jump Lévy process with Lévy density $\nu(x).$ This is a
special type of the model introduced in Bates (2005). In our numerical example
we take $Z(t)$ to be $\alpha$-stable Lévy process with stability index
$\alpha<1$, i.e.
$\displaystyle\nu(x)=C/|x|^{1+\alpha},\quad\rho(x)=2\left(1-\frac{\sin
x}{x}\right)\nu(x)$
for some constant $C>0.$ For the sake of simplicity we consider a fixed design
and simulate a set of i.i.d pairs
(24) $(X^{(n)}(0),X^{(n)}(\Delta)),\quad n=1,\ldots,N$
with some fixed $\Delta>0$, where
$(X^{(1)}(0),V^{(1)}(0))=\ldots=(X^{(N)}(0),V^{(N)}(0)).$
Without loss of generality we set $(X^{(1)}(0),V^{(1)}(0))=(0,1).$ Our aim is
to reconstruct $\rho$ using sample (24). First compute
$\displaystyle\widehat{\psi}_{s,N}(u|0)=\widehat{\phi}_{s,N}(u|0)$
$\displaystyle:=$
$\displaystyle\frac{1}{N\Delta}\sum_{n=1}^{N}\left[\exp\left(\mathfrak{i}u^{\top}X^{(n)}(\Delta)\right)-\exp\left(\mathfrak{i}u^{\top}X^{(n)}(0)\right)\right]$
and
$\displaystyle\widehat{\Psi}_{N}(u)$ $\displaystyle:=$
$\displaystyle\int_{-1}^{1}\left(\widehat{\psi}_{s,N}(u|0)-\widehat{\psi}_{s,N}(u+w|0)\right)\,dw.$
###### Remark 6.1.
In the case of high-frequency data observations are usually available for
different frequency scales $\Delta$ and the choice of an appropriate frequency
for estimation procedure should be done depending on $N$, the number of points
available for the given frequency scale. If $\Delta$ is too small then the
variance of $\widehat{\phi}_{s,N}(u|0)$ explodes. On the other hand, if
$\Delta$ is too large than the approximation error of
$\widehat{\phi}_{s,N}(u|0)$ becomes large.
Next define a parametric family of functions
$\displaystyle\widetilde{\rho}^{(0)}_{N}(x;U)$ $\displaystyle=$
$\displaystyle\operatorname{Re}\left\\{\frac{1}{2\pi}\int_{-U}^{U}e^{-\mathfrak{i}ux}\left[\widehat{\Psi}_{N}(u)-\Psi_{N}(U)\right]\,du\right\\},\quad
U>0$
and find $U$ by solving the following minimization problem
$\displaystyle\widehat{U}=\operatorname{arginf}_{U}\left\\{\int_{\\{|u|>U\\}}\left|\widehat{\Psi}_{N}(u)-\widehat{\Psi}_{N}(U)\right|^{2}\,du+\pi\int|\partial_{xx}\widetilde{\rho}^{(0)}_{N}(x;U)|\,dx\right\\},$
where $\pi>0$ is a regularization parameter. In fact, this approach for
choosing $U$ employs additional information about smoothness of $\rho$ and
turns out to be rather efficient in practice. In Figure 2 typical results of
estimation based on $N=1000$ samples (24) with $\Delta=0.1$ are shown for two
specifications of the process $Z_{t}.$
Figure 2: Typical estimates for transformed Lévy density $\rho$ (dashed black
line) together with the true $\rho$ (solid red line) in the Bates stochastic
volatility model with symmetric stable process $Z_{t}$ and different stability
indexes $\alpha$.
As can be seen the overall quality of estimation is good taking into account
severely ill-posedness of the underlying estimation problem. However, the
behavior of the transformed Lévy density $\rho$ at zero is not captured by
estimation method. In order to correct $\rho(x)$ at $x=0$ we separately
estimate the stability index $\alpha$ using a modification of the spectral
algorithm proposed in Belomestny (2009) for Lévy processes. Motivated by
relations (2) and (3), we define for any $a\in(0,1)$
(25)
$\displaystyle\mathcal{O}(a):=\min_{(l_{0},l_{1},l_{2},l_{3})}\int_{0}^{\widehat{U}}(\widehat{\psi}_{s,N}(u|0)-l_{3}u^{a}-l_{2}u^{2}-l_{1}u-l_{0})^{2}\,du$
and estimate $\alpha$ via
$\widetilde{\alpha}:=\operatorname{argmin}_{a\in(0,1)}\mathcal{O}(a).$ In
Figure 3 functions $\mathcal{O}(a)$ based on the same samples as in Figure 2
are shown.
Figure 3: Function $\mathcal{O}(a)$ in the case of symmetric stable process
$Z_{t}$ with stability indexes $0.5$ (left) and $0.8$ (right) respectively.
The resulting estimates for $\alpha$ are $\widetilde{\alpha}=0.451$ and
$\widetilde{\alpha}=0.783$ respectively. Now we correct estimate
$\widetilde{\rho}^{(0)}_{N}(x;\widehat{U})$ by setting
$\displaystyle\widehat{\rho}^{(0)}_{N}(x;\widehat{U})=\begin{cases}c(\varepsilon)\left(1-\frac{\sin
x}{x}\right)|x|^{-(1+\widetilde{\alpha})},&|x|\leq\varepsilon,\\\
\widetilde{\rho}^{(0)}_{N}(x;\widehat{U}),&|x|>\varepsilon.\end{cases}$
where for any $\varepsilon>0$ the constant $c(\varepsilon)$ is chosen in such
a way that function $\widehat{\rho}^{(0)}_{N}(x;\widehat{U})$ is continuous.
Finally, we find small enough $\varepsilon>0$ which minimizes the integral
$\int|\partial_{xx}\widehat{\rho}^{(0)}_{N}(x;\widehat{U})|\,dx$. Here again
the smoothness of $\rho$ is used. A corrected estimate
$\widehat{\rho}^{(0)}_{N}(x;\widehat{U})$ is shown in Figure 4.
Figure 4: Corrected estimates for transformed Lévy density $\rho$ (dashed
black line) together with the true $\rho$ (solid red line) in the Bates
stochastic volatility model with symmetric stable process $Z_{t}$ and
different stability indexes $\alpha$.
## 7 Proofs
### 7.1 Proof of Lemma 2.2
Due to (6) all derivatives of functions $F_{0}(z)$ and $F_{1}(z)$ up to order
$k$ exist for
$\operatorname{Re}z\in\mathbb{R}^{m}_{-}\times\\{0\\}\times\ldots\times\\{0\\}.$
Fix some $s\in\mathcal{S}$ and $x\in\mathcal{D}$. Using (2) and (3), we get
for $1\leq l\leq k+1$
(26) $\frac{\partial^{l}\phi(u|s,x)}{\partial
s^{l}}=H\left(F_{0}(\psi_{1}(u,s)),F_{0}^{(1)}(\psi_{1}(u,s)),\ldots,F_{0}^{(l-1)}(\psi_{1}(u,s)),\right.\\\
\left.F_{1}(\psi_{1}(u,s)),F_{1}^{(1)}(\psi_{1}(u,s)),\ldots,F_{1}^{(l-1)}(\psi_{1}(u,s))\right)\phi(u|s,x),$
where $H$ is a polynomial of order $l$ in $\mathbb{R}^{2l}.$ Note that under
admissibility conditions and (4) functions $F_{0}$ and $F_{1}$ simplify to
$\displaystyle F_{0}(z)$ $\displaystyle=$
$\displaystyle\sum_{i,j=m+1}^{d}\alpha_{ij}^{(0)}z_{i}{z}_{j}+(z,\beta^{(0)})-\gamma^{(0)}+\int_{\mathcal{D}}\left(e^{z^{\top}u}-1-(\chi(u),z)\right)\,\nu^{(0)}(du)$
$\displaystyle F_{1,j}(z)$ $\displaystyle=$
$\displaystyle(\alpha^{(1)}_{j}z,z)+\sum_{i=1}^{m}z_{i}\beta_{j,i}^{(1)}-\gamma_{j}^{(1)},\quad
j=1,\ldots,m,$ $\displaystyle F_{1,j}(z)$ $\displaystyle=$
$\displaystyle\sum_{i=m+1}^{d}z_{i}\beta_{j,i}^{(1)},\quad j=m+1,\ldots,d.$
Solving the system of linear equations
$\displaystyle\frac{\partial\psi_{1,j}(u,s)}{\partial
s}=F_{1,j}(\psi_{1}(u,s)),\quad\psi_{1,j}(u,s)=\mathfrak{i}u_{j},\quad
j=m+1,\ldots,d,$
we get
$(\psi_{1,m+1}(u,s),\ldots,\psi_{1,d}(u,s))=e^{s\mathfrak{B}}(\mathfrak{i}u_{m+1},\ldots,\mathfrak{i}u_{d})^{\top}$.
Furthermore, it follows from the theory of Riccati ODE that the solution of a
system
$\displaystyle\frac{\partial\psi_{1,j}(u,s)}{\partial
s}=F_{1,j}(\psi_{1}(u,s)),\quad\psi_{1,j}(u,s)=\mathfrak{i}u_{j}$
can have at most linear growth in $u.$ Since
$\displaystyle\max\\{\|F_{1}^{(l)}(z)\|,|F_{0}^{(l)}(z)|\\}\lesssim\|z\|_{2}^{2},\quad\|z\|_{2}\to\infty,\quad
l=0,\ldots,k$
it holds
$\displaystyle\max\\{\|F_{1}^{(l)}(\psi_{1}(u,s))\|,|F_{0}^{(l)}(\psi_{1}(u,s))|\\}\lesssim\|\psi_{1}(u,s)\|_{2}^{2}\lesssim\|u\|_{2}^{2},\quad\|u\|_{2}\to\infty,\quad
l=0,\ldots,k+1.$
Combining the last inequality with (26), we get (7).
### 7.2 Proof of Proposition 3.3
The analysis of the system of Riccati ODEs (2) and (3) performed in the proof
of Lemma 2.2 leads to bounds
$\displaystyle\operatorname{Re}[\psi_{0}(u,s)]$ $\displaystyle=$
$\displaystyle\int_{0}^{s}\operatorname{Re}[F_{0}(\psi_{1}(u,t))]\,dt\gtrsim-\|u\|_{2}^{2}\,\lambda^{2}_{\max}(\mathfrak{A})\int_{0}^{s}\lambda_{\max}(e^{t\mathfrak{B}})\,dt,$
$\displaystyle\operatorname{Re}[\psi_{1}(u,s)]$ $\displaystyle=$
$\displaystyle o\left(\|u\|^{2}_{2}\right),\quad\|u\|_{2}\to\infty.$
### 7.3 Proof of Lemma 4.1
We prove only a more involved bound (13). Since
$\displaystyle\partial_{s}\phi-\widehat{\psi}_{s,N}\widehat{\phi}_{N}$
$\displaystyle=$
$\displaystyle[\partial_{s}\psi]\phi-\widehat{\psi}_{s,N}\widehat{\phi}_{N}$
$\displaystyle=$
$\displaystyle(\partial_{s}\psi-\widehat{\psi}_{s,N})\phi+\widehat{\psi}_{s,N}(\phi-\widehat{\phi}_{N}),$
we have
$\displaystyle\partial_{s}\psi-\widehat{\psi}_{s,N}=\phi^{-1}\left[(\partial_{s}\phi-\widehat{\psi}_{s,N}\widehat{\phi}_{N})-\widehat{\psi}_{s,N}(\phi-\widehat{\phi}_{N})\right]$
and hence
(27)
$\displaystyle|\partial_{s}\psi-\widehat{\psi}_{s,N}|\leq\varrho_{0}^{-1}\left[|\partial_{s}\phi-\widehat{\psi}_{s,N}\widehat{\phi}_{N}|+\varrho_{1}|\phi-\widehat{\phi}_{N}|\right].$
Furthermore, it holds
(28)
$\displaystyle|\widehat{\phi}_{s,N}-\widehat{\psi}_{s,N}\widehat{\phi}_{N}|$
$\displaystyle\leq$
$\displaystyle\left|\widehat{\phi}_{s,N}-\varrho_{1}|\widehat{\phi}_{N}|\frac{\widehat{\phi}_{s,N}}{|\widehat{\phi}_{s,N}|}\right|\mathbf{1}(|\widehat{\phi}_{s,N}|/|\widehat{\phi}_{N}|>\varrho_{1})$
$\displaystyle=$
$\displaystyle\left||\widehat{\phi}_{s,N}|-\varrho_{1}|\widehat{\phi}_{N}|\right|\mathbf{1}(|\widehat{\phi}_{s,N}|/|\widehat{\phi}_{N}|>\varrho_{1})$
$\displaystyle\leq$
$\displaystyle\left||\widehat{\phi}_{s,N}|-\frac{|\partial_{s}\phi|}{|\phi|}|\widehat{\phi}_{N}|\right|$
$\displaystyle\leq$
$\displaystyle|\partial_{s}\psi||\phi-\widehat{\phi}_{N}|+|\widehat{\phi}_{s,N}-\partial_{s}\phi|.$
Combining (27) and (28), we get
$\displaystyle|\partial_{s}\psi-\widehat{\psi}_{s,N}|\leq\varrho_{0}^{-1}(u)[2\varrho_{1}(u)|\phi-\widehat{\phi}_{N}|+|\widehat{\phi}_{s,N}-\partial_{s}\phi|]$
Now the assumptions (8) and (9) imply
(29)
$\sup_{x\in\mathcal{X}}\sup_{u\in\mathbb{R}^{d}}\left[w_{1}(\|u\|)|\widehat{\phi}_{s,N}(u|s,x)-\partial_{s}\phi(u|s,x)|\right]=O_{a.s.}(\zeta_{N}).$
with $w_{1}(u)=\varrho_{0}(u)(2^{-1}\varrho^{-1}_{1}(u)\wedge 1)w(u)$.
### 7.4 Proof of Proposition 4.3
Lemma 4.1 implies
$\displaystyle\sup_{u\in\mathbb{R}^{d}}\left[w_{1}(\|u\|)|\widehat{\Psi}_{N}(u)-\Psi(u)|\right]$
$\displaystyle=$ $\displaystyle O_{a.s}(\zeta_{N}).$
Since
(30) $\displaystyle\mathcal{L}_{U,N}-\mathcal{L}_{0}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\left(\widehat{\Psi}_{N}(u)-\Psi(u)\right)\,du$
$\displaystyle+\int_{\mathbb{R}^{d}}\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\mathcal{F}[\rho^{(0)}](u)\,du$
and $w_{1}(z)$ is monotone decreasing, we have for any $U>0$
$|\mathcal{L}_{U,N}-\mathcal{L}|\leq
w_{1}^{-1}(U)\int_{\mathbb{R}^{d}}\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\\\
\times\left[w_{1}(\|u\|)\left|\widehat{\Psi}_{N}(u)-\Psi(u)\right|\right]\,du\\\
+\int_{\mathbb{R}^{d}}\left|\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\mathcal{F}[\rho^{(0)}](u)\right|\,du.$
Then conditions (14) and (15) imply
(31)
$\int_{\mathbb{R}^{d}}\left|\left[\mathcal{K}^{U}(u_{1})\times\ldots\times\mathcal{K}^{U}(u_{d})\right]\mathcal{F}[\rho^{(0)}](u)\right|\,du\leq\\\
R\int_{\mathbb{R}^{d}}|u_{1}|^{-\varkappa}|\mathcal{K}^{U}(u_{1})|\times\ldots\times|u_{d}|^{-\varkappa}|\mathcal{K}^{U}(u_{d})|\,du\leq
CU^{-d\varkappa}$
with some constant $C>0$. Combining (31) with (30), we get
$\displaystyle|\mathcal{L}_{U,N}-\mathcal{L}|=O_{a.s}\left(w_{1}^{-1}(U)\zeta_{N}+U^{-d\varkappa}\right).$
Furthermore, using the Fourier inversion formula, we get
$\displaystyle\sup_{x\in\mathcal{D}}|\rho^{(0)}(x)-\widetilde{\rho}^{(0)}_{N}(x;U)|$
$\displaystyle\lesssim$ $\displaystyle U^{d}\left[\sup_{\|u\|\leq
U}\left|\widehat{\Psi}_{N}(u)-\Psi(u)\right|+|\mathcal{L}_{U,N}-\mathcal{L}|\right]$
$\displaystyle+\left|\int_{\\{\|u\|>U\\}}e^{-\mathfrak{i}u^{\top}x}\mathcal{F}[\rho^{(0)}](u)\,du\right|$
$\displaystyle\lesssim$ $\displaystyle
U^{d}\left[w_{1}^{-1}(\|U\|)\sup_{\|u\|\leq
U}\left[w_{1}(\|u\|)\left|\widehat{\Psi}_{N}(u)-\Psi(u)\right|\right]+|\mathcal{L}_{U,N}-\mathcal{L}|\right]$
$\displaystyle+U^{-(\varkappa-1)},\quad U\to\infty.$
Recalling definition of $w_{1}$ (see Lemma 4.1) and using Lemma 2.2, we get
(16).
### 7.5 Proof of Proposition 4.4
In order to prove minimax lower bounds we apply general results from Tsybakov
(2008). Let $\Theta$ be a semi-parametric class of models. Consider a family
$\\{P_{\theta},\theta\in\Theta\\}$ of probability measures, indexed by
$\Theta$. For any $\theta_{1},\,\theta_{2}\in\Theta$ let
$d(\theta_{1},\theta_{2})$ be a semi-distance between two models $\theta_{1}$
and $\theta_{2}.$
###### Lemma 7.1.
Suppose that $\Theta$ contains two elements $\theta_{1}$ and $\theta_{2}$ such
that $d(\theta_{1},\theta_{2})>2s$ for some $s>0$ and $\chi^{2}(P^{\otimes
N}_{\theta_{1}},P^{\otimes N}_{\theta_{2}})\leq\tau<1/2,$ where
$\chi^{2}(P,Q)=:\begin{cases}\int\left(\frac{dP}{dQ}-1\right)^{2}\,dQ,&\mbox{
if }P\ll Q\\\ +\infty,&\mbox{otherwise}\end{cases}$
for any two measures $P$ and $Q$. Then
$\displaystyle\inf_{\widehat{\theta}}\sup_{\theta\in\Theta}\operatorname{P}_{\theta}(d(\widehat{\theta},\theta)\geq
s)\geq c(\tau)>0,$
where $c(\tau)$ is constant depending on $\tau$ and infimum is taken over all
estimates $\widehat{\theta}$ of $\theta$ based on $N$ observations under
$P_{\theta}$.
Turn now to the construction of models $\theta_{1}$ and $\theta_{2}$ from the
class $\mathcal{A}(\Lambda,\varkappa,R)$. Let us consider a symmetric stable
Lévy model with a nonzero diffusion part ($\sigma>0$)
$\psi(u)=\mathfrak{i}\mu
u-\sigma^{2}u^{2}/2+\vartheta(u),\quad\vartheta(u)=-\eta|u|^{\alpha},\quad
0<\alpha\leq 1,\quad u\in\mathbb{R}.$
For any $\delta$ satisfying $0<\delta<\alpha$ and $M>0$ define
$\psi_{\delta}(u)=\mathfrak{i}\mu u-\sigma^{2}u^{2}/2+\vartheta_{\delta}(u),$
with
$\vartheta_{\delta}(u)=-\eta|u|^{\alpha}\mathbf{1}_{\\{|u|\leq M\\}}-\eta
M^{\delta}|u|^{\alpha-\delta}\mathbf{1}_{\\{|u|>M\\}}.$
Then $\phi_{\delta}(u)=\exp(\psi_{\delta}(u))$ is a characteristic function of
some Lévy process and
$\phi_{\delta}(u)=\phi(u),\quad|u|\leq M,$
where $\phi(u)=\exp(\psi(u))$. Indeed, $\vartheta_{\delta}(u)$ is continuous,
non-positive, symmetric function which is convex on $\mathbb{R}_{+}$ for large
enough $M$. According to the well known Pólya criteria (see e.g. Ushakov
(1999)), the function $\exp(\xi\vartheta_{\delta}(u))$ is the c. f. of some
absolutely continuous distribution for any $\xi>0$. In particular, for any
natural $n$ the function $\exp(\vartheta_{\delta}(u)/n)$ is the c. f. of some
absolutely continuous distribution. Hence, $\exp(\vartheta_{\delta}(u))$ is
the c.f. of some infinitely divisible distribution. Define now two affine (in
fact, Lévy) models $\theta_{1}$ and $\theta_{2}$ corresponding to the
characteristic exponents $\psi$ and $\psi_{\delta}$ respectively. Let
$\nu_{\theta_{1}}$ and $\nu_{\theta_{2}}$ be the corresponding Lévy measures.
It holds
$\displaystyle\chi^{2}(P^{\otimes N}_{\theta_{1}},P^{\otimes
N}_{\theta_{2}})=N\chi^{2}(p_{\theta_{1}},p_{\theta_{2}})$ $\displaystyle=$
$\displaystyle
N\int_{\mathbb{R}}\frac{|p_{\theta_{1}}(y)-p_{\theta_{2}}(y)|^{2}}{p_{\theta_{1}}(y)}\,dy,$
where $p_{\theta_{1}}$ and $p_{\theta_{2}}$ are densities corresponding to
c.f. $\phi_{\theta_{1}}$ and $\phi_{\theta_{2}}$ respectively. Using the
asymptotic inequality
$p_{\theta_{1}}(y)\gtrsim|y|^{-(\alpha+1)},\quad|y|\to\infty$
and the fact that the density of stable law does not vanish on any compact set
in $\mathbb{R}$, we derive
$\displaystyle N\chi^{2}(p_{\theta_{1}},p_{\theta_{2}})$ $\displaystyle\leq$
$\displaystyle NC_{1}\int_{|y|\leq
A}|p_{\theta_{1}}(y)-p_{\theta_{2}}(y)|^{2}\,dy$
$\displaystyle+NC_{2}\int_{|y|>A}|y|^{\alpha+1}|p_{\theta_{1}}(y)-p_{\theta_{2}}(y)|^{2}\,dy=NC_{1}I_{1}+NC_{2}I_{2}$
for large enough $A>0$ and some constants $C_{1},C_{2}>0$. Parseval’s identity
implies
$\displaystyle I_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}}|\phi_{\theta_{1}}(u)-\phi_{\theta_{2}}(u)|^{2}\,du$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|u|>M}e^{-\sigma^{2}|u|^{2}}\,du\lesssim
M^{-1}e^{-\sigma^{2}M^{2}},\quad M\to\infty$ $\displaystyle I_{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|u|>M}|(\phi_{\theta_{1}}(u)-\phi_{\theta_{2}}(u))^{\prime}|^{2}\,du$
$\displaystyle\lesssim$
$\displaystyle\int_{|u|>M}|u|^{2}e^{-\sigma^{2}|u|^{2}}\,du\lesssim
Me^{-\sigma^{2}M^{2}},\quad M\to\infty.$
The choice
$M\asymp\left[\sigma^{-2}\log\left(N\log^{\beta}N\right)\right]^{1/2}$ with
some $\beta>0$ yields
$N\chi^{2}(p_{\theta_{1}},p_{\theta_{2}})<1/2$
for large enough $N.$ On the other hand
$\bar{\vartheta}(u)-\bar{\vartheta}_{\delta}(u)=-\eta\int_{-1}^{1}\left[|u|^{\alpha}\mathbf{1}_{\\{|u|>M\\}}-|u+w|^{\alpha}\mathbf{1}_{\\{|u+w|>M\\}}\right]\,dw\\\
+\eta
M^{\delta}\int_{-1}^{1}\left[|u|^{\alpha-\delta}\mathbf{1}_{\\{|u|>M\\}}-|u+w|^{\alpha-\delta}\mathbf{1}_{\\{|u+w|>M\\}}\right]\,dw$
with
$\displaystyle\bar{\vartheta}(u)$ $\displaystyle:=$
$\displaystyle\int_{-1}^{1}\left[\vartheta(u)-\vartheta(u+w)\right]\,dw,$
$\displaystyle\bar{\vartheta}_{\delta}(u)$ $\displaystyle:=$
$\displaystyle\int_{-1}^{1}\left[\vartheta_{\delta}(u)-\vartheta_{\delta}(u+w)\right]\,dw.$
Using the identity
(32)
$\int_{-1}^{1}\left[|u|^{\alpha}\mathbf{1}_{\\{|u|>M\\}}-|u+w|^{\alpha}\mathbf{1}_{\\{|u+w|>M\\}}\right]\,dw=\\\
|u|^{\alpha}\int_{-1}^{1}\left[1-|1+w/u|^{\alpha}\right]\,dw=2\sum_{k=1}^{\infty}{\alpha\choose
2k}\frac{|u|^{\alpha-2k}}{2k+1},$
which holds for any $|u|>M+1$ and $M>1$, we get
$\displaystyle\left|\int_{\mathbb{R}}[\bar{\vartheta}(u)-\bar{\vartheta}_{\delta}(u)]\,du\right|\,du\gtrsim
M^{\alpha-1},\quad M\to\infty$
for any $0<\alpha<1.$ Denote
$\displaystyle\rho_{\theta_{1}}(x)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}}e^{\mathfrak{i}ux}\bar{\vartheta}(u)\,du=\left(1-\frac{\sin
x}{x}\right)\nu_{\theta_{1}}(x),$ $\displaystyle\rho_{\theta_{2}}(x)$
$\displaystyle:=$
$\displaystyle\int_{\mathbb{R}}e^{\mathfrak{i}ux}\bar{\vartheta}_{\delta}(u)\,du=\left(1-\frac{\sin
x}{x}\right)\nu_{\theta_{2}}(x),$
then the Fourier inversion formula implies that
$\displaystyle\sup_{x\in\mathbb{R}}|\rho_{\theta_{1}}(x)-\rho_{\theta_{2}}(x)|\geq\left|\int_{\mathbb{R}}[\bar{\vartheta}(u)-\bar{\vartheta}_{\delta}(u)]\,du\right|\gtrsim
M^{\alpha-1},\quad M\to\infty.$
Asymptotic expansion (32) shows that there is a constant $R$ depending on
$\eta$ such that
$\displaystyle|u|^{2-\alpha}\bar{\vartheta}(u)\leq
R,\quad|u|^{2-\alpha}\bar{\vartheta}_{\delta}(u)\leq R,\quad u\in\mathbb{R}.$
Hence, taking $\Lambda=\sigma^{2}/2,$ $\varkappa=2-\alpha$, we conclude that
both models $\theta_{1}$ and $\theta_{2}$ are in
$\mathcal{A}(\Lambda,\varkappa,R)$.
### 7.6 Proof of Proposition 5.3
The smallest eigenvalue $\lambda_{\min}(\Gamma)$ of the matrix $\Gamma$
satisfies
(33) $\displaystyle\lambda_{\min}(\Gamma)$ $\displaystyle=$
$\displaystyle\min_{\|W\|=1}W^{\top}\Gamma W$ $\displaystyle\geq$
$\displaystyle\min_{\|W\|=1}W^{\top}\bar{\Gamma}W+\min_{\|W\|=1}W^{\top}(\bar{\Gamma}-\Gamma)W$
$\displaystyle\geq$
$\displaystyle\min_{\|W\|=1}W^{\top}\bar{\Gamma}W-\sum_{1\leq k,l\leq
2}|\Gamma_{k,l}-\bar{\Gamma}_{k,l}|.$
According to the assumption (AS) it holds
$\min_{\|W\|=1}W^{\top}\bar{\Gamma}W\geq\gamma_{0}.$ Fix some $s>0$,
$k,l\in\\{0,1\\}$ and denote
$\displaystyle\Delta_{n}:=\frac{1}{h_{N}}\left(\frac{\delta_{n}-s}{h_{N}}\right)^{k+l}K\left(\frac{\delta_{n}-s}{h_{N}}\right)-\int_{\mathbb{R}}z^{k+l}K(z)p_{\delta}(s+h_{N}z)\,dz.$
We have $\mathbb{E}[\Delta_{n}]=0$,
$\Delta_{n}\leq
h_{N}^{-1}\sup_{z\in\mathbb{R}}\left[(1+|z|^{2})K(z)\right]=:D_{1}h_{N}^{-1}$
and
$\displaystyle\mathbb{E}[\Delta_{n}]^{2}$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}z^{2l+2k}K^{2}(z)p_{\delta}(s+h_{N}z)\,dz$
$\displaystyle\leq$
$\displaystyle\frac{p_{\max}}{h_{N}}\int_{\mathbb{R}}(1+|z|^{4})K^{2}(z)\,dz=:D_{2}h_{N}^{-1}$
with $p_{\max}=\max_{s\in\mathcal{S}}p_{\delta}(s)$ and some positive constant
$D_{1}$ and $D_{2}.$ Hence, due to the Bernstein inequality we get for any
$\delta>0$
(34)
$\displaystyle\operatorname{P}\left(|\Gamma_{k,l}-\bar{\Gamma}_{k,l}|\geq\delta\right)=\operatorname{P}\left(\frac{1}{N}\left|\sum_{n=1}^{N}\Delta_{n}\right|\geq\delta\right)\leq
L\exp(-\delta^{2}BNh_{N})$
with some positive constants $L$ and $B$. Combining (33) and (34), we get
(35)
$\displaystyle\operatorname{P}\left(\lambda_{\min}(\Gamma)\leq\gamma_{0}/2\right)\leq
4L\exp(-\gamma^{2}_{0}BNh_{N}/4).$
Due to the definition of estimates $\widetilde{\phi}_{N}(u)$ and
$\widetilde{\phi}_{s,N}(u)$
(36)
$\displaystyle\operatorname{P}\left(w(\|u\|)|\widetilde{\phi}_{N}(u)-\phi(u)|\geq\delta\right)$
$\displaystyle\leq$
$\displaystyle\operatorname{P}\left(\lambda_{\min}(\Gamma)\leq\gamma_{0}/2\right)$
$\displaystyle+\operatorname{P}\left(w(\|u\|)|\widehat{\phi}_{N}(u)-\phi(u)|\geq\delta,\,\lambda_{\min}(\Gamma)>\gamma_{0}/2\right),$
(37)
$\displaystyle\operatorname{P}\left(w(\|u\|)|\widetilde{\phi}_{s,N}(u)-\partial_{s}\phi(u)|\geq\delta\right)$
$\displaystyle\leq$
$\displaystyle\operatorname{P}\left(\lambda_{\min}(\Gamma)\leq\gamma_{0}/2\right)$
$\displaystyle+\operatorname{P}\left(w(\|u\|)|\widehat{\phi}_{s,N}(u)-\partial_{s}\phi(u)|\geq\delta,\,\lambda_{\min}(\Gamma)>\gamma_{0}/2\right).$
Furthermore, the following representation holds on the set
$\\{\lambda_{\min}(\Gamma)>\gamma_{0}/2\\}$
$\displaystyle(\widehat{\phi}_{N}(u|s,x)-\phi(u|s,x),h_{N}(\widehat{\phi}_{s,N}(u|s,x)-\partial_{s}\phi(u|s,x)))=\Gamma^{-1}\varepsilon_{N}(u)$
with
$\varepsilon_{N,k}(u):=\frac{1}{N}\sum_{n=1}^{N}\left[\exp(\mathfrak{i}u^{\top}X(t_{n}+\delta_{n}))-\phi(u|s,x)-\partial_{s}\phi(u|s,x)(s-\delta_{n})\right]\pi_{n,k}(s)$
and
$\pi_{n,k}(s)=\frac{1}{h_{N}}\left(\frac{\delta_{n}-s}{h_{N}}\right)^{k}K\left(\frac{\delta_{n}-s}{h_{N}}\right),\quad
k=0,1.$
We have $|\pi_{n,k}|\leq\pi^{*}_{1}h_{N}^{-1}$ and
$\mathbb{E}\left[\pi^{2}_{n,k}\log^{2(1+\varepsilon)}\pi^{2}_{n,k}\right]\leq\pi^{*}_{2}h_{N}^{-1}\log^{2(1+\varepsilon)}(h_{N}),\quad
k=0,1,\quad\varepsilon>0,$
with some positive constants $\pi_{1}^{*}$ and $\pi^{*}_{2}$. The following
lemma holds
###### Lemma 7.2.
Fix some $(s,x)\in\mathcal{S}\times\mathcal{D}$ and denote
$\displaystyle\mathcal{W}_{N,l}(u):=\frac{1}{N}\sum_{n=1}^{N}\pi_{n,l}(s)w(\|u\|)D_{n}(u),\quad
l=0,1,$
where
$D_{n}(u):=\exp(\mathfrak{i}u^{\top}\bar{X}_{n})-e^{-\mathfrak{i}u^{\top}x}\phi(u|\delta_{n},x).$
Let $w$ be a Lipschitz continuous weighting function satisfying
$0<w(z)\leq\min\\{1,\log^{-1/2-\delta}(z)\\},\quad z\in\mathbb{R}_{+}$
for arbitrary small $\delta>0$. Then for large enough $\zeta>0$
(38)
$\displaystyle\operatorname{P}\left(\sup_{u\in\mathbb{R}^{d}}|\mathcal{W}_{N,l}(u)|\geq\frac{\zeta}{2}\sqrt{\frac{\log^{1+r}N}{Nh_{N}}}\right)\lesssim\log^{(r-1)d/2}(N)N^{-\kappa},\quad
N\to\infty$
with some $\kappa>1$, $r>0$ and $l=0,1.$
###### Proof.
Consider the sequence $A_{k}=e^{k},\,k\in\mathbb{N}$ and cover each cube
$[-A_{k},A_{k}]^{d}$ by
$M_{k}=\left(\lfloor(2d^{1/2}A_{k})/\gamma\rfloor+1\right)^{d}$ disjoint small
cubes $\Lambda_{k,1},\ldots,\Lambda_{k,M_{k}}$, the edges of each cube being
of the length $\gamma/d^{1/2}.$ Let $u_{k,1},\ldots,u_{k,M_{k}}$ be the
centers of these cubes. We have for any natural $K>0$
$\displaystyle\max_{k=1,\ldots,K}\sup_{A_{k-1}<\|u\|\leq
A_{k}}|\mathcal{W}_{N,0}(u)|$ $\displaystyle\leq$
$\displaystyle\max_{k=1,\ldots,K}\max_{\|u_{k,m}\|>A_{k-1}}|\mathcal{W}_{N,0}(u_{k,m})|$
$\displaystyle+\max_{k=1,\ldots,K}\max_{1\leq m\leq
M_{k}}\sup_{u\in\Lambda_{k,m}}|\mathcal{W}_{N,0}(u)-\mathcal{W}_{N,0}(u_{k,m})|.$
Hence
(39) $\operatorname{P}\left(\max_{k=1,\ldots,K}\sup_{A_{k-1}<\|u\|\leq
A_{k}}|\mathcal{W}_{N,0}(u)|>\lambda\right)\leq\sum_{k=1}^{K}\sum_{\\{\|u_{k,m}\|>A_{k-1}\\}}\operatorname{P}(|\mathcal{W}_{N,0}(u_{k,m})|>\lambda/2)+\\\
\operatorname{P}\left(\sup_{\|u-v\|<\gamma}|\mathcal{W}_{N,0}(v)-\mathcal{W}_{N,0}(u)|>\lambda/2\right).$
It holds for any $u,v\in\mathbb{R}^{d}$
(40) $\displaystyle|\mathcal{W}_{N,0}(v)-\mathcal{W}_{N,0}(u)|$
$\displaystyle\leq$ $\displaystyle 2\pi_{1}^{*}h_{N}^{-1}|w(\|v\|)-w(\|u\|)|$
$\displaystyle+\frac{\pi_{1}^{*}}{Nh_{N}}\sum_{n=1}^{N}\left|\exp(\mathfrak{i}v^{\top}\bar{X}_{n})-\exp(\mathfrak{i}u^{\top}\bar{X}_{n})\right|$
$\displaystyle+\frac{\pi_{1}^{*}}{Nh_{N}}\sum_{n=1}^{N}\left|e^{-\mathfrak{i}v^{\top}x}\phi(v|\delta_{n},x)-e^{-\mathfrak{i}u^{\top}x}\phi(u|\delta_{n},x)\right|$
$\displaystyle\leq$ $\displaystyle
2\pi_{1}^{*}h_{N}^{-1}\|u-v\|\left[L_{w}+\frac{1}{N}\sum_{n=1}^{N}\|\bar{X}_{n}\|+\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\|\bar{X}_{n}\|\right],$
where $L_{\omega}$ is the Lipschitz constant of $w$. The Markov inequality
implies
$\displaystyle\operatorname{P}\left(\frac{1}{N}\sum_{n=1}^{N}[\|\bar{X}_{n}\|-\mathbb{E}\|\bar{X}_{n}\|]>c\right)\leq
c^{-p}N^{-p}\mathbb{E}\left|\sum_{n=1}^{N}[\|\bar{X}_{n}\|-\mathbb{E}\|\bar{X}_{n}\|]\right|^{p}$
for any $c>0.$ Using now Dedecker and Rio inequalities (see Dedecker and et
al. (2007)) and taking into account assumptions (AX1)-(AX2), we get
$\displaystyle\mathbb{E}\left|\sum_{n=1}^{N}[\|\bar{X}_{n}\|-\mathbb{E}\|\bar{X}_{n}\|]\right|^{p}\leq
C_{p}(\beta)N^{p/2},$
where $C_{p}(\beta)$ is some constant depending on $\beta$ and $p$ from
assumptions (AX1) and (AX2) respectively. Hence,
(41)
$\displaystyle\operatorname{P}\left(\frac{1}{N}\sum_{n=1}^{N}\|\bar{X}_{n}\|>2\beta_{1,N}\right)\leq
C_{p}(\beta)N^{-p/2}/\beta^{p}_{1,N},$
where $\beta_{1,N}=\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\|\bar{X}_{n}\|$. Note
that since $X$ is ergodic it holds
$\beta_{1,N}\to\beta^{*}_{1}=\int\|x\|\pi(dx)<\infty,\quad N\to\infty,$
where $\pi$ is a unique invariant distribution. Setting
$\gamma=\lambda/(24\pi_{1}^{*}h_{N}^{-1}\max\\{\beta_{1,N},L_{w}\\})$ and
combining (40) with the inequality (41), we obtain
(42)
$\displaystyle\operatorname{P}\left(\sup_{\|u-v\|<\gamma}|\mathcal{W}_{N,0}(v)-\mathcal{W}_{N,0}(u)|>\lambda/2\right)\leq
B_{1}N^{-p/2}$
with some constant $B_{1}$ not depending on $\lambda$ and $N$. Let us turn now
to the first term on the right-hand side of (39). If $\|u_{k,m}\|>A_{k-1}$
then it follows from Theorem 8.1 and Corollary 8.2 (see Appendix)
$\operatorname{P}\left(|\operatorname{Re}\left[\mathcal{W}_{N,0}(u_{k,m})\right]|>\lambda/4\right)\\\
\leq
B_{2}\exp\left(-\frac{B_{3}\lambda^{2}N}{4w^{2}(A_{k-1})\log^{2(1+\varepsilon)}(h_{N}w(A_{k-1}))\pi_{2}^{*}/h_{N}+\lambda\log^{2}(N)w(A_{k-1})\pi_{1}^{*}/h_{N}}\right),$
$\operatorname{P}\left(|\operatorname{Im}\left[\mathcal{W}_{N,0}(u_{k,m})\right]|>\lambda/4\right)\\\
\leq
B_{4}\exp\left(-\frac{B_{3}\lambda^{2}N}{4w^{2}(A_{k-1})\log^{2(1+\varepsilon)}(h_{N}w(A_{k-1}))\pi_{2}^{*}/h_{N}+\lambda\log^{2}(N)w(A_{k-1})\pi_{1}^{*}/h_{N}}\right)$
with some constants $B_{2}$, $B_{3}$ and $B_{4}$ depending only on the
characteristics of the process $X$. Taking $\lambda=\zeta
h_{N}^{-1/2}\log^{(1+\varepsilon)}(h_{N})N^{-1/2}\log^{1/2}N$ with $\zeta>0$
and using the fact that $h^{-1}_{N}=O(N)$, we get
$\sum_{\\{\|u_{k,m}\|>A_{k-1}\\}}\operatorname{P}(|\mathcal{W}_{N,0}(u_{k,m})|>\lambda/2)\leq\left(\lfloor(2d^{1/2}A_{k})/\gamma\rfloor+1\right)^{d}\\\
\times\exp\left(-\frac{B_{3}\lambda^{2}N}{4w^{2}(A_{k-1})\log^{2(1+\varepsilon)}(h_{N}w(A_{k-1}))\pi_{2}^{*}/h_{N}+\lambda\log^{2}(N)w(A_{k-1})\pi_{1}^{*}/h_{N}}\right)\\\
\lesssim
A_{k}^{d}h^{-d/2}_{N}N^{d/2}\exp\left(-\frac{B\zeta^{2}\log(N)}{w^{2}(A_{k-1})\log^{2(1+\varepsilon)}(w(A_{k-1}))}\right)\log^{(r-1)d/2}(N),\quad
N\to\infty$
with $r=2(1+\varepsilon)$ and some constant $B>0.$ Fix $\theta>0$ such that
$B\theta>d$ and compute
$\displaystyle\sum_{\\{\|u_{k,m}\|>A_{k-1}\\}}\operatorname{P}(|\mathcal{W}_{N,0}(u_{k,m})|>\lambda/2)$
$\displaystyle\lesssim$ $\displaystyle h_{N}^{-d/2}e^{kd-\theta
B(k-1)}N^{d/2}\log^{(r-1)d/2}(N)e^{-B(k-1)(\zeta^{2}\log N-\theta)}$
$\displaystyle\lesssim$ $\displaystyle e^{k(d-\theta
B)}\log^{(r-1)d/2}(N)e^{-B(k-1)(\zeta^{2}\log N-\theta)+d\log(N)}.$
If $\zeta^{2}\log N>\theta$ we get asymptotically
$\displaystyle\sum_{k=2}^{K}\sum_{\\{\|u_{k,m}\|>A_{k-1}\\}}\operatorname{P}(|\mathcal{W}_{N,0}(u_{k,m})|>\lambda/2)$
$\displaystyle\lesssim$
$\displaystyle\log^{(r-1)d/2}(N)e^{-(B\zeta^{2}-d)\log(N)}.$
Taking large enough $\zeta>0$, we get (38). ∎
Combining Lemma 7.2 with the inequality
$\displaystyle|\phi(u|\delta_{n},x)-\phi(u|s,x)-\partial_{s}\phi(u|s,x)(s-\delta_{n})|\leq
h_{N}^{2}\sup_{s\in\mathcal{S}}|\partial_{ss}\phi(u|s,x)|,\quad|s-\delta_{n}|\leq
h_{N}$
and using Lemma 2.2, we obtain for large enough $\zeta>0$
$\displaystyle\operatorname{P}\left(\sup_{u\in\mathbb{R}^{d}}\left[w(\|u\|)|\varepsilon_{k}(u)|\right]>\zeta\left[\sqrt{\frac{\log^{1+r}N}{Nh_{N}}}+h_{N}\right]\right)\lesssim\log^{-d/2}(N)N^{-\kappa},\quad
N\to\infty,\quad k=0,1.$
Finally, inequalities (35),(36) and (37) together with the Borel-Cantelli
lemma entail (21) and (22).
## 8 Appendix
For any two $\sigma$ algebras $\mathcal{A}$ and $\mathcal{B}$, define the
$\alpha$-mixing coefficient by
$\displaystyle\alpha_{Z}(\mathcal{A},\mathcal{B})=\sup_{A\in\mathcal{A},\,B\in\mathcal{B}}|\operatorname{P}(A\cap
B)-\operatorname{P}(A)\operatorname{P}(B)|.$
Let $(Z_{k},\,k\geq 1)$ be a sequence of real random variables defined on
$(\Omega,\mathcal{F},\operatorname{P})$. This sequence is called strongly
mixing if
$\displaystyle\alpha_{Z}(n)=\sup_{k\geq
1}\alpha(\mathcal{M}_{k},\mathcal{G}_{k+n})\to 0,\quad n\to\infty,$
where $\mathcal{M}_{j}=\sigma(Z_{i},\,i\leq j)$ and
$\mathcal{G}_{j}=\sigma(Z_{i},\,i\geq j)$ for $j\geq 1.$ The following theorem
can be found in Merlevède, Peligrad and Rio (2009).
###### Theorem 8.1.
Let $(Z_{k},\,k\geq 1)$ be a strongly mixing sequence of centered real valued
random variables on the probability space $(\Omega,\cal F,P)$ with the mixing
coefficients satisfying
(43) $\displaystyle\alpha(n)\leq\bar{\alpha}\exp(-cn),\quad n\geq
1,\quad\bar{\alpha}>0,\quad c>0.$
Assume that $\sup_{k\geq 1}|Z_{k}|\leq M$ a.s., then there is a positive
constant $C$ depending on $c$ and $\bar{\alpha}$ such that
$\operatorname{P}\left\\{\sum_{i=1}^{N}Z_{i}\geq\zeta\right\\}\leq\exp\left[-\frac{C\zeta^{2}}{Nv^{2}+M^{2}+M\zeta\log^{2}(N)}\right].$
for all $\zeta>0$ and $N\geq 4,$ where
$\displaystyle v^{2}:=\sup_{i}\left(\mathbb{E}[Z_{i}]^{2}+2\sum_{j\geq
i}\operatorname{Cov}(Z_{i},Z_{j})\right).$
###### Corollary 8.2.
Denote
$\rho_{j}=\mathbb{E}\left[Z_{j}^{2}\log^{2(1+\varepsilon)}\left(|Z_{j}|^{2}\right)\right],\quad
j=1,2,\ldots,$
with some $\varepsilon>0$ and suppose that all $\rho_{j}$ are finite. Then
$\displaystyle\sum_{j\geq i}\operatorname{Cov}(Z_{i},Z_{j})\leq
C\max_{i}\rho_{Z_{i}}$
for some constant $C>0$ provided that (43) holds. Consequently the following
inequality holds
$\displaystyle
v^{2}\leq\sup_{i}\left(\mathbb{E}[Z_{i}]^{2}+C\max_{i}\rho_{Z_{i}}\right).$
###### Proof.
Due to Rio inequality
$\left|\operatorname{Cov}(Z_{i},Z_{j})\right|\leq
2\int_{0}^{\alpha(|j-i|)}Q_{Z_{i}}(u)Q_{Z_{j}}(u)du,$
where for any random variable $X$ we denote by $Q_{X}$ the quantile function
of $X.$ Define
$\rho_{X}=\mathbb{E}\left[X^{2}\log^{2(1+\varepsilon)}\left(|X|^{2}\right)\right]$
for some $\varepsilon>0.$ Markov inequality implies for small enough $u>0$
$\displaystyle\operatorname{P}\left(|X|>\frac{\rho_{X}^{1/2}}{u^{1/2}|\log(u)|^{(1+\varepsilon)}}\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[X^{2}\log^{2(1+\varepsilon)}\left(|X|^{2}\right))\right]\frac{\rho_{X}^{-1}}{u^{-1}\log^{-2(1+\varepsilon)}(u)}$
$\displaystyle\times\log^{-2(1+\varepsilon)}\left(\frac{\rho_{X}}{u\log^{2(1+\varepsilon)}(u)}\right)$
$\displaystyle=$ $\displaystyle
u\log^{-2(1+\varepsilon)}\left(\rho_{X}\log^{-2(1+\varepsilon)}(u)\right)\leq
u$
and therefore
$Q_{X}(u)\leq\frac{\rho_{X}^{1/2}}{u^{1/2}|\log(u)|^{(1+\varepsilon)}}.$
Hence
$\left|\operatorname{Cov}(Z_{i},Z_{j})\right|\leq
2\int_{0}^{\alpha(|j-i|)}\frac{\sqrt{\rho_{Z_{i}}\rho_{Z_{j}}}}{u\log^{2(1+\varepsilon)}(u)}du\leq
2\sqrt{\rho_{Z_{i}}\rho_{Z_{j}}}\log^{-1-2\varepsilon}(\alpha(|j-i|))$
and
$\sum_{j\geq i}\operatorname{Cov}(Z_{i},Z_{j})\leq
C\sqrt{\rho_{Z_{i}}\rho_{Z_{j}}}\sum_{j>i}\frac{1}{|j-i|^{1+2\varepsilon}}$
with some constant $C>0$ depending on $\bar{\alpha}.$ ∎
##### Acknowledgements
I would like to thank Olivier Lopez for remarks and helpful discussions.
## References
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|
arxiv-papers
| 2009-07-28T08:57:08 |
2024-09-04T02:49:04.235206
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Denis Belomestny",
"submitter": "Denis Belomestny",
"url": "https://arxiv.org/abs/0907.4865"
}
|
0907.4866
|
# Stochastic Flows of SDEs with Irregular Coefficients and Stochastic
Transport Equations
Xicheng Zhang Department of Mathematics, Huazhong University of Science and
Technology
Wuhan, Hubei 430074, P.R.China,
School of Mathematics and Statistics
The University of New South Wales, Sydney, 2052, Australia
Email: XichengZhang@gmail.com
###### Abstract.
In this article we study (possibly degenerate) stochastic differential
equations (SDE) with irregular (or discontiuous) coefficients, and prove that
under certain conditions on the coefficients, there exists a unique almost
everywhere stochastic (invertible) flow associated with the SDE in the sense
of Lebesgue measure. In the case of constant diffusions and BV drifts, we
obtain such a result by studying the related stochastic transport equation. In
the case of non-constant diffusions and Sobolev drifts, we use a direct
method. In particular, we extend the recent results on ODEs with non-smooth
vector fields to SDEs. Moreover, we also give a criterion for the existence of
invariant measures for the associated transition semigroup.
Keywords: Stochastic flow, DiPerna-Lions flow, Hardy-Littlewood maximal
function, Stochastic transport equation, Invariant measure
## 1\. Introduction
Consider the following Itô’s stochastic differential equation (SDE):
$\displaystyle{\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm
d}}}t+\sigma(X_{t}){\mathord{{\rm d}}}W_{t},\ \ X_{0}=x,$ (1.1)
where $b:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ and
$\sigma:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}\times{\mathbb{R}}^{m}$ are two
Borel measurable functions, and $(W_{t})_{t\geqslant 0}$ is the
$m$-dimensional standard Brownian motion on the classical Wiener space
$(\Omega,{\mathcal{F}},P;({\mathcal{F}}_{t})_{t\geqslant 0})$, i.e., $\Omega$
is the space of all continuous functions from ${\mathbb{R}}_{+}$ to
${\mathbb{R}}^{m}$ with locally uniform convergence topology, ${\mathcal{F}}$
is the Borel $\sigma$-field, $P$ is the Wiener measure,
$({\mathcal{F}}_{t})_{t\geqslant 0}$ is the natural filtration generated by
the coordinate process $W_{t}(\omega)=\omega(t)$.
It is by now a classical result that if $b$ and $\sigma$ are globally
Lipschitz continuous, then there exists a unique bi-continuous solution
$(t,x)\mapsto X_{t}(x)$ to SDE (1.1) such that for almost all $\omega$ and any
$t\geqslant 0$, $x\mapsto X_{t}(\omega,x)$ is a homeomorphism. Thus,
$\\{X_{t}(x),x\in{\mathbb{R}}^{d}\\}_{t\geqslant 0}$ forms a stochastic
homeomorphism flow (cf. [16]). Recently, there are increasing interests for
studying the stochastic homeomorphism flow property associated with SDE (1.1)
under various non-Lipschitz assumptions on $b$ and $\sigma$ (cf. [21, 1, 20,
25, 8, 9, 10, 11, 28], etc.). Here, the non-Lipschitz conditions may be less
smooth or not global Lipschitz .
On the other hand, when $\sigma$ is non-degenerate and $b$ is not continuous
and even singular, SDE (1.1) may have a unique strong solution for each
starting point $x\in{\mathbb{R}}^{d}$ (cf. [14, 17, 26], etc.). But it is not
known whether it still defines a stochastic homeomorphism flow. In the
completely degenerate case ($\sigma=0$), a celebrated theory established by
DiPerna and Lions [6] says that ordinary differential equation (ODE)
$\displaystyle{\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm d}}}t,\ \
X_{0}=x$ (1.2)
defines a regular Lagrangian flow in the sense of Lebesgue measure when $b$ is
a Sobolev vector field with bounded divergence. This theory was later extended
to the case of BV vector fields by Ambrosio [2]. The central of DiPerna and
Lions’ theory are based on the connection between ODE and the Cauchy problem
for the transport equation:
$\displaystyle\partial_{t}u+b^{i}\partial_{i}u=0,\ \ u|_{t=0}=u_{0}.$ (1.3)
Here and below, we use the usual convention: the repeated indices will be
summed. By introducing a new notion of renormalized solutions, DiPerna and
Lions showed the uniqueness and stability of $L^{\infty}$-distributional
solutions for (1.3) when $b$ is Sobolev regular so that they can go back to
ODE and show the well posedness of (1.2) with Sobolev vector field $b$ in the
distributional sense.
We now back to SDE (1.1). It is also well known that SDE (1.1) is connected
with the following stochastic transport equation (cf. [16, 24]):
$\displaystyle{\mathord{{\rm
d}}}u=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u-(b^{i}-\sigma^{jl}\partial_{j}\sigma^{il})\partial_{i}u{\mathord{{\rm
d}}}t-\sigma^{il}\partial_{i}u{\mathord{{\rm d}}}W^{l}_{t},\ \
u|_{t=0}=u_{0}.$ (1.4)
Thus, it is natural to ask whether we can extend the DiPerna and Lions theory
to the case of SDEs. Notice that (1.4) is always a degenerate second order
stochastic parabolic equation whatever $\sigma$ is or not degenerate. More
general second order linear stochastic partial differential equation has been
recently studied in [28]. In general, it is hard to solve equation (1.4) if
$b$ and $\sigma$ are not smooth (cf. [24]). The source of difficulty clearly
comes from the degeneracy. Nevertheless, we can extend the well known theory
about the transport equation to the case of constant $\sigma$ and BV vector
field $b$. In this case, it will be shown that we can also go back to SDE
(1.1) from stochastic transport equation (1.4) and obtain the well posedness
of SDE (1.1) with BV drift. We remark that in another direction, Flandoli,
Gubinelli and Priola [13] studied the well posedness of (1.4) when $b$ is
Hölder continuous and $\sigma$ is the unit matrix, where their proofs benefit
from the stochastic flow associated with SDE (1.1). We emphasize that when
$\sigma$ is constant, SDE (1.1) can be directly solved by transferring it to a
time dependent ODE. But, this will lose some “stochastic flavor”.
Recently, Crippa and De Lellis [5] derived some new estimates for ODEs with
Sobolev coefficients. These estimates allowed them to give a direct and simple
treatment for DiPerna-Lions flows. The key ingredient of their method is to
give some control for the following quantity in terms of $\|\nabla
b\|_{L^{p}}$ ($p>1$):
$\int_{B_{R}}\sup_{t\in[0,T]}\sup_{r\in[0,2R]}\left[\fint_{B_{r}}\log\left(\frac{|X_{t}(x)-X_{t}(x+y)|}{\delta}+1\right){\mathord{{\rm
d}}}y\right]^{p}{\mathord{{\rm d}}}x,$
where $B_{r}:=\\{x\in{\mathbb{R}}^{d}:|x|\leqslant r\\}$ denotes the ball with
radius $r$ and center $0$. For estimating this quantity, the Hardy-Littlewood
maximal function was used to control the difference
$|b(X_{s}(x))-b(X_{s}(x+y))|$. Moreover, the stability was also derived in [5]
by using a similar quantity. We remark that the above quantity was first
introduced in [3] in order to prove the approximative differentiability of
regular Lagrangian flows. The second part of this paper is to extend Crippa
and De Lellis’ result to the stochastic case so that $\sigma$ can be non-
constant.
We also mention that Figalli [12] has already developed a stochastic
counterpart for DiPerna-Lions theory. Therein, the martingale solution (or
weak solution) in the sense of Stroock-Varadhan was considered corresponding
to the Fokker-Planck equation. Moreover, the non-degenerate condition on
$\sigma$ is required when $\sigma$ is non-constant. Compared with [12], we can
directly construct the “strong” solution of SDE (1.1) with Sobolev drift and
possibly degenerate diffusion coefficients in the sense of Lebesgue measure.
Moreover, as an easy consequence, we can uniquely solve the SDE in the
classical sense when the initial value is an absolutely continuous
${\mathcal{F}}_{0}$-measurable random variable (see Corollary 6.4 and
Corollary 6.5 below). It should be noted that for the simplicity, we only
consider the time independent coefficients in the present paper. Clearly, our
results can be extended to the time dependent case by requiring some
integrability in the time variable.
In the study of stochastic dynamical systems, an important problem is to prove
the existence of equilibrium point (invariant measure). Since we are dealing
with non-smooth stochastic differential equations, it is not expected to have
the Feller property for the associated transition semigroup. Thus, it seems
that the classical coercivity condition is not enough to guarantee the
existence of an invariant probability measure for SDE (1.1) (cf. [4, 16]). In
the present paper, we shall give a criterion for the existence of an invariant
probability measure in terms of the classical coercivity condition as well as
some divergence condition (see Theorem 2.8 below). We want to emphasize that
in our result, such an invariant measure is indeed absolutely continuous with
respect to the Lebesgue measure.
This paper is organized as follows: in Section 2, after introducing the notion
of almost everywhere stochastic (invertible) flow, we give two direct
consequences of this notion and then state our main results. In Section 3, we
give some necessary preliminaries for later use. In Section 4, we study
stochastic transport equation (1.4) in case that $b\in\mathrm{BV}_{loc}$ has
bounded divergence and $\sigma$ is constant. In Section 5, we apply the
results of Section 4 to the study of stochastic flows of SDE with BV drift and
constant diffusion coefficients. In Section 6, we extend the result of [5] to
the stochastic case. Here, an SDE with discontinuous coefficients is provided
to show our result. This section can be read independently of Sections 4 and
5. In Section 7, we prove our main results. In the appendix, we give a
detailed proof about the flow property as well as the Markov property when SDE
(1.1) admits a unique almost everywhere stochastic flow in the sense of
Definition 2.1 below.
## 2\. Main Results
We first introduce some necessary notations. Let $(E,{\mathcal{E}},\mu)$ be a
measure space and ${\mathscr{T}}:E\to E$ a measurable transformation. We shall
use $\mu\circ{\mathscr{T}}$ to denote the image measure of $\mu$ under
${\mathscr{T}}$, i.e., for any nonnegative measurable function $\varphi$,
$\int_{E}\varphi(x)\mu\circ{\mathscr{T}}({\mathord{{\rm
d}}}x):=\int_{E}\varphi({\mathscr{T}}(x))\mu({\mathord{{\rm d}}}x).$
By $\mu\circ{\mathscr{T}}\ll\mu$ we mean that $\mu\circ{\mathscr{T}}$ is
absolutely continuous with respect to $\mu$. Let
$C^{\infty}_{c}({\mathbb{R}}^{d})$ be the set of all smooth functions on
${\mathbb{R}}^{d}$ with compact supports, $C_{b}({\mathbb{R}}^{d})$ the set of
all bounded continuous functions, and ${\mathcal{L}}^{+}({\mathbb{R}}^{d})$
the set of all nonnegative Borel measurable functions. Below, we shall denote
the Lebesgue measure by ${\mathscr{L}}({\mathord{{\rm d}}}x)$ or
${\mathord{{\rm d}}}x$.
Convention: The repeated indices will be summed. The letter $C$ with or
without subscripts will denote a positive constant whose value is not
important and may change in different occasions. Moreover, all the
derivatives, gradients and divergences are taken in the distributional sense.
We introduce the following notion of almost everywhere stochastic (invertible)
flows, which is inspired by LeBris and Lions [18] and Ambrosio [2].
###### Definition 2.1.
Let $X_{t}(\omega,x)$ be a ${\mathbb{R}}^{d}$-valued measurable stochastic
field on ${\mathbb{R}}_{+}\times\Omega\times{\mathbb{R}}^{d}$. We say $X$ an
almost everywhere stochastic flow of (1.1) corresponding to $(b,\sigma)$ if
1. (A)
For ${\mathscr{L}}$-almost all $x\in{\mathbb{R}}^{d}$, $t\mapsto X_{t}(x)$ is
a continuous (${\mathcal{F}}_{t}$)-adapted stochastic process satisfying that
for any $T>0$
$\int^{T}_{0}|b(X_{s}(x))|{\mathord{{\rm
d}}}s+\int^{T}_{0}|\sigma(X_{s}(x))|^{2}{\mathord{{\rm d}}}s<+\infty,\ \
P-a.s.,$
and solves
$X_{t}(x)=x+\int^{t}_{0}b(X_{s}(x)){\mathord{{\rm
d}}}s+\int^{t}_{0}\sigma(X_{s}(x)){\mathord{{\rm d}}}W_{s},\ \ \forall
t\geqslant 0.$
2. (B)
For any $t\geqslant 0$ and $P$-almost all $\omega\in\Omega$,
${\mathscr{L}}\circ X_{t}(\omega,\cdot)\ll{\mathscr{L}}$. Moreover, for any
$T>0$, there exists a constant $K_{T,b,\sigma}>0$ such that for all
$\varphi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$
$\displaystyle\sup_{t\in[0,T]}{\mathbb{E}}\int_{{\mathbb{R}}^{d}}\varphi(X_{t}(x)){\mathord{{\rm
d}}}x\leqslant K_{T,b,\sigma}\int_{{\mathbb{R}}^{d}}\varphi(x){\mathord{{\rm
d}}}x.$ (2.1)
We say $X$ an almost everywhere stochastic invertible flow of (1.1)
corresponding to $(b,\sigma)$ if in addition to the above (A) and (B),
1. (C)
For any $t\geqslant 0$ and $P$-almost all $\omega\in\Omega$, there exists a
measurable inverse $X^{-1}_{t}(\omega,\cdot)$ of $X_{t}(\omega,\cdot)$ so that
${\mathscr{L}}\circ
X^{-1}_{t}(\omega,\cdot)=\rho_{t}(\omega,\cdot){\mathscr{L}}$, where the
density $\rho_{t}(x)$ is given by
$\displaystyle\rho_{t}(x):=\exp\left\\{\int^{t}_{0}\Big{[}\mathord{{\rm
div}}b-\frac{1}{2}\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}\Big{]}(X_{s}(x)){\mathord{{\rm
d}}}s+\int^{t}_{0}\mathord{{\rm div}}\sigma(X_{s}(x)){\mathord{{\rm
d}}}W_{s}\right\\}.$ (2.2)
Here, $\mathord{{\rm div}}\sigma^{\cdot l}:=\partial_{i}\sigma^{il}$ and we
require that for any $T>0$ and ${\mathscr{L}}$-almost all
$x\in{\mathbb{R}}^{d}$,
$\int^{T}_{0}\Big{[}|\mathord{{\rm
div}}b|+|\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}|+|\mathord{{\rm
div}}\sigma|^{2}\Big{]}(X_{s}(x)){\mathord{{\rm d}}}s<+\infty,\ \ P-a.s.$
###### Remark 2.2.
If $\sigma=constant$ and $\mathord{{\rm div}}b\in
L^{\infty}({\mathbb{R}}^{d})$, then (C) clearly implies (B). In fact, in this
case we have
${\mathscr{L}}\circ
X_{t}(\omega,\cdot)=\rho^{-1}_{t}(\omega,X^{-1}_{t}(\omega,\cdot)){\mathscr{L}}$
and by (2.2)
$|\rho^{-1}_{t}(\omega,X^{-1}_{t}(\omega,x))|\leqslant e^{t\|\mathord{{\rm
div}}b\|_{\infty}}.$
In what follows, for the simplicity of notations, we shall drop the time
variable $t$ and the spatial variable $x$ if there are no confusions. For
examples, for a function $f_{s}(x)$, we simply write
$\int^{t}_{0}\\!\\!\\!\int
f:=\int^{t}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}f_{s}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$
and
$\int^{t}_{0}\\!\\!\\!\int f{\mathord{{\rm
d}}}W_{s}:=\int^{t}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}f_{s}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}W_{s}.$
The following result is an easy consequence of Definition 2.1.
###### Proposition 2.3.
Assume that $b\in L^{1}_{loc}({\mathbb{R}}^{d})$ with $\mathord{{\rm div}}b\in
L^{1}_{loc}({\mathbb{R}}^{d})$ and $\sigma\in C^{2}({\mathbb{R}}^{d})$. Let
$X$ be an almost everywhere stochastic invertible flow of (1.1) in the sense
of Definition 2.1. Let $u_{0}\in L^{\infty}({\mathbb{R}}^{d})$ and set
$u_{t}(x):=u_{0}(X^{-1}_{t}(x))$. Then $u_{t}(x)$ solves the following
stochastic transport equation in the distributional sense:
${\mathord{{\rm
d}}}u=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u-b^{i}_{\sigma}\partial_{i}u{\mathord{{\rm
d}}}t-\sigma^{il}\partial_{i}u{\mathord{{\rm d}}}W^{l}_{t},$
where $b^{i}_{\sigma}:=b^{i}-\sigma^{jl}\partial_{j}\sigma^{il}$. In
particular, $\bar{u}_{t}(x):={\mathbb{E}}u_{0}(X^{-1}_{t}(x))$ is a
distributional solution of the following second order parabolic differential
equation:
$\partial_{t}\bar{u}=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\bar{u}-b^{i}_{\sigma}\partial_{i}\bar{u}.$
###### Proof.
Let $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d})$. By (C) of Definition 2.1,
we have
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int(b^{i}_{\sigma}\partial_{i}u)\varphi$
$\displaystyle=$ $\displaystyle\int^{t}_{0}\\!\\!\\!\int
u_{0}(X^{-1})\cdot\mathord{{\rm
div}}(b_{\sigma}\varphi)=\int^{t}_{0}\\!\\!\\!\int u_{0}\cdot\mathord{{\rm
div}}(b_{\sigma}\varphi)(X)\cdot\rho=$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(b^{i}_{\sigma}\partial_{i}\varphi)(X)\cdot\rho+\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(\varphi\mathord{{\rm div}}b_{\sigma})(X)\cdot\rho.$
Similarly,
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int(\sigma^{il}\partial_{i}u)\varphi{\mathord{{\rm
d}}}W^{l}_{s}=\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(\sigma^{il}\partial_{i}\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}W^{l}_{s}+\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(\partial_{i}\sigma^{il}\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}W^{l}_{s}$
and
$\displaystyle\frac{1}{2}\int^{t}_{0}\\!\\!\\!\int\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u\varphi=\frac{1}{2}\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot[\partial^{2}_{ij}(\sigma^{il}\sigma^{jl}\varphi)](X)\cdot\rho.$
Moreover, by stochastic Fubini’s theorem, we have
$\displaystyle\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(b^{i}_{\sigma}\partial_{i}\varphi)(X)\cdot\rho+\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(\sigma^{il}\partial_{i}\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}W^{l}_{s}$ $\displaystyle\quad=\int
u_{0}\left(\int^{t}_{0}(b^{i}_{\sigma}\partial_{i}\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}s+\int^{t}_{0}(\sigma^{il}\partial_{i}\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}W^{l}_{s}\right)$
and
$\displaystyle\int^{t}_{0}\\!\\!\\!\int u_{0}\cdot(\mathord{{\rm
div}}b_{\sigma}\cdot\varphi)(X)\cdot\rho+\int^{t}_{0}\\!\\!\\!\int
u_{0}\cdot(\mathord{{\rm div}}\sigma\cdot\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}W_{s}$ $\displaystyle\quad=\int u_{0}\left(\int^{t}_{0}(\mathord{{\rm
div}}b_{\sigma}\cdot\varphi)(X)\cdot\rho{\mathord{{\rm
d}}}s+\int^{t}_{0}(\mathord{{\rm
div}}\sigma\cdot\varphi)(X)\cdot\rho{\mathord{{\rm d}}}W_{s}\right).$
On the other hand, by (2.2) and Itô’s formula, we have
$\displaystyle\rho_{t}=1+\int^{t}_{0}\rho_{s}\Big{[}\mathord{{\rm
div}}b_{\sigma}+\frac{1}{2}\partial^{2}_{ij}(\sigma^{il}\sigma^{jl})\Big{]}(X_{s}){\mathord{{\rm
d}}}s+\int^{t}_{0}\rho_{s}\partial_{i}\sigma^{il}(X_{s}){\mathord{{\rm
d}}}W^{l}_{s},$
and
$\displaystyle{\mathord{{\rm d}}}[\varphi(X_{t})\rho_{t}]$ $\displaystyle=$
$\displaystyle\Big{[}b^{i}\partial_{i}\varphi+\frac{1}{2}\sigma^{ik}\sigma^{jk}\partial^{2}_{ij}\varphi\Big{]}(X_{t})\rho_{t}{\mathord{{\rm
d}}}t+(\sigma^{il}\partial_{i}\varphi)(X_{t})\rho_{t}{\mathord{{\rm
d}}}W^{l}_{t}$ $\displaystyle+\Big{[}\varphi\mathord{{\rm
div}}b_{\sigma}+\frac{1}{2}\varphi\partial^{2}_{ij}(\sigma^{il}\sigma^{jl})\Big{]}(X_{t})\rho_{t}{\mathord{{\rm
d}}}t+[\varphi\partial_{i}\sigma^{il}](X_{t})\rho_{t}{\mathord{{\rm
d}}}W^{l}_{t}$
$\displaystyle+[\sigma^{il}\partial_{i}\varphi\partial_{j}\sigma^{jl}](X_{t})\rho_{t}{\mathord{{\rm
d}}}t$ $\displaystyle=$ $\displaystyle
b^{i}_{\sigma}\partial_{i}\varphi(X_{t})\rho_{t}{\mathord{{\rm
d}}}t+(\sigma^{il}\partial_{i}\varphi)(X_{t})\rho_{t}{\mathord{{\rm
d}}}W^{l}_{t}$ $\displaystyle+\varphi\mathord{{\rm
div}}b_{\sigma}(X_{t})\rho_{t}{\mathord{{\rm
d}}}t+[\varphi\partial_{i}\sigma^{il}](X_{t})\rho_{t}{\mathord{{\rm
d}}}W^{l}_{t}$
$\displaystyle+\frac{1}{2}\partial_{ij}^{2}(\sigma^{il}\sigma^{jl}\varphi)(X_{t})\rho_{t}{\mathord{{\rm
d}}}t.$
Combining the above calculations, we get
$\displaystyle\frac{1}{2}\int^{t}_{0}\\!\\!\\!\int\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u\varphi-\int^{t}_{0}\\!\\!\\!\int(b^{i}_{\sigma}\partial_{i}u)\varphi-\int^{t}_{0}\\!\\!\\!\int(\sigma^{il}\partial_{i}u)\varphi{\mathord{{\rm
d}}}W^{l}_{s}$ $\displaystyle\qquad=\int u_{0}\left(\int^{t}_{0}{\mathord{{\rm
d}}}[\varphi(X_{s})\rho_{s}]\right)=\int
u_{0}[\varphi(X_{t})\rho_{t}-\varphi]$ $\displaystyle\qquad=\int
u_{0}(X^{-1}_{t})\varphi-\int u_{0}\varphi=\int u_{t}\varphi-\int
u_{0}\varphi.$
The proof is complete. ∎
The following proposition is much technical. We shall prove it in the
appendix.
###### Proposition 2.4.
Assume that SDE (1.1) admits a unique almost everywhere stochastic (or
invertible) flow. Then the following flow property holds: for any $s\geqslant
0$ and $(P\times{\mathscr{L}})$-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\displaystyle X_{t+s}(\omega,x)=X_{t}(\theta_{s}\omega,X_{s}(\omega,x)),\
\forall t\geqslant 0,$ (2.3)
where $\theta_{s}\omega:=\omega(s+\cdot)-\omega(s)$. Moreover, for any bounded
measurable function $\varphi$ on ${\mathbb{R}}^{d}$, define
${\mathbb{T}}_{t}\varphi(x):={\mathbb{E}}\varphi(X_{t}(x)),$
then for any $t,s\geqslant 0$
$\displaystyle{\mathbb{E}}(\varphi(X_{t+s}(x))|{\mathcal{F}}_{s})={\mathbb{T}}_{t}\varphi(X_{s}(x)),\
\ (P\times{\mathscr{L}})-a.e.$ (2.4)
In particular, $({\mathbb{T}}_{t})_{t\geqslant 0}$ forms a bounded linear
operator semigroup on $L^{p}({\mathbb{R}}^{d})$ for any $p\geqslant 1$.
###### Remark 2.5.
Here, an open question is that whether the following stronger flow property
holds: For $(P\times{\mathscr{L}})$-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\displaystyle X_{t+s}(\omega,x)=X_{t}(\theta_{s}\omega,X_{s}(\omega,x)),\
\forall t,s\geqslant 0.$ (2.5)
In the language of random dynamical systems (cf. [4, Definition 1.1.1]),
property (2.3) is called “crude”, and property (2.5) is called “perfect”. A
deep result of Arnold and Scheotzow (cf. [4, p.17, Theorem 1.3.2]) asserted
that a crude cocycle admits an indistinguishable and perfect version. But, it
seems that we can not use their result to deduce (2.5) since it is not clear
how to endow a structure on the set of all measurable transformations so that
it becomes a Hausdorff topological group with countable topological base.
Our main result of this paper is:
###### Theorem 2.6.
Assume that
$\displaystyle\frac{|b(x)|}{1+|x|},\ \mathord{{\rm div}}b(x)\in
L^{\infty}({\mathbb{R}}^{d})$ (2.6)
and one of the following conditions holds:
$\displaystyle b(x)\in\mathrm{BV}_{loc}\mbox{ and }\sigma\mbox{ is independent
of $x$};$ (2.7) $\displaystyle\left\\{\begin{aligned} &|\nabla b(x)|\in(L\log
L)_{loc}({\mathbb{R}}^{d}),\\\ &|\nabla\sigma(x)|,\ \ \sup_{|z|\leqslant
1}|\sigma(x-z)|\cdot|\nabla\mathord{{\rm div}}\sigma|(x)\in
L^{\infty}({\mathbb{R}}^{d}).\end{aligned}\right.$ (2.8)
Then there exists a unique almost everywhere stochastic invertible flow of
(1.1) corresponding to $(b,\sigma)$ in the sense of Definition 2.1.
###### Remark 2.7.
By definitions, $b\in BV_{loc}$ means that $\nabla b$ is a locally finite
vector valued Radon measure on ${\mathbb{R}}^{d}$; and $|\nabla b|\in(L\log
L)_{loc}({\mathbb{R}}^{d})$ means that $|\nabla b|\log(|\nabla b|+1)\in
L^{1}_{loc}({\mathbb{R}}^{d})$. In particular, for any $p>1$,
$L^{p}_{loc}({\mathbb{R}}^{d})\subset(L\log L)_{loc}({\mathbb{R}}^{d})\subset
L^{1}_{loc}({\mathbb{R}}^{d}).$
In (2.8), the second condition on $\sigma$ is certain growth restriction of
$\sigma$ and $\nabla\mathord{{\rm div}}\sigma$.
About the existence of invariant measure of $({\mathbb{T}}_{t})_{t\geqslant
0}$, we have the following criterion.
###### Theorem 2.8.
Assume that SDE (1.1) admits a unique almost everywhere stochastic flow with
$K_{T,b,\sigma}=K_{b,\sigma}$ in (2.1) independent of $T$, and $(b,\sigma)$
satisfies
$\displaystyle{\langle}x,b(x){\rangle}_{{\mathbb{R}}^{d}}+\|\sigma(x)\|_{H.S.}^{2}\leqslant
0(\mbox{ or $-C_{1}|x|^{2}+C_{2}$}),$ (2.9)
where $C_{1},C_{2}>0$, and $\|\sigma(x)\|_{H.S.}$ denotes the Hilbert-Schmidt
norm of matrix $\sigma(x)$. Then $({\mathbb{T}}_{t})_{t\geqslant 0}$ admits an
invariant probability measure $\mu({\mathord{{\rm
d}}}x)=\gamma(x){\mathord{{\rm d}}}x$ with $\gamma\in
L^{\infty}({\mathbb{R}}^{d})\cap L^{1}({\mathbb{R}}^{d})$ so that for all
$\varphi\in L^{1}({\mathbb{R}}^{d})$ and $t\geqslant 0$
$\displaystyle\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{t}\varphi(x)\gamma(x){\mathord{{\rm
d}}}x=\int_{{\mathbb{R}}^{d}}\varphi(x)\gamma(x){\mathord{{\rm d}}}x.$ (2.10)
###### Remark 2.9.
It is well known that if ${\mathbb{T}}_{t}$ is a Feller semigroup, then under
(2.9), there exists an invariant probability measure for ${\mathbb{T}}_{t}$.
In our case, ${\mathbb{T}}_{t}$ may be not a Feller semigroup. In Theorem 6.3
below, we shall give a condition such that $K_{T,b,\sigma}=K_{b,\sigma}$ in
(2.1) is independent of $T$.
These two theorems will be proved in Section 7.
## 3\. Preliminaries
In this section, we prepare some lemmas for later use. Below, we consider SDE
(1.1) and assume that $b,\sigma\in C^{\infty}_{b}({\mathbb{R}}^{d})$ are
$C^{\infty}$-smooth, which together with their derivatives of all orders are
bounded. It is well known that the family of solutions $\\{X_{t}(x),t\geqslant
0\\}_{x\in{\mathbb{R}}^{d}}$ to SDE (1.1) forms a $C^{\infty}$-diffeomorphism
flow (cf. [15, 16]). We have the following simple result about the Jacobian
determinant of stochastic flow.
###### Lemma 3.1.
Let $\rho_{t}(x)$ be defined by (2.2). Then
$\displaystyle\det(\nabla X_{t}(x))=\rho_{t}(x)$ (3.1)
and for any $T>0$ and $p\geqslant 1$,
$\displaystyle{\mathbb{E}}|\det(\nabla
X^{-1}_{T}(x))|^{p}\leqslant\exp\left\\{pT\Big{(}\|[-\mathord{{\rm
div}}b+\frac{1}{2}\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}+\sigma^{il}\partial_{ij}^{2}\sigma^{jl}+\frac{p}{2}|\mathord{{\rm
div}}\sigma|^{2}]^{+}\|_{\infty}\Big{)}\right\\},$ (3.2)
where for a real number $a$, $a^{+}:=a\vee 0:=\max(a,0)$.
###### Proof.
Let $\tilde{b}^{i}:=b^{i}-\frac{1}{2}\sigma^{jl}\partial_{j}\sigma^{il}$. We
write equation (1.1) as Stratonovich form:
${\mathord{{\rm d}}}X=\tilde{b}(X){\mathord{{\rm
d}}}t+\sigma(X)\circ{\mathord{{\rm d}}}W_{t},\ \ X_{0}=x.$
Let $W^{n}_{t}$ be the linearized approximation of $W_{t}$. Consider the
following ODE:
${\mathord{{\rm d}}}X_{n}(x)=\tilde{b}(X_{n}){\mathord{{\rm
d}}}t+\sigma(X_{n})\dot{W}^{n}_{t}{\mathord{{\rm d}}}t.$
Then,
$\det(\nabla X_{n,t}(x))=\exp\left\\{\int^{t}_{0}\mathord{{\rm
div}}\tilde{b}(X_{n,s}(x)){\mathord{{\rm d}}}s+\int^{t}_{0}\mathord{{\rm
div}}\sigma(X_{n,s}(x))\dot{W}^{n}_{s}{\mathord{{\rm d}}}s\right\\}.$
By the limit theorem (cf. [15, 16]), we get
$\det(\nabla X_{t}(x))=\exp\left\\{\int^{t}_{0}\mathord{{\rm
div}}\tilde{b}(X_{s}(x)){\mathord{{\rm d}}}s+\int^{t}_{0}\mathord{{\rm
div}}\sigma(X_{s}(x))\circ{\mathord{{\rm d}}}W_{s}\right\\}.$
(3.1) then follows by rewriting the Stratonovich integral as Itô’s integral.
On the other hand, fix $T>0$ and let $Y_{s}$ solve the following SDE:
${\mathord{{\rm d}}}Y_{t}=-\tilde{b}(Y_{t}){\mathord{{\rm
d}}}t+\sigma(Y_{t})\circ{\mathord{{\rm d}}}W^{T}_{t},\ \ Y_{0}=x,$
where $W^{T}_{t}:=W_{T-t}-W_{T}$. It is well known that (cf. [15, 16])
$X_{T}^{-1}(x)=Y_{T}(x).$
As above, we have
$\displaystyle\det(\nabla Y_{T})$ $\displaystyle=$
$\displaystyle\exp\left\\{-\int^{T}_{0}\mathord{{\rm
div}}\tilde{b}(Y_{s}){\mathord{{\rm d}}}s+\int^{T}_{0}\mathord{{\rm
div}}\sigma(Y_{s})\circ{\mathord{{\rm d}}}W^{T}_{s}\right\\}$ $\displaystyle=$
$\displaystyle\exp\left\\{\int^{T}_{0}\Big{[}-\mathord{{\rm
div}}\tilde{b}+\frac{1}{2}\sigma^{il}\partial^{2}_{ij}\sigma^{jl}\Big{]}(Y_{s}){\mathord{{\rm
d}}}s+\int^{T}_{0}\mathord{{\rm div}}\sigma(Y_{s}){\mathord{{\rm
d}}}W^{T}_{s}\right\\}.$
Note that for any $p\geqslant 1$
$t\mapsto\exp\left\\{p\int^{t}_{0}\mathord{{\rm
div}}\sigma(Y_{s}){\mathord{{\rm
d}}}W^{T}_{s}-\frac{p^{2}}{2}\int^{t}_{0}|\mathord{{\rm
div}}\sigma(Y_{s})|^{2}{\mathord{{\rm d}}}s\right\\}$
is a continuous exponential martingale. Estimate (3.2) then follows by
Hölder’s inequality. ∎
Let $C^{\infty}_{p}({\mathbb{R}}^{d})$ be the set of all smooth functions with
polynomial growth. The following proposition is an easy consequence of
Proposition 2.3 (see also [24, p.180, Theorem 1]).
###### Proposition 3.2.
For any $u_{0}\in C^{\infty}_{p}({\mathbb{R}}^{d})$, let
$u_{t}(x):=u_{0}(X^{-1}_{t}(x))$. Then $u_{t}(x)$ solves the following
stochastic transport equation in the classical sense:
${\mathord{{\rm
d}}}u=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u{\mathord{{\rm
d}}}t-b^{i}_{\sigma}\partial_{i}u{\mathord{{\rm
d}}}t-\sigma^{il}\partial_{i}u{\mathord{{\rm d}}}W^{l}_{t},\ \
u|_{t=0}=u_{0},$
where $b^{i}_{\sigma}:=b^{i}-\sigma^{jl}\partial_{j}\sigma^{il}$.
The following result can be found in [16] and [24, p. 180, Theorem 1].
###### Proposition 3.3.
Let $X_{s,t}(x)$ solve
$X_{s,t}(x)=x+\int^{t}_{s}b(X_{s,r}){\mathord{{\rm
d}}}r+\int^{t}_{s}\sigma(X_{s,r}){\mathord{{\rm d}}}W_{r},\ \ t\geqslant
s\geqslant 0.$
Fix $t>0$. For any $v_{0}\in C^{\infty}_{p}({\mathbb{R}}^{d})$, let
$v_{s,t}(x):=v_{0}(X_{s,t}(x))$, where $s\in[0,t]$. Then $v_{s,t}(x)$ solves
the following backward stochastic Kolmogorov equation in the classical sense:
${\mathord{{\rm
d}}}v+\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}v{\mathord{{\rm
d}}}s+b^{i}\partial_{i}v{\mathord{{\rm
d}}}s+\sigma^{il}\partial_{i}v*{\mathord{{\rm d}}}W_{s}=0,\ \ v|_{s=t}=v_{0},$
where the asterisk denotes the backward Itô’s integral.
Let $C^{+}_{c}({\mathbb{R}}^{d})$ be the set of all non-negative continuous
functions on ${\mathbb{R}}^{d}$ with compact support and ${\mathscr{C}}$ a
countable and dense subset of $C^{+}_{c}({\mathbb{R}}^{d})$ with respect to
the uniform norm
$\|\varphi\|_{\infty}:=\sup_{x\in{\mathbb{R}}^{d}}|\varphi(x)|$. We need the
following simple lemma.
###### Lemma 3.4.
Let $X,Y:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be two measurable
transformations.
(i) Let $\gamma\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})\cap
L^{1}_{loc}({\mathbb{R}}^{d})$. Assume that for any $\varphi\in{\mathscr{C}}$,
$\displaystyle\int\varphi(X)\leqslant\int\varphi\cdot\gamma.$ (3.3)
Then this inequality still holds for all
$\varphi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$. In particular,
${\mathscr{L}}\circ X\ll{\mathscr{L}}$.
(ii) Let $\rho:{\mathbb{R}}^{d}\to{\mathbb{R}}^{+}$ be a positive measurable
function with $\rho\in L^{1}_{loc}({\mathbb{R}}^{d})$. Assume that for any
$\varphi,\psi\in{\mathscr{C}}$,
$\displaystyle\int\varphi(Y)\cdot\psi=\int\varphi\cdot\psi(X)\cdot\rho.$ (3.4)
Then $X$ admits a measurable invertible $Y$, i.e., $X^{-1}(x)=Y(x)$ a.e..
Moreover,
${\mathscr{L}}\circ X^{-1}=\rho{\mathscr{L}},\ \ \ {\mathscr{L}}\circ
X=\rho^{-1}(X^{-1}){\mathscr{L}}.$
###### Proof.
(i) Thanks to the density of ${\mathscr{C}}$ in $C^{+}_{c}({\mathbb{R}}^{d})$,
by Fatou’s lemma and the dominated convergence theorem, one sees that (3.3)
holds for all $\varphi\in C^{+}_{c}({\mathbb{R}}^{d})$. Now, let
$O\subset{\mathbb{R}}^{d}$ be a bounded open set. Define
$\varphi_{n}(x):=1-\left(\frac{1}{1+\mathrm{distance}(x,O^{c})}\right)^{n}.$
Then $\varphi_{n}\in C^{+}_{c}({\mathbb{R}}^{d})$ and for every
$x\in{\mathbb{R}}^{d}$,
$\varphi_{n}(x)\uparrow 1_{O}(x)\mbox{ as $n\to\infty$}.$
By the monotone convergence theorem, we find that (3.3) holds for
$\varphi=1_{O}$. Thus, the desired conclusion follows by the monotone class
theorem.
(ii) As above, one sees that (3.4) holds for all
$\varphi,\psi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$. Thus, we have for all
$\varphi,\psi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$,
$\int\varphi(X\circ
Y)\cdot\psi=\int\varphi(X)\cdot\psi(X)\cdot\rho=\int\varphi\cdot\psi$
and
$\int\varphi\cdot\psi(Y\circ
X)\cdot\rho=\int\varphi(Y)\cdot\psi(Y)=\int\varphi\cdot\psi\cdot\rho.$
By the monotone class theorem, we obtain that for any Borel measurable set
$A\subset{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$,
$\int_{{\mathbb{R}}^{d}}1_{A}(X\circ Y(x),x)\cdot e^{-|x|}{\mathord{{\rm
d}}}x=\int_{{\mathbb{R}}^{d}}1_{A}(x,x)\cdot e^{-|x|}{\mathord{{\rm d}}}x$
and
$\int_{{\mathbb{R}}^{d}}1_{A}(x,Y\circ X(x))\cdot e^{-|x|}{\mathord{{\rm
d}}}x=\int_{{\mathbb{R}}^{d}}1_{A}(x,x)\cdot e^{-|x|}{\mathord{{\rm d}}}x.$
Hence, letting $A=\\{(x,y):x\not=y\\}$ yields that $X\circ Y(x)=x$ and $Y\circ
X(x)=x$ for ${\mathscr{L}}$-almost all $x\in{\mathbb{R}}^{d}$. The result
follows. ∎
The following lemma will play a crucial role for taking limits below.
###### Lemma 3.5.
Let
$X_{n}(\omega,x):\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},n\in{\mathbb{N}}$
be a family of measurable mappings, which are uniformly bounded in
$L^{\infty}_{loc}({\mathbb{R}}^{d};L^{p}(\Omega))$ for any $p\geqslant 1$.
Suppose that for $P$-almost all $\omega\in\Omega$, ${\mathscr{L}}\circ
X_{n}(\omega,\cdot)\ll{\mathscr{L}}$ and the density $\gamma_{n}(\omega,x)$
satisfies
$\displaystyle\sup_{n}~{}\mathrm{ess.}\sup_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|\gamma_{n}(x)|^{2}\leqslant
C_{1}.$ (3.5)
If for ($P\times{\mathscr{L}}$)-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$, $X_{n}(\omega,x)\to X(\omega,x)$
as $n\to\infty$, then for $P$-almost all $\omega\in\Omega$,
${\mathscr{L}}\circ X(\omega,\cdot)\ll{\mathscr{L}}$ and the density $\gamma$
also satisfies
$\displaystyle\mathrm{ess.}\sup_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|\gamma(x)|^{2}\leqslant
C_{1}.$ (3.6)
Moreover, let $(\psi_{n})_{n\in{\mathbb{N}}}$ be a family of measurable
functions on ${\mathbb{R}}^{d}$ and satisfy that for some $C_{2}>0$ and
$\alpha\geqslant 1$
$\displaystyle\sup_{n\in{\mathbb{N}}}~{}\mathrm{ess.}\sup_{x\in{\mathbb{R}}^{d}}\frac{|\psi_{n}(x)|}{1+|x|^{\alpha}}\leqslant
C_{2}.$ (3.7)
If $\psi_{n}$ converges to some $\psi$ in $L^{1}_{loc}({\mathbb{R}}^{d})$,
then for any $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int_{B_{N}}|\psi_{n}(X_{n})-\psi(X)|=0.$
(3.8)
###### Proof.
Fix $\varphi\in{\mathscr{C}}\subset C^{+}_{c}({\mathbb{R}}^{d})$ with support
contained in $B_{N}$ for some $N>0$. Then by Fubini’s theorem and Fatou’s
lemma, we have for $P$-almost all $\omega\in\Omega$,
$\displaystyle\int\varphi(X(\omega))\leqslant\varliminf_{n\to\infty}\int\varphi(X_{n}(\omega))=\varliminf_{n\to\infty}\int\varphi\cdot\gamma_{n}(\omega)=:\varliminf_{n\to\infty}J^{\varphi}_{n}(\omega).$
(3.9)
By (3.5), there exists a subsequence still denoted by $n$ and a $\gamma_{0}\in
L^{\infty}({\mathbb{R}}^{d};L^{2}(\Omega))$ satisfying (3.6) such that
$\gamma_{n}\mbox{ weakly * converges to $\gamma_{0}$ in
$L^{\infty}({\mathbb{R}}^{d};L^{2}(\Omega))$}.$
Since $\gamma_{n}$ also weakly converges to $\gamma_{0}$ in
$L^{2}(B_{N}\times\Omega)$, by Banach-Saks’ theorem, there is another
subsequence still denoted by $n$ such that its Cesàro mean
$\bar{\gamma}_{n}:=\frac{1}{n}\sum_{k=1}^{n}\gamma_{k}$ strongly converges to
$\gamma_{0}$ in $L^{2}(B_{N}\times\Omega)$. Thus, there is another subsequence
still denoted by $n$ such that for $P$-almost all $\omega\in\Omega$,
$\bar{\gamma}_{n}(\omega)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\gamma_{0}(\omega)\mbox{ in $L^{2}(B_{N})$}.$
Hence,
$\bar{J}^{\varphi}_{n}(\omega):=\frac{1}{n}\sum_{k=1}^{n}J^{\varphi}_{k}(\omega)=\int\varphi\cdot\bar{\gamma}_{n}(\omega)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int\varphi\cdot\gamma_{0}(\omega),$
which together with (3.9) yields that for $P$-almost all $\omega$,
$\int\varphi(X(\omega))\leqslant\varliminf_{n\to\infty}J^{\varphi}_{n}(\omega)\leqslant\lim_{n\to\infty}\bar{J}^{\varphi}_{n}(\omega)=\int\varphi\cdot\gamma_{0}(\omega).$
Since ${\mathscr{C}}$ is countable, we may find a common null set
$\Omega^{\prime}\subset\Omega$ such that the above inequality holds for all
$\omega\notin\Omega^{\prime}$ and $\varphi\in{\mathscr{C}}$. The first
conclusion then follows by (i) of Lemma 3.4.
We now prove (3.8). We make the following decomposition:
$\displaystyle\int_{B_{N}}|\psi_{n}(X_{n})-\psi(X)|\leqslant\int_{B_{N}}|\psi_{n}(X_{n})-\psi(X_{n})|$
$\displaystyle\qquad+\int_{B_{N}}|\psi(X_{n})-\psi(X)|=:I_{n}+J_{n}.$
By (3.7) and $\psi_{n}\to\psi$ in $L^{1}_{loc}({\mathbb{R}}^{d})$, we also
have
$\mathrm{ess.}\sup_{x\in{\mathbb{R}}^{d}}\frac{|\psi(x)|}{1+|x|^{\alpha}}\leqslant
C_{2}.$
Let $(\phi_{m})_{m\in{\mathbb{N}}}$ be a family of bounded continuous
functions such that $\phi_{m}\to\psi$ in $L^{1}_{loc}({\mathbb{R}}^{d})$ as
$m\to\infty$ and
$\displaystyle\sup_{m\in{\mathbb{N}}}~{}\mathrm{ess.}\sup_{x\in{\mathbb{R}}^{d}}\frac{|\phi_{m}(x)|}{1+|x|^{\alpha}}\leqslant
C_{2}.$ (3.10)
We have
$\displaystyle J_{n}$ $\displaystyle\leqslant$
$\displaystyle\int_{B_{N}}|\phi_{m}(X_{n})-\psi(X_{n})|+\int_{B_{N}}|\phi_{m}(X)-\psi(X)|$
$\displaystyle+\int_{B_{N}}|\phi_{m}(X_{n})-\phi_{m}(X)|=:J_{1nm}+J_{2m}+J_{3nm}.$
For any $R>0$, we may write
$\displaystyle J_{1nm}=\int_{B_{N}\cap\\{|X_{n}|\leqslant
R\\}}|\phi_{m}(X_{n})-\psi(X_{n})|$
$\displaystyle\qquad+\int_{B_{N}\cap\\{|X_{n}|>R\\}}|\phi_{m}(X_{n})-\psi(X_{n})|=:J_{1nm}^{1,R}+J^{2,R}_{1nm}.$
By the change of variable and (3.5), we have
${\mathbb{E}}J_{1nm}^{1,R}\leqslant\int_{B_{R}}|\phi_{m}-\psi|\cdot{\mathbb{E}}\gamma_{n}\leqslant
C_{1}\int_{B_{R}}|\phi_{m}-\psi|.$
By Chebyshev’s inequality and (3.10), we have
${\mathbb{E}}J^{2,R}_{2nm}\leqslant\frac{C_{N,\alpha}}{R}\sup_{x\in
B_{N}}{\mathbb{E}}(1+|X_{n}(x)|^{2\alpha})\leqslant\frac{C_{N,\alpha}}{R}.$
Combining the above two estimates, we obtain
$\displaystyle\lim_{m\to\infty}\sup_{n\in{\mathbb{N}}}{\mathbb{E}}J_{1nm}=0.$
(3.11)
Similarly, we also have
$\lim_{m\to\infty}{\mathbb{E}}J_{2m}=0$
and for fixed $m\in{\mathbb{N}}$, by the dominated convergence theorem,
$\lim_{n\to\infty}{\mathbb{E}}J_{3nm}=0.$
Hence,
$\lim_{n\to\infty}{\mathbb{E}}J_{n}=0.$
As proving (3.11), we also have
$\lim_{n\to\infty}{\mathbb{E}}I_{n}=0.$
The proof is then complete. ∎
The following lemma will be used to prove the strong convergence in Theorem
4.7 below.
###### Lemma 3.6.
Let ${\mathbb{B}}$ be a separable and uniformly convex Banach space. Let
$(u_{n})_{n\in{\mathbb{N}}}$ be a bounded sequence in
$L^{1}(\Omega;C([0,T];{\mathbb{B}}))$. Assume that for some $u\in
L^{1}(\Omega;C([0,T];{\mathbb{B}}))$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T]}|{}_{{\mathbb{B}}^{*}}{\langle}\phi,u_{n}(t)-u(t){\rangle}_{\mathbb{B}}|\right)=0,\
\ \forall\phi\in{\mathbb{B}}^{*},$ (3.12)
where ${\mathbb{B}}^{*}$ is the dual space of ${\mathbb{B}}$, and
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T]}\big{|}\|u_{n}(t)\|_{{\mathbb{B}}}-\|u(t)\|_{\mathbb{B}}\big{|}\right)=0.$
(3.13)
Then $\sup_{t\in[0,T]}\|u_{n}(t)-u(t)\|_{{\mathbb{B}}}$ converges to zero in
probability as $n\to\infty$.
###### Proof.
It is enough to prove that for any subsequence $n_{k}$, there exists a
subsubsequence $n^{\prime}_{k}$ such that
$\sup_{t\in[0,T]}\|u_{n_{k}^{\prime}}(t)-u(t)\|_{{\mathbb{B}}}$ converges to
zero $P$-almost surely as $k\to\infty$. We now fix a subsequence $n_{k}$
below. Since ${\mathbb{B}}^{*}$ is separable, by (3.12) and (3.13), we may
find a subsubsequence $n_{k}^{\prime}$ and a measurable set
$\Omega^{\prime}\subset\Omega$ with $P(\Omega^{\prime})=1$ such that for all
$\omega\in\Omega^{\prime}$, $u(\omega,\cdot)\in C([0,T];{\mathbb{B}})$ and
$\displaystyle\lim_{k\to\infty}\sup_{t\in[0,T]}|{}_{{\mathbb{B}}^{*}}{\langle}\phi,u_{n^{\prime}_{k}}(\omega,t)-u(\omega,t){\rangle}_{\mathbb{B}}|=0,\
\ \forall\phi\in{\mathbb{B}}^{*}$ (3.14)
and
$\displaystyle\lim_{k\to\infty}\sup_{t\in[0,T]}\big{|}\|u_{n^{\prime}_{k}}(\omega,t)\|_{{\mathbb{B}}}-\|u(\omega,t)\|_{\mathbb{B}}\big{|}=0.$
(3.15)
We want to show that for such $\omega\in\Omega^{\prime}$,
$\lim_{k\to\infty}\sup_{t\in[0,T]}\|u_{n^{\prime}_{k}}(\omega,t)-u(\omega,t)\|_{\mathbb{B}}=0.$
Suppose that this is not true. Then, there exist a $\delta>0$ and a sequence
$(t_{k})_{k\in{\mathbb{N}}}\subset[0,T]$ such that
$\displaystyle\|u_{n^{\prime}_{k}}(\omega,t_{k})-u(\omega,t_{k})\|_{\mathbb{B}}\geqslant\delta,\
\ \forall k\in{\mathbb{N}}.$ (3.16)
Without loss of generality, we assume that $t_{k}$ converges to $t_{0}$. By
(3.14), (3.15) and $u(\omega,\cdot)\in C([0,T];{\mathbb{B}})$, we have
$\lim_{k\to\infty}\|u_{n^{\prime}_{k}}(\omega,t_{k})-u(\omega,t_{0})\|_{\mathbb{B}}=0,$
which together with $u(\omega,\cdot)\in C([0,T];{\mathbb{B}})$ yields
$\lim_{k\to\infty}\|u_{n^{\prime}_{k}}(\omega,t_{k})-u(\omega,t_{k})\|_{\mathbb{B}}=0.$
This is a contradiction with (3.16). The proof is complete. ∎
We also recall some facts about local maximal functions. Let $f$ be a locally
integrable function on ${\mathbb{R}}^{d}$. For every $R>0$, the local maximal
function is defined by
$M_{R}f(x):=\sup_{0<r<R}\frac{1}{|B_{r}|}\int_{B_{r}}f(x+y){\mathord{{\rm
d}}}y=:\sup_{0<r<R}\fint_{B_{r}}f(x+y){\mathord{{\rm d}}}y.$
The following result can be found in [7, p.143, Theorem 3] and [5, Appendix
A].
###### Lemma 3.7.
(i) (Morrey’s inequality) Let $f\in L^{1}_{loc}({\mathbb{R}}^{d})$ be such
that $\nabla f\in L^{q}_{loc}({\mathbb{R}}^{d})$ for some $q>d$. Then there
exist $C_{q,d}>0$ and a negligible set $A$ such that for all $x,y\in A^{c}$
with $|x-y|\leqslant R$,
$\displaystyle|f(x)-f(y)|$ $\displaystyle\leqslant$ $\displaystyle
C_{q,d}\cdot|x-y|\cdot\left(\fint_{B_{|x-y|}}|\nabla f|^{q}(x+z){\mathord{{\rm
d}}}z\right)^{1/q}$ (3.17) $\displaystyle\leqslant$ $\displaystyle
C_{q,d}\cdot|x-y|\cdot(M_{R}|\nabla f|^{q}(x))^{1/q}.$
(ii) Let $f\in L^{1}_{loc}({\mathbb{R}}^{d})$ be such that $\nabla f\in
L^{1}_{loc}({\mathbb{R}}^{d})$. Then there exist $C_{d}>0$ and a negligible
set $A$ such that for all $x,y\in A^{c}$ with $|x-y|\leqslant R$,
$\displaystyle|f(x)-f(y)|\leqslant C_{d}\cdot|x-y|\cdot(M_{R}|\nabla
f|(x)+M_{R}|\nabla f|(y)).$ (3.18)
(iii) Let $f\in(L\log L)_{loc}({\mathbb{R}}^{d})$. Then for any $N,R>0$ and
some $C_{d,N},C_{d}>0$,
$\displaystyle\int_{B_{N}}M_{R}|f|\leqslant
C_{d,N}+C_{d}\int_{B_{N+R}}|f|\log(|f|+1).$ (3.19)
(iv) Let $f\in L^{p}_{loc}({\mathbb{R}}^{d})$ for some $p>1$. Then for some
$C_{d,p}>0$ and any $N,R>0$,
$\displaystyle\left(\int_{B_{N}}(M_{R}|f|)^{p}\right)^{1/p}\leqslant
C_{d,p}\left(\int_{B_{N+R}}|f|^{p}\right)^{1/p}.$ (3.20)
## 4\. Stochastic Transport Equations
In this section we work on $[0,T]$ and mainly study the following stochastic
transport equation:
$\displaystyle{\mathord{{\rm
d}}}u=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u+b^{i}\partial_{i}u\Big{]}{\mathord{{\rm
d}}}t+\sigma^{il}\partial_{i}u{\mathord{{\rm d}}}W^{l}_{t},\ \
u|_{t=0}=u_{0},$ (4.1)
where $\sigma\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{m}$ does not depend on
$x$, and $b$ is a BV vector field and satisfies
$\displaystyle\frac{b(x)}{1+|x|},\ \mathord{{\rm div}}b(x)\in
L^{\infty}({\mathbb{R}}^{d}),\ \ b\in\mathrm{BV}_{loc}.$ (4.2)
We first introduce the following notion of renormalized solutions for equation
(4.4).
###### Definition 4.1.
A measurable and (${\mathcal{F}}_{t}$)-adapted stochastic field
$u:[0,T]\times\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ is called a
renormalized solution of (4.1) if for any $\beta\in C^{2}({\mathbb{R}})$,
$v_{t}(\omega,x):=\beta(\arctan u_{t}(\omega,x))$
solves (4.1) in the distributional sense, i.e., for any $\phi\in
C^{\infty}_{c}({\mathbb{R}}^{d})$
$\displaystyle\int v_{t}\phi=\int
v_{0}\phi+\frac{1}{2}\int^{t}_{0}\\!\\!\\!\int
v\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\phi-\int^{t}_{0}\\!\\!\\!\int
v(\mathord{{\rm div}}b\phi+b^{i}\partial_{i}\phi)-\int^{t}_{0}\\!\\!\\!\int
v\sigma^{il}\partial_{i}\phi{\mathord{{\rm d}}}W^{l}_{s}.$ (4.3)
###### Remark 4.2.
Since $v$ is bounded, it is clear that both sides of (4.3) are well defined.
Our main result in this section is that
###### Theorem 4.3.
Assume that condition (4.2) holds.
(Existence and Uniqueness) For any measurable function $u_{0}$, there exists a
unique renormalized solution $u$ to stochastic transport equation (4.1) with
$u|_{t=0}=u_{0}$ in the sense of Definition 4.1. Moreover, for any $p>1$ and
$N>0$,
$\arctan u\in L^{p}(\Omega;C([0,T];L^{p}(B_{N}))).$
(Stability) Let $b_{n}\in L^{1}_{loc}({\mathbb{R}}^{d})$ be such that
$\mathord{{\rm div}}b_{n}\in L^{1}_{loc}({\mathbb{R}}^{d})$ and
$b_{n},\mathord{{\rm div}}b_{n}$ converge to $b,\mathord{{\rm div}}b$
respectively in $L^{1}_{loc}({\mathbb{R}}^{d})$. Let $u^{n}_{0}$
${\mathscr{L}}$-almost everywhere converge to $u_{0}$. Let $u^{n}$ and $u$ be
the renormalized solutions corresponding to $(b^{n},u^{n}_{0})$ and
$(b,u_{0})$ in the sense of Definition 4.1. Then for any $p>1$ and $N>0$,
$\arctan u^{n}\to\arctan u\ \ \mbox{strongly in
$L^{p}(\Omega;C([0,T];L^{p}(B_{N})))$}.$
For proving this theorem, we first study the following more general stochastic
partial differential equation:
$\displaystyle{\mathord{{\rm
d}}}u=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u+b^{i}\partial_{i}u+cu\Big{]}{\mathord{{\rm
d}}}t+(\sigma^{il}\partial_{i}u+h^{l}u){\mathord{{\rm d}}}W^{l}_{t},\ \
u|_{t=0}=u_{0},$ (4.4)
where $\sigma$ and $b$ are as above and
$\displaystyle c,h,\nabla h\in L^{\infty}({\mathbb{R}}^{d}).$ (4.5)
As Definition 4.1, we also introduce the following notion about the
renormalized solutions for equation (4.4).
###### Definition 4.4.
We say $u\in L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ a
renormalized solution of (4.4) if for any $\beta\in C^{2}({\mathbb{R}})$, it
holds that in the distributional sense
$\displaystyle{\mathord{{\rm d}}}\beta(u)$ $\displaystyle=$
$\displaystyle\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\beta(u)+b^{i}\partial_{i}\beta(u)+cu\beta^{\prime}(u)\Big{]}{\mathord{{\rm
d}}}t$
$\displaystyle+\Big{[}\frac{1}{2}|h|^{2}\beta^{\prime\prime}(u)u^{2}+h^{l}\sigma^{il}\partial_{i}\beta^{\prime}(u)u\Big{]}{\mathord{{\rm
d}}}t$
$\displaystyle+(\sigma^{il}\partial_{i}\beta(u)+h^{l}u\beta^{\prime}(u)){\mathord{{\rm
d}}}W^{l}_{t}.$
We remark that for equation (4.4), the renormalized solution is a nonlinear
notion, whereas the distributional solution is a linear notion. However, under
(4.2) and (4.5), we can show that these two notions are equivalent. For this
aim, we need the following class of regularized functions:
$\displaystyle{\mathcal{N}}:=\left\\{\varrho\in C^{\infty}_{c}(B_{1}),\ \
\varrho\geqslant 0,\ \ \int\varrho=1\right\\}.$ (4.6)
We now establish the following equivalence between the distributional solution
and renormalized solution.
###### Proposition 4.5.
Let $u\in L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ be a
distributional solution of (4.4). Then under (4.2) and (4.5), $u$ is also a
renormalized solution of (4.4) in the sense of Definition 4.4.
###### Proof.
Let $\varrho\in{\mathcal{N}}$ and set
$\varrho_{\varepsilon}(x):=\varepsilon^{-d}\varrho(x/\varepsilon)$. Define
$u_{\varepsilon}:=u_{t,\varepsilon}(x):=u_{t}*\varrho_{\varepsilon}(x)=\int
u_{t}(y)\varrho_{\varepsilon}(x-y){\mathord{{\rm d}}}y.$
Taking convolutions for both sides of (4.4), we obtain
${\mathord{{\rm
d}}}u_{\varepsilon}=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u_{\varepsilon}+(b^{i}\partial_{i}u)*\varrho_{\varepsilon}+(cu)*\varrho_{\varepsilon}\Big{]}{\mathord{{\rm
d}}}t+[\sigma^{il}\partial_{i}u_{\varepsilon}+(h^{l}u)*\varrho_{\varepsilon}]{\mathord{{\rm
d}}}W^{l}_{t}.$
Let $\beta\in C^{2}({\mathbb{R}})$. By Itô’s formula, we have
$\displaystyle{\mathord{{\rm d}}}\beta(u_{\varepsilon})$ $\displaystyle=$
$\displaystyle\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\beta(u_{\varepsilon})+((b^{i}\partial_{i}u)*\varrho_{\varepsilon}+(cu)*\varrho_{\varepsilon})\cdot\beta^{\prime}(u_{\varepsilon})\Big{]}{\mathord{{\rm
d}}}t$
$\displaystyle+\Big{[}\frac{1}{2}\beta^{\prime\prime}(u_{\varepsilon})((h^{l}u)*\varrho_{\varepsilon})^{2}+\beta^{\prime\prime}(u_{\varepsilon})\sigma^{il}\partial_{i}u_{\varepsilon}\cdot(h^{l}u)*\varrho_{\varepsilon}\Big{]}{\mathord{{\rm
d}}}t$
$\displaystyle+[\sigma^{il}\partial_{i}\beta(u_{\varepsilon})+(h^{l}u)*\varrho_{\varepsilon}\cdot\beta^{\prime}(u_{\varepsilon})]{\mathord{{\rm
d}}}W^{l}_{t}.$
Write
$r^{\rho}_{\varepsilon}:=((b^{i}\partial_{i}u)*\varrho_{\varepsilon}-b^{i}\partial_{i}(u*\varrho_{\varepsilon}))\cdot\beta^{\prime}(u_{\varepsilon})$
and
$[\varrho_{\varepsilon},h^{l}](u):=(h^{l}u)*\varrho_{\varepsilon}-h^{l}(u*\varrho_{\varepsilon}).$
Let $\phi\in C^{\infty}_{c}({\mathbb{R}}^{d})$. Multiplying both sides by
$\phi$ and integrating over ${\mathbb{R}}^{d}$, by the integration by parts
formula, we get
$\displaystyle\int\beta(u_{t,\varepsilon})\phi$ $\displaystyle=$
$\displaystyle\int\beta(u_{0,\varepsilon})\phi+\int^{t}_{0}\\!\\!\\!\int\beta(u_{\varepsilon})\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\phi-\mathord{{\rm
div}}b\phi-b^{i}\partial_{i}\phi\Big{]}$
$\displaystyle+\int^{t}_{0}\\!\\!\\!\int
r^{\varrho}_{\varepsilon}\phi+\int^{t}_{0}\\!\\!\\!\int(cu)*\varrho_{\varepsilon})\cdot\beta^{\prime}(u_{\varepsilon})\phi$
$\displaystyle+\int^{t}_{0}\\!\\!\\!\int\frac{1}{2}\beta^{\prime\prime}(u_{\varepsilon})((h^{l}u)*\varrho_{\varepsilon})^{2}\phi+\int^{t}_{0}\\!\\!\\!\int\Big{[}\beta(u_{\varepsilon})-u_{\varepsilon}\beta^{\prime}(u_{\varepsilon})\Big{]}\sigma^{il}\partial_{i}(h^{l}\phi)$
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int\beta^{\prime}(u_{\varepsilon})\sigma^{il}\Big{(}\partial_{i}[\varrho_{\varepsilon},h^{l}](u)\phi+[\varrho_{\varepsilon},h^{l}](u)\partial_{i}\phi\Big{)}$
$\displaystyle+\int^{t}_{0}\\!\\!\\!\int[(h^{l}u)*\varrho_{\varepsilon}\cdot\beta^{\prime}(u_{\varepsilon})\phi-\sigma^{il}\partial_{i}\phi\beta(u_{\varepsilon})]{\mathord{{\rm
d}}}W^{l}_{s}.$
Now taking limits $\varepsilon\to 0$ and using [6, p.516, Lemma II.1], we find
that
$\displaystyle\Bigg{|}\int\beta(u_{t})\phi-\int\beta(u_{0})\phi-\int^{t}_{0}\\!\\!\\!\int\beta(u)\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\phi-\mathord{{\rm
div}}b\phi-b^{i}\partial_{i}\phi\Big{]}$
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int(cu)\cdot\beta^{\prime}(u)\phi-\int^{t}_{0}\\!\\!\\!\int\frac{1}{2}\beta^{\prime\prime}(u)(h^{l}u)^{2}\phi-\int^{t}_{0}\\!\\!\\!\int\Big{[}\beta(u)-u_{\varepsilon}\beta^{\prime}(u)\Big{]}\sigma^{il}\partial_{i}(h^{l}\phi)$
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int[(h^{l}u)\cdot\beta^{\prime}(u)\phi-\sigma^{il}\partial_{i}\phi\beta(u)]{\mathord{{\rm
d}}}W^{l}_{s}\Bigg{|}\leqslant\limsup_{\varepsilon\to
0}\left|\int^{t}_{0}\\!\\!\\!\int r^{\varrho}_{\varepsilon}\phi\right|.$
Since the left hand side of the above inequality does not depend on $\varrho$,
it suffices to show that
$\inf_{\varrho\in{\mathcal{N}}}\limsup_{\varepsilon\to
0}\left|\int^{t}_{0}\\!\\!\\!\int r^{\varrho}_{\varepsilon}\phi\right|=0.$
This has been proved in the proof of [2, Theorem 3.5]. ∎
Using Proposition 4.5, we can prove the uniqueness of distributional
solutions.
###### Proposition 4.6.
Let $u\in L^{\infty}([0,T]\times\Omega;L^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d}))$ be a distributional solution of (4.4). If
$u|_{t=0}=0$, then
$u_{t}(\omega,x)=0,\ \ a.e.$
###### Proof.
Let $\chi\in C^{\infty}_{c}({\mathbb{R}}^{d})$ be a nonnegative cutoff
function with
$\|\chi\|_{\infty}\leqslant 1,\ \ \chi(x)=\left\\{\begin{aligned} &1,\ \
|x|\leqslant 1,\\\ &0,\ \ |x|\geqslant 2.\end{aligned}\right.$ (4.7)
Set $\chi_{n}(x):=\chi(x/n)$. By Proposition 4.5 and Definition 4.4, we have
$\displaystyle{\mathbb{E}}\int u^{2}_{t}\chi_{n}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\int^{t}_{0}\\!\\!\\!\int\Big{[}\frac{1}{2}u^{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\chi_{n}-u^{2}b^{i}\partial_{i}\chi_{n}\Big{]}$
$\displaystyle+{\mathbb{E}}\int^{t}_{0}\\!\\!\\!\int\Big{[}-u^{2}\mathord{{\rm
div}}b\chi_{n}+2cu^{2}\chi_{n}\Big{]}$
$\displaystyle+{\mathbb{E}}\int^{t}_{0}\\!\\!\\!\int\Big{[}|h|^{2}u^{2}\chi_{n}-u^{2}\partial_{i}(h^{l}\chi_{n})\sigma^{il}\Big{]}.$
Observe that by (4.2)
$\displaystyle|b^{i}\partial_{i}\chi_{n}|\leqslant\frac{|b|\cdot
1_{\\{n\leqslant|x|\leqslant 2n\\}}\cdot\|\nabla\chi\|_{\infty}}{n}\leqslant
C_{1}\cdot 1_{\\{|x|\geqslant n\\}},$ (4.8)
where $C_{1}=3\|b/(1+|x|)\|_{\infty}\cdot\|\nabla\chi\|_{\infty}$, and
$|h^{l}\partial_{i}\chi_{n}|\leqslant\|h\|_{\infty}\cdot\|\nabla\chi\|_{\infty}.$
Since $u^{2}\in L^{\infty}([0,T]\times\Omega;L^{1}({\mathbb{R}}^{d}))$, by
letting $n\to\infty$, we obtain
$\displaystyle{\mathbb{E}}\int u_{t}^{2}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\int^{t}_{0}\\!\\!\\!\int\Big{[}(-\mathord{{\rm
div}}b+2c+|h|^{2}-\sigma^{il}\partial_{i}h^{l})u^{2}_{s}\Big{]}$
$\displaystyle\leqslant$ $\displaystyle\|2c+|h|^{2}-\mathord{{\rm
div}}b-\sigma^{il}\partial_{i}h^{l}\|_{\infty}\int^{t}_{0}\left({\mathbb{E}}\int
u^{2}_{s}\right){\mathord{{\rm d}}}s,$
which gives by Gronwall’s inequality that
${\mathbb{E}}\int u_{t}^{2}=0.$
The uniqueness follows. ∎
In general, it is not expected to have a bounded solution for SPDE (4.4)
because of the presence of stochastic integral
$\int^{t}_{0}h^{l}u{\mathord{{\rm d}}}W^{l}_{s}$ (cf. [24]). We now turn back
to stochastic transport equation (4.1), and prove the existence-uniqueness and
stability of $L^{\infty}$-distributional solutions when the initial value
belongs to $L^{\infty}({\mathbb{R}}^{d})$.
###### Theorem 4.7.
Assume that condition (4.2) holds.
(Existence and Uniqueness) For any $u_{0}\in L^{\infty}({\mathbb{R}}^{d})$,
there exists a unique distributional solution $u\in
L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ (also a renormalized
solution in the sense of Definition 4.4) to stochastic transport equation
(4.1) satisfying
$\displaystyle\|u_{t}(\omega)\|_{\infty}\leqslant\|u_{0}\|_{\infty}.$ (4.9)
Moreover, there is a version still denoted by $u$ such that for any $p>1$ and
$N>0$
$\displaystyle u\in L^{p}(\Omega;C([0,T];L^{p}(B_{N}))).$ (4.10)
(Stability) Let $b_{n}\in L^{1}_{loc}({\mathbb{R}}^{d})$ and $u^{n}_{0}\in
L^{\infty}({\mathbb{R}}^{d})$ be such that $\mathord{{\rm div}}b_{n}\in
L^{1}_{loc}({\mathbb{R}}^{d})$ and $b_{n},\mathord{{\rm div}}b_{n},u^{n}_{0}$
converge to $b,\mathord{{\rm div}}b,u_{0}$ respectively in
$L^{1}_{loc}({\mathbb{R}}^{d})$. Let $u_{n},u\in
L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ be the distributional
solutions of (4.1) corresponding to $(b_{n},u^{n}_{0})$ and $(b,u_{0})$ and
satisfy (4.10). Assume that
$\displaystyle\sup_{n}\|u_{n}\|_{L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})}<+\infty.$
(4.11)
Then for any $p>1$ and $N>0$,
$\displaystyle u_{n}\to u\ \ \mbox{strongly in
$L^{p}(\Omega;C([0,T];L^{p}(B_{N})))$}.$ (4.12)
###### Proof.
(Existence) Fix a $\varrho\in{\mathcal{N}}$ and a cutoff function $\chi$
satisfying (4.7). Let
$\varrho_{n}(x):=n^{d}\varrho(nx),\ \ \chi_{n}(x)=\chi(x/n)$
and define
$\displaystyle b_{n}=b*\varrho_{n}\cdot\chi_{n}.$ (4.13)
Let $X_{n}$ solve the following SDE:
${\mathord{{\rm d}}}X_{n}=-b_{n}(X_{n}){\mathord{{\rm
d}}}t-\sigma{\mathord{{\rm d}}}W_{t},\ \ X_{n}|_{t=0}=x.$
By Proposition 3.2, $u_{n,t}:=u_{0}(X^{-1}_{n,t})$ solves the following SPDE:
${\mathord{{\rm
d}}}u_{n}=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u_{n}+b^{i}_{n}\partial_{i}u_{n}\Big{]}{\mathord{{\rm
d}}}t+\sigma^{il}\partial_{i}u_{n}{\mathord{{\rm d}}}W^{l}_{t},\ \
u_{n}|_{t=0}=u_{0}.$
Clearly, $u_{n}\in L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ and
$\|u_{n,t}(\omega,\cdot)\|_{\infty}\leqslant\|u_{0}\|_{\infty}.$
Therefore, for some $u\in L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$
and some subsequence $n_{k}$,
$u_{n_{k}}\to u\mbox{ weakly* in
$L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$}.$
Taking weakly* limits, it is easy to see that $u$ is a distributional solution
of (4.1). Moreover, (4.9) holds. As for (4.10), it can be seen from the proof
of the following stability.
(Uniqueness) Let $u$ and $\hat{u}$ be two distributional solutions of (4.1)
with the same initial value. Then $v:=u-\hat{u}\in
L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ is still a distributional
solution of (4.1) with zero initial value. Since $v$ does not belong to
$L^{\infty}([0,T]\times\Omega;L^{1}({\mathbb{R}}^{d}))$, we can not directly
use Proposition 4.6 to obtain $v=0$. Below, we use a simple trick. Let
$\lambda(x):=\frac{1}{(1+|x|^{2})^{d}},\ \ \hat{v}_{t}:=v_{t}\cdot\lambda.$
It is easy to see that
$\hat{v}_{t}\in L^{\infty}([0,T]\times\Omega;L^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})).$
Moreover, noting that
$\partial_{i}\lambda(x)=-\frac{2dx_{i}}{1+|x|^{2}}\lambda(x),\ \
\partial_{i}\partial_{j}\lambda(x)=\left(\frac{4d(d+1)x_{i}x_{j}}{(1+|x|^{2})^{2}}-\frac{2d\delta_{ij}}{1+|x|^{2}}\right)\lambda(x),$
we can check that $\hat{v}_{t}$ is a distributional solution of
${\mathord{{\rm
d}}}\hat{v}=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\hat{v}+\hat{b}^{i}\partial_{i}\hat{v}+c\hat{v}\Big{]}{\mathord{{\rm
d}}}t+(\sigma^{il}\partial_{i}\hat{v}+h^{l}\hat{v}){\mathord{{\rm
d}}}W^{l}_{t},\ \ \hat{v}|_{t=0}=0,$
where
$\hat{b}^{i}(x)=b^{i}(x)+\frac{2dx_{j}\sigma^{il}\sigma^{jl}}{1+|x|^{2}},\ \
h^{l}(x)=\frac{2dx_{i}\sigma^{il}}{1+|x|^{2}}$
and
$c(x)=\frac{2dx_{i}b^{i}(x)}{1+|x|^{2}}+\left(\frac{2d(d-1)x_{i}x_{j}}{(1+|x|^{2})^{2}}+\frac{d\delta_{ij}}{1+|x|^{2}}\right)\sigma^{il}\sigma^{jl}.$
By (4.2), one sees that $\hat{b}$ still satisfies (4.2) and $c,h$ satisfy
(4.5). Thus, we can use Proposition 4.6 to get $\hat{v}=0$. The uniqueness
follows.
(Stability) We follow DiPerna-Lions’ argument [6, p.523]. Fix an even number
$p>1$ and let $v_{n}:=u_{n}^{p}$. Then, by Definition 4.4, $v_{n}$ is a
distributional solution of
${\mathord{{\rm
d}}}v_{n}=\Big{[}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}v_{n}+b^{i}_{n}\partial_{i}v_{n}\Big{]}{\mathord{{\rm
d}}}t+\sigma^{il}\partial_{i}v_{n}{\mathord{{\rm d}}}W^{l}_{t},\ \
v_{n}|_{t=0}=(u^{n}_{0})^{p}.$
By (4.11), we have
$\sup_{n}\|u_{n}\|_{L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})}+\sup_{n}\|v_{n}\|_{L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})}<+\infty.$
Without loss of generality, we may assume that $u_{n}$ and $v_{n}$ converges
weakly* in $L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$ to $u$ and
$v$, which are distributional solutions of (4.1) corresponding to
$u|_{t=0}=u_{0}$ and $v|_{t=0}=u_{0}^{p}$ by the assumptions. By Proposition
4.5 and the uniqueness proved above, we have
$u^{p}=v.$
Thus,
$u_{n}^{p}\to u^{p}\ \ \mbox{weakly* in
$L^{\infty}([0,T]\times\Omega\times{\mathbb{R}}^{d})$}.$
Hence, for any $N>0$,
${\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{B_{N}}u_{n}^{p}\to{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{B_{N}}u^{p}.$
By virtue of
$u_{n}\to u\ \ \mbox{weakly in $L^{p}([0,T]\times\Omega\times B_{N})$},$
we thus obtain that for any $N>0$,
$\displaystyle u_{n}\to u\ \ \mbox{strongly in $L^{p}([0,T]\times\Omega\times
B_{N})$}.$ (4.14)
We now strengthen this convergence to (4.12). Let $w_{n}=u_{n}-u$. Then we
have for any $N>0$ and $\phi\in C^{\infty}_{c}(B_{N})$,
$\displaystyle\left|\int w_{n,t}\phi\right|$ $\displaystyle=$
$\displaystyle\Bigg{|}\int
w_{n,0}\phi+\int^{t}_{0}\\!\\!\\!\int\Big{[}w_{n}\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}\phi-
w_{n}(\mathord{{\rm div}}b\phi+b^{i}\partial_{i}\phi)\Big{]}$
$\displaystyle-\int^{t}_{0}\\!\\!\\!\int u_{n}\Big{[}\mathord{{\rm
div}}(b_{n}-b)\phi+(b^{i}_{n}-b^{i})\partial_{i}\phi\Big{]}-\int^{t}_{0}\left(\int
w_{n}\sigma^{il}\partial_{i}\phi\right){\mathord{{\rm d}}}W^{l}_{s}\Bigg{|}$
$\displaystyle\leqslant$ $\displaystyle
C\int_{B_{N}}|w_{n,0}|+C\int^{t}_{0}\\!\\!\\!\int_{B_{N}}|w_{n}|+C\int^{t}_{0}\\!\\!\\!\int_{B_{N}}\Big{[}|\mathord{{\rm
div}}(b_{n}-b)|+|b^{i}_{n}-b^{i}|\Big{]}$
$\displaystyle+\left|\int^{t}_{0}\left(\int
w_{n}\sigma^{il}\partial_{i}\phi\right){\mathord{{\rm d}}}W^{l}_{s}\right|.$
Hence, by BDG’s inequality, (4.14) and the assumptions, we get
$\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T]}\left|\int
w_{n,t}\phi\right|\right)\leqslant
C\lim_{n\to\infty}{\mathbb{E}}\left(\int^{T}_{0}\left(\int_{B_{N}}|w_{n}|\right)^{2}{\mathord{{\rm
d}}}s\right)^{1/2}=0.$
By another approximation, we further have for any $N>0$ and $\phi\in
L^{p/(p-1)}(B_{N})$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T]}\left|\int_{B_{N}}w_{n,t}\phi\right|\right)=0.$
(4.15)
Similarly, we also have for any $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T]}\left|\int_{B_{N}}(u^{p}_{n,t}-u^{p}_{t})\right|\right)=0.$
(4.16)
Combining (4.11), (4.15) and (4.16), we then obtain (4.12) by Lemma 3.6. ∎
We are now in a position to give:
Proof of Theorem 4.3: (Uniqueness) Let $u$ and $\hat{u}$ be two renormalized
solutions of SPDE (4.1) corresponding to the initial value $u_{0}$ in the
sense of Definition 4.1. Then $\arctan u$ and $\arctan\hat{u}$ are two
distributional solutions of SPDE (4.1) corresponding to the initial value
$\arctan u_{0}$. By Proposition 4.6, we have
$\arctan u=\arctan\hat{u}.$
Hence,
$u=\hat{u}.$
(Existence) Let $v$ be the unique renormalized solution of SPDE (4.1) given in
Proposition 4.7 corresponding to the initial value $\arctan
u_{0}\in[-\pi/2,\pi/2]$. Since
$\|v_{t}(\omega)\|_{\infty}\leqslant\|\arctan u_{0}\|_{\infty}\leqslant\pi/2,$
we may define
$u_{t}(\omega,x)=\tan v_{t}(\omega,x)$
so that $u$ is a renormalized solution of (4.1) in the sense of Definition
4.1.
(Stability) It follows from the stability in Theorem 4.7.
## 5\. Stochastic Flows with BV Drifts and Constant Diffusions
Consider the following SDE:
$\displaystyle{\mathord{{\rm d}}}X_{t}(x)=b(X_{t}(x)){\mathord{{\rm
d}}}t+\sigma{\mathord{{\rm d}}}W_{t},\ \ X_{0}=x.$ (5.1)
In this section, we use Theorem 4.3 to prove the following result.
###### Theorem 5.1.
Assume that $b$ is a BV vector field and satisfies
$\frac{b(x)}{1+|x|},\ \mathord{{\rm div}}b(x)\in
L^{\infty}({\mathbb{R}}^{d}),\ \ b\in\mathrm{BV}_{loc}.$
Then there exists a unique almost everywhere stochastic invertible flow to SDE
(5.1) in the sense of Definition 2.1.
###### Proof.
(Existence): Define $b_{n}$ as in (4.13). Let $X_{n,s,t}(x)$ solve the
following SDE:
$\displaystyle X_{n,s,t}(x)=x+\int^{t}_{s}b_{n}(X_{n,s,r}(x)){\mathord{{\rm
d}}}r+\sigma(W_{t}-W_{s}),\ \ \forall t\geqslant s\geqslant 0.$ (5.2)
We divide the proof into two steps.
(Step 1): Fix $t>0$. By Proposition 3.3, $v^{k}_{n,s,t}(x):=X^{k}_{n,s,t}(x)$
solves the following backward stochastic Kolmogorov equation:
${\mathord{{\rm
d}}}v^{k}_{n}+\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}v^{k}_{n}{\mathord{{\rm
d}}}s+b^{i}_{n}\partial_{i}v^{k}_{n}{\mathord{{\rm
d}}}s+\sigma^{il}\partial_{i}v^{k}_{n}*{\mathord{{\rm d}}}W_{s}=0,\ \
v^{k}_{n}|_{s=t}=x^{k},$
and by Proposition 3.2, $u^{k}_{n,t}(x):=[X^{-1}_{n,0,t}(x)]^{k}$ solves the
following equation:
${\mathord{{\rm
d}}}u^{k}_{n}=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u^{k}_{n}-b^{i}_{n}\partial_{i}u^{k}_{n}{\mathord{{\rm
d}}}t-\sigma^{il}\partial_{i}u^{k}_{n}{\mathord{{\rm d}}}W^{l}_{t},\ \
u^{k}_{n}|_{t=0}=x^{k},$
where $x^{k}$ is the $k$-th coordinate of spatial variable $x$.
By Theorem 4.3, let $v^{k}_{s,t}$ and $u^{k}_{s}$ be the unique renormalized
solutions of the following SPDEs in the sense of Definition 4.1
$\displaystyle{\mathord{{\rm
d}}}v^{k}+\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}v^{k}{\mathord{{\rm
d}}}s+b^{i}\partial_{i}v^{k}{\mathord{{\rm
d}}}s+\sigma^{il}\partial_{i}v^{k}*{\mathord{{\rm d}}}W_{s}=0,\ \
v^{k}|_{s=t}=x^{k},$ $\displaystyle{\mathord{{\rm
d}}}u^{k}=\frac{1}{2}\sigma^{il}\sigma^{jl}\partial^{2}_{ij}u^{k}-b^{i}\partial_{i}u^{k}{\mathord{{\rm
d}}}s-\sigma^{il}\partial_{i}u^{k}{\mathord{{\rm d}}}W^{l}_{s},\ \
u^{k}|_{s=0}=x^{k}.$
Then by the stability result in Theorem 4.3, we have for any $p>1$ and $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{s\in[0,t]}\int_{B_{N}}|\arctan
v^{k}_{n,s,t}-\arctan v^{k}_{s,t}|^{p}\right)=0$ (5.3)
and
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{s\in[0,t]}\int_{B_{N}}|\arctan
u^{k}_{n,s}-\arctan u^{k}_{s}|^{p}\right)=0.$ (5.4)
Define
$X_{t}(\omega,x):=v_{0,t}(\omega,x),\ \ Y_{t}(\omega,x):=u_{t}(\omega,x)$
Below, we want to show that $X_{t}(x)$ satisfies (A), (B) and (C) of
Definition 2.1 and $X_{t}^{-1}(\omega,x)=Y_{t}(\omega,x)$.
(Step 2): By (5.3), we have for any $p>1$ and $N>0$
$\lim_{n\to\infty}{\mathbb{E}}\left(\int^{t}_{0}\\!\\!\\!\int_{B_{N}}|\arctan
v^{k}_{n,0,s}-\arctan v^{k}_{0,s}|^{p}\right)=0.$
Hence, there exists a subsequence still denoted by $n$ such that for almost
all $(s,\omega,x)\in[0,t]\times\Omega\times{\mathbb{R}}^{d}$ and any
$k=1,\cdots,d$
$\lim_{n\to\infty}\arctan v^{k}_{n,0,s}(\omega,x)=\arctan
v^{k}_{0,s}(\omega,x),$
i.e.,
$\lim_{n\to\infty}X_{n,0,s}(\omega,x)=X_{s}(\omega,x),$
as well as for ($P\times{\mathscr{L}}$)-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\displaystyle\lim_{n\to\infty}X_{n,0,t}(\omega,x)=X_{t}(\omega,x).$ (5.5)
Note that by (3.2) and (4.8), for any $p\geqslant 1$,
${\mathbb{E}}|\det(\nabla X^{-1}_{n,0,t}(x))|^{p}\leqslant
e^{C_{p}t\|\mathord{{\rm div}}b_{n}\|_{\infty}}\leqslant
e^{C_{p}t(\|\mathord{{\rm div}}b\|_{\infty}+\|b/(1+|x|)\|_{\infty})}.$
By Lemma 3.5, it is easy to see that (A) and (B) of Definition 2.1 hold, and
for any $N>0$,
$\lim_{n\to\infty}{\mathbb{E}}\int^{t}_{0}\\!\\!\\!\int_{B_{N}}|\mathord{{\rm
div}}b_{n}(X_{n,0,s})-\mathord{{\rm div}}b(X_{s})|=0.$
Thus, for ($P\times{\mathscr{L}}$)-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\displaystyle\det(\nabla X_{n,0,t}(\omega,x))$ $\displaystyle=$
$\displaystyle\exp\left\\{\int^{t}_{0}\mathord{{\rm
div}}b_{n}(X_{n,0,s}(\omega,x)){\mathord{{\rm d}}}s\right\\}$ (5.6)
$\displaystyle\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}$
$\displaystyle\exp\left\\{\int^{t}_{0}\mathord{{\rm
div}}b(X_{s}(\omega,x)){\mathord{{\rm d}}}s\right\\}=:\rho_{t}(\omega,x).$
On the other hand, for fixed $t\geqslant 0$ and $P$-almost all
$\omega\in\Omega$, it holds that for all $\varphi,\psi\in
C_{c}^{+}({\mathbb{R}}^{d})$,
$\displaystyle\int\varphi(u_{n,t}(\omega))\cdot\psi=\int\varphi(X^{-1}_{n,0,t}(\omega))\cdot\psi=\int\varphi\cdot\psi(X_{n,0,t}(\omega))\cdot\det(\nabla
X_{n,0,t}(\omega)).$ (5.7)
If necessary, by extracting a subsequence and then taking limits $n\to\infty$
for both sides of (5.7), by (5.4), (5.5) and (5.6), we obtain that for
$P$-almost all $\omega\in\Omega$ and all $\varphi,\psi\in
C_{c}^{+}({\mathbb{R}}^{d})$,
$\int\varphi(Y_{t}(\omega))\cdot\psi=\int\varphi\cdot\psi(X_{t}(\omega))\cdot\rho_{t}(\omega).$
Thus, by (ii) of Lemma 3.4, one sees that (C) of Definition 2.1 holds.
(Uniqueness): It follows from Propositions 2.3 and 4.7. ∎
## 6\. Stochastic Flows with Sobolev Drifts and Non-Constant Diffusions
We first prove the following key estimate.
###### Lemma 6.1.
Let $X_{t}(x)$ and $\hat{X}_{t}(x)$ be two almost everywhere stochastic flows
of (1.1) corresponding to $(b,\sigma)$ and $(\hat{b},\hat{\sigma})$ in the
sense of Definition 2.1, where
$b,\hat{b}\in L^{1}_{loc}({\mathbb{R}}^{d}),\ \ |\nabla\hat{b}|\in(L\log
L)_{loc}({\mathbb{R}}^{d})$
and
$\sigma,\hat{\sigma}\in L^{2}_{loc}({\mathbb{R}}^{d}),\ \
|\nabla\hat{\sigma}|\in L^{2}_{loc}({\mathbb{R}}^{d}).$
Then, for any $T,N,R>0$, there exist constants $C_{1},C_{2}$ given below such
that for all $\delta>0$,
$\displaystyle{\mathbb{E}}\int_{B_{N}\cap
G^{R}_{T}}\log\left(\frac{\sup_{t\in[0,T]}|X_{t}-\hat{X}_{t}|^{2}}{\delta^{2}}+1\right)\leqslant$
$\displaystyle\qquad\leqslant
C_{1}+\frac{C_{2}}{\delta}\left(\int_{B_{R}}|b-\hat{b}|+\left[\int_{B_{R}}|\sigma-\hat{\sigma}|^{2}\right]^{1/2}\right),$
where
$\displaystyle
G_{T}^{R}(\omega):=\Big{\\{}x\in{\mathbb{R}}^{d}:\sup_{t\in[0,T]}|X_{t}(\omega,x)|\vee|\hat{X}_{t}(\omega,x)|\leqslant
R\Big{\\}},$ $\displaystyle C_{1}:=C_{d,R,N}\cdot
T\cdot(K_{T,b,\sigma}+K_{T,\hat{b},\hat{\sigma}})\left(1+\int_{B_{2R}}|\nabla\hat{b}|\log(|\nabla\hat{b}|+1)+\left[\int_{B_{2R}}|\nabla\hat{\sigma}|^{2}\right]^{\frac{1}{2}}\right),$
and $C_{2}:=C_{N}\cdot T\cdot K_{T,b,\sigma}$. Here, $K_{T,b,\sigma}$ is from
(2.1), $C_{d,R,N}$ only depends on $d,R,N$, and $C_{N}$ only depends on $N$.
###### Proof.
Set
$Z_{t}(x):=X_{t}(x)-\hat{X}_{t}(x).$
By Itô’s formula, we have
$\displaystyle\log\left(\frac{|Z_{t}|^{2}}{\delta^{2}}+1\right)$
$\displaystyle=$ $\displaystyle
2\int^{t}_{0}\frac{{\langle}Z,b(X)-\hat{b}(\hat{X}){\rangle}}{|Z|^{2}+\delta^{2}}{\mathord{{\rm
d}}}s+2\int^{t}_{0}\frac{{\langle}Z,(\sigma(X)-\hat{\sigma}(\hat{X})){\mathord{{\rm
d}}}W_{s}{\rangle}}{|Z|^{2}+\delta^{2}}$
$\displaystyle+\int^{t}_{0}\frac{\|\sigma(X)-\hat{\sigma}(\hat{X})\|^{2}}{|Z|^{2}+\delta^{2}}{\mathord{{\rm
d}}}s-2\int^{t}_{0}\frac{|(\sigma(X)-\hat{\sigma}(\hat{X}))^{\mathrm{t}}\cdot
Z|^{2}}{(|Z|^{2}+\delta^{2})^{2}}{\mathord{{\rm d}}}s$ $\displaystyle=:$
$\displaystyle I_{1}(t)+I_{2}(t)+I_{3}(t)+I_{4}(t).$
For $I_{1}(t)$, we have
$I_{1}(t)\leqslant\frac{1}{\delta}\int^{t}_{0}|b(X)-\hat{b}(X)|{\mathord{{\rm
d}}}s+2\int^{t}_{0}\frac{|\hat{b}(X)-\hat{b}(\hat{X})|}{\sqrt{|Z|^{2}+\delta^{2}}}{\mathord{{\rm
d}}}s=:I_{11}(t)+I_{12}(t).$
Below, we write for a continuous function $f:{\mathbb{R}}_{+}\to{\mathbb{R}}$,
$f^{*}(T):=\sup_{t\in[0,T]}|f(t)|.$
Noting that
$G_{T}^{R}(\omega)\subset\\{x:|X_{t}(\omega,x)|\leqslant
R\\}\cap\\{x:|\hat{X}_{t}(\omega,x)|\leqslant R\\},\ \ \forall t\in[0,T],$
by (2.1), we have
${\mathbb{E}}\int_{G_{T}^{R}}|I^{*}_{11}(T)|\leqslant\frac{1}{\delta}{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{\\{|X|\leqslant
R\\}}|b(X)-\hat{b}(X)|\leqslant\frac{\tilde{K}_{T,b,\sigma}}{\delta}\int_{B_{R}}|b-\hat{b}|,$
where $\tilde{K}_{T,b,\sigma}:=T\cdot K_{T,b,\sigma}$, and by
${\mathscr{L}}\circ X\ll{\mathscr{L}}$ and
${\mathscr{L}}\circ\hat{X}\ll{\mathscr{L}}$,
$\displaystyle{\mathbb{E}}\int_{G_{T}^{R}}|I^{*}_{12}(T)|$
$\displaystyle\stackrel{{\scriptstyle(\ref{Es2})}}{{\leqslant}}$
$\displaystyle
C_{d}{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{G_{T}^{R}}\Big{(}[M_{R}|\nabla\hat{b}|](X)+[M_{R}|\nabla\hat{b}|](\hat{X})\Big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{d}{\mathbb{E}}\int^{T}_{0}\left(\int_{\\{|X|\leqslant
R\\}}[M_{R}|\nabla\hat{b}|](X)+\int_{\\{|\hat{X}|\leqslant
R\\}}[M_{R}|\nabla\hat{b}|](\hat{X})\right)$ $\displaystyle\leqslant$
$\displaystyle
C_{d}\cdot(\tilde{K}_{T,b,\sigma}+\tilde{K}_{T,\hat{b},\hat{\sigma}})\int_{B_{R}}M_{R}|\nabla\hat{b}|$
$\displaystyle\stackrel{{\scriptstyle(\ref{Es3})}}{{\leqslant}}$
$\displaystyle
C_{d,R}\cdot(\tilde{K}_{T,b,\sigma}+\tilde{K}_{T,\hat{b},\hat{\sigma}})\left(1+\int_{B_{2R}}|\nabla\hat{b}|\log(|\nabla\hat{b}|+1)\right).$
Hence,
$\displaystyle{\mathbb{E}}\int_{G_{T}^{R}}|I^{*}_{1}(T)|$
$\displaystyle\leqslant$ $\displaystyle
C_{d,R}\cdot(\tilde{K}_{T,b,\sigma}+\tilde{K}_{T,\hat{b},\hat{\sigma}})\left(1+\int_{B_{2R}}|\nabla\hat{b}|\log(|\nabla\hat{b}|+1)\right)$
$\displaystyle+\frac{\tilde{K}_{T,b,\sigma}}{\delta}\int^{T}_{0}\\!\\!\\!\int_{B_{R}}|b-\hat{b}|.$
For $I_{2}(t)$, set
$\tau_{R}(\omega,x):=\inf\Big{\\{}t\geqslant
0:|X_{t}(\omega,x)|\vee\hat{X}_{t}(\omega,x)>R\Big{\\}},$
then
$G_{T}^{R}(\omega)=\\{x:\tau_{R}(\omega,x)>T\\}.$
By BDG’s inequality, we have
$\displaystyle{\mathbb{E}}\int_{B_{N}\cap G_{T}^{R}}|I^{*}_{2}(T)|$
$\displaystyle\leqslant$
$\displaystyle\int_{B_{N}}{\mathbb{E}}\left(\sup_{t\in[0,T\wedge\tau_{R}]}\left|\int^{t}_{0}\frac{{\langle}Z,(\sigma(X)-\hat{\sigma}(\hat{X})){\mathord{{\rm
d}}}W_{s}{\rangle}}{|Z|^{2}+\delta^{2}}\right|\right)$
$\displaystyle\leqslant$ $\displaystyle
C\int_{B_{N}}{\mathbb{E}}\left[\int^{T\wedge\tau_{R}}_{0}\frac{|\sigma(X)-\hat{\sigma}(\hat{X})|^{2}}{|Z|^{2}+\delta^{2}}{\mathord{{\rm
d}}}s\right]^{\frac{1}{2}}$ $\displaystyle\leqslant$ $\displaystyle
C_{N}\left[{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{\\{x:\tau_{R}(x)\geqslant
s\\}}\frac{|\sigma(X)-\hat{\sigma}(\hat{X})|^{2}}{|Z|^{2}+\delta^{2}}\right]^{\frac{1}{2}}.$
As the treatment of $I_{1}(t)$, we can prove that
$\displaystyle{\mathbb{E}}\int_{B_{N}\cap G_{T}^{R}}|I^{*}_{2}(T)|\leqslant
C_{d}\cdot(\tilde{K}_{T,b,\sigma}+\tilde{K}_{T,\hat{b},\hat{\sigma}})\left[\int_{B_{2R}}|\nabla\hat{\sigma}|^{2}\right]^{\frac{1}{2}}+\frac{C_{N}\cdot\tilde{K}_{T,b,\sigma}}{\delta}\left[\int_{B_{R}}|\sigma-\hat{\sigma}|^{2}\right]^{\frac{1}{2}}.$
$I_{3}(t)$ is dealt with similarly and $I_{4}(t)$ is negative and abandoned.
The proof is thus complete. ∎
###### Lemma 6.2.
Let
$\Phi(\omega,x):=\sup_{t\in[0,T]}|X_{t}(\omega,x)-\hat{X}_{t}(\omega,x)|^{2}$.
Assume that for some $M>0$,
$\int_{B_{N}\cap
G_{T}^{R}(\omega)}\log\left(\frac{|\Phi(\omega)|}{\delta^{2}}+1\right)\leqslant
M,$
where $G^{R}_{T}(\omega)$ is as in Lemma 6.1. Then,
$\int_{B_{N}\cap
G_{T}^{R}(\omega)}|\Phi(\omega)|\leqslant\frac{4R^{2}}{M}+\delta^{2}(e^{M^{2}}-1)|B_{N}|,$
where $|B_{N}|$ denotes the volume of the ball $B_{N}$.
###### Proof.
It follows from
$\log\left(\frac{|\Phi(\omega,x)|}{\delta^{2}}+1\right)\leqslant
M^{2}\Longrightarrow|\Phi(\omega,x)|\leqslant\delta^{2}(e^{M^{2}}-1)$
and Chebyshev’s inequality. ∎
We introduce the following assumptions on $b$ and $\sigma$:
1. (H1)
$b\in L^{1}_{loc}({\mathbb{R}}^{d}),|\nabla b|\in(L\log
L)_{loc}({\mathbb{R}}^{d})$ and $\sigma\in
L^{2}_{loc}({\mathbb{R}}^{d}),|\nabla\sigma|\in
L^{2}_{loc}({\mathbb{R}}^{d})$.
2. (H2)
There exist $b_{n},\sigma_{n}\in C^{\infty}_{b}({\mathbb{R}}^{d})$ such that
1. (i)
For any $R>0$
$\displaystyle\lim_{n\to\infty}\int_{B_{R}}|b_{n}-b|=0,\ \
\lim_{n\to\infty}\int_{B_{R}}|\sigma_{n}-\sigma|^{2}=0$ (6.1)
and
$\displaystyle\sup_{n}\left(\int_{B_{R}}|\nabla b_{n}|(\log(|\nabla
b_{n}|+1))+\int_{B_{R}}|\nabla\sigma_{n}|^{2}\right)<+\infty.$ (6.2)
2. (ii)
For some $C_{1},C_{2}>0$ independent of $n$,
$\displaystyle\|[-\mathord{{\rm
div}}b_{n}+\frac{1}{2}\partial_{i}\sigma^{jl}_{n}\partial_{j}\sigma^{il}_{n}+\sigma_{n}^{il}\partial_{ij}^{2}\sigma^{jl}_{n}+|\mathord{{\rm
div}}\sigma_{n}|^{2}]^{+}\|_{\infty}\leqslant C_{1}$ (6.3)
and
$\displaystyle{\langle}x,b_{n}(x){\rangle}_{{\mathbb{R}}^{d}}+2\|\sigma_{n}(x)\|_{H.S.}^{2}\leqslant
C_{2}(|x|^{2}+1),\ \ \forall x\in{\mathbb{R}}^{d}.$ (6.4)
We are now in a position to prove our main result of this section.
###### Theorem 6.3.
Assume that (H1) and (H2) hold. Then there exists a unique almost everywhere
stochastic flow of (1.1) in the sense of Definition 2.1. Moreover, the
constant $K_{T,b,\sigma}$ in (2.1) is less than $e^{C_{1}T}$, where $C_{1}$ is
from (6.3). In particular, if $C_{1}=0$, then $K_{T,b,\sigma}\leqslant 1$.
###### Proof.
(Existence): Let $b_{n}$ and $\sigma_{n}$ be as in (H2). Let $X_{n}$ solve the
following SDE
${\mathord{{\rm d}}}X_{n}=b_{n}(X_{n}){\mathord{{\rm
d}}}t+\sigma_{n}(X_{n}){\mathord{{\rm d}}}W_{t},\ \ X_{n}|_{t=0}=x.$
We want to prove that for any $T,N>0$ and $q\in[1,2)$,
$\displaystyle\lim_{n,m\to\infty}{\mathbb{E}}\int_{B_{N}}\sup_{t\in[0,T]}|X_{n,t}(x)-X_{m,t}(x)|^{q}{\mathord{{\rm
d}}}x=0.$ (6.5)
First of all, by (6.4), it is standard to prove that
$\displaystyle\sup_{n}\sup_{x\in
B_{N}}{\mathbb{E}}\left(\sup_{t\in[0,T]}|X_{n,t}(x)|^{2}\right)<+\infty.$
(6.6)
Thus, for proving (6.5), it suffices to prove that for any $\eta>0$,
$\displaystyle\lim_{n,m\to\infty}P\left\\{\omega:\int_{B_{N}}\sup_{t\in[0,T]}|X_{n,t}(\omega,x)-X_{m,t}(\omega,x)|^{2}{\mathord{{\rm
d}}}x\geqslant 2\eta\right\\}=0.$ (6.7)
Fix $\varepsilon,\eta,T>0$ below and set
$\Phi_{n,m}(\omega,x):=\sup_{t\in[0,T]}|X_{n,t}(\omega,x)-X_{m,t}(\omega,x)|^{2}$
and
$G^{R}_{n,m}(\omega):=\Big{\\{}x\in{\mathbb{R}}^{d}:\sup_{t\in[0,T]}|X_{n,t}(\omega,x)|\vee|X_{m,t}(\omega,x)|\leqslant
R\Big{\\}}.$
Then,
$\displaystyle P\left\\{\omega:\int_{B_{N}}\Phi_{n,m}(\omega)\geqslant
2\eta\right\\}\leqslant P\left\\{\omega:\int_{B_{N}\cap
G^{R}_{n,m}(\omega)^{c}}\Phi_{n,m}(\omega)\geqslant\eta\right\\}$
$\displaystyle\qquad\qquad+P\left\\{\omega:\int_{B_{N}\cap
G^{R}_{n,m}(\omega)}\Phi_{n,m}(\omega)\geqslant\eta\right\\}=:I_{n,m}^{R}+J_{n,m}^{R}.$
(6.8)
For $I_{n,m}^{R}$, by Chebyshev’s inequality and (6.6), we may choose $R>0$
large enough such that for all $n,m\in{\mathbb{N}}$,
$\displaystyle
I_{n,m}^{R}\leqslant\frac{1}{\eta}{\mathbb{E}}\int_{B_{N}\cap(G^{R}_{n,m})^{c}}\Phi_{n,m}\leqslant\frac{1}{\eta}\int_{B_{N}}\Big{(}{\mathbb{E}}\Phi_{n,m}^{2}\cdot
P\\{\omega:x\notin
G^{R}_{n,m}(\omega)\\}\Big{)}^{\frac{1}{2}}\leqslant\varepsilon.$ (6.9)
Fixing such a $R$, we look at $J_{n,m}^{R}$. Set
$\xi^{\delta}_{n,m}:=\int_{B_{N}\cap
G^{R}_{n,m}}\log\left(\frac{\Phi_{n,m}}{\delta^{2}}+1\right).$
By (3.2) and (6.3), we have
$\displaystyle\sup_{n}\sup_{t\in[0,T],x\in{\mathbb{R}}^{d}}{\mathbb{E}}|\det(\nabla
X^{-1}_{n,t})(x)|^{2}\leqslant e^{2TC_{1}},$ (6.10)
which yields that the constant $K_{T,b_{n},\sigma_{n}}$ in (2.1) is bounded by
$e^{C_{1}T}$. Hence, in Lemma 6.1, if we choose
$\delta=\delta_{n,m}=\int_{B_{R}}|b_{n}-b_{m}|+\left[\int_{B_{R}}|\sigma_{n}-\sigma_{m}|^{2}\right]^{\frac{1}{2}},$
then by (6.2), we have for some $C_{T,R,N}$ independent of $n,m$,
${\mathbb{E}}\xi^{\delta_{n,m}}_{n,m}\leqslant C_{T,R,N}.$
Thus, there exists an $M_{1}>0$ such that for all $M\geqslant M_{1}$ and all
$n,m$,
$P(\xi^{\delta_{n,m}}_{n,m}>M)\leqslant\varepsilon.$
Now, by Lemma 6.2 and (6.1), we may choose $M>M_{1}\vee 8R^{2}/\eta$ and $n,m$
large enough such that
$\delta_{n,m}<\sqrt{\frac{\eta}{4(e^{M^{2}}-1)|B_{N}|}},$
which leads to
$\Omega^{M}_{n,m}:=\left\\{\omega:\int_{B_{N}\cap
G^{R}_{n,m}(\omega)}\Phi_{n,m}(\omega)\geqslant\eta;\xi^{\delta_{n,m}}_{n,m}(\omega)\leqslant
M\right\\}=\emptyset.$
Hence, first letting $M$ large enough and then $n,m$ large enough, we obtain
$\displaystyle J_{n,m}^{R}\leqslant
P(\Omega^{M}_{n,m})+P(\xi^{\delta_{n,m}}_{n,m}>M)\leqslant\varepsilon.$ (6.11)
Combining (6.8), (6.9) and (6.11), by the arbitrariness of $\varepsilon$, we
get (6.7) as well as (6.5). So, for $q\in(1,2)$, there exists a stochastic
field $X\in L^{q}_{loc}({\mathbb{R}}^{d};L^{q}(\Omega;C([0,T])))$ such that
for any $N>0$
$\lim_{n\to\infty}{\mathbb{E}}\int_{B_{N}}\sup_{t\in[0,T]}|X_{n,t}(x)-X_{t}(x)|^{q}{\mathord{{\rm
d}}}x=0.$
In particular, there is a subsequence still denoted by $n$ such that for
$(P\times{\mathscr{L}})$-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$
$\displaystyle\lim_{n\to\infty}\sup_{t\in[0,T]}|X_{n,t}(\omega,x)-X_{t}(\omega,x)|=0.$
(6.12)
In view of (6.6), (6.10) and (6.12), by Lemma 3.5 and (6.1), it is easy to
check that $X_{t}(\omega,x)$ satisfies (A) and (B) of Definition 2.1.
(Uniqueness): Let $X_{t}(x)$ and $\hat{X}_{t}(x)$ be two almost everywhere
stochastic flows of (1.1). Then, by Lemma 6.1, we have for any $T,N,R>0$ and
$\delta>0$,
${\mathbb{E}}\int_{B_{N}\cap
G_{T}^{R}}\log\left(\frac{\sup_{t\in[0,T]}|X_{t}-\hat{X}_{t}|^{2}}{\delta^{2}}+1\right)\leqslant
C_{T,N,R},$
where $C_{T,N,R}$ is independent of $\delta$. Letting $\delta$ go to zero, we
obtain
$1_{G_{T}^{R}(\omega)}(x)\cdot\sup_{t\in[0,T]}|X_{t}(\omega,x)-\hat{X}_{t}(\omega,x)|=0\
\ \mbox{a.e. on $\Omega\times B_{N}$}$
The uniqueness then follows by letting $R\to\infty$. ∎
The following example is inspired by [22, 19].
Example: Let $d\geqslant 3$. Consider the following SDE in ${\mathbb{R}}^{d}$
with discontinuous and degenerate coefficients:
${\mathord{{\rm d}}}X_{t}=\frac{\beta X_{t}}{|X_{t}|^{2}}{\mathord{{\rm
d}}}t+\frac{X_{t}\otimes X_{t}}{|X_{t}|^{2}}{\mathord{{\rm d}}}W_{t},\ \
X_{0}=x,$
where $\beta\geqslant(4d^{2}+5d)/(d-2)$. Define
$b(x):=\frac{\beta x}{|x|^{2}},\ \ \ \ \sigma(x):=\frac{x\otimes x}{|x|^{2}}$
and
$b_{n}(x):=\frac{\beta x}{|x|^{2}+1/n},\ \ \sigma_{n}(x):=\frac{x\otimes
x}{|x|^{2}+1/n}.$
By virtue of $d\geqslant 3$, one sees that for any $q\in(1,3/2)$
$|\nabla b|\in L^{q}_{loc}({\mathbb{R}}^{d})\subset(L\log
L)_{loc}({\mathbb{R}}^{d}),\ \ |\nabla\sigma|\in
L^{2}_{loc}({\mathbb{R}}^{d}).$
Thus, (H1) is true for $b$ and $\sigma$.
Let us verify (H2). First of all, (6.1), (6.2) and (6.4) are easily checked.
We look at (6.3). Noting that
$\partial_{i}\sigma^{jl}_{n}(x)=\frac{\partial_{i}(x^{j}x^{l})(|x|^{2}+1/n)-2x^{i}x^{j}x^{l}}{(|x|^{2}+1/n)^{2}},$
we have
$\mathord{{\rm div}}\sigma^{\cdot
l}=\partial_{i}\sigma^{il}(x)=\frac{((d-1)|x|^{2}+(d+1)/n)x^{l}}{(|x|^{2}+1/n)^{2}}.$
Hence,
$\displaystyle\sum_{l}|\mathord{{\rm div}}\sigma^{\cdot l}_{n}(x)|^{2}$
$\displaystyle=$
$\displaystyle\frac{((d-1)|x|^{2}+(d+1)/n)^{2}|x|^{2}}{(|x|^{2}+1/n)^{4}}$
$\displaystyle\leqslant$
$\displaystyle\frac{((d-1)|x|^{2}+(d+1)/n)^{2}}{(|x|^{2}+1/n)^{3}}\leqslant\frac{4d^{2}}{|x|^{2}+1/n}$
and
$\partial_{i}\sigma^{jl}_{n}(x)\partial_{j}\sigma^{il}_{n}(x)=\frac{(d+3)|x|^{2}(|x|^{2}+1/n)^{2}-8|x|^{4}/n-4|x|^{6}}{(|x|^{2}+1/n)^{4}}\leqslant\frac{d+3}{|x|^{2}+1/n}.$
Similarly, we have
$\sigma^{il}_{n}(x)\partial_{i}\mathord{{\rm div}}\sigma^{\cdot
l}_{n}(x)=\frac{3(d-1)|x|^{4}+(d+1)|x|^{2}/n}{(|x|^{2}+1/n)^{3}}-\frac{4|x|^{4}((d-1)|x|^{2}+(d+1)/n)}{(|x|^{2}+1/n)^{4}}\leqslant\frac{4d-2}{|x|^{2}+1/n}.$
Moreover,
$\mathord{{\rm
div}}b_{n}(x)=\frac{\beta(d-2)}{|x|^{2}+1/n}+\frac{2\beta}{n(|x|^{2}+1/n)^{2}},$
Thus, combining the above calculations and by
$\beta\geqslant(4d^{2}+5d)/(d-2)$, we have
$-\mathord{{\rm
div}}b_{n}+\frac{1}{2}\partial_{i}\sigma^{jl}_{n}\partial_{j}\sigma^{il}_{n}+\sigma_{n}^{il}\partial_{ij}^{2}\sigma^{jl}_{n}+|\mathord{{\rm
div}}\sigma_{n}|^{2}\leqslant 0,$
and so, (6.3) holds. Thus, (H2) is also true.
We now give two corollaries of Theorem 6.3.
###### Corollary 6.4.
Assume that (H1) and (H2) hold. Let $Y_{0}\in L^{2}(\Omega,{\mathcal{F}}_{0})$
be such that $P\circ Y_{0}\ll{\mathscr{L}}$ and the density $\gamma_{0}\in
L^{\infty}({\mathbb{R}}^{d})$. Then there exists a unique continuous
(${\mathcal{F}}_{t}$)-adapted process $Y_{t}(\omega)$ such that
$P\circ Y_{t}\ll{\mathscr{L}}$ with the density $\gamma_{t}\in
L^{\infty}_{loc}({\mathbb{R}}_{+};L^{\infty}({\mathbb{R}}^{d}))$ (6.13)
and $Y_{t}$ solves
$\displaystyle Y_{t}=Y_{0}+\int^{t}_{0}b(Y_{s}){\mathord{{\rm
d}}}s+\int^{t}_{0}\sigma(Y_{s}){\mathord{{\rm d}}}W_{s},\ \ \forall t\geqslant
0.$ (6.14)
Moreover,
$Y_{t}(\omega)=X_{t}(\omega,Y_{0}(\omega)),$
where $X_{t}(x)$ is the unique almost everywhere stochastic flow given in
Theorem 6.3.
###### Proof.
As in the proof in the appendix, we can check that
$Y_{t}(\omega):=X_{t}(\omega,Y_{0}(\omega))$ solves equation (6.14). Moreover,
since $X_{t}(x)$ is independent of $Y_{0}$, by (2.1), we have for any
$\varphi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$ and $t\in[0,T]$,
$\displaystyle{\mathbb{E}}\varphi(Y_{t})$ $\displaystyle=$
$\displaystyle{\mathbb{E}}({\mathbb{E}}\varphi(X_{t}(x))|_{x=Y_{0}})=\int_{{\mathbb{R}}^{d}}{\mathbb{E}}\varphi(X_{t}(x))\gamma_{0}(x){\mathord{{\rm
d}}}x$ $\displaystyle\leqslant$
$\displaystyle\|\gamma_{0}\|_{\infty}\int_{{\mathbb{R}}^{d}}{\mathbb{E}}\varphi(X_{t}(x)){\mathord{{\rm
d}}}x\leqslant\|\gamma_{0}\|_{\infty}\cdot
K_{T,b,\sigma}\int_{{\mathbb{R}}^{d}}\varphi(x){\mathord{{\rm d}}}x,$
which implies that $P\circ Y_{t}\ll{\mathscr{L}}$ and the density $\gamma_{t}$
satisfies
$\sup_{t\in[0,T]}\|\gamma_{t}\|_{\infty}\leqslant\|\gamma_{0}\|_{\infty}\cdot
K_{T,b,\sigma}.$
Let us now look at the uniqueness. Let $\hat{Y}_{t}$ be another solution of
(6.14) with $\hat{Y}_{0}=Y_{0}$ and satisfy that
$\displaystyle P\circ\hat{Y}_{t}\ll{\mathscr{L}}\ \ \mbox{with the density
$\hat{\gamma}_{t}\in
L^{\infty}_{loc}({\mathbb{R}}_{+};L^{\infty}({\mathbb{R}}^{d}))$}.$ (6.15)
It is now standard to prove that for any $T>0$,
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,T]}|Y_{t}|^{2}\right)+{\mathbb{E}}\left(\sup_{t\in[0,T]}|\hat{Y}_{t}|^{2}\right)<+\infty.$
(6.16)
Set
$Z_{t}:=Y_{t}-\hat{Y}_{t}$
and for $R>0$
$\tau_{R}:=\inf\\{t\geqslant 0:|Y_{t}|\vee|\hat{Y}_{t}|\geqslant R\\}.$
Then by (6.16), we have
$P\left\\{\omega:\lim_{R\to\infty}\tau_{R}(\omega)=+\infty\right\\}=1.$
As in the proof of Lemma 6.1, we have
$\displaystyle{\mathbb{E}}\log\left(\frac{|Z_{t\wedge\tau_{R}}|^{2}}{\delta^{2}}+1\right)$
$\displaystyle\leqslant$ $\displaystyle
2{\mathbb{E}}\int^{t\wedge\tau_{R}}_{0}\frac{{\langle}Z,b(Y)-b(\hat{Y}){\rangle}}{|Z|^{2}+\delta^{2}}{\mathord{{\rm
d}}}s+{\mathbb{E}}\int^{t\wedge\tau_{R}}_{0}\frac{\|\sigma(Y)-\sigma(\hat{Y})\|^{2}}{|Z|^{2}+\delta^{2}}{\mathord{{\rm
d}}}s$ $\displaystyle\stackrel{{\scriptstyle(\ref{Es2})}}{{\leqslant}}$
$\displaystyle C{\mathbb{E}}\int^{t\wedge\tau_{R}}_{0}([M_{R}|\nabla
b|](Y)+[M_{R}|\nabla b|](\hat{Y})){\mathord{{\rm d}}}s+C_{T}$
$\displaystyle\leqslant$ $\displaystyle
C\int^{t}_{0}({\mathbb{E}}(1_{|Y|\leqslant R}\cdot[M_{R}|\nabla
b|](Y))+{\mathbb{E}}(1_{|\hat{Y}|\leqslant R}\cdot[M_{R}|\nabla
b|](\hat{Y}))){\mathord{{\rm d}}}s+C_{T}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Lp7})(\ref{Lp8})}}{{\leqslant}}$
$\displaystyle C_{T}\int_{|y|\leqslant R}[M_{R}|\nabla b|](y){\mathord{{\rm
d}}}y+C_{T}$ $\displaystyle\stackrel{{\scriptstyle(\ref{Es3})}}{{\leqslant}}$
$\displaystyle C_{T}\int_{|y|\leqslant R}|\nabla b|(y)\log(|\nabla
b(y)|+1){\mathord{{\rm d}}}y+C_{T},$
which yields the uniqueness by first letting $\delta\to 0$ and then
$R\to\infty$. ∎
###### Corollary 6.5.
In addition to (H1) and (H2), we also assume that for some $q>d$,
$|\nabla b|\in L^{q}_{loc}({\mathbb{R}}^{d}).$
Let $Y_{0}\in L^{2}(\Omega,{\mathcal{F}}_{0})$ be such that $P\circ
Y_{0}\ll{\mathscr{L}}$ and the density $\gamma_{0}\in
L^{\infty}({\mathbb{R}}^{d})$. Then
$Y_{t}(\omega):=X_{t}(\omega,Y_{0}(\omega))$ uniquely solves SDE (6.14), where
$X_{t}(x)$ is the unique almost everywhere stochastic flow given in Theorem
6.3.
###### Proof.
Following the proof of Corollary 6.4, we only need to prove the uniqueness.
Let $\hat{Y}$ be another solution of SDE (6.14) with the same initial value
$\hat{Y}_{0}=Y_{0}$. Choosing $q^{\prime}\in(d,q)$, and using (3.17) in (6),
we have
$\displaystyle{\mathbb{E}}\log\left(\frac{|Z_{t\wedge\tau_{R}}|^{2}}{\delta^{2}}+1\right)$
$\displaystyle\leqslant$ $\displaystyle
C_{q^{\prime}}{\mathbb{E}}\int^{t\wedge\tau_{R}}_{0}[M_{R}|\nabla
b|^{q^{\prime}}]^{1/q^{\prime}}(Y){\mathord{{\rm d}}}s+C_{T}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Lp7})}}{{\leqslant}}$
$\displaystyle C_{q^{\prime},T}\int_{|y|\leqslant R}[M_{R}|\nabla
b|^{q^{\prime}}]^{1/q^{\prime}}(y){\mathord{{\rm d}}}y+C_{T}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Es30})}}{{\leqslant}}$
$\displaystyle C_{q^{\prime},T,q,R}\int_{|y|\leqslant R}|\nabla
b|^{q}(y){\mathord{{\rm d}}}y+C_{T},$
which in turn implies the uniqueness as Corollary 6.4. ∎
## 7\. Proofs of Main Results
We first give:
Proof of Theorem 2.6: Under (2.6) and (2.7), it has been proven in Theorem
5.1. We now consider the case of (2.6) and (2.8). Let us define
$b_{n}:=b*\varrho_{n}\cdot\chi_{n}$ and
$\sigma_{n}:=\sigma*\varrho_{n}\cdot\chi_{n}$ as in (4.13). Note that as in
estimating (4.8),
$\displaystyle|\nabla b_{n}|$ $\displaystyle\leqslant$ $\displaystyle|\nabla
b|*\varrho_{n}\cdot\chi_{n}+|b|*\varrho_{n}\cdot|\nabla\chi_{n}|$
$\displaystyle\leqslant$ $\displaystyle|\nabla
b|*\varrho_{n}+2\|\nabla\chi\|_{\infty}\cdot\|b/(1+|x|)\|_{\infty}$
$\displaystyle=:$ $\displaystyle|\nabla b|*\varrho_{n}+C_{1}.$
If we define
$\Psi(r):=(r+C_{1})\log(r+C_{1}+1),$
then $r\to\Psi(r)$ is a convex function on ${\mathbb{R}}_{+}$. Thus, by
Jensen’s inequality, we have for any $R>0$,
$\displaystyle\int_{B_{R}}|\nabla b_{n}|\log(|\nabla
b_{n}|+1)\leqslant\int_{B_{R}}\Psi(|\nabla
b|*\varrho_{n})\leqslant\int_{B_{R}}\Psi(|\nabla
b|)*\varrho_{n}\leqslant\int_{B_{R}}\Psi(|\nabla b|).$ (7.1)
Moreover, by (2.6) and (2.8), it is easy to check that
$\displaystyle\sup_{n}\left(\big{\|}\frac{|b_{n}|}{1+|x|}\big{\|}_{\infty}+\|\mathord{{\rm
div}}b_{n}\|_{\infty}+\|\nabla\sigma_{n}\|_{\infty}+\||\sigma_{n}|\cdot|\nabla\mathord{{\rm
div}}\sigma_{n}|\|_{\infty}\right)<+\infty.$ (7.2)
Hence, (H1) and (H2) hold.
By Theorem 6.3, there exists a unique almost everywhere stochastic flow.
Following the proof of Theorem 6.3, we only need to check (C) of Definition
2.1.
Fix a $T>0$ and let
$\rho_{n}:=\exp\left\\{\int^{T}_{0}\Big{(}\mathord{{\rm
div}}b_{n}-\frac{1}{2}\partial_{i}\sigma^{jl}_{n}\partial_{j}\sigma^{il}_{n}\Big{)}(X_{n}){\mathord{{\rm
d}}}s+\int^{T}_{0}\mathord{{\rm div}}\sigma_{n}(X_{n}){\mathord{{\rm
d}}}W_{s}\right\\}.$
As in Lemma 3.1 and by (7.2), we have for any $p\geqslant 1$,
$\displaystyle\sup_{n\in{\mathbb{N}}}\sup_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|\rho_{n}(x)|^{p}<+\infty.$
(7.3)
In view of (6.6), (6.10) and (6.12), by Lemma 3.5, we have for any $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{B_{N}}|\mathord{{\rm
div}}b_{n}(X_{n})-\mathord{{\rm div}}b(X)|=0,$
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int^{T}_{0}\\!\\!\\!\int_{B_{N}}|\partial_{i}\sigma^{jl}_{n}\partial_{j}\sigma^{il}_{n}(X_{n})-\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}(X)|=0,$
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int_{B_{N}}\left|\int^{T}_{0}(\mathord{{\rm
div}}\sigma_{n}(X_{n})-\mathord{{\rm div}}\sigma(X)){\mathord{{\rm
d}}}W_{s}\right|=0.$
So, there is a subsequence still denoted by $n$ such that for almost all
$(\omega,x)$,
$\displaystyle\lim_{n\to\infty}\rho_{n}(\omega,x)=\rho_{T}(\omega,x),$ (7.4)
where $\rho_{T}(x)$ is defined by (2.2). By (7.3) and (7.4), we further have
for any $p\geqslant 1$ and $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int_{B_{N}}|\rho_{n}-\rho_{T}|^{p}=0.$
(7.5)
Now, let $Y_{n}$ solve the following SDE
${\mathord{{\rm d}}}Y_{n}=-\hat{b}_{n}(Y_{n}){\mathord{{\rm
d}}}t+\sigma_{n}(Y_{n}){\mathord{{\rm d}}}W^{T}_{t},\ \ Y_{n}|_{t=0}=x,$
where $\hat{b}^{i}_{n}=b^{i}_{n}-\sigma^{jl}_{n}\partial_{j}\sigma^{il}_{n}$
and $W^{T}_{t}:=W_{T-t}-W_{T}$. As in the proof of Theorem 6.3, there exists
$Y\in L^{2}_{loc}({\mathbb{R}}^{d};L^{2}(\Omega;C([0,T])))$
such that for any $N>0$,
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\int_{B_{N}}\sup_{t\in[0,T]}|Y_{n,t}(x)-Y_{t}(x)|^{2}{\mathord{{\rm
d}}}x=0.$ (7.6)
Note that for any $\varphi,\psi\in C_{c}^{+}({\mathbb{R}}^{d})$ (see the proof
of Lemma 3.1),
$\displaystyle\int\varphi(Y_{n,T}(\omega))\cdot\psi=\int\varphi\cdot\psi(X_{n,T}(\omega))\cdot\rho_{n}(\omega),\
\ P-a.s.$ (7.7)
By (6.12), (7.5) and (7.6), if necessary, extracting a subsequence and then
taking limits $n\to\infty$ in $L^{1}(\Omega)$ for both sides of (7.7), we get
that for all $\varphi,\psi\in{\mathscr{C}}\subset C^{+}_{c}({\mathbb{R}}^{d})$
and $P$-almost all $\omega\in\Omega$,
$\displaystyle\int\varphi(Y_{T}(\omega))\cdot\psi=\int\varphi\cdot\psi(X_{T}(\omega))\cdot\rho_{T}(\omega).$
(7.8)
Since ${\mathscr{C}}$ is countable, one may find a common null set
$\Omega^{\prime}\subset\Omega$ such that (7.8) holds for all
$\omega\notin\Omega^{\prime}$ and $\varphi,\psi\in{\mathscr{C}}$. Thus, by
(ii) of Lemma 3.4, one sees that (C) of Definition 2.1 holds.
We next give:
Proof of Theorem 2.8: We follow the classical Krylov-Bogoliubov’s method. Let
$Y_{0}$ be an ${\mathcal{F}}_{0}$-measurable ${\mathbb{R}}^{d}$-valued random
variable. Suppose that the probability law of $Y_{0}$ is absolutely continuous
with respect to ${\mathscr{L}}$ with the density $\gamma_{0}\in
L^{\infty}({\mathbb{R}}^{d})$. Define
$Y_{t}(\omega):=X_{t}(\omega,Y_{0}(\omega))$ and
$\mu_{n}(\varphi):=\frac{1}{n}\int^{n}_{0}{\mathbb{E}}\varphi(Y_{s}){\mathord{{\rm
d}}}s=\frac{1}{n}\int^{n}_{0}{\mathbb{E}}[({\mathbb{T}}_{s}\varphi)(X_{0})]{\mathord{{\rm
d}}}s,$
where $\\{X_{s}(x),x\in{\mathbb{R}}^{d}\\}_{t\geqslant 0}$ is the unique
almost everywhere stochastic flow of (1.1).
Noting that $Y_{t}(\omega)$ solves the following SDE (see Corollary 6.4)
$Y_{t}=Y_{0}+\int^{t}_{0}b(Y_{s}){\mathord{{\rm
d}}}s+\int^{t}_{0}\sigma(Y_{s}){\mathord{{\rm d}}}W_{s},$
by (2.9) and Itô’s formula, it is standard to prove that
${\mathbb{E}}|Y_{t}|^{2}\leqslant{\mathbb{E}}|Y_{0}|^{2}\ \mbox{ or }\
\frac{1}{t}\int^{t}_{0}{\mathbb{E}}|Y_{s}|^{2}{\mathord{{\rm
d}}}s\leqslant\frac{{\mathbb{E}}|Y_{0}|^{2}}{C_{1}t}+\frac{C_{2}}{C_{1}}.$
From this, we derive that the family of probability measures $\mu_{n}$ is
tight.
On the other hand, for any $\varphi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$, we
have
$\displaystyle\mu_{n}(\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\int^{n}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{s}\varphi(x)\cdot\gamma_{0}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$ $\displaystyle\leqslant$
$\displaystyle\|\gamma_{0}\|_{\infty}\frac{1}{n}\int^{n}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{s}\varphi(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$
$\displaystyle\stackrel{{\scriptstyle(\ref{Den})}}{{\leqslant}}$
$\displaystyle\|\gamma_{0}\|_{\infty}\cdot
K_{b,\sigma}\cdot\int_{{\mathbb{R}}^{d}}\varphi(x){\mathord{{\rm d}}}x,$
which means that
$\mu_{n}\ll{\mathscr{L}}$
and the density $\gamma_{n}$ satisfies
$\|\gamma_{n}\|_{\infty}\leqslant\|\gamma_{0}\|_{\infty}\cdot K_{b,\sigma}.$
Hence, there exists a subsequence $n_{k}$, $\gamma\in
L^{\infty}({\mathbb{R}}^{d})$ and a probability measure $\mu$ such that
$\gamma_{n_{k}}\mbox{ weakly $*$ converges to $\gamma$ in
$L^{\infty}({\mathbb{R}}^{d})$}$
and $\mu_{n_{k}}$weakly converges to $\mu$ in the sense that for any
$\varphi\in C_{b}({\mathbb{R}}^{d})$
$\lim_{k\to\infty}\int_{{\mathbb{R}}^{d}}\varphi(x)\mu_{n_{k}}({\mathord{{\rm
d}}}x)=\int_{{\mathbb{R}}^{d}}\varphi(x)\mu({\mathord{{\rm d}}}x).$
Since for all $\varphi\in C_{c}({\mathbb{R}}^{d})$,
$\int_{{\mathbb{R}}^{d}}\varphi(x)\mu({\mathord{{\rm
d}}}x)=\int_{{\mathbb{R}}^{d}}\varphi(x)\gamma(x){\mathord{{\rm d}}}x.$
we have $\mu({\mathord{{\rm d}}}x)=\gamma(x){\mathord{{\rm d}}}x$.
Let us verify (2.10). For $\varphi\in L^{1}({\mathbb{R}}^{d})$ and $t\geqslant
0$, since ${\mathbb{T}}_{t}\varphi\in L^{1}({\mathbb{R}}^{d})$, we have
$\displaystyle\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{t}\varphi(x)\gamma(x){\mathord{{\rm
d}}}x$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{t}\varphi(x)\gamma_{n_{k}}(x){\mathord{{\rm
d}}}x=\lim_{k\to\infty}\frac{1}{n_{k}}\int^{n_{k}}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{s}{\mathbb{T}}_{t}\varphi(x)\gamma_{0}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}\frac{1}{n_{k}}\int^{n_{k}}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}{\mathbb{T}}_{t+s}\varphi(x)\gamma_{0}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}\frac{1}{n_{k}}\left(\int^{n_{k}}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}+\int^{n_{k}+t}_{n_{k}}\\!\\!\\!\int_{{\mathbb{R}}^{d}}-\int^{t}_{0}\\!\\!\\!\int_{{\mathbb{R}}^{d}}\right){\mathbb{T}}_{s}\varphi(x)\gamma_{0}(x){\mathord{{\rm
d}}}x{\mathord{{\rm d}}}s$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}\int_{{\mathbb{R}}^{d}}\varphi(x)\gamma_{n_{k}}(x){\mathord{{\rm
d}}}x=\int_{{\mathbb{R}}^{d}}\varphi(x)\gamma(x){\mathord{{\rm d}}}x.$
The proof is thus complete.
## 8\. Appendix
Before proving Proposition 2.4, we need the following simple lemma.
###### Lemma 8.1.
Let ${\mathscr{G}}$ and ${\mathscr{A}}$ be two independent
$\sigma$-subalgebras of ${\mathcal{F}}$. Let
$G:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a bounded
${\mathscr{G}}\times{\mathcal{B}}({\mathbb{R}}^{d})$-measurable function and
$X:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ a
${\mathscr{A}}\times{\mathcal{B}}({\mathbb{R}}^{d})$-measurable mapping.
Suppose that for $P$-almost all $\omega$, ${\mathscr{L}}\circ
X(\omega,\cdot)\ll{\mathscr{L}}$. Then for ${\mathscr{L}}$-almost all
$x\in{\mathbb{R}}^{d}$,
$\displaystyle{\mathbb{E}}(G(\cdot,X(\cdot,x))|{\mathscr{A}})=({\mathbb{E}}G(\cdot,y))|_{y=X(\cdot,x)}.$
(8.1)
###### Proof.
Define $G_{\varepsilon}(\omega,y):=G(\omega,\cdot)*\varrho_{\varepsilon}(y)$,
where $\varrho_{\varepsilon}$ is a family of regularized kernel functions as
in Section 4. It is easy to see that
$\displaystyle{\mathbb{E}}(G_{\varepsilon}(\cdot,X(\cdot,x))|{\mathscr{A}})=({\mathbb{E}}G_{\varepsilon}(\cdot,y))|_{y=X(\cdot,x)}.$
(8.2)
Since for ($P\times{\mathscr{L}}$)-almost all
$(\omega,y)\in\Omega\times{\mathbb{R}}^{d}$,
$\lim_{\varepsilon\to 0}G_{\varepsilon}(\omega,y)=G(\omega,y)$
and
$(P\times{\mathscr{L}})\circ(\cdot,X(\cdot,\cdot))\ll P\times{\mathscr{L}}.$
By taking limits $\varepsilon\to 0$ for both sides of (8.2), we get (8.1). ∎
Proof of Proposition 2.4: Consider the case of almost everywhere stochastic
invertible flow. Fix an $s>0$ below. By (B) of Definition 2.1, one sees that
$\displaystyle(P\times{\mathscr{L}})\circ(\theta_{s}(\cdot),X_{s}(\cdot,\cdot))\ll
P\times{\mathscr{L}}.$ (8.3)
Therefore, there exists a null set $A_{s}\subset\Omega\times{\mathbb{R}}^{d}$
such that for all $(\omega,x)\notin A_{s}$,
$\tilde{X}_{t}(\omega,x):=\left\\{\begin{aligned} &X_{t}(\omega,x),\
&t\in[0,s],\ \ \\\ &X_{t-s}(\theta_{s}\omega,X_{s}(\omega,x)),\ &\ \
t\in[s,\infty)\end{aligned}\right.$
is well defined. We now check that $\tilde{X}$ still satisfies (A), (B) and
(C) of Definition 2.1.
Verification of (A) for $\tilde{X}$: It is clear that for
${\mathscr{L}}$-almost all $x\in{\mathbb{R}}^{d}$, $t\mapsto\tilde{X}_{t}(x)$
is a continuous and (${\mathcal{F}}_{t}$)-adapted process. We just need to
show that for any $t>s$,
$\displaystyle\int^{t}_{s}|b(\tilde{X}_{r}(x))|{\mathord{{\rm
d}}}r+\int^{t}_{s}|\sigma(\tilde{X}_{r}(x))|^{2}{\mathord{{\rm d}}}r<+\infty,\
\ (P\times{\mathscr{L}})-a.e.,$ (8.4)
and for ${\mathscr{L}}$-almost all $x\in{\mathbb{R}}^{d}$,
$\displaystyle\tilde{X}_{t}(x)=X_{s}(x)+\int^{t}_{s}b(\tilde{X}_{r}(x)){\mathord{{\rm
d}}}r+\int^{t}_{s}\sigma(\tilde{X}_{r}(x)){\mathord{{\rm d}}}W_{s},\ \ P-a.s.$
(8.5)
First of all, by (8.3) it is easy to see that (8.4) is true. We look at (8.5).
Write
$Y_{s,t}(\omega,x):=X_{t-s}(\theta_{s}\omega,x),\ \ t\geqslant s$
and for $M>0$, set
$\tau_{M}(\omega,x):=\inf\left\\{t\geqslant
0:\int^{t}_{0}|\sigma(X_{r}(\omega,x))|^{2}{\mathord{{\rm d}}}r>M\right\\}.$
Then for ${\mathscr{L}}$-almost all $x,y\in{\mathbb{R}}^{d}$,
$\tau_{M}(\theta_{s}(\cdot),y)$ and $Y_{s,t}(\omega,y)$ are independent of
$X_{s}(x)$. (8.6)
By (A) for $X$ and (8.3), we have for ($P\times{\mathscr{L}}$)-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\displaystyle\lim_{M\to\infty}\tau_{M}(\theta_{s}(\omega),X_{s}(\omega,x))=+\infty.$
(8.7)
Observe that $Y_{s,t}(x)$ solves
$Y_{s,t}(x)=x+\int^{t}_{s}b(Y_{s,r}(x)){\mathord{{\rm
d}}}r+\int^{t}_{s}\sigma(Y_{s,r}(x)){\mathord{{\rm d}}}W_{r},\ \ t\geqslant
s.$
For verifying (8.5), by (8.7) it suffices to show that for
${\mathscr{L}}$-almost all $x\in{\mathbb{R}}^{d}$,
$\displaystyle\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}\sigma(Y_{s,r}(y)){\mathord{{\rm
d}}}W_{r}\Big{|}_{y=X_{s}(x)}=\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}\sigma(Y_{s,r}(X_{s}(x))){\mathord{{\rm
d}}}W_{r},\ P-a.s.$ (8.8)
We extend $\sigma(Y_{s,r}(y))=0$ for $r<s$ and define for $h>0$
$f^{h}_{r}(y):=\frac{1}{h}\int^{r}_{r-h}\sigma(Y_{s,r^{\prime}}(y)){\mathord{{\rm
d}}}r^{\prime}.$
Then $r\to f^{h}_{r}(y)$ is a continuous and (${\mathcal{F}}_{t}$)-adapted
process and
$\displaystyle\int^{t}_{s}|f^{h}_{r}(y)|^{2}{\mathord{{\rm
d}}}r\leqslant\int^{t}_{s}|\sigma(Y_{s,r}(y))|^{2}{\mathord{{\rm d}}}r,\ \
\lim_{h\to 0}\int^{t}_{s}|f^{h}_{r}(y)-\sigma(Y_{s,r}(y))|^{2}{\mathord{{\rm
d}}}r=0.$ (8.9)
Hence, for any $R>0$, by (8.6) and Lemma 8.1, we have
$\displaystyle{\mathbb{E}}\int_{|X_{s}(x)|\leqslant
R}\left|\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}(f^{h}_{r}(X_{s}(x))-\sigma(Y_{s,r}(X_{s}(x)))){\mathord{{\rm
d}}}W_{r}\right|^{2}{\mathord{{\rm d}}}x$
$\displaystyle\quad=\int{\mathbb{E}}\left|\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}(f^{h}_{r}(X_{s}(x))-\sigma(Y_{s,r}(X_{s}(x))))\cdot
1_{\\{|X_{s}(x)|\leqslant R\\}}{\mathord{{\rm
d}}}W_{r}\right|^{2}{\mathord{{\rm d}}}x$
$\displaystyle\quad=\int{\mathbb{E}}\left(\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}|f^{h}_{r}(X_{s}(x))-\sigma(Y_{s,r}(X_{s}(x)))|^{2}\cdot
1_{\\{|X_{s}(x)|\leqslant R\\}}{\mathord{{\rm d}}}r\right){\mathord{{\rm
d}}}x$
$\displaystyle\quad=\int{\mathbb{E}}\left({\mathbb{E}}\left(\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}|f^{h}_{r}(y)-\sigma(Y_{s,r}(y))|^{2}\cdot
1_{\\{|y|\leqslant R\\}}{\mathord{{\rm
d}}}r\right)\Bigg{|}_{y=X_{s}(x)}\right){\mathord{{\rm d}}}x$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{Den})}}{{\leqslant}}K_{T,b,\sigma}\int_{B_{R}}{\mathbb{E}}\left(\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}|f^{h}_{r}(y)-\sigma(Y_{s,r}(y))|^{2}{\mathord{{\rm
d}}}r\right){\mathord{{\rm d}}}y$ $\displaystyle\quad\to 0,\ \ \mbox{ as $h\to
0$},$ (8.10)
where the last step is due to (8.9) and the dominated convergence theorem.
Similarly, we can prove that
$\lim_{h\to 0}{\mathbb{E}}\int_{|X_{s}(x)|\leqslant
R}\left(\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}(f^{h}_{r}(y)-\sigma(Y_{s,r}(y))){\mathord{{\rm
d}}}W_{r}\Bigg{|}_{y=X_{s}(x)}\right)^{2}{\mathord{{\rm d}}}x=0.$
Thus, for proving (8.8), we only need to prove that for fixed $h>0$,
$\displaystyle\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}f^{h}_{r}(y){\mathord{{\rm
d}}}W_{r}\Big{|}_{y=X_{s}(x)}=\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}f^{h}_{r}(X_{s}(x)){\mathord{{\rm
d}}}W_{r},\ P-a.s.$ (8.11)
Let $\Delta_{n}=\\{s=r_{0}<r_{1}<\cdots<r_{n}=t\\}$ be a division of $[s,t]$.
Write
$F^{h}_{n}(y):=\sum_{r_{k}\in\Delta_{n}\setminus\\{r_{n}\\}}f^{h}_{r_{k}}(y)(W_{r_{k+1}}-W_{r_{k}})\cdot
1_{r_{k}\leqslant\tau_{M}(\theta_{s}(\cdot),y)}$
and
$F^{h}(y):=\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}f^{h}_{r}(y){\mathord{{\rm
d}}}W_{r}.$
Then $F^{h}_{n}(y)$ and $F^{h}(y)$ are independent of $X_{s}(x)$ and for
${\mathscr{L}}$-almost all $y\in{\mathbb{R}}^{d}$,
$\displaystyle{\mathbb{E}}|F_{n}^{h}(y)|^{2}\leqslant C_{h,M},\ \
\lim_{|\Delta_{n}|\to 0}{\mathbb{E}}|F^{h}_{n}(y)-F^{h}(y)|^{2}=0,$ (8.12)
where $|\Delta_{n}|:={\mathord{{\rm
min}}}_{r_{k}\in\Delta_{n}\setminus\\{r_{n}\\}}|r_{k+1}-r_{k}|$. Thus, as in
estimating (8.10), by (2.1) and (8.12), we have
$\lim_{|\Delta_{n}|\to 0}{\mathbb{E}}\int_{|X_{s}(x)|\leqslant
R}\left|F^{h}_{n}(X_{s}(x))-\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),X_{s}(x))}_{s}f^{h}_{r}(X_{s}(x)){\mathord{{\rm
d}}}W_{r}\right|^{2}{\mathord{{\rm d}}}x=0$
and
$\lim_{|\Delta_{n}|\to 0}{\mathbb{E}}\int_{|X_{s}(x)|\leqslant
R}\left(F^{h}_{n}(X_{s}(x))-\int^{t\wedge\tau_{M}(\theta_{s}(\cdot),y)}_{s}f^{h}_{r}(y){\mathord{{\rm
d}}}W_{r}\Bigg{|}_{y=X_{s}(x)}\right)^{2}{\mathord{{\rm d}}}x=0,$
which in turn yields (8.11).
Verification of (B) for $\tilde{X}$: By (8.6) and Lemma 8.1, we have for any
bounded measurable function $\varphi$,
${\mathbb{E}}\varphi(Y_{s,t}(X_{s}(x)))={\mathbb{E}}\big{(}{\mathbb{E}}\varphi(Y_{s,t}(y))|_{y=X_{s}(x)}\big{)}.$
Hence, by (2.1), we have for any $s\leqslant t\leqslant T$
$\int_{{\mathbb{R}}^{d}}{\mathbb{E}}\varphi(\tilde{X}_{t}(x)){\mathord{{\rm
d}}}x=\int_{{\mathbb{R}}^{d}}{\mathbb{E}}\varphi(Y_{s,t}(X_{s}(x))){\mathord{{\rm
d}}}x\leqslant
K_{T,b,\sigma}\int_{{\mathbb{R}}^{d}}{\mathbb{E}}\varphi(Y_{s,t}(y)){\mathord{{\rm
d}}}y\leqslant
K_{T,b,\sigma}^{2}\int_{{\mathbb{R}}^{d}}\varphi(x){\mathord{{\rm d}}}x.$
Verification of (C) for $\tilde{X}$: Fixing $t\geqslant s$, we have for
$\varphi\in{\mathcal{L}}^{+}({\mathbb{R}}^{d})$,
$\displaystyle\int_{{\mathbb{R}}^{d}}\varphi(\tilde{X}^{-1}_{s}(\omega,x)){\mathord{{\rm
d}}}x$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}\varphi(X^{-1}_{s}(\omega,X^{-1}_{t-s}(\theta_{s}\omega,x))){\mathord{{\rm
d}}}x$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}\varphi(X^{-1}_{s}(\omega,x))\rho_{t-s}(\theta_{s}\omega,x){\mathord{{\rm
d}}}x$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}\varphi(x)\rho_{t-s}(\theta_{s}\omega,X_{s}(\omega,x))\rho_{s}(\omega,x){\mathord{{\rm
d}}}x.$
Noticing that
$\rho_{t-s}(\theta_{s}\cdot,x)=\exp\left\\{\int^{t}_{s}\Big{[}\mathord{{\rm
div}}b-\frac{1}{2}\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}\Big{]}(Y_{s,r}(x)){\mathord{{\rm
d}}}r+\int^{t}_{s}\mathord{{\rm div}}\sigma(Y_{s,r}(x)){\mathord{{\rm
d}}}W_{r}\right\\},$
as in verifying (8.5), we have
$\rho_{t-s}(\theta_{s}\cdot,X_{s}(x))=\exp\left\\{\int^{t}_{s}\Big{[}\mathord{{\rm
div}}b-\frac{1}{2}\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}\Big{]}(\tilde{X}_{r}(x)){\mathord{{\rm
d}}}r+\int^{t}_{s}\mathord{{\rm div}}\sigma(\tilde{X}_{r}(x)){\mathord{{\rm
d}}}W_{r}\right\\}.$
Thus,
$\displaystyle\qquad\tilde{\rho}_{t}(x)=\rho_{t-s}(\theta_{s}\cdot,X_{s}(x))\rho_{s}(x)=$
$\displaystyle=\exp\left\\{\int^{t}_{0}\Big{[}\mathord{{\rm
div}}b-\frac{1}{2}\partial_{i}\sigma^{jl}\partial_{j}\sigma^{il}\Big{]}(\tilde{X}_{s}(x)){\mathord{{\rm
d}}}s+\int^{t}_{0}\mathord{{\rm div}}\sigma(\tilde{X}_{s}(x)){\mathord{{\rm
d}}}W_{s}\right\\}.$
Finally, by the uniqueness, we have for $(P\times{\mathscr{L}})$-almost all
$(\omega,x)\in\Omega\times{\mathbb{R}}^{d}$,
$\tilde{X}_{t}(\omega,x)=X_{t}(\omega,x),\ \forall t\geqslant 0,$
that is, (2.3) holds.
Markov Property (2.4): It follows from (2.3) and Lemma 8.1 as well as the
independence of $X_{t}(\theta_{s}\cdot,x)$ and ${\mathcal{F}}_{s}$.
Acknowledgements:
The author would like to thank Professor Benjamin Goldys for providing him an
excellent environment to work in the University of New South Wales. His work
is supported by ARC Discovery grant DP0663153 of Australia and NSF of China
(No. 10871215).
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|
arxiv-papers
| 2009-07-28T08:49:21 |
2024-09-04T02:49:04.245231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xicheng Zhang",
"submitter": "Xicheng Zhang",
"url": "https://arxiv.org/abs/0907.4866"
}
|
0907.4965
|
11institutetext: 1: Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
2: Department of Physics, Washington University, St. Louis, Missouri 63160,
USA
3: CINT, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
# The glassy response of solid 4He to torsional oscillations
M. J. Graf1 Z. Nussinov2 A. V. Balatsky1,3
(07.23.2009)
###### Abstract
We calculated the glassy response of solid 4He to torsional oscillations
assuming a phenomenological glass model. Making only a few assumptions about
the distribution of glassy relaxation times in a small subsystem of otherwise
rigid solid 4He, we can account for the magnitude of the observed period shift
and concomitant dissipation peak in several torsion oscillator experiments.
The implications of the glass model for solid 4He are threefold: (1) The
dynamics of solid 4He is governed by glassy relaxation processes. (2) The
distribution of relaxation times varies significantly between different
torsion oscillator experiments. (3) The mechanical response of a torsion
oscillator does not require a supersolid component to account for the observed
anomaly at low temperatures, though we cannot rule out its existence.
###### Keywords:
Torsion oscillator solid 4He glass supersolid
###### pacs:
67.80.B-, 64.70.Q-, 67.80.bd
††journal: Journal of Low Temperature Physics
## 1 Introduction
Torsion oscillators have been used successfully to measure an anomalous change
in resonant period and accompanying dissipation in solid 4He.1, 2, 3, 4, 5, 6,
7 It has been speculated that the observed signature is due to Bose-Einstein
condensation of vacancies or interstitials forming a novel supersolid state in
otherwise crystalline 4He.8, 9, 10, 11, 12, 13 Early on, the change in
resonant period has been attributed to a (nonclassical) decoupling of a
supersolid component from the (classical) normal moment of inertia. This is
not surprising, since the observed change in period is in agreement with
similar observations of onset of superfluidity in liquid 4He, measured a long
time ago by torsion oscillators.14, 15, 16, 17 However, it is important to
remember that for liquid 4He it was already well established, long before the
torsion oscillator (TO) experiments were performed, that it undergoes a
transition from liquid to superfluid with no viscosity. It was natural to use
the connection between superflow and period drop. Status of a search of
supersolidity in solid 4He is different: a change in period has been reported,
but no evidence of mass superflow18, 19, 20, 21, 22, 23, 24 or condensation
25, 26, 27 has been seen below the expected transition temperature. It is
therefore necessary to ask the question what is the relationship between the
change in period and superflow, and what alternate physical mechanisms can
explain the change in period and concomitant peak in dissipation.
Figure 1: (Color online) Hysteresis of rim velocity of nonclassical rotational
inertia fraction (NCRIf) by Aoki et al.4 Measurements were taken at 19 mK as
the oscillator drive was increased (circle) and decreased (square).
A potential contender for an alternate explanation is the glass scenario,
which automatically accounts for hysteresis effects, annealing dependence,
linear term in specific heat, and long relaxation times in many observables.
For example, the strong hysteresis effect reported by Aoki et al.4 for the rim
velocity dependence of the torsion oscillator, shown in Fig. 1, is consistent
with a glassy response. However, it is difficult to explain within a
supersolid scenario how increasing the rim velocity of the cell does not
change the reported nonclassical rotational inertia: between slow and fast rim
velocity one increases kinetic energy of solid, $E_{kin}\propto v^{2}$, and
the ratio of kinetic energies is on the order of $\sim(600/10)^{2}=3600$. Fast
rim velocity exceeds the supposed critical velocity of the condensate, $\sim
10\,{\rm\mu m/s}$, by several orders of magnitude.
In this article, we explore if a phenomenological glass model can account for
the change in period without postulating a nonclassical moment of inertia.
Similar to the supersolid picture, we assume that only a small subsystem of
solid 4He exhibits glassy properties that dominate the response at low
temperatures. This is an important point, since it has been argued before that
the observed large change in dissipation cannot be described by uniform Bose-
Einstein condensation.28, 29, 30 It remains to be seen if nonuniform Bose-
Einstein condensation alone either along grain boundaries 31 or along the axis
of screw dislocations 32, 33, 34 can explain the dynamic response of TOs. A
discussion on the role of a glassy component does not rule in or out the
presence of a supersolid component. We emphasize that our analysis addresses a
glassy contribution regardless of the magnitude of a supersolid component.
## 2 Glass model for torsion oscillator
Glass model: In previous work, we argued for the possibility of a glass phase
at low temperatures, roughly below $\sim 150$ mK, to explain the observed
anomalous linear temperature dependence in the specific heat of the otherwise
perfect Debye solid 4He.35, 29 Below we will give an extended oscillator
analysis for which the exact nature of the glass is not crucial. For example,
it may be caused by two-level systems (TLS) of pinned dislocation lines,
vortex excitations, etc. However, it is important to point out that the
amplitude of the period shift and dissipation peak can be changed dramatically
by the growth history or annealing process of the crystal. In order to explain
these puzzling features of solid 4He, we conjectured earlier35 that structural
defects, e.g., localized dislocation segments, form a set of TLS observable at
low temperatures. These immobile crystal defects affect the thermodynamics35,
29 of bulk 4He and the mechanics 28 of the TO loaded with 4He. For the
analysis of the specific heat, we used independent TLS to obtain the universal
glass signature of a linear-in-temperature specific heat term at low
temperatures. In parallel, we used a phenomenological glass model, that may
originate from an ensemble of TLS, to describe the mechanical response of the
TO.
Torsion oscillator and rotational susceptibility: To set the discussion, we
note that TO experiments measure the period and dissipation. One applies a
force and generates a displacement of the oscillator. The relationship between
the force and displacement (angle) is controlled by the TO susceptibility. In
order to extract information from such an experiment, one needs a model to
determine the relation between observable period and dissipation of the TO and
the corresponding moment of inertia, damping and effective stiffness of the
media. We start with the general equation of motion for a harmonic TO defined
by an angular coordinate $\theta$ in the presence of an external and internal
torque,28
$\displaystyle
I_{osc}\ddot{\theta}(t)+\gamma_{osc}\dot{\theta}(t)+\alpha_{osc}\theta(t)={M}_{ext}(t)+{M}_{int}(t).$
(1)
Here, $I_{osc}$ is the moment of inertia of the (empty) TO chassis,
$\alpha_{osc}$ is the restoring (stiffness) coefficient of the torsion rod,
and $\gamma_{osc}$ is its dissipative coefficient. $M_{ext}(t)$ is the
externally imposed torque by the drive.
${M}_{int}(t)=\int{g}(t-t^{\prime})\theta(t^{\prime})dt^{\prime}$ is the
internal torque exerted by solid 4He on the oscillator for a system with time
translation invariance. In general, the backaction $g(t-t^{\prime})$ is
temperature, $T$, dependent. The experimentally measured quantity is the
angular motion of the TO - not that of bulk helium, which is enclosed in it.
Ab initio, we cannot assume that the medium moves as one rigid body. If the
non-solid subsystem “freezes” into a glass, the medium will move with greater
uniformity and speed. This leads to an effect similar to that of the
nonclassical rotational moment of inertia, although its physical origin is
completely different. Therefore, we argue for an alternate physical picture,
namely that of softening of the oscillator’s stiffness. The angular coordinate
$\theta(t)$ of the TO is a convolution of the applied external torque
${M}_{ext}(t)$ with the TO susceptibility ${\chi}(t,t^{\prime})$. Under
Fourier transformation the angular response of the TO is
$\displaystyle\chi_{0}^{-1}(\omega)\theta(\omega)=M_{ext}(\omega)+M_{int}(\omega).$
(2)
Defining the total angular susceptibility as
$\chi^{-1}=\chi_{0}^{-1}-M_{int}$, we write
$\displaystyle\chi^{-1}(\omega)=\alpha_{osc}-i\gamma_{osc}\omega-
I_{osc}\omega^{2}-g(\omega),$ (3)
where $g(\omega)$ is the Fourier transform of the backaction due to the added
solid 4He. In what follows, we will treat the backaction as a small
perturbation to the TO chassis.
## 3 Period and dissipation of torsion oscillator
We now determine the experimental consequences of the phenomenological glass
model, where a small glassy subsystem of solid 4He gives rise to the observed
dynamic behavior. Glass is generally defined as a supercooled liquid out of
equilibrium on measurable time scales: its equilibration time becomes
extremely large (and unmeasurable) at low temperatures. Any glass former is a
liquid at high temperature and becomes an amorphous solid (the glass) at
sufficiently low temperatures. In this context bulk solid 4He is not a glass
former; we are talking about glass forming within a small fraction of 4He
sample. We note that our analysis will remain qualitatively unchanged for a
general description of the system by a ”freezing” at low temperature of the
appropriate component (defect or other) that is dynamic at high temperatures
(see the appendix of Nussinov et al.28).
We start by reviewing results for an underdamped harmonic torsion oscillator.
The resonant period is obtained from the angular coordinate $\theta(t)={\rm
Re}\\{\theta_{0}\exp[-i\omega t]\\}$, with a complex amplitude $\theta_{0}$
and complex frequency $\omega=\omega_{0}-i\kappa$. In the case of an
underdamped TO with $\kappa\ll\omega_{0}$, the resonant period is
$P={2\pi}/{\omega_{0}}$, and the quality factor $Q$ or dissipation is
$Q^{-1}={2\kappa}/{\omega_{0}}$, with resonant frequency
$\omega_{0}=\sqrt{\alpha_{osc}/I_{osc}}$.
In the remainder, we use effective oscillator parameters, which are defined as
the sum of parameters describing the chassis, $\chi_{0}^{-1}$, and the added
solid 4He given by
$g(\omega)=i\gamma_{He}\omega+I_{He}\omega^{2}+{\cal G}(\omega).$ (4)
Thus, we write the net moment of inertia $I=I_{osc}+I_{He}$ and dissipation
$\gamma=\gamma_{osc}+\gamma_{He}$. The total response function of the TO is
given by Eqns. (3) and (4). The term ${\cal G}(\omega)$ captures the dynamics
of a glass component and is a function of temperature and frequency. In the
limit $\omega\to 0$ the term ${\cal G}(\omega)\to 0$ as the mechanical motion
of any glass component will be the same as that of the surrounding solid.
Hence there will be no relative motion and no transient overdamped modes for
$\omega=0$. However, at any finite frequency $\omega$, we can approximate the
glass response by ${\cal G}(\omega)\approx g_{0}G(\omega)$, where the
coefficient $g_{0}$ measures the glassy contribution of the solid and is
evaluated at the resonant frequency $\omega_{0}$ of the TO. The dynamic
response function $G(\omega)$ of a glass can be approximated by a distribution
of overdamped oscillators with different relaxation times $\tau$. Two popular
relaxation time distributions used in the literature are the Cole-Cole (CC)
and Davidson-Cole (DC) functions. Both describe a superposition of overdamped
oscillators.36, 37 The CC distribution gives
$G(\omega)=1/[1-(i\omega\tau)^{\alpha}]$, while the DC distribution results in
$G(\omega)=1/[1-i\omega\tau]^{\beta}$.
By comparison to Eq. (3) for the TO chassis system with no helium, the glassy
part of the backaction of 4He, ${\cal G}(\omega)$, renormalizes the effective
spring stiffness 28, 38
$\displaystyle\alpha^{eff}\simeq(\alpha_{osc}-g_{0}),~{}~{}~{}\mbox{for}~{}\omega\tau\ll
1,$ (5)
$\displaystyle\alpha^{eff}\simeq\alpha_{osc},~{}~{}~{}~{}\mbox{for}~{}\omega\tau\gg
1.$ (6)
These expressions flesh out the dependence of the medium response on the
applied driving frequency. When the driving frequency is far more rapid,
$\omega\gg\tau^{-1}$, then the transient response of the medium is that of a
liquid. In that limit, the transient modes within the medium cannot “keep up”
with the driving torque and only the bare stiffness of the TO remains
augmented by the solid helium contribution. The effective spring stiffness is
that of the driving oscillator, $\alpha^{eff}=\alpha_{osc}$, see Eq. (6). The
limit $\tau^{-1}\to 0$ corresponds to that of an ideal rigid low-temperature
glass in which no transient liquid-like response of the system is present. By
contrast, for slow oscillations $\omega\ll\tau^{-1}$, the excited modes within
4He are of characteristic transient time $\tau$ that is long enough to respond
to the driving torque and lead to an additional backaction and effective
reduction of the spring stiffness, see Eq. (5). From this discussion it is
clear that the maximum relative shift in period or frequency will depend on
the glassy fraction $g_{0}$ given by $\Delta\omega_{max}/\omega_{0}\sim
g_{0}/(2\alpha_{osc})$, which can vary widely between different torsion
oscillators, growth and annealing procedures.
The resonant frequency of the TO with backaction is given by the root of
$\displaystyle\chi^{-1}(\omega)=\alpha-i\gamma\omega-I\omega^{2}-g_{0}G(\omega)\equiv
0.$ (7)
We anticipate that when the relaxation time is similar to the period of the
underdamped TO, the dissipation will be maximal. Here, the glassy component
responds with the same frequency as the “normal” solid component. The glassy
part merely renormalizes the effective spring constant $\alpha$, but does not
lead to additional transient modes, which closely interfere with the
oscillations of the “normal” part of the TO. We look for the largest magnitude
of the imaginary part of the root and see when it is maximal as a function of
$\tau$. A larger imaginary part implies a shorter decay time and a smaller
value of $Q^{-1}$. Since the homogeneous Eq. (7) is scale invariant, we
normalize all oscillator quantities by the effective moment of inertia $I$,
i.e., $\bar{\alpha}=\alpha/I$, $\bar{\gamma}=\gamma/I$, and
$\bar{g}_{0}=g_{0}/I$.
As can be seen from Eq. (7), for an ideal dissipationless oscillator,
$\bar{\gamma}=0$, the resonant frequency $\omega_{0}=\sqrt{\bar{\alpha}}$ is
the pole of $\chi(\omega)$ in the limit $\tau^{-1}\to 0$. If we expand
$\chi^{-1}$ about this root, $\omega=\omega_{0}+\delta\omega$, with
$\delta\omega=\omega_{1}-i\kappa$, then we find to leading order in
$\delta\omega$
$\displaystyle\delta\omega\approx-\frac{i\bar{\gamma}\omega_{0}+\bar{g}_{0}G(\omega_{0})}{i\bar{\gamma}+2\omega_{0}}.$
(8)
Therefore, the root attains an imaginary component $\kappa$ and the
dissipation becomes
$\displaystyle
Q^{-1}=\frac{2\kappa}{\omega_{0}}\approx\frac{A}{\omega_{0}}{\rm Im\
}G(\omega_{0})+Q_{\infty}^{-1},$ (9)
with $A=\bar{g}_{0}/\omega_{0}$ and
$Q_{\infty}^{-1}={\bar{\gamma}}/{\omega_{0}}$. As $\omega_{0}$ increases for
fixed $\alpha_{osc}$, $Q_{\infty}^{-1}$ increases. For
$\alpha\simeq\beta\simeq 1$, the dissipation peaks near $\omega_{0}\tau\sim
1$. Similarly the resonant frequency becomes
$\displaystyle 2\pi f\equiv\frac{2\pi}{P}$ $\displaystyle\approx$
$\displaystyle\omega_{0}-\frac{A}{4\omega_{0}}\Big{(}2\omega_{0}\,{\rm Re\
}G(\omega_{0})+\bar{\gamma}\,{\rm Im\ }G(\omega_{0})\Big{)},$ (10)
which increases monotonically when $T$ is lowered. For the special case of
Debye relaxation processes ( $\alpha=\beta=1$), we find ${\rm Re\
}G(\omega_{0})=[1+(\omega_{0}\tau)^{2}]^{-1}$ and ${\rm Im\
}G(\omega_{0})=\omega_{0}\tau[1+(\omega_{0}\tau)^{2}]^{-1}$ and recover
results reported earlier,28 except for the additional contribution
proportional to $\bar{\gamma}$ in Eq. (10). It follows that the changes in
dissipation, $\Delta Q^{-1}=Q^{-1}-Q^{-1}_{\infty}$, and frequency,
$\Delta\omega=\omega_{0}-2\pi/P$, determine the glass relaxation time $\tau$.
Combining Eqns. (9) and (10) we arrive at a general relation between shift in
dissipation and frequency for $\bar{\gamma}\tau\ll 1$:
$\frac{\Delta Q^{-1}}{\Delta\omega}=\frac{4{\rm Im\ }G}{2\omega_{0}{\rm Re\
}G+\bar{\gamma}{\rm Im\ }G}\approx\frac{2}{\omega_{0}}\frac{{\rm Im\ }G}{{\rm
Re\ }G}.$ (11)
For example, for a DC glass distribution this becomes
$\frac{\Delta
Q^{-1}}{\Delta\omega}\approx\frac{2}{\omega_{0}}\tan\big{(}\beta\,{\rm
arctan}(\omega_{0}\tau)\big{)}\sim\left\\{\begin{array}[]{ll}2\beta\tau,&\omega_{0}\tau\to
0,\\\ \frac{2}{\omega_{0}}\tan(\beta\pi/4),&\omega_{0}\tau=1,\\\
\frac{2}{\omega_{0}}\tan(\beta\pi/2),&\omega_{0}\tau\to\infty.\end{array}\right.$
(12)
In the past, there have been several reports of large experimental ratios
${\Delta Q^{-1}}\frac{\omega_{0}}{\Delta\omega}\sim 3-12$.2, 4, 5 Because of
$\beta\leq 1$ and for $\omega_{0}\tau\sim 1$ the ratio is limited to ${\Delta
Q^{-1}}\frac{\omega_{0}}{\Delta\omega}\leq 2$, this requires diverging
relaxation times close to the temperature where the dissipation peaks. For
such cases, $\omega_{0}\tau\to\infty$ in Eq. (12) and ratios of order 10 can
be obtained for values of $\beta\sim 0.6$. On the other hand, for $\beta=1$,
Eq. (11) simplifies even further with ${\Delta Q^{-1}}\approx
2\omega_{0}\tau({\Delta\omega}/\omega_{0})$. Similar results for the ratio
were obtained for other phenomenological models with dissipative channels.30,
38 For example, Huse and Khandker 30 assumed a simple phenomenological two-
fluid model, where the supersolid is dissipatively coupled to a normal solid
resulting in a ratio of ${\Delta Q^{-1}}\frac{\omega_{0}}{\Delta\omega}\approx
1$. Yoo and Dorsey38 developed a viscoelastic model and Korshunov39 derived a
TLS glass model for solid 4He that captures the results of the general
phenomenological glass model originally proposed by Nussinov et al.28 Here, we
like to emphasize that it is challenging to reconcile a large dissipative
$\Delta Q/Q$ ratio with uniform Bose-Einstein condensation.28, 29, 30
We now make further assumptions about the glassy relaxation time $\tau$. In
many glass formers $\tau$ follows the phenomenological Vogel-Fulcher-Tamman
(VFT) expression $\tau(T)=\tau_{0}\exp[DT_{0}/(T-T_{0})]$ for $T>T_{0}$. Here,
$T_{0}$ is the temperature at which an ideal glass transition occurs, which is
below the temperature where the peak in dissipation occurs. The parameter $D$
is a measure of the fragility of the glass ($D\lesssim 10$ for fragile glasses
40, 41). Finally, at temperatures $T<T_{0}$ the glassy subsystem freezes out
and $\tau$ becomes infinite.
## 4 Results and discussion
Figure 2: (Color online) The period shift $\Delta P=P-P_{0}$ (black, left
axis) and dissipation (red, right axis) vs. temperature for solid 4He. The
experimental data are from Rittner and Reppy, Fig. 2 of Ref. 2. A Davidson-
Cole (DC) and Cole-Cole (CC) fit are shown. The DC fit was performed with
parameters $\beta=0.60$, $Q_{\infty}^{-1}=11.4\cdot 10^{-6}$, $f_{0}=184.2305$
Hz, $\delta f=69\ \mu{\rm Hz}$, $A=33.8$ mHz, $\tau_{0}=0.439\ \mu{\rm s}$,
$DT_{0}=1.173$ K, $T_{0}=0$ K, $\alpha_{T}=2.0\cdot 10^{-5}$ K-1. The CC fit
used parameters $\alpha=1.15$, $Q_{\infty}^{-1}=11.1\cdot 10^{-6}$,
$f_{0}=184.2305$ Hz, $\delta f=20\ \mu{\rm Hz}$, $A=34.7$ mHz, $\tau_{0}=1.95\
\mu{\rm s}$, $DT_{0}=0.868$ K, $T_{0}=0$ K, $\alpha_{T}=1.6\cdot 10^{-5}$ K-1.
Figure 3: (Color online) The resonant frequency (black, left axis) and
dissipation (red, right axis) vs. temperature. The experimental data are from
Hunt et al.7 with Cole-Cole (CC) parameters $\alpha=1.85$,
$Q_{\infty}^{-1}=1.23\cdot 10^{-6}$, $f_{0}=574.4768$ Hz, $\delta f=1.489$
mHz, $A=347$ mHz, $\tau_{0}=2.52\ \mu{\rm s}$, $DT_{0}=0.408$ K, $T_{0}=-44$
mK, $\alpha_{T}=2.43\cdot 10^{-5}$ K-1.
Figure 4: (Color online) The period shift (black, left axis) and dissipation
(red, right axis) vs. temperature. The experimental data are from the in-phase
mode (495.8 Hz) of the coupled double oscillator by Aoki et al.4 The
experimental data are already corrected for temperature dependence. The DC
parameters are $\beta=0.12$, $Q_{\infty}^{-1}=0.824\cdot 10^{-6}$,
$f_{0}=495.829$ Hz, $\delta f=4.4\ \mu{\rm Hz}$, $A=8.19$ mHz, $\tau_{0}=2.15\
\mu{\rm s}$, $DT_{0}=0.306$ K, $T_{0}=23.6$ mK, $\alpha_{T}=0$ K-1. The CC
parameters are $\alpha=1.70$, $Q_{\infty}^{-1}=0.793\cdot 10^{-6}$,
$f_{0}=495.829$ Hz, $\delta f=-19$ $\mu$Hz, $A=8.97$ mHz, $\tau_{0}=13.2\
\mu{\rm s}$, $DT_{0}=0.248$ K, $T_{0}=-17$ mK, $\alpha_{T}=0$ K-1.
All samples of solid 4He used in this study had in common that they were grown
with the blocked capillary method using commercial grade helium (3He impurity
level $\sim 0.3$ ppm). Also, it is important to remember that both glass
models (CC and DC) use only five fit parameters: $g_{0}$, $\tau_{0}$,
$DT_{0}$, $T_{0}$, and either an exponent $\alpha$ or $\beta$. All other
oscillator parameters are determined by normal state properties of the TO
loaded with solid 4He. In addition, we noticed during our analysis of the TO
experiments that in order to fit the glass models to the experimental data, we
had to correct the resonant frequency by a small amount, $f=f_{0}+\delta f$,
because in many reports $f_{0}$ is not available to desired absolute accuracy
or data are only reported relative to a high-temperature resonant frequency.
Furthermore, the fits were complicated by the experimental observation of a
slight temperature dependence of the resonant frequency at higher
temperatures. To account for this drift in frequency of yet unknown origin, we
approximated $f^{2}_{0}(T)\approx f^{2}_{0}(0)[1-\alpha_{T}T]$ by a small
linear-in-$T$ correction.
The TO experiment reported by Rittner and Reppy,2 see Fig. 2, is in excellent
agreement with the proposed glass models. Both CC and DC glass distributions
require exponents different from unity, which means that there is a spread of
relaxation times $\tau$.
In Fig. 3, we report an analysis of the measured data by Hunt et al.7 assuming
a CC distribution of relaxation times. As can be seen, an excellent fit is
obtained. For comparison, we also tried a DC distribution for relaxation
times, but found only fair agreement. It is worth pointing out that unlike in
the Debye relaxation analysis by Hunt and coworkers (a single overdamped
mode), we do not require a supersolid component to simultaneously account for
frequency shift and concomitant dissipation peak.
Finally, in Fig. 4, we report a DC and CC analysis of the measured data by
Aoki et al.4 for the in-phase mode of their double resonance compound TO. We
obtain excellent agreement between experiment and glass model assuming a CC
distribution, while a DC distribution for glassy relaxation times results only
in fair agreement.
## 5 Conclusions
To summarize, we have shown that a phenomenological glass model describing a
small subsystem of solid 4He can simultaneously account for the experimentally
observed change in resonant period (frequency) and the concomitant peak in
dissipation.
Our analysis of TO experiments reveals that most are better described by a
Cole-Cole distribution for glassy relaxation times. Unlike for conventional
structural or dielectric glasses, where the CC exponent $\alpha$ is usually
less than unity, we find consistently $\alpha>1$. This may reflect on the
possible nature of a quantum or superglass in solid helium. Further, we
derived a simple relation for the ratio of change in dissipation and change in
resonant frequency (period) that can explain the large ratios of order $\sim
10$ observed in experiments. The values for the glass exponents $\alpha$ or
$\beta$ required to fit the experiments by the Rutgers and Cornell groups
point toward broad distributions of glassy relaxation times. This invalidates
any attempt to describe these experiments by a single overdamped mode (Debye
relaxation). These glassy relaxation processes should also have significant
effects on thermodynamics and dynamics of solid 4He. The key result of this
work is that many TO experiments can be described assuming that a small
fraction of solid 4He undergoes a glass transition at low temperatures.
Whether or not there is a supersolid fraction present in solid 4He is beyond
this analysis. A frequency-tunable TO may differentiate between a glassy
contribution leading to an increase in the maximum frequency shift,
$\Delta\omega_{max}\sim\omega_{0}g_{0}/\alpha_{osc}$, and no change in the
dissipation shift, $\Delta Q^{-1}\sim g_{0}/\alpha_{osc}$, with increasing
$\omega_{0}$, while the frequency shift for a supersolid should decrease with
increasing $\omega_{0}$. Our study shows that the unequivocal identification
of supersolidity in solid4He is challenging and does require clear
understanding of normal state dynamics. Clearly, more dynamic studies probing
the frequency or time response to a stimulus and detailed bulk
characterization of samples are necessary to investigate the differences
between small subsystems of glassy, supersolid or superglassy origin.
###### Acknowledgements.
This work was partially supported by the by the US Dept. of Energy at Los
Alamos National Laboratory under contract No. DE-AC52-06NA25396 and by the
Center for Materials Innovation (CMI) of Washington University, St. Louis. We
are grateful to A.T. Dorsey, S.E. Korshunov, J. Beamish, J.M. Goodkind, H.
Kojima and J.C. Davis for many stimulating discussions on this topic.
## References
* 1 E. Kim and M. H. W. Chan, Nature (London) 427, 225 (2004); Science 305, 1941 (2005); Phys. Rev. Lett. 97, 115302 (2006).
* 2 A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97, 165301 (2006); ibid. 98, 175302 (2007).
* 3 M. Kondo, S. Takada, Y. Shibayama, and K. Shirahama, J. Low Temp. Phys. 148, 695 (2007).
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* 6 A. Penzev, Y. Yasuta, and M. Kubota, J. Low Temp. Phys. 148, 677 (2007); Phys. Rev. Lett. 101, 065301 (2008).
* 7 B. Hunt et al., Science 324, 632 (2009).
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* 25 S. O. Diallo et al., Phys. Rev. Lett. 98, 205301 (2007).
* 26 E. Blackburn et al., Phys. Rev. B 76, 024523 (2007).
* 27 O. Kirichek, JPCS 150, 032042 (2009).
* 28 Z. Nussinov, M. J. Graf, A. V. Balatsky, and S. A. Trugman, Phys. Rev. B 76, 014530 (2007).
* 29 M. J. Graf et al., JPCS 150, 032025 (2009).
* 30 D. A. Huse and Z. U. Khandker, Phys. Rev. B 75, 212504 (2007).
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|
arxiv-papers
| 2009-07-28T17:20:29 |
2024-09-04T02:49:04.258852
|
{
"license": "Public Domain",
"authors": "M.J. Graf, Z. Nussinov, A.V. Balatsky",
"submitter": "Matthias J. Graf",
"url": "https://arxiv.org/abs/0907.4965"
}
|
0907.4970
|
# Sampling Distributions of Random Electromagnetic Fields in
Mesoscopic or Dynamical Systems
Luk R. Arnaut
Time, Quantum and Electromagnetics Division
National Physical Laboratory, Teddington, United Kingdom
and
Department of Electrical and Electronic Engineering
Imperial College of Science, Technology and Medicine
South Kensington Campus, London, United Kingdom
###### Abstract
We derive the sampling probability density function (pdf) of an ideal
localized random electromagnetic field, its amplitude and intensity in an
electromagnetic environment that is quasi-statically time-varying
statistically homogeneous or static statistically inhomogeneous. The results
allow for the estimation of field statistics and confidence intervals when a
single spatial or temporal stochastic process produces randomization of the
field. Sampling distributions are particularly significant when the number of
degrees of freedom $\nu$ is relatively small (typically, $\nu<40$), e.g., in
mesoscopic systems when the sample set size $N$ is relatively small by choice
or by force. Results for both coherent and incoherent detection techniques are
derived, for Cartesian, planar and full-vectorial fields. We show that the
functional form of the sampling pdf depends on whether the random variable is
dimensioned (e.g., the sampled electric field proper) or is expressed in
dimensionless standardized or normalized form (e.g., the sampled electric
field divided by its sampled standard deviation). For dimensioned quantities,
the electric field, its amplitude and intensity exhibit different types of
Bessel $K$ sampling pdfs, which differ significantly from the asymptotic Gauss
normal and $\chi^{(2)}_{2p}$ ensemble pdfs when $\nu$ is relatively small. By
contrast, for the corresponding standardized quantities, Student $t$, Fisher-
Snedecor $F$ and root-$F$ sampling pdfs are obtained that exhibit heavier
tails than comparable Bessel $K$ pdfs. Statistical uncertainties obtained from
classical small-sample theory for dimensionless quantities are shown to be
overestimated compared to dimensioned quantities. Differences in the sampling
pdfs arising from de-normalization versus de-standardization are obtained.
###### pacs:
PACS: 02.50.-r, 03.50.De, 05.10.Gg, 06.30.Ka, 41.20.-q, 41.20.Jb, 42.25.Bs
## I Introduction
In the application of statistical methods to disordered and chaotic systems,
particularly nonergodic and/or mesoscopic systems [1]–[6], the role played by
sampling distributions [7] is of fundamental importance because, in practice,
data sets are necessarily of limited (and often small) size $N$. Strictly, the
Central Limit Theorem (CLT) is inapplicable to finite sample sets, a fortiori
to small sets. Therefore, sampling distributions, rather than their underlying
(parent) ensemble distributions, should always be used in any proper
comparison between theoretical and empirical probability distributions based
on numerical data from practical measurement (experiment) or simulation
(computation), particularly when detecting or validating unexpected phenomena.
The finiteness of the number of degrees of freedom for sample sets, $\nu$, can
have a profound effect not only on the statistical moments, but also on the
functional form and shape of the probability density function (pdf) –
particularly near its extremities – and on second- or higher-order statistics,
such as auto- or cross-correlation functions and associated spectral
densities, etc. As a matter of fact, sampling pdfs draw on properties of both
first- and second-order ensemble statistics: while the sampling distributions
are representations for local instantaneous sampled fields, the value of $\nu$
as a sampling distribution parameter is governed by the correlation distance
between points. This dependence on nonlocal properties of the field applies to
any physically limited realizable sampled region. For sample sets in which all
$N$ values are statistically independent (as we shall further assume
throughout), $\nu=N-1$.
Ensemble pdfs for ideal random 3-D electromagnetic (EM) fields are well known
and have been amply investigated in various physical applications, whether in
unbounded space [8] or in the presence of an impedance boundary [9]. By
contrast, their associated sampling pdfs have received little attention yet
are pertinent and require characterization. Idealized random (stochastic)
classical EM fields offer a paradigm for characterizing wave propagation or
transport governed by dynamic multiple scattering (time-varying configuration
or boundaries). Examples include fields inside acoustic or EM mode-tuned or
mode-stirred reverberation chambers (MT/MSRCs) [8]–[12]; mesoscopic
structures; random or turbulent media; polarization and anisotropy in, e.g.,
the cosmic microwave background radiation [13]–[15]; static or dynamic optical
diffusion using random phase screens; diffusing wave spectroscopy [16]–[21];
etc.
For the estimation of statistics of the field, amplitude, intensity, energy
density and power for such ‘wave chaos’, the ensemble pdf of the field is
usually adequate when the number of statistically independent partial
contributions and, hence, $\nu$ approaches infinity. For example, for
MT/MSRCs, the prerequisites are that the wavelength is very short relative to
the physical dimensions of the cavity and that the observation time is long
compared to the correlation time of the stirring process because, physically,
a random EM field is governed by the spatial or temporal extent of the space-
or time-limited process. On occasion, however, $\nu$ is too small for these
ensemble pdfs to be sufficiently accurate. In this case, the CLT or maximum-
entropy principle may not be applicable. Nevertheless, in such low-dimensional
cases, spatial and/or temporal averaging increases the value of $\nu$ [22],
which may have a substantial effect on the measured or perceived distribution
of the field.
A key issue addressed in this paper is that statistically homogeneous random
fields are characterized by pdfs with distribution parameters (e.g., average,
standard deviation, number of degrees of freedom, etc.) whose values have to
be estimated from the same set of data of the fluctuating field. As a result,
these parameters themselves show sampling fluctuations because of the finite
value of $\nu$. These variations give rise to bi- or multivariate fluctuations
and increased uncertainty (i.e., wider confidence intervals) for the sampled
field, compared to the corresponding ensemble distribution where these
parameters are known exactly.
The physical origin of the limitation of the value of $\nu$ can be twofold.
First, the potential (i.e., maximum attainable) number of statistically
independent realizations, $N_{\rm max}$, may be practically restricted, even
if an unlimited number of different states of the statistical ensemble were
physically realized. For example, in a MT/MSRC, this case corresponds to the
so-called undermoded regime (i.e., multi-mode operation but with less than,
say, ten cavity modes being excited simultaneously); typically occurring at
wavelengths where modal overlap is relatively small. As another example, in a
relatively small sample of a random medium, the limitation on the value of
$\nu$ refers to the case of a relatively small loading fraction of inclusions.
In this case, even the ensemble distribution does not possess Gauss normal
statistics. Secondly, $N_{\rm max}$ may be unlimited but, for economical or
other reasons, the sample size may have to be severely restricted ($N\ll
N_{\rm max}$). In this case, while $\nu_{\rm max}$ is potentially large, the
value of $\nu$ actually realized in a sample set is relatively small.
In this paper, we investigate sampling pdfs of the complex value, amplitude
and intensity of 1-D, 2-D and 3-D random EM fields that have ideal Gaussian
ensemble probability distributions, for both the actual (non-standardized)
observed EM quantities and for their standardized forms. Here, the notions of
standarized and normalized random variables refer to a random field quantity
divided by its own (sample or ensemble) standard deviation or mean value,
respectively. Coherent as well as incoherent detection techniques are
considered in each case. However, specific nonstationary effects associated
with the dynamics of multiple scattering, which are relevant to certain
aspects of diffusing wave spectroscopy [17, 18], are not addressed here: each
realized state of the system is considered in its quasi-stationary
approximation. The scenario investigated here is also relevant, in particular,
to several practical applications in wireless communications (e.g., mobile-to-
fixed and mobile-to-mobile transmission), signal processing, wave propagation
in turbulent media, modal noise in optical fibres under restricted-mode launch
conditions [20], etc.
## II Electric or magnetic field
### II.1 Coherent detection
#### II.1.1 Non-standardized field
Consider a modulated local analytic electric field $E=E^{\prime}-{\rm
j}E^{\prime\prime}$ received by an antenna (sensor) or scatterer immersed in a
time-varying multi-scattering environment. A harmonic time dependence
$\exp({\rm j}\omega t)$ is assumed and suppressed. If this field is made up of
an arbitrarily large (theoretically infinite) number of fluctuating partial
fields (i.e., modal or angular plane-wave spectral components) whose
realization forms a random walk in the complex plane [23], then, on account of
the CLT (valid under very general but definite conditions [24]), the
associated conditional probability of $E^{\prime(\prime)}$ coincides with the
ensemble pdf, given by
$\displaystyle
f_{E^{\prime(\prime)}|(S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}})}(e^{\prime(\prime)}|(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}}))$
$\displaystyle=\frac{\exp\left[-\frac{\left(e^{\prime(\prime)}-m_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace
s^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi}~{}s_{E^{\prime(\prime)}}}.$
(1)
Here, $M_{E^{\prime(\prime)}}$ and $S_{E^{\prime(\prime)}}$ represent random
variables induced by the sample mean and sample standard deviation of
$E^{\prime(\prime)}$, respectively, whereas lowercase symbols
$m_{E^{\prime(\prime)}}$ and $s_{E^{\prime(\prime)}}$ represent their
corresponding sample values. If the mean and standard deviation of
$E^{\prime(\prime)}$ are known with certainty, with respective constant
ensemble values $\mu_{E^{\prime(\prime)}}$ and $\sigma_{E^{\prime(\prime)}}$,
i.e.,
$\displaystyle f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}})$
$\displaystyle=$
$\displaystyle\delta(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}})$ (2)
$\displaystyle f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}})$
$\displaystyle=$
$\displaystyle\delta(s_{E^{\prime(\prime)}}-\sigma_{E^{\prime(\prime)}})$ (3)
then the conditional pdf (1) coincides with the marginal pdf
$f_{E^{\prime(\prime)}}({e^{\prime(\prime)}})$ because, in general,
$\displaystyle f_{E^{\prime(\prime)}}({e^{\prime(\prime)}})$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{-\infty}{\rm
d}m_{E^{\prime(\prime)}}\int^{+\infty}_{0}{\rm d}s_{E^{\prime(\prime)}}$
$\displaystyle\times
f_{E^{\prime(\prime)},S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(e^{\prime(\prime)},s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})$
$\displaystyle=$ $\displaystyle\int^{+\infty}_{-\infty}{\rm
d}m_{E^{\prime(\prime)}}\int^{+\infty}_{0}{\rm d}s_{E^{\prime(\prime)}}$
$\displaystyle\times
f_{E^{\prime(\prime)}|(S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}})}(e^{\prime(\prime)}|(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}}))$
$\displaystyle\times
f_{S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})$
(5)
and because in (and only in) the case of a Gauss normal ensemble distribution
of $E^{\prime(\prime)}$ are the sample mean and sample standard deviation
independent random variables [7], i.e.,
$\displaystyle
f_{S_{E^{\prime(\prime)}},M_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}},m_{E^{\prime(\prime)}})=f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}})f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}}).$
(6)
From (1) and (2)–(6), we obtain
$\displaystyle
f_{E^{\prime(\prime)}}(e^{\prime(\prime)})=\frac{\exp\left[-\frac{\left(e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi}~{}\sigma_{E^{\prime(\prime)}}}.$
(7)
Here,
$\sigma_{E^{\prime(\prime)}}=\sigma_{E}/\sqrt{2}=\sqrt{p}\thinspace\sigma_{E^{(\prime)\prime}_{\alpha}}=\sqrt{p/2}\thinspace\sigma_{E_{\alpha}}$
where $\alpha=x,y$ or $z$ for the three Cartesian components $E_{\alpha}$ of
${\bf E}$ and where the value of $p$ corresponds to the number of spatial
dimensions in which ${\bf E}$ is being considered.
Of the possible values of $p$, viz., $1$, $2$, or $3$, its specific value is
governed by the polarization direction of the electric-field sensor and/or by
any EM excitation or boundary conditions that may enforce a certain fixed
(deterministic) polarization state of the field. Thus, the Cartesian component
of a vector field and the 3-D vector field itself correspond to the cases
$p=1$ and $p=3$, respectively. For a paraxially propagating polarized or
unpolarized optical random field, $p=1$ or $p=2$, respectively. The case $p=2$
refers to an unpolarized electric field that is transverse to the local
wavevector (i.e., randomly elliptically polarized field) and will be denoted
by a subscript ‘t’, whereas $p=1$ corresponds to its polarized detection or to
a randomly modulated linearly polarized field, denoted by a subscript
‘$\alpha$’. These values of $p$ apply to local fields detected by an
electrically small sensor, whose characteristic length is less than the
spatial coherence length of the field, i.e., typically $\lambda/2$ in an
unbounded medium. Larger values of $p$ apply to electrically large detectors
whose receiving cross-section (whether physical or as a synthetic aperture) is
larger than $\lambda/2$ in one or more dimensions. For example, for currents
induced in a linear antenna of length $L$, we have $p={\rm
max}[1,L/(\lambda/2)]$.
If, instead of (3), $S_{E^{\prime(\prime)}}$ exhibits random fluctuations or
if its value is not known precisely, the sampling pdf of
$S_{E^{\prime(\prime)}}$ for Gaussian
$E^{\prime(\prime)}|S_{E^{\prime(\prime)}}$ is a $\chi_{pN-1}$ pdf, i.e.,
$\displaystyle f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}};N)$
$\displaystyle=$
$\displaystyle\frac{C_{S_{E^{\prime(\prime)}}}}{\sigma_{S_{E^{\prime(\prime)}}}}\left(\frac{s_{E^{\prime(\prime)}}}{\sigma_{S_{E^{\prime(\prime)}}}}\right)^{pN-2}$
(8) $\displaystyle\times\exp\left[-{\cal
N}\left(\frac{s_{E^{\prime(\prime)}}}{\sigma_{S_{E^{\prime(\prime)}}}}\right)^{2}\right]$
with
$\displaystyle C_{S_{E^{\prime(\prime)}}}$
$\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{2~{}{\cal
N}^{\frac{pN-1}{2}}}{\Gamma\left(\frac{pN-1}{2}\right)},$ (9)
$\displaystyle{\cal N}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle{\frac{pN-1}{2}-\left[\frac{\Gamma\left(\frac{pN}{2}\right)}{\Gamma\left(\frac{pN-1}{2}\right)}\right]^{2}}.$
(10)
The sampling pdf (8) is in its self-sufficient form [22], i.e., it contains
the standard deviation of $S_{E^{\prime(\prime)}}$ itself as a distribution
parameter. Alternatively, (8) can be re-expressed in terms of the ensemble
standard deviation $\sigma_{E^{\prime(\prime)}}$ of $E^{\prime(\prime)}$,
because both statistics are related via
$\displaystyle\sigma_{S_{E^{\prime(\prime)}}}$ $\displaystyle=$
$\displaystyle\sigma_{E^{\prime(\prime)}}\sqrt{\frac{2~{}{\cal N}}{pN-1}}.$
(11)
With the aid of [25, (3.471.9)], the sampling pdf of $E^{\prime(\prime)}$ for
$\mu_{E^{\prime(\prime)}}=0$ in (5) is obtained as a marginal pdf of the joint
pdf
$f_{E^{\prime(\prime)},S_{E^{\prime(\prime)}}}(e^{\prime(\prime)},s_{E^{\prime(\prime)}};N)$,
viz.,
$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}(e^{\prime(\prime)}|s_{E^{\prime(\prime)}})f_{S_{E^{\prime(\prime)}}}(s_{E^{\prime(\prime)}};N){\rm
d}s_{E^{\prime(\prime)}}$ $\displaystyle=$
$\displaystyle\frac{(pN-1)^{\frac{pN-1}{2}}}{2^{\frac{pN}{2}}\sqrt{\pi}\thinspace\Gamma\left(\frac{pN-1}{2}\right)~{}\sigma^{pN-1}_{E^{\prime(\prime)}}}$
$\displaystyle\times\int^{+\infty}_{0}t^{\frac{pN}{2}-2}\exp\left(-\frac{e^{{\prime(\prime)}^{2}}}{2t}-\frac{pN-1}{2}\frac{t}{\sigma^{2}_{E^{\prime(\prime)}}}\right){\rm
d}t,$
i.e.,
$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)=$ (13)
$\displaystyle\frac{C_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\left(\frac{|e^{\prime(\prime)}|}{\sigma_{E^{\prime(\prime)}}}\right)^{\frac{pN}{2}-1}K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}|}{\sigma_{E^{\prime(\prime)}}}\right)$
with
$\displaystyle
C_{E^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{(pN-1)^{\frac{pN}{4}}}{2^{\frac{pN}{2}}\sqrt{\pi}\thinspace\Gamma\left(\frac{pN-1}{2}\right)}.$
(14)
The pdf (13) belongs to McKay’s class of Bessel $K$ distributions [26]. These
pdfs have also been obtained in a variety of applied statistical problems, in
electromagnetics and elsewhere; cf., e.g., [27], [28] and references in [23].
It is further worth noting that the pdf (13) has also been obtained in a
different case, viz., as an ensemble pdf associated with an ad hoc Bessel $K$
distributed field amplitude (implying non-Gaussian/non-Rayleigh ensemble
statistics) and derived via a Blanc-Lapierre transformation [4]. Results that
are at least qualitatively compliant with those obtained from the pdf (13)
have been observed recently in experiments involving correlated scattering
[6]. The pdf (13) is also included in the class of Meijer $G$ limit pdfs for
complex fields in undermoded MT/MSRCs [23], which in turn is a special case of
more general Fox $H$ distributions [29].
It is emphasized that the result (13) strictly holds for a physically ideal,
viz., Gaussian random field if it is subjected to a finite-sized sampling
(detection) process. Only in this case is a Gaussian marginal pdf
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ compatible with a non-delta-
distributed ${S_{E^{\prime(\prime)}}}$. For the physical field itself, a
Gaussian $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ implies the number of
degrees of freedom approaching infinity, hence resulting in a delta
distribution for $f_{S_{E^{\prime(\prime)}}}$, and vice versa.
Fig. 1a shows the pdf (13) for selected values of $N$. Because of symmetry of
$f_{E^{\prime(\prime)}}$ with respect to its mean value, only the positive
half of the distribution is shown.
If $E^{\prime(\prime)}$ exhibits a deterministic, i.e., constant bias
$\mu_{E^{\prime(\prime)}}$ as in (7), then
$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\frac{C_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\left(\frac{|e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right)^{\frac{pN}{2}-1}$
$\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}-\mu_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right).$
For nonnegligible “slow” fluctuations of $m_{E^{\prime(\prime)}}$ in (1),
i.e., for a random bias (trend) of $E^{\prime(\prime)}$, the results are
easily generalized as follows. On account of linearity, the sampling variable
$M_{E^{\prime(\prime)}}$ for Gaussian $E^{\prime(\prime)}$ is also Gaussian
with the same expected value but standard deviation
$\sigma_{E^{\prime(\prime)}}/\sqrt{N}$. Therefore, if the bias is itself
random with Gauss normal distribution, i.e.,
$\displaystyle
f_{M_{E^{\prime(\prime)}}}(m_{E^{\prime(\prime)}};N)=\frac{\exp\left[-\frac{N\left(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]}{\sqrt{2\pi/N}~{}\sigma_{E^{\prime(\prime)}}}$
(16)
instead of (2), then with (5)–(6) we arrive at
$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)=$ (17)
$\displaystyle\frac{C^{\prime}_{E^{\prime(\prime)}}}{\sigma_{E^{\prime(\prime)}}}\int^{+\infty}_{-\infty}\left|e^{\prime(\prime)}-m_{E^{\prime(\prime)}}\right|^{\frac{pN}{2}-1}$
$\displaystyle\times\exp\left[-\frac{N\left(m_{E^{\prime(\prime)}}-\mu_{E^{\prime(\prime)}}\right)^{2}}{2\thinspace\sigma^{2}_{E^{\prime(\prime)}}}\right]$
$\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|e^{\prime(\prime)}-m_{E^{\prime(\prime)}}|}{\sigma_{E^{\prime(\prime)}}}\right){\rm
d}m_{E^{\prime(\prime)}}.$
For practical numerical integration of (17), the double-infinite range of
$m_{E^{\prime(\prime)}}$ can be limited to, say,
$[\mu_{E^{\prime(\prime)}}-5\sigma_{E^{\prime(\prime)}}/\sqrt{N},\mu_{E^{\prime(\prime)}}+5\sigma_{E^{\prime(\prime)}}/\sqrt{N}]$.
For sufficiently large $N$, the field $E^{\prime(\prime)}$ and its sample mean
$M_{E^{\prime(\prime)}}$ are approximately independent (sharing $N-1$ degrees
of freedom), whence $E^{\prime(\prime)}-M_{E^{\prime(\prime)}}$ is then also
Gaussian with zero mean and standard deviation
$\sigma^{*}_{E^{\prime(\prime)}}\simeq\sigma_{E^{\prime(\prime)}}\sqrt{1+1/N}$.
Thus, (LABEL:eq:pdfE_BesselK_determbias) with $\sigma_{E^{\prime(\prime)}}$
replaced by $\sigma^{*}_{E^{\prime(\prime)}}$ serves as a first-order
approximation to the pdf (17) for large $N$.
#### II.1.2 Standardized field
For comparison of an experimental distribution against a theoretical sampling
distribution, it is necessary to standardize the experimentally obtained
$E^{\prime(\prime)}$. Then, rather than determining the pdf of
$E^{\prime(\prime)}$, we require the pdf of the dimensionless variate
$\displaystyle
X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{E^{\prime(\prime)}-M_{E^{\prime(\prime)}}}{S_{E^{\prime(\prime)}}}$
(18)
i.e., for uncertain sample values of $M_{E^{\prime(\prime)}}$ and
$S_{E^{\prime(\prime)}}$ that are unknown a priori. Their values are often to
be estimated from the same limited sample data set of $E^{\prime(\prime)}$.
For Gaussian $E^{\prime(\prime)}$, the $M_{E^{\prime(\prime)}}$ and
$S_{E^{\prime(\prime)}}$ are statistically independent and their sampling
distributions are Gaussian and $\chi_{pN-1}$, respectively [30]. We further
assume that the fluctuations of $M_{E^{\prime(\prime)}}$ are small compared to
those of $S_{E^{\prime(\prime)}}$ and, a fortiori, $E^{\prime(\prime)}$.
It is shown in Sec. A.1 that $X^{\prime(\prime)}$ for
$M_{E^{\prime(\prime)}}=0$ exhibits a Student $t$ distribution [31] with
$pN-1$ degrees of freedom, i.e.,
$\displaystyle
f_{X^{\prime(\prime)}}(x^{\prime(\prime)};N)=C_{X^{\prime(\prime)}}\left(1+\frac{{x^{\prime(\prime)}}^{2}}{pN-1}\right)^{-{pN}/{2}}$
(19)
with
$X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}E^{\prime(\prime)}/S_{E^{\prime(\prime)}}$
for $\langle E^{\prime(\prime)}\rangle=0$, where the sample value
$x^{\prime(\prime)}=e^{\prime(\prime)}/s_{E^{\prime(\prime)}}$ is estimated as
a (dimensionless) ratio of the two sample values $e^{\prime(\prime)}$ and
$s_{E^{\prime(\prime)}}$, and where
$\displaystyle
C_{X^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\Gamma\left(\frac{pN}{2}\right)}{\Gamma\left(\frac{pN-1}{2}\right)\sqrt{(pN-1)\pi}}.$
(20)
The pdf (19) can be used to compare the empirical distribution of the measured
standardized complex field against its ideal theoretical sampling
distribution, particularly when $N$ is relatively small ($N\sim 40$ or less).
For multi-scattering contributions, the value of $N$ can be associated with
the number of scattering centers surrounding the receiver and/or the states
during the motion of the environment around a momentarily stationary receiver.
Without loss of generality, we further consider a deterministically unbiased
field ($\mu_{E^{\prime(\prime)}}=0$). Fig. 1b shows the pdf of
$X^{\prime(\prime)}$ for selected values of $N$. Again, only the positive half
of the distribution is shown. It can be seen that, compared to the asymptotic
Gauss normal pdf, differences with $f_{E^{\prime(\prime)}}$ in Fig. 1a occur
mainly near the origin and in the tails.
### II.2 Incoherent detection
Of significant practical importance, particular for measurements at optical
wavelengths, is the case where the random field is sampled incoherently, i.e.,
by square-law detection of the electric or magnetic power, energy or intensity
devoid of phase information. Yet one may wish to extract the sampling pdf of
the complex-valued electric or magnetic field from such a scalar measurement.
To this end, we borrow results derived in Sec. III.1, and restrict further
analysis to the case of a circular $f_{E}(e)$. (For an elliptic $f_{E}(e)$,
data for a second linearly independent quadratic form of $E^{\prime}$ and
$E^{\prime\prime}$ are needed, e.g.,
${E^{\prime}}^{2}-{E^{{\prime\prime}}}^{2}$.) From the derivation in Sec.
III.1, it follows that the sampling pdf of
$U^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}{E^{{\prime(\prime)}}}^{2}$
is (LABEL:eq:samplepdfU_final) after replacing $p$ with $p/2$, because the
ensemble pdf of $U^{\prime(\prime)}|S_{U^{\prime(\prime)}}$ is $\chi^{2}_{p}$
whereas $f_{U^{\prime(\prime)}}$ has a $\chi^{2}_{pN-1}$ sampling pdf. With
the variate transformation
$f_{E^{\prime(\prime)}}(e^{\prime(\prime)})=2|e^{\prime(\prime)}|f_{U^{\prime(\prime)}}(u^{\prime(\prime)}=e^{{\prime(\prime)}^{2}})$,
we arrive at
$\displaystyle f_{E^{\prime(\prime)}}(e^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle
C^{\prime\prime}_{E^{\prime(\prime)}}\left(\frac{|e^{\prime(\prime)}|}{\sqrt{\sigma_{U}}}\right)^{\frac{1}{2}[p(N+1)-3]}$
$\displaystyle\times
K_{\frac{1}{2}[p(N-1)-1]}\left(\sqrt{\sqrt{{2p}}\left({p}N-1\right)}\frac{|e^{\prime(\prime)}|}{\sqrt{\sigma_{U}}}\right).$
For detection of a Cartesian component of intensity or power, i.e., $p=1$,
(LABEL:eq:samplepdfE_incoh) is equivalent with (13).
## III Field intensity (energy density, power)
For the field intensity $U=U^{\prime}+U^{\prime\prime}=|E|^{2}$ – as well as
for the energy density or power, which are proportional to $U$ – we can
determine its pdf either on the basis of measured in-phase and/or quadrature
components of the field (i.e., coherent detection, e.g., using a vector
network analyzer), or measured directly using a square-law detector (i.e.,
incoherent detection, e.g., using a power meter, spectrum analyzer, field
probe, etc.). We determine sampling pdfs for both cases.
### III.1 Incoherent detection
#### III.1.1 Non-standardized intensity
In many practical cases, the intensity or energy density is measured or
perceived through a square-law detector or perception process. Compared to
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ for coherent detection, the
pertinent conditional pdf (cpdf) is now $f_{U|S_{U}}$. On account of the CLT,
the sampling cpdf for $U|S_{U}$ is $\chi^{2}_{2p}$, i.e.,
$\displaystyle
f_{U|S_{U}}(u|s_{U})=\frac{p^{p/2}}{\Gamma(p)~{}s_{U}}\left(\frac{u}{s_{U}}\right)^{p-1}\exp\left(-\sqrt{p}\frac{u}{s_{U}}\right).$
(22)
The sampling pdf of $S_{U}$, associated with an underlying circular Gauss
normal $E$, is a $\chi^{2}_{2pN-1}$ pdf, i.e.,
$\displaystyle f_{S_{U}}(s_{U};N)$ $\displaystyle=$
$\displaystyle\frac{\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left(pN-\frac{1}{2}\right)}}{\Gamma\left(pN-\frac{1}{2}\right)\thinspace\sigma_{S_{U}}}\left(\frac{s_{U}}{\sigma_{S_{U}}}\right)^{pN-\frac{3}{2}}$
(23)
$\displaystyle\times\exp\left(-\sqrt{pN-\frac{1}{2}}~{}\frac{s_{U}}{\sigma_{S_{U}}}\,\right)$
with
$\displaystyle\sigma_{S_{U}}=\frac{\sigma_{U}}{\sqrt{pN-\frac{1}{2}}}.$ (24)
Performing a similar calculation as in Sec. II.1.1 and again using [25,
(3.471.9)] yields the sampling pdf of $U$ as a marginal pdf of
$f_{U,S_{U}}(u,s_{U};N)$. A Bessel $K$ distribution is again obtained, but
compared to (13) it is now of a different type (i.e., the exponent of the
power and the order of the Bessel function differ by a different amount),
viz.,
$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{U|S_{U}}(u|s_{U})f_{S_{U}}(s_{u};N){\rm
d}s_{U}$ $\displaystyle=$
$\displaystyle\frac{C_{U}}{\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{\frac{1}{2}\left[p(N+1)-\frac{5}{2}\right]}$
$\displaystyle\times
K_{p(N-1)-\frac{1}{2}}\left(2\sqrt{\sqrt{p}\left(pN-\frac{1}{2}\right)}\sqrt{\frac{u}{\sigma_{U}}}\right)$
with normalization constant $C_{U}$ given by
$\displaystyle C_{U}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{2}{\Gamma(p)\Gamma\left(pN-\frac{1}{2}\right)}p^{\frac{1}{4}\left[p(N+1)-\frac{1}{2}\right]}$
(27)
$\displaystyle\times\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left[p(N+1)-\frac{1}{2}\right]}.$
Again, (LABEL:eq:samplepdfU_incoh_BesselK) is one of McKay’s Bessel $K$
distributions. It is remarkable that for the case $p=1$, the same type of
Bessel $K$ distribution (LABEL:eq:samplepdfU_incoh_BesselK) has been obtained
in the context of sea echo for microwave radar [2], but as an ensemble pdf of
$U_{\alpha}$ based on different starting assumptions, viz., for a finite
random walk in the complex plane that assumes a randomly fluctuating and large
but finite number of steps (independent scattering contributions) that is
distributed according to a negative binomial distribution, as a discretization
of a gamma distribution for continuous stepping. For general values of $p$,
the pdf (LABEL:eq:samplepdfU_incoh_BesselK) was also obtained in [23] as a
limit distribution for imperfect reverberation based on a Bayesian model for a
physical process of omnidirectional scattering. Hence, the present derivation
of (LABEL:eq:samplepdfU_incoh_BesselK) shows that Bessel $K$ distributions are
far more universal than previously thought, as they can arise under the much
less restrictive condition of a mere small-sample effect for an underlying
Gaussian field, as opposed to a need for any functional form of a priori
distributions for a fluctuating $\nu$, whether chosen ad hoc or otherwise.
However, any apparent departure from an underlying Gauss normal field
distribution must be interpreted with due care and does not necessarily point
to physical nonlinearity. The identification of Bessel $K$ distributions as
sampling distributions for small sample sets of received power is supported by
measurements of the evolution of the distribution function of pulsed energy in
a reverberant cavity [32]. Such evolution is characterized by a steady growth
in the number of multipath components and, hence, $\nu$.
Figs. 2a, 3a, and 4a show the pdf (LABEL:eq:pdfU_BesselK) of incoherently
detected Cartesian ($p=1$), planar ($p=2$) and total ($p=3$) field
intensities, respectively, for selected values of $N$. The heavier tail and
the sharper peak (mode) of the distribution at smaller values of $N$ are
characteristic features. Fig. 5a shows the standard deviation of
(LABEL:eq:pdfU_BesselK) for these three cases.
#### III.1.2 Standardized intensity
For comparison with the empirical pdf, we consider the sampling pdf of the
ratio of the two random variables $U$ and $S_{U}$, viz.,
$\displaystyle W\stackrel{{\scriptstyle\Delta}}{{=}}\frac{U}{S_{U}}.$ (28)
The calculation of $f_{W}(w;N)$ is detailed in Sec. A.2. The result is a
Fisher-Snedecor $F$ distribution
$\displaystyle f_{W}(w;N)$ $\displaystyle=$
$\displaystyle\frac{C_{W}}{s_{W}}\left(\frac{w}{s_{W}}\right)^{p-1}$ (29)
$\displaystyle\times\left(1+\frac{\sqrt{p}}{pN-\frac{1}{2}}\thinspace
w\right)^{-[p(N+1)-\frac{1}{2}]}$
as a counterpart of (19), where
$\displaystyle
C_{W}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{p^{p/2}}{\left(pN-\frac{1}{2}\right)^{p}}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$
(30)
The pdf (29) for the incoherently detected and sample-standardized Cartesian
($p=1$), planar ($p=2$) and total ($p=3$) field intensities $W_{\alpha}$,
$W_{\rm t}$, and $W$ is shown in Figs. 2b, 3b, and 4b, respectively, at
selected values of $N$. Comparison between the $K$ distributed intensities
with their $F$ distributed sample-standarized values in Figs. 2–4 shows that,
for a given value of $N$, the sampling pdf of $W$ exhibits a more pronounced
spread than for $U$, owing to the larger uncertainty of $W$ caused by
fluctuations of $S_{U}$. Fig. 5b shows the standard deviation of (29) for the
three cases, indicating much larger standard deviations for small $N$ compared
to those in Fig. 5a for nonstandardized intensities.
#### III.1.3 Non-standardized vs. non-normalized intensity
In Sec. III.1.1, $f_{U}(u;N)$ was obtained via the sample-standardized variate
$U/S_{U}$ in the cpdf (22). However, since $\mu_{U}\not=0$, we could have
equally used the sample-normalized variate $U/M_{U}$ for the cpdf to reference
the data in this case. To this end, instead of (III.1.1), we now use
$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{0}f_{U|M_{U}}(u|m_{U})f_{M_{U}}(m_{u};N){\rm
d}m_{U}~{}~{}~{}~{}$ (31)
with [30]
$\displaystyle f_{U|M_{U}}(u|m_{U})$ $\displaystyle=$
$\displaystyle\frac{p^{p}}{\Gamma(p)~{}m_{U}}\left(\frac{u}{m_{U}}\right)^{p-1}\exp\left(-{p}\frac{u}{m_{U}}\right)~{}~{}~{}~{}~{}$
(32) $\displaystyle f_{M_{U}}(m_{U};N)$ $\displaystyle=$
$\displaystyle\frac{\left(pN\right)^{\frac{pN}{2}-1}}{\Gamma\left(pN\right)\thinspace\sigma_{M_{U}}}\left(\frac{m_{U}}{\sigma_{M_{U}}}\right)^{pN-1}$
(33)
$\displaystyle\times\exp\left(-\sqrt{pN}~{}\frac{m_{U}}{\sigma_{M_{U}}}\,\right)$
and $\mu_{U}=m_{U}$, $\sigma_{U}=\sqrt{N}\sigma_{M_{U}}$, whence (31) becomes
$\displaystyle f_{U}(u;N)$ $\displaystyle=$
$\displaystyle\frac{C^{\prime}_{U}}{\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{\frac{1}{2}p(N+1)-1}$
(34) $\displaystyle\times
K_{p(N-1)}\left(2p^{\frac{3}{4}}\sqrt{N}\sqrt{\frac{u}{\sigma_{U}}}\right)~{}~{}~{}~{}$
with normalization constant $C^{\prime}_{U}$ given by
$\displaystyle
C^{\prime}_{U}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2~{}p^{\frac{3}{4}p(N+1)}~{}N^{\frac{1}{2}p(N+1)}}{\Gamma(p)\Gamma\left(pN\right)}.$
(35)
The form (34) is readily re-expressed in terms of $\mu_{U}$ by replacing
$\sigma_{U}$ with $\mu_{U}/\sqrt{p}$. Comparing (34)–(35) to
(LABEL:eq:pdfU_BesselK)–(27) shows that the non-normalized cpdf (34) is
retrieved by replacing $N-(2p)^{-1}$ in the non-standardized cpdf
(LABEL:eq:pdfU_BesselK) by $N$, i.e., resulting in a marginal increase of the
number of independent samples. In other words, the non-normalized cpdf is
marginally closer to the asymptotic ensemble pdf ($N\rightarrow+\infty$) than
the non-standardized cpdf, resulting in slightly smaller uncertainties. This
result is made plausible by the fact that the uncertainty of the sampling mean
value is smaller than for the sampling standard deviation. It is worth
emphasizing that, even though $f_{U}(u;N)$ is for the (dimensioned) energy
density $U$, the chosen route for arriving at this pdf – i.e., whether via
intermediary standardization or normalization in the cpdf of $U$ – has an
effect on the number of degrees of freedom of the end result but not on the
functional form. In summary, whenever $pN\gg 1$ (i.e., for most practical
cases), standardization and normalization yield indistinguishable final
results.
#### III.1.4 Intensity of biased field
Instead of (22), the sampling cpdf of $U|S_{U}$ for the incoherently detected
intensity of a biased field is a generalization of the so-called modified
Nakagami–Rice $m$ distribution, given in self-sufficient form by
$\displaystyle f_{U_{\alpha}|S_{U_{\alpha}}}(u_{\alpha}|s_{U_{\alpha}})$
$\displaystyle=$
$\displaystyle\frac{\sqrt{1+2k_{U_{\alpha}}}}{s_{U_{\alpha}}}$ (36)
$\displaystyle\times\exp\left(-\sqrt{1+2k_{U_{\alpha}}}\frac{u_{\alpha}+u_{\alpha
0}}{s_{U_{\alpha}}}\right)$ $\displaystyle\times
I_{0}\left(2\sqrt{1+2k_{U_{\alpha}}}\frac{\sqrt{u_{\alpha
0}u_{\alpha}}}{s_{U_{\alpha}}}\right)$
with
$\displaystyle
k_{U_{\alpha}}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{|E_{\alpha
0}|^{2}}{\langle|E_{\alpha}-E_{\alpha
0}|^{2}\rangle}=\frac{1-n^{2}_{U_{\alpha}}+\sqrt{1-n^{2}_{U_{\alpha}}}}{n^{2}_{U_{\alpha}}}$
(37)
where $n_{U_{\alpha}}$ is a sample value of
$\nu_{U_{\alpha}}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma_{U_{\alpha}}/\mu_{U_{\alpha}}$.
Since $\sigma_{U_{\alpha}}=\sigma_{U_{\alpha}-u_{\alpha 0}}$ for any constant
(deterministic) value $u_{\alpha 0}$, the pdf
$f_{S_{U_{\alpha}}}(s_{U_{\alpha}};N)$ is still given by (23) with $p=1$.
Thus,
$\displaystyle f_{U_{\alpha}}(u_{\alpha};N)$ $\displaystyle=$
$\displaystyle\frac{C^{\prime}_{U_{\alpha}}}{\sigma_{U_{\alpha}}}\int^{+\infty}_{0}\sqrt{1+2k_{U_{\alpha}}}\left(\frac{s_{U_{\alpha}}}{\sigma_{{U_{\alpha}}}}\right)^{N-\frac{5}{2}}$
(38)
$\displaystyle\times\exp\left[-\left({N-\frac{1}{2}}\right)~{}\frac{s_{U_{\alpha}}}{\sigma_{{U_{\alpha}}}}\,\right]$
$\displaystyle\times\exp\left(-\sqrt{1+2k_{U_{\alpha}}}\frac{u_{\alpha}+u_{\alpha
0}}{s_{U_{\alpha}}}\right)$ $\displaystyle\times
I_{0}\left(2\sqrt{1+2k_{U_{\alpha}}}\frac{\sqrt{u_{\alpha
0}u_{\alpha}}}{s_{U_{\alpha}}}\right){\rm d}s_{U_{\alpha}}.~{}~{}~{}~{}~{}$
Note that $k_{U_{\alpha}}$ depends implicitly on $s_{U_{\alpha}}$: to first
approximation, $k_{U_{\alpha}}\simeq s_{U_{\alpha}}/\mu_{U_{\alpha}}$.
### III.2 Coherent detection
In the case of estimating $f_{U}(u;N)$ from measurements of $E$, the sampling
pdf takes a different form from that for incoherent detection, as we show
next. Knowing the pdf of $E_{\alpha}=E^{\prime}_{\alpha}-{\rm
j}E^{\prime\prime}_{\alpha}$, we can derive the pdf of
$U=|E|^{2}=p|E_{\alpha}|^{2}$, where
$|E_{\alpha}|^{2}\equiv{E^{\prime}_{\alpha}}^{2}+{E^{\prime\prime}_{\alpha}}^{2}=U_{\alpha}$
is the intensity of a Cartesian field component, and similarly for the
electric or magnetic power $P\propto U$. Since
$f_{U^{\prime(\prime)}_{\alpha}}(u^{\prime(\prime)}_{\alpha})\propto
f_{E^{\prime(\prime)}_{\alpha}}(e^{\prime(\prime)}_{\alpha}=\sqrt{u^{\prime(\prime)}_{\alpha}})/\sqrt{u^{\prime(\prime)}_{\alpha}}$,
we obtain
$\displaystyle f_{U_{\alpha}}(u_{\alpha};N)$ $\displaystyle=$ $\displaystyle
C_{U_{\alpha}}\int^{u_{\alpha}}_{0}f_{U^{\prime}_{\alpha}}(x;N)f_{U^{\prime\prime}_{\alpha}}(u_{\alpha}-x;N){\rm
d}x$ $\displaystyle=$ $\displaystyle
C_{U_{\alpha}}\int^{u_{\alpha}}_{0}\frac{f_{E^{\prime}_{\alpha}}(\sqrt{x};N)f_{E^{\prime\prime}_{\alpha}}(\sqrt{u_{\alpha}-x};N)}{\sqrt{x}\sqrt{u_{\alpha}-x}}{\rm
d}x$
where $C_{U_{\alpha}}$ is a normalization constant. The general expression
(LABEL:eq:fUCart) makes allowance for the fact that $E^{\prime}_{\alpha}$ and
$E^{\prime\prime}_{\alpha}$ may, in principle, have different pdfs
(functionally and/or parametrically), although in most cases the pdf of
$E_{\alpha}$ is circular, i.e.,
$f_{E^{\prime}_{\alpha}}=f_{E^{\prime\prime}_{\alpha}}$. The sample pdfs of
$U_{\rm t}\equiv U_{x}+U_{y}$ and $U\equiv U_{x}+U_{y}+U_{z}$ follow similarly
from two- and threefold convolutions of (LABEL:eq:fUCart), respectively. Fig.
6 shows the pdf (LABEL:eq:fUCart) for selected values of $N$.
If ${U_{\alpha}}$ is to be compared with measured data, then the sampling pdf
can be similarly calculated from $f_{X^{\prime}_{\alpha}}$ and
$f_{X^{\prime\prime}_{\alpha}}$ as
$\displaystyle f_{W_{\alpha}}(w_{\alpha})$ $\displaystyle=$ $\displaystyle
C_{W_{\alpha}}\int^{w_{\alpha}}_{0}\frac{f_{X^{\prime}_{\alpha}}(\sqrt{x};N)f_{X^{\prime\prime}_{\alpha}}(\sqrt{w_{\alpha}-x};N)}{\sqrt{x}\sqrt{w_{\alpha}-x}}{\rm
d}x$
where
$W_{\alpha}\stackrel{{\scriptstyle\Delta}}{{=}}U_{\alpha}/S_{U_{\alpha}}$.
## IV Field amplitude
### IV.1 Incoherent detection
#### IV.1.1 Non-standardized field amplitude
The sampling pdf of
$A\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{{E^{\prime}}^{2}+{E^{\prime\prime}}^{2}}$
follows in a manner similar to that for $U$. In this case, $A|S_{A}$ has a
$\chi_{2p}$ cpdf, given in self-sufficient form as
$\displaystyle f_{A|S_{A}}(a|s_{A})$ $\displaystyle=$
$\displaystyle\frac{2\left[p-\left(\frac{\Gamma(p+\frac{1}{2})}{\Gamma(p)}\right)^{2}\right]^{p}\thinspace}{\Gamma(p)\thinspace
s_{A}}\left(\frac{a}{s_{A}}\right)^{2p-1}$
$\displaystyle\times\exp\left\\{-\left[p-\left(\frac{\Gamma(p+\frac{1}{2})}{\Gamma(p)}\right)^{2}\right]\left(\frac{a}{s_{A}}\right)^{2}\right\\},$
which has the Rayleigh distribution as a special case for $p=1$, while $S_{A}$
has the $\chi_{2pN-1}$ distribution
$\displaystyle f_{S_{A}}(s_{A};N)$ $\displaystyle=$
$\displaystyle\frac{2\thinspace\left[pN-\frac{1}{2}-\left(\frac{\Gamma(pN)}{\Gamma(pN-\frac{1}{2})}\right)^{2}\right]^{pN-\frac{1}{2}}}{(pN-\frac{1}{2})~{}\Gamma(pN-\frac{1}{2})\thinspace\sigma_{S_{A}}}\left(\frac{s_{A}}{\sigma_{S_{A}}}\right)^{2(pN-1)}$
$\displaystyle\times\exp\left\\{-\left[pN-\frac{1}{2}-\left(\frac{\Gamma(pN)}{\Gamma(pN-\frac{1}{2})}\right)^{2}\right]\left(\frac{s_{A}}{\sigma_{S_{A}}}\right)^{2}\right\\}$
with $\sigma_{S_{A}}$ and $\sigma_{A}$ related via
$\displaystyle\sigma_{S_{A}}=\sigma_{A}\sqrt{1-\frac{1}{pN-\frac{1}{2}}\left(\frac{\Gamma(pN)}{\Gamma\left(pN-\frac{1}{2}\right)}\right)^{2}}.$
(44)
The sampling pdf of $A$ is obtained as a marginal pdf of
$f_{A,S_{A}}(a,s_{A};N)$ and is again a Bessel $K$ distribution, but of yet
another type compared to (13) and (LABEL:eq:samplepdfU_incoh_BesselK), viz.,
$\displaystyle
f_{A}(a;N)=\int^{+\infty}_{0}f_{A|S_{A}}(a|s_{A})f_{S_{A}}(s_{A};N){\rm
d}s_{A}$
$\displaystyle=\frac{C_{A}}{\sigma_{A}}\left(\frac{a}{\sigma_{A}}\right)^{p(N+1)-\frac{3}{2}}$
$\displaystyle\times
K_{p(N-1)-\frac{1}{2}}\left(2\sqrt{\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]\left(pN-\frac{1}{2}\right)}\frac{a}{\sigma_{A}}\right)$
where $C_{A}$ is obtained, with the aid of [25, (6.561.16)], as
$\displaystyle C_{A}$ $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}$
$\displaystyle\frac{4}{\Gamma(p)\Gamma\left(pN-\frac{1}{2}\right)}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{\frac{1}{2}\left[p(N+2)-1\right]}$
(46)
$\displaystyle\times\left(pN-\frac{1}{2}\right)^{\frac{1}{2}\left[p(N+1)-1\right]}.$
The pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $p=1$ has been obtained in [3]
for the abovementioned scenario of a random walk with fluctuating number of
steps with negative binomial distribution. For general $p$, this pdf has been
retrieved as a limit distribution for imperfect reverberation in [23].
The pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $p=1,2$ and $3$ is shown in
Figs. 7a, 8a, and 9a, respectively, for selected values of $N$. Fig. 10a shows
the corresponding standard deviations of (29). Compared to Fig. 5a, the
increase of the standard deviations with decreasing $N$ is generally smaller
and less dependent on dimensionality.
#### IV.1.2 Standarized field amplitude
For comparison with the empirical pdf of a field amplitude measured by an
electric or magnetic field probe, we consider the sampling pdf of the ratio of
the variates $A$ and $S_{A}$, viz.,
$\displaystyle V\stackrel{{\scriptstyle\Delta}}{{=}}\frac{A}{S_{A}}.$ (47)
The calculation of $f_{V}(v;N)$ is detailed in Sec. A.3, where the final
result is shown to be
$\displaystyle f_{V}(v;N)$ $\displaystyle=$ $\displaystyle C_{V}~{}{v^{2p-1}}$
(48)
$\displaystyle\times{\left[1+\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{pN-\frac{1}{2}}\thinspace
v^{2}\right]^{-[p(N+1)-\frac{1}{2}]}}$
with
$\displaystyle
C_{V}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2}{\left(pN-\frac{1}{2}\right)^{p}}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{p}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$
(49)
The sampling pdf (48) can be referred to as a root-$F$ distribution and
constitutes a counterpart of (19) and (29). Fig. 10b shows its standard
deviation.
#### IV.1.3 Amplitude of biased field
Unfortunately, unlike (36), the cpdf
$f_{A_{\alpha}|S_{A_{\alpha}}}(a_{\alpha}|s_{A_{\alpha}})$ cannot be expressed
in self-sufficient closed form, because $\sigma_{A_{\alpha}}$ for the
Nakagami–Rice cpdf of $A$ depends in a complicated manner on
$\sigma_{E^{\prime(\prime)}}$. Therefore, we do not further pursue this case.
A simpler approach is to derive $f_{A}(a;N)$ via variate transformation of
(38).
### IV.2 Coherent detection
Using the variate transformation $A=\sqrt{U}$, the pdf of the local field
magnitude $A=\sqrt{A^{2}_{x}+A^{2}_{y}+A^{2}_{z}}\equiv|E|\propto\sqrt{U}$
follows. With
$A_{\alpha}\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{{E^{\prime}_{\alpha}}^{2}+{E^{\prime\prime}_{\alpha}}^{2}}$,
(LABEL:eq:fUCart) yields
$\displaystyle f_{A_{\alpha}}(a_{\alpha};N)$ (50) $\displaystyle=$
$\displaystyle a_{\alpha}\thinspace f_{U_{\alpha}}(u_{\alpha}=a^{2}_{\alpha})$
$\displaystyle=$ $\displaystyle C_{A_{\alpha}}\thinspace
a_{\alpha}\int^{a^{2}_{\alpha}}_{0}\frac{f_{E^{\prime}_{\alpha}}(\sqrt{x};N)f_{E^{\prime\prime}_{\alpha}}(\sqrt{a^{2}_{\alpha}-x};N)}{\sqrt{x}\sqrt{a^{2}_{\alpha}-x}}{\rm
d}x~{}~{}~{}~{}~{}$
where $C_{A_{\alpha}}$ is a normalization constant. The pdf for $A_{\alpha}$
is shown for selected values of $N$ in Fig. 11 and for $A$ in Fig. 12.
In fact, one may consider the amplitude of the sampling field itself as a
marginal of the joint sampling pdf
$f_{E^{\prime},E^{\prime\prime}}(e^{\prime},e^{\prime\prime};N)$, followed by
variate transformation to the amplitude
$A=\sqrt{{E^{\prime}}^{2}+{E^{\prime\prime}}^{2}}$ and phase
$\Phi=\tan^{-1}(E^{\prime\prime}/E^{\prime})$. Assuming that $E^{\prime}$ and
$E^{\prime\prime}$ are statistically independent (which, strictly, requires
them to be normally distributed) so that
$f_{E^{\prime},E^{\prime\prime}}(e^{\prime},e^{\prime\prime};N)=f_{E^{\prime}}(e^{\prime};N)f_{E^{\prime\prime}}(e^{\prime\prime};N)$,
then
$\displaystyle f_{A}(a;N)$ $\displaystyle=$
$\displaystyle\frac{C_{A}}{\sigma_{A}}a^{{pN}-{1}}\int^{\pi}_{-\pi}\left({|\cos\phi-(a_{0}/a)|}{|\sin\phi|}\right)^{\frac{pN}{2}-1}$
(51) $\displaystyle\times K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|a\cos\phi-
a_{0}|}{\sigma_{E^{\prime}}}\right)$ $\displaystyle\times
K_{\frac{pN}{2}-1}\left(\sqrt{pN-1}\frac{|a\sin\phi|}{\sigma_{E^{\prime\prime}}}\right){\rm
d}\phi$
where $a_{0}\stackrel{{\scriptstyle\Delta}}{{=}}\mu_{E^{\prime}}$ with
$\mu_{E^{\prime\prime}}=0$.
## V Extension to non-Gaussian ensemble distributions of the field: Iterative
Bayesian scheme
From the theorem of total probability, it follows that
$\displaystyle f_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}\propto
f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}~{}f_{S_{E^{\prime(\prime)}}}.$
(52)
For non-Gaussian $E^{\prime(\prime)}$, for which the prior pdf
$f_{S_{E^{\prime(\prime)}}}$ is more difficult to determine than in the
exposition given before, (52) can be used in an iterative process by means of
an update equation for $f_{S_{E^{\prime(\prime)}}}$, where the latter can be
assigned an initial $\chi_{N-1}$ distribution
$f^{(0)}_{S_{E^{\prime(\prime)}}}$. This prior distribution, together with the
non-Gaussian $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$, then allows for
calculating the posterior $f_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}$ that
serves as the “prior” pdf for the next iteration. Explicitly, denoting the
$i$th iteration in this scheme by the superscript ‘$(i)$’, the iteration
process is specified by
$\displaystyle
f^{(i+1)}_{S_{E^{\prime(\prime)}}}=f^{(i)}_{S_{E^{\prime(\prime)}}|E^{\prime(\prime)}}$
(53)
for $i=0,1,\ldots$, thus yielding
$f^{(n)}_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$ after $n$ iterations,
with similar relations for the intensity and amplitude.
In general, (52)–(53) can be used even with empirical pdfs
$f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$. A similar equation and
procedure can be used for determining the intensity or magnitude for
incoherent detection. This outline of a procedure is sketched here for sake of
completeness and guidance. However, no explicit results are included because
they require knowledge of $f_{E^{\prime(\prime)}|S_{E^{\prime(\prime)}}}$
which must follow from a separate investigation, e.g., from experiment.
Further study is needed with respect to the demonstration of convergence of
(53) to a stable pdf.
## VI Confidence intervals
$95\%$-confidence interval boundaries from sampling pdfs of $E$, $U$, $A$ as
well as $X$, $W$, $V$ and their components, are compared to corresponding
boundaries for the ensemble pdfs in Figs. 13, 14 and 15, respectively. The
Figures show that $N\sim 10$ or larger is needed in order for the sampling
confidence interval boundaries to be close to the ensemble boundaries. For
example, for an overmoded Fabry-Pérot resonator, a resonator length of the
order of five wavelengths or larger is needed to achieve this.
## VII Conclusion
In this paper, we studied sampling distributions for the analytic complex-
valued field, the intensity (energy density, power) and the magnitude for
Gaussian statistically homogeneous random electromagnetic waves. The main
results are (13)–(LABEL:eq:samplepdfE_coh_BesselK_determbias), (17), (19),
(LABEL:eq:samplepdfU_incoh_BesselK), (29), (LABEL:eq:samplepdfA_incoh_BesselK)
and (48). A common feature is that lowering the number of degrees of freedom
characterizing sampling distributions results in their tails becoming heavier
and a consequent associated increase of widths of confidence intervals
compared to the Gauss normal, $\chi^{2}_{2p}$ and $\chi_{2p}$ ensemble
distributions, respectively. Furthermore, it was shown that standardized
quantities (i.e., the sampled random field, magnitude or intensity divided by
its own sample standard deviation, whereby the latter is itself considered as
a random variable whose sample value is calculated from the sample data set
itself) exhibit sampling pdfs that are characterized by a wider spread than
for the case where that quantity has an a priori known (ensemble) standard
deviation. The reason is that in the former case the ratio of random variables
defines a bivariate sampling distribution whose (univariate) marginal pdf is
sought, whereas in the latter case the sampling pdf is univariate from the
outset. The differences in sampling distributions arising from choosing
normalization rather than standardization were shown to be negligible in all
practical cases.
While the focus in this paper was on the simplest of a priori chosen
conditional pdfs for the local instantaneous field in unbounded media,
extension to some more general cases and boundary-value problems is
straightforward. For example, compound exponential distributions for
anisotropic ideal random fields near a conducting or dielectric boundary [9,
33] can be used instead, leading to the total (vectorial) intensity and
amplitude of 3-D random fields near a planar isotropic surface by simple
superposition.
## VIII Acknowledgements
This work was supported in part by the Physical Programme of the U.K. National
Measurement System Policy Unit 2006–2009. I wish to thank Dr. P. Harris (NPL)
for comments to the manuscript, Mr. S. Burbridge (Imperial College) for
assistance with SGI Altix high-performance computation relating to Fig. 6, and
Dr. M. Little (University of Oxford, U.K.) for discussions relating to ref.
[29].
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## Appendix A Sampling distributions of field, intensity and amplitude
### A.1 Field
Here we derive the one-dimensional sampling distribution for the intensity
$|E|^{2}$ or energy density $U\propto|E|^{2}$ of a random statistically
homogeneous Cartesian or vector electric (or, by extension, magnetic) field
$E$.
The pdf of an ideal random field $E$ is circular with independent and
identically distributed (i.i.d.) real (in-phase) and imaginary (quadrature)
components $E^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}{\rm Re}(E)$ and
$E^{\prime\prime}\stackrel{{\scriptstyle\Delta}}{{=}}{\rm Im}(E)$. As a
result, these components can be studied in isolation from each other. The
parent (ensemble) distribution of $E$ is a central Gauss normal distribution
($\langle E\rangle=0$). Therefore, the standardized $N$-point sample variance
$D_{N-1}\stackrel{{\scriptstyle\Delta}}{{=}}(N{-}1)\thinspace
S^{2}_{E^{\prime(\prime)}}/\sigma^{2}_{E^{\prime(\prime)}}=\sum^{N}_{i=1}(E^{\prime(\prime)}_{i}{-}M_{E^{\prime(\prime)}})^{2}/\sigma^{2}_{E^{\prime(\prime)}}$
exhibits a $\chi^{2}_{N{-}1}$ pdf, while
$D_{1}\stackrel{{\scriptstyle\Delta}}{{=}}(E^{\prime(\prime)}-M_{E^{\prime(\prime)}})^{2}/\sigma^{2}_{E^{\prime(\prime)}}$
has a $\chi^{2}_{1}$ pdf. Consequently, the ratio
$\displaystyle
X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{E^{\prime(\prime)}-M_{E^{\prime(\prime)}}}{S_{E^{\prime(\prime)}}}=\sqrt{\frac{D_{1}}{D_{N-1}/(N{-}1)}}$
(54)
has a Student $t$ sampling distribution with $N{-}1$ degrees of freedom,
whence for $E^{\prime(\prime)}$ itself,
$\displaystyle f_{E^{\prime(\prime)}}\left(e^{\prime(\prime)};N\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{N}{2}\right)}{\sqrt{\pi}~{}\Gamma\left(\frac{N-1}{2}\right)~{}\sqrt{N-1}}$
(55) $\displaystyle\times$
$\displaystyle\left[1+\frac{1}{N-1}\left(\frac{e^{\prime(\prime)}-m_{E^{\prime(\prime)}}}{s_{E^{\prime(\prime)}}}\right)^{2}\right]^{-N/2}$
for $N>1$, where $e^{\prime(\prime)}$, $m_{E^{\prime(\prime)}}$ and
$s_{E^{\prime(\prime)}}$ are sample values. Compared to the sample value
$s_{E^{\prime(\prime)}}$ of the parent pdf, the sampling pdf exhibits an
increased sampling standard deviation, viz.,
$\sqrt{(N{-}1)/(N{-}3)}s_{E^{\prime(\prime)}}$ for $N>3$.
Typically, the Student $t$ distribution arises in the characterization of the
sample mean, i.e., for $[(M_{E^{\prime(\prime)}}-\langle
E^{\prime(\prime)}\rangle)/(\sigma_{E^{\prime(\prime)}}/\sqrt{N})]/(S_{E^{\prime(\prime)}}/\sigma_{E^{\prime(\prime)}})$,
instead of
$[({E^{\prime(\prime)}}-M_{E^{\prime(\prime)}})/\sigma_{E^{\prime(\prime)}}]/(S_{E^{\prime(\prime)}}/\sigma_{E^{\prime(\prime)}})$
as in (54). Without lack of generality, however, we further use the simplified
definition
$X^{\prime(\prime)}\stackrel{{\scriptstyle\Delta}}{{=}}{E^{\prime(\prime)}}/{S_{E^{\prime(\prime)}}}$,
i.e., $M_{E^{\prime(\prime)}}=0$, because each sample set can always be
centralized by its own sample mean value while maintaining its sampling pdf.
The square of $X^{\prime(\prime)}$, represented as
$Y^{\prime(\prime)}/S_{Y^{\prime(\prime)}}\stackrel{{\scriptstyle\Delta}}{{=}}X^{{\prime(\prime)}^{2}}={E^{\prime(\prime)}}^{2}/S^{2}_{E^{\prime(\prime)}}$,
exhibits a $t^{2}$ sampling distribution that follows from (55) as
$\displaystyle f_{Y^{\prime(\prime)}}(y^{\prime(\prime)};N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{N}{2}\right)}{2\sqrt{\pi}~{}\Gamma\left(\frac{N-1}{2}\right)~{}\sqrt{N-1}}$
(56)
$\displaystyle\times\left[\frac{y^{\prime(\prime)}}{s_{Y^{\prime(\prime)}}}\left(1+\frac{y^{\prime(\prime)}}{(N-1)\thinspace
s_{Y^{\prime(\prime)}}}\right)^{N}\right]^{-1/2}.$
which is a Fisher-Snedecor $F_{1,N-1}$ pdf of the ratio of the standard
$\chi^{2}_{1}$ variate $D_{1}$ and the standard $\chi^{2}_{N-1}$ variate
$D_{N-1}$ (both being statistically independent, because
$M_{E^{\prime(\prime)}}$ and $S_{E^{\prime(\prime)}}$ are independent for
Gauss normal $E^{\prime(\prime)}$), whereby each variate is divided by its
corresponding number of degrees of freedom, i.e.,
$\displaystyle t^{2}=F_{1,N-1}=\frac{\chi^{2}_{1}/1}{\chi^{2}_{N-1}/(N-1)}.$
(57)
### A.2 Field intensity
To find the sampling distribution of a squared Cartesian or vectorial EM field
with $2p$ i.i.d. components $Y^{\prime(\prime)}_{i}$ (corresponding to a
$p$-dimensional analytic complex-valued field), i.e.,
$\sigma_{Y^{\prime(\prime)}_{i}}=\sigma^{2}_{X}$, we consider the ratio
$Z/D_{Z}$, where
$\displaystyle
Z\stackrel{{\scriptstyle\Delta}}{{=}}\frac{Y^{\prime}+Y^{\prime\prime}}{\sigma_{Y^{\prime(\prime)}}}=\sum^{p}_{i=1}\left(\frac{Y^{\prime}_{i}}{\sigma_{Y^{\prime}}}+\frac{Y^{\prime\prime}_{i}}{\sigma_{Y^{\prime\prime}}}\right)\sum^{2p}_{i=1}\frac{(X_{i}-m_{X})^{2}}{\sigma^{2}_{X}}~{}~{}~{}$
(58)
has a standard $\chi^{2}_{2p}$ distribution, on account of the addition
theorem for $2p$ i.i.d. standard $\chi^{2}_{1}$ variates, and
$\displaystyle
D_{Z}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{(2pN-1)~{}S^{2}_{X^{\prime(\prime)}}}{\sigma^{2}_{X^{\prime(\prime)}}}$
(59)
exhibits a standard $\chi^{2}_{2pN{-}1}$ distribution, with
$S^{2}_{X^{\prime(\prime)}}=$$\sum^{2pN}_{i=1}(X^{\prime(\prime)}_{i}{-}M_{X^{\prime(\prime)}})^{2}/(2pN{-}1)$.
Hence, their ratio $G$ is a scaled Fisher-Snedecor $F$ variate with
($2p,2pN-1)$ degrees of freedom:
$\displaystyle G\stackrel{{\scriptstyle\Delta}}{{=}}\frac{Z}{D_{Z}}$
$\displaystyle\equiv$
$\displaystyle\frac{2p}{2pN-1}\frac{Z/(2p)}{D_{Z}/(2pN-1)}$ (60)
$\displaystyle=$ $\displaystyle\frac{2p}{2pN-1}F_{2p,2pN-1}.$
With $\sigma_{|E|^{2}}=2\sqrt{p}\thinspace\sigma^{2}_{E^{\prime(\prime)}}$,
the ratio $Z/D_{Z}$ can be related to the standardized power or field
intensity as
$\displaystyle\frac{U}{S_{U}}=\frac{\sum^{2p}_{i=1}{E^{\prime}_{i}}^{2}}{2\sqrt{p}~{}S^{2}_{E^{\prime}}}$
$\displaystyle=$
$\displaystyle\frac{2pN-1}{2\sqrt{p}}\frac{\sum^{2p}_{i=1}(E^{\prime}_{i}/\sigma_{E^{\prime}})^{2}}{(2pN-1)~{}S^{2}_{E^{\prime}}/\sigma^{2}_{E^{\prime}}}$
(61) $\displaystyle=$ $\displaystyle\frac{2pN-1}{2\sqrt{p}}\frac{Z}{D_{Z}}.$
Thus, combining (60) and (61) yields
$\displaystyle
W\stackrel{{\scriptstyle\Delta}}{{=}}\frac{U}{S_{U}}=\sqrt{p}~{}F_{2p,2pN-1}.$
(62)
With (60), upon scaling the standard $F_{2p,2pN-1}$ distribution, the sampling
pdf of $G$ is
$\displaystyle f_{G}(g;N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}~{}\frac{g^{p-1}}{\left(1+g\right)^{pN-\frac{1}{2}+p}}~{}~{}~{}.$
(63)
For the square of the Cartesian in-phase field component $[{\rm Re}(E)]^{2}$
or the quadrature component $[{\rm Im}(E)]^{2}$ (i.e., for $p=1/2$), (63)
reduces to (56) as expected. Finally, from (62), the sampling distribution of
$W=U/S_{U}$ follows as
$\displaystyle f_{W}(w;N)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}~{}\frac{p^{p/2}}{\left(pN-\frac{1}{2}\right)^{p}}$
(64)
$\displaystyle\times\frac{w^{p-1}}{\left(1+\frac{\sqrt{p}}{pN-\frac{1}{2}}\thinspace
w\right)^{pN-\frac{1}{2}+p}}.$
This represents the pdf of the standardized received sampled power $W$ at $w$,
i.e., for the ratio of sampled values $u$ and $s_{U}$.
In the limit $N\rightarrow+\infty$, the exponent $p(N+1)-\frac{1}{2}$ in (64)
reduces to $pN-\frac{1}{2}$; the prefactor
$\Gamma(pN-\frac{1}{2}+p)/\Gamma(pN-\frac{1}{2})=(pN-\frac{1}{2})\cdot\ldots\cdot(pN-\frac{3}{2}+p)$
for $p\geq 1$ becomes $(pN-\frac{1}{2})^{p}$; and $s_{U}$ approaches the
ensemble statistic $\sigma_{U}$, whence
$\displaystyle f_{W}(w;N)\rightarrow
f_{U}(u)=\frac{p^{p/2}}{\Gamma(p)\sigma_{U}}\left(\frac{u}{\sigma_{U}}\right)^{p-1}\exp\left(-\sqrt{p}\thinspace\frac{u}{\sigma_{U}}\right)$
which is the ensemble $\chi^{2}_{2p}$ limit pdf, as expected. The result
agrees with a well-known limit theorem from probability theory [7] stating
that $F_{m,n}(x)$ converges to $\chi^{2}_{m}(mx)$ when
$n/m{\rightarrow}+\infty$, i.e., $F_{q}(m,n)\rightarrow\chi^{2}_{q}(m)/m$ for
the corresponding quantiles. In the same limit $N{\rightarrow}{+}\infty$, the
asymptotic mean value of $W$ is
$\displaystyle\langle
W\rangle=\frac{2pN{-}1}{2pN{-}3}\sqrt{p}\rightarrow\sqrt{p}$ (66)
and the standard deviation for $pN>5/2$ is
$\displaystyle\sigma_{W}=\sqrt{\frac{2(2pN{-}1)^{2}(2pN{+}2p{-}3)}{2p(2pN{-}3)^{2}(2pN{-}5)}{p}}\rightarrow
1.$ (67)
Thus, we find that the coefficient of variation, i.e.,
$\displaystyle\frac{\sigma_{W}}{\langle{W}\rangle}=\sqrt{\frac{2pN{+}2p{-}3}{p(2pN{-}5)}}~{}\rightarrow\frac{1}{\sqrt{p}}$
(68)
has its value reduced to that for the $\chi^{2}_{2p}$ ensemble distribution
when $N{\rightarrow}+\infty$, whereas for small $N$ its value is substantially
larger, indicating larger relative uncertainty.
### A.3 Field amplitude
For the standardized field magnitude
$V\stackrel{{\scriptstyle\Delta}}{{=}}A/S_{A}=\sqrt{(S^{2}_{V}/S_{W})~{}W}$,
with
$\displaystyle\frac{\sigma^{2}_{Z}}{\sigma_{W}}=\frac{\sigma^{2}_{A}}{\sigma_{A^{2}}}=\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{\sqrt{p}}$
(69)
valid for $\chi^{(2)}_{2p}$ distributions, from variate transformation of (64)
and noting that
$\displaystyle\frac{w}{s_{W}}=\left(\frac{v}{s_{V}}\right)^{2}=\frac{v^{2}}{s_{W}}\cdot\frac{s_{W}}{s^{2}_{V}},~{}~{}{\rm
d}w=2v\frac{s_{W}}{s^{2}_{V}}{\rm d}v,~{}~{}~{}$ (70)
we obtain the sampling pdf of $V$ at $v=a/s_{A}$ as
$\displaystyle f_{V}(v;N)$ $\displaystyle=$ $\displaystyle C_{V}{v^{2p-1}}$
(71)
$\displaystyle\times{\left[1+\frac{p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}}{pN-\frac{1}{2}}\thinspace
v^{2}\right]^{-(pN+p-\frac{1}{2})}}$
which can be referred to as a root-$F$ distribution, where
$\displaystyle
C_{V}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{2}{\left(pN-\frac{1}{2}\right)^{p}}\left[p-\left(\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma(p)}\right)^{2}\right]^{p}\frac{\Gamma\left(pN-\frac{1}{2}+p\right)}{\Gamma(pN-\frac{1}{2})\Gamma(p)}.$
## Figures
|
---|---
Figure 1: (color on-line) Sampling probability density function for real or
imaginary Cartesian component of electric field ($p{=}1$) at selected values
of $N$: (a) Bessel $K$ sampling pdfs of $E^{\prime(\prime)}_{\alpha}$ [eq.
(13)]; (b) Student $t$ sampling pdfs of
$X^{\prime(\prime)}=E^{\prime(\prime)}_{\alpha}/S_{{E^{\prime(\prime)}_{\alpha}}}$
[eq. (19)]. In both plots, the right tail becomes thinner for increasing
values of $N$.
|
---|---
Figure 2: (color on-line) Sampling probability density function for intensity
or energy density of Cartesian component of field ($p{=}1$) at selected values
of $N$ based on incoherent detection: (a) Bessel $K$ sampling pdfs of
$U_{\alpha}$ [eq. (LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor
$F$ sampling pdfs of $W_{\alpha}=U_{\alpha}/S_{U_{\alpha}}$ [eq. (29)]. In
both plots, the right tail becomes thinner for increasing values of $N$.
|
---|---
Figure 3: (color on-line) Probability density function of intensity or energy
density of planar field ($p{=}2$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdfs of $U_{\rm t}$ [eq.
(LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor $F$ sampling pdfs of
$W_{\rm t}=U_{\rm t}/S_{U_{\rm t}}$ [eq. (29)]. In both plots, the right tail
becomes thinner for increasing values of $N$.
|
---|---
Figure 4: (color on-line) Probability density function of intensity or energy
density of total (vector) field ($p{=}3$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdfs of $U$ [eq.
(LABEL:eq:samplepdfU_incoh_BesselK)]; (b) Fisher-Snedecor $F$ sampling pdfs of
$W=U/S_{U}$ [eq. (29)]. In both plots, the right tail becomes thinner for
increasing values of $N$.
|
---|---
(a) | (b)
Figure 5: (color on-line) Sampling standard deviations as a function of $N$
(a) for Bessel $K$ sampling distributions (LABEL:eq:samplepdfU_incoh_BesselK)
of $U_{\alpha}$, $U_{\rm t}$, and $U$; (b) for Fisher-Snedecor $F$ sampling
distributions (29) of $U_{\alpha}/S_{U_{\alpha}}$, and $U_{\rm t}/S_{U_{\rm
t}}$, $U/S_{U}$.
---
Figure 6: (color on-line) Probability density function of intensity or energy
density of Cartesian component of field ($p{=}1$) derived from coherent
detection at selected values of $N$: Bessel $K$ sampling pdf of $U_{\alpha}$
[eqn. (LABEL:eq:samplepdfU_coh_BesselK)].
|
---|---
Figure 7: (color on-line) Sampling probability density function of amplitude
of Cartesian component of field ($p=1$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ sampling pdf
(LABEL:eq:samplepdfA_incoh_BesselK) for $A_{\alpha}$; (b) root-$F$ pdf (48)
for $V_{\alpha}=A_{\alpha}/S_{A_{\alpha}}$. In both plots, the right tail
becomes thinner for increasing values of $N$.
|
---|---
Figure 8: (color on-line) Sampling probability density function of amplitude
of planar field ($p{=}2$) at selected values of $N$ based on incoherent
detection: (a) Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK) for $A_{\rm
t}$; (b) root-$F$ pdf (48) for $V_{\rm t}=A_{\rm t}/S_{A_{\rm t}}$. In both
plots, the right tail becomes thinner for increasing values of $N$.
|
---|---
Figure 9: (color on-line) Sampling probability density function of amplitude
of total (vector) field ($p{=}3$) at selected values of $N$ based on
incoherent detection: (a) Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK)
for $A$; (b) root-$F$ pdf (48) for $V=A/S_{A}$. In both plots, the right tail
becomes thinner for increasing values of $N$.
|
---|---
(a) | (b)
Figure 10: (color on-line) Sampling standard deviation as a function of $N$
(a) for Bessel $K$ pdf (LABEL:eq:samplepdfA_incoh_BesselK) of $A_{\alpha}$,
$A_{\rm t}$, and $A$; (b) for Fisher-Snedecor $F$ pdf (48) of
$A_{\alpha}/S_{A_{\alpha}}$, $A_{\rm t}/S_{A_{\rm t}}$, and $A/S_{A}$.
---
Figure 11: (color on-line) Sampling probability density function of magnitude
of Cartesian component of field ($p{=}1$) at selected values of $N$, derived
for coherent detection. The right tail becomes thinner for increasing values
of $N$.
---
Figure 12: (color on-line) Sampling probability density function of magnitude
of total (vector) field ($p{=}3$) at selected values of $N$, derived for
coherent detection. In both plots, the right tail becomes thinner for
increasing values of $N$.
---
Figure 13: (color on-line) Upper boundaries of $95\%$-confidence intervals
for real or imaginary parts of $E_{\alpha}$, $E_{\rm t}$, or $E$. The lower
boundary $F^{-1}(0.025)\equiv-F^{-1}(0.975)$ is symmetric with respect to
$e=0$.
---
(a)
(b)
(c)
Figure 14: (color on-line) Lower and upper boundaries of $95\%$-confidence
intervals (a) for a 1D Cartesian component $U_{\alpha}$ ($p{=}1$), (b) for a
2D planar field $U_{\rm t}$ ($p{=}2$), and (c) for the 3D vectorial $U$
($p{=}3$), normalized by the respective sampling standard deviations. For each
line type, the lower and upper curves represent $2.5\%$ and $97.5\%$
percentiles, respectively.
---
(a)
(b)
(c)
Figure 15: (color on-line) Lower and upper boundaries of $95\%$-confidence
intervals (a) for a 1D Cartesian component $A_{\alpha}$ ($p{=}1$), (b) for a
2D planar field $A_{\rm t}$ ($p{=}2$), and (c) for the 3D vectorial $A$
($p{=}3$), normalized by the respective sampling standard deviations. For each
line type, the lower and upper curves represent $2.5\%$ and $97.5\%$
percentiles, respectively.
|
arxiv-papers
| 2009-07-28T17:40:16 |
2024-09-04T02:49:04.264862
|
{
"license": "Public Domain",
"authors": "L. R. Arnaut",
"submitter": "Luk Arnaut",
"url": "https://arxiv.org/abs/0907.4970"
}
|
0907.5144
|
# Interacting agegraphic dark energy models in non-flat universe
Ahmad Sheykhi 111 sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
A so-called “agegraphic dark energy”, was recently proposed to explain the
dark energy-dominated universe. In this Letter, we generalize the agegraphic
dark energy models to the universe with spatial curvature in the presence of
interaction between dark matter and dark energy. We show that these models can
accommodate $w_{D}=-1$ crossing for the equation of state of dark energy. In
the limiting case of a flat universe, i.e. $k=0$, all previous results of
agegraphic dark energy in flat universe are restored.
## I Introduction
The dark energy problem constitute a major puzzle of modern cosmology. A great
variety of cosmological observations suggest that our universe is currently
undergoing a phase of accelerated expansion likely driven by some unknown
energy component whose main feature is to possess a negative pressure Rie .
Although the nature of such dark energy is still speculative, an overwhelming
flood of papers has appeared which attempt to describe it by devising a great
variety of models. Among them are cosmological constant, exotic fields such as
phantom or quintessence, modified gravity, etc, see Pad for a recent review.
Recently, a new dark energy candidate, based not in any specific field but on
the holographic principle, was proposed Hor ; Hsu . According to the
holographic principle, the number of degrees of freedom of physical systems
scale with their bounding area rather than with their volume Suss1 . On these
basis, Cohen et al. Coh suggested that in quantum field theory a short
distance cutoff is related to a long distance cutoff due to the limit set by
formation of a black hole, which results in an upper bound on the zero-point
energy density. The extension of the holographic principle to a general
cosmological setting was addressed by Fischler and Susskind Suss2 . Following
this line, Li Li argued that zero-point energy density could be viewed as the
holographic dark energy density satisfying $\rho_{D}\leq
3c^{2}m^{2}_{p}/L^{2}$, the equality sign holding only when the holographic
bound is saturated. Here $c^{2}$ is a constant, the coefficient $3$ is for
convenience, $L$ is an IR cutoff and $m^{2}_{p}=(8\pi G)^{-1}$. The
holographic models of dark energy have been proposed and studied widely in the
literature Huang ; HDE ; Xin ; Setare . It is fair to claim that simplicity
and reasonability of holographic model of dark energy provides more reliable
frame to investigate the problem of dark energy rather than other models
proposed in the literature Seta2 . For example the coincidence problem can be
easily solved in some models of holographic dark energy Pav1 .
More recently, a new dark energy model, called agegraphic dark energy (ADE)
was proposed by Cai Cai1 . This model is based on the uncertainty relation of
quantum mechanics together with the gravitational effect in general
relativity. Following the line of quantum fluctuations of spacetime,
Karolyhazy et al. Kar1 argued that the distance $t$ in Minkowski spacetime
cannot be known to a better accuracy than $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$
where $\beta$ is a dimensionless constant of order unity. Based on Karolyhazy
relation, Maziashvili discussed that the energy density of metric fluctuations
of the Minkowski spacetime is given by Maz
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (1)
where $t_{p}$ is the reduced Planck time. Throughout this Letter we use the
units $c=\hbar=k_{b}=1$. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$
and $m_{p}$ are the reduced Planck length and mass, respectively. The
agegraphic dark energy model assumes that the observed dark energy comes from
the spacetime and matter field fluctuations in the universe Wei1 ; Wei2 . The
agegraphic models of dark energy have been examined age1 ; shey1 ; seta3 and
constrained by various astronomical observations age2 .
On the other hand, lacking a fundamental theory, most discussions on dark
energy rely on the fact that its evolution is independent of other matter
fields. Given the unknown nature of both dark matter and dark energy there is
nothing in principle against their mutual interaction and it seems very
special that these two major components in the universe are entirely
independent. Indeed, this possibility is receiving growing attention in the
literature Ame ; Zim (see also Seta1 ; wang1 and references therein) and
appears to be compatible with SNIa and CMB data Oli . Furthermore, the
interacting holographic dark energy models have also been extended to the
universe with spacial curvature Seta2 ; wang2 .
Besides, it is generally believed that inflation practically washes out the
effect of curvature in the early stages of cosmic evolution. However, it does
not necessarily imply that the curvature has to be wholly neglected at
present. Indeed, aside from the sake of generality, there are sound reasons to
include it: (i) Inflation drives the $k/a^{2}$ ratio close to zero but it
cannot set it to zero if $k\neq 0$ initially. (ii) The closeness to perfect
flatness depends on the number of e-folds and we can only speculate about the
latter. (iii) After inflation the absolute value of the $k/a^{2}$ term in the
field equations may increase with respect to the matter density term, thereby
the former should not be ignored when studying the late universe. (iv)
Observationally there is room for a small but non-negligible spatial curvature
spe . For instance, the tendency of preferring a closed universe appeared in a
suite of CMB experiments Sie . The improved precision from WMAP provides
further confidence, showing that a closed universe with positively curved
space is marginally preferred Uzan . In addition to CMB, recently the spatial
geometry of the universe was probed by supernova measurements of the cubic
correction to the luminosity distance Caldwell , where a closed universe is
also marginally favored.
In the light of all mentioned above, it becomes obvious that the investigation
on the interacting agegraphic dark energy models in the universe with spacial
curvature is well motivated. In this Letter, we would like to generalize,
following Seta2 , the agegraphic dark energy models to the universe with
spacial curvature in the presence of interaction between the dark matter and
dark energy. We will also show that the equation of state of dark energy can
accommodate $w=-1$ crossing. The plan of the work is as follows: In section
II, we study the original agegraphic model of dark energy in a non-flat
universe where the time scale is chosen to be the age of the universe. In
section III, we consider the new model of agegraphic dark energy while the
time scale is chosen to be the conformal time instead of the age of the
universe. Finally, in section IV we summarize our results.
## II THE ORIGINAL ADE IN NONFLAT UNIVERSE
The original agegraphic dark energy density has the form (1) where $t$ is
chosen to be the age of the universe
$T=\int{dt}=\int_{0}^{a}{\frac{da}{Ha}},$ (2)
where $a$ is the scale factor and $H=\dot{a}/a$ is the Hubble parameter. Thus,
the energy density of the agegraphic dark energy is given by Cai1
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (3)
where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time (since the energy density is derived for Minkowski
space-time), and so on. The dark energy density (3) has the same form as the
holographic dark energy, but the length measure is chosen to be the age of the
universe instead of the horizon radius of the universe. Thus the causality
problem in the holographic dark energy is avoided Cai1 .
The total energy density is $\rho=\rho_{m}+\rho_{D}$, where $\rho_{m}$ and
$\rho_{D}$ are the energy density of dark matter and dark energy,
respectively. The total energy density satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0.$ (4)
However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (5)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (6)
Here $w_{D}$ is the equation of state parameter of agegraphic dark energy and
$Q$ denotes the interaction term and can be taken as $Q=3b^{2}H\rho$ with
$b^{2}$ being a coupling constant. This expression for the interaction term
was first introduced in the study of the suitable coupling between a
quintessence scalar field and a pressureless cold dark matter field Ame . In
the context of holographic dark energy model, this form of interaction was
derived from the choice of Hubble scale as the IR cutoff Pav1 . Although at
this point the interaction may look purely phenomenological but different
Lagrangians have been proposed in support of it (see Tsu and references
therein). Besides, in the absence of a symmetry that forbids the interaction
there is nothing, in principle, against it. Further, the interacting dark
mater dark energy (the latter in the form of a quintessence scalar field and
the former as fermions whose mass depends on the scalar field) has been
investigated at one quantum loop with the result that the coupling leaves the
dark energy potential stable if the former is of exponential type but it
renders it unstable otherwise Dor . Thus, microphysics seems to allow enough
room for the coupling; however, this point is not fully settled and should be
further investigated. The difficulty lies, among other things, in that the
very nature of both dark energy and dark matter remains unknown whence the
detailed form of the coupling cannot be elucidated at this stage.
For the Friedmann-Robertson-Walker (FRW) universe filled with dark energy and
dust (dark matter), the corresponding Friedmann equation takes the form
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right),$ (7)
where $k$ is the curvature parameter with $k=-1,0,1$ corresponding to open,
flat, and closed universes, respectively. A closed universe with a small
positive curvature ($\Omega_{k}\simeq 0.02$) is compatible with observations
spe . If we introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (8)
then the Friedmann equation can be written
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (9)
Using Eq. (3), we have
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (10)
Differentiating Eq. (10) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(11)
where the dot is the derivative with respect to the cosmic time and the prime
denotes the derivative with respect to $x=\ln{a}$. Taking the derivative of
both side of the Friedman equation (7) with respect to the cosmic time, and
using Eqs. (3), (5), (9) and (10), it is easy to find that
$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{n}-\frac{\Omega_{k}}{2}+\frac{3}{2}b^{2}(1+\Omega_{k}).$
(12)
Substituting this relation into Eq. (11), we obtain the equation of motion of
agegraphic dark energy
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)\right.$
(13) $\displaystyle\left.-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
Inserting $\Omega_{k}=0=b$, this equation reduces to Eq. (12) of Ref. Cai1 .
Using Eqs. (3) and (6), as well as Eq. (10), we can obtain the equation of
state parameter for the interacting agegraphic dark energy
$\displaystyle
w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}-b^{2}\Omega^{-1}_{D}(1+\Omega_{k}).$
(14)
The total equation of state parameter is given by
$\displaystyle
w_{\mathrm{tot}}=\frac{p}{\rho}=\frac{\Omega_{D}}{1+\Omega_{k}}w_{D}.$ (15)
For completeness, we give the deceleration parameter
$\displaystyle q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (16)
which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. Substituting Eq. (12) in Eq. (16) we get
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{3}{2}{\Omega_{D}}+\frac{\Omega^{3/2}_{D}}{n}-\frac{3}{2}b^{2}+\frac{1}{2}\Omega_{k}(1-3b^{2}).$
(17)
It is worth noting that in the absence of interaction between dark energy and
dark matter, $b^{2}=0$, from Eq. (14) we see that $w_{D}$ is always larger
than $-1$ and cannot cross the phantom divide $w_{D}=-1$. In addition the
condition $n>1$ is necessary to derive the present accelerated expansion Cai1
. However, the situation is changed as soon as the interaction term is taken
into account. In this case ($b^{2}\neq 0$), from Eq. (14) one can easily see
that $w_{D}$ can cross the phantom divide provided
$3nb^{2}(1+\Omega_{k})>2{\Omega^{3/2}_{D}}$. If we take $\Omega_{D}=0.72$ and
$\Omega_{k}=0.02$ for the present time, the phantom-like equation of state can
be achieved only if $nb^{2}>0.4$. The best fit result for agegraphic dark
energy which is consistent with most observations like WMAP and SDSS data,
shows that $n=3.4$ age2 . Thus, the condition $w_{D}<-1$ leads to $b^{2}>0.12$
for the coupling between dark energy and dark matter. For instance, if we take
$b^{2}=0.15$ we get $w_{D}=-1.05$. This indicates that one can generate
phantom-like equation of state from an interacting agegraphic dark energy
model in the universe with any spacial curvature. Putting $\Omega_{k}=0$ in
Eqs. (14) and (17), these equations reduce to their respective equations of
original interacting agegraphic dark energy model in flat universe Wei1 . The
original interacting agegraphic dark energy has laso some interesting
features. From Eq. (14), it is easy to see that $w_{D}<-1$ is necessary in the
early time where $\Omega_{D}\rightarrow 0$, while in the late time where
$\Omega_{D}\rightarrow 1$ ($\Omega_{k}\simeq 0$), we have $w_{D}>-1$ for
$nb^{2}<0.67$ and $w_{D}<-1$ for $nb^{2}>0.67$.
It is important to note that in the absence of interaction, the original
agegraphic dark energy model has a drawback to describe the matter-dominated
universe in the far past where $a\ll 1$ and $\Omega_{D}\ll 1$. On one side,
from Eq. (14) with $b^{2}=0$ we have $w_{D}\rightarrow-1$ as
$\Omega_{D}\rightarrow 0$. This means that in the matter-dominated epoch the
agegraphic dark energy behaves like a cosmological constant. On the other
side, Eq. (13) with $b^{2}=0$, $\Omega_{D}\ll 1$ and $\Omega_{k}\ll 1$
approximately becomes
$\displaystyle\frac{d\Omega_{D}}{da}\simeq\frac{\Omega_{D}}{a}\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(18)
which has a solution of the form $\Omega_{D}=9n^{2}/4$. Substituting this
relation into Eq. (14) with $b^{2}=0$, we get $w_{D}=0$. Therefore, the dark
energy behaves as pressureless matter. Obviously, pressureless matter cannot
generate accelerated expansion, which seems to rule out the choice $t=T$. Thus
there is a confusion in the original agegraphic dark energy model. This issue
is similar to the holographic dark energy model when choosing the Hubble
parameter as the IR cutoff Hsu . It was argued by Pavon and Zimdahl that the
problem can be solved as soon as an interaction between the dark energy and
dark matter is taken into account Pav1 . Similarly, in the agegraphic model of
dark energy the inconsistency in the original version can be removed with the
interaction between dark energy and dark matter. To see this, consider the
matter-dominated epoch where $a\ll 1$ and $\Omega_{D}\ll 1$ for interacting
agegraphic dark energy. In this case Eq. (13) with $\Omega_{k}\ll 1$
approximately becomes
$\displaystyle\frac{d\Omega_{D}}{da}\simeq\frac{\Omega_{D}}{a}\left(3-\frac{2}{n}\sqrt{\Omega_{D}}-3b^{2}\right),$
(19)
which has a solution of the form $\Omega_{D}=9n^{2}(1-b^{2})^{2}/4$.
Substituting this relation into Eq. (14), we obtain
$\displaystyle w_{D}=-b^{2}\left(1+\frac{4}{9n^{2}(1-b^{2})^{2}}\right).$ (20)
Therefore for $b\neq 0$ we have $w_{D}<0$, and the agegraphic model of dark
energy can generate accelerated expansion. The presence of the spatial
curvature does not seriously modify the above discussion. Thus, the confusion
in the original agegraphic dark energy model without interaction is removed.
Nevertheless, Wei and Cai Wei2 proposed a new model of agegraphic dark energy
to resolve the contradiction in the far past of original non-interacting
agegraphic dark energy model.
## III THE NEW MODEL OF ADE IN NONFLAT UNIVERSE
As we argued the original agegraphic dark energy model has some difficulties
Cai1 . Therefore one may seek for another agegraphic dark energy model. Wei
and Cai proposed a new model of agegraphic dark energy Wei2 , while the time
scale is chosen to be the conformal time $\eta$ instead of the age of the
universe, which is defined by $dt=ad\eta$, where $t$ is the cosmic time. It is
worth noting that the Karolyhazy relation $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$
was derived for Minkowski spacetime $ds^{2}=dt^{2}-d\mathrm{x^{2}}$ Kar1 ; Maz
. In the case of the FRW universe, we have
$ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$.
Therefore, it is more reasonable to choose the time scale in Eq. (3) to be the
conformal time $\eta$ Wei2 . Taking this into account, the energy density of
the new agegraphic dark energy can be written
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (21)
where the conformal time is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (22)
If we write $\eta$ to be a definite integral, there will be an integral
constant in addition. Thus, we have $\dot{\eta}=1/a$. Let us again consider a
FRW universe with spatial curvature containing the new agegraphic dark energy
and pressureless matter. The Friedmann equation can be written
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right),$
(23)
where can be rewritten as
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (24)
The fractional energy density of the agegraphic dark energy is now given by
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (25)
We can also find the equation of motion for $\Omega_{D}$ by taking the
derivative of Eq. (25). The result is
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{na}\sqrt{\Omega_{D}}\right).$
(26)
Taking the derivative of both side of the Friedman equation (23) with respect
to the cosmic time $t$, and using Eqs. (5), (21), (24) and (25), we obtain
$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{na}-\frac{\Omega_{k}}{2}+\frac{3}{2}b^{2}(1+\Omega_{k}).$
(27)
Substituting this relation into Eq. (26), we obtain the evolution behavior of
the new agegraphic dark energy
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)\right.\
$ (28) $\displaystyle\left.-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
The equation of state parameter of the interacting new agegraphic dark energy
can be obtained as
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}\Omega^{-1}_{D}(1+\Omega_{k}),$
(29)
Again we see that $w_{D}$ can cross the phantom divide provided
$3nab^{2}(1+\Omega_{k})>2{\Omega^{3/2}_{D}}$. Taking $\Omega_{D}=0.72$,
$\Omega_{k}=0.02$, and $a=1$ for the present time, the phantom-like equation
of state can be achieved only if $nb^{2}>0.4$. The joint analysis of the
astronomical data for the new agegraphic dark energy gives the best-fit value
(with $1\sigma$ uncertainty) $n=2.7$ age2 . Thus, the condition $w_{D}<-1$
leads to $b^{2}>0.15$ for the coupling between dark energy and dark matter.
For instance, if we take $b^{2}=0.2$ we get $w_{D}=-1.07$. The deceleration
parameter is now given by
$\displaystyle q$ $\displaystyle=$
$\displaystyle-1-\frac{\dot{H}}{H^{2}}=\frac{1}{2}-\frac{3}{2}{\Omega_{D}}+\frac{\Omega^{3/2}_{D}}{na}$
(30) $\displaystyle-\frac{3}{2}b^{2}+\frac{1}{2}\Omega_{k}(1-3b^{2}).$
Comparing Eqs. (27)-(30) with Eqs. (12), (13), (14) and (17) in the previous
section, we see that the scale factor $a$ enters Eqs. (27)-(30) explicitly. In
the late time where $a\rightarrow\infty$ and $\Omega_{D}\rightarrow 1$, from
Eq. (29) with $b^{2}=0$ we have $w_{D}\rightarrow-1$; thus the new agegraphic
dark energy mimics a cosmological constant in the late time. Let us now
consider the matter-dominated epoch where $a\ll 1$ and $\Omega_{D}\ll 1$. In
this case Eq. (28) with $b^{2}=0$ and $\Omega_{k}\ll 1$ approximately becomes
$\displaystyle\frac{d\Omega_{D}}{da}\simeq\frac{\Omega_{D}}{a}\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right).$
(31)
Solving this equation we find $\Omega_{D}=n^{2}a^{2}/4$. Substituting this
relation into Eq. (29) with $b^{2}=0$, we obtain $w_{D}=-2/3$. On the other
hand, in the matter-dominated epoch, $H^{2}\propto\rho_{m}\propto a^{-3}$. So
$\sqrt{a}da\propto dt=ad\eta$. Thus $\eta\propto\sqrt{a}$. From Eq. (3) we
have $\rho_{D}\propto a^{-1}$. Putting this in conservation law,
$\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0$, we obtain $w_{D}=-{2}/{3}$.
Substituting this $w_{D}$ in Eq. (29) with $b^{2}=0$ and $\Omega_{k}\ll 1$ we
find that $\Omega_{D}=n^{2}a^{2}/4$ as expected. Therefore, all things are
consistent. The confusion in the original agegraphic dark energy model does
not exist in this new model. These results are regardless of the value of $n$.
Again one can see that in the absence of interaction between dark energy and
dark matter, $b^{2}=0$, $w_{D}$ in Eq. (29) is always larger than $-1$ and
cannot cross the phantom divide. However, in the presence of interaction,
$b^{2}\neq 0$, it is quite possible that $w_{D}$ cross the phantom divide. In
the limiting case $\Omega_{k}=0$, Eqs. (28)-(30), restore their respective
equations in interacting new agegraphic dark energy model in flat universe
Wei2 .
## IV Conclusions
There is a wide consensus among cosmologists that the universe has entered a
phase of accelerated expansion likely driven by dark energy. However, the
nature and the origin of such dark energy is still the source of much debate.
Indeed, until now we don’t know what might be the best candidate for dark
energy to explain the accelerated expansion. Therefore, cosmologists have
attended to various models of dark energy by considering all the possibilities
they have. In this regard, based on the uncertainty relation of quantum
mechanics together with the gravitational effect in general relativity, Cai
proposed an agegraphic dark energy model to explain the acceleration of the
cosmic expansion Cai1 . However, the original agegraphic dark energy model had
some difficulties. In particular it fails to describe the matter-dominated
epoch properly Cai1 . Thus, Wei and Cai Wei2 proposed a new model of
agegraphic dark energy, while the time scale is chosen to be the conformal
time $\eta$ instead of the age of the universe. In this Letter, we extended
these agegraphic dark energy models, in the presence of interaction between
dark energy and dark matter, to the universe with spatial curvature. Although
it is believed that our universe is spatially flat, a contribution to the
Friedmann equation from spatial curvature is still possible if the number of
e-foldings is not very large Huang . Besides, some experimental data has
implied that our universe is not a perfectly flat universe and recent papers
have favored the universe with spatial curvature spe . We obtained the
equation of state for interacting agegraphic energy density in a non-flat
universe. When the interaction between dark matter and agegraphic dark energy
is taken into account, the equation of state parameter of dark energy,
$w_{D}$, can cross the phantom divide in the universe with any spacial
curvature.
In interacting agegraphic models of dark energy, the properties of agegraphic
dark energy is determined by the parameters $n$ and $b$ together. These
parameters would be obtained by confronting with cosmic observational data. In
this work we just restricted ourselves to limited observational data. Giving
the wide range of cosmological data available, in the future we expect to
further constrain our model parameter space and test the viability of our
model.
Finally I would like to mention that, as we found, the spatial curvature does
not seriously modify the qualitative picture of the interacting agegraphic
dark energy models but it may however affect the time of onset of the
acceleration. For an open universe ($\Omega_{k}<0$) the acceleration sets in
earlier whereas in a closed universe ($\Omega_{k}>0$) the accelerated phase is
delayed.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, Iran.
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|
arxiv-papers
| 2009-07-29T14:33:39 |
2024-09-04T02:49:04.275842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0907.5144"
}
|
0907.5214
|
# Weighted projective embeddings, stability of orbifolds and constant scalar
curvature Kähler metrics
Julius Ross and Richard Thomas
###### Abstract
We embed polarised orbifolds with cyclic stabiliser groups into weighted
projective space via a weighted form of Kodaira embedding. Dividing by the
(non-reductive) automorphisms of weighted projective space then formally gives
a moduli space of orbifolds. We show how to express this as a _reductive_
quotient and so a GIT problem, thus defining a notion of stability for
orbifolds.
We then prove an orbifold version of Donaldson’s theorem: the existence of an
orbifold Kähler metric of constant scalar curvature implies K-semistability.
By extending the notion of slope stability to orbifolds we therefore get an
explicit obstruction to the existence of constant scalar curvature orbifold
Kähler metrics. We describe the manifold applications of this orbifold result,
and show how many previously known results (Troyanov, Ghigi-Kollár, Rollin-
Singer, the AdS/CFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-
Yau) fit into this framework.
######
1. 0 Introduction
1. 1 Extensions
2. 1 Orbifold embeddings in weighted projective space
1. 1 Orbibasics
2. 2 Codimension one stabilisers
3. 3 Weighted projective spaces
4. 4 Orbifold line bundles and $\mathbb{Q}$-divisors
5. 5 Orbifold polarisations
6. 6 Orbifold Kodaira embedding
7. 7 OrbiProj
8. 8 Orbifold Riemann-Roch
1. 1 Equivariant case
9. 9 Reducing to the reductive quotient
3. 2 Metrics and balanced orbifolds
4. 3 Limits of Fubini-Study metrics
5. 4 Limits of balanced metrics
6. 5 K-stability as an obstruction to orbifold cscK metrics
1. 1 Definition of orbifold K-stability
2. 2 Orbifold version of Donaldson’s theorem
7. 6 Slope stability of orbifolds
8. 7 Applications and further examples
1. 1 Orbifold Riemann surfaces
2. 2 Index obstruction to stability
3. 3 Orbifold ruled surfaces
4. 4 Slope stability of canonically polarised orbifolds
*
## Chapter 0 Introduction
The problem of finding canonical Kähler metrics on complex manifolds is
central in Kähler geometry. Much of the recent work in this area centres
around the conjecture of Yau, Tian and Donaldson that the existence of a
constant scalar curvature Kähler (cscK) metric should be equivalent to an
algebro-geometric notion of stability. This notion, called “K-stability”,
should be understood roughly as follows. Suppose we are looking for such a
metric on $X$ whose Kähler form lies in the first Chern class of an ample line
bundle $L$. Then using sections of $L^{k}$ one can embed $X$ in a large
projective space $\mathbb{P}^{N_{k}}$ for $k\gg 0$, and stability is taken in
a Geometric Invariant Theory (GIT) sense with respect to the automorphisms of
these projective spaces as $k\to\infty$. By the Hilbert-Mumford criterion this
in turn can be viewed as a statement about numerical invariants coming from
one-parameter degenerations of $X$. The connection with metrics is through the
Kempf-Ness theorem, that a stable orbit contains a zero of the moment map.
Here this says that a (Chow) stable $X$ can be moved by an automorphism of
$\mathbb{P}^{N_{k}}$ to be _balanced_ , and then the restriction of the
Fubini-Study metric on $\mathbb{P}^{N_{k}}$ approximates a cscK metric for
$k\gg 0$.
In this paper we formulate and study a Yau-Tian-Donaldson correspondence for
orbifolds. On the algebro-geometric side this involves orbifold line bundles,
embeddings in weighted projective space, and a notion of stability for
orbifolds. This is related in differential geometry to orbifold Kähler metrics
(those which pull back to a genuine Kähler metric upstairs in an orbifold
chart; downstairs these are Kähler metrics with cone angles $2\pi/m$ about
divisors with stabiliser group $\mathbb{Z}/m\mathbb{Z}$) and their scalar
curvature. So we restrict to the case of orbifolds with cyclic quotient
singularities, but importantly we do allow the possibility of orbifold
structure in codimension one.
Our motivation is not the study of orbifolds per se, but their applications to
manifolds. Orbifold metrics are often the starting point for constructions of
metrics on manifolds (see for instance GK , and the gluing construction of RS
) or arise naturally as quotients of manifolds (for instance quasi-regular
Sasaki-Einstein metrics on odd dimensional manifolds correspond to orbifold
Kähler-Einstein metrics on the leaf space of their Reeb vector fields). What
first interested us in this subject was the remarkable work of GMSY finding
new obstructions to the existence of Ricci-flat cone metrics on cones over
singularities, Sasaki-Einstein metrics on the links of the singularities, and
orbifold Kähler-Einstein metrics on the quotient. We wanted to understand
their results in terms of stability. In fact we found that most known results
concerning orbifold cscK metrics could be understood through an extension of
the “slope stability” of RT [1, 2] to orbifolds.
The end product is a theory very similar to that of manifolds, but with a few
notable differences requiring new ideas:
* •
Embedding an orbifold into projective space loses the information of the
stabilisers, so instead we show how to embed them faithfully into weighted
projective space. This requires the correct notion of ampleness for an orbi-
line bundle $L$, and we are forced to use sections of _more than one_ power
$L^{k}$ – in fact at least as many as the order of the orbifold (defined in
Section 1). Then the relevant stability problem is taken not with respect to
the full automorphism group of weighted projective space (which is not
reductive) but with respect to its reductive part (a product of general linear
groups). This later quotient exactly reflects the ambiguity given by the
choice of sections used in the embedding and, it turns out, gives the same
moduli problem.
* •
By considering the relevant moment maps we define the Fubini-Study Kähler
metrics on weighted projective space required for stability. A difference
between this and the smooth case is that the curvature of the natural
hermitian metric on the hyperplane line bundle is not the Fubini-Study Kähler
metric, though we prove that the difference becomes negligible asymptotically.
* •
A key tool connecting metrics of constant scalar curvature to stability is the
asymptotic expansion of the Bergman kernel. To ensure an expansion on
orbifolds similar to that on manifolds we consider not just the sections of
$L^{k}$ but sections of $L^{k+i}$ as $i$ ranges over one or more periods.
Moreover these sections must be taken with appropriate weights to ensure
contributions from the orbifold locus add up to give a global expansion. This
is the topic of the companion paper RT [3], which also contains a discussion
of the exact weights needed.
This choice of weights can also be seen from the moment map framework. The
stability we consider is with respect to a product of unitary groups acting on
a weighted projective space, and since the centraliser of this group is large
the moment map is only defined up to some arbitrary constants. These
correspond exactly to the weights required for the Bergman kernel expansion,
and the main result of RT [3] is that there is a choice of weights (and thus a
choice of stability notion) that connects with scalar curvature.
* •
The numerical invariants associated to orbifolds and their 1-parameter
degenerations are not polynomial but instead consist of a of polynomial
“Riemann-Roch” term plus periodic terms coming from the orbifold strata. The
definition of the numerical invariants needed for stability (such as the
Futaki invariant) will be made by normalising these periodic terms so they
have average zero, and then only using the Riemann-Roch part. Then
calculations involving stability become identical to the manifold case, only
with the canonical divisor replaced with the orbifold canonical divisor.
After setting up this general framework, our main result is one direction of
the Yau-Tian-Donaldson conjecture for orbifolds.
###### Theorem 1.
Let $(X,L)$ be a polarised orbifold with cyclic quotient singularities. If
$c_{1}(L)$ admits an orbifold Kähler metric of constant scalar curvature then
$(X,L)$ is K-semistable.
Our approach follows the proof given for manifolds by Donaldson in Don [2]. An
improvement by Stoppa Sto says that, as long as one assumes a discrete
automorphism group, the existence of a cscK metric actually implies
K-stability – it is natural to ask if this too can be extended to orbifolds.
Finally we give an orbifold version of the slope semistability of RT [1, 2],
which we show is implied by orbifold K-semistability. Together with Theorem 1
it gives an obstruction to the existence of orbifold cscK metrics. We use this
to interpret some of the known obstructions in terms of stability, for
instance the work of Troyanov on orbifold Riemann surfaces, Ghigi-Kollár on
orbifold projective spaces, and Rollin-Singer on projectivisations of
parabolic bundles. A particularly important class for this theory is Fano
orbifolds, where cscK metrics are Kähler-Einstein and equivalent to certain
quasi-regular Sasaki-Einstein metrics on odd dimensional manifolds. In this
vein we interpret the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-
Yau in terms of stability.
### 1 Extensions
Non-cyclic orbifolds. We have restricted our attention purely to orbifold with
cyclic quotient singularities. It should extend easily to orbifolds whose
stabilisers are products of cyclic groups by using several ample (in the sense
of Section 5) line bundles to embed in a product of weighted projective
spaces. To encompass also non-abelian orbifolds one should replace the line
bundle with a bundle of higher rank so that the local stabiliser groups can
act effectively on the fibre over a fixed point, to give a definition of local
ampleness mirroring 7 in the cyclic case. Then one would hope to embed into
weighted Grassmannians. We thank Dror Varolin for this suggestion.
More general cone angles and ramifolds. It would be nice to extend our results
from orbifold Kähler metrics – which have cone angles of the form $2\pi/p,\
p\in\mathbb{N}$, along divisors $D$ – to metrics with cone angles which are
any positive rational multiple of $2\pi$. It should be possible to study these
within the framework of algebro-geometric stability as well.
The one dimensional local model transverse to $D$ is as follows. In this paper
to get cone angle $2\pi/m$ along $x=0$ we introduce extra local functions
$x^{\frac{k}{m}}$ (by passing the local $m$-fold cover and working with
orbifolds). Therefore to produce cone angles $2\pi p$ it makes sense to
discard the local functions $x,x^{2},\ldots x^{p-1}$ and use only
$1,x^{p},x^{p+1},\ldots$ . (We could use $1,x^{p},x^{2p},\ldots$, i.e. pass to
a $p$-fold quotient instead of an $m$-fold cover, but this would be less
general, producing metrics invariant under $\mathbb{Z}/p\mathbb{Z}$ rather
than those with this invariance on only the tangent space at $x=0$.)
The map $(x^{p},x^{p+1})$ from $\mathbb{C}$ to $\mathbb{C}^{2}$ is a set-
theoretic injection with image $\\{v^{p}=u^{p+1}\\}\subset\mathbb{C}^{2}$. For
very small $x$ (so that $x^{p+1}$ is negligible compared to $x^{p}$) it is
very close to the $p$-fold cover $x\mapsto x^{p}$. More precisely,
$\\{v^{p}=u^{p+1}\\}$ has $p$ local branches (interchanged by monodromy) all
tangent to the $u$-axis. Going once round $x=0$ through angle $2\pi$ we go $p$
times round $u=0$ through angle $2\pi p$. Therefore if we restrict a Kähler
metric from $\mathbb{C}^{2}$ to $\\{v^{p}=u^{p+1}\\}$ and pullback to
$\mathbb{C}$ we get a smooth Kähler metric away from $x=0$ which has cone
angle $2\pi p$ at the origin. Similarly the map
$(x^{p},x^{p+1},\ldots,x^{p+k})$ to $\mathbb{C}^{k+1}$ has the same property.
To work globally one has to pick a splitting $H^{0}(X,L^{k})\cong
H^{0}(D,L^{k})\oplus H^{0}(X,L^{k}(-D))$ and discard those functions in the
second summand which do not vanish to at least order $p$ along $D$. That is,
we take the obvious map
$X\to\mathbb{P}\big{(}H^{0}(D,L^{k})^{*}\oplus H^{0}(L^{k}(-pD))^{*}\big{)}.$
(2)
So instead of Kodaira embedding, we take an injection which fails to be an
embedding in the normal directions to $D$ just as in the local model above.
(More generally, to get cone angles $2\pi p/q$ one should apply the above
description to an orbifold with $\mathbb{Z}/q\mathbb{Z}$ stabilisers along $D$
and injections instead into weighted projective spaces.) One might hope for a
relation between balanced injections of $X$ (2) and cscK metrics with
prescribed cone angles along $D$. We thank Dmitri Panov for discussions about
these “ramifolds”. He has also pointed out that it is too ambitious to expect
the full theory for manifolds and orbifolds to carry over verbatim to this
setting since cscK metrics with cone angles greater than $2\pi$ can be non-
unique. We hope to return to this in future work.
Zero cone angles, cuspidal metrics and stability of pairs. It would be
fruitful to consider the limit of large orbifold order. By this we mean fixing
the underlying space $X$ and a divisor $D$, then putting
$\mathbb{Z}/m\mathbb{Z}$-stabilisers along $D$ (as in Section 2) and
considering $m\gg 0$. Then, formally at least, stability in the limit
$m\to\infty$ is the same as stability of the underlying space where the
numerical invariants are calculated with $K_{X}$ replaced with $K_{X}+D$. This
has been studied by Székelyhidi Szé under the name of “relative stability” of
the pair $(X,D)$, which he conjectures to be linked via a Yau-Tian-Donaldson
conjecture to the existence of complete “cuspidal” cscK metrics on
$X\backslash D$. And indeed one can think of orbifold metrics with cone angle
$2\pi/m$ along a divisor $D$ as tending (as $m\to\infty$) to a complete metric
on $X\backslash D$ (thanks to Simon Donaldson and Dmitri Panov for explaining
this to us).
Pairs. In principle this paper gives many other ways of forming moduli spaces
of pairs $(X,D)$. Initially one should take $X$ smooth projective and $D$ a
simple normal crossings divisor which is a union of smooth divisors $D_{i}$.
Labelling the $D_{i}$ by integers $m_{i}>0$ satisfying the conditions of
Section 2 we get a natural orbifold structure on $X$ from which we recover $D$
as the locus with nontrivial stabiliser group. Taking (for instance) the
orbifold line bundle produced by tensoring a polarisation on $X$ by
$\mathcal{O}(\sum_{i}D_{i}/m_{i})$ gives an orbifold line bundle which is
ample in the sense of Section 5. Embedding in weighted projective space as in
Section 6 and dividing the resulting Hilbert scheme by the reductive group
described in Section 9 gives a natural GIT problem and notion of stability.
One should then analyse which orbischemes appear in the compactification that
this produces (in this paper we mainly study only smooth orbifolds and their
cscK metrics). It is quite possible that the resulting stable pairs will form
a new interesting class. Studying moduli and stability of varieties using GIT
fell out of favour, not least because the singularities it allows are not
those that arise naturally in birational geometry, but interesting recent work
of Odaka suggests a relationship between the newer notion of K-stability
(rather than Chow stability) and semi-log-canonical singularities. It is
therefore natural to wonder if orbifold K-stability of $(X,D)$ is related to
some special types of singularity of pairs (perhaps this is most likely in the
$m\to\infty$ limit of the last section). In fact the recent work of
Abramovich-Hassett AH precisely studies moduli of varieties and pairs using
orbischemes, birational geometry and the minimal model programme (but not
GIT).
An obvious special case is curves with weighted marked points, as studied by
Hassett Has and constructed using GIT by Swinarski Swi . It is possible that
Swinarski’s construction can be simplified by using embeddings in weighted
projective space instead of projective space, and even that his (difficult)
stability argument might follow from the existence of an orbifold cscK metric.
##### Acknowledgements
We thank Dan Abramovich, Simon Donaldson, Alessandro Ghigi, Hiroshi Iritani,
Johan de Jong, Dmitri Panov, Miles Reid, Yann Rollin, James Sparks, Balázs
Szendrői and Dror Varolin for useful conversations. Abramovich and Brendan
Hassett have also recently studied moduli of orbifolds and weighted projective
embeddings AH , though from a very different and much more professional point
of view. In particular they do not use GIT and are mainly interested in the
singularities that occur in the compactification; here we are only concerned
with smooth orbifolds for the link to differential geometry. JR received
support from NSF Grant DMS-0700419 and Marie Curie Grant PIRG-GA-2008-230920
and RT held a Royal Society University Research Fellowship while this work was
carried out.
## Chapter 1 Orbifold embeddings in weighted projective space
The proper way to write this paper would be using Deligne-Mumford stacks, but
this would alienate much of its potential readership (as well as the two
authors). Most of our DM stacks are smooth, so there is an elementary
description in terms of orbifolds, and it therefore makes sense to use it.
However at points (such as when we consider the central fibre of a
degeneration of orbifolds) DM stacks, or orbischemes, are unavoidable. At this
point most of the results we need (such as the appropriate version of Riemann-
Roch) are only available in the DM stacks literature. So we adopt the
following policy. Where possible we phrase things in elementary terms using
only orbifolds. We state the results we need in this language, even when the
only proofs available are in the DM stacks literature. Where we do something
genuinely new we give proofs using the orbifold language, even though they of
course apply more generally to orbischemes or DM stacks.
### 1 Orbibasics
We sketch some of the basics of the theory of orbifolds and refer the reader
to BG [2]; GK for more details. An orbifold consists of a variety $X$ (either
an algebraic variety or, for us, an analytic space) with only finite quotient
singularities that is covered by orbifold charts of the form $U\to U/G\cong
V\subset X$, where $V$ is an open set in $X$, $U$ is an open set in
$\mathbb{C}^{n}$ and $G$ is a finite group acting effectively on $U$. We also
insist on a minimality condition, that the subgroups of $G$ given by the
stabilisers of points of $U$ generate $G$ (otherwise one should make both $U$
and $G$ smaller – it is important that we are using the analytic topology
here).
The gluing condition on charts is the following. If $V^{\prime}\subset V$ are
open sets in $X$ with charts $U^{\prime}/G^{\prime}\cong V^{\prime}$ and
$U/G\cong V$ then there should exist a monomorphism $G^{\prime}\hookrightarrow
G$ and an injection $U^{\prime}\hookrightarrow U$ commuting with the given
$G^{\prime}$-action on $U^{\prime}$ and its action through
$G^{\prime}\hookrightarrow G$ on $U$.
Notice that these injections are _not_ in general unique, so the charts do not
have to satisfy a cocycle condition upstairs, though of course they do
downstairs where the open sets $V$ glue to give the variety $X$. That is, the
orbifold charts need not glue since an orbifold need not be a global quotient
by a finite group, though we will see in Remark 16 that they _are_ global
$\mathbb{C}^{*}$-quotients under a mild condition.
It follows from the gluing condition that the _order_ of a point $x\in X$ –
the size of the stabiliser of any lift of $x$ is any orbifold chart – is well
defined. The _order of $X$_ is defined to be the least common multiple of the
order of its points (which is finite if $X$ is compact). The _orbifold locus_
is the set of points with nontrivial stabiliser group.
In this paper we will mostly consider only compact orbifolds with cyclic
stabiliser groups, so that each $G$ is always cyclic.
By an _embedding_ $f\colon X\to Y$ of orbifolds we shall mean an embedding of
the underlying spaces of $X$ and $Y$ such that for every $x\in X$ there exist
orbifold charts $U^{\prime}\to U^{\prime}/G\ni x$ and $U\to U/G\ni f(x)$ such
that $f$ lifts to an equivariant embedding $U^{\prime}\hookrightarrow U$. We
say that the orbifold structure on $X$ is pulled back from that on $Y$.
Similarly we get a notion of isomorphism of orbifolds.
Given a point in the orbifold locus with stabiliser group
$\mathbb{Z}/m\mathbb{Z}$, call its preimage in a chart $p$, with maximal ideal
$\mathfrak{m}_{p}$. Split its cotangent space
$\mathfrak{m}_{p}/\mathfrak{m}_{p}^{2}$ into weight spaces under the group
action (and use the fact that the ring of formal power series about that point
is $\oplus_{i}S^{i}(\mathfrak{m}_{p}/\mathfrak{m}_{p}^{2})$) to see that
locally analytically there is a chart $U\to U/(\mathbb{Z}/m\mathbb{Z})$ of the
form
$(z_{1},z_{2},\ldots,z_{n})\mapsto(z_{1}^{a_{1}},z_{2}^{a_{2}},\cdots
z_{k}^{a_{k}},z_{k+1},\ldots,z_{n}),$ (1)
for some integers $a_{i}$ which divide $m$. We call this an _orbifold point_
of type $\frac{1}{m}(\lambda_{1},\ldots,\lambda_{k})$ if
$\zeta\in\mathbb{Z}/m\mathbb{Z}$ acts111Here $\lambda_{i}$ is a multiple of
$m/a_{i}$, of course. We are disobeying Miles Reid and picking the usual
identification of $\mathbb{Z}/m\mathbb{Z}$ with the $m$th roots of unity. as
$\zeta\cdot(z_{1},\ldots,z_{k})=(\zeta^{\lambda_{1}}z_{1},\ldots,\zeta^{\lambda_{k}}z_{k}).$
The general principle is that any local object (e.g. a tensor) on an orbifold
is defined to be an invariant object on a local chart (rather than an object
downstairs on the underlying space). So an _orbifold Kähler metric_ is an
invariant Kähler metric on $U$ for each orbifold chart $U\to U/G$ which glues:
its pullback under an injection $U^{\prime}\hookrightarrow U$ of charts above
is the corresponding metric on $U^{\prime}$. Such a metric descends to give a
Kähler metric on the underlying space $X$, but with possible singularities
along the orbifold locus.
For instance the standard orbifold Kähler metric on
$\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ is given by $\frac{i}{2}dz\,d\bar{z}$,
where $z$ is the coordinate on $\mathbb{C}$ upstairs and $x=z^{m}$ is the
coordinate on the scheme theoretic quotient $\mathbb{C}$. Downstairs this
takes the form $\frac{i}{2}m^{-2}|x|^{\frac{2}{m}-2}dx\,d\bar{x}$, which is a
singular Kähler metric on $\mathbb{C}$. The circumference of the circle of
radius $r$ about the origin is easily calculated to be $2\pi r/m$, so the
metric has cone angle $2\pi/m$ at the origin, whereas usual Kähler metrics
have cone angle $2\pi$. More generally for any divisor $D$ in the orbifold
locus with stabiliser group $\mathbb{Z}/m\mathbb{Z}$, orbifold Kähler metrics
on $X$ have cone angle $2\pi/m$ along $D$. So it is important for us to think
of $\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ as an orbifold, and not as its scheme
theoretic quotient $\mathbb{C}$.
Even when the stabilisers have codimension two (so that the orbifold is
determined by the underlying variety with quotient singularities, and one “can
forget” the orbifold structure if only interested in the algebraic or analytic
structure) an orbifold metric is very different from the usual notion of a
Kähler metric over the singularities (i.e. one which is locally the
restriction of a Kähler metric from an embedding in a smooth ambient space).
### 2 Codimension one stabilisers
The cyclic orbifolds which will most interest us will be those for which the
orbifold locus has codimension one. These are the orbifolds whose local model
(1) has coprime weights $a_{i}$.
Therefore globally the orbifold is described by the pair $(X,\Delta)$, where
* •
$X$ is a _smooth_ variety,
* •
$\Delta$ is a $\mathbb{Q}$-divisor of the form
$\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_{i}$,
* •
the $D_{i}$ are distinct smooth irreducible effective divisors,
* •
$D=\sum D_{i}$ has normal crossings, and
* •
the $m_{i}$ are positive integers such that $m_{i}$ and $m_{j}$ are coprime if
$D_{i}$ and $D_{j}$ intersect.
Then the stabiliser group of points in the intersection of several components
$D_{i}$ will be the product of groups $\mathbb{Z}/m_{i}\mathbb{Z}$, and this
is cyclic by the coprimality assumption.
Here $\Delta$ is the ramification divisor of the orbifold charts; see Example
8 for the expression of this in terms of the orbifold canonical bundle.
Notice that above we are also claiming the converse: that given such a pair
$(X,\Delta)$ it is an easy exercise to construct an orbifold with stabiliser
groups $\mathbb{Z}/m_{i}\mathbb{Z}$ along the $D_{i}$, and this is unique.
This can be generalised to Deligne-Mumford stacks Cad ; we give a global
construction in (15).
Orbifolds with codimension one stabilisers were called “not well formed” in
the days when “we were doing the wrong thing” (Miles Reid, Alghero 2006). Then
orbifolds were studied as a means to produce schemes, so only the quotient was
relevant. The orbifold locus could be removed, since the quotient is smooth.
Hence in much of the literature (e.g. Dol ) the not well formed case is
unfortunately ignored.
More generally, _any_ orbifold can be dealt with in much the same way: it can
be described by a pair $(X,\Delta)$ just as above, but where $X$ has at worst
finite cyclic quotient singularities. This is the point of view taken by GK .
### 3 Weighted projective spaces
The standard source of examples of orbifolds is weighted projective spaces. A
graded vector space $V=\oplus_{i}V^{i}$ is equivalent to a vector space $V$
with a $\mathbb{C}^{*}$-action, acting on $V^{i}$ with weight $i$. Throughout
this paper $V$ will always be finite dimensional, with all weights strictly
positive. We can therefore form the associated weighted projective space
$\mathbb{P}(V):=(V\backslash\\{0\\})/\mathbb{C}^{*}$. This is sometimes
denoted $\mathbb{P}(\lambda_{1},\ldots\lambda_{n})$, where $n=\dim V$ and the
$\lambda_{j}$ are the weights (so the number of $\lambda_{j}$ that equal $i$
is $\dim V^{i}$).
Let $x_{j},\ j=1,\ldots,n$, be coordinates on $V$ such that $x_{j}$ has weight
$-\lambda_{j}$. Then $\mathbb{P}(V)$ is covered by the orbifold charts
$\displaystyle\\{x_{j}=1\\}$ $\displaystyle\\!\\!\cong\mathbb{C}^{n-1}$ (2)
$\displaystyle\downarrow\ \ \ \,$ $\displaystyle\mathbb{P}(V).\ $
The $\lambda_{j}$th roots of unity
$\mathbb{Z}/\lambda_{j}\mathbb{Z}\subset\mathbb{C}^{*}$ act trivially on the
$x_{j}$ coordinate, preserving the above $\mathbb{C}^{n-1}$ slice. The
vertical arrow is the quotient by this $\mathbb{Z}/\lambda_{j}\mathbb{Z}$; the
generator $\exp(2\pi i/\lambda_{j})\in\mathbb{C}^{*}$ acting by
$(x_{i})\mapsto(\exp(2\pi i\lambda_{i}/\lambda_{j})x_{i}).$ (3)
The order of $\mathbb{P}(V)$ is the least common multiple of the weights
$\lambda_{j}$. If the $\lambda_{j}$ have highest common factor $\lambda>1$
then $\mathbb{P}(V)$ has generic stabilisers: every point is stabilised by the
$\lambda$th roots of unity, and we will usually assume that this is not the
case, so $\mathbb{P}(V)$ inherits the structure of an orbifold with cyclic
stabiliser groups.
The orbifold points of $\mathbb{P}(V)$ are as follows. Each vertex
$P_{i}:=[0,\dots,1,\dots,0]$ is of type
$\frac{1}{\lambda_{i}}(\lambda_{1},\ldots,\widehat{\lambda}_{i},\ldots,\lambda_{N})$.
The general points along the line $P_{i}P_{j}$ are orbifold points of type
$\frac{1}{\operatorname{hcf}(\lambda_{i},\lambda_{j})}(\lambda_{1},\ldots,\widehat{\lambda}_{i},\ldots,\widehat{\lambda}_{j},\ldots,\lambda_{N})$,
with similar orbifold types along higher dimensional strata.
Thus if for some $j$ the $\lambda_{i},\,i\neq j$, have highest common factor
$\lambda>1$ then $\mathbb{P}(V)$ is not well formed: it has a divisor of
orbifold points with stabiliser group containing
$\mathbb{Z}/\lambda\mathbb{Z}$ along $x_{j}=0$. Replacing the
$\lambda_{i},\,i\neq j$, by $\lambda_{i}/\lambda$ gives a well formed weighted
projective space Dol ; Fle which is just the underlying variety without the
divisor of orbifold points. As discussed in the last section, it is important
for us _not_ to mess with the orbifold structure in this way.
Similarly the map
$\mathbb{P}^{n-1}\to\mathbb{P}(\lambda_{1},\ldots,\lambda_{n}),\
[x_{1},\ldots,x_{n}]\mapsto[x_{1}^{\lambda_{1}},\ldots,x_{n}^{\lambda_{n}}]$
exhibits the underlying variety of weighted projective space as a global
finite quotient of ordinary projective space. Again this does _not_ give the
right orbifold structure of (2), so we do not use it.
### 4 Orbifold line bundles and $\mathbb{Q}$-divisors
Locally an orbifold line bundle is simply an equivariant line bundle on an
orbifold chart. This differs from an ordinary line bundle pulled back from
downstairs which satisfies the property that the $G$-action on the line over
any fixed point is trivial. In other words (the pull back to an orbifold chart
of) an ordinary line bundle has a local invariant trivialisation, which an
orbifold line bundle may not. So in general orbifold line bundles are _not_
locally trivial.
To define them globally we need some notation. Suppose that
$V_{i},V_{j},V_{k}$ are open sets in $X$ with charts $U_{i}/G_{i}\cong V_{i}$,
etc. Then by the definition of an orbifold the overlaps $V_{ij}:=V_{i}\cap
V_{j}$, etc, also have charts $U_{ij}/G_{ij}\cong V_{ij}$ and inclusions
$U_{ij}\hookrightarrow U_{i},\ G_{ij}\hookrightarrow G_{i}$, etc.
Given local equivariant line bundles $L_{i}$ over each $U_{i}$, the gluing (or
cocycle) condition to define a global orbifold line bundle is the following.
Pulling back $L_{j}$ and $L_{i}$ to $U_{ij}$ (via its inclusions in
$U_{j},\,U_{i}$ respectively) there should be isomorphisms $\phi_{ij}$ from
the former to the latter, intertwining the actions of $G_{ij}$. Pulling back
further to $U_{ijk}$ we call this isomorphism $\phi_{ij}\in L_{i}\otimes
L_{j}^{*}$ (suppressing the pullback maps for clarity). The cocycle condition
is that over $U_{ijk}$,
$\phi_{ij}\phi_{jk}\phi_{ki}\ \in\ L_{i}\otimes L_{j}^{*}\otimes L_{j}\otimes
L_{k}^{*}\otimes L_{k}\otimes L_{i}^{*}$
should be precisely the identity element $1$.
The standard example is the orbifold canonical bundle $K_{\text{orb}}$, which
is defined to be $K_{U}$ on the chart $U$ (with the obvious $G$-action induced
from that on $U$) and which glues automatically.
###### Example 4.
Take $X$ a smooth space with a smooth divisor $D$ along which we put
$\mathbb{Z}/(m\mathbb{Z})$ stabiliser group to form the orbifold
$(X,(1-1/m)D)$. Then the orbifold line bundle
$\mathcal{O}\big{(}\\!-\\!\frac{1}{m}D\big{)}$ is easily defined as the ideal
sheaf of the reduced pullback of $D$ to any chart. In this way it glues
automatically.
Locally it has generator $z$, a local coordinate upstairs cutting out the
reduced pullback of $D$. But this has weight one under the
$\mathbb{Z}/m\mathbb{Z}$-action; it is not an invariant section, so does _not_
define a section of the orbifold line bundle downstairs ($z^{km-1}$ times this
generator does, for all $k\geq 0$). Therefore this orbifold bundle is _not_
locally trivial: it is locally the trivial line bundle with the weight one
_nontrivial_ $\mathbb{Z}/m\mathbb{Z}$-action.
Away from $D$, the section which is $z^{-1}$ times by this weight one
generator is both regular and invariant, so can be glued to the trivial line
bundle. In this way one can give an equivalent definition of
$\mathcal{O}\big{(}\\!-\\!\frac{1}{m}D\big{)}$ via transition functions, much
as in the manifold case.
Taking tensor powers we can form $\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ for
any integer $n$. This is an ordinary line bundle only for $n/m$ an integer.
The inclusion $\mathcal{O}(-\frac{1}{m}D)\hookrightarrow\mathcal{O}_{X}$
defines a canonical section $s_{D/m}$ of $\mathcal{O}(\frac{1}{m}D)$ which in
the orbifold chart above looks like $z$ vanishing on $D$.
The pushdown to the underlying manifold $X$ of
$\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ is the ordinary line bundle given by
the round down
$\mathcal{O}\left(\left\lfloor\frac{n}{m}\right\rfloor D\right).$ (5)
That is to say that the (invariant) sections of
$\mathcal{O}\big{(}\frac{n}{m}D\big{)}$ are of the form
$s_{D/m}^{\frac{n}{m}-\left\lfloor\\!\frac{n}{m}\\!\right\rfloor}t$, where $t$
is any section of the ordinary line bundle
$\mathcal{O}\big{(}\\!\left\lfloor\frac{n}{m}\right\rfloor\\!\big{)}$ on $X$.
Since tensor product does not commute with round down, we lose information by
pushing down to $X$: the natural consequence of orbifold line bundles not
being locally trivial.
More generally on any orbifold given by a pair $(X,\Delta)$ as in Section 2,
orbifold line bundles and their sections correspond to $\mathbb{Q}$-divisors
such that the denominator of the coefficient of $D_{i}$ must divide $m_{i}$,
and any irreducible divisor $D$ not in the list of $D_{i}$ must have integral
coefficients. The space of global sections of the orbifold line bundle is the
space of sections of the round down. Care must be taken however; for instance
if $D_{1}$ and $D_{2}$ have $\mathbb{Z}/m\mathbb{Z}$-stabilisers along them
and $\mathcal{O}(D_{1})\cong\mathcal{O}(D_{2})$ this certainly does _not_
imply that $\mathcal{O}(D_{1}/m)\cong\mathcal{O}(D_{2}/m)$.
The _tautological line bundle_ $\mathcal{O}_{\mathbb{P}(V)}(-1)$ over the
weighted projective space $\mathbb{P}(V)$ is the orbi-line bundle over
$\mathbb{P}(V)$ with fibre over $[v]$ the union of the orbit
$\mathbb{C}^{*}.v\subset V$ and $0\in V$. (Any two elements in a fibre can be
written $w_{i}=t_{i}.v$ for $t_{i}\in\mathbb{C},\ i=1,2$, so we can define the
linear structure by $aw_{1}+bw_{2}:=(at_{1}+bt_{2}).v$. Ordinarily this is not
the linear structure on $V$ and the fibre $\mathcal{O}_{[v]}(-1)\subset V$ is
not a linear subspace.) Over the orbi-chart (2) this is the trivial line
bundle $\mathbb{C}^{n-1}\times\mathbb{C}$ with the weight one
$\mathbb{Z}/\lambda_{j}\mathbb{Z}$-action on the line $\mathbb{C}$ times by
its action (3) on $\mathbb{C}^{n-1}$. In other words the map
$\displaystyle\mathbb{C}^{n-1}\times\mathbb{C}$ $\displaystyle\to$
$\displaystyle\mathbb{C}^{n}$ (6)
$\displaystyle(x_{1},\ldots,\widehat{x}_{j},\ldots,x_{n},t)$
$\displaystyle\mapsto$
$\displaystyle(t^{\lambda_{1}}x_{1},\ldots,t^{\lambda_{j}},\ldots,t^{\lambda_{n}}x_{n})$
becomes $(\mathbb{Z}/\lambda_{j}\mathbb{Z})$-equivariant when we use the
action (3) on $\mathbb{C}^{n-1}$, the standard weight-one action on
$\mathbb{C}$, and the original weighted $\mathbb{C}^{*}$-action on
$\mathbb{C}^{n}$. The map (6) is defined in order to take the trivialisation
$1$ of $\mathbb{C}$ to the tautological trivialisation of the pullback of the
orbit to the chart (2) (a point of the chart (2) is a point of its own orbit
and so trivialises it).
Note that Dolgachev Dol uses the same notation
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ to denote the push forward of our
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ to the underlying space, thus rounding down
fractional divisors. Therefore
$\mathcal{O}_{\mathbb{P}(V)}(a+b)=\mathcal{O}_{\mathbb{P}(V)}(a)\otimes\mathcal{O}_{\mathbb{P}(V)}(b)$
does not hold for his sheaves, but is true almost by definition for our
orbifold line bundles.
As a trivial example, consider $\mathcal{O}(k)$ over the weighted projective
line $\mathbb{P}(1,m)$. The first coordinate $x$ on $\mathbb{C}^{2}$ has
weight one, so restricts to a linear functional on orbits (the fibres of
$\mathcal{O}(-1)$). It therefore defines a section of $\mathcal{O}(1)$ which
vanishes at the orbifold point $x=0$. Since $x$ is the coordinate upstairs in
the chart (2) and $x^{m}$ the coordinate downstairs, this is $\frac{1}{m}$
times by a real manifold point. The coordinate $y$ has weight $m$ on the
fibres of $\mathcal{O}(-1)$ so defines a section of $\mathcal{O}(m)$ which
vanishes at the manifold point $y=0$.
The underlying variety is the projective space on the degree $m$ variables
$x^{m},y$, i.e. it is $\mathbb{P}^{1}$ with reduced points $0$ and $\infty$
where these two variables vanish. Thus
$\mathcal{O}_{\mathbb{P}(1,m)}(k)=\mathcal{O}\left(\frac{k}{m}(0)\right)=\mathcal{O}\left(\left\lfloor\frac{k}{m}\right\rfloor(\infty)+\left(\frac{k}{m}-\left\lfloor\frac{k}{m}\right\rfloor\right)(0)\right).$
Similarly on $\mathbb{P}(a,b)$ with $pa+qb=1$, the underlying variety is the
usual Proj of the graded ring on the degree $ab$ generators $x^{b}$ and
$y^{a}$. Denote by $0$ and $\infty$ the zeros of $x^{b}$ and $y^{a}$
respectively. Then it is a nice exercise to check that the orbifold line
bundle $\mathcal{O}_{\mathbb{P}(a,b)}(1)$ is isomorphic to
$\mathcal{O}\left(\frac{p}{b}(0)+\frac{q}{a}(\infty)\right),$
of degree $\frac{1}{ab}$.
### 5 Orbifold polarisations
To define orbifold polarisations we need the right notion of ampleness or
positivity. For manifolds (or schemes) this is engineered to ensure that the
global sections of $L$ generate the local ring of functions at each point. For
orbifolds, this requires also a _local_ condition on an orbifold line bundle
$L$, as we explain using the simplest example. Consider the orbifold
$\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ with local coordinate $z$ on
$\mathbb{C}$ acted on by $\mathbb{Z}/2\mathbb{Z}$ via $z\mapsto-z$. Then
$x=z^{2}$ is a local coordinate on the quotient thought of as a manifold. Any
line bundle pulled back from the quotient (i.e. which has trivial
$\mathbb{Z}/2\mathbb{Z}$-action upstairs when considered as a trivial line
bundle there) has invariant sections $\mathbb{C}[x]=\mathbb{C}[z^{2}]$.
Therefore it sees the quotient only as a manifold, missing the extra functions
of $\sqrt{x}=z$ that the orbifold sees. So we do not think of it as locally
ample: if we tried to embed using its sections we would “contract” the
stabilisers, leaving us with the underlying manifold.
Conversely the trivial line bundle upstairs with nontrivial
$\mathbb{Z}/2\mathbb{Z}$-action (acting as $-1$ on the trivialisation) has
invariant sections $\sqrt{x}\,\mathbb{C}[x]=z\mathbb{C}[z^{2}]$. Its square
has trivial $\mathbb{Z}/2\mathbb{Z}$-action and has sections
$\mathbb{C}[x]=\mathbb{C}[z^{2}]$ as above. Therefore its sections and those
of its powers generate the entire ring of functions
$\mathbb{C}[\sqrt{x}]=\mathbb{C}[z]$ upstairs, and see the full orbifold
structure.
###### Definition 7.
An orbifold line bundle $L$ over a cyclic orbifold $X$ is _locally ample_ if
in an orbifold chart around $x\in X$, the stabiliser group acts faithfully on
the line $L_{x}$. We say $L$ is _orbi-ample_ if it is both locally ample and
globally positive. (By globally positive here we mean
$L^{\operatorname{ord}(X)}$ is ample in the usual sense when thought of as a
line bundle on the underlying space of $X$; from the Kodaira-Baily embedding
theorem Bai one can equivalently ask that $L$ admits a hermitian metric with
positive curvature.)
By a _polarised_ orbifold we mean a pair $(X,L)$ where $L$ is an orbi-ample
line bundle on $X$.
Note that ordinary line bundles on the underlying space are never ample on
genuine orbifolds. Some care needs to be taken when applying the usual theory
to orbi-ample line bundles. For instance it is not necessarily the case that
the tensor product of locally ample line bundles remain locally ample, but if
$L$ is locally ample then so is $L^{-1}$. One can easily check that $L$ is
orbi-ample if and only if $L^{k}$ is ample for one (or all) $k>0$ coprime to
$\operatorname{ord}(X)$.
###### Example 8.
The orbifold canonical bundle $K_{\text{orb}}$ is locally ample along divisors
of orbifold points, but not necessarily at codimension two orbifold points.
For instance the quotient of $\mathbb{C}^{2}$ by the scalar action of $\pm 1$
has trivial canonical bundle, so local ampleness is not determined in
codimension one.
Suppose that $X$ is smooth but with a divisor $D$ with stabiliser group
$\mathbb{Z}/m\mathbb{Z}$. Locally write $D$ as $x=0$ and pick a chart with
coordinate $z$ such that $z^{m}=x$. Then the identity
$dx=mz^{m-1}dz=mx^{1-\frac{1}{m}}dz$ shows that $X$ has orbifold canonical
bundle
$K_{\text{orb}}\ =\ K_{X}+\left(1-\frac{1}{m}\right)D\ =\ K_{X}+\Delta,$
where $K_{X}$ is the canonical divisor of the variety underlying $X$. More
generally if the orbifold locus is a union of divisors $D_{i}$ with stabiliser
groups $\mathbb{Z}/m_{i}\mathbb{Z}$ then $K_{\text{orb}}=K_{X}+\Delta$, where
$\Delta=\sum_{i}\big{(}1-\frac{1}{m_{i}}\big{)}D_{i}$ as in Section 2.
###### Example 9.
The hyperplane bundle $\mathcal{O}_{\mathbb{P}(V)}(1)$ on any weighted
projective space $\mathbb{P}(V)$ is locally ample, and it is actually orbi-
ample since some power is ample [Dol, , Proposition 1.3.3] (we shall also show
below that it admits a hermitian metric with positive curvature). The pullback
of an orbi-ample bundle along an orbifold embedding is also orbi-ample, and
thus any orbifold embedded in weighted projective space admits an orbi-ample
line bundle. If $(X,\Delta)$ is an orbifold, $X$ is smooth and $H$ is an ample
divisor on $X$ then the orbifold bundle $H+\Delta$ of Section 2 is orbi-ample
if and only if $H+\Delta$ is an ample $\mathbb{Q}$-divisor on $X$.
### 6 Orbifold Kodaira embedding
Now fix a polarised orbifold $(X,L)$ and $k\gg 0$. Let $i$ run throughout a
fixed indexing set ${0,1,\ldots,M}$, where $M\geq\operatorname{ord}(X)$, and
let $V$ be the graded vector space
$V=\bigoplus_{i}V^{k+i}:=\bigoplus_{i}H^{0}(L^{k+i})^{*}.$
We give the $i$th summand weight $k+i$. Map $X$ to the weighted projective
space $\mathbb{P}(V)$ by
$\phi_{k}(x):=\big{[}\oplus_{i}\operatorname{ev}^{k+i}_{x}\big{]}.$ (10)
Here we fix a trivialisation of $L_{x}$ on an orbifold chart, inducing
trivialisations of all powers $L_{x}^{k+i}$, and then
$\operatorname{ev}^{k+i}_{x}$ is the element of $H^{0}(L^{k+i})^{*}$ which
takes a section $s\in H^{0}(L^{k+i})$ to $s(x)\in L^{k+i}_{x}\cong\mathbb{C}$.
The weights are chosen so that a change in trivialisation induces a change in
$\oplus_{i}\operatorname{ev}^{k+i}_{x}$ that differs only by the action of
$\mathbb{C}^{*}$ on $V$.
Picking a basis $s_{j}^{k+i}$ for $H^{0}(L^{k+i})$, then, the map can be
described by
$\phi_{k}(x)=\big{[}(s_{j}^{k+i}(x))_{i,j}\big{]}.$
This map is well defined at all points $x$ for which there exists a global
section of some $L^{k+i}$ not vanishing at $x$.
###### Proposition 11.
If $(X,L)$ is a polarised orbifold then for $k\gg 0$ the map (10) is an
embedding of orbifolds (i.e. the orbifold structure on $X$ is pulled back from
that on the weighted projective space $\mathbb{P}(V)$) and
$\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(1)\cong L.$
###### Proof.
Fix $x\in X$. It has stabiliser group $\mathbb{Z}/m\mathbb{Z}$ for some $m\geq
1$, and a local orbifold chart $U/(\mathbb{Z}/m\mathbb{Z})$. Let $y\in U$
(with maximal ideal $\mathfrak{m}_{y}$) map to $x$, and decompose
$\mathfrak{m}_{y}/\mathfrak{m}_{y}^{2}=\oplus_{l}V^{l}$ into weight spaces.
Since we have chosen the indexing set for $i$ to range over at least a full
period of length $m$, at least one of the $L^{k+i}_{y}$ has weight 0 and, for
each $l$, there is at least one $i_{l}$ in the indexing set such that
$L^{k+i}_{y}\otimes V^{i_{l}}$ has weight 0.
Therefore each of these $\mathbb{Z}/m\mathbb{Z}$-modules has invariant local
generators, defining local sections of the appropriate power of $L$ on $X$.
For $k\gg 0$ these extend to global sections, by ampleness. (The pushdowns of
the powers of $L$ from the orbifold to the underlying scheme give sheaves
which all come from a finite collection of sheaves tensored by a line bundle.
For $k\gg 0$ this line bundle becomes very positive, and so eventually has no
cohomology. This value of $k$ can be chosen uniformly for all $y$ by
cohomology vanishing for a _bounded_ family of sheaves on a scheme.)
Therefore, trivialising $L$ locally, the sections generate $\mathcal{O}_{y}$
and $\mathfrak{m}_{y}/\mathfrak{m}_{y}^{2}$, so the pullback of the local
functions on $\mathbb{P}(V)$ (the polynomials in $(x_{i})_{i\neq j}$ on the
orbifold chart (2)) generate the local functions on $U$. It follows that the
map is an embedding for large $k$.
Invariantly, the map (10) can be described as follows. Any lift $\tilde{x}\in
L^{-1}_{x}$ of $x$ is a linear functional on $L_{x}$. Similarly
$\tilde{x}^{\otimes(k+1)}$ is a linear functional on $L_{x}^{k+i}$. Composed
with the evaluation map $\operatorname{ev}_{x}^{k+i}\colon H^{0}(L^{k+i})\to
L^{k+i}_{x}$ gives
$\tilde{x}^{\otimes(k+1)}\circ\operatorname{ev}_{x}^{k+i}\,\colon\
H^{0}(L^{k+i})\to\mathbb{C}.$
Therefore
$\oplus_{i}\big{(}\tilde{x}^{\otimes(k+1)}\circ\operatorname{ev}_{x}^{k+i}\big{)}\,\in\
\bigoplus_{i}H^{0}(L^{k+i})^{*}=V$
is a well defined point, with no $\mathbb{C}^{*}$-scaling ambiguities or
choices. In other words (10) lifts to a natural $\mathbb{C}^{*}$-equivariant
embedding of the orbi-line
$L^{-1}_{x}\ \hookrightarrow\ \bigoplus_{i}H^{0}(L^{k+i})^{*}$ (12)
onto the $\mathbb{C}^{*}$-orbit over the point (10). This makes it clear that
under this weighted Kodaira embedding, the pullback of the
$\mathcal{O}_{\mathbb{P}(V)}(-1)$ orbifold line bundle over $\mathbb{P}(V)$ is
$L^{-1}$. ∎
###### Remark 13.
That $\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(-1)=L^{-1}$, even though the
embedding uses the sections of $L^{k},\ldots,L^{k+M}$ and not those of $L$,
follows from the fact that we give $H^{0}(L^{k+i})^{*}$ weight $k+i$. This
might come as a surprise and appear to contradict what we know about Kodaira
embedding for manifolds. For instance, suppose we embed the _manifold_
$\mathbb{P}^{1}$ using $\mathcal{O}(2)$. Under the normal Kodaira embedding we
get a conic in
$\mathbb{P}^{2}=\mathbb{P}(H^{0}(\mathcal{O}_{\mathbb{P}^{1}}(2)^{*})$ such
that the pullback of $\mathcal{O}_{\mathbb{P}^{2}}(-1)$ is
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$.
However, from the above orbifold perspective, this is _not_ an embedding of
$\mathbb{P}^{1}$, but of the orbifold
$\mathbb{P}^{1}/(\mathbb{Z}/2\mathbb{Z})$, where the
$\mathbb{Z}/2\mathbb{Z}$-action is trivial. We see this as follows. At the
level of line bundles (12), it is an embedding of
$\mathcal{O}_{\mathbb{P}^{1}}(-1)\big{/}(\mathbb{Z}/2\mathbb{Z})$ into
$\mathcal{O}_{\mathbb{P}^{2}}(-2)$, where the $\mathbb{Z}/2\mathbb{Z}$-action
is by $-1$ on each fibre. As a manifold this quotient is indeed
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$, but as an orbifold it is instead an
orbifold line bundle over the orbifold
$\mathbb{P}^{1}/(\mathbb{Z}/2\mathbb{Z})$, where the
$\mathbb{Z}/2\mathbb{Z}$-action is trivial.
###### Remark 14.
When we began this project in early 2006 we were using a different, perhaps
more natural, weighted projective embedding. We embedded in the same way in
$\mathbb{P}\Big{(}\bigoplus_{i}H^{0}(L^{ik})^{*}\Big{)},$
where we give $H^{0}(L^{ik})^{*}$ weight $i$ (_not_ $ik$). (Notice how this
cures the problem with Veronese embeddings described in Remark 13 above.) This
can also be shown to pull back the orbifold structure of weighted projective
space to that of $X$ when $L$ is ample, and to pull $\mathcal{O}(1)$ back to
$L^{k}$. However the corresponding Bergman kernel turns out not to be relevant
to constant scalar curvature orbifold Kähler metrics. We learnt about the
related alternative embedding (10) from Dan Abramovich; see AH . The idea of
using weighted projective embeddings certainly goes back further to Miles
Reid; see for instance Rei [1].
### 7 OrbiProj
It is similarly simple to write down an orbifold version of the Proj
construction, using the whole graded ring $\oplus_{k}H^{0}(L^{k})$ at once.
Given a finitely generated graded ring $R=\oplus_{k\geq 0}R_{k}$ (not
necessarily generated in degree 1!) we can form the scheme
$\operatorname{Proj}\,R$ in the usual way [Har, , Proposition II.2.5]. However
this loses information (for instance we could throw away all the graded pieces
except the $R_{nk},\,k\gg 0$, and get the same result).
We endow $\operatorname{Proj}\,R$ with an orbischeme structure by describing
the orbischeme charts. Fix a homogeneous element $r\in R_{+}$ and consider the
Zariski-open subset
$\operatorname{Spec}\,R_{(r)}=(\operatorname{Proj}\,R)\backslash\\{r=0\\}$.
(As usual $R_{(r)}$ is the degree zero part of the localised ring $r^{-1}R$.)
Then
$\operatorname{Spec}\,\,\frac{R}{(r-1)}\ \longrightarrow\
\operatorname{Spec}\,R_{(r)}$
is our orbichart. Here $R/(r-1)$ is the quotient of $R$ (thought of as a ring
and forgetting the grading) by the ideal $(r-1)$. The map from $R_{(r)}$ sets
$r$ to $1$.
More simply but less invariantly, pick homogeneous generators and relations
for the graded ring $R$. Then $\operatorname{Proj}\,R$ is embedded in the
weighted projective space on the generators, cut out by the equations defined
by the relations.
Given a projective scheme $(X,L)$ and a Cartier divisor $D\subset X$, this
gives a very direct way to produce Cadman’s $r$th root orbischeme
$\big{(}X,\big{(}1-\frac{1}{r}\big{)}D\big{)}$ Cad . This has underlying
scheme $X$ but with stabilisers $\mathbb{Z}/r\mathbb{Z}$ along $D$, and in the
above notation it is simply
$\left(X,\Big{(}1-\frac{1}{r}\Big{)}D\right)\ =\ \operatorname{Proj}\,\
\bigoplus_{k\geq
0}H^{0}\Big{(}X,\mathcal{O}\left(\left\lfloor\frac{k}{r}\right\rfloor
D\right)\otimes L^{k}\Big{)}.$ (15)
The hyperplane line bundle $\mathcal{O}(1)\otimes L^{-1}$ on this Proj is
$\mathcal{O}\big{(}\frac{1}{r}D\big{)}$. Picking generators and relations for
the above graded ring we see the $r$th root orbischeme very concretely, cut
out by equations in weighted projective space.
###### Remark 16.
Although orbifolds need not be global quotients by _finite_ groups, we see
that polarised orbifolds _are_ global quotients of varieties by
$\mathbb{C}^{*}$-actions. In terms of the weighted Kodaira embedding of
Proposition 11, we take the total space of $L^{-1}$ over $X$, minus the zero
section, and divide by the natural $\mathbb{C}^{*}$-action on the fibres.
Equivalently, we express the orbifold Proj of the graded ring $R$ as the
quotient of $\operatorname{Spec}\,(R)\backslash\\{0\\}$ by the action of
$\mathbb{C}^{*}$ induced by the grading.
### 8 Orbifold Riemann-Roch
Suppose that $L$ is an orbifold polarisation on $X$. We will need the
asymptotics of $h^{0}(L^{k})$ for $k\gg 0$. These follow from Kawazaki’s
orbifold Riemann-Roch theorem Kaw , or Toën’s for Deligne-Mumford stacks Toë ,
and some elementary algebra (see for example Rei [2] in the well formed case).
Alternatively they follow from the weighted Bergman kernel expansion (see [RT,
3, Corollary 1.12]), or by embedding in weighted projective space and taking
hyperplane sections in the usual way. The result is that
$h^{0}(L^{k})\ =\ \frac{\int_{X}c_{1}(L)^{n}}{n!}\,k^{n}\ -\
\frac{\int_{X}c_{1}(L)^{n-1}.c_{1}(K_{\text{orb}})}{2(n-1)!}\,k^{n-1}\ +\
\tilde{o}(k^{n-1}).$ (17)
Here and in what follows we define $\tilde{o}(k^{n-1})$ to mean functions of
$k$ that can be written as $r(k)\delta(k)+O(k^{n-2})$, where $r(k)$ is a
polynomial of degree $n-1$ and $\delta(k)$ is periodic in $k$ with period
$m=\operatorname{ord}(X)$ and _average zero_ :
$\delta(k)=\delta(k+m),\qquad\sum_{u=1}^{m}\delta(u)=0.$
Therefore the average of $\tilde{o}(k^{n-1})$ over a period is in fact
$O(k^{n-2})$, and we think of it as being a lower order term than the two
leading ones of (17).
Here we are also using integration of Chern-Weil forms on orbifolds (or
intersection theory on DM stacks). Of course integration works for orbifolds
just as it does for manifolds; it is defined in local charts, but then the
local integral is divided by the size of the group. It also extends easily to
orbischemes, just as usual integration works on schemes once we weight by
local multiplicities.
We give a simple example which nonetheless illustrates a number of the issues
we have been considering.
###### Example 18.
Let $\mathbb{Z}/m\mathbb{Z}$ act on ordinary $\mathbb{P}^{1}$ and the
tautological line bundle over it by $\lambda\cdot[x,y]=[\lambda x,y]$. Then
the quotient $X$ is naturally an orbifold with $\mathbb{Z}/m\mathbb{Z}$
stabilisers at the two points $x=0$ and $y=0$. And the quotient of
$\mathcal{O}(-1)$ is naturally an orbifold line bundle $L_{X}^{-1}$ over $X$.
However $L_{X}$ is not locally ample at $x=0$, since the above action is
trivial on the fibre over $x=0$. So we “contract” the orbifold structure of
$X$ at this point to produce another orbifold $Y$ by ignoring the stabiliser
group at $x=0$ and thinking of it locally as a manifold. Only the orbifold
point $y=0$ survives, and $L_{X}$ automatically descends to an ample orbifold
line bundle $L_{Y}$ on $Y$, to which orbifold Riemann-Roch (17) should
therefore apply.
The sections of $L_{Y}^{k}$ (or those of $L_{X}^{k}$; they are the same) are
the invariant sections of $\mathcal{O}_{\mathbb{P}^{1}}(k)$, a basis for which
is
$y^{k},y^{k-m}x^{m},\ldots,y^{k-m\left\lfloor\\!\frac{k}{m}\\!\right\rfloor}x^{m\left\lfloor\\!\frac{k}{m}\\!\right\rfloor}$.
In particular $Y=\mathbb{P}\langle x^{m},y\rangle=\mathbb{P}(m,1)$ and
$h^{0}(L^{k})=\left\lfloor\frac{k}{m}\right\rfloor+1$.
Writing this as $\frac{k}{m}+1-\frac{m-1}{2m}+\delta(k)$, where $\delta$ is
periodic with average zero, we find
$h^{0}(L^{k})=\frac{k}{m}-\frac{1}{2}\left(-2+\left(1-\frac{1}{m}\right)\right)+\delta(k)=k\deg
L-\frac{1}{2}\deg K_{\text{orb}}+\delta(k).$
Hence, as expected, the single $\mathbb{Z}/m\mathbb{Z}$-orbifold point of $Y$
adds $1-1/m$ to the degree of $K_{\text{orb}}$, and the other orbifold point
of $X$ does not show up.
#### 1 Equivariant case
Fix a polarised orbifold $(X,L)$ as above, but now with a
$\mathbb{C}^{*}$-action on $L$ linearising one on $X$. We need a similar
expansion for the weight of a $\mathbb{C}^{*}$-action on $H^{0}(L^{k})$.
Instead of using the full equivariant Riemann-Roch theorem we follow Donaldson
in deducing what we need by using $\mathbb{P}^{1}$ to approximate
$B\mathbb{C}^{*}=\mathbb{P}^{\infty}$ and applying the above orbifold Riemann-
Roch asymptotics to the total space of the associated bundle over
$\mathbb{P}^{1}$.
So let $\mathcal{O}_{\mathbb{P}^{1}}(1)^{*}$ denote the principal
$\mathbb{C}^{*}$-bundle over $\mathbb{P}^{1}$ given by the complement of the
zero-section in $\mathcal{O}(1)$. Form the associated $(X,L)$-bundle
$(\mathcal{X},\mathcal{L}):=\mathcal{O}_{\mathbb{P}^{1}}(1)^{*}\times^{\
}_{\mathbb{C}^{*}}(X,L).$
Let $\pi\colon\mathcal{X}\to\mathbb{P}^{1}$ denote the projection. Then it is
clear that $\pi_{*}\mathcal{L}^{k}$ is the associated bundle of the
$\mathbb{C}^{*}$-representation $H^{0}(X,L^{k})$. Splitting the latter into
one dimensional weight spaces splits the former into line bundles. A line with
weight $i$ becomes the line bundle $\mathcal{O}(i)$. It follows that the
_total weight_ (i.e. the weight of the induced action on the determinant) of
the $\mathbb{C}^{*}$-action on $H^{0}(X,L^{k})$ is the first Chern class of
$\pi_{*}\mathcal{L}^{k}$. Therefore
$w(H^{0}(X,L^{k}))=\chi(\mathbb{P}^{1},\pi_{*}\mathcal{L}^{k})-\operatorname{rank}(\pi_{*}\mathcal{L}^{k})=\chi(\mathcal{X},\mathcal{L}^{k})-\chi(X,L^{k}).$
In particular orbifold Riemann-Roch on $\mathcal{X}$ and $X$ show that this
has an expansion $b_{0}k^{n+1}+b_{1}k^{n}+\tilde{o}(k^{n})$, where
$b_{0}=\int_{\mathcal{X}}\frac{c_{1}(\mathcal{L})^{n}}{(n+1)!}$ .
We can express $b_{0}$ as an integral over $X$ as follows. Take a hermitian
metric $h$ on $L$ which is invariant under the action of
$S^{1}\subset\mathbb{C}^{*}$ and which has positive curvature $2\pi\omega$.
Let $\sigma$ denote the resulting connection 1-form on the principal
$S^{1}$-bundle given by the unit sphere bundle $S(L)$ of $L$.
Differentiating the $S^{1}$-action gives a vector field $v$ on $S(L)$. Then
$\sigma(v)$ is the pullback of a function $H$ on $X$. With respect to the
symplectic form $\omega$, this $H$ is a Hamiltonian for the $S^{1}$-action on
$X$.
Write $(\mathcal{X},\mathcal{L})$ as the associated bundle to the
$S^{1}$-principal bundle $S(\mathcal{O}_{\mathbb{P}^{1}}(1))$ as follows,
$(\mathcal{X},\mathcal{L})=S(\mathcal{O}_{\mathbb{P}^{1}}(1))\times^{\
}_{S^{1}}(X,L).$
The Fubini-Study connection on $\mathcal{O}_{\mathbb{P}^{1}}(1)$ and the
connection $\sigma$ on $L$ induce natural connections on
$\mathcal{X}\to\mathbb{P}^{1}$ and on $\mathcal{L}\to\mathcal{X}$. In [Don, 2,
Section 5.1] Donaldson shows that the latter has curvature
$H\omega_{FS}+\omega$. (Here $\omega_{FS}$ is pulled back from
$\mathbb{P}^{1}$, and we think of $\omega$ as a form on $\mathcal{X}$ by using
its natural connection over $\mathbb{P}^{1}$ to split its tangent bundle as
$T\mathcal{X}=T\mathbb{P}^{1}\oplus TX$.) Therefore $b_{0}$ equals
$\frac{1}{(n+1)!}\int_{\mathcal{X}}(H\omega_{FS}+\omega)^{n+1}=\frac{n+1}{(n+1)!}\int_{\mathbb{P}^{1}}\omega_{FS}\int_{X}H\omega^{n}=\int_{X}H\frac{\omega^{n}}{n!}.$
This proves
###### Proposition 19.
The total weight of the $\mathbb{C}^{*}$-action on $H^{0}(L^{k})$ is
$w(H^{0}(X,L^{k}))=b_{0}k^{n+1}+b_{1}k^{n}+\tilde{o}(k^{n}),\quad\text{where
}\ b_{0}=\int_{X}H\,\frac{\omega^{n}}{n!}\,.$
We will apply this to weighted projective space $X=\mathbb{P}(V)$ and also to
its sub-orbischemes, where the integral on the right must then take into
account scheme-theoretic multiplicities and the possibility of generic
stabiliser (so if an irreducible component of $X$ has generic stabiliser
$\mathbb{Z}/m\mathbb{Z}$ then the integral over it is $\frac{1}{m}$ times the
integral over the underlying scheme).
Finally we remark that working with $\mathcal{O}_{\mathbb{P}^{2}}(1)$ in place
of $\mathcal{O}_{\mathbb{P}^{1}}(1)$ replaces the trace of the infinitesimal
action on $H^{0}(X,L^{k})$ (i.e. the total weight) by the trace of the
_square_ of the infinitesimal action on $H^{0}(X,L^{k})$, proving it equals
$c_{0}k^{n+2}+O(k^{n+1})\quad\text{where}\
c_{0}=\int_{X}H^{2}\frac{\omega^{n}}{n!}\,.$ (20)
### 9 Reducing to the reductive quotient
To form a moduli space of polarised varieties $(X,L)$ one first embeds $X$ in
projective space $\mathbb{P}$ with a high power of $L$, thus identifying $X$
with a point of the relevant Hilbert scheme of subvarieties of $\mathbb{P}$.
It is easy to see that two points of the Hilbert scheme correspond to
abstractly isomorphic polarised varieties if and only if they differ by an
automorphism of $\mathbb{P}$. Therefore a moduli space of varieties can be
formed by taking the GIT quotient of the Hilbert scheme by the special linear
group. (Different choices of linearisations of the action give different
notions of stability of varieties.)
By Proposition 11 we can now mimic this for polarised orbifolds, first
embedding in a _weighted_ projective space $\mathbb{P}$. The Hilbert scheme of
suborbischemes of $\mathbb{P}$ has been constructed in OS . Therefore we are
left with the problem of quotienting this by the action of
$\operatorname{Aut}(\mathbb{P})$.
At first sight this seems difficult because $\operatorname{Aut}(\mathbb{P})$
is not reductive. Classical GIT works only for reductive groups (though a
remarkable amount of the theory has now been pushed through in the
nonreductive case DK ).
As a trivial example consider $\mathbb{P}(1,2)$ embedded by the identity map
in itself. The automorphisms contain a nonreductive piece $\mathbb{C}$ in
which $t\in\mathbb{C}$ acts by
$[x,y]\ \mapsto\ [x,y+tx^{2}].$ (21)
However this arises because $\mathbb{P}(1,2)$ has not been Kodaira embedded as
described in Section 6. Using _all_ sections of $H^{0}(\mathcal{O}(1))=\langle
x\rangle$ and $H^{0}(\mathcal{O}(2))=\langle x^{2},y\rangle$ (not just $x$ and
$y$) we embed instead via
$\mathbb{P}(1,2)\hookrightarrow\mathbb{P}(1,2,2),\qquad[x,y]\mapsto[x,x^{2},y].$
Then the nonreductive $\mathbb{C}$ lies in a bigger, reductive subgroup of
$\operatorname{Aut}(\mathbb{P}(1,2,2))$. Namely (21) can be realised as the
restriction to $\mathbb{P}(1,2)$ of the automorphism
$[A,B,C]\mapsto[A,B,C+tB]$
lying in the reductive subgroup
$SL(H^{0}(\mathcal{O}(2)))\subset\operatorname{Aut}\mathbb{P}(1,2,2)$. Of
course it can also be seen as the restriction of
$[A,B,C]\mapsto[A,B,C+tA^{2}]$, another nonreductive $\mathbb{C}$ subgroup,
but the point is that our embedding has a stabiliser in
$\operatorname{Aut}(\mathbb{P}(1,2,2))$, and this causes the two copies of
$\mathbb{C}$ restrict to the same action.
Having seen an example, the general case is actually simpler. Given a
polarised variety $(X,L)$, pick an isomorphism from $H^{0}(X,L^{k+i})$ to a
fixed vector space $V^{k+i}$. Then from Section 6 we get an embedding of $X$
into $\mathbb{P}(\oplus_{i}(V^{i+k})^{*})$. This embedding is _normal_ – the
restriction map $H^{0}(\mathcal{O}_{\mathbb{P}}(k+i))\to
H^{0}(\mathcal{O}_{X}(k+i))$ is an isomorphism by construction. The next
result says that the resulting point of the Hilbert scheme of $\mathbb{P}$ is
unique up to the action of the _reductive_ group $\prod_{i}GL(V^{k+i})$.
###### Proposition 22.
Two normally embedded orbifolds
$X_{j}\subset\mathbb{P}(\oplus_{i}(V^{i+k})^{*})$ are abstractly isomorphic
polarised varieties if and only if there is $g\in\prod_{i}GL(V^{k+i})$ such
that $g.X_{1}=X_{2}$.
###### Proof.
If the $(X_{j},\mathcal{O}_{X_{j}}(1)))$ are abstractly isomorphic then their
spaces of sections $H^{0}(\mathcal{O}_{X_{j}}(k+i))$ are isomorphic vector
spaces. Under this isomorphism, the two identifications
$H^{0}(\mathcal{O}_{X_{j}}(k+i))\cong V^{i+k},\ j=1,2$, therefore differ by an
element $g_{k+i}\in GL(V^{k+i})$. Then $g:=\oplus_{i\,}g_{k+i}$ takes
$X_{1}\subset\mathbb{P}$ to $X_{2}$.
The converse is of course trivial, needing only the fact that the action of
$\prod_{i}GL(V^{k+i})$ preserves the polarisation
$\mathcal{O}_{\mathbb{P}}(1)$. ∎
Therefore one can set up a GIT problem to form moduli of orbifolds, just as
Mumford did for varieties.
Firstly one needs Matsusaka’s big theorem for orbifolds, to ensure that for a
fixed $k\gg 0$, uniform over all smooth polarised orbifolds of the same
topological type, the orbifold line bundles $L^{k+i}$ have the number of
sections predicted by orbifold Riemann-Roch. This follows by pushing down to
the underlying variety, which has only quotient, and so rational,
singularities, to which [Mat, , Theorem 2.4] applies.
We can thus embed them all in the same weighted projective space. Then one can
remove those suborbifolds of weighted projective space whose embedding is non-
normal, since they are easily seen to be unstable for the action of
$\prod_{i}GL(V^{k+i})$. Thus by the above result, orbits on the Hilbert scheme
really corresponds to isomorphism classes of polarised orbifolds. Finally one
should compactify with orbischemes (or Deligne-Mumford stacks) to get proper
moduli spaces of stable objects. We do not pursue this here as only smooth
orbifolds and their stability are relevant to cscK metrics, but many of the
foundations are worked out in AH . (Their point of view is slightly different
from ours – their notion of stability is related to the minimal model
programme rather than GIT, and they form moduli using the machinery of
stacks.)
## Chapter 2 Metrics and balanced orbifolds
Our next point of business is to generalise the Fubini-Study metric to
weighted projective space. Anticipating the application we have in mind, fix
some $k\geq 0$ and let $V=\oplus_{i=1}^{M}V^{k+i}$ be a finite dimensional
graded vector space. By a metric $|\cdot|_{V}$ on a $V$ we will mean a
hermitian metric which makes the vector spaces $V^{p}$ and $V^{q}$ orthogonal
for $p\neq q$. Thus a metric on $V$ is simply given by a hermitian metric
$|\cdot|_{V^{p}}$ on each $V^{p}$. By a graded orthonormal basis
$\\{t_{\alpha}^{p}\\}$ for $V$ we mean an orthonormal basis
$\\{t^{p}_{1},\ldots,t^{p}_{\dim V^{p}}\\}$ for $V^{p}$ for each
$p=k+1,\ldots,k+M$.
As usual let $\mathbb{P}(V)$ be the weighted projective space obtained by
declaring that $V^{k+i}$ has weight $k+i$. The unitary group
$U:=\prod_{i}U(V^{k+i})$ acts on $V$ with moment map
$\mu^{\ }_{U}(v)=\frac{1}{2}\left(v\otimes
v^{*}-\bigoplus_{i}c_{i}\operatorname{Id}_{V^{k+i}}\right)\ \in\
\oplus_{i}\,\mathfrak{u}(V^{k+i})^{*}.$ (1)
Here the $c_{i}$ are arbitrary real constants, which we will take to be
positive, and $v^{*}\in V^{*}$ is the linear functional corresponding to $v$
under the hermitian inner product. Therefore the $U(1)$ action on $V$ which
acts on $V^{k+i}$ with weight $k+i$ has moment map
$\mu_{U(1)}=\operatorname{Tr}_{w}\circ\,\mu_{U}$, where
$\operatorname{Tr}_{w}\colon\mathfrak{u}^{*}\to\mathfrak{u}(1)^{*}$ is the
projection
$\operatorname{Tr}_{w}(\oplus_{i}A^{i})=\sum_{i}(k+i)\operatorname{tr}(A^{i})$.
Thus if $v=\oplus_{i}v_{k+i}$, then
$\mu^{\
}_{U(1)}(v)=\frac{1}{2}\\!\left(\sum_{i}(k+i)|v_{k+i}|^{2}-c\right)\\!\\!,\quad\text{where
}\ c:=\sum_{i}(k+i)c_{i}\dim V^{k+i}.$ (2)
###### Definition 3.
The _Fubini-Study_ orbifold Kähler metric $\omega_{FS}$ associated to
$|\cdot|_{V}$ is $\frac{1}{c}$ times the metric on $\mathbb{P}(V)$ which
results from viewing it as the symplectic quotient $\mu_{U(1)}^{-1}(0)/U(1)$
and taking the Kähler reduction of the metric $|\cdot|_{V}$ under the
isometric action of $U$.
This is an orbifold Kähler metric: on the orbifold chart (2) it pulls back to
a genuine Kähler metric on $\mathbb{C}^{n-1}$. In fact it follows from Lemma 6
below that it is the curvature of a hermitian metric $h_{1}$ on the orbifold
line bundle $\mathcal{O}_{\mathbb{P}(V)}(1)$. The dual of this hermitian
metric is the one of three natural candidates for the name of Fubini-Study
metric on $\mathcal{O}_{\mathbb{P}(V)}(-1)$. A second natural choice $h_{2}$
is given by $|v|^{2}_{h_{2}}=\sum_{i}|v_{k+i}|^{\frac{2}{k+i}}$ (note that
$|v|^{2}=\sum_{i}|v_{k+i}|^{2}$ does not scale correctly under the action of
$\mathbb{C}^{*}$ to define a hermitian metric). However it is the third
candidate $h_{3}=h_{FS}$ below that we choose. It should be noted that only on
an unweighted projective space do all three agree and metrics. It seems that
$h_{FS}$ is a special case of the more general metrics on line bundles over
toric varieties constructed by Batyrev-Tschinkel [BT, , Section 2.1].
###### Definition 4.
The _Fubini-Study metric_ $h_{FS}$ on $\mathcal{O}_{\mathbb{P}(V)}(-1)$ is the
hermitian metric defined by setting the points of $\mu_{U(1)}^{-1}(0)$ to have
norm 1\. Therefore
$|v|_{h_{FS}}:=\frac{1}{\lambda(v)}\,,$
where $\lambda(v).v$ is the unique point of $\mu_{U(1)}^{-1}(0)$ in the orbit
$(0,\infty).v$. That is, by (2), $\lambda(v)$ is the unique positive real
solution to
$\sum_{i}(k+i)\lambda(v)^{2(k+i)}|v_{k+i}|^{2}=c.$ (5)
We also use $h_{FS}$ to denote the induced metrics on
$\mathcal{O}_{\mathbb{P}(V)}(i)$.
The discrepancy between $\omega_{FS}$ and the curvature
$2\pi\omega_{h_{FS}}:=i\partial\overline{\partial}\log h_{FS}$ of the metric
$h_{FS}$ on $\mathcal{O}_{\mathbb{P}(V)}(1)$ can be deduced from a result in
BG [1].
###### Lemma 6.
We have
$\omega_{FS}=\omega_{h_{FS}}+\frac{i}{2c}\partial\overline{\partial}f,$ (7)
where $f\colon\mathbb{P}(V)\to\mathbb{R}$ is the function
$f:=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|^{2}_{h_{FS}}.$ (8)
Here $\\{t_{\alpha}^{i}\\}$ is a $|\cdot|_{V}$-orthonormal basis of $V^{*}$,
so each $t_{\alpha}^{i}$ defines a section of
$\mathcal{O}_{\mathbb{P}(V)}(i)$, whose pointwise $h_{FS}$-norm is what
appears in (8).
###### Proof.
Let $p\colon V\backslash\\{0\\}\to\mathbb{P}(V)$ be projection to the
quotient. We use [BG, 1, 3.1]; in their notation we set $\chi$ to be the $c$
th power homomorphism from $S^{1}$ to itself and shift our moment map by
$\frac{c}{2}$ to agree with theirs. The result is that the pullback of the
Kähler form produced by symplectic reduction is
$p^{*}(c\,\omega_{FS})=\frac{i}{2}\partial\overline{\partial}\,|\lambda(v).v|_{V}^{2}+\frac{i}{2\pi}\partial\overline{\partial}\log\lambda(v)^{c},$
(9)
where $\lambda(v)\in(0,\infty)$ is defined as in (5) so that
$\lambda(v).v\in\mu_{U(1)}^{-1}(0)$.
Over an open set of $\mathbb{P}(V)$ pick a holomorphic section, or
multisection, of $p$, lifting $x$ to $v=v(x)$. Then the curvature of $h_{FS}$
on $\mathcal{O}_{\mathbb{P}(V)}(-1)$ is
$i\partial\overline{\partial}\log|v|_{h_{FS}}$, which by Definition 4 is
$i\partial\overline{\partial}\log\lambda(v)^{-1}$. Therefore the curvature of
$\mathcal{O}_{\mathbb{P}(V)}(1)$ is
$i\partial\overline{\partial}\log\lambda(v)$ and we can rewrite (9) (divided
through by $c$) as
$p^{*}(\omega_{FS})=\frac{i}{2c}\partial\overline{\partial}\,|\lambda(v).v|_{V}^{2}+p^{*}\omega_{h_{FS}}.$
Then at $v\in V\backslash\\{0\\}$ lying over a point $x\in\mathbb{P}(V)$ we
calculate $|\lambda(v).v|_{V}^{2}$ as
$\sum_{i}|\lambda(v).v|_{V_{i}}^{2}=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}(\lambda(v).v)|^{2}=\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|^{2}_{h_{FS},x}\,.$
The last equality follows from the definition of $h_{FS}$ (4), since
$\lambda(v).v$ lies in $\mu_{U(1)}^{-1}(0)$. ∎
The restriction of $\mu_{U}$ (1) to $\mu_{U(1)}^{-1}(0)$ descends to
$\mathbb{P}(V)$ as the moment map $m$ for the induced action of $U/U(1)$ on
$\mathbb{P}(V)$:
$m([v])=\frac{1}{2}\bigoplus_{i}\left(\lambda^{2(k+i)}(v)\,v_{k+i}\otimes
v_{k+i}^{*}-c_{i}\operatorname{Id}_{V^{k+i}}\right),$ (10)
with $\lambda(v)$ is defined in (5). Integrating this allows us to define a
notion of balanced orbifolds.
###### Definition 11.
Given an orbifold embedding $X\subset\mathbb{P}(V)$ define
$M(X)=\int_{X}m\,\frac{\omega_{FS}^{n}}{n!}\,,$
where $m$ is the moment map from (10). We say that an orbifold
$X\subset\mathbb{P}(V)$ is _balanced_ if $M(X)=0$.
###### Remark 12.
The balanced condition depends on $|\cdot|_{V}$ and on the choice of constants
$c_{i}$. Later we will choose specific constants to ensure a connection with
scalar curvature.
Just as in the manifold situation Don [1]; Wan , $M$ is the moment map for the
action of $U/U(1)$ on Olsson and Starr’s Hilbert scheme OS of sub-orbischemes
of $\mathbb{P}(V)$ endowed with its natural $L^{2}$-symplectic form. To make
sense of this statement one can either work purely formally, make a precise
statement at smooth points, or restrict attention to a single orbit of
Aut$(\mathbb{P}(V))$; the latter is smooth and all we will need in the
application to constant scalar curvature. For $X\subset\mathbb{P}(V)$ and
$v,w$ sections of $T\mathbb{P}(V)|_{X}$ their pairing with the symplectic form
is defined to be
$\Omega(v,w):=\int_{X}v\lrcorner\left(w\lrcorner\frac{\omega^{n+1}}{(n+1)!}\right).$
The moment map calculation is the following. We let $A=\oplus_{i}A^{k+i}$ be a
graded Hermitian matrix generating the 1-parameter subgroup $\exp(tA)$ of
automorphisms of $\mathbb{P}(V)$, inducing the vector field $v_{A}$ on
$\mathbb{P}(V)$. Since $m_{A}:=\operatorname{tr}(mA)$ is a hamiltonian for
$v_{A}$, we have $v_{A}\lrcorner\,\omega=dm_{A}$. Moving in the Hilbert scheme
down a vector field $v$ on $\mathbb{P}(V)$ we have
$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\\!\\!\operatorname{tr}(M(X)A)$
$\displaystyle=$
$\displaystyle\int_{X}\mathcal{L}_{v}\left(m_{A}\frac{\omega^{n}}{n!}\right)$
(13) $\displaystyle=$
$\displaystyle\int_{X}v(m_{A})\frac{\omega^{n}}{n!}+\int_{X}m_{A}d\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}\omega(v,v_{A})\frac{\omega^{n}}{n!}-\int_{X}d(m_{A})\wedge\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}\omega(v,v_{A})\frac{\omega^{n}}{n!}-\int_{X}(v_{A}\lrcorner\,\omega)\wedge\left(v\lrcorner\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{X}v\lrcorner\left(v_{A}\lrcorner\left(\frac{\omega^{n+1}}{(n+1)!}\right)\right)=\Omega(v,v_{A}).$
To express the balanced condition in terms of sections of line bundles, fix a
polarised orbifold $(X,L)$ with cyclic stabiliser groups. Embed $X$ in
weighted projective space with $k\gg 0$ as in Section 6:
$\phi_{k}\colon X\hookrightarrow\mathbb{P}(V)\ \text{ where }\
V=\bigoplus_{i=1}^{M}H^{0}(L^{k+i})^{*}\text{ and
}L=\phi_{k}^{*}\mathcal{O}_{\mathbb{P}(V)}(1).$ (14)
A metric $|\cdot|_{V}$ on $V$ induces by Definition 4 a Fubini-Study metric on
$\mathcal{O}(1)$, and so one on $L$ which we also denote by $h_{FS}$. The next
Lemma expresses the balanced condition in terms of coordinates on $V$ given by
a graded $|\cdot|_{V}$-orthonormal basis $\\{t_{\alpha}^{i}\\}$, where
$t^{i}_{\alpha}\in H^{0}(L^{k+i})$. To ease notation we write
$\operatorname{vol}:=\int_{X}\frac{c_{1}(L)^{n}}{n!}\,.$
###### Lemma 15.
With respect to these coordinates the matrix $M(X)=\oplus_{i}M^{i}(X)$ has
entries
$(M^{i}(X))_{\alpha\beta}=\frac{1}{2}\left(\int_{X}(t^{i}_{\alpha},t^{i}_{\beta})_{h_{FS}}\,\frac{\omega_{FS}^{n}}{n!}-c_{i}\operatorname{vol}\delta_{\alpha\beta}\right).$
###### Proof.
Given a point $x$ in $X$ let $\tilde{x}\in L_{x}^{-1}$ be any non-zero lift,
and write $t_{\alpha}^{i}(\tilde{x})$ for the complex number
$(\tilde{x}^{\otimes(k+i)},t_{\alpha}^{i}(x))$. Then
$(t_{\alpha},t_{\beta})_{h_{FS}}=\lambda(\tilde{x})^{2(k+i)}t_{\alpha}^{i}(\tilde{x})\overline{t_{\beta}^{i}(\tilde{x})}$
where $\lambda(\tilde{x})$ is the positive solution to
$\sum_{i}(k+i)\lambda(\tilde{x})^{2(k+i)}\sum_{\alpha}|t_{\alpha}^{i}(\tilde{x})|^{2}=c.$
Now the embedding of $X$ maps $x$ to the point with coordinates
$[t_{\alpha}^{i}(x)]$, so in these coordinates $m(x)=\oplus_{i}m^{i}(x)$ where
$(m^{i}(x))_{\alpha\beta}=\frac{1}{2}\left(\lambda(\tilde{x})^{2(k+i)}t_{\alpha}^{i}(\tilde{x})\overline{t_{\beta}^{i}(\tilde{x})}-c_{i}\delta_{\alpha\beta}\right)=\frac{1}{2}\left((t^{i}_{\alpha},t^{i}_{\beta})_{h_{FS}}-c_{i}\delta_{\alpha\beta}\right)$
and the result follows by integrating over $X$. ∎
The balanced condition can also be expressed in terms of hermitian metrics on
$L$. Let $\mathcal{K}(c_{1}(L))$ denote the orbifold Kähler metrics on $X$
which are $(2\pi)^{-1}$ times the curvature of an orbifold hermitian metric on
$L$. Define maps as follows:
* •
If $|\cdot|_{V}$ is a metric on $V:=\oplus_{i}H^{0}(L^{k+i})^{*}$ then
$FS(|\cdot|_{V})=(\phi_{k}^{*}h_{FS},\phi_{k}^{*}\omega_{FS}),$
where $h_{FS}$ and $\omega_{FS}$ are Fubini-Study metrics associated to
$|\cdot|_{V}$.
* •
If $h$ is a hermitian metric on $L$ and $\omega$ a Kähler metric in
$\mathcal{K}(c_{1}(L))$ the metric $\operatorname{Hilb}(h,\omega)$ on $V$ is
defined by requiring that for $s\in H^{0}(L^{k+i})$
$|s|^{2}_{\operatorname{Hilb}(h,\omega)}=\frac{1}{c_{i}\operatorname{vol}}\int_{X}|s|^{2}_{h}\frac{\omega^{n}}{n!}\,.$
(16)
Notice this differs from the usual $L^{2}$-metric by the
$c_{i}\operatorname{vol}$ factors. Obviously $V$ and the maps
$\operatorname{Hilb}$ and $\operatorname{FS}$ depend on $k$, and this will
always be clear from the context.
###### Definition 17.
We say that the pair $(h,\omega)$ is _balanced_ at level $k$ if it is a fixed
point of $\operatorname{FS}\circ\operatorname{Hilb}$. A metric $|\cdot|_{V}$
is said to be _balanced_ if it is a fixed point of
$\operatorname{Hilb}\circ\operatorname{FS}$.
###### Proposition 18.
A metric $|\cdot|_{V}$ on $V$ is balanced if and only if $\phi_{k}\colon
X\subset\mathbb{P}(V)$ is a balanced orbifold.
###### Proof.
Given a metric $|\cdot|_{V}$ let $(h_{FS},\omega_{FS})=FS(|\cdot|_{V})$ and
$\\{t_{\alpha}^{i}\\}$ be a graded $|\cdot|_{V}$-orthonormal basis for $V$.
Then by Lemma 15, $M(X)=0$ if and only if
$\frac{1}{c_{i}\operatorname{vol}}\int_{X}(t_{\alpha}^{i},t^{i}_{\beta})_{h_{FS}}\,\frac{\omega_{FS}^{n}}{n!}=\delta_{\alpha\beta}\quad\text{
for all }i,\alpha,\beta,$
if and only if $\\{t_{\alpha}^{i}\\}$ is orthonormal with respect to the
$\operatorname{Hilb}(h_{FS},\omega_{FS})$ metric, if and only if it is the
same metric as $|\cdot|_{V}$. ∎
Another way to express the balanced condition is through Bergman kernels.
###### Definition 19.
Let $h$ be a hermitian metric on $L$ and $\omega$ be a Kähler metric on $X$.
The _weighted Bergman kernel_ is the function
$B_{k}=B_{k}(h,\omega):=\operatorname{vol}\sum_{i}c_{i}(k+i)\sum_{\alpha}|s_{\alpha}^{i}|_{h}^{2}$
where $\\{s_{\alpha}^{i}\\}$ is graded basis of $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the $L^{2}$-metric defined by
$(h,\omega)$. Equivalently
$B_{k}=\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}|_{h}^{2}$
where $\\{t_{\alpha}^{i}\\}$ is orthonormal with respect to the
$\operatorname{Hilb}(h,\omega)$ metric. Of course $B_{k}$ is independent of
these choices of basis.
If $B_{k}$ is constant over $X$, then we see by integrating over $X$ that this
constant is necessarily $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$. In the
unweighted case, $B_{k}$ can be written invariantly in terms of the ratio of
the hermitian metrics $h$ and $h_{FS}$ on $L$. We have the following analogue
here.
###### Proposition 20.
Fix a hermitian metric $h$ on $L$ and a Kähler metric
$\omega\in\mathcal{K}(c_{1}(L))$ and let
$(h_{FS},\omega_{FS})=FS\circ\operatorname{Hilb}(h,\omega)$. Then $h=h_{FS}$
if and only if $B_{k}(h,\omega)\equiv c$ is constant on $X$.
###### Proof.
Let $\\{t_{\alpha}^{i}\\}$ be a graded basis for $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the
$\operatorname{Hilb}(h,\omega)$-metric. For $x\in X$ let $\tilde{x}$ be any
non-zero lift in $L^{-1}|_{x}$. Then
$\displaystyle\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h_{FS}}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\sum_{\alpha}|(\tilde{x},t_{\alpha}^{i}(x))|^{2}|\tilde{x}|^{-2(k+i)}_{h_{FS}}$
(21) $\displaystyle=$
$\displaystyle\sum_{i}(k+i)\sum_{\alpha}\lambda(\tilde{x})^{2(k+i)}|(\operatorname{ev}_{\tilde{x}}^{i},t_{\alpha}^{i})|^{2}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\lambda(\tilde{x})^{2{i}}|\operatorname{ev}_{\tilde{x}}^{i}|^{2}=c,$
from the definition of dual norms, the fact that
$\lambda(\tilde{x})=|\tilde{x}|_{h_{FS}}^{-1}$ and the defining equation for
$\lambda(\tilde{x})$ (5). Thus if $h=h_{FS}$ then $B_{k}$ is constant.
Conversely, if $\beta:=h_{FS}/h$ we have
$c=\sum_{i}(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h_{FS}}=\sum_{i}^{m}(k+i)\sum_{\alpha}\beta(x)^{(k+i)}|t_{\alpha}^{i}(x)|^{2}_{h}.$
(22)
Now note that for fixed $x$, the quantity
$u_{i}=(k+i)\sum_{\alpha}|t_{\alpha}^{i}(x)|^{2}_{h}$ is nonnegative for each
$i$, so there is a unique positive real solution to the equation
$\sum_{i=1}\beta^{2(k+i)}(x)u_{i}=c$. If $B_{k}\equiv c$ is constant then
$\beta(x)=1$ is one solution, and thus the unique solution, so $h=h_{FS}$. ∎
## Chapter 3 Limits of Fubini-Study metrics
The connection between constant scalar curvature metrics and stability comes
through the asymptotics of Fubini-Study metrics. The crucial ingredient is the
asymptotics, as $k\to\infty$, of the weighted Bergman kernel of Definition 19:
$B_{k}=\operatorname{vol}\,\sum_{i}c_{i}(k+i)\sum_{\alpha}|s^{i}_{\alpha}|^{2}_{h}.$
Here $\\{s_{\alpha}^{i}\\}$ is a basis of $H^{0}(L^{k+i})$ that is orthonormal
with respect to the $L^{2}$-metric induced by $h$ and $\omega$. Ensuring that
this is related to scalar curvature requires a particular choice of $c_{i}$,
so for concreteness assume from now on they are chosen by requiring
$\sum_{i}c_{i}t^{i}:=(t^{\operatorname{ord}(X)-1}+t^{\operatorname{ord}(X)-2}+\cdots+1)^{p+1}$
(1)
for some sufficiently large integer $p$. We prove in [RT, 3, 1.7 and 4.13]
that with this choice of $c_{i}$ there is an asymptotic expansion
$B_{k}=b_{0}k^{n+1}+b_{1}k^{n}+\cdots\quad\mathrm{as}\ k\to\infty$ (2)
for some smooth functions $b_{i}$. Taking larger values of $p$ yields a
stronger expansion: in fact if $p\geq r+q$ for integers $r,q\geq 0$ then (2)
holds up to terms of order $O(k^{n+1-r})$ in the $C^{q}$-norm. By this we mean
that there is a constant $C$ such that for all $k$,
$\left|\\!\left|B_{k}-b_{0}k^{n+1}-b_{1}k^{n}-\cdots-
b_{r-1}k^{n+1-(r-1)}\right|\\!\right|\leq Ck^{n+1-r},$
where the norm is the $C^{q}$-norm taken over $X$ in the orbifold sense, with
the pointwise norm of the derivatives measured with respect to the metric
defined by $\omega$. Moreover the constant $C$ can be taken to be uniform for
$(h,\omega)$ in a compact set.
To achieve what we need in this paper it is sufficient to select $p=5$, so in
particular there is a $C^{2}$-expansion involving the top two terms $b_{0}$
and $b_{1}$; however nothing is lost if the reader prefers to take a larger
$p$ for simplicity. Moreover if $2\pi\omega_{h}$ denotes the curvature
$i\partial\overline{\partial}\log h$ of $h$, the top two coefficients are
given by [RT, 3, 1.11]
$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i},$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i}\left((n+1)i+\operatorname{tr}_{\omega_{h}}(\operatorname{Ric}(\omega))-\frac{1}{2}\operatorname{Scal}(\omega_{h})\right).$
In particular if $2\pi\omega$ is in fact the curvature of $h$ this simplifies
to
$b_{0}=\operatorname{vol}\,\sum_{i}c_{i},\qquad
b_{1}=\operatorname{vol}\,\sum_{i}c_{i}\left((n+1)i+\frac{1}{2}\operatorname{Scal}(\omega)\right).$
(3)
Observe that in this case the top order term, $b_{0}$, is constant over $X$.
Now integrating the expansion over $X$ shows the quantity
$c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is polynomial modulo small terms (this is
shown directly in Lemma 5). In fact
$c=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left((n+1)i+\frac{\,\overline{\\!S}}{2}\right)k^{n}\right]+O(k^{n-1}),$
(4)
where $\overline{\\!S}$ denotes the average of the scalar curvature of any
Kähler metric in $\mathcal{K}(c_{1}(L))$.
Similarly [RT, 3, Remark 4.13] there is also an asymptotic expansion
$\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}|^{2}_{h}\ =\
b_{0}k^{n}+b_{1}^{\prime}k^{n-1}+\cdots$ (5)
for some function $b_{1}^{\prime}$, and where $b_{0}$ is as above. Here the
choice of $c_{i}$ is as above (1), and if $p\geq r+q$ the expansion is in the
$C^{q}$-norm up to terms of order $O(k^{n-r})$.
In what follows fix a hermitian metric $h$ on $L$ and Kähler metric
$\omega\in\mathcal{K}(c_{1}(L))$, and let $(h_{FS,k},\omega_{FS,k})$ be the
pair $FS\circ\operatorname{Hilb}(h,\omega)$ coming from the embedding
$X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$. For embeddings of
manifolds in ordinary projective space, the asymptotics of $h/h_{FS,k}$ are
those of the Bergman kernel. For orbifolds, the fact that the Fubini-Study
fibre metric is defined implicitly in Definition 4 means that we have to work
harder.
###### Theorem 6.
Suppose that $2\pi\omega$ is the curvature of $h$. Then
$(h_{FS,k},\omega_{FS,k})$ converges to $(h,\omega)$ as $k$ tends to infinity.
In fact if $\,\overline{\\!S}$ denotes the average of the scalar curvature
then
$\frac{h_{FS,k}}{h}=1+\frac{\,\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+O({k^{-3}})$
(7)
in the $C^{2}$-norm, and
$\omega=\omega_{FS,k}+O(k^{-2})$ (8)
in $C^{0}$. In particular the set of Fubini-Study Kähler metrics is dense in
$\mathcal{K}(c_{1}(L))$.
###### Remark 9.
The Theorem can be generalised to the case that $\omega$ is not the curvature
of $h$, in which case there will be an additional $O(k^{-1})$ term appearing
in the expansion of $h_{FS,k}/h$.
###### Proof of (7).
The aim is to find an asymptotic expansion of
$\displaystyle\alpha_{k}:=\frac{h_{FS,k}}{h}\,.$
Set $\mathcal{B}_{r}:=\sum_{\alpha}|t^{r}_{\alpha}|_{h}^{2}$ where
$\\{t^{r}_{\alpha}\\}$ is a basis of $H^{0}(L^{r})$ that is orthonormal with
respect to the $\operatorname{Hilb}(h,\omega)$-norm from (16), so that
$B_{k}=\sum_{i}(k+i)\mathcal{B}_{k+i}$. Then if $0\neq\tilde{x}\in
L_{x}^{-1}$,
$\displaystyle\sum_{i}(k+i)\alpha_{k}^{k+i}\mathcal{B}_{{k+i}}$
$\displaystyle=$
$\displaystyle\sum_{i}(k+i)\|\tilde{x}^{k+i}\|^{-2}_{h_{FS,k}}\sum_{\alpha}|t_{\alpha}^{k+i}(\tilde{x})|_{h}^{2}$
(10) $\displaystyle=$
$\displaystyle\sum_{i}(k+i)\|\tilde{x}^{k+i}\|_{h_{FS,k}}^{-2}\sum_{\alpha}\|\tilde{x}\|^{2}_{\operatorname{Hilb}(h,\omega)}$
$\displaystyle=$ $\displaystyle c,$
where the second equality uses the fact that the $t_{\alpha}^{k+i}$ are
orthonormal, the third inequality comes from the definition of the $FS$-norm
(5), and as in (2), $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is constant over $X$.
We aim first for an asymptotic expansion of $\alpha_{k}$ that holds in
$C^{0}$. Say a sequence $a_{k}$ of real numbers is of order $\Omega(k^{p})$ if
there is a $\delta>0$ such that $a_{k}\geq\delta k^{p}$ for $p\gg 0$. A
sequence of real-valued functions $f_{k}$ on $X$ is of order $\Omega(k^{p})$
if there is a $\delta>0$ with $f_{k}\geq\delta k^{p}$ uniformly on $X$ for all
$p\gg 0$.
Step 1: We show $\alpha_{k}=1+O(k^{-1})$ in $C^{0}$. Observe that from (3) and
(4),
$B_{k}=\sum_{i}(k+i)\mathcal{B}_{k+i}=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+O(k^{n})=c+O(k^{n}).$
Taking the difference with (10) gives
$\sum_{i}(k+i)\Big{(}\alpha_{k}^{k+i}-1\Big{)}\mathcal{B}_{k+i}=O(k^{n}),$
and so
$(\alpha_{k}-1)\sum_{i}(k+i)\Big{[}1+\alpha_{k}+\alpha_{k}^{2}+\dots+\alpha_{k}^{k+i-1}\Big{]}\mathcal{B}_{k+i}=O(k^{n}).$
(11)
Now $\alpha_{k}$ is pointwise positive, so the term in square brackets is at
least $1$, and $\sum_{i}(k+i)\mathcal{B}_{k+i}=\Omega(k^{n+1})$, so the sum on
the left hand side is $\Omega(k^{n+1})$. Thus $\alpha_{k}-1=O(k^{-1})$ as
claimed.
Step 2: There are positive constants $C_{1},C_{2}$ such that
$\displaystyle C_{1}$ $\displaystyle\leq$
$\displaystyle\alpha_{k}^{j}\quad\text{ for all }\frac{k}{2}\leq j\leq k.$
$\displaystyle\alpha_{k}^{j}$ $\displaystyle\leq$ $\displaystyle
C_{2}\quad\text{ for all }0\leq j\leq k.$ (12)
###### Proof.
As $\alpha_{k}=1+O(k^{-1})$ we have $C_{1}^{2}\leq\alpha_{k}^{k}\leq C_{2}$
for some $C_{1}\in(0,1),\ C_{2}>1$ and all $k\gg 0$. Thus for
$j\geq\frac{k}{2}$ we have $\alpha_{k}^{j}\geq C_{1}$ and for $j\leq k$ we
have $\alpha_{k}^{j}\leq C_{2}$. ∎
Using this we can improve on Step 1 by observing that the term in square
brackets in (11) is of order $\Omega(k)$ since each power of $\alpha_{k}$ is
nonnegative, and there are at least $k/2$ terms bounded from below by $C_{1}$.
Hence
$\alpha_{k}-1=O(k^{-2})\quad\text{in }C^{0}.$
Step 3: Next define
$\beta_{k}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}.$
(13)
We claim that
$\displaystyle\sum_{i}(k+i)\beta_{k}^{k+i}\mathcal{B}_{k+i}=c+O(k^{n-1})\quad\text{
in }C^{0}.$ (14)
That is, the $\beta_{k}$ satisfy an implicit equation very close to the one
(10) satisfied by the $\alpha_{k}$, which we shall use to deduce they are
approximately equal.
###### Proof.
Note
$\beta_{k}^{k+i}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-1}+O(k^{-2}).$
So using the asymptotic expansion (2, 3) of the weighted Bergman kernel
$B_{k}=\sum(k+i)\mathcal{B}_{k+i}$,
$\displaystyle\sum_{i}$
$\displaystyle(k+i)\beta_{k}^{k+i}\mathcal{B}_{{k+i}}=\sum_{i}(k+i)\left(1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2k}+O(k^{-2})\right)\mathcal{B}_{k+i}$
$\displaystyle=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left(\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}+(n+1)i+\frac{\operatorname{Scal}(\omega)}{2}\right)k^{n}\right]+O(k^{n-1})$
$\displaystyle=\operatorname{vol}\sum_{i}c_{i}\left[k^{n+1}+\left((n+1)i+\frac{\overline{\\!S}}{2}\right)k^{n}\right]+O(k^{n-1})\quad\text{in
}C^{0},$ (15)
since $\mathcal{B}_{k+i}=O(k^{n})$ in $C^{0}$. Comparing with (4) proves the
claim. ∎
Step 4: To simplify notation set
$\gamma_{k}:=\alpha_{k}^{k+i-1}+\alpha_{k}^{k+i-2}\beta_{k}+\dots+\beta_{k}^{k+i-1}.$
Taking the difference between the implicit equations (10) and (14) for
$\alpha_{k}$ and $\beta_{k}$ yields
$(\alpha_{k}-\beta_{k})\sum_{i}(k+i)\gamma_{k}\mathcal{B}_{k+i}=O(k^{n-1})\quad\text{in
}C^{0}.$ (16)
From (12) and the definition (13) of $\beta_{k}$ we see that
$\gamma_{k}=\Omega(k)$. Therefore by (16),
$\alpha_{k}=\beta_{k}+O(k^{-3})=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+O(k^{-3}),$
(17)
which is the expansion we wanted at the level of $C^{0}$-norms.
Step 5: To extend this to the $C^{2}$-norm we actually require an expansion in
the $C^{0}$-norm to higher order (this is because although the pieces of the
Bergman kernel $\mathcal{B}_{k+i}$ are of order $O(k^{n})$, their derivatives
$D^{p}\mathcal{B}_{k+i}$ are of order $O(k^{n+p})$ [RT, 3, Corollary 4.10],
resulting in a loss of a factor of $k$ for each derivative we take). To
achieve this replace $\beta_{k}$ with
$\beta_{k}=1+\frac{\overline{\\!S}-\operatorname{Scal}(\omega)}{2}k^{-2}+\tau_{1}k^{-3}+\tau_{2}k^{-4},$
where the $\tau_{i}$ are smooth functions independent of $k$. Then the
coefficient of $k^{n-1}$ in (15) is $b_{0}\tau_{1}+f$, where $f$ is
independent of $k$ and the $\tau_{i}$. Similarly the coefficient of $k^{n-2}$
is $b_{0}\tau_{2}+g$, where $g$ is independent of $k$ and $\tau_{2}$.
So setting $\tau_{1}=-f/b_{0}$ and $\tau_{2}=-g/b_{0}$ we may assume that the
$k^{n-1}$ and $k^{n-2}$ terms in (15) vanish. Therefore
$\sum_{i}(k+i)\beta_{k}^{k+i}\mathcal{B}_{k+i}=c+O(k^{n-3+p})\text{ in
}C^{p}\text{ for }p=0,1,2,$ (18)
where we have used $\mathcal{B}_{k+i}=O(k^{n+p})$ in $C^{p}$ in place of the
original argument using $\mathcal{B}_{k+i}=O(k^{n})$ in $C^{0}$. Thus
$(\alpha_{k}-\beta_{k})\sum_{i}(k+i)\gamma_{k}\mathcal{B}_{k+i}=O(k^{n-3+p})\text{
in }C^{p},\ p=0,1,2.$ (19)
In particular, $\alpha_{k}=\beta_{k}+O(k^{-5})$ in $C^{0}$.
Now to bound $D\alpha_{k}$, differentiate (10) to get
$D{\alpha_{k}}\sum_{i}(k+i)^{2}\alpha_{k}^{k+i-1}\mathcal{B}_{k+i}=-\sum_{i}(k+i)\alpha_{k}^{k+i}D\mathcal{B}_{k+i}.$
(20)
Since powers of $\alpha_{k}$ are bounded above uniformly (12) and
$D\mathcal{B}_{k+i}=O(k^{n+1})$, the sum on the right hand side is
$O(k^{n+2})$. On the other hand, using the lower bound in (12), the sum on the
left hand side is of order $\Omega(k^{n+2})$, and hence $D\alpha_{k}=O(1)$. We
claim that $D\gamma_{k}=O(k^{2})$. In fact both $\alpha_{k}^{j}$ and
$\beta_{k}^{j}$ are uniformly bounded from above for all $k$ and all $j\leq
k+i$. Thus if $u+v\leq k+i$,
$D(\alpha_{k}^{u}\beta_{k}^{v})=u\alpha_{k}^{u-1}\beta_{k}^{v}D{\alpha_{k}}+v\alpha_{k}^{u}\beta_{k}^{v-1}D\beta_{k}=O(k),$
(21)
since $D\alpha_{k}=O(1)$ and $D\beta_{k}=O(k^{-2})$. Thus $D\gamma_{k}$ is a
sum of $O(k)$ terms each of order $O(k)$ and so $D\gamma_{k}=O(k^{2})$ as
claimed.
So we know $\gamma_{k}\mathcal{B}_{k+i}=O(k^{n+1})$ and
$D(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+2})$. Differentiating the $p=1$
statement of (19) and using $\gamma_{k}=\Omega(k)$ yields
$(D\alpha_{k}-D\beta_{k})\Omega(k^{n+2})=-(\alpha_{k}-\beta_{k})\sum_{i}(k+i)D(\gamma_{k}\mathcal{B}_{k+i})+O(k^{n-2})=O(k^{n-2})$
as $\alpha_{k}-\beta_{k}=O(k^{-5})$. Hence $D\alpha_{k}=D\beta_{k}+O(k^{-4})$,
and thus we have $\alpha_{k}-\beta_{k}=O(k^{-4})$ in $C^{1}$. In particular
$D\alpha_{k}=O(k^{-2})$.
A similar argument applies to the second derivative. Differentiating (20)
yields
$\displaystyle(D^{2}\alpha_{k})\Omega(k^{n+2})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-2\sum_{i}(k+i)^{2}\alpha_{k}^{k+i-1}D\alpha_{k}D\mathcal{B}_{k+i}-\sum_{i}(k+i)\alpha_{k}^{k+i}D^{2}\mathcal{B}_{k+i}$
$\displaystyle-\sum_{i}(k+i)^{2}(k+i-1)\alpha_{k}^{k+i-2}(D{\alpha_{k}})^{2}\mathcal{B}_{k+i}$
which is $O(k^{n+3})$. Thus $D^{2}\alpha_{k}=O(k)$. If $u+v\leq k+i$ then
$D^{2}(\alpha_{k}^{u}\beta_{k}^{v})=u\alpha_{k}^{u-1}\beta_{k}^{v}D^{2}\alpha_{k}+v\alpha_{k}^{u}\beta_{k}^{v-1}D^{2}\beta_{k}+O(k^{-2})$
since $D\alpha_{k}$ and $D\beta_{k}$ are both $O(k^{-2})$. Therefore
$D^{2}(\alpha_{k}^{u}\beta_{k}^{v})=O(k^{2})$ which implies that
$D^{2}\gamma_{k}=O(k^{3})$ and hence
$D^{2}(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+3})$.
Now taking the second derivative of the $p=2$ statement in (19),
$\displaystyle(D^{2}\alpha_{k}-D^{2}\beta_{k})\Omega(k^{n+2})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!-(\alpha_{k}-\beta_{k})\sum_{i}(k+i)D^{2}(\gamma_{k}\mathcal{B}_{k+i})$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2(D\alpha_{k}-D\beta_{k})\sum_{i}(k+i)D(\gamma_{k}\mathcal{B}_{k+i})+O(k^{n-1}).$
Since $\alpha_{k}-\beta_{k}=O(k^{-5})$, $D\alpha_{k}-D\beta_{k}=O(k^{-4})$ and
$D(\gamma_{k}\mathcal{B}_{k+i})=O(k^{n+2})$, this is $O(k^{n-1})$. Hence
$D^{2}\alpha_{k}=D^{2}\beta_{k}+O(k^{-3})$ as required. ∎
###### Proof of (8).
From Lemma 6 we have
$\omega_{FS,k}=\omega_{h_{FS,k}}+\frac{i}{2c}\partial\overline{\partial}f_{k}$,
where
$f_{k}=\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}|_{h_{FS,k}}^{2}$
and the $\\{s^{i}_{\alpha}\\}$ is a graded basis of $\oplus_{i}H^{0}(L^{k+i})$
that is orthonormal with respect to the $L^{2}$-norm defined by $(h,\omega)$.
(So $t^{i}_{\alpha}:=\sqrt{c_{i}\operatorname{vol}\,}s^{i}_{\alpha}$ is an
orthonormal basis with respect to the $\operatorname{Hilb}(h,\omega)$ metric.)
Applying $\partial\overline{\partial}\log$ to (7) shows that
$\omega_{h_{FS,k}}=\omega_{h}+O(k^{-2})=\omega+O(k^{-2})$ in $C^{0}$, so
$\omega_{FS,k}=\omega+\frac{i}{2c}\partial\overline{\partial}f_{k}+O(k^{-2})$.
So since $c$ is of order $\Omega(k^{n+1})$, to prove (8) it will be sufficient
to show that $f_{k}$ is constant on $X$ to $O(k^{n-1})$ in $C^{2}$-norm.
Applying the expansion (7),
$\displaystyle f_{k}(x)$ $\displaystyle=$
$\displaystyle\operatorname{vol}\,\sum_{i}c_{i}\frac{h_{FS,k}^{k+i}}{h^{k+i}}\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$
$\displaystyle=$
$\displaystyle\operatorname{vol}\,\sum_{i}c_{i}(1+\frac{\operatorname{Scal}(\omega)-\overline{\\!S}}{2k}+O(k^{-2}))\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$
$\displaystyle=$ $\displaystyle b_{0}k^{n}+O(k^{n-1}),$
by (5), where $b_{0}$ is constant. ∎
## Chapter 4 Limits of balanced metrics
We digress in this section from our proof of Donaldson’s Theorem to give
another application of the weighted Bergman kernel that illustrates the
connection between balanced metrics and metrics of constant scalar curvature.
###### Theorem 1.
Let $(h_{k},\omega_{k})$ be a pair that is balanced for the embedding
$X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$, and suppose this sequence
converges in $C^{2}$ to a limit $(h,\omega)$. Then $2\pi\omega$ is the
curvature of $h$ and $\operatorname{Scal}(\omega)$ is constant.
###### Proof.
Letting $2\pi\omega_{h_{k}}$ denote the curvature of $h_{k}$, by Lemma 6 we
have
$\omega_{k}=\omega_{h_{k}}+\frac{i}{2c}\partial\overline{\partial}f_{k},$ (2)
where
$f_{k}(x)=\operatorname{vol}\,\sum_{i}c_{i}\sum_{\alpha}|s^{i}_{\alpha}(x)|^{2}_{h_{k}}$
and $\\{s^{i}_{\alpha}\\}$ is a graded orthonormal basis of
$\oplus_{i}H^{0}(L^{k+i})$ with respect to the $L^{2}$-metric defined by
$(h_{k},\omega_{k})$. (Here we are using the balanced condition: that
$(h_{k},\omega_{k})$ is the Fubini-Study metric induced from this
$L^{2}$-metric.)
By (5) we have the $C^{4}$-estimate
$f_{k}=\operatorname{vol}\frac{\omega_{h_{k}}^{n}}{\omega_{k}^{n}}\sum_{i}c_{i}k^{n}+O(k^{n-1})$
(3)
(The estimate is in $C^{4}$ rather than $C^{2}$ since we only require it to
top order. Moreover we have used here that the sequence $(h_{k},\omega_{k})$
converges so lies in a compact set, and thus the $O(k^{n-1})$ can be taken
uniformly.) Since $c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ is of order
$\Omega(k^{n+1})$, we deduce from (2) that
$\omega_{k}=\omega_{h_{k}}+O(k^{-1})$ in $C^{2}$.
In turn this implies that $\omega_{h_{k}}^{n}/\omega_{k}^{n}=1+O(k^{-1})$,
which we can feed back into (3) to give
$\partial\overline{\partial}f_{k}=O(k^{n-1})$. Hence in fact
$\omega_{k}=\omega_{h_{k}}+O(k^{-2}).$
In particular, taking the limit as $k\to\infty$ implies that
$\omega=\omega_{h}$, i.e. that $2\pi\omega$ is the curvature of $h$.
Therefore $\omega_{h_{k}}^{n}/\omega_{k}^{n}=1+O(k^{-2})$ and
$\operatorname{tr}_{\omega_{h_{k}}}(\operatorname{Ric}(\omega_{k}))=\operatorname{tr}_{\omega_{k}}(\operatorname{Ric}(\omega_{k}))+O(k^{-2})=\operatorname{Scal}(\omega_{k})+O(k^{-2}).$
Thus the asymptotic expansion (2) for the weighted Bergman kernel becomes
$B_{k}=\operatorname{vol}\sum_{i}(k+i)c_{i}\sum_{\alpha}|s^{i}_{\alpha}|^{2}=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+b_{1}k^{n}+O(k^{n-1}),$
(4)
where
$b_{1}=\operatorname{vol}\sum_{i}c_{i}\left((n+1)i+\frac{1}{2}\operatorname{Scal}(\omega_{k})\right)$.
But by Proposition 20 the balanced condition implies that this weighted
Bergman kernel is the constant
$c=\operatorname{vol}\sum_{i}c_{i}k^{n+1}+O(k^{n}).$
So the coefficient of $k^{n+1}$ agrees with that of (4). Taking coefficients
of $k^{n}$ gives, after some rearranging, a constant $\,\overline{\\!S}$
independent of $k$ such that
$\operatorname{Scal}(\omega_{k})-\,\overline{\\!S}=O(k^{-1}).$
Taking $k$ to infinity yields $\operatorname{Scal}(\omega)=\,\overline{\\!S}$
as required. ∎
###### Remark 5.
The previous theorem was first observed by Donaldson Don [1] in the case of
manifolds embedded in projective space. In the same paper Donaldson also
proves a much harder converse: a cscK metric implies the existence of balanced
metrics for large $k$. We expect that this converse can also be generalised to
orbifolds embeddings in weighted projective space, but have not attempted to
prove it.
## Chapter 5 K-stability as an obstruction to orbifold cscK metrics
We now have the tools required to prove the orbifold version of Donaldson’s
Theorem, and start with the precise definition of stability.
### 1 Definition of orbifold K-stability
Fix a compact $n$-dimensional polarised orbifold with cyclic quotient
singularities $(X,L)$.
###### Definition 1.
A test configuration for $(X,L)$ consists of a pair
$(\pi\colon\mathcal{X}\to\mathbb{C},\mathcal{L})$ where $\mathcal{X}$ is an
orbischeme, $\pi$ is flat and $\mathcal{L}$ is an ample orbi-line bundle along
with a $\mathbb{C}^{*}$-action such that (1) the action is linear and covers
the usual action on $\mathbb{C}$ and (2) the general fibre $\pi^{-1}(t)$ of
the test configuration is $(X,L)$.
Test configurations arise from the action of a one parameter
$\mathbb{C}^{*}$-subgroup of the automorphisms of weighted projective space
$\mathbb{P}$ on an orbifold embedded in $\mathbb{P}$. In general the limit
$\mathcal{X}_{0}=\pi^{-1}(0)$ will not itself be an orbifold, as it may have
scheme structure or entire components consisting of points with nontrivial
stabilisers. In general one should allow $\mathcal{X}$ to be a Deligne-Mumford
stack, but for most of the applications in this paper $\mathcal{X}$ will
itself be an orbifold.
Conversely, we can realise an abstract test configuration via a
$\mathbb{C}^{*}$-action on weighted projective space, just as in the manifold
case [RT, 2, Proposition 3.7]. Using the orbi-ampleness of $\mathcal{L}$, we
can embed $\mathcal{X}$ into the weighted projective bundle
$\mathbb{P}(\oplus_{i}(\pi_{*}\mathcal{L}^{k+i})^{*})$ over the base curve
$\mathbb{C}$ for $k\gg 0$, such that the pullback of
$\mathcal{O}_{\mathbb{P}}(1)$ is $\mathcal{L}$. Pick a trivialisation of the
bundle, making it isomorphic to $\mathbb{P}(V)\times\mathbb{C}$, where
$V=\oplus_{i}H^{0}(\mathcal{X}_{0},\mathcal{L}^{k+i}|_{\mathcal{X}_{0}})^{*}$.
Thus the $\mathbb{C}^{*}$-action on $V$ arising from the one on the central
fibre $(\mathcal{X}_{0},\mathcal{L}_{0})$ induces a diagonal
$\mathbb{C}^{*}$-action on $\mathbb{P}(V)\times\mathbb{C}\supset\mathcal{X}$
giving the original test configuration.
By Proposition 19 we can write the total weight of the $\mathbb{C}^{*}$-action
on $H^{0}(L^{k})$ as
$w(H^{0}(L^{k}))=w(k)+\tilde{o}(k^{n}),$ (2)
where $w(k)$ is a polynomial $b_{0}k^{n+1}+b_{1}k^{n}$ of degree $n+1$.
Similarly
$h^{0}(L^{k})=h(k)+\tilde{o}(k^{n-1}),$ (3)
where $h(k)=a_{0}k^{n}+a_{1}k^{n-1}$.
###### Definition 4.
The _Futaki invariant_ of the test configuration $(\mathcal{X},\mathcal{L})$
is the $F_{1}=\frac{a_{0}b_{1}-a_{1}b_{0}}{a_{0}^{2}}$ term in the expansion
$\frac{w(k)}{kh(k)}=F_{0}+\frac{F_{1}}{k}+O\left(\frac{1}{k^{2}}\right).$
We say $(X,L)$ is _K-semistable_ if $F_{1}\geq 0$ for any test configuration
with general fibre $(X,L)$. We say it is _K-polystable_ if in addition
$F_{1}=0$ only if the test configuration is a product
$\mathcal{X}=X\times\mathbb{C}$, i.e. it arises from a $\mathbb{C}^{*}$-action
on $X$.
In other words we are simply ignoring the non-polynomial terms in the Hilbert
and weight functions, and then defining stability exactly as for manifolds.
One reason this is a sensible stability notion related to scalar curvature is
given by our next result. This shows that taking a weighted sum with our
choice of $c_{i}$ kills the periodic terms, a result we will apply later to
both $w$ (2) and $h$ (3).
###### Lemma 5.
Let $H$ be a function of the form
$H(k)=h(k)+\epsilon_{h}(k),$
where $h$ is a polynomial of degree $n$ and $\epsilon_{h}$ is a sum of terms
of the form $r(k)\delta(k)$ where $r$ is a polynomial of degree $n-1$ and
$\delta(k)$ is periodic with period $m$ and average zero. Then
$\sum_{i}c_{i}H(k+i)=\sum_{i}c_{i}h(k+i)+O(k^{n-4}).$
###### Proof.
First we claim that if $0\leq p\leq 3$ then $\sum_{i\equiv u}c_{i}i^{p}$ is
independent of $u$. To see this, let $m=\operatorname{ord}(X)$ and observe
that by (1), $\sum_{i}c_{i}t^{i}$ has a root of order at least 4 at every non-
trivial $m$th root of unity. Thus if $\sigma^{m}=1$ with $\sigma\neq 1$ then
$\sum_{i}c_{i}i^{p}\sigma^{ri}=0$ for $1\leq r\leq m-1$. So given any $u$,
$\sum_{i}i^{p}c_{i}=\sum_{r=0}^{m-1}\sigma^{-ru}\sum_{i}i^{p}c_{i}\sigma^{ri}=\sum_{i}i^{p}c_{i}\left(\sum_{r=0}^{m-1}\sigma^{(i-u)r}\right)=m\sum_{i\equiv
u}i^{p}c_{i},$
which proves the claim.
We have to show that $\sum_{i}c_{i}r(k+i)\delta(k+i)=O(k^{n-4})$. By the
claim,
$\displaystyle\sum_{i}c_{i}i^{p}\delta(k+i)$ $\displaystyle=$
$\displaystyle\sum_{u=1}^{m}\sum_{i\equiv u-k\text{ mod
}m}c_{i}i^{p}\delta(u)$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{u=1}^{m}\delta(u)\sum_{i}c_{i}i^{p}\ =0$
for $0\leq p\leq 3$. Hence the $k^{d},\ldots,k^{d-3}$ terms in
$\sum_{i}c_{i}(k+i)^{d}\delta(k+i)$ vanish, and the sum is $O(k^{n-4})$ if
$d\leq n$. The result for general polynomials $r$ follows by linearity. ∎
### 2 Orbifold version of Donaldson’s theorem
To recall the general setup, let $h$ be a hermitian metric on $L$ with
positive curvature $2\pi\omega$ and for $k\gg 0$ consider the
$\operatorname{Hilb}(h,\omega)$ metric on $\oplus_{i}H^{0}(L^{k+i})$ from
(16). From the embedding $X\subset\mathbb{P}(\oplus_{i}H^{0}(L^{k+i})^{*})$ we
produced in Definition 11 a hermitian matrix $M(X)=M_{k}(X)$ (and defined the
embedding to be balanced at level $k$ when $M_{k}(X)$ vanishes). Using the
norm $\|A\|^{2}=\operatorname{tr}(AA^{*})$ on hermitian matrices, the
following is the key estimate.
###### Theorem 6.
There is a constant $C$ such that
$\lVert M_{k}(X)\rVert\leq
Ck^{\frac{n-2}{2}}\lVert\operatorname{Scal}(\omega)-\,\overline{\\!S}\rVert_{L^{2}}+O(k^{\frac{n-4}{2}}),$
where the $L^{2}$-norm is taken with respect to the volume form determined by
$\omega$.
###### Proof.
To ease notation we write $M=M_{k}(X)$. Since $\|M\|$ is unchanged by a
unitary transformation we may pick $\operatorname{Hilb}(h,\omega)$-orthonormal
coordinates $\\{t_{\alpha}^{i}\\}$ such that $M=\oplus M^{i}$ with each
$M^{i}$ diagonal. Thus $M^{i}$ has entries (15)
$M^{i}_{\alpha\alpha}=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h_{FS,k}}^{2}\frac{\omega^{n}_{FS,k}}{n!}-c_{i}\operatorname{vol}\right)=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h}^{2}\frac{h_{FS,k}^{k+i}}{h^{k+i}}\frac{\omega_{FS,k}^{n}}{n!}-c_{i}\operatorname{vol}\right)\\!,$
where $h_{FS,k}$ and $\omega_{FS,k}$ are the induced Fubini-Study metrics.
Using the expansion of $h_{FS,k}/h=1+(\,\overline{\\!S}-S)/2k^{2}+O(k^{-3})$
of Theorem 6 we can write $M=A+B$ where $B^{i}_{\alpha\alpha}=O(k^{-2})$ and
$\displaystyle A^{i}_{\alpha\alpha}$
$\displaystyle=\frac{1}{2}\left(\int_{X}|t^{i}_{\alpha}|_{h}^{2}\left(1+\frac{\,\overline{\\!S}-S}{2k}\right)\frac{\omega^{n}}{n!}-c_{i}\operatorname{vol}\right)=\frac{1}{4k}\int_{X}|t^{i}_{\alpha}|_{h}^{2}(\,\overline{\\!S}-S)\frac{\omega^{n}}{n!}\,.$
Here we have used $\|t^{i}_{\alpha}\|^{2}_{L^{2}}=c_{i}\operatorname{vol}$
from the definition of the $\operatorname{Hilb}(h,\omega)$-norm. Using the
Cauchy-Schwarz inequality,
$\displaystyle|A^{i}_{\alpha\alpha}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{16k^{2}}\int_{X}|t^{i}_{\alpha}|_{h}^{2}\frac{\omega^{n}}{n!}\int_{X}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!}$
$\displaystyle\leq$
$\displaystyle\frac{C^{\prime}}{k^{2}}\int_{X}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!},$
for some constant $C^{\prime}$. Thus from the weak form of the expansion
$\sum_{i}\sum_{\alpha}|t^{i}_{\alpha}|_{h}^{2}=O(k^{n})$,
$\displaystyle\|A\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C^{\prime}}{k^{2}}\int_{X}\sum_{i,\alpha}|t^{i}_{\alpha}|_{h}^{2}|\,\overline{\\!S}-S|^{2}\frac{\omega^{n}}{n!}$
$\displaystyle\leq$ $\displaystyle
C^{\prime\prime}k^{n-2}\|S-\,\overline{\\!S}\|_{L^{2}}^{2}.$
Therefore $\|M\|\leq\|A\|+\|B\|\leq
Ck^{\frac{n-2}{2}}\|S-\,\overline{\\!S}\|_{L^{2}}+\|B\|$, where
$C=\sqrt{C^{\prime\prime}}$. But $B$ is diagonal with $O(k^{n})$ entries of
size $O(k^{-2})$, so $\|B\|^{2}=O(k^{n-4})$. ∎
Now let $(\mathcal{X},\mathcal{L})$ be a nontrivial test configuration for
$(X,L)$, embedded in $\mathbb{P}(V)\times\mathbb{C}$ (with
$V=\oplus_{i}H^{0}(X_{0},L_{0}^{k+i})^{*}$) as before, induced by a
$\mathbb{C}^{*}$-action on $\mathbb{P}(V)$ that takes $X$ to the limit
$X_{0}$. Suppose that $L$ has a metric $h$ with positive curvature
$2\pi\omega$, inducing the Hilb$(h,w)$-metric on $\oplus_{i}H^{0}(X,L^{k+i})$.
Applying [Don, 2, Lemma 2] to each of the spaces $H^{0}(X,L^{k+i})$, we get a
metric on $V$ such that the induced $S^{1}$-action is unitary. Therefore the
infinitesimal generator $A^{k+i}$ of the induced action on
$H^{0}(L_{0}^{k+i})$ is hermitian. We set
$A:=\bigoplus_{i}A^{k+i}.$ (7)
As in Section 1, we get a hamiltonian $H_{A}$ for the $S^{1}$-action on
$\mathbb{P}(V)$ by contracting its vector field on the circle bundle
$S(\mathcal{O}_{\mathbb{P}(V)}(1))$ with the connection 1-form whose curvature
is $\omega_{FS}$. It is
$H_{A}\colon\mathbb{P}(V)\to\mathbb{R},\qquad
H_{A}([v])=\frac{1}{c}\sum_{i}\lambda^{2(k+i)}(v)\langle
A^{k+i}v_{k+i},v_{k+i}\rangle,$
using the inner product $\langle\cdot,\cdot\rangle$ on $V$ and $\lambda$ as
defined in (5). This differs from our usual hamiltonian
$m_{A}=\operatorname{tr}(mA)$ of (10) by the additive constant
$\sum_{i}c_{i}\dim V^{k+i}$, and by the multiplicative factor $\frac{1}{c}$.
(The latter scaling compensates for the fact that $m_{A}$ is the hamiltonian
for $c\omega_{FS}$; see Definition 3.)
By Proposition 19, then, the polynomial part of the total weight of the
$\mathbb{C}^{*}$-action on $V^{*}$ is $w(k)=b_{0}k^{n+1}+b_{1}k^{n}$, where
$b_{0}=\int_{X_{0}}H_{A}\frac{\omega_{FS}^{n}}{n!}$. From this we can define
the Futaki invariant $F_{1}(\mathcal{X},\mathcal{L})$ of the test
configuration $(\mathcal{X},\mathcal{L})$ as in Definition 4.
###### Theorem 8.
In the set-up as above, suppose that $\omega$ has constant scalar curvature.
Then $F_{1}(\mathcal{X},\mathcal{L})\geq 0$.
###### Proof.
In the notation above, set $s=\log t$ and let $X_{t}=\exp(sA).X$ denote the
fibre of the given test configuration over $t\in\mathbb{C}$, with central
fibre $X_{0}$ the limit of $\exp(sA).X$ as $s\to-\infty$.
For fixed $k$, $\operatorname{tr}(M_{k}(X_{s})A)$ is an increasing function of
$s\in\mathbb{R}$, because $\operatorname{tr}(M(X)A)$ is a hamiltonian for the
action of $\exp(sA)$ on the space of sub-orbifolds of $\mathbb{P}(V)$.
Explicitly, substituting $v=Jv_{A}$ into (13) shows that the derivative of
$\operatorname{tr}(M_{k}(X_{s})A)$ is $\Omega(Jv_{A},v_{A})>0$. Therefore
$\operatorname{tr}(AM_{k}(X))\ =\ \operatorname{tr}(AM_{k}(X_{1}))\ \geq\
\lim_{s\to-\infty}\operatorname{tr}(AM_{k}(X_{t}))\ =\
\operatorname{tr}(AM_{k}(X_{0})).$
Recalling the definition of $M_{k}(X)$ (11), this gives
$\displaystyle\|A\|\|M_{k}(X)\|\ \geq\
\int_{X_{0}}m_{A}\frac{\omega^{n}_{FS}}{n!}$ $\displaystyle=$
$\displaystyle\int_{X_{0}}cH_{A}\frac{\omega^{n}_{FS}}{n!}-\operatorname{vol}\sum_{i}c_{i}\operatorname{tr}(A^{k+i})$
(9) $\displaystyle=$ $\displaystyle
cb_{0}-a_{0}\sum_{i}c_{i}w(H^{0}(L^{k+i})),$
by Proposition 19. Here we are writing
$h^{0}(L^{k})=a_{0}k^{n}+a_{1}k^{n-1}+\widetilde{o}(k^{n-1})$ and
$w(H^{0}(L^{k}))=b_{0}k^{n+1}+b_{1}k^{n}+\widetilde{o}(k^{n})$. Lemma 5 then
gives
$\displaystyle c=\sum_{i}c_{i}(k+i)h^{0}(L^{k+i})$ $\displaystyle=$
$\displaystyle\tilde{a}_{0}k^{n+1}+\tilde{a}_{1}k^{n}+O(k^{n-1}),$
$\displaystyle\text{and}\quad\sum_{i}c_{i}w(H^{0}(L^{k+i}))$ $\displaystyle=$
$\displaystyle\tilde{b}_{0}k^{n+1}+\tilde{b}_{1}k^{n}+O(k^{n-1}),$
where
$\displaystyle\tilde{a}_{0}=a_{0}\sum_{i}c_{i}\ $ and $\displaystyle\
\tilde{a}_{1}=\sum_{i}c_{i}(a_{0}i(n+1)+a_{1}),$
$\displaystyle\tilde{b}_{0}=b_{0}\sum_{i}c_{i}\ $ and $\displaystyle\
\tilde{b}_{1}=\sum_{i}c_{i}(b_{0}i(n+1)+b_{1}).$
Therefore (9) becomes
$\displaystyle\lVert A\rVert\lVert M_{k}(X)\rVert$ $\displaystyle\geq$
$\displaystyle
c\left(b_{0}-a_{0}\frac{\tilde{b}_{0}k^{n+1}+\tilde{b}_{1}k^{n}+O(k^{n-1})}{\tilde{a}_{0}k^{n+1}+\tilde{a}_{1}k^{n}+O(k^{n-1})}\right)$
$\displaystyle=$ $\displaystyle
ca_{0}\left(k^{-1}\frac{\tilde{b}_{0}\tilde{a}_{1}-\tilde{b}_{1}\tilde{a}_{0}}{\tilde{a}_{0}^{2}}+O(k^{-2})\right)$
$\displaystyle=$ $\displaystyle
ca_{0}\left(k^{-1}\frac{b_{0}a_{1}-b_{1}a_{0}}{a_{0}^{2}}+O(k^{-2})\right)$
$\displaystyle=$ $\displaystyle ca_{0}\left(-k^{-1}F_{1}+O(k^{-2})\right).$
Now $\|A\|^{2}=|\operatorname{tr}A^{2}|=O(k^{n+2})$ by (20), and $c$ is
strictly of order $O(k^{n+1})$. So Theorem 6 now gives
$k^{\frac{n-2}{2}}\lVert\operatorname{Scal}(\omega)-\,\overline{\\!S}\rVert_{L^{2}}+O(k^{\frac{n-4}{2}})\geq
Ck^{\frac{n}{2}}\left(-k^{-1}F_{1}+O(k^{-2})\right)$
for some constant $C>0$. Hence when $\operatorname{Scal}(\omega)$ is constant
(and therefore equal to $\,\overline{\\!S}$) we see that $F_{1}\geq 0$. ∎
###### Corollary 10.
Let $(X,L)$ be a polarised orbifold with cyclic stabiliser groups. If $X$
admits an orbifold Kähler metric $\omega\in\mathcal{K}(c_{1}(L))$ with
constant scalar curvature then $(X,L)$ is K-semistable.
## Chapter 6 Slope stability of orbifolds
To get examples where K-stability obstructs the existence of constant scalar
curvature metrics we need a supply of test configurations for which we can
calculate the Futaki invariant. To this end we briefly describe the notion of
slope stability. The detailed descriptions in RT [1, 2] extend easily from
manifolds to orbifolds with a few minor changes.
Fix an $n$-dimensional polarised orbifold $(X,L)$ and a sub-orbischeme (or
substack) $Z\subset X$: an invariant subscheme $Z_{U}$ in each orbifold chart
$U\to U/G\subset X$ such that for each injection of charts
$U^{\prime}\hookrightarrow U$, the subscheme $Z_{U^{\prime}}$ is the scheme-
theoretic intersection $Z_{U}\cap U^{\prime}$. In most of our examples $Z$
will be smooth but with generic stabilisers. Working equivariantly in charts
one can produce a new orbischeme, the blowup $\pi\colon\hat{X}\to X$ of $X$
along $Z$. Locally this is the blow up of $U$ in $Z_{U}$ divided by the
induced action of the Galois group on this blow up. The exceptional divisors
glue to give an orbifold exceptional divisor $E\subset\hat{X}$.
For large $N$, $\pi^{*}L^{N}(-E)$ is positive. (From now on we will suppress
$\pi^{*}$.) Thus we can define the _Seshadri constant_ by
$\epsilon_{\text{orb}}(Z)=\text{sup}\left\\{x\in\mathbb{Q}_{+}\colon(L(-xE))^{M}\text{
is ample for some }M\in\mathbb{N}\right\\}.$
For example, if we put $\mathbb{Z}/m\mathbb{Z}$ stabilisers along a smooth
divisor $D\subset X$ then as in Section 4 there is a well defined orbi-divisor
$D/m$ whose Seshadri constant $\epsilon_{\text{orb}}(D/m)=m\epsilon(D)$ is
$m$-times the usual Seshadri constant of $D$ in the underlying space of $X$.
To get a test configuration from $Z$, consider the suborbifold
$Z\times\\{0\\}\subset X\times\mathbb{C}$. Blowing this up gives the
degeneration $\mathcal{X}\to X\times\mathbb{C}\to\mathbb{C}$ to the normal
cone of $Z$ with exceptional divisor $P$. As shown in [RT, 2, Proposition 4.1]
for schemes (and the same results go through easily for orbifolds),
$\epsilon_{\text{orb}}(Z\times\\{0\\})=\epsilon_{\text{orb}}(Z)$. Let
$p\colon\mathcal{X}\to X$ be the projection. Then for generic
$c\in(0,\epsilon_{\text{orb}}(Z))\cap\mathbb{Q}$ general integer powers of
$\mathcal{L}_{c}:=p^{*}L(-cP)$ define a polarisation of $\mathcal{X}$. The
natural action of $\mathbb{C}^{*}$ on $X\times\mathbb{C}$ (trivial on $(X,L)$,
weight one on $\mathbb{C}$) lifts naturally to a linearised action on
$(\mathcal{X},\mathcal{L})$, and thus for such $c$ we have a test
configuration $(\mathcal{X},\mathcal{L}_{c})$ with general fibre $(X,L)$. The
central fibre is $\mathcal{X}_{0}=\hat{X}\cup_{E}P$ consisting of the blowup
$\hat{X}\to X$ along $Z$ glued to $P$ along $E$, and the induced
$\mathbb{C}^{*}$ action is trivial on $\hat{X}$ and acts by scaling $P$ along
the normal to $E$.
As usual we write
$h^{0}(L^{k})=a_{0}k^{n}+a_{1}k^{n-1}+\tilde{o}(k^{n-1}),$ (1)
and then define the _slope_ of $(X,L)$ to be
$\mu(X,L)=\frac{a_{1}}{a_{0}}=-\frac{n\int_{X}c_{1}(K_{\text{orb}}).c_{1}(L)^{n-1}}{2\int_{X}c_{1}(L)^{n}}\,,$
by (17). To define the slope of $Z\subset X$, we work on the orbifold blowup
$\pi\colon\hat{X}\to X$ along $Z$ with exceptional divisor $E$. Then orbifold
Riemann-Roch to $L^{k}(-\frac{j}{k}E)$ for fixed $j$ (and $k=jK$ for some
integer $K$) takes the form
$h^{0}(L^{k}(-jE))=p(k,j)+\epsilon_{p}(k,j).$ (2)
Here $p$ is a polynomial of two variables of total degree $n$ and
$\epsilon_{p}$ is a sum of terms of the form $r_{p}\delta^{\prime}$ where
$r_{p}$ is a polynomial of two variables of total degree $n-1$ and
$\delta^{\prime}=\delta^{\prime}(k,j)$ is periodic in each variable with
average $\sum_{k,j=1}^{M}\delta^{\prime}(k,j)=0$. Define polynomials
$a_{i}(x)$ by
$p(k,xk)=a_{0}(x)k^{n}+a_{1}(x)k^{n-1}+O(k^{n-2})\quad\text{for
}kx\in\mathbb{N}.$
Then the _slope_ of $Z$ (with respect to $c$) is
$\mu_{c}(\curly
I_{Z}):=\frac{\int_{0}^{c}a_{1}(x)+\frac{a^{\prime}_{0}(x)}{2}dx}{\int_{0}^{c}a_{0}(x)dx}\,.$
(3)
The only difference from the manifold case is that we ignored the periodic
terms in the relevant Hilbert functions. This amounts to replacing $K_{X}$ by
$K_{\text{orb}}$.
###### Definition 4.
We say that $(X,L)$ is _slope semistable with respect to $Z$_ if
$\mu_{c}(\curly I_{Z})\leq\mu(X)\quad\text{ for all
}0<c<\epsilon_{\text{orb}}(Z).$
We say that $X$ is _slope semistable_ if it is slope semistable with respect
to all sub-orbischemes $Z\subset X$.
Alternatively, just as in the manifold case [RT, 1, Definition 3.13] we can
put $\tilde{a}_{i}(x):=a_{i}-a_{i}(x)$ and define the _quotient slope_ of $Z$
as
$\mu_{c}(\mathcal{O}_{Z}):=\frac{\int_{0}^{c}\tilde{a}_{1}(x)+\frac{\tilde{a}^{\prime}_{0}(x)}{2}dx}{\int_{0}^{c}\tilde{a}_{0}(x)dx}\,,$
(5)
and $(X,L)$ is slope semistable with respect to $Z$ if and only if
$\mu(X)\leq\mu_{c}(\mathcal{O}_{Z})$ for all $0<c<\epsilon_{\text{orb}}(Z)$.
One can check easily that slope semistability is invariant upon replacing $L$
by a positive power. The point of these definitions is that the sign of the
Futaki invariant of the test configuration given by deformation to the normal
cone of $Z$ is the same as the sign of $\mu(X)-\mu_{c}(\curly I_{Z})$,
resulting in the following slope obstruction to stability.
###### Theorem 6.
If $(X,L)$ is K-semistable then it is slope semistable.
###### Proof.
The argument is essentially the same as that in the smooth case [RT, 2,
Section 4]; only the Riemann-Roch formula changes. Since being not slope
semistable is an open condition, we may without loss of generality assume that
$c<\epsilon_{\text{orb}}(Z)$ is general, and so by rescaling $L$ be may assume
that $c$ is integral and coprime to $m$, making
$(\mathcal{X},\mathcal{L}_{c})$ a test configuration. The space of sections on
the central fibre of the test configuration splits as
$H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k})=H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck})\
\oplus\,\bigoplus_{j=1}^{ck}t^{j}\frac{H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j})}{H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j+1})}\,.$
(7)
Here $H^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{j})$ is the space of sections of
$L^{k}$ which vanish to order $j$ on $Z$ (in an orbi chart). The coordinate
$t$ is pulled back from the base $\mathbb{C}$, and is acted on by
$\mathbb{C}^{*}$ with weight $-1$. Therefore (7) is the weight space
decomposition of $H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k})$, with total
weight
$\displaystyle w(H^{0}_{\mathcal{X}_{0}}(\mathcal{L}_{c}^{k}))$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{ck}j\Big{(}h^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j})-h^{0}_{X}(L^{k}\otimes\mathcal{I}_{Z}^{ck-j+1})\Big{)}.$
Some manipulation, and the vanishing of higher cohomology of the pushdowns of
these sheaves to the underlying scheme, give
$\displaystyle\sum_{j=1}^{ck}h_{X}^{0}(L^{k}\otimes\mathcal{I}_{Z}^{j})-ckh_{X}^{0}(L^{k})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{ck}h^{0}_{\hat{X}}(L^{k}(-jE))-ckh^{0}(L^{k})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{ck}p(k,j)+\epsilon_{p}(k,j)-ck[h(k)+\tilde{o}(k^{n-1})].$
By Lemma 8 below, the periodic terms to not contribute to the top two order
parts of this sum, so the leading order polynomial parts of the weight are
$w(k)=\sum_{j=1}^{ck}p(j,k)-ckh(k)+\tilde{o}(k^{n}).$
The calculation of the Futaki invariant is now exactly as in the smooth case
[RT, 2, Proposition 4.14 and Equation 4.19], yielding
$F_{1}(\mathcal{X},\mathcal{L}_{c})=(\mu(X)-\mu_{c}(\mathcal{I}_{Z}))\frac{\int_{0}^{c}a_{0}(x)dx}{a_{0}}\,.$
This is nonnegative if and only if $X$ is slope semistable with respect to
$Z$. ∎
###### Lemma 8.
Suppose $\delta(k,j)$ is periodic in each variable with period $m$ and average
$\sum_{k,j=1}^{m}\delta(k,j)=0$. Suppose also that $r(k,j)$ is a polynomial of
two variables of total degree $n-1$ and $c$ is a fixed integer. Then
$\sum_{j=1}^{ck}r(k,j)\delta(k,j)=\epsilon(k)+O(k^{n-1}),$
where $\epsilon(k)$ is a sum of terms of the form $r(k)\delta^{\prime}(k)$,
with $r$ a polynomial of degree $n$ and $\delta^{\prime}$ periodic of average
zero.
###### Proof.
By linearity it is sufficient to consider the case where $r(k,j)=j^{n-1}$. Set
$j_{0}=\left\lfloor\frac{ck}{m}\right\rfloor m$ and split the sum into two
pieces depending on whether $j\leq j_{0}$ or $j\geq j_{0}+1$. In the first
case, writing $j=mu+v$,
$\sum_{j=0}^{j_{0}}j^{n-1}\delta(k,j)=\sum_{v=1}^{m}\delta(k,v)\sum_{u}(um+v)^{n-1}.$
This splits into pieces of the form $P(k)\epsilon(k)$ where $\epsilon(k)$ is
periodic and $P$ is a polynomial of degree at most $n$, with the degree being
equal to $n$ only when $\epsilon(k)=\sum_{v=1}^{m}\delta(v,k)$ in which case
$\epsilon$ has average zero. Then note that if $P$ has degree $n-1$ there is a
constant $a$ so that $\epsilon-a$ has average zero, and
$P\epsilon=P(\epsilon-a)+aP=P(\epsilon-a)+O(k^{n-1})$. Thus, after some
rearrangement, this part of the sum is of the form claimed. The sum for $j\geq
j_{0}+1$ immediately splits into terms of the form $P(k)\epsilon(k)$ where $P$
has degree at most $n-1$ and $\epsilon$ is periodic, so by the same argument
these terms are also of the required form. ∎
We can calculate the slope of (sufficiently nice) suborbifolds much as in the
manifold case. For instance let $\hat{X}\stackrel{{\scriptstyle\pi}}{{\to}}X$
be the orbifold blowup along a smooth $Z$ of codimension $r\geq 2$, with
orbifold exceptional divisor $E$. Then
$K_{\text{orb},\hat{X}}=\pi^{*}K_{\text{orb},X}+(r-1)E$, and
$\displaystyle a_{0}(x)$ $\displaystyle=$
$\displaystyle-\frac{\int_{\hat{X}}c_{1}(L(-xE))^{n}}{n!}\,,$ $\displaystyle
a_{1}(x)$ $\displaystyle=$
$\displaystyle-\frac{\int_{\hat{X}}c_{1}(K_{\text{orb},\hat{X}})c_{1}(L(-xE))^{n-1}}{2(n-1)!}\,.$
(9)
So the formulae only differ from those in RT [1] in replacing $K_{X}$ by
$K_{\text{orb}}$. For example if $Z$ is as small as possible – the invariant
subvariety defined by a reduced fixed point upstairs in an orbifold chart –
then these quickly imply
$\mu_{c}(\mathcal{O}_{Z})=\frac{n(n+1)}{2c}\,.$ (10)
Notice the order of the stabiliser group at this point does not feature;
however it enters into the Seshadri constant of $Z$ and so does affect slope
stability.
Similarly if $Z$ is an orbifold divisor in an orbifold surface then
$\mu_{c}(\mathcal{O}_{Z})=\frac{3(2L.Z-c(K_{\text{orb}}.Z+Z^{2})}{2c(3L.Z-cZ^{2})}\,.$
## Chapter 7 Applications and further examples
### 1 Orbifold Riemann surfaces
By an orbifold Riemann surface we mean an orbifold of complex dimension one.
This is equivalent to the data of a Riemann surface of genus $g\geq 0$ and $r$
points $p_{1},\dots,p_{r}\in X$ marked by orders of stabiliser groups
$m_{1},\ldots,m_{r}\geq 2$. We assume $r\geq 1$.
###### Theorem 1.
A polarised Riemann surface $(X,L)$ is slope semistable if and only if
$2g+\sum_{i=1}^{r}\left(\frac{m_{i}-1}{m_{i}}\right)\geq
2\max_{i=1,\ldots,r}\left\\{\frac{m_{i}-1}{m_{i}}\right\\}$ (2)
###### Proof.
The orbifold canonical bundle of $X$ is
$K_{\text{orb}}=K_{X}+\sum_{i=1}^{r}\left(1-\frac{1}{m_{i}}\right)p_{i}$ so
the slope of $X$ is
$\mu(X,L)=-\frac{\deg K_{\text{orb}}}{2\deg
L}=\frac{1-g-\frac{1}{2}\sum_{i=1}^{r}\left(1-\frac{1}{m_{i}}\right)}{\deg
L}\,.$ (3)
Without loss of generality assume $m_{1}\geq m_{2}\geq\dots\geq m_{r}$. Let
$Z=\\{p_{1}/m\\}$ be the orbifold point of order $m_{1}$ with a reduced lift
upstairs. Then
$\mu_{c}(\mathcal{O}_{Z})=c^{-1}\quad\text{and}\quad\epsilon_{\text{orb}}(Z,X,L)=m_{1}\deg
L.$
Now if $(X,L)$ is semistable then $\mu_{\epsilon}(\mathcal{O}_{Z})\geq\mu(X),$
so
$\frac{1}{m_{1}}\geq 1-g-\frac{1}{2}\sum_{i=1}^{r}\frac{m_{i}-1}{m_{i}}$
which rearranges to give the inequality (2).
For the converse suppose that (2) holds and consider an orbifold subspace
$Z\subset X$. This $Z$ is an orbifold divisor whose degree is a rational
number $q\geq\frac{1}{m_{1}}$. Thus
$\epsilon:=\epsilon_{\text{orb}}(Z,L)=\frac{1}{q}\deg L\leq m_{1}\deg L$. As
$\mu_{c}(\mathcal{O}_{Z})=c^{-1}$ is decreasing with respect to $c$, we get
$\mu_{c}(\mathcal{O}_{Z})\geq\epsilon^{-1}\geq(m_{1}\deg L)^{-1}$. But (2)
implies that this is greater than or equal to $\mu(X,L)$ (3), so $X$ is slope
semistable. ∎
###### Remark 4.
1. 1.
It is hard to either violate or achieve equality in the inequality (2). Either
would imply that $2g+0\leq 2\max(1-\frac{1}{m_{i}})<2$ and so $g=0$. Then
since each integer $m_{i}\geq 2$, we find there are only three cases in which
an orbifold Riemann surface is not strictly slope stable:
1. (a)
$g=0,\ r=1$ (this gives $\mathbb{P}(1,m)$),
2. (b)
$g=0,\ r=2,\ m_{1}\neq m_{2}$ (giving $\mathbb{P}(m_{1},m_{2})$ if
$\operatorname{hcf}(m_{1},m_{2})=1$), and
3. (c)
$g=0,\ r=2,\ m_{1}=m_{2}$.
In the first two cases $(X,L)$ is not slope semistable and so not cscK. In the
third case $(X,L)$ is actually slope polystable, as we now describe. The only
way in which $\mu_{c}(\mathcal{O}_{Z})=\mu(X)$ can occur in the proof of (1)
is if $Z=\\{p_{1}/m_{1}\\}$ or $Z=\\{p_{2}/m_{2}\\}$ and $c=\epsilon(Z)=2$. In
this case deformation to the normal cone $\mathcal{X},\mathcal{L}_{c})$ has
$\mathcal{L}_{c}$ only semi-ample, pulled back from the contraction of the
proper transform of the central fibre $X\times\\{0\\}$. This contraction is in
fact a product configuration $X\times\mathbb{C}$ (with a nontrivial
$\mathbb{C}^{*}$-action).
2. 2.
The stability condition (2) is actually a special case of (10) and thus an
manifestation of the index obstruction which we discuss below.
Slope stability of orbifold Riemann surfaces fits perfectly into the known
theory of orbifold cscK metrics which has been studied by several authors
including Picard Pic , McOwen McO and Troyanov Tro [1, 2]. In the terminology
of this paper Troyanov’s results can be paraphrased as follows:
###### Corollary 5 (Troyanov).
Let $(X,L)$ be a polarised orbifold Riemann surface. Then $c_{1}(L)$ admits an
orbifold cscK metric if and only if it is slope polystable.
###### Proof.
The main theorems in Tro [2] imply that $X$ admits a cscK metric when strict
inequality holds in (2) i.e. as long as $(X,L)$ is slope stable. (To compare
our notation with Troyanov’s set $\theta_{i}=\frac{2\pi}{m_{i}}$ and
$\chi_{\text{orb}}=-\deg K_{\text{orb}}$. Then this is Theorem A in Tro [2]
when $\chi_{\text{orb}}<0$, Proposition 2 when $\chi_{\text{orb}}=0$ and
Theorem C when $\chi_{\text{orb}}>0$.) The only way that $(X,L)$ can be slope
polystable and not slope stable is case c): if $g=0,r=2$ and $m_{1}=m_{2}$. In
this case $X$ is the global quotient $\mathbb{P}^{1}/(\mathbb{Z}/m\mathbb{Z})$
with orbifold cscK metric descended from the Fubini-Study metric on
$\mathbb{P}^{1}$.
For the converse, if $(X,L)$ is not slope polystable then $g=0$ and either a)
$r=1$ or b) $r=2$ and $m_{1}\neq m_{2}$. In these two cases $(X,L)$ is not
slope semistable, which by Corollary 10 and Theorem 6 implies that $X$ does
not admit an orbifold cscK metric. ∎
###### Remark 6.
The statement that if $g=0$ and a) $r=1$ or b) $r=2$ and $m_{1}\neq m_{2}$
then $X$ does not admit a cscK metric has also been proved by Troyanov [Tro,
1, Theorem I]. Troyanov’s work applies much more generally to cone angles not
necessarily of the form $2\pi/m$, and this is also studied further in Che ; CL
; LT . We hope to return to cone angles in $2\pi\mathbb{Q}_{+}$ using the
method described in the Introduction.
### 2 Index obstruction to stability
Recall that the index $\operatorname{ind}(X)$ of a Fano manifold $X$ is
defined to be the largest integer $r$ such that $K^{-1}_{X}$ is linearly
equivalent to $rD$ for some Cartier divisor $D\subset X$, and it is well known
that if $X$ is smooth then $\operatorname{ind}(X)\leq n+1$ with equality if
and only if $X=\mathbb{P}^{n}$. By contrast for an Fano orbifold $(X,\Delta)$
it is possible that $K^{-1}_{\text{orb}}\cong\mathcal{O}(rD)$ where $D$ is an
orbi divisor and $r\in\mathbb{N}$ is larger than $n+1$. We will show that this
prevents $(X,\Delta)$ from being K-stable. In fact the same is true under the
weaker condition that $K_{\text{orb}}^{k}\cong\mathcal{O}(krD)$ for some
$k\in\mathbb{N}$ and $n+1\leq r\in\mathbb{Q}$.
###### Theorem 7.
(Index Obstruction) Let $(X,K^{-1}_{\text{orb}})\supset D$ be a Fano orbifold
and an orbi divisor. Suppose that $K^{-k}_{\text{orb}}\cong\mathcal{O}(krD)$
for some $k>0$ and $r\in\mathbb{Q}_{+}$. If $r>n+1$ then
$(X,K^{-1}_{\text{orb}})$ is slope unstable, and thus does not admit an
orbifold Kähler-Einstein metric.
###### Proof.
Set $L=K^{-1}_{\text{orb}}$. Using (6) to calculate the slope,
$\displaystyle a_{0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\int_{X}(c_{1}(L)-xc_{1}(D))^{n}=\frac{1}{n!}(r-x)^{n}\int_{X}c_{1}(D)^{n},$
$\displaystyle a_{1}(x)+\frac{a_{0}^{\prime}(x)}{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2(n-1)!}(r-1)(r-x)^{n-1}\int_{X}c_{1}(D)^{n}.$
Now $a_{0}=\frac{r^{n}}{n!}\int_{X}c_{1}(D)^{n}$ and
$a_{1}=\frac{r^{n}}{2(n-1)!}\int_{X}c_{1}(D)^{n}$, so $\mu(X,L)=\frac{n}{2}$.
The Seshadri constant of $D$ is $r$. Using the definition of the slope (5),
$\mu_{r}(\mathcal{O}_{D})=\frac{(n+1)((n-1)r+1)}{2nr}\,,$
which is less that $\mu(X)=n/2$ if and only if $r>n+1$. ∎
###### Remark 8.
At the level of Kähler-Einstein metrics the analogous result has already been
proved by Gauntlett-Martelli-Sparks-Yau using the “Lichnerowicz obstruction”
to the existence of Sasaki-Einstein metrics with non-regular Reeb vector
fields [GMSY, , Section 2.2]. In fact it was their work that originally
motivated this paper. In the same paper the authors discuss the “Bishop
obstruction” which we have been unable to interpret in terms of stability.
###### Example 9.
(Weighted projective space) Consider weighted projective space
$\mathbb{W}\mathbb{P}=\mathbb{P}(\lambda_{0},\ldots,\lambda_{n})$, with
$\lambda_{0}\leq\lambda_{1}\leq\ldots\leq\lambda_{n}$ _not all equal_. Then
$\\{x_{0}=0\\}$ defines an effective divisor in $\mathcal{O}(\lambda_{0})$,
while $K^{-1}_{\text{orb}}\cong\mathcal{O}(\sum_{i}\lambda_{i})$. Since
$\sum_{i}\lambda_{i}>(n+1)\lambda_{0}$ the index obstruction shows that
$\mathbb{W}\mathbb{P}$ is unstable, recovering the well known fact that it
does not admit an orbifold cscK metric.
###### Example 10.
(Orbifold projective space) Let $X=\mathbb{P}^{n}$ and take $n+2$ hyperplanes
$H_{1},\ldots,H_{n+2}$ in general position, and integers $m_{i}\geq 2$.
Setting
$\Delta=\sum_{i=1}^{n+2}\left(1-\frac{1}{m_{i}}\right)H_{i},$
we consider the orbifold $(\mathbb{P}^{n},\Delta)$. Then
$K^{-1}_{\text{orb}}=K^{-1}_{\mathbb{P}^{n}}(-\Delta)$ becomes equivalent
after passing to powers to
$\mathcal{O}\bigg{(}\\!n+1-\sum_{i=0}^{n+2}\Big{(}1-\frac{1}{m_{i}}\Big{)}\bigg{)}=\mathcal{O}\bigg{(}\\!\\!-1+\sum_{i=1}^{n+2}\frac{1}{m_{i}}\bigg{)}.$
(11)
Thus $(\mathbb{P}^{n},\Delta)$ is a Fano orbifold as long as
$\sum_{i=1}^{n+2}\frac{1}{m_{i}}>1$.
The right hand side of (11) can be written
$\mathcal{O}\bigg{(}m_{j}\bigg{(}\\!\\!-1+\sum_{i=1}^{n+2}\frac{1}{m_{i}}\bigg{)}D_{j}\bigg{)},$
where $D_{j}=\frac{1}{m_{j}}H_{j}$ is an orbi divisor. Thus by the index
obstruction, if $(X,\Delta)$ is a semistable Fano orbifold then
$\sum_{i=1}^{n+2}\frac{1}{m_{i}}\leq 1+(n+1)\min_{1\leq i\leq
n+2}\left(\frac{1}{m_{i}}\right).$ (12)
###### Remark 13.
The previous example is considered by Ghigi-Kollár [GK, , Example 43]. They
show that as long as
$1<\sum_{i=1}^{n+2}\frac{1}{m_{i}}<1+(n+1)\min_{1\leq i\leq
n+2}\left(\frac{1}{m_{i}}\right)$
then $(X,\Delta)$ admits a Kähler-Einstein metric. Thus the previous example
suggests this condition is strict (our slightly weaker inequality comes from
only having a proof that a cscK metric implies semistability rather than
polystability). We remark that Ghigi-Kollár also prove a much more general
condition under which a Kähler-Einstein Fano manifold with boundary divisor
$\Delta$ yields a Kähler-Einstein orbifold $(X,\Delta)$ [GK, , Theorem 41]. It
is not the case that this condition is simply the index obstruction, and we
have not been able to determine if this condition is related to slope
stability or if it is also strict.
### 3 Orbifold ruled surfaces
Let $(\Sigma,L)$ be a polarised orbifold Riemann surface and $\pi\colon
E\to\Sigma$ be an orbifold vector bundle of rank $r$. Then $\mathbb{P}(E)$ is
itself naturally an orbifold: on a chart $U\to U/G$ of $\Sigma$, the $G$
action on $E|_{U}$ induces an action on $\mathbb{P}(E|_{U})$ (which is
effective as the action on $U$ is) and these give orbifold charts on
$\mathbb{P}(E)$. Suppose that the $G$-action on the fibres $E$ over points of
$\Sigma$ with stabiliser group $G$ has distinct eigenvalues, so that
$\mathbb{P}(E)$ has codimension two orbifold locus and all fibres are finite
quotients of $\mathbb{P}^{r-1}$. The hyperplane bundle
$\mathcal{O}_{\mathbb{P}(E)}(1)$ is both locally ample and relatively ample,
so $L_{m}:=\mathcal{O}_{\mathbb{P}(E)}(1)\otimes\pi^{*}L^{m}$ is ample for $m$
sufficiently large.
We claim that stability of $\mathbb{P}(E)$ is connected to stability of the
underlying bundle $E$. Here stability of a bundle is to be taken in the sense
of Mumford, so define
$\mu_{E}:=\frac{\deg E}{\operatorname{rank}E}$
where the degree is taken in the orbifold sense. Then $E$ is defined to be
stable if for all orbifold bundles $F$ with a proper injection $F\subset E$ we
have $\mu_{F}<\mu_{E}$.
Now if $F\subset E$ then $\mathbb{P}(F)$ is a suborbifold of $\mathbb{P}(E)$.
Using $\pi_{*}\mathcal{O}_{\mathbb{P}(E)}(k)=S^{k}E^{*}$ one can use orbifold
Riemann-Roch to compute the slope of each in exactly the same way as in the
manifold case [RT, 1, Section 5.4]. The upshot is that the Seshadri constant
of $\mathbb{P}(F)$ is $\epsilon_{\text{orb}}(\mathbb{P}(F))=1$ and
$\mu_{1}(\mathcal{O}_{\mathbb{P}(F)})-\mu(\mathbb{P}(E))=C(\mu_{E}-\mu_{F})\Big{(}rm+(r-1)\mu(\Sigma)-r\mu_{E}\Big{)}$
for some $C>0$, where $\mu(\Sigma)=-\deg K_{\text{orb}}/2\deg L$ is the
orbifold slope of $(\Sigma,L)$. The term inside the last set of brackets is
positive for any $m$ sufficiently positive that $L_{m}$ is ample (it is
essentially the volume of $(\mathbb{P}(E),L_{m})$). Therefore if $E$ is
unstable as an orbifold vector bundle then $(\mathbb{P}(E),L_{m})$ is slope
unstable as an orbifold. This result also generalises to higher dimensional
base as long as one works near the adiabatic limit of sufficiently large $m$,
just as in the manifold case.
If $E$ is polystable then $\mathbb{P}(E)$ carries an orbifold cscK metric; see
for example RS . We therefore get a (partial) converse – for strictly unstable
bundles, $(\mathbb{P}(E),L_{m})$ does _not_ carry an orbifold cscK metric for
any $m$. (The discrepancy lies in strictly semistable, but not polystable,
bundles.)
In fact Rollin and Singer phrase their results in terms of parabolic bundles,
but there is a complete correspondence between orbifold bundles $E$ on
$\Sigma$ and parabolic vector bundles $E^{\prime}$ on the underlying space of
$\Sigma$. In the notation of Section 4, the bundle $E^{\prime}$ is the
pushdown of $E$ from the orbifold to its underlying space; this is therefore
the vector bundle analogue of rounding down of $\mathbb{Q}$-divisors in the
line bundle case. The information lost is then encoded via the parabolic
structure on $E^{\prime}$ at each of the orbifold points $x$, with rational
weights of the form $p_{j}/\\!\operatorname{ord}(x)$ for
$p_{j}<\operatorname{ord}(x)$ corresponding to the weights of the action on
$E_{x}$. See for example [FS, , Section 5]. Moreover this correspondence
preserves subobjects and their degrees, where the parabolic degree of
$E^{\prime}$ is defined as
$\operatorname{pardeg}E^{\prime}=\deg
E^{\prime}+\sum_{x,j}m_{x,j}\frac{p_{j}}{\operatorname{ord}(x)}\,.$
Here the sum is over all orbifold points $x$, and if the parabolic structure
over $x$ is given by the flag $F_{0}\subset F_{1}\subset\ldots\subset
E^{\prime}_{x}$ then $m_{x,j}=\dim F_{j}/F_{j+1}$. Thus orbifold stability of
$E$ corresponds precisely to the parabolic stability of $E^{\prime}$.
Rollin and Singer RS use such orbifold cscK metrics as a starting point to
produce ordinary cscK metrics (with zero scalar curvature, in fact) on small
blow ups of the orbifolds $\mathbb{P}(E)$, using a gluing method. Our results
suggest that if $E$ is unstable, destabilised by $F$, then one should be able
to slope destabilise such blow ups using the pullback (or proper transform) of
$\mathbb{P}(F)$.
### 4 Slope stability of canonically polarised orbifolds
By the orbifold version of the Aubin-Yau theorems, orbifolds which have
positive or trivial canonical bundle admit orbifold Kähler-Einstein metrics.
Therefore by Corollary 10 they are K-semistable, and so by Theorem 6 are also
slope semistable. In fact this can be proved directly. That is, suppose that
$(X,L)$ is a polarised orbifold and either
1. 1.
$K_{\text{orb}}$ is numerically trivial and $L$ is arbitrary _or_
2. 2.
$L=K_{\text{orb}}$.
Then $(X,L)$ is slope stable. The proof is the same as the manifold case (see
[RT, 2, Theorem 8.4] or [RT, 1, Theorem 5.4]), with $K_{X}$ replaced by
$K_{\text{orb}}$, so we do not repeat it here.
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|
arxiv-papers
| 2009-07-30T08:05:37 |
2024-09-04T02:49:04.285690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Ross and R. P. Thomas",
"submitter": "Julius Ross",
"url": "https://arxiv.org/abs/0907.5214"
}
|
0907.5215
|
# Weighted Bergman kernels on orbifolds
Julius Ross and Richard Thomas
###### Abstract
We describe a notion of ampleness for line bundles on orbifolds with cyclic
quotient singularities that is related to embeddings in weighted projective
space, and prove a global asymptotic expansion for a weighted Bergman kernel
associated to such a line bundle.
††volume: :††volume: B††volume: missing††volume: missing
## Chapter 0 Introduction
Let $(X,\omega)$ be a compact $n$-dimensional Kähler manifold and $L$ be a
positive line bundle on $X$ equipped with a hermitian metric $h$ whose
curvature form is $2\pi\omega$. These induce an $L^{2}$-metric on the space of
sections $H^{0}(L^{k})$, and given an orthonormal basis $\\{s_{\alpha}\\}$ the
Bergman kernel is the smooth function
$B_{k}(x)=\sum_{\alpha}|s_{\alpha}(x)|^{2},$
where $|s_{\alpha}(x)|$ is the pointwise norm induced by $h$ (more precisely,
its $k$th power considered as a metric on $L^{k}$). More invariantly, the
$L^{2}$-metric on $H^{0}(L^{k})$ induces a Fubini-Study metric on
$\mathbb{P}(H^{0}(L^{k})^{*})\supset X$ and $\mathcal{O}(1)$ over it;
restricting to $X$ we get a metric $h_{FS}$ on $\mathcal{O}_{X}(1)\cong L^{k}$
which may not equal the first $h$. Then $B_{k}=h/h_{FS}$ is their ratio.
A well known theorem of Tian-Zelditch-Catlin describes the asymptotic
behaviour of this function for large $k$: there exist smooth functions
$b_{1},\ldots,b_{N}$ and a global expansion
$B_{k}=k^{n}+b_{1}k^{n-1}+\cdots+b_{N}k^{n-N}+O(k^{n-N-1})$ (1)
for $k\gg 0$, where $O(k^{n-N-1})$ can even be taken in the $C^{\infty}$ norm
[Rua]. Moreover the $b_{j}$ can be expressed in terms of the derivatives of
the metric. In particular $b_{1}$ is half the scalar curvature of $\omega$,
leading to the importance of $B_{k}$ in the theory of constant scalar
curvature Kähler metrics.
Throughout this paper, $X$ will instead be a orbifold (usually compact, except
when we work locally) with cyclic stabiliser groups and an orbifold line
bundle $L$ over it. We let $\operatorname{ord}_{x}$ denote the order of a
point $x\in X$: the size of the cyclic stabiliser group of any lift of this
point in an orbifold chart. When $X$ is compact, $\operatorname{ord}(X)$
denotes the least common multiple of this finite collection of integers.
Then the same expansion of $B_{k}$ holds away from the orbifold locus [DLM,
Son] but not over it. We illustrate the situation locally using the simplest
example: the orbifold $\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ with local
coordinate $z$ on $\mathbb{C}$ acted on by $\mathbb{Z}/2\mathbb{Z}$ via
$z\mapsto-z$. Then $x=z^{2}$ is a local coordinate on the quotient thought of
as a manifold. An ordinary (non-orbifold) line bundle is one pulled back from
the quotient, i.e. one which has trivial $\mathbb{Z}/2\mathbb{Z}$-action
upstairs when considered as a trivial line bundle there. This has invariant
sections $\mathbb{C}[x]=\mathbb{C}[z^{2}]$. We do not consider such line
bundles as their sections only see the quotient manifold
$\mathrm{Spec}\,\mathbb{C}[x]$ (to which the usual Bergman expansion applies),
missing the extra functions of $\sqrt{x}=z$ that the orbifold sees.
Instead we use the nontrivial _orbifold line bundle_ given by the trivial line
bundle upstairs with nontrivial $\mathbb{Z}/2\mathbb{Z}$-action (acting as
$-1$ on the trivialisation). This has invariant sections
$\sqrt{x}\,\mathbb{C}[x]=z\mathbb{C}[z^{2}]$, while its square has trivial
$\mathbb{Z}/2\mathbb{Z}$-action and has sections
$\mathbb{C}[x]=\mathbb{C}[z^{2}]$ as above. Therefore the sections of its
powers generate the entire ring of functions
$\mathbb{C}[\sqrt{x}]=\mathbb{C}[z]$ upstairs, and see the full orbifold
structure.
###### Definition 2.
[RT] An orbifold line bundle $L$ over a cyclic orbifold $X$ is _locally ample_
if in an orbifold chart around $x\in X$, the stabiliser group acts faithfully
on the line $L_{x}$. We say $L$ is _orbi-ample_ if it is both locally ample
and globally positive. (That is, $L^{\operatorname{ord}(X)}$ is ample in the
usual sense when thought of as a line bundle on the underlying space of $X$;
by the Kodaira-Baily embedding theorem [Bai] one can equivalently ask that $L$
admits a hermitian metric with positive curvature.)
In [RT, Sections 2.5–2.6] this notion of ampleness is shown to be equivalent
to the existence of an orbifold embedding
$(X,L)\hookrightarrow(\mathbb{WP},\mathcal{O}(1))$
of $X$ into a weighted projective space $\mathbb{W}\mathbb{P}$. That is, the
orbifold structure of $X$ is pulled back from that of $\mathbb{WP}$, and $L$
is the pullback of the orbifold hyperplane bundle
$\mathcal{O}_{\mathbb{WP}}(1)$.
Just as in our example above, for any orbifold line bundle $L$ which has
nontrivial stabiliser action on the line $L_{x}$ over an orbifold point $x$,
the sections of $L$ (i.e. the invariant sections upstairs) all vanish at $x$.
Therefore for most powers $L^{k}$ of an orbi-ample line bundle, the Bergman
kernel $B_{k}$ will vanish at orbifold points. Conversely for powers divisible
by $\operatorname{ord}_{x}$, $B_{k}(x)$ will be nonzero, so in general
$B_{k}(x)$ will have some periodic behaviour in $k$ at orbifold points. The
purpose of this paper is to get a smooth global expansion by taking weighted
sums of Bergman kernels associated to various powers of $L$, flattening out
the periodicity. In the companion paper [RT] we apply this to orbifold Kähler
metrics of constant scalar curvature and their relationship to stability.
The Bergman kernel measures the local density of holomorphic sections, so to
get an expansion on an orbifold we first discuss the general local situation –
the quotient of an open set in $\mathbb{C}^{n}$ by a linear action of the
cyclic group $\mathbb{Z}/m\mathbb{Z}$. The reader will not lose much by
considering only the $\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ example above, with
the orbifold line bundle with trivialisation which has weight $\pm 1$ under
the $\mathbb{Z}/2\mathbb{Z}$-action.
By the local ampleness condition there exists an identification between
$\mathbb{Z}/m\mathbb{Z}$ and the group of $m$th roots of unity with respect to
which there exists a local equivariant (not invariant!) trivialisation of $L$
of weight $-1$. Then, with respect to this trivialisation, sections of $L^{k}$
downstairs, i.e. invariant sections upstairs on $\mathbb{C}^{n}$, are the same
thing as functions upstairs of $\mathbb{Z}/m\mathbb{Z}$-weight $k$ mod $m$.
(In particular they vanish unless $k\equiv 0$ mod $m$.)
We consider the weighted Bergman kernel
$B^{\mathrm{orb}}_{k}:=\sum_{i}c_{i}B_{k+i}\,,$ (3)
where $c_{i}$ are positive constants and $i$ runs over some fixed finite index
set of nonnegative integers. This is the global expression downstairs on the
quotient; upstairs locally we are taking a similar expression, but only
include the sections of $L^{k+i}$ that have $\mathbb{Z}/m\mathbb{Z}$-weight
$0$ mod $m$ (or, using the trivialisation, the functions of weight $k+i$ mod
$m$).
Since functions of nonzero weight vanish at the orbifold point, the ordinary
Bergman kernel at the origin upstairs is equal to the sum of the
$|s_{\alpha}|^{2}$ over an orthonormal basis of _only the invariant sections_
– i.e. it equals the “downstairs” Bergman kernel $B_{k+i}$ at the origin.
(Using the trivialisation, this corresponds to summing over only the functions
of weight $k+i$.) Therefore by (1) applied to the ordinary upstairs Bergman
kernel at the origin, the downstairs kernel $B_{k+i}(0)$ has the expansion
$(k+i)^{n}+\frac{1}{2}\operatorname{Scal}(\omega)(k+i)^{n-1}+\cdots.$
Summing over $i$ we find that in (3) the functions upstairs of
$\mathbb{Z}/m\mathbb{Z}$-weight $u$ contribute, to leading order in $k$,
$\sum_{i\equiv u-k}c_{i}k^{n}$ (4)
to $B_{k}^{\mathrm{orb}}(0)$. (Here and throughout the paper the subscript
means that the sum is taken over all $i$ equal to $u-k$ mod $m$.)
For instance, in our $\mathbb{C}/(\mathbb{Z}/2\mathbb{Z})$ example above with
the nontrivial orbifold line bundle, the functions of
$\mathbb{Z}/2\mathbb{Z}$-weight $0$ (respectively $1$) mod $2$ contribute only
to the $B_{k+i}(0)$s with $k+i$ even (respectively odd). To even things out
and ensure that $B_{k}^{\mathrm{orb}}$ has a smooth global expansion we want
to use _all_ local functions of all $\mathbb{Z}/m\mathbb{Z}$-weights equally
(thus returning us to something close to the original Bergman kernel
_upstairs_). So we choose the $c_{i}$ so that the functions of each
$\mathbb{Z}/m\mathbb{Z}$-weight contribute to the sum (3) with the same
coefficient.
We therefore want for each $k$ for (4) to be equal for all $u$, i.e.
$\sum_{i\equiv u}c_{i}\quad\text{is independent of }u.$
More generally if $N\leq n$ and all of the coefficients of
$k^{n},\ldots,k^{n-N}$ in
$\sum_{i}c_{i}(k+i)^{n}$ (5)
have equal contributions in each weight (mod $m$), i.e.
$\sum_{i\equiv u}i^{p}c_{i}\quad\text{is independent of }u\text{ for
}p=0,..,N,$ (6)
then the same will be true of each $\sum_{i}c_{i}(k+i)^{n-j}$ for
$j=0,\ldots,N$, and $B_{k}^{\mathrm{orb}}$ will admit a polynomial expansion
of order $N$. In fact we impose a slightly stronger condition than (6) in
order to get the expansion at the $C^{r}$ level.
###### Theorem 7.
Let $(X,\omega)$ be a compact $n$-dimensional Kähler orbifold with cyclic
quotient singularities, and $L$ be an orbi-ample line bundle on $X$ equipped
with a hermitian metric whose curvature form is $2\pi\omega$. Fix $0\leq N\leq
n$ and $r\geq 0$ and suppose $c_{i}$ are a finite number of positive constants
chosen so that if $X$ has an orbifold point of order $m$ then
$\displaystyle\frac{1}{m}\sum_{i}i^{p}c_{i}$ $\displaystyle=$
$\displaystyle\mathop{\sum{i^{p}c_{i}}}_{i\equiv u\text{ mod }m}\quad\text{for
all }u\text{ and all }p=0,\ldots,N+r.$ (8)
Then the function
$B^{\mathrm{orb}}_{k}:=\sum_{i}c_{i}B_{k+i}$
admits a global $C^{r}$-expansion of order $N$. That is, there exist smooth
functions $b_{0},\ldots,b_{N}$ on $X$ such that
$B^{\mathrm{orb}}_{k}=b_{0}k^{n}+b_{1}k^{n-1}+\cdots+b_{N}k^{n-N}+O(k^{n-N-1}),$
where the $O(k^{n-N-1})$ term is to be taken in the $C^{r}$-norm. Furthermore
the $b_{j}$ are universal polynomials in the constants $c_{i}$ and the
derivatives of the Kähler metric $\omega$; in particular
$b_{0}=\sum_{i}c_{i}\quad\text{ and }\quad
b_{1}=\sum_{i}c_{i}\left(ni+\frac{1}{2}\operatorname{Scal}(\omega)\right),$
where $\operatorname{Scal}(\omega)$ is the scalar curvature of $\omega$.
###### Remark 9.
If condition (8) holds for some $m$, then it also holds if $m$ is replaced by
any factor; hence for the Theorem it is sufficient to assume it holds when
$m=\operatorname{ord}(X)$ is the lowest common multiple of the orders of the
stabiliser groups of all points of $X$. So in particular, the $N=r=0$ case
yields a top-order $C^{0}$-expansion for $B_{k}^{\mathrm{orb}}$ when $c_{i}=1$
for $i=1,\ldots,\operatorname{ord}(X)$ and $c_{i}=0$ otherwise. It turns out
(Lemma 5) that (8) is equivalent to asking that the function
$\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at each nontrivial $m$-th
root of unity. Thus there is little loss in assuming the $c_{i}$ are defined
by
$\sum_{i}c_{i}z^{i}:=(z^{m-1}+z^{m-2}+\cdots+1)^{N+r+1}.$
In fact the condition (8) is natural in the sense that it is also necessary
for the existence of an expansion; see Remark 8. Also, if $X$ is a manifold
then there is no condition on the $c_{i}$, and the Theorem is equivalent to
the expansion (1) for manifolds stated above.
###### Remark 10.
By a $C^{r}$-expansion of order $N$ we mean there is a constant $C_{r,N}$ such
that
$\left\|B_{k}^{\mathrm{orb}}-\left(b_{0}k^{n}+b_{1}k^{n-1}+\dots+b_{N}k^{n-N}\right)\right\|_{C^{r}}\leq
C_{r,N}k^{n-N-1},$
where the norm $\|f\|_{C^{r}}$ is the sum of the supremum norms of $f$ and its
first $r$ derivatives over all of $X$ in the orbifold sense (i.e. we measure
the norm of derivatives using the metric upstairs on orbifold charts). The
Theorem generalises to hermitian metrics $h$ on $L$ with curvature form
$2\pi\omega_{h}$ not necessarily equal to $\omega$, the only change being that
the coefficients in the expansion become
$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i},$ $\displaystyle
b_{1}$ $\displaystyle=$
$\displaystyle\frac{\omega_{h}^{n}}{\omega^{n}}\sum_{i}c_{i}[ni+\operatorname{tr}_{\omega_{h}}(\operatorname{Ric}(\omega))-\frac{1}{2}\operatorname{Scal}(\omega_{h})].$
(11)
Moreover the constants $C_{r,N}$ can be taken to be uniform as $(h,\omega)$
runs over a compact set.
The strategy of our proof is to use known results on manifolds. For global
quotients one can obtain what we want rather easily by averaging the expansion
upstairs. To apply this to general orbifolds we need to be able to work
locally, which is what the approach of Berman-Berndtsson-Sjöstrand [BBS] to
the expansion (1) allows us to to do. This starts by proving a local expansion
for the Bergman kernel, and then uses the Hörmander estimate to pass to a
global expansion. In Section 2 we average these local expansions to get local
orbifold expansion, and use it in Section 3 to get a global expansion in much
the same way as for manifolds.
One consequence is a Riemann-Roch expression for orbi-ample line bundles
obtained by integrating the expansion over $X$. The difference between this
and the general Kawasaki-Riemann-Roch formula for orbifolds [Kaw] is that the
periodic term coming from the orbifold singularities vanishes due to the
weighted sum over $i$. We denote the orbifold canonical bundle by
$K_{\mathrm{orb}}$ (defined, for instance, in [RT, Section 2.4]).
###### Corollary 12.
Let $X$ be an $n$-dimensional orbifold with cyclic quotient singularities and
$L$ be an orbi-ample line bundle. Let $m=\operatorname{ord}(X)$ and suppose
$\frac{1}{m}\sum_{i}c_{i}=\mathop{\sum{c_{i}}}_{i\equiv u\text{ mod
m}}\quad\text{ and
}\,\,\quad\frac{1}{m}\sum_{i}ic_{i}=\mathop{\sum{ic_{i}}}_{i\equiv u\text{ mod
m}}\quad\text{ for all }u.$
Then
$\sum_{i}c_{i}h^{0}(L^{k+i})=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2})$
with
$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}c_{i}}{n!}\int_{X}c_{1}(L)^{n},$ $\displaystyle
a_{1}$ $\displaystyle=$ $\displaystyle\frac{\sum
ic_{i}}{(n-1)!}\int_{X}c_{1}(L)^{n}-\frac{\sum_{i}c_{i}}{2(n-1)!}\int_{X}c_{1}(K_{\mathrm{orb}})c_{1}(L)^{n-1}.$
(13)
###### Proof.
This follows from $\int_{X}B_{k}\frac{\omega^{n}}{n!}=h^{0}(L^{k})$ and the
fact that the integral of the scalar curvature of any orbifold Kähler metric
is the topological quantity
$-\frac{1}{(n-1)!}\int_{X}c_{1}(K_{\mathrm{orb}})c_{1}(L)^{n-1}$. ∎
As an indication of the connection with constant scalar curvature metrics we
give an analogue of Donaldson’s Theorem [Don, Theorem 2] for orbifolds (for a
more applicable generalisation see [RT, Theorem 5.1]).
###### Corollary 14.
Suppose that (8) is satisfied for $N=1$ and $r=2$. For each $k\gg 0$ let
$h_{k}$ be a hermitian metric on $L$ with curvature $2\pi\omega_{k}$. Consider
the diagonal sequence of Bergman kernels $B_{k}^{\mathrm{orb}}$ associated to
the metrics $(h_{k},\omega_{k})$ and the power $L^{k}$ of $L$, and suppose
that for each $k$ this function is constant over $X$. If also $\omega_{k}$
converges in $C^{2}$ to a Kähler metric $\omega_{\infty}$ then
$\operatorname{Scal}(\omega_{\infty})$ is constant.
###### Proof.
Let $=\int_{X}\frac{c_{1}(L)^{n}}{n!}$ be the volume of $L$. Notice that since
the Bergman kernel is constant, by the previous corollary and integrating over
$X$ we have ${}_{k}^{\mathrm{orb}}=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2})$ with
$a_{i}$ as in (12) . Hence applying Theorem 7 to the pair
$(h_{k},\omega_{k})$,
$\frac{1}{}(a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2}))=B_{k}^{\mathrm{orb}}=b_{0}k^{n}+b_{1}k^{n-1}+O(k^{n-2}),$
where the $O(k^{n-2})$ term is independent of $(h_{k},\omega_{k})$ since they
lie in a compact set. Furthermore $b_{0}=\sum_{i}c_{i}=\frac{a_{0}}{}$ and
$b_{1}=\sum_{i}c_{i}(ni+\frac{1}{2}\operatorname{Scal}(\omega_{k}))$. Thus the
$k^{n}$ terms cancel, so $b_{1}$ tends to $a_{1}{-1}$ uniformly over $X$ as
$k$ tends to infinity. Since $\omega_{k}$ tends to $\omega_{\infty}$ in
$C^{2}$, $\operatorname{Scal}(\omega_{k})$ tends to
$\operatorname{Scal}(\omega_{\infty})$ and thus
$\operatorname{Scal}(\omega_{\infty})$ is constant. ∎
### Interpretation using Peaked Sections
Intuitively one can understand what is going on as follows. On a manifold
$B_{k}(x)$ is the pointwise norm square of a _peaked section_ $s_{x}$ at $x$
of unit $L^{2}$-norm. This is glued from zero outside a small ball of radius
$O(k^{-1/2})$ centred on $x$ and the standard local model (with Gaussian norm)
inside the ball. (If $s_{x}$ is constructed to have zero $L^{2}$-inner product
with any section vanishing at $x$ then the Bergman kernel at $x$ is
_precisely_ $|s_{x}(x)|^{2}$; if not then this is still a good enough
approximation for the discussion here.)
To describe the orbifold case consider the simple model consisting of a single
orbifold chart $\mathbb{C}\to\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$. Suppose
that $x$ is a smooth point, which in this model case is any point other than
the origin. Then for large $k$ we can pick a ball of radius $O(k^{-1/2})$
centred at $x$ and disjoint from the origin. It therefore admits a copy
upstairs in $\mathbb{C}$, i.e. a ball around one of its preimages. Then just
as in the smooth case, there is a peaked section supported in this ball, and
moving it around $\mathbb{C}$ by the group action gives an invariant section
with $m$ peaks. This can then be equivariantly glued to zero, giving an
invariant section that has $L^{2}$-norm approximately $m$, but considered as a
section downstairs it has norm $1$ (since the volume downstairs is defined to
be the volume upstairs divided by the order of the chart). Therefore its
pointwise norm squared at $x$ is approximately the value of the Bergman
kernel.
Now instead consider an orbifold point, which here is the origin. A section
upstairs on $\mathbb{C}$, peaked about the origin, can be equivariant _only_
when $k\equiv 0$ mod $m$. Therefore for most $k$, the Bergman kernel will have
value zero at the origin. But when $k$ is a multiple of $m$, we get an
invariant section with $L^{2}$-norm 1 upstairs, and therefore $L^{2}$-norm
only $\frac{1}{m}$ when considered as an orbifold section downstairs.
Therefore we multiply the section by $\sqrt{m}$ to get one of unit
$L^{2}$-norm, and find that the Bergman kernel of the orbifold at the origin
is approximately $m$ times as big as at manifold points.
Thus when we sum the Bergman kernels over a period $L^{k},L^{k+1},\ldots$ of
length at least $m$ we stand a chance of them averaging out to something which
is again approximately constant to top order in $k$. The condition that they
do is that the average of the coefficients $c_{i}$ with which we make each
$L^{k+i}$ contribute is equal to the sum of the coefficients $c_{i}$ which
contribute at the origin (i.e. those for which $k+i\equiv 0$ mod $m$). But
this is precisely the $p=0$ condition in (8). A similar analysis for $p=1$
means that we can ensure that we smooth out all of the $i$-dependence of the
orbifold Bergman kernel except the part coming from scalar curvature.
Conventions: We refer to [RT] for detailed definitions and conventions. For
this paper we need only that an orbifold with cyclic stabiliser groups is an
analytic space covered by charts of the form $U\to U/G\subset X$, where $U$ is
an open set in $\mathbb{C}^{n}$ and $G$ is a finite cyclic group acting
effectively and _linearly_ on $U$; see for example [RT, Section 2.1]. It is
important for our applications that we allow orbifold structure in codimension
one. Thus if $m\geq 2$ then $\mathbb{C}/(\mathbb{Z}/m\mathbb{Z})$ is
considered a distinct orbifold from $\mathbb{C}$ even though they have the
same underlying space. Quantities on $X$ (hermitian metrics, Kähler metrics,
sections, supremum norms etc.) are always taken in the orbifold sense, as an
invariant quantity upstairs in orbifold charts. An orbifold line bundle is an
equivariant line bundle on orbifold charts; the gluing condition is described
in [RT].
Acknowledgements. We thank Florin Ambro, Bo Berndtsson and Yanir Rubenstein
for helpful conversations. We also wish to acknowledge our debt to the paper
[BBS], which allowed for a significant improvement of ours.
## Chapter 1 Bergman Kernels
From now on let $(X,\omega)$ be a compact $n$-dimensional Kähler orbifold and
$L$ be an orbifold line bundle with a hermitian metric $h$ whose curvature
form is $2\pi\omega$. By abuse of notation we denote the induced metric on
$L^{k}$ also by $h$, which along with the volume form defined by $\omega$
yields an $L^{2}$-inner product on $H^{0}(L^{k})$ given by
$(s,t)_{L^{2}}=\int_{X}(s,t)_{h}\frac{\omega^{n}}{n!}\quad\text{for }s,t\in
H^{0}(L^{k}).$
The Bergman kernel of $L^{k}$ is defined to be
$B_{k}(x)=\sum_{\alpha}|s_{\alpha}(x)|_{h}^{2}$, where $\\{s_{\alpha}\\}$ is
any $L^{2}$-orthonormal basis for $H^{0}(L^{k})$. Recall that the _reproducing
kernel_ of $L^{k}$ is the section $K=K_{k}$ of $L^{k}\boxtimes\bar{L}^{k}$ on
$X\times X$ given by
$K(y,x)=\sum_{\alpha}s_{\alpha}(y)\boxtimes\overline{s}_{\alpha}(x),$
and therefore
$B_{k}(x)=|K(x,x)|_{h}.$
More invariantly, for $x\in X$ let $K_{x}$ be the section of
$L^{k}\otimes\bar{L}_{x}^{k}$ on $X$ given by $K_{x}(y):=K(y,x)$, so that
$(s,K_{x})_{L^{2}}$ is an element of the line $L_{x}^{k}$ for any section $s$
of $L^{k}$. Then the defining property of the reproducing kernel is that this
should equal $s(x)\in L_{x}^{k}$:
$s(x)=(s,K_{x})_{L^{2}}\quad\text{ for all }s\in H^{0}(L^{k}).$
As discussed in the introduction, when $X$ is a manifold the functions $B_{k}$
admit a global $C^{\infty}$-expansion as $k$ tends to infinity. This has been
worked out and extended in various ways, e.g. [Fef, Tia, Zel, Cat, Rua, DLM];
moreover the associated functions $b_{j}$ are geometric and depend on the
curvature of $\omega$ [Lu]. Our proof is based on the approach [BBS] that
starts by constructing a local expansion, as we now describe.
Let $U$ be an open ball in $\mathbb{C}^{n}$ centred at the origin, and
$\phi\colon U\to\mathbb{R}$ be smooth and strictly plurisubharmonic. Setting
$\omega=\frac{i}{2\pi}\partial\bar{\partial}\phi$, define an inner product on
the space of smooth functions by
$(u,v)^{2}_{\phi}=\int_{U}u\bar{v}\,e^{-\phi}\frac{\omega^{n}}{n!},$
and let $H(U)_{\phi}$ be the space of holomorphic functions of finite norm.
(The reader should of course think of this situation as arising when $U$ is a
chart of a manifold over which the hermitian line bundle $L|_{U}$ has
curvature $2\pi\omega$ and a holomorphic trivialisation of norm
$e^{-\phi/2}$.) Let $\chi$ be a smooth cutoff function supported on $U$ that
takes the value $1$ on $\frac{1}{2}U$.
###### Definition 1.
We say that a sequence $\mathcal{K}_{k},\ k\gg 0,$ of smooth functions on
$U\times U$ are _local reproducing kernels_ mod $O(k^{-N-1})$ for
$H(U)_{k\phi}$ if there is a neighbourhood $U_{0}\subset U$ of the origin such
that for any $x\in U_{0}$ and any $u\in H(U)_{k\phi}$,
$u(x)=(\chi
u,\mathcal{K}_{k,x})_{k\phi}+O(k^{-N-1}e^{k\phi(x)/2})\|u\|_{k\phi},$
where $\mathcal{K}_{k,x}(y):=\mathcal{K}_{k}(y,x)$, and the
$O(k^{-N-1}e^{k\phi(x)/2})$ term is uniform on $U_{0}$.
So whereas the reproducing kernels $K_{k}$ are globally and uniquely defined,
the local kernels $\mathcal{K}_{k}$ are far from unique. The next theorem
exhibits preferred ones, which will be shown later to be local approximations
to the global kernels.
We need the notion of a symmetric almost sesqui-holomorphic extension
$\psi\colon U\times U\to\mathbb{C}$ of $\phi$. That is,
$\psi(x,y)=\overline{\psi(y,x)},\ \psi(x,x)=\phi(x),$ and
$\bar{\partial}(\psi(x,\bar{y}))$ vanishes to all orders on $\\{x=y\\}$, i.e.
$D^{\alpha}(\bar{\partial}(\psi(x,\bar{y})))|_{\\{x=y\\}}=0$ for all $\alpha$.
(This notion is discussed in [BBS, Section 2.6] and [Rub, Section 3.3.3.1]; if
$\phi$ is analytic then one can take $\psi$ to be holomorphic in the first
variable and anti-holomorphic in the second.) By shrinking $U$ if necessary we
can ensure there is a $\delta>0$ such that
$\operatorname{Re}(\psi(x,y))\leq\phi(x)/2+\phi(y)/2-\delta\|x-y\|^{2}$ (2)
for $x,y\in U$ [BBS, Equation 2.7], [Rub, Lemma 3.8]. We write
$f=O(k^{-\infty})$ to mean $f=O(k^{-M})$ for any $M$.
###### Theorem 3 (Berman-Berndtsson-Sjöstrand).
For fixed $N,r\geq 0$ there exist smooth functions $\tilde{b}_{j}$ defined on
$U\times U$ such that
$\mathcal{K}_{k}(y,x)=\left(k^{n}+\tilde{b}_{1}(y,x)k^{n-1}+\dots+\tilde{b}_{N+r}(y,x)k^{n-N-r}\right)e^{k\psi(y,x)}$
is a local reproducing kernel mod $O(k^{-N-r-1})$ for $H(U)_{k\phi}$. Each
$\tilde{b}_{j}$ can be written as a polynomial in the derivatives
$\partial_{x}^{\alpha}\bar{\partial}^{\beta}_{y}\psi(x,y)$, and in particular
$\tilde{b}_{1}(x,x)=\frac{1}{2}\operatorname{Scal}(\omega)$.
Moreover, if $D_{x,y}$ is a differential operator of any order in $x$ and $y$
then $e^{-k(\phi(x)/2+\phi(y)/2)}D_{x,y}\bar{\partial}_{x}\mathcal{K}_{k}$ and
$e^{-k(\phi(x)/2+\phi(y)/2)}D_{x,y}\partial_{y}\mathcal{K}_{k}$ are both
$O(k^{-\infty})$.
###### Proof.
This is [BBS, Proposition 2.5 and Section 2.4] when $\phi$ is analytic, and
generalised to the smooth case in _loc. cit._ Proposition 2.7. (The cited work
is stated in the case that $U$ is a ball of radius $1$, but case for general
radius follows immediately on rescaling; there is also an extra factor of
$\pi^{-n}$ due to differences in conventions for the curvature form.) ∎
Observe that if a finite group $G$ acts on $U$ preserving $\phi$ then we can
take $\psi$ and $\chi$ to be $G$-invariant, in which case
$\mathcal{K}_{k}(\zeta x,\zeta x)=\mathcal{K}_{k}(x,x)$ for all $\zeta\in G$.
## Chapter 2 Local expansion
We next extend the local expansion of the previous section to orbifold charts,
which begins by averaging the local reproducing kernels $\mathcal{K}_{k}$.
Building on the notation of the previous section, suppose that a cyclic group
$G$ of order $m$ acts linearly and faithfully on $U$ and that our
plurisubharmonic function $\phi$ and cutoff function $\chi$ are invariant. The
inner product on the space of functions is replaced with
$(u,v)_{\phi,m}=\frac{1}{m}\int_{U}u\bar{v}\,e^{-\phi}\frac{\omega^{n}}{n!},$
and $H(U)_{\phi,m}$ is the space of holomorphic functions of finite norm. Fix
a generator $\zeta$ of $G$ and suppose we are given a character of $G$ which
maps $\zeta$ to a primitive $m$-th root of unity $\lambda$. We say a function
$u$ on $U$ _has weight $j$_ if $u(\zeta x)=\lambda^{j}u(x)$ for all $x\in U$.
(The reader should now think of $U$ as arising from an orbifold chart $U\to
U/G$, and the character as describing the action of $G$ on $L|_{U}$ under some
(noninvariant) trivialisation; the condition that $\lambda$ is primitive comes
from the local condition for orbi-ample line bundles, and the sections of
$L^{k}|_{U}$ now arise from functions on $U$ of weight $k$ mod $m$.)
Let
$\mathcal{K}_{k}=(k^{n}+\tilde{b}_{1}k^{n-1}+\cdots+\tilde{b}_{N+r}k^{n-N-r})e^{k\psi}$
be the preferred local reproducing kernel for $H(U)_{k\phi}$ from Theorem 3
and define
$\mathcal{K}_{k}^{\mathrm{av}}(y,x):=\frac{1}{m}\sum_{u,v=0}^{m-1}\lambda^{k(v-u)}\mathcal{K}_{k}(\zeta^{u}y,\zeta^{v}x).$
(1)
One sees that $\mathcal{K}_{k}^{\mathrm{av}}$ has weight $k$ in the first
variable and weight $-k$ in the second variable (as indeed it must if it is to
be a section of $L^{k}|_{U}\boxtimes\bar{L}^{k}|_{U}$).
###### Lemma 2.
$\mathcal{K}_{k}^{\mathrm{av}}$ is a local reproducing kernel mod
$O(k^{-N-r-1})$ for the subspace of $H(U)_{k\phi,m}$ consisting of functions
that have weight $k$.
###### Proof.
Let $u$ have weight $k$. Then by the change of variables
$y^{\prime}=\zeta^{u}y$ and the invariance of $\phi$ and $\chi$ we have, up to
terms of order $O(e^{k\phi(x)/2}k^{-N-r-1})\|u\|_{k\phi}$,
$\int_{U}\chi(y)u(y)\overline{\lambda^{k(v-u)}\mathcal{K}_{k,\zeta^{v}x}(\zeta^{u}y)}e^{-k\phi(y)}\frac{\omega^{n}}{n!}=\lambda^{-kv}u(\zeta^{v}x)=u(x),$
so
$(\chi
u,\mathcal{K}_{k,x}^{\mathrm{av}})_{k\phi,m}=\frac{1}{m}\int_{U}\chi(y)u(y)\overline{\mathcal{K}_{k,x}^{\mathrm{av}}}(y)e^{-k\phi(y)}\frac{\omega^{n}}{n!}=\frac{1}{m^{2}}\sum_{u,v=0}^{m-1}u(x)=u(x).$
∎
We show in the next section that downstairs on the orbifold, the Bergman
kernel $B_{k}$ is locally approximated by
$\mathcal{K}_{k}^{\mathrm{av}}(x,x)e^{-k\phi(x)}$, and thus the weighted
Bergman kernel $B^{\mathrm{orb}}_{k}=\sum_{i}c_{i}B_{k+i}$ is approximated by
$\mathcal{B}_{k}^{\mathrm{orb}}(x):=\sum_{i}c_{i}\mathcal{K}_{k+i}^{\mathrm{av}}(x,x)e^{-(k+i)\phi(x)},$
where, we recall, the sum is over a fixed finite index of nonnegative integers
$i$.
###### Theorem 3.
Suppose the $c_{i}$ satisfy
$\frac{1}{m}\sum_{i}i^{p}c_{i}=\sum_{i\equiv u}i^{p}c_{i}\quad\text{for }0\leq
u\leq m-1,0\leq p\leq N+r.$ (4)
Then there is a local $C^{r}$-expansion of $\mathcal{B}_{k}^{\mathrm{orb}}$ of
order $N$
$\mathcal{B}_{k}^{\mathrm{orb}}(x)=b_{0}(x)k^{n}+b_{1}(x)k^{n-1}+\cdots+b_{N}(x)k^{n-N}+O(k^{n-N-1})$
on $U$. Moreover the $b_{j}$ depend only on $c_{i}$ and the derivatives of the
metric; in particular $b_{0}=\sum_{i}c_{i}$ and
$b_{1}=\sum_{i}c_{i}(ni+\frac{1}{2}\operatorname{Scal}(\omega))$.
Our proof requires a reformulation of the condition made on the $c_{i}$.
###### Lemma 5.
The constants $c_{i}$ satisfy (4) if and only if the function
$\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at every $m$-th root of unity
other than $1$.
###### Proof.
If (4) holds then for each $0\leq p\leq N+r$ the quantity $c=\sum_{i\equiv
u}i^{p}c_{i}$ is independent of $u$. Hence if $\sigma^{m}=1$ with $\sigma\neq
1$ then
$\sum_{i}i^{p}c_{i}\sigma^{i}=\sum_{u=0}^{m-1}\sum_{i\equiv
u}i^{p}c_{i}\sigma^{i}=\sum_{u=0}^{m-1}\sigma^{u}\sum_{i\equiv
u}i^{p}c_{i}=c\sum_{u=0}^{m-1}\sigma^{u}=0,$
proving that $\sum_{i}c_{i}z^{i}$ has a root of order $N+r+1$ at $\sigma$. For
the converse let $\sigma$ be a primitive root of unity, so the hypothesis is
that $\sum_{i}i^{p}c_{i}\sigma^{si}=0$ for $1\leq s\leq m-1$ and $0\leq p\leq
N+r$, so for any given $u$,
$\frac{1}{m}\sum_{i}i^{p}c_{i}=\frac{1}{m}\sum_{s=0}^{m-1}\sigma^{-su}\sum_{i}i^{p}c_{i}\sigma^{si}=\frac{1}{m}\sum_{i}i^{p}c_{i}\sum_{s=0}^{m-1}\sigma^{(i-u)s}=\sum_{i\equiv
u}i^{p}c_{i}.$
∎
###### Proof of Theorem 3.
Write
$\displaystyle\mathcal{B}_{k}^{\mathrm{orb}}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{i}c_{i}\sum_{u,v=0}^{m-1}\lambda^{(k+i)(v-u)}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{v}x)e^{-(k+i)\phi(x)}$
$\displaystyle=$ $\displaystyle S_{1}+S_{2},$
where $S_{1}$ consists of the terms with $u=v$ and $S_{2}$ consists of the
terms with $u\neq v$. We show below that $S_{2}$ is $O(k^{n-N-1})$. Observe
that
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{i}c_{i}\sum_{u=0}^{m-1}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{u}x)e^{-(k+i)\phi(x)}$
$\displaystyle=$
$\displaystyle\sum_{i}c_{i}\mathcal{K}_{k+i}(x,x)e^{-(k+i)\phi(x)},$
by the invariance of $\mathcal{K}_{k+i}$, and since each
$\mathcal{K}_{k+i}e^{-(k+i)\phi}$ is a polynomial in $k$ the same is true of
$S_{1}$. In fact,
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}\sum_{j=0}^{N}(k+i)^{n-j}\tilde{b}_{j}(x,x)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{N}b_{j}(x)k^{n-j},$
where
$b_{j}(x)=\sum_{q=0}^{j}\left(\tilde{b}_{q}(x,x)\binom{n-q}{j-q}\sum_{i}c_{i}i^{j-q}\right).$
In particular $b_{0}$ and $b_{1}$ are as claimed in the statement of the
Theorem.
Now to bound $S_{2}$, write $S_{2}=\frac{1}{m}\sum_{u\neq v}S_{u,v}$ where
$S_{u,v}=\sum_{i}c_{i}\mathcal{K}_{k+i}(\zeta^{u}x,\zeta^{v}x)\lambda^{(k+i)(v-u)}e^{-(k+i)\phi(x)}.$
Fixing $u\neq v$, let $\sigma=\lambda^{v-u}$ and note that since $\lambda$ is
primitive, $\sigma\neq 1$. Furthermore set
$\eta=\eta(x)=e^{\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x)},$
so
$\displaystyle S_{u,v}$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}\sum_{j=0}^{N}(k+i)^{n-j}\tilde{b}_{j}(\zeta^{u}x,\zeta^{v}x)\sigma^{k+i}\eta^{k+i}$
$\displaystyle=$
$\displaystyle\sigma^{k}\sum_{j=0}^{N}k^{n-j}\sum_{q=0}^{j}\tilde{b}_{q}(\zeta^{u}x,\zeta^{v}x)\binom{n-q}{j-q}\sum_{i}c_{i}i^{j-q}\sigma^{i}\eta^{k+i}.$
Therefore to prove the theorem it is sufficient to show that for $0\leq l\leq
N$,
$u_{l,k}:=\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{k+i}=O(k^{l-N-1})\text{ in
}C^{r}.$ (7)
In fact since $du_{l,k}=\frac{d\eta}{\eta}(ku_{l,k}+u_{l+1,k})$, one sees by
induction that it is sufficient to prove $u_{l,k}=O(k^{l-N-r-1})$ in $C^{0}$.
To this end write
$\displaystyle u_{l,k}$ $\displaystyle=$
$\displaystyle\left[\frac{\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{i}}{(\eta-1)^{N+r-l+1}}\right](\eta-1)^{N+r-l+1}\eta^{k}.$
From Lemma 5 the function $\eta\mapsto\sum_{i}c_{i}i^{l}\sigma^{i}\eta^{i}$
has a root of order $N+r-l+1$ at $\eta=1$, and so the term is square brackets
is bounded. So it is sufficient to prove the following:
_Claim:_ Let $s\geq 1$ and $u\neq v$. Then
$(\eta-1)^{s}\eta^{k}=O(k^{-s}).$
_Proof:_ Since $G$ acts linearly on $U$, so we can write $U=U_{1}\times U_{2}$
where the action is faithful on $U_{1}$ and trivial on $U_{2}$. For $x\in U$
write $x=(x_{1},x_{2})$ under this decomposition. Then there exist positive
constants $c,c^{\prime}$ such that
$c\|x_{1}\|^{2}\leq\|\zeta^{u}x-\zeta^{v}x\|^{2}\leq c^{\prime}\|x_{1}\|^{2}$
for all $x\in U$. Hence from (2), there is a $\delta^{\prime}>0$ such that
$\operatorname{Re}(\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x))\leq-\delta^{\prime}\|x_{1}\|^{2}$
for all $x\in U$. If $x_{1}=0$ then $\eta=1$ and we are done. Assuming
$x_{1}\neq 0$, recall that if $z$ is a complex number with
$\operatorname{Re}z<0$ then
$|e^{kz}|\leq\frac{s!}{(-k\operatorname{Re}z)^{s}}$ for all $k$. Thus there is
a constant $C$ such that
$\left|(\eta(x)-1)^{s}\eta(x)^{k}\right|\leq\frac{C}{k^{s}}\left|\frac{\eta(x)-1}{\|x_{1}\|^{2}}\right|^{s}.$
But $\psi(\zeta^{u}x,\zeta^{v}x)-\phi(x)=O(\|x_{1}\|^{2})$, so
$\frac{\eta(x)-1}{\|x_{1}\|^{2}}$ is bounded and the claim follows. ∎
###### Remark 8.
Conversely, suppose that $\mathcal{B}_{k}^{\mathrm{orb}}$ admits an asymptotic
expansion in $C^{0}$ of order $N$ at the point $x=0$ in $U$ which is fixed by
the group action. We will show that the $c_{i}$ must satisfy the conditions
(4) for $r=0$.
Since $\lambda$ is primitive and sections of orbi-ample line bundles vanish at
orbifold points,
$\mathcal{K}_{k+i}^{\mathrm{av}}(0,0)=\left\\{\begin{array}[]{ll}m\mathcal{K}_{k+i}(0,0)&\text{
if }k+i\equiv 0\text{ mod }m,\\\ 0&\text{ otherwise.}\end{array}\right.$
Therefore
$\displaystyle\mathcal{B}_{k}^{\mathrm{orb}}(0)$ $\displaystyle=$
$\displaystyle
m\sum_{i\equiv-k}c_{i}\mathcal{K}^{\mathrm{av}}_{k+i}(0,0)=m\sum_{i\equiv-k}c_{i}\sum_{p=0}^{N}\tilde{b}_{p}(0,0)(k+i)^{n-p}$
$\displaystyle=$ $\displaystyle\sum_{p=0}^{N}b_{p}k^{n-p},$
where
$b_{p}=\sum_{q=0}^{p}\left(\tilde{b}_{q}(0,0)\binom{n-q}{p-q}\sum_{i\equiv-k}c_{i}i^{p-q}\right).$
(9)
Each $b_{p}$ is periodic in $k$; the existence of an asymptotic expansion of
$\mathcal{B}_{k}^{\mathrm{orb}}(0)$ implies that in fact each $b_{p}$ is
independent of $k$.
We now use induction on $p$ to show the conditions (4), i.e. that
$\sum_{i\equiv-k}c_{i}i^{p}$ is also independent of $k$. For $p=0$ the sum (9)
reduces to $b_{0}=\sum_{i\equiv-k}c_{i}$ since $\tilde{b}_{0}(0,0)=1$.
Therefore $b_{0}$’s independence of $k$ implies the $p=0$ case of (4). For
general $p$ the sum (9) is $\binom{n}{p}\sum_{i\equiv-k}c_{i}i^{p}$ plus terms
shown inductively to be independent of $k$. Therefore we recover the
conditions (4) for $r=0$.
## Chapter 3 Global Expansion
We will require the following standard estimate, which we prove for
completeness. At first we do not require $X$ to be compact.
###### Lemma 1.
Let $X$ be a Kähler manifold of dimension $n$. There exists a constant $C$
such that for any $p\in X$ we have $|f(p)|\leq
C(a\|\bar{\partial}f\|_{C^{0}}+a^{-n}\|f\|_{L^{2}})$ for all smooth functions
$f$ and all sufficiently small $a>0$. Moreover if $X$ is compact then we can
choose $a>0$ uniformly over all $p\in X$.
###### Proof.
We first prove the one dimensional case. Let $\epsilon=\sup|\bar{\partial}f|$,
pick a local coordinate $z=re^{i\theta}$ about the point $p=0$, and let
$D_{a}$ denote the ball $|z|=r\leq a$ in this coordinate. By the fundamental
theorem of calculus,
$\frac{1}{2\pi a}\int_{\partial D_{a}}\frac{\partial
f}{\partial\theta}d\theta=0.$ (2)
We know that
$\bar{\partial}f=\frac{1}{2}\left(\frac{\partial f}{\partial
r}+\frac{i}{r}\frac{\partial f}{\partial\theta}\right)(dr-ird\theta)$
has a pointwise bound on its norm of $O(\epsilon)$. Here we measure norms of
1-forms in the standard metric, which by the compactness of $X$ is boundedly
close to the Kähler metric in $D_{a}$.
Then $\frac{1}{r}\frac{\partial f}{\partial\theta}$ is within $\epsilon$ of
$i\frac{\partial f}{\partial r}$, and (2) gives
$\frac{1}{2\pi}\left|\int_{\partial D_{a}}\frac{\partial f}{\partial
r}d\theta\right|=\left|\frac{1}{2\pi}\frac{d}{da}\int_{\partial
D_{a}}fd\theta\right|\leq\epsilon.$
Integrating with respect to $a$ gives the estimate
$|f(0)|\leq\frac{1}{2\pi}\left|\int_{\partial
D_{a}}fd\theta\right|+a\epsilon.$
Multiplying by $a$ and integrating again yields
$\frac{a^{2}}{2}|f(0)|\leq\frac{1}{2\pi}\int_{D_{a}}|f|rdrd\theta+\frac{a^{3}}{3}\epsilon.$
By the Cauchy-Schwarz inequality this gives the desired bound on $|f(0)|$ in
terms of the $L^{2}$-norm of $f$ and the pointwise supremum of
$|\bar{\partial}f|$.
Now we pass to the general case. Pick local coordinates $z_{i}$ in which $p$
is the origin, and apply the above argument over one dimensional discs in the
$z_{n}$ direction to bound
$|f(z_{1},\ldots,z_{n-1},0)|\leq\frac{1}{\pi a^{2}}\int_{|z_{n}|\leq
a}|f(z_{1},\ldots,z_{n_{1}},z_{n})|+\frac{2a}{3}\epsilon.$ (3)
Similarly
$|f(z_{1},\ldots,z_{n-2},0,0)|\leq\frac{1}{\pi a^{2}}\int_{|z_{n-1}|\leq
a}|f(z_{1},\ldots,z_{n_{2}},z_{n-1},0)|+\frac{2a}{3}\epsilon,$
which by (3) can be bounded by
$|f(z_{1},\ldots,z_{n-2},0,0)|\leq\frac{1}{\pi^{2}a^{4}}\int_{|z_{n}|,|z_{n-1}|\leq
a}|f(z_{1},\ldots,z_{n})|+\frac{2a}{3}\epsilon+\frac{2a}{3}\epsilon.$
Inductively we find that
$|f(0)|\leq\frac{1}{\pi^{n}a^{2n}}\int_{|z_{i}|\leq
a}|f(z_{1},\ldots,z_{n})|+\frac{2na}{3}\epsilon.$
Again an application of Cauchy-Schwartz gives the result. ∎
###### Corollary 4.
Let $X$ be a compact Kähler orbifold and $L$ be an orbifold line bundle with
hermitian metric. There exists a constant $C$ such that for any section
$s\in\Gamma(L^{k})$ and $x\in X$ we have $|s(x)|\leq
C(k^{-\frac{1}{2}}\|\bar{\partial}s\|_{C^{0}}+k^{\frac{n}{2}}\|s\|_{L^{2}})$.
###### Proof.
This follows from the previous result, since we can work locally in an
orbifold chart where upstairs we have a smooth Kähler metric. We trivialise
$L$ (and so each $L^{k}$) upstairs with a holomorphic section of norm $1$ at
$x$ which is possibly not invariant under $G$. In a ball of radius
$a=k^{-1/2}$ about $x$, the hermitian metric is then boundedly close to taking
absolute values. Picking the coordinates in the above proof to be invariant
under the finite group gives, for $k\gg 0$, a bound in terms of an integral
upstairs over a $G$-invariant ball. The actual integral on the orbifold
differs from this by dividing by the order of the group; since this is finite
we get the same order bound. ∎
We now apply the results from the previous section to orbifold charts. As
usual, $X$ is an orbifold with cyclic quotient singularities, and $L$ is an
orbi-ample line bundle with hermitian metric with curvature $2\pi\omega$.
Suppose that $U\to U/G$ is a small orbifold chart in $X$, where $U$ is a small
ball centred at the origin in $\mathbb{C}^{n}$ and $G$ is cyclic of order $m$.
By the orbi-ampleness condition, $L$ has a trivialisation with weight $-1$
under the action of $G$ via an identification between $G$ and the $m$th roots
of unity. Then sections of $L^{k}|_{U}$ are given locally by holomorphic
functions $f$ of weight $k$. The pointwise norm of such a section is
$|f(z)|e^{-k\phi(z)/2}$, where the plurisubharmonic function $\phi$ is the
norm squared of the trivialising section.
As previously mentioned, the local reproducing kernel
$\mathcal{K}_{k}^{\mathrm{av}}(y,x)$ on $U\times U$ defined in (1) has weight
$k$ in the first variable, and weight $-k$ in the second and $\chi$ is an
invariant cutoff function supported on $U$ and identically 1 on
$\frac{1}{2}U$. Thus multiplying by the local trivialisation, we think of
$\chi\mathcal{K}_{k,x}^{\mathrm{av}}$ as a smooth section of
$L^{k}\otimes\bar{L}^{k}_{x}$. The local reproducing property for
$\mathcal{K}_{k}^{\mathrm{av}}$ proved in Lemma 2 says there is a
neighbourhood $U_{0}\subset U$ of the origin such that if $x\in U_{0}$ and
$t\in H^{0}(L^{k})\otimes\bar{L}^{k}_{x}$ then
$t(y)=(t,\chi\mathcal{K}^{\mathrm{av}}_{k,y})_{L^{2}}+O(k^{-N-r-1})\|t\|_{L^{2}}\quad\text{for
}y\in U_{0},$
where the error term is measured using the hermitian metric on
$L_{y}^{k}\otimes\bar{L}_{x}^{k}$. In what follows $U$ will be fixed, but
$U_{0}$ will be allowed to shrink as required. If $s\in
H^{0}(L\boxtimes\bar{L})$ then applying this to $t=s_{x}=s(\ \cdot\ ,x)$ gives
$s(y,x)=(s_{x},\chi\mathcal{K}^{\mathrm{av}}_{k,y})_{L^{2}}+O(k^{-N-r-1})\|s_{x}\|_{L^{2}}.$
(5)
Now recall that $K_{k}=\sum_{\alpha}s_{\alpha}\boxtimes\overline{s}_{\alpha}$
denotes the global reproducing kernel discussed at the start of Section 1. The
next proposition shows how this is approximated by the local reproducing
kernels $\mathcal{K}^{\mathrm{av}}_{k}$.
###### Proposition 6.
There is a neighbourhood $U_{0}\subset\frac{1}{4}U$ of the origin such that
$K_{k}|_{U_{0}\times U_{0}}=\mathcal{K}_{k}^{\mathrm{av}}+O(k^{n-N-r-1}).$
###### Proof.
This is identical to the manifold case [BBS, Theorem 3.1], and for convenience
we sketch the details. The Bergman kernel has the extremal characterisation
$B_{k}(x)=\sup_{\|s\|=1}|s(x)|^{2}$ where the supremum is over all sections of
$L^{k}$ of unit $L^{2}$-norm. Using this, it is shown in [Ber, Theorem 1.1]
that there is a constant $C$ such that $B_{k}(x)\leq Ck^{n}$ uniformly on $X$
(the cited work is for noncompact manifolds; what is important is that it is
local, so the inequality only improves if we apply it upstairs on an orbifold
chart and restrict to invariant sections).
By the Cauchy-Schwarz inequality applied to
$K_{k}=\sum_{\alpha}s_{\alpha}\boxtimes\bar{s}_{\alpha}$, we have
$|K_{k}(y,x)|_{h}\leq\sqrt{B_{k}(x)B_{k}(y)}=O(k^{n})$, so in particular
$\|K_{k,x}\|_{L^{2}}=O(k^{n})$ uniformly in $x$. Putting $s:=K_{k}$ (which is
holomorphic) into the local reproducing property, (5) gives
$\displaystyle K_{k}(y,x)$ $\displaystyle=$
$\displaystyle(K_{k,x},\chi\mathcal{K}_{k,y}^{\mathrm{av}})_{L^{2}}+O(k^{n-N-r-1})$
on $U_{0}\times U_{0}$. Then for $x,y\in U_{0}$ we can write
$\displaystyle K_{k}(y,x)$ $\displaystyle=$
$\displaystyle\mathcal{K}^{\mathrm{av}}_{k}(y,x)-\overline{w(y,x)}+O(k^{n-N-r-1}),$
(7)
where
$\displaystyle w(y,x)$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{K}^{\mathrm{av}}_{k}(y,x)}-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,x})_{L^{2}}$
$\displaystyle=$
$\displaystyle\chi(x)\mathcal{K}^{\mathrm{av}}_{k,y}(x)-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,x})_{L^{2}},$
as $\mathcal{K}_{k}^{\mathrm{av}}$ is hermitian. With the aim of bounding $w$,
define $u_{y}(\,\cdot\,):=w(y,\,\cdot\,)$ so
$u_{y}(\,\cdot\,)=\chi\mathcal{K}^{\mathrm{av}}_{k,y}-(\chi\mathcal{K}^{\mathrm{av}}_{k,y},K_{k,\,\cdot\,})_{L^{2}}.$
Notice that the inner product in this expression is precisely the projection
of $\chi\mathcal{K}^{\mathrm{av}}_{k,y}$ onto the space of holomorphic
sections of $L^{k}$. So said another way, $u_{y}$ is the $L^{2}$-minimal
solution of the equation
$\bar{\partial}u_{y}=\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y}).$ (8)
Now
$\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y})(t)=\mathcal{K}^{\mathrm{av}}_{k,y}(t)\bar{\partial}\chi(t)+\chi(t)\bar{\partial}\mathcal{K}^{\mathrm{av}}_{k,y}(t),$
which we claim is $O(k^{-\infty})$ in $C^{0}$. Since we are assuming that
$U_{0}\subset\frac{1}{4}U$, $\chi$ is identically $1$ on $2U_{0}$ and
supported on $U$ so $\bar{\partial}\chi(t)$ vanishes for $t\in 2U_{0}$ and for
$t\notin U$. On the other hand by (2) there is a $\delta_{1}>0$ such that
$\operatorname{Re}[2\psi(t,y)-\phi(t)-\phi(y)]\leq-\delta_{1}\|t-y\|^{2}$
on $U\times U$. If $t\in U\backslash 2U_{0}$, so that $t$ is a bounded
distance away from $y$, then by Theorem 3
$\displaystyle|\mathcal{K}_{k,y}(t)|_{h}^{2}$ $\displaystyle=$ $\displaystyle
O(k^{2n})e^{2k\psi(t,y)}e^{-\phi(t)}e^{-\phi(y)}$ $\displaystyle=$
$\displaystyle O(k^{2n})e^{k[2\psi(t,y)-\phi(t)-\phi(y)]}$ $\displaystyle\leq$
$\displaystyle O(k^{2n})e^{-\delta_{2}k}\ =\ O(k^{-\infty}).$
Thus $\mathcal{K}_{k,y}\bar{\partial}\chi=O(k^{-\infty})$ in $C^{0}$ on all of
$X$, and furthermore the $O(k^{-\infty})$ term is independent of $y\in U_{0}$.
Applying this to $\xi^{v}y$ as $v$ ranges over a period we get
$\mathcal{K}^{\mathrm{av}}_{k,y}\bar{\partial}\chi=O(k^{-\infty})$ as well.
Now from the second statement in Theorem 3,
$\chi\bar{\partial}\mathcal{K}^{\mathrm{av}}_{k,y}=O(k^{-\infty})$, hence
$\bar{\partial}(\chi\mathcal{K}^{\mathrm{av}}_{k,y})=O(k^{-\infty})$ in
$C^{0}$, and therefore in $L^{2}$ as well.
Thus applying the Hörmander estimate to sections of $L\otimes\bar{L}_{y}$ we
conclude that $\|u_{y}\|_{L^{2}}=O(k^{-\infty})$. Using this and that the
bound on $\bar{\partial}u_{y}$ is uniform in $x$ and $y$, Corollary 4 gives
the pointwise estimate
$|w(y,x)|=|u_{y}(x)|=O(k^{-\infty})\quad\text{ on }U_{0}\times U_{0}.$
Therefore (7) becomes $K_{k}=\mathcal{K}^{\mathrm{av}}_{k}+O(k^{n-N-r-1})$ on
$U_{0}\times U_{0}$, as required. ∎
So $\mathcal{K}_{k}^{\mathrm{av}}$ approximates $K_{k}$ to order
$O(k^{n-N-r-1})$ in the $C^{0}$ norm. We next use this to get a $C^{r}$
expansion at the expense of a factor of $O(k^{r})$.
###### Lemma 9.
Suppose $f_{k}(x,y)$ is a sequence of functions on $U_{0}\times U_{0}$ such
that $(\bar{\partial}_{x}f_{k})e^{-k(\phi(x)/2+\phi(y)/2)}$ and
$(\partial_{y}f_{k})e^{-k(\phi(x)/2+\phi(y)/2)}$ are both of order
$O(k^{-\infty})$. Suppose also that
$f_{k}(x,y)e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{q})$ uniformly on $U_{0}\times
U_{0}$. Then for any differential operator in $x$ and $y$ of total order $p$,
$(Df_{k})e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{q+p})\text{ on
}\frac{1}{2}U_{0}\times\frac{1}{2}U_{0}.$
###### Proof.
Fix $(x,y)\in\frac{1}{2}U_{0}\times\frac{1}{2}U_{0}$. Consider first the case
that $f(x,y)=f_{k}(x,y)$ is holomorphic in the first variable and anti-
holomorphic in the second. Suppose $x=(x_{1},\ldots x_{n})$,
$y=(y_{1},\ldots,y_{n})$ and let
$F(t)=f(x_{1},\ldots,t,\ldots,x_{n},y_{1},\ldots y_{n})$ and similarly
$\Phi(t)=\phi(x_{1},\ldots,t,\ldots,x_{n})$, where the $t$ lies in the $i$-th
coordinate. If $B$ denotes the ball of radius $k^{-1}$ around $x_{i}$, then
for $k$ sufficiently large $(x_{1},\ldots,t,\ldots,x_{n})\in U$ for all $t\in
B$. So applying the Cauchy formula to $B$ gives
$\displaystyle\left.\frac{\partial f}{\partial
x_{i}}\right|_{(x,y)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{-k(\phi(x)/2+\phi(y)/2)}$
$\displaystyle=\\!\\!$ $\displaystyle\frac{1}{2\pi i}\int_{\partial
B}\\!\\!\\!\frac{F(t)e^{-k(\Phi(t)/2+\phi(y)/2)}}{(t-x_{i})^{2}}e^{-k(\phi(x)/2-\Phi(t)/2)}dt$
$\displaystyle=$ $\displaystyle\int_{\partial
B}\frac{O(k^{q})}{(t-x_{i})^{2}}dt$ $\displaystyle=$ $\displaystyle
O(k^{q+1}),$
where the second inequality uses $\phi(x)-\Phi(t)=O(|x_{i}-t|)=O(k^{-1})$ on
$\partial B$, so the exponential term is $O(1)$. The other derivatives are
treated similarly.
More generally if we assume only $\bar{\partial}_{x}f=O(k^{-\infty})$ then the
calculation above holds up to a term
$e^{-k(\phi(x)/2+\phi(y)/2)}\int_{B}\partial F/\partial\bar{t}\,dtd\bar{t}$
which is of order $O(k^{-\infty})$, so the conclusion still holds. ∎
Recall that the Bergman kernel is $B_{k}(x)=|K_{k}(x,x)|_{h}$.
###### Corollary 10.
For $k$ sufficiently large,
$B_{k}(x)=\mathcal{K}^{\mathrm{av}}_{k}(x,x)e^{-k\phi(x)}+O(k^{n-N-1})$ in
$C^{r}$ on $\frac{1}{2}U_{0}$. Moreover if $D$ is a differential operator of
order $p$ then
$DB_{k}=O(k^{n+p})\text{ in }C^{0}$ (11)
uniformly on $X$.
###### Proof.
Let $g_{k}$ be the local expression of $K_{k}$ in our trivialisation of
$L|_{U}$, so $B_{k}(x)=|K_{k}(x,x)|_{h}=g_{k}(x,x)e^{-k\phi(x)}$. Now
Proposition 6 restricted to the diagonal implies
$(g_{k}-\mathcal{K}_{k}^{\mathrm{av}})e^{-k\phi(x)}=O(k^{n-N-r-1})$ in $C^{0}$
on $U_{0}$. So we can apply Lemma 9 to
$f_{k}:=g_{k}-\mathcal{K}_{k}^{\mathrm{av}}$ to deduce that if $D$ is a
differential operator of order $q$ in $x$ and $y$ then
$D(g_{k}-\mathcal{K}_{k}^{\mathrm{av}})e^{-k(\phi(x)/2+\phi(x)/2)}=O(k^{n-N-r-1+q})$
on $\frac{1}{2}U_{0}$. Therefore taking the first $r$ derivatives of
$B_{k}(x)-\mathcal{K}_{k}^{\mathrm{av}}(x,x)e^{-k\phi(x)}=\big{(}g_{k}(x,x)-\mathcal{K}_{k}^{\mathrm{av}}(x,x)\big{)}e^{-k\phi(x)}$
with respect to $x$ using the Leibnitz rule proves the first statement.
For the second statement, Proposition 6 implies
$g_{k}e^{-k(\phi(x)/2+\phi(y)/2)}=O(k^{n})$ on $U_{0}\times U_{0}$. Thus from
Lemma 9 we get that if $D$ has order at most $q$ then
$(Dg_{k})e^{-k(\phi(x)/2+\phi(x)/2)}=O(k^{n+q})$ on $\frac{1}{2}U_{0}$. Hence
taking the derivatives of $B_{k}(x)=g_{k}(x,x)e^{-k\phi(x)}$ with the Leibnitz
rule gives $DB_{k}=O(k^{n+p})$ on $\frac{1}{2}U_{0}$ for any differential
operator of order $p$. So by compactness the same holds uniformly on all of
$X$. ∎
The proof of our main theorem is now immediate. Each $x$ is contained in some
open set $U_{0}$ for which Theorem 3 and Corollary 10 hold, and by compactness
there is a finite cover by such sets. Thus
$\displaystyle B_{k}^{\mathrm{orb}}(x)$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}B_{k+i}(x)=\sum_{i}c_{i}\mathcal{K}_{k+i}^{\mathrm{av}}(x,x)e^{-(k+i)\phi(x)}+O(k^{n-N-1})$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{N}b_{j}(x)k^{n-j}+O(k^{n-N-1})$
in $C^{r}$, as required.
In [RT] we use a variant of our main theorem, which for convenience we record
here.
###### Corollary 12.
Let $\gamma$ be a homogeneous polynomial in two variables of degree $d\leq N$,
and suppose that as usual the $c_{i}$ are chosen to satisfy (8). Then there is
a $C^{r}$-expansion
$\sum_{i}c_{i}\gamma(k,i)B_{k+i}=b_{0}k^{n+d}+b_{1}k^{n+d-1}+\cdots+O(k^{n-N+d-1}).$
Moreover if $\gamma(i,k)=Ak^{d}+Bk^{d-1}i+\cdots$ then
$b_{0}=A\sum_{i}c_{i}\quad\text{ and }\quad
b_{1}=\sum_{i}c_{i}(A[ni+\operatorname{Scal}(\omega)/2]+iB).$
###### Proof.
By linearity we may suppose $\gamma(i,k)=i^{a}k^{d-a}$ for some $a\leq d$.
Since $c_{i}$ satisfy (8) for $p=0,\ldots,N+r$, we get that
$c_{i}^{\prime}:=i^{a}c_{i}$ satisfy this condition for $p=0,\ldots,N+r-a$.
Applying our main Theorem with the constants $c_{i}^{\prime}$ proves the
corollary. ∎
###### Remark 13.
For example
$\sum_{i}(k+i)c_{i}B_{k+i}=b_{0}k^{n+1}+b_{1}k^{n}+\cdots$
where $b_{0}=\sum_{i}c_{i}$ and
$b_{1}=\sum_{i}c_{i}[(n+1)i+\operatorname{Scal}(\omega)/2]$.
## References
* [Bai] W. L. Baily. On the imbedding of $V$-manifolds in projective space. Amer. J. Math., 79:403–430, 1957.
* [BBS] R. Berman, B. Berndtsson, and J. Sjöstrand. A direct approach to Bergman kernel asymptotics for positive line bundles. Arkiv för Matematik, 46(2):197–217, 2008. arXiv:math/0506367.
* [Ber] R. Berman. Bergman kernels and local holomorphic Morse inequalities. Math. Z., 248(2):325–344, 2004. arXiv:math/0211235.
* [Cat] D. Catlin. The Bergman kernel and a theorem of Tian. In Analysis and geometry in several complex variables (Katata, 1997), Trends Math., pages 1–23. Birkhäuser Boston, Boston, MA, 1999.
* [DLM] X. Dai, K. Liu, and X. Ma. On the asymptotic expansion of Bergman kernel. J. Differential Geom., 72(1):1–41, 2006. arXiv:math/0404494.
* [Don] S. Donaldson. Scalar curvature and projective embeddings. I. J. Differential Geom., 59(3):479–522, 2001.
* [Fef] C. Fefferman. The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math., 26:1–65, 1974.
* [Kaw] T. Kawasaki. The Riemann-Roch theorem for complex $V$-manifolds. Osaka J. Math., 16(1):151–159, 1979.
* [Lu] Z. Lu. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math., 122(2):235–273, 2000. arXiv:math/9811126.
* [RT] J. Ross and R. P. Thomas. Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics. 2009, arXiv:0907.5214.
* [Rua] W.-D. Ruan. Canonical coordinates and Bergmann metrics. Comm. Anal. Geom., 6(3):589–631, 1998. arXiv:dg-ga/9610011.
* [Rub] Y. Rubenstein. Geometric quantization and dynamical constructions on the space of Kähler metrics. 2008, Ph.D. Thesis. Massachusetts Institute of Technology, Dept. of Mathematics.
* [Son] J. Song. The Szegö kernel on an orbifold circle bundle. arXiv:math/0405071.
* [Tia] G. Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32(1):99–130, 1990.
* [Zel] S. Zelditch. Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices, (6):317–331, 1998. arXiv:math-ph/0002009.
|
arxiv-papers
| 2009-07-30T17:27:44 |
2024-09-04T02:49:04.301817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Ross and R. P. Thomas",
"submitter": "Julius Ross",
"url": "https://arxiv.org/abs/0907.5215"
}
|
0907.5229
|
# Computation of the $p^{6}$ order chiral Lagrangian coefficients
from the underlying theory of QCD
Shao-Zhou Jiang1,2111Email:jsz@mails.tsinghua.edu.cn., Ying Zhang3222Email:
hepzhy@mail.xjtu.edu.cn., Chuan Li1,2333Email:lcsyhshy2008@yahoo.com.cn., and
Qing Wang1,2444Email: wangq@mail.tsinghua.edu.cn.555corresponding author
1Center for High Energy Physics, Tsinghua University, Beijing 100084, P.R.
China
2Department of Physics, Tsinghua University, Beijing 100084, P.R.
China666mailing address
3School of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R. China
###### Abstract
We present results of computing the $p^{6}$ order low energy constants in the
normal part of chiral Lagrangian both for two and three flavor pseudo-scalar
mesons. This is a generalization of our previous work on calculating the
$p^{4}$ order coefficients of the chiral Lagrangian in terms of the quark self
energy $\Sigma(p^{2})$ approximately from QCD. We show that most of our
results are consistent with those we can find in the literature.
###### pacs:
12.39.Fe, 11.30.Rd, 12.38.Aw, 12.38.Lg
††preprint: TUHEP-TH-09169
## I Introduction
Chiral Lagrangian for low lying pseudoscalar mesonsweinberg GS as the most
successful effective field theory is now widely used in various strong, weak
and electromagnetic processes. To match the increasing demand for higher
precision in low energy description of QCD, the applications of the low energy
expansion of the chiral Lagrangian is extended from early time discussions on
the leading $p^{2}$ and next to leading $p^{4}$ orders to present $p^{6}$
order. For the latest review, see Ref.Review . In the chiral Lagrangian, there
are many unknown phenomenological low energy constants (LECs) which appear in
front of each Goldstone field dependent operators and the number of the LECs
increases rapidly when we go to the higher orders of the low energy expansion.
For example for the three flavor case, the $p^{2}$ and $p^{4}$ order chiral
Lagrangian have 2 and 10 LECs respectively, while the normal part of $p^{6}$
order chiral Lagrangian have 90 LECs. Such a large number of LECs is very
difficult to fix from the experiment data. This badly reduces the predictive
power of the chiral Lagrangian and blur the check of its convergence. The area
of estimating $p^{6}$ order LECs is where most improvement is needed in the
future of higher order chiral Lagrangian calculations.
A way to increase the precision of the low energy expansion and improve the
present embarrassed situation is studying the relation between the chiral
Lagrangian and the fundamental principles of QCD. We expect that this relation
will be helpful for understanding the origin of these LECs and further offer
us their values. In previous paper WQ1 , based on a more earlier study of
deriving the chiral Lagrangian from the first principles of QCD WQ0 in which
LECs are defined in terms of certain Green’s functions in QCD, we have
developed techniques and calculated the $p^{2}$ and $p^{4}$ order LECs
approximately from QCD. Our simple approach involves the approximations of
taking the large-$N_{c}$ limit, the leading order in dynamical perturbation
theory, and the improved ladder approximation, thereby the relevant Green’s
functions relate to LECs are expressed in terms of the quark self energy
$\Sigma(p^{2})$. The result chiral Lagrangian in terms of the quark self
energy is proved equivalent to a gauge invariant, nonlocal, dynamical (GND)
quark modelWQ2 . By solving the Schwinger-Dyson equation (SDE) for
$\Sigma(p^{2})$, we obtain the approximate QCD predicted LECs which are
consistent with the experimental values. With these results, generalization of
the calculations to $p^{6}$ order LECs becomes the next natural step.
Considering that the algebraic derivations for those formulae to express LECs
in terms of the quark self energy at $p^{4}$ order are lengthy (they need at
least several months of handwork), it is almost impossible to achieve the
similar works for the $p^{6}$ order calculations just by hand. Therefore, to
realize the calculations for the $p^{6}$ order LECs, we need to computerize
the original calculations and this is a very hard task. The key difficulty
comes from that the formulation developed in Ref.det0 and exploited in
Ref.WQ1 not automatically keeps the local chiral covariance of the theory and
one has to adjust the calculation procedure by hand to realize the covariance
of the results. To match with the computer program, we need to change the
original formulation to a chiral covariant one. In Ref.covariant ; covariant1
; covariant2 , we have built and developed such a formulation, followed by
next several year’s efforts, we now successfully encode the formulation into
computer programs. With the help of these computer codes we can reproduce
analytical results on the computer originally derived by hand in Ref.WQ1
within 15 minutes now. This not only confirms the reliability of the program
itself, but also checks the correctness of our original formulae. Based on
these progresses, in this paper, we generalize our previous works on
calculating the $p^{4}$ order LECs to computing the $p^{6}$ order LECs of
chiral Lagrangian both for two and three flavor pseudo-scalar mesons. This
generalization not only produces new numerical predictions for the $p^{6}$
order LECs, but also forces us to reexamine our original formulation from a
new angle in dealing with $p^{2}$ and $p^{4}$ order LECs.
This paper is organized as follows: In Sec.II, we review our previous
calculations on the $p^{2}$ and $p^{4}$ order LECs. Then, in Sec.III, based on
the technique developed in Ref.covariant , we reformulate the original low
energy expansion used in Ref.WQ1 into a chiral covariant one suitable for
computer derivation. In Sec.IV, from present $p^{6}$ order viewpoint, we
reexamine the formulation we taken before and show that if we sum all higher
order anomaly part contributions terms together, their total contributions to
the normal part of the chiral Lagrangian vanish. This leads a change the role
of finite $p^{4}$ order anomaly part contributions which originally are
subtracted in the chiral Lagrangian in Ref.WQ1 and now must be used to cancel
divergent higher order anomaly part contributions. We reexhibit the numerical
result of the $p^{4}$ order LECs without subtraction of $p^{4}$ order anomaly
part contributions. In Sec.V, we present general $p^{6}$ order chiral
Lagrangian in terms of rotated sources and express the $p^{6}$ order LECs in
terms of the quark self energy. Sec.VI is a part where we give numerical
results for $p^{6}$ order LECs in the normal part of chiral Lagrangian both
for two and three flavor pseudo scalar mesons. In Sec. VII, we apply and
compare with our results to some individuals and combinations of LECs proposed
and estimated in the literature, checking the correctness of our numerical
predictions. Sec.VIII is a summary. In Appendices, we list some necessary
formulae and relations.
## II Review of the Calculations on the $p^{2}$ and $p^{4}$ Order LECs
Theoretically, the action of the chiral Lagrangian at large $N_{c}$ limit
derived from the first principle of QCD takes form WQ0
$\displaystyle S_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle-
iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Pi_{\Omega
c}]+iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J]+N_{c}\mathrm{Tr}[\Phi_{\Omega
c}\Pi^{T}_{\Omega c}]$
$\displaystyle+N_{c}\sum^{\infty}_{n=2}{\int}d^{4}x_{1}\cdots
d^{4}x_{n}^{\prime}\frac{(-i)^{n}(N_{c}g_{s}^{2})^{n-1}}{n!}\bar{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})\Phi^{\sigma_{1}\rho_{1}}_{\Omega
c}(x_{1},x^{\prime}_{1})\cdots\Phi^{\sigma_{n}\rho_{n}}_{\Omega
c}(x_{n},x^{\prime}_{n})+O(\frac{1}{N_{c}})\,$
in which $J_{\Omega}$ is external source $J$ including currents and densities
after Goldstone field dependent chiral rotation $\Omega$
$\displaystyle J_{\Omega}=[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][J+i\not{\partial}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]=\not{v}_{\Omega}+\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}\hskip
28.45274ptJ=\not{v}+\not{a}\gamma_{5}-s+ip\gamma_{5}\hskip
28.45274ptU=\Omega^{2}\;.~{}~{}~{}~{}$ (2)
$\Phi_{\Omega c}$ and $\Pi_{\Omega c}$ are two point rotated quark Green’s
function and interaction part of two point rotated quark vertex in presence of
external sources respectively, $\Phi_{\Omega c}$ is defined by
$\displaystyle\Phi_{\Omega
c}^{\sigma\rho}(x,y)\equiv\frac{1}{N_{c}}\langle\overline{\psi}_{\Omega}^{\sigma}(x)\psi_{\Omega}^{\rho}(y)\rangle=-i[(i\not{\partial}+J_{\Omega}-\Pi_{\Omega
c})^{-1}]^{\rho\sigma}(y,x)\hskip
56.9055pt\psi_{\Omega}(x)\equiv[\Omega(x)P_{L}+\Omega^{\dagger}(x)P_{R}]\psi(x)$
(3)
with subscript c denoting the classical field and $\psi(x)$ being light quark
fields.
$\bar{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})$
is effective gluon n-point Green’s function and $g_{s}$ is coupling constant
of QCD. It can be shown that the last term in the first line and the term in
the second line of the r.h.s. of Eq.(II) are independent of pseudo scalar
meson field $U$ or $\Omega$ and therefore are just irrelevant constants in the
effective action. While the second and third terms in the first line of the
r.h.s. of Eq.(II) are anomaly part contributions, since they represent the
variations of the path integral measure for light quark fields $\psi$. The
remaining first term is called normal part contributions which relies on
$\Pi_{\Omega c}$. The $\Phi_{\Omega c}$ and $\Pi_{\Omega c}$ are related by
the first equation of (3) and determined by
$\displaystyle[\Phi_{\Omega
c}+\tilde{\Xi}]^{\sigma\rho}+\sum^{\infty}_{n=1}{\int}d^{4}x_{1}d^{4}x^{\prime}_{1}\cdots{d^{4}}x_{n}d^{4}x^{\prime}_{n}\frac{(-i)^{n+1}(N_{c}g_{s}^{2})^{n}}{n!}\overline{G}^{\sigma\sigma_{1}\cdots\sigma_{n}}_{\rho\rho_{1}\cdots\rho_{n}}(x,y,x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})$
$\displaystyle\times\Phi_{\Omega
c}^{\sigma_{1}\rho_{1}}(x_{1},x^{\prime}_{1})\cdots\Phi_{\Omega
c}^{\sigma_{n}\rho_{n}}(x_{n},x^{\prime}_{n})=O(\frac{1}{N_{c}}),$ (4)
where $\tilde{\Xi}$ is a Lagrangian multiplier which insures the constraint
$\mathrm{tr}_{l}[\gamma_{5}\Phi_{\Omega c}^{T}(x,x)]=0$. Eq.(4) is the SDE in
presence of the rotated external source. In Ref.WQ1 , we have assumed the
solution of (4) approximately by
$\displaystyle\Pi^{\sigma\rho}_{\Omega
c}(x,y)=[\Sigma(\overline{\nabla}^{2}_{x})]^{\sigma\rho}\delta^{4}(x-y)\hskip
85.35826pt\overline{\nabla}^{\mu}_{x}=\partial^{\mu}_{x}-iv_{\Omega}^{\mu}(x)\;,$
(5)
where $\Sigma$ is the quark self energy which satisfy SDE (4) with vanishing
rotated external source. Under the ladder approximation, this SDE in Euclidean
space-time is reduced to the standard form of
$\displaystyle\Sigma(p^{2})-3C_{2}(R)\int\frac{d^{4}q}{4\pi^{3}}\frac{\alpha_{s}[(p-q)^{2}]}{(p-q)^{2}}\frac{\Sigma(q^{2})}{q^{2}+\Sigma^{2}(q^{2})}=0\;,$
(6)
where $C_{2}(R)$ is the second order Casimir operator of the quark
representation R, in our case, quark is belong to $SU(N_{c})$ fundamental
representation, therefore $C_{2}(R)=(N_{c}^{2}-1)/2N_{c}$ and in the large
$N_{c}$ limit, we will neglect the second term of it. $\alpha_{s}(p^{2})$ is
the running coupling constant of QCD which depends on $N_{c}$ and quark
flavor. With these approximations, the result action (II) of the chiral
Lagrangian becomes the GND model introduced in Ref.WQ2 ,
$\displaystyle S_{\mathrm{eff}}\approx
S_{\mathrm{GND}}+O(\frac{1}{N_{c}})\hskip 28.45274ptS_{\mathrm{GND}}\equiv-
iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]+iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J]\;.~{}~{}~{}~{}~{}~{}$
(7)
In which the third term at r.h.s. of (7) is independent of pseudo scalar field
$U$, therefore it only affects contact term of the chiral Lagrangian. In fact,
for the contact term part, we can take $\Omega=1$ in (7), then
$\displaystyle S_{\mathrm{eff}}\bigg{|}_{\mathrm{contact}}\approx-
iN_{c}\mathrm{Tr}\ln\\{i\not{\partial}+J-\Sigma[(\partial-
iv)^{2})]\\}+O(\frac{1}{N_{c}})\;.~{}~{}~{}~{}~{}~{}$ (8)
For the non-contact terms concerned in this paper, we can ignore the third
term at r.h.s. of (7) and the next key element is to compute term
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$.
The remaining term $\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]$ in our
previous work is obtained by further taking limit $\Sigma\rightarrow 0$ in
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$
777This will cause some confusions and we are going to discuss them in section
IV.. Since anomaly terms are at least the $p^{4}$ order and at this order,
anomaly is the well known Wess-Zumino terms which have no unknown LECs (In
Ref.WQma , we have derived such terms from QCD). All unknown LECs at $p^{2}$
and $p^{4}$ orders are in the normal part of chiral Lagrangian, so to
calculate the $p^{2}$ and $p^{4}$ orders LECs, we only need to discuss the
normal part of chiral Lagrangian which is in fact the real part of
$\mathrm{Tr}\ln(\cdots)$. With the help of Schwinger proper time method det0 ,
this real part in Euclidean space-time888Our extension from Minkovski space to
Euclidean space takes $x^{0}|_{M}\rightarrow-ix^{4}|_{E}$,
$x^{i}|_{M}\rightarrow x^{i}|_{M}$,
$\gamma^{0}|_{M}\rightarrow\gamma^{4}|_{E}$, $\gamma^{i}|_{M}\rightarrow
i\gamma^{i}|_{E}$, with $i=1,2,3$ being space indices and there
$\gamma_{E}^{\mu}$ are hermitian. $v^{\mu}_{\Omega},a^{\mu}_{\Omega}$
transform as $x^{\mu}$. $\gamma_{5}|_{M}\rightarrow\gamma_{5}|_{E}$,
$s|_{M}\rightarrow-s|_{E}$, $p|_{M}\rightarrow-p|_{E}$. with metric tensor
$g^{\mu\nu}=\mbox{diag}(1,1,1,1)$, can be written as
$\displaystyle\mathrm{ReTr}\ln[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\Sigma(-\bar{\nabla}^{2})]$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\ln\Big{[}[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\Sigma(-\bar{\nabla}^{2})]^{\dagger}[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\Sigma(-\bar{\nabla}^{2})]\Big{]}$
$\displaystyle=-\frac{1}{2}\lim_{\Lambda\to\infty}\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int{d^{4}x}~{}\mathrm{tr}\langle
x|\exp\bigg{[}-\tau[\bar{E}-(\bar{\nabla}-ia_{\Omega})^{2}+\Sigma^{2}(-\bar{\nabla}^{2})+\hat{I}_{\Omega}\Sigma(-\bar{\nabla}^{2})+\Sigma(-\bar{\nabla}^{2})\tilde{I}_{\Omega}-d\\!\\!\\!/\;\Sigma(-\bar{\nabla}^{2})]\bigg{]}|x\rangle$
$\displaystyle=-\frac{1}{2}\lim_{\Lambda\to\infty}\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int{d^{4}x}\int\frac{d^{4}k}{(2\pi)^{4}}~{}\mathrm{tr}\exp\bigg{[}-\tau[\bar{E}+(k+i\bar{\nabla}_{x}+a_{\Omega})^{2}+\Sigma^{2}((k+i\bar{\nabla}_{x})^{2})+\hat{I}_{\Omega}\Sigma((k+i\bar{\nabla}_{x})^{2})$
$\displaystyle\hskip
14.22636pt+\Sigma((k+i\bar{\nabla}_{x})^{2})\tilde{I}_{\Omega}-d\\!\\!\\!/\;\Sigma((k+i\bar{\nabla}_{x})^{2})]\bigg{]}\;,~{}~{}~{}~{}~{}~{}$
(9)
in which
$\displaystyle\bar{E}=\frac{i}{4}[\gamma_{\mu},\gamma_{\nu}]R^{\mu\nu}+\gamma_{\mu}d^{\mu}(s_{\Omega}-ip_{\Omega}\gamma_{5})+i\gamma_{\mu}[a^{\mu}_{\Omega}\gamma_{5}(s_{\Omega}-ip_{\Omega}\gamma_{5})+(s_{\Omega}-ip_{\Omega}\gamma_{5})a_{\Omega}^{\mu}\gamma_{5}]+s_{\Omega}^{2}+p_{\Omega}^{2}-[s_{\Omega},p_{\Omega}]i\gamma_{5}$
$\displaystyle d^{\mu}{\cal O}\equiv\partial^{\mu}{\cal
O}-i[v^{\mu}_{\Omega},{\cal O}]\hskip 8.5359pt({\cal O}=\mbox{any
operator})\hskip
19.91684ptR^{\mu\nu}\\!\\!=V_{\Omega}^{\mu\nu}\\!-i[a_{\Omega}^{\mu},a_{\Omega}^{\nu}]+(d^{\mu}a_{\Omega}^{\nu}-d^{\nu}a_{\Omega}^{\mu})\gamma_{5}\hskip
19.91684ptV_{\Omega,\mu\nu}\\!\\!=i[\bar{\nabla}_{\mu},\bar{\nabla}_{\nu}]\hskip
19.91684pt$
$\displaystyle\bar{\nabla}^{\mu}_{x}=\partial^{\mu}\\!\\!-iv_{\Omega}^{\mu}(x)\hskip
19.91684pt\hat{I}_{\Omega}=-ia\\!\\!\\!/_{\Omega}\gamma_{5}-s_{\Omega}-ip_{\Omega}\gamma_{5}\hskip
19.91684pt\tilde{I}_{\Omega}=-ia\\!\\!\\!/_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}\hskip
19.91684ptd\\!\\!\\!/\;\Sigma(-\bar{\nabla}^{2})=\gamma_{\mu}d^{\mu}\Sigma(-\bar{\nabla}^{2})\;.$
In (9), a cutoff $\Lambda$ is introduced into the theory to regularize the
possible ultraviolet divergences. In practical calculations, we treat it as
the physical cutoff of the theory. Taking the low energy expansion for (9), we
can finally express
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$ in
terms of power expansion of external sources with coefficients being $\Sigma$
dependent functions. Further vanishing $\Sigma$, we obtain
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]$. Then the r.h.s. of (7) is
expressed in terms of power expansion of rotated external sources, compare the
result with the parametrization of the effective action without applying the
equations of motion for pseudo scalar mesons,
$\displaystyle S_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x~{}\mathrm{tr}_{f}\bigg{[}F_{0}^{2}a_{\Omega}^{2}+F_{0}^{2}B_{0}s_{\Omega}-{\cal
K}_{1}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}[d_{\mu}a_{\Omega}^{\mu}]^{2}-{\cal K}_{2}^{(\mathrm{norm},\Pi_{\Omega
c}\neq
0)}(d^{\mu}a_{\Omega}^{\nu}-d^{\nu}a_{\Omega}^{\mu})(d_{\mu}a_{\Omega,\nu}-d_{\nu}a_{\Omega,\mu})$
(10) $\displaystyle+{\cal K}_{3}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}[a_{\Omega}^{2}]^{2}+{\cal K}_{4}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}a_{\Omega}^{\mu}a_{\Omega}^{\nu}a_{\Omega,\mu}a_{\Omega,\nu}+{\cal
K}_{5}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}a_{\Omega}^{2}\mathrm{tr}_{f}[a_{\Omega}^{2}]$ $\displaystyle+{\cal
K}_{6}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}a_{\Omega}^{\mu}a_{\Omega}^{\nu}\mathrm{tr}_{f}[a_{\Omega,\mu}a_{\Omega,\nu}]+{\cal
K}_{7}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}s_{\Omega}^{2}+{\cal
K}_{8}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}s_{\Omega}\mathrm{tr}_{f}[s_{\Omega}]+{\cal
K}_{9}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}p_{\Omega}^{2}$
$\displaystyle+{\cal K}_{10}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}p_{\Omega}\mathrm{tr}_{f}[p_{\Omega}]+{\cal
K}_{11}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}s_{\Omega}a_{\Omega}^{2}+{\cal
K}_{12}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}s_{\Omega}\mathrm{tr}_{f}[a_{\Omega}^{2}]-{\cal
K}_{13}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}V_{\Omega}^{\mu\nu}V_{\Omega,\mu\nu}$ $\displaystyle+i{\cal
K}_{14}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}V_{\Omega}^{\mu\nu}a_{\Omega,\mu}a_{\Omega,\nu}+{\cal
K}_{15}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}p_{\Omega}d_{\mu}a^{\mu}_{\Omega}\bigg{]}+O(p^{6})+U\mbox{-independent~{}source~{}terms}.$
We can read out $F_{0}^{2},B_{0}$ and
$\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}$ for $i=1,\ldots,15$
as functions of $\Sigma$. $\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}$s relate to the conventional $p^{4}$ order LECs through (25) of Ref.WQ1 .
A superscript ${}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}$ on each of
$\mathcal{K}_{i}$ denotes the property that when $\Pi_{\Omega c}=\Sigma=0$,
all $\mathcal{K}_{i}$ vanish, i.e.
$\displaystyle\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}=\mathcal{K}_{i}^{(\mathrm{norm})}-\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega
c}=0)}\hskip
56.9055pt\mathcal{K}_{i}^{(\mathrm{norm})}\stackrel{{\scriptstyle\Sigma=0}}{{====}}\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega
c}=0)}\hskip 56.9055pti=1,\ldots,15\;,$ (11)
where $\mathcal{K}_{i}^{(\mathrm{norm})}$ and
$-\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega c}=0)}$ are the contributions to
the effective action from the first, second and third terms in the r.h.s. of
(7) respectively. Replacing superscript ${}^{(\mathrm{norm},\Pi_{\Omega c}\neq
0)}$ with (norm) in the r.h.s. of (10), we obtain term
$-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$.
And replacing superscript ${}^{(\mathrm{norm},\Pi_{\Omega}\neq 0)}$ with
${}^{(\mathrm{norm},\Pi_{\Omega}=0)}$ and vanishing $F_{0}^{2}$ in the r.h.s.
of (10) , we obtain term
$-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]+iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J]$.
The result formulae for $F_{0}^{2}B_{0},F_{0}^{2}$ and
$\mathcal{K}_{i}^{(\mathrm{norm})}$ expressed in terms of $\Sigma$ are
explicitly given in (34), (35) and (36) in Ref.WQ1 .
With the analytical formulae for LECs of $F_{0}^{2},B_{0}$ and
$\mathcal{K}_{i}^{(\mathrm{norm},\Pi_{\Omega c}\neq 0)}$ for $i=1,\ldots,15$
as functions of $\Sigma$, we can suitably choose running coupling constant
$\alpha_{s}(p^{2})$, solve SDE (6) numerically obtaining quark self energy
$\Sigma$, then calculate the numerical values of all $p^{2}$ and $p^{4}$ order
LECs. To obtain the final numerical result in Ref.WQ1 , we have assumed
$F_{0}=f_{\pi}=93$MeV as input999Later we will use a changed value of
$F_{0}=87$MeV for two-flavour case. For detail, see the discussion of Eq.(58)
to fix the dimensional parameter $\Lambda_{\mathrm{QCD}}$ appear in running
coupling constant $\alpha_{s}(p^{2})$ and taken cutoff parameter $\Lambda$
appear in (9) equal to infinity and 1GeV respectively. The final obtained
values are consistent with those fixed phenomenologically.
## III Chiral Covariant Low Energy Expansion
Eq.(9) is the starting point of our reformulation in this section. In Ref.WQ1
, we expand (9) up to the $p^{4}$ order and obtain analytical result. This
expansion is not explicitly chiral covariant, since the operator
$\bar{\nabla}_{x}^{\mu}$ appears in the formula is not always covariant under
the local chiral symmetry transformations. For example, when
$\bar{\nabla}_{x}^{\mu}$ acts on a constant number 1, it gives
$\bar{\nabla}_{x}^{\mu}~{}1=-iv_{\Omega}^{\mu}(x)$ which is not covariant
since $v_{\Omega}^{\mu}(x)$ itself behaves as the gauge field in the local
chiral symmetry transformations. Only when they combined into commutators,
such as $[\bar{\nabla}_{x}^{\mu},\bar{\nabla}_{x}^{\nu}]$ or
$[\bar{\nabla}_{x}^{\mu},a_{\Omega}^{\nu}(x)]$, the covariance recovers back.
Therefore in the detail calculation, we need to confirm that all
$\bar{\nabla}_{x}^{\mu}$s appear in the result do can be arranged into some
commutators. This is a conjecture. In the original work of Ref.WQ1 , we have
found that this conjecture is valid up to some terms with coefficients being
expressed as integration over some total derivatives, i.e. form of $\int
d^{4}k\frac{\partial}{\partial k^{\mu}}g(k)$. If we ignore these total
derivative terms, up to order of $p^{4}$, we can explicitly prove the
conjecture. At the stage of our earlier works, we do not question the reason
that why we can drop out those total derivative terms (In fact, in Eq.(74) of
Ref.WQ0 , we have shown that in order to obtain the well-known Pagels-Stokar
formula, a total derivative term must be dropped out). This leads the further
discussions on the role of total derivative terms in the quantum field theory
covariant2 . Later in this section, we will give the correct reason of
dropping out those total derivative terms. Arranging various
$\bar{\nabla}_{x}^{\mu}$ into commutators is a very tricky and complex task
which is very hard to be achieved by computer. In order to computerize the
calculation, we need to find a way which can automatically arrange all
$\bar{\nabla}_{x}^{\mu}$s into some commutators. This leads the developments
given in Ref.covariant ; covariant1 ; covariant2 , where we have introduced
$\displaystyle
k^{\mu}+i\bar{\nabla}_{x}^{\mu}=e^{i\bar{\nabla}_{x}\cdot\frac{\partial}{\partial
k}}\Big{(}k^{\mu}+\tilde{F}^{\mu}(\bar{\nabla},\frac{\partial}{\partial
k})\Big{)}e^{-i\bar{\nabla}_{x}\cdot\frac{\partial}{\partial k}}\;,$ (12)
in which
$\displaystyle\tilde{F}^{\mu}(\bar{\nabla},\frac{\partial}{\partial k})$
$\displaystyle\equiv$
$\displaystyle-e^{Ad(-i\bar{\nabla}_{x}\\!\\!\cdot\\!\frac{\partial}{\partial
k})}\bigg{(}F\Big{[}Ad(i\bar{\nabla}_{x}\cdot\frac{\partial}{\partial
k})\Big{]}(i\bar{\nabla}_{x}^{\mu})\bigg{)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\nu\mu)\frac{\partial}{\partial
k^{\nu}}-\frac{i}{3}(\lambda\nu\mu)\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\nu}}-\frac{1}{8}(\rho\lambda\nu\mu)\frac{\partial^{3}}{\partial
k^{\rho}\partial k^{\lambda}\partial
k^{\nu}}+\frac{i}{30}(\sigma\rho\lambda\nu\mu)\frac{\partial^{4}}{\partial
k^{\sigma}\partial k^{\rho}\partial k^{\lambda}\partial k^{\nu}}$
$\displaystyle+\frac{1}{144}(\delta\sigma\rho\lambda\nu\mu)\frac{\partial^{5}}{\partial
k^{\delta}\partial k^{\sigma}\partial k^{\rho}\partial k^{\lambda}\partial
k^{\nu}}+O(p^{7})\;,$ $\displaystyle
F(z)=\sum^{\infty}_{n=2}\frac{z^{n-1}}{n!}\hskip
85.35826pt[Ad(B)]^{n}(C)\equiv[\underbrace{B,[B,\cdots,[B}_{\mbox{n
times}},C]\cdots]]\;,$
$\displaystyle(\mu_{n}\mu_{n-1}\cdots\mu_{2}\mu_{1})\equiv[\overline{\nabla}_{x}^{\mu_{n}},[\overline{\nabla}_{x}^{\mu_{n-1}},\cdots,[\overline{\nabla}_{x}^{\mu_{2}},\overline{\nabla}_{x}^{\mu_{1}}]\cdots]]\;,$
where the default set of Lorentz indices for
$(\mu_{n}\mu_{n-1}\cdots\mu_{2}\mu_{1})$ is the supperscripts, in some cases,
we need subscript, we will use $\underline{\mu}$ to denote the corresponding
subscript for $\mu$. Note that in present notation for
$(\mu_{n}\mu_{n-1}\cdots\mu_{2}\mu_{1})$, we don’t explicitly write
$\overline{\nabla}_{x}$s, but only their Greek superscripts for short. If we
use other symbols, such as $s_{\Omega}$ appeared in $(\mu s_{\Omega})$ and
$a_{\Omega}^{\nu}$ in $(\mu a_{\Omega}^{\nu})$, then we take definition that
$(\mu s_{\Omega})\equiv[\overline{\nabla}_{x}^{\mu},s_{\Omega}]$ and $(\mu
a_{\Omega}^{\nu})\equiv[\overline{\nabla}_{x}^{\mu},a_{\Omega}^{\nu}]$.
Substitute (12) into (9), we change (9) to
$\displaystyle\mathrm{ReTr}\ln[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\Sigma(-\bar{\nabla}^{2})]$
$\displaystyle=-\frac{1}{2}\lim_{\Lambda\to\infty}\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int{d^{4}x}\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{tr}~{}e^{i\bar{\nabla}_{x}^{\mu}\cdot\frac{\partial}{\partial
k}}\exp\bigg{\\{}-\tau\bigg{[}\tilde{E}+(k+\tilde{F})^{2}+\tilde{a}_{\mu}\gamma_{5}(k^{\mu}+\tilde{F}^{\mu})+(k^{\mu}+\tilde{F}^{\mu})\tilde{a}_{\mu}\gamma_{5}+\tilde{a}^{2}$
$\displaystyle\hskip
11.38092pt+\Sigma^{2}\big{(}(k+\tilde{F})^{2}\big{)}+\tilde{J}\Sigma\big{(}(k+\tilde{F})^{2}\big{)}+\Sigma\big{(}(k+\tilde{F})^{2}\big{)}\tilde{K}-\gamma_{\mu}\Big{[}\tilde{\bar{\nabla}}^{\mu}_{x},\Sigma\big{(}(k+\tilde{F})^{2}\big{)}\Big{]}\bigg{]}\bigg{\\}}\cdot
1$ (14)
with tilde operation defined as
$\displaystyle\tilde{\mathcal{O}}$ $\displaystyle\equiv$
$\displaystyle\mathcal{O}-i(\nu\mathcal{O})\frac{\partial}{\partial
k^{\nu}}-\frac{1}{2}(\lambda\nu\mathcal{O})\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\nu}}+\frac{i}{6}(\rho\lambda\nu\mathcal{O})\frac{\partial^{3}}{\partial
k^{\rho}\partial k^{\lambda}\partial
k^{\nu}}+\frac{1}{24}(\sigma\rho\lambda\nu\mathcal{O})\frac{\partial^{4}}{\partial
k^{\sigma}\partial k^{\rho}\partial k^{\lambda}\partial k^{\nu}}$
$\displaystyle-\frac{i}{120}(\delta\sigma\rho\lambda\nu\mathcal{O})\frac{\partial^{5}}{\partial
k^{\delta}\partial k^{\sigma}\partial k^{\rho}\partial k^{\lambda}\partial
k^{\nu}}+O(p^{7})\;,$
where
$\tilde{\mathcal{O}}\equiv(\tilde{E},\tilde{J},\tilde{K},\tilde{a}^{\mu},\tilde{a}^{2},\tilde{\bar{\nabla}}^{\mu}_{x})^{T}$
and
$\mathcal{O}\equiv(E,\hat{I},\tilde{I},a_{\Omega}^{\mu},a_{\Omega}^{2},\bar{\nabla}^{\mu}_{x})^{T}$.
Note that for finite cutoff $\Lambda$, the value of parameter $\tau$ must be
real and larger than zero, the term $e^{-\tau k^{2}}$ in (14) then provides a
natural suppression factor for the momentum integration and this leads the
convergence of the integration. For a converged integration, we can replace
the term $e^{i\bar{\nabla}_{x}^{\mu}\cdot\frac{\partial}{\partial k}}$ in
front of the integration kernel in (14) by $1$, since the difference
$(e^{i\bar{\nabla}_{x}^{\mu}\cdot\frac{\partial}{\partial k}}-1)\cdots$ are
some momentum total derivative terms which vanish as long as we have
nontrivial suppression factor $e^{-\tau k^{2}}$. With these considerations,
(14) becomes
$\displaystyle\mathrm{ReTr}\ln[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\Sigma(-\bar{\nabla}^{2})]=-\frac{1}{2}\lim_{\Lambda\to\infty}\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int{d^{4}x}\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{tr}~{}e^{B}\cdot
1\;,~{}~{}~{}~{}~{}$ (16) $\displaystyle
B\equiv-\tau\bigg{[}\tilde{E}+(k+\tilde{F})^{2}+\tilde{a}_{\mu}\gamma_{5}(k^{\mu}+\tilde{F}^{\mu})+(k^{\mu}+\tilde{F}^{\mu})\tilde{a}_{\mu}\gamma_{5}+\tilde{a}^{2}+\Sigma^{2}\big{(}(k+\tilde{F})^{2}\big{)}+\tilde{J}\Sigma\big{(}(k+\tilde{F})^{2}\big{)}+\Sigma\big{(}(k+\tilde{F})^{2}\big{)}\tilde{K}$
$\displaystyle\hskip
8.5359pt-\gamma_{\mu}\Big{[}\tilde{\bar{\nabla}}^{\mu}_{x},\Sigma\big{(}(k+\tilde{F})^{2}\big{)}\Big{]}\bigg{]}\;.$
(17)
From (17), we see that all $\bar{\nabla}^{\mu}$ in (16) appear as commutators,
therefore (16) and (17) offer a covariant formulation which matches the
general result that the real part of $\mathrm{Trln}\cdots$ should be invariant
under local chiral transformations. The price is that we need to handle many
momentum derivatives on the exponential and the result computations become
extremely lengthy. But as long as our reformulation is suitable to
computerize, it is worth to pay such a price. To deal the next problem of
derivatives on the exponential, we first take the low energy expansion on $B$
$\displaystyle
B=B_{0}+B_{1}+\frac{1}{2}B_{2}+\frac{1}{3!}B_{3}+\frac{1}{4!}B_{4}+\frac{1}{5!}B_{5}+\frac{1}{6!}B_{6}+\cdots$
(18)
with $\frac{1}{n!}B_{n}$ is the $p^{n}$ order part of $B$. Further introduce a
parameter $t$ dependent $B(t)$ as
$\displaystyle
B(t)=B_{0}+tB_{1}+\frac{t^{2}}{2}B_{2}+\frac{t^{3}}{3!}B_{3}+\frac{t^{4}}{4!}B_{4}+\frac{t^{5}}{5!}B_{5}+\frac{t^{6}}{6!}B_{6}+\cdots\hskip
56.9055ptB=B(t)\bigg{|}_{t=1}\;.$ (19)
Then take Taylor expansion of $e^{B(t)}$ at point $t=0$,
$\displaystyle
e^{B}=e^{B(t)}\bigg{|}_{t=1}=e^{B_{0}}+[\frac{d}{dt}e^{B(t)}]_{t=0}+\frac{1}{2!}[\frac{d^{2}}{dt^{2}}e^{B(t)}]_{t=0}+\frac{1}{3!}[\frac{d^{3}}{dt^{3}}e^{B(t)}]_{t=0}+\frac{1}{4!}[\frac{d^{4}}{dt^{4}}e^{B(t)}]_{t=0}+\ldots\;.~{}~{}~{}~{}~{}$
(20)
With the help of identities
$\displaystyle[\frac{d}{d\tau}e^{B}]e^{-B}=f(Ad(B))(\frac{dB}{d\tau})\hskip
85.35826ptf(z)=\frac{e^{z}-1}{z}=1+\frac{z}{2!}+\frac{z^{2}}{3!}+\cdots\;.$
(21)
One can explicitly work out
$\frac{1}{n!}[\frac{d^{n}}{dt^{n}}e^{B(t)}]_{t=0}$, for several lowest orders
$\displaystyle\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle e^{B_{0}}f[Ad(-B_{0})](B_{1})\;,$ (22)
$\displaystyle\frac{d^{2}}{dt^{2}}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}~{}f[Ad(-B_{0})](B_{1})+e^{B_{0}}\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+e^{B_{0}}f[Ad(-B_{0})](B_{2})\;,$
(23)
where
$\displaystyle\frac{d^{m}f}{dt^{m}}[Ad(-B(t))]={\displaystyle\sum_{n=0}^{\infty}}\frac{1}{(n+1)!}\frac{d^{m}}{dt^{m}}[Ad(-B(t))]^{n}\;.$
(24)
For more higher orders needed in our computations, we list the results of
$\frac{d^{3}}{dt^{3}}e^{B(t)}\bigg{|}_{t=0}$,
$\frac{d^{4}}{dt^{4}}e^{B(t)}\bigg{|}_{t=0}$,
$\frac{d^{5}}{dt^{5}}e^{B(t)}\bigg{|}_{t=0}$ and
$\frac{d^{6}}{dt^{6}}e^{B(t)}\bigg{|}_{t=0}$ in Appendix A.
With the help of (22), (23) and (229)-(232) , as long as the
$B_{0},B_{1},B_{2},B_{3},B_{4},B_{5},B_{6}$ are known, (20) is known and we
can substitute it back into (16) to calculate the real part of
$-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$
order by orders up to the $p^{6}$ order in the low energy expansion. To obtain
$B_{i}$, (17) tells us that the difficulty is the low energy expansion for
$\Sigma\big{(}(k+\tilde{F})^{2}\big{)}$. To achieve it, we expand the argument
of $\Sigma\big{(}(k+\tilde{F})^{2}\big{)}$ as
$\displaystyle(k+\tilde{F})^{2}$ $\displaystyle=$ $\displaystyle
k^{2}+\frac{1}{2}A_{2}+\frac{1}{6}A_{3}+\frac{1}{24}A_{4}+\frac{1}{120}A_{5}+\frac{1}{720}A_{6}+O(p^{5,6})|_{\mathrm{traceless}}+O(p^{7})\;,$
(25)
in which
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle-2(\mu\nu)k_{\mu}\frac{\partial}{\partial k^{\nu}}\;,$ (26)
$\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle
4i(\mu\nu\lambda)k_{\nu}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+2i(\mu\underline{\mu}\nu)\frac{\partial}{\partial k^{\nu}}\;,$
(27) $\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle
6(\mu\nu\lambda\rho)k_{\lambda}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial
k^{\rho}}+6(\mu\nu)(\underline{\mu}\lambda)\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+3(\mu\nu\underline{\nu}\lambda)\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+3(\mu\nu\underline{\mu}\lambda)\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}\;,$ (28) $\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle 0\;,$ (29) $\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle-90(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}-80(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}\;.$ (30)
Since we are only interested in the terms not higher than $p^{6}$, we find
that those traceless terms of $p^{5}$ and $p^{6}$ orders will not make
contributions to the final result. So to save space and simplify the
computations, we do not explicitly write down the detail structure of them,
just represent these terms with symbol $O(p^{5,6})|_{\mathrm{traceless}}$ and
remove traceless term in $A_{5}$ and $A_{6}$. Further introduce $A(t)$ as,
$\displaystyle A(t)\equiv
k^{2}+\frac{t^{2}}{2}A_{2}+\frac{t^{3}}{6}A_{3}+\frac{t^{4}}{24}A_{4}+\frac{t^{5}}{120}A_{5}+\frac{t^{6}}{720}A_{6}+O(p^{5,6})|_{\mathrm{traceless}}+O(p^{7})\hskip
28.45274ptA(t)=\left\\{\begin{array}[]{lll}k^{2}&&t=0\\\ &&\\\
(k+\tilde{F})^{2}&&t=1\end{array}.\right.~{}~{}~{}~{}$ (34)
Then
$\displaystyle\Sigma\big{(}(k+\tilde{F})^{2}\big{)}=\Sigma(k^{2})+\bigg{[}\frac{d}{dt}\Sigma[A(t)]\bigg{]}_{t=0}+\frac{1}{2!}\bigg{[}\frac{d^{2}}{dt^{2}}\Sigma[A(t)]\bigg{]}_{t=0}+\frac{1}{3!}\bigg{[}\frac{d^{3}}{dt^{3}}\Sigma[A(t)]\bigg{]}_{t=0}+\frac{1}{4!}\bigg{[}\frac{d^{4}}{dt^{4}}\Sigma[A(t)]\bigg{]}_{t=0}+\ldots\;.$
(35)
Now, we need to know $\bigg{[}\frac{d^{m}}{dt^{m}}\Sigma[A(t)]\bigg{]}_{t=0}$,
using the following formula
$\displaystyle\Sigma[A(t)]$ $\displaystyle=$
$\displaystyle\Sigma[s+A(t)]\bigg{|}_{s=0}=e^{A(t)\frac{\partial}{\partial
s}}\Sigma(s)e^{-A(t)\frac{\partial}{\partial
s}}\bigg{|}_{s=0}=e^{A(t)\frac{\partial}{\partial
s}}\Sigma(s)\bigg{|}_{s=0}\;,$ (36)
then
$\displaystyle\bigg{[}\frac{d^{m}}{dt^{m}}\Sigma[A(t)]\bigg{]}_{t=0}=\bigg{[}\frac{d^{m}}{dt^{m}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{]}_{t=0}\Sigma(s)\bigg{|}_{s=0}\;.$ (37)
Therefore to compute $\bigg{[}\frac{d^{m}}{dt^{m}}\Sigma[A(t)]\bigg{]}_{t=0}$,
we only need to calculate
$\bigg{[}\frac{d^{m}}{dt^{m}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{]}_{t=0}\Sigma(s)\bigg{|}_{s=0}$ which is just equivalent to replace
$B_{l}\rightarrow A_{l}\frac{\partial}{\partial s}$ in (22),(23) and
(229)-(232), followed by multiplying an extra factor $\Sigma(s)$ at the r.h.s.
and vanishing parameter $s$ after finishing all differential operations.
Follow this calculation road map, the detail calculation gives
$\displaystyle\bigg{[}\frac{d}{dt}\Sigma[A(t)]\bigg{]}_{t=0}$ $\displaystyle=$
$\displaystyle e^{Ad(A_{0}\frac{\partial}{\partial
s})}(f[Ad(-A_{0}\frac{\partial}{\partial
s})]A_{1})\Sigma^{\prime}(s+A_{0})\bigg{|}_{s=0}=0\;,$ (38)
$\displaystyle\frac{1}{2}\bigg{[}\frac{d^{2}}{dt^{2}}\Sigma[A(t)]\bigg{]}_{t=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2}e^{Ad(A_{0}\frac{\partial}{\partial
s})}\bigg{[}\bigg{(}e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d}{dt}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{1}\frac{\partial}{\partial
s})+\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$ (39)
$\displaystyle+f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{2}\frac{\partial}{\partial
s})\bigg{)}\bigg{]}\Sigma(s+A(t))\bigg{|}_{s=0}=-(\mu,\nu)k_{\mu}\Sigma^{\prime}_{k}\frac{\partial}{\partial
k^{\nu}}\;,$
where $\Sigma_{k}\equiv\Sigma(k^{2})$. For more higher orders, we list the
results of $\bigg{[}\frac{d^{3}}{dt^{3}}\Sigma[A(t)]\bigg{]}_{t=0}$,
$\bigg{[}\frac{d^{4}}{dt^{4}}\Sigma[A(t)]\bigg{]}_{t=0}$,
$\bigg{[}\frac{d^{5}}{dt^{5}}\Sigma[A(t)]\bigg{]}_{t=0}$ and
$\bigg{[}\frac{d^{6}}{dt^{6}}\Sigma[A(t)]\bigg{]}_{t=0}$ in Appendix A. With
these results, we finally obtain the low energy expansion of $B$,
$\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle-\tau(k^{2}+\Sigma_{k}^{2})\;,$ (40) $\displaystyle B_{1}$
$\displaystyle=$ $\displaystyle
2\tau(-a^{\mu}_{\Omega}k_{\mu}+ia_{\Omega}^{\mu}\gamma_{\mu}\Sigma_{k})\gamma_{5}\;,$
(41) $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-2a_{\Omega}^{2}\tau+(\mu\nu)\gamma_{\mu}\gamma_{\nu}\tau-
a_{\Omega}^{\mu}a_{\Omega}^{\nu}[\gamma_{\mu},\gamma_{\nu}]\tau-i(d^{\mu}a_{\Omega}^{\nu}-d^{\nu}a_{\Omega}^{\mu})\gamma_{\mu}\gamma_{\nu}\gamma_{5}\tau+4s_{\Omega}\tau\Sigma_{k}+2(\mu\nu)\tau
k_{\mu}\frac{\partial}{\partial k^{\nu}}+4(\mu
a_{\Omega}^{\nu})\gamma_{\nu}\gamma_{5}\tau k_{\mu}\Sigma_{k}^{\prime}$ (42)
$\displaystyle+4(\mu
a_{\Omega}^{\nu})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}\frac{\partial}{\partial
k^{\mu}}+2i(\mu a_{\Omega\mu})\gamma_{5}\tau+4i(\mu
a_{\Omega}^{\nu})\gamma_{5}\tau k_{\nu}\frac{\partial}{\partial
k^{\mu}}+4i(\mu\nu)\gamma_{\mu}\tau k_{\nu}\Sigma_{k}^{\prime}+4(\mu\nu)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}\;.$
We list $B_{3},B_{4},B_{5},B_{6}$ in Appendix.A. With these explicit
expressions for $B_{0},B_{1},B_{2},B_{3},B_{4},B_{5},B_{6}$, using (22), (23)
and (229)-(232) , we get (20) and further substitute (20) back into (16), we
can obtain the real part of
$-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$
order by orders up to the $p^{6}$ order in the low energy expansion. The
analytical results of $p^{2}$ and $p^{4}$ orders are the same as those given
by (34),(35) and (36) in Ref.WQ1 , except some total derivative terms which,
as we mentioned before, can be ignored as long as we take finite cutoff
$\Lambda$.
## IV Ambiguities in the Anomaly Part Contributions to the Chiral Lagrangian
In the last section, we have introduced a chiral covariant method to calculate
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$
which is already computerized now. With the help of computer, for the $p^{2}$
and $p^{4}$ order analytical formulae in the low energy expansion, we can get
results within 15 minutes, while for the $p^{6}$ order terms, we need roughly
13 hours to output all expansion results. From our general result (7), the
term
$-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$
is the normal part. To get the full result of the chiral Lagrangian, we need
to calculate the remaining anomaly part contributions
$iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J]$.
As the discussion of Ref.WQ3 , in 1980s there is a class of works (see
references given in WQ3 ) identifying this part as the full chiral Lagrangian,
and in Ref.WQ3 we refer them as the anomaly approach of calculating LECs. In
our previous work WQ1 , we pointed out that this anomaly part contributions
are completely canceled by the normal part contribution, left nontrivial pure
$\Sigma$ dependent terms contribute to the chiral Lagrangian.
For the anomaly part contributions, the key is to calculate the $U$ field
dependent term $\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]$ which, as we
mentioned before, can be obtained by vanishing $\Sigma$ in
$\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]$. In
practice, the limit was taken by first assuming $\Sigma$ being a constant mass
$m$ and then letting $m\rightarrow 0$. For $p^{2}$ order, this operation gives
null result, while for $p^{4}$ order, it gives the result originally presented
in anomaly approach. Now in this work, naively what we need to do is to
generalize the calculation to $p^{6}$ order. But to our surprise, we get many
terms with divergent coefficients. Checking the calculation carefully, we find
that the reason of appearance of infinities is due to the fact that most of
the coefficients in front of the $p^{6}$ order operators have dimension of
$1/m^{2}$ which goes to infinity when we take limit $\Sigma=m\rightarrow 0$.
Note that the $p^{6}$ terms may also have coefficients of $1/\Lambda^{2}$
which are finite in the limit of $m\rightarrow 0$ , although they vanish when
we take $\Lambda\rightarrow\infty$. These terms are irrelevant to our
discussion on the divergence of $p^{6}$ order terms and therefore we do not
need to care about them. Applying the argument on $1/m^{2}$ dependence of the
$p^{6}$ order coefficients back to the $p^{2}$ and $p^{4}$ order results we
discussed before, coefficients in front of $p^{2}$ order operators have
dimension of $m^{2}$ which goes to zero, this explains the phenomena that
anomaly approach can not produce $p^{2}$ order terms. For $p^{4}$ order, the
coefficients in front of operators are dimensionless and therefore the $m$
dependence is at most logarithmic of form $\ln m/\Lambda$ which implies
existence of a logarithmic ultraviolet divergence. Since we know that in the
large $N_{c}$ limit, the $p^{4}$ order LECs (non-contact coefficients) are not
divergent, the $\ln m/\Lambda$ term then can not appear in the final
expression of these LECs, therefore in $p^{4}$ order, anomaly approach leads
finite result LECs. In general for a $p^{2n}$ order operator, the
corresponding coefficient should has dimension $1/m^{2(n-2)}$. This implies
that the infinity in the anomaly part contributions will be a general
phenomena, when we go to the higher orders of the low energy expansion, since
the more higher the order is, the more negative powers of $m$ dependence the
coefficient will have and these negative powers of $m$ will result in
infinities as we take limit $\Sigma=m\rightarrow 0$.
The appearance of these high order infinities provides another evidence that
the anomaly approach is not a correct formulation in calculating LECs, at
least not for the $p^{6}$ and more higher order LECs. Since high order
divergence term is as an addition part of our general result (7), we can not
avoid them in our computations. How to deal with these high order infinities
from negative powers of $m$? There exists an alternative way, not relying on
the low energy expansion, to examine this anomaly part contributions in which
we must exploit the first equation of (2) and we find
$\displaystyle\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}]-\mathrm{Tr}\ln[i\not{\partial}+J]$
$\displaystyle=$
$\displaystyle\ln\mathrm{Det}[i\not{\partial}+J_{\Omega}]-\ln\mathrm{Det}[i\not{\partial}+J]$
(43) $\displaystyle=$ $\displaystyle\ln\mathrm{Det}\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][J+i\not{\partial}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]\bigg{]}-\ln\mathrm{Det}[i\not{\partial}+J]$
$\displaystyle=$ $\displaystyle\ln\mathrm{Det}\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]\bigg{]}=\mathrm{Tr}\ln\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][\Omega P_{R}+\Omega^{\dagger}P_{L}]\bigg{]}\;.$
For our interests, we are only interested in the real part of it, then
$\displaystyle\mathrm{ReTr}\ln[i\not{\partial}+J_{\Omega}]-\mathrm{ReTr}\ln[i\not{\partial}+J]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\mathrm{Tr}\ln\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]\bigg{]}+\frac{1}{2}\mathrm{Tr}\ln\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}]^{\dagger}[\Omega
P_{R}+\Omega^{\dagger}P_{L}]^{\dagger}\bigg{]}$ (44) $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Tr}\ln\bigg{[}[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]\bigg{]}+\frac{1}{2}\mathrm{Tr}\ln\bigg{[}[\Omega^{\dagger}P_{R}+\Omega
P_{L}][\Omega^{\dagger}P_{R}+\Omega P_{L}]\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Tr}\ln\bigg{[}[P_{R}+P_{L}][P_{R}+P_{L}]\bigg{]}=\frac{1}{2}\mathrm{Tr}\ln
1=0\;.$
Which shows that the compact form of anomaly part contributions to normal part
of the chiral Lagrangian is zero !
How can this null result be consistent with another divergent result obtained
from the low energy expansion ? The only possible explanation is that the
$p^{4}$ order finite term plus all those higher order infinities results a
zero ! Is it possible ? A well-known positive example is the expansion
$1/(1+x)=1-x+x^{2}-x^{3}+x^{4}-x^{5}+x^{6}-\cdots$ goes to zero when $x$ is
very large, which implies that the summation of series
$x-x^{2}+x^{3}-x^{4}+x^{5}-x^{6}+\cdots$ converges to $1$ when $x$ is very
large and each individual term in the series diverges. In the following we
take a more realistic but simplified example to show that this really happens
in our formulation. Our example starts from (9) for the case of $\Sigma$ equal
to a constant mass $m$
$\displaystyle\mathrm{ReTr}\ln[\not{\partial}-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+m]=-\frac{1}{2}\lim_{\Lambda\to\infty}\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int{d^{4}x}\int\frac{d^{4}k}{(2\pi)^{4}}~{}\mathrm{tr}e^{-\tau(k^{2}+k\cdot
b^{\prime}+m^{2}+bm+C)}\;,~{}~{}~{}~{}~{}~{}$ (45)
$\displaystyle{b^{\prime}}^{\mu}\equiv
2i\bar{\nabla}_{x}^{\mu}+2a_{\Omega}^{\mu}\hskip
56.9055ptb\equiv\hat{I}_{\Omega}+\tilde{I}_{\Omega}\hskip
56.9055ptC\equiv\bar{E}+(i\bar{\nabla}_{x}+a_{\Omega})^{2}\;.$ (46)
For simplicity, we ignore the contributions from $b^{\prime}$ which does not
change the key result of our discussion. Then our example becomes to
investigate following integration
$\displaystyle
I\equiv\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau}\int_{0}^{\infty}k^{2}dk^{2}e^{-\tau
k^{2}-\tau m^{2}-\tau bm-\tau C}$ (47)
with $b$ and $C$ not commuting each other. We will show that high order terms
in the low energy expansion of the above integration $I$ will go to infinity
when we take $m\rightarrow 0$, but if summing all the expansion terms
together, we get finite result which corresponds to previous null result of
summing all higher order terms of anomaly part contributions into a compact
form. We use three different methods to finish above integration and explain
our point. The first method is to vanish $m$ firstly and then to finish the
integration, i.e.
$\displaystyle
I\bigg{|}_{m=0}=\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau}\int_{0}^{\infty}k^{2}dk^{2}e^{-\tau
k^{2}-\tau
C}=\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau^{3}}e^{-\tau
C}=\frac{\Lambda^{4}}{2}e^{-\frac{C}{\Lambda^{2}}}-\frac{\Lambda^{2}}{2}Ce^{-\frac{C}{\Lambda^{2}}}-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{C}{\Lambda^{2}})\;,~{}~{}~{}~{}~{}~{}$
(48)
where
$\displaystyle\mathbf{Ei}(-x)\equiv-\int_{x}^{\infty}\frac{e^{-u}}{u}du=\gamma+\ln
x+\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n!n}x^{n}\hskip 85.35826pt|x|<\infty\;.$
(49)
The second method is first finishing the integration and then vanishing $m$,
$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau}\int_{0}^{\infty}k^{2}dk^{2}e^{-\tau(k^{2}+m^{2}+bm+C)}=\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau^{3}}e^{-\tau(m^{2}+bm+C)}$
(50) $\displaystyle=$
$\displaystyle\frac{\Lambda^{4}}{2}e^{-\frac{(m^{2}\\!+bm+C)^{2}}{\Lambda^{2}}}\\!-\frac{\Lambda^{2}(m^{2}\\!+\\!bm\\!+\\!C)}{2}e^{-\frac{(m^{2}\\!+bm+C)^{2}}{\Lambda^{2}}}\\!\\!-\frac{1}{2}(m^{2}\\!+\\!bm\\!+\\!C)^{2}\mathbf{Ei}(-\frac{(m^{2}\\!+\\!bm\\!+\\!C)}{\Lambda^{2}})$
$\displaystyle\stackrel{{\scriptstyle m\rightarrow 0}}{{====}}$
$\displaystyle\frac{\Lambda^{4}}{2}e^{-\frac{C}{\Lambda^{2}}}-\frac{\Lambda^{2}}{2}Ce^{-\frac{C}{\Lambda^{2}}}-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{C}{\Lambda^{2}})\;.$
We obtain the same result as that obtained in the first method, therefore
interchange the order of integration and $m\rightarrow 0$ limit does not
change the result.
The third method is first taking Taylor expansion in terms the power of $b$
and $C$ which corresponds performing the low energy expansion and then
finishing integration, finally vanishing $m$,
$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau}\int_{0}^{\infty}k^{2}dk^{2}e^{-\tau
k^{2}-\tau m^{2}}\sum^{\infty}_{n=0}\frac{1}{n!}(-\tau bm-\tau
C)^{n}=\int_{\frac{1}{\Lambda^{2}}}^{\infty}d\tau e^{-\tau
m^{2}}\sum^{\infty}_{n=0}\frac{1}{n!}\tau^{n-3}(-bm-C)^{n}$ (51)
$\displaystyle=$ $\displaystyle
m^{4}\left(\frac{\Lambda^{4}}{2m^{4}}e^{-\frac{m^{2}}{\Lambda^{2}}}-\frac{\Lambda^{2}}{2m^{2}}e^{-\frac{m^{2}}{\Lambda^{2}}}-\frac{1}{2}\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})\right)-(bm^{3}+Cm^{2})\bigg{(}\frac{\Lambda^{2}}{m^{2}}e^{-\frac{m^{2}}{\Lambda^{2}}}+\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})\bigg{)}$
$\displaystyle-\frac{1}{2}(bm+C)^{2}\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})+\sum^{\infty}_{n=0}\frac{(-\frac{b}{m}-\frac{C}{m^{2}})^{n+3}}{(n+3)!}m^{4}\Gamma(n+1,\frac{m^{2}}{\Lambda^{2}})$
$\displaystyle\stackrel{{\scriptstyle m\rightarrow 0}}{{====}}$
$\displaystyle\frac{\Lambda^{4}}{2}-C\Lambda^{2}-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})+\sum^{\infty}_{n=0}\frac{n!}{(n+3)!}(-\frac{b}{m}-\frac{C}{m^{2}})^{n+3}m^{4}e^{-\frac{m^{2}}{\Lambda^{2}}}\sum_{k=0}^{n}\frac{1}{k!}\left(\frac{m^{2}}{\Lambda^{2}}\right)^{k}\bigg{|}_{m\rightarrow
0}\;.$
We see that there are negative power of $m$ terms which will cause divergence
when we take limit $m\rightarrow 0$. This is just what has happened for the
high order terms in the anomaly part contributions. So if we calculate term by
terms in above expansion, we will meet infinities which seems contradict with
results obtained in first two methods. The only way left to escape this
contradiction is to sum all these divergences together, to see that what will
happen after the summation, we introduce a series
$\displaystyle g(x,c)$ $\displaystyle\equiv$
$\displaystyle\sum^{\infty}_{n=0}\frac{n!}{(n+3)!}x^{n+3}\sum_{k=0}^{n}\frac{c^{k}}{k!}\;,$
(52)
in which $c=m^{2}/\Lambda^{2}$ and $x=-\frac{b}{m}-\frac{C}{m^{2}}$ which will
go to negative infinity when $m\rightarrow 0$. With the help of relation
$\frac{d}{dx}\mathbf{Ei}(-x)=\frac{e^{-x}}{x}$ and boundary condition
$g^{\prime\prime}(0,c)=g^{\prime}(0,c)=g(0,c)=0$, we find
$\displaystyle
g^{\prime\prime\prime}(x,c)=\sum^{\infty}_{n=0}x^{n}\sum_{k=0}^{n}\frac{c^{k}}{k!}=\frac{e^{cx}}{1-x}\;,\hskip
85.35826ptg^{\prime\prime}(x,c)=e^{c}[-\mathbf{Ei}(cx-c)+\mathbf{Ei}(-c)]\;,$
$\displaystyle
g^{\prime}(x,c)=(x-1)e^{c}[-\mathbf{Ei}(cx-c)+\mathbf{Ei}(-c)]+\frac{1}{c}(e^{cx}-1)\;,$
$\displaystyle
g(x,c)=\frac{1}{2}(x-1)^{2}e^{c}[-\mathbf{Ei}(cx-c)+\mathbf{Ei}(-c)]+\frac{x-1}{2c}e^{cx}+\frac{1}{2c}+\frac{1}{2c^{2}}(e^{cx}-1)-\frac{x}{c}\;.$
(53)
Then (51) becomes
$\displaystyle I\bigg{|}_{m=0}$ $\displaystyle=$
$\displaystyle\lim_{m\rightarrow
0}\bigg{[}\frac{\Lambda^{4}}{2}-C\Lambda^{2}-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})+m^{4}e^{-\frac{m^{2}}{\Lambda^{2}}}g(-\frac{b}{m}-\frac{C}{m^{2}},\frac{m^{2}}{\Lambda^{2}})\bigg{]}$
(54) $\displaystyle=$ $\displaystyle\lim_{m\rightarrow
0}\bigg{[}\frac{\Lambda^{4}}{2}-C\Lambda^{2}-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})+\frac{1}{2}(bm+C+m^{2})^{2}[-\mathbf{Ei}(\frac{-bm-
C-m^{2}}{\Lambda^{2}})+\mathbf{Ei}(-\frac{m^{2}}{\Lambda^{2}})]$
$\displaystyle+\frac{1}{2}\Lambda^{2}(-bm-C-m^{2})e^{\frac{-bm-
C-m^{2}}{\Lambda^{2}}}+\frac{1}{2}\Lambda^{2}m^{2}e^{-\frac{m^{2}}{\Lambda^{2}}}+\frac{1}{2}\Lambda^{4}(e^{\frac{-bm-
C-m^{2}}{\Lambda^{2}}}-e^{-\frac{m^{2}}{\Lambda^{2}}})+\Lambda^{2}(bm+C)\bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}C^{2}\mathbf{Ei}(-\frac{C}{\Lambda^{2}})-\frac{1}{2}\Lambda^{2}Ce^{-\frac{C}{\Lambda^{2}}}+\frac{1}{2}\Lambda^{4}e^{-\frac{C}{\Lambda^{2}}}\;.$
It is the same as the results obtained from first two methods, i.e. summing
all those infinities together, we obtain correct finite result.
With above discussion, our result now is that total anomaly part contributions
to the normal part of the chiral Lagrangian vanish ! Just take several
individual terms can not reflect the true result of the full action. In fact,
finite result of the $p^{4}$ order plays a role to cancel that summations of
all higher order terms which results in the final total null result. In this
sense, in order to make sense for the $p^{6}$ and more higher order divergent
terms, we must sum them together and then we get $p^{4}$ order result with an
extra minus sign. To avoid the appearance of divergences in $p^{6}$ and higher
orders terms, what we need to do is to drop out all anomaly part
contributions, since divergences from high order terms are finally canceled by
$p^{4}$ order terms. In this view, our general result (7) must be changed to
$\displaystyle S_{\mathrm{eff}}\bigg{|}_{\mathrm{normal~{}part}}\approx-
iN_{c}\mathrm{ReTr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]\;.~{}~{}~{}~{}~{}~{}$
(55)
In fact in Ref.WQma , we already show that including in the anomalous part,
the total effective action takes the form (see Eq.(21) in Ref.WQma ),
$\displaystyle
S_{\mathrm{GND}}=-iN_{c}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]+\mbox{Wess-
Zumino terms}\;.$ (56)
With this new viewpoint on all anomaly part contributions, we need to modify
our original numerical results on $p^{4}$ order LECs, since it takes into
account of the finite values of anomaly part contributions and now we know
that these nontrivial values must be used to cancel the infinities come from
all higher order terms. In Table I, we list our modified $p^{4}$ order LECs
for cutoff $\Lambda=$1000${}^{+100}_{-100}$MeV. The $10\%$ variation of the
cutoff is considered in our calculation to examine the effects of cutoff
dependence and the result change can be treated as the error of our
calculations. The result LECs are taken the values at $\Lambda=1$GeV with
superscript the difference caused at $\Lambda=1.1$GeV and subscript the
difference caused at $\Lambda=0.9$GeV, i.e.,
$\displaystyle
L_{\Lambda=1\mathrm{GeV}}\bigg{|}^{L_{\Lambda=1.1\mathrm{GeV}}-L_{\Lambda=1\mathrm{GeV}}}_{L_{\Lambda=0.9\mathrm{GeV}}-L_{\Lambda=1\mathrm{GeV}}}\hskip
56.9055pt\bar{l}_{\Lambda=1\mathrm{GeV}}\bigg{|}^{\bar{l}_{\Lambda=1.1\mathrm{GeV}}-\bar{l}_{\Lambda=1\mathrm{GeV}}}_{\bar{l}_{\Lambda=0.9\mathrm{GeV}}-\bar{l}_{\Lambda=1\mathrm{GeV}}}\hskip
14.22636pt\mbox{or}\hskip
14.22636ptl_{\Lambda=1\mathrm{GeV}}\bigg{|}^{l_{\Lambda=1.1\mathrm{GeV}}-l_{\Lambda=1\mathrm{GeV}}}_{l_{\Lambda=0.9\mathrm{GeV}}-l_{\Lambda=1\mathrm{GeV}}}\;.$
(57)
TABLE I. The obtained values of the $p^{4}$ coefficients $L_{1}\cdots,L_{10}$
for three flavor quarks and $\bar{l}_{1}\cdots,\bar{l}_{6},l_{7}$ for two
flavor quarks
where
$l_{i}\\!=\\!\frac{1}{32\pi^{2}}\gamma_{i}(\bar{l}_{i}\\!+\\!\ln\frac{M^{2}_{\pi}}{\mu^{2}})$
for $i=1,\ldots,7$, $\mu\\!=\\!$770MeV and $\gamma_{i}$ are given in Ref.GS .
Since $\gamma_{7}\\!=\\!0$, we calculate $l_{7}$ instead of $\bar{l}_{7}$.
Together with the experimental values given in Ref.GS and our old result
given in Ref.WQ1 for comparisons.
$\Lambda_{\mathrm{QCD}}$, $\Lambda$ and
$-\langle\bar{\psi}\psi\rangle^{\frac{1}{3}}$ are in units of MeV, and
$L_{1}\cdots,L_{10},l_{7}$ are in units of $10^{-3}$. $\Lambda_{\mathrm{QCD}}$
-$\langle\bar{\psi}\psi\rangle^{\frac{1}{3}}$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$
$L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$
$\Lambda=$1000${}^{+100}_{-100}$ 453${}^{-6}_{+12}$ 260${}^{-8}_{+9}$
1.23${}^{+0.03}_{-0.04}$ 2.46${}^{+0.05}_{-0.08}$ -6.85${}^{-0.14}_{+0.21}$
0.0${}^{+0.0}_{-0.0}$ 1.48${}^{-0.01}_{-0.03}$ 0.0${}^{+0.0}_{-0.0}$
-0.51${}^{+0.05}_{-0.06}$ 1.02${}^{-0.06}_{+0.06}$ 8.86${}^{+0.24}_{-0.37}$
-7.40${}^{-0.29}_{+0.44}$ Ref.WQ1 : 484 296 0.403 0.805 -3.47 0 1.47 0 -0.792
1.83 2.28 -4.08 Expt: 250 $0.9\pm 0.3$ $1.7\pm 0.7$ -$4.4\pm 2.5$ $0\pm 0.5$
$2.2\pm 0.5$ $0\pm 0.3$ -$0.4\pm 0.15$ $1.1\pm 0.3$ $7.4\pm 0.7$ -$6.0\pm 0.7$
$\Lambda_{\mathrm{QCD}}$ -$\langle\bar{\psi}\psi\rangle^{\frac{1}{3}}$
$\bar{l}_{1}$ $\bar{l}_{2}$ $\bar{l}_{3}$ $\bar{l}_{4}$ $\bar{l}_{5}$
$\bar{l}_{6}$ $l_{7}$ $\Lambda=$1000${}^{+100}_{-100}$ 465${}^{-6}_{+12}$
227${}^{-6}_{+8}$ -4.77${}^{-0.17}_{+0.24}$ 8.01${}^{+0.09}_{-0.14}$
1.97${}^{+0.29}_{-0.35}$ 4.34${}^{-0.01}_{-0.02}$ 17.35${}^{+0.53}_{-0.80}$
19.98${}^{+0.44}_{-0.67}$ -8.18${}^{+0.50}_{-0.43}$ Expt: 250 $-2.3\pm 3.7$
$6.0\pm 1.3$ $2.9\pm 2.4$ $4.3\pm 0.9$ $13.9\pm 1.3$ $16.5\pm 1.1$ $O(5)$
In obtaining the result, we have taken the running coupling constant as model
A given by (40) of Ref.WQ1 and the low energy value of this $\alpha_{s}$ is
already chosen well above the critical value to trigger the S$\chi$SB of the
theory. It should be noted that $\alpha_{s}$ depends on the number of quark
flavors, so does for $\Sigma$ from SDE. In fixing the $\Lambda_{\mathrm{QCD}}$
we have taken input $F_{0}=87$MeV. The reason of taking this value is that if
the final $F_{\pi}$ is around value of 93MeV, then our formula shows $F_{0}$
must be located around 87MeV. In Sec.VI, we will exhibit this phenomena
explicitly.
## V $p^{6}$ order of chiral Lagrangian: normal part
The general form of $p^{6}$ order chiral Lagrangian was first introduced in
Ref.p6-0 and then discussed in Ref.p6-1 . Now we can express the normal part
of it in terms of our rotated sources as what we have done in (10) for the
$p^{4}$ order terms. Considering that our computation is done under large
$N_{c}$ limit, within this approximation, terms in the chiral Lagrangian with
two and more traces vanish when we not apply the equation of motion. To avoid
unnecessary complicities, in this paper we only write down those terms with
one trace
$\displaystyle S_{\mathrm{eff}}\bigg{|}_{p^{6},~{}\mathrm{normal}}=\int
d^{4}x~{}\bigg{[}~{}{\displaystyle\sum_{n=1}^{94}}~{}\mathcal{Z}_{n}\mathrm{tr}_{f}[\bar{O}_{n}]~{}+~{}O(\frac{1}{N_{c}})\bigg{]}$
(58)
with $\bar{O}_{n}$ being $p^{6}$ order operator we could get in our
calculation and $\mathcal{Z}_{n}$ being corresponding coefficient,
$O(\frac{1}{N_{c}})$ are consist of the most multi-traces terms. Our
computations then give the explicit expressions of $\mathcal{Z}_{n}$ in terms
of quark self energy. The detail expressions are given in (241). And the
definitions of operators $\bar{O}_{n}$ for $n=1,2,\ldots,94$ are given in
Table.II,
TABLE II. $p^{6}$ order operators
$\displaystyle\begin{array}[]{|r|c|r|c|r|c|}\hline\cr
n&\bar{O}_{n}&n&\bar{O}_{n}&n&\bar{O}_{n}\\\ \hline\cr
1&(a_{\Omega}^{2})^{3}&33&a_{\Omega}^{\mu}a_{\Omega}^{\nu}a_{\Omega\mu}d_{\nu}p_{\Omega}&65&d^{2}a_{\Omega}^{\nu}d_{\nu}p_{\Omega}\\\
2&a_{\Omega}^{2}a_{\Omega}^{\nu}a_{\Omega}^{\lambda}a_{\Omega\nu}a_{\Omega\lambda}&34&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\mu}a_{\Omega\nu}p_{\Omega}+p_{\Omega}d_{\nu}a_{\Omega\mu})&66&d^{\mu}d^{\nu}a_{\Omega\nu}d_{\mu}p_{\Omega}\\\
3&a_{\Omega}^{2}a_{\Omega}^{\nu}a_{\Omega}^{2}a_{\Omega\nu}&35&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\nu}a_{\Omega\mu}p_{\Omega}+p_{\Omega}d_{\mu}a_{\Omega\nu})&67&d^{\mu}s_{\Omega}d_{\mu}s_{\Omega}\\\
4&a_{\Omega}^{\mu}a_{\Omega}^{\nu}a_{\Omega\mu}a_{\Omega}^{\lambda}a_{\Omega\nu}a_{\Omega\lambda}&36&a_{\Omega}^{\mu}p_{\Omega}a_{\Omega\mu}d^{\nu}a_{\Omega\nu}&68&d^{\mu}p_{\Omega}d_{\mu}p_{\Omega}\\\
5&a_{\Omega}^{\mu}a_{\Omega}^{\nu}a_{\Omega}^{\lambda}a_{\Omega\mu}a_{\Omega\nu}a_{\Omega\lambda}&37&a_{\Omega}^{\mu}p_{\Omega}a_{\Omega}^{\nu}(d_{\mu}a_{\Omega\nu}+d_{\nu}a_{\Omega\mu})&69&iV_{\Omega}^{\mu\nu}V_{\Omega\mu}^{~{}~{}\lambda}V_{\Omega\nu\lambda}\\\
6&a_{\Omega}^{2}(d^{\nu}a_{\Omega\nu})^{2}&38&a_{\Omega}^{\mu}(d_{\mu}a_{\Omega}^{\nu}d_{\nu}s_{\Omega}+d^{\nu}s_{\Omega}d_{\mu}a_{\Omega\nu})&70&V_{\Omega}^{\mu\nu}V_{\Omega\mu\nu}a_{\Omega}^{2}\\\
7&a_{\Omega}^{2}d^{\nu}a_{\Omega}^{\lambda}d_{\nu}a_{\Omega\lambda}&39&a_{\Omega}^{\mu}(d^{\nu}a_{\Omega\mu}d_{\nu}s_{\Omega}+d^{\nu}s_{\Omega}d_{\nu}a_{\Omega\mu})&71&V_{\Omega}^{\mu\nu}V_{\Omega\mu}^{~{}~{}\lambda}a_{\Omega\nu}a_{\Omega\lambda}\\\
8&a_{\Omega}^{2}d_{\nu}a_{\Omega}^{\lambda}d_{\lambda}a_{\Omega\nu}&40&a_{\Omega}^{\mu}(d^{\nu}a_{\Omega\nu}d_{\mu}s_{\Omega}+d_{\mu}s_{\Omega}d^{\nu}a_{\Omega\nu})&72&V_{\Omega}^{\mu\nu}V_{\Omega\mu}^{~{}~{}\lambda}a_{\Omega\lambda}a_{\Omega\nu}\\\
9&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\mu}a_{\Omega\nu}d^{\lambda}a_{\Omega\lambda}+d^{\lambda}a_{\Omega\lambda}d_{\nu}a_{\Omega\mu})&41&(a_{\Omega}^{2})^{2}s_{\Omega}&73&V_{\Omega}^{\mu\nu}(a_{\Omega\mu}V_{\Omega\nu}^{~{}~{}\lambda}a_{\Omega\lambda}-a_{\Omega}^{\lambda}V_{\Omega\mu\lambda}a_{\Omega\nu})\\\
10&a_{\Omega}^{\mu}a_{\Omega}^{\nu}d_{\mu}a_{\Omega}^{\lambda}d_{\nu}a_{\Omega\lambda}&42&a_{\Omega}^{\mu}a_{\Omega}^{\nu}a_{\Omega\mu}a_{\Omega\nu}s_{\Omega}&74&V_{\Omega}^{\mu\nu}a_{\Omega}^{\lambda}V_{\Omega\mu\nu}a_{\Omega\lambda}\\\
11&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\mu}a_{\Omega}^{\lambda}d_{\lambda}a_{\Omega\nu}+d^{\lambda}a_{\Omega\mu}d_{\nu}a_{\Omega\lambda})&43&a_{\Omega}^{\mu}a_{\Omega}^{2}a_{\Omega\mu}s_{\Omega}&75&V_{\Omega}^{\mu\nu}V_{\Omega\mu\nu}s_{\Omega}\\\
12&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\nu}a_{\Omega\mu}d^{\lambda}a_{\Omega\lambda}+d^{\lambda}a_{\Omega\lambda}d_{\mu}a_{\Omega\nu})&44&ia_{\Omega}^{\mu}(d_{\mu}a_{\Omega}^{\nu}d^{\lambda}V_{\Omega\nu\lambda}+d^{\nu}V_{\Omega\nu}^{~{}~{}\lambda}d_{\mu}a_{\Omega\lambda})&76&iV_{\Omega}^{\mu\nu}(a_{\Omega\mu}d_{\nu}p_{\Omega}+d_{\mu}p_{\Omega}a_{\Omega\nu})\\\
13&a_{\Omega}^{\mu}a_{\Omega}^{\nu}d_{\nu}a_{\Omega}^{\lambda}d_{\mu}a_{\Omega\lambda}&45&ia_{\Omega}^{\mu}(d^{\nu}a_{\Omega\mu}d^{\lambda}V_{\Omega\nu\lambda}+d^{\nu}V_{\Omega\nu}^{~{}~{}\lambda}d_{\lambda}a_{\Omega\mu})&77&iV^{\mu\nu}(p_{\Omega}d_{\mu}a_{\Omega\nu}-d_{\mu}a_{\Omega\nu}p_{\Omega})\\\
14&a_{\Omega}^{\mu}a_{\Omega}^{\nu}(d_{\nu}a_{\Omega}^{\lambda}d_{\lambda}a_{\Omega\mu}+d^{\lambda}a_{\Omega\nu}d_{\mu}a_{\Omega\lambda})&46&ia_{\Omega}^{\mu}(d^{\nu}a_{\Omega\nu}d^{\lambda}V_{\Omega\mu\lambda}-d^{\nu}V_{\Omega\mu\nu}d^{\lambda}a_{\Omega\lambda})&78&iV_{\Omega}^{\mu\nu}(d_{\mu}a_{\Omega\nu}d^{\lambda}a_{\Omega\lambda}-d^{\lambda}a_{\Omega\lambda}d_{\mu}a_{\Omega\nu})\\\
15&a_{\Omega}^{\mu}a_{\Omega}^{\nu}d^{\lambda}a_{\Omega\mu}d_{\lambda}a_{\Omega\nu}&47&ia_{\Omega}^{\mu}(d^{\nu}a_{\Omega}^{\lambda}d_{\mu}V_{\Omega\nu\lambda}-d_{\mu}V_{\Omega}^{\nu\lambda}d_{\nu}a_{\Omega\lambda})&79&iV_{\Omega}^{\mu\nu}d_{\mu}a_{\Omega}^{\lambda}d_{\nu}a_{\Omega\lambda}\\\
16&a_{\Omega}^{\mu}a_{\Omega}^{\nu}d^{\lambda}a_{\Omega\nu}d_{\lambda}a_{\Omega\mu}&48&ia_{\Omega}^{\mu}(d^{\nu}a_{\Omega}^{\lambda}d_{\nu}V_{\Omega\mu\lambda}-d^{\nu}V_{\Omega\mu}^{~{}~{}\lambda}d_{\nu}a_{\Omega\lambda})&80&iV_{\Omega}^{\mu\nu}(d_{\mu}a_{\Omega}^{\lambda}d_{\lambda}a_{\Omega\nu}+d^{\lambda}a_{\Omega\mu}d_{\nu}a_{\Omega\lambda})\\\
17&a_{\Omega}^{\mu}(d_{\mu}a_{\Omega}^{\nu}a_{\Omega\nu}+d^{\nu}a_{\Omega\mu}a_{\Omega\nu})d^{\lambda}a_{\Omega\lambda}&49&ia_{\Omega}^{\mu}(d^{\nu}a_{\Omega}^{\lambda}d_{\lambda}V_{\Omega\mu\nu}-d^{\nu}V_{\Omega\mu}^{~{}~{}\lambda}d_{\lambda}a_{\Omega\nu})&81&iV_{\Omega}^{\mu\nu}d^{\lambda}a_{\Omega\mu}d_{\lambda}a_{\Omega\nu}\\\
18&a_{\Omega}^{\mu}(d_{\mu}a_{\Omega}^{\nu}a_{\Omega}^{\lambda}d_{\nu}a_{\Omega\lambda}+d^{\nu}a_{\Omega}^{\lambda}a_{\Omega\nu}d_{\lambda}a_{\Omega\mu})&50&d^{\mu}V_{\Omega\mu}^{~{}\nu}d^{\lambda}V_{\Omega\nu\lambda}&82&iV_{\Omega}^{\mu\nu}(a_{\Omega\mu}a_{\Omega\nu}s_{\Omega}+s_{\Omega}a_{\Omega\mu}a_{\Omega\nu})\\\
19&a_{\Omega}^{\mu}(d_{\mu}a_{\Omega}^{\nu}a_{\Omega}^{\lambda}d_{\lambda}a_{\Omega\nu}+d^{\nu}a_{\Omega}^{\lambda}a_{\Omega\nu}d_{\mu}a_{\Omega\lambda})&51&d^{\mu}V_{\Omega}^{\nu\lambda}d_{\mu}V_{\Omega\nu\lambda}&83&iV_{\Omega}^{\mu\nu}a_{\Omega\mu}s_{\Omega}a_{\Omega\nu}\\\
20&a_{\Omega}^{\mu}(d^{\nu}a_{\Omega\mu}a_{\Omega}^{\lambda}d_{\nu}a_{\Omega\lambda}+d^{\nu}a_{\Omega}^{\lambda}a_{\Omega\lambda}d_{\nu}a_{\Omega\mu})&52&d^{\mu}V_{\Omega}^{\nu\lambda}d_{\nu}V_{\Omega\mu\lambda}&84&iV_{\Omega}^{\mu\nu}(a_{\Omega\mu}a_{\Omega\nu}a_{\Omega}^{2}+a_{\Omega}^{2}a_{\Omega\mu}a_{\Omega\nu})\\\
21&a_{\Omega}^{\mu}d^{\nu}a_{\Omega\nu}a_{\Omega\mu}d^{\lambda}a_{\Omega\lambda}&53&d^{2}a_{\Omega}^{\nu}d_{\nu}d^{\lambda}a_{\Omega\lambda}&85&iV_{\Omega}^{\mu\nu}(a_{\Omega\mu}a_{\Omega}^{\lambda}a_{\Omega\nu}a_{\Omega\lambda}+a_{\Omega}^{\lambda}a_{\Omega\mu}a_{\Omega\lambda}a_{\Omega\nu})\\\
22&a_{\Omega}^{\mu}d^{\nu}a_{\Omega}^{\lambda}a_{\Omega\mu}d_{\nu}a_{\Omega\lambda}&54&d^{2}a_{\Omega}^{\nu}d^{\lambda}d_{\nu}a_{\Omega\lambda}&86&iV_{\Omega}^{\mu\nu}a_{\Omega\mu}a_{\Omega}^{2}a_{\Omega\nu}\\\
23&a_{\Omega}^{\mu}d^{\nu}a_{\Omega}^{\lambda}a_{\Omega\mu}d_{\lambda}a_{\Omega\nu}&55&d^{2}a_{\Omega}^{\nu}d^{2}a_{\Omega\nu}&87&iV_{\Omega}^{\mu\nu}a_{\Omega}^{\lambda}a_{\Omega\mu}a_{\Omega\nu}a_{\Omega\lambda}\\\
24&a_{\Omega}^{2}s_{\Omega}^{2}&56&d^{\mu}d^{\nu}a_{\Omega\mu}d_{\nu}d^{\lambda}a_{\Omega\lambda}&88&s_{\Omega}^{3}\\\
25&a_{\Omega}^{2}p_{\Omega}^{2}&57&d^{\mu}d^{\nu}a_{\Omega\mu}d^{\lambda}d_{\nu}a_{\Omega\lambda}&89&s_{\Omega}p_{\Omega}^{2}\\\
26&a_{\Omega}^{\mu}s_{\Omega}a_{\Omega\mu}s_{\Omega}&58&d^{\mu}d^{\nu}a_{\Omega\nu}d_{\mu}d^{\lambda}a_{\Omega\lambda}&90&s_{\Omega}p_{\Omega}d^{\mu}a_{\Omega\mu}\\\
27&a_{\Omega}^{\mu}p_{\Omega}a_{\Omega\mu}p_{\Omega}&59&d^{\mu}d^{\nu}a_{\Omega}^{\lambda}d_{\mu}d_{\nu}a_{\Omega\lambda}&91&s_{\Omega}d^{\mu}a_{\Omega\mu}p_{\Omega}\\\
28&a_{\Omega}^{\mu}(s_{\Omega}d_{\mu}p_{\Omega}+d_{\mu}p_{\Omega}s_{\Omega})&60&d^{\mu}d^{\nu}a_{\Omega}^{\lambda}d_{\mu}d_{\lambda}a_{\Omega\nu}&92&s_{\Omega}d^{\mu}a_{\Omega}^{\nu}d_{\mu}a_{\Omega\nu}\\\
29&a_{\Omega}^{\mu}(p_{\Omega}d_{\mu}s_{\Omega}+d_{\mu}s_{\Omega}p_{\Omega})&61&d^{\mu}d^{\nu}a_{\Omega}^{\lambda}d_{\nu}d_{\mu}a_{\Omega\lambda}&93&s_{\Omega}d^{\mu}a_{\Omega}^{\nu}d_{\nu}a_{\Omega\mu}\\\
30&a_{\Omega}^{2}d^{\nu}a_{\Omega\nu}p_{\Omega}&62&d^{\mu}d^{\nu}a_{\Omega}^{\lambda}d_{\nu}d_{\lambda}a_{\Omega\mu}&94&s_{\Omega}(d^{\mu}a_{\Omega\mu})^{2}\\\
31&a_{\Omega}^{2}p_{\Omega}d^{\nu}a_{\Omega\nu}&63&d^{\mu}d^{\nu}a_{\Omega}^{\lambda}d_{\lambda}d_{\nu}a_{\Omega\mu}&&\\\
32&a_{\Omega}^{2}a_{\Omega}^{\nu}d_{\nu}p_{\Omega}+a_{\Omega}^{\mu}a_{\Omega}^{2}d_{\mu}p_{\Omega}&64&d^{\mu}d^{\nu}a_{\Omega\mu}d_{\nu}p_{\Omega}&&\\\
\hline\cr\end{array}$ (92)
where some operators have $i$ in front of them to insure their coefficients
being real. In Ref.p6-1 , $p^{6}$ order operator was denoted by $Y_{i}$ for
the case of n flavor with coefficient $K_{i}$ p6-2 , $O_{i}$ for the case of
three flavor with coefficient $C_{i}$ and $P_{i}$ for the case of two flavors
with coefficient $c_{i}$,
$\displaystyle S_{\mathrm{eff}}\bigg{|}_{p^{6},~{}\mathrm{normal}}=\int
d^{4}x~{}\left\\{\begin{array}[]{lll}{\displaystyle\sum_{i=1}^{112}}~{}K_{i}Y_{i}+\mbox{3
contact terms}&&\mbox{n flavors}\\\
{\displaystyle\sum_{i=1}^{90}}~{}C_{i}O_{i}+\mbox{4 contact
terms}&&\mbox{three flavors}\\\
{\displaystyle\sum_{i=1}^{53}}~{}c_{i}P_{i}+\mbox{4 contact terms}&&\mbox{two
flavors}\end{array}\right.\;.$ (96)
Consider that our parametrization of the $p^{6}$ order chiral Lagrangian (58)
is general to the case of n flavor quarks, there exist some relations among
our coefficients and n flavor coefficients given in (96). With the help of
computer derivations, we have worked out these relations and list them in
ApendixC. As a check, we vanish the quark self energy $\Sigma$ in the codes
which corresponds to take $m=0$ before other further calculations and find
null $p^{6}$ result. This verify the analytical result discussed in Sec.IV
that the anomaly part contributions do not the contribute to normal part of
chiral Lagrangian. Another consistency check is done for those operators which
have two terms combined together by $C$ and $P$ symmetry requirements. For $n$
flavor case, such operators are $\bar{O}_{9}$, $\bar{O}_{11}$, $\bar{O}_{12}$,
$\bar{O}_{14}$, $\bar{O}_{17}$, $\bar{O}_{18}$, $\bar{O}_{19}$,
$\bar{O}_{20}$, $\bar{O}_{28}$, $\bar{O}_{29}$, $\bar{O}_{32}$,
$\bar{O}_{34}$, $\bar{O}_{35}$, $\bar{O}_{37}$, $\bar{O}_{38}$,
$\bar{O}_{39}$, $\bar{O}_{40}$, $\bar{O}_{44}$, $\bar{O}_{45}$,
$\bar{O}_{46}$, $\bar{O}_{47}$, $\bar{O}_{48}$, $\bar{O}_{49}$,
$\bar{O}_{73}$, $\bar{O}_{76}$, $\bar{O}_{77}$, $\bar{O}_{78}$,
$\bar{O}_{80}$, $\bar{O}_{82}$, $\bar{O}_{84}$, $\bar{O}_{85}$. Since in each
of these operators, there are two terms, we can compute the coefficients in
front of each terms and check if they are same. We have done the checks for
all these operators and all obtain the same analytical expressions for the two
terms in the same operator. This partly verifies the correctness of our result
given in (241). From n flavors to three flavors, there are some extra
constraints (see (B1) in Ref.p6-1 )which make some operators depending on
others. Further from three flavors to two flavors, there are also some extra
constraints (see (B3) in Ref.p6-1 )which make some more operators depending on
others. The sequence number for n flavors, three flavors and two flavors are
different, their comparisons are list in Table 2 in Ref.p6-1 .
## VI Numerical Values of $p^{6}$ order LECs: Normal Part
With all above preparations in previous sections, we now come to the stage of
giving numerical values to the $p^{6}$ order LECs in the normal part of the
chiral Lagrangian. Note the necessary input and process of the present
computations for the $p^{6}$ order LECs are the same as those for the $p^{2}$
and $p^{4}$ order LECs given in the end of Sec.IV, we list the numerical
result in Table.III, as done in Table.I, the result LECs are taken for the
values at $\Lambda=1$GeV with superscript the difference caused at
$\Lambda=1.1$GeV and subscript the difference caused at $\Lambda=0.9$GeV,
$\displaystyle\hskip
28.45274ptC_{\Lambda=1\mathrm{GeV}}\bigg{|}^{C_{\Lambda=1.1\mathrm{GeV}}-C_{\Lambda=1\mathrm{GeV}}}_{C_{\Lambda=0.9\mathrm{GeV}}-C_{\Lambda=1\mathrm{GeV}}}\hskip
113.81102ptc_{\Lambda=1\mathrm{GeV}}\bigg{|}^{c_{\Lambda=1.1\mathrm{GeV}}-c_{\Lambda=1\mathrm{GeV}}}_{c_{\Lambda=0.9\mathrm{GeV}}-c_{\Lambda=1\mathrm{GeV}}}\;.$
We further list result of the $p^{6}$ order LECs at $\Lambda=\infty$ in
Table.IV. Consider that in the limit of $\Lambda=\infty$, dropping out
momentum total derivative terms in Eq.(14) is problematic, we only take result
LECs at $\Lambda=\infty$ as a reference. Since the terms of three flavors and
two flavors may have different sequence numbers, as done in Ref.p6-1 , we put
them in the same line in our table. Since the number of independent operators
in the two flavors is smaller than that in the three flavors, there are some
operators in three flavors being independent operators, but being dependent in
two flavors, then these operators will not have their two flavor counter parts
in our table, these leave the r.h.s. some empty blanks in the corresponding
two flavor columns. For two flavor case, Ref.NewRelation further propose a
new relation among operators,
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
8P_{1}-2P_{2}+6P_{3}-12P_{13}+8P_{14}-3P_{15}-2P_{16}-20P_{24}+8P_{25}+12P_{26}-12P_{27}-28P_{28}+8P_{36}-8P_{37}$
(97) $\displaystyle-8P_{39}+2P_{40}+8P_{41}-8P_{42}-6P_{43}+4P_{48}\;,$
which implies that one of the operators appears in above formula should be
further dependent operator. Due to ignorance of the values of the coefficients
in front of these operators, Ref.NewRelation arbitrarily chooses this
operator being $P_{27}$. Now our computations show that $c_{27}\neq 0$, so
original choice is not suitable. Considering that $c_{37}=0$ in our
computation, we instead now take $P_{37}$ as that dependent operator. $P_{37}$
now is a dependent operator, its name then is deleted in our Table.III.
To verify our choice of $F_{0}=87$MeV will really results in experimental
value of $F_{\pi}$, we exploit the relation between $F_{0}$ and $F_{\pi}$
given in Ref.Fpi
$\displaystyle\frac{F_{\pi}}{F_{0}}=1+x_{2}(l_{4}^{r}-L)+x_{2}^{2}\bigg{[}\frac{1}{16\pi^{2}}(-\frac{1}{2}l_{1}^{r}-l_{2}^{r}+\frac{29}{12}L)-\frac{13}{192}\frac{1}{(16\pi^{2})^{2}}+\frac{7}{4}k_{1}+k_{2}-2l_{3}^{r}l_{4}^{r}+2(l_{4}^{r})^{2}-\frac{5}{4}k_{4}+r_{F}^{r}\bigg{]}+O(x_{2}^{3})\;,~{}~{}~{}~{}~{}$
(98) $\displaystyle x_{2}=\frac{M_{\pi}^{2}}{F_{\pi}^{2}}\hskip
28.45274ptL=\frac{1}{16\pi^{2}}\ln\frac{M_{\pi}^{2}}{\mu^{2}}\hskip
42.67912ptk_{i}=(4l^{r}_{i}-\gamma_{i}L)L\hskip
42.67912ptr_{F}^{r}=(8c_{7}+16c_{8}+8c_{9})F_{0}^{2}~{}~{}~{}~{}\mbox{from
Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$ (99)
in which $l^{r}_{i}$ and $\gamma_{i}$ for $i=1,2,\ldots,7$ are defined in
Ref.GS . Scale $\mu$ is taken to be $\rho$ mass $\mu=M_{\rho}=$770MeV.
Numerical calculations show that for $\Lambda=1000^{+100}_{-100}$MeV, the
contributions up to order of $p^{4}$ (ignoring $x_{2}^{2}$ terms in (98)) give
result $F_{\pi}=92.99^{+.00}_{-.03}$MeV and the contributions up to order of
$p^{6}$ (ignoring $x_{2}^{3}$ terms in (98)) give result
$F_{\pi}=92.97^{+.00}_{-.04}$MeV with
$r_{F}^{r}=-5.036^{+0.730}_{-1.290}\times 10^{-5}$. We see that the $p^{6}$
order contributions to $F_{\pi}$ are very small and $F_{0}=87$MeV is directly
relate to $F_{\pi}=93$MeV.
TABLE III. The obtained values of the $p^{6}$ order LECs $C_{1}\cdots,C_{90}$
for three flavor and $c_{1}\cdots,c_{53}$ for two flavors. The LECs are in
units of $10^{-3}\mathrm{GeV}^{-2}$. The result LECs are taken the values at
$\Lambda=1$GeV with superscript the difference caused at $\Lambda=1.1$GeV and
subscript the difference caused at $\Lambda=0.9$GeV. The value $\equiv 0$
means that the constants vanish at large $N_{c}$ limit. $\displaystyle\hskip
14.22636pt\begin{array}[]{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline\cr
i&C_{i}&j&c_{j}&i&C_{i}&j&c_{j}&i&C_{i}&j&c_{j}\\\ \hline\cr
1&3.79^{+0.10}_{-0.17}&1&3.58^{+0.09}_{-0.15}&31&-0.63^{+0.05}_{-0.09}&17&-1.10^{+0.12}_{-0.19}&61&2.88^{-0.22}_{+0.26}&34&2.84^{-0.22}_{+0.26}\\\
2&\equiv 0&&&32&0.18^{-0.03}_{+0.04}&18&0.43^{-0.07}_{+0.08}&62&\equiv 0&&\\\
3&-0.05^{+0.01}_{-0.01}&2&-0.03^{+0.01}_{-0.01}&33&0.09^{-0.00}_{+0.03}&19&0.41^{-0.06}_{+0.10}&63&2.99^{-0.24}_{+0.30}&&\\\
4&3.10^{+0.09}_{-0.15}&3&2.89^{+0.08}_{-0.13}&34&1.59^{-0.10}_{+0.16}&20&1.56^{-0.10}_{+0.17}&64&\equiv
0&&\\\
5&-1.01^{+0.08}_{-0.11}&4&1.21^{-0.07}_{+0.06}&35&0.17^{-0.12}_{+0.17}&21&0.29^{-0.18}_{+0.24}&65&-2.43^{+0.15}_{-0.16}&35&3.39^{-0.32}_{+0.41}\\\
6&\equiv 0&&&36&\equiv 0&&&66&1.71^{+0.07}_{-0.12}&36&1.57^{+0.06}_{-0.10}\\\
7&\equiv 0&&&37&-0.56^{+0.09}_{-0.11}&&&67&\equiv 0&&\\\
8&2.31^{-0.16}_{+0.18}&&&38&0.41^{-0.08}_{+0.07}&22&-1.32^{+0.18}_{-0.25}&68&\equiv
0&&\\\ 9&\equiv 0&&&39&\equiv
0&23&0.86^{-0.12}_{+0.15}&69&-0.86^{-0.04}_{+0.06}&38&-0.68^{-0.03}_{+0.05}\\\
10&-1.05^{+0.08}_{-0.09}&5&-0.98^{+0.07}_{-0.09}&40&-6.35^{-0.18}_{+0.32}&24&-4.84^{-0.14}_{+0.25}&70&1.73^{-0.08}_{+0.07}&39&1.81^{-0.08}_{+0.07}\\\
11&\equiv 0&&&41&\equiv 0&&&71&\equiv 0&&\\\
12&-0.34^{+0.02}_{-0.01}&6&-0.33^{+0.01}_{-0.01}&42&0.60^{+0.00}_{-0.00}&&&72&-3.30^{+0.05}_{-0.00}&40&-3.17^{+0.05}_{-0.02}\\\
13&\equiv 0&&&43&\equiv 0&&&73&0.50^{+0.43}_{-0.56}&41&0.30^{+0.42}_{-0.54}\\\
14&-0.83^{+0.12}_{-0.19}&7&-1.72^{+0.25}_{-0.35}&44&6.32^{+0.20}_{-0.36}&25&6.03^{+0.19}_{-0.33}&74&-5.07^{-0.16}_{+0.27}&42&-4.74^{-0.14}_{+0.24}\\\
15&\equiv 0&8&0.86^{-0.12}_{+0.15}&45&\equiv 0&&&75&\equiv 0&&\\\ 16&\equiv
0&&&46&-0.60^{-0.02}_{+0.04}&26&-1.14^{-0.05}_{+0.07}&76&-1.44^{-0.23}_{+0.31}&43&-1.29^{-0.23}_{+0.30}\\\
17&0.01^{-0.01}_{-0.01}&9&-0.84^{+0.12}_{-0.17}&47&0.08^{+0.01}_{-0.00}&&&77&\equiv
0&&\\\
18&-0.56^{+0.09}_{-0.11}&&&48&3.41^{+0.06}_{-0.10}&&&78&17.51^{+1.02}_{-1.59}&44&16.16^{+0.94}_{-1.45}\\\
19&-0.48^{+0.09}_{-0.13}&10&-0.37^{+0.07}_{-0.10}&49&\equiv
0&&&79&-0.56^{-0.30}_{+0.40}&45&0.26^{-0.26}_{+0.34}\\\
20&0.18^{-0.03}_{+0.04}&11&\equiv
0&50&8.71^{+0.78}_{-1.12}&27&13.57^{+1.41}_{-2.00}&80&0.87^{-0.04}_{+0.03}&46&0.85^{-0.04}_{+0.02}\\\
21&-0.06^{+0.01}_{-0.01}&&&51&-11.49^{+0.18}_{-0.09}&28&0.93^{+0.98}_{-1.25}&81&\equiv
0&&\\\
22&0.27^{+0.19}_{-0.25}&12&0.15^{+0.18}_{-0.24}&52&-5.04^{-0.67}_{+0.93}&&&82&-7.13^{-0.32}_{+0.51}&47&-6.73^{-0.29}_{+0.47}\\\
23&\equiv
0&&&53&-11.99^{-0.87}_{+1.33}&29&-11.01^{-0.81}_{+1.23}&83&0.07^{+0.20}_{-0.27}&48&-0.22^{+0.18}_{-0.25}\\\
24&1.62^{+0.04}_{-0.07}&&&54&\equiv 0&&&84&\equiv 0&&\\\
25&-5.98^{-0.49}_{+0.72}&13&-5.39^{-0.45}_{+0.66}&55&16.79^{+0.96}_{-1.49}&30&15.72^{+0.89}_{-1.38}&85&-0.82^{+0.03}_{-0.02}&49&-0.78^{+0.03}_{-0.01}\\\
26&3.35^{+0.29}_{-0.47}&14&4.17^{+0.30}_{-0.49}&56&19.34^{+0.52}_{-0.98}&31&17.57^{+0.42}_{-0.82}&86&\equiv
0&&\\\
27&-1.54^{+0.15}_{-0.18}&15&-2.71^{+0.21}_{-0.25}&57&7.92^{+1.34}_{-1.85}&32&7.18^{+1.28}_{-1.76}&87&7.57^{+0.37}_{-0.60}&50&7.18^{+0.34}_{-0.55}\\\
28&0.30^{+0.01}_{-0.01}&&&58&\equiv
0&&&88&-5.47^{-0.73}_{+1.03}&51&-4.85^{-0.69}_{+0.97}\\\
29&-3.08^{-0.26}_{+0.32}&16&-2.22^{-0.22}_{+0.27}&59&-22.49^{-1.21}_{+1.89}&33&-21.19^{-1.12}_{+1.76}&89&34.74^{+1.61}_{-2.62}&52&32.19^{+1.46}_{-2.37}\\\
30&0.60^{+0.02}_{-0.03}&&&60&\equiv
0&&&90&2.44^{-0.38}_{+0.46}&53&2.51^{-0.37}_{+0.46}\\\ \hline\cr\end{array}$
(131)
TABLE IV. The obtained values of the $p^{6}$ order LECs $C_{1}\cdots,C_{90}$
for three flavor and $c_{1}\cdots,c_{53}$ for two flavors. The LECs are in
units of $10^{-3}\mathrm{GeV}^{-2}$ and are taken the values at
$\Lambda=\infty$. The value $\equiv 0$ means that the constants vanish at
large $N_{c}$. $\displaystyle\hskip
71.13188pt\begin{array}[]{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline\cr
i&C_{i}&j&c_{j}&i&C_{i}&j&c_{j}&i&C_{i}&j&c_{j}\\\ \hline\cr
1&3.61&1&3.39&31&-0.22&17&-0.22&61&1.36&34&1.45\\\ \hline\cr 2&\equiv
0&&&32&0.02&18&0.09&62&\equiv 0&&\\\ \hline\cr
3&-0.01&2&0.00&33&0.08&19&0.09&63&1.41&&\\\ \hline\cr
4&2.98&3&2.77&34&1.03&20&0.97&64&\equiv 0&&\\\ \hline\cr
5&-0.51&4&0.66&35&-0.40&21&-0.46&65&-1.28&35&1.56\\\ \hline\cr 6&\equiv
0&&&36&\equiv 0&&&66&1.73&36&1.58\\\ \hline\cr 7&\equiv
0&&&37&-0.06&&&67&\equiv 0&&\\\ \hline\cr 8&1.16&&&38&-0.01&22&-0.25&68&\equiv
0&&\\\ \hline\cr 9&\equiv 0&&&39&\equiv 0&23&0.20&69&-0.90&38&-0.71\\\
\hline\cr 10&-0.49&5&-0.49&40&-6.10&24&-4.70&70&0.91&39&1.08\\\ \hline\cr
11&\equiv 0&&&41&\equiv 0&&&71&\equiv 0&&\\\ \hline\cr
12&-0.19&6&-0.20&42&0.49&&&72&-2.43&40&-2.37\\\ \hline\cr 13&\equiv
0&&&43&\equiv 0&&&73&2.47&41&2.08\\\ \hline\cr
14&-0.26&7&-0.42&44&6.17&25&5.86&74&-4.96&42&-4.61\\\ \hline\cr 15&\equiv
0&8&0.20&45&\equiv 0&&&75&\equiv 0&&\\\ \hline\cr 16&\equiv
0&&&46&-0.58&26&-1.11&76&-2.33&43&-2.08\\\ \hline\cr
17&-0.15&9&-0.29&47&0.08&&&77&\equiv 0&&\\\ \hline\cr
18&-0.06&&&48&3.13&&&78&18.97&44&17.41\\\ \hline\cr
19&-0.08&10&-0.09&49&\equiv 0&&&79&-1.81&45&-0.89\\\ \hline\cr
20&0.02&11&\equiv 0&50&10.73&27&17.28&80&0.49&46&0.52\\\ \hline\cr
21&-0.01&&&51&-8.65&28&4.93&81&\equiv 0&&\\\ \hline\cr
22&1.11&12&0.91&52&-7.24&&&82&-7.27&47&-6.83\\\ \hline\cr 23&\equiv
0&&&53&-13.65&29&-12.49&83&0.96&48&0.59\\\ \hline\cr 24&1.55&&&54&\equiv
0&&&84&\equiv 0&&\\\ \hline\cr
25&-7.21&13&-6.46&55&18.10&30&16.83&85&-0.47&49&-0.49\\\ \hline\cr
26&3.93&14&4.68&56&17.99&31&16.33&86&\equiv 0&&\\\ \hline\cr
27&-0.60&15&-1.42&57&12.69&32&11.45&87&7.83&50&7.39\\\ \hline\cr
28&0.29&&&58&\equiv 0&&&88&-7.83&51&-6.96\\\ \hline\cr
29&-3.81&16&-2.78&59&-23.88&33&-22.35&89&35.69&52&32.93\\\ \hline\cr
30&0.58&&&60&\equiv 0&&&90&0.25&53&0.51\\\ \hline\cr\end{array}$ (163)
## VII Comparisons with Experiment and model results
As we have mentioned in the introduction of this paper, present experiment
data is far enough to fix the $p^{6}$ order LECs. But there do exist some
combinations of the LECs which already have their experiment or model
calculation values. Usually, these LECs are labeled by dimensionless
parameters with convention 101010An alternative convention is that $C_{i}^{r}$
and $c^{r}_{i}$ are used to denote the renormalized LECs in some
literatures.of $C_{i}^{r}\equiv C_{i}F_{0}^{2}$ or $c_{i}^{r}\equiv
c_{i}F_{0}^{2}$. In this section, we collect those combinations of LECs in the
literature which have their experiment or model calculation values and compare
them with our numerical results obtained in the last section with finite
cutoff 111111If the LECs at finite cutoff are replaced with those at
$\Lambda=\infty$, we have checked that qualitative feature of the comparisons
results of this section will not change. as the check of our computations.
### VII.1 $\pi\pi$ and $\pi K$ scattering
From the investigation of $\pi\pi$ scattering amplitudes, one can work out the
values of some combinations of $p^{6}$ order LECs. Ref.p6-2 introduces
following combinations,
$\displaystyle r^{r}_{1}$ $\displaystyle=$ $\displaystyle
64c^{r}_{1}-64c^{r}_{2}+32c^{r}_{4}-32c^{r}_{5}+32c^{r}_{6}-64c^{r}_{7}-128c^{r}_{8}-64c^{r}_{9}+96c^{r}_{10}+192c^{r}_{11}-64c^{r}_{14}+64c^{r}_{16}+96c^{r}_{17}+192c^{r}_{18}$
$\displaystyle r^{r}_{2}$ $\displaystyle=$
$\displaystyle-96c^{r}_{1}+96c^{r}_{2}+32c^{r}_{3}-32c^{r}_{4}+32c^{r}_{5}-64c^{r}_{6}+32c^{r}_{7}+64c^{r}_{8}+32c^{r}_{9}-32c^{r}_{13}+32c^{r}_{14}-64c^{r}_{16}$
$\displaystyle r^{r}_{3}$ $\displaystyle=$ $\displaystyle
48c^{r}_{1}-48c^{r}_{2}-40c^{r}_{3}+8c^{r}_{4}-4c^{r}_{5}+8c^{r}_{6}-8c^{r}_{12}+20c^{r}_{13}$
$\displaystyle r^{r}_{4}$ $\displaystyle=$
$\displaystyle-8c^{r}_{3}+4c^{r}_{5}-8c^{r}_{6}+8c^{r}_{12}-4c^{r}_{13}$
$\displaystyle r^{r}_{5}$ $\displaystyle=$
$\displaystyle-8c^{r}_{1}+10c^{r}_{2}+14c^{r}_{3}$ $\displaystyle r^{r}_{6}$
$\displaystyle=$ $\displaystyle 6c^{r}_{2}+2c^{r}_{3}$ (164)
and gives the values of them by two theoretical methods of the resonance-
saturation (RS)pipi and pure dimensional analysis (ND) which only accounts
for the order of magnitude and in Table.V.
TABLE V. The obtained values for the combinations of the $p^{6}$ order LECs
from $\pi\pi$ scattering and our work. The coefficients in the table are in
units of $10^{-4}$ $\displaystyle\hskip
56.9055pt\begin{array}[]{lcccccc}\hline\cr&r^{r}_{1}&r^{r}_{2}&r^{r}_{3}&r^{r}_{4}&r^{r}_{5}&r^{r}_{6}\\\
\hline\cr\mbox{RS in Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}&-0.6&1.3&-1.7&-1.0&1.1&0.3\\\
\mbox{ND in Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}&80&40&20&3&6&2\\\
\hline\cr\mbox{ours}&-9.32^{-2.62}_{+3.51}&8.93^{+3.12}_{-4.27}&-3.06^{-0.81}_{+1.11}&-0.12^{+.022}_{-0.29}&0.87^{+0.04}_{-0.06}&0.42^{+0.02}_{-0.03}\\\
\hline\cr\end{array}$ (169)
We see that all coefficients obtained from our calculations are consistent
with those more precise RS results given in Ref.p6-2 . With our predictions
for $p^{4}$ and $p^{6}$ order LECs, we can directly calculate the scattering
lengths $a_{l}^{I}$ and slope parameters $b_{l}^{I}$ which relate to $p^{4}$
and $p^{6}$ order LECs through formulae given in Appendix C and D. of Ref.pipi
. We list experimental and our results in Table.VI. In our results, as done in
Table.III, we take $\mu=770$MeV, but to match the result given in Ref.pipi
where $\mu$ is taken at $\mu=1$GeV, we also take $\mu=1000$MeV for comparison.
We take two options, one only includes $p^{4}$ order contributions and the
other combines in $p^{6}$ order contributions. For $p^{6}$ order
contributions, for comparisons, we consider the cases of without and with
$r_{i}^{r}$ coefficients.
TABLE VI. The obtained values for $a_{l}^{I}$ and $b_{l}^{I}$ in $\pi\pi$
scattering from experimental values given by Ref.pipi1 and our work.
$\displaystyle\begin{array}[]{ccccccccc}\hline\cr&a_{0}^{0}&b_{0}^{0}&-10a_{0}^{2}&-10b_{0}^{2}&10a_{1}^{1}&10^{2}b_{1}^{1}&10^{2}a_{2}^{0}&10^{3}a_{2}^{2}\\\
\hline\cr\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{pipi1}{\@@citephrase{(}}{\@@citephrase{)}}}}&.26\pm.05&.25\pm.03&.28\pm.12&.82\pm.08&.38\pm.02&&.17\pm.03&.13\pm.30\\\
\hline\cr
p^{4}~{}\mu=10^{3}\mathrm{MeV}&.210^{-.000}_{+.000}&.260^{-.000}_{+.000}&.406^{+.001}_{-.001}&.662^{-.002}_{+.003}&.405^{+.001}_{-.002}&.772^{+.015}_{-.020}&.264^{+.002}_{-.003}&.195^{-.009}_{+.012}\\\
p^{4}~{}\mu=770\mathrm{MeV}&.204^{-.000}_{+.000}&.248^{-.000}_{+.000}&.411^{+.001}_{-.001}&.685^{-.002}_{+.003}&.401^{+.001}_{-.002}&.772^{+.015}_{-.020}&.235^{+.002}_{-.003}&.076^{-.009}_{+.012}\\\
\hline\cr p^{6}~{}\mu=10^{3}\mathrm{MeV}~{}r^{r}_{i}\neq
0&.237^{-.000}_{+.000}&.307^{-.000}_{+.000}&.394^{+.001}_{-.001}&.637^{-.004}_{+.005}&.447^{+.002}_{-.003}&1.255^{+.029}_{-.037}&.421^{+.005}_{-.008}&.339^{-.011}_{+.011}\\\
p^{6}~{}\mu=10^{3}\mathrm{MeV}~{}r^{r}_{i}=0&.237^{-.000}_{+.000}&.305^{-.000}_{-.000}&.392^{+.001}_{-.001}&.629^{-.003}_{+.004}&.445^{+.002}_{-.002}&1.217^{+.019}_{-.024}&.409^{+.004}_{-.006}&.337^{-.005}_{+.003}\\\
p^{6}~{}\mu=770\mathrm{MeV}~{}r^{r}_{i}\neq
0&.228^{-.000}_{+.000}&.287^{-.000}_{+.000}&.402^{+.001}_{-.001}&.665^{-.003}_{+.005}&.435^{+.002}_{-.003}&1.164^{+.028}_{-.037}&.363^{+.005}_{-.008}&.212^{-.012}_{+.012}\\\
p^{6}~{}\mu=770\mathrm{MeV}~{}r^{r}_{i}=0&.227^{-.000}_{+.000}&.285^{-.000}_{-.000}&.400^{+.001}_{-.001}&.657^{-.003}_{+.004}&.433^{+.002}_{-.002}&1.125^{+.018}_{-.023}&.352^{+.004}_{-.006}&.210^{-.006}_{+.004}\\\
\hline\cr\end{array}$ (178)
We see that the contributions from $p^{6}$ order LECs are rather small and
only change the third digit of the result.
Further, Ref.piK0 introduce coefficients in $\pi K$ scattering
$\displaystyle c_{01}^{-}$ $\displaystyle=$ $\displaystyle
32m_{K^{+}}^{3}(-C_{1}^{r}+2C_{3}^{r}+2C_{4}^{r})\;,\hskip
85.35826ptc_{20}^{-}=6m_{K^{+}}(-C_{1}^{r}+2C_{3}^{r}+2C_{4}^{r})\;,$
$\displaystyle c_{11}^{+}$ $\displaystyle=$ $\displaystyle
8m_{K^{+}}^{2}(3C_{1}^{r}+6C_{3}^{r}-2C_{4}^{r})\;,\hskip
93.89418ptc_{30}^{+}=\frac{1}{2}(-7C_{1}^{r}-32C_{2}^{r}+2C_{3}^{r}+10C_{4}^{r})\;,$
$\displaystyle c_{01}^{+}$ $\displaystyle=$ $\displaystyle
16m^{2}_{K^{+}}m^{2}_{\pi^{+}}(C_{6}^{r}+C_{8}^{r}+C^{r}_{10}+2C^{r}_{11}-2C^{r}_{12}-2C^{r}_{13}+2C^{r}_{22}+4C^{r}_{23})+16m^{4}_{K^{+}}(C^{r}_{5}+2C^{r}_{6}+C^{r}_{10}$
$\displaystyle+4C^{r}_{11}-2C^{r}_{12}-4C^{r}_{13}+2C^{r}_{22}+4C^{r}_{23})\;,$
$\displaystyle c_{10}^{-}$ $\displaystyle=$ $\displaystyle
8m_{K^{+}}m^{2}_{\pi^{+}}(-4C_{4}^{r}-C_{6}^{r}-C_{8}^{r}+C^{r}_{10}+2C_{11}^{r}-2C_{12}^{r}-6C_{13}^{r}+2C^{r}_{22}-2C^{r}_{25})+8m_{K^{+}}^{3}(-4C_{4}^{r}-C_{5}^{r}$
$\displaystyle-2C_{6}^{r}+C_{10}^{r}+4C_{11}^{r}-2C_{12}^{r}-12C_{13}^{r}+2C_{22}^{r}-2C_{25}^{r})\;,$
$\displaystyle c_{20}^{+}$ $\displaystyle=$ $\displaystyle
m_{K^{+}}^{2}(12C_{1}^{r}+48C_{2}^{r}-8C_{4}^{r}+C_{5}^{r}+10C_{6}^{r}+8C_{7}^{r}+4C_{8}^{r}+C_{10}^{r}+4C_{11}^{r}-2C_{12}^{r}-4C_{13}^{r}+2C_{22}^{r}$
(179)
$\displaystyle-4C_{23}^{r}+4C_{25}^{r})+m_{\pi^{+}}^{2}(12C_{1}^{r}+48C_{2}^{r}-8C_{4}^{r}+4C_{5}^{r}+5C_{6}^{r}+8C_{7}^{r}+C_{8}^{r}+C_{10}^{r}+2C_{11}^{r}-2C_{12}^{r}$
$\displaystyle-10C_{13}^{r}+2C_{22}^{r}-4C_{23}^{r}+4C_{25}^{r})\;.$
In the table 1. of Ref.piK , in terms of
$c^{+}_{30},c^{+}_{11},c^{-}_{20},c^{-}_{01}$, three constraints of $p^{6}$
order LECs are fixed from $\pi K$ subthreshold parameters, $\pi\pi$ amplitude
and a resonance model. And in the table 2. of Ref.piK , in terms of
$c^{+}_{20},c^{+}_{01},c^{-}_{10}$, another three constraints of $p^{6}$ order
LECs are fixed from the dispersive calculations and a resonance model,
TABLE VII. The obtained values for the combinations of the $p^{6}$ order LECs
from $\pi K$, $\pi\pi$ scattering and our work. The coefficients in the l.h.s.
of the table are in units of $10^{-4}\mathrm{GeV}^{-2}$
$\displaystyle\begin{array}[]{lcccc|lccc}\hline\cr&C_{1}+4C_{3}&C_{2}&C_{4}+3C_{3}&C_{1}+4C_{3}+2C_{2}&&c_{20}^{+}\frac{m_{\pi}^{4}}{F_{\pi}^{4}}&c_{01}^{+}\frac{m_{\pi}^{2}}{F_{\pi}^{4}}&c_{10}^{-}\frac{m_{\pi}^{3}}{F_{\pi}^{4}}\\\
\hline\cr\mbox{input~{}}c^{+}_{30},c^{+}_{11},c^{-}_{20}&20.7\pm 4.9&-9.2\pm
4.9&9.9\pm 2.5&2.3\pm 10.8&&&\\\
\mbox{input~{}}c^{+}_{30},c^{+}_{11},c^{-}_{01}&28.1\pm 4.9&-7.4\pm
4.9&21.0\pm 2.5&13.4\pm 10.8&\mbox{Dispersive}&0.024\pm 0.006&2.07\pm
0.10&0.31\pm 0.01\\\ \pi\pi\mbox{~{}amplitude}&&&23.5\pm 2.3&18.8\pm 7.2&&&\\\
\mbox{Resonance
model}&7.2&-0.5&10.0&6.2&\mbox{Resonance~{}model}&0.003&3.8&0.09\\\
\hline\cr\mbox{ours}&35.9^{+1.3}_{-2.1}&0.0^{+0.0}_{-0.0}&29.5^{+1.1}_{-1.9}&35.9^{+1.3}_{-2.1}&\mbox{ours}&0.006^{-0.002}_{+0.003}&-0.159^{+0.133}_{-0.178}&0.020^{+0.037}_{-0.050}\\\
\hline\cr\end{array}$ (186)
In which for l.h.s. of the Table VII., except $C_{2}$, all other LECs or
combinations of LECs obtained by us have the same signs and orders of
magnitudes as those from Ref.piK . While for r.h.s. of the table, our results
are not consistent with those obtained from the dispersive calculations.
### VII.2 Form factors
In Ref.p6-2 , in dealing with the vector form factor of the pion, $r_{V1}^{r}$
and $r_{V2}^{r}$ are introduced into theory which relate to $p^{6}$ order LECs
through
$\displaystyle r_{V1}^{r}=-16c_{6}^{r}-4c_{35}^{r}-8c_{53}^{r}\;,\hskip
85.35826ptr_{V2}^{r}=-4c_{51}^{r}-4c_{53}^{r}\;.$ (187)
While for the scalar form factor, people introduce $r_{S2}^{r}$ and
$r_{S3}^{r}$ relate to $p^{6}$ order LECs by
$\displaystyle
r_{S2}^{r}=32c_{6}^{r}+16c_{7}^{r}+32c_{8}^{r}+16c_{9}^{r}+16c_{20}^{r}\;,\hskip
85.35826ptr_{S3}^{r}=-8c_{6}^{r}\;.$ (188)
In Ref.p6-2 , discussion of the decay of $\pi(p)\rightarrow e\nu\gamma(q)$
further introduces $r_{A1}^{r}$ and $r_{A2}^{r}$ relate to $p^{6}$ order LECs
by
$\displaystyle
r^{r}_{A1}=48c^{r}_{6}-16c^{r}_{34}+8c^{r}_{35}-8c^{r}_{44}+16c^{r}_{46}-16c^{r}_{47}+8c^{r}_{50}\;,\hskip
56.9055ptr^{r}_{A2}=8c^{r}_{44}-16c^{r}_{50}+4c^{r}_{51}\;.$ (189)
In Ref.Kl3 , a naive estimation of $C_{12}^{r}$ is made from scalar meson
dominance (SMD) of the pion scalar form-factor and $2C_{12}^{r}+C_{34}^{r}$ is
estimated through $\lambda_{0}$ in $K_{l3}$ measurements (see Eq.(8.11) in
Ref.Kl3 ). While in Ref.Kpi , $C^{r}_{12}$ and $C^{r}_{12}+C^{r}_{34}$ are
also estimated from the $\pi K$ form factors. In Table.VIII, we list the
numerical results for above combinations of LECs given by our calculations
based on Table.III in last section and by Ref.p6-2 ,Kl3 ,Kpi .
TABLE VIII. The obtained values for the combinations of the $p^{6}$ order LECs
appear in vector and scalar form factor of pion. the coefficients in the table
are in units of $10^{-4}$ $\displaystyle\hskip
71.13188pt\begin{array}[]{ccc|ccc|ccc}\hline\cr&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}&&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}&&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-2}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
r^{r}_{V1}&-2.13^{+0.30}_{-0.39}&-2.5&r^{r}_{S2}&0.07^{+0.05}_{-0.08}&-0.3&r_{A1}^{r}&1.14^{+0.07}_{-0.09}&-0.5\\\
r^{r}_{V2}&2.23^{+0.10}_{-0.16}&2.6&r^{r}_{S3}&0.20^{-0.01}_{+0.01}&0.6&r_{A2}^{r}&-0.38^{-0.06}_{+0.08}&1.1\\\
\hline\cr\end{array}$ (193) $\displaystyle\hskip
51.21504pt\begin{array}[]{ccc|ccc}\hline\cr&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kl3}{\@@citephrase{(}}{\@@citephrase{)}}}}&&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kpi}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
C_{12}^{r}&-0.026^{+0.001}_{-0.001}&-0.1&C^{r}_{12}&-0.026^{+0.001}_{-0.001}&(0.3\pm
5.4)\times 10^{-3}\\\
2C_{12}^{r}\\!\\!+C_{34}^{r}&0.068^{-0.006}_{+0.010}&-0.10\pm
0.17&C_{12}^{r}\\!\\!+C_{34}^{r}&0.094^{-0.007}_{+0.011}&(3.2\pm 1.5)\times
10^{-2}\\\ \hline\cr\end{array}$ (197)
From which we see that among ten parameters between our predictions and values
given in the literature, four of them have the same orders of magnitudes and
signs ($r^{r}_{V1}$, $r^{r}_{V2}$, $r^{r}_{S3}$ and
$C_{12}^{r}\\!\\!+C_{34}^{r}$), another one of them has different orders of
magnitudes but the same signs ($C_{12}^{r}$ in Ref.Kl3 ) , the left five of
them have opposite signs ($r^{r}_{S2}$, $r^{r}_{A1}$, $r^{r}_{A2}$,
$2C_{12}^{r}\\!\\!+C_{34}^{r}$ and $C_{12}^{r}$ in Ref.Kpi ).
Further in Fig.1, we compare the experimental data for vector form factors
collected in Figure 4. and Figure 5. of Ref.Fpi with our results. In
obtaining our numerical predictions, we have exploited the formula given by
Eq.(3.16) in Ref.Fpi which especially depends on $p^{6}$ order LECs through
$r_{V1}^{r},r_{V2}^{r}$ defined in (187) and we input the formula $p^{4}$ and
$p^{6}$ LECs obtained in Table.III of the last section.
Figure 1: The space like and time like data for the vector form factor.
The red solid curve corresponds to predictions from chiral perturbation up to
$p^{6}$ order with LECs obtained in Table.III of this paper. The red dashed
line is the result by vanishing $p^{6}$ order LECs in corresponding red solid
curve. The blue dot-dashed curve corresponds to predictions from chiral
perturbation up to $p^{4}$ order with LECs obtained in Table.I of this paper.
The blue dotted line is the result by vanishing $p^{4}$ order LECs in
corresponding blue dot-dashed curve.The black x-axis of with
$|F_{\pi}^{V}|^{2}=1.0$ corresponds to predictions from $p^{2}$ order chiral
perturbation.
From Fig.1, we see that $p^{6}$ order LECs explicitly improve the $p^{4}$ and
$p^{2}$ order chiral perturbation predictions and making them being more
consistent with experimental data.
### VII.3 Photon-Photon Collisions
In Ref.pigamma , discussion of the photon-photon collision
$\gamma\gamma\rightarrow\pi^{0}\pi^{0}$ introduces $a^{r}_{1},a_{2}^{r}$ and
$b^{r}$ relate to $p^{6}$ order LECs by
$\displaystyle a^{r}_{1}=4096\pi^{4}(-c^{r}_{29}-c^{r}_{30}+c^{r}_{34})\hskip
17.07182pta^{r}_{2}=256\pi^{4}(8c^{r}_{29}+8c^{r}_{30}+c^{r}_{31}+c^{r}_{32}+2c^{r}_{33})\hskip
17.07182ptb^{r}=-128\pi^{4}(c^{r}_{31}+c^{r}_{32}+2c^{r}_{33})\;.~{}~{}~{}~{}~{}~{}$
(198)
While in Ref.pigamma1 , calculation of the photon-photon collision
$\gamma\gamma\rightarrow\pi^{+}\pi^{-}$ introduces another type of
$a^{r}_{1},a_{2}^{r}$ and $b^{r}$ relate to $p^{6}$ order LECs by
$\displaystyle a^{r}_{1}$ $\displaystyle=$
$\displaystyle-4096\pi^{4}(6c^{r}_{6}+c^{r}_{29}-c^{r}_{30}-3c^{r}_{34}+c^{r}_{35}+2c^{r}_{46}-4c^{r}_{47}+c^{r}_{50})\;,$
$\displaystyle a^{r}_{2}$ $\displaystyle=$ $\displaystyle
256\pi^{4}(8c^{r}_{29}-8c^{r}_{30}+c^{r}_{31}+c^{r}_{32}-2c^{r}_{33}+4c^{r}_{44}+8c^{r}_{50}-4c^{r}_{51})\;,$
(199) $\displaystyle b^{r}$ $\displaystyle=$
$\displaystyle-128\pi^{4}(c^{r}_{31}+c^{r}_{32}-2c^{r}_{33}-4c^{r}_{44})\;.$
In Table.IX, we list the numerical results for above combinations of LECs
given by our calculations based on Table.III in last section and by
Ref.pigamma and pigamma1 .
TABLE IX. The obtained values for the combinations of the $p^{6}$ order LECs
appear in photon-photon collisions. $\displaystyle\hskip
113.81102pt\begin{array}[]{ccc|ccc}\hline\cr&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{pigamma}{\@@citephrase{(}}{\@@citephrase{)}}}}&&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{pigamma1}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr a^{r}_{1}&-5.65^{-0.91}_{+1.23}&-14\pm
5&a^{r}_{1}&-5.86^{-0.49}_{+0.58}&-3.2\\\ a^{r}_{2}&3.79^{+0.02}_{-0.05}&7\pm
3&a^{r}_{2}&-0.98^{-0.07}_{+0.12}&0.7\\\ b^{r}&1.66^{+0.05}_{-0.09}&3\pm
1&b^{r}&-0.23^{-0.01}_{+0.02}&0.4\\\ \hline\cr\end{array}$ (204)
For which we see that among six parameters between our predictions and values
given in the literature, except two have opposite signs, other four all have
the same orders of magnitudes and signs.
### VII.4 Radiative pion decay
In Ref.PiDecay , through reanalysis of the radiative pion decay, a group of
$p^{6}$ order LECs are fixed.
TABLE X. The obtained values for the combinations of the $p^{6}$ order LECs
from pion radiative decay and our work. The coefficients in the table are in
units of $10^{-5}$ $\displaystyle\hskip
42.67912pt\begin{array}[]{lcccccc}\hline\cr&C^{r}_{12}&C^{r}_{13}&C^{r}_{61}&C^{r}_{62}&2C^{r}_{63}-C^{r}_{65}&C^{r}_{64}\\\
\hline\cr\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PiDecay}{\@@citephrase{(}}{\@@citephrase{)}}}}&-0.6\pm
0.3&0\pm 0.2&1.0\pm 0.3&0\pm 0.2&1.8\pm 0.7&0\pm 0.2\\\
\hline\cr\mbox{ours}&-0.26^{+0.01}_{-0.01}&0.0^{+0.0}_{-0.0}&2.18^{-0.17}_{+0.20}&0.0^{+0.0}_{-0.0}&6.36^{-0.42}_{+0.56}&0.0^{+0.0}_{-0.0}\\\
\hline\cr\hline\cr&C^{r}_{78}&C^{r}_{80}&C^{r}_{81}&C^{r}_{82}&C^{r}_{87}&C^{r}_{88}\\\
\hline\cr\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PiDecay}{\@@citephrase{(}}{\@@citephrase{)}}}}&10.0\pm
3.0&1.8\pm 0.4&0\pm 0.2&-3.5\pm 1.0&3.6\pm 1.0&-3.5\pm 1.0\\\
\hline\cr\mbox{ours}&13.26^{+0.77}_{-1.20}&0.66^{-0.03}_{+0.02}&0.0^{+0.0}_{-0.0}&-5.39^{-0.24}_{+0.39}&5.73^{+0.28}_{-0.45}&-4.14^{-0.55}_{+0.78}\\\
\hline\cr\end{array}$ (211)
We find that all LECs and combination of LECs from our predictions have the
same signs and orders of magnitudes as those from experiment values.
### VII.5 Model calculations
Except above phenomenological estimations on the values of some LECs, there
are model calculations for some others of them and most of these analysis use
a (single) resonance approximation. In contrast, our calculations do not rely
on the assumption of existence of resonances. In this subsection, we list down
these calculation values we can collect from the literature and compare with
our results.
Ref.TwoPoint estimates values of some LECs.
TABLE XI. The obtained values for the $p^{6}$ order LEC in Ref.TwoPoint and
our works The coefficients in the table are in units of
$10^{-3}\mathrm{GeV}^{-2}$ $\displaystyle\hskip
56.9055pt\begin{array}[]{ccccccc}\hline\cr&C^{r}_{14}&C^{r}_{19}&C^{r}_{38}&C^{r}_{61}&C^{r}_{80}&C^{r}_{87}\\\
\hline\cr\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{TwoPoint}{\@@citephrase{(}}{\@@citephrase{)}}}}&-4.3&-2.8&1.2&1.9&1.9&7.6\\\
\mbox{ours}&-0.83^{+0.12}_{-0.19}&-0.48^{+0.09}_{-0.13}&0.41^{-0.08}_{+0.07}&2.88^{-0.22}_{+0.26}&0.87^{-0.04}_{+0.03}&7.57^{+0.37}_{-0.60}\\\
\hline\cr\end{array}\hskip 85.35826pt$ (215)
For $C_{63}^{r}$ and $C_{65}^{r}$, Ref.C63-65 gives the value for their
combination $2C_{63}^{r}-C_{65}^{r}=(1.8\pm 0.7)\times 10^{-5}$ which,
compares to our result of $6.36^{-0.48}_{+0.56}\times 10^{-5}$, is at the same
order of magnitude and has the same sign.
For $C^{r}_{87}$, there are several works to estimate its values, we list them
in Table.XII.
TABLE XII. The obtained values for the $p^{6}$ order LEC $C^{r}_{87}$ The
coefficients in the table are in units of $10^{-5}$ $\displaystyle\hskip
56.9055pt\begin{array}[]{ccccc}\hline\cr&\mbox{ours}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{C87-1}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{C87-2}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{C87-3}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
C^{r}_{87}&5.73^{+0.28}_{-0.45}&3.1\pm 1.1&4.3\pm 0.4&3.70\pm 0.14\\\
\hline\cr\end{array}\hskip 85.35826pt$ (218)
where $C_{87}^{r}$ given in Ref.C87-2 and C87-3 are in form of $C_{87}$ in
unit of GeV-2, we have transformed them into our expression of $C^{r}_{87}$
with $C_{87}^{r}=C_{87}F_{0}^{2}$.
Further, Ref.RL exploits resonance Lagrangian estimates values of LECs
$C_{78}$, $C_{82}$, $C_{87}$, $C_{88}$, $C_{89}$, $C_{90}$.
TABLE XIII. The obtained values for the $p^{6}$ order LEC from resonance
Lagrangian given by Ref.RL and our work The coefficients in the table are in
units of $10^{-4}/F_{0}^{2}$
$\displaystyle\begin{array}[]{ccccccc}\hline\cr&C_{78}&C_{82}&C_{87}&C_{88}&C_{89}&C_{90}\\\
\hline\cr\mbox{Lowest Meson Dominance}&1.09&-0.36&0.40&-0.52&1.97&0.0\\\
\mbox{Resonance Lagrangian I }&1.09&-0.29&0.47&-0.16&2.29&0.33\\\
\mbox{Resonance Lagrangian II}&1.49&-0.39&0.65&-0.14&3.22&0.51\\\
\hline\cr\mbox{ours}&1.326^{+0.077}_{-0.120}&-0.539^{-0.024}_{+0.039}&0.573^{+0.028}_{-0.045}&-0.414^{-0.055}_{+0.078}&2.630^{+0.122}_{-0.198}&0.185^{-0.029}_{+0.035}\\\
\hline\cr\end{array}$ (224)
We find that our results are consistent with those obtained from resonance
Lagrangian.
Ref.C38 estimates the value of $C_{38}$ and gives $C^{r}_{38}=(2\pm 6)\times
10^{-6}$ which is also consistent with our result of
$C^{r}_{38}=3.1^{-0.6}_{+0.6}\times 10^{-6}$.
In terms of resonance exchange, Ref.Relation proposes some relations among
different $p^{6}$ order LECs,
$\displaystyle C_{20}=-3C_{21}=C_{32}=\frac{1}{6}C_{35}\hskip
85.35826ptC_{24}=6C_{28}=3C_{30}\;.$ (225)
To check the validity of these relations for our results, in Table. XIV, we
write corresponding values obtained in our calculations
TABLE XIV. The obtained values for the $p^{6}$ order LEC from our work The
coefficients in the table are in units of $10^{-3}\mathrm{GeV}^{-2}$
$\displaystyle\begin{array}[]{cccc|ccc}\hline\cr
C_{20}&-3C_{21}&C_{32}&\frac{1}{6}C_{35}&C_{24}&6C_{28}&3C_{30}\\\ \hline\cr
0.18^{-0.03}_{+0.04}&0.18^{-0.03}_{+0.03}&0.18^{-0.03}_{+0.04}&0.028^{-0.020}_{+0.028}&1.62^{+0.04}_{-0.07}&1.80^{+0.06}_{-0.06}&1.80^{+0.06}_{-0.09}\\\
\hline\cr\end{array}$ (228)
We see that except $C_{35}$, all the other LECs satisfy the relations.
## VIII Summary
In this paper, we revise our original formulation of calculating LECs from the
first principle of QCD to a chiral covariant one suitable to computerize. With
the help of computer, we successfully obtain the analytical expressions for
all the $p^{6}$ order LECs in the normal part of chiral Lagrangian for pseudo
scalar mesons on the quark self energy $\Sigma(k^{2})$. The ambiguities for
anomaly part contributions to the normal part of the chiral Lagrangian are
clarified and we prove that this part totally should vanish and therefore need
not to be considered in our computations. Since our calculation is done under
large $N_{c}$ limit, only operators of $p^{6}$ order with one trace and some
multi-traces from the equation of motion survive in our formulation. We set up
relations among the coefficients in front of these operators and LECs defined
in Ref.p6-1 . Then with input of $F_{0}=$87MeV to fix the
$\Lambda_{\mathrm{QCD}}$ in the running coupling constant of
$\alpha_{s}(k^{2})$ appear in the kernel of SDE and choose cutoff of the
theory being $\Lambda=1000^{+100}_{-100}$MeV and $\Lambda=\infty$, we
calculate all $p^{6}$ order LECs numerically both for two flavor and three
flavor cases. Compare our result LECs with those combinations which we can
find experimental or model calculation values in the literature, we find that
except few of them have wrong signs, most of our predicted combinations of
$p^{6}$ order LECs have the same signs and orders of magnitudes with
experiment or model calculation values. This sets the solid basis for our
$p^{6}$ order computations. For those combinations with wrong signs or wrong
order of magnitudes with experiment values, we need further investigations.
Based on these obtained $p^{6}$ order LECs, we expect a very large number of
predictions for various pseudo scalar meson physics in the near future.
## Acknowledgments
This work was supported by National Science Foundation of China (NSFC) under
Grant No.10875065. We thank Prof. Y.P.Kuang for the helpful discussions.
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## Appendix A Low Energy Expansion for $e^{B}$,
$\Sigma\big{(}(k+\tilde{F})^{2}\big{)}$ and $B$
In this appendix, we first list down the $p^{3}$, $p^{4}$, $p^{5}$ and $p^{6}$
order low energy expansion result for $e^{B}$ used in (20).
$\displaystyle\frac{d^{3}}{dt^{3}}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle\frac{d^{2}}{dt^{2}}e^{B(t)}\bigg{|}_{t=0}f[Ad(-B_{0})](B_{1})+\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}2\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+2f[Ad(-B_{0})](B_{2})\bigg{\\}}$
(229)
$\displaystyle+e^{B_{0}}\bigg{\\{}\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+2\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+f[Ad(-B_{0})](B_{3})\bigg{\\}}\;,$
$\displaystyle\frac{d^{4}}{dt^{4}}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle\frac{d^{3}}{dt^{3}}e^{B(t)}\bigg{|}_{t=0}~{}f[Ad(-B_{0})](B_{1})+\frac{d^{2}}{dt^{2}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}3\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+3f[Ad(-A_{0})](B_{2})\bigg{\\}}$
(230)
$\displaystyle+\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}3\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+6\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+3f[Ad(-B_{0})](B_{3})\bigg{\\}}$
$\displaystyle+e^{B_{0}}\bigg{\\{}\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+3\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})$
$\displaystyle+3\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})+f[Ad(-B_{0})](B_{4})\bigg{\\}}\;,$
$\displaystyle\frac{d^{5}}{dt^{5}}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle\frac{d^{4}}{dt^{4}}e^{B(t)}\bigg{|}_{t=0}~{}f[Ad(-B_{0})](B_{1})+\frac{d^{3}}{dt^{3}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}4\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+4f[Ad(-A_{0})](B_{2})\bigg{\\}}$
(231)
$\displaystyle+\frac{d^{2}}{dt^{2}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}6\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+12\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+6f[Ad(-B_{0})](B_{3})\bigg{\\}}$
$\displaystyle+\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}4\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+12\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+12\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})$
$\displaystyle+4f[Ad(-B_{0})](B_{4})\bigg{\\}}+e^{B_{0}}\bigg{\\{}\frac{d^{4}f}{dt^{4}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+4\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})$
$\displaystyle+6\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})+4\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{4})+f[Ad(-B_{0})](B_{5})\bigg{\\}}\;,$
$\displaystyle\frac{d^{6}}{dt^{6}}e^{B(t)}\bigg{|}_{t=0}$ $\displaystyle=$
$\displaystyle\frac{d^{5}}{dt^{5}}e^{B(t)}\bigg{|}_{t=0}f[Ad(-B_{0})](B_{1})+\frac{d^{4}}{dt^{4}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}5\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+5f[Ad(-B_{0})](B_{2})\bigg{\\}}$
(232)
$\displaystyle+\frac{d^{3}}{dt^{3}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}10\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+20\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+10f[Ad(-B_{0})](B_{3})\bigg{\\}}$
$\displaystyle+\frac{d^{2}}{dt^{2}}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}10\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+30\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})$
$\displaystyle+30\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})+10f[Ad(-B_{0})](B_{4})\bigg{\\}}+\frac{d}{dt}e^{B(t)}\bigg{|}_{t=0}\bigg{\\{}5\frac{d^{4}f}{dt^{4}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})$
$\displaystyle+20\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})+30\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})+20\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{4})$
$\displaystyle+5f[Ad(-B_{0})](B_{5})\bigg{\\}}+e^{B_{0}}\bigg{\\{}\frac{d^{5}f}{dt^{5}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{1})+5\frac{d^{4}f}{dt^{4}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{2})$
$\displaystyle+10\frac{d^{3}f}{dt^{3}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{3})+10\frac{d^{2}f}{dt^{2}}[Ad(-B(t))]\bigg{|}_{t=0}(B_{4})+5\frac{df}{dt}[Ad(-B(t))]\bigg{|}_{t=0}(B_{5})$
$\displaystyle+f[Ad(-B_{0})](B_{6})\bigg{\\}}\;.$
Then, we list down the $p^{3}$, $p^{4}$, $p^{5}$ and $p^{6}$ order low energy
expansion result for $\Sigma\big{(}(k+\tilde{F})^{2}\big{)}$ used in (35).
Note traceless terms in $p^{5}$ and $p^{6}$ orders are dropped out.
$\displaystyle\frac{1}{6}\bigg{[}\frac{d^{3}}{dt^{3}}\Sigma[A(t)]\bigg{]}_{t=0}$
$\displaystyle=$ $\displaystyle\frac{1}{6}e^{Ad(A_{0}\frac{\partial}{\partial
s})}\bigg{[}e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{2}}{dt^{2}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{1}\frac{\partial}{\partial s})$ (233)
$\displaystyle+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d}{dt}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}2\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+2f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{2}\frac{\partial}{\partial
s})\bigg{\\}}$
$\displaystyle+\bigg{\\{}\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+2\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial s})$
$\displaystyle+f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{3}\frac{\partial}{\partial
s})\bigg{\\}}\bigg{]}\Sigma(s+A(t))\bigg{|}_{s=0}$ $\displaystyle=$
$\displaystyle-\frac{i}{3}(\mu\underline{\mu}\nu)k_{\nu}\Sigma_{k}^{\prime\prime}+\frac{2i}{3}(\mu\nu\lambda)k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{2i}{3}(\mu\nu\lambda)k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+\frac{i}{3}(\mu\underline{\mu}\nu)\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}\;,$
$\displaystyle\bigg{[}\frac{d^{4}}{dt^{4}}\Sigma[A(t)]\bigg{]}_{t=0}$
$\displaystyle=$ $\displaystyle e^{Ad(A_{0}\frac{\partial}{\partial
s})}\bigg{[}e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{3}}{dt^{3}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}~{}f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{1}\frac{\partial}{\partial s})$ (234)
$\displaystyle+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{2}}{dt^{2}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}3\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+3f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{2}\frac{\partial}{\partial
s})\bigg{\\}}$ $\displaystyle+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d}{dt}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}3\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+6\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial s})$
$\displaystyle+3f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{3}\frac{\partial}{\partial
s})\bigg{\\}}+\bigg{\\{}\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+3\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial s})$
$\displaystyle+3\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial
s})+f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{4}\frac{\partial}{\partial
s})\bigg{\\}}\bigg{]}_{t=0}\Sigma(s+A(t))\bigg{|}_{s=0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}(\mu\nu)(\underline{\mu}\lambda)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{1}{2}(\mu\nu)(\lambda\rho)k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-\frac{1}{8}(\mu\nu\underline{\nu}\lambda)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle-\frac{1}{8}(\mu\nu\underline{\mu}\lambda)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+\frac{1}{4}(\mu\underline{\mu}\nu\lambda)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{1}{4}(\mu\nu\lambda\rho)k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle+\frac{1}{4}(\mu\nu\lambda\rho)k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+\frac{1}{4}(\mu\nu\lambda\rho)k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial
k^{\rho}}+\frac{1}{4}(\mu\nu)(\underline{\mu}\underline{\nu})\Sigma_{k}^{\prime\prime}$
$\displaystyle+\frac{1}{4}(\mu\nu)(\underline{\mu}\lambda)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+\frac{1}{4}(\mu\nu)(\underline{\mu}\lambda)\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+\frac{1}{8}(\mu\nu\underline{\nu}\lambda)k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+\frac{1}{8}(\mu\nu\underline{\nu}\lambda)\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+\frac{1}{8}(\mu\nu\underline{\mu}\lambda)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{1}{8}(\mu\nu\underline{\mu}\lambda)\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$
$\displaystyle-\frac{1}{6}(\mu\nu\underline{\nu}\lambda)k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-\frac{1}{6}(\mu\nu\underline{\mu}\lambda)k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{3}(\mu\nu\lambda\rho)k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{3}(\mu\nu)(\underline{\mu}\lambda)k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\;,$
$\displaystyle\frac{1}{5!}\bigg{[}\frac{d^{5}}{dt^{5}}\Sigma[A(t)]\bigg{]}_{t=0}$
$\displaystyle=\frac{1}{5!}e^{Ad(A_{0}\frac{\partial}{\partial
s})}\bigg{[}e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{4}}{dt^{4}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}~{}f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{1}\frac{\partial}{\partial s})+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{3}}{dt^{3}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}4\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$
$\displaystyle+4f[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{2}\frac{\partial}{\partial
s})\bigg{\\}}+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{2}}{dt^{2}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}6\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$
$\displaystyle+12\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})+6f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{3}\frac{\partial}{\partial
s})\bigg{\\}}+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d}{dt}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}4\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$
$\displaystyle+12\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})+12\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial
s})+4f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{4}\frac{\partial}{\partial
s})\bigg{\\}}$
$\displaystyle+\bigg{\\{}\frac{d^{4}f}{dt^{4}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+4\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})+6\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial s})$
$\displaystyle+4\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{4}\frac{\partial}{\partial
s})+f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{5}\frac{\partial}{\partial
s})\bigg{\\}}\bigg{]}\Sigma(s+A(t))\bigg{|}_{s=0}$
$\displaystyle=\mbox{traceless terms}\;,$ (235)
$\displaystyle\bigg{[}\frac{d^{6}}{dt^{6}}\Sigma[A(t)]\bigg{]}_{t=0}$
$\displaystyle=e^{Ad(A_{0}\frac{\partial}{\partial
s})}\bigg{[}e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{5}}{dt^{5}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}[Ad(-A_{0}\frac{\partial}{\partial
s})](A_{1}\frac{\partial}{\partial s})+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{4}}{dt^{4}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}5\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$
$\displaystyle+5f[Ad(-A_{0}\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})\bigg{\\}}+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{3}}{dt^{3}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}10\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial s})$
$\displaystyle+20\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})+10f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{3}\frac{\partial}{\partial
s})\bigg{\\}}$ $\displaystyle+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d^{2}}{dt^{2}}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}10\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+30\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial s})$
$\displaystyle+30\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial
s})+10f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{4}\frac{\partial}{\partial
s})\bigg{\\}}$ $\displaystyle+e^{-A_{0}\frac{\partial}{\partial
s}}\frac{d}{dt}e^{A(t)\frac{\partial}{\partial
s}}\bigg{|}_{t=0}\bigg{\\{}5\frac{d^{4}f}{dt^{4}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+20\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial s})$
$\displaystyle+30\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial
s})+20\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{4}\frac{\partial}{\partial
s})+5f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{5}\frac{\partial}{\partial
s})\bigg{\\}}$
$\displaystyle+\bigg{\\{}\frac{d^{5}f}{dt^{5}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{1}\frac{\partial}{\partial
s})+5\frac{d^{4}f}{dt^{4}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{2}\frac{\partial}{\partial
s})+10\frac{d^{3}f}{dt^{3}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{3}\frac{\partial}{\partial s})$
$\displaystyle+10\frac{d^{2}f}{dt^{2}}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{4}\frac{\partial}{\partial
s})+5\frac{df}{dt}[Ad(-A(t)\frac{\partial}{\partial
s})]\bigg{|}_{t=0}(A_{5}\frac{\partial}{\partial
s})+f[Ad(-A_{0}\frac{\partial}{\partial s})](A_{6}\frac{\partial}{\partial
s})\bigg{\\}}\bigg{]}\Sigma(s+A_{0})\bigg{|}_{s=0}$
$\displaystyle=\frac{1}{9}(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}-\frac{1}{6}(\mu\underline{\mu}\nu\lambda)(\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}-\frac{1}{6}(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}-\frac{1}{6}(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}-\frac{4}{27}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}-\frac{4}{27}(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\underline{\lambda})\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-\frac{1}{24}(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\rho)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{1}{24}(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\rho)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{1}{18}(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{1}{4}(\mu\nu\lambda\rho)(\underline{\lambda}\underline{\rho})k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+\frac{1}{4}(\mu\nu\lambda\rho)(\underline{\nu}\underline{\lambda})k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{1}{4}(\mu\nu\lambda\rho)(\underline{\mu}\underline{\lambda})k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{1}{4}(\mu\underline{\mu}\nu\lambda)(\underline{\nu}\rho)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{7}{54}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle-\frac{2}{9}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{2}{9}(\mu\nu\lambda)(\rho\underline{\mu}\underline{\nu})k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{2}{9}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{2}{9}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+\frac{11}{54}(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{5}{24}(\mu\nu\underline{\nu}\lambda)(\underline{\lambda}\rho)k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{5}{24}(\mu\nu\underline{\mu}\lambda)(\underline{\lambda}\rho)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}-\frac{1}{12}(\mu\nu)(\underline{\mu}\lambda)(\underline{\lambda}\rho)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle-\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\mu}\underline{\lambda})k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+\frac{1}{8}(\mu\nu\underline{\nu}\lambda)(\underline{\lambda}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}+\frac{1}{12}(\mu\nu\underline{\nu}\lambda)(\underline{\lambda}\rho)k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{24}(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+\frac{1}{12}(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{8}(\mu\nu\underline{\mu}\lambda)(\underline{\lambda}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+\frac{1}{12}(\mu\nu\underline{\mu}\lambda)(\underline{\lambda}\rho)k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{24}(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+\frac{1}{12}(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{18}(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+\frac{1}{18}(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\lambda}\underline{\rho})k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\lambda}\underline{\rho})k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\nu}\underline{\lambda})k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}+\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\nu}\underline{\lambda})k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\mu}\underline{\lambda})k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\mu}\underline{\lambda})k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{12}(\mu\underline{\mu}\nu\lambda)(\underline{\nu}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-\frac{2}{27}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\mu}\underline{\nu})k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\mu}\underline{\nu})k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+\frac{4}{27}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}+\frac{4}{27}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{4}{27}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{4}{27}(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+\frac{2}{27}(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{2}{27}(\mu\nu\lambda)(\rho\underline{\mu}\underline{\rho})k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{4}{27}(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{2}{27}(\mu\underline{\mu}\nu)(\underline{\nu}\lambda\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-\frac{1}{6}(\mu\nu)(\underline{\mu}\underline{\nu})(\lambda\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\mu}\underline{\lambda})k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\lambda}\underline{\rho})k_{\mu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-\frac{1}{12}(\mu\nu)(\underline{\mu}\lambda)(\underline{\lambda}\rho)k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+\frac{1}{16}(\mu\nu\underline{\nu}\lambda)(\underline{\lambda}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}-\frac{1}{16}(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+\frac{1}{16}(\mu\nu\underline{\mu}\lambda)(\underline{\lambda}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-\frac{1}{16}(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle-\frac{1}{18}(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\lambda}\underline{\rho})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\nu}}+\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}+\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\mu}\underline{\lambda})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle-\frac{1}{8}(\mu\underline{\mu}\nu\lambda)(\underline{\nu}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}+\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\mu}\underline{\nu})\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle+\frac{1}{9}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}-\frac{1}{9}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-\frac{1}{6}(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{8}(\mu\nu)(\underline{\mu}\lambda)(\underline{\lambda}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle-\frac{1}{8}(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-\frac{16}{45}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime\prime}-\frac{2}{5}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime\prime}$
$\displaystyle-\frac{1}{4}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-\frac{1}{4}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-\frac{2}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle-\frac{2}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\nu}\sigma)k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\mu}\sigma)k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+\frac{1}{6}(\mu\nu\underline{\nu}\lambda)(\rho\sigma)k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+\frac{1}{6}(\mu\nu\underline{\mu}\lambda)(\rho\sigma)k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+\frac{1}{27}(\mu\nu\lambda)(\rho\underline{\mu}\sigma)k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\rho}\sigma)k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+\frac{1}{9}(\mu\underline{\mu}\nu)(\lambda\rho\sigma)k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle-\frac{1}{4}(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-\frac{5}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+\frac{5}{27}(\mu\nu\lambda)(\underline{\nu}\rho\sigma)k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle-\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}+\frac{2}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle+\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\nu}\sigma)k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\nu}\sigma)k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\sigma}}+\frac{1}{12}(\mu\nu\lambda\rho)(\underline{\mu}\sigma)k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\mu}\sigma)k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\sigma}}+\frac{1}{8}(\mu\nu\underline{\nu}\lambda)(\rho\sigma)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu\underline{\nu}\lambda)(\rho\sigma)k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+\frac{1}{8}(\mu\nu\underline{\mu}\lambda)(\rho\sigma)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu\underline{\mu}\lambda)(\rho\sigma)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\mu}\sigma)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\rho}\sigma)k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\rho}\sigma)k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-\frac{1}{9}(\mu\underline{\mu}\nu)(\lambda\rho\sigma)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+\frac{1}{9}(\mu\underline{\mu}\nu)(\lambda\rho\sigma)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-\frac{1}{6}(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}-\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+\frac{1}{12}(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}-\frac{1}{4}(\mu\underline{\mu}\nu\lambda)(\rho\sigma)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+\frac{4}{27}(\mu\nu\lambda)(\underline{\nu}\rho\sigma)k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}$
$\displaystyle-\frac{4}{27}(\mu\nu\lambda)(\underline{\mu}\rho\sigma)k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-\frac{2}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-\frac{4}{27}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-\frac{1}{6}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$
$\displaystyle-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\sigma}}+\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\nu}\sigma)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\mu}\sigma)k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu\underline{\nu}\lambda)(\rho\sigma)k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-\frac{1}{8}(\mu\nu\underline{\mu}\lambda)(\rho\sigma)k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}-\frac{2}{9}(\mu\nu\lambda)(\rho\underline{\mu}\sigma)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\rho}\sigma)k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-\frac{1}{9}(\mu\underline{\mu}\nu)(\lambda\rho\sigma)k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}-\frac{1}{8}(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle-\frac{1}{8}(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}-\frac{1}{9}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}+\frac{1}{6}(\mu\nu)(\lambda\rho)(\underline{\mu}\sigma)k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle-\frac{1}{3}(\mu\nu\lambda\rho)(\sigma\delta)k_{\mu}k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\delta}}-\frac{2}{9}(\mu\nu\lambda)(\rho\sigma\delta)k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\delta}}-\frac{1}{4}(\mu\nu\lambda\rho)(\sigma\delta)k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle-\frac{1}{4}(\mu\nu\lambda\rho)(\sigma\delta)k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\delta}}-\frac{2}{9}(\mu\nu\lambda)(\rho\sigma\delta)k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\delta}}-\frac{2}{9}(\mu\nu\lambda)(\rho\sigma\delta)k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle-\frac{1}{4}(\mu\nu\lambda\rho)(\sigma\delta)k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial
k^{\delta}}-\frac{2}{9}(\mu\nu\lambda)(\rho\sigma\delta)k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}-\frac{1}{6}(\mu\nu)(\lambda\rho)(\sigma\delta)k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$ $\displaystyle+\mbox{traceless
terms}\;.$ (236)
Finally, we list down the $p^{3}$, $p^{4}$, $p^{5}$, $p^{6}$ order low energy
expansion result for $B$ used in (18),
$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle-6ds_{\Omega}^{\mu}\gamma_{\mu}\tau+6idp_{\Omega}^{\mu}\gamma_{\mu}\gamma_{5}\tau-6ia_{\Omega}^{\mu}s\gamma_{\mu}\gamma_{5}\tau-6a_{\Omega}^{\mu}p\gamma_{\mu}\tau-6is_{\Omega}a_{\Omega}^{\mu}\gamma_{\mu}\gamma_{5}\tau-6p_{\Omega}a_{\Omega}^{\mu}\gamma_{\mu}\tau$
$\displaystyle-3i(\mu\nu\lambda)\gamma_{\nu}\gamma_{\lambda}\tau\frac{\partial}{\partial
k^{\mu}}+3ia_{\Omega}^{\mu}(\nu
a_{\Omega}^{\lambda})\gamma_{\mu}\gamma_{\lambda}\tau\frac{\partial}{\partial
k^{\nu}}+3i(\mu
a_{\Omega}^{\nu})a_{\Omega}^{\lambda}\gamma_{\nu}\gamma_{\lambda}\tau\frac{\partial}{\partial
k^{\mu}}-3ia_{\Omega}^{\mu}(\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{\mu}\tau\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle-3i(\mu
a_{\Omega}^{\nu})a_{\Omega}^{\lambda}\gamma_{\lambda}\gamma_{\nu}\tau\frac{\partial}{\partial
k^{\mu}}-3(\mu{\nabla_{a}}_{\Omega}^{\nu}{\nabla_{a}}_{\Omega}^{\lambda})\gamma_{\nu}\gamma_{\lambda}\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}-2i(\mu\underline{\mu}\nu)\tau\frac{\partial}{\partial
k^{\nu}}+6ia_{\Omega}^{\mu}(\nu a_{\Omega\mu})\tau\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle+6i(\mu
a_{\Omega}^{\nu})a_{\Omega\nu}\tau\frac{\partial}{\partial k^{\mu}}-12i(\mu
s_{\Omega})\tau k_{\mu}\Sigma_{k}^{\prime}+12(\mu p_{\Omega})\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}-12i(\mu
s_{\Omega})\tau\Sigma_{k}\frac{\partial}{\partial k^{\mu}}$
$\displaystyle+4i(\mu\underline{\mu}\nu)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime}+4i(\mu\underline{\mu}\nu)\tau
k_{\nu}\Sigma_{k}^{\prime 2}-4i(\mu\nu\lambda)\tau
k_{\nu}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}-4i(\mu\underline{\mu}\nu)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle-6i(\mu\underline{\mu}a_{\Omega}^{\nu})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}-12i(\mu\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}-6i(\mu\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}-6i(\mu\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\mu}}$
$\displaystyle-6i(\mu\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\nu}}+3(\mu\nu)a_{\Omega\mu}\gamma_{5}\tau\frac{\partial}{\partial
k^{\nu}}+3a_{\Omega}^{\mu}(\underline{\mu}\nu)\gamma_{5}\tau\frac{\partial}{\partial
k^{\nu}}-6i(\mu\nu)a_{\Omega}^{\lambda}\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle+3(\mu\nu a_{\Omega\mu})\gamma_{5}\tau\frac{\partial}{\partial
k^{\nu}}+3(\mu\nu a_{\Omega\nu})\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}+6(\mu\nu a_{\Omega}^{\lambda})\gamma_{5}\tau
k_{\lambda}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-6(\mu\nu\lambda)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-6(\mu\underline{\mu}\nu)\gamma_{\nu}\tau\Sigma_{k}^{\prime}+12(\mu\nu\lambda)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}+6(\mu\nu\lambda)\gamma_{\nu}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}+6(\mu\nu\lambda)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\mu}}$
$\displaystyle-6ia_{\Omega}^{\mu}(\nu\lambda)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}-8i(\mu\nu\lambda)\tau
k_{\mu}k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-8i(\mu\nu\lambda)\tau k_{\mu}k_{\nu}\Sigma_{k}^{\prime
2}\frac{\partial}{\partial k^{\lambda}}-8i(\mu\nu\lambda)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}\;,$ $\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle-24s_{\Omega}^{2}\tau-24p_{\Omega}^{2}\tau+24i[s_{\Omega},p_{\Omega}]\gamma_{5}\tau+24i(\mu
d^{\nu}s_{\Omega})\gamma_{\nu}\tau\frac{\partial}{\partial k^{\mu}}+24(\mu
d^{\nu}p_{\Omega})\gamma_{\nu}\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}-24a_{\Omega}^{\mu}(\nu
s_{\Omega})\gamma_{\mu}\gamma_{5}\tau\frac{\partial}{\partial k^{\nu}}$ (238)
$\displaystyle-24(\mu
a_{\Omega}^{\nu})s_{\Omega}\gamma_{\nu}\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}+24ia_{\Omega}^{\mu}(\nu
p_{\Omega})\gamma_{\mu}\tau\frac{\partial}{\partial k^{\nu}}+24i(\mu
a_{\Omega}^{\nu})p_{\Omega}\gamma_{\nu}\tau\frac{\partial}{\partial
k^{\mu}}-24s_{\Omega}(\mu
a_{\Omega}^{\nu})\gamma_{\nu}\gamma_{5}\tau\frac{\partial}{\partial k^{\mu}}$
$\displaystyle-24(\mu
s_{\Omega})a_{\Omega}^{\nu}\gamma_{\nu}\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}+24ip_{\Omega}(\mu
a_{\Omega}^{\nu})\gamma_{\nu}\tau\frac{\partial}{\partial k^{\mu}}+24i(\mu
p_{\Omega})a_{\Omega}^{\nu}\gamma_{\nu}\tau\frac{\partial}{\partial
k^{\mu}}-6(\mu\nu\lambda\rho)\gamma_{\lambda}\gamma_{\rho}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$ $\displaystyle+6a_{\Omega}^{\mu}(\nu\lambda
a_{\Omega}^{\rho})\gamma_{\mu}\gamma_{\rho}\tau\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+12(\mu a_{\Omega}^{\nu})(\lambda
a_{\Omega}^{\rho})\gamma_{\nu}\gamma_{\rho}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}+6(\mu\nu
a_{\Omega}^{\lambda})a_{\Omega}^{\rho}\gamma_{\lambda}\gamma_{\rho}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$ $\displaystyle-6a_{\Omega}^{\mu}(\nu\lambda
a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{\mu}\tau\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}-12(\mu a_{\Omega}^{\nu})(\lambda
a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{\nu}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}-6(\mu\nu
a_{\Omega}^{\lambda})a_{\Omega}^{\rho}\gamma_{\rho}\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$
$\displaystyle+6i(\mu\nu(d^{\lambda}a_{\Omega}^{\rho}-d^{\rho}a_{\Omega}^{\lambda}))\gamma_{\lambda}\gamma_{\rho}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\nu}}-3(\mu\nu\underline{\nu}\lambda)\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-3(\mu\nu\underline{\mu}\lambda)\tau\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+12a_{\Omega}^{\mu}(\nu\lambda
a_{\Omega\mu})\tau\frac{\partial^{2}}{\partial k^{\nu}\partial k^{\lambda}}$
$\displaystyle+24(\mu a_{\Omega}^{\nu})(\lambda
a_{\Omega\nu})\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+12(\mu\nu
a_{\Omega}^{\lambda})a_{\Omega\lambda}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\nu}}-12(\mu\nu)(\underline{\mu}\underline{\nu})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}-12(\mu\nu)(\underline{\mu}\underline{\nu})\tau\Sigma_{k}^{\prime
2}$
$\displaystyle-6(\mu\nu)(\underline{\mu}\lambda)\tau\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+8(\mu\underline{\mu}\nu)a_{\Omega}^{\lambda}\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}-8(\mu\underline{\mu}\nu)a_{\Omega}^{\lambda}\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-24(\mu\underline{\mu}s_{\Omega})\tau\Sigma_{k}^{\prime}-24i(\mu\underline{\mu}p)\gamma_{5}\tau\Sigma_{k}^{\prime}$
$\displaystyle-48(\mu\nu s_{\Omega})\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}-48i(\mu\nu p_{\Omega})\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}-24(\mu\nu s_{\Omega})\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}-24i(\mu\nu
p_{\Omega})\gamma_{5}\tau k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle-24(\mu\nu s_{\Omega})\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\mu}}-24i(\mu\nu
p_{\Omega})\gamma_{5}\tau k_{\nu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\mu}}-24(\mu\nu s_{\Omega})\tau\Sigma_{k}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}-8i(\mu\nu\underline{\nu}\lambda)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}$
$\displaystyle+8i(\mu\nu\underline{\nu}\lambda)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}+8a_{\Omega}^{\mu}(\nu\underline{\nu}\lambda)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}-8a_{\Omega}^{\mu}(\nu\underline{\nu}\lambda)\gamma_{\mu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}+8(\mu\nu\underline{\nu}\lambda)\tau
k_{\mu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+8(\mu\nu\underline{\mu}\lambda)\tau
k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}+24(\mu\nu\underline{\nu}\lambda)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+24(\mu\nu\underline{\mu}\lambda)\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+6(\mu\nu\underline{\nu}\lambda)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$ $\displaystyle+6(\mu\nu\underline{\mu}\lambda)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-12(\mu\underline{\mu}\nu\lambda)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-6(\mu\nu\underline{\nu}\lambda)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-6(\mu\nu\underline{\mu}\lambda)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$ $\displaystyle+6(\mu\nu\underline{\nu}\lambda)\tau
k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial}{\partial
k^{\mu}}+6(\mu\nu\underline{\mu}\lambda)\tau k_{\lambda}\Sigma_{k}^{\prime
2}\frac{\partial}{\partial k^{\nu}}-12(\mu\underline{\mu}\nu\lambda)\tau
k_{\nu}\Sigma_{k}^{\prime 2}\frac{\partial}{\partial
k^{\lambda}}-6(\mu\nu\underline{\nu}\lambda)\tau k_{\mu}\Sigma_{k}^{\prime
2}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-6(\mu\nu\underline{\mu}\lambda)\tau k_{\nu}\Sigma_{k}^{\prime
2}\frac{\partial}{\partial k^{\lambda}}-24(\mu\nu)s_{\Omega}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}+24i(\mu\nu)p_{\Omega}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-6(\mu\nu\lambda\rho)\tau k_{\lambda}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}}$
$\displaystyle-6(\mu\nu\underline{\nu}\lambda)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-6(\mu\nu\underline{\mu}\lambda)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}-12i(\mu\nu)(\lambda
a_{\Omega\mu})\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}-16(\mu\nu\underline{\nu}a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}$
$\displaystyle-8(\mu\nu\underline{\nu}a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\mu}}-16(\mu\nu\underline{\mu}a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}-16(\mu\underline{\mu}\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}-32(\mu\nu\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-16(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}-8(\mu\nu\underline{\mu}a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-16(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-8(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-8(\mu\underline{\mu}\nu
a_{\Omega}^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-16(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-8(\mu\nu\lambda a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}-8(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}-8(\mu\nu\lambda
a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}}$
$\displaystyle-8i(\mu\nu\lambda)a_{\Omega\nu}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}-12i(\mu
a_{\Omega}^{\nu})(\underline{\nu}\lambda)\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}-8ia_{\Omega}^{\mu}(\nu\underline{\mu}\lambda)\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}-24s_{\Omega}(\mu\nu)\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-24ip_{\Omega}(\mu\nu)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-16(\mu\nu)(\underline{\mu}\lambda)\tau
k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}-48(\mu\nu)(\underline{\mu}\lambda)\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+12(\mu\nu)(\underline{\mu}\lambda)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$ $\displaystyle-12(\mu\nu)(\underline{\mu}\lambda)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+12(\mu\nu)(\underline{\mu}\lambda)\tau k_{\nu}\Sigma_{k}^{\prime
2}\frac{\partial}{\partial k^{\lambda}}-12(\mu\nu)(\underline{\mu}\lambda)\tau
k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-12(\mu\nu)(\underline{\mu}\lambda)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}-16(\mu\nu\lambda)a_{\Omega}^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-24(\mu\nu)(\lambda a_{\Omega}^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}$
$\displaystyle-16(\mu\nu\lambda)a_{\Omega}^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}-24(\mu
a_{\Omega}^{\nu})(\underline{\mu}\lambda)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}-48(\mu a_{\Omega}^{\nu})(\lambda\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-24(\mu a_{\Omega}^{\nu})(\lambda\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\rho}}-4i(\mu\nu\lambda
a_{\Omega\mu})\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}-4i(\mu\nu\lambda
a_{\Omega\nu})\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}$ $\displaystyle-4i(\mu\nu\lambda
a_{\Omega\lambda})\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-8i(\mu\nu\lambda a_{\Omega}^{\rho})\gamma_{5}\tau
k_{\rho}\frac{\partial^{3}}{\partial k^{\mu}\partial k^{\nu}\partial
k^{\lambda}}+16i(\mu\nu\lambda\rho)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle+16i(\mu\nu\lambda\rho)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}-24i(\mu\nu)(\lambda\rho)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}+24i(\mu\nu)(\lambda\rho)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}$ $\displaystyle-16i(\mu\underline{\mu}\nu\lambda)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}-8i(\mu\underline{\mu}\nu\lambda)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}+16i(\mu\nu\underline{\mu}\lambda)\gamma_{\lambda}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}+16i(\mu\nu\underline{\nu}\lambda)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}$
$\displaystyle-32i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\mu}k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-16i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+8i(\mu\nu\underline{\mu}\lambda)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-16i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-8i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}+8i(\mu\nu\underline{\nu}\lambda)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\mu}}-16i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}-8i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\rho}}$ $\displaystyle-8i(\mu\nu\lambda\rho)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-16a^{\mu}(\nu\lambda\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-16a^{\mu}(\nu\lambda\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}$ $\displaystyle-16(\mu\nu\lambda\rho)\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-48(\mu\nu\lambda\rho)\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-12(\mu\nu\lambda\rho)\tau
k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$ $\displaystyle-12(\mu\nu\lambda\rho)\tau
k_{\mu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}-12(\mu\nu\lambda\rho)\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}-12(\mu\nu\lambda\rho)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle-12(\mu\nu\lambda\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial
k^{\rho}}+24i(\mu\nu)(\underline{\mu}\lambda)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\lambda}}-48i(\mu\nu)(\lambda\rho)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-24(\mu\nu)(\lambda\rho)\tau
k_{\mu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}-24(\mu\nu)(\lambda\rho)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}\;,$ $\displaystyle B_{5}$ $\displaystyle=$
$\displaystyle 120is(\mu s)\tau\frac{\partial}{\partial k^{\mu}}+120i(\mu
s)s\tau\frac{\partial}{\partial k^{\mu}}+120ip(\mu
p)\tau\frac{\partial}{\partial k^{\mu}}+120i(\mu
p)p\tau\frac{\partial}{\partial k^{\mu}}+120s(\mu
p)\gamma_{5}\tau\frac{\partial}{\partial k^{\mu}}+120(\mu
s)p\gamma_{5}\tau\frac{\partial}{\partial k^{\mu}}$ (239)
$\displaystyle-120p(\mu s)\gamma_{5}\tau\frac{\partial}{\partial
k^{\mu}}-120(\mu p)s\gamma_{5}\tau\frac{\partial}{\partial k^{\mu}}+60(\mu\nu
ds^{\lambda})\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-60i(\mu\nu
dp^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$ $\displaystyle+60ia^{\mu}(\nu\lambda
s)\gamma_{\mu}\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}+120i(\mu a^{\nu})(\lambda
s)\gamma_{\nu}\gamma_{5}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+60i(\mu\nu
a^{\lambda})s\gamma_{\lambda}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}+60a^{\mu}(\nu\lambda
p)\gamma_{\mu}\tau\frac{\partial^{2}}{\partial k^{\nu}\partial k^{\lambda}}$
$\displaystyle+120(\mu a^{\nu})(\lambda
p)\gamma_{\nu}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+60(\mu\nu
a^{\lambda})p\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}+60is(\mu\nu
a^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}+120i(\mu s)(\nu
a^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}$ $\displaystyle+60i(\mu\nu
s)a^{\lambda}\gamma_{\lambda}\gamma_{5}\tau\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}+60p(\mu\nu
a^{\lambda})\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}+120(\mu p)(\nu
a^{\lambda})\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}+60(\mu\nu
p)a^{\lambda}\gamma_{\lambda}\tau\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}$
$\displaystyle+30i(\mu\nu)(\underline{\mu}\underline{\nu})a^{\lambda}\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}-40i(\mu\underline{\mu}\nu)s\tau
k_{\nu}\Sigma_{k}^{\prime\prime}-40(\mu\underline{\mu}\nu)p\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}+40i(\mu\underline{\mu}\nu)s\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}+40(\mu\underline{\mu}\nu)p\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+10i(\mu\nu\lambda\rho\sigma)\gamma_{\rho}\gamma_{\sigma}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}}-10ia^{\mu}(\nu\lambda\rho
a^{\sigma})\gamma_{\mu}\gamma_{\sigma}\tau\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}}-30i(\mu a^{\nu})(\lambda\rho
a^{\sigma})\gamma_{\nu}\gamma_{\sigma}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}}$ $\displaystyle-30i(\mu\nu
a^{\lambda})(\rho
a^{\sigma})\gamma_{\lambda}\gamma_{\sigma}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}}-10i(\mu\nu\lambda
a^{\rho})a^{\sigma}\gamma_{\rho}\gamma_{\sigma}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}}+10ia^{\mu}(\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{\mu}\tau\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}}$ $\displaystyle+30i(\mu
a^{\nu})(\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{\nu}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}}+30i(\mu\nu a^{\lambda})(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{\lambda}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}}+10i(\mu\nu\lambda
a^{\rho})a^{\sigma}\gamma_{\sigma}\gamma_{\rho}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}}$
$\displaystyle+10(\mu\nu\lambda{\nabla_{a}}^{\rho}{\nabla_{a}}^{\sigma})\gamma_{\rho}\gamma_{\sigma}\gamma_{5}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}}-20ia^{\mu}(\nu\lambda\rho
a_{\mu})\tau\frac{\partial^{3}}{\partial k^{\nu}\partial k^{\lambda}\partial
k^{\rho}}-60i(\mu a^{\nu})(\lambda\rho
a_{\nu})\tau\frac{\partial^{3}}{\partial k^{\mu}\partial k^{\lambda}\partial
k^{\rho}}$ $\displaystyle-60i(\mu\nu a^{\lambda})(\rho
a_{\lambda})\tau\frac{\partial^{3}}{\partial k^{\mu}\partial k^{\nu}\partial
k^{\rho}}-20i(\mu\nu\lambda a^{\rho})a_{\rho}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial
k^{\lambda}}+30(\mu\nu)(\lambda\underline{\mu}\underline{\nu})\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}+30(\mu\nu\lambda)(\underline{\nu}\underline{\lambda})\gamma_{\mu}\tau\Sigma_{k}^{\prime\prime}$
$\displaystyle+30ia^{\mu}(\nu\lambda)(\underline{\nu}\underline{\lambda})\gamma_{\mu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}-40is(\mu\underline{\mu}\nu)\tau
k_{\nu}\Sigma_{k}^{\prime\prime}+40p(\mu\underline{\mu}\nu)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}+40is(\mu\underline{\mu}\nu)\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-40p(\mu\underline{\mu}\nu)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+20i(\mu\nu)(\lambda\underline{\mu}\rho)\tau\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\rho}}+20i(\mu\nu\lambda)(\underline{\nu}\rho)\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\rho}}-20i(\mu\nu\underline{\nu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-20i(\mu\nu\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-40i(\mu\underline{\mu}\nu)(\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-15i(\mu\nu\underline{\nu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}-15i(\mu\nu\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle+30i(\mu\underline{\mu}\nu\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+15i(\mu\nu\underline{\nu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+15i(\mu\nu\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+40i(\mu\underline{\mu}\nu)(\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+15i(\mu\nu\underline{\nu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+15i(\mu\nu\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$ $\displaystyle+40i(\mu
a^{\nu})(\lambda\underline{\mu}\underline{\lambda})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}-80i(\mu
a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-40i(\mu
a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\mu}}$
$\displaystyle+80i(\mu
a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}+40i(\mu
a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+80i(\mu\nu\underline{\nu}s)\tau
k_{\mu}\Sigma_{k}^{\prime\prime}-80(\mu\nu\underline{\nu}p)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}$
$\displaystyle+40i(\mu\nu\underline{\nu}s)\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\mu}}-40(\mu\nu\underline{\nu}p)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\mu}}+80i(\mu\nu\underline{\mu}s)\tau
k_{\nu}\Sigma_{k}^{\prime\prime}-80(\mu\nu\underline{\mu}p)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}+80i(\mu\underline{\mu}\nu s)\tau
k_{\nu}\Sigma_{k}^{\prime\prime}$ $\displaystyle-80(\mu\underline{\mu}\nu
p)\gamma_{5}\tau k_{\nu}\Sigma_{k}^{\prime\prime}+160i(\mu\nu\lambda s)\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-160(\mu\nu\lambda
p)\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+80i(\mu\nu\lambda
s)\tau k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$ $\displaystyle-80(\mu\nu\lambda p)\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+40i(\mu\nu\underline{\mu}s)\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-40(\mu\nu\underline{\mu}p)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}+80i(\mu\nu\lambda s)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-80(\mu\nu\lambda p)\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+40i(\mu\nu\lambda s)\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}-40(\mu\nu\lambda p)\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\nu}}+40i(\mu\underline{\mu}\nu
s)\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-40(\mu\underline{\mu}\nu
p)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}+80i(\mu\nu\lambda s)\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-80(\mu\nu\lambda p)\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+40i(\mu\nu\lambda s)\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}$ $\displaystyle-40(\mu\nu\lambda p)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+40i(\mu\nu\lambda s)\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}-40(\mu\nu\lambda p)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}$ $\displaystyle+40i(\mu\nu\lambda
s)\tau\Sigma_{k}\frac{\partial^{3}}{\partial k^{\mu}\partial k^{\nu}\partial
k^{\lambda}}-20(\mu\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-20(\mu\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-15(\mu\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle-15(\mu\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+30(\mu\nu\underline{\nu}\lambda\rho)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+15(\mu\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+15(\mu\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle+40(\mu\nu)(\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-40(\mu\underline{\mu}\nu)(\lambda\rho)\gamma_{\lambda}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+15(\mu\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+15(\mu\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle-40(\mu\nu)(\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+40(\mu\underline{\mu}\nu)(\lambda\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-20ia^{\mu}(\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-20ia^{\mu}(\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-15ia^{\mu}(\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-15ia^{\mu}(\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+30ia^{\mu}(\nu\underline{\nu}\lambda\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+15ia^{\mu}(\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+15ia^{\mu}(\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle+15ia^{\mu}(\nu\lambda\underline{\lambda}\rho)\gamma_{\mu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+15ia^{\mu}(\nu\lambda\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+40i(\mu\nu)(\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-30i(\mu\nu)(\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+30i(\mu\nu)(\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+30i(\mu\nu)(\underline{\mu}\lambda)a^{\rho}\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$ $\displaystyle+80i(\mu\nu\lambda)s\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+80(\mu\nu\lambda)p\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+120i(\mu\nu)(\lambda s)\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}$ $\displaystyle+120(\mu\nu)(\lambda p)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}+80i(\mu\nu\lambda)s\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}+80(\mu\nu\lambda)p\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}$ $\displaystyle+120i(\mu
s)(\underline{\mu}\nu)\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}-120(\mu
p)(\underline{\mu}\nu)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial}{\partial
k^{\nu}}+240i(\mu s)(\nu\lambda)\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-240(\mu p)(\nu\lambda)\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+120i(\mu s)(\nu\lambda)\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}-120(\mu p)(\nu\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}$ $\displaystyle-30(\mu\nu)(\lambda\rho
a_{\mu})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}}-40(\mu\nu\lambda)(\rho
a_{\nu})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial
k^{\rho}}+20i(\mu\underline{\mu}\nu\underline{\nu}a^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}$
$\displaystyle+40i(\mu\nu\lambda\underline{\lambda}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}+20i(\mu\nu\lambda\underline{\lambda}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+20i(\mu\nu\lambda\underline{\lambda}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle+10i(\mu\nu\lambda\underline{\lambda}a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\nu}}+20i(\mu\nu\underline{\nu}\underline{\mu}a^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}+40i(\mu\nu\underline{\nu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+20i(\mu\nu\underline{\nu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+20i(\mu\nu\underline{\mu}\underline{\nu}a^{\lambda})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}+40i(\mu\nu\lambda\underline{\nu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+20i(\mu\nu\lambda\underline{\nu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+40i(\mu\nu\underline{\mu}\lambda a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+40i(\mu\nu\lambda\underline{\mu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+40i(\mu\underline{\mu}\nu\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+80i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime\prime}+40i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}$ $\displaystyle+20i(\mu\nu\underline{\mu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+20i(\mu\nu\lambda\underline{\mu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+40i(\mu\nu\lambda\rho a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}+20i(\mu\nu\underline{\nu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+10i(\mu\nu\underline{\nu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$ $\displaystyle+20i(\mu\nu\underline{\mu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+20i(\mu\underline{\mu}\nu\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+40i(\mu\nu\lambda\rho a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+10i(\mu\nu\underline{\mu}\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle+10i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial
k^{\rho}}+20i(\mu\nu\lambda\underline{\nu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+10i(\mu\nu\lambda\underline{\nu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$
$\displaystyle+20i(\mu\nu\lambda\underline{\mu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+20i(\mu\underline{\mu}\nu\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+40i(\mu\nu\lambda\rho a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$ $\displaystyle+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+10i(\mu\nu\lambda\underline{\mu}a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$ $\displaystyle+10i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\lambda}}+10i(\mu\underline{\mu}\nu\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+20i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$ $\displaystyle+10i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}}+10i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}}+10i(\mu\nu\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau\Sigma_{k}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle-15(\mu\nu\lambda\rho)a_{\lambda}\gamma_{5}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}}-40(\mu
a^{\nu})(\lambda\underline{\nu}\rho)\gamma_{5}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}}-30(\mu\nu
a^{\lambda})(\underline{\lambda}\rho)\gamma_{5}\tau\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}}$
$\displaystyle-15a^{\mu}(\nu\lambda\underline{\mu}\rho)\gamma_{5}\tau\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\rho}}+40(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\lambda}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}+40(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-30(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\lambda}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-30(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+30(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+30(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+30(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+30(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+40(\mu\nu)(\lambda\underline{\mu}\underline{\lambda})\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}+80(\mu\nu)(\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-80(\mu\nu)(\lambda\underline{\lambda}\rho)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+40ia^{\mu}(\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}-30ia^{\mu}(\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle+30ia^{\mu}(\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+30ia^{\mu}(\nu\lambda)(\underline{\nu}\rho)\gamma_{\mu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}+80is(\mu\nu\lambda)\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-80p(\mu\nu\lambda)\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+80is(\mu\nu\lambda)\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}-80p(\mu\nu\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\lambda}}$
$\displaystyle+40i(\mu\nu\lambda\rho)a^{\sigma}\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+80i(\mu\nu\lambda)(\rho a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+30i(\mu\nu\lambda\rho)a^{\sigma}\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle+30i(\mu\nu\lambda\rho)a^{\sigma}\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+60i(\mu\nu)(\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}}$ $\displaystyle+80i(\mu\nu\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial
k^{\rho}}+30i(\mu\nu\lambda\rho)a^{\sigma}\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle+80i(\mu
a^{\nu})(\underline{\mu}\lambda\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}+80i(\mu
a^{\nu})(\lambda\underline{\mu}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+160i(\mu a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+80i(\mu
a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}+80i(\mu
a^{\nu})(\lambda\underline{\mu}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}+160i(\mu
a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$ $\displaystyle+80i(\mu
a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\sigma}}+120i(\mu\nu
a^{\lambda})(\underline{\mu}\rho)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}+60i(\mu\nu
a^{\lambda})(\underline{\mu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle+120i(\mu\nu
a^{\lambda})(\underline{\nu}\rho)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+120i(\mu\underline{\mu}a^{\nu})(\lambda\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+240i(\mu\nu a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+120i(\mu\nu
a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+60i(\mu\nu
a^{\lambda})(\underline{\nu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$ $\displaystyle+120i(\mu\nu
a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}+60i(\mu\nu
a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\sigma}}$ $\displaystyle-5(\mu\nu\lambda\rho
a_{\nu})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}}-5(\mu\nu\lambda\rho
a_{\lambda})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}-5(\mu\nu\lambda\rho
a_{\rho})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\lambda}}-5(\mu\nu\lambda\rho
a_{\mu})\gamma_{5}\tau\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}}$ $\displaystyle-10(\mu\nu\lambda\rho
a^{\sigma})\gamma_{5}\tau k_{\sigma}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}\partial
k^{\rho}}+40(\mu\nu\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+30(\mu\nu\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+30(\mu\nu\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}-80(\mu\nu)(\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+80(\mu\nu\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+30(\mu\nu\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}-80(\mu\nu)(\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}+80(\mu\nu\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+60(\mu\nu)(\lambda\rho\sigma)\gamma_{\rho}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}-60(\mu\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial
k^{\sigma}}+40ia^{\mu}(\nu\lambda\rho\sigma)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+30ia^{\mu}(\nu\lambda\rho\sigma)\gamma_{\mu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}+30ia^{\mu}(\nu\lambda\rho\sigma)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+30ia^{\mu}(\nu\lambda\rho\sigma)\gamma_{\mu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+60i(\mu\nu)(\lambda\rho)a^{\sigma}\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+60(\mu\nu)(\lambda\rho\sigma)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+60(\mu\nu\lambda)(\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+80(\mu\nu)(\underline{\mu}\lambda\rho)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+80(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-160(\mu\nu)(\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+80(\mu\nu)(\lambda\underline{\mu}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}-160(\mu\nu)(\lambda\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-120(\mu\nu\lambda)(\underline{\mu}\rho)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-60(\mu\nu\lambda)(\underline{\mu}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+120(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+120(\mu\underline{\mu}\nu)(\lambda\rho)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-240(\mu\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-120(\mu\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+60(\mu\nu\lambda)(\underline{\nu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle-120(\mu\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\sigma}}+60ia^{\mu}(\nu\lambda)(\rho\sigma)\gamma_{\mu}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}\;,$ $\displaystyle B_{6}$ $\displaystyle=$
$\displaystyle 240(\mu
s)(\nu\underline{\mu}\underline{\nu})\tau\Sigma_{k}^{\prime\prime}-160(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}-240(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}-240(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}$
(240)
$\displaystyle+240(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\underline{\nu})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}+\frac{640}{3}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}+\frac{640}{3}(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}-480(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\underline{\lambda})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-720(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\underline{\lambda})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-720(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\underline{\lambda})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+720(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\underline{\nu})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+640(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\underline{\lambda})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle+640(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\underline{\lambda})\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+360(\mu\nu)(\underline{\mu}\underline{\nu})s\tau\Sigma_{k}^{\prime\prime}+240i(\mu
p)(\nu\underline{\mu}\underline{\nu})\gamma_{5}\tau\Sigma_{k}^{\prime\prime}-480(\mu
s)(\nu\underline{\nu}\lambda)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$ $\displaystyle-480(\mu
s)(\nu\underline{\nu}\lambda)\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\mu}}+480(\mu
s)(\nu\underline{\nu}\lambda)\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}+480(\mu
s)(\underline{\mu}\nu\lambda)\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}+480(\mu
s)(\nu\underline{\mu}\lambda)\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+720(\mu\nu)(\underline{\mu}\lambda s)\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+720(\mu\nu)(\lambda\underline{\mu}s)\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+720(\mu\nu)(\lambda\underline{\lambda}s)\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}+480(\mu
s)(\nu\underline{\nu}\lambda)\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$ $\displaystyle+480(\mu
s)(\nu\underline{\mu}\lambda)\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+360(\mu\nu)(\underline{\mu}\lambda
s)\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}+360(\mu\nu)(\lambda\underline{\mu}s)\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}-180i(\mu\nu)(\underline{\mu}\underline{\nu})p\gamma_{5}\tau\Sigma_{k}^{\prime\prime}$
$\displaystyle+180i(\mu\nu)(\underline{\mu}\underline{\nu})\gamma_{5}p\tau\Sigma_{k}^{\prime\prime}+120i(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}-120i(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+120i(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-120i(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}+240i(\mu\nu\lambda)(\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-240i(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime\prime\prime}+240i(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-90i(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-90i(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-180i(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-90i(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-90i(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+120i(\mu\nu\lambda)(\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-120i(\mu\nu\lambda)(\rho\underline{\nu}\underline{\rho})\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle-240i(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-240i(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-240i(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-240i(\mu\nu\underline{\nu}\lambda)(\underline{\mu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-240i(\mu\nu\underline{\mu}\lambda)(\underline{\nu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-240i(\mu\underline{\mu}\nu\lambda)(\underline{\nu}\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle-240(\mu\nu\underline{\nu}\lambda)s\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-240(\mu\nu\underline{\mu}\lambda)s\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-180(\mu\nu\underline{\nu}\lambda)s\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}-180(\mu\nu\underline{\mu}\lambda)s\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\nu}}$
$\displaystyle+360(\mu\underline{\mu}\nu\lambda)s\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180(\mu\nu\underline{\nu}\lambda)s\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180(\mu\nu\underline{\mu}\lambda)s\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180(\mu\nu\underline{\nu}\lambda)s\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$
$\displaystyle+180(\mu\nu\underline{\mu}\lambda)s\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+360i(\mu\nu)(\lambda\rho)(\underline{\lambda}\underline{\rho})\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime\prime}+240i(\mu\nu)(\underline{\mu}\lambda)(\underline{\lambda}\rho)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+240i(\mu\nu)(\lambda\rho)(\underline{\mu}\underline{\lambda})\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}-180i(\mu\nu)(\underline{\mu}\underline{\nu})(\lambda\rho)\gamma_{\lambda}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180i(\mu\nu)(\lambda\rho)(\underline{\lambda}\underline{\rho})\gamma_{\mu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle-180i(\mu\nu)(\underline{\mu}\lambda)(\underline{\lambda}\rho)\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180i(\mu\nu)(\lambda\rho)(\underline{\mu}\underline{\lambda})\gamma_{\nu}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+300(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+300(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}+60(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}+60(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle-480(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}+80(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}+360(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+360(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}-360(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}-360(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+320(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}-640(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}+320(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+240(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+1200(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+240(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+1200(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}-1920(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+320(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+1440(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+1440(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}-1440(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-1440(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+1280(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}-2560(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+1280(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}+240(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime
2}-1440(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle+180(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime
2}+180(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime
2}+900(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime
2}+900(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle+1080(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime
2}+1080(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime
2}-1080(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle-1080(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime
2}+960(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime
2}-1920(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime
2}+960(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle-160(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+120(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-60(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-60(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-120(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-120(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+120(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+120(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+240(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+240(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-240(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-240(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-240(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle-240(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-320(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-320(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+\frac{1280}{3}(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1280}{3}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{1280}{3}(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle+\frac{640}{3}(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-\frac{640}{3}(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+540(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle+540(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+360(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-180(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-180(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-480(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau
k_{\nu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-960(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle-640(\mu\nu\lambda)(\underline{\nu}\underline{\mu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+640(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-960(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-360(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-360(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+360(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+360(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+720(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+720(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-720(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-720(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-720(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$
$\displaystyle-720(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+1280(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau
k_{\mu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-1280(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$
$\displaystyle-1280(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau
k_{\mu}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+90(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+90(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+80(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+90(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+90(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+180(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+180(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}-180(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle-180(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-160(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+160(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle-320(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\rho}}+160(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}-160(\mu\underline{\mu}\nu)(\lambda\underline{\nu}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle+80(\mu\underline{\mu}\nu)(\lambda\underline{\lambda}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}+90(\mu\nu)(\lambda\rho\underline{\mu}\underline{\rho})\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\lambda}}+90(\mu\nu)(\underline{\mu}\lambda\underline{\lambda}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial k^{\rho}}$
$\displaystyle+90(\mu\nu)(\lambda\rho\underline{\mu}\underline{\lambda})\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}+90(\mu\nu)(\lambda\underline{\mu}\underline{\lambda}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}+180(\mu\nu)(\lambda\rho\underline{\mu}\underline{\nu})\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle+180(\mu\nu)(\lambda\underline{\lambda}\underline{\mu}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\rho}}-180(\mu\nu)(\lambda\underline{\mu}\underline{\nu}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial
k^{\rho}}-180(\mu\nu)(\underline{\mu}\lambda\underline{\nu}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle+160(\mu\nu\lambda)(\rho\underline{\nu}\underline{\lambda})\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\rho}}-320(\mu\nu\lambda)(\underline{\nu}\underline{\lambda}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\rho}}+160(\mu\nu\lambda)(\underline{\mu}\underline{\nu}\rho)\tau\Sigma_{k}^{\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial k^{\rho}}$
$\displaystyle+90(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau\frac{\partial^{4}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}+80(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}+480(\mu\nu)(\underline{\mu}\lambda)s\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-360(\mu\nu)(\underline{\mu}\lambda)s\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+360(\mu\nu)(\underline{\mu}\lambda)s\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+360(\mu\nu)(\underline{\mu}\lambda)s\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$ $\displaystyle-480i(\mu
p)(\nu\underline{\nu}\lambda)\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+240i(\mu\underline{\mu}\nu)(\lambda
p)\gamma_{5}\tau k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-240i(\mu p)(\nu\underline{\nu}\lambda)\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\mu}}$
$\displaystyle+480i(\mu p)(\nu\underline{\nu}\lambda)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}+480i(\mu
p)(\underline{\mu}\nu\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}+480i(\mu
p)(\nu\underline{\mu}\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+720i(\mu\nu p)(\underline{\mu}\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+720i(\mu\nu p)(\underline{\nu}\lambda)\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+720i(\mu\underline{\mu}p)(\nu\lambda)\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-240i(\mu\underline{\mu}\nu)(\lambda
p)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+240i(\mu
p)(\nu\underline{\nu}\lambda)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}+480i(\mu
p)(\nu\underline{\mu}\lambda)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$ $\displaystyle+360i(\mu\nu
p)(\underline{\mu}\lambda)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+360i(\mu\nu
p)(\underline{\nu}\lambda)\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}-120(\mu
a^{\nu})(\underline{\mu}\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$ $\displaystyle+120(\mu
a^{\nu})(\lambda\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-120(\mu
a^{\nu})(\lambda\underline{\mu}\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}+120(\mu
a^{\nu})(\lambda\rho\underline{\mu}\underline{\lambda})\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\underline{\mu}\underline{\rho})\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime\prime}+240(\mu\nu
a^{\lambda})(\rho\underline{\nu}\underline{\rho})\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime\prime}-240(\mu\underline{\mu}a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$ $\displaystyle+90(\mu
a^{\nu})(\lambda\rho\underline{\mu}\underline{\rho})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+90(\mu
a^{\nu})(\lambda\rho\underline{\mu}\underline{\lambda})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180(\mu
a^{\nu})(\lambda\underline{\lambda}\underline{\mu}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+90(\mu
a^{\nu})(\underline{\mu}\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+90(\mu
a^{\nu})(\lambda\underline{\mu}\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+120(\mu\nu
a^{\lambda})(\rho\underline{\mu}\underline{\rho})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle+120(\mu\nu
a^{\lambda})(\rho\underline{\nu}\underline{\rho})\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+240(\mu\underline{\mu}a^{\nu})(\lambda\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+240(\mu\nu
a^{\lambda})(\underline{\mu}\underline{\nu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\underline{\nu}\underline{\mu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+240(\mu\underline{\mu}\nu
a^{\lambda})(\underline{\nu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+240(\mu\nu\underline{\mu}a^{\lambda})(\underline{\nu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+240(\mu\nu\underline{\nu}a^{\lambda})(\underline{\mu}\rho)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+120i(\mu\nu\underline{\nu}\lambda)p\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+120i(\mu\nu\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle+90i(\mu\nu\underline{\nu}\lambda)p\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}+90i(\mu\nu\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-180i(\mu\underline{\mu}\nu\lambda)p\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-90i(\mu\nu\underline{\nu}\lambda)p\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-90i(\mu\nu\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-90i(\mu\nu\underline{\nu}\lambda)p\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$
$\displaystyle-90i(\mu\nu\underline{\mu}\lambda)p\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}-120i(\mu\nu\underline{\nu}\lambda)\gamma_{5}p\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}-120i(\mu\nu\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-90i(\mu\nu\underline{\nu}\lambda)\gamma_{5}p\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}-90i(\mu\nu\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+180i(\mu\underline{\mu}\nu\lambda)\gamma_{5}p\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle+90i(\mu\nu\underline{\nu}\lambda)\gamma_{5}p\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+90i(\mu\nu\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+90i(\mu\nu\underline{\nu}\lambda)\gamma_{5}p\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}$
$\displaystyle+90i(\mu\nu\underline{\mu}\lambda)\gamma_{5}p\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+360(\mu\nu)(\underline{\mu}\underline{\nu})(\lambda\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+360(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-120(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+1080(\mu\nu)(\underline{\mu}\underline{\nu})(\lambda\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+1080(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle-360(\mu\nu)(\underline{\mu}\lambda)(\underline{\nu}\rho)\tau
k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+360(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\underline{\rho})\gamma_{\nu}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime\prime}-240(\mu
a^{\nu})(\underline{\mu}\lambda)(\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}$ $\displaystyle-240(\mu
a^{\nu})(\lambda\rho)(\underline{\mu}\underline{\lambda})\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime\prime}+180(\mu\nu)(\underline{\mu}\underline{\nu})(\lambda
a^{\rho})\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\underline{\rho})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\mu}}$ $\displaystyle+180(\mu
a^{\nu})(\underline{\mu}\lambda)(\underline{\lambda}\rho)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}-180(\mu
a^{\nu})(\lambda\rho)(\underline{\mu}\underline{\lambda})\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\rho}}+960(\mu s)(\nu\lambda\rho)\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+1440(\mu\nu)(\lambda\rho s)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+960(\mu s)(\nu\lambda\rho)\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+960(\mu s)(\nu\lambda\rho)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle+720(\mu\nu)(\lambda\rho s)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+720(\mu\nu)(\lambda\rho s)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+720(\mu\nu)(\lambda\rho s)\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}}$ $\displaystyle+960(\mu s)(\nu\lambda\rho)\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}-240i(\mu\nu)(\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}+180i(\mu\nu)(\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial k^{\lambda}}$
$\displaystyle-180i(\mu\nu)(\underline{\mu}\lambda)p\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}-180i(\mu\nu)(\underline{\mu}\lambda)p\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}+60i(\mu\nu)(\lambda\rho\sigma
a_{\mu})\gamma_{5}\tau\frac{\partial^{4}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+120i(\mu\nu\lambda)(\rho\sigma
a_{\nu})\gamma_{5}\tau\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}+90i(\mu\nu\lambda\rho)(\sigma
a_{\lambda})\gamma_{5}\tau\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}+90i(\mu
a^{\nu})(\lambda\rho\underline{\nu}\sigma)\gamma_{5}\tau\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+120i(\mu\nu
a^{\lambda})(\rho\underline{\lambda}\sigma)\gamma_{5}\tau\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial k^{\sigma}}+60i(\mu\nu\lambda
a^{\rho})(\underline{\rho}\sigma)\gamma_{5}\tau\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+240i(\mu\nu)(\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}$
$\displaystyle-180i(\mu\nu)(\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+180i(\mu\nu)(\underline{\mu}\lambda)\gamma_{5}p\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial}{\partial
k^{\nu}}+180i(\mu\nu)(\underline{\mu}\lambda)\gamma_{5}p\tau\Sigma_{k}^{\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$
$\displaystyle-240i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}-240i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}-480i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle-120i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+120i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-120i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle+120i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-180i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-180i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$
$\displaystyle+360i(\mu\nu)(\lambda\underline{\lambda}\rho\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+180i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+180i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle-240i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}-240i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}+480i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle-240i(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-240i(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-240i(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+480i(\mu\nu\lambda)(\underline{\mu}\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480i(\mu\nu\lambda)(\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-480i(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle-480i(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-480i(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480i(\mu\nu\lambda\rho)(\underline{\mu}\sigma)\gamma_{\lambda}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+480i(\mu\nu\lambda\rho)(\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480i(\mu\underline{\mu}\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-480i(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\rho}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle-480i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\lambda}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-480i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-90i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$
$\displaystyle+90i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\sigma}}-90i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+90i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+180i(\mu\nu)(\lambda\underline{\lambda}\rho\sigma)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}-180i(\mu\underline{\mu}\nu\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+90i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle-90i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+90i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}-90i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-120i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\lambda}}+120i(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+180i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+180i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}+240i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+240i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}$
$\displaystyle-180i(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-180i(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-180i(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-180i(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\sigma}}+240i(\mu\nu\lambda)(\underline{\mu}\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+240i(\mu\nu\lambda)(\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-240i(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}-240i(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+480i(\mu\nu\lambda)(\rho\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle-480i(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}-480i(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+240i(\mu\nu\lambda\rho)(\underline{\mu}\sigma)\gamma_{\lambda}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}$
$\displaystyle+240i(\mu\nu\lambda\rho)(\underline{\mu}\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}+240i(\mu\nu\lambda\rho)(\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}+240i(\mu\underline{\mu}\nu\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-240i(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\rho}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}-240i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}+240i(\mu\nu\lambda\rho)(\underline{\nu}\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}$
$\displaystyle-240i(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}-240i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\lambda}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+90i(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-90i(\mu\nu\underline{\nu}\lambda)(\rho\sigma)\gamma_{\rho}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\sigma}}+90i(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}-90i(\mu\nu\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+120i(\mu\nu\lambda)(\rho\underline{\rho}\sigma)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\sigma}}-120i(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\gamma_{\rho}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}-180i(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+240i(\mu\nu\lambda)(\rho\underline{\mu}\sigma)\gamma_{\nu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}-240i(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\sigma}}+120i(\mu\nu\lambda\rho)(\underline{\mu}\sigma)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+120i(\mu\nu\lambda\rho)(\underline{\nu}\sigma)\gamma_{\lambda}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\sigma}}-120i(\mu\nu\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\rho}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\sigma}}+480(\mu\nu\lambda\rho)s\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+360(\mu\nu\lambda\rho)s\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+360(\mu\nu\lambda\rho)s\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+360(\mu\nu\lambda\rho)s\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+480i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}-240i(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+240i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$
$\displaystyle-360i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+360i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180i(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-180i(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}-180i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}+180i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+360i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}-360i(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\nu}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-360i(\mu\nu)(\lambda\rho)(\underline{\mu}\sigma)\gamma_{\nu}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle-180i(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\rho}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+180i(\mu\nu)(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\mu}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}+512(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime\prime}$
$\displaystyle+576(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime\prime}+2880(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime\prime}+2560(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+5760(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\Sigma_{k}^{\prime\prime\prime}+5120(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\Sigma_{k}^{\prime\prime\prime}+240(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle-240(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-240(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}+320(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle-120(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-120(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-320(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+360(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+360(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+640(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle-320(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-960(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-960(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}$ $\displaystyle-480(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-480(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+960(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle-1280(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-1280(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+1280(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+1440(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}+1440(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+2560(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle-960(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}-960(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}+960(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle-720(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime
2}\frac{\partial}{\partial
k^{\nu}}-720(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\nu}}+1080(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\lambda}}$ $\displaystyle+720(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}+1080(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime
2}\frac{\partial}{\partial
k^{\rho}}+1920(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime
2}\frac{\partial}{\partial k^{\rho}}$
$\displaystyle-360(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}-360(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}+120(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+120(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\sigma}}-180(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\lambda}}+180(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle-180(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+180(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}-320(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+320(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}+360(\mu\nu)(\lambda\underline{\lambda}\rho\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}-120(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-120(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+240(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+\frac{640}{3}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$
$\displaystyle+240(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+240(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}+\frac{640}{3}(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-\frac{640}{3}(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial
k^{\sigma}}+\frac{640}{3}(\mu\nu\lambda)(\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+240(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+\frac{640}{3}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+\frac{640}{3}(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-540(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\lambda}}$
$\displaystyle-540(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+1080(\mu\nu)(\lambda\underline{\lambda}\rho\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}-360(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+360(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}-360(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+360(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle+540(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}+540(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}-960(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+960(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}-640(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}+640(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle+640(\mu\nu\lambda)(\underline{\mu}\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+640(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+720(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+720(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+720(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}+640(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+720(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\rho}}+640(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+180(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\lambda}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+320(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+180(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+320(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\sigma}}+320(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\rho\underline{\rho}\sigma)\tau
k_{\mu}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu)(\lambda\rho\underline{\lambda}\sigma)\tau
k_{\mu}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\rho}\partial k^{\sigma}}+320(\mu\nu\lambda)(\underline{\nu}\rho\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\lambda}\partial
k^{\sigma}}+320(\mu\underline{\mu}\nu)(\lambda\rho\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu)(\lambda\underline{\mu}\rho\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial
k^{\sigma}}+180(\mu\nu)(\underline{\mu}\lambda\rho\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau
k_{\lambda}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\rho}\partial k^{\sigma}}+320(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau
k_{\mu}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\lambda}\partial
k^{\rho}\partial
k^{\sigma}}+180(\mu\nu)(\lambda\rho\underline{\mu}\sigma)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+160(\mu\nu\lambda)(\rho\underline{\nu}\sigma)\tau\Sigma_{k}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\sigma}}+720(\mu\nu)(\lambda\rho)s\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+960i(\mu p)(\nu\lambda\rho)\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle+1440i(\mu\nu p)(\lambda\rho)\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-480i(\mu\nu\lambda)(\rho p)\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}+480i(\mu p)(\nu\lambda\rho)\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$ $\displaystyle+960i(\mu
p)(\nu\lambda\rho)\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+720i(\mu\nu p)(\lambda\rho)\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+720i(\mu\nu p)(\lambda\rho)\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}$ $\displaystyle-360i(\mu\nu)(\lambda\rho
p)\gamma_{5}\tau k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}}-480i(\mu\nu\lambda)(\rho
p)\gamma_{5}\tau k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}}+480i(\mu
p)(\nu\lambda\rho)\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}$ $\displaystyle+360i(\mu\nu
p)(\lambda\rho)\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}-240(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}-240(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle-480(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}-120(\mu\nu\underline{\nu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-120(\mu\nu\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle-120(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-120(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-180(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\lambda}}$ $\displaystyle-180(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+360(\mu
a^{\nu})(\lambda\underline{\lambda}\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+180(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}-240(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\nu}}-240(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}$ $\displaystyle+480(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+240(\mu
a^{\nu})(\underline{\mu}\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+240(\mu
a^{\nu})(\lambda\underline{\mu}\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+240(\mu
a^{\nu})(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu
a^{\lambda})(\underline{\mu}\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu
a^{\lambda})(\rho\underline{\mu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+480(\mu\nu
a^{\lambda})(\underline{\nu}\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu
a^{\lambda})(\rho\underline{\nu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\underline{\mu}a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+480(\mu\nu\lambda
a^{\rho})(\underline{\lambda}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu\lambda
a^{\rho})(\underline{\nu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu\underline{\nu}a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+480(\mu\nu\lambda
a^{\rho})(\underline{\mu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\nu\underline{\mu}a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+480(\mu\underline{\mu}\nu
a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle-120(\mu\underline{\mu}\nu)(\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}-90(\mu\nu\underline{\nu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}-90(\mu\nu\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+180(\mu\underline{\mu}\nu\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}+90(\mu\nu\underline{\nu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}+90(\mu\nu\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$ $\displaystyle-90(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\lambda}}-90(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+180(\mu
a^{\nu})(\lambda\underline{\lambda}\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}$ $\displaystyle+90(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}+90(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$ $\displaystyle+180(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}-120(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\nu}}+240(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+180(\mu
a^{\nu})(\lambda\underline{\mu}\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$ $\displaystyle+180(\mu
a^{\nu})(\underline{\mu}\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}+240(\mu\nu
a^{\lambda})(\underline{\mu}\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\underline{\mu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}+240(\mu\nu
a^{\lambda})(\underline{\nu}\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+240(\mu\nu
a^{\lambda})(\rho\underline{\nu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}$ $\displaystyle+480(\mu\nu
a^{\lambda})(\rho\underline{\mu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}+480(\mu\nu
a^{\lambda})(\rho\underline{\nu}\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}+480(\mu\underline{\mu}a^{\nu})(\lambda\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$ $\displaystyle+240(\mu\nu\lambda
a^{\rho})(\underline{\lambda}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+240(\mu\nu\lambda
a^{\rho})(\underline{\lambda}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}+240(\mu\nu\lambda
a^{\rho})(\underline{\nu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\sigma}}$
$\displaystyle+240(\mu\nu\underline{\nu}a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}+240(\mu\nu\lambda
a^{\rho})(\underline{\mu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}+240(\mu\nu\underline{\mu}a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}$ $\displaystyle+240(\mu\nu\lambda
a^{\rho})(\underline{\nu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}+240(\mu\nu\lambda
a^{\rho})(\underline{\mu}\sigma)\gamma_{\rho}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}+240(\mu\underline{\mu}\nu
a^{\lambda})(\rho\sigma)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\sigma}}$ $\displaystyle+120(\mu\underline{\mu}\nu)(\lambda\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\rho}}+90(\mu\nu\underline{\nu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\rho}}+90(\mu\nu\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}}$ $\displaystyle+90(\mu
a^{\nu})(\lambda\rho\underline{\rho}\sigma)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\sigma}}+90(\mu
a^{\nu})(\lambda\rho\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\sigma}}+120(\mu\nu
a^{\lambda})(\rho\underline{\rho}\sigma)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\sigma}}$ $\displaystyle+180(\mu
a^{\nu})(\lambda\rho\underline{\mu}\sigma)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}+240(\mu\nu
a^{\lambda})(\rho\underline{\mu}\sigma)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\underline{\nu}\sigma)\gamma_{\lambda}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\sigma}}+120(\mu\nu\lambda
a^{\rho})(\underline{\lambda}\sigma)\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\sigma}}$ $\displaystyle+120(\mu\nu\lambda
a^{\rho})(\underline{\nu}\sigma)\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\sigma}}+120(\mu\nu\lambda
a^{\rho})(\underline{\mu}\sigma)\gamma_{\rho}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}-240i(\mu\nu\lambda\rho)p\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}$ $\displaystyle-180i(\mu\nu\lambda\rho)p\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}-180i(\mu\nu\lambda\rho)p\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}-180i(\mu\nu\lambda\rho)p\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+240i(\mu\nu\lambda\rho)\gamma_{5}p\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+180i(\mu\nu\lambda\rho)\gamma_{5}p\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+180i(\mu\nu\lambda\rho)\gamma_{5}p\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}$
$\displaystyle+180i(\mu\nu\lambda\rho)\gamma_{5}p\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}}+480(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}+1920(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$
$\displaystyle+1440(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\nu}k_{\lambda}k_{\rho}\Sigma_{k}^{\prime\prime 2}\frac{\partial}{\partial
k^{\sigma}}+120(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\sigma}}-120(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}$
$\displaystyle-360(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\nu}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\sigma}}+360(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\lambda}k_{\rho}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\sigma}}$
$\displaystyle+360(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\rho}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\sigma}}+360(\mu\nu)(\underline{\mu}\lambda)(\rho\sigma)\tau
k_{\rho}\Sigma_{k}^{\prime 2}\frac{\partial^{3}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\sigma}}+480(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}$
$\displaystyle+240(\mu\nu)(\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}+240(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\mu}}-360(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\sigma}}$ $\displaystyle+360(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial}{\partial
k^{\rho}}-180(\mu\nu)(\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\rho}}$
$\displaystyle+180(\mu\nu)(\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}-180(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\mu}\partial
k^{\sigma}}$ $\displaystyle+180(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\rho}}+360(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial k^{\rho}\partial
k^{\sigma}}$ $\displaystyle+360(\mu
a^{\nu})(\underline{\mu}\lambda)(\rho\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\sigma}}+360(\mu
a^{\nu})(\lambda\rho)(\underline{\mu}\sigma)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu)(\underline{\mu}\lambda)(\rho
a^{\sigma})\gamma_{\sigma}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}}+180(\mu
a^{\nu})(\lambda\rho)(\underline{\lambda}\sigma)\gamma_{\nu}\gamma_{5}\tau\Sigma_{k}^{\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle-360i(\mu\nu)(\lambda\rho)p\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\rho}}+360i(\mu\nu)(\lambda\rho)\gamma_{5}p\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\rho}}+480i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}$
$\displaystyle+960i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}+960i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}$
$\displaystyle+240i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\delta}}-240i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\sigma}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}$
$\displaystyle+360i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\delta}}+360i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}$
$\displaystyle+480i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial
k^{\delta}}+480i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\delta}}$
$\displaystyle+960i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\delta}}+480i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}$
$\displaystyle+480i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial
k^{\delta}}+480i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\delta}}$
$\displaystyle+180i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\delta}}-180i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\sigma}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+180i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\delta}}-180i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\sigma}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+240i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial
k^{\delta}}-240i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\sigma}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+360i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}+480i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+480i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\delta}}+240i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+240i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial
k^{\delta}}+240i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\delta}}$
$\displaystyle+180i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\mu}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}-180i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\sigma}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+240i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\nu}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}-240i(\mu\nu\lambda)(\rho\sigma\delta)\gamma_{\sigma}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle-120i(\mu\nu)(\lambda\rho\sigma\delta)\gamma_{\sigma}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}+120i(\mu\nu\lambda\rho)(\sigma\delta)\gamma_{\lambda}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+720i(\mu\nu)(\lambda\rho)(\sigma\delta)\gamma_{\mu}\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial
k^{\delta}}-360i(\mu\nu)(\lambda\rho)(\sigma\delta)\gamma_{\sigma}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+360i(\mu\nu)(\lambda\rho)(\sigma\delta)\gamma_{\mu}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\delta}}+480(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\delta}}$
$\displaystyle+320(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\delta}}+1920(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\delta}}$
$\displaystyle+1280(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\delta}}+960(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime
2}\frac{\partial^{2}}{\partial k^{\lambda}\partial k^{\delta}}$
$\displaystyle+1440(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime
2}\frac{\partial^{2}}{\partial k^{\nu}\partial
k^{\delta}}+360(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\delta}}$
$\displaystyle+360(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\delta}}+640(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+1080(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial
k^{\delta}}+1080(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+1920(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}+360(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{4}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+320(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\nu}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial
k^{\delta}}+360(\mu\nu)(\lambda\rho\sigma\delta)\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime 2}\frac{\partial^{4}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$
$\displaystyle+320(\mu\nu\lambda)(\rho\sigma\delta)\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime 2}\frac{\partial^{4}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\delta}}+480(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}$ $\displaystyle+960(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}+960(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime\prime}\frac{\partial}{\partial
k^{\delta}}$ $\displaystyle+240(\mu\nu\lambda\rho)(\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\sigma}}+240(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\delta}}$ $\displaystyle+360(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\delta}}+360(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}$ $\displaystyle+480(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\delta}}+480(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\delta}}$ $\displaystyle+960(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}+480(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\mu}\partial k^{\delta}}$ $\displaystyle+480(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\nu}\partial k^{\delta}}+480(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\lambda}\partial k^{\delta}}$ $\displaystyle+240(\mu\nu\lambda)(\rho\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\mu}k_{\nu}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}+180(\mu\nu\lambda\rho)(\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\nu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu\lambda\rho)(\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\delta}}$ $\displaystyle+180(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\delta}}+360(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\rho}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\delta}}+480(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$ $\displaystyle+480(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\delta}}+240(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\nu}\partial k^{\delta}}$ $\displaystyle+240(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\nu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\lambda}\partial k^{\delta}}+240(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\mu}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\lambda}\partial k^{\delta}}$
$\displaystyle+120(\mu\nu)(\lambda\rho\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\mu}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\nu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}+240(\mu\nu\lambda)(\rho\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\nu}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\sigma}}$
$\displaystyle+180(\mu\nu\lambda\rho)(\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\lambda}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}+180(\mu
a^{\nu})(\lambda\rho\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\lambda}\partial k^{\rho}\partial k^{\delta}}$ $\displaystyle+240(\mu\nu
a^{\lambda})(\rho\sigma\delta)\gamma_{\lambda}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}+120(\mu\nu\lambda
a^{\rho})(\sigma\delta)\gamma_{\rho}\gamma_{5}\tau
k_{\sigma}\Sigma_{k}^{\prime}\frac{\partial^{4}}{\partial k^{\mu}\partial
k^{\nu}\partial k^{\lambda}\partial k^{\delta}}$
$\displaystyle+240(\mu\nu)(\lambda\rho)(\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial
k^{\delta}}+720(\mu\nu)(\lambda\rho)(\sigma\delta)\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\delta}}$ $\displaystyle+720(\mu
a^{\nu})(\lambda\rho)(\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\mu}k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime\prime}\frac{\partial^{2}}{\partial
k^{\rho}\partial k^{\delta}}+360(\mu\nu)(\lambda\rho)(\sigma
a^{\delta})\gamma_{\delta}\gamma_{5}\tau
k_{\mu}k_{\lambda}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\nu}\partial k^{\rho}\partial k^{\sigma}}$ $\displaystyle+360(\mu
a^{\nu})(\lambda\rho)(\sigma\delta)\gamma_{\nu}\gamma_{5}\tau
k_{\lambda}k_{\sigma}\Sigma_{k}^{\prime\prime}\frac{\partial^{3}}{\partial
k^{\mu}\partial k^{\rho}\partial k^{\delta}}\;.$
## Appendix B Analytical expressions of $\mathcal{Z}_{i}$ on $\Sigma(k^{2})$
$\displaystyle\mathcal{Z}_{1}=\int
dK\bigg{[}-\frac{2}{3}\tau^{3}+\frac{1}{3}\tau^{4}k^{2}+4\tau^{4}\Sigma_{k}^{2}-\frac{1}{18}\tau^{5}k^{4}-\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{8}{3}\tau^{5}\Sigma_{k}^{4}+\frac{1}{540}\tau^{6}k^{6}+\frac{2}{45}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{4}{15}\tau^{6}k^{2}\Sigma_{k}^{4}+\frac{16}{45}\tau^{6}\Sigma_{k}^{6}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{2}=\int
dK\bigg{[}2\tau^{3}-\frac{1}{3}\tau^{4}k^{2}-12\tau^{4}\Sigma_{k}^{2}-\frac{1}{18}\tau^{5}k^{4}+\frac{7}{3}\tau^{5}k^{2}\Sigma_{k}^{2}+8\tau^{5}\Sigma_{k}^{4}+\frac{1}{180}\tau^{6}k^{6}-\frac{2}{45}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{4}{5}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{16}{15}\tau^{6}\Sigma_{k}^{6}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{3}=\int
dK\bigg{[}-\tau^{3}+\frac{1}{2}\tau^{4}k^{2}+6\tau^{4}\Sigma_{k}^{2}-\frac{1}{18}\tau^{5}k^{4}-\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{2}-4\tau^{5}\Sigma_{k}^{4}+\frac{1}{360}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{2}{5}\tau^{6}k^{2}\Sigma_{k}^{4}+\frac{8}{15}\tau^{6}\Sigma_{k}^{6}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{4}=\int
dK\bigg{[}-\tau^{3}+\frac{1}{6}\tau^{4}k^{2}+6\tau^{4}\Sigma_{k}^{2}-\tau^{5}k^{2}\Sigma_{k}^{2}-4\tau^{5}\Sigma_{k}^{4}+\frac{1}{360}\tau^{6}k^{6}-\frac{1}{45}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{2}{5}\tau^{6}k^{2}\Sigma_{k}^{4}+\frac{8}{15}\tau^{6}\Sigma_{k}^{6}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{5}=\int
dK\bigg{[}\frac{1}{3}\tau^{3}-\frac{1}{6}\tau^{4}k^{2}-2\tau^{4}\Sigma_{k}^{2}+\frac{1}{3}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{3}\tau^{5}\Sigma_{k}^{4}+\frac{1}{1080}\tau^{6}k^{6}+\frac{1}{45}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{2}{15}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{8}{45}\tau^{6}\Sigma_{k}^{6}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{6}=\int
dK\bigg{[}-\frac{4}{15}\tau^{3}+\frac{1}{3}\tau^{4}k^{2}+\frac{2}{5}\tau^{4}\Sigma_{k}^{2}-\frac{1}{10}\tau^{5}k^{4}+\frac{2}{3}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{7}=\int
dK\bigg{[}-\frac{11}{5}\tau^{3}+3\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{13}{10}\tau^{4}k^{2}+\frac{92}{15}\tau^{4}\Sigma_{k}^{2}-18\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{17}{90}\tau^{5}k^{4}-\frac{11}{5}\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{8}{5}\tau^{5}\Sigma_{k}^{4}$
$\displaystyle+\frac{14}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+12\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{120}\tau^{6}k^{6}+\frac{2}{15}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{2}{5}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{8}{5}\tau^{6}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{8}=\int
dK\bigg{[}\frac{4}{15}\tau^{3}+\frac{1}{5}\tau^{4}k^{2}-\frac{2}{5}\tau^{4}\Sigma_{k}^{2}-\frac{1}{10}\tau^{5}k^{4}+\frac{2}{3}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{9}=\int
dK\bigg{[}-\frac{23}{15}\tau^{3}+\frac{59}{60}\tau^{4}k^{2}+\frac{17}{15}\tau^{4}\Sigma_{k}^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{7}{45}\tau^{5}k^{4}+\frac{8}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{10}=\int
dK\bigg{[}-\frac{4}{5}\tau^{3}+\frac{19}{30}\tau^{4}k^{2}+\frac{8}{15}\tau^{4}\Sigma_{k}^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{7}{45}\tau^{5}k^{4}+\frac{8}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{11}=\int
dK\bigg{[}-\frac{2}{5}\tau^{3}+\frac{1}{30}\tau^{4}k^{2}+\frac{4}{15}\tau^{4}\Sigma_{k}^{2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{1}{30}\tau^{5}k^{4}-\frac{2}{3}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}-\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{12}=\int
dK\bigg{[}\frac{16}{15}\tau^{3}-\frac{2}{5}\tau^{4}k^{2}-\frac{14}{15}\tau^{4}\Sigma_{k}^{2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{1}{30}\tau^{5}k^{4}-\frac{2}{3}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}-\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{13}=\int
dK\bigg{[}-\frac{28}{15}\tau^{3}+\frac{9}{10}\tau^{4}k^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{4}{5}\tau^{4}\Sigma_{k}^{2}-\frac{7}{45}\tau^{5}k^{4}+\frac{8}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{14}=\int
dK\bigg{[}\frac{13}{15}\tau^{3}+\frac{23}{60}\tau^{4}k^{2}-\frac{7}{15}\tau^{4}\Sigma_{k}^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{7}{45}\tau^{5}k^{4}+\frac{8}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{120}\tau^{6}k^{6}+\frac{1}{15}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{15}=\int
dK\bigg{[}\frac{11}{5}\tau^{3}-3\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{19}{30}\tau^{4}k^{2}-\frac{92}{15}\tau^{4}\Sigma_{k}^{2}+18\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{11}{5}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{8}{5}\tau^{5}\Sigma_{k}^{4}-\frac{4}{3}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-12\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{120}\tau^{6}k^{6}-\frac{2}{15}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{2}{5}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{8}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{8}{5}\tau^{6}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{16}=\int
dK\bigg{[}-\frac{11}{5}\tau^{3}+3\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{13}{10}\tau^{4}k^{2}+\frac{92}{15}\tau^{4}\Sigma_{k}^{2}-18\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{11}{45}\tau^{5}k^{4}-\frac{11}{5}\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{8}{5}\tau^{5}\Sigma_{k}^{4}$
$\displaystyle\hskip
11.38092pt+\frac{16}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+12\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{120}\tau^{6}k^{6}+\frac{2}{15}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{2}{5}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{1}{30}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{15}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{8}{5}\tau^{6}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{17}=\int
dK\bigg{[}-\frac{11}{15}\tau^{3}+\frac{49}{60}\tau^{4}k^{2}+\frac{3}{5}\tau^{4}\Sigma_{k}^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{17}{90}\tau^{5}k^{4}+\frac{10}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{90}\tau^{6}k^{6}+\frac{4}{45}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{2}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{16}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{18}=\int
dK\bigg{[}-\frac{29}{15}\tau^{3}+\frac{67}{60}\tau^{4}k^{2}+\frac{11}{15}\tau^{4}\Sigma_{k}^{2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{17}{90}\tau^{5}k^{4}+\frac{10}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{90}\tau^{6}k^{6}+\frac{4}{45}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{2}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{16}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{19}=\int
dK\bigg{[}\frac{4}{3}\tau^{3}-\frac{2}{5}\tau^{4}k^{2}-\frac{2}{3}\tau^{4}\Sigma_{k}^{2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{1}{45}\tau^{5}k^{4}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{180}\tau^{6}k^{6}-\frac{2}{45}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{8}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{20}=\int
dK\bigg{[}-\frac{4}{5}\tau^{3}+2\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{7}{10}\tau^{4}k^{2}+\frac{58}{15}\tau^{4}\Sigma_{k}^{2}-12\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{13}{90}\tau^{5}k^{4}-\frac{22}{15}\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{16}{15}\tau^{5}\Sigma_{k}^{4}$
$\displaystyle\hskip
11.38092pt+\frac{10}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+8\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{180}\tau^{6}k^{6}+\frac{4}{45}\tau^{6}k^{4}\Sigma_{k}^{2}+\frac{4}{15}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{1}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{16}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{16}{15}\tau^{6}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{21}=\int
dK\bigg{[}\frac{1}{15}\tau^{3}-\frac{1}{30}\tau^{4}k^{2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{4}\Sigma_{k}^{2}-\frac{1}{45}\tau^{5}k^{4}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{180}\tau^{6}k^{6}-\frac{2}{45}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{8}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{22}=\int
dK\bigg{[}-2\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{4}{5}\tau^{3}-\frac{58}{15}\tau^{4}\Sigma_{k}^{2}-\frac{1}{5}\tau^{4}k^{2}+12\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{3}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{22}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{16}{15}\tau^{5}\Sigma_{k}^{4}-\frac{8}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-8\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{180}\tau^{6}k^{6}-\frac{4}{45}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{4}{15}\tau^{6}k^{2}\Sigma_{k}^{4}-\frac{1}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{16}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{16}{15}\tau^{6}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{23}=\int
dK\bigg{[}-\frac{1}{15}\tau^{3}-\frac{1}{10}\tau^{4}k^{2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{4}\Sigma_{k}^{2}-\frac{1}{45}\tau^{5}k^{4}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{1}{180}\tau^{6}k^{6}-\frac{2}{45}\tau^{6}k^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{45}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{8}{45}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{24}=\int
dK\bigg{[}4\tau^{2}-\tau^{3}k^{2}-16\tau^{3}\Sigma_{k}^{2}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}^{2}+\frac{16}{3}\tau^{4}\Sigma_{k}^{4}\bigg{]}\;,\hskip
85.35826pt\mathcal{Z}_{25}=\int
dK\bigg{[}4\tau^{2}-\tau^{3}k^{2}-4\tau^{3}\Sigma_{k}^{2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{26}=\int
dK\bigg{[}2\tau^{2}-8\tau^{3}\Sigma_{k}^{2}+\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}^{2}+\frac{8}{3}\tau^{4}\Sigma_{k}^{4}\bigg{]}\;,\hskip
28.45274pt\mathcal{Z}_{27}=\int dK\bigg{[}2\tau^{2}\bigg{]}\;,\hskip
28.45274pt\mathcal{Z}_{28}=\int
dK\bigg{[}-4\tau^{2}+\tau^{3}k^{2}+4\tau^{3}\Sigma_{k}^{2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{29}=\int dK\bigg{[}2\tau^{2}\bigg{]}\;,\hskip
71.13188pt\mathcal{Z}_{30}=\int
dK\bigg{[}-\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}\bigg{]}\;,\hskip
71.13188pt\mathcal{Z}_{31}=\int
dK\bigg{[}-\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}\bigg{]};,$
$\displaystyle\mathcal{Z}_{32}=\int
dK\bigg{[}-6\tau^{3}\Sigma_{k}+\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{8}{3}\tau^{4}\Sigma_{k}^{3}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{33}=\int
dK\bigg{[}4\tau^{3}\Sigma_{k}+\frac{4}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}-\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}-\frac{8}{3}\tau^{4}\Sigma_{k}^{3}-\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}\bigg{]}\;,\hskip
56.9055pt\mathcal{Z}_{34}=\int
dK\bigg{[}\frac{4}{3}\tau^{3}\Sigma_{k}+\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{35}=\int
dK\bigg{[}-2\tau^{3}\Sigma_{k}+\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}\bigg{]}\;,\hskip
56.9055pt\mathcal{Z}_{36}=\int
dK\bigg{[}-\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}\bigg{]}\;,\hskip
56.9055pt\mathcal{Z}_{37}=\int
dK\bigg{[}\frac{2}{3}\tau^{3}\Sigma_{k}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{38}=\int
dK\bigg{[}-\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{3}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{39}=\int
dK\bigg{[}6\tau^{3}\Sigma_{k}-10\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-2\tau^{4}k^{2}\Sigma_{k}-\frac{8}{3}\tau^{4}\Sigma_{k}^{3}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{3}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}+\frac{40}{3}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}$ $\displaystyle\hskip
11.38092pt+\frac{2}{3}\tau^{5}k^{2}\Sigma_{k}^{3}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}-\frac{8}{3}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{40}=\int
dK\bigg{[}-\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{4}{3}\tau^{3}\Sigma_{k}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{3}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{41}=\int
dK\bigg{[}10\tau^{3}\Sigma_{k}-\frac{8}{3}\tau^{4}k^{2}\Sigma_{k}-\frac{40}{3}\tau^{4}\Sigma_{k}^{3}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}+\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{3}+\frac{8}{3}\tau^{5}\Sigma_{k}^{5}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{42}=\int
dK\bigg{[}-10\tau^{3}\Sigma_{k}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{40}{3}\tau^{4}\Sigma_{k}^{3}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}-\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{3}-\frac{8}{3}\tau^{5}\Sigma_{k}^{5}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{43}=\int
dK\bigg{[}10\tau^{3}\Sigma_{k}-2\tau^{4}k^{2}\Sigma_{k}-\frac{40}{3}\tau^{4}\Sigma_{k}^{3}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}+\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{3}+\frac{8}{3}\tau^{5}\Sigma_{k}^{5}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{44}=\int
dK\bigg{[}-\frac{1}{12}\tau^{3}-\frac{1}{6}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{4}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{29}{120}\tau^{4}k^{2}+\frac{1}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{29}{540}\tau^{5}k^{4}+\frac{19}{90}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{135}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{45}=\int
dK\bigg{[}-\frac{1}{3}\tau^{2}\Sigma_{k}^{\prime
2}+\frac{2}{3}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{5}{6}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{2}{5}\tau^{3}+\frac{4}{3}\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{7}{45}\tau^{4}k^{2}-\frac{8}{45}\tau^{4}\Sigma_{k}^{2}+\frac{4}{9}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}$ $\displaystyle\hskip
11.38092pt-\frac{4}{9}\tau^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{9}\tau^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{270}\tau^{5}k^{4}+\frac{1}{45}\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{1}{45}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{135}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{2}{15}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}$
$\displaystyle\hskip
11.38092pt-\frac{4}{45}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime
3}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{46}=\int
dK\bigg{[}\frac{7}{36}\tau^{3}+\frac{5}{6}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{5}{18}\tau^{4}k^{2}-\frac{2}{9}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{18}\tau^{5}k^{4}-\frac{2}{9}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{47}=\int
dK\bigg{[}\frac{11}{12}\tau^{3}-\frac{1}{18}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{2}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{113}{360}\tau^{4}k^{2}-\frac{1}{9}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{31}{540}\tau^{5}k^{4}-\frac{7}{30}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{135}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{48}=\int
dK\bigg{[}\frac{35}{18}\tau^{3}-\frac{1}{9}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{14}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{211}{180}\tau^{4}k^{2}-\frac{4}{3}\tau^{4}\Sigma_{k}^{2}+\frac{85}{18}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt+\frac{59}{540}\tau^{5}k^{4}+\frac{1}{3}\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{13}{30}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{135}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{49}=\int
dK\bigg{[}\frac{1}{36}\tau^{3}-\frac{1}{18}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{5}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{3}{20}\tau^{4}k^{2}-\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{7}{135}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt-\frac{1}{5}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{135}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{50}=\int
dK\bigg{[}-\frac{2}{45}\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}-\frac{2}{15}\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{2}{15}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{7}{15}\tau^{2}\Sigma_{k}^{\prime
2}+\frac{4}{15}\tau^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{4}{15}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime 2}-\frac{1}{45}\tau^{3}$
$\displaystyle\hskip
11.38092pt-\frac{2}{15}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{4}{15}\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{8}{45}\tau^{3}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{4}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}-\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{4}{27}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{51}=\int
dK\bigg{[}\frac{1}{135}\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{45}\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{1}{45}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{1}{45}\tau^{2}\Sigma_{k}^{\prime
2}-\frac{2}{45}\tau^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{2}{45}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime
2}-\frac{22}{135}\tau^{3}$ $\displaystyle\hskip
11.38092pt+\frac{1}{45}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{2}{45}\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{135}\tau^{3}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{4}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}+\frac{1}{24}\tau^{4}k^{2}+\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{1}{6}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime 2}$
$\displaystyle\hskip
11.38092pt+\frac{4}{27}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{52}=\int
dK\bigg{[}\frac{1}{135}\tau\Sigma_{k}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{45}\tau\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{1}{45}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{1}{45}\tau^{2}\Sigma_{k}^{\prime
2}-\frac{2}{45}\tau^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{2}{45}\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime 2}+\frac{1}{270}\tau^{3}$
$\displaystyle\hskip
11.38092pt+\frac{1}{45}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{2}{45}\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{135}\tau^{3}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{4}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}+\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{4}{27}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{53}=\int
dK\bigg{[}-\frac{1}{18}\tau^{3}+\frac{17}{180}\tau^{4}k^{2}-\frac{1}{18}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{54}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{36}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{29}{270}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{29}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{540}\tau^{6}k^{6}-\frac{2}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{54}=\int
dK\bigg{[}-\frac{1}{6}\tau^{3}+\frac{5}{36}\tau^{4}k^{2}-\frac{1}{6}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{54}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{17}{540}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{7}{54}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{7}{81}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{540}\tau^{6}k^{6}-\frac{2}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{55}=\int
dK\bigg{[}\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime
2}-\frac{1}{5}\tau^{3}+\frac{1}{2}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{11}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}+\frac{2}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}+\frac{37}{180}\tau^{4}k^{2}+\frac{11}{45}\tau^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip 11.38092pt-\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{14}{9}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{26}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{44}{27}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle\hskip
11.38092pt-\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}-\frac{1}{30}\tau^{5}k^{4}-\frac{1}{9}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{41}{135}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{22}{45}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{182}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt+\frac{44}{135}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{8}{27}\tau^{5}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}+\frac{1}{135}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{135}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}\Sigma_{k}^{\prime 3}$ $\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{32}{405}\tau^{6}k^{4}\Sigma_{k}^{5}-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}-\frac{16}{405}\tau^{6}k^{4}\Sigma_{k}^{6}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{56}=\int
dK\bigg{[}-\frac{1}{18}\tau^{3}+\frac{5}{72}\tau^{4}k^{2}-\frac{1}{18}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{54}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{45}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{2}{27}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{81}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{540}\tau^{6}k^{6}-\frac{2}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{57}=\int
dK\bigg{[}-\frac{1}{36}\tau^{3}+\frac{5}{144}\tau^{4}k^{2}-\frac{1}{36}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{90}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{1}{27}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{81}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{58}=\int
dK\bigg{[}-\frac{1}{36}\tau^{3}+\frac{5}{144}\tau^{4}k^{2}-\frac{1}{36}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{90}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{1}{27}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{81}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{59}=\int
dK\bigg{[}\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime
2}-\frac{2}{5}\tau^{3}+\frac{2}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}+\frac{31}{180}\tau^{4}k^{2}-\frac{5}{9}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{27}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip 11.38092pt-\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{26}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{2}{45}\tau^{4}\Sigma_{k}^{2}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}+\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle\hskip
11.38092pt-\frac{1}{45}\tau^{5}k^{4}+\frac{32}{135}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{164}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{8}{27}\tau^{5}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}-\frac{2}{45}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{45}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{8}{135}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}+\frac{1}{135}\tau^{6}k^{4}\Sigma_{k}^{2}$ $\displaystyle\hskip
11.38092pt-\frac{8}{135}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{32}{405}\tau^{6}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime
3}-\frac{16}{405}\tau^{6}k^{4}\Sigma_{k}^{6}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{60}=\int
dK\bigg{[}\frac{5}{18}\tau^{3}-\frac{1}{20}\tau^{4}k^{2}+\frac{5}{18}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{108}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{7}{270}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{7}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{61}=\int
dK\bigg{[}\frac{1}{18}\tau^{2}k^{4}\Sigma_{k}^{\prime\prime
2}+\frac{2}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{15}\tau^{3}+\frac{2}{27}\tau^{3}k^{4}\Sigma_{k}^{\prime
4}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}+\frac{4}{45}\tau^{4}k^{2}-\frac{2}{9}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-\frac{8}{27}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{26}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{2}{45}\tau^{4}\Sigma_{k}^{2}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime 2}$
$\displaystyle\hskip
11.38092pt+\frac{2}{27}\tau^{4}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime\prime
2}-\frac{1}{45}\tau^{5}k^{4}+\frac{32}{135}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{164}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{8}{27}\tau^{5}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}-\frac{2}{45}\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{45}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt+\frac{8}{135}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}$
$\displaystyle\hskip
11.38092pt-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime
4}+\frac{1}{135}\tau^{6}k^{4}\Sigma_{k}^{2}-\frac{8}{135}\tau^{6}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{32}{405}\tau^{6}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime
3}-\frac{16}{405}\tau^{6}k^{4}\Sigma_{k}^{6}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{62}=\int
dK\bigg{[}\frac{1}{45}\tau^{4}k^{2}-\frac{4}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{54}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{2}{135}\tau^{5}k^{4}+\frac{4}{135}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt+\frac{8}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{2}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{540}\tau^{6}k^{6}-\frac{2}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{8}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{63}=\int
dK\bigg{[}\frac{1}{18}\tau^{3}-\frac{1}{90}\tau^{4}k^{2}+\frac{1}{18}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{2}{81}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{324}\tau^{4}k^{6}\Sigma_{k}^{\prime
4}+\frac{1}{108}\tau^{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime
2}-\frac{1}{180}\tau^{5}k^{4}$ $\displaystyle\hskip
11.38092pt+\frac{1}{270}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{405}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{162}\tau^{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{81}\tau^{5}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
4}+\frac{1}{1080}\tau^{6}k^{6}-\frac{1}{135}\tau^{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt-\frac{4}{405}\tau^{6}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{2}{405}\tau^{6}k^{6}\Sigma_{k}^{4}\Sigma_{k}^{\prime 4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{64}=\int
dK\bigg{[}-\frac{1}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{4}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{\prime
3}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{65}=\int
dK\bigg{[}\frac{4}{3}\tau^{3}\Sigma_{k}-\frac{7}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime}-\frac{20}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{\prime
3}-\frac{1}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{66}=\int
dK\bigg{[}-\frac{4}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime}-\frac{2}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{9}\tau^{3}k^{4}\Sigma_{k}^{\prime
3}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{67}=\int
dK\bigg{[}\frac{5}{3}\tau^{2}-\tau^{2}k^{2}\Sigma_{k}^{\prime
2}-\frac{4}{3}\tau^{3}\Sigma_{k}^{2}-\frac{1}{3}\tau^{3}k^{2}+4\tau^{3}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{3}\tau^{4}k^{2}\Sigma_{k}^{2}-\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{68}=\int
dK\bigg{[}\tau^{2}+\tau^{2}k^{2}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{69}=\int
dK\bigg{[}\frac{1}{3}\tau^{3}+\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{70}=\int
dK\bigg{[}-\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\tau^{2}\Sigma_{k}^{\prime
2}+\frac{5}{3}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{2}\tau^{3}+4\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{2}{3}\tau^{3}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{6}\tau^{4}k^{2}+\frac{1}{3}\tau^{4}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{4}{3}\tau^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{4}{3}\tau^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{71}=\int
dK\bigg{[}\frac{13}{6}\tau^{3}+\frac{1}{3}\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{7}{12}\tau^{4}k^{2}-\frac{4}{3}\tau^{4}\Sigma_{k}^{2}-\frac{5}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{72}=\int
dK\bigg{[}-2\tau^{3}+\tau^{3}\Sigma_{k}\Sigma_{k}^{\prime}-3\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{13}{24}\tau^{4}k^{2}+\frac{4}{3}\tau^{4}\Sigma_{k}^{2}+\frac{11}{6}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{73}=\int
dK\bigg{[}-\frac{11}{12}\tau^{3}+\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{23}{48}\tau^{4}k^{2}+\frac{2}{3}\tau^{4}\Sigma_{k}^{2}+\frac{1}{12}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{74}=\int
dK\bigg{[}\frac{1}{12}\tau^{4}k^{2}+\frac{1}{3}\tau^{4}\Sigma_{k}^{2}-\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{75}=\int
dK\bigg{[}-\frac{1}{3}\tau\Sigma_{k}^{\prime\prime}+\frac{2}{3}\tau^{2}\Sigma_{k}^{\prime}+2\tau^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}+\frac{2}{3}\tau^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}+2\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{4}{3}\tau^{3}\Sigma_{k}^{2}\Sigma_{k}^{\prime}+\frac{2}{3}\tau^{3}\Sigma_{k}-\frac{4}{3}\tau^{3}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{76}=\int
dK\bigg{[}-\frac{2}{3}\tau^{2}\Sigma_{k}^{\prime}+2\tau^{3}\Sigma_{k}-\frac{8}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{20}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{77}=\int
dK\bigg{[}-\frac{2}{3}\tau^{2}\Sigma_{k}^{\prime}+\frac{1}{9}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{2}{9}\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{78}=\int
dK\bigg{[}-\frac{2}{3}\tau^{3}-\frac{1}{6}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{43}{120}\tau^{4}k^{2}-\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}-\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{11}{180}\tau^{5}k^{4}+\frac{23}{90}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{45}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{79}=\int
dK\bigg{[}\frac{31}{15}\tau^{3}-\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+2\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{13}{30}\tau^{4}k^{2}+\frac{2}{5}\tau^{4}\Sigma_{k}^{2}-\frac{8}{3}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{1}{45}\tau^{5}k^{4}-\frac{2}{15}\tau^{5}k^{2}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt+\frac{2}{15}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{4}{5}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{4}{45}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{8}{15}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{80}=\int
dK\bigg{[}-\frac{4}{3}\tau^{3}+\frac{5}{6}\tau^{3}k^{2}\Sigma_{k}^{\prime
2}+\frac{3}{40}\tau^{4}k^{2}-\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{18}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{7}{180}\tau^{5}k^{4}-\frac{11}{90}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}$ $\displaystyle\hskip
11.38092pt+\frac{1}{15}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{81}=\int
dK\bigg{[}\frac{8}{3}\tau^{3}-2\tau^{3}k^{2}\Sigma_{k}^{\prime
2}-\frac{6}{5}\tau^{4}k^{2}-\frac{4}{3}\tau^{4}\Sigma_{k}^{2}+5\tau^{4}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{4}k^{4}\Sigma_{k}^{\prime
2}+\frac{2}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{1}{10}\tau^{5}k^{4}+\frac{1}{3}\tau^{5}k^{2}\Sigma_{k}^{2}$
$\displaystyle\hskip
11.38092pt-\frac{17}{45}\tau^{5}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{2i}{45}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{4}{3}\tau^{5}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{82}=\int
dK\bigg{[}-6\tau^{3}\Sigma_{k}-\frac{2}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{8}{3}\tau^{4}\Sigma_{k}^{3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{83}=\int
dK\bigg{[}-\frac{20}{3}\tau^{3}\Sigma_{k}+\frac{4}{3}\tau^{3}k^{2}\Sigma_{k}^{\prime}+\frac{4}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{8}{3}\tau^{4}\Sigma_{k}^{3}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{84}=\int
dK\bigg{[}\frac{17}{15}\tau^{3}-\frac{47}{60}\tau^{4}k^{2}-\frac{73}{15}\tau^{4}\Sigma_{k}^{2}+\frac{1}{9}\tau^{5}k^{4}+\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{3}\tau^{5}\Sigma_{k}^{4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{85}=\int
dK\bigg{[}-\frac{8}{5}\tau^{3}+\frac{11}{30}\tau^{4}k^{2}+\frac{76}{15}\tau^{4}\Sigma_{k}^{2}-\tau^{5}k^{2}\Sigma_{k}^{2}-\frac{4}{3}\tau^{5}\Sigma_{k}^{4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{86}=\int
dK\bigg{[}\frac{46}{15}\tau^{3}-\frac{43}{30}\tau^{4}k^{2}-\frac{28}{5}\tau^{4}\Sigma_{k}^{2}+\frac{1}{9}\tau^{5}k^{4}+\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{3}\tau^{5}\Sigma_{k}^{4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{87}=\int
dK\bigg{[}\frac{4}{5}\tau^{3}+\frac{2}{15}\tau^{4}k^{2}-\frac{26}{5}\tau^{4}\Sigma_{k}^{2}-\frac{1}{9}\tau^{5}k^{4}+\tau^{5}k^{2}\Sigma_{k}^{2}+\frac{4}{3}\tau^{5}\Sigma_{k}^{4}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{88}=\int
dK\bigg{[}-4\tau^{2}\Sigma_{k}+\frac{8}{3}\tau^{3}\Sigma_{k}^{3}\bigg{]}\;,\hskip
54.06006pt\mathcal{Z}_{89}=\int
dK\bigg{[}-4\tau^{2}\Sigma_{k}\bigg{]}\;,\hskip
54.06006pt\mathcal{Z}_{90}=0\hskip 54.06006pt\mathcal{Z}_{91}=0\;,$
$\displaystyle\mathcal{Z}_{92}=\int
dK\bigg{[}6\tau^{3}\Sigma_{k}-10\tau^{3}k^{2}\Sigma_{k}\Sigma_{k}^{\prime
2}-2\tau^{4}k^{2}\Sigma_{k}-\frac{8}{3}\tau^{4}\Sigma_{k}^{3}+\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}+\frac{40}{3}\tau^{4}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}+\frac{2}{3}\tau^{5}k^{2}\Sigma_{k}^{3}$
$\displaystyle\hskip
11.38092pt-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}-\frac{8}{3}\tau^{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime 2}\bigg{]}\;,$
$\displaystyle\mathcal{Z}_{93}=\int
dK\bigg{[}-\frac{2}{3}\tau^{3}\Sigma_{k}+\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ $\displaystyle\mathcal{Z}_{94}=\int
dK\bigg{[}\frac{2}{3}\tau^{3}\Sigma_{k}+\frac{4}{9}\tau^{4}k^{4}\Sigma_{k}\Sigma_{k}^{\prime
2}-\frac{2}{3}\tau^{4}k^{2}\Sigma_{k}+\frac{1}{9}\tau^{5}k^{4}\Sigma_{k}-\frac{4}{9}\tau^{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime
2}\bigg{]}\;,$ (241)
where
$\displaystyle\int dK\equiv
N_{c}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\int_{\frac{1}{\Lambda^{2}}}^{\infty}\frac{d\tau}{\tau}\;,\hskip
56.9055ptX=\frac{1}{k^{2}+\Sigma_{k}^{2}}\;,$ (242)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau=Xe^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;,$
(243)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{2}=\left(X^{2}+\frac{X}{\Lambda^{2}}\right)e^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;,$
(244)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{3}=\left(2X^{3}+2\frac{X^{2}}{\Lambda^{2}}+\frac{X}{\Lambda^{4}}\right)e^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;,$
(245)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{4}=\left(6X^{4}+6\frac{X^{3}}{\Lambda^{2}}+3\frac{X^{2}}{\Lambda^{4}}+\frac{X}{\Lambda^{6}}\right)e^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;,$
(246)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{5}=\left(24X^{5}+24\frac{X^{4}}{\Lambda^{2}}+12\frac{X^{3}}{\Lambda^{4}}+4\frac{X^{2}}{\Lambda^{6}}+\frac{X}{\Lambda^{8}}\right)e^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;,$
(247)
$\displaystyle\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{6}=\left(120X^{6}+120\frac{X^{5}}{\Lambda^{2}}+60\frac{X^{4}}{\Lambda^{4}}+20\frac{X^{3}}{\Lambda^{6}}+5\frac{X^{2}}{\Lambda^{8}}+\frac{X}{\Lambda^{10}}\right)e^{-\frac{(k^{2}+\Sigma_{k}^{2})}{\Lambda^{2}}}\;.$
(248)
As we mentioned before, since the momentum integrations in (241) are
convergent due to suppression factor $e^{-\tau k^{2}}$, the integrand is only
accurate up to some total derivatives , but if we insist on taking the limit
of $\Lambda\rightarrow\infty$, the dropping out momentum space total
derivatives becomes problematic. We choose these momentum space total
derivatives to reduce the high order self energy derivatives as much as
possible. For example, following relation with
$m,m_{0},m_{1},m_{2},m_{3},m_{4}$ being zero or positive integers can be used
to reduce the term with a $\Sigma_{k}^{\prime\prime\prime\prime\prime}$ to the
terms with lower order self energy derivatives,
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}(k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime
m_{4}}\Sigma_{k}^{\prime\prime\prime\prime\prime})$ (249) $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}\Big{(}-\frac{4+m}{2(m_{4}+1)}k^{m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime m_{4}+1}$
$\displaystyle-\frac{m_{0}}{m_{4}+1}k^{2+m}\Sigma_{k}^{m_{0}-1}\Sigma_{k}^{\prime
m_{1}+1}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime
m_{4}+1}-\frac{m_{1}}{m_{4}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}-1}\Sigma_{k}^{\prime\prime m_{2}+1}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime m_{4}+1}$
$\displaystyle-\frac{m_{2}}{m_{4}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}-1}\Sigma_{k}^{\prime\prime\prime
m_{3}+1}\Sigma_{k}^{\prime\prime\prime\prime
m_{4}+1}-\frac{m_{3}}{m_{4}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}-1}\Sigma_{k}^{\prime\prime\prime\prime m_{4}+2}$
$\displaystyle+\frac{1}{m_{4}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime m_{1}}\Sigma_{k}^{\prime\prime
m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime m_{4}+1}+\frac{2}{m_{4}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}+1}\Sigma_{k}^{\prime m_{1}+1}\Sigma_{k}^{\prime\prime
m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime m_{4}+1}\Big{)}$
Similarly, we have a series relations to reduce the term with a
$\Sigma_{k}^{\prime\prime\prime\prime}$, or a
$\Sigma_{k}^{\prime\prime\prime}$, a $\Sigma_{k}^{\prime\prime}$, a
$\Sigma_{k}^{\prime}$ to the terms with lower order self energy derivatives,
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}(k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}}\Sigma_{k}^{\prime\prime\prime\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}\Big{(}-\frac{4+m}{2(m_{3}+1)}k^{m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}+1}-\frac{m_{0}}{m_{3}+1}k^{2+m}\Sigma_{k}^{m_{0}-1}\Sigma_{k}^{\prime
m_{1}+1}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime
m_{3}+1}$
$\displaystyle-\frac{m_{1}}{m_{3}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}-1}\Sigma_{k}^{\prime\prime m_{2}+1}\Sigma_{k}^{\prime\prime\prime
m_{3}+1}-\frac{m_{2}}{m_{3}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}-1}\Sigma_{k}^{\prime\prime\prime
m_{3}+2}$ $\displaystyle+\frac{1}{m_{3}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime m_{1}}\Sigma_{k}^{\prime\prime
m_{2}}\Sigma_{k}^{\prime\prime\prime m_{3}+1}+\frac{2}{m_{3}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}+1}\Sigma_{k}^{\prime m_{1}+1}\Sigma_{k}^{\prime\prime
m_{2}}\Sigma_{k}^{\prime\prime\prime m_{3}+1}\Big{)}$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}(k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime m_{2}}\Sigma_{k}^{\prime\prime\prime})$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}\Big{(}-\frac{4+m}{2(m_{2}+1)}k^{m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime
m_{2}+1}-\frac{m_{0}}{m_{2}+1}k^{2+m}\Sigma_{k}^{m_{0}-1}\Sigma_{k}^{\prime
m_{1}+1}\Sigma_{k}^{\prime\prime m_{2}+1}$
$\displaystyle-\frac{m_{1}}{m_{2}+1}k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}-1}\Sigma_{k}^{\prime\prime m_{2}+2}+\frac{1}{m_{2}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime m_{1}}\Sigma_{k}^{\prime\prime
m_{2}+1}+\frac{2}{m_{2}+1}\tau k^{2+m}\Sigma_{k}^{m_{0}+1}\Sigma_{k}^{\prime
m_{1}+1}\Sigma_{k}^{\prime\prime m_{2}+1}\Big{)}$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}(k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}}\Sigma_{k}^{\prime\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}\Big{(}-\frac{4+m}{2(m_{1}+1)}k^{m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime
m_{1}+1}-\frac{m_{0}}{m_{1}+1}k^{2+m}\Sigma_{k}^{m_{0}-1}\Sigma_{k}^{\prime
m_{1}+2}$ $\displaystyle+\frac{1}{m_{1}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}}\Sigma_{k}^{\prime m_{1}+1}+\frac{2}{m_{1}+1}\tau
k^{2+m}\Sigma_{k}^{m_{0}+1}\Sigma_{k}^{\prime m_{1}+2}\Big{)}$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\tau^{n+1}k^{2+m}\Sigma_{k}^{m_{0}+1}\Sigma_{k}^{\prime}$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\Sigma_{k}^{2})}\tau^{n}\Big{(}(1+\frac{m}{4})k^{m}\Sigma_{k}^{m_{0}}+\frac{1}{2}m_{0}k^{m+2}\Sigma_{k}^{m_{0}-1}\Sigma_{k}^{\prime}-\frac{1}{2}\tau
k^{2+m}\Sigma_{k}^{m_{0}}\Big{)}\;.$ (253)
With the help of above relations, the highest self energy derivatives appear
in the final result (241) is $\Sigma_{k}^{\prime\prime\prime}$.
## Appendix C Relations among $K_{i}$ and $\mathcal{Z}_{i}$
$\displaystyle K_{2}$ $\displaystyle=$ $\displaystyle
K_{4}=K_{6}=K_{8}=K_{9}=K_{10}=K_{12}=K_{14}=K_{15}=K_{16}=K_{18}=K_{20}=K_{21}=K_{22}=K_{26}=K_{27}=K_{29}$
$\displaystyle=$ $\displaystyle
K_{32}=K_{35}=K_{38}=K_{40}=K_{45}=K_{48}=K_{50}=K_{51}=K_{53}=K_{55}=K_{56}=K_{57}=K_{59}=K_{61}=K_{62}=K_{63}$
$\displaystyle=$ $\displaystyle
K_{65}=K_{69}=K_{70}=K_{72}=K_{74}=K_{77}=K_{79}=K_{80}=K_{82}=K_{84}=K_{87}=K_{88}=K_{91}=K_{93}=K_{96}=K_{98}$
$\displaystyle=$ $\displaystyle K_{99}=K_{103}=K_{106}=K_{108}=0\;,$
$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{7}+\frac{1}{64}\mathcal{Z}_{8}-\frac{1}{32}\mathcal{Z}_{10}-\frac{1}{64}\mathcal{Z}_{13}-\frac{1}{32}\mathcal{Z}_{14}+\frac{1}{32}\mathcal{Z}_{15}-\frac{1}{64}\mathcal{Z}_{16}+\frac{1}{32}\mathcal{Z}_{44}+\frac{1}{32}\mathcal{Z}_{45}-\frac{1}{16}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{32}\mathcal{Z}_{49}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{50}-\frac{1}{16}\mathcal{Z}_{51}-\frac{1}{32}\mathcal{Z}_{52}-\frac{1}{8}\mathcal{Z}_{54}-\frac{1}{4}\mathcal{Z}_{55}-\frac{7}{32}\mathcal{Z}_{59}-\frac{7}{32}\mathcal{Z}_{60}-\frac{7}{32}\mathcal{Z}_{61}-\frac{5}{32}\mathcal{Z}_{62}-\frac{3}{32}\mathcal{Z}_{63}+\frac{1}{64}\mathcal{Z}_{79}-\frac{1}{32}\mathcal{Z}_{80}$
$\displaystyle-\frac{3}{64}\mathcal{Z}_{81}\;,$ $\displaystyle K_{3}$
$\displaystyle=$
$\displaystyle-\frac{1}{64}\mathcal{Z}_{11}-\frac{1}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{22}+\frac{1}{64}\mathcal{Z}_{23}+\frac{1}{64}\mathcal{Z}_{47}-\frac{1}{64}\mathcal{Z}_{49}+\frac{1}{32}\mathcal{Z}_{54}+\frac{1}{16}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{59}+\frac{1}{16}\mathcal{Z}_{60}+\frac{1}{16}\mathcal{Z}_{61}+\frac{3}{64}\mathcal{Z}_{62}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{63}+\frac{1}{64}\mathcal{Z}_{80}+\frac{1}{64}\mathcal{Z}_{81}\;,$
$\displaystyle K_{5}$ $\displaystyle=$
$\displaystyle-\frac{3}{128}\mathcal{Z}_{10}-\frac{1}{128}\mathcal{Z}_{13}-\frac{1}{64}\mathcal{Z}_{14}+\frac{3}{128}\mathcal{Z}_{15}-\frac{1}{128}\mathcal{Z}_{16}+\frac{1}{64}\mathcal{Z}_{18}+\frac{1}{64}\mathcal{Z}_{19}+\frac{1}{64}\mathcal{Z}_{20}+\frac{1}{32}\mathcal{Z}_{44}+\frac{1}{32}\mathcal{Z}_{45}-\frac{3}{64}\mathcal{Z}_{47}$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{64}\mathcal{Z}_{49}+\frac{1}{32}\mathcal{Z}_{50}-\frac{1}{16}\mathcal{Z}_{51}-\frac{1}{32}\mathcal{Z}_{52}-\frac{3}{32}\mathcal{Z}_{54}-\frac{3}{16}\mathcal{Z}_{55}-\frac{5}{32}\mathcal{Z}_{59}-\frac{5}{32}\mathcal{Z}_{60}-\frac{5}{32}\mathcal{Z}_{61}-\frac{7}{64}\mathcal{Z}_{62}$
$\displaystyle-\frac{1}{16}\mathcal{Z}_{63}+\frac{1}{64}\mathcal{Z}_{79}-\frac{1}{64}\mathcal{Z}_{80}-\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{64B_{0}}\mathcal{Z}_{41}+\frac{1}{64B_{0}}\mathcal{Z}_{83}\;,$
$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{64B_{0}}\mathcal{Z}_{43}+\frac{1}{32B_{0}}\mathcal{Z}_{75}+\frac{1}{32B_{0}}\mathcal{Z}_{82}\;,$
$\displaystyle K_{13}$ $\displaystyle=$
$\displaystyle-\frac{1}{16B_{0}}\mathcal{Z}_{38}-\frac{1}{16B_{0}}\mathcal{Z}_{39}+\frac{1}{64B_{0}}\mathcal{Z}_{42}-\frac{1}{32B_{0}}\mathcal{Z}_{75}-\frac{1}{32B_{0}}\mathcal{Z}_{82}-\frac{1}{64B_{0}}\mathcal{Z}_{83}\;,$
$\displaystyle K_{17}$ $\displaystyle=$
$\displaystyle-\frac{1}{32B_{0}}\mathcal{Z}_{38}-\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{64B_{0}}\mathcal{Z}_{92}+\frac{1}{64B_{0}}\mathcal{Z}_{93}\;,$
$\displaystyle K_{19}$ $\displaystyle=$
$\displaystyle\frac{1}{64B_{0}^{2}}\mathcal{Z}_{24}-\frac{1}{32}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{32B_{0}}\mathcal{Z}_{66}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{68}\;,$
$\displaystyle K_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{64B_{0}^{2}}\mathcal{Z}_{26}-\frac{1}{32}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{32B_{0}}\mathcal{Z}_{66}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{68}\;,$
$\displaystyle K_{24}$ $\displaystyle=$
$\displaystyle\frac{1}{16N_{f}}\sum_{n=53}^{63}\mathcal{Z}_{n}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{64}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{65}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{66}\;,$
$\displaystyle K_{25}$ $\displaystyle=$
$\displaystyle\frac{1}{64B_{0}^{3}}\mathcal{Z}_{88}\;,$ $\displaystyle K_{28}$
$\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{9}+\frac{1}{128}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{11}+\frac{1}{64}\mathcal{Z}_{12}+\frac{1}{128}\mathcal{Z}_{13}+\frac{1}{64}\mathcal{Z}_{14}+\frac{1}{128}\mathcal{Z}_{15}+\frac{1}{128}\mathcal{Z}_{16}+\frac{1}{64B_{0}}\mathcal{Z}_{32}+\frac{1}{64B_{0}}\mathcal{Z}_{33}$
$\displaystyle-\frac{1}{64B_{0}}\mathcal{Z}_{34}-\frac{1}{64B_{0}}\mathcal{Z}_{35}-\frac{1}{64}\mathcal{Z}_{46}-\frac{1}{64}\mathcal{Z}_{48}-\frac{1}{64}\mathcal{Z}_{49}+\frac{1}{64}\mathcal{Z}_{50}-\frac{1}{32}\mathcal{Z}_{51}-\frac{1}{64}\mathcal{Z}_{52}-\frac{1}{64}\mathcal{Z}_{53}-\frac{1}{32}\mathcal{Z}_{54}$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{55}-\frac{1}{64}\mathcal{Z}_{56}-\frac{1}{32}\mathcal{Z}_{57}-\frac{1}{32}\sum_{n=59}^{63}\mathcal{Z}_{n}+\frac{1}{64B_{0}}\mathcal{Z}_{64}+\frac{1}{64B_{0}}\mathcal{Z}_{65}-\frac{1}{64B_{0}}\mathcal{Z}_{76}\;,$
$\displaystyle K_{30}$ $\displaystyle=$
$\displaystyle-\frac{1}{32N_{f}}\mathcal{Z}_{9}-\frac{1}{16N_{f}}\mathcal{Z}_{10}-\frac{1}{32N_{f}}\mathcal{Z}_{11}-\frac{1}{32N_{f}}\mathcal{Z}_{12}-\frac{1}{32N_{f}}\mathcal{Z}_{13}-\frac{1}{16N_{f}}\mathcal{Z}_{14}+\frac{1}{32N_{f}}\mathcal{Z}_{15}-\frac{1}{32N_{f}}\mathcal{Z}_{16}$
$\displaystyle-\frac{1}{32N_{f}}\mathcal{Z}_{17}+\frac{1}{16N_{f}}\mathcal{Z}_{44}+\frac{1}{16N_{f}}\mathcal{Z}_{45}-\frac{3}{32N_{f}}\mathcal{Z}_{47}-\frac{1}{16N_{f}}\mathcal{Z}_{48}+\frac{1}{32N_{f}}\mathcal{Z}_{49}+\frac{1}{16N_{f}}\mathcal{Z}_{50}-\frac{1}{8N_{f}}\mathcal{Z}_{51}$
$\displaystyle-\frac{1}{16N_{f}}\mathcal{Z}_{52}-\frac{3}{16N_{f}}\mathcal{Z}_{54}-\frac{3}{8N_{f}}\mathcal{Z}_{55}-\frac{5}{16N_{f}}\mathcal{Z}_{59}-\frac{5}{16N_{f}}\mathcal{Z}_{60}-\frac{5}{16N_{f}}\mathcal{Z}_{61}-\frac{7}{32N_{f}}\mathcal{Z}_{62}-\frac{1}{8N_{f}}\mathcal{Z}_{63}$
$\displaystyle+\frac{1}{32N_{f}}\mathcal{Z}_{79}-\frac{1}{32N_{f}}\mathcal{Z}_{80}-\frac{1}{16N_{f}}\mathcal{Z}_{81}\;,$
$\displaystyle K_{31}$ $\displaystyle=$
$\displaystyle\frac{3}{64}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{32}\mathcal{Z}_{14}-\frac{3}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}+\frac{1}{32}\mathcal{Z}_{17}+\frac{1}{32B_{0}}\mathcal{Z}_{32}-\frac{1}{32B_{0}}\mathcal{Z}_{37}+\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}-\frac{1}{16}\mathcal{Z}_{44}$
$\displaystyle-\frac{1}{16}\mathcal{Z}_{45}+\frac{1}{32}\mathcal{Z}_{46}+\frac{3}{32}\mathcal{Z}_{47}+\frac{3}{32}\mathcal{Z}_{48}-\frac{3}{32}\mathcal{Z}_{50}+\frac{3}{16}\mathcal{Z}_{51}+\frac{3}{32}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{4}\mathcal{Z}_{54}+\frac{7}{16}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{57}$
$\displaystyle+\frac{3}{8}\mathcal{Z}_{59}+\frac{3}{8}\mathcal{Z}_{60}+\frac{3}{8}\mathcal{Z}_{61}+\frac{9}{32}\mathcal{Z}_{62}+\frac{3}{16}\mathcal{Z}_{63}-\frac{1}{32B_{0}}\mathcal{Z}_{64}-\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{3}{64B_{0}}\mathcal{Z}_{76}-\frac{1}{32}\mathcal{Z}_{79}+\frac{1}{32}\mathcal{Z}_{80}+\frac{1}{16}\mathcal{Z}_{81}\;,$
$\displaystyle K_{33}$ $\displaystyle=$
$\displaystyle-\frac{1}{64}\mathcal{Z}_{6}-\frac{1}{32}\mathcal{Z}_{10}-\frac{1}{64}\mathcal{Z}_{13}-\frac{1}{32}\mathcal{Z}_{14}+\frac{1}{32}\mathcal{Z}_{15}-\frac{1}{64}\mathcal{Z}_{16}-\frac{1}{64B_{0}^{2}}\mathcal{Z}_{25}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{28}+\frac{1}{32B_{0}^{2}}\mathcal{Z}_{29}+\frac{1}{64B_{0}}\mathcal{Z}_{30}$
$\displaystyle+\frac{1}{64B_{0}}\mathcal{Z}_{31}-\frac{1}{32B_{0}}\mathcal{Z}_{32}-\frac{1}{32B_{0}}\mathcal{Z}_{38}-\frac{1}{32B_{0}}\mathcal{Z}_{39}-\frac{1}{32B_{0}}\mathcal{Z}_{40}+\frac{1}{32}\mathcal{Z}_{44}+\frac{1}{32}\mathcal{Z}_{45}-\frac{1}{32}\mathcal{Z}_{46}-\frac{1}{16}\mathcal{Z}_{47}-\frac{1}{16}\mathcal{Z}_{48}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{50}-\frac{1}{8}\mathcal{Z}_{51}-\frac{1}{16}\mathcal{Z}_{52}-\frac{3}{32}\mathcal{Z}_{53}-\frac{1}{4}\mathcal{Z}_{54}-\frac{3}{8}\mathcal{Z}_{55}-\frac{3}{32}\mathcal{Z}_{56}-\frac{1}{8}\mathcal{Z}_{57}-\frac{1}{16}\mathcal{Z}_{58}-\frac{11}{32}\mathcal{Z}_{59}-\frac{11}{32}\mathcal{Z}_{60}-\frac{11}{32}\mathcal{Z}_{61}$
$\displaystyle-\frac{9}{32}\mathcal{Z}_{62}-\frac{7}{32}\mathcal{Z}_{63}+\frac{3}{32B_{0}}\mathcal{Z}_{64}+\frac{3}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{16B_{0}}\mathcal{Z}_{66}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{67}-\frac{1}{16B_{0}^{2}}\mathcal{Z}_{68}-\frac{1}{32B_{0}}\mathcal{Z}_{76}+\frac{1}{64}\mathcal{Z}_{79}$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{80}-\frac{3}{64}\mathcal{Z}_{81}\;,$
$\displaystyle K_{34}$ $\displaystyle=$
$\displaystyle\frac{1}{32N_{f}}\mathcal{Z}_{6}+\frac{1}{16N_{f}}\mathcal{Z}_{10}+\frac{1}{32N_{f}}\mathcal{Z}_{11}+\frac{1}{32N_{f}}\mathcal{Z}_{13}+\frac{1}{16N_{f}}\mathcal{Z}_{14}-\frac{1}{32N_{f}}\mathcal{Z}_{15}+\frac{1}{32N_{f}}\mathcal{Z}_{16}+\frac{1}{32N_{f}}\mathcal{Z}_{21}$
$\displaystyle-\frac{1}{64N_{f}B_{0}}\mathcal{Z}_{30}-\frac{1}{64N_{f}B_{0}}\mathcal{Z}_{31}+\frac{1}{32N_{f}B_{0}}\mathcal{Z}_{32}+\frac{1}{64N_{f}B_{0}}\mathcal{Z}_{33}-\frac{1}{64N_{f}B_{0}}\mathcal{Z}_{36}+\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{38}+\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{39}$
$\displaystyle+\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{40}-\frac{1}{16N_{f}}\mathcal{Z}_{44}-\frac{1}{16N_{f}}\mathcal{Z}_{45}+\frac{3}{32N_{f}}\mathcal{Z}_{47}+\frac{1}{16N_{f}}\mathcal{Z}_{48}-\frac{1}{32N_{f}}\mathcal{Z}_{49}-\frac{1}{16N_{f}}\mathcal{Z}_{50}+\frac{1}{8N_{f}}\mathcal{Z}_{51}$
$\displaystyle+\frac{1}{16N_{f}}\mathcal{Z}_{52}+\frac{1}{8N_{f}}\mathcal{Z}_{53}+\frac{5}{16N_{f}}\mathcal{Z}_{54}+\frac{1}{2N_{f}}\mathcal{Z}_{55}+\frac{1}{8N_{f}}\mathcal{Z}_{56}+\frac{1}{8N_{f}}\mathcal{Z}_{57}+\frac{1}{8N_{f}}\mathcal{Z}_{58}+\frac{7}{16N_{f}}\mathcal{Z}_{59}+\frac{7}{16N_{f}}\mathcal{Z}_{60}$
$\displaystyle+\frac{7}{16N_{f}}\mathcal{Z}_{61}+\frac{11}{32N_{f}}\mathcal{Z}_{62}+\frac{1}{4N_{f}}\mathcal{Z}_{63}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{64}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{65}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{66}-\frac{1}{32N_{f}}\mathcal{Z}_{79}+\frac{1}{32N_{f}}\mathcal{Z}_{80}$
$\displaystyle+\frac{1}{16N_{f}}\mathcal{Z}_{81}\;,$ $\displaystyle K_{36}$
$\displaystyle=$
$\displaystyle-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{6}-\frac{1}{32N_{f}^{2}}\mathcal{Z}_{10}-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{11}-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{13}-\frac{1}{32N_{f}^{2}}\mathcal{Z}_{14}+\frac{1}{64N_{f}^{2}}\mathcal{Z}_{15}-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{16}-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{21}$
$\displaystyle+\frac{1}{32N_{f}^{2}}\mathcal{Z}_{44}+\frac{1}{32N_{f}^{2}}\mathcal{Z}_{45}-\frac{3}{64N_{f}^{2}}\mathcal{Z}_{47}-\frac{1}{32N_{f}^{2}}\mathcal{Z}_{48}+\frac{1}{64N_{f}^{2}}\mathcal{Z}_{49}+\frac{1}{32N_{f}^{2}}\mathcal{Z}_{50}-\frac{1}{16N_{f}^{2}}\mathcal{Z}_{51}-\frac{1}{32N_{f}^{2}}\mathcal{Z}_{52}-\frac{3}{32N_{f}^{2}}\mathcal{Z}_{54}$
$\displaystyle-\frac{3}{16N_{f}^{2}}\mathcal{Z}_{55}-\frac{5}{32N_{f}^{2}}\mathcal{Z}_{59}-\frac{5}{32N_{f}^{2}}\mathcal{Z}_{60}-\frac{5}{32N_{f}^{2}}\mathcal{Z}_{61}-\frac{7}{64N_{f}^{2}}\mathcal{Z}_{62}-\frac{1}{16N_{f}^{2}}\mathcal{Z}_{63}+\frac{1}{64N_{f}^{2}}\mathcal{Z}_{79}-\frac{1}{64N_{f}^{2}}\mathcal{Z}_{80}-\frac{1}{32N_{f}^{2}}\mathcal{Z}_{81}\;,$
$\displaystyle K_{37}$ $\displaystyle=$
$\displaystyle-\frac{1}{64}\mathcal{Z}_{11}-\frac{1}{64}\mathcal{Z}_{15}-\frac{1}{64}\mathcal{Z}_{21}-\frac{1}{64B_{0}^{2}}\mathcal{Z}_{27}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{28}+\frac{1}{32B_{0}^{2}}\mathcal{Z}_{29}-\frac{1}{64B_{0}}\mathcal{Z}_{33}+\frac{1}{64B_{0}}\mathcal{Z}_{36}-\frac{1}{16B_{0}}\mathcal{Z}_{38}$
$\displaystyle-\frac{1}{16B_{0}}\mathcal{Z}_{39}-\frac{1}{32B_{0}}\mathcal{Z}_{40}+\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{64}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{64}\mathcal{Z}_{49}-\frac{1}{32}\mathcal{Z}_{50}+\frac{1}{16}\mathcal{Z}_{51}+\frac{1}{32}\mathcal{Z}_{52}-\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{32}\mathcal{Z}_{54}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{55}-\frac{1}{32}\mathcal{Z}_{56}-\frac{1}{16}\mathcal{Z}_{58}+\frac{1}{16}\mathcal{Z}_{59}+\frac{1}{16}\mathcal{Z}_{60}+\frac{1}{16}\mathcal{Z}_{61}+\frac{3}{64}\mathcal{Z}_{62}+\frac{1}{32}\mathcal{Z}_{63}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}$
$\displaystyle+\frac{1}{16B_{0}}\mathcal{Z}_{66}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{67}-\frac{1}{16B_{0}^{2}}\mathcal{Z}_{68}+\frac{1}{32B_{0}}\mathcal{Z}_{76}+\frac{1}{64}\mathcal{Z}_{80}+\frac{1}{64}\mathcal{Z}_{81}\;,$
$\displaystyle K_{39}$ $\displaystyle=$
$\displaystyle-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{28}-\frac{1}{32B_{0}}\mathcal{Z}_{38}-\frac{1}{32B_{0}}\mathcal{Z}_{39}-\frac{1}{16}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{16B_{0}}\mathcal{Z}_{64}+\frac{1}{16B_{0}}\mathcal{Z}_{65}+\frac{1}{16B_{0}}\mathcal{Z}_{66}-\frac{1}{16B_{0}^{2}}\mathcal{Z}_{68}$
$\displaystyle-\frac{1}{64B_{0}^{3}}\mathcal{Z}_{89}+\frac{1}{64B_{0}^{2}}\mathcal{Z}_{90}+\frac{1}{64B_{0}^{2}}\mathcal{Z}_{91}-\frac{1}{64B_{0}}\mathcal{Z}_{94}\;,$
$\displaystyle K_{41}$ $\displaystyle=$
$\displaystyle\frac{1}{32N_{f}B_{0}^{2}}\mathcal{Z}_{28}+\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{38}+\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{39}+\frac{1}{8N_{f}}\sum_{n=53}^{63}\mathcal{Z}_{n}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{64}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{65}-\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{66}$
$\displaystyle+\frac{1}{16N_{f}B_{0}^{2}}\mathcal{Z}_{68}-\frac{1}{64N_{f}B_{0}^{2}}\mathcal{Z}_{90}-\frac{1}{64N_{f}B_{0}^{2}}\mathcal{Z}_{91}+\frac{1}{32N_{f}B_{0}}\mathcal{Z}_{94}\;,$
$\displaystyle K_{42}$ $\displaystyle=$
$\displaystyle-\frac{1}{32N_{f}^{2}B_{0}}\mathcal{Z}_{38}-\frac{1}{32N_{f}^{2}B_{0}}\mathcal{Z}_{39}-\frac{1}{16N_{f}^{2}}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{16N_{f}^{2}B_{0}}\mathcal{Z}_{64}+\frac{1}{16N_{f}^{2}B_{0}}\mathcal{Z}_{65}+\frac{1}{16N_{f}^{2}B_{0}}\mathcal{Z}_{66}$
$\displaystyle-\frac{1}{64N_{f}^{2}B_{0}}\mathcal{Z}_{94}\;,$ $\displaystyle
K_{43}$ $\displaystyle=$
$\displaystyle\frac{1}{32B_{0}^{2}}\mathcal{Z}_{28}-\frac{1}{32B_{0}^{2}}\mathcal{Z}_{29}+\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{32B_{0}}\mathcal{Z}_{40}+\frac{1}{16}\sum_{n=53}^{63}\mathcal{Z}_{n}-\frac{1}{16B_{0}}\mathcal{Z}_{64}-\frac{1}{16B_{0}}\mathcal{Z}_{65}$
$\displaystyle-\frac{1}{16B_{0}}\mathcal{Z}_{66}+\frac{1}{16B_{0}^{2}}\mathcal{Z}_{67}+\frac{1}{16B_{0}^{2}}\mathcal{Z}_{68}\;,$
$\displaystyle K_{44}$ $\displaystyle=$
$\displaystyle-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{38}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{39}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{40}-\frac{1}{8N_{f}}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{64}+\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{65}+\frac{1}{8N_{f}B_{0}}\mathcal{Z}_{66}\;,$
$\displaystyle K_{46}$ $\displaystyle=$
$\displaystyle\frac{1}{16N_{f}}\sum_{n=53}^{63}\mathcal{Z}_{n}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{64}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{65}-\frac{1}{16N_{f}B_{0}}\mathcal{Z}_{66}\;,$
$\displaystyle K_{47}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}\sum_{n=53}^{63}\mathcal{Z}_{n}+\frac{1}{16B_{0}}\sum_{n=64}^{67}\mathcal{Z}_{n}-\frac{1}{16B_{0}^{2}}\mathcal{Z}_{68}\;,$
$\displaystyle K_{49}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{1}+\frac{1}{32}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{32}\mathcal{Z}_{14}-\frac{1}{32}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}-\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{32}\mathcal{Z}_{45}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{16}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{50}+\frac{1}{8}\mathcal{Z}_{51}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{52}+\frac{5}{32}\mathcal{Z}_{54}+\frac{9}{32}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{57}+\frac{1}{4}\mathcal{Z}_{59}+\frac{1}{4}\mathcal{Z}_{60}+\frac{1}{4}\mathcal{Z}_{61}+\frac{3}{16}\mathcal{Z}_{62}+\frac{1}{8}\mathcal{Z}_{63}+\frac{1}{64}\mathcal{Z}_{69}+\frac{1}{64}\mathcal{Z}_{72}-\frac{1}{64}\mathcal{Z}_{79}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{80}+\frac{3}{64}\mathcal{Z}_{81}+\frac{1}{64}\mathcal{Z}_{86}\;,$
$\displaystyle K_{52}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{3}+\frac{1}{64}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{32}\mathcal{Z}_{14}-\frac{1}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}-\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{32}\mathcal{Z}_{45}+\frac{1}{32}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{50}+\frac{1}{16}\mathcal{Z}_{51}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{52}+\frac{3}{32}\mathcal{Z}_{54}+\frac{5}{32}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{57}+\frac{3}{16}\mathcal{Z}_{59}+\frac{3}{16}\mathcal{Z}_{60}+\frac{5}{32}\mathcal{Z}_{61}+\frac{1}{8}\mathcal{Z}_{62}+\frac{3}{32}\mathcal{Z}_{63}+\frac{1}{32}\mathcal{Z}_{70}+\frac{1}{32}\mathcal{Z}_{73}+\frac{1}{32}\mathcal{Z}_{80}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{81}+\frac{1}{32}\mathcal{Z}_{84}\;,$
$\displaystyle K_{54}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{2}-\frac{3}{64}\mathcal{Z}_{10}+\frac{1}{32}\mathcal{Z}_{11}-\frac{1}{32}\mathcal{Z}_{13}-\frac{1}{16}\mathcal{Z}_{14}+\frac{5}{64}\mathcal{Z}_{15}-\frac{1}{32}\mathcal{Z}_{16}+\frac{1}{16}\mathcal{Z}_{44}+\frac{1}{16}\mathcal{Z}_{45}-\frac{1}{8}\mathcal{Z}_{47}-\frac{5}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{49}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{50}-\frac{5}{16}\mathcal{Z}_{51}-\frac{5}{32}\mathcal{Z}_{52}-\frac{3}{8}\mathcal{Z}_{54}-\frac{5}{8}\mathcal{Z}_{55}-\frac{1}{8}\mathcal{Z}_{57}-\frac{5}{8}\mathcal{Z}_{59}-\frac{5}{8}\mathcal{Z}_{60}-\frac{19}{32}\mathcal{Z}_{61}-\frac{15}{32}\mathcal{Z}_{62}-\frac{11}{32}\mathcal{Z}_{63}-\frac{3}{64}\mathcal{Z}_{69}$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{70}+\frac{1}{64}\mathcal{Z}_{71}-\frac{1}{32}\mathcal{Z}_{72}-\frac{1}{16}\mathcal{Z}_{73}+\frac{1}{64}\mathcal{Z}_{79}-\frac{3}{32}\mathcal{Z}_{80}-\frac{7}{64}\mathcal{Z}_{81}-\frac{1}{32}\mathcal{Z}_{84}+\frac{1}{32}\mathcal{Z}_{85}-\frac{1}{64}\mathcal{Z}_{86}\;,$
$\displaystyle K_{58}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{5}-\frac{1}{64}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{15}-\frac{1}{32}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{16}\mathcal{Z}_{51}-\frac{1}{32}\mathcal{Z}_{52}-\frac{1}{16}\mathcal{Z}_{54}-\frac{1}{8}\mathcal{Z}_{55}-\frac{1}{16}\mathcal{Z}_{59}-\frac{1}{16}\mathcal{Z}_{60}-\frac{3}{32}\mathcal{Z}_{61}$
$\displaystyle-\frac{1}{16}\mathcal{Z}_{62}-\frac{1}{32}\mathcal{Z}_{63}-\frac{1}{64}\mathcal{Z}_{69}+\frac{1}{64}\mathcal{Z}_{71}-\frac{1}{32}\mathcal{Z}_{74}+\frac{1}{64}\mathcal{Z}_{79}-\frac{1}{64}\mathcal{Z}_{81}-\frac{1}{64}\mathcal{Z}_{87}\;,$
$\displaystyle K_{60}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{4}+\frac{1}{64}\mathcal{Z}_{10}-\frac{1}{32}\mathcal{Z}_{11}-\frac{3}{64}\mathcal{Z}_{15}+\frac{1}{16}\mathcal{Z}_{47}+\frac{3}{32}\mathcal{Z}_{48}+\frac{1}{32}\mathcal{Z}_{49}+\frac{3}{16}\mathcal{Z}_{51}+\frac{3}{32}\mathcal{Z}_{52}+\frac{3}{16}\mathcal{Z}_{54}+\frac{5}{16}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{57}$
$\displaystyle+\frac{1}{4}\mathcal{Z}_{59}+\frac{1}{4}\mathcal{Z}_{60}+\frac{9}{32}\mathcal{Z}_{61}+\frac{7}{32}\mathcal{Z}_{62}+\frac{5}{32}\mathcal{Z}_{63}+\frac{3}{64}\mathcal{Z}_{69}-\frac{1}{32}\mathcal{Z}_{71}+\frac{1}{64}\mathcal{Z}_{72}+\frac{1}{32}\mathcal{Z}_{73}+\frac{1}{32}\mathcal{Z}_{74}-\frac{1}{64}\mathcal{Z}_{79}+\frac{1}{32}\mathcal{Z}_{80}$
$\displaystyle+\frac{3}{64}\mathcal{Z}_{81}-\frac{1}{32}\mathcal{Z}_{85}+\frac{1}{64}\mathcal{Z}_{87}\;,$
$\displaystyle K_{64}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{32}\mathcal{Z}_{14}-\frac{1}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}-\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{32}\mathcal{Z}_{45}+\frac{1}{32}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{50}+\frac{1}{16}\mathcal{Z}_{51}+\frac{1}{32}\mathcal{Z}_{52}$
$\displaystyle+\frac{1}{8}\mathcal{Z}_{54}+\frac{3}{16}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{57}+\frac{9}{32}\mathcal{Z}_{59}+\frac{9}{32}\mathcal{Z}_{60}+\frac{7}{32}\mathcal{Z}_{61}+\frac{3}{16}\mathcal{Z}_{62}+\frac{5}{32}\mathcal{Z}_{63}+\frac{1}{16}\mathcal{Z}_{70}+\frac{1}{16}\mathcal{Z}_{73}+\frac{1}{64}\mathcal{Z}_{79}+\frac{1}{16}\mathcal{Z}_{80}$
$\displaystyle+\frac{3}{64}\mathcal{Z}_{81}+\frac{1}{32}\mathcal{Z}_{84}\;,$
$\displaystyle K_{66}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{10}-\frac{1}{32}\mathcal{Z}_{15}-\frac{1}{16}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{16}\mathcal{Z}_{48}+\frac{1}{8}\mathcal{Z}_{50}+\frac{1}{8}\mathcal{Z}_{51}+\frac{1}{16}\mathcal{Z}_{52}-\frac{1}{16}\mathcal{Z}_{53}+\frac{1}{8}\mathcal{Z}_{55}-\frac{1}{16}\mathcal{Z}_{56}-\frac{1}{8}\mathcal{Z}_{57}$
$\displaystyle-\frac{1}{8}\mathcal{Z}_{59}-\frac{1}{8}\mathcal{Z}_{60}-\frac{1}{16}\mathcal{Z}_{62}-\frac{1}{8}\mathcal{Z}_{63}+\frac{1}{16B_{0}}\mathcal{Z}_{64}+\frac{1}{16B_{0}}\mathcal{Z}_{65}+\frac{3}{32}\mathcal{Z}_{69}-\frac{1}{16}\mathcal{Z}_{71}+\frac{1}{8}\mathcal{Z}_{74}-\frac{1}{16B_{0}}\mathcal{Z}_{76}-\frac{1}{16}\mathcal{Z}_{79}$
$\displaystyle-\frac{1}{16}\mathcal{Z}_{80}+\frac{1}{32}\mathcal{Z}_{87}\;,$
$\displaystyle K_{67}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{10}+\frac{1}{32}\mathcal{Z}_{13}+\frac{1}{16}\mathcal{Z}_{14}-\frac{1}{16}\mathcal{Z}_{15}+\frac{1}{32}\mathcal{Z}_{16}-\frac{1}{16}\mathcal{Z}_{44}-\frac{1}{16}\mathcal{Z}_{45}-\frac{1}{16}\mathcal{Z}_{46}+\frac{1}{8}\mathcal{Z}_{47}+\frac{1}{8}\mathcal{Z}_{48}+\frac{1}{16}\mathcal{Z}_{50}+\frac{1}{4}\mathcal{Z}_{51}$
$\displaystyle+\frac{1}{8}\mathcal{Z}_{52}-\frac{1}{16}\mathcal{Z}_{53}+\frac{1}{4}\mathcal{Z}_{54}+\frac{1}{2}\mathcal{Z}_{55}-\frac{1}{16}\mathcal{Z}_{56}+\frac{7}{16}\mathcal{Z}_{59}+\frac{7}{16}\mathcal{Z}_{60}+\frac{7}{16}\mathcal{Z}_{61}+\frac{5}{16}\mathcal{Z}_{62}+\frac{3}{16}\mathcal{Z}_{63}+\frac{1}{16B_{0}}\mathcal{Z}_{64}+\frac{1}{16B_{0}}\mathcal{Z}_{65}$
$\displaystyle+\frac{3}{32}\mathcal{Z}_{69}+\frac{1}{16}\mathcal{Z}_{72}-\frac{1}{16B_{0}}\mathcal{Z}_{76}-\frac{1}{32}\mathcal{Z}_{79}+\frac{1}{16}\mathcal{Z}_{80}+\frac{3}{32}\mathcal{Z}_{81}+\frac{1}{32}\mathcal{Z}_{86}\;,$
$\displaystyle K_{68}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{11}+\frac{1}{32}\mathcal{Z}_{15}+\frac{1}{16}\mathcal{Z}_{46}-\frac{1}{32}\mathcal{Z}_{47}-\frac{1}{16}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{49}-\frac{1}{8}\mathcal{Z}_{50}-\frac{1}{8}\mathcal{Z}_{51}-\frac{1}{16}\mathcal{Z}_{52}+\frac{1}{16}\mathcal{Z}_{53}-\frac{1}{16}\mathcal{Z}_{54}-\frac{1}{8}\mathcal{Z}_{55}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{56}-\frac{1}{8}\mathcal{Z}_{59}-\frac{1}{8}\mathcal{Z}_{60}-\frac{1}{8}\mathcal{Z}_{61}-\frac{3}{32}\mathcal{Z}_{62}-\frac{1}{16}\mathcal{Z}_{63}-\frac{1}{16B_{0}}\mathcal{Z}_{64}-\frac{1}{16B_{0}}\mathcal{Z}_{65}-\frac{3}{32}\mathcal{Z}_{69}+\frac{1}{32}\mathcal{Z}_{71}-\frac{1}{32}\mathcal{Z}_{72}-\frac{1}{16}\mathcal{Z}_{73}$
$\displaystyle+\frac{1}{16B_{0}}\mathcal{Z}_{76}-\frac{1}{32}\mathcal{Z}_{80}-\frac{1}{32}\mathcal{Z}_{81}+\frac{1}{32}\mathcal{Z}_{85}\;,$
$\displaystyle K_{71}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{49}-\frac{1}{4}\mathcal{Z}_{50}+\frac{1}{2}\mathcal{Z}_{51}+\frac{1}{4}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{8}\mathcal{Z}_{54}+\frac{1}{8}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{57}+\frac{1}{4}\mathcal{Z}_{59}$
$\displaystyle+\frac{1}{4}\mathcal{Z}_{60}+\frac{3}{16}\mathcal{Z}_{61}+\frac{3}{16}\mathcal{Z}_{62}+\frac{1}{8}\mathcal{Z}_{63}-\frac{1}{32B_{0}}\mathcal{Z}_{64}-\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{16}\mathcal{Z}_{70}+\frac{1}{32B_{0}}\mathcal{Z}_{76}+\frac{1}{32}\mathcal{Z}_{79}+\frac{1}{16}\mathcal{Z}_{80}+\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{73}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{46}-\frac{1}{16}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{32}\mathcal{Z}_{49}+\frac{1}{4}\mathcal{Z}_{50}-\frac{1}{2}\mathcal{Z}_{51}-\frac{1}{4}\mathcal{Z}_{52}-\frac{1}{32}\mathcal{Z}_{53}-\frac{1}{8}\mathcal{Z}_{54}-\frac{1}{8}\mathcal{Z}_{55}-\frac{1}{32}\mathcal{Z}_{56}-\frac{1}{16}\mathcal{Z}_{57}-\frac{1}{4}\mathcal{Z}_{59}$
$\displaystyle-\frac{1}{4}\mathcal{Z}_{60}-\frac{3}{16}\mathcal{Z}_{61}-\frac{3}{16}\mathcal{Z}_{62}-\frac{1}{8}\mathcal{Z}_{63}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{16}\mathcal{Z}_{74}-\frac{1}{32B_{0}}\mathcal{Z}_{76}-\frac{1}{32}\mathcal{Z}_{79}-\frac{1}{16}\mathcal{Z}_{80}-\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{75}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{46}-\frac{1}{8}\mathcal{Z}_{47}-\frac{1}{16}\mathcal{Z}_{48}+\frac{1}{16}\mathcal{Z}_{49}+\frac{1}{4}\mathcal{Z}_{50}-\frac{5}{4}\mathcal{Z}_{51}-\frac{5}{8}\mathcal{Z}_{52}+\frac{1}{16}\mathcal{Z}_{53}+\frac{1}{16}\mathcal{Z}_{56}+\frac{1}{8}\mathcal{Z}_{57}+\frac{1}{8}\mathcal{Z}_{63}-\frac{1}{16B_{0}}\mathcal{Z}_{64}$
$\displaystyle-\frac{1}{16B_{0}}\mathcal{Z}_{65}-\frac{3}{16}\mathcal{Z}_{69}+\frac{1}{16}\mathcal{Z}_{71}+\frac{1}{16B_{0}}\mathcal{Z}_{76}\;,$
$\displaystyle K_{76}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\mathcal{Z}_{46}+\frac{3}{8}\mathcal{Z}_{50}-\frac{1}{8}\sum_{n=53}^{57}\mathcal{Z}_{n}-\frac{1}{8}\sum_{n=59}^{63}\mathcal{Z}_{n}+\frac{1}{8B_{0}}\mathcal{Z}_{64}+\frac{1}{8B_{0}}\mathcal{Z}_{65}+\frac{3}{16}\mathcal{Z}_{69}+\frac{1}{16}\mathcal{Z}_{72}-\frac{1}{8B_{0}}\mathcal{Z}_{76}\;,$
$\displaystyle K_{78}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{49}-\frac{5}{16}\mathcal{Z}_{50}+\frac{5}{8}\mathcal{Z}_{51}+\frac{5}{16}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{16}\mathcal{Z}_{54}+\frac{1}{16}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{59}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{60}+\frac{1}{16}\mathcal{Z}_{61}+\frac{1}{16}\mathcal{Z}_{62}-\frac{1}{32B_{0}}\mathcal{Z}_{64}-\frac{1}{32B_{0}}\mathcal{Z}_{65}-\frac{1}{16}\mathcal{Z}_{73}+\frac{1}{32B_{0}}\mathcal{Z}_{76}\;,$
$\displaystyle K_{81}$ $\displaystyle=$
$\displaystyle\frac{1}{16B_{0}}\mathcal{Z}_{75}\;,$ $\displaystyle K_{83}$
$\displaystyle=$
$\displaystyle\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{16B_{0}}\mathcal{Z}_{75}+\frac{1}{32B_{0}}\mathcal{Z}_{82}\;,$
$\displaystyle K_{85}$ $\displaystyle=$
$\displaystyle\frac{1}{16B_{0}}\mathcal{Z}_{38}+\frac{1}{16B_{0}}\mathcal{Z}_{39}+\frac{1}{32B_{0}}\mathcal{Z}_{83}\;,$
$\displaystyle K_{86}$ $\displaystyle=$
$\displaystyle-\frac{1}{64}\mathcal{Z}_{10}-\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}+\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{32}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{32}\mathcal{Z}_{50}-\frac{1}{16}\mathcal{Z}_{51}-\frac{1}{32}\mathcal{Z}_{52}-\frac{1}{32}\mathcal{Z}_{54}-\frac{1}{16}\mathcal{Z}_{55}$
$\displaystyle-\frac{1}{16}\mathcal{Z}_{59}-\frac{1}{16}\mathcal{Z}_{60}-\frac{1}{16}\mathcal{Z}_{61}-\frac{1}{32}\mathcal{Z}_{62}\;,$
$\displaystyle K_{89}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{19}+\frac{1}{32}\mathcal{Z}_{20}-\frac{1}{32}\mathcal{Z}_{44}+\frac{1}{32}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{50}+\frac{1}{16}\mathcal{Z}_{51}+\frac{1}{32}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{54}+\frac{1}{16}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{59}+\frac{1}{16}\mathcal{Z}_{60}$
$\displaystyle+\frac{1}{16}\mathcal{Z}_{61}+\frac{1}{32}\mathcal{Z}_{62}\;,$
$\displaystyle K_{90}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{7}-\frac{1}{64}\mathcal{Z}_{8}+\frac{1}{32}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{13}+\frac{1}{32}\mathcal{Z}_{14}-\frac{1}{32}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{16}-\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{32}\mathcal{Z}_{45}+\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{16}\mathcal{Z}_{48}$
$\displaystyle+\frac{1}{32}\mathcal{Z}_{50}-\frac{1}{16}\mathcal{Z}_{51}-\frac{1}{32}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{53}+\frac{3}{16}\mathcal{Z}_{54}+\frac{5}{16}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{57}+\frac{9}{32}\mathcal{Z}_{59}+\frac{9}{32}\mathcal{Z}_{60}+\frac{9}{32}\mathcal{Z}_{61}+\frac{7}{32}\mathcal{Z}_{62}$
$\displaystyle+\frac{5}{32}\mathcal{Z}_{63}-\frac{1}{32B_{0}}\mathcal{Z}_{64}-\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{32B_{0}}\mathcal{Z}_{76}-\frac{1}{64}\mathcal{Z}_{79}+\frac{1}{32}\mathcal{Z}_{80}+\frac{3}{64}\mathcal{Z}_{81}\;,$
$\displaystyle K_{92}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{11}+\frac{1}{64}\mathcal{Z}_{15}+\frac{1}{64}\mathcal{Z}_{22}-\frac{1}{64}\mathcal{Z}_{23}-\frac{1}{32}\mathcal{Z}_{46}-\frac{1}{64}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{64}\mathcal{Z}_{49}-\frac{1}{16}\mathcal{Z}_{50}+\frac{1}{8}\mathcal{Z}_{51}+\frac{1}{16}\mathcal{Z}_{52}-\frac{1}{32}\mathcal{Z}_{53}$
$\displaystyle-\frac{3}{32}\mathcal{Z}_{54}-\frac{1}{8}\mathcal{Z}_{55}-\frac{1}{32}\mathcal{Z}_{56}-\frac{1}{16}\mathcal{Z}_{57}-\frac{1}{8}\mathcal{Z}_{59}-\frac{1}{8}\mathcal{Z}_{60}-\frac{1}{8}\mathcal{Z}_{61}-\frac{7}{64}\mathcal{Z}_{62}-\frac{3}{32}\mathcal{Z}_{63}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}$
$\displaystyle-\frac{1}{32B_{0}}\mathcal{Z}_{76}-\frac{1}{64}\mathcal{Z}_{80}-\frac{1}{64}\mathcal{Z}_{81}\;,$
$\displaystyle K_{94}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{15}-\frac{1}{16}\mathcal{Z}_{46}-\frac{1}{16}\mathcal{Z}_{48}-\frac{1}{16}\mathcal{Z}_{49}-\frac{1}{16}\mathcal{Z}_{53}-\frac{1}{4}\mathcal{Z}_{54}-\frac{1}{4}\mathcal{Z}_{55}-\frac{1}{16}\mathcal{Z}_{56}-\frac{1}{8}\mathcal{Z}_{57}-\frac{1}{4}\mathcal{Z}_{59}-\frac{1}{4}\mathcal{Z}_{60}-\frac{1}{4}\mathcal{Z}_{61}$
$\displaystyle-\frac{1}{4}\mathcal{Z}_{62}-\frac{1}{4}\mathcal{Z}_{63}+\frac{1}{16B_{0}}\mathcal{Z}_{64}+\frac{1}{16B_{0}}\mathcal{Z}_{65}-\frac{1}{16B_{0}}\mathcal{Z}_{76}-\frac{1}{16}\mathcal{Z}_{81}\;,$
$\displaystyle K_{95}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{10}+\frac{1}{32}\mathcal{Z}_{13}-\frac{1}{32}\mathcal{Z}_{15}+\frac{1}{32}\mathcal{Z}_{16}-\frac{1}{16}\mathcal{Z}_{44}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{16}\mathcal{Z}_{48}-\frac{1}{16}\mathcal{Z}_{50}+\frac{1}{8}\mathcal{Z}_{51}+\frac{1}{16}\mathcal{Z}_{52}+\frac{3}{16}\mathcal{Z}_{54}+\frac{1}{4}\mathcal{Z}_{55}$
$\displaystyle+\frac{1}{4}\mathcal{Z}_{59}+\frac{1}{4}\mathcal{Z}_{60}+\frac{1}{4}\mathcal{Z}_{61}+\frac{3}{16}\mathcal{Z}_{62}+\frac{1}{8}\mathcal{Z}_{63}+\frac{1}{16}\mathcal{Z}_{81}\;,$
$\displaystyle K_{97}$ $\displaystyle=$
$\displaystyle\frac{3}{128}\mathcal{Z}_{10}+\frac{1}{128}\mathcal{Z}_{13}+\frac{1}{64}\mathcal{Z}_{14}-\frac{3}{128}\mathcal{Z}_{15}+\frac{1}{128}\mathcal{Z}_{16}-\frac{1}{64}\mathcal{Z}_{18}+\frac{1}{64}\mathcal{Z}_{19}+\frac{1}{64}\mathcal{Z}_{20}-\frac{1}{32}\mathcal{Z}_{45}+\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{64}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}$
$\displaystyle+\frac{1}{64}\mathcal{Z}_{49}+\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{8}\mathcal{Z}_{54}+\frac{3}{16}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{57}+\frac{5}{32}\mathcal{Z}_{59}+\frac{5}{32}\mathcal{Z}_{60}+\frac{5}{32}\mathcal{Z}_{61}+\frac{9}{64}\mathcal{Z}_{62}+\frac{1}{8}\mathcal{Z}_{63}-\frac{1}{32B_{0}}\mathcal{Z}_{64}$
$\displaystyle-\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{1}{32B_{0}}\mathcal{Z}_{76}-\frac{1}{64}\mathcal{Z}_{79}+\frac{1}{64}\mathcal{Z}_{80}+\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{100}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}\mathcal{Z}_{46}-\frac{1}{16}\mathcal{Z}_{47}-\frac{1}{32}\mathcal{Z}_{48}+\frac{1}{32}\mathcal{Z}_{49}+\frac{5}{16}\mathcal{Z}_{50}-\frac{5}{8}\mathcal{Z}_{51}-\frac{5}{16}\mathcal{Z}_{52}-\frac{1}{32}\mathcal{Z}_{53}-\frac{3}{16}\mathcal{Z}_{54}-\frac{5}{16}\mathcal{Z}_{55}-\frac{1}{32}\mathcal{Z}_{56}-\frac{1}{16}\mathcal{Z}_{57}$
$\displaystyle-\frac{5}{16}\mathcal{Z}_{59}-\frac{5}{16}\mathcal{Z}_{60}-\frac{5}{16}\mathcal{Z}_{61}-\frac{1}{4}\mathcal{Z}_{62}-\frac{3}{16}\mathcal{Z}_{63}+\frac{1}{32B_{0}}\mathcal{Z}_{64}+\frac{1}{32B_{0}}\mathcal{Z}_{65}-\frac{1}{32B_{0}}\mathcal{Z}_{76}-\frac{1}{16}\mathcal{Z}_{80}-\frac{1}{16}\mathcal{Z}_{81}\;,$
$\displaystyle K_{101}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{47}+\frac{1}{32}\mathcal{Z}_{48}-\frac{1}{32}\mathcal{Z}_{49}+\frac{1}{16}\mathcal{Z}_{50}+\frac{5}{8}\mathcal{Z}_{51}+\frac{5}{16}\mathcal{Z}_{52}+\frac{1}{32}\mathcal{Z}_{53}+\frac{1}{16}\mathcal{Z}_{54}+\frac{3}{16}\mathcal{Z}_{55}+\frac{1}{32}\mathcal{Z}_{56}+\frac{1}{16}\mathcal{Z}_{57}$
$\displaystyle+\frac{3}{16}\mathcal{Z}_{59}+\frac{3}{16}\mathcal{Z}_{60}+\frac{3}{16}\mathcal{Z}_{61}+\frac{1}{8}\mathcal{Z}_{62}+\frac{1}{16}\mathcal{Z}_{63}-\frac{1}{32B_{0}}\mathcal{Z}_{64}-\frac{1}{32B_{0}}\mathcal{Z}_{65}+\frac{3}{16}\mathcal{Z}_{69}+\frac{1}{32B_{0}}\mathcal{Z}_{76}+\frac{1}{16}\mathcal{Z}_{80}\;,$
$\displaystyle K_{102}$ $\displaystyle=$
$\displaystyle\frac{1}{32B_{0}}\mathcal{Z}_{38}+\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{64B_{0}}\mathcal{Z}_{92}-\frac{1}{64B_{0}}\mathcal{Z}_{93}\;,$
$\displaystyle K_{104}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\mathcal{Z}_{46}+\frac{1}{32}\mathcal{Z}_{47}+\frac{1}{64}\mathcal{Z}_{48}-\frac{1}{64}\mathcal{Z}_{49}-\frac{3}{32}\mathcal{Z}_{50}+\frac{3}{16}\mathcal{Z}_{51}+\frac{3}{32}\mathcal{Z}_{52}-\frac{3}{64}\mathcal{Z}_{53}-\frac{1}{32}\mathcal{Z}_{54}-\frac{1}{32}\mathcal{Z}_{55}-\frac{1}{64}\mathcal{Z}_{56}-\frac{1}{32}\mathcal{Z}_{57}$
$\displaystyle-\frac{1}{32}\sum_{n=59}^{63}\mathcal{Z}_{n}+\frac{1}{64B_{0}}\mathcal{Z}_{64}+\frac{3}{64B_{0}}\mathcal{Z}_{65}-\frac{1}{64B_{0}}\mathcal{Z}_{76}+\frac{1}{32B_{0}}\mathcal{Z}_{77}+\frac{1}{32}\mathcal{Z}_{78}\;,$
$\displaystyle K_{105}$ $\displaystyle=$
$\displaystyle-\frac{1}{64}\mathcal{Z}_{9}-\frac{1}{128}\mathcal{Z}_{10}+\frac{1}{64}\mathcal{Z}_{11}+\frac{1}{64}\mathcal{Z}_{12}-\frac{1}{128}\mathcal{Z}_{13}-\frac{1}{64}\mathcal{Z}_{14}+\frac{3}{128}\mathcal{Z}_{15}-\frac{1}{128}\mathcal{Z}_{16}-\frac{1}{64B_{0}}\mathcal{Z}_{32}+\frac{1}{64B_{0}}\mathcal{Z}_{33}$
$\displaystyle+\frac{1}{64B_{0}}\mathcal{Z}_{34}-\frac{1}{64B_{0}}\mathcal{Z}_{35}-\frac{1}{32B_{0}}\mathcal{Z}_{39}+\frac{1}{32}\mathcal{Z}_{44}-\frac{1}{64}\mathcal{Z}_{46}-\frac{1}{32}\mathcal{Z}_{47}-\frac{3}{64}\mathcal{Z}_{48}-\frac{1}{64}\mathcal{Z}_{49}+\frac{3}{64}\mathcal{Z}_{50}-\frac{3}{32}\mathcal{Z}_{51}-\frac{3}{64}\mathcal{Z}_{52}$
$\displaystyle-\frac{5}{64}\mathcal{Z}_{53}-\frac{3}{16}\mathcal{Z}_{54}-\frac{9}{32}\mathcal{Z}_{55}-\frac{3}{64}\mathcal{Z}_{56}-\frac{3}{32}\mathcal{Z}_{57}-\frac{9}{32}\mathcal{Z}_{59}-\frac{9}{32}\mathcal{Z}_{60}-\frac{9}{32}\mathcal{Z}_{61}-\frac{7}{32}\mathcal{Z}_{62}-\frac{5}{32}\mathcal{Z}_{63}+\frac{3}{64B_{0}}\mathcal{Z}_{64}$
$\displaystyle+\frac{5}{64B_{0}}\mathcal{Z}_{65}-\frac{3}{64B_{0}}\mathcal{Z}_{76}+\frac{1}{32B_{0}}\mathcal{Z}_{77}+\frac{1}{32}\mathcal{Z}_{78}-\frac{1}{32}\mathcal{Z}_{80}-\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{107}$ $\displaystyle=$
$\displaystyle-\frac{1}{16B_{0}}\mathcal{Z}_{39}\;,$ $\displaystyle K_{109}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\mathcal{Z}_{50}-\frac{1}{4}\mathcal{Z}_{51}-\frac{1}{8}\mathcal{Z}_{52}+\frac{1}{8}\mathcal{Z}_{55}+\frac{1}{8}\mathcal{Z}_{59}+\frac{1}{8}\mathcal{Z}_{61}\;,$
$\displaystyle K_{110}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{48}+\frac{1}{16}\mathcal{Z}_{49}-\frac{1}{8}\mathcal{Z}_{50}+\frac{1}{4}\mathcal{Z}_{51}+\frac{1}{8}\mathcal{Z}_{52}+\frac{1}{16}\mathcal{Z}_{53}+\frac{1}{8}\mathcal{Z}_{54}+\frac{1}{8}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{56}+\frac{1}{8}\mathcal{Z}_{57}+\frac{1}{8}\mathcal{Z}_{59}$
$\displaystyle+\frac{1}{8}\mathcal{Z}_{60}+\frac{1}{8}\mathcal{Z}_{61}+\frac{1}{8}\mathcal{Z}_{62}+\frac{1}{8}\mathcal{Z}_{63}-\frac{1}{16B_{0}}\mathcal{Z}_{64}-\frac{1}{16B_{0}}\mathcal{Z}_{65}+\frac{1}{16B_{0}}\mathcal{Z}_{76}\;,$
$\displaystyle K_{111}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{44}-\frac{1}{16}\mathcal{Z}_{45}-\frac{1}{8}\mathcal{Z}_{47}-\frac{1}{16}\mathcal{Z}_{48}+\frac{1}{16}\mathcal{Z}_{49}+\frac{5}{8}\mathcal{Z}_{50}-\frac{5}{4}\mathcal{Z}_{51}-\frac{5}{8}\mathcal{Z}_{52}-\frac{1}{8}\mathcal{Z}_{54}-\frac{1}{4}\mathcal{Z}_{55}-\frac{5}{16}\mathcal{Z}_{59}-\frac{5}{16}\mathcal{Z}_{60}$
$\displaystyle-\frac{5}{16}\mathcal{Z}_{61}-\frac{3}{16}\mathcal{Z}_{62}-\frac{1}{16}\mathcal{Z}_{63}-\frac{1}{32}\mathcal{Z}_{79}-\frac{1}{16}\mathcal{Z}_{80}-\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{112}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mathcal{Z}_{44}+\frac{1}{16}\mathcal{Z}_{45}+\frac{1}{16}\mathcal{Z}_{46}+\frac{1}{16}\mathcal{Z}_{53}+\frac{1}{8}\mathcal{Z}_{54}+\frac{1}{8}\mathcal{Z}_{55}+\frac{1}{16}\mathcal{Z}_{56}+\frac{1}{8}\mathcal{Z}_{57}+\frac{3}{16}\mathcal{Z}_{59}+\frac{3}{16}\mathcal{Z}_{60}+\frac{3}{16}\mathcal{Z}_{61}+\frac{3}{16}\mathcal{Z}_{62}$
$\displaystyle+\frac{3}{16}\mathcal{Z}_{63}-\frac{1}{16B_{0}}\mathcal{Z}_{64}-\frac{1}{16B_{0}}\mathcal{Z}_{65}+\frac{1}{16B_{0}}\mathcal{Z}_{76}+\frac{1}{32}\mathcal{Z}_{79}+\frac{1}{16}\mathcal{Z}_{80}+\frac{1}{32}\mathcal{Z}_{81}\;,$
$\displaystyle K_{113}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{n=53}^{63}\mathcal{Z}_{n}-\frac{1}{4B_{0}}\mathcal{Z}_{64}-\frac{1}{4B_{0}}\mathcal{Z}_{65}-\frac{1}{4B_{0}}\mathcal{Z}_{66}+\frac{1}{4B_{0}^{2}}\mathcal{Z}_{68}\;,$
$\displaystyle K_{114}$ $\displaystyle=$
$\displaystyle\mathcal{Z}_{50}+\frac{1}{2}\mathcal{Z}_{69}\;,$ $\displaystyle
K_{115}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mathcal{Z}_{50}+\frac{1}{2}\mathcal{Z}_{51}+\frac{1}{4}\mathcal{Z}_{52}\;.$
(254)
To help understanding mutual relation between the definition of symbols in our
formulation and those in Ref.p6-1 , in Table XIII, we give a comparison.
TABLE XIV. Comparisons between the symbols introduce in Ref.p6-1 (the first
and third columns) and corresponding ones defined in present paper (the second
and fourth columns).
$\displaystyle\begin{array}[]{|c|c||c|c|}\hline\cr\mbox{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-1}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{Present
paper}&\mbox{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6-1}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{Present
paper}\\\ \hline\cr
g_{R}&V_{R}&\chi_{-}^{\mu}&4iB_{0}d^{\mu}p_{\Omega}-4iB_{0}s_{\Omega}a_{\Omega}^{\mu}-4iB_{0}a_{\Omega}^{\mu}s_{\Omega}\\\
\hline\cr g_{L}&V_{L}&F_{L,R}^{\mu\nu}&f_{L,R}^{\mu\nu}\\\ \hline\cr
u(\varphi)&\Omega&f_{+}^{\mu\nu}&2V^{\mu\nu}-2i(a_{\Omega}^{\mu}a_{\Omega}^{\nu}-a_{\Omega}^{\nu}a_{\Omega}^{\mu})\\\
\hline\cr
u^{\mu}&2a_{\Omega}^{\mu}&\nabla^{\lambda}f_{+}^{\mu\nu}&2d^{\lambda}V^{\mu\nu}-2id^{\lambda}(a_{\Omega}^{\mu}a_{\Omega}^{\nu}-a_{\Omega}^{\nu}a_{\Omega}^{\mu})\\\
\hline\cr u\cdot
u=u^{\mu}u^{\mu}&4a_{\Omega}^{2}&f_{-}^{\mu\nu}&-2(d^{\mu}a_{\Omega}^{\nu}-d^{\nu}a_{\Omega}^{\mu})\\\
\hline\cr\chi&\chi&\nabla^{\lambda}f_{-}^{\mu\nu}&-2(d^{\lambda}d^{\mu}a_{\Omega}^{\nu}-d^{\lambda}d^{\nu}a_{\Omega}^{\mu})\\\
\hline\cr\chi_{+}&4B_{0}s_{\Omega}&h^{\mu\nu}&2(d^{\mu}a_{\Omega}^{\nu}+d^{\nu}a_{\Omega}^{\mu})\\\
\hline\cr\chi_{+}^{\mu}&4B_{0}d^{\mu}s_{\Omega}+4B_{0}p_{\Omega}a_{\Omega}^{\mu}+4B_{0}a_{\Omega}^{\mu}p_{\Omega}&\nabla^{\mu}&d^{\mu}\\\
\hline\cr\chi_{-}&4iB_{0}p_{\Omega}&\Gamma^{\mu}&-iv_{\Omega}^{\mu}\\\
\hline\cr\end{array}$ (265)
In obtaining (254), some dependent terms must be reduced into independent
terms, in Table XV, the first column is the dependent operators and the second
column is the sum of its corresponding independent operators.
TABLE XV. Expand dependent operator in terms of independent operators
$\displaystyle\begin{array}[]{|c|c|}\hline\cr\mbox{dependent
operator}&\mbox{corresponding independent operators}\\\
\hline\cr\mathrm{tr}(u_{\mu}u_{\nu}h^{\mu\lambda}h^{\nu\lambda})&-\frac{1}{2}Y_{3}+\frac{1}{2}Y_{28}-\frac{1}{N_{f}}Y_{30}+\frac{1}{N_{f}}Y_{34}-\frac{1}{2N_{f}^{2}}Y_{36}-\frac{1}{2}Y_{37}+Y_{54}-Y_{60}\\\
&+Y_{68}+\frac{1}{2}Y_{92}+Y_{94}+\frac{1}{2}Y_{105}\\\
\hline\cr\mathrm{tr}(u_{\mu}u_{\nu}h^{\mu\lambda}f_{-}^{\nu\lambda}&2Y_{1}-\frac{1}{2}Y_{3}+\frac{3}{2}Y_{5}+\frac{3}{N_{f}}Y_{30}-3Y_{31}+2Y_{33}-\frac{3}{N_{f}}Y_{34}+\frac{3}{2N_{f}^{2}}Y_{36}-\frac{1}{2}Y_{37}\\\
+u_{\mu}u_{\nu}f_{-}^{\mu\lambda}h^{\nu\lambda})&-2Y_{49}-Y_{52}+4Y_{54}+Y_{58}-2Y_{60}-Y_{64}-2Y_{66}-4Y_{67}+Y_{68}\\\
&+Y_{86}-2Y_{90}+\frac{1}{2}Y_{92}+2Y_{94}-2Y_{95}-\frac{3}{2}Y_{97}+Y_{105}\\\
\hline\cr\mathrm{tr}(u_{\mu}u_{\nu}h^{\nu\lambda}h^{\mu\lambda})&-Y_{1}-\frac{1}{2}Y_{5}+\frac{1}{2}Y_{28}-\frac{2}{N_{f}}Y_{30}+Y_{31}-Y_{33}+\frac{2}{N_{f}}Y_{34}-\frac{1}{N_{f}^{2}}Y_{36}\\\
&+Y_{49}+Y_{52}-2Y_{54}+Y_{64}+2Y_{67}+Y_{90}+Y_{95}+\frac{1}{2}Y_{97}-\frac{1}{2}Y_{105}\\\
\hline\cr\mathrm{tr}(u^{\mu}h_{\mu\nu}\chi_{+}^{\nu}&Y_{7}+Y_{11}-2Y_{13}-Y_{17}-\frac{1}{2}Y_{28}-Y_{33}+\frac{2}{N_{f}}Y_{34}-2Y_{37}-Y_{39}+\frac{2}{N_{f}}Y_{41}\\\
+u^{\mu}\chi_{+}^{\nu}h_{\mu\nu})&-\frac{1}{N_{f}^{2}}Y_{42}+Y_{43}-\frac{2}{N_{f}}Y_{44}+Y_{83}+2Y_{85}+Y_{102}-\frac{1}{2}Y_{105}-Y_{107}\\\
\hline\cr\mathrm{tr}(i\nabla^{\mu}f_{+}^{\nu\lambda}{f_{-}}_{\nu\lambda}u_{\mu}&Y_{71}-Y_{73}-2Y_{75}+Y_{78}-Y_{100}+Y_{101}+\frac{1}{2}Y_{104}-2Y_{111}\\\
-i\nabla^{\mu}f_{+}^{\nu\lambda}u_{\mu}{f_{-}}_{\nu\lambda})&\\\
\hline\cr\mathrm{tr}(\nabla_{\mu}f_{+}^{\mu\nu}u_{\nu}\chi_{-}&-Y_{66}-Y_{67}+Y_{68}+\frac{1}{2}Y_{71}-\frac{1}{2}Y_{73}+Y_{75}-2Y_{76}+\frac{1}{2}Y_{78}\\\
-\nabla_{\mu}f_{+}^{\mu\nu}\chi_{-}u_{\nu})&+\frac{1}{2}Y_{90}-\frac{1}{2}Y_{92}-Y_{94}+\frac{1}{2}Y_{97}-\frac{1}{2}Y_{100}+\frac{1}{2}Y_{101}+Y_{110}+\frac{1}{4}Y_{104}+Y_{112}\\\
\hline\cr\mathrm{tr}(i\nabla^{\mu}f_{+}^{\nu\lambda}{f_{-}}_{\mu\nu}u_{\lambda}&-\frac{1}{2}Y_{71}+\frac{1}{2}Y_{73}+Y_{75}-\frac{1}{2}Y_{78}+\frac{1}{2}Y_{90}-\frac{1}{2}Y_{92}-Y_{94}+\frac{1}{2}Y_{97}\\\
+i\nabla^{\mu}f_{+}^{\nu\lambda}u_{\nu}{f_{-}}_{\mu\lambda})&+\frac{1}{2}Y_{100}-\frac{1}{2}Y_{101}-\frac{1}{4}Y_{104}+Y_{111}\\\
\hline\cr\mathrm{tr}(if_{+}^{\mu\nu}h_{\mu\lambda}h_{\nu}^{~{}\lambda})&\frac{1}{2}Y_{64}-Y_{66}+Y_{71}-Y_{73}-Y_{100}+\frac{1}{2}Y_{101}-Y_{111}+Y_{112}\\\
\hline\cr\mathrm{tr}({f_{+}}_{\mu\nu}u^{\mu}\chi_{-}^{\nu}&-Y_{66}-Y_{67}+Y_{68}+\frac{1}{2}Y_{71}-\frac{1}{2}Y_{73}+Y_{75}-2Y_{76}+\frac{1}{2}Y_{78}-\frac{1}{2}Y_{83}-Y_{85}\\\
+{f_{+}}_{\mu\nu}\chi_{-}^{\mu}u^{\nu})&+\frac{1}{2}Y_{90}-\frac{1}{2}Y_{92}-Y_{94}+\frac{1}{2}Y_{97}-\frac{1}{2}Y_{100}+\frac{1}{2}Y_{101}-\frac{1}{4}Y_{104}+Y_{110}+Y_{112}\\\
\hline\cr\mathrm{tr}(\nabla_{\mu}f_{+}^{\mu\nu}\nabla^{\lambda}{f_{+}}_{\lambda\nu})&\frac{3}{4}Y_{71}-\frac{3}{4}Y_{73}-\frac{3}{2}Y_{75}-\frac{1}{2}Y_{76}+Y_{78}-\frac{1}{2}Y_{90}+\frac{1}{2}Y_{92}+\frac{1}{2}Y_{94}\\\
&-\frac{1}{4}Y_{97}-Y_{100}-\frac{1}{2}Y_{101}+\frac{1}{4}Y_{104}-\frac{1}{2}Y_{109}-2Y_{111}-4Y_{114}+Y_{115}\\\
\hline\cr\mathrm{tr}(\nabla_{\mu}f_{-}^{\mu\nu}\nabla^{\lambda}{f_{-}}_{\lambda\nu})&-\frac{1}{4}Y_{71}+\frac{1}{4}Y_{73}+\frac{1}{2}Y_{75}-\frac{1}{4}Y_{78}+\frac{1}{2}Y_{94}-\frac{1}{2}Y_{95}+\frac{1}{2}Y_{101}+\frac{1}{2}Y_{109}\\\
\hline\cr\mathrm{tr}(\nabla^{\mu}f_{-}^{\nu\lambda}\nabla_{\nu}{f_{-}}_{\mu\lambda})&-\frac{1}{4}Y_{71}+\frac{1}{4}Y_{73}+\frac{1}{2}Y_{75}-\frac{1}{4}Y_{78}+\frac{1}{2}Y_{109}\\\
\hline\cr\mathrm{tr}(iu_{\mu}\chi_{-}^{\mu}\chi_{+}&Y_{19}+Y_{23}+Y_{33}+Y_{37}+Y_{39}-\frac{1}{N_{f}}Y_{41}-Y_{43}\\\
+iu_{\mu}\chi_{+}\chi_{-}^{\mu})&\\\
\hline\cr\mathrm{tr}(iu_{\mu}\chi_{+}\langle\chi_{-}^{\mu}\rangle)&Y_{24}+Y_{34}+\frac{1}{2}Y_{41}-\frac{1}{2N_{f}}Y_{42}-Y_{44}\\\
\hline\cr\mathrm{tr}(\nabla^{\mu}f_{+}^{\nu\lambda}\nabla_{\mu}{f_{+}}_{\nu\lambda})&\frac{3}{2}Y_{71}-\frac{3}{2}Y_{73}-4Y_{75}+2Y_{78}-\frac{1}{2}Y_{90}+\frac{1}{2}Y_{92}\\\
&-2Y_{100}+2Y_{101}+\frac{1}{2}Y_{104}-Y_{109}-4Y_{111}+2Y_{115}\\\
\hline\cr\end{array}$ (284)
|
arxiv-papers
| 2009-07-29T22:10:46 |
2024-09-04T02:49:04.316843
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shao-Zhou Jiang, Ying Zhang, Chuan Li, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/0907.5229"
}
|
0907.5264
|
§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§**University
of Maryland * Center for String and Particle Theory* Physics
Department***University of Maryland *Center for String and Particle Theory**
**University of Maryland * Center for String and Particle Theory* Physics
Department***University of Maryland *Center for String and Particle Theory**
July 2009 UMDEPP 09-043
hep-th/0907.5264
Ectoplasm & Superspace Integration Measure for
$2$D Supergravity with Four Spinorial Supercurrents111Supported in part by
National Science Foundation Grant PHY-0354401.
S. James Gates, Jr.222gatess@wam.umd.edu and Gabriele Tartaglino-
Mazzucchelli333gtm@umd.edu
Center for String and Particle Theory
Department of Physics, University of Maryland
College Park, MD 20742-4111 USA
ABSTRACT
> Building on a previous derivation of the local chiral projector for a two
> dimensional superspace with eight real supercharges, we provide the complete
> density projection formula required for locally supersymmetrical theories in
> this context. The derivation of this result is shown to be very efficient
> using techniques based on the Ectoplasmic construction of local measures in
> superspace.
>
>
> PACS: 04.65.+e, 11.30.Pb
## 1 Introduction
Some years ago, a formulation of a 2D supergravity theory which included off-
shell closure of the local supersymmetry algebra with four real spinorial
supercharges and a necessary set of auxiliary fields was introduced into the
literature [1]. In a subsequent development there was made a proposal (called
‘Ectoplasm’ [2]) for a conceptual framework leading to efficient derivations
of local superspace integration measures (density projection operators)444 A
mathematical construction giving the formal bases for the Ectoplasm methods
can be found
in the theory of integration over surfaces in supermanifolds developed in [3,
4, 5].. In addition, about the same time there was put forward an alternative
general framework for the derivation of density projection operators based on
the use of superspace normal coordinate expansions first introduced in [6] and
rediscovered555Previous approaches for component reduction, ultimately related
to normal coordinates
expansions, can be found in [8, 9, 10, 11, 12, 13, 14]. in [7]; see [15] for
recent reformulations and improvements of the normal coordinates techniques.
The ectoplasm and normal coordinates frameworks have been found to be closely
related [16, 17].
Prior to the introduction of the ectoplasmic and normal coordinate approaches,
the question of how to construct local superspace supergravity densities had
been approached by two other and more cumbersome methods. Both of these can be
seen in two books on the subject. In the first, Superspace [18], an approach
was taken to reproduce, at the level of superfields, a Noether approach thus
leading to the density projector. In the second Ideas [19], an approach that
was taken to utilize the prepotential formulation of supergravity theory to
derive the density projector.
It has been argued from its inception that the ectoplasmic concept is not only
extremely efficient but also likely applies to even more complicated theories
such as string theory. Though there was no such evidence at the time of the
introduction of the ectoplasm approach, later it was shown that integration
measures in the ‘pure spinor formulation’ of superstrings follow precisely
from the extension of the ectoplasmic concept to this realm of theories [20].
The off-shell formulation of a 2D, $\cal N$ = 4 supergravity theory implies
the existence of a straightforward way to completely develop an efficient
local integration theory for the associated local Salam-Strathdee superspace.
We will complete such a construction in the current work by use of the
ectoplasmic suggestion.
This paper is organized as follows. In section 2 we review the 2D, $\cal N$=4
supergravity formulation of [1]. Section 3 is devoted to the presentation of a
new super 2-form multiplet. In section 4 we make use of the ectoplasmic
approach to build the density projector for the 2D, $\cal N$=4 supergravity of
[1]; this is the main result of the paper. Section 5 collects some
conclusions. The paper includes two appendices. Appendix A contains the
derivation of the result of section 3. Then, the appendix B is a collection of
formulas used in the paper.
## 2 An Off-Shell 2D Supergravity Geometry With Eight Real Local
Supersymmetries
In this section we review some aspects of the off-shell 2D, ${\cal N}=4$
minimal supergravity multiplet first introduced in [1]. We focus on the curved
superspace geometry underlining the minimal supergravity that will be used in
the computations of this paper.
The work in [1] showed there exists component fields
$(e_{a}{}^{m},~{}\psi_{a}{}^{\alpha i},~{}A_{ai}{}^{j},~{}B,~{}G,~{}H)$ which
describe an off-shell 2D supergravity theory possessing eight real local (or
four real spinorial) supercharges. The previous list of component fields
contains the graviton, the gravitini, SU(2) connection, a complex scalar $B$,
one real scalar $G$ and one real pseudoscalar $H$. These are the components
associated with the following constraints on the 2D, ${\cal N}$ = 4 superspace
supergravity covariant derivative algebra666In the present paper we adopt the
Lorentz and SU(2) notations collected in Appendix
A of [21] and consistent with the conventions of [18].
$\displaystyle\\{\,\nabla{}_{\alpha i}~{},~{}\nabla{}_{\beta j}\,\\}$
$\displaystyle=$ $\displaystyle 2\overline{B}[\,C{}_{\alpha\beta}C{}_{ij}{\cal
M}~{}-~{}(\gamma{}^{3}){}_{\alpha\beta}{\cal
Y}{}_{ij}\,]\,~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
(1)
$\displaystyle\\{\,{\overline{\nabla}}_{\alpha}{}^{i}~{},~{}{\overline{\nabla}}_{\beta}{}^{j}\,\,\\}$
$\displaystyle=$ $\displaystyle 2B{[}\,C_{\alpha\beta}C^{ij}{\cal
M}~{}-~{}(\gamma{}^{3})_{\alpha\beta}{\cal
Y}^{ij}\,{]}\,~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
(2) $\displaystyle\\{\,\nabla{}_{\alpha
i}~{},~{}\overline{\nabla}{}_{\beta}{}^{j}\,\\}$ $\displaystyle=$
$\displaystyle 2{\rm
i}\,\delta{}_{i}{}^{j}(\gamma{}^{c}){}_{\alpha\beta}\nabla{}_{c}~{}+~{}2\delta{}_{i}{}^{j}\phi{}_{\alpha}{}^{\gamma}(\gamma^{3})_{\gamma\beta}{\cal
M}~{}-~{}2\phi{}_{\alpha\beta}{\cal
Y}{}_{i}{}^{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$ (3)
$\displaystyle{[}\,\nabla{}_{\alpha i}~{},~{}\nabla{}_{b}\,{]}$
$\displaystyle=$ $\displaystyle\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\phi{}_{\alpha}{}^{\gamma}(\gamma{}_{b}){}_{\gamma}{}^{\beta}\nabla{}_{\beta
i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}(\gamma^{3}\gamma{}_{b}){}_{\alpha}{}^{\beta}\overline{B}C{}_{ij}\overline{\nabla}{}_{\beta}{}^{j}$
(4) $\displaystyle~{}-~{}{\rm
i}(\gamma^{3}\gamma_{b})_{\alpha\beta}\bar{\Sigma}^{\beta}{}_{i}{\cal
M}~{}+~{}{\rm i}(\gamma{}_{b})_{\alpha\beta}\bar{\Sigma}^{\beta}{}_{j}{\cal
Y}_{i}{}^{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle~{}{[}\,{\overline{\nabla}}_{\alpha}{}^{i}~{},~{}\nabla{}_{b}\,{]}$
$\displaystyle=$ $\displaystyle-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\bar{\phi}_{\alpha}{}^{\gamma}(\gamma_{b})_{\gamma}{}^{\beta}{\overline{\nabla}}_{\beta}{}^{i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}(\gamma^{3}\gamma_{b})_{\alpha}{}^{\beta}BC^{ij}\nabla_{\beta
j}$ (5) $\displaystyle~{}~{}-~{}{\rm
i}(\gamma^{3}\gamma_{b})_{\alpha\beta}\Sigma^{\beta i}{\cal M}~{}-~{}{\rm
i}(\gamma{}_{b})_{\alpha\beta}\Sigma^{\beta j}{\cal
Y}^{i}{}_{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle~{}{[}\,\nabla{}_{a}~{},~{}\nabla{}_{b}\,{]}$ $\displaystyle=$
$\displaystyle-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\varepsilon_{ab}[(\gamma{}^{3}){}_{\alpha}{}^{\beta}\Sigma{}^{\alpha
i}\nabla{}_{\beta
i}~{}+~{}(\gamma{}^{3}){}_{\alpha}{}^{\beta}\overline{\Sigma}{}^{\alpha}{}_{i}\overline{\nabla}{}_{\beta}{}^{i}~{}+~{}{\cal
R}{\cal M}~{}+~{}{\rm i}{\cal F}{}_{i}{}^{j}{\cal
Y}{}_{j}{}^{i}]~{}.~{}~{}~{}~{}$ (6)
where
$\displaystyle(B)^{*}$ $\displaystyle=$
$\displaystyle\bar{B}~{},~{}~{}~{}(G)^{*}~{}=\,G~{},~{}~{}~{}(H)^{*}~{}=\,H~{},~{}~{}~{}(\Sigma_{\alpha}{}^{i})^{*}~{}=\,\bar{\Sigma}_{\alpha
i}~{},$ (7) $\displaystyle\phi_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle C_{\alpha\beta}G+{\rm i}(\gamma^{3})_{\alpha\beta}H~{},$ (8)
$\displaystyle\bar{\phi}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle(\phi_{\alpha\beta})^{*}=-C_{\alpha\beta}G+{\rm
i}(\gamma^{3})_{\alpha\beta}H=\phi_{\beta\alpha}~{}.$ (9)
In writing these, we have corrected some coefficients that appear in the work
of [21] in the terms that appear in (3) - (5). These corrected coefficients do
not affect (1) and (2). Thus the result in the work of [21] is unaffected by
this change.
In the previous algebra, the covariant derivatives are
$\nabla_{A}=(\nabla_{a},\nabla_{\alpha i},{\overline{\nabla}}_{\alpha}{}^{i})$
$\displaystyle\nabla_{A}~{}=~{}E_{A}{}^{M}\partial_{M}+\omega_{A}{\cal M}+{\rm
i}\,\Gamma_{A}{}_{k}{}^{l}{\cal Y}_{l}{}^{k}~{}.$ (10)
The 2D, ${\cal N}=4$ curved superspace is locally parametrized by the
coordinates $z^{M}=(x^{m},\theta^{\mu i},\bar{\theta}^{\mu}{}_{i})$ with the
Grassmann variables $\theta^{\mu i}$ and $\bar{\theta}^{\mu}{}_{i}$ related by
complex conjugation $\bar{\theta}^{\mu}{}_{i}=(\theta^{\mu i})^{*}$; the
bosonic coordinates will be also denoted as $x^{m}=(\tau,\sigma)$. In (10),
$E_{A}{}^{M}$ is the inverse of the vielbein $E_{M}{}^{A}$
($E_{M}{}^{A}E_{A}{}^{N}=\delta_{M}^{N}$,
$E_{A}{}^{M}E_{M}{}^{B}=\delta_{A}^{B}$) with $\partial_{M}=\partial/\partial
z^{M}$, $\omega_{A}$ the 2D Lorentz connection and $\Gamma_{A}{}_{k}{}^{l}$ is
the SU(2) connection. The torsion $T_{AB}{}^{C}$, Lorentz curvature $R_{AB}$
and SU(2) curvature $R_{AB}{}_{k}{}^{l}$ superfields are defined by (1) - (6)
and
$\displaystyle[\nabla_{A},\nabla_{B}\\}~{}=~{}T_{AB}{}^{C}\nabla_{C}+R_{AB}{\cal
M}+{\rm i}\,R_{AB}{}_{k}{}^{l}{\cal Y}_{l}{}^{k}~{}.$ (11)
The action of the local 2D Lorentz generator ${\cal M}$ and of the local SU(2)
generator ${\cal Y}_{k}{}^{l}$ on the spinor covariant derivatives are the
following (${\cal Y}_{kl}={\cal Y}_{k}{}^{p}C_{pl}$)
$\displaystyle{[}{\cal M},\nabla_{\alpha i}{]}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}(\gamma^{3})_{\alpha}{}^{\beta}\nabla_{\beta
i}~{},~{}~{}~{}{[}{\cal
M},{\overline{\nabla}}_{\alpha}{}^{i}{]}={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}(\gamma^{3})_{\alpha}{}^{\beta}{\overline{\nabla}}_{\beta}{}^{i}~{},~{}~{}~{}$
(12) $\displaystyle{[}{\cal Y}_{kl},\nabla_{\alpha i}{]}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{i(k}\nabla_{\beta
l)}~{},~{}~{}~{}{[}{\cal
Y}_{kl},{\overline{\nabla}}_{\alpha}{}^{i}{]}=-{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta^{i}_{(k}{\overline{\nabla}}_{\beta
l)}~{}.$ (13)
It is worthy to recall that the consistency of the Bianchi identities
constructed from the commutator algebra above requires the conditions [1]
$\displaystyle\overline{\nabla}{}_{\alpha}{}^{i}B$ $\displaystyle=$
$\displaystyle 0~{}~{}~{}~{}~{}~{},~{}~{}~{}\nabla{}_{\alpha
i}B~{}=\,-2C{}_{ij}(\gamma{}^{3}){}_{\alpha\beta}\Sigma{}^{\beta j}~{}~{},$
(14) $\displaystyle\nabla{}_{\alpha i}G$ $\displaystyle=$
$\displaystyle\overline{\Sigma}{}_{\alpha
i}~{}~{}~{},~{}~{}~{}\nabla{}_{\alpha i}H~{}=\,{\rm
i}(\gamma{}^{3}){}_{\alpha}{}^{\beta}\overline{\Sigma}{}_{\beta i},~{}~{}~{},$
(15) $\displaystyle\overline{\nabla}{}_{\alpha}{}^{i}\Sigma{}^{\beta j}$
$\displaystyle=$ $\displaystyle{\rm
i}C{}^{ij}(\gamma{}^{3}\gamma{}^{a}){}_{\alpha}{}^{\beta}\nabla{}_{a}B~{}~{},$
(16) $\displaystyle\nabla{}_{\alpha i}\Sigma{}^{\beta j}$ $\displaystyle=$
$\displaystyle{1\over 2}\delta{}_{\alpha}{}^{\beta}\delta{}_{i}{}^{j}[{\cal
R}~{}-~{}2G{}^{2}~{}-~{}2H{}^{2}~{}-~{}2B\overline{B}]~{}+~{}{\rm
i}(\gamma{}^{3}){}_{\alpha}{}^{\beta}{\cal F}{}_{i}{}^{j}$ (18)
$\displaystyle+~{}{\rm
i}\delta{}_{i}{}^{j}(\gamma{}^{a}){}_{\alpha}{}^{\beta}(\nabla{}_{a}G)-~{}\delta{}_{i}{}^{j}(\gamma{}^{3}\gamma{}^{a}){}_{\alpha}{}^{\beta}(\nabla{}_{a}H)~{}~{}~{}.$
The component gauge fields occur in the above supertensors in the following
manner777Given a superfield $\Psi(\tau,\sigma,\theta,\bar{\theta})$, we denote
as usual with $\Psi|:=\Psi|_{\theta=0}$ the field obtained by
setting to zero all the Grassmanian coordinates.
$\begin{array}[]{lll}{\cal R}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}{{\cal
R}}_{ab}(\hat{\omega})~{}+~{}[~{}2{\rm
i}(\gamma^{3}\gamma_{a})_{\alpha\beta}\psi_{b}{}^{\alpha
i}\overline{\Sigma}^{\beta}{}_{i}~{}+~{}{\rm{h.c.}}~{}]\\\
&&~{}~{}~{}~{}~{}~{}+~{}4\phi_{\alpha}{}^{\gamma}(\gamma^{3})_{\gamma\beta}\psi_{a}{}^{\alpha
i}{\overline{\psi}}_{b}{}^{\beta}{}_{i}~{}-~{}2[~{}C_{ij}\overline{B}\psi_{a}{}^{\alpha
i}\psi_{b\alpha}{}^{j}~{}+~{}{\rm{h.c.}}~{}]~{}{\large\\}}~{}~{},\\\ &&\\\
\Sigma^{\alpha i}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}\psi_{ab}{}^{\beta
i}(\gamma^{3})_{\beta}{}^{\alpha}~{}-~{}{\rm i}\psi_{a}{}^{\beta
i}{\phi}_{\beta}{}^{\gamma}(\gamma^{3}\gamma_{b})_{\gamma}{}^{\alpha}~{}+~{}{\rm
i}C^{ij}B\overline{\psi}_{a}{}^{\beta}{}_{j}(\gamma_{b})_{\beta}{}^{\alpha}~{}{\large\\}}~{}~{},\\\
&&\\\ {\cal F}_{i}{}^{j}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}{\rm
F}_{ab}(A)_{i}{}^{j}~{}-~{}2{\rm
i}(\gamma_{a})_{\alpha\beta}[~{}\psi_{b}{}^{\alpha
j}\overline{\Sigma}^{\beta}{}_{i}~{}+~{}\overline{\psi}_{b}{}^{\alpha}{}_{i}\Sigma^{\beta
j}\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}(\psi_{b}{}^{\alpha
k}\overline{\Sigma}^{\beta}{}_{k}~{}+~{}\overline{\psi}_{b}{}^{\alpha}{}_{k}\Sigma^{\beta
k})~{}]\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}4\phi_{\alpha\beta}[\psi_{a}{}^{\alpha
j}\overline{\psi}_{b}{}^{\beta}{}_{i}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}\psi_{a}{}^{\alpha
k}\overline{\psi}_{b}{}^{\beta}{}_{k}]\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}2(\gamma^{3})_{\alpha\beta}[~{}\overline{B}(C_{ik}\psi_{a}{}^{\alpha
k}\psi_{b}{}^{\beta}{}^{k}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}C_{kl}\psi_{a}{}^{\alpha
k}\psi_{b}{}^{\beta l})\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+~{}B(C^{jk}\overline{\psi}_{a}{}^{\alpha}{}_{i}\overline{\psi}_{b}{}^{\beta}{}_{k}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}C^{kl}\overline{\psi}_{a}{}^{\alpha}{}_{k}\overline{\psi}_{b}{}^{\beta}{}_{l})~{}]~{}{\large\\}}~{}~{},\end{array}$
(19)
where $\varepsilon^{ab}\,{{\cal R}}_{ab}(\hat{\omega})$ is the usual two-
dimensional curvature in terms of the inverse of the vielbein ${e}_{a}{}^{m}$
and of the Lorentz connection $\hat{\omega}_{a}$;
$\varepsilon^{ab}\psi_{ab}{}^{\beta i}$ is the gravitini field strength;
$\varepsilon^{ab}F_{ab}(A)$ is the SU(2) field strength function of
${e}_{a}{}^{m}$ and of the SU(2) connection $A_{a}{}_{k}{}^{l}$ [1]. The
component gauge fields $e_{a}{}^{m},\,\hat{\omega}_{a},\,A_{a}{}_{k}{}^{l}$
are easily related to the gauge superfields
$E_{A}{}^{M},\,\omega_{A},\,\Gamma_{A}{}_{k}{}^{l}$ in (10) by using standard
Wess-Zumino gauge reduction techniques [18, 19].
## 3 Defining A Closed 2D, ${\cal N}=4$ Super Two-Form
In this section we are going to present a new closed 2D, ${\cal N}=4$ super
two-form defined in terms of an unconstrained scalar chiral superfield. The
result contained in the Theorem 1 is crucial to build the measure of the local
superspace integration theory for 2D, ${\cal N}=4$ supergravity theories as we
will see in section 4.
The work in [21] established that the fourth-order spinorial derivatives
operator ${\cal{\overline{D}}}{}^{(4)}$, defined by
$\displaystyle{{\cal{\overline{D}}}{}^{(4)}~{}=~{}\left[\,{\overline{\nabla}}{}^{(2)\,\alpha\,\beta}~{}-~{}2\,B\,(\gamma^{3}){}^{\alpha\,\beta}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\alpha\,\beta}~{}~{},}$
---
(20)
is the chiral projection operator satisfying
$\displaystyle{{\overline{\nabla}}{}_{\gamma}^{i}\,{\cal{\overline{D}}}{}^{(4)}\,\Psi~{}=~{}{\overline{\nabla}}{}_{\gamma}^{i}\,\left[\,{\overline{\nabla}}{}^{(2)\,\alpha\,\beta}~{}-~{}2\,B\,(\gamma^{3}){}^{\alpha\,\beta}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\alpha\,\beta}\,\Psi~{}=~{}0}$
---
(21)
for any general scalar superfield $\Psi$. We note the derivation of (20) and
(21) given in [21] follows solely from algebraic manipulations of the
derivatives that appear in (2).
In a later section we will exploit the fact that a closed 2D, $\cal N$ = 4
super two-form is sufficient to determine the local integration measure for an
appropriate curved superspace. For this purpose it is necessary to define the
components of a 2D, $\cal N$ = 4 super two-form. The general framework for the
construction of such forms was presented some time ago [22] which implies for
the present consideration we should introduce a super 2-form whose component
superfields may be written in the form $J_{A}{}_{B}~{}=~{}\big{(}\,J_{\alpha
i}{}_{\beta j},~{}J_{\alpha
i}{}_{\beta}{}^{j},~{}J_{\alpha}{}^{i}{}_{\beta}{}^{j},~{}J_{\gamma
k}{}_{a},~{}J_{\gamma}{}^{k}{}_{a},~{}J_{ab}\,\big{)}$. We refer the reader to
[22, 18] for the notations we adopt in the use of super p-forms. In general,
given a super p-form $\Omega$, described by the component superfields
$\Omega_{A_{1}\cdots A_{p}}$, its exterior derivative $F=d\Omega$ has
components $F_{A_{1}\cdots A_{p+1}}$ given by 888With $[\cdots)$ we denote the
complete graded symmetrization of indeces.
$\displaystyle F_{A_{1}\cdots A_{p}A_{p+1}}={1\over
p!}\nabla_{[A_{1}}\Omega_{A_{2}\cdots A_{p+1})}-{1\over
2((p-1)!)}T_{[A_{1}A_{2}|}{}^{B}\Omega_{B|A_{3}\cdots A_{p+1})}~{}.$ (22)
The superform $\Omega$ is closed if $F_{A_{1}\cdots A_{p+1}}=0$. We can now
state a theorem.
Theorem 1
If $U$ is a chiral superfield, i.e. satisfies
$\,{\overline{\nabla}}{}_{\alpha}^{i}\,U\,=\,0\,,$ the components defined by
$\displaystyle{J_{\alpha i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}0~{}~{}~{},}$
---|---
$\displaystyle{J_{\alpha i}{}_{\beta j}}$ | $\displaystyle{{}=~{}2(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}-\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\overline{U}~{}~{}~{},}$
$\displaystyle{J_{\alpha}{}^{i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}\,2\,(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}{U}~{}-~{}\,C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}{U}~{}~{}~{},}$
$\displaystyle{J_{\gamma k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\overline{U}~{}~{}~{},}$
$\displaystyle{J_{\gamma}{}^{k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}\nabla_{\delta p}\nabla^{(2)\,kp}{U}~{}~{}~{},}$
$\displaystyle{J_{ab}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}\,\Big{(}\nabla^{(4)}~{}-~{}2\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}\Big{)}{U}~{}+~{}\Big{(}{\overline{\nabla}}^{(4)}~{}-~{}2\,B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\Big{)}\overline{U}\,\Big{]}~{},}$
(23)
describe a closed 2D, $\cal N$ = 4 super two-form with respect to the
supergravity
commutator algebra in Eq. (1) - Eq. (6). The superfield $\bar{U}:=(U)^{*}$ is
antichiral $\nabla_{\alpha i}\bar{U}=0$. In writing these results, we have
introduced second and fourth order spinorial derivative operators via the
equations
|
$\displaystyle{{}\nabla{}^{(2)}_{\alpha\,\beta}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{i\,j}\left[\,\nabla_{\alpha\,i}\,\nabla_{\beta\,j}~{}+~{}\nabla_{\beta\,i}\,\nabla_{\alpha\,j}\,\right]~{},~{}\nabla{}^{(2)}_{i\,j}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{\alpha\,\beta}\left[\,\nabla_{\alpha\,i}\,\nabla_{\beta\,j}~{}+~{}\nabla_{\alpha\,j}\,\nabla_{\beta\,i}\,\right]~{},}$
---|---
|
$\displaystyle{{}{\overline{\nabla}}{}^{(2)}_{\alpha\,\beta}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C_{i\,j}\left[\,{\overline{\nabla}}_{\alpha}{}^{i}\,{\overline{\nabla}}_{\beta}{}^{j}~{}+~{}{\overline{\nabla}}_{\beta}{}^{i}\,{\overline{\nabla}}_{\alpha}{}^{j}\,\right]~{},~{}{\overline{\nabla}}{}^{(2)}{}^{i\,j}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{\alpha\,\beta}\left[\,{\overline{\nabla}}_{\alpha}{}^{i}\,{\overline{\nabla}}_{\beta}{}^{j}~{}+~{}{\overline{\nabla}}_{\alpha}{}^{j}\,{\overline{\nabla}}_{\beta}{}^{i}\,\right]~{},}$
|
$\displaystyle{{}\nabla{}^{(4)}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\nabla{}^{(2)}{}^{k\,l}\,\nabla{}^{(2)}_{k\,l}~{}~{}~{},~{}~{}~{}~{}{\overline{\nabla}}{}^{(4)}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\overline{\nabla}}{}^{(2)}{}^{k\,l}\,{\overline{\nabla}}{}^{(2)}_{k\,l}~{}~{}~{}.}$
(24)
The proof of the theorem involves using the above equations to show that the
Bianchi identities for this two-form vanish. This is relegated to an appendix.
We next note that the chiral superfield $U$ above may be replaced using the
result from (21) according to $U=\bar{\cal D}^{(4)}{\cal L}$ ($\bar{U}={\cal
D}^{(4)}\bar{\cal L}=(\bar{\cal D}^{(4)}{\cal L})^{*}$) where the 2D, $\cal N$
= 4 superfield $\cal L$, ($\bar{\cal L}:=({\cal L})^{*}$), is not subject to
any algebraic nor differential restrictions. Stated another way, this implies
that an arbitrary 2D, $\cal N$ = 4 superfield $\cal L$ can be used to create a
closed super 2-form whose components are defined by $J_{A\,B}$ above. We
conclude with a result that will be needed in the next section. Defining the
component vierbein $E_{m}{}^{a}|=e_{m}{}^{a}$
($e_{m}{}^{a}e_{a}{}^{n}=\delta_{m}^{n}$,
$e_{a}{}^{m}e_{m}{}^{b}=\delta_{a}^{b}$), and the gravitini $E_{m}{}^{\alpha
i}|=-\psi_{m}{}^{\alpha i}$ ($\psi_{a}{}^{\alpha
i}=e_{a}{}^{m}\psi_{m}{}^{\alpha i}$),
$E_{m}{}^{\alpha}_{i}|=-\bar{\psi}_{m}{}^{\alpha}_{i}$
($\bar{\psi}_{a}{}^{\alpha}_{i}=e_{a}{}^{m}\bar{\psi}_{m}{}^{\alpha}_{i}$), by
a general result given in [18, 2] taking the limit as all Grassmann
coordinates go to zero one obtains
$\begin{array}[]{lll}~{}~{}\varepsilon^{ab}{J}_{ab}{\Big{|}}&=&\varepsilon^{ab}{\cal
J}_{ab}{\Big{|}}~{}+~{}2\,\varepsilon^{ab}(\psi_{a}{}^{\alpha i}{J}_{\alpha
i~{}b}{\Big{|}}~{}+~{}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{J}_{\alpha}{}^{i}{}_{~{}b}{\Big{|}})~{}+~{}2\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}_{i}{}_{\beta}{}^{j}{\Big{|}}\\\
&&\\\ &}{&+~{}\varepsilon^{ab}\psi_{a}{}^{\alpha i}\psi_{b}{}^{\beta
j}{J}_{\alpha\,i\,\beta\,j}{\Big{|}}~{}+~{}\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}{\Big{|}}~{}~{}~{},\end{array}$
(25)
where ${\cal J}_{ab}{\Big{|}}$ describe an ordinary space closed 2-form.
## 4 A 2D, ${\cal N}=4$ Density Projection Operator
It remains for us to calculate the explicit form of the density projection
operator (that we will denote by ${\Delta}^{(4)}$) which is the main purpose
of this work. As we are going to describe in this section, by using
${\Delta}^{(4)}$ and the chiral projector $\overline{{\cal D}}^{(4)}$, we can
build the integration measure of component actions in 2D, ${\cal N}=4$ minimal
supergravity.
As noted by Siegel [17], the ‘secret’ to the ectoplasmic approach is to
realize that the integration theory of superspace can be totally cast into the
language of closed super-forms. Indeed it was argued in the work of [2] that
the requirement that the topology of a superspace be totally determined by the
topology of its purely bosonic sub-manifold naturally provides a reason for
the appearance of super-forms in constructing integration measures of
superspace.
In the work of Ref. [18] it was noted that the derivation of component results
follows efficiently from replacing the integration of fermionic coordinates by
a process using first application of the superspace covariant derivative
followed by taking the limit as the Grassmann coordinates are taken to zero.
In the 2D, $\cal N$=4 case, this is described in the form of an equation
$\displaystyle{\int d^{2}\sigma\,d^{4}\theta\,d^{4}{\overline{\theta}\ }{\rm E}^{-1}~{}{\cal L}}$ | $\displaystyle{{}\to~{}\int d^{2}\sigma\ \,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,{\rm e}{}^{-1}\,{\Big{[}}\,{\Delta}{}^{(4)}\,{{\cal{\overline{D}}}{}^{(4)}}\,\,{\cal L}~{}+~{}{\rm h}.\,{\rm c}.\,{\Big{]}}\,{\Big{|}}}$
---|---
(26)
in terms of two differential operators, ${{\Delta}}^{(4)}$ (the density
projection operator) and ${\cal{\overline{D}}}{}^{(4)}$ (the chiral projection
operator) which may be expanded as
${{\Delta}}{}^{(4)}~{}=~{}\sum_{i=0}^{4}\,b_{(4-i)}\,\cdot\,\,\left[\,(\nabla)\,\times\,\,\cdots\,\,\times\,(\nabla)^{4-i}\,\right]~{}~{}~{},$
(27)
${\cal{\overline{D}}}{}^{(4)}~{}=~{}\sum_{i=0}^{4}\,a_{(4-i)}\,\cdot\,\,\left[\,(\overline{\nabla})\,\times\,\cdots\,\,\times\,({\overline{\nabla}})^{4-i}\,\right]~{}~{}~{},$
(28)
in terms of some field-dependent coefficients $a_{(4-i)}$ and $b_{(4-i)}$ and
powers of the spinorial superspace supergravity covariant derivatives
$\nabla_{\alpha\,i}$ and ${\overline{\nabla}}{}_{\alpha}{}^{i}$. In (26) we
have the expressions ${\rm E}^{-1}=[{\rm Ber}\,{E_{A}{}^{M}}]^{-1}$ and ${\rm
e}^{-1}=[\det{e_{a}{}^{m}}]^{-1}$ which are functions respectively of the
supervielbein and the component vielbein and $d^{2}\sigma$ denotes the measure
over the two-dimensional bosonic space. A further consequence of (26) - (28)
is that the superfield Lagrangian $\cal L$ need not be hermitian as it is the
linear combination of terms that appear in the action formula that must
satisfy this requirement. In the present context these spinorial superspace
supergravity covariant derivatives satisfy the relations given in section 2.
The basis for the ectoplasmic derivations of local supergravity measures and
projections operators, lies in a proposition for how to integrate an arbitrary
super $p$-form. This was proposed in the work of [2]. Given a curved
superspace with ${\rm N}_{B}$ bosonic coordinates (labelled by $\underline{m}$
indeces) and ${\rm N}_{F}$ fermionic coordinates (labelled by
$\underline{\mu}$ indices), we have Proposition 1
If $J{}_{{\underline{A}}_{1}\dots{\underline{A}}_{p}}$ is a closed super
$p$-form superfield whose Bianchi identities
vanish and $d\Omega{}^{{\underline{m}}_{1}\dots{\underline{m}}_{p}}$ is a co-
chain of dimension $p$ $\leq$ NB (where NB is
the dimensionality of the bosonic subspace), then the integral of the su-
per $p$-form over the co-chain is given by
$\displaystyle{{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}\equiv~{}(p!)^{-1}\,\int~{}d\Omega^{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}\,{\cal J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}~{}~{}~{}.}$
---|---
(29)
and this is a supersymmetrical invariant. In (29) we note the quantity ${\cal
J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}$ is related
to the super $p$-form $J{}_{{\underline{A}}_{1}\dots{\underline{A}}_{p}}$ via
$\displaystyle{\Big{(}\,J_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}\,\Big{)}~{}\equiv~{}\Big{[}}$ | $\displaystyle{{}~{}{\cal J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}\,+\,\lambda^{(p,1)}\psi_{[\,{\@@underline a}_{1}|}{}^{{\@@underline\alpha}_{1}}\Big{(}\,J_{{\@@underline\alpha}_{1}|\,{\@@underline a}_{2}\cdots{\@@underline a}_{p}\,]}{\Big{|}}\,\Big{)}}$
---|---
| $\displaystyle{{}+\,\lambda^{(p,2)}\psi_{[\,{\@@underline
a}_{1}|}{}^{{\@@underline\alpha}_{1}}\psi_{|{\@@underline
a}_{2}|}{}^{{\@@underline\alpha}_{2}}\Big{(}\,J_{{\@@underline\alpha}_{1}{\@@underline\alpha}_{2}|\,{\@@underline
a}_{3}\cdots{\@@underline a}_{p}\,]}{\Big{|}}\,\Big{)}\cdots}$
| $\displaystyle{{}+\,\lambda^{(p,p)}\,[\,\psi_{{\@@underline
a}_{1}}{}^{{\@@underline\alpha}_{1}}\cdots\psi_{{\@@underline
a}_{p}}{}^{{\@@underline\alpha}_{p}}\,]\Big{(}\,J_{{\@@underline\alpha}_{1}{\@@underline\alpha}_{2}\cdots{\@@underline\alpha}_{p}}{\Big{|}}\,\Big{)}~{}\Big{]}~{}~{}~{}.}$
(30)
where $\psi_{{\@@underline a}}{}^{{\@@underline\alpha}}$ denotes the
gravitino. The quantities ${\cal J}^{(p)}_{{\@@underline
a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}$ and coefficients
$\lambda^{(p,1)}\,$ $\cdots$ $\lambda^{(p,p)}\,$ are determined by taking the
limit as the Grassmann coordinates go to zero in $J_{{\@@underline
a}_{1}\cdots{\@@underline a}_{p}}$. In the 2D, ${\cal N}=4$ case with $J_{ab}$
the component of a super 2-form, the equation (25) informs us about the
$\lambda$-coefficients.”
We next observe that upon setting $p$ = NB the proposition takes the form
$\displaystyle{{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}=~{}\int~{}d^{N_{B}}x\,{\rm e}^{-1}\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{~{}{\rm N}{}_{B}!~{}}}$}}}}}$}}\,\varepsilon^{{\@@underline a}_{1}\cdots{\@@underline a}_{N_{B}}}\,{\cal J}^{({\rm N}{}_{B})}_{{\@@underline a}_{1}\cdots{\@@underline a}_{{\rm N}{}_{B}}}{\Big{|}}~{}~{}~{},}$
---|---
(31)
where e-1 denotes the determinant of the vielbein for the bosonic subspace. In
the case considered in this paper, we thus reach the result
$\displaystyle{{~{}~{}~{}~{}~{}~{}}{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm e}^{-1}\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{a\,b}\,{\cal J}^{(2)}_{a\,b}{\Big{|}}}$
---|---
| $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm
e}^{-1}\,{\Big{[}}~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{a\,b}\,{J}_{a\,b}{\Big{|}}~{}-~{}\varepsilon^{ab}(\psi_{a}{}^{\alpha
i}{J}_{\alpha
i~{}b}{\Big{|}}~{}+~{}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{J}_{\alpha}{}^{i}{}_{~{}b}{\Big{|}})}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}_{i}{}_{\beta}{}^{j}{\Big{|}}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}\psi_{b}{}^{\beta
j}{J}_{\alpha\,i\,\beta\,j}{\Big{|}}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\,\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}{\Big{|}}~{}{\Big{]}}~{}~{}~{}.}$
(32)
More explicitly the equations in (23) are expressed as
$\displaystyle{{~{}~{}}J_{\alpha i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}0~{}~{}~{},}$
---|---
$\displaystyle{J_{\alpha i}{}_{\beta j}}$ | $\displaystyle{{}=~{}2(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal L}}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}-\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal
L}}~{}~{}~{},}$
$\displaystyle{J_{\alpha}{}^{i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}\,2\,(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}-~{}\,C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal
L}~{}~{}~{},}$
$\displaystyle{J_{\gamma k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal L}}~{}~{}~{},}$
$\displaystyle{J_{\gamma}{}^{k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}\nabla_{\delta p}\nabla^{(2)\,kp}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}~{}~{}~{},}$
$\displaystyle{J_{ab}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}\nabla^{(4)}~{}-~{}2\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}\Big{]}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}~{}{\overline{\nabla}}^{(4)}~{}-~{}2\,B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\Big{]}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal
L}}~{}~{}~{}.}$
(33)
Finally, the results in (33) can be substituted into equation (32) to reach
the main result of this presentation. Given an arbitrary 2D, $\cal N$ = 4
superfield Lagrangian $\cal L$, a local supersymmetrical invariant is given by
$\displaystyle{{\cal S}}$ | $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm e}^{-1}\,\Delta^{(4)}\,\bar{\cal D}^{(4)}\,{\cal L}{\Big{|}}~{}+~{}{\rm h.~{}c.}}$
---|---
| $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm
e}^{-1}\,{\Big{\\{}}~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\nabla^{(4)}~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\bar{\psi}}_{a}{}^{\gamma}{}_{i}(\gamma^{a})_{\gamma}{}^{\delta}\nabla_{\delta
j}\nabla^{(2)\,ij}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}~{}{\Big{\\}}}\times}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal
L}{\Big{|}}~{}+~{}{\rm h.~{}c.}}$
(34)
in the presence of the off-shell supergravity theory described in section 2.
## 5 Conclusion
With this present work, we have completed the task of developing an efficient
local superspace integration theory for two dimensional theories that possess
eight real supercharges. We believe that the result given in (34) is
unexpectedly elegant and simple given that the general form of the eigth-order
spinorial differential operator defined by (26), (27) and (28) could, in
principle, take a more complicated form. Perhaps one of most surprising
features of this derivation has been the use of the closed 2D, $\cal N$ = 4
super 2-form used in Theorem 1. The superfield $U$ that appeared in equation
(23) is not required to describe any irreducible supermultiplet. The only
requirement imposed on the superfield $U$ is its chirality.
As proved in [21], the chiral superfield $U$ can be expressed in terms of the
chiral projector $\overline{\cal{D}}^{(4)}$ and an unconstrained superfield
${\cal{L}}$ as $U=\overline{\cal{D}}^{(4)}{\cal{L}}$. This result has been
used in sections 3 and 4. According to the discussion of section 4, the main
result of this paper is the computation of the density projector operator
$\Delta^{(4)}$ of (26), (27) and (34), which, together with
$\overline{\cal{D}}^{(4)}$, allows to define the component supergravity
integration measure (34). In deriving for the first time $\Delta^{(4)}$ we
used the Ectoplasmic techniques and the new super 2-form of Theorem 1 (23).
One other point we wish to emphasize is the efficiency of the Ectoplasmic
approach in the case we considered here. It would be interesting to re-derive
the integration measure (34) via the normal coordinate expansion technique [7,
16, 15] (in particular using its last version [15]) even if we do not expect
that the latter approach would require shorter computations. This is
especially true considering that in 2D the number of Bianchi identities to be
solved for a closed super 2-form is relatively low. This once more emphasizes
the important role of forms as a basis for superspace integration theory as
advocated in the ectoplasmic approach. The success of this also points to the
generality of using this as a tool in all cases to derive superspace local
integration measures.
Note also that here we focused on the 2D, ${\cal N}=4$ minimal superspace
geometry of [1] as described in section 2. In general, it is known that there
could exist different off-shell superspace supergravities. We expect that the
Ectoplasm paradigm and the results of our paper can be extended to any
covariant superspace formulation of 2D, ${\cal N}=4$ supergravity. For
example, in the first paper of [1] a variant central charge formulation of the
minimal multiplet was given; once noticed that the Lagrangian ${\cal L}$ in
(34) has to be neutral for the central charges, one can see that the results
of our paper apply without modifications to the variant formulation. Moreover,
recently a new extended covariant formulation of 2D, ${\cal N}=4$ supergravity
in superspace was given [24]. The ectoplasm techniques to compute the chiral
action in components apply straightforward if one consider the geometry of
[24] even if in this case longer computations are expected due to the more
involved structure of the torsion multiplet. Other superspace formulations of
2D, ${\cal N}=4$ supergravity [25] are known in the bi-harmonic superspace of
[26]. Being those superspace supergravities based on a prepotential approach
the definition of a covariant components reduction is not clear. However, on
the ground of the related bi-projective formalism [27], recently extended to
covariantly study 2D, ${\cal N}=4$ matter-couplet supergravity, it would be of
interest and well defined to find by using Ectoplasm techniques the bi-
projective density operator analogously to the chiral action studied here.
“Where the senses fail us, reason must step in.”
\- Galileo Galilei
Acknowledgements
This research was supported in part by the endowment of the John S. Toll
Professorship, the University of Maryland Center for String & Particle Theory,
National Science Foundation Grant PHY-0354401.
Appendix A: Consistency of Bianchi Identities & Constraints For 2-Form
In this appendix, we will present the explicit proof that the Bianchi
identities associated with the results in (23) imply that it is a closed 2D,
$\cal N$ = 4 super 2-form. We begin by writing an ansatz for the lowest
components of a 2D, $\cal N$ = 4 super 2-form under the assumption that these
component should:
(a.) be linear in a (anti)chiral superfield $U$ $(\bar{U})$;
${\overline{\nabla}}_{\alpha}{}^{i}U=0$ $(\nabla_{\alpha i}\bar{U}=0)$,
(b.) depend on the superspace supergravity covariant derivative,
(c.) and are local functions of the superspace supergravity field strengths
$B$, $\overline{B}$, $G$ and $H.$
Under the previous assumptions we will begin with an ansatz given by 999The
ansatz we are using can also be guessed by: (i) consider the flat 4D, $\cal
N$=2 “chiral” closed
super 4-form introduced in [23]; (ii) perform a dimensional reduction of the
4D, $\cal N$=2 super
4-form to derive a 2D, $\cal N$=4 closed super 2-form; (iii) extend the
resulting dimension-1
components of the flat 2D, $\cal N$=4 2-form to the curved case by modifying
the flat derivatives to
the curved covariant derivatives and by adding torsion dependent terms.
$\displaystyle{J_{\alpha i}{}_{\beta
j}=a(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\bar{U}+b\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\bar{U}+C_{\alpha\beta}C_{ij}F\bar{U}~{},~{}~{}~{}}$
---
$None$ $\displaystyle{{J}_{\alpha
i}{}_{\beta}{}^{j}=0~{},~{}~{}~{}~{}~{}~{}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}=-(J_{\alpha
i}{}_{\beta j})^{*}~{},}$
---
$None$
where
$\displaystyle{F=F(B,\bar{B},G,H)=b_{1}B+b_{2}\bar{B}+gG+hH~{},}$
---
$None$
and $a,\,b,\,b_{1},\,b_{2},\,g,\,h$ are constants to be fixed.
The task is to study the Bianchi identities that derive from the closure of
the 2-form $J$
$dJ=0~{},~{}~{}~{}~{}~{}~{}\Longleftrightarrow~{}~{}~{}~{}~{}~{}0={1\over
2}\nabla_{[A}J_{BC)}-{1\over 2}T_{[AB|}{}^{D}J_{D|C)}~{},$ $None$
with $J_{A}{}_{B}=\big{(}\,J_{\alpha i}{}_{\beta j},~{}J_{\alpha
i}{}_{\beta}{}^{j},~{}J_{\alpha}{}^{i}{}_{\beta}{}^{j},~{}J_{\gamma
k}{}_{a},~{}J_{\gamma}{}^{k}{}_{a},~{}J_{ab}\,\big{)}$ and the lowest
components satisfying (A.1) - (A.3).
Substituting the results of (A.1), (A.2) into the identity (A.4) with
$A=\alpha i,\,B=\beta j,\,C=\gamma k$ one obtains
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}0}$ | $\displaystyle{{}=~{}a(\gamma^{3})_{\beta\gamma}[\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{jk}]\overline{U}+a(\gamma^{3})_{\gamma\alpha}[\nabla_{\beta j},{\overline{\nabla}}^{(2)}_{ki}]\overline{U}+a(\gamma^{3})_{\alpha\beta}[\nabla_{\gamma k},{\overline{\nabla}}^{(2)}_{ij}]\overline{U}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}+b\,C_{\beta\gamma}C_{jk}(\gamma^{3})^{\delta\rho}[\nabla_{\alpha
i},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}+b\,C_{\gamma\alpha}C_{ki}(\gamma^{3})^{\delta\rho}[\nabla_{\beta
j},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}}$
|
$\displaystyle{{}~{}~{}~{}~{}+b\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\delta\rho}[\nabla_{\gamma
k},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}+C_{\beta\gamma}C_{jk}\,(\nabla_{\alpha
i}F)\overline{U}+C_{\gamma\alpha}C_{ki}\,(\nabla_{\beta j}F)\overline{U}}$
| $\displaystyle{{}~{}~{}~{}~{}+C_{\alpha\beta}C_{ij}\,(\nabla_{\gamma
k}F)\overline{U}~{},~{}~{}~{}~{}~{}~{}}$
$None$
where we have used the fact that $\bar{U}$ is antichiral to write this. At
this point, there are two useful identities to note
${[}\nabla_{\alpha
i},{\overline{\nabla}}^{(2)}_{ij}{]}\overline{U}~{}=~{}\Big{(}-2{\rm
i}C_{i(j}(\gamma^{c})_{\alpha}{}^{\delta}\nabla_{c}{\overline{\nabla}}_{\delta
k)}\Big{)}\overline{U}$ $None$ ${[}\nabla_{\alpha
i},(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}{]}~{}=~{}\Big{(}-4{\rm
i}\varepsilon^{bc}(\gamma_{b})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta
i}\Big{)}\overline{U}$ $None$
which shows that in principle there are terms containing spacetime derivatives
in (A.5). In order to satisfy the Bianchi identity, two sets of conditions are
required:
| $\displaystyle{{}(a.)~{}~{}a~{}=~{}-2b~{}~{}{\rm{and}}}$
---|---
|
$\displaystyle{{}(b.)~{}~{}b_{1}~{}=~{}b_{2}~{}=~{}g~{}=~{}h~{}=~{}0~{}~{}.}$
$None$
For simplicity we also set
$a=1~{}.$ $None$
The next Bianchi identity encountered takes the form
$\displaystyle{0}$ | $\displaystyle{{}=~{}{\overline{\nabla}}_{\alpha}^{i}J_{\beta j}{}_{\gamma k}+T_{\alpha}{}^{i}{}_{\beta j}{}^{a}J_{\gamma k}{}_{a}+T_{\alpha}{}^{i}{}_{\gamma k}{}^{a}J_{\beta j}{}_{a}~{}~{}~{}.}$
---|---
$None$
The result in (A.1), subject to (A.8), (A.9), can be substituted into this
equation. To satisfy this, it is useful to use the following identities
$\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{jk}\bar{U}}$ | $\displaystyle{{}=~{}-\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{i(j}{\overline{\nabla}}_{\alpha}^{p}{\overline{\nabla}}^{(2)}_{k)p}\overline{U}~{}~{},}$
---|---
$\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\bar{U}}$ | $\displaystyle{{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{\alpha(\beta}{\overline{\nabla}}_{\gamma)}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}-~{}\hbox{\large{${{\textstyle{{4}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,BC_{\alpha(\beta}(\gamma^{3})_{\gamma)}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,B(\gamma^{3})_{(\alpha\beta}{\overline{\nabla}}_{\gamma)i}\overline{U}~{}~{},}$
$\displaystyle{(\gamma^{3})^{\beta\gamma}{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\bar{U}}$ | $\displaystyle{{}=~{}-\,\hbox{\large{${{\textstyle{{2}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,(\gamma^{3})_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}~{}~{}.}$
$None$
Then, to completely satisfy (A.10) one has to impose
$J{}_{\gamma k}{}_{a}~{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\rho}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\bar{U}~{}~{}~{}.$
$None$
Note that it holds
$J{}_{\gamma}{}^{k}{}_{a}~{}=~{}-(J{}_{\gamma
k}{}_{a})^{*}~{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\rho}\nabla_{\rho
p}\nabla^{(2)}{}^{kp}{U}~{}~{}~{}.$ $None$
We can continue our deliberations by considering the Bianchi identity given by
$0~{}=~{}\nabla_{a}J_{\beta j\gamma k}~{}+~{}\nabla_{\beta j}J_{\gamma
k}{}_{a}~{}+~{}\nabla_{\gamma k}J_{\beta j}{}_{a}~{}-~{}T_{a\beta j}{}^{\delta
l}J_{\delta l\gamma k}~{}-~{}T_{a\gamma k}{}^{\delta l}J_{\delta l\beta
j}~{}~{}~{},$ $None$
and into this are substituted the results (A.1), (A.8), (A.9) and (A.12). When
this is done, a differential equation on $\overline{U}$ of the form
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}0~{}=}$ | $\displaystyle{{}~{}\nabla_{a}\Big{(}2(\gamma^{3})_{\beta\gamma}{\overline{\nabla}}^{(2)}_{jk}-\,C_{\beta\gamma}C_{jk}(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\nabla_{\beta
j}\Big{(}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\overline{U}\Big{)}~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\nabla_{\gamma
k}\Big{(}\varepsilon_{ab}(\gamma^{b})_{\beta}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{jp}\overline{U}{\Big{)}}}$
| $\displaystyle{{}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\delta_{j}^{l}\phi_{\beta}{}^{\rho}(\gamma_{a})_{\rho}{}^{\delta}\Big{(}2(\gamma^{3})_{\delta\gamma}{\overline{\nabla}}^{(2)}_{lk}-\,C_{\delta\gamma}C_{lk}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
| $\displaystyle{{}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\delta_{k}^{l}\phi_{\gamma}{}^{\rho}(\gamma_{a})_{\rho}{}^{\delta}\Big{(}2(\gamma^{3})_{\delta\beta}{\overline{\nabla}}^{(2)}_{lj}-\,C_{\delta\beta}C_{lj}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
$None$
emerges. Further progress is possible by using the identity
$\displaystyle{{~{}~{}}\\{\nabla_{\alpha i},{\overline{\nabla}}_{\delta}^{p}{\overline{\nabla}}^{(2)}_{kp}\\}\overline{U}}$ | $\displaystyle{{}=~{}\Bigg{(}3{\rm i}(\gamma^{a})_{\alpha\delta}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}-3{\rm i}C_{ik}(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---|---
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}+~{}\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{ik}{\phi}^{\tau}{}_{\delta}(\gamma^{3})_{\alpha\tau}(\gamma^{3})^{\rho\beta}{\overline{\nabla}}^{(2)}_{\beta\rho}-3\phi_{\alpha\delta}{\overline{\nabla}}^{(2)}_{ik}}$
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}}~{}+~{}6C_{ik}C_{\alpha\delta}\Sigma^{\beta
p}{\overline{\nabla}}_{\beta
p}~{}+~{}6C_{ik}(\gamma^{3})_{\alpha\delta}(\gamma^{3})^{\beta\rho}\Sigma_{\beta}{}^{p}{\overline{\nabla}}_{\rho
p}\Bigg{)}\overline{U}~{}~{}.}$
$None$
This result is substituted into (A.15) and after some algebra, the
$\Sigma$-dependent terms are seen to cancel leaving
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}0\,}$ | $\displaystyle{{}=\Bigg{(}2(\gamma^{3})_{\alpha\gamma}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}~{}-~{}C_{\alpha\gamma}C_{ik}(\gamma^{3})^{\delta\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}-2(\gamma^{3})_{\alpha\gamma}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}+C_{ik}C_{\alpha\gamma}(\gamma^{3})^{\delta\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
| $\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}-~{}{\rm
i}\varepsilon_{ab}\phi_{\beta\delta}(\gamma^{b})_{\alpha\gamma}C^{\beta\delta}{\overline{\nabla}}^{(2)}_{ik}~{}-~{}{\rm
i}\phi_{\beta\delta}(\gamma^{3})^{\beta\delta}(\gamma_{a})_{\alpha\gamma}{\overline{\nabla}}^{(2)}_{ik}}$
| $\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}+~{}{\rm
i}\varepsilon_{ab}\phi_{\alpha^{\prime}\delta}(\gamma^{b})_{\alpha\delta}C^{\alpha^{\prime}\delta}{\overline{\nabla}}^{(2)}_{ik}~{}+~{}{\rm
i}\phi_{\alpha^{\prime}\delta}(\gamma_{c})_{\alpha\delta}(\gamma^{3})^{\alpha^{\prime}\delta}{\overline{\nabla}}^{(2)}_{ik}\Bigg{)}\overline{U}~{}~{}}$
$None$
which is clearly identically satisfied.
There is a second dimension-2 Bianchi identity of the form
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}0\,}$ | $\displaystyle{{}=~{}-~{}{\overline{\nabla}}_{\alpha}{}^{i}J_{\gamma k}{}_{b}~{}-~{}\nabla_{\gamma k}J_{\alpha}{}^{i}{}_{b}~{}+~{}T_{b}{}_{\gamma k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}+~{}T_{b}{}_{\alpha}{}^{i}{}^{\delta l}J_{\delta l}{}_{\gamma k}~{}+~{}T_{\gamma k}{}_{\alpha}{}^{i}{}^{c}J_{cb}~{}~{}.}$
---|---
$None$
One may substitute from results derived previously to cast this into the form
of
$\displaystyle{2{\rm i}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}J_{bc}}$ | $\displaystyle{{}=~{}{\rm i}\Big{(}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{bc}(\gamma^{c})_{\gamma}{}^{\rho}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\,+\,B(\gamma_{b})_{\alpha\gamma}{\overline{\nabla}}^{(2)}{}^{i}{}_{k}\,-\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}B\delta^{i}_{k}(\gamma^{3}\gamma_{b})_{\alpha\gamma}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}~{}~{}~{}-~{}\nabla_{\gamma
k}J_{\alpha}{}^{i}{}_{b}~{}+~{}T_{b}{}_{\gamma
k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}~{},}$
$None$
and progress is achieved in analyzing this identity by noting that it holds
$\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}~{}=~{}\Big{(}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ik}{\overline{\nabla}}^{(2)}_{jk}~{}+~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C^{ik}{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{jk}~{}-~{}2B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\bar{U}~{}~{}~{}.}$
---
$None$
One other identity tells us
$\displaystyle{{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\bar{U}~{}=~{}-2B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}~{}~{}~{},}$
---
$None$
so that (A.20) becomes
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}~{}}$ | $\displaystyle{{}=~{}\Big{(}{\textstyle{1\over\vphantom{2}\smash{\raise 0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ik}{\overline{\nabla}}^{(2)}_{jk}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}}$
---|---
|
$\displaystyle{{}=~{}\Big{(}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\delta_{j}{}^{i}\,C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{kl}{\overline{\nabla}}^{(2)}_{kl}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}}$
|
$\displaystyle{{}=~{}\Big{(}\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\delta_{j}{}^{i}\,C_{\alpha\beta}\,{\overline{\nabla}}^{(4)}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}~{}~{}~{}.}$
$None$
where on the first term we have used a sequence of identities (see also the
final appendix). The final line of (A.22) can now be substituted into (A.19)
to yield after a bit of algebra
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}2{\rm i}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}J_{bc}~{}}$ | $\displaystyle{{}=~{}{\rm i}\Big{(}\,-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\varepsilon_{bc}(\gamma^{c})_{\alpha\gamma}\delta^{i}_{k}{\overline{\nabla}}^{(4)}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}B\varepsilon_{bc}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\,\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}~{}~{}~{}-\nabla_{\gamma
k}J_{\alpha}{}^{i}{}_{b}+T_{b}{}_{\gamma
k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}~{}~{}.}$
$None$
Finally this equation informs us that
$\displaystyle{J_{ab}~{}=~{}\varepsilon_{ab}\Big{(}\,-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}{\overline{\nabla}}^{(4)}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\,\Big{)}\overline{U}~{}+~{}{\rm
h.c.}}$
---
$None$
There remains one final Bianchi Identity of the form
$\displaystyle{{~{}~{}~{}}0~{}}$ | $\displaystyle{{}=~{}\nabla_{\alpha i}J_{bc}-\nabla_{b}J_{\alpha i}{}_{c}~{}+~{}\nabla_{c}J_{\alpha i}{}_{b}~{}+~{}T_{\alpha ib}{}^{D}J_{Dc}~{}+~{}T_{\alpha ic}{}^{D}J_{Db}~{}-~{}T_{bc}{}^{\delta l}J_{\delta l\alpha i}~{}~{}~{},}$
---|---
$\displaystyle{0~{}}$ | $\displaystyle{{}=~{}\varepsilon^{ab}\Big{(}\nabla_{\alpha i}J_{ab}~{}+~{}2\nabla_{a}J_{b\alpha i}~{}-~{}2T_{\alpha ia}{}^{\delta l}J_{\delta l}{}_{b}~{}-~{}2T_{\alpha ia}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{b}~{}-~{}T_{ab}{}^{\delta l}J_{\delta l\alpha i}\Big{)}~{}~{}~{}.}$
$None$
To prove this identity is satisfied requires a calculation of some length. The
key to its satisfaction requires one final identity
$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}}[\nabla_{\alpha i},{\overline{\nabla}}^{(4)}]\bar{U}}$ | $\displaystyle{{}=~{}\Big{(}~{}-~{}\hbox{\large{${{\textstyle{{8{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{ip}~{}-~{}8{\rm i}B\varepsilon_{bc}(\gamma^{b})_{\alpha}{}^{\beta}\nabla^{c}{\overline{\nabla}}_{\beta i}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}+\hbox{\large{${{\textstyle{{8}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\phi_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{p}{\overline{\nabla}}^{(2)}_{ip}~{}+~{}8\Sigma_{\alpha}{}^{l}{\overline{\nabla}}^{(2)}_{il}~{}\Big{)}\overline{U}}$
$None$
that is valid for the supergravity covariant derivative acting on a anti-
chiral scalar superfield such as $\bar{U}$.
Other Bianchi identities, not explicitly mentioned here, are identically
solved by complex conjugation of the results obtained in this section.
Appendix B: Miscellaneous Identities
For the reader convenience, here we also collect some useful formulas used in
the derivations provided in this paper and especially in Appendix A (we remind
that $\bar{U}$ is anti-chiral)
$\displaystyle{{~{}~{}~{}~{}~{}~{}}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{j}\,}$ | $\displaystyle{{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ij}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C^{ij}{\overline{\nabla}}^{(2)}_{\alpha\beta}~{}+~{}BC_{\alpha\beta}C^{ij}{\cal M}~{}-~{}B(\gamma^{3})_{\alpha\beta}{\cal Y}^{ij}~{}~{}~{},}$
---|---
$None$ $\displaystyle{{[}\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{jk}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}-2{\rm i}C_{i(j}(\gamma^{c})_{\alpha}{}^{\delta}\nabla_{c}{\overline{\nabla}}_{\delta k)}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{[}\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{\delta\rho}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}\Big{(}-2{\rm i}(\gamma^{c})_{\alpha(\delta}\nabla_{c}{\overline{\nabla}}_{\rho)i}-G(\gamma^{3})_{\alpha(\delta}(\gamma^{3})_{\rho)}{}^{\gamma}{\overline{\nabla}}_{\gamma i}+GC_{\alpha(\delta}{\overline{\nabla}}_{\rho)i}}$
---|---
| $\displaystyle{{}{~{}~{}~{}~{}~{}~{}\,}-\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}HC_{\alpha(\delta}(\gamma^{3})_{\rho)}{}^{\tau}{\overline{\nabla}}_{\tau
i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}H(\gamma^{3})_{\alpha(\delta}{\overline{\nabla}}_{\rho)i}-{\rm
i}H(\gamma^{3})_{\delta\rho}{\overline{\nabla}}_{\alpha
i}\Big{)}\overline{U}~{},}$
$None$ $\displaystyle{{[}\nabla_{\alpha i},(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}-4{\rm i}\varepsilon^{bc}(\gamma_{b})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta i}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{jk}\overline{U}\,}$ | $\displaystyle{{}=~{}-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{i(j}{\overline{\nabla}}_{\alpha}{}^{p}{\overline{\nabla}}^{(2)}_{k)p}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}{1\over 3}C_{\alpha(\beta}{\overline{\nabla}}_{\gamma)}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}-\hbox{\large{${{\textstyle{{4}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}BC_{\alpha(\beta}(\gamma^{3})_{\gamma)}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}+\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}B(\gamma^{3})_{(\alpha\beta}{\overline{\nabla}}_{\gamma)i}\overline{U}~{},}$
---|---
$None$ $\displaystyle{(\gamma^{3})^{\beta\gamma}{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}-\hbox{\large{${{\textstyle{{2}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}(\gamma^{3})_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{\gamma}{}_{i}{\overline{\nabla}}^{(2)}_{\alpha\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}-{\overline{\nabla}}_{\alpha}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}+4B(\gamma^{3})_{\alpha}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{}\\{\nabla_{\alpha
i},{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\\}\overline{U}=\Bigg{(}3{\rm
i}(\gamma^{a})_{\alpha\delta}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}-3{\rm
i}C_{ik}(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}+\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{ik}{\phi}^{\tau}{}_{\delta}(\gamma^{3})_{\alpha\tau}(\gamma^{3})^{\rho\beta}{\overline{\nabla}}^{(2)}_{\beta\rho}-3\phi_{\alpha\delta}{\overline{\nabla}}^{(2)}_{ik}}$
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}+6C_{ik}C_{\alpha\delta}\Sigma^{\beta
p}{\overline{\nabla}}_{\beta
p}+6C_{ik}(\gamma^{3})_{\alpha\delta}(\gamma^{3})^{\beta\gamma}\Sigma_{\beta}{}^{p}{\overline{\nabla}}_{\gamma
p}\Bigg{)}\overline{U}~{},}$
$None$ $\displaystyle{{~{}}{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}\,}$ | $\displaystyle{{}=~{}-2B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{(2)}{}^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\overline{U}\,}$ | $\displaystyle{{}=~{}-{\overline{\nabla}}^{(2)}{}^{ij}{\overline{\nabla}}^{(2)}_{ij}\overline{U}-4B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{(4)}\overline{U}\,}$ | $\displaystyle{{}:=~{}-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\overline{\nabla}}^{(2)}{}^{kl}{\overline{\nabla}}^{(2)}_{kl}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}\,}$ | $\displaystyle{{}=~{}\Big{(}\,\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}C_{\alpha\beta}\delta^{i}_{j}{\overline{\nabla}}^{(4)}-3B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{i}{}_{j}\Big{)}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$
$\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}\big{(}{\overline{\nabla}}^{(2)}{}^{\gamma\delta}-2{B}(\gamma^{3})^{\gamma\delta}\big{)}{\overline{\nabla}}^{(2)}_{\gamma\delta}\overline{U}\,~{}=~{}0~{},}$
---
$None$ $\displaystyle{[\nabla_{\alpha i},{\overline{\nabla}}^{(4)}]\overline{U}\,}$ | $\displaystyle{{}=~{}\Bigg{(}-\hbox{\large{${{\textstyle{{8{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}(\gamma^{c})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{ik}-8{\rm i}B\varepsilon^{ab}(\gamma_{a})_{\alpha}{}^{\delta}\nabla_{b}{\overline{\nabla}}_{\delta i}}$
---|---
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}}+8\Sigma_{\alpha}{}^{j}{\overline{\nabla}}^{(2)}_{ij}+\hbox{\large{${{\textstyle{{8}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\phi_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{k}{\overline{\nabla}}^{(2)}_{ik}\Bigg{)}\overline{U}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$None$
By complex conjugation, the reader can derive an analogue set of equations for
the chiral superfield $U$.
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* [21] S. J. Gates, Jr. and A. Morrison, J. Phys. A 42, 442002 (2009) [arXiv:0901.4165 [hep-th]].
* [22] S. J. Gates, Jr., Nucl. Phys. B184 (1981) 381.
* [23] T. Biswas and W. Siegel, JHEP 0111, 004 (2001) [arXiv:hep-th/0105084].
* [24] G. Tartaglino-Mazzucchelli, arXiv:0911.2546 [hep-th].
* [25] S. Bellucci and E. Ivanov, Nucl. Phys. B 587 (2000) 445 [arXiv:hep-th/0003154].
* [26] E. Ivanov and A. Sutulin, Nucl. Phys. B 432 (1994) 246 [Erratum-ibid. B 483 (1997) 531] [arXiv:hep-th/9404098]; Class. Quant. Grav. 14, 843 (1997) [arXiv:hep-th/9604186].
* [27] T. Buscher, U. Lindström and M. Roček, Phys. Lett. B 202, 94 (1988); M. Roček, K. Schoutens and A. Sevrin, Phys. Lett. B 265, 303 (1991); U. Lindström, I. T. Ivanov and M. Roček, Phys. Lett. B 328, 49 (1994) [arXiv:hep-th/9401091].
|
arxiv-papers
| 2009-07-30T19:57:11 |
2024-09-04T02:49:04.349654
|
{
"license": "Public Domain",
"authors": "S. James Gates Jr., Gabriele Tartaglino-Mazzucchelli",
"submitter": "Gabriele Tartaglino-Mazzucchelli",
"url": "https://arxiv.org/abs/0907.5264"
}
|
0907.5357
|
# Quantum logic as superbraids of entangled qubit world lines
Jeffrey Yepez 1Air Force Research Laboratory, Hanscom Air Force Base,
Massachusetts 01731
###### Abstract
Presented is a topological representation of quantum logic that views
entangled qubit spacetime histories (or qubit world lines) as a generalized
braid, referred to as a superbraid. The crossing of world lines is purely
quantum in nature, most conveniently expressed analytically with ladder-
operator-based quantum gates. At a crossing, independent world lines can
become entangled. Complicated superbraids are systematically reduced by
recursively applying novel quantum skein relations. If the superbraid is
closed (e.g. representing quantum circuits with closed-loop feedback, quantum
lattice gas algorithms, loop or vacuum diagrams in quantum field theory), then
one can decompose the resulting superlink into an entangled superposition of
classical links. In turn, for each member link, one can compute a link
invariant, e.g. the Jones polynomial. Thus, a superlink possesses a unique
link invariant expressed as an entangled superposition of classical link
invariants.
topological quantum logic, entangled world lines, quantum skein relation,
superbraid, superlink
###### pacs:
03.67.-a,02.10.Kn,03.67.Lx
††preprint: Version 1.0
In topological quantum computing Zanardi and Lloyd (2003); Nayak et al.
(2008), a quantum gate operation derives from braiding quasiparticles, e.g.
two Majoranna zero-energy vortices made of entangled Cooper-pair states in a
$p+ip$ superconductor where the vortex-vortex phase interaction has a non-
abelian SU(2) gauge group Ivanov (2001); Tewari et al. (2007). Dynamically
braiding such quantum vortices (point defects in a planar cross-section of the
condensate) induces phase shifts in the quantum fluid’s multiconnected wave
function. Local nonlinear interactions between deflects (vortex-vortex
straining) is otherwise neglected, i.e. the separation distance $\delta$ of
the zero mode vortices is much greater then the vortex core size, which scales
as the coherence length $\xi\ll\delta$ in quantum fluids. The braiding occurs
adiabatically so the quantum fluid remains in local equilibrium and the number
of deflects (qubits) remains fixed. For implementations, the usual question is
how can quantum logic gates, and in turn quantum algorithms, be represented by
braiding deflects, quasiparticles with a nonabelian gauge group.
This Rapid Communication addresses the related fundamental question about the
relationship between quantum entanglement, tangled strands, and quantum logic
Kauffman and Lomonaco (2004). How can a quantum logic gate, and in turn a
quantum algorithm, be decomposed into a linear combination (entangled
superposition) of classical braid operators? The goal is to comprehend and
categorize quantum algorithms topologically. This is done by first viewing a
quantum gate as a braid of two qubit spacetime histories or world lines.
Qubit-qubit interaction associated with a quantum gate is rendered as a tree-
level scattering diagram, a form of ribbon graph. Then, a quantum algorithm is
represented as a weave of such graphs, a superbraid of qubit world lines.
Finally, one closes a superbraid to form a superlink. In fact, quantum lattice
gas algorithms, e.g. employed for the simulation of superfluids themselves
Yepez et al. (2009), are a good superlink archetype, hence the shared
nomenclature.
With this technology we can calculate superlink invariants (Laurent series in
linear combination). In principle, each algorithm has its own unique linear
combination of invariant Laurent polynomials; e.g. two competing quantum
circuit implementations of a particular algorithm can be judged equivalent,
irrespective of circuit schemes and number of gates and wires. If two quantum
algorithms, first and second-order accurate, are topologically equivalent,
then the simpler one can be used for analytical predictions of their common
effective theory while the latter for faster simulations on smaller grids.
In short, presented is a quantum generalization of the Temperley-Lieb algebra
TLQ and Artin braid group $B_{Q}$: a superbraid and its closure, a superlink,
is formed out of the world lines of $Q$ qubits (strands) undergoing dynamics
generated by quantum gates. Furthermore, the superbraid representation of
quantum dynamics works equally well for either bosonic or fermionic quantum
simulations. There exists a classical limit where the quantum Temperley-Lieb
algebra and the superbraid group, defined below, reduce to the usual
Temperley-Lieb algebra and braid group.
Classical braid operators (nearest neighbor permutations), represented in
terms of Temperley-Lieb algebra Temperley and Lieb (1971), were originally
discovered in six-vertex Potts models and statistical mechanical treatments of
two-dimensional lattice systems Baxter (1982); Levy (1990). The quantum
algorithm to compute the Jones polynomial Aharonov et al. (2006); Kauffman and
Samuel J. Lomonaco (2007) employs unitary gate operators that are mapped to
unitary representations of the braid group, i.e. generated by hermitian
representations of the Temperley-Lieb algebra. To prepare for our presentation
of superbraids as a novel topological representation of the quantum logic
underlying quantum information dynamics, let us first briefly mention some
basics of knot theory and some basic quantum gate technology using qubit
ladder operators.
A link comprising $Q$ strands, denoted by $L$ say, is the closure of a braid.
The Jones polynomial $V_{L}(A)$ is an invariant of $L$ Jones (1985), where $A$
is a complex parameter associated with the link whose physical interpretation
will be presented below. $V_{L}(A)$ is a Laurent series in $A$. The Jones
polynomial is defined for a link embedded in three space–an oriented link. One
projects $L$ onto a plane. In the projected image, in general crossing of
strands occurs but is disambiguated by its sign $\pm 1$, i.e. one assigns
over-crossings the sign of $+1$ and under-crossings $-1$. The writhe $w(L)$ is
sum of the signs of all the crossings, i.e. the net sign of a link’s planar
projection. The Jones polynomial is computed as follows
$V_{L}(A)=\frac{1}{d}\,(-A^{3})^{w(L)}\,K_{L}(A),$ (1)
where $K_{L}(A)$ is the Kauffman bracket of the link. $K_{L}(A)$ is determined
by summing over all possible planar projections of $L$. In the simplest case
of an unknotted link (or unknot), the Kauffman bracket is
$\textstyle{=K_{\bigcirc}(A)=d=-A^{2}-A^{-2}.}$ (2)
The Kauffman bracket of a disjoint union of $n$ unknots has the value $d^{n}$,
e.g. $\textstyle{\ =d^{2}.}$
$K_{L}(A)$ for a link with crossings can be computed recursively using a skein
relation that equates it to the weighted sum of two links, each with one less
crossing:
$\textstyle{=A}$$\textstyle{+\ A^{-1}}$ (3a) $\textstyle{=A}$$\textstyle{+\
A^{-1}}$ (3b)
where $A$ and its inverse are the weighting factors. As an example, let us
recursively apply (3) to prove an intuitively obvious link identity
$\textstyle{=}$ . One reduces the relevant braid as follows
$\displaystyle\stackrel{{\scriptstyle(\ref{hcross_neg_rule})}}{{=}}$
$\textstyle{A}$ $\textstyle{+\ A^{-1}}$ (4a)
$\displaystyle\stackrel{{\scriptstyle(\ref{hcross_rule})}}{{=}}$
$\textstyle{A^{2}}$$\textstyle{+}$$\textstyle{+\ A^{-1}}$ (4b)
$\displaystyle\stackrel{{\scriptstyle(\ref{hcross_rule})}}{{=}}$
$\textstyle{A^{2}}$$\textstyle{+}$$\textstyle{+}$$\textstyle{+\ A^{-2}}$ (4c)
$\displaystyle\stackrel{{\scriptstyle(\ref{unknot_rule})}}{{=}}$
$\textstyle{A^{2}}$$\textstyle{+}$$\textstyle{\ +\ d}$$\textstyle{+\ A^{-2}}$
(4d) $\displaystyle=$ $\textstyle{+\left(d+A^{2}+A^{-2}\right)}$ (4e)
$\displaystyle\stackrel{{\scriptstyle(\ref{unknot_rule})}}{{=}}$ (4f)
A quantum gate represents the qubit-qubit coupling that occurs at the crossing
of world lines of a pair of qubits, say $|q_{\alpha}\rangle$ and
$|q_{\gamma}\rangle$ in a system of $Q$ qubits. Every quantum gate is
generated by an hermitian operator, ${\cal E}_{\alpha\gamma}$ say, and whose
action on the quantum state may be expressed as
$|\dots q_{\alpha}\dots q_{\gamma}\dots\rangle^{\prime}=e^{i\zeta{\cal
E}_{\alpha\gamma}}|\dots q_{\alpha}\dots q_{\gamma}\dots\rangle,$ (5)
where $\zeta$ is a real parameter. The archetypal case considered here is
${\cal E}_{\alpha\gamma}^{2}={\cal E}_{\alpha\gamma}$; the generator is
idempotent.
Suppose the system of qubits is employed to model the quantum dynamics of
fermions or bosons. Is there an analytical form of the generator ${\cal
E}_{\alpha\gamma}$ that allows one to easily distinguish between the two
cases? It is natural to begin by treating fermion statistics. With the logical
one state of a qubit ${\scriptsize|1\rangle=\begin{pmatrix}0\\\
1\end{pmatrix}},$ notice that $\sigma_{z}|1\rangle=-|1\rangle,$ so one can
count the number of preceding bits that contribute to the overall phase shift
due to fermionic bit exchange involving the $\gamma$th qubit with tensor
product operator,
$\sigma_{z}^{\otimes\gamma-1}|\psi\rangle=(-1)^{N_{\gamma}}|\psi\rangle.$ The
phase factor is determined by the number of bit crossings
$N_{\gamma}=\sum_{k=1}^{\gamma-1}n_{k}$ in the state $|\psi\rangle$ and where
the boolean number variables are $n_{k}\in[0,1]$. Hence, an annihilation
operator is decomposed into a tensor product known as the Jordan-Wigner
transformation Jordan and Wigner (1928)
$a_{\gamma}=\sigma_{z}^{\otimes\gamma-1}\otimes\,a\otimes\bm{1}^{\otimes
Q-\gamma}$ (6)
for integer $\gamma\in[1,Q]$ and here the singleton operator is
$a=\left.\frac{1}{2}\middle(\sigma_{x}+i\sigma_{y}\right)$, where $\sigma_{i}$
for $i=x,y,z$ are the Pauli matrices. See page 17 of Fetter and Walecka (1971)
for the usual development for determining $N_{\gamma}$. (6) and its transpose,
the creation operator $a^{\dagger}_{\gamma}=a_{\gamma}^{\text{\tiny T}}$,
satisfy the anti-commutation relations
$\displaystyle\\{a_{\gamma},a^{\dagger}_{\beta}\\}$ $\displaystyle=$
$\displaystyle\delta_{\gamma\beta},\quad\\{a_{\gamma},a_{\beta}\\}=0,\quad\\{a^{\dagger}_{\gamma},a^{\dagger}_{\beta}\\}=0.\quad$
(7)
The hermitian generator of a quantum gate can be analytically expressed in
terms of qubit creation and annihilation operators. A novel generator that is
manifestly hermitian is the following
$\begin{split}{\cal
E}_{\Delta\alpha\gamma}&=d^{-1}\Big{[}-A^{2}\,n_{\alpha}-A^{-2}\,n_{\gamma}\\\
&+i\,\left(e^{i\xi}a^{\dagger}_{\alpha}a_{\gamma}-e^{-i\xi}a^{\dagger}_{\gamma}a_{\alpha}\right)+d\,(\Delta-1)n_{\alpha}n_{\gamma}\Big{]},\end{split}$
(8)
where $d=-A^{2}-A^{-2}$ is real, and $\xi$ is an internal e-bit phase angle.
The parameter $\Delta$ is boolean, and it allows one to select between
fermionic ($\Delta=1$) or bose ($\Delta=0$) statistics of the modeled quantum
particles.
The coefficients in (8) can be parameterized by a real angle $\mu$: ${\cal
E}_{\Delta\alpha\gamma}={\cal E}_{\Delta\alpha\gamma}(\mu)$ with
$A^{2}=-\frac{\cos\mu+1}{\sin\mu}$ and $d=2\csc\mu$. The quantum logic gate
generated by ${\cal E}_{\Delta\alpha\gamma}$ is
$e^{i\zeta\,{\cal E}_{\Delta\alpha\gamma}}=\mathbf{1}^{\otimes\text{\tiny\it
Q}}+(e^{i\zeta}-1){\cal E}_{\Delta\alpha\gamma}.$ (9)
The state evolution (5) by the quantum logic gate (9) can be understood as
scattering between two qubits
$\displaystyle|\psi^{\prime}\rangle=e^{i\zeta\,{\cal
E}_{\Delta\alpha\gamma}}|\psi\rangle\quad$ $\displaystyle\Longleftrightarrow$
$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
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(10a) $\displaystyle|\psi\rangle=e^{-i\zeta\,{\cal
E}_{\Delta\alpha\gamma}}|\psi^{\prime}\rangle\quad$
$\displaystyle\Longleftrightarrow$
$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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(10b)
where the “gauge field” that couples the external qubit world-lines is
represented by an internal double wavy line (or ribbon). The external lines
either over-cross or under-cross and are assigned $+1$ and $-1$ multiplying
the action, i.e. $\pm\zeta{\cal E}_{\Delta}$. This sign disambiguates between
a quantum gate and its adjoint, respectively, as shown in (10a) and (10b). Let
us denote a qubit graphically $|q_{\alpha}\rangle\equiv
u_{\alpha}\uparrow+d_{\alpha}\downarrow$, with complex amplitudes constrained
by conservation of probability $|u_{\alpha}|^{2}+|d_{\alpha}|^{2}=1$.
Starting, for example, with a separable input state
$|\psi\rangle=|q_{\alpha}\rangle|q_{\gamma}\rangle$, a scattering diagram is a
quantum superposition of four oriented graphs
$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.28076pt\hbox{\ignorespaces{}{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.06467pt\raise 0.64874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\kern 6.0pt\vrule
height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.28076pt\raise
6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.00589pt\raise-5.73688pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.70251pt\raise 8.12135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$}}}}}}}}}+d_{\alpha}d_{\gamma}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 17.54016pt\hbox{{\hbox{\kern-17.54016pt\raise
1.42262pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
0.28076pt\hbox{\ignorespaces{}{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.06467pt\raise 0.64874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\kern 6.0pt\vrule
height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.28076pt\raise
6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.00589pt\raise-5.73688pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.70251pt\raise 8.12135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$}}}}}}}}}.\end{split}$ (11)
Each oriented scattering graph can be reduced to a quantum superposition of
classical graphs, or just a single classical graph, as the case may be. There
are four quantum skein relations representing dynamics generated by (8)
$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
0.28076pt\hbox{\ignorespaces{}{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.06467pt\raise 0.64874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\kern 6.0pt\vrule
height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.28076pt\raise
6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.00589pt\raise-5.73688pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.70251pt\raise 8.12135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$ $\displaystyle=$
$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
0.0pt\hbox{\kern-16.05917pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}\ignorespaces{\hbox{\kern
16.05917pt\raise-3.00307pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\ignorespaces\hbox{\kern-0.55489pt\raise
0.83192pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{}{{}{{}{{}}{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}\ignorespaces{\hbox{\kern
31.9409pt\raise-3.00307pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\ignorespaces\hbox{\kern 0.56047pt\raise
0.82817pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}$
(12a) $\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
0.28076pt\hbox{\ignorespaces{}{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.06467pt\raise 0.64874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\kern 6.0pt\vrule
height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.28076pt\raise
6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.00589pt\raise-5.73688pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.70251pt\raise 8.12135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$ $\displaystyle=$
$\frac{-A^{2}-A^{-2}e^{i\zeta}}{d}$$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
0.0pt\hbox{\kern-16.05917pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}\ignorespaces{\hbox{\kern
31.9409pt\raise-14.99702pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\ignorespaces\hbox{\kern
0.56047pt\raise-0.82817pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{}{{}{{}{{}}{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}\ignorespaces{\hbox{\kern
16.05917pt\raise-3.00307pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\ignorespaces\hbox{\kern-0.56047pt\raise
0.82817pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces}}}}}}$$\
+\frac{ie^{-i\xi}\left(e^{i\zeta}-1\right)}{d}$
$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
0.0pt\hbox{\kern-18.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{}{{}{{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}\ignorespaces{\hbox{\kern
32.20499pt\raise-3.47339pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\ignorespaces\hbox{\kern-0.43806pt\raise-0.89894pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{{}}{}{}{{}{{}}}{\hbox{\kern
24.5pt\raise-9.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{\hbox
to5.0pt{\hfill\vrule
height=2.5pt,depth=2.5pt,width=0.0pt}}$}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}{}{{}}{}{}{{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{}{{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{}}}}}}}}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}{{}}{}{}{{{}}{{{}}{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}}}}}}}}}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}\ignorespaces{\hbox{\kern
32.20499pt\raise-14.52669pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\ignorespaces\hbox{\kern-0.44301pt\raise
0.89651pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}\kern
3.0pt}}}}}}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces}}}}}}$
(12b) $\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
0.28076pt\hbox{\ignorespaces{}{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.06467pt\raise 0.64874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\kern 6.0pt\vrule
height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.28076pt\raise
6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
0.00589pt\raise-5.73688pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.70251pt\raise 8.12135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$ $\displaystyle=$
$\frac{-A^{2}e^{i\zeta}-A^{-2}}{d}$$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
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0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{}{{}{{}{{}}{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}\ignorespaces{\hbox{\kern
16.05917pt\raise-3.00307pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{}{}{}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{}{}{{}{}{{}}{}{}{{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{{{}}{}{{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{}}}}}}}}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}}}}}}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{{}{}{{}}{}{}{{{}}{{{}}{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}{{{}}}{{}}{{{}}{{{}}{{{}}{{{}}{{{}}}{{}}{{{}}}{{}}}}}}}}}}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}}{{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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(12c) $\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.4107pt\raise-5.96811pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}$ $\displaystyle=$
$\displaystyle\text{\scriptsize$1+\left(e^{i\zeta}-1\right)\Delta$}\lx@xy@svg{\hbox{\raise
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0.0pt\hbox{{}{}{}{{}}\ignorespaces\ignorespaces{{}{}{}{}\ignorespaces{}\lx@xy@spline@}{{}{}{}\ignorespaces{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{{}{}}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}}}}\ignorespaces{\hbox{\kern
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(12d)
These are the quantum analog of (3). Adjoint quantum skein relations are
obtained simply by taking $\zeta\rightarrow-\zeta$ in the amplitudes in the
diagrams in (12). All superbraids can be reduced to a quantum superposition of
classical braids. The closure of a superbraid forms a superlink. Hence, a
superlink can be reduced to a quantum superposition of classical links, and
consequently, for each superlink one can compute a superlink invariant, for
example a superposition of Jones polynomials.
In the context of quantum information dynamics, a physical interpretation of
the parameter $A$ can be rendered as follows. If the strands in $L$ are
considered closed spacetime histories of $Q$ qubits (e.g. qubit states
evolving in a quantum circuit with closed-loop feedback), then the L.H.S. of
(12) represent a trajectory configuration within a piece of the superlink
where entanglement is generated by a qubit-qubit coupling that occurs at a
quantum-gate (i.e. generalized crossing point). For the 1-body cases (12b) and
(12c), the R.H.S. represents classical alternatives in quantum superposition:
$d^{-1}(-A^{2}-A^{-2}e^{i\zeta})$ is the amplitude for no interaction (non-
swapping of qubit states) whereas the amplitude of a swap interaction
(interchanging of qubit states) goes as $d^{-1}(e^{i\zeta}-1)$.
As an example of reducing a superbraid, let us recursively apply (12) to prove
an obvious evolution identity: the composition of a quantum gate with its
adjoint is the identity operator, i.e. $UU^{\dagger}=1$. For simplicity, we
start with $|q_{\alpha}\rangle=\uparrow$ and $|q_{\gamma}\rangle=\downarrow$,
so the initially oriented superbraid is reduced to a superposition of
classical braids as follows:
U$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise 0.0pt\hbox{\kern
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6.55443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}\ignorespaces{{}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}}}}}}}$U†$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
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0.0pt\hbox{{}{{}{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
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$\displaystyle\stackrel{{\scriptstyle(\ref{quantum_skein_relations_b})}}{{=}}$
$\frac{-A^{2}-A^{-2}e^{i\zeta}}{d}$$\textstyle{\mathord{\lx@xy@svgnested{\hbox{\raise
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(13)
$\displaystyle\stackrel{{\scriptstyle(\ref{quantum_skein_relations_b})^{\dagger}}}{{=}}$
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It is easy to verify the same result occurs for inputs
$|q_{\alpha}\rangle=\downarrow$ and $|q_{\gamma}\rangle=\uparrow$.
Furthermore, the identity trivially follows for $|q_{\alpha}\rangle=\uparrow$
and $|q_{\gamma}\rangle=\uparrow$ and for $|q_{\alpha}\rangle=\downarrow$ and
$|q_{\gamma}\rangle=\downarrow$ since $\Delta$ is boolean.
With adjacent indices, e.g. $\gamma=\alpha+1$ in (9), we need write the first
index only (i.e. suppress the second indice), ${\cal
E}_{\Delta\alpha}\equiv{\cal E}_{\Delta\alpha,\alpha+1}.$ Using this
compressed notation, (8) satisfies the following quantum Temperley-Lieb
algebra
$\displaystyle{\cal E}_{\Delta\alpha}^{2}$ $\displaystyle=$
$\displaystyle{\cal E}_{\Delta\alpha},\ \ \qquad\alpha=1,2,\dots,Q-1\qquad$
(14a) $\displaystyle{\cal E}_{\Delta\alpha}{\cal E}_{\Delta\alpha\pm 1}{\cal
E}_{\Delta\alpha}$ $\displaystyle-$ $\displaystyle{\cal E}_{\Delta\alpha\pm
1}{\cal E}_{\Delta\alpha}{\cal E}_{\Delta\alpha\pm 1}=$ (14b) $\displaystyle+$
$\displaystyle d^{-2}{\cal E}_{\Delta\alpha}-d^{-2}{\cal E}_{\Delta\alpha\pm
1}$ $\displaystyle{\cal E}_{\Delta\alpha}{\cal E}_{\Delta\beta}$
$\displaystyle=$ $\displaystyle{\cal E}_{\Delta\beta}{\cal
E}_{\Delta\alpha},\quad|\alpha-\beta|\geq 2.$ (14c)
To help understand this algebra, we may write (14b) as follows
$\displaystyle{\cal E}_{\Delta\alpha}{\cal E}_{\Delta\alpha+1}{\cal
E}_{\Delta\alpha}-d^{-2}{\cal E}_{\Delta\alpha}$ $\displaystyle=$
$\displaystyle d^{-2}X_{\alpha,\alpha+1}\,$ (15a) $\displaystyle{\cal
E}_{\Delta\alpha+1}{\cal E}_{\Delta\alpha}{\cal
E}_{\Delta\alpha+1}-d^{-2}{\cal E}_{\Delta\alpha+1}$ $\displaystyle=$
$\displaystyle d^{-2}Y_{\alpha,\alpha+1},\quad$ (15b)
where $X_{\alpha,\alpha+1}$ and $Y_{\alpha,\alpha+1}$ are introduced solely
for the purpose of separating (14b) into two equations. For (15) to be
equivalent to (14b), one must demonstrate that
$X_{\alpha,\alpha+1}=Y_{\alpha,\alpha+1}$. Inserting (8) into the L.H.S. of
(15), after considerable ladder operator algebra, one finds that the
difference of the R.H.S. of (15) is
$\displaystyle X_{\alpha,\alpha+1}-Y_{\alpha,\alpha+1}$ $\displaystyle=$
$\displaystyle\Delta(\Delta-1)\big{[}(A^{4}-A^{-4})n_{\alpha}n_{\alpha+1}n_{\alpha+2}$
(16) $\displaystyle-$ $\displaystyle
A^{4}n_{\alpha}n_{\alpha+1}+A^{-4}n_{\alpha+1}n_{\alpha+2}\big{]},$
vanishing for boolean $\Delta$. Thus, (14b) follows from (8).
One finds $X$ and $Y$ are proportional to $\Delta$, so a remarkable reduction
of (14) occurs for the $\Delta=0$ case:
$\displaystyle{\cal E}_{0\alpha}^{2}$ $\displaystyle=$ $\displaystyle{\cal
E}_{0\alpha},\ \qquad\alpha=1,2,\dots,Q-1\qquad$ (17a) $\displaystyle{\cal
E}_{0\alpha}{\cal E}_{0\alpha\pm 1}{\cal E}_{0\alpha}$
$\displaystyle\stackrel{{\scriptstyle(\ref{interleaving_separation})}}{{=}}$
$\displaystyle d^{-2}{\cal E}_{0\alpha}$ (17b) $\displaystyle{\cal
E}_{0\alpha}{\cal E}_{0\beta}$ $\displaystyle=$ $\displaystyle{\cal
E}_{0\beta}{\cal E}_{0\alpha},\quad|\alpha-\beta|\geq 2.$ (17c)
This is the Temperley-Lieb algebra over a system of $Q$ qubits (TLQ). Thus,
entangled bosonic states generated by ${\cal E}_{0\alpha}$ are isomorphic to
links generated by ${\cal E}_{0\alpha}$. So (14) is a generalization of TLQ.
We now consider the generalized braid that it generates: a superbraid.
A general superbraid operator is an amalgamation of both a classical braid
operator and a quantum gate
$b^{\text{s}}_{\Delta\alpha\beta}\equiv A\,e^{z\,{\cal
E}_{\Delta\alpha\beta}},$ (18)
where $A$ and $z$ are complex parameters. (18) can be applied to any two
qubits, $\alpha$ and $\beta$, in a system of qubits (i.e. we do not impose a
restriction to the adjacency case when $\beta=\alpha+1$). (18) can be written
in several different ways, each way useful in its own right.
Letting $z\equiv i\zeta+\ln\tau$, the superbraid operator has the following
exponential form
$\displaystyle b^{\text{s}}_{\Delta\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle\tau^{4}\,e^{(i\zeta+\ln\tau)\,{\cal
E}_{\Delta\alpha\beta}}=\tau^{4}\left(e^{i\zeta}\tau\right)^{{\cal
E}_{\Delta\alpha\beta}},\qquad$ (19a)
where $\tau^{4}\equiv A$. The superbraid operator can be written linearly in
its generator
$\displaystyle b^{\text{s}}_{\Delta\alpha\beta}$ $\displaystyle=$
$\displaystyle A\left[\mathbf{1}_{\text{\tiny\it
Q}}+(A^{-4}\,e^{i\zeta}-1){\cal E}_{\Delta\alpha\beta}\right]$ (20a)
$\displaystyle=$ $\displaystyle A\,\mathbf{1}_{\text{\tiny\it
Q}}+A^{-1}d\left(\frac{1-e^{i\zeta}\tau}{1+\tau}\right){\cal
E}_{\Delta\alpha\beta}.\qquad$ (20b)
A non-trivial classical limit of quantum logic gates represented as (9) occurs
at $\zeta=\pi$ (swap operator). Consequently, the superbraid operator in
product form is
$\displaystyle b^{\text{s}}_{\Delta\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle\tau^{4}\,e^{(\ln\tau+i\pi)\,{\cal
E}_{\Delta\alpha\beta}}\,e^{(i\zeta-i\pi)\,{\cal E}_{\Delta\alpha\beta}}$
(21a) $\displaystyle=$ $\displaystyle
b_{\Delta\alpha\beta}\,e^{i(\zeta-\pi)\,{\cal E}_{\Delta\alpha\beta}},$ (21b)
where $b_{\Delta\alpha\beta}=\tau^{4}\,e^{(\ln\tau+i\pi)\,{\cal
E}_{\Delta\alpha\beta}}$ is the conventional braid operator. (21b) is useful
for comprehending the physical behavior of the superbraid operator. It
classically braids world lines $\alpha$ and $\beta$ and quantum mechanically
entangles these world lines according to the deficit angle $\zeta-\pi$.
The superbraid group is defined by
$\displaystyle b^{\text{s}}_{\alpha}\,b^{\text{s}}_{\beta}$ $\displaystyle=$
$\displaystyle b^{\text{s}}_{\beta}\,b^{\text{s}}_{\alpha},\ \
\,\quad\text{for}\quad|\alpha-\beta|>1\quad$ (22a) $\displaystyle
b^{\text{s}}_{\alpha}\,b^{\text{s}}_{\alpha+1}\,b^{\text{s}}_{\alpha}+\gamma\,b^{\text{s}}_{\alpha}$
$\displaystyle=$ $\displaystyle
b^{\text{s}}_{\alpha+1}b^{\text{s}}_{\alpha}b^{\text{s}}_{\alpha+1}+\gamma\,b^{\text{s}}_{\alpha+1},$
$\displaystyle\ \quad\qquad\quad\text{for}\quad 1\leq\alpha<Q,\qquad$
where $\gamma$ is a constant that depends on the representation. For (8), we
have
$\gamma=\left(A^{4}+A^{-4}e^{i\zeta}\middle)\middle(1+e^{i\zeta}\right)A^{-2}d^{-2}.$
In the classical limit $\zeta=\pi$, the superbraid operator reduces to the
classical braid operator, $b_{\alpha}\equiv b^{\text{s}}_{\alpha}(\pi,\tau)$,
and (22) reduces to the Artin braid group
$\displaystyle b_{\alpha}\,b_{\beta}$ $\displaystyle=$ $\displaystyle
b_{\beta}\,b_{\alpha},\ \ \,\qquad\quad\text{for}\quad|\alpha-\beta|>1$ (23a)
$\displaystyle b_{\alpha}\,b_{\alpha+1}\,b_{\alpha}$ $\displaystyle=$
$\displaystyle b_{\alpha+1}b_{\alpha}b_{\alpha+1},\quad\text{for}\quad
1\leq\alpha<Q.\qquad$ (23b)
(23) follows from (22) because $\gamma=0$ for $\zeta=\pi$. Also, in this
classical limit, (20a) reduces to the braid operator
$b_{\alpha}=A\,\mathbf{1}_{\text{\tiny\it Q}}+A^{-1}d\,{\cal
E}_{\Delta\alpha\beta},$ for $\alpha=1,2,\dots,Q-1$. After some ladder
operator algebra, one finds that
$\begin{split}&b_{\alpha}b_{\alpha+1}b_{\alpha}-b_{\alpha+1}b_{\alpha}b_{\alpha+1}=\\\
&A^{-1}(A^{4}-A^{-4})d^{-2}\Delta(\Delta-1)\,(1-n_{\alpha})n_{\alpha+1}n_{\alpha+2},\end{split}$
(24)
where $n_{\alpha}\equiv a^{\dagger}_{\alpha}a_{\alpha}$. Since $\Delta$ is
boolean, the R.H.S. vanishes, and this is just (23b).
## References
* Zanardi and Lloyd (2003) P. Zanardi et al., Phys. Rev. Lett. 90, 067902 (2003).
* Nayak et al. (2008) C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008)
* Ivanov (2001) D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).
* Tewari et al. (2007) S. Tewari et al., Phys. Rev. Lett. 98, 010506 (2007).
* Kauffman and Lomonaco (2004) L. Kauffman et al. New J. Phys. 6, 134 (2004).
* Yepez et al. (2009) J. Yepez et al., Phys. Rev. Lett. (2009). To appear.
arXiv: quant-ph/0905.0159v1
* Temperley and Lieb (1971) H. N. V. Temperley and E. H. Lieb, Proc. Roy. Soc. Lond. A. Math. Phys. Sci. 322, 251 (1971).
* Baxter (1982) R. J. Baxter, _Exactly Solved Models in Statistical Mechanics_ (Academic Press, London, 1982).
* Levy (1990) D. Levy, Phys. Rev. Lett. 64, 499 (1990).
* Aharonov et al. (2006) D. Aharonov, V. Jones, Z. Landau, in _STOC ’06: Proc. 38th ACM Sym. Theory of Comp._ 2006), pp. 427–436.
* Kauffman and Samuel J. Lomonaco (2007) L. H. Kauffman et al., in _Quantum information theory_ , (SPIE, 2007), vol. 6573, p. 65730T.
* Jones (1985) V. Jones, Bull., New Ser., Am. Math. Soc. 12, 103 (1985).
* Jordan and Wigner (1928) P. Jordan and E. Wigner, Zeits. Physik A 47, 631 (1928).
* Fetter and Walecka (1971) A. L. Fetter and J. D. Walecka, _Quantum Theory of Many-Particle Systems_ , (McGraw-Hill Book Company, New York, 1971).
|
arxiv-papers
| 2009-07-30T14:23:10 |
2024-09-04T02:49:04.360888
|
{
"license": "Public Domain",
"authors": "Jeffrey Yepez",
"submitter": "Jeffrey Yepez",
"url": "https://arxiv.org/abs/0907.5357"
}
|
0907.5458
|
# Interacting holographic dark energy in Brans-Dicke theory
Ahmad Sheykhi 111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
We study cosmological application of interacting holographic energy density in
the framework of Brans-Dicke cosmology. We obtain the equation of state and
the deceleration parameter of the holographic dark energy in a non-flat
universe. As system’s IR cutoff we choose the radius of the event horizon
measured on the sphere of the horizon, defined as $L=ar(t)$. We find that the
combination of Brans-Dicke field and holographic dark energy can accommodate
$w_{D}=-1$ crossing for the equation of state of noninteracting holographic
dark energy. When an interaction between dark energy and dark matter is taken
into account, the transition of $w_{D}$ to phantom regime can be more easily
accounted for than when resort to the Einstein field equations is made.
## I Introduction
Recent data from type Ia supernova, cosmic microwave background (CMB)
radiation, and other cosmological observations suggest that our universe is
currently experiencing a phase of accelerated expansion and nearly three
quarters of the universe consists of dark energy with negative pressure Rie .
Nevertheless, the nature of such a dark energy is still the source of much
debate. Despite the theoretical difficulties in understanding dark energy,
independent observational evidence for its existence is impressively robust.
Explanations have been sought within a wide range of physical phenomena,
including a cosmological constant, exotic fields, a new form of the
gravitational equation, new geometric structures of spacetime, etc, see Pad
for a recent review. One of the dramatic candidate for dark energy, that arose
a lot of enthusiasm recently, is the so-called “Holographic Dark Energy” (HDE)
proposal. This model is based on the holographic principle which states that
the number of degrees of freedom of a physical system should scale with its
bounding area rather than with its volume Suss1 and it should be constrained
by an infrared cutoff Coh . On these basis, Li Li suggested the following
constraint on its energy density $\rho_{D}\leq 3c^{2}m^{2}_{p}/L^{2}$, the
equality sign holding only when the holographic bound is saturated. In this
expression $c^{2}$ is a dimensionless constant, $L$ denotes the IR cutoff
radius and $m^{2}_{p}=(8\pi G)^{-1}$ stands for the reduced Plank mass. Based
on cosmological state of holographic principle, proposed by Fischler and
Susskind Suss2 , the HDE models have been proposed and studied widely in the
literature Huang ; Hsu ; HDE ; Setare ; Seta1 ; Setare1 . The HDE model has
also been tested and constrained by various astronomical observations Xin ;
Feng as well as by the anthropic principle Huang1 . It is fair to claim that
simplicity and reasonability of HDE model provides more reliable frame to
investigate the problem of dark energy rather than other models proposed in
the literature. For example, the coincidence problem can be easily solved in
some models of HDE based on the fundamental assumption that matter and HDE do
not conserve separately Pav1 .
On the other side, scalar-tensor theories of gravity have been widely applied
in cosmology Fara . Scalar-tensor theories are not new and have a long
history. The pioneering study on scalar-tensor theories was done by Brans and
Dicke several decades ago who sought to incorporate Mach’s principle into
gravity BD . In recent years this theory got a new impetus as it arises
naturally as the low energy limit of many theories of quantum gravity such as
superstring theory or Kaluza-Klein theory. Because the holographic energy
density belongs to a dynamical cosmological constant, we need a dynamical
frame to accommodate it instead of general relativity. Therefore it is
worthwhile to investigate the HDE model in the framework of the Brans-Dicke
theory. The studies on the HDE model in the framework of Brans-Dicke cosmology
have been carried out in Pavon2 ; Setare2 ; other . The purpose of the present
paper is to construct a cosmological model of late acceleration based on the
Brans-Dicke theory of gravity and on the assumption that the pressureless dark
matter and HDE do not conserve separately but interact with each other. Given
the unknown nature of both dark matter and dark energy there is nothing in
principle against their mutual interaction and it seems very special that
these two major components in the universe are entirely independent. Indeed,
this possibility is receiving growing attention in the literature Ame ; Zim ;
wang1 ; wang2 ; wang3 and appears to be compatible with SNIa and CMB data Oli
. On the other hand, although it is believed that our universe is spatially
flat, a contribution to the Friedmann equation from spatial curvature is still
possible if the number of e-foldings is not very large Huang . Besides, some
experimental data has implied that our universe is not a perfectly flat
universe and recent papers have favored the universe with spatial curvature
spe .
In the light of all mentioned above, it becomes obvious that the investigation
on the interacting HED in the framework of non-flat Brans-Dicke cosmology is
well motivated. We will show that the equation of state of dark energy can
accommodate $w_{D}=-1$ crossing. As systems’s IR cutoff we shall choose the
radius of the event horizon measured on the sphere of the horizon, defined as
$L=ar(t)$. Our work differs from that of Ref. Pavon2 in that we take
$L=ar(t)$ as the IR cutoff not the Hubble radius $L=H^{-1}$. It also differs
from that of Ref. Setare2 , in that we assume the pressureless dark matter and
HDE do not conserve separately but interact with each other, while the author
of Setare2 assumes that the dark components do not interact with each other.
This paper is outlined as follows: In section II, we consider noninteracting
HDE model in the framework of Brans-Dicke cosmology in a non-flat universe. In
section III, we extend our study to the case where there is an interaction
term between dark energy and dark matter. We summarize our results in section
IV.
## II HDE in Branse-Dicke cosmology
The action of Brans-Dicke theory is given by
$S=\int{d^{4}x\sqrt{g}\left(-\varphi{R}+\frac{\omega}{\varphi}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+L_{M}\right)}.$
(1)
The above action can be transformed into the standard canonical form by re-
defining the scalar field $\varphi$ and introducing a new field $\phi$, in
such a way that
$\varphi=\frac{\phi^{2}}{8\omega}.$ (2)
Therefore, in the canonical form, the action of Brans-Dicke theory can be
written Arik
$S=\int{d^{4}x\sqrt{g}\left(-\frac{1}{8\omega}\phi^{2}{R}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+L_{M}\right)},$
(3)
where ${R}$ is the scalar curvature and $\phi$ is the Brans-Dicke scalar
field. The non-minimal coupling term $\phi^{2}R$ replaces with the Einstein-
Hilbert term ${R}/{G}$ in such a way that
$G^{-1}_{\mathrm{eff}}={2\pi\phi^{2}}/{\omega}$, where $G_{\mathrm{eff}}$ is
the effective gravitational constant as long as the dynamical scalar field
$\phi$ varies slowly. The signs of the non-minimal coupling term and the
kinetic energy term are properly adopted to $(+---)$ metric signature. The HDE
model will be accommodated in the non-flat Friedmann-Robertson-Walker (FRW)
universe which is described by the line element
$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(4)
where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . Varying action (3) with respect to metric
(4) for a universe filled with dust and HDE yields the following field
equations
$\displaystyle\frac{3}{4\omega}\phi^{2}\left(H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{2}\dot{\phi}^{2}+\frac{3}{2\omega}H\dot{\phi}\phi=\rho_{M}+\rho_{D},$
(5)
$\displaystyle\frac{-1}{4\omega}\phi^{2}\left(2\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{\omega}H\dot{\phi}\phi-\frac{1}{2\omega}\ddot{\phi}\phi-\frac{1}{2}\left(1+\frac{1}{\omega}\right)\dot{\phi}^{2}=p_{D},$
(6)
$\displaystyle\ddot{\phi}+3H\dot{\phi}-\frac{3}{2\omega}\left(\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)\phi=0,$
(7)
where $H=\dot{a}/a$ is the Hubble parameter, $\rho_{D}$ and $p_{D}$ are,
respectively, the energy density and pressure of dark energy. We further
assume the energy density of pressureless matter can be separated as
$\rho_{M}=\rho_{BM}+\rho_{DM}$, where $\rho_{BM}$ and $\rho_{DM}$ are the
energy density of baryonic and dark matter, respectively. We also assume the
holographic energy density has the following form
$\rho_{D}=\frac{3c^{2}\phi^{2}}{4\omega L^{2}},$ (8)
where $\phi^{2}={\omega}/({2\pi G_{\mathrm{eff}}})$. In the limit of Einstein
gravity where $G_{\mathrm{eff}}\rightarrow G$, the above expression reduces to
the holographic energy density in standard cosmology
$\rho_{D}=\frac{3c^{2}}{8\pi GL^{2}}=\frac{3c^{2}m^{2}_{p}}{L^{2}}.$ (9)
The radius $L$ is defined as
$L=ar(t),$ (10)
where the function $r(t)$ can be obtained from the following relation
$\int_{0}^{r(t)}{\frac{dr}{\sqrt{1-kr^{2}}}}=\int_{0}^{\infty}{\frac{dt}{a}}=\frac{R_{h}}{a}.$
(11)
It is important to note that in the non-flat universe the characteristic
length which plays the role of the IR-cutoff is the radius $L$ of the event
horizon measured on the sphere of the horizon and not the radial size $R_{h}$
of the horizon. Solving the above equation for the general case of the non-
flat FRW universe, we have
$r(t)=\frac{1}{\sqrt{k}}\sin y,$ (12)
where $y=\sqrt{k}R_{h}/a$. Now we define the critical energy density,
$\rho_{\mathrm{cr}}$, and the energy density of the curvature, $\rho_{k}$, as
$\displaystyle\rho_{\mathrm{cr}}=\frac{3\phi^{2}H^{2}}{4\omega},\hskip
22.76228pt\rho_{k}=\frac{3k\phi^{2}}{4\omega a^{2}}.$ (13)
We also introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{M}$ $\displaystyle=$
$\displaystyle\frac{\rho_{M}}{\rho_{\mathrm{cr}}}=\frac{4\omega\rho_{M}}{3\phi^{2}H^{2}},$
(14) $\displaystyle\Omega_{k}$ $\displaystyle=$
$\displaystyle\frac{\rho_{k}}{\rho_{\mathrm{cr}}}=\frac{k}{H^{2}a^{2}},$ (15)
$\displaystyle\Omega_{D}$ $\displaystyle=$
$\displaystyle\frac{\rho_{D}}{\rho_{\mathrm{cr}}}=\frac{c^{2}}{H^{2}L^{2}}.$
(16)
For latter convenience we rewrite Eq. (16) in the form
$\displaystyle HL=\frac{c}{\sqrt{\Omega_{D}}}.$ (17)
Taking derivative with respect to the cosmic time $t$ from Eq. (10) and using
Eqs. (12) and (17) we obtain
$\displaystyle\dot{L}=HL+a\dot{r}(t)=\frac{c}{\sqrt{\Omega_{D}}}-\cos y.$ (18)
Consider the FRW universe filled with dark energy and pressureless matter
which evolves according to their conservation laws
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (19)
$\displaystyle\dot{\rho}_{M}+3H\rho_{M}=0,$ (20)
where $w_{D}$ is the equation of state parameter of dark energy. We shall
assume that Brans-Dicke field can be described as a power law of the scale
factor, $\phi\propto a^{\alpha}$. A case of particular interest is that when
$\alpha$ is small whereas $\omega$ is high so that the product $\alpha\omega$
results of order unity Pavon2 . This is interesting because local astronomical
experiments set a very high lower bound on $\omega$ Will ; in particular, the
Cassini experiment implies that $\omega>10^{4}$ Bert ; Aca . Taking the
derivative with respect to time of relation $\phi\propto a^{\alpha}$ we get
$\displaystyle\dot{\phi}=\alpha H\phi,$ (21)
$\displaystyle\ddot{\phi}=\alpha^{2}H^{2}\phi+\alpha\phi\dot{H}.$ (22)
Taking the derivative of Eq. (8) with respect to time and using Eqs. (18) and
(21) we reach
$\displaystyle\dot{\rho}_{D}=2H\rho_{D}\left(\alpha-1+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right).$ (23)
Inserting this equation in conservation law (19), we obtain the equation of
state parameter
$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos y.$
(24)
It is important to note that in the limiting case $\alpha=0$
($\omega\rightarrow\infty$), the Brans-Dicke scalar field becomes trivial and
Eq. (24) reduces to its respective expression in non-flat standard cosmology
Huang
$\displaystyle w_{D}=-\frac{1}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos y.$ (25)
We will see that the combination of the Brans-Dicke field and HDE brings rich
physics. For $\alpha\geq 0$, $w_{D}$ is bounded from below by
$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}.$ (26)
If we take $\Omega_{D}=0.73$ for the present time and choosing $c=1$ c , the
lower bound becomes $w_{D}=-\frac{2\alpha}{3}-0.9$. Thus for $\alpha=0.15$ we
have $w_{D}=-1$. The cases with $\alpha>0.15$ and $\alpha<0.15$ should be
considered separately. In the first case where $\alpha>0.15$ we have
$w_{D}<-1$. This is an interesting result and shows that, theoretically, the
combination of Brans-Dicke scalar field and HDE can accommodate $w_{D}=-1$
crossing for the equation of state of dark energy. Therefore one can generate
phantom-like equation of state from a noninteracting HDE model in the Brans-
Dicke cosmology framework. This is in contrast to the general relativity where
the equation of state of a noninteracting HDE cannot cross the phantom divide
Li . In the second case where $0\leq\alpha<0.15$ we have $-1<w_{D}\leq-0.9$.
Since $\alpha\approx 1/{\omega}$ and for $\omega\geq 10^{4}$ the Brans-Dicke
theory is consistent with solar system observations Bert , thus practically
$\alpha\simeq 10^{-4}$ is compatible with recent cosmological observations
which implies $w_{D}\simeq-0.903$ for the present time in this model. In both
cases discussed above $w_{D}<-1/3$ and the universe undergoing a phase of
accelerated expansion. It is worthwhile to note that since $\alpha\approx
1/\omega$ and $\omega>10^{4}$, therefore for all practical purposes Brans-
Dicke theory reduces to Einstein gravity as one can see from the above
discussion.
For completeness, we give the deceleration parameter
$\displaystyle q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (27)
which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. Dividing Eq. (6) by $H^{2}$, and using Eqs. (8),
(17), (21) and (22), we find
$\displaystyle
q=\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}+3\Omega_{D}w_{D}\right].$
(28)
Substituting $w_{D}$ from Eq. (24), we get
$\displaystyle
q=\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+1)\Omega_{D}-\frac{2}{c}{\Omega^{3/2}_{D}}\cos
y\right].$ (29)
If we take $\Omega_{D}=0.73$ and $\Omega_{k}\approx 0.01$ for the present time
and choosing $c=1$, $\alpha\omega\approx 1$, $\omega=10^{4}$ and $\cos y\simeq
1$, we obtain $q=-0.48$ for the present value of the deceleration parameter
which is in good agreement with recent observational results Daly . When
$\alpha\rightarrow 0$, Eq. (29) restores the deceleration parameter for HDE
model in Einstein gravity wang2
$\displaystyle
q=\frac{1}{2}(1+\Omega_{k})-\frac{\Omega_{D}}{2}-\frac{\Omega^{3/2}_{D}}{c}\cos
y.$ (30)
## III Interacting HDE in Branse-Dicke cosmology
In this section we extend our study to the case where both dark components-
the pressureless dark matter and the HDE- do not conserve separately but
interact with each other. Although at this point the interaction may look
purely phenomenological but different Lagrangians have been proposed in
support of it Tsu . Besides, in the absence of a symmetry that forbids the
interaction there is nothing, in principle, against it. Further, the
interacting dark energy has been investigated at one quantum loop with the
result that the coupling leaves the dark energy potential stable if the former
is of exponential type but it renders it unstable otherwise Dor . Therefore,
microphysics seems to allow enough room for the coupling. With the interaction
between the two dark constituents of the universe, we explore the evolution of
the universe. The total energy density satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0.$ (31)
where $\rho=\rho_{M}+\rho_{D}$ and $p=p_{D}$. However, since we consider the
interaction between dark energy and dark matter, $\rho_{DM}$ and $\rho_{D}$ do
not conserve separately. They must rather enter the energy balances Pav1
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (32)
$\displaystyle\dot{\rho}_{DM}+3H\rho_{DM}=Q,$ (33)
$\displaystyle\dot{\rho}_{BM}+3H\rho_{BM}=0,$ (34)
where we have assumed the baryonic matter does not interact with dark energy.
Here $Q$ denotes the interaction term and we take it as
$Q=3b^{2}H(\rho_{DM}+\rho_{D})$ with $b^{2}$ is a coupling constant. This
expression for the interaction term was first introduced in the study of the
suitable coupling between a quintessence scalar field and a pressureless cold
dark matter field Ame ; Zim . The choice of the interaction between both
components was meant to get a scaling solution to the coincidence problem such
that the universe approaches a stationary stage in which the ratio of dark
energy and dark matter becomes a constant. In the context of HDE models, this
form of interaction was derived from the choice of Hubble scale as the IR
cutoff Pav1 .
Combining Eqs. (13) and (21) with the first Friedmann equation (5), we can
rewrite this equation as
$\displaystyle\rho_{\mathrm{cr}}+\rho_{k}=\rho_{BM}+\rho_{DM}+\rho_{D}+\rho_{\phi},$
(35)
where we have defined
$\displaystyle\rho_{\phi}\equiv\frac{1}{2}\alpha
H^{2}\phi^{2}\left(\alpha-\frac{3}{\omega}\right).$ (36)
Dividing Eq. (35) by $\rho_{\mathrm{cr}}$, this equation can be written as
$\displaystyle\Omega_{BM}+\Omega_{DM}+\Omega_{D}+\Omega_{\phi}=1+\Omega_{k},$
(37)
where
$\displaystyle\Omega_{\phi}=\frac{\rho_{\phi}}{\rho_{\mathrm{cr}}}=-2\alpha\left(1-\frac{\alpha\omega}{3}\right).$
(38)
Thus, we can rewrite the interaction term $Q$ as
$\displaystyle Q=3b^{2}H(\rho_{DM}+\rho_{D})=3b^{2}H\rho_{D}(1+r),$ (39)
where $r={\rho_{DM}}/{\rho_{D}}$ is the ratio of the energy densities of two
dark components,
$\displaystyle
r=\frac{\Omega_{DM}}{\Omega_{D}}=-1+\frac{1}{\Omega_{D}}\left[1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(40)
Using the continuity equation (34), it is easy to show that
$\displaystyle\Omega_{BM}=\Omega_{BM0}a^{-3}=\Omega_{BM0}(1+z)^{3},$ (41)
where $\Omega_{BM0}\approx 0.04$ is the present value of the fractional energy
density of the baryonic matter and $z=a^{-1}-1$ is the red shift parameter.
Inserting Eqs. (23), (39) and (40) in Eq. (32) we obtain the equation of state
parameter
$\displaystyle
w_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y-b^{2}{\Omega^{-1}_{D}}\left[1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(42)
If we define, following Setare1 , the effective equation of state as
$\displaystyle w^{\mathrm{eff}}_{D}=w_{D}+\frac{\Gamma}{3H},$ (43)
where $\Gamma=3b^{2}(1+r)H$. Then, the continuity equation (32) for the dark
energy can be written in the standard form
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w^{\mathrm{eff}}_{D})=0.$ (44)
Substituting Eq. (42) in Eq. (43), we find
$\displaystyle
w^{\mathrm{eff}}_{D}=-\frac{1}{3}-\frac{2\alpha}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y,$ (45)
From Eq. (45) we see that with the combination of Brans-Dicke field and HDE,
the effective equation of state, $w^{\mathrm{eff}}_{D}$, can cross the phantom
divide. For instance, taking $\Omega_{D}=0.73$ for the present time and $c=1$,
the lower bound of Eq. (45) is $w^{\mathrm{eff}}_{D}=-\frac{2\alpha}{3}-0.9$.
Thus for $\alpha>0.15$ we have $w^{\mathrm{eff}}_{D}<-1$. Therefore, the
Brans- Dicke field plays a crucial role in determining the behaviour of the
effective equation of state of interacting HDE. It is important to note that
in standard HDE ($\alpha=0$) it is impossible to have $w^{\mathrm{eff}}_{D}$
crossing $-1$ Setare1 . Returning to the general case (42), we see that when
the interacting HDE is combined with the Brans-Dicke scalar field the
transition from normal state where $w_{D}>-1$ to the phantom regime where
$w_{D}<-1$ for the equation of state of interacting dark energy can be more
easily achieved for than when resort to the Einstein field equations is made.
In the absence of the Brans- Dicke field ($\alpha=0$), Eq. (42) restores its
respective expression in non-flat standard cosmology wang2
$\displaystyle w_{D}=-\frac{1}{3}-\frac{2\sqrt{\Omega_{D}}}{3c}\cos
y-b^{2}{\Omega_{D}}^{-1}\left(1+\Omega_{k}-\Omega_{BM}\right).$ (46)
Next, we examine the deceleration parameter, $q=-\ddot{a}/(aH^{2})$.
Substituting $w_{D}$ from Eq. (42) in Eq. (28), one can easily show
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+1)\Omega_{D}-\frac{2}{c}{\Omega^{3/2}_{D}}\cos
y\right.\ $ (47)
$\displaystyle\left.-3b^{2}\left(1+\Omega_{k}-\Omega_{BM}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right)\right].$
If we take $\Omega_{D}=0.73$ and $\Omega_{k}\approx 0.01$ for the present time
and $c=1$, $\alpha\approx 1/\omega$, $\omega=10^{4}$, $\cos y\simeq 1$,
$\Omega_{BM}\approx 0.04$ and $b=0.1$, we obtain $q=-0.5$ which is again
compatible with recent observational data Daly . When $\alpha=0$, Eq. (47)
reduces to the deceleration parameter of the interacting HDE in Einstein
gravity wang2
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\Omega_{k})-\frac{\Omega_{D}}{2}-\frac{{\Omega^{3/2}_{D}}}{c}\cos
y-\frac{3b^{2}}{2}\left(1+\Omega_{k}-\Omega_{BM}\right).$ (48)
We can also obtain the evolution behavior of the dark energy. Taking the
derivative of Eq. (16) and using Eq. (18) and relation
${\dot{\Omega}_{D}}=H{\Omega^{\prime}_{D}}$, we find
$\displaystyle{\Omega^{\prime}_{D}}=2\Omega_{D}\left(-\frac{\dot{H}}{H^{2}}-1+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right),$ (49)
where the dot is the derivative with respect to time and the prime denotes the
derivative with respect to $x=\ln{a}$. Using relation
$q=-1-\frac{\dot{H}}{H^{2}}$, we have
$\displaystyle{\Omega^{\prime}_{D}}=2\Omega_{D}\left(q+\frac{\sqrt{\Omega_{D}}}{c}\cos
y\right),$ (50)
where $q$ is given by Eq. (47). This equation describes the evolution behavior
of the interacting HDE in Brans-Dicke cosmology framework. In the limit of
standard cosmology ($\alpha=0$), Eq. (50) reduces to its respective expression
in HDE model wang2
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left[(1-\Omega_{D})\left(1+\frac{2\sqrt{\Omega_{D}}}{c}\cos
y\right)-3b^{2}(1+\Omega_{k}-\Omega_{BM})+\Omega_{k}\right].$ (51)
For flat universe, $\Omega_{k}=0$, and Eq. (51) recovers exactly the result of
wang1 .
## IV Summary and discussion
In summary, we studied the interacting holographic model of dark energy in the
framework of Brans-Dicke cosmology where the HDE density
$\rho_{D}={3c^{2}}/(8\pi GL^{2})$ is replaced with
$\rho_{D}={3c^{2}\phi^{2}}/({4\omega L^{2}})$. Here $\phi^{2}={\omega}/({2\pi
G_{\mathrm{eff}}})$, where $G_{\mathrm{eff}}$ is the time variable Newtonian
constant. In the limit of Einstein gravity, $G_{\mathrm{eff}}\rightarrow G$.
With this replacement in Brans-Dicke theory, we found that the accelerated
expansion will be more easily achieved for than when the standard HDE is
considered. We obtained the equation of state and the deceleration parameter
of the HDE in a non-flat universe enclosed by the event horizon measured on
the sphere of the horizon with radius $L=ar(t)$. Interestingly enough, we
found that the combination of Brans-Dicke and HDE can accommodate $w_{D}=-1$
crossing for the equation of state of noninteracting dark energy. For
instance, taking $\Omega_{D}=0.73$ for the present time and $c=1$, the lower
bound of $w_{D}$ becomes $w_{D}=-\frac{2\alpha}{3}-0.9$. Thus for
$\alpha>0.15$ we have $w_{D}<-1$. This is in contrast to Einstein gravity
where the equation of state of noninteracting HDE cannot cross the phantom
divide $w_{D}=-1$ Li . When the interaction between dark energy and dark
matter is taken into account, the transition from normal state where
$w_{D}>-1$ to the phantom regime where $w_{D}<-1$ for the equation of state of
HDE can be more easily accounted for than when resort to the Einstein field
equations is made.
In Brans-Dicke theory of interacting HDE, the properties of HDE is determined
by the parameters $c$, $b$ and $\alpha$ together. These parameters would be
obtained by confronting with cosmic observational data. In this work we just
restricted our numerical fitting to limited observational data. Giving the
wide range of cosmological data available, in the future we expect to further
constrain our model parameter space and test the viability of our model. The
issue is now under investigation and will be addressed elsewhere.
###### Acknowledgements.
I thank the anonymous referee for constructive comments. This work has been
supported by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.
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|
arxiv-papers
| 2009-07-31T02:41:18 |
2024-09-04T02:49:04.369878
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0907.5458"
}
|
0907.5463
|
# Polarized Emission of Sagittarius A*
Lei Huang11affiliation: Key Laboratory for Research in Galaxies and Cosmology,
The University of Sciences and Technology of China, Chinese Academy of
Sciences, Hefei 230026, China; mlhuang@ustc.edu.cn, yfyuan@ustc.edu.cn
33affiliation: Key Laboratory for Research in Galaxies and Cosmology, Shanghai
Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
zshen@shao.ac.cn 44affiliation: Academia Sinica, Institute of Astronomy and
Astrophysics, Taipei 106, Taiwan; mike@asiaa.sinica.edu.tw , Siming
Liu22affiliation: Department of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, UK; sliu@astro.gla.ac.uk , Zhi-Qiang Shen33affiliation: Key
Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
zshen@shao.ac.cn , Ye-Fei Yuan11affiliation: Key Laboratory for Research in
Galaxies and Cosmology, The University of Sciences and Technology of China,
Chinese Academy of Sciences, Hefei 230026, China; mlhuang@ustc.edu.cn,
yfyuan@ustc.edu.cn , Mike J. Cai44affiliation: Academia Sinica, Institute of
Astronomy and Astrophysics, Taipei 106, Taiwan; mike@asiaa.sinica.edu.tw , Hui
Li55affiliation: Los Alamos National Laboratory, Los Alamos, NM 87545;
hli@lanl.gov, fryer@lanl.gov , and Christopher L. Fryer55affiliation: Los
Alamos National Laboratory, Los Alamos, NM 87545; hli@lanl.gov, fryer@lanl.gov
6 6affiliationmark:
###### Abstract
We explore the parameter space of the two temperature pseudo-Newtonian
Keplerian accretion flow model for the millimeter and shorter wavelength
emission from Sagittarius A*. A general relativistic ray-tracing code is used
to treat the radiative transfer of polarized synchrotron emission from the
flow. The synchrotron self-Comptonization and bremsstrahlung emission
components are also included. It is shown that the model can readily account
for the millimeter to sub-millimeter emission characteristics with an
accretion rate of $\sim 6\times 10^{17}\mathrm{g}\cdot\mathrm{s}^{-1}$ and an
inclination angle of $\sim 40^{\circ}$. However, the corresponding model
predicted near-infrared and X-ray fluxes are more than one order of magnitude
lower than the observed ‘quiescent’ state values. While the extended
quiescent-state X-ray emission has been attributed to thermal emission from
the large-scale accretion flow, the NIR emission and flares are likely
dominated by emission regions either within the last stable orbit of a
Schwarzschild black hole or associated with outflows. With the viscous
parameter derived from numerical simulations, there is still a degeneracy
between the electron heating rate and the magnetic parameter. A fully general
relativistic treatment with the black hole spin incorporated will resolve
these issues.
black hole physics — Galaxy: center — plasmas — polarization — radiative
transfer — sub-millimeter
66affiliationtext: Physics Department, The University of Arizona, Tucson, AZ
85721
## 1 INTRODUCTION
Sagittarius (Sgr) A*, the compact radio source associated with the super-
massive black hole at the Galactic center, is perhaps the best source for the
study of physical processes near the event horizon of a black hole (Schödel et
al., 2002; Ghez et al., 2005). Although its luminosity is relatively low, with
the high resolution and sensitivity of modern instruments, the source has been
routinely observed from radio to X-rays. It may also play a role in the
production of TeV gamma-ray emission from the Galactic center region (Liu et
al., 2006; Aharonian et al., 2006). The low luminosity also renders the source
optically thin at millimeter and shorter wavelengths. One therefore can
observe emission from the very inner region close to the black hole directly
at these wavelengths.
Most of Sgr A* emission is released in the sub-millimeter band, where a strong
linear polarization is observed. The flux density also varies with the
variation amplitude and rate decreasing with the decrease of the observation
frequency (Herrnstein et al., 2004). The variation timescale of sub-millimeter
emission can be as short as a few hours, slightly longer than the dynamical
time near the black hole. The source is highly variable at shorter wavelengths
with the peak luminosity of some near infrared (NIR) and X-ray flares
comparable to the sub-millimeter luminosity (Baganoff et al., 2001; Genzel et
al., 2003) The variation timescale of these flares is consistent with events
occurring within a few Schwarzschild radii of the black hole. And correlated
flare activities in the X-ray, NIR, millimeter, and radio bands suggest
outflows during some flares (Zhao et al., 2004; Yusef-Zadeh et al., 2007;
Marrone et al., 2007; Eckart et al., 2008). The flares also contain rich
temporal, spectral, and polarization information (Porquet, 2008; Falanga et
al., 2008; Eckart et al., 2008). Detailed modelling of these emission
characteristics is expected to probe the geometry near the black hole,
processes of the general relativistic magnetohydrodynamics (GRMHD) (Gammie et
al., 2003), and the kinetics of electron heating and acceleration in a
magnetized relativistic plasma (Liu et al., 2004, 2007b).
Motivated by MHD simulations of radiatively inefficient accretion flows
(Hawley & Balbus, 1991, 2002), we proposed a Keplerian accretion flow model
for the time averaged millimeter and shorter wavelength emission from Sgr A*
(Melia et al., 2000, 2001). The model plays important roles in constraining
the time averaged properties of the plasma near the black hole (Yuan et al.,
2003) for detailed GRMHD studies (Noble et al., 2007). It also treats the
polarization characteristics quantitatively (Bromley et al., 2001). The model
is later generalized to a two temperature accretion flow in a pseudo-Newtonian
potential (Liu et al., 2007a, b). Huang et al. (2008) first carried out a
self-consistent treatment of the transfer of polarized emission through the
accretion flow and predicted distinct linear and circular polarization
characteristics between the sub-millimeter and NIR band due to general
relativistic light bending and birefringence effects. In this paper, we
present the details of the calculations (§ 2), explore the model parameter
space (§ 3), and discuss future developments (§ 4).
## 2 RADIATIVE TRANSFER OF SYNCHROTRON RADIATION
Although the importance of a self-consistent treatment of polarized radiative
transfer through relativistic plasmas near black holes was recognized a while
ago (Melrose, 1997), there are still significant uncertainties in the
quantitative details (Shcherbakov, 2008). The thermal synchrotron emission and
absorption coefficients have been derived independently by several authors
(Legg & Westfold, 1968; Sazonov, 1969; Melrose, 1971), which in general show
agreement. The derivation of the Faraday rotation and conversion coefficients
was mostly done by (Melrose, 1997). Shcherbakov (2008) recently derived
expressions appropriate for the transition from non-relativistic temperatures
to relativistic ones. In this section, we have a brief discussion of the
general theory of polarized radiation transfer and show that the Faraday
rotation and conversion coefficients are not independent. The radio of the two
is equal to the ratio of the circular and linear emission coefficients.
Therefore our approach of deriving the Faraday conversion coefficient from the
emission coefficients and Faraday rotation coefficient is appropriate (Huang
et al., 2008).
### 2.1 Properties and Transfer of Partially Polarized Radiation
The synchrotron radiation from an individual particle is elliptically
polarized. The major axis of the polarization ellipse e1 is perpendicular to
the plane spanned by the magnetic vector B and the wave vector k, and the
minor axis e${}^{2}\propto$ k$\times$e1 is perpendicular to the major axis and
wave vector. The electric field component of the radiation can be projected
along the major and minor axes, namely the e- and o-component, respectively.
Figure 1 shows the corresponding geometry in the three dimensional (3D)
Cartesian coordinates $(x,y,z)$:
$\displaystyle{\rm\bf B}/B$ $\displaystyle=$
$\displaystyle(0,\,1,\,0),\qquad{\rm\bf
k}/k\quad=\quad(0,\,\cos\theta_{B},\,\sin\theta_{B}),$ $\displaystyle{\rm\bf
e}^{1}$ $\displaystyle=$ $\displaystyle(1,\,0,\,0),\qquad\quad{\rm\bf
e}^{2}\quad=\quad(0,\,\sin\theta_{B},\,-\cos\theta_{B}),$ (1)
where $\theta_{B}$ is the angle between B and k, and $k$ and $B$ are the wave
number and the amplitude of the magnetic field, respectively. For a population
of relativistic particles with a smooth pitch angle distribution, the circular
polarization (CP) component of synchrotron emission almost cancels out, and
the emissivity of the e-component is generally much greater than that of the
o-component. Therefore, the synchrotron radiation from a population of
relativistic particles is often treated as partially linearly polarized along
e1. The e-component base ${\rm\bf e}^{1}$ then indicates the electric vector
of the linearly polarized (LP) radiation. Since the CP component does not
cancel out exactly, sometimes it is more convenient to use the right-handed
and left-handed CP bases
$\displaystyle\mathrm{\bf e}^{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mathrm{\bf e}^{1}+i\mathrm{\bf
e}^{2}),\qquad\mathrm{\bf e}^{L}\quad=\quad\frac{1}{\sqrt{2}}(\mathrm{\bf
e}^{1}-i\mathrm{\bf e}^{2}).$ (2)
According to the intensity matrix $\mathcal{I}^{ij}$ defined by Melrose (1971)
with an arbitrary orthogonal bases, the total intensity is given by
$\displaystyle I$ $\displaystyle=$
$\displaystyle\mathrm{trace}(\mathcal{I}^{ij})\quad=\quad\mathcal{I}^{11}+\mathcal{I}^{22}\quad=\quad\mathcal{I}^{RR}+\mathcal{I}^{LL}.$
(3)
Then one has the formal polarization vector $\vec{p}=(p_{Q},p_{U},p_{V})$
introduced by Landi Degl’Innocenti & Landolfi (2004)
$\displaystyle p_{Q}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{I}^{11}-\mathcal{I}^{22}}{\mathcal{I}^{11}+\mathcal{I}^{22}},$
$\displaystyle p_{U}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{I}^{12}+\mathcal{I}^{21}}{\mathcal{I}^{11}+\mathcal{I}^{22}},$
$\displaystyle p_{V}$ $\displaystyle=$
$\displaystyle\frac{i(\mathcal{I}^{12}-\mathcal{I}^{21})}{\mathcal{I}^{11}+\mathcal{I}^{22}}\quad=\quad\frac{\mathcal{I}^{RR}-\mathcal{I}^{LL}}{\mathcal{I}^{RR}+\mathcal{I}^{LL}},$
(4)
The total polarization fraction $\Pi$, CP fraction $\Pi_{C}$, LP fraction
$\Pi_{L}$, and the electric vector position angle (EVPA) $\chi_{0}$ are then
given by
$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\left[1-\frac{4\mathrm{det}(\mathcal{I}^{ij})}{\mathrm{trace}(\mathcal{I}^{ij})^{2}}\right]^{1/2},$
$\displaystyle\Pi_{C}$ $\displaystyle=$ $\displaystyle p_{V},$
$\displaystyle\Pi_{L}$ $\displaystyle=$
$\displaystyle(\Pi^{2}-\Pi_{C}^{2})^{1/2}\quad=\quad(p_{Q}^{2}+p_{U}^{2})^{1/2},$
$\displaystyle\tan 2\chi_{0}$ $\displaystyle=$
$\displaystyle\frac{p_{U}}{p_{Q}}\,.$ (5)
The normalized polarization vector can be rewritten in the form of the 4D
Stokes vector as $\vec{S}=(I,Q,U,V)^{T}=I(1,p_{Q},p_{U},p_{V})^{T}$. The
Stokes vector, the intensity matrix, and ($I$, $\Pi_{C}$, $\Pi_{L}$,
$\chi_{0}$) all give a complete description of the properties of partially
polarized emission.
In order to describe the polarized radiative transfer in highly-magnetized
plasma, Landi Degl’Innocenti & Landolfi (2004) introduce another three 3D
formal vector $\vec{\epsilon},\vec{\eta},\vec{\rho}$, where $\vec{\epsilon}$
is the normalized emission vector, which is related to the 4D emission
coefficient
$\vec{\varepsilon}=(\varepsilon_{I},\varepsilon_{Q},\varepsilon_{U},\varepsilon_{V})^{T}$:
$\vec{\epsilon}\equiv(\epsilon_{Q},\epsilon_{U},\epsilon_{V})=(\varepsilon_{Q},\varepsilon_{U},\varepsilon_{V})/I$,
$\vec{\eta}\equiv(\eta_{Q},\eta_{U},\eta_{V})$ is the absorption vector, and
$\vec{\rho}$ is the Faraday rotation vector. The total emission and average
absorption coefficients are represented by $\varepsilon_{I}=\epsilon_{I}I$ and
$\eta_{I}$, respectively. Then the transfer of polarized emission is described
with
$\displaystyle{{\rm d}I\over{\rm d}\rm s}$ $\displaystyle=$
$\displaystyle-(\eta_{I}+\vec{\eta}\cdot\vec{p}-\epsilon_{I})I,$
$\displaystyle{{\rm d}\vec{p}\over{\rm d}\rm s}$ $\displaystyle=$
$\displaystyle-\vec{\eta}+(\vec{\eta}\cdot{}\vec{p})\vec{p}+\vec{\rho}\times\vec{p}+\vec{\epsilon}-\epsilon_{I}\vec{p}\,,$
(6)
which govern the evolution of the polarization properties of radiation through
a magnetized plasma. These equations can be rewritten in the form of Stokes
vector as
$\displaystyle\frac{\rm d}{\rm d\rm s}\vec{S}$ $\displaystyle=$
$\displaystyle\vec{\varepsilon}\quad-\quad\mathbf{K}\vec{S},$ (7)
or explicitly
$\displaystyle\frac{\rm d}{\rm d\rm s}\left(\begin{array}[]{cccc}I\\\ Q\\\
U\\\ V\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\varepsilon_{I}\\\ \varepsilon_{Q}\\\
\varepsilon_{U}\\\
\varepsilon_{V}\end{array}\right)\quad-\quad\left(\begin{array}[]{cccc}\eta_{I}&\eta_{Q}&\eta_{U}&\eta_{V}\\\
\eta_{Q}&\eta_{I}&\rho_{V}&-\rho_{U}\\\
\eta_{U}&-\rho_{V}&\eta_{I}&\rho_{Q}\\\
\eta_{V}&\rho_{U}&-\rho_{Q}&\eta_{I}\end{array}\right)\left(\begin{array}[]{cccc}I\\\
Q\\\ U\\\ V\end{array}\right).$ (24)
One can use the rotation matrix
$\displaystyle\mathrm{R}(\chi)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&\cos 2\chi&\sin
2\chi&0\\\ 0&-\sin 2\chi&\cos 2\chi&0\\\ 0&0&0&1\end{array}\right),$ (29)
to transform the Stokes vector from the coordinates $({\rm\bf e}^{1},{\rm\bf
e}^{2})$ to the reference coordinates $({\bf a},{\bf b})$, where a corresponds
to the North at the observer and b corresponds to the East (see Figure 1).
Then we have
$\displaystyle\frac{\rm d}{\rm d\rm s}\vec{S}^{\prime}$ $\displaystyle=$
$\displaystyle\vec{\varepsilon}^{\prime}\quad-\quad\mathbf{K}^{\prime}\quad\vec{S}^{\prime}\,,$
(30)
where $\vec{S}^{\prime}=\mathrm{R}(\chi)\vec{S}$,
$\vec{\varepsilon}^{\prime}=\mathrm{R}(\chi)\vec{\varepsilon}$, and
$\mathbf{K}^{\prime}=\mathrm{R}(\chi)\mathbf{K}\mathrm{R}(-\chi)$.
### 2.2 Synchrotron Emission, Absorption, and Faraday Coefficients
For a population of relativistic electrons in the Maxwell distribution, i.e.,
relativistic thermal distribution, with the number density $N_{0}$ and
temperature $T_{e}\gg m_{e}c^{2}/k$, the distribution function with respect to
the electron energy $E$ is
$\displaystyle N(E)$ $\displaystyle=$ $\displaystyle
N_{0}\frac{E^{2}}{2(kT_{e})^{3}}\mathrm{exp}(-E/kT_{e})\,,$ (31)
where $m_{e}$, $c$, and $k$ are the electron mass, speed of light, and
Boltzmann constant, respectively. In the coordinates of (e1, e2), the four
synchrotron emission coefficients are (Sazonov, 1969; Pacholczyk, 1970;
Melrose, 1971)
$\displaystyle\varepsilon_{I}$ $\displaystyle=$
$\displaystyle\frac{e^{2}m_{e}^{2}c^{3}}{\sqrt{3}}\frac{N_{0}}{2(kT_{e})^{2}}\nu
I_{I}(x_{M}),$ $\displaystyle\varepsilon_{Q}$ $\displaystyle=$
$\displaystyle\frac{e^{2}m_{e}^{2}c^{3}}{\sqrt{3}}\frac{N_{0}}{2(kT_{e})^{2}}\nu
I_{Q}(x_{M}),$ $\displaystyle\varepsilon_{U}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\varepsilon_{V}$ $\displaystyle=$
$\displaystyle\frac{4e^{2}m_{e}^{3}c^{5}\mathrm{cot}\theta_{B}}{3\sqrt{3}}\frac{N_{0}}{2(kT_{e})^{3}}\nu
I_{V}(x_{M})\,,$ (32)
where $e$ is the elementary charge units and $x_{M}$ is the ratio of the
emission frequency $\nu$ to the characteristic frequency
$\nu_{c}={3eB\mathrm{sin}\theta_{B}(kT_{e})^{2}}/{4\pi m_{e}^{3}c^{5}}$, and
the integrations $I_{I},I_{Q}$ and $I_{V}$ can be calculated with the aid of
the modified Bessel functions $K_{\alpha}(z)$:
$\displaystyle F(x)$ $\displaystyle=$ $\displaystyle
x\int_{x}^{\infty}K_{5/3}(z)dz\,,$ $\displaystyle G(x)$ $\displaystyle=$
$\displaystyle xK_{2/3}(x)\,,$ $\displaystyle H(x)$ $\displaystyle=$
$\displaystyle xK_{1/3}(x)+\int_{x}^{\infty}K_{1/3}(z)dz,\quad$ $\displaystyle
I_{I}(x_{M})$ $\displaystyle=$
$\displaystyle\frac{1}{x_{M}}\int_{0}^{\infty}z^{2}\mathrm{exp}(-z)F\left(\frac{x_{M}}{z^{2}}\right)dz\,,$
$\displaystyle I_{Q}(x_{M})$ $\displaystyle=$
$\displaystyle\frac{1}{x_{M}}\int_{0}^{\infty}z^{2}\mathrm{exp}(-z)G\left(\frac{x_{M}}{z^{2}}\right)dz\,,$
$\displaystyle I_{V}(x_{M})$ $\displaystyle=$
$\displaystyle\frac{1}{x_{M}}\int_{0}^{\infty}z^{\quad}\mathrm{exp}(-z)H\left(\frac{x_{M}}{z^{2}}\right)dz\,.$
(33)
The emission coefficients along the two axes ${\rm\bf e}^{1}$ and ${\rm\bf
e}^{2}$ are given, respectively, by
$\displaystyle\varepsilon_{1}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{I}+\varepsilon_{Q}}{2},$
$\displaystyle\varepsilon_{2}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{I}-\varepsilon_{Q}}{2}.$ (34)
The absorption coefficient can be calculated with the Kirchhoff’s law
$\displaystyle\frac{S}{2}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{1}}{\eta_{1}}=\frac{\varepsilon_{2}}{\eta_{2}}\,,$
$\displaystyle\eta_{I}$ $\displaystyle=$
$\displaystyle\frac{\eta_{1}+\eta_{2}}{2}\,,$ $\displaystyle\vec{\eta}$
$\displaystyle=$
$\displaystyle\frac{(\varepsilon_{Q},\,\varepsilon_{U},\,\varepsilon_{V})}{S}=\frac{\vec{\epsilon}I}{S}\,,$
(35)
where $S=\varepsilon_{I}/\eta_{I}$ is the source function,
$S=B_{\nu}=2h\nu^{3}/c^{2}[\exp{(h\nu/kT_{e})}-1]$ for a thermal distribution,
and $h$ is the Planck constant.
According to Melrose (1997), the high-frequency waves may be treated as two
transverse nature wave modes with dispersion relations
$k_{\pm}^{2}c^{2}-\omega^{2}=\frac{1}{2}\\{\alpha^{11}+\alpha^{22}\pm[(\alpha^{11}-\alpha^{22})^{2}+4\alpha^{12}\alpha^{21}]^{1/2}\\},$
where $\omega=2\pi\nu$, $k$ is the wave number, and the response tensor
$\alpha^{ij}$ satisfies $\alpha^{12}=-\alpha^{21}$. The polarization vectors
of the two natural modes are then given by
$\displaystyle{\rm\bf e}^{\pm}=\frac{T_{\pm}{\rm\bf e}^{1}+i{\rm\bf
e}^{2}}{\sqrt{1+T_{\pm}^{2}}},$ $\displaystyle
T_{\pm}=\frac{\alpha^{11}-\alpha^{22}\mp[(\alpha^{11}-\alpha^{22})^{2}+4\alpha^{12}\alpha^{21}]^{1/2}}{2i\alpha^{12}},$
(36)
which are two orthogonal elliptically polarized modes with axial ratios
$T_{\pm}$. Note that $T_{+}T_{-}=-1$ implying that the two natural modes have
identical ellipticity and opposite handedness. The electric vector ${\rm\bf
E}$ of any wave can be decomposed on the bases of the two natural modes as
$E_{+}{\rm\bf e}^{+}+E_{-}{\rm\bf e}^{-}$ or on the bases of the two LP modes
as $E_{1}{\rm\bf e}^{1}+E_{2}{\rm\bf e}^{2}$. Then
$E_{1}=E_{+}\frac{T_{+}}{\sqrt{1+T_{+}^{2}}}+E_{-}\frac{T_{-}}{\sqrt{1+T_{-}^{2}}}$
and
$E_{2}=E_{+}\frac{1}{\sqrt{1+T_{+}^{2}}}+E_{-}\frac{1}{\sqrt{1+T_{-}^{2}}}.$
Using the definitions in equation (2), a third decomposition can be made on
two CP modes as $E_{R}{\rm e}^{R}+E_{L}{\rm e}^{L}$, with
$E_{R}=E_{+}\frac{T_{+}+1}{\sqrt{2(1+T_{+}^{2})}}+E_{-}\frac{T_{-}+1}{\sqrt{2(1+T_{-}^{2})}}$
and
$E_{L}=E_{+}\frac{T_{+}-1}{\sqrt{2(1+T_{+}^{2})}}+E_{-}\frac{T_{-}-1}{\sqrt{2(1+T_{-}^{2})}}.$
The total emission coefficient $\varepsilon_{I}$ is proportional to $E^{2}$,
i.e.,
$\displaystyle\varepsilon_{I}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}\begin{array}[]{ccc}\varepsilon_{+}+\varepsilon_{-}\quad\propto\quad
E_{+}^{2}+E_{-}^{2}\\\ \varepsilon_{1}+\varepsilon_{2}\quad\propto\quad
E_{1}^{2}+E_{2}^{2}\\\ \varepsilon_{R}+\varepsilon_{L}\quad\propto\quad
E_{R}^{2}+E_{L}^{2}\end{array}.$ (40)
Then the LP and CP emission coefficients ($\varepsilon_{Q}$ and
$\varepsilon_{V}$) are given by
$\displaystyle\varepsilon_{Q}$ $\displaystyle=$
$\displaystyle\varepsilon_{1}-\varepsilon_{2}=\varepsilon_{Q+}+\varepsilon_{Q-}=\frac{T_{+}^{2}-1}{1+T_{+}^{2}}\varepsilon_{+}+\frac{T_{-}^{2}-1}{1+T_{-}^{2}}\varepsilon_{-}=\frac{T_{+}+T_{-}}{T_{+}-T_{-}}(\varepsilon_{+}-\varepsilon_{-})$
$\displaystyle\varepsilon_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{R}-\varepsilon_{L}=\varepsilon_{V+}+\varepsilon_{V-}=\frac{2T_{+}}{1+T_{+}^{2}}\varepsilon_{+}+\frac{2T_{-}}{1+T_{-}^{2}}\varepsilon_{-}=\frac{2}{T_{+}-T_{-}}(\varepsilon_{+}-\varepsilon_{-}),$
(41)
which implies that
$\displaystyle\frac{\varepsilon_{V}}{\varepsilon_{Q}}$ $\displaystyle=$
$\displaystyle\frac{2}{T_{+}+T_{-}}=\frac{2i\alpha^{12}}{\alpha^{11}-\alpha^{22}}=\frac{\rho_{\rm
RM}}{\rho_{\rm RRM}},$ (42)
where $\rho_{\rm RM}=i\alpha^{12}/2\omega c$ is called the rotation measure,
or the Faraday rotation coefficient, and $\rho_{\rm
RRM}=(\alpha^{11}-\alpha^{22})/4\omega c$ is called the relativistic rotation
measure, or the Faraday conversion coefficient. Note that
$\varepsilon_{\pm}^{2}=\varepsilon_{Q\pm}^{2}+\varepsilon_{V\pm}^{2}$,
$\varepsilon_{\pm}/\eta_{\pm}=S/2$,
$\eta_{Q}=\varepsilon_{Q}/S=(\eta_{Q+}+\eta_{Q-})/2$,
$\eta_{V}=\varepsilon_{V}/S=(\eta_{V+}+\eta_{V-})/2$.
In a uniform plasma, $\vec{\rho}=(2\rho_{\rm RRM},0,2\rho_{\rm RM})$ and the
three formal vector $\vec{\epsilon}$, $\vec{\eta}$, and $\vec{\rho}$ are
parallel to each other. If there is no external radiation, the term
$(\vec{\rho}\times\vec{p})$ in equations (2.1) vanishes (Landi Degl’Innocenti
& Landolfi, 2004; Bekefi, 1966; Kennett & Melrose, 1998) and one has the
solution with $\vec{p}$ also parallel to these vectors. Then equations (2.1)
become
$\displaystyle{{\rm d}I\over{\rm d}\rm s}$ $\displaystyle=$
$\displaystyle-[(1+\Pi\Pi_{0})\eta_{I}-\epsilon_{I}]I=-\left[(1+\Pi\Pi_{0})\frac{I}{S}-1\right]\varepsilon_{I}\,,$
$\displaystyle{{\rm d}\Pi\over{\rm d}\rm s}$ $\displaystyle=$
$\displaystyle\eta_{I}\Pi_{0}(\Pi^{2}-1)+\epsilon_{I}(\Pi_{0}-\Pi)=\left[\frac{\Pi_{0}(\Pi^{2}-1)}{S}+\frac{\Pi_{0}-\Pi}{I}\right]\varepsilon_{I}\,,$
(43)
where
$\Pi_{0}=(\varepsilon_{Q}^{2}+\varepsilon_{U}^{2}+\varepsilon_{V}^{2})^{1/2}/\varepsilon_{I}$
is the polarization fraction of an optically thin source. Note that in this
case, if we denote the unit vector along $\vec{\epsilon}$ as $\vec{u}$, then
$\vec{p}=\Pi\vec{u}$, $\vec{\epsilon}=\Pi_{0}\epsilon_{I}\vec{u}$,
$\vec{\eta}=\vec{\epsilon}I/S=\Pi_{0}\eta_{I}\vec{u}$. With the increase of
the optical depth, we see that $\Pi$ decreases, and the existence of $\Pi$
enhances the self-absorption by a factor of $1+\Pi\Pi_{0}$.
Therefore, in a uniform plasma, besides the source function that is determined
by the particle distribution, there are four independent coefficients. From
the three emission coefficients given by equations (2.2) and the Faraday
rotation coefficient $\rho_{V}=2\rho_{\rm RM}$, one can derive the absorption
and Faraday conversion coefficients with equations (2.2) and (42),
respectively. Melrose (1997) derived the three emission coefficients and two
Faraday coefficients separately. It appears that his results are not exactly
in agreement with equation (42). Neither is the $\rho_{\rm RM}$ and $\rho_{\rm
RRM}$ obtained by Shcherbakov (2008). All these calculations invoke some
approximations to simplify the Trubnikov’s linear response tensor. While
equation (42) is derived from very general theoretical considerations without
any approximations. We therefore stand by our approach of deriving $\rho_{\rm
RRM}$ from $\rho_{\rm RM}$ and the emission coefficients in a previous paper
(Huang et al., 2008).
Melrose (1997) gives $\rho_{\rm RM}$ for thermal plasmas in the cold
($\gamma_{C}=kT_{e}/m_{e}c^{2}+1\approx 1$) and extremely relativistic (ER)
($\gamma_{C}\gg 1$) limits:
$\displaystyle\rho_{\rm RM}^{{\rm C}}$ $\displaystyle=$
$\displaystyle\frac{e^{3}N_{0}B\cos\theta_{B}}{2\pi\gamma_{c}m_{e}^{2}c^{2}\nu^{2}},$
$\displaystyle\rho_{\rm RM}^{{\rm ER}}$ $\displaystyle=$
$\displaystyle\frac{\ln\gamma_{c}}{2\gamma_{c}}\frac{e^{3}N_{0}B\cos\theta_{B}}{2\pi\gamma_{c}m_{e}^{2}c^{2}\nu^{2}}.$
(44)
We use the following extrapolation to obtain $\rho_{\rm RM}$ for arbitrary
electron temperatures
$\displaystyle\rho_{\rm RM}$ $\displaystyle=$
$\displaystyle\gamma_{c}^{-1}(\rho_{\rm RM}^{{\rm C}}-\rho_{\rm RM}^{{\rm
ER}})+\rho_{\rm RM}^{{\rm ER}}\,,$ (45)
which is simple and accurate enough according to Shcherbakov (2008). For cold
plasmas, cyclotron emission dominates,
$\varepsilon_{I}>\varepsilon_{V}\gg\varepsilon_{Q}$, and
$\rho_{V}\gg\rho_{Q}$. For hot plasmas, synchrotron emission dominates,
$\epsilon_{I}>\epsilon_{Q}\gg\epsilon_{V}$, and $\rho_{Q}\gg\rho_{V}$.
For a given ray, the magnetic field structure of the accretion flow determines
the coordinates (e1, e2). For given reference directions of the North and
East, we also obtain the coordinates (a, b). One therefore can use the
coefficients obtained above in the coordinates of (e1, e2) to obtain the
contributions of this ray to the Stokes parameters in the coordinates of (a,
b).
### 2.3 General Relativistic Radiative Transfer
In magnetized plasmas around black holes, photons experience both the general
relativistic light bending and plasma birefringence effects and change their
propagation directions. Fanton et al. (1997) showed how photons propagate
along the null geodesics near black holes. Broderick & Blandford (2003)
discussed how the plasma effect causes refraction, which makes the photon
propagation path deviate from the null geodesics. However, such refraction is
significant only if either the electron cyclotron frequency $\nu_{B}$ or the
electron plasma frequency $\nu_{P}$ is comparable to the observation frequency
$\nu_{\mathrm{obs}}$. In the observational band (from millimeter to near-
infrared band) we are interested in, we always have
$\nu_{B}/\nu_{\mathrm{obs}}<10^{-3}$, and
$\nu_{P}/\nu_{\mathrm{obs}}<10^{-4}$. Therefore, although we consider the
Faraday conversion and rotation effects of magnetized plasmas, we still assume
photons of both natural modes propagating along the null geodesics.
We use the ray-tracing code discussed in Huang et al. (2007) to determine the
photon trajectory from a specific direction as observed at infinity. Along the
trajectory which crosses the emission region, we record the four-wave-vector
$k^{\mu}$, accretion flow velocity $u^{\mu}$, electron temperature
$T_{e}=\gamma_{c}m_{e}c^{2}/k$, number density $n$, and three magnetic vector
$\mathrm{B}^{i}$ at each line element. We convert the three magnetic vector
$\mathrm{B}^{i}$ into four-vector $b^{\mu}$ following e.g., Gammie et al.
(2003). The four-vectors of the elliptical axes of synchrotron emission
$(e^{1\mu},e^{2\mu})$ are calculated the same way as that discussed by
Broderick & Blandford (2004). The four-vectors of the reference coordinates
$(a^{\mu},b^{\mu})$ are calculated according to the parallel transport in the
general relativistic theory with $(a^{t},b^{t})$ arbitrarily set as $(0,0)$
(Chandrasekhar, 1983). The Stokes parameters are not conserved along the
photon trajectories due to gravitational effect. However, the photon
occupation numbers
$\mathcal{N}_{S}=(\mathcal{N}_{I},\mathcal{N}_{Q},\mathcal{N}_{U},\mathcal{N}_{V})$,
defined as $\mathcal{N}_{S}=\vec{S}/\nu^{3}$, are Lorentz invariants.
Therefore, the radiative transfer equation (7) can be rewritten as
$\displaystyle\frac{\rm d^{\quad}}{\rm d\ell^{\quad}}\mathcal{N}_{S}$
$\displaystyle=$
$\displaystyle\mathcal{E}\quad-\quad\bf{\mathcal{K}}\mathcal{N}_{S},\qquad(\mathcal{E}=\frac{\vec{\varepsilon}}{\nu^{2}},\qquad\mathcal{K}=\nu\mathbf{K}),$
(46)
with the differential affine parameter $\rm d\ell=\rm ds/\nu_{\mathrm{obs}}$
and the emission frequency $\nu=-u^{\mu}k_{\mu}$.
## 3 SIMULATION RESULTS OF POLARIZATIONS
In this section, we present our simulation results in detail. The data we
adopt are mainly from (multi-epoch) linear polarization observations reported
by Aitken et al. (2000); Bower et al. (2005); Macquart et al. (2006); Marrone
et al. (2006), and Eckart et al. (2006), marked with crosses in Panels (a),
(b), and (c) of Figure 2. Two circular polarization data in Panel (d) are from
Bower et al. (2001) and Marrone et al. (2006). The ties in X-ray band are from
Baganoff et al. (2001). Other data of luminosity shown in Panel (a) are the
same adopted in Huang et al. (2008).
### 3.1 Configuration of the Magnetic Field
We adopt the two-temperature magneto-rotational instability (MRI) driven
Keplerian accretion flow model of magnetized plasmas around the central black
hole in Sgr A* [see in Liu et al. (2007a, b) for details]. A black hole mass
of $M_{\mathrm{BH}}=4.1\times 10^{6}M_{\odot}$, where $M_{\odot}$ is the solar
mass, is recently derived from observations of the orbital motion of the
short-period star S0-2 (Ghez et al., 2008). The corresponding model parameters
include the ratio of the viscous stress to the magnetic field energy density
$\beta_{\nu}$ [fixed at $0.7$, see Pessah et al. (2006)], the ratio of the
magnetic field energy density to the gas pressure $\beta_{p}$, the electron
heating rate indicated by a dimensionless parameter $C_{1}$, the mass
accretion rate $\dot{M}$, the inclination angle $i$ and the position angle
$\Theta$ of the axis perpendicular to the equatorial plane of the accretion
flow. In this model, thermal electrons are energised by sound waves with the
heating rate given by $\tau_{\rm ac}^{-1}=c_{S}^{2}/3C_{1}H\langle
v_{e}\rangle$, where $c_{S}$, $H$, and $\langle v_{e}\rangle$ are the sound
speed, the scale height of the accretion flow, and the mean speed of the
electrons, respectively, and we have assumed that the scattering mean free
path of the electrons by the sound waves is scaled with $H$. A comparison of
this heating model with other two-temperature models (Yuan et al., 2003) can
be found in Liu et al. (2007b).
MHD simulations of the MRI show that the magnetic field is dominated by its
azimuthal component due to the shearing motion of the Keplerian accretion
flow. Previous pseudo-Newtonian modelling of the sub-millimeter to NIR
polarization of Sgr A* has assumed that the magnetic field does not have
vertical and radial components (Melia et al., 2001; Bromley et al., 2001; Liu
et al., 2007a). Here we consider more realistic configurations. Most MRI
simulations adopt poloidal or toroidal magnetic field loops confined to an
initial torus of matter (Hawley & Balbus, 2002; Gammie et al., 2003).
Simulations with initial magnetic field lines perpendicular to the disk plane
tend to produce stronger magnetic fields in the final quasi-steady state
(Pessah et al., 2006). The field lines are dragged and twisted due to the
Keplerian motion of the accretion flow. If there is a large scale poloidal
magnetic field, the field lines will be nearly parallel (anti-parallel) to the
velocity field in the upper half to the equatorial plane and anti-parallel
(parallel) in the lower half. We will adopt such a mean magnetic field
configuration in the following.
### 3.2 The Fiducial Model
We first obtain a fiducial model to the spectrum, linear polarization (LP)
fraction, electric vector position angle (EVPA), and circular polarization
(CP) fraction with $\beta_{p}=0.4$, $C_{1}=0.47$, $\dot{M}=6\times
10^{17}\mathrm{g}\cdot\mathrm{s}^{-1}$, $i=40^{\circ}$, and
$\Theta=115^{\circ}$. No external Faraday rotation measure is introduced. The
corresponding results are shown by the solid lines in Panels (a),(b),(c), and
(d) of Figure 2, respectively. It is consistent with the fiducial model in
Huang et al. (2008). This Keplerian accretion flow reproduces the emission in
the millimeter to sub-millimeter bump of Sgr A*, but underestimates the
emission in the NIR and X-ray bands. The thick solid lines in Figure 2 (a)
represent the synchrotron (to the left) and synchrotron self-Comptonization
(SSC, in the middle) radiation components, and the thin solid line to the
right represents the bremsstrahlung radiation. This result is different from
that of Liu et al. (2007), where they are able to reproduce the observed NIR
flux level. In their model, an un-polarized jet/outflow component were
introduced, which effectively suppresses the LP below 100 GHz (Liu et al.,
2007a). We don’t have such a component here. The onset frequency of the LP
therefore constrains the model parameter space so that the model predicted NIR
and X-ray fluxes are more than one order of magnitude lower than the observed
values. The quiescent-state X-ray emission is extended and has been attributed
to thermal emission from plasmas near the capture radius (Baganoff et al.,
2003; Quataert, 2004; Xu et al., 2006). The X-ray emission from our accretion
flow at small radii should be lower than the observed value. However, there is
evidence for a quiescent-state NIR emission from the direction of Sgr A* (Do
et al., 2009). This emission can be caused by smaller flares or a quasi-
stationary emission component. If it is indeed powered by the accretion flow
at small radii, it must be dominated by emission regions within $6GM_{\rm
BH}/c^{2}$, where $G$ is the gravitational constant and $c$ is the speed of
light, or associated with outflows, which may also be responsible for the long
wavelength radio emissions. 111The black hole does not have a spin in our
pseudo-Newtonian model. The accretion disk has an inner boundary radius of
$6GM_{\rm BH}/c^{2}$.
As shown in Figure 2 (b) and (c), the LP fraction is a few percent in the
centimeter bands. It can be reduced to the observed level either by
depolarization of an external medium or through the dominance of an un-
polarized emission component from a jet/outflow component. In this band, only
the red-shifted side of the accretion flow is optically thin. Hence, the EVPA
is perpendicular to the accretion flow axis projection, since the magnetic
field is toroidal and the emission is dominated by the extraordinary mode.
When the observational frequency increases to $\sim$100GHz, the LP degree
increases because the Faraday rotation and Faraday conversion coefficients
decrease with the observation frequency $\nu$ as $\nu^{-2}$ and $\nu^{-3}$,
respectively. At the same frequency, the near and far sides of the accretion
flow become optically thin and dominate the polarized emission so that EVPA
preforms a $\sim 90^{\circ}$ flip to be parallel to the disk axis projection.
In the sub-millimeter band, the LP degree increases to $\sim 10\%$, explaining
observations well. Furthermore, the EVPA has a small increase of $\sim
20^{\circ}$ in this band. With further increasing of the observational
frequency, the blue-shifted side finally become optically thin at $\sim 2-3$
THz, making the whole disk optically thin with its blue-shifted side being the
dominant emission region. Thus, the EVPA flips back to be perpendicular to the
axis projection again. In the quiescent state discussed in this paper, there
is almost zero polarization in the NIR band because the non-polarized SSC
emission component dominates. However, the EVPA is still meaningful for
flaring activities dominated by the synchrotron emission.
The predicted CP fraction is shown in Figure 2 (d). The CP degree is nearly
zero in the centimeter bands, so that the observed CP fraction of several
percents (Bower et al. 2002) might be attributed to a jet/outflow component
(Beckert & Falcke 2002). At sub-millimeter and shorter wavelengths, however,
the CP amplitude increases gradually, then significantly to greater than
$10\%$ at $\sim$1 THz. We show the CP fraction in different scales for clarity
of presentation here. Notice that the LP fraction also has some oscillations
in the same band. These high LP and CP may be due to the combination of the
general relativity and birefringence effects, and can be tested with future
THz polarization observations.
Interestingly in the millimeter to sub-millimeter band, the CP degree is
mostly negative, i.e., the polarization is left-handed. Here, we adopt two
observational data for CPs, a limit of $-1.8\%$ at 112 GHz reported by Bower
et al. (2001) and a detection of $\sim-0.5\%\pm 0.3\%$ at 340GHz by Marrone et
al. (2006) (both marked with red crosses). Although they mentioned that the
detections were quite uncertain due to weather conditions and instrumental
effects, the left-handed properties seemed to be real. Our model therefore
reproduces the CP observations. However, we notice that the CP degree is very
sensitive to the magnetic field structure and model parameters. Thus, our
predictions here should be interpreted with precautions.
### 3.3 The Inclination Angle Dependence
The dependence of these results on the inclination angle $i$ is also shown in
Figure 2 with the corresponding models indicated in the legend of Panel (b).
The other model parameters have been fixed here. Results for $i=20^{\circ}$,
$30^{\circ}$, $50^{\circ}$, and $60^{\circ}$ are shown with the 3-dots-dashed,
long dashed, dot-dashed, and dotted lines, respectively. Different from the
pseudo-Newtonian cases [see Liu et al. (2007a)], the changes in the
inclination angle slightly affect the centimeter polarization and high
frequency flux density. This is due to two effects: absorption of emission
from the edge of accretion flow and amplification of emission by the GR light
bending.
However, the inclination angle affects the LP degree significantly. Generally
speaking, the onset frequency of high LP decreases and the LP degree in sub-
millimeter band increases with the increase of the inclination angle. This is
because the projected magnetic field lines are close to parallel lines in the
nearly edge-on case while becoming circular helices in the nearly face-on
case. Parallel magnetic field lines result in the highest LP degree (up to
$30\%$ in sub-millimeter band), while circular helices result in a zero LP
degree due to the rotation symmetry of the flow. The observed LP degrees in
the sub-millimeter band remain stable at $\sim 10\%$ since its discovery
(Aitken et al., 2000), which provides a constraint of $\sim 40^{\circ}\pm
10^{\circ}$ on the inclination angle of the Keplerian accretion flow.
For the EVPA, it flips at a lower frequency for the first time and a higher
frequency again. This is because the near side of the inclined disk dominates
the emission in a wider band. For the $i=60^{\circ}$ case, the near side
dominates after the first flip, and the second flip doesn’t appear until the
frequency increases to the NIR band. The second flip may disappear in the band
we are interested in for higher values of the inclination angle. The second
flip between sub-millimeter band to NIR band is quite important. A stable
difference of $\sim 80^{\circ}$ in the EVPAs between 230GHz and $\lambda$
2.2$\mu$m has been observed, which suggests a stable geometry for the
accretion plasma around Sgr A* if the NIR emission also originates from the
disk. Comparing the fiducial model and the other four cases shown in Fig.2, a
mildly-inclined $i\sim 40^{\circ}$ accretion flow appears to better reproduce
such a difference in the EVPA caused by changes in optical depth across the
emission structure. However, Meyer et al. (2007) derived a highly-inclined
$i\gtrsim 70^{\circ}$ disk to obtain significantly variable light-curves in
the NIR band. The NIR flare emission was associated with a bright spot
orbiting the black hole at the last stable orbit of a disk with a toroidal
magnetic field similar to our magnetic field configuration. They also assumed
that the sub-millimeter emission comes from a jet to explain the EVPA
difference, while the jet/outflow component in our model only dominates the
emission in millimeter and longer wavelengths. Therefore, the difference of
inclination angle predictions is caused by different assumptions of the
geometry for the emitting region, which can be checked by future VLBI
observations in the sub-millimeter band.
### 3.4 The Position Angle Dependence
The observed EVPA is related to the position angle ($\Theta$) of the accretion
flow axis. Observationally, the EVPAs in the sub-millimeter band show frequent
variability, presumably associated with flares. At 230 GHz, however, it was
detected as $\sim 117^{\circ}\pm 24^{\circ}$, showing relatively stable
values, during several epochs within more than one year (Bower et al., 2005).
Marrone et al. (2007) also reported comparable values of the EVPA at 230GHz.
At $\lambda$ 2.2$\mu$m, on the other hand, the EVPA is reported to be stable
with a value of $60^{\circ}\pm 20^{\circ}$ (Meyer et al., 2007). In the
fiducial model, we find $\Theta=114^{\circ}$ so that the predicted EVPA can
give a best fit to both measurements at 230GHz and $\lambda$ 2.2 $\mu$m. In
practice, values in the range of $\sim 115^{\circ}\pm 20^{\circ}$ are all
acceptable. Interestingly, we find that the predicted first flip in the EVPA
also fits the multi-epoch detections at 86 GHz well (Macquart et al., 2006).
Moreover, Marrone et al. (2007) observed the EVPA at 340GHz simultaneously
with that at 230GHz and reported an averaged $\sim 30^{\circ}$ increase from
230 GHz to 340 GHz. Our model also reproduces this increase in the sub-
millimeter band. Meyer et al. (2007) derived a $\Theta$ of $\sim 110^{\circ}$,
which is consistent with ours, although their emission model is quite
different.
Notice that we have assumed that all the observed EVPAs are intrinsic to the
accretion flow, i.e., the depolarization and Faraday rotation only occur in
the accretion region of Sgr A*. It is likely that the emission from the
accretion flow may experience depolarization and Faraday rotation by an
external medium described with a rotation measure (RM) $=4.4\times
10^{5}\mathrm{rad}\cdot\mathrm{m}^{2}$ derived from millimeter to sub-
millimeter observations (Macquart et al., 2006). In Panel (a) of Figure 3, we
plot the corresponding EVPAs of three well-fit models in Figure 2, with
different position angles. Multi-epoch observations of the EVPA from
$\lambda=$ 3 mm to $\lambda=$ 2.2 $\mu$m can be reproduced with
$\Theta=147^{\circ}$ for the fiducial model. Fits with another RM of
$5.6\times 10^{5}\mathrm{rad}\cdot\mathrm{m}^{2}$ derived by Marrone et al.
(2007) from their simultaneous observations at $230$ GHz and $350$ GHz are
plotted in Panel (b). $\Theta=159^{\circ}$ for the fiducial model. Obviously,
the position angle $\Theta$ depends on the external rotation measure. More
observations in the sub-millimeter and NIR bands will show whether an external
rotation measure is necessary.
### 3.5 Dependence on the Mass Accretion Rate
For these radiatively inefficient accretion flows, the gas pressure is
dominated by proton and ions, which have a relatively high temperature Liu et
al. (2007). Pessah et al. (2006) showed $\beta_{\nu}\simeq 0.7$. Then both the
viscosity (and therefore the mass accretion rate) and the magnetic field
energy density are proportional to $\beta_{p}$. The mass accretion rate
therefore determines the amplitude of the magnetic field. For a given mass
accretion rate, the density is inversely proportional to $\beta_{p}$. Figure 4
shows the dependence of the results on the mass accretion rate, where
$\beta_{p}$ and $C_{1}$ are also adjusted to reproduce the millimeter and sub-
millimeter spectrum. For $\dot{M}=1.5\times
10^{18}\mathrm{g}\cdot\mathrm{s}^{-1}$, $\beta_{p}=0.2$, $C_{1}=0.94$. The
density profile increases by a factor of 5 with respect to the fiducial model.
And for a lower accretion rate of $\dot{M}=2\times
10^{17}\mathrm{g}\cdot\mathrm{s}^{-1}$, $\beta_{p}=0.7$, $C_{1}=0.269$. The
density profile decreases by a factor of 5.3 with respect to the fiducial
model. The inclination angle of the accretion flow is fixed at $40^{\circ}$
and $\Theta$ is obtained by fitting observations at $1.3$ mm and $2.2\mu$ m
wavelengths. It can be seen that with the increase of $\dot{M}$ and therefore
the magnetic field $B$ and $n$, a lower electron temperature is needed to
reproduce the spectrum. This leads to a lower cutoff frequency for both the
synchrotron and SSC components. The emission is also strongly self-absorbed so
that the onset frequency of strong LP occurs at about 150 GHz. Opposite
effects can be seen with the decrease of $\dot{M}$ and there is a higher X-ray
emission flux due to the SSC. However, the X-ray flux is still one order of
magnitude lower than the observed quiescent-state thermal X-ray emission from
a large scale accretion flow (Xu et al. 2005). Even higher NIR and X-ray
emission may be produced with further decrease of the mass accretion rate and
increase of the electron temperature. However, radio emission will show strong
LP. An un-polarized emission component needs to be introduced to suppress the
LP to the observed level (Liu et al. 2007).
Moreover, for lower values of $\dot{M}$, the high amplitude of the CP degree
starts at lower frequencies, similar to the high LP degrees, e.g., the dash-
dotted line shown in Figure 4 (d) with $\beta_{p}=0.7$. Current observations
of the LP and CP therefore constrain the $\dot{M}$ to be near $6\times
10^{17}$ g s-1 with an uncertainty of a factor of $\sim 2$. 222This result is
for the pseudo-Newtonian potential without a black hole spin. The range of
$\dot{M}$ will be different in a Kerr metric with a significant spin.
### 3.6 Degeneracy Between the Electron Heating and Magnetic Field
We have shown that the emission spectrum and polarization give good
constraints on the orientation of the accretion flow and the mass accretion
rate. However, the constraint on the electron heating rate and the magnetic
field parameter $\beta_{p}$ is generally poor even with $\beta_{\nu}$ fixed.
Figure 5 shows several reproductions to the observed spectrum and polarization
with $\dot{M}=6\times 10^{17}$ g s-1, $\beta_{p}=0.2$; $C_{1}=0.8$ (long-
dashed lines), and $\beta_{p}=0.7$; $C_{1}=0.33$ (dashed-dotted lines). The
current observations will not be able to distinguish these models. However,
there are changes in the SSC and bremsstrahlung components. These can be
understood as the following.
For a given $\dot{M}$, the magnetic field is fixed because both of them are
proportional to $\beta_{p}N_{0}T_{p}$, where $T_{p}$ is the proton temperature
and does not change with $\beta_{p}$. With the increase $\beta_{p}$, we
increase the viscosity and therefore the radial velocity. The density will
decrease as $N_{0}\propto 1/\beta_{p}$. For a given optically thin synchrotron
luminosity required to reproduce the millimeter and sub-millimeter spectral
hump, $N_{0}B^{2}T_{e}^{2}$ should not change. Therefore the electron
temperature $T_{e}$ should be proportional to $\beta_{p}^{1/2}$. Then with the
increase of $\beta_{p}$, the synchrotron spectrum should be slightly harder,
i.e., its spectral cutoff $\propto BT_{e}^{2}\propto\beta_{p}$ should shift
toward higher frequencies. We therefore expect less low frequency SSC flux
density due to the decrease of the electron density. However the SSC component
should cut off at a higher frequency, which is proportional to
$T_{e}^{2}\propto\beta_{p}$. The SSC luminosity, which is proportional to
$N_{0}T_{e}^{2}$, however does not change. To make the SSC cut off in the
Chandra X-ray band, we will need to increase $\beta_{p}$ by a factor of $\sim
10$. However, this will reduce the flux density by the same factor. Therefore
the SSC component cannot have significant contribution to the observed
quiescent state X-ray emission. The bremsstrahlung flux density scales with
$N_{0}^{2}/T_{e}^{1/2}\propto\beta_{p}^{-9/4}$. A very low value of
$\beta_{p}$ is required for this component to have significant contribution in
the X-ray band. There is therefore a degeneracy between the electron heating
rate and the magnetic parameter. By introducing a spin parameter to the black
hole, the emission volume will be reduced and therefore enhancing the SSC
X-ray flux. One may be able to break this degeneracy with the quiescent state
NIR and X-ray flux densities.
### 3.7 Polarized Images in the Millimeter and Sub-Millimeter Bands
Sgr A* is embedded in the inter-stellar medium (ISM) on the Galactic plane.
Emission from the accretion flow will experience significant scattering, which
smoothes the image without changing its total flux density and polarization
properties. We adopt an elliptical Gaussian structure for the scattering
screen with a full width of half maximum (FWHM) along its major and minor axis
being $\vartheta_{\mathrm{maj}}=(1.39\pm 0.02)(\lambda/1{\rm\ cm})^{2}$ mas
and $\vartheta_{\mathrm{min}}=(0.69\pm 0.06)(\lambda/1{\rm\ cm})^{2}$ mas,
respectively, and a position angle $\sim 80^{\circ}$ (Shen et al., 2005). The
size of the scattering screen becomes smaller with the decrease of the
observation wavelength. In practice, the black hole shadow structure is
significantly washed out in the millimeter and longer wavelength band by this
scattering. However, the scattering becomes less important in sub-millimeter
band. We plot images of the total emission $I$, LP :$\sqrt{Q^{2}+U^{2}}$, and
CP: $V$ of the fiducial model at several wavelengths $\lambda=$ 3.5mm, 1.3mm,
0.86mm, and 0.6mm in Figure 6. The black dashes in the LP emission represent
the EVPAs $\bar{\chi}$. The left-handed and right-handed regions in the CP
emission are respectively represented with grey and red color.
The image of the total emission at $\lambda=$ 3.5 mm is smeared by the
screening and has an $\sim 80^{\circ}$-oriented elliptical Gaussian structure
with the major FWHM of $\sim(0.189\pm 0.003)$mas. This size is consistent with
but somewhat smaller than the observation of $\sim 0.21$ mas reported in Shen
et al. (2005). The LP and CP emission at $\lambda$ 3.5 mm are also smoothed
significantly. The spurs mark the EVPAs, mostly in the radial direction as a
result of the toroidal magnetic field configuration. The CP emission is
dominated by a grey region representing a left-handed polarization.
Clear black hole shadow structures can be found in images of the total
emission at 1.3 mm or shorter wavelengths. Interestingly, clumpy patterns are
seen in the sub-millimeter LP with bright and faint regions, and in the CP
emission with their left-handed and right-handed regions. Such complex
patterns are results of the gravitational light bending and the birefringence
effects posing new challenges for future polarization-sensitive VLBI
observations.
## 4 SUMMARY AND DISCUSSION
In this paper we explore the parameter space of the two-temperature MRI driven
Keplerian accretion flow model in a non-spin pseudo-Newtonian potential for
the time averaged millimeter and shorter wavelength emission from Sgr A* (Liu
et al. 2007). The model reproduces the observed emission spectrum and
polarization with an inclination angle of $40\pm 10^{\circ}$ and a mass
accretion rate of $\sim 6\times 10^{17}$ g s-1. The former is mostly
determined by the amplitude of the LP fraction and the latter is well-
constrained by the onset frequency of prominent LP. The LP is low in the
centimeter wavelength due to strong self-absorption in the emission region.
The orientation of the accretion flow projected on the sky depends on the
amplitude of an external Faraday rotation measure. And the model predicted NIR
and X-ray fluxes are more than one order of magnitude lower than the observed
low flux levels. Although nearly identical millimeter and sub-millimeter
spectrum and polarization can be obtained by adjusting the electron heating
rate and the magnetic parameter, none of these models can produce NIR and
X-ray fluxes comparable to the observed values. This strongly suggests that
the black hole is rotating so that the last stable orbit can be smaller,
resulting in more SSC and bremsstrahlung emission. We are in the process of
this study and the results will be reported in a separate paper.
The time averaged source size measurements provide critical constraints on the
model. Although we showed that our model are consistent with current
observations by simulating the images of the accretion flow at different
wavelengths. Given the challenges in imaging the black hole with the global
VLBA, a direct fit to the observed visibility is needed to fully utilize the
observed information (Doeleman et al., 2008).
Very rich information is contained in the flare observations. To fully explore
the implication of these observations, one needs to carry out GRMHD
simulations of the accretion flow. Our currently study of the time averaged
properties will lead to good constraints on the properties of the plasmas near
the black hole. These will be helpful for setting up the simulations to
recover the essential observations. Besides GRMHD simulations, the kinetics of
electron heating and acceleration must also be addressed (Liu et al. 2004).
The GR ray-tracing treatment of polarized radiation transfer we have here in
this paper, GRMHD simulations, and a kinetic model for the electron
acceleration should be combined to develop a self-consistent model for the
emission structure near the black hole. In light of continuous high resolution
and sensitivity observations, these theoretical developments will be essential
to uncover the nature of the black hole and its interaction with the plasmas
feeding it.
This work was supported in part by the National Natural Science Foundation of
China (grants 10573029, 10625314, 10633010, 10821302, 10733010, 10673010, and
10573016) and the National Key Basic Research Development Program of China
(No.2007CB815405 and 2009CB824800). SL is supported by a Marie Curie
Fellowship under the EC’s SOLAIRE Network at the University of Glasgow (MTRN-
CT-2006-035484) and an IGPP grant from LANL. ZQS is supported by the Knowledge
Innovation Program of the Chinese Academy of Sciences (Grant No. KJCX2-YW-T03)
and the Program of Shanghai Subject Chief Scientist (06XD14024). YFY is
supported by Program for New Century Excellent Talents in University. LH
thanks G.-X. Li for the assit in programming.
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Figure 1: Sketch map of polarization in synchrotron radiation: vectors ${\rm
k}$, ${\rm B}$, ${\rm e}^{1}$, and ${\rm e}^{2}$ represent the wave vector,
magnetic field, e-mode, and o-mode, respectively. $\chi$ and $d\chi$
respectively represent the EVPA defined in arbitrary coordinate system $(a,b)$
and the change of EVPA due to radiation transfer effect. See text for detail.
Figure 2: Simulated results from the fiducial model with
$\beta_{\nu}=0.7,\beta_{p}=0.4,C_{1}=0.47,\dot{M}=6\times 10^{17}{\rm g\cdot
s^{-1}},i=40^{\circ}$, and $\Theta=115^{\circ}$ (solid lines). Models with the
same parameters except $i=20^{\circ}$ (3-dots-dashed lines), $30^{\circ}$
(long dashed lines), $i=50^{\circ}$ (dash-dotted lines), and $i=60^{\circ}$
(dotted lines) are shown for comparison. Data are from Aitken et al. (2000);
Bower et al. (2005); Macquart et al. (2006); Marrone et al. (2006); Eckart et
al. (2006), Bower et al. (2001); Marrone et al. (2006), Baganoff et al.
(2001),etc.. (a) spectrum of synchrotron, SSC radiation (thick) and
bremsstrahlung radiation (thin); (b) linear polarization degrees; (c) EVPA;
(d) circular polarization degrees. Figure 3: Results of EVPA three models in
Fig.2 with external rotation measure of (a) $4.4\times 10^{5}{\rm
rad}\cdot{\rm m}^{-2}$ (Macquart et al., 2006) and (b) $5.6\times 10^{5}{\rm
rad}\cdot{\rm m}^{-2}$ (Marrone et al., 2007) considered. Figure 4:
Simulated results for models with $\beta_{p}=0.2$, $\dot{M}=1.5\times
10^{18}$g s-1 (long-dashed), $\beta_{p}=0.4$ (solid), and $\beta_{p}=0.7$,
$\dot{M}=2\times 10^{17}$g s-1 (dash-dotted), where $\beta_{p}\cdot
C_{1}=0.4\cdot 0.47$, and the luminosity in sub-millimeter band does not
change significantly. Figure 5: Simulated results from models with
$\beta_{p}=0.2$, $C_{1}=0.8$ (dashed) and $\beta_{p}=0.7$ $C_{1}=0.33$ (dot-
dashed) with $\dot{M}=6\times 10^{17}{\rm g\cdot s^{-1}}$ and $i=40^{\circ}$
unchanged.
$\lambda=3.5$mm
$\lambda=1.3$mm
$\lambda=0.86$mm
$\lambda=0.6$mm
Figure 6: Images of total intensity ($I$), LP emission
($\sqrt{Q^{2}+U^{2}}$), and CP emission ($V$) predicted for Sgr A* at
observational wavelengths of 3.5 mm, 1.3 mm, 0.86 mm, and 0.6 mm, from top to
bottom, respectively. In images of LP emission, black spurs represent the
averaged EVPAs in the nearby regions. In images of CP emission, the left-
handed and right-handed regions are respectively presented in grey and red.
|
arxiv-papers
| 2009-07-31T03:51:43 |
2024-09-04T02:49:04.376192
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Huang, Siming Liu, Zhi-Qiang Shen, Ye-Fei Yuan, Mike J. Cai, Hui\n Li, and Christopher L. Fryer",
"submitter": "Lei Huang",
"url": "https://arxiv.org/abs/0907.5463"
}
|
0907.5545
|
# Elliptic Pseudo-Differential Equations and Sobolev Spaces over $p$-adic
Fields
J. J. Rodríguez-Vega Departamento de Matemáticas, Universidad Nacional de
Colombia, Ciudad Universitaria, Bogotá D.C., Colombia. and W. A. Zúñiga-
Galindo Centro de Investigación y de Estudios Avanzados del I.P.N.,
Departamento de Matemáticas, Av. Instituto Politécnico Nacional 2508, Col. San
Pedro Zacatenco, México D.F., C.P. 07360, México.
###### Abstract.
We study the solutions of equations of type $f(D,\alpha)u=v$, where
$f(D,\alpha)$ is a $p$-adic pseudo-differential operator. If $v$ is a Bruhat-
Schwartz function, then there exists a distribution $E_{\alpha}$, a
fundamental solution, such that $u=E_{\alpha}\ast v$ is a solution. However,
it is unknown to which function space $E_{\alpha}\ast v$ belongs. In this
paper, we show that if $f(D,\alpha)$ is an elliptic operator, then
$u=E_{\alpha}\ast v$ belongs to a certain Sobolev space. Furthermore, we give
conditions for the continuity and uniqueness of $u$. By modifying the Sobolev
norm, we can establish that $f(D,\alpha)$ gives an isomorphism between certain
Sobolev spaces.
###### Key words and phrases:
$p$-adic numbers,
###### Key words and phrases:
$p$-adic fields, p-adic pseudo-differential operators, fundamental solutions,
$p$-adic Sobolev spaces.
###### 2000 Mathematics Subject Classification:
primary ; secondary .
###### 2000 Mathematics Subject Classification:
Primary: 46S10, 47S10; Secondary: 35S05, 11S80.
## 1\. Introduction
In recent years $p-$adic analysis has received a lot of attention due to its
applications in mathematical physics, see e.g. [1], [2], [3], [9], [10], [12],
[16], [19], [20] and references therein. As a consequence new mathematical
problems have emerged, among them, the study of $p$-adic pseudo-differential
equations, see e.g. [4], [6], [11], [8], [12], [13], [14], [15], [17], [20],
[21], [22], [23] and references therein. In this paper, we study the solutions
of $p$-adic elliptic pseudo-differential equations on Sobolev spaces.
A pseudo-differential operator $f(D,\beta)$ is an operator of the form
$\left(f(D,\alpha)\varphi\right)(x)=\mathcal{F}_{\xi\rightarrow
x}^{-1}\left(|f(\xi)|_{p}^{\alpha}\mathcal{F}_{x\rightarrow\xi}\phi(x)\right),\text{
\ }\phi\in S,$
where $\mathcal{F}$ denotes the Fourier transform, $\alpha$ is a positive real
number, $S$ denotes the $\mathbb{C}$-vector space of Bruhat-Schwartz functions
over $\mathbb{Q}_{p}^{n}$, and
$f(\xi)\in\mathbb{Q}_{p}[\xi_{1},\dotsc,\xi_{n}]$. If $f(\xi)$ is a
homogeneous polynomial of degree $d$ satisfying
$f(\xi)=0\text{ if and only if }\xi=0,$
then the corresponding operator is called an elliptic pseudo-differential
operator. At any case, the operator $f(D,\beta)$ is continuous and has a self-
adjoint extension with dense domain in $L^{2}(\mathbb{Q}_{p}^{n})$. This
operator is considered to be a $p$-adic analogue of a linear partial elliptic
differential operator with constant coefficients. A $p$-adic pseudo-
differential equation is an equation of type
$f(D,\alpha)u=v.$
If $v\in S$, then there exists a distribution $E_{\alpha}$, a fundamental
solution, such that $u=E_{\alpha}\ast v$ is a solution. The existence of a
fundamental solution for general pseudo-differential operators was established
by the second author in [21] by adapting the proof given by Atiyah for the
Archimedean case [5]. However, it is unknown to which function space
$E_{\alpha}\ast v$ belongs. In this paper, we show that if $f(D,\alpha)$ is an
elliptic operator, then $u=E_{\alpha}\ast v$ belongs to a certain Sobolev
space (see Theorem 3). Furthermore, we give conditions for the continuity and
uniqueness of $u$. By modifying the Sobolev norm, we can establish that
$f(D,\alpha)$ gives an isomorphism between certain Sobolev spaces, (see
Propositions 1, 2 and Theorem 4). Our approach is based on the explicit
calculation of fundamental solutions of pseudo-differential operators on
certain function spaces and the fact that elliptic pseudo-differential
operators behave like the Taibleson operator when acting on certain function
spaces (see Theorems 1, 2).
Acknowledgement. The authors wish to thank the referee for his/her careful
reading of the original manuscript.
## 2\. Preliminary Results
We summarize some basic facts about $p$-adic analysis that will be used in
this paper. For a complete exposition, we refer the reader to [18], [20].
Let $\mathbb{Q}_{p}$ be the field of the $p$-adic numbers, and let
$\mathbb{Z}_{p}$ be the ring of $p$-adic integers. For $x\in\mathbb{Q}_{p}$,
let $v(x)\in\mathbb{Z}\cup\left\\{\infty\right\\}$ denote the valuation of $x$
normalized by the condition $v(p)=1$. By definition $v(x)=\infty$ if and only
if $x=0$. Let $|x|_{p}=p^{-v(x)}$ be the normalized absolute value. Here, by
definition $|x|_{p}=0$ if and only if $x=0$. We extend the $p$-adic absolute
value to $\mathbb{Q}_{p}^{n}$ as follows:
$||x||_{p}:=\text{max}\\{|x_{1}|_{p},\dotsc,|x_{n}|_{p}\\},\text{ for
}x=(x_{1},\dotsc,x_{n})\in\mathbb{Q}_{p}^{n}.$
We define the exponent of local constancy of $\varphi(x)\in
S(\mathbb{Q}_{p}^{n})$ as the smallest integer, $l\geq 0$, with the property
that, for any $x\in\mathbb{Q}_{p}^{n}$,
$\varphi(x+x^{\prime})=\varphi(x)\text{ if }||x^{\prime}||_{p}\leq p^{-l}.$
For $x$, $y$ in $\mathbb{Q}_{p}^{n}$, we put $x\cdot
y=\sum_{i=1}^{n}x_{i}y_{i}$.
Let $\Psi$ denote an additive character of $\mathbb{Q}_{p}$, trivial on
$\mathbb{Z}_{p}$, but not on $p^{-1}\mathbb{Z}_{p}$. For $\varphi\in
S(\mathbb{Q}_{p}^{n})$, we define its Fourier transform as
$(\mathcal{F}\varphi)(\xi)=\int_{\mathbb{Q}_{p}^{n}}\Psi(-x\cdot\xi)\varphi(x)\,dx,$
where $dx$ denotes the Haar measure of $\mathbb{Q}_{p}^{n}$ normalized in such
a way that $\mathbb{Z}_{p}^{n}$ has measure one.
We denote by $\chi_{r}$, $r\in\mathbb{Z}$, the characteristic function of the
polydisc $B_{r}(0):=(p^{r}\mathbb{Z}_{p})^{n}$. For any $\varphi\in S$, we set
$r_{\varphi}:=\text{min}\\{r\in\mathbb{N}\mid\varphi|_{B_{r}\left({0}\right)}=\varphi(0)\\}.$
###### Definition 1.
We set $\mathcal{L}:=\mathcal{L}(\mathbb{Q}_{p}^{n})=\\{\varphi\in
S\mid\int_{\mathbb{Q}_{p}^{n}}\varphi(x)\,dx=0\\}$, and
$\mathcal{W}:=\mathcal{W}(\mathbb{Q}_{p}^{n})$ to be the $\mathbb{C}$-vector
space generated by the functions $\chi_{r}$, $r\in\mathbb{Z}$.
We note that any $\varphi\in S$ can be written uniquely as
$\varphi_{\mathcal{L}}+\varphi_{\mathcal{W}}$, where
$\varphi_{\mathcal{W}}=p^{r_{\varphi}n}\left(\int_{\mathbb{Q}_{p}^{n}}\phi(x)\,dx\right)\chi_{r_{\varphi}}\in\mathcal{W}$,
and $\varphi_{\mathcal{L}}=\varphi-\varphi_{\mathcal{W}}\in\mathcal{L}$.
However, $S$ is not the direct sum of $\mathcal{L}$ and $\mathcal{W}$. The
space $\mathcal{W}$ was introduced in [22], and
$\\{\mathcal{F}(\varphi)\mid\varphi\in\mathcal{L}\\}$ is a Lizorkin space of
second class [4].
### 2.1. Elliptic Pseudo-differential Operators
Let $f(\xi)\in\mathbb{Q}_{p}[\xi_{1},\dotsc,\xi_{n}]$ be a nonconstant
polynomial. A pseudo-differential operator $f(D,\alpha)$, $\alpha>0$, with
symbol $|f(\xi)|_{p}^{\alpha}$, is an operator of the form
$\left(f(D,\alpha)\varphi\right)=\mathcal{F}^{-1}\left(|f|_{p}^{\alpha}\mathcal{F}\varphi\right),$
where $\varphi\in S$.
###### Definition 2.
Let $f(\xi)\in\mathbb{Q}_{p}[\xi_{1},\dotsc,\xi_{n}]$ be a nonconstant
polynomial. We say that $f(\xi)$ is an elliptic polynomial of degree $d$, if
it satisfies: $(i)$ $f(\xi)$ is a homogeneous polynomial of degree $d$, and
$(ii)$ $f(\xi)=0\Leftrightarrow\xi=0$.
###### Lemma 1.
[23, Lemma 1] Let $f(\xi)\in\mathbb{Q}_{p}[\xi_{1},\dotsc,\xi_{n}]$ be an
elliptic polynomial of degree $d$. There exist positive constants, $C_{0}(f)$
and $C_{1}(f)$, such that
$C_{0}(f)||\xi||_{p}^{d}\leq|f(\xi)|_{p}\leq C_{1}(f)||\xi||_{p}^{d},\text{
for every }\xi\in\mathbb{Q}_{p}^{n}.$
We note that if $f(\xi)$ is elliptic, then $cf(\xi)$ is elliptic for any
$c\in\mathbb{Q}_{p}^{\times}$. For this reason, we will assume from now on
that the elliptic polynomials have coefficients in $\mathbb{Z}_{p}$.
###### Lemma 2.
[23, Lemma 3] Let $f(\xi)\in\mathbb{Q}_{p}[\xi_{1},\dotsc,\xi_{n}]$ be an
elliptic polynomial of degree $d$. Let $A\subset\mathbb{Q}_{p}^{n}$ be a
compact subset such that $0\notin A$. Then there exists a positive integer
$m=m(A,f)$ such that $|f(\xi)|_{p}\geq p^{-m}$, for any $\xi\in A$.
Furthermore, for any covering of $A$ of the form $\cup_{i=1}^{L}B_{i}$, with
$B_{i}=z_{i}+(p^{m}\mathbb{Z}_{p})^{n}$, we have $|f(\xi)|_{p}=|f(z_{i})|_{p}$
for any $\xi\in B_{i}$.
###### Definition 3.
Let $f(\xi)\in\mathbb{Z}_{p}[\xi_{1},\dotsc,\xi_{n}]$ be an elliptic
polynomial of degree $d$. We will say that $|f|_{p}^{\beta}$ is an elliptic
symbol, and that $f(D,\beta)$ is an elliptic pseudo-differential operator of
order $d$.
### 2.2. Igusa’s local zeta functions
Let $g(x)\in\mathbb{Q}_{p}[x]$, $x=(x_{1},\dotsc,x_{n})$, be a non-constant
polynomial. Igusa’s local zeta function associated to $g(x)$ is the
distribution
$\langle|g|_{p}^{s},\varphi\rangle=\int\limits_{\mathbb{Q}_{p}^{n}\smallsetminus
g^{-1}(0)}|g(x)|_{p}^{s}\varphi(x)\,dx,$
for $s\in\mathbb{C}$, $\text{Re}(s)>0$, where $\varphi\in S$, and $dx$ denotes
the normalized Haar measure of $\mathbb{Q}_{p}^{n}$. The local zeta functions
were introduced by Weil and their basic properties for general $g(x)$ were
first studied by Igusa. A central result in the theory of local zeta functions
established that $|g|_{p}^{s}$ admits a meromorphic continuation to the
complex plane such that $\langle|g|_{p}^{s},\varphi\rangle$ is rational
function of $p^{-s}$ for each $\varphi\in S$. Furthermore, there exists a
finite set $\cup_{E\in\mathcal{E}}\\{(N_{E},n_{E})\\}$ of pairs of positive
integers such that
$\prod\limits_{E\in\mathcal{E}}(1-p^{-n_{E}-N_{E}s})|g|_{p}^{s}$
is a holomorphic distribution on $S$. In particular, the real parts of the
poles of $|g|_{p}^{s}$ are negative rational numbers see [7, Chap. 8]. The
existence of a meromorphic continuation for the distribution $|g|_{p}^{s}$
implies the existence of a fundamental solution for the pseudo-differential
operator with symbol $|g|_{p}^{\alpha}$, [21].
For a fixed $\varphi\in S$, we denote the integral
$\langle|g|_{p}^{s},\varphi\rangle$ by $Z_{\varphi}(s,g)$. In particular,
$Z(s,g)=Z_{\chi_{0}}(s,g)$.
###### Lemma 3.
Let $f(x)\in\mathbb{Z}_{p}[x]$, $x=(x_{1},\dotsc,x_{n})$, be an elliptic
polynomial of degree $d$. Then
$Z(s,f)=\dfrac{L(p^{-s})}{1-p^{-ds-n}},$
where $L(p^{-s})$ is a polynomial in $p^{-s}$ with rational coefficients.
Furthermore, $s=-n/d$ is a pole of $Z(s,f)$.
###### Proof.
Let $A=\\{x\in\mathbb{Z}_{p}^{n}\mid\text{ord}(x_{i})\geq d,\quad
i=1,\dotsc,n\\}$, and
$A^{\prime}=\\{x\in\mathbb{Z}_{p}^{n}\mid\text{ord}(x_{i})<d,\text{ for some
}i\\}$. Then $\mathbb{Z}_{p}^{n}$ is the disjoint union of $A$ and
$A^{\prime}$ and
$\displaystyle Z(s,f)$
$\displaystyle=\int_{A}|f(x)|_{p}^{s}\,dx+\int_{A^{\prime}}|f(x)|_{p}^{s}\,dx$
$\displaystyle=p^{-ds-n}Z(s,f)+\int_{A^{\prime}}|f(x)|_{p}^{s}\,dx,$
i.e., $Z(s,f)=\frac{1}{1-p^{-ds-n}}\int_{A^{\prime}}|f(x)|_{p}^{s}\,dx$. Since
$A^{\prime}$ is compact, by applying Lemma 2, we find a covering of
$A^{\prime}=\cup_{i=1}^{L}B_{i}$, where $|f|_{p}$ is constant on each $B_{i}$.
Hence,
$\int_{A^{\prime}}|f(x)|_{p}^{s}\,dx=p^{-nm}\sum_{i=1}^{L}|f(z_{i})|_{p}^{s},$
and
$Z(s,f)=\dfrac{p^{-nm}\sum_{i=1}^{L}|f(z_{i})|_{p}^{s}}{1-p^{-ds-n}}.$
∎
### 2.3. The Riesz Kernel
We collect some well-know results about the Riesz kernel that will be used in
the next sections, we refer the reader to [18] or [20] for further details.
The $p$-adic _Gamma function_ $\Gamma_{p}^{(n)}(s)$ is defined as follows:
$\Gamma_{p}^{(n)}(s)=\frac{1-p^{s-n}}{1-p^{-s}}\text{, }s\in\mathbb{C},{\
}s\neq 0.$
The Gamma function is meromorphic with simple zeros at $n+\frac{2\pi i}{\ln
p}\mathbb{Z}$ and unique simple pole at $s=0$. In addition, it satisfies
$\Gamma_{p}^{(n)}(s)\Gamma_{p}^{(n)}(n-s)=1,\text{ for
}s\notin\\{0\\}\cup\\{n+\frac{2\pi i}{\ln p}\mathbb{Z}\\}.$
The _Riesz kernel_ $\mathit{R}_{s}$ is the distribution determined by the
function
$\mathit{R}_{s}(x)=\frac{||x||_{p}^{s-n}}{\Gamma_{p}^{(n)}(s)},\quad\text{Re}(s)>0,{\
}s\notin n+\frac{2\pi i}{\ln p}\mathbb{Z},\quad x\in\mathbb{Q}_{p}^{n}.$
The Riesz kernel has, as a distribution, a meromorphic continuation to
$\mathbb{C}$ given by
$\displaystyle\left\langle\mathit{R}_{s}(x),\varphi(x)\right\rangle$
$\displaystyle=\frac{1-p^{-n}}{1-p^{s-n}}\varphi(0)+\frac{1-p^{-s}}{1-p^{s-n}}\int_{||x||_{p}>1}||x||_{p}^{s-n}\varphi(x)\,dx$
$\displaystyle+\frac{1-p^{-s}}{1-p^{s-n}}\int_{||x||_{p}\leq
1}||x||_{p}^{s-n}(\varphi(x)-\varphi(0))\,dx,$
with poles at $n+\frac{2\pi i}{\ln p}\mathbb{Z}$. In particular, for
$\text{Re}(s)>0$,
$\langle\mathit{R}_{s}(x),\varphi(x)\rangle=\frac{1-p^{-s}}{1-p^{s-n}}\int_{\mathbb{Q}_{p}^{n}}\varphi(x)||x||_{p}^{s-n}\,dx,\quad
s\notin n+\frac{2\pi i}{\ln p}\mathbb{Z},$ (2.1)
$\langle\mathit{R}_{-s}(x),\varphi(x)\rangle=\frac{1-p^{s}}{1-p^{-s-n}}\int_{\mathbb{Q}_{p}^{n}}(\varphi(x)-\varphi(0))||x||_{p}^{-s-n}\,dx.$
In the case $s=0$, by passing to the limit, we obtain
$\langle\mathit{R}_{0}(x),\varphi(x)\rangle:=\lim_{s\rightarrow
0}\left\langle\mathit{R}_{s}(x),\varphi(x)\right\rangle=\varphi(0),$
i.e., $\mathit{R}_{0}(x)=\delta\left(x\right)$, the Dirac delta function.
Therefore, $\mathit{R}_{s}\in S^{\prime}(\mathbb{Q}_{p}^{n})$, for
$s\in\mathbb{C\setminus}\left\\{n+\frac{2\pi i}{\ln p}\mathbb{Z}\right\\}$.
###### Remark 1.
The distribution $||x||_{p}^{s}$, $\text{Re}(s)>0$, admits the following
meromorphic continuation,
$\displaystyle\left\langle||x||_{p}^{s},\varphi(x)\right\rangle$
$\displaystyle=\frac{1-p^{-n}}{1-p^{-s-n}}\varphi(0)+\int_{||x||_{p}>1}||x||_{p}^{s}\varphi(x)\,dx$
$\displaystyle+\int_{||x||_{p}\leq
1}||x||_{p}^{s}(\varphi(x)-\varphi(0))\,dx,\quad\varphi\in S.$
In particular, all the poles of $||x||_{p}^{s}$ have real part equal to $-n$.
###### Lemma 4 ([18, Chap. III, Theorem 4.5]).
As element of $S^{\prime}(\mathbb{Q}_{p}^{n})$, $\left(\mathcal{F}\mathit{\
R}_{s}\right)(x)$ equals $||x||_{p}^{-s}$, for $s\notin n+\frac{2\pi i}{\ln
p}\mathbb{Z}$.
The following explicit formula will be used in the next sections.
###### Lemma 5.
Let $f(x)\in\mathbb{Q}_{p}[x]$, $x=(x_{1},\dotsc,x_{n})$, be an elliptic
polynomial of degree $d$. Then
$|f|_{p}^{s}=\frac{(1-p^{ds})L(p^{-s})}{(1-p^{-n})(1-p^{-ds-n})}\mathit{R}_{ds+n},\quad
s\in\mathbb{C}$
as distributions on $\mathcal{W}$. Here $L(p^{-s})$ is the numerator of
$Z(s,f)$ which is a polynomial in $p^{-s}$ with rational coefficients.
###### Proof.
Let $\varphi\in\mathcal{W}$, then
$\varphi(x)=\sum_{i}c_{i}\chi_{r_{i}}(x),$
where $c_{i}\in\mathbb{C}$, $r_{i}\in\mathbb{Z}$ (recall that
$\mathcal{F}(\chi_{r})=p^{-nr}\chi_{-r}$). The action of $|f|_{p}^{s}$ on
$\mathcal{F}\varphi$ can be explicitly described as follows:
$\langle|f|_{p}^{s},\mathcal{F}\varphi\rangle=\sum_{i}c_{i}\langle|f|_{p}^{s},p^{-nr_{i}}\chi_{-r_{i}}\rangle,$
but
$\langle|f|_{p}^{s},p^{-nr_{i}}\chi_{-r_{i}}\rangle=p^{-nr_{i}}\int_{\mathbb{Q}_{p}^{n}}|f(x)|_{p}^{s}\chi_{-r_{i}}(x)\,dx=p^{dr_{i}s}Z(s,f),$
for $\text{Re}(s)>0$, thus
$\langle|f|_{p}^{s},\mathcal{F}\varphi\rangle=Z(s,f)\sum_{i}c_{i}p^{dr_{i}s},\quad\text{Re}(s)>0.$
On the other hand,
$\langle\dfrac{1-p^{ds}}{1-p^{-n}}\mathit{R}_{ds+n},p^{-nr_{i}}\chi_{-r_{i}}\rangle=\langle\dfrac{1-p^{-ds-n}}{1-p^{-n}}||x||_{p}^{ds},p^{-nr_{i}}\chi_{-r_{i}}\rangle=p^{dr_{i}s},$
for every $r_{i}\in\mathbb{Z}$ and $\text{Re}(s)>0$. Then we have
$\langle|f|_{p}^{s},\mathcal{F}\varphi\rangle=\dfrac{1-p^{ds}}{1-p^{-n}}Z(s,f)\langle\mathit{R}_{ds+n},\mathcal{F}\varphi\rangle,$
for $\text{Re}(s)>0$. Now $Z(s,f)$ and $\mathit{R}_{ds+n}$ have a meromorphic
continuation to the complex plane, therefore this formula extends to
$\mathbb{C}$. Finally, since the Fourier transform establishes a
$\mathbb{C}$-isomorphism on $\mathcal{W}$, it is possible remove the Fourier
transform symbol. ∎
### 2.4. The Taibleson Operator
###### Definition 4.
The Taibleson pseudo-differential operator $D_{T}^{\alpha}$, $\alpha>0$, is
defined as
$(D_{T}^{\alpha}\varphi)(x)=\mathcal{F}_{\xi\rightarrow
x}^{-1}\left(||\xi||_{p}^{\alpha}\mathcal{F}_{x\rightarrow\xi}\varphi\right)\text{,
for }\varphi\in S.$
As a consequence of the Lemma 4 and (2.1), one gets
$\displaystyle\left(D_{T}^{\alpha}\varphi\right)\left(x\right)$
$\displaystyle=\left(\mathit{k}_{-\alpha}\ast\varphi\right)\left(x\right)=$
$\displaystyle\frac{1-p^{\alpha}}{1-p^{-\alpha-n}}\int_{\mathbb{Q}_{p}^{n}}||y||_{p}^{-\alpha-n}(\varphi(x-y)-\varphi(x))\,dy.$
The right-hand side of previous formula makes sense for a wider class of
functions than $S(\mathbb{Q}_{p})$, for example, for the class
$\mathfrak{E}_{\alpha}(\mathbb{Q}_{p}^{n})$ of locally constant functions
$\varphi(x)$ satisfying
$\int_{||x||_{p}\geq 1}||x||_{p}^{-\alpha-n}|\varphi(x)|\,dx<\infty.$
###### Remark 2.
As a consequence of the previous observations we may assume that the constant
functions are contained in the domain of $D_{T}^{\alpha}$, and that
$D_{T}^{\alpha}\varphi=0$, for any constant function.
## 3\. Fundamental Solutions for the Taibleson Operator
We now consider the following pseudo-differential equation:
(3.1) $D_{T}^{\alpha}u=v,\quad\text{with $v\in S$},\text{ and }\alpha>0.$
We say that $E_{\alpha}\in S^{\prime}$ is a fundamental solution of (3.1) if
$E_{\alpha}\ast v$ is a solution.
###### Lemma 6.
If $E_{\alpha}$ is a fundamental solution of (3.1), then for any constant $c$,
$E_{\alpha}+c$ is also a fundamental solution.
###### Proof.
Let $E_{\alpha}$ a fundamental solution for (3.1), then
$\displaystyle D_{T}^{\alpha}((E_{\alpha}+c)\ast v)$
$\displaystyle=D_{T}^{\alpha}((E_{\alpha}\ast v)+(c\ast v))$
$\displaystyle=u+D_{T}^{\alpha}(c\ast v)=u,$
because $u$ and the constant function, $c\ast v$, are in the domain of
$D_{T}^{\alpha}$. ∎
###### Theorem 1.
A fundamental solution of (3.1) is
$E_{\alpha}(x)=\begin{cases}\dfrac{1-p^{-\alpha}}{1-p^{\alpha-n}}||x||_{p}^{\alpha-n}&\text{
if $\alpha\neq n$}\\\ \dfrac{1-p^{n}}{p^{n}\ln p}\ln(||x||_{p})&\text{ if
$\alpha=n$.}\end{cases}$
###### Proof.
The proof is based on the ideas introduced in [21]. The existence of a
fundamental solution $E_{\alpha}$ is equivalent to the existence of a
distribution $\mathcal{F}E_{\alpha}$ satisfying
(3.2) $||x||_{p}^{\alpha}\mathcal{F}E_{\alpha}=1,$
as distributions. Let
$||x||_{p}^{s}=\sum\limits_{m\in\mathbb{Z}}c_{m}(s+\alpha)^{m}$ be the Laurent
expansion at $-\alpha$ with $c_{m}\in S^{\prime}$ for all $m$. The existence
of this expansion is a consequence of the completeness of $S^{\prime}$ (see
e.g. [7, pp. 65-66]). Since the real parts of the poles of the meromorphic
continuation of $||x||_{p}^{s}$ are negative rational numbers (cf. Remark 1),
$||x||_{p}^{s+\alpha}=||x||_{p}^{\alpha}||x||_{p}^{s}$ is holomorphic at
$s=-\alpha$. Therefore, $||x||_{p}^{\alpha}c_{m}=0$ for all $m<0$ and
$||x||_{p}^{s+\alpha}=||x||_{p}^{\alpha}c_{0}+\sum_{m=1}^{\infty}||x||_{p}^{\alpha}c_{m}(s+\alpha)^{m}.$
By using the Lebesgue dominated convergence theorem, one verifies that
$\underset{s\rightarrow-\alpha}{\text{lim}}\langle||x||_{p}^{s+\alpha},\phi\rangle=\int_{\mathbb{Q}_{p}^{n}}\phi(x)\,dx=\langle
1,\phi\rangle,$
and then we can take $\mathcal{F}E_{\alpha}=c_{0}$. Furthermore, if $-\alpha$
is not a pole of $||x||_{p}^{s}$,
(3.3)
$\mathcal{F}E_{\alpha}=\underset{s\rightarrow-\alpha}{\text{lim}}||x||_{p}^{s}.$
To calculate $c_{0}$, consider the following two cases.
Case $\mathbf{\alpha\neq n}$.
We use (3.3) and the Lemma 4, i.e.,
$\int_{(\mathbb{Q}_{p}^{\times})^{n}}||x||_{p}^{s}\mathcal{F}(\varphi)(x)\,dx=\dfrac{1-p^{s}}{1-p^{-s-n}}\int_{(\mathbb{Q}_{p}^{\times})^{n}}||x||_{p}^{-s-n}\varphi(x)\,dx,$
for $s\neq n+(2\pi i/\ln p)\mathbb{Z}$. If $\alpha\neq n$, by (3.3),
$\langle
E_{\alpha},\mathcal{F}(\varphi)\rangle=\underset{s\rightarrow-\alpha}{\text{lim}}\int_{(\mathbb{Q}_{p}^{\times})^{n}}||x||_{p}^{s}\mathcal{F}(\varphi)(x)\,dx.$
If $\alpha>n$, by the Lebesgue dominated convergence theorem, we can
interchange the limit and the integral. If $0<\alpha<n$, by taking into
account that
$\int_{||x||_{p}\leq 1}||x||_{p}^{\alpha-n}\,dx<+\infty,\quad\text{for
}0<\alpha<n,$
and by using Lebesgue dominated convergence theorem, we can exchange the limit
and the integral. Therefore,
$\displaystyle\langle E_{\alpha},\varphi\rangle$
$\displaystyle=\frac{1-p^{-\alpha}}{1-p^{\alpha-n}}\int_{(\mathbb{Q}_{p}^{\times})^{n}}||x||_{p}^{\alpha-n}\varphi(x)\,dx$
$\displaystyle=\frac{1-p^{-\alpha}}{1-p^{\alpha-n}}\int_{\mathbb{Q}_{p}^{n}}||x||_{p}^{\alpha-n}\varphi(x)\,dx.$
Set $\overset{\sim}{\varphi}(x)=\varphi(-x)$, with $\varphi\in S$. The results
follow by replacing $\varphi$ by $\mathcal{F}(\overset{\sim}{\varphi})$
because
$\mathcal{F}\bigl{(}\mathcal{F}(\overset{\sim}{\varphi})\bigr{)}=\varphi$.
Case $\mathbf{\alpha=n}$.
We compute the constant term, $c_{0}$, in the expansion
$\langle||x||_{p}^{s},\mathcal{F}(\varphi)\rangle=\sum_{m\in\mathbb{Z}}\langle
c_{m},\mathcal{F}(\varphi)\rangle(s+n)^{m}.$
Since
$\displaystyle\langle||x||_{p}^{s},\mathcal{F}(\varphi)\rangle$
$\displaystyle=\frac{1-p^{s}}{1-p^{-s-n}}\int_{\mathbb{Q}_{p}^{n}}||x||_{p}^{-s-n}\varphi(x)\,dx$
$\displaystyle=(1-p^{s})\int_{\mathbb{Q}_{p}^{n}}\frac{p^{v(x)(s+n)}}{1-p^{-s-n}}\varphi(x)\,dx,$
where $x=(x_{1},\dotsc,x_{n})$, $v(x):=\underset{1\leq i\leq
n}{\min}v(x_{i})$, and $||x||_{p}=p^{-v(x)}$, by expanding
$\displaystyle\dfrac{(1-p^{s})p^{v(x)(s+n)}}{1-p^{-s-n}}$
$\displaystyle=\dfrac{1-p^{-n}}{\ln p}(s+n)^{-1}$
$\displaystyle+\dfrac{(1-p^{-1})v(x)\ln p-\frac{\ln p}{p}+\frac{(p-1)}{2p}\ln
p}{\ln p}+O(\left(s+n\right)),$
one gets
$\langle E_{n},\varphi\rangle=\langle
c_{0},\varphi\rangle=\int_{\mathbb{Q}_{p}^{n}}\Bigl{(}\frac{1-p^{n}}{p^{n}\ln(p)}\ln(||x||_{p})+\frac{p^{n}-3}{2p^{n}}\Bigr{)}\varphi(x)\,dx.$
The announced results follow by replacing $\varphi$ by
$\mathcal{F}(\overset{\sim}{\varphi})$, $\varphi\in S$, and using the fact
that the fundamental solution is determined up to the addition of a constant
(cf. Lemma 6). ∎
In the case $n=1$, the previous result is already known, see e.g. [14, Theorem
2.1].
## 4\. Fundamental Solutions for Elliptic Operators
###### Theorem 2.
Let $f(D,\alpha)$ be an elliptic operator of order $d$. Then, a fundamental
solution $E_{\alpha}$ of $f(D,\alpha)u=v$, $\alpha>0$, and $v\in\mathcal{W}$,
is given by
$E_{\alpha}(x)=\begin{cases}\dfrac{L(p^{\alpha})(1-p^{-d\alpha})}{(1-p^{-n})(1-p^{d\alpha-n})}||x||_{p}^{d\alpha-n}&\text{
as a distribution on $\mathcal{W}$, with $\alpha\neq n/d$}\\\ \,&\\\
\dfrac{L(p^{n/d})(1-p^{n})}{(1-p^{-n})(p^{n}\ln p)}\ln(||x||_{p})&\text{ as a
distribution on $\mathcal{W}$, with $\alpha=n/d$},\end{cases}$
where $L(p^{-s})$ is the numerator of $Z(s,f)$.
###### Proof.
As we mention before, the problem of the existence of a fundamental solution,
$E_{\alpha}$, is equivalent to the existence of a distribution
$\mathcal{F}E_{\alpha}$ satisfying
$|f|_{p}^{\alpha}\mathcal{F}E_{\alpha}=1\text{ in }S^{\prime}.$
By Lemma 5,
$\langle|f|_{p}^{\alpha},\varphi\rangle=\langle\frac{(1-p^{d\alpha})L(p^{-\alpha})}{(1-p^{-n})(1-p^{-d\alpha-n})}\mathit{R}_{d\alpha+n},\varphi\rangle$
$\varphi\in\mathcal{W}$, $s\in\mathbb{C}$. The result follows by reasoning as
in the proof of Theorem 1, and by the fact that the space $\mathcal{W}$ is
invariant under the Fourier transform. ∎
###### Corollary 1.
With the hypotheses of the previous theorem, and assuming that $\alpha\neq
n/d$, we have
$|\mathcal{F}(E_{\alpha}\ast\varphi)(x)|\leq
C(\alpha)||x||_{p}^{-d\alpha}|\mathcal{F}(\varphi)(x)|,$
for all $x\in\mathbb{Q}_{p}^{n}$, and $\varphi\in\mathcal{W}$.
## 5\. Solutions of Elliptic Pseudo-Differential Equations in Sobolev Spaces
Given $\phi\in\mathcal{S}$ and $l$ a non-negative number, we define
$||\phi||_{H^{l}}^{2}=\int_{\mathbb{Q}_{p}^{n}}[\text{max}(1,||\xi||_{p})]^{2l}|\mathcal{F}(\phi)(\xi)|^{2}\,d\xi.$
We call the completion of $\mathcal{S}$ with respect to $||\cdot||_{H^{l}}$
the $l$-Sobolev space $H^{l}:=H^{l}(\mathbb{Q}_{p}^{n})$.
We note that $H^{l}$ contains properly the space of test functions, $S$.
Indeed, consider the function
$f(x)=\begin{cases}0&\text{ if }||x||_{p}\leq 1\\\ ||x||_{p}^{-\beta}&\text{
if }||x||_{p}>1\end{cases}$
with $\beta>n$. A direct calculation shows that
$||f||_{H^{l}}^{2}=\int\limits_{||\xi||_{p}\leq
1}\Bigl{|}\frac{(1-p^{-n})(1-||\xi||_{p}^{\beta-n}p^{n-\beta})}{(1-p^{n-\beta})}-p^{-\beta}||\xi||_{p}^{\beta-n}\Bigr{|}^{2}\,d\xi.$
Thus, $||f||_{H^{l}}^{2}<\infty$, but $f$ does not have compact support.
###### Lemma 7.
If $l>n/2$, then there exists an embedding of $H^{l}$ into the space of
uniformly continuous functions.
###### Proof.
Let $\phi\in H^{l}$. Since the Fourier transform of a function in $L^{1}$ is
uniformly continuous, it is sufficient to show that $\mathcal{F}(\phi)\in
L^{1}$. By using the Hölder inequality and the fact that
$\int_{\mathbb{Q}_{p}^{n}}(\text{max}(1,||\xi||_{p}))^{-2l}\,d\xi<+\infty\text{,
for }l>n/2\text{,}$
we have
$\int_{\mathbb{Q}_{p}^{n}}|\mathcal{F}(\phi)(\xi)|\,d\xi=\int_{\mathbb{Q}_{p}^{n}}\frac{(\text{max}(1,||\xi||_{p}))^{l}}{(\text{max}(1,||\xi||_{p}))^{l}}|\mathcal{F}(\phi)(\xi)|\,d\xi\leq
C||\phi||_{H^{l}}.$
∎
###### Lemma 8.
For any $\alpha>0$ and $l\geq 0$, the mapping
$f(D,\alpha):H^{l+d\alpha}\rightarrow H^{l}$ is a well-defined continuous
mapping between Banach spaces.
###### Proof.
Let $\phi\in S$. Since $f(D,\alpha)$ is an elliptic operator, by Lemma 1, we
have that
$\displaystyle||f(D,\alpha)\phi||_{H^{l}}^{2}$
$\displaystyle=\int_{\mathbb{Q}_{p}^{n}}[\text{max}(1,||\xi||_{p})]^{2l}|f(\xi)|^{2\alpha}|\mathcal{F}(\phi)(\xi)|^{2}\,d\xi$
$\displaystyle\leq
C_{1}\int_{\mathbb{Q}_{p}^{n}}[\text{max}(1,||\xi||_{p})]^{2(l+\alpha)}|\mathcal{F}(\phi)(\xi)|^{2}\,d\xi=C_{1}||\phi||_{H^{l+\alpha}}^{2}.$
The result follows from the fact that $S$ is dense in $H^{l+d\alpha}$. ∎
###### Remark 3.
Let $\beta$ be a positive real number, and let
$I(\beta):=\int\limits_{||\varepsilon||_{p}\leq
1}||\varepsilon||_{p}^{\beta}\,d\varepsilon.$
Then
$I(\beta)=\dfrac{1-p^{-n}}{1-p^{-n-n\beta}},\text{ for }\beta>-n.$
Indeed,
$\displaystyle I(\beta)$
$\displaystyle=\int\limits_{||\varepsilon||_{p}<1}||\varepsilon||_{p}^{\beta}\,d\varepsilon+\int\limits_{||\varepsilon||_{p}=1}\,d\varepsilon$
$\displaystyle=\int\limits_{||\varepsilon||_{p}<1}||\varepsilon||_{p}^{\beta}\,d\varepsilon+1-p^{-n}.$
By making the change of variables $\varepsilon_{i}=px_{i}$, $i=1,\dotsc,n$, we
have
$I(\beta)=p^{-n-n\beta}I(\beta)+1-p^{-n}.$
###### Theorem 3.
Let $f(D,\alpha)$, $0<\alpha<n/2d$ be an elliptic pseudo-differential operator
of order $d$. Let $l$ be a positive real number satisfying $l>n/2$. Then, the
equation
$f(D,\alpha)u=v\quad\left(v\in S\right),$
has a unique uniformly continuous solution $u\in H^{l+d\alpha}$.
###### Proof.
Let $v\in\mathcal{S}$, then $v=v_{\mathcal{W}}+v_{\mathcal{L}}$, where
$v_{\mathcal{W}}\in\mathcal{W}$ and $v_{\mathcal{L}}\in\mathcal{L}$. Thus, in
order to prove the existence of a solution $u$, it is sufficient to show that
the two following equations have solutions:
(5.1) $f(D,\alpha)u_{\mathcal{W}}=v_{\mathcal{W}},$ (5.2)
$f(D,\alpha)u_{\mathcal{L}}=v_{\mathcal{L}}.$
We first consider equation (5.1). By Theorem 2,
$u_{\mathcal{W}}=E_{\alpha}\ast v_{\mathcal{W}}$ is a solution of (5.1), and
by Corollary 1, we have
$\displaystyle||u_{\mathcal{W}}||_{H^{l+d\alpha}}^{2}$
$\displaystyle=\int_{\mathbb{Q}_{p}^{n}}[\text{max}(1,||\xi||_{p})]^{2(l+d\alpha)}|\mathcal{F}(E_{\alpha}\ast
v_{\mathcal{W}})(\xi)|^{2}\,d\xi$
$\displaystyle=C(\alpha,d,n)\int_{\mathbb{Q}_{p}^{n}}[\text{max}(1,||\xi||_{p})]^{2(l+d\alpha)}|\left\|\xi\right\|_{p}^{-2d\alpha}|\mathcal{F}(v_{\mathcal{W}})(\xi)|^{2}\,d\xi$
$\displaystyle=C(\alpha,d,n)\Bigl{\\{}\int_{||\xi||_{p}\leq
1}||\xi||_{p}^{-2d\alpha}|\mathcal{F}(v_{\mathcal{W}})(\xi)|^{2}\,d\xi$
$\displaystyle\hskip
28.45274pt+\int_{||\xi||_{p}>1}||\xi||_{p}^{2l}|\mathcal{F}(v_{\mathcal{W}})(\xi)|^{2}\,d\xi\Bigr{\\}}.$
We now recall that $v_{\mathcal{W}}(\xi)=p^{rn}C\chi_{r}(\xi)$, with $r>0$.
Then, $\mathcal{F}(v_{\mathcal{W}})(\xi)=C\chi_{-r}(\xi)$ and
$\displaystyle||u_{\mathcal{W}}||_{H^{l+d\alpha}}^{2}$ $\displaystyle\leq
C(\alpha,d,n)\Bigl{\\{}C^{2}p^{2rn}\int\limits_{||\varepsilon||_{p}\leq
1}||\varepsilon||_{p}^{-2d\alpha}\,d\varepsilon$ $\displaystyle\hskip
71.13188pt+||v_{\mathcal{W}}||_{H^{l}}^{2}\Bigr{\\}}$ $\displaystyle\leq
C(\alpha,d,n)\Bigl{\\{}C_{1}(\alpha,d,n)+||v_{\mathcal{W}}||_{H^{l}}^{2}\Bigr{\\}},$
since $-2d\alpha>-n$, cf. Remark 3. Therefore $u_{\mathcal{W}}\in
H^{l+d\alpha}$.
We now consider equation (5.2). Since
$\mathcal{F}(u_{\mathcal{L}})=\mathcal{F}(v_{\mathcal{L}})|f|_{p}^{-\alpha},$
and $f$ is elliptic,
$|\mathcal{F}(u_{\mathcal{L}})(\xi)|\leq
C||\xi||^{-d\alpha}|\mathcal{F}(v_{\mathcal{L}})(\xi)|,\text{(cf. Lemma
\ref{Lemma 1})}.$
Then,
$\displaystyle||u_{\mathcal{L}}||_{H^{l+d\alpha}}^{2}$
$\displaystyle\leq\int_{||\xi||_{p}\leq
1}||\xi||_{p}^{-2d\alpha}|\mathcal{F}(v_{\mathcal{L}})(\xi)|^{2}\,d\xi$
$\displaystyle\hskip
28.45274pt+\int_{||\xi||_{p}>1}||\xi||_{p}^{2l}|\mathcal{F}(v_{\mathcal{L}})(\xi)|^{2}\,d\xi.$
The second integral is bounded by $||v_{\mathcal{L}}||_{H^{l}}^{2}$. For the
first integral, we observe that if $0<\alpha<n/2d$, then
$\int_{||\xi||_{p}\leq 1}||\xi||_{p}^{-2d\alpha}|\varphi(\xi)|^{2}\,d\xi\leq
C||\varphi||_{L^{2}},$
for any $\varphi\in S$. Therefore,
$||u_{\mathcal{L}}||_{H^{l+d\alpha}}^{2}\leq
C||\mathcal{F}(v_{\mathcal{L}})||_{L^{2}}+||v_{\mathcal{L}}||_{H^{l}}^{2}.$
In this way, we established the existence of $u\in H^{l+d\alpha}$ which is
uniformly continuous, by Lemma 7, such that $f(D,\alpha)u=v$, for any $v\in
S$. Finally, we show that $u$ is unique. Indeed, if $f(D,\alpha)u^{\prime}=v$,
then
$f(D,\alpha)(u-u^{\prime})=0,\qquad\text{i.e.,
}|f|_{p}^{\alpha}\mathcal{F}(u-u^{\prime})=0,$
and thus $\mathcal{F}(u-u^{\prime})(\xi)=0$ if $\xi\neq 0$, since $f$ is
elliptic. Then $\Psi(x\cdot\xi)(u-u^{\prime})(\xi)=0$ almost everywhere, and a
fortiori $(u-u^{\prime})(\xi)=0$ almost everywhere, and by the continuity of
$u-u^{\prime}$, $u(\xi)=u^{\prime}(\xi)$ for any $\xi\in\mathbb{Q}_{p}^{n}$. ∎
## 6\. Solutions of Elliptic Pseudo-Differential Equations in Singular
Sobolev Spaces
In this section, we modify the Sobolev norm to obtain spaces of functions on
which $f(D,\alpha)$ gives a surjective mapping.
###### Definition 5.
Given $\varphi\in S$ and $l$ a non-negative number, we set
$||\varphi||_{\mathcal{H}^{l}}^{2}:=\int_{\mathbb{Q}_{p}^{n}}||\xi||_{p}^{2l}|\mathcal{F}(\varphi)(\xi)|^{2}\,d\xi.$
We call the completion of $S$ with respect to $||\cdot||_{\mathcal{H}^{l}}$
the _$l$ -singular Sobolev Space_
$\mathcal{H}^{l}:=\mathcal{H}^{l}(\mathbb{Q}_{p}^{n})$. Note that
$H^{l}\subseteq\mathcal{H}^{l}$, $l\geq 0$, since
$||\varphi||_{\mathcal{H}^{l}}\leq||\varphi||_{H^{l}}$.
###### Lemma 9.
For any, $\alpha>0$, $l\geq 0$, the mapping
$f(D,\alpha):\mathcal{H}^{l+d\alpha}\rightarrow\mathcal{H}^{l}$ is a well-
defined continuous mapping between Banach Spaces.
###### Proof.
Similar to the proof of Lemma 8. ∎
We denote by $\mathcal{L}^{l}$ and $\mathcal{W}^{l}$, the respective
completions of $\mathcal{L}$ and $\mathcal{W}$ with respect to
$||\cdot||_{\mathcal{H}^{l}}$; furthermore, we set
$\mathcal{H}_{0}^{l}:=\mathcal{L}^{l}+\mathcal{W}^{l}\subseteq\mathcal{H}^{l}.$
###### Proposition 1.
Let $f(D,\alpha)$, $\alpha>0$, be an elliptic pseudo-differential operator of
order $d$, and let $l$ be a non-negative real number. Then
$f(D,\alpha):\mathcal{H}^{l+d\alpha}\rightarrow\mathcal{W}^{l}$, is a
surjective mapping between Banach spaces.
###### Proof.
By Lemma 9, the mapping is well-defined. Let $v\in\mathcal{W}^{l}$, and let
$\\{v_{n}\\}$ a Cauchy sequence in $\mathcal{W}$ converging to $v$. By Theorem
2, there exits a sequence $\\{u_{n}\\}$ in $H^{l+d\alpha}$ such that
$f(D,\alpha)u_{n}=v_{n}$. We now show that $\\{u_{n}\\}$ is a Cauchy sequence
in $\mathcal{H}^{l+d\alpha}$ as follows:
$\displaystyle||u_{n}-u_{m}||_{\mathcal{H}^{l+d\alpha}}^{2}$
$\displaystyle\leq
C\int_{\mathbb{Q}_{p}^{n}}||\xi||_{p}^{2(l+d\alpha)}||\xi||_{p}^{-2d\alpha}|\mathcal{F}(v_{n}-v_{m})(\xi)|^{2}\,d\xi$
$\displaystyle\leq C||v_{n}-v_{m}||_{\mathcal{H}^{l}}^{2}.$
Thus, there exists $u\in\mathcal{H}^{l+d\alpha}$ such that $u_{n}\rightarrow
u$, and by the continuity of $f(D,\alpha)$, $f(D,\alpha)u=v$. ∎
###### Proposition 2.
Let $f(D,\alpha)$, $\alpha>0$, be an elliptic pseudo-differential operator of
order $d$, and let $l$ be a non-negative real number. Then,
$f(D,\alpha):\mathcal{H}^{l+d\alpha}\rightarrow\mathcal{L}^{l}$ is a
surjective mapping between Banach spaces.
###### Proof.
By Lemma 9, the mapping is well-defined. Let $v\in\mathcal{L}^{l}$, and let
$\\{v_{n}\\}$ a Cauchy sequence in $\mathcal{L}$ converging to $v$. By the
same reasoning given in proof Theorem 3 for establishing the existence of a
solution for equation (5.2), we obtain a sequence $\\{u_{n}\\}$ in
$H^{l+d\alpha}$ such that $f(D,\alpha)u_{n}=v_{n}$. We now show that
$\\{u_{n}\\}$ is a Cauchy sequence in $\mathcal{H}^{l+d\alpha}$.
By using
$|\mathcal{F}(u_{n})(\xi)|\leq C||\xi||^{-d\alpha}|\mathcal{F}(v_{n})(\xi)|,$
one gets
$\displaystyle||u_{n}-u_{m}||_{\mathcal{H}^{l+d\alpha}}^{2}$
$\displaystyle\leq
C\int_{\mathbb{Q}_{p}^{n}}||\xi||_{p}^{2(l+d\alpha)}||\xi||_{p}^{-2d\alpha}|\mathcal{F}(v_{n}-v_{m})(\xi)|^{2}\,d\xi$
$\displaystyle\leq C||v_{n}-v_{m}||_{\mathcal{H}^{l}}^{2}.$
Thus, there exists $u\in\mathcal{H}^{l+d\alpha}$ such that $u_{n}\rightarrow
u$, and by the continuity of $f(D,\alpha)$, $f(D,\alpha)u=v$. ∎
From the previous two lemmas we obtain the following result.
###### Theorem 4.
Let $f(D,\alpha)$ be an elliptic pseudo-differential operator of order $d$.
Let $l$ be a positive real number. Then the equation
$f(D,\alpha)u=v,\quad v\in\mathcal{H}_{0}^{l}$
has a unique solution $u\in\mathcal{H}^{l+d\alpha}$.
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|
arxiv-papers
| 2009-07-31T14:46:59 |
2024-09-04T02:49:04.385337
|
{
"license": "Public Domain",
"authors": "J. J. Rodriguez-Vega and W. A. Zuniga-Galindo",
"submitter": "W. A. Zuniga-Galindo",
"url": "https://arxiv.org/abs/0907.5545"
}
|
0908.0194
|
# QCD Phase Boundary and Critical Point in a Bag Model Calculation
C. P. Singh, P. K. Srivastava, S. K. Tiwari Department of Physics, Banaras
Hindu University, Varanasi 221005, INDIA
###### Abstract
Location of critical point and mapping the QCD phase boundary still exists as
one of the most interesting and studied problems of heavy-ion physics. A new
equation of state(EOS) for a gas of extended baryons and pointlike mesons is
presented here which accounts for the repulsive hard-core interactions arising
due to the geometrical size of the baryons. A first order deconfining phase
transition is obtained using Gibbs’ equilibrium criteria and a bag model EOS
for the weakly interacting quark matter. It is interesting to find that the
phase transition line ends at a critical point beyond which a cross-over
region exists between hot-dense meson gas and quark-antiquark gluon matter.
Our curve resembles in shape closely with the predicitions of the available
lattice gauge calculations and also reproduces the conjectured phase boundary.
PACS numbers: 12.38 Gc, 25.75 Nq, 24.10 Pa
The existence of critical point in the studies of QCD phase diagram has
attracted considerable theoretical and experimental attention recently. The
phase diagram of quark matter is still not understood either experimentally or
theoretically. The conjectured phase boundary between quark gluon plasma (QGP)
and hot, dense hadron gas (HG) represents a first-order phase transition line
for nonzero and moderate values of temperature T and baryon chemical potential
($\mu_{B}$) [1-3]. At extremely high baryon densities (i.e., large $\mu_{B}$),
we expect a colour-flavour-locked (CFL) phase involving colour-superconducting
quark matter. As we increase T and decrease $\mu_{B}$, it is also expected
that the first-order phase transition line ends at a critical point beyond
which there exists a cross-over region since thermal fluctations at
temperature (say $T>170$ MeV) break up mesons ( mostly pions) which are densly
populated in this region and this thus results into a gas of quarks,
antiquarks, gluons etc. The existence of such a critical point was proposed a
long time ago [4,5] and more recently its properties were investigated in
detail with the help of several models [6,7]. We are hopeful that the
experiments with different colliding beam energies at Relatvistic Heavy-Ion
Collider (RHIC) will provide a suitable experimental window [8] for the search
of QCD critical point as well as for mapping the QCD phase boundary. Indeed we
have gained reliable insight into the thermodynamics of QGP from lattice QCD
calculation and our knowledge about its dynamics is particularly helpful in
the high temperature limit where it becomes weakly coupled, However, RHIC has
given us result that, at least, at temperatures within a factor of two of that
at which hadrons melt, the dynamics of QGP is closer to an ideal liquid limit
rather than to the ideal gas limit indicating the presence of a strongly
coupled QGP. Confusions still prevail about the theoretical understanding of
the QCD phase transition. We do not know whether the conjectured phase
boundary is an outcome of deconfinement and/or chiral symmetry restoration.
The purpose of this paper is to determine the phase boundary and to locate the
critical point in a first order deconfining phase transition obtained by using
EOS for the interacting quark matter and HG, separately.
We proposed a new thermodynamically consistent EOS for the HG where the
geometrical size of the baryons is explicitly incorporated as the excluded
volume correction and our model uses full quantum statistics in the partition
function of the grand canonical ensemble so that there arises no problem in
dealing with large $\mu_{B}$ and low T region and thus the full phase boundary
in T, $\mu_{B}$ plane can be investigated easily. In the earlier version of
the model [9,10], we have simplified the calculations by using Boltzmann
approximation and we have noticed that the model successfully describes the
observed particle yields, particle ratios etc at the chemical freezeout of the
HG fireball in the heavy-ion collisions [10]. In order to determine the
thermodynamic properties of the weakly interacting quark matter, we use a
simple bag model EOS with the perturbative corrections of the order of
$\alpha_{S}^{3/2}$ in strong interaction coupling constant $\alpha_{S}$. The
advantage of the bag model clearly lies in determining the thermodynamic
parameters in the region of nonzero as well as large baryon chemical potential
$\mu_{B}$ which is still not properly accessible in the lattice calculations.
We obtain the full phase boundary by Gibbs’ construction of equlibrium phase
transition between QGP and HG. We find a significant result that the first-
order phase transition line ends at a point beyond which there occurs no phase
transition and the cross-over region is only present. Thus we determine the
precise coordinates of the QCD critical point in the phase diagram and we
compare the location with the predictions of other models including lattice
calculations. The construction of the QCD phase boundary by comparing EOS of
weakly interacting QGP with a bag pressure term to EOS of hadron gas with an
excluded volume correction is not new and was done by several authors [11-14].
Excluded volume corrections in many of these approaches have usually been
incorporated in a thermodynamically incosistent manner. Our approach has
following new and interesting features :(1) Our EOS for the HG is
thermodynamically consistent and we have obtained the chemical freeze out
curve using the same formulation.(2) We have used quantum statistics in our
formulation so that we can determine the phase boundary in the entire
(T,$\mu_{B}$) plane.(3) We find that our calculated phase diagram almost
reproduces the conjectured QCD phase diagram and the coordinates of the
critical point matches well with the lattice prediction. No other model
reproduces the features so well.(4) Most importantly, we get a first-order
deconfining phase transition line where other models including lattice
calculations reveal chiral phase transition.(5) The chemical freeze out curve
obtained from our formulation lies in close proximity to the critical point
and this supports the suggestions of previous authors [3].
Let us first consider QGP and we assume that it consists of massless quarks
(u,d), their antiquarks and gluons only. So the pressure of QGP can be written
as [15]:
$\begin{split}P_{QGP}=\frac{37}{90}\pi^{2}T^{4}+\frac{1}{9}\mu_{B}^{2}T^{2}+\frac{\mu_{B}^{4}}{162\pi^{2}}\\\
-\alpha_{S}\left[\frac{11}{9}\pi
T^{4}+\frac{2}{9\pi^{2}}\mu_{B}^{2}T^{2}+\frac{1}{81\pi^{3}}\mu_{B}^{4}\right]\\\
+\frac{8\alpha_{S}^{3/2}T}{3\pi^{2}\sqrt{2\pi}}{\left[\frac{8\pi^{2}T^{2}}{3}+\frac{2}{9}\mu_{B}^{2}\right]^{3/2}}-B\end{split}$
(1)
where $\mu_{B}$, T dependence of $\alpha_{S}$ can be given as [15]:
$\alpha_{S}=\frac{12\pi}{29}\left[ln(\frac{0.089\mu_{B}^{2}+15.622T^{2}}{\Lambda^{2}})\right]^{-1}$
(2)
Here we have used $B^{1/4}=216~{}MeV$ and $\Lambda=100~{}MeV$ in our
calculation.
The grand canonical partition function for the HG with full quantum statistics
and after incorporating excluded-volume correction in a thermodynamically
consistent manner, can be written as [16]:
$\begin{split}lnZ_{i}^{ex}=\frac{g_{i}}{6\pi^{2}T}\int_{V_{i}^{0}}^{V-\sum_{j}N_{j}V_{j}^{0}}dV\\\
\int_{0}^{\infty}\frac{k^{4}dk}{\sqrt{k^{2}+m_{i}^{2}}}\frac{1}{[exp\left(\frac{E_{i}-\mu_{i}}{T}\right)+1]}\end{split}$
(3)
where $g_{i}$ is the degeneracy factor of ith species of baryons, E is the
energy of the particle ($E=\sqrt{k^{2}+m_{i}^{2}}$), $V_{i}^{0}$ is the eigen
volume of one ith species baryon and $\sum_{j}N_{j}V_{j}^{0}$ is the total
occupied volume.
We can write Eq.(3) as:
$lnZ_{i}^{ex}=V(1-\sum_{j}n_{j}^{ex}V_{j}^{0})I_{i}\lambda_{i}$ (4)
where
$I_{i}=\frac{g_{i}}{6\pi^{2}T}\int_{0}^{\infty}\frac{k^{4}dk}{\sqrt{k^{2}+m_{i}^{2}}}\frac{1}{\left[exp(\frac{E_{i}}{T})+\lambda_{i}\right]}$
(5)
and $\lambda_{i}=exp(\frac{\mu_{i}}{T})$ is the fugacity of the particle,
$n_{j}^{ex}$ is the number density of jth type of baryons after excluded
volume correction and can be obtained from Eq.(4) as:
$n_{i}^{ex}=\frac{\lambda_{i}}{V}\left(\frac{\partial{lnZ_{i}^{ex}}}{\partial{\lambda_{i}}}\right)_{T,V}$
(6)
This leads to a transcedental equation as
$n_{i}^{ex}=(1-R)I_{i}\lambda_{i}-I_{i}\lambda_{i}^{2}\frac{\partial{R}}{\partial{\lambda_{i}}}+\lambda_{i}^{2}(1-R)I_{i}^{{}^{\prime}}$
(7)
where
$I_{i}^{{}^{\prime}}=\frac{\partial{I_{i}}}{\partial{\lambda_{i}}}=-\frac{g_{i}}{6\pi^{2}T}\int_{0}^{\infty}\frac{k^{4}dk}{\sqrt{k^{2}+m_{i}^{2}}}\frac{1}{\left[exp(\frac{E_{i}}{T})+\lambda_{i}\right]^{2}}$
(8)
and $R=\sum_{i}n_{i}^{ex}V_{i}^{0}$ is the fractional occupied volume. We can
write R in an operator equation:
$R=\hat{R}+\Omega R$ (9)
where $\hat{R}=\frac{R^{0}}{1+R^{0}}$ with $R^{0}=\sum n_{i}^{0}V_{i}^{0}+\sum
I_{i}^{{}^{\prime}}V_{i}^{0}\lambda_{i}^{2}$; $n_{i}^{0}$ is the density of
pointlike baryons of ith species and the operator $\Omega$ is:
$\Omega=-\frac{1}{1+R^{0}}\sum_{i}n_{i}^{0}V_{i}^{0}\lambda_{i}\frac{\partial}{\partial{\lambda_{i}}}$
(10)
Using Neumann iteration method in Eq.(9) and retaining the series upto
$\Omega^{2}$ term, we get
$R=\hat{R}+\Omega\hat{R}+\Omega^{2}\hat{R}$ (11)
Eq.(11) can be solved numerically. The total pressure [15] of the hadron gas
after excluded volume correction is:
$P_{HG}^{ex}=T(1-R)\sum_{i}I_{i}\lambda_{i}+\sum_{i}P_{i}^{meson}$ (12)
In (12), the first term represents the pressure due to all types of baryons
and the second term gives the total pressure due to all mesons having
pointlike size only. This makes it clear that we consider the hard-core
repulsion existing between two baryons only.
We have considered all the baryons and the mesons as well as their resonances
having masses upto 2 GeV/$c^{2}$ in our calculation. In order to conserve
strangeness quantum number, we have used the criterion of equating the net
strangeness equal to zero, i.e., $\sum_{i}S_{i}(n_{i}^{S}-n_{i}^{\bar{S}})=0$
where $S_{i}$ is the strangeness quantum number of ith hadron, $n_{i}^{S}$ and
$n_{i}^{\bar{S}}$ are the strange hadron density of ith hadron and ith anti-
hadron, respectively. Strangeness neutrality condition yields the value of
strange chemical potential in terms of $\mu_{B}$. We have considered mesons as
pointlike particles in this calculation. Furthermore, we have taken an equal
volume $V^{0}=\frac{4\pi r^{3}}{3}$ for each type of baryon with a hard-core
radius r=0.8 fm. The first-order phase transition boundary is determined by
using the Gibbs’ equilibrium condition
$P_{HG}^{ex}(T_{c},\mu_{c})=P_{QGP}(T_{c},\mu_{c})$.
In Fig.1, we have shown the phase boundary between QGP and HG as obtained from
our calculations. We start from a low but nonzero value of T and large value
of $\mu_{B}$ and we move towards large T and small( and nonzero) $\mu_{B}$. We
find a first order phase transition line and it ends at a QCD critical
point.The coordinates are $T_{c}$=160 MeV and $\mu_{c}$=156 MeV. The critical
point as obtained by us lies close to the lattice result LTE04 [17]. Since the
value of $\mu_{c}$ is the lowest in comparision to all other models, we hope
that this point can be reached in the RHIC experiments. From this critical
point to the $\mu_{B}=0$ line , we find a transitional cross-over region which
cannot be described or modelled analytically. Hadronic degrees of freedom are
insufficient to give a valid description [18] of this region whereas free
quark and gluons start playing a significant role only at much higher
temperatures. We have compared our prediction regarding the location of the
QCD critical point with those obtained from different models. We find that
critical point in our curve lies closer to the lattice gauge predictions. It
should be emphasized that the location of the critical point in our
calculation means the end point of the first order phase transition line and
beyond this point, Gibbs’ equilibrium condition does not remain valid. Here we
stress that the dependence of our results on the values of two parameters, the
bag constant B and the hard core radius r is small. We have shown in fig.1 by
the curves $P_{1}$, $P_{2}$ and $P_{3}$ respectively. We find that the
location of the critical point shifts from $C_{1}$ to $C_{2}$ as we decrease
the value of the bag constant B. Furthermore, the variation in the QCD scale
factor $\Lambda$ do not give any substantial change.
It is very difficult to predict the coordinates of the critical point reliably
and this is also evident by the plot in Fig.1 where we find that the
predictions of different models vary wildly [2]. However, it has been
suggested that the present heavy-ion experiments can be used to locate the QCD
critical point [7]. We find that the critical point as obtained from our
calculation lies in the region of the phase diagram accessible at the current
energy of
Figure 1: The location of the QCD critical point in the QCD phase diagram as
calculated in our model.$P_{2}$ is the phase boundary with $B^{1/4}=216MeV$
and r=0.8 fm, $P_{1}$ is with $B^{1/4}=216MeV$ and r=0.6 fm, and $P_{3}$ is
with $B^{1/4}=200MeV$ and r=0.8 fm. $F_{1}$ is the chemical freeze out line
obtained in our model. $C_{1}(T_{c}=160MeV,\mu_{c}=156MeV)$ and
$C_{2}(T_{c}=146MeV,\mu_{c}=156MeV)$ are the critical end points on
$P_{1},orP_{2}andP_{3}$, respectively. Critical points denoted by LR04[24],
LR01[25], LTE03[26], LTE04[17] are lattice model results and NJLinst
[5],LSM,NJL [20],NJL/I,NJL/II [21],RM [22] are in other models and the points
and the notations have been taken from ref. [2].
200 GeV/nucleon at which RHIC explores the cross-over region. Near the
critical point, chemical freezeout points are also helpful in finding its
location. It has been suggested that the experimental observables should show
nonmonotonic behaviour as a function of centre-of-mass energy $\sqrt{s}$ when
the freeze out point lies close to the critical point [1,19]. We have
determined the locations of freeze out points in various heavy-ion experiments
by measuring the ratios of particle yields and fitting to our excluded volume
HG model with T and $\mu_{B}$ as parameters. We have plotted in Fig.1, the
freezeout curve as obtained from our model. We find that the critical point
lies almost on (or near) the freeze out curve.This endorses the usefulness of
the finding of Stephanov, Rajagopal and Shuryak [3] because they have shown
that a non-monotonic behaviour of fluctations (eg, of multiplicity) can be
considered as a signal for the critical point. However after the critical
point, the difference between the freeze out curve and the phase transition
line increases as $\mu_{B}$ increases and T decreases. The freeze out points
tend to cluster near the QCD critical point.
In conclusion, we have demonstrated the first order phase transition boundary
in a simple bag like model describing the deconfining phase transition of
quarks and gluons. It should be stressed that our excluded volume model
proposed in this paper is thermodynamically consistent and also incorporates
full quantum statistics. We have also determined the precise location of QCD
critical point using new EOS for the HG proposed by us. The results are in
agreement with what we expect from lattice calculations [17]. It should be
emphasized that the lattice calculations [23-26] have failed so far to
converge on a prediction for the location of the critical point. However, our
interpretation differs from other QCD models which are all based on the chiral
dynamics. Obviously the existence of the critical point in all these
calculations follow from the basic assumption, that the finite $\mu_{B}$
chiral phase transition is first order.However, the picture based on the
chiral dynamics in the baryon-dense region casts a shadow of doubt as CFL
phase breaks chiral symmetry. The occurrence of a novel phase of dense quarks,
named as quarkyonic phase was recently proposed based on the large $N_{c}$
argument where $N_{c}$ denotes number of colours [27]. This phase occurs above
$\mu_{B}=M_{B}$ where $M_{B}$ is the baryon mass and is characterized by non-
vanishing baryon number density and confinement. The clear separation of the
quarkyonic phase from the hadronic phase is lost in a system with finite
$N_{c}$ but any large change in the baryon number density can reveal a
quarkyonic transition. Our model endorses the deconfining nature of the first
order phase transition. We also find the existence of a cross-over region
lying beyond the critical point where the meson dominant HG pressure is always
less than the QGP pressure.. This region can be interpreted in terms of the
dual description of mostly the quarks and gluons together with $\pi$ mesons.
The fundamental assumption in our model is that the baryons in the HG possess
a hard-core size and there exists a repulsive interaction between two baryons
[13]. However, mesons are not subjected to any such force because they do not
have any hard-core size. In constructing a first order phase transition it is
essential to include the excluded-volume corrections for baryons in the HG and
the EOS for QGP phase should also include QCD interaction terms [16]. Mesons
at high temperature can fuse into one another but baryons retain their space.
So at large $\mu_{B}$, the fractional occupied volume R is finite and hence
mobility of baryons is affected. Therefore, for any low temperature T, we find
a corresponding $\mu_{B}$ at which the QGP pressure becomes equal to the HG
pressure and beyond which the QGP pressure dominates. At higher T also, this
continues unless we reach the end point at which the QGP pressure is always
larger than the HG pressure. This is defined as the critical end point in our
model. The physical mechanism in this calculation is analogous to the
percolation model [14] where a first order transition is obtained through
’jamming’ of baryons without any comparison to the QGP. Recent progress and
results are encouraging in this direction, but much more work still needs to
be done further before this picture becomes conclusive.
CPS and PKS acknowledge the financial support from Department of Science and
Tecnology (DST), New Delhi and SKT is grateful to CSIR, New Delhi for the
financial support.
## References
* (1) M. A. Stephanov,Phys. Rev. Lett. 102, 032301 (2009).
* (2) M. A. Stephanov,Acta Physica Polonica B35, 2939 (2004), M. A. Stephanov, Int. J. Mod. Phys. A20, 4387 (2005).
* (3) M. A. Stephanov, K. Rajagopal and E.V. Shuryak, Phys. Rev. Lett. 81,4816 (1998)
* (4) A. Barducci, R. Cascalbuoni, S. DeCurtis, R. Gatto and G. Pettini, Phys. Rev. D41, 1610 (1910)
* (5) J. Berges and K. Rajagopal, Nucl. Phys B538, 215 (1999)
* (6) M. A. Stephanov, Prog. Theor. Phys. Suppl. 153, 139 (2004)
* (7) M. A. Stephanov, Int. J. Mod. Phys. A20, 4387 (2005)
* (8) Dam T. Son, Physics 2, 5 (2009)
* (9) M. Mishra and C. P. Singh, Phys. Rev. C76, 024908 (2007); M. Mishra and C. P. Singh, Phys. Lett. B651, 119 (2007)
* (10) M. Mishra and C. P. Singh, Phys. Rev. C78, 024910 (2008)
* (11) F. Karsch and H. Satz, Phys. Rev. D21, 1168 (1980)
* (12) V. V. Dixit, F. Karsch and H. Satz, Phys. Lett. B101, 412 (1981), J. Cleymans, K. Redlich, H. Satz and E. Suhonen, Z. Phys. C33, 151 (1986)
* (13) V. Magas and H. Satz, Eur. Phys. J. C32, 115 (2003)
* (14) P. Castorina, K. Redlich and H. Satz, Eur. Phys. J. C59, 67 (2009)
* (15) B. K. Patra and C. P. Singh, Phys. Rev. D54, 3551 (1996)
* (16) C. P. Singh,B. K. Patra and K. K. Singh, Phys. Lett. B387, 680 (1996); S. Uddin and C. P. Singh, Zeit. f. Phys. C63, 147 (1994)
* (17) R. V. Gavai and S. Gupta, Phys. Rev. D71, 114014 (2005)
* (18) L. Ferroni and V. Koch, Phys. Rev. C79, 034905 (2009)
* (19) V. Koch, J. Phys. G: Nucl. Part. Phys: 35, 104030 (2008)
* (20) O. Scavenius, A. Mocsy, I. N. Mishustin and D. H. Rischke, Phys. Rev. C64, 045202 (2001)
* (21) M. Asakawa and K. Yazaki, Nucl. Phys. A504, 668 (1989)
* (22) M. A. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov and J. J. M. Verbaarschot, Phys. Rev. D58, 096007 (1998)
* (23) Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature 443, 675 (2006)
* (24) Z. Fodor and S. D. Katz, J. High Energy Phys. 0404, 050 (2004)
* (25) Z. Fodor and S. D. Katz, J. High Energy Phys. 0203, 014 (2002)
* (26) S. Ejiri, C. R. Alton, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Prog. Theor. Phys. Suppl. 153, 118 (2004)
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|
arxiv-papers
| 2009-08-03T11:49:45 |
2024-09-04T02:49:04.398466
|
{
"license": "Public Domain",
"authors": "C. P. Singh, P. K. Srivastava, S. K. Tiwari",
"submitter": "Swatantra Tiwari",
"url": "https://arxiv.org/abs/0908.0194"
}
|
0908.0219
|
# Critical Dimension for Stable Self-gravitating Stars in $AdS$
Zhong-Hua Li sclzh888@163.com Department of Physics, China West Normal
University, Nanchong 637002, China Rong-Gen Cai cairg@itp.ac.cn Key
Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, P.O.Box 2735, Beijing 100190, China
###### Abstract
Abstract
We study the self-gravitating stars with a linear equation of state,
$P=a\rho$, in AdS space, where $a$ is a constant parameter. There exists a
critical dimension, beyond which the stars are always stable with any central
energy density; below which there exists a maximal mass configuration for a
certain central energy density and when the central energy density continues
to increase, the configuration becomes unstable. We find that the critical
dimension depends on the parameter $a$, it runs from $d=11.1429$ to $10.1291$
as $a$ varies from $a=0$ to $1$. The lowest integer dimension for a
dynamically stable self-gravitating configuration should be $d=12$ for any
$a\in[0,1]$ rather than $d=11$, the latter is the case of self-gravitating
radiation configurations in AdS space.
## I Introduction
Self-gravitating configuration is a subject of long-standing interest in
general relativity since its thermodynamic behavior is quite different from
that of a usual thermal system without gravity, due to the attractive long-
range, unshielded nature of gravitational potential. For example, it is well
known that the canonical ensemble is not defined in asymptotically flat space.
This is because having thermal radiation at constant temperature at infinity
is not compatible with asymptotic flatness. One can avoid this problem by
enclosing the system in a box, which is unphysical, or by working in anti-de
Sitter (AdS) space which needs not any unphysical perfectly reflecting walls
at finite radius, the rising gravitational potential in AdS space, plus
natural boundary conditions at infinity, acts to confine whatever is inside.
On the other hand, due to the conjecture of AdS/CFT correspondence Mald ,
which says that string theory/M theory on an AdS space (times a compact
manifold) is dual to a strong coupling conformal field theory (CFT) residing
on the boundary of the AdS space, over the past decade, a lot of attention has
been focused on AdS space and relevant physics. Further, Witten Witten argued
that thermodynamics of black holes in AdS space can be identified to that of
dual strong coupling CFTs. Therefore one could study thermodynamics and phase
structure of strong coupling CFTs by investigating thermodynamics and phase
structure of AdS black holes. It is well-known that thermodynamics of AdS
black holes is quite different from that of their counterparts in
asymptotically flat space and that there is a Hawking-Page phase transition
between large black hole and thermal radiation in AdS space HP . Therefore it
is also of great interest to study self-gravitating configurations in AdS
space and their thermodynamics.
Self-gravitating radiation gas has been investigated thoroughly. Sorkin, Wald
and Zhang SWZ have studied the equilibrium configurations of self-gravitating
radiation in a spherical box of radius $R$ in asymptotically flat space. It
was found that for locally stable configuration, the total gravitational mass
of radiation obeys the inequality $M<\mu_{\rm max}R$, where $\mu_{\rm
max}=0.246$. In AdS space, Page and Philips PP examined the self-gravitating
configuration of radiation in four dimensional space-time. The configuration
can be labeled as its mass, entropy and temperature versus the central energy
density. They found that there exist locally stable radiation configurations
all the way up to a maximum red-shifted temperature, above which there are no
solutions; there is also a maximum mass and maximum entropy configuration
occurring at a higher central density than the maximal temperature
configuration. Beyond their peaks the temperature, mass and entropy undergo an
infinite series of damped oscillations, which indicates the configurations in
this regime are unstable. The self-gravitating radiation in five dimensional
AdS space has been studied in HLR (see also Hemm ) with similar conclusions.
Recently, Vaganov Vaga and Hammersley Hamm independently discussed the self-
gravitating radiation configurations in higher dimensional AdS spaces. They
found that in the case of $4\leq d\leq 10$, the situation is qualitatively
similar to the case in four dimensions, while $d\geq 11$, the oscillation
behavior disappears. Namely, there is a critical dimension, $d_{c.ads}=11$
(very close, but not exact), beyond which, the temperature, mass and entropy
of the self-gravitating configuration are monotonic functions of the central
energy density, asymptoting to their maxima as the central density goes to
infinity. The equilibrium configurations of self-gravitating radiation gas in
AdS space are quite different from those of their counterparts in
asymptotically flat space.
Recently Chavanis investigated spherically symmetric equilibrium
configurations of relativistic stars with a linear equation of state $P=a\rho$
in flat space Chav . The equation of state $P=a\rho$ says the pressure $P$ is
proportional to the energy density $\rho$, where $a$ is a ratio coefficient.
Theoretically, different coefficients correspond to different stars, although
it is impossible to define names for all the stars corresponding to each
coefficient. The speed of sound is given by $(dP/d\rho)^{2}c=a^{2}c$, where
$c$ is the speed of light. Thus causality requires $a^{2}\leq 1$. As a result,
$0\leq a\leq 1$ is considered only. One could identify relativistic stars by
an equation of state with their different ratio coefficients. In the Newtonian
limit $a\rightarrow 0$, it returns the classical isothermal equation of state.
On the other hand, the case with $a=1/(d-1)$, where $d$ is the dimension of
space-time, corresponds to a gas of self-gravitating radiation; the core of
neutron stars or so called “photon stars” where the pressure is entirely due
to radiation. A gas of baryons interacting through a vector meson field for
the case of $a=1$ is called the “stiffest” star. Chavanis found that the
structure of the system is highly dependent on the dimensionality of space-
time and that the oscillations in the mass-central density profile disappear
above a critical dimension $d_{c.flat}(a)$ depending on the coefficient $a$.
Above this dimension, the equilibrium configurations are stable for $any$
central density, contrary to the case $d<d_{c.flat}$. For Newtonian isothermal
stars ($a\rightarrow 0$), the critical dimension $d_{c.flat}(0)=11$. For the
stiffest stars ($a=1$), $d_{c.flat}(1)=10$ and for a self-gravitating
radiation($a=1/(d-1)$), $d_{c.flat}=1+9.96404372...$ very close to $11$. The
oscillations exist for any $a\in[0,1]$ when $d\leq 10$ and they cease to exist
for any $a\in[0,1]$ when $d\geq 11$.
In a previous paper LHC , we have studied self-gravitating radiation
configurations with plane symmetry in AdS space and found that the situation
is quite different from the one in spherically symmetric AdS space. In the
present paper we continue this study and consider the self-gravitating
configurations with a linear equation of state $P=a\rho$ in higher ($d\geq
4$)-dimensional spherically symmetric AdS space, in order to see the
dependence of the critical dimension on the parameter $a$ and to see whether
there is any essential difference between the configurations in flat space and
the configurations in AdS space.
The organization of the paper is as follows. In the next section, we give a
general formulism to describe the relativistic stars with a linear equation of
state $P=a\rho$ in AdS space. The numerical results are given in Sec. III. The
Sec. IV is devoted to the conclusions.
## II relativistic stars with linear equation of state in AdS space
Consider a $d$-dimensional asymptotically AdS space with metric
$ds^{2}=-e^{2\delta(r)}h(r)dt^{2}+h^{-1}(r)dr^{2}+r^{2}\gamma_{ij}dx^{i}dx^{j},$
(1)
where $\delta$ and $h$ are two functions of the radial coordinate $r$, and
$\gamma_{ij}$ is the metric of a $(d-2)$-dimensional Einstein manifold with
constant scalar curvature $(d-2)(d-3)$. We take the gauge
$\lim_{r\to\infty}\delta(r)=0$, and rewrite the metric function $h(r)$ as
$h(r)=1+\frac{r^{2}}{l^{2}}-\frac{16\pi Gm(r)}{(d-2)\Omega r^{d-3}},$ (2)
where $l$ denotes the radius of the AdS space with cosmological constant
$\Lambda=-(d-1)(d-2)/2l^{2}$, $\Omega$ is the volume of the Einstein manifold,
and $m(r)$ is the mass function of the solution. In this gauge, the total
gravitational mass of the solution is just
$M=\lim_{r\to\infty}m(r).$ (3)
The Einstein field equations with the cosmological constant and energy-
momentum tensor $T_{\mu\nu}$ are
$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{(d-1)(d-2)}{2l^{2}}g_{\mu\nu}=8\pi
GT_{\mu\nu}.$ (4)
The metric function $\delta(r)$ and the mass function $m(r)$ satisfy the
following equations
$\displaystyle\delta^{\prime}(r)=-\frac{8\pi Gr}{(d-2)h(r)}\left(T^{t}_{\
t}-T^{r}_{\ r}\right),$ $\displaystyle m^{\prime}(r)=-\Omega r^{d-2}T^{t}_{\
t}.$ (5)
The stress-energy tensor of perfect fluid is
$T_{\mu\nu}=(\rho+P)U_{\mu}U_{\nu}+Pg_{\mu\nu}$, where $U_{\mu}$ is the four-
velocity of the fluid. For self-gravitating radiation, the equation of state
obeys $P=\rho/(d-1)$. For linear relativistic stars, $P=a\rho$ with
$a\in[0,1]$. Thus the equations in (II) reduce to
$\displaystyle\delta^{\prime}(r)=\frac{8\pi Gr}{(d-2)h(r)}(1+a)\rho,$ (7)
$\displaystyle m^{\prime}(r)=\Omega r^{d-2}\rho.$
In this paper we are particularly interested in the relation between the total
mass of configuration and the central energy density. To integrate (7) and
(7), we make a scaling transformation as follows,
$r\to l\,r,\ \ \ \rho\to l^{-2}\rho,\ \ \ m(r)\to l^{d-3}m(r),$ (8)
so that $r$, $\rho$ and $m$ become dimensionless. In the numerical
integration, we will adopt the units $8\pi G=1$ and $l=1$, and rescale the
mass function as
$16\pi Gm(r)/(d-2)\Omega\to m(r).$
In that case, the gravitational “mass” $M$ in the plots in the next section in
fact is the gravitational mass density, $16\pi GM/(d-2)\Omega$, of
corresponding relativistic star configurations. From the conservation of the
energy-momentum tensor, one can derive the energy density $\rho(r)$ satisfies
the following equation
$\frac{d\rho}{dr}=\frac{(a+1)\rho
r}{2a}\left(\frac{d-3}{r^{2}}-\frac{d-1}{h(r)}-\frac{d-3}{r^{2}h(r)}-\frac{2a\rho}{(d-2)h(r)}\right).$
(9)
Further, note that the equations (7) and (9) are singular at $r=0$. To avoid
this, in the numerical calculations, we will start the integration from
$r=\epsilon=10^{-5}$ to $r=L=100$. Obviously, the accuracy of the numerical
calculations depends on the values of $\epsilon$ and $L$.
## III Numerical results
Figure 1: The self-gravitating radiation in AdS space: The mass of the self-
gravitating radiation configuration versus the central energy density from
dimension $d=4$ (top curve) to $d=12$ (bottom curve). Figure 2: The case of
$a=0.5$ in AdS space: The mass of relativistic star configurations versus the
central energy density from dimension $d=4$ (bottom curve) to $d=12$ (top
curve).
To be more clear to discuss all kinds of critical dimensions of relativistic
stars with a linear equation of state in AdS space, let us first revisit the
self-gravitating radiation case. In Hamm Hammersley found that in the case of
$4\leq d\leq 10$, there exist locally stable radiation configurations all the
way up to a maximum mass, beyond their peaks the curves undergo an infinite
series of damped oscillations, which indicates the configurations in this
regime are unstable. However, with $d\geq 11$, the oscillation behavior
disappears. The configurations turn to be monotonic functions of the central
energy density, asymptoting to their maxima as the central density goes to
infinity. Namely, there is a critical dimension. By numerical analysis,
Hammersley Hamm found a semi-empirical model
$\log\rho_{c}\approx 0.50d+\frac{5.75}{\sqrt{11.0-d}}-2.20,$ (10)
which gives a critical dimension $d_{c.ads}=11$, and a similar conclusion was
also reached by Vaganov Vaga independently. We reproduce the result in Fig.
1.
Now we turn to the case of relativistic stars with a linear equation of state
$P=a\rho$. Due to the requirement of causality, we consider the cases with
$0\leq a\leq 1$. In Fig. 2, we plot the mass of relativistic stars
configurations versus the central energy density from dimension $d=4$ to
$d=12$ for the case of $a=0.5$. Note that in Fig. 2 we rescale the mass $M$
with scale $10^{8-2d}(d-3)^{2}$ in each dimension for better displaying all
curves in one figure. We notice that for a certain value of $a$, the
configurations of different dimensions from $4$ to $12$ are similar to the
case of self-gravitating radiation stars. For the case of lower dimensions,
there exist obvious oscillations which can be seen from Fig 3. As for the
higher dimensions, the oscillations become weaker and weaker, and finally
beyond some critical dimensions, configurations become monotonic functions of
the central energy density. Fig. 4 is the case of $d=11$. Note that in Fig. 3
and Fig. 4 each curve corresponds to each relativistic star with coefficient
$a$ which undergoes a continuous variation from 0.01 to 1.0 with step 0.1, and
also we rescale the mass density $M$ to $M/M_{\rm max}$ in each case of
coefficient $a$ for better comparing those different cases.
Figure 3: The case of d=4 in AdS space: The mass of relativistic star
configurations versus the central energy density from the coefficient $a=1$
(left curve) to $a=0.01$ (right curve). Figure 4: The case of d=11 in AdS
space: The mass of relativistic star configurations versus the central energy
density from the coefficient $a=1$ (left curve) to $a=0.01$ (right curve).
As a matter of fact, it is not really accurate to determine the critical
dimensions only by the naked eyes to watch these curves. To be more accurate
to determine the critical dimensions, we take the numerical analysis approach.
In Fig. 5 one can see clearly that for each curve there exists a saturation
point, $\log\rho_{c}$ (the red dot in the figure), which is the location of
the first local maximum. The saturation point moves towards the right side as
the dimension $d$ increases. Beyond the saturation points, the curves undergo
an infinite series of damped oscillations. When the dimension increases, the
oscillations become weaker and weaker, while the saturation point
$\log\rho_{c}$ becomes larger and larger, and finally, beyond a critical
dimension $d_{c.ads}$, the saturation point $\log\rho_{c}$ goes to infinity.
So we can determine the critical dimension by analyzing the variation of the
saturation points.
Some data of saturation points $\log\rho_{c}$ are listed in Table 1.
Obviously, $\log\rho_{c}$ depends on the coefficient $a$ and dimension $d$, so
namely $\log\rho_{c}(a,d)$. We use a formula like (10) and obtain the critical
dimensions $d_{c}(a)$ for different coefficient $a$, which are listed in Table
2. As can be seen from Table 1, in the case of $d=11$ some data singularity
appears. So we only use data from $d=4$ to $d=10$ to fit.
Figure 5: The self-gravitating radiation in AdS space: The mass of the self-
gravitating radiation configuration versus the central energy density from
dimension $d=4$ (top curve) to $d=10$ (bottom curve). The red dots are the
saturation points $\log\rho_{c}$. Table 1: Saturation Points
$\log\rho_{c}(a,d)$
a d=4 d=5 d=6 d=7 d=8 d=9 d=10 d=11 0.01 3.90 3.60 3.90 4.20 5.10 6.00 8.10
21.00 0.1 3.00 3.00 3.30 3.90 4.80 6.00 8.40 21.90 0.2 2.40 2.40 3.00 3.90
4.80 6.30 8.70 $\infty$ 0.3 1.80 2.10 3.00 3.90 4.80 6.30 9.00 $\infty$ 0.4
1.50 2.10 3.00 3.90 5.10 6.60 9.60 $\infty$ 0.5 1.20 2.10 3.00 3.90 5.10 6.60
10.20 $\infty$ 0.6 1.20 2.10 3.00 4.20 5.10 6.90 11.10 $\infty$ 0.7 1.20 2.10
3.00 4.20 5.40 7.20 12.30 $\infty$ 0.8 1.20 2.10 3.30 4.20 5.40 7.50 14.10
$\infty$ 0.9 1.20 2.10 3.30 4.20 5.70 7.50 15.60 $\infty$ 1.0 1.20 2.10 3.30
4.20 5.70 7.80 16.50 $\infty$
Table 2: Critical Dimensions $d_{c.ads}(a)$
a 0.01 0.1 0.2 0.3 0.4 0.5 $d_{c.ads}(a)$ 11.1061 11.0952 10.9989 10.9254
10.7465 10.6333 a 0.6 0.7 0.8 0.9 1.0 $d_{c.ads}(a)$ 10.4928 10.3707 10.2602
10.2023 10.1763
Figure 6: The critical dimensions $d_{c.ads}(a)$ versus the coefficient $a$.
The blue hollow dots are the scattered data of the critical dimensions
$d_{c.ads}(a)$. The red dashed curve is the case of linear fitting. The blue
dashed curve is the case of the third-order fitting. The black thick curve is
the case of average fitting. Table 3: Numerical Fitting Results
$d_{c.ads}(a)$ linear second-order third-order average low bound 10.1063
10.1048 10.1763 10.1291 up bound 11.1601 11.1591 11.1097 11.1429
In Fig. 6 we plot the scattered diagram of the critical dimensions
$d_{c.ads}(a)$ versus the coefficient $a$. It can be seen that the value of
$d_{c.ads}(a)$ becomes smaller and smaller as the value of the coefficient $a$
goes from $0$ to $1$. So $d_{c.ads}(a)$ can be considered to be a monotonic
function of the coefficient $a$. The red dashed line in Fig. 6 is the result
of a linear numerical fitting for these scattered data. It could be seen that
for all kinds of relativistic stars with a linear equation of state $P=a\rho$
$(a\in[0,1])$, the critical dimensions $d_{c.ads}(a)\in[10.1063,11.1601]$.
Obviously, the lower and upper bounds of $d_{c.ads}(a)$ for this kind of
fitting are a little wider because these scattered data are not really linear
related. To obtain more accurate results, we need to adopt non-linear fitting
which could better fit the scattered data. The results of the second and
third-order fitting are listed in Table 3. The blue dashed curve in Fig. 6 is
the case of the third-order fitting. Note that the curve of the second-order
fitting is not plotted in Fig. 6 because it is overlapped with the most of the
linear fitting curve. It can be seen from Fig. 6 that the curve of the third-
order fitting can better fit the scattered data. Namely, in AdS space for all
kinds of relativistic stars with a linear equation of state $P=a\rho$
$(a\in[0,1])$ the critical dimensions $d_{c.ads}(a)\in[10.1763,11.1097]$. The
minimum for the stiffest star in AdS space is $d_{c.ads}(1)=10.1763$ and the
maximum for the Newtonian isothermal star is $d_{c.ads}(0)=11.1097$. In Fig. 7
we plot the saturation points $\log\rho_{c}$ versus the dimension $d$ for each
relativistic star (corresponding to each coefficient $a$ ) from $a=0.1$ (right
blue curve) to $a=1.0$ (left red curve) in the range $9.4\leq d\leq 11.4$. The
low and up bounds of the critical dimensions
$d_{c.ads}(a)\in[10.1763,11.1097]$ are also plotted in this figure (the blue
dashed curves). One can find that for the stiffest stars $(a=1)$, the left red
curve increases steeply and finally up to a critical dimension $d=10.1763$.
However, for Newtonian isothermal stars $a\rightarrow 0$, the right blue curve
changes more slowly, up to the critical dimension $d=11.1097$.
Figure 7: The saturation points $\rho_{c}$ versus the dimension $d$ from
$a=0.1$ (right blue curve) to $a=1.0$ (left red curve). The blue dashed curves
are the low and up bounds of the critical dimensions
$d_{c.ads}(a)\in[10.1763,11.1097]$, respectively. The black thick curves are
the low and up bounds of the critical dimensions
$d_{c.ads.aver}(a)\in[10.1291,11.1429]$, respectively.
By an analysis of asymptotic behavior, Chavanis Chav found that the critical
dimension runs from $d=10$ to $11$ when the coefficient $a$ decreases from
$a=1$ to $0$ in flat space. Namely the difference of dimension $\triangle
d_{flat}=d_{c.flat}(0)-d_{c.flat}(1)=1$. In our case, namely in AdS space, the
critical dimension runs from $d=11.1097$ to $10.1763$ with $\triangle
d_{ads}=d_{c.ads}(0)-d_{c.ads}(1)=0.9334$ when $a$ varies from $0$ to $1$. If
taking the averaged values, then one has $\overline{\triangle
d}_{ads}=d_{c.ads.aver}(0)-d_{c.ads.aver}(1)=11.1429-10.1291=1.0138$. There
exists some difference between the flat and AdS cases. Therefore we guess that
for all kinds of relativistic stars with a linear equation of state regardless
of in AdS and flat spaces, the critical dimensions $d_{c.ads}(a)$ and
$d_{c.flat}(a)$ vary just within $1$ dimension when $a$ changes from $0$ to
$1$, and the inaccuracy between the two cases is very small, within $1.38\%$.
Therefor the critical dimension for a stable relativistic stars with a linear
equation of state is $12$, rather than $11$, which is the case of self-
gravitating radiation configurations.
## IV Conclusion
There exists a critical dimension for self-gravitating configurations in
general relativity, beyond which the configurations with any central energy
density are always stable; below which there exists a maximal mass
configuration for a certain central energy density, when the central energy
density increases, the configuration becomes unstable. In this paper we
studied the self-gravitating configurations (relativistic stars) with a linear
equation of state $P=a\rho$ in AdS space, where $a$ is a constant parameter
within $a\in[0,1]$. We found that the critical dimension depends on the
parameter $a$, it runs from $d=11.1097$ to $10.1763$ in the third order
fitting as $a$ varies from $a=0$ to $1$. The result is a little different from
the case in flat space. It runs from $d=11.1429$ to $10.1291$ if one takes the
averaged fitting. In that case, it is very close to the case in flat space
within the inaccuracy $1.38\%$. Therefore it is of interest to study the self-
gravitating configurations in de Sitter space and to see whether there exist
any differences among the three cases.
In Vaga and Hamm , it was found that the critical dimension of self-
gravitating radiation configurations is $d=11$. Combing the results obtained
in LHC and in the present paper, we see that in fact, the dimension $d=11$
might not have any other special physical meaning, except for the stability
divide. Note that the fact that no special happens for thermodynamics of AdS
Schwarzschild black holes in $d\geq 4$, it is of interest to understand the
meaning of the existence of the critical dimension in the AdS/CFT
correspondence.
###### Acknowledgements.
This work was supported partially by grants from NSFC, China (No. 10821504 and
No. 10525060), a grant from the Chinese Academy of Sciences with No.
KJCX3-SYW-N2.
## References
* (1) J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1998)] [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].
* (2) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) [arXiv:hep-th/9803131].
* (3) S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983).
* (4) R. Sorkin, R. Wald and Z.J. Zhang, Gen. Rel. Grav. 13, 1127 (1981).
* (5) D.N. Page and K.C. Philips, Gen. Rel. Grav. 17, 1029 (1985).
* (6) V. E. Hubeny, H. Liu and M. Rangamani, JHEP 0701, 009 (2007) [arXiv:hep-th/0610041].
* (7) S. Hemming and L. Thorlacius, JHEP 0711, 086 (2007) [arXiv:0709.3738 [hep-th]].
* (8) V. Vaganov, arXiv:0707.0864 [gr-qc].
* (9) J. Hammersley, Class. Quantum Grav. 25, 205010 (2008) [arXiv:0707.0961 [hep-th]].
* (10) P. H. Chavanis, Astron. Astrophys. 483, 673 (2008) [arXiv:0707.2292 [astro-ph]].
* (11) Z. H. Li, B. Hu and R. G. Cai, Phys. Rev. D 77, 104032 (2008) [arXiv:0804.3233 [gr-qc]].
|
arxiv-papers
| 2009-08-03T10:19:56 |
2024-09-04T02:49:04.403851
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhong-Hua Li, Rong-Gen Cai",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/0908.0219"
}
|
0908.0359
|
# On the determination of the boundary impedance
from the far field pattern
Yuri A. Godin Department of Mathematics and Statistics, University of North
Carolina at Charlotte, Charlotte, NC 28223 (ygodin@uncc.edu,
brvainbe@uncc.edu). Boris Vainberg∗
( 14:54 )
###### Abstract
We consider the Helmholtz equation in the half space and suggest two methods
for determining the boundary impedance from knowledge of the far field pattern
of the time-harmonic incident wave. We introduce a potential for which the far
field patterns in specially selected directions represent its Fourier
coefficients. The boundary impedance is then calculated from the potential by
an explicit formula or from the WKB approximation. Numerical examples are
given to demonstrate efficiency of the approaches. We also discuss the
validity of the WKB approximation in determining the impedance of an obstacle.
## 1 Introduction
Various impurities such as gases, dust,cracks, etc., on the surface of a body
subject to an incident wave can be modeled by the impedance boundary condition
[1]. The detection of these inhomogeneities using nondestructive testing is
then reduced to the reconstruction of the impedance from the measurements of
scattering field [2]. Optical scanning of the surface of silicon wafers used
for quality control in semiconductor industry [3] is one of the possible
applications of this method.
We consider the scattering of an incident time-harmonic plane wave from the
boundary of the half-space
$\mathbb{R}^{3}_{+}=\\{{\boldsymbol{x}}=(x_{1},x_{2},x_{3}),\,x_{3}>0\\}$. The
problem is described by the Helmholtz equation
$-\Delta u=k^{2}u,\quad x_{3}>0,$ (1)
with the impedance boundary condition
$\left.u_{x_{3}}+ik\gamma({\boldsymbol{x}^{\prime}})\,u\right|_{x_{3}=0}=0,\quad{\boldsymbol{x}^{\prime}}=(x_{1},x_{2},0),$
(2)
where $\gamma({\boldsymbol{x}^{\prime}})$ is the surface impedance with a
bounded support $\mathop{\rm supp}\gamma\subset[-1,1]\times[-1,1]$ and
$u=u({\boldsymbol{x}})$ is the superposition of the incident, reflected, and
scattered waves
$u({\boldsymbol{x}})=e^{i{\boldsymbol{k}}\cdot{\boldsymbol{x}}}+e^{i{\boldsymbol{k}^{\ast}}\cdot{\boldsymbol{x}}}+\psi({\boldsymbol{x}}).$
(3)
Here ${\boldsymbol{k}}=(k_{1},k_{2},k_{3})$ is a vector such that
$|{\boldsymbol{k}}|=k$, ${\boldsymbol{k}^{\ast}}=(k_{1},k_{2},-k_{3})$, and
function $\psi({\boldsymbol{x}})$ satisfies the radiation condition
$\psi({\boldsymbol{x}})=\frac{e^{ik|{\boldsymbol{x}}|}}{|{\boldsymbol{x}}|}\left[f({\boldsymbol{k}},\widehat{\boldsymbol{x}})+O\left(\frac{1}{|{\boldsymbol{x}}|}\right)\right],\quad{\widehat{\boldsymbol{x}}}=\frac{{\boldsymbol{x}}}{|{\boldsymbol{x}}|}\in\mathbb{S}^{2}.$
(4)
The inverse scattering problem for (1)-(2) consists in determining the
impedance $\gamma({\boldsymbol{x}^{\prime}})$ by the far field pattern
$f=f(\boldsymbol{k},\widehat{\boldsymbol{x}})$ when $\boldsymbol{k}$ is fixed
and $\widehat{\boldsymbol{x}}\in\mathbb{S}^{2}$.
In the next section we introduce a modified potential $v$ and express the
impedance $\gamma$ through $v$ using an explicit formula. The mapping $v\to f$
is linear. Hence, the initial nonlinear inverse problem is split into two
steps: solution of a linear problem (restoring $v$ from $f$) and application
the explicit formula. The similar approach was used in the discrete
counterpart of the problem [4]. This approach does not formally require $k\gg
1$. We also modify it for large $k$ using the WKB method.
In the case of a bounded obstacle, the WKB method allows one to connect the
impedance $\gamma$ with the asymptotic expansion of the far field pattern (see
[5]). We perform a simple numerical calculations in order to find the range of
parameters for which the WKB method can be used to determine the impedance.
The inverse impedance problem has been considered in [6]-[7] for general
bounded obstacles. Our assumptions simplify the problem, and as a result its
analytical and numerical solutions become easier.
Note the difference in WKB approach in the inverse impedance problem for the
half space and a bounded obstacle. In the latter case the asymptotic expansion
of the far field in a given direction is determined by the value of the
impedance in a specific point if the obstacle is convex. This is not true for
the half space.
## 2 Explicit formula for the impedance
We reduce the problem to the whole space $\mathbb{R}^{3}$ by extending
function $u$ evenly through the boundary $x_{3}=0$ for $x_{3}<0$. Then
equations (1)-(2) are replaced by the Schrödinger equation
$(-\Delta+q)\,u=k^{2}u,\quad{\boldsymbol{x}}\in\mathbb{R}^{3},$ (5)
where potential
$q({\boldsymbol{x}})=-2ik\gamma({\boldsymbol{x}^{\prime}})\delta(x_{3})$, and
$\delta(x)$ is the Dirac delta-function.
Substituting (3) into (5), we obtain that the scattering solution
$\psi(\boldsymbol{x})$ satisfies the equation
$(-\Delta+q(\boldsymbol{x})-k^{2})\psi=-q(\boldsymbol{x})\left(e^{i\boldsymbol{k}\cdot\boldsymbol{x}}+e^{i\boldsymbol{k}^{\ast}\cdot\boldsymbol{x}}\right)=-2q(\boldsymbol{x})e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}.$
(6)
Equation (6) is uniquely solvable if $\psi$ satisfies the radiation conditions
(4). From (6) it follows
$(-\Delta-k^{2})\psi=-q(\boldsymbol{x})\left(2e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}+\psi\right)=2ik\gamma(\boldsymbol{x}^{\prime})\left(2e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}+\psi(\boldsymbol{x}^{\prime})\right)\delta(x_{3}).$
(7)
Let us denote by $c(\boldsymbol{x}^{\prime})$ the coefficient of
$\delta(\boldsymbol{x}^{\prime})$ in the right hand side of (7)
$c(\boldsymbol{x}^{\prime})=2ik\gamma(\boldsymbol{x}^{\prime})\left(2e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}+\psi(\boldsymbol{x}^{\prime})\right).$
(8)
In this notation equation (7) has the form
$(-\Delta-k^{2})\psi=c(\boldsymbol{x}^{\prime})\delta(x_{3}).$ (9)
Observe that coefficient $c(\boldsymbol{x}^{\prime})$ vanishes outside the
support of $\gamma(\boldsymbol{x}^{\prime})$ and hence solution of equation
(9) can be written as
$\psi(\boldsymbol{x})=\int_{\mathop{\rm
supp}\,\gamma}G(\boldsymbol{x}-\boldsymbol{y}^{\prime})c(\boldsymbol{y}^{\prime})\,d\boldsymbol{y}^{\prime},$
(10)
where $\displaystyle
G(\boldsymbol{x}-\boldsymbol{y})=\frac{1}{4\pi}\frac{e^{ik|\boldsymbol{x}-\boldsymbol{y}|}}{|\boldsymbol{x}-\boldsymbol{y}|}$
is the Green’s function of (9). Form (10) and (8) we obtain equation for
determining $c(\boldsymbol{x}^{\prime})$
$c(\boldsymbol{x}^{\prime})+q(\boldsymbol{x}^{\prime})\int_{\mathop{\rm
supp}\,\gamma}G(\boldsymbol{x}^{\prime}-\boldsymbol{y}^{\prime})c(\boldsymbol{y}^{\prime})\,d\boldsymbol{y}^{\prime}=-2q(\boldsymbol{x}^{\prime})e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}.$
(11)
Finally, it is convenient to introduce a modified potential
$v(\boldsymbol{x}^{\prime})$ as
$v(\boldsymbol{x}^{\prime})=\frac{1}{\pi}\,c(\boldsymbol{x}^{\prime})e^{-i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}}.$
(12)
Then (11) becomes
$v(\boldsymbol{x}^{\prime})2ik\gamma(\boldsymbol{x}^{\prime})\int_{\mathop{\rm
supp}\,\gamma}G(\boldsymbol{x}^{\prime}-\boldsymbol{y}^{\prime})e^{i\boldsymbol{k}^{\prime}\cdot(\boldsymbol{y}^{\prime}-\boldsymbol{x}^{\prime})}v(\boldsymbol{y}^{\prime})\,d\boldsymbol{y}^{\prime}=\frac{4ik}{\pi}\,\gamma(\boldsymbol{x}^{\prime}).$
(13)
Thus, if $v(\boldsymbol{x}^{\prime})$ is known, one can find
$\gamma(\boldsymbol{x})$ from (13) from the formula
$\gamma(\boldsymbol{x}^{\prime})=-\frac{iv(\boldsymbol{x}^{\prime})k^{-1}}{4\pi^{-1}+2\int_{\mathop{\rm
supp}\,\gamma}G(\boldsymbol{x}^{\prime}-\boldsymbol{y}^{\prime})e^{i\boldsymbol{k}^{\prime}\cdot(\boldsymbol{y}^{\prime}-\boldsymbol{x}^{\prime})}v(\boldsymbol{y}^{\prime})\,d\boldsymbol{y}^{\prime}},\quad\boldsymbol{x}^{\prime}\in\mathop{\rm
supp}\,\gamma.$ (14)
In the next section, we will describe a method of determining
$v(\boldsymbol{x}^{\prime})$ from the far field pattern
$f(\boldsymbol{k},\widehat{\boldsymbol{x}})$ with fixed $\boldsymbol{k}$, and
this will complete the solution of the inverse impedance problem.
## 3 Calculation of the modified potential
Equation (10) contains Green’s function of a shifted argument whose asymptotic
behavior has the form
$G(\boldsymbol{x}-\boldsymbol{y})=\frac{1}{4\pi}\frac{e^{ik|\boldsymbol{x}|}}{|\boldsymbol{x}|}\,e^{-ik\widehat{\boldsymbol{x}}\cdot\boldsymbol{y}}\left[1+O\left(\frac{1}{|\boldsymbol{x}|}\right)\right],\quad|\boldsymbol{x}|\to\infty.$
(15)
Substituting it into (4) and (10), we obtain the following representation for
the far field pattern $f(\boldsymbol{k},\widehat{\boldsymbol{x}})$
$f(\boldsymbol{k},\widehat{\boldsymbol{x}})=\frac{1}{4}\int_{\mathop{\rm
supp}\,\gamma}e^{-i(k\widehat{\boldsymbol{x}}-\boldsymbol{k})\cdot\boldsymbol{y}^{\prime}}v(\boldsymbol{y}^{\prime})\,d\boldsymbol{y}^{\prime},$
(16)
where $\boldsymbol{y}^{\prime}=(y_{1},y_{2},0)$ and
$\widehat{\boldsymbol{x}}=\boldsymbol{x}/|\boldsymbol{x}|$. Our next goal is
to select directions $\widehat{\boldsymbol{x}}$ so that the integral (16)
would represent the Fourier coefficients of function
$v(\boldsymbol{y}^{\prime})$.
To this end, we write down the incident vector as
$\boldsymbol{k}=k(\cos\varphi_{1},\cos\varphi_{2},\cos\varphi_{3})$, while
${\widehat{\boldsymbol{x}}}=(\cos\theta_{1},\cos\theta_{2},\cos\theta_{3})$.
Then (16) becomes
$f({\boldsymbol{k},\widehat{\boldsymbol{x}}})=\frac{1}{4}\int_{-1}^{1}\int_{-1}^{1}e^{-ik\left[x(\cos\theta_{1}-\cos\varphi_{1})+y(\cos\theta_{2}-\cos\varphi_{2})\right]}\,v(x,y)\,dxdy.$
(17)
Expression (17) can be associated with the Fourier coefficients of $v(x,y)$ if
angles $\theta_{1}=\theta_{1,n1}$ and $\theta_{2,n2}$ are chosen in such a way
that
$\displaystyle k(\cos\theta_{1,n1}-\cos\varphi_{1})$ $\displaystyle=\pi
n_{1},$ (18) $\displaystyle k(\cos\theta_{2,n2}-\cos\varphi_{2})$
$\displaystyle=\pi n_{2},$ (19)
where $n_{1},n_{2}=0,\pm 1,\pm 2,\ldots$, and
$\displaystyle-\frac{k}{\pi}\,(1+\cos\varphi_{i})\leqslant
n_{i}\leqslant\frac{k}{\pi}\,(1-\cos\varphi_{i}),\quad i=1,2.$ (20)
For those directions defined by the angles $\theta_{1,n_{1}}$ and
$\theta_{2,n_{2}}$, the measured far field pattern $f_{n_{1},n_{2}}$ will be
the Fourier coefficient in the expansion of the modified potential $v(x,y)$
$f_{n_{1},n_{2}}=\frac{1}{4}\int_{-1}^{1}\int_{-1}^{1}e^{-\pi
i\left(n_{1}x+n_{2}y\right)}\,v(x,y)\,dxdy.$ (21)
Hence, $v(x,y)$ has the following Fourier series representation
$\displaystyle v(x,y)$ $\displaystyle=\sum_{n_{1},n_{2}}f_{n_{1}n_{2}}\,e^{\pi
i\left(n_{1}x+n_{2}y\right)}.$ (22)
Formula (14) along with (22) provides the solution of the inverse impedance
problem.
## 4 Asymptotic solution
Now we are going to modify the previous approach assuming $k\gg 1$ and using
the WKB approximation. The scattered wave $\psi(\boldsymbol{x})$ satisfies the
Helmholtz equation in the half space
$\mathbb{R}^{3}_{+}=\\{{\boldsymbol{x}}=(x_{1},x_{2},x_{3}),\,x_{3}>0\\}$
$-\Delta\psi=k^{2}\psi,\quad x_{3}>0,$ (23)
and the boundary condition
$\left.\rule[11.38109pt]{0.0pt}{0.0pt}\psi_{x_{3}}+ik\gamma({\boldsymbol{x}^{\prime}})\psi({\boldsymbol{x}^{\prime}})\right|_{x_{3}=0}=-2ik\gamma({\boldsymbol{x}^{\prime}})\,e^{i\boldsymbol{k}^{\prime}\cdot\boldsymbol{x}^{\prime}},\quad{\boldsymbol{x}^{\prime}}=(x_{1},x_{2},0).$
(24)
In order to find asymptotic behavior of $\psi(\boldsymbol{x})$ for large $k$,
we will use the WKB approximation of $\psi(\boldsymbol{x})$ in a neighborhood
of support of $\gamma({\boldsymbol{x}^{\prime}})$. We will be looking for
expansion of $\psi$ in the form
$\psi(\boldsymbol{x})=e^{i{\boldsymbol{k}^{\ast}}\cdot{\boldsymbol{x}}}\sum_{n=0}^{\infty}\Psi_{n}(\boldsymbol{x})\left(ik\right)^{-n},\quad\boldsymbol{k}^{\ast}=(k_{1},k_{2},-k_{3}).$
(25)
Coefficients in this expansion can be found explicitly. Substituting (25) into
equation (23) and equating the coefficients of like powers of $k$, we obtain a
recurrence system of differential equation for $\Psi_{n}(\boldsymbol{x})$
$\displaystyle\widehat{\boldsymbol{k}^{\ast}}\cdot\nabla\Psi_{0}$
$\displaystyle=0;$ (26) $\displaystyle
2\widehat{\boldsymbol{k}^{\ast}}\cdot\nabla\Psi_{n}+\Delta\Psi_{n-1}$
$\displaystyle=0,\;\;n\geqslant 1,$ (27)
where $\widehat{\boldsymbol{k}^{\ast}}$ denotes the unit vector in the
direction of vector $\boldsymbol{k}^{\ast}$. From the boundary condition (24),
one can find the initial condition for $\Psi_{n}(\boldsymbol{x})$ and thus
determine all the coefficients $\Psi_{n}(\boldsymbol{x})$. In particular, (26)
implies
$\Psi_{0}(x_{1},x_{2},x_{3})=\Phi_{0}\left(x_{1}+\frac{k_{1}x_{3}}{k_{3}},x_{2}+\frac{k_{2}x_{3}}{k_{3}}\right),$
(28)
where $\Phi_{0}$ is an arbitrary differentiable function. From (24) it follows
that
$\Psi_{0}(x_{1},x_{2},0)=-\frac{2\gamma(x_{1},x_{2})}{\gamma(x_{1},x_{2})-k_{3}k^{-1}},$
(29)
and hence
$\Psi_{0}(x_{1},x_{2},x_{3})=-\frac{2\gamma(x_{1}+k_{1}k_{3}^{-1}x_{3},x_{2}+k_{2}k_{3}^{-1}x_{3})}{\gamma(x_{1}+k_{1}k_{3}^{-1}x_{3},x_{2}+k_{2}k_{3}^{-1}x_{3})-k_{3}k^{-1}}.$
(30)
Thus, using expansion (25) and relations (8) and (12), we obtain the following
asymptotic representation for the scattering amplitude $f$ (16) through the
boundary impedance $\gamma$
$f(\boldsymbol{k},\widehat{\boldsymbol{x}})=-\frac{ik_{3}}{\pi}\int_{\mathop{\rm
supp}\,\gamma}e^{-i(k\widehat{\boldsymbol{x}}-\boldsymbol{k})\cdot\boldsymbol{y}^{\prime}}\frac{\gamma(\boldsymbol{y}^{\prime})}{\gamma(\boldsymbol{y}^{\prime})-k_{3}k^{-1}}d\boldsymbol{y}^{\prime}+O\left(k^{-1}\right).$
(31)
If we choose the direction of measurements $\widehat{\boldsymbol{x}}$ of the
far field pattern the same as before in (18)-(20), then the value
$f(\boldsymbol{k},\widehat{\boldsymbol{x}})$ becomes proportional to the
Fourier coefficient of $\gamma(\gamma-k_{3}k^{-1})^{-1}$
$f_{n_{1},n_{2}}=-\frac{ik_{3}}{\pi}\int_{-1}^{1}\int_{-1}^{1}e^{-\pi
i(n_{1}y_{1}+n_{2}y_{2})}\,\frac{\gamma(y_{1},y_{2})}{\gamma(y_{1},y_{2})-k_{3}k^{-1}}\,dy_{1}dy_{2}+O\left(k^{-1}\right).$
(32)
Applying the inverse Fourier transform, we obtain
$\frac{\gamma(x_{1},x_{2})}{\gamma(x_{1},x_{2})-k_{3}k^{-1}}=\frac{\pi
i}{4k_{3}}\sum_{m,n}f_{m,n}\,e^{\pi i(mx_{1}+nx_{2})}+O\left(k^{-1}\right).$
(33)
This equation can be solved for $\gamma$. Hence the boundary impedance can be
restored using the values of the far field pattern in the specific directions
given by (18)-(19).
## 5 Scattering from sphere
In the case of convex body $\Omega$, there is a direct asymptotic relation
between the far field pattern and the boundary impedance for large values of
$k$ [5]
$\gamma({\boldsymbol{y}}^{+})=\frac{\mbox{\tenscr
K}^{\;\;-\frac{1}{2}}({\boldsymbol{y}}^{+})+2|f({\boldsymbol{k}},\widehat{\boldsymbol{x}})|}{\mbox{\tenscr
K}^{\;\;-\frac{1}{2}}({\boldsymbol{y}}^{+})-2|f({\boldsymbol{k}},\widehat{\boldsymbol{x}})|}\;{\boldsymbol{n}}\cdot\widehat{\boldsymbol{x}}+O(k^{-1}),$
(34)
where ${\boldsymbol{y}}^{+}(\widehat{\boldsymbol{x}})\in\partial\Omega$ is the
preimage of
${\boldsymbol{n}}=(\widehat{\boldsymbol{x}}-\widehat{\boldsymbol{k}})/|\widehat{\boldsymbol{x}}-\widehat{\boldsymbol{k}}|$
under the Gauss map, and $\mbox{\tenscr K}\;({\boldsymbol{y}}^{+})$ is the
Gauss curvature at ${\boldsymbol{y}}^{+}\in\Omega$. Formula (34) has
asymptotic character, and we want to figure out the range of values of $k$
that give a good approximation of $\gamma$. We also analyze the approximation
of the far field pattern $f({\boldsymbol{k}},\widehat{\boldsymbol{x}})$ by the
measurement of the scattered field at the distance $r$.
In order to conduct a numerical experiment, we restrict ourselves to the case
where the direct problem can be easily solved. For that purpose we consider
the problem that has an exact solution – scattering of plane wave $e^{ikz}$
from a sphere of radius $a$ with constant boundary impedance. Then the formula
(34) takes the form
$\gamma=\frac{a+2|f({\boldsymbol{k}},\widehat{\boldsymbol{x}})|}{a-2|f({\boldsymbol{k}},\widehat{\boldsymbol{x}})|}\sin\frac{\theta}{2}+O(k^{-1}),\quad\frac{\pi}{2}\leqslant\theta\leqslant\pi,$
(35)
where $\theta$ is the polar angle on sphere.
Similar to (1), we need to solve the boundary value problem for the Helmholtz
equation
$\displaystyle-\Delta u=k^{2}u,\quad r>a,$ (36)
$\displaystyle\left.u_{r}-ik\gamma\,u\right|_{r=a}=0,$ (37)
where $\gamma>0$ is a constant surface impedance and
$u({\boldsymbol{x}})=e^{ikz}+\varphi({\boldsymbol{x}})$ (38)
with $\varphi({\boldsymbol{x}})$ satisfying the radiation condition (4).
Solution of the problem (36)-(38) is given by
$u=e^{ikz}-\sum_{n=0}^{\infty}(2n+1)i^{n}\frac{nj_{n-1}(ka)-(n+1)j_{n+1}(ka)-i\gamma
j_{n}(ka)}{nh_{n-1}(ka)-(n+1)h_{n+1}(ka)-i\gamma
h_{n}(ka)}\,h^{(1)}_{n}(kr)P_{n}(\cos\theta),$ (39)
where $j_{n}(z)$ and $h^{(1)}_{n}(z)$ are spherical Bessel functions of the
first and third kind, respectively, and $P_{n}(x)$ are Legendre polynomials
[8]. From this formula we can determine the far field pattern
$f({\boldsymbol{k}},\widehat{\boldsymbol{x}})$ (4) and calculate the surface
impedance $\gamma$ using it asymptotics (35) for large values of $k$.
## 6 Numerical examples
(a) (b)
Figure 1: Reconstruction of the boundary impedance $\gamma=2$ of the
illuminated part of the sphere of radius $a=1$ from the formula (35). Left:
the far field pattern $f(k,\widehat{\boldsymbol{x}})$ with the wave number
$k=200$ is approximated by the amplitude of the scattered wave at different
distances $r$ from the center of the sphere. Right: the far field pattern
$f(k,\widehat{\boldsymbol{x}})$ is approximated as before with $r=200a$, wave
numbers $k$ vary. Deviation from $\gamma=2$ increases as angle $\theta$
approached $90^{\circ}$ where incident rays are tangent to the sphere.
Figure 1 shows the reconstructed boundary impedance $\gamma=2$ of a unit
sphere based on the asymptotic formula (35). In the left figure, the far field
pattern with the wave number $k=200$ was determined from the exact solution
using (4)
$|f({\boldsymbol{k}},\widehat{\boldsymbol{x}})|\approx|\boldsymbol{x}|\psi(\boldsymbol{x})$
(40)
for various distances $|\boldsymbol{x}|$ from the sphere. The accuracy of
approximation monotonically improves as the polar angle $\theta$ increases and
does not exceed about $0.5\%$ for the distances beyond $r=100a$. The right
figure shows the dependence of the restored impedance on the wave number of
the incident plane wave while at the distance $r=200a$ from the sphere. As the
wave number $k$ decreases, not only approximation of $\gamma$ deteriorates,
but it also starts to exhibit oscillatory behavior. Approximation of $\gamma$
is also improving for larger values of $\theta$ and remains below $0.5\%$ as
long as $k>50$.
The above approach leads to a good approximation of the impedance if the far
field is measured at a distance by order of magnitude greater than the
diameter of the sphere and for the wave length that is by order of magnitude
lesser than the diameter of the sphere. Similar results are observed in
restoring a compactly supported boundary impedance of a half space. Using the
measurements of the far field pattern and both formulas (33), (14), and (12),
we reconstructed the boundary impedance 2(a). In figure 2, we used explicit
formula (14) with $k=10$ in (b). Then the Fourier coefficients of the far
field pattern were perturbed by random numbers uniformly distributed in the
interval $[-1,1]$. Figures 2(c)-(d) show reconstructed boundary impedance for
$k=15$ when the amplitudes of the additive random noise were $1\%$ and $5\%$
of the greatest Fourier coefficient, respectively.
(a) (b)
(c) (d)
Figure 2: Reconstruction of the boundary impedance $\gamma(x)$ (a) using
explicit formula formula (14) for different values of the wave vector $k$. In
(b) $k=10$. $k=15$ with $1\%$ (c) and $5\%$ (d) additive random noise.
Reconstruction of the boundary impedance by asymptotic formula (33) gives
slightly lesser accuracy as compared with exact formula (14). In figure 3 we
reconstructed the boundary impedance from figure 2a using asymptotic formula
(33) and $k=15$. The Fourier coefficients of the far field pattern are
corrupted by an additive uniformly distributed random noise form the interval
$[-1,1]$ with the amplitude $1\%$ (a) and $5\%$ (b) of the largest Fourier
coefficient. The accuracy of reconstruction in this case is slightly less as
compared with exact formula (14).
(a) (b)
Figure 3: Reconstruction of the same boundary impedance $\gamma(x)$ from
figure 2a using asymptotic formula (33) when $k=15$ and the amplitude of the
additive random noise is $1\%$ (a) and $5\%$ (b), respectively.
Finally, in figure 4 we reconstruct the impedance in the presence of a $1\%$
additive random noise when the wavenumber is as small as $k=5$. Although in
this case there is a significant error in the restored amplitude, it captures
qualitatively the shape and the location of the inhomogeneity of the
impedance.
Figure 4: Reconstructed boundary impedance $\gamma(x)$ from figure 2a using
either explicit (14) or asymptotic formula (33) when $k=5$ and the amplitude
of the additive random noise is $1\%$.
## 7 Conclusions
We have considered the problem of determining a compactly supported boundary
impedance from knowledge of the time harmonic incident wave and its far field
pattern. The approach is based on a special selection of the directions in
which the far field pattern is measured. Then the boundary impedance is
expressed through a potential using a simple exact formula, while the Fourier
coefficients of the potential equal the measured far field patterns.
Efficiency of the approach is illustrated by numerical examples.
## References
* [1] M. Shimoda and M. Miyoshi, Estimation of surface impedance for inhomogeneous half-space using far fields, IEICE Trans. Electron. 88-C (2005), pp. 2199-2207.
* [2] D. Hellin et al., Trends in total reflection X-ray fluorescence spectrometry for metallic contamination control in semiconductor nanotechnology, Spectrochimica Acta Part B 61 (2006), pp. 496–514.
* [3] L. Berquez, D. Marty-Dessus and J. L. Franceschi, Defect detection in dilicon wafer by photoacoustic imaging, Jpn. J. Appl. Phys. 42 (2003), pp.L1198-L1200.
* [4] Yu. Godin and B. Vainberg, A simple method for solving the inverse scattering problem for the difference Helmholtz equation, Inverse Problems 24 (2008), 025007.
* [5] A. Majda, High-frequency asymptotics for scattering matrix and inverse problem of acoustical scattering, Communications on Pure and Applied Mathematics 29 (1976), pp.261-291.
* [6] D. Colton and A. Kirsch, The determination of the surface impedance of an obstacle from measurements of the far field pattern, SIAM J. Appl. Math. 41 (1981), pp.8-15.
* [7] F. Cakoni and D. Colton, The determination of the surface impedance of a partially coated obstacle from far field data, SIAM J. Appl. Math. 64 (2004), pp.709-723.
* [8] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover 1965.
|
arxiv-papers
| 2009-08-03T22:32:51 |
2024-09-04T02:49:04.410544
|
{
"license": "Public Domain",
"authors": "Yuri A. Godin, Boris Vainberg",
"submitter": "Yuri Godin",
"url": "https://arxiv.org/abs/0908.0359"
}
|
0908.0373
|
11institutetext: T-5, Center for Nonlinear Studies
Los Alamos National Laboratory
Los Alamos, New Mexico 87545
marko@lanl.gov
# A Reflection on the Structure and Process of the Web of Data
Marko A. Rodriguez
11footnotetext: Rodriguez, M.A., “A Reflection on the Structure and Process of
the Web of Data,” Bulletin of the American Society for Information Science and
Technology, volume 35, number 6, pages 38–43, ISSN:1550-8366, LA-UR-09-03724,
September 2009.
The Web community has introduced a set of standards and technologies for
representing, querying, and manipulating a globally distributed data structure
known as the Web of Data. The proponents of the Web of Data envision much of
the world’s data being interrelated and openly accessible to the general
public. This vision is analogous in many ways to the Web of Documents of
common knowledge, but instead of making documents and media openly accessible,
the focus is on making data openly accessible. In providing data for public
use, there has been a stimulated interest in a movement dubbed Open Data
opendata:miller2008 . Open Data is analogous in many ways to the Open Source
movement. However, instead of focusing on software, Open Data is focused on
the legal and licensing issues around publicly exposed data. Together, various
technological and legal tools are laying the groundwork for the future of
global-scale data management on the Web. As of today, in its early form, the
Web of Data hosts a variety of data sets that include encyclopedic facts, drug
and protein data, metadata on music, books and scholarly articles, social
network representations, geospatial information, and many other types of
information. The size and diversity of the Web of Data is a demonstration of
the flexibility of the underlying standards and the overall feasibility of the
project as a whole. The purpose of this article is to provide a review of the
technological underpinnings of the Web of Data as well as some of the hurdles
that need to be overcome if the Web of Data is to emerge as the defacto medium
for data representation, distribution, and ultimately, processing.
Technically, on the Web of Data, Uniform Resource Identifiers (URI) are used
to identify resources uri:berners2005 . For example, depending on what is
being modeled, a URI can denote a city, a protein, a music album, a scholarly
article, a person, etc. In general, any thing can be assigned a URI. An
example URI is http://www.lanl.gov#marko. This URI denotes the author of this
article, Marko. The URI has information pertaining to the what (marko), where
(www.lanl.gov), and how (http) of a resource. The URI is more general than the
URL of common knowledge as URIs are not required to resolve to retrievable
digital objects such as documents and media. Instead, URIs can denote abstract
concepts such as the person Marko, the class of dogs, or the notion of
friendship. Finally, the space of all URIs is an inherently distributed and
theoretically infinite space. This makes the URI space fit to represent
massive amounts of data distributed world wide. A convenient consequence of
this space is that the Web of Data can emerge atop it. However, while URIs can
denote things, they can not denote how things relates to each other. Relating
URIs is necessary in order to give greater meaning and context to datum.
Moreover, relating URIs is necessary to create the “web” aspect of the Web of
Data.
The Resource Description Framework (RDF) is a standardized data model for
linking URIs in order to create a network/graph of space of all URIs
rdfintro:miller1998 . RDF also supports the linking of URIs to primitive
literals such as strings, integers, floating point values, etc. An example RDF
statement to denote the fact that “Marko knows Fluffy” is
$\langle$http://www.lanl.gov#marko, http://xmlns.com/foaf/0.1/knows,
http://www.lanl.gov#fluffy$\rangle$. In order to make long URIs more readable,
namespace prefixes are generally used. With namespace prefixes, the previous
statement can be represented as
$\langle\texttt{lanl:marko},\texttt{foaf:knows},\texttt{lanl:fluffy}\rangle$.
All RDF statements have this three component form, where there exists a
subject (lanl:marko), a predicate (foaf:knows), and an object (lanl:fluffy).
As such, an RDF statement is also known as a triple. A URI can be involved in
multiple statements. For example, it is possible to state that, while being
known by Marko, Fluffy is also a dog, 5 years old, and lives in Santa Fe, New
Mexico. Data on Santa Fe and data on Fluffy become merged when statements
involving their two URIs are joined (directly or indirectly through multiple
links). The Web of Data becomes powerful when seemingly different data sets
are interlinked. The fact that Fluffy lives in Santa Fe automatically connects
data about Fluffy to geographic and encyclopedic data about Santa Fe, New
Mexico—its geospatial coordinates, nearby cities, culture, population, etc. As
more and more statements are added to the Web of Data, the Web of Data serves,
in a sense, as a global database of interlinked heterogeneous data. The
combination of both the URI and RDF has moved the World Wide Web beyond a Web
of Documents to that of a Web of Data, where every minutia of information can
be represented and interlinked for consumption by both man and machine.
RDF’s original use case has evolved beyond that of a logic-language for
knowledge representation and reasoning on the Semantic Web
webinterpret:rodriguez2009 . As the foundational technology of the Web of
Data, RDF can be seen as a general-purpose data model. It can be used to model
formal knowledge (the Semantic Web), graph/networks (the Giant Global Graph),
and software and abstract virtual machines (the Web of Programs) to name a
few. In many ways, URIs and RDF afford a memory structure analogous to the
local memory of a physical machine except that this memory structure is
distributed over physical machines world wide. Each physical machine stores
and manages a subset representation of the full Web of Data. RDF can be stored
on a physical machine in many ways. A simple, straightforward way is to
represent RDF statements in a file—an RDF document. A common misconception is
that RDF and RDF/XML are one in the same. RDF is a data model that has various
serialized representations with RDF/XML being one such serialization. Other
popular serializations include N3 and N-Triple. Thus, there are many types of
RDF documents. For the small-scale exposure of RDF data, an RDF document
suffices. For the large-scale exposure and processing of RDF data, an RDF
repository known as a triple store or graph database is usually the chosen
solution. The expanded use of RDF has been greatly facilitated by the
continued increase in the capacity and speed of RDF triple stores. Modern
high-end RDF triple stores can hold and process on the order of 10 billion
triples. Example high-end triple stores include Neo4j111Neo4j is available at
http://neo4j.org/. and AllegroGraph222AllegroGraph is available at
http://www.franz.com/agraph/allegrograph/.. What has been the sole territory
of relational database technologies may soon be displaced by the use of RDF
and the triple store. Moreover, because RDF is the common data model utilized
by triple stores, it is possible to integrate data sets across different
triple stores—across different RDF data providers. This integration is
conveniently afforded by the URI and RDF as Web standards and is a function
foreign to the relational database domain. With the Web of Data, no longer is
information isolated in individual inaccessible data silos, but is instead
exposed in an open and interconnected environment—the Web environment. The
means to integrate RDF data across different RDF data sets is explained next.
## 1 Linked Data and a Distributed Data Structure
In an effort to provide a seamless integration between the data provided by
different RDF data providers, the Linked Data community is focused on
developing the specifications and tools for linking RDF data sets into a
single, global “web of data” linkeddata:bizer2008 . Two RDF data sets link
together when one data set uses a URI maintained by another. For example,
suppose the URI lanl:fluffy minted and maintained by the Los Alamos National
Laboratory (LANL). As previously explained, this URI is denoting something in
the world—namely Fluffy. However, it is possible for someone other than LANL
to express statements about Fluffy. Assume that the Rensselaer Polytechnic
Institute (RPI) mints their own URI rpi:fluffy to denote Fluffy, where rpi is
the namespace prefix that resolves to http://www.rpi.edu#. At this point, both
lanl:fluffy and rpi:fluffy denote the same thing—they both denote the same
real-world object known as Fluffy. This idea is diagrammed in Figure 1a, where
the dashed lines identify which worldly things the URIs stand in reference to.
In order to link the LANL data set with the RPI data set, LANL can add the RDF
statement
$\langle\texttt{lanl:fluffy},\texttt{owl:sameAs},\texttt{rpi:fluffy}\rangle$
to its data set. This statement states that both lanl:fluffy and rpi:fluffy
denote the same real-world thing, Fluffy. This idea is diagrammed in Figure
1b. Given this statement, it is possible to traverse from lanl:fluffy (LANL)
to rpi:fluffy (RPI) and thus, migrate from the LANL data set to the RPI data
set. When two data sets denote the same thing, they can be linked.
Figure 1: Two RDF repositories utilize different URIs to denote the same
thing. By making it explicit that two URIs denote the same thing (i.e.
$\langle\texttt{lanl:fluffy},\texttt{owl:sameAs},\texttt{rpi:fluffy}\rangle$),
it is possible to merge data sets together. This merging of data sets is what
creates the Web of Data.
The Linked Data community is interested in both unifying RDF data sets as well
as specifying the behaviors associated with URI resolution. A Linked Data-
compliant data provider should return data when a URI is dereferenced—when a
representation of the resource being identified by the URI is requested. More
specifically, when a URI is dereferenced, a collection of statements
associated with that URI should be returned in some RDF serialization such as
RDF/XML. Given the example above, if a machine dereferences lanl:fluffy, it
will get the statement $\langle$lanl:fluffy, owl:sameAs, rpi:fluffy$\rangle$
returned to it. In other words, LANL returns all the RDF statements for which
lanl:fluffy is the subject of the triple (i.e. the outgoing edges from
lanl:fluffy). Now, the machine knows that lanl:fluffy and rpi:fluffy denote
the same thing. Thus, if it wants to know what RPI has stated about Fluffy, it
will dereference rpi:fluffy. Upon doing so, it should get the statement
$\langle$rpi:fluffy, rdf:type, rpi:Dog$\rangle$ returned to it. To the
machine, the Web of Data is one expansive interlinked web of URIs. To the
underlying servers, the Web of Data is broken up into multiple RDF subgraphs
(multiple data providers) and linked together when one data provider
references a URI minted and maintained by another data provider. It is noted
that resolving a Linked Data-compliant store’s URI is one way of getting data
from the Web of Data. For more complicated data gathering situations, many RDF
data providers expose SPARQL end-points to their triple stores. SPARQL is a
query language similar to SQL, but focused on graph queries as opposed to
table queries sparql:prud2004 .
An interesting consequence of the Web of Data is that it can greatly shift the
role of application and data providers. Currently, web applications are
required to maintain their own data source. For example, Amazon.com maintains
its database of books, Springer its database of journal articles, and iTunes
its database of music metadata. In order for users to utilize this data in
interesting ways, these same data providers must provide a front-end
application to interact with the data. In this way, data providers and
application developers are one in the same entity. This idea is diagrammed in
Figure 2a, where each application utilizes its own back-end database to
provide its front-end application with data. With the Web of Data, this model
is significantly altered. On the Web of Data, application providers and data
providers are cleanly separated. Data providers can provide and interlink
book, article, and music data on the Web of Data and application providers can
develop software to utilize this data for different end-user services—book
recommendations, citation analysis, and music metadata population. Moreover,
this same data can be utilized by multiple different application developers
and thus, this can yield many ways for the end-user to interact with the Web
of Data. In other words, Amazon.com’s data may be more efficiently presented
and processed if it was open for any developer to create a front-end
application for it. This idea is diagrammed in Figure 2b. The clean separation
between data and application providers is already taking place as plenty of
interlinked heterogeneous data currently exists on the Web of Data. A few
examples are provided here. Book data can be found at Amazon.com’s RDF
BookMashup and the RDF representation of Project Guttenburg. Scholarly data is
provided by the Digital Bibliography and Library Project (DBLP), ACM, IEEE,
amongst many others. Finally, various music data sets exist such as
MusicBrainz and AudioScrobbler. This data is leveraged, as mentioned
previously, by resolving URIs. For example, if one were to dereference this
URI in a standard web browser
http://rdf.freebase.com/ns/guid.9202a8c04000641f800000000001a49d, what is
returned is a set of RDF statements (as an RDF/XML document) linking this URI
to other URIs and literals. Accessible, interlinked, structured data is the
point of the Web of Data. An ecology of applications leveraging this data may
greatly advance applications and algorithms for processing data as no longer
are application developers burdened by the “cold start” problem of requiring
large-amounts of data to initiate a successful service faith2:rodriguez2009 .
No longer will consumers be confined to use certain web applications as no
longer are applications and data so tightly coupled.
Figure 2: a.) The typical web application requires its own data source on
which to provide its service. b.) On the Web of Data, application providers
are cleanly separated from the data providers.
## 2 Linked Process and a Distributed Process Infrastructure
While the Web of Data and the efforts expended by the Linked Data community
have provided a path towards global-scale data management, this model is
lacking one important component: an infrastructure for data processing. A
significant hurdle to overcome for this community is that of distributed
processing on this distributed data structure. Traversing the Web of Data is
not quite the same as traversing the Web of Documents. For the human, it is
reasonable to traverse from URI to URI exploring the Web of Data in a manner
similar to how the Web of Documents is traversed. That is, a human, using
their Web browser, can resolve URIs and view the RDF data returned. Moreover,
various human-friendly RDF browsers exist (usually in the form of a Web
browser plugin) to make it easy for humans to view and traverse the data on
the Web of Data. However, for a machine (i.e. an application, an algorithm),
the Web of Data can be traversed much faster than what a human can do by
manually clicking from URI to URI. Moreover, there will be orders of magnitude
more resources and links on the Web of Data than what is found on the Web of
Documents. While a machine can crawl and pull the data to its local
environment for processing, this becomes inefficient when the data
requirements span large parts of the Web of Data. Again, note that every time
a URI is dereferenced, the resolving server prepares an RDF subgraph and
returns it (over the wire) to the requesting machine. Thus, “traversing” the
Web of Data requires data to be migrated to the traversing machine and
processed remotely from the data source. This architecture is analogous to the
current Web of Documents whereby “traversing” the Web of Documents pulls HTML
documents and media to the requesting machine. For human consumption, this is
necessary as data/documents must be rendered where the human is physically
located—remote from the data source. For a machine (a virtual machine) its
physical location need not be a factor in how data is consumed and processed.
Thus, for processing large parts of a distributed data structure, a more
efficient mechanism would be to migrate the process between data providers so
that information is not pulled over the wire, but instead, processed where the
data is maintained. In other words, an efficient mechanism for processing the
Web of Data would be to move the process to the data, not the data to the
process rodriguez:gpsemnet2009 .
For the Web of Documents, the search engine philosophy of “download and index”
has made it possible for end users to find information in a more efficient
manner than by simply surfing and bookmarking. With modern commercial triple
stores scaling to the order of 10 billion triples, centralized indexing
repositories will have to contend with not only the volume of data, but also
the computational complexities of analyzing such data in sophisticated ways.
The Web of Data provides a much richer machine processable data structure than
what is provided by HTML and thus, antiquated keyword search simply does not
take significant advantage of what the Web of Data is providing. The future of
the Web of Data will be rife with algorithms from many schools of
thought—formal logic, graph analysis, object-oriented programming, etc.
webinterpret:rodriguez2009 . Many of these algorithms will compute across
various underlying stores of the Web of Data and will require a distributed
Turing complete infrastructure to do so. For any algorithm of sufficient
complexity, there is simply too much data to pull over the wire and thus, the
Web of Data in its current form greatly reduces what is possible. This is an
unfortunate state of affairs. Given the potential role of the Web of Data as
the defacto medium for interconnecting data, a distributed computing
environment is necessary. The Linked Data community needs a parallel Linked
Process effort. In a sense, data providers already expose their processors for
public use by way of their SPARQL-endpoints. SPARQL serves as an on-site data
processing language. However, this language, being a query language, is not
sufficient for representing complex algorithms. What is needed is a framework
that is more general-purpose and respective of the three following basic
requirements:
1. 1.
safe: applications must not be able to destroy the integrity of the open
processor or its data set when using this infrastructure.
2. 2.
efficient: applications must run faster in this infrastructure than what is
possible when pulling the required data over the wire.
3. 3.
easy to use: application developers must be be able to utilize common
programming languages and packages and be relatively blind to the underlying
infrastructure.
Developing a distributed process infrastructure that accounts for these three
factors will ultimately drive its adoption. With the widespread adoption of
such a processing infrastructure by RDF data providers, the Web of Data will
reach a new level of functionality. No longer will the Web of Data be only a
database serving data over the wire to third-party applications, but instead,
a distributed computing environment supporting complex algorithms that can
leverage rich data in ways not previously possible in the history of
computing. The unification of Linked Data and Linked Process in many ways is
similar to “cloud computing.” However, with the integration of data sets and
hardware processors world wide, this “cloud” will be much richer and more
decentralized than what exists with other cloud providers. In this form, the
Web of Data will afford the world a democratization of both data and process
and may perhaps enjoy a frenzied adoption similar to what has occurred with
its predecessor, the Web of Documents.
## 3 Conclusion
The Web of Data provides an infrastructure that supports an instantiation of a
distributed graph of Web resources. This distributed graph is created by many
data providers representing and interrelating their data. What emerges from
this collective effort is a publicly accessible global database that can be
leverage by both man and machine to any end they deem appropriate. However,
the current instantiation of the Web of Data lacks one crucial component: a
distributed processing infrastructure. For the Web of Documents of common
knowledge, the solution to the issue of processing the vast amount of
information has been to literally download the entire Web and index and
process it in a single environment. While the content on the Web of Documents
is distributed, the means by which the information on the Web of Documents is
analyzed is not. The Web of Data need not fall into this same model. With the
nearly limitless ways in which RDF data can be processed, it would be a
disappointment if the data on the Web of Data was left solely to centralized
repositories to store, index, and provide query functionality. Beyond
disappointment, it would reduce the potential utility the Web of Data would
have given a distributed process infrastructure. By extending the work of the
Linked Data community with Linked Process, the Web of Data may one day rise to
become the defacto medium for representing and processing data much like the
Web of Documents is the defacto medium for storing and sharing documents.
## References
* (1) Tim Berners-Lee, Roy T. Fielding, Day Software, Larry Masinter, and Adobe Systems. Uniform Resource Identifier (URI): Generic Syntax, January 2005.
* (2) Christian Bizer, Tom Heath, Kingsley Idehen, and Tim Berners-Lee. Linked data on the web. In Proceedings of the International World Wide Web Conference, Linked Data Workshop, Beijing, China, April 2008.
* (3) Eric Miller. An introduction to the Resource Description Framework. Bulletin of the American Society for Information Science and Technology, 25(1):15–19, November 1998.
* (4) Paul Miller, Rob Styles, and Tom Heath. Open Data commons: A license for Open Data. In Proceedings of the Workshop on Linked Data on the Web, New York, NY, April 2008. ACM Press.
* (5) Eric Prud’hommeaux and Andy Seaborne. SPARQL query language for RDF. Technical report, World Wide Web Consortium, October 2004.
* (6) Marko A. Rodriguez. Data Management in the Semantic Web, chapter Interpretations of the Web of Data. Nova Publishing, 2009.
* (7) Marko A. Rodriguez. Emergent Web Intelligence, chapter General-Purpose Computing on a Semantic Network Substrate. Springer-Verlag, Berlin, DE, 2009.
* (8) Marko A. Rodriguez and Jennifer H. Watkins. Faith in the algorithm, part 2: Computational eudaemonics. In Juan D. Velásquez, Robert J. Howlett, and Lakhmi C. Jain, editors, Proceedings of the International Conference on Knowledge-Based and Intelligent Information & Engineering Systems, Lecture Notes in Artificial Intelligence. Springer-Verlag, 2009.
|
arxiv-papers
| 2009-08-04T01:59:07 |
2024-09-04T02:49:04.415751
|
{
"license": "Public Domain",
"authors": "Marko A. Rodriguez",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/0908.0373"
}
|
0908.0421
|
# Depolarization for quantum channels with higher symmetries
A B Klimov1 and L L Sánchez-Soto2 1 Departamento de Física, Universidad de
Guadalajara, 44420 Guadalajara, Jalisco, Mexico 2 Departamento de Óptica,
Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
###### Abstract
The depolarization channel is usually modelled as a quantum operation that
destroys all input information, replacing it by a completely chaotic state.
For qubits this has a quite intuitive interpretation as a shrinking of the
Bloch sphere. We propose a way to deal with depolarizing dynamics (in the
Markov approximation) for systems with arbitrary symmetries.
###### pacs:
03.65.Yz,03.67.Hk,42.50.Lc
Quantum systems evolve via unitary operations determined by the Schrödinger
equation. This is also true for composite systems, e.g., a small quantum
system $\mathcal{S}$ that is surrounded by some environment $\mathcal{E}$,
with which it interacts. However, while the evolution of the total system is
described by a unitary operation $\hat{U}_{\mathcal{SE}}(t)$, the dynamics of
the system $\mathcal{S}$ alone (obtained by tracing out the uncontrollable
degrees of freedom of $\mathcal{E}$) is, in general, no longer unitary
[Breuer:2007, Weiss:2008, Alicki:2007]. The system-environment interaction
leads to entanglement between them, and this is reflected into the fact that
$\hat{U}_{\mathcal{SE}}(t)\neq\hat{U}_{\mathcal{S}}(t)\otimes\hat{U}_{\mathcal{E}}(t)$.
From the perspective of quantum information processing, such an interaction is
undesirable and causes errors and noise in the system [Nielsen:2000].
The dynamics of $\mathcal{S}$ can be described by a finite-time trace-
preserving completely positive map (CPM) [Stinespring:1955, Choi:1972] that
transforms input states $\hat{\varrho}_{\mathrm{in}}=\hat{\varrho}(0)$ into
output states $\hat{\varrho}_{\mathrm{out}}=\hat{\varrho}(t)$, i.e.,
$\varrho_{\mathrm{in}}\mapsto\hat{\varrho}_{\mathrm{out}}=\mathcal{E}_{t}(\hat{\varrho}_{\mathrm{in}})\,,$
(1)
which is also known as a quantum channel. One may think of the environment as
extracting information from the system, as it will typically map pure states
into mixed states.
This noise process can be also described by a quantum operation involving only
operators of the system of interest. This is called a Kraus decomposition
[Kraus:1983] and has the form (omitting all the unnecessary subscripts)
$\mathcal{E}(\hat{\varrho})=\sum_{r}\hat{K}_{r}\,\hat{\varrho}\,\hat{K}_{r}^{\dagger}\,,$
(2)
where the Kraus operators satisfy the condition
$\sum_{r}\hat{K}_{r}\,\hat{K}_{r}^{\dagger}=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,$ (3)
which ensures that unit trace is preserved for all times.
A channel is Markovian when the coupling coupling of the system $\mathcal{S}$
with the environment $\mathcal{E}$ can be treated under the Markov and Born
approximations. The evolution of the state at a given instant is then fully
determined by the state at that instant, so the process ha no “memory” of its
past. This is a commonly used approximation in quantum optics and leads to the
well-known Lindblad form of a master equation [Lindblad:1976, Gorini:1976].
For these Markovian channels, one can always write
$\mathcal{E}_{t}(\hat{\varrho})=e^{\mathcal{L}t}\,\hat{\varrho}(0)\,,$ (4)
where the Lindblad superoperator $\mathcal{L}$ is
$\mathcal{L}(\hat{\varrho})=-i[\hat{H},\hat{\varrho}]+\frac{1}{2}\sum_{r}a_{r}\left([\hat{L}_{r},\hat{\varrho}\hat{L}_{r}^{\dagger}]+[\hat{L}_{r}\hat{\varrho},\hat{L}_{r}^{\dagger}]\right)\,.$
(5)
Here $\hat{H}$ is the Hamiltonian of the undamped system $\mathcal{S}$,
$\hat{L}_{r}$ are system operators defined to model the effective dissipative
interaction with the environment, and $a_{r}\geq 0$ are constants that account
for decoherence rates. In this form, $\mathcal{L}$ appears as the generator of
a CMP. In fact, expanding equation (5) to first order in the short-time
interval $\tau$ one can immediately find the corresponding Kraus operators
[Shabani:2005]
$\displaystyle\hat{K}_{0}$ $\displaystyle=$
$\displaystyle\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}-\tau\left(i\hat{H}+\frac{1}{2}\sum_{r}\hat{L}_{r}^{\dagger}\,\hat{L}_{r}\right)\,$
$\displaystyle\hat{K}_{r}$ $\displaystyle=$
$\displaystyle\sqrt{\tau}\,\hat{L}_{r}\,.$
For obvious reasons, the characterization and classification of these maps
have attracted a lot of interest in recent years [Keyl:2002]. The majority of
the results obtained so far relate to two specific classes: the qubit channels
and the bosonic Gaussian channels [Caruso:2007]. In this paper, we shall be
mainly concerned with the former.
In classical computation, the only error that can occur is the bit flip
$0\leftrightarrow 1$. In quantum computation, however, the existence of
superposition states brings also the possibility of other basic errors for a
single qubit. They are the phase flip and the bit-phase flip. The first
changes the phase of the state, and the latter combines phase and bit flips.
The set of Kraus operators for each one of these channels is given by
$\hat{K}_{0}=\sqrt{1-p/2}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,\qquad\hat{K}^{j}_{1}=\sqrt{p/2}\,\hat{\sigma}_{j}\,,$
(7)
where $j=x$ gives the bit flip, $j=z$ the phase flip and $j=y$ the phase-bit
flip. They can also interpreted as corresponding to a probability $1-p/2$ of
remaining in the same state, and a probability $p/2$ of having an error. It is
not difficult to represent these channels in terms of a master equation of the
form (5): the associated Lindblad superoperator $\hat{L}^{j}_{1}$ turns out to
be the corresponding Pauli matrix $\hat{\sigma}_{j}$.
The environment-induced noise is modelled using various simplified approaches.
Here, we concentrate in the depolarization channel, which employs unbiased
noise generating bit flip errors and phase flip errors and is represented by
the CPM [Bennett:1997]
$\mathcal{E}(\hat{\varrho})=(1-\hat{\varrho})+\frac{p}{3}(\hat{\sigma}_{x}\,\hat{\varrho}\,\hat{\sigma}_{x}+\hat{\sigma}_{y}\,\hat{\varrho}\,\hat{\sigma}_{y}+\hat{\sigma}_{z}\,\hat{\varrho}\,\hat{\sigma}_{z})\,.$
(8)
Note, however, that, since the three interaction channels corresponding to bit
error, flip error, and phase error do not commute, one could rightly argue
that the incoherent addition of these channels in the CMP (8) is, at least,
questionable.
The Kraus decomposition of this channel reads as
$\displaystyle\hat{K}_{0}=\sqrt{1-p}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},$ (9)
$\displaystyle\displaystyle\hat{K}_{1}=\frac{p}{3}\,\hat{\sigma}_{x},\qquad\hat{K}_{2}=\frac{p}{3}\,\hat{\sigma}_{y},\qquad\hat{K}_{3}=\frac{p}{3}\,\hat{\sigma}_{z}\,,$
while the associated Lindblad equation is
$\dot{\hat{\varrho}}=-\Gamma\left(\hat{\varrho}-\frac{1}{2}\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\right)\,,$ (10)
where we have omitted the free evolution of the system, since it is irrelevant
for our purposes here, and $\Gamma$ is a constant. If we use the standard
Bloch parametrization for the density matrix
$\hat{\varrho}=\frac{1}{2}\left(\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}+\bm{s}\cdot\hat{\bm{\sigma}}\right)\,,$ (11)
we immediately get
$\dot{\bm{s}}=-\Gamma\bm{s}\,,$ (12)
which clearly shows that the action of the channel is to contract the sphere
with a lifetime $\Gamma^{-1}$. This is precisely the idea behind
depolarization: for long times the system will end in a fully depolarized or
chaotic state, whose density matrix is diagonal
$\hat{\varrho}_{\mathrm{unpol}}=\frac{1}{2}\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}$.
For $n$ qubits, a possible extension is to assume that the error operators can
be represented by the $n$-qubit Pauli group [Nielsen:2000]
$\mathcal{P}_{n}=\\{\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\\}^{\otimes
n}\,,$ (13)
where $\otimes n$ denotes the $n$-fold tensor product. This means that each
qubit is acted by the identical independent depolarizing channels. However,
alternative models, such as collective or correlated depolarization, have been
proposed [Reina:2002, Banaszek:2004, Ball:2004, Ball:2005, Ban:2006], which do
not fit in this simple extension. The point we wan to stress is that the final
result depends obviously of the symmetry of the channel and this, in turn,
depends on physical considerations.
The purpose of this work is to address this problem and identify the proper
depolarizing channel for more general situations. To this end, consider a
system whose Hamiltonian is a function of a symmetry Lie algebra
$\mathfrak{A}$. In order to maintain the discussion as simple as possible, we
assume that this algebra is semisimple and denote by $\Delta$ the set of
nonzero roots [Erdmann:2006dg]. We use the standard Cartan-Weyl basis
$\\{\hat{h}_{i},\hat{e}_{\alpha}\\}$ in terms of which we have the commutation
relations
$[\hat{h}_{i},\hat{h}_{j}]=0\,,\qquad[\hat{h}_{i},\hat{e}_{\alpha}]=\alpha(\hat{h}_{i})\,\hat{e}_{\alpha}\,,\qquad[\hat{e}_{\alpha},\hat{e}_{\beta}]=N_{\alpha\beta}\,\hat{e}_{\alpha+\beta}\,,$
(14)
where the last one is valid only when $\alpha+\beta\in\Delta$. The operators
$\\{\hat{h}_{i}\\}$ ($i$ runs from 1 to $\ell$, where $\ell$ is the rank of
the group) constitute the Cartan subalgebra and may be taken diagonal in any
irreducible representation. On the other hand,
$\\{\hat{e}_{\alpha},\hat{e}_{-\alpha}\\}$ are raising and lowering operators
and we can always choose $\hat{e}_{\alpha}^{\dagger}=\hat{e}_{-\alpha}$.
The Hilbert space decomposes also into finite-dimensional subspaces
$\mathcal{H}=\bigoplus_{\lambda}\mathcal{H}_{\lambda}$ (15)
so that in each $\mathcal{H}_{\lambda}$ the operators of $\mathfrak{A}$ act
irreducibly. Let $|\mathbf{h};\lambda\rangle$ be an orthonormal basis in
$\mathcal{H}_{\lambda}$, where
$\mathbf{h}=(h_{1},\ldots,h_{j},\ldots,h_{\ell})$. Then, we have that
$\hat{h}_{j}|\mathbf{h};\lambda\rangle=h_{j}|\mathbf{h};\lambda\rangle\,.$
(16)
Notice also that the density operator of any quantum unpolarized state can be
written in terms of these invariant subspaces
$\hat{\varrho}_{\mathrm{unpol}}=\bigoplus_{\lambda}r_{\lambda}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}_{\lambda}\,,$ (17)
where $\hat{\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}}_{\lambda}$
denotes the unity in the corresponding subspace and all the coefficients
$r_{\lambda}$ are real and nonnegative and they fulfill the unit-trace
condition.
Since all the error operators are in the algebra $\mathfrak{A}$, the only
possible Kraus operators are also elements of $\mathfrak{A}$, and they induce
a local Lindblad equation fully analogous to (5). Very different processes can
be described in terms of this master equation. The first one is what can be
called a pure dephasing channel, represented by
$\hat{L}_{r}=\hat{h}_{r}\,.$ (18)
Such a map obviously preserves the diagonal operators:
$\mathcal{L}(\hat{h}_{r})=0$, and the asymptotic limit of the channel is given
by
$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\mathbf{h},\lambda}\varrho_{\mathbf{h}\mathbf{h}}(\lambda)\,|\mathbf{h};\lambda\rangle\langle\mathbf{h};\lambda|\,,$
(19)
where $\varrho_{\mathbf{h}\mathbf{h}}(\lambda)$ are just precisely the
occupation probabilities in each invariant subspace. For the case a symmetry
algebra su(2), generated by $\\{\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}\\}$, the
archetypal example of this situation is
$\mathcal{L}(\hat{\varrho})=\hat{S}_{z}\,\hat{\varrho}\,\hat{S}_{z}-\frac{1}{2}(\hat{S}_{z}^{2}\,\hat{\varrho}+\hat{\varrho}\,\hat{S}_{z}^{2})=\hat{S}_{z}\,\hat{\varrho}\,\hat{S}_{z}-\hat{\varrho}\,,$
(20)
which is a standard way of accounting for process that lead to a loss of
coherence without changing the level populations [Briegel:1993xy].
Next we consider a generalized amplitude-damping channel, for which
$\hat{L}_{r}=\hat{e}_{-\alpha_{r}}\,.$ (21)
In the asymptotic limit, this leads to the ground state in each invariant
subspace
$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\lambda}|\mathbf{h}_{\mathrm{min}};\lambda\rangle\langle\mathbf{h}_{\mathrm{min}};\lambda|\,,$
(22)
where $|\mathbf{h}_{\mathrm{min}};\lambda\rangle$ is the lowest-weight state
that is annihilated by all the lowering operators:
$\hat{e}_{-\alpha}|\mathbf{h}_{\mathrm{min}};\lambda\rangle=0$. It is clear
that now no Lindblad (neither Kraus) map preserves the diagonal operators. In
particular, for su(2) symmetry, the textbook example is
$\mathcal{L}(\hat{\varrho})=\hat{S}_{-}\hat{\varrho}\hat{S}_{+}-\frac{1}{2}(\hat{S}_{+}\hat{S}_{-}\hat{\varrho}+\hat{\varrho}\hat{S}_{+}\hat{S}_{-})\,.$
(23)
which is a standard model for damping [Agarwal:1974rm].
While these generalizations are more or less obvious, the corresponding one
for a depolarization channel is far from trivial. We claim that such a
situation must be described by the action of a Lindblad operator proportional
to $\hat{e}_{-\alpha_{r}}$ followed by other proportional to
$\hat{e}_{+\alpha_{r}}$. One can check that only in this way we asymptotically
get an unpolarized state
$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\lambda}\Tr[\hat{\varrho}(\lambda)]\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}_{\lambda}=\hat{\varrho}_{\mathrm{unpol}},$ (24)
where the trace operation is taken in each invariant subspace. The associated
Lindblad operator is
$\mathcal{L}(\hat{\varrho})=\frac{1}{2}\sum_{r}a_{r}\left(2\hat{e}_{-\alpha_{r}}\,\hat{\varrho}\,\hat{e}_{\alpha_{r}}+2\hat{e}_{\alpha_{r}}\,\hat{\varrho}\hat{e}_{-\alpha_{r}}-\\{\hat{e}_{\alpha_{r}},\hat{e}_{-\alpha_{r}}\\}\,\hat{\varrho}-\hat{\varrho}\,\\{\hat{e}_{\alpha_{r}},\hat{e}_{-\alpha_{r}}\\}\right)\,,$
(25)
and one can check that $\mathcal{L}(\hat{\varrho}_{\mathrm{unpol}})=0$. An
example of this situation for qubit systems parallels completely (25), but
with the corresponding spin-like operators is
$\mathcal{L}(\hat{\varrho})=\hat{S}_{-}\hat{\varrho}\hat{S}_{+}+\hat{S}_{+}\hat{\varrho}\hat{S}_{-}-\frac{1}{2}\left(\\{\hat{S}_{+},\hat{S}_{-}\\}\hat{\varrho}+\hat{\varrho}\\{\hat{S}_{+},\hat{e}_{-}\\}\right)\,.$
(26)
It is worth noting that such a Linblad operator appears as a limit case of
decaying into a bath at infinite temperature.
An effective depolarization channel is actually a common situation in driven
dissipative systems. Consider a typical evolution equation
$\dot{\hat{\varrho}}=-ig[\hat{S}_{x},\hat{\varrho}]+\gamma[\hat{S}_{-}\hat{\varrho}\hat{S}_{+}-\frac{1}{2}\left(\\{\hat{S}_{+},\hat{S}_{-}\\}\,\hat{\varrho}+\hat{\varrho}\,\\{\hat{S}_{+},S_{-}\\}\right)\,,$
(27)
which describes the decay (with rate $\gamma$) of an externally-driven (with
coupling constant $\kappa$) collective spin into a zero-temperature bath
[Agarwal:1974rm]. In the strong-pumping limit ($g\gg\gamma$), the above
equation can be easily diagonalized [Klimov:2000la]: it is enough to apply the
rotation $\hat{U}=\exp(i\pi\hat{S}_{y}/2)$ and to make the rotating wave
approximation. The final result is
$\displaystyle\dot{\hat{\varrho}}_{d}$ $\displaystyle=$ $\displaystyle-
ig[\hat{S}_{z},\hat{\varrho}_{d}]+\frac{\gamma}{2}\left(2\hat{S}_{z}\hat{\varrho}\hat{S}_{z}-\hat{S}_{z}^{2}\hat{\varrho}-\hat{\varrho}\hat{S}_{z}^{2}\right)$
(28) $\displaystyle+$
$\displaystyle\frac{\gamma}{2}\left(2\hat{S}_{-}\hat{\varrho}\hat{S}_{+}+2\hat{S}_{+}\hat{\varrho}\hat{S}_{-}-\\{\hat{S}_{+},\hat{S}_{-}\\}\,\hat{\varrho}-\hat{\varrho}\,\\{\hat{S}_{+},\hat{S}_{-}\\}\right)\,,$
where $\hat{\varrho}_{d}=\hat{U}\,\hat{\varrho}\,\hat{U}^{\dagger}$ is the
density matrix in the rotated frame. We can clearly observe the emergence a
pure dephasing (first line) and a depolarizing channel (second line).
In summary, what we expect to have accomplished in this paper is to provide a
construction of the depolarizing channel for systems with arbitrary
symmetries. This may be more than an academic curiosity for more involved
systems currently under investigation as candidates for quantum information
processing.
This work was supported by the Grant No 45704 of Consejo Nacional de Ciencia y
Tecnología (CONACyT) and the Spanish Research Directorate (Grant
FIS2005-06714). A. B. K. was also supported by the Spanish Sabbatical Program
(Grant SAB2006-0064).
## References
* [1] Agarwal1974Agarwal:1974rm Agarwal G 1974 Quantum Statistical Theories of Spontaneous Emission and their Relation to other Approaches Vol. 70 of Springer Tracts in Modern Physics Springer.
http://dx.doi.org/10.1007/BFb0042382
* [2] Alicki Lendi2007Alicki:2007 Alicki R Lendi K 2007 Quantum Dynamical Semigroups and Applications Vol. 286 of Lecture Notes in Physics Springer.
* [3] Ball et al.2004Ball:2004 Ball J, Dragan A Banaszek K 2004 Phys. Rev. A 69, 042324.
* [4] Ball Banaszek2005Ball:2005 Ball J L Banaszek K 2005 Open Sys. Inf. Dyn. 12, 121–131.
* [5] Ban Shibata2006Ban:2006 Ban M Shibata F 2006 Phys. Lett. A 354, 35–39.
* [6] Banaszek et al.2004Banaszek:2004 Banaszek K, Dragan A, Wasilewski W Radzewicz C 2004 Phys. Rev. Lett. 92, 257901.
* [7] Bennett et al.1997Bennett:1997 Bennett C H, DiVincenzo D P Smolin J A 1997 Phys. Rev. Lett. 78, 3217–3220.
* [8] Breuer Petruccione2007Breuer:2007 Breuer H P Petruccione F 2007 The Theory of Open Quantum Systems Oxford University Press Oxford.
* [9] Briegel Englert1993Briegel:1993xy Briegel H J Englert B G 1993 Phys. Rev. A 47, 3311–3329.
* [10] Caruso Giovannetti2007Caruso:2007 Caruso F Giovannetti V 2007 Phys. Rev. A 76, 042331.
* [11] Choi1972Choi:1972 Choi M D 1972 Can. J. Math. 24, 520—529.
* [12] Erdmann Wildon2006Erdmann:2006dg Erdmann K Wildon M 2006 Introduction to Lie Algebras Springer Berlin.
* [13] Gorini et al.1976Gorini:1976 Gorini V, Kossakowski A Sudarshan E C G 1976 J. Math. Phys. 17, 821–825.
* [14] Keyl2002Keyl:2002 Keyl M 2002 Phys. Rep. 369, 431–548.
* [15] Klimov Sánchez-Soto2000Klimov:2000la Klimov A B Sánchez-Soto L L 2000 Phys. Rev. A 61, 063802.
* [16] Kraus1983Kraus:1983 Kraus K 1983 States, Effects and Operations: Fundamental No- tions of Quantum Theorytions of Quantum Theory Springer Berlin.
* [17] Lindblad1976Lindblad:1976 Lindblad G 1976 Commun. Math. Phys. 48, 119–130.
* [18] Nielsen Chuang2000Nielsen:2000 Nielsen M A Chuang I L 2000 Quantum Computation and Quantum Information Cambridge University Press Cambridge.
* [19] Reina et al.2002Reina:2002 Reina J H, Quiroga L Johnson N F 2002 Phys. Rev. A 65, 032326.
* [20] Shabani Lidar2005Shabani:2005 Shabani A Lidar D A 2005 Phys. Rev. A 71, 020101.
* [21] Stinespring1955Stinespring:1955 Stinespring W F 1955 Proc. Am. Math. Soc. 6, 211–216.
* [22] Weiss2008Weiss:2008 Weiss U 2008 Quantum Dissipative Systems third edn World Scientific Singapore.
* [23]
|
arxiv-papers
| 2009-08-04T10:20:05 |
2024-09-04T02:49:04.421642
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. B. Klimov and L. L. Sanchez-Soto",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0908.0421"
}
|
0908.0501
|
# Zipf’s law from a Fisher variational-principle
A. Hernando alberto@ecm.ub.es D. Puigdomènech puigdomenech@ecm.ub.es D.
Villuendas diego@ffn.ub.es C. Vesperinas cristina.vesperinas@sogeti.com A.
Plastino plastino@fisica.unlp.edu.ar Departament ECM, Facultat de Física,
Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain Departament
FFN, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028
Barcelona, Spain Sogeti España, WTCAP 2, Plaça de la Pau s/n, 08940 Cornellà,
Spain National University La Plata, IFCP-CCT-CONICET, C.C. 727, 1900 La
Plata, Argentina
###### Abstract
Zipf’s law is shown to arise as the variational solution of a problem
formulated in Fisher’s terms. An appropriate minimization process involving
Fisher information and scale-invariance yields this universal rank
distribution. As an example we show that the number of citations found in the
most referenced physics journals follows this law.
###### keywords:
Fisher information , scale-invariance , Zipf’s law
††journal: Physics Letters A
## 1 Introduction
This work discusses the application of Fisher’s information measure to some
scale-invariant phenomena. We thus begin our considerations with a brief
review of the pertinent ingredients.
### 1.1 Scale-invariant phenomena
The study of scale-invariant phenomena has unravelled interesting and somewhat
unexpected behaviours in systems belonging to disciplines of different nature,
from physical and biological to technological and social sciences [1]. Indeed,
empirical data from percolation theory and nuclear multifragmentation [2]
reflect scale-invariant behaviour, and so do the abundances of genes in
various organisms and tissues [3], the frequency of words in natural languages
[4], scientific collaboration networks [5], the Internet traffic [6], Linux
packages links [7], as well as electoral results [8], urban agglomerations [9,
10] and firm sizes all over the world [11].
The common feature in these systems is the lack of a characteristic size,
length or frequency for an observable $k$ at study. This lack generally leads
to a power law distribution $p(k)$, valid in most of the domain of definition
of $k$,
$p(k)\sim 1/k^{1+\gamma},$ (1)
with $\gamma\geq 0$. Special attention has been paid to the class of
universality defined by $\gamma=1$, which corresponds to Zipf’s law in the
cumulative distribution or the rank-size distribution [2, 3, 4, 6, 7, 9, 10,
11, 12]. Recently, Maillart et al. [7] have studied the evolution of the
number of links to open source software projects in Linux packages, and have
found that the link distribution follows Zipf’s law as a consequence of
stochastic proportional growth. In its simplest formulation, the stochastic
proportional growth model, or namely the geometric Brownian motion, assumes
the growth of an element of the system to be proportional to its size $k$, and
to be governed by a stochastic Wiener process. The class $\gamma=1$ emerges
from the condition of stationarity, i.e., when the system reaches a dynamic
equilibrium [12]. Together with geometric Brownian motion, there is a variety
of models arising in different fields that yield Zipf’s law and other power
laws on a case-by-case basis [9, 10, 12, 13, 14], as preferential attachment
[6] and competitive cluster growth [15] in complex networks, used to explain
many of the scale-free properties of social, technological and biological
networks.
### 1.2 Fisher’s information measure
Much effort has recently been devoted to Fisher’s information measure (FIM),
usually denoted as $I$. The work of Frieden and co-workers [16, 17, 18, 19,
20, 21, 22, 23, 24, 25], Silver [26], and Plastino et al. [27, 28, 29, 30],
among many others, has shed much light upon the manifold physical applications
of $I$. As a small sample we mention that Frieden and Soffer have shown that
FIM provides a powerful variational principle, called EPI (extreme physical
information) that yields the canonical Lagrangians of theoretical physics
[24]. Additionally, $I$ has been proved to characterize an arrow of time with
reference to the celebrated Fokker-Planck equation [28]. Moreover, there exist
interesting relations that connect FIM and the relative Shannon information
measure invented by Kullback [31, 32]. These can be shown to have some bearing
on the time evolution of arbitrary systems governed by quite general
continuity equations [29, 30]. Additionally, a rather general $I$-based H
theorem has recently been proved [33, 34]. As for Hamiltonian systems [35],
EPI allows to describe the behaviour of complex systems, as the allometric or
power laws found in biological sciences [36]. The pertinent list could be
extended quite a bit. $I$ is then an important quantity, involved in many
aspects of the theoretical description of nature.
For our present purposes it is of the essence to mention that Frieden et al.
[37] have also shown that equilibrium and non-equilibrium thermodynamics can
be derived from a principle of minimum Fisher information, with suitable
constraints (MFI). Here $I$ is specialized to the particular but important
case of _translation families_ , i.e., distribution functions whose form does
not change under translational transformations. In this case, Fisher measure
becomes _shift-invariant_. It is shown in [37] than such minimizing of
Fisher’s measure leads to a Schrödinger-like equation for the probability
amplitude, where the ground state describes equilibrium physics and the
excited states account for non-equilibrium situations.
### 1.3 Goals and motivation
Scale-invariant phenomena are generally addressed by appeal to ad-hoc models
(see the references citing in 1.1). In spite of the success of these models,
the intrinsic complexity involved therein makes their study at a macroscopic
level a rather difficult task. One sorely misses a general formulation of the
thermodynamics of scale-invariant physics, which is not quite established yet.
It is our goal here to show, in such a vein, that minimization of Fisher
information provides a unifying framework that allows these phenomena to be
understood as arising from an MFI variational principle, entirely analogous to
how termodynamics is generated in [34].
## 2 Minimum Fisher Information approach (MFI)
The Fisher information measure $I$ for a system described by a set of
coordinates $\mathbf{q}$ and physical parameters $\mathbf{\theta}$, has the
form [34]
$I(F)=\int_{\Omega}d\mathbf{q}F(\mathbf{q}|\mathbf{\theta})\sum_{ij}c_{ij}\frac{\partial}{\partial\theta_{i}}\ln
F(\mathbf{q}|\mathbf{\theta})\frac{\partial}{\partial\theta_{j}}\ln
F(\mathbf{q}|\mathbf{\theta}),$ (2)
where $F(\mathbf{q}|\mathbf{\theta})$ is the density distribution in a
configuration space ($\mathbf{q}$) of volume $\Omega$ conditioned by the
physical parameters ($\mathbf{\theta}$). The constants $c_{ij}$ account for
dimensionality, and take the form $c_{ij}=c_{i}\delta_{ij}$ if $q_{i}$ and
$q_{j}$ are uncorrelated. The equilibrium state of the system minimizes $I$
subject to prior conditions, like the normalization of $F$ or any constraint
on the mean value of an observable $\langle A_{i}\rangle$ [37]. The MFI is
then written as a variation problem of the form
$\delta\left\\{I(F)-\sum_{i}\mu_{i}\langle A_{i}\rangle\right\\}=0,$ (3)
where $\mu_{i}$ are appropriate Lagrange multipliers.
### 2.1 One-dimensional system with discrete coordinate
Because of the nature of the systems to be addressed we consider now a one-
dimensional system with a physical parameter $\theta$ and a discrete
coordinate $k=k_{1},k_{2},\ldots,k_{i},\ldots$ where $k_{i+1}-k_{i}=\Delta k$
for a certain value of the interval $\Delta k$. This scenario arises, for
instance, in the case of nuclear multifragmentation [2], the abundances of
genes [3], the frequency of words [4], scientific collaboration networks [5],
the Internet traffic [6], Linux packages links [7], electoral results [8],
urban agglomerations [9, 10], firm sizes [11], etc.
In the continuous limit ($\Delta k\rightarrow dk$), the Fisher information
measure is cast as
$I(F)=c_{k}\int_{k_{1}}^{\infty}dkF(k|\theta)\left|\frac{\partial}{\partial\theta}\ln
F(k|\theta)\right|^{2}.$ (4)
Instead of using translation invariance à la Frieden-Soffer [24], we will
appeal to scaling invariance [38] so that we can anticipate some new physics.
All members of the family $F(k/\theta)$ possess identical shape —there are no
characteristic size, length or frequency for the observable $k$— namely
$dkF(k/\theta)=dk^{\prime}F(k^{\prime})$ under the transformation
$k^{\prime}=k/\theta$.
To deal with this new symmetry it is convenient to change to the new
coordinate $u=\ln k$ and parameter $\Theta=\ln\theta$. Why? Because then the
scale invariance becomes again translational invariance, and we are entitled
to use one essential result of [34], namely, that MFI leads to a Schroedinger-
like equation. Note that the new coordinate $u^{\prime}=\ln k^{\prime}$
transforms as $u^{\prime}=u-\Theta$. Defining $f(u)=F(e^{u})$ and taking into
account the fact that the Jacobian of the transformation is $|dx/du|=e^{u}$
and $\partial/\partial\theta=e^{-\Theta}\partial/\partial\Theta$, the Fisher
information measure acquires now the form
$I(F)=c_{k}e^{-2\Theta}\int_{u_{1}}^{\infty}du~{}e^{u}f(u)\left|\frac{\partial\ln
f(u)}{\partial u}\right|^{2},$ (5)
where $u_{1}=\ln k_{1}$, and the factor $e^{-2\Theta}$ guaranties the
invariance of the associated Cramer-Rao inequality as shown in [38].
For reasons that will become apparent below, we will apply the MFI without any
constraint. This is tantamount to posing no bound to the physical “sizes” that
characterize the system. The extremization of Fisher information with no
constraints ($\mu_{i}=0$) is written as
$\delta\left\\{\int_{u_{1}}^{\infty}du~{}e^{u}f(u)\left|\frac{\partial\ln
f(u)}{\partial u}\right|^{2}\right\\}=0.$ (6)
Introducing $f(u)=e^{-u}\Psi^{2}(u)$, and varying with respect to $\Psi$ and
$\partial\Psi/\partial u$ as in [37] one is easily led to a (real)
Schrödinger-like equation of the form
$\left[-4\frac{\partial^{2}}{\partial u^{2}}+1\right]\Psi(u)=0.$ (7)
Notice that the lack of normalization constraints implies zero eigenvalue,
since the Lagrange multiplier associated with the normalization is the energy
eigenvalue [37]. At this point we introduce boundary conditions to guaranty
convergence of the Fisher measure (5) and thus compensate for the lack of
constraints in (6). We impose $\lim_{u\rightarrow\infty}\Psi(u)=0$ and
$\Psi(u_{1})=\sqrt{N}$, where $N$ is an dimensionless constant the meaning of
which will become clear later. The solution to (7) with these boundary
conditions is $\Psi(u)=\sqrt{N}e^{-(u-u_{1})/2}$, which leads to
$f(u)=Ne^{-(2u-u_{1})}$ and to the density distribution
$F(k)dk=N\frac{k_{1}}{k^{2}}dk$ (8)
with $N=1$ for a density normalized to unity. This distribution is just the
Zipf’s law (universal class $\gamma=1$) of Refs. [2, 3, 4, 6, 7, 9, 11, 12].
This result is remarkable: _Zipf’s law has been here derived from first
principles_.
## 3 Applications
A common representation of empirical data is the so-called rank-plot or Zipf
plot [4, 10, 13], where the $j$th element of the system is represented by its
size, length or frequency $k_{j}$ against its rank, sorted from the largest to
the smallest one. This process just renders the inverse function of the
ensuing cumulative distribution, normalized to the number of elements. We call
$r$ the rank that ranges from 1 to $N$. Thus, the constant $N$ arising from
the boundary conditions is the total number of elements considered in building
up the distribution (8), as will be illustrated in the examples bellow. This
rank-distribution takes the form
$k(r)=N\frac{k_{1}}{r}$ (9)
which yields a straight line in a logarithmic representation with slope $-1$.
In Fig. 1a we depict the known behavior [12] of the rank size distribution for
the top 100 largest cities of the United States [39], which shows a slope near
$-1$ ($\gamma=1$) in the logarithmic representation of the rank-plot.
We have also studied the system formed by the most referenced physics journals
[40], using their total number of cites as coordinate $k$. If a journal
receives more cites due to its popularity, it becomes even more popular and,
therefore, receives still more cites, etc. Under such conditions, proportional
growth and scale invariance are expected, as we depict in Fig. 1b, where the
slope’s value can be regarded as illustrating the universality of the
underlying law.
Figure 1: a. Rank-plot of the 100 largest cities of the United States, from
most-populated to less-populated, in logarithm scale. b. Rank-plot of the
total number of cites of the 30 most cited physics journals, from most-cited
to less-cited, in logarithm scale.
## 4 Conclusions
We have here shown that Zipf’s law results from the scaling invariance of the
Crammer-Rao inequality derived in [35]. This entails that the relevant
probability distribution, usually called the rank-distribution, has to be
size-invariant. Consequently, it should be derivable from a minimization
process in which Fisher’s information measure is the protagonist. No
constraints are needed in the concomitant variational problem because, a
priori, our sizes have no upper bound. A physical analogy is the non-
normalizability of plane waves. The universal character of our demonstration
thus resides in the universal form to be minimized (Fisher’s), with no
constraints.
## Acknowledgments
We would like to thank M. Barranco, R. Frieden, and B. H. Soffer for useful
discussions. This work has been partially performed under grant
FIS2008-00421/FIS from DGI, Spain (FEDER).
## References
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* [15] A. A. Moreira, D. R. Paula, R. N. Costa Filho, and J. S. Andrade, Phys. Rev. E 73 (2006) 065101(R).
* [16] B. R. Frieden, Science from Fisher Information, Cambridge University Press, Cambridge, England, 2004.
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* [26] R. N. Silver, in E. T. Jaynes: Physics and Probability, edited by W. T. Grandy, Jr. and P. W. Milonni, Cambridge University Press, Cambridge, England, 1992.
* [27] A. Plastino, A. R. Plastino, H. G. Miller, and F. C. Khana, Phys. Lett. A 221 (1996) 29.
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* [33] A. R. Plastino, M. Casas, and A. Plastino, Phys. Lett. A 246 (1998) 498.
* [34] B. R. Frieden, Physics from Fisher Information, Cambridge University Press, Cambridge, England, 1998.
* [35] F. Pennini and A. Plastino, Phys. Lett. A 349 (2006) 15.
* [36] B. R. Frieden and R. A. Gatenby, Phys. Rev. E 72 (2005) 036101.
* [37] B. R. Frieden, A. Plastino, A. R. Plastino, and B. H. Soffer, Phys. Rev. E 60 (1999) 48; 66 (2002) 046128.
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* [40] Journal Citation Reports (JCR), Thomson Reuters, 2007.
|
arxiv-papers
| 2009-08-04T17:52:34 |
2024-09-04T02:49:04.427579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Hernando, D. Puigdomenech, D. Villuendas, C. Vesperinas, A.\n Plastino",
"submitter": "Alberto Hernando",
"url": "https://arxiv.org/abs/0908.0501"
}
|
0908.0504
|
# Fisher-information and the thermodynamics of scale-invariant systems
A. Hernando alberto@ecm.ub.es C. Vesperinas cristina.vesperinas@sogeti.com
A. Plastino plastino@fisica.unlp.edu.ar Departament ECM, Facultat de Física,
Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain Sogeti España,
WTCAP 2, Plaça de la Pau s/n, 08940 Cornellà, Spain National University La
Plata, IFCP-CCT-CONICET, c.c. 727, 1900 La Plata, Argentina
###### Abstract
We present a thermodynamic formulation for scale-invariant systems based on
the minimization with constraints of Fisher’s information measure. In such a
way a clear analogy between these systems’s thermal properties and those of
gases and fluids is seen to emerge in natural fashion. We focus attention on
the non-interacting scenario, speaking thus of _scale-free ideal gases_
(SFIGs) and present some empirical evidences regarding such disparate systems
as electoral results, city populations and total citations in Physics
journals, that seem to indicate that SFIGs do exist. We also illustrate the
way in which Zipf’s law can be understood in a thermodynamical context as the
surface of a finite system. Finally, we derive an equivalent microscopic
description of our systems which totally agrees with previous numerical
simulations found in the literature.
###### keywords:
Fisher information , scale-invariance
††journal: Physica A
## 1 Introduction
Scale-invariant phenomena are rather abundant in Nature and display somewhat
unexpected features. Examples can be found that range from physical and
biological to technological and social sciences [1]. One may cite, among many
possibilities, that empirical data from percolation theory and nuclear
multifragmentation [2] reflect scale-invariant behaviour, as does the
abundance of genes in various organisms and tissues [3]. Additionally, we can
speak of the frequency of words in natural languages [4], scientific
collaboration networks [5], the Internet traffic [6], Linux packages links
[7], as well as of electoral results [8, 9], urban agglomerations [10, 11] and
firm sizes all over the world [12]. What characterizes these disparate systems
is the lack of a characteristic size, length or frequency for an observable
$k$ under scrutiny. This fact usually leads to a power law distribution
$p(k)$, valid in most of the domain of definition of $k$,
$p(k)\sim 1/k^{1+\gamma},$ (1)
with $\gamma\geq 0$. Special attention deserves the class of universality
defined by $\gamma=1$, which corresponds to the so-called Zipf’s law in the
cumulative distribution or the rank-size distribution [2, 3, 4, 6, 7, 10, 11,
12, 13]. Recently, Maillart et al. [7] have found that links’ distributions
follow Zipf’s law as a consequence of stochastic proportional growth. In its
simplest formulation such kind of growth assumes that an element of the system
becomes enlarged proportionally to its size $k$, being governed by a Wiener
process. The class $\gamma=1$ emerges from the condition of stationarity,
i.e., when the system reaches a dynamic equilibrium [13]. We will as well
propose to consider the case $\gamma=0$ as representative of a second class of
universality, since the ensuing behavior, empirically found by Costa Filho et
al. [8] with regards to the vote-distribution in Brazilian electoral results,
emerges as the result of multiplicative processes in complex networks [9].
In this paper we attempt to formulate a thermodynamic treatment common to
these systems. Our efforts are based on the minimization with appropriate
constraints of Fisher’s information measure (FIM), abbreviated as the MFI
approach. It is shown in [14] that MFI leads to a (real) Schreodinger-like
equation whose “potential” function is given by the constraints employed to
constrain the variational process. The interplay between constraints and
associated Lagrange multipliers turns our to be Legendre-invariant [14] and
leads to all known thermodynamic relations. Such result constitutes the
essential ingredient of our considerations here. We will first consider the
MFI treatment of the ideal gas (Seccion 3), not given elsewhere as far as we
are aware of, since it is indispensable to deal with it in order to fully
understand the methodology employed for scale-free systems, which is tackled
in Section 4. Applicatioms are discussed in Section 5 and some conclusiones
drawn in Section 6. We begin our considerations in Section 2 with a brief
Fisher’s sketch.
## 2 Minimum Fisher Information approach (MFI)
The Fisher information measure $I$ for a system described by a set of
coordinates $\mathbf{q}$ and physical parameters $\mathbf{\theta}$, has the
form [15]
$I(F)=\int_{\Omega}d\mathbf{q}F(\mathbf{q}|\mathbf{\theta})\sum_{ij}c_{ij}\frac{\partial}{\partial\theta_{i}}\ln
F(\mathbf{q}|\mathbf{\theta})\frac{\partial}{\partial\theta_{j}}\ln
F(\mathbf{q}|\mathbf{\theta}),$ (2)
where $F(\mathbf{q}|\mathbf{\theta})$ is the density distribution in a
configuration space ($\mathbf{q}$) of volume $\Omega$ conditioned by the
physical parameters collectively represented by the variable
$\mathbf{\theta}$. The constants $c_{ij}$ account for dimensionality, and take
the form $c_{ij}=c_{i}\delta_{ij}$ if $q_{i}$ and $q_{j}$ are uncorrelated,
where $\delta_{ij}$ is the Kronecker delta. As shown in [14], the thermal-
equilibrium state of the system can be determined by minimizing $I$ subject to
adequate prior conditions (MFI), like the normalization of $F$ or by any
constraint on the mean value of an observable $\langle A_{i}\rangle$ [14]. The
MFI is then cast as a variation problem of the form
$\delta\left\\{I(F)-\sum_{i}\mu_{i}\langle A_{i}\rangle\right\\}=0,$ (3)
where $\mu_{i}$ are appropriate Lagrange multipliers.
## 3 MFI treatment of the ideal gas
As a didactic introductory example, not discussed in [14], we will here
rederive, via MFI (something original as far as we know), the density
distribution, in configuration space, of the (translational invariant) ideal
gas (IG) [16], that describes non-interacting classical particles of mass $m$
with coordinates $\mathbf{q}=(\mathbf{r},\mathbf{p})$, where
$md\mathbf{r}/dt=\mathbf{p}$. The translational invariance is described by the
translational family of distributions
$F(\mathbf{r},\mathbf{p}|\mathbf{\theta}_{r},\mathbf{\theta}_{p})=F(\mathbf{r}^{\prime},\mathbf{p}^{\prime})$
whose form does not change under the transformations
$\mathbf{r}^{\prime}=\mathbf{r}-\mathbf{\theta}_{r}$ and
$\mathbf{p}^{\prime}=\mathbf{p}-\mathbf{\theta}_{p}$. We assume that these
coordinates are canonical [17] and uncorrelated. This assumption is introduced
into the information measure (2) setting $c_{ij}=c_{i}\delta_{ij}$, where
$c_{i}=c_{r}$ for space coordinates, $c_{i}=c_{p}$ for momentum coordinates.
The density can obviously be factorized in the fashion
$F(\mathbf{r},\mathbf{p})=\rho(\mathbf{r})\eta(\mathbf{p})$, and then [15] it
follows from the additivity of the information measure that $I=I_{r}+I_{p}$.
If $D$ is the dimensionality we have
$\begin{array}[]{rl}I_{r}=&\displaystyle c_{r}\int
d^{D}\mathbf{r}~{}\rho(\mathbf{r})\left|\mathbf{\nabla}_{r}\ln\rho(\mathbf{r})\right|^{2}\\\
I_{p}=&\displaystyle c_{p}\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})\left|\mathbf{\nabla}_{p}\ln\eta(\mathbf{p})\right|^{2}.\end{array}$
(4)
In extremizing FIM we constrain the normalization of $\rho(\mathbf{r})$ and
$\eta(\mathbf{p})$ to the total number of particles $N$ and to $1$,
respectively, i.e.,
$\int d^{D}\mathbf{r}~{}\rho(\mathbf{r})=N,\qquad\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})=1.$ (5)
In addition, we penalize infinite values for the particle momentum with a
constraint on the variance of $\eta(\mathbf{p})$ to a given empirically
obtained value, namely,
$\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})(\mathbf{p}-\overline{\mathbf{p}})^{2}=D\sigma_{p}^{2},$
(6)
where $\overline{\mathbf{p}}$ is the mean value of $\mathbf{p}$. For each
degree of freedom it is known from the Virial Theorem that the variance is
related to the temperature $T$ as $\sigma_{p}^{2}=mk_{B}T$, with $k_{B}$ the
Boltzmann constant. Variation thus yields
$\displaystyle\delta\left\\{c_{r}\int
d^{D}\mathbf{r}~{}\rho\left|\mathbf{\nabla}_{r}\ln\rho\right|^{2}+\mu\int
d^{D}\mathbf{r}~{}\rho\right\\}=0$ (7)
and
$\delta\left\\{c_{p}\int
d^{D}\mathbf{p}~{}\eta\left|\mathbf{\nabla}_{p}\ln\eta\right|^{2}+\lambda\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p}-\overline{\mathbf{p}})^{2}+\nu\int
d^{D}\mathbf{p}~{}\eta\right\\}=0,$ (8)
where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing now
$\rho(\mathbf{r})=\Psi^{2}(\mathbf{r})$ and varying (7) with respect to $\Psi$
leads to a Schroedinger-like equation [14, 18]
$\left[-4\nabla_{r}^{2}+\mu^{\prime}\right]\Psi(\mathbf{r})=0,$ (9)
where $\mu^{\prime}=\mu/c_{r}$. To fix the boundary conditions, we first
assume that the $N$ particles are confined in a box of volume $V$, and next we
take the thermodynamic limit $N,V\rightarrow\infty$ with $N/V$ finite. The
equilibrium state compatible with this limit corresponds to the ground state
solution ($\mu^{\prime}=0$), which is the uniform density
$\rho(\mathbf{r})=N/V$.
Introducing $\eta(\mathbf{p})=\Phi^{2}(\mathbf{p})$ and varying (8) with
respect to $\Phi$ leads to the quantum harmonic oscillator-like equation [18]
$\left[-4\nabla_{p}^{2}+\lambda^{\prime}(\mathbf{p}-\overline{\mathbf{p}})^{2}+\nu^{\prime}\right]\Phi(\mathbf{p})=0,$
(10)
where $\lambda^{\prime}=\lambda/c_{p}$ and $\nu^{\prime}=\nu/c_{p}$. The
equilibrium configuration corresponds to the ground state solution, which is
now a gaussian distribution. Using (6) to identify
$|\lambda^{\prime}|^{-1/2}=\sigma_{p}^{2}$ we get the Maxwell-Boltzmann
distribution, which leads to a density distribution in configuration space of
the form
$f(\mathbf{r},\mathbf{p})=\frac{N}{V}\frac{\exp\left[-(\mathbf{p}-\overline{\mathbf{p}})^{2}/2\sigma_{p}^{2}\right]}{(2\pi\sigma_{p}^{2})^{D/2}}.$
(11)
If $H$ is the elementary volume in phase space, the total number of
microstates is
$Z=N!H^{DN}\prod_{i=1}^{N}F_{1}(\mathbf{r}_{i},\mathbf{p}_{i})$, where
$F_{1}=F/N$ is the monoparticular distribution and $N!$ counts all possible
permutations for distinguishable particles. The entropy $S=-k_{B}\ln Z$ gets
then written in the form
$S=Nk_{B}\left\\{\ln\frac{V}{N}\left(\frac{2\pi\sigma_{p}^{2}}{H^{2}}\right)^{D/2}+\frac{2+D}{2}\right\\},$
(12)
where we have used the Stirling approximation for $N!$. This expression
agrees, of course with the known value entropic expression for the IG [16],
illustrating on the predictive power of the MFI formulation advanced in [14].
## 4 Scale invariant systems
We pass now to the leit-motif of the present communication and consider a one-
dimensional system with dynamical coordinates $\mathbf{q}=(k,v)$ where
$dk/d\tau=v$, with $\tau$ the time variable. We define $k$ as a discrete
coordinate, i.e. $k=k_{1},k_{2},\ldots,k_{M}$, where $k_{i}=i\Delta k$ and
$M\gg 1$, is the total number of bins of width $\Delta k$ in our system. In
order to address the scale-invariance behaviour of $k$ we change variables
passing to new coordinates $u=\ln k$ and $w=du/dt$. We work under the
hypothesis that $u$ and $w$ are canonically conjugated [17] and uncorrelated.
This assumption immediately leads to proportional growth since
$dk/dt=v=kw.$ (13)
For constant $w$ this equation yields an exponential growth $k=k_{0}e^{wt}$,
which represents uniform linear motion in $u$, that is, $u=wt+u_{0}$, with
$u_{0}=\ln k_{0}$ 111This exponential growth allows to identify the systems
that we study in this work in macroscopic fashion with those addressed in
[19].. It is easy to check that the scale transformation
$k^{\prime}=k/\theta_{k}$ leaves invariant the coordinate $w$, whereas the
coordinate $u$ transforms translationally as $u^{\prime}=u-\Theta_{k}$, where
$\Theta_{k}=\ln\theta_{k}$. Thus, the physics does not depend on scale and the
system is translationally invariant with respect to the coordinates $u$ and
$w$, entailing that the distribution of physical elements can be described by
the monoparametric translation families
$f(u,w|\Theta_{k},\Theta_{w})=f(u^{\prime},w^{\prime})$. By analogy with the
IG, we will call our system a “scale-free ideal gas” (SFIG), i.e., a system of
$N$ non-interacting elements. Taking into account that i) $u$ and $w$ are
canonical and uncorrelated ($c_{ii}=c_{i}\neq 0$ and $c_{uw}=c_{wu}=0$), so
the density distribution can be factorized as $f(u,w)=g(u)h(w)$, and ii) that
the Jacobian for our change of variables is $dkdv=e^{2u}dudw$, the information
measure $I=I_{u}+I_{w}$ can be obtained in the continuous limit as
$\begin{array}[]{rl}I_{u}=&\displaystyle
c_{u}\int_{\Omega}du~{}e^{2u}g(u)\left|\frac{\partial\ln g(u)}{\partial
u}\right|^{2}\\\ I_{w}=&\displaystyle
c_{w}\int_{-\infty}^{\infty}dw~{}h(w)\left|\frac{\partial\ln h(w)}{\partial
w}\right|^{2},\end{array}$ (14)
where $\Omega=\ln(k_{M}/k_{1})=\ln M$ is the volume defined in “$u$”-space.
### 4.1 MFI treatment of the scale-free ideal gas
The constraints to the given observables $\langle A_{i}\rangle$ in the
extremization problem determine the behaviour of the system. For the general
case, we constrain the normalization of $g(u)$ and $h(w)$ to the total number
of particles $N$ and to $1$, respectively
$\int_{\Omega}du~{}e^{2u}g(u)=N,\qquad\int_{-\infty}^{\infty}dw~{}h(w)=1.$
(15)
In addition, we penalize infinite values for $w$ with a constraint on the
variance of $h(w)$ to a given measured value
$\int_{-\infty}^{\infty}dw~{}h(w)(w-\overline{w})^{2}=\sigma_{w}^{2},$ (16)
where $\overline{w}$ is the average growth. The variation yields
$\delta\left\\{c_{u}\int_{\Omega}du~{}e^{2u}g\left|\frac{\partial\ln
g}{\partial u}\right|^{2}+\mu\int_{\Omega}du~{}e^{2u}g\right\\}=0$ (17)
and
$\delta\left\\{c_{w}\int_{-\infty}^{\infty}dw~{}h\left|\frac{\partial\ln
h}{\partial
w}\right|^{2}+\lambda\int_{-\infty}^{\infty}dw~{}h(w-\overline{w})^{2}+\nu\int_{-\infty}^{\infty}dw~{}h\right\\}=0,$
(18)
where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing
$g(u)=e^{-2u}\Psi^{2}(u)$, and varying (17) with respect to $\Psi$ leads, as
is always the case with the MFI [14], to the Schroedinger-like equation
$\left[-4\frac{\partial^{2}}{\partial u^{2}}+4+\mu^{\prime}\right]\Psi(u)=0,$
(19)
where $\mu^{\prime}=\mu/c_{u}$. Analogously to the IG, we impose solutions
compatible with a finite normalization of $g$ in the thermodynamic limit
$N,\Omega\rightarrow\infty$ with $N/\Omega=\rho_{0}$ finite, where $\rho_{0}$
is defined as the _bulk density_. Solutions compatible with the normalization
of (15) are given by $\Psi(u)=A_{\alpha}e^{-\alpha u/2}$, where $A_{\alpha}$
is the normalization constant and $\alpha=\sqrt{4+\mu^{\prime}}$. In this
general case, the density distribution as a function of $k$ takes the form of
a power law: $g_{\alpha}(\ln k)=A^{2}/k^{2+\alpha}$. The equilibrium is always
defined for the MFI as the ground state solution [14], which corresponds to
the lowest allowed value $\alpha=0$.
Introducing now $h(w)=\Phi^{2}(w)$ and varying (18) with respect to $\Phi$
leads to the quantum harmonic oscillator-like equation [14, 18]
$\left[-4\frac{\partial^{2}}{\partial
w^{2}}+\lambda^{\prime}(w-\overline{w})^{2}+\nu^{\prime}\right]\Phi(w)=0,$
(20)
where $\lambda^{\prime}=\lambda/c_{w}$ and $\nu^{\prime}=\nu/c_{w}$. The
equilibrium configuration corresponds to the ground state solution, which is
now a Gaussian distribution. Using (16) to identify
$|\lambda^{\prime}|^{-1/2}=\sigma_{w}^{2}$ we get the Maxwell-Boltzmann
distribution
$h(w)=\frac{\exp\left[-(w-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$
(21)
The density distribution in configuration space $F(k,v)dkdv=f(u,w)e^{2u}dudw$
is then
$F(k,v)=\frac{N}{\Omega
k^{2}}\frac{\exp\left[-(v/k-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$
(22)
If we define $H=\Delta k^{2}/\Delta\tau$ as the elementary volume in phase
space, where $\Delta\tau$ is the time element, the total number of microstates
is $Z=N!H^{N}\prod_{i=1}^{N}F_{1}(k_{i},v_{i})$, where $F_{1}=F/N$ is the
monoparticular distribution function and $N!$ counts all possible permutations
for distinguishable elements. The entropy equation of state $S=-\kappa\ln Z$
reads
$S=N\kappa\left\\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_{w}}{H^{\prime}}+\frac{3}{2}\right\\},$
(23)
where $\kappa$ is a constant that accounts for dimensionality and
$H^{\prime}=H/(k_{M}k_{1})=H/(M\Delta k^{2})=1/(M\Delta\tau)$. Remarkably,
this expression has the same form as the one-dimensional IG ($D=1$ in (12));
instead of the thermodynamical variables $(N,V,T)$, here we deal with the
variables $(N,\Omega,\sigma_{w})$, which make the entropy scale-invariant as
well.
Figure 1: (colour on-line) a, rank-size distribution of the cities of the
province of Huelva, Spain (2008), sorted from largest to smallest, compared
with the result of a simulation with Brownian walkers (green squares). b,
rank-plot of the 2008 General Elections results in Spain. c, rank-plot of the
2005 General Elections results in the United Kingdom. (Red dots: empirical
data; blue lines: fit to (25)).
The total density distribution for $k$ is obtained integrating for all $v$ the
density distribution in configuration space. Accordingly, from (22) we get
$F(k)=\int dvF(k,v)=\frac{N}{\Omega}\frac{1}{k}=\frac{\rho_{0}}{k}.$ (24)
It can be shown that this it is just a uniform density-distribution in
$u$-space of the bulk density: $F(k)dk=f(u)e^{u}du=N/\Omega du=\rho_{0}du$.
## 5 Social examples of scale-free ideal gases
A common representation of empirical data is the so-called rank-plot or Zipf
plot [4, 11, 20], where the $j$th element of the system is represented by its
size, length or frequency $k_{j}$ against its rank, sorted from the largest to
the smallest one. This process just renders the inverse function of the
ensuing cumulative distribution, normalized to the number of elements. We call
$r$ the rank that ranges from 1 to $N$. For large $N$, the density
distribution (24) correspond to an exponential rank-size distribution
$k(r)=k_{M}\exp\left[-\frac{r-1}{\rho_{0}}\right].$ (25)
This behaviour, which corresponds to the class of universality $\gamma=0$ in
(1), is that empirically found by Costa Filho et al. [8] in the distribution
of votes in the Brazilian electoral results. We have found such a behaviour in
i) the city-size distribution of small regions (as in the province of Huelva
(Spain) [21]) and ii) electoral results (as in the 2008 Spanish General
Elections results [22]). We depict in figures 1a and 1b the pertinent rank-
sizes distributions in semi-logarithmic scale, where a straight line
corresponds to a distribution of type (25). A large portion of the
distributions can be fitted to (25), with a correlation coefficient of $0.994$
and $0.998$, respectively. From these fits we have obtain a bulk density of
$\rho_{0}=0.058$ for the General Elections results, and in the case of Huelva
of $\rho_{0}=17.1$ ($N=77$, $\Omega=4.5$). Using historical data for the later
[21], we have used the backward differentiation formula to calculate the
relative growth rate of the $i$-th city as
$w_{i}=\frac{\ln k_{i}^{(2008)}-\ln k_{i}^{(2007)}}{\Delta t}$ (26)
where $k_{i}^{(2007)}$ and $k_{i}^{(2008)}$ are the number of inhabitants of
the $i$-th city in $2007$ and $2008$, respectively while $\Delta t=1$ year. We
show in figure 2 the empirical rank-plot of the relative growth, where we have
obtained $\overline{w}=0.011$ years-1 and $\sigma_{w}=0.030$ years-1, compared
with the rank-plot of a Maxwell-Boltzmann distribution with the same mean
value and standard deviation.
Figure 2: (colour on-line) Rank-plot of the growth rate $w$ of the province of
Huelva between 2007 and 2008 (red dots) compared with a Boltzmann distribution
with the same mean value and standard deviation (blue line).
However, these regularities are not always obvious to the naked eye, as shown
for the case of the most voted parties in Spain’08 or for the whole
distribution of the 2005 General Elections results in the United Kingdom [23]
(figure 1c). In both cases, the competition between parties seems to play an
important role, and the assumption of non-interacting elements can be
unrealistic 222The effects of interaction are studied in [24], where we go
beyond the non-interacting system using a microscopic description based on
complex networks..
### 5.1 Bulk and Zipf regimes
The situation in which $N/\Omega\rightarrow\,{\rm constant}\,\neq 0$ as
$N,\Omega\rightarrow\infty$ will be referred to herefrom as the _bulk regime_.
Now, in a recent communication [25], we show that Zipf’s law ($\gamma=1$ in
(1) with a slope of $-1$ in the rank-plot) can be derived from the
extremization of Fisher’s information with no constraints. In the
thermodynamic context studied here, the absence of normalization can be
understood as the inability of the system to reach the thermodynamic limit,
i.e. $N/\Omega\rightarrow 0$ as $N,\Omega\rightarrow\infty$. In this case the
system can not follow (24). Zipf’s law emerges as this behaviour of the
density in what we will accordingly denominate the _Zipf regime_
($N/\Omega\rightarrow 0$). We digress on the conditions for both regimes in
the example discussed below.
We have studied the system formed by all Physics journals [26] ($N=310$) using
their total number of cites as coordinate $k$. If a journal receives more
cites due to its popularity, it becomes even more popular and therefore it
will receive more cites. Under such conditions proportional growth and scale
invariance are expected. Since we consider all sub-fields of Physics,
correlation effects are much lower than they would be should we only consider
journals pertaining to an specific sub-field. Accordingly, the non-interacting
approximation seems to be realistic in this instance. In figure 3 we depict
the rank-plot of the number of citations in Physic journals, and find a slope
approaching $-1$ for the most-cited journals in the logarithmic representation
(figure 3a) and an slope in the vicinity of r $+1$ for the less-cited journals
(figure 3b). For the central part of the distribution, the bulk density
reaches a value of $\rho_{0}\sim 57$ (figure 3c).
Figure 3: (colour on-line) a, rank-plot of the total number of cites of
physics journal, from most-cited to less-cited, in logarithm scale. b, sorted
from less-cited to most cited c, same as a, in semi-logarithm scale. (Red
dots: empirical data; blue line: fit to (25)).
This distribution shows a notably symmetric behaviour under the change
$k\rightarrow 1/k$ ($u\rightarrow-u$). We exhibit in figure 4 the raw
empirical data as compared with the distribution obtained from the
transformation $k^{\prime}=c/k$ ($u^{\prime}=-u+\ln c$), where $c=3.3\times
10^{6}$. The main part of the density distribution reaches the bulk density
obeying (24), whereas Zipf’s law emerges at the edges, which could be
understood as constituting the _surface_ of the system, since they explain how
the density (exponentially) falls from its bulk-value to zero in $u$-space
when the system is exposed to an infinitely empty volume. This effect is
clearly visible in figure 5, where the empirical density distribution $p(u)du$
in $u$-space is compared with the “fitted” density
$p(u)=\left\\{\begin{array}[]{ll}\rho_{Z}e^{u-u_{1}}&\mathrm{if~{}}u<u_{1}\\\
\rho_{0}&\mathrm{if~{}}u_{1}<u<u_{2}\\\
\rho_{Z}e^{u_{2}-u}&\mathrm{if~{}}u>2\end{array}\right.$ (27)
where $\rho_{Z}=18$, $\rho_{0}=57$, $u_{1}=5.2$ and $u_{2}=10$. These findings
lead us to conclude that the system consisting of Physics journals, when
sorted by total number of citations, is a perfect example of a finite scale-
free ideal gas at equilibrium.
Figure 4: (colour online) Rank-plot of the total number of cites of Physics
journal, from most-cited to less-cited, compared with the distribution
obtained from the inverse transformation $k^{\prime}=3.3\times 10^{6}/k$ where
$k$ is the number of cites. Figure 5: (colour online) Empirical density
distribution in $u$-space of the total number of cites of Physics journals,
compared with (27). The bulk regime and the Zipf regime at the edges is
clearly visible.
### 5.2 An accompanying microscopic description
The dynamics of the system under scrutiny here can be microscopically
described as a stochastic process using (13) together with the density
distribution (21). Treating $w$ as a random variable, the pertinent stochastic
equation of motion is written in the guise of a geometrical Brownian motion,
i.e., [19]
$dk=k\overline{w}dt+k\sigma_{w}dW,$ (28)
where $dW$ represents a Wiener process. In the $u$-space, this equation reads
$du=\overline{w}dt+\sigma_{w}dW,$ (29)
and is known to describe the celebrated Brownian motion (28), which exactly
describes the dynamical condition found empirically in [7] and also the
(stochastic) proportional growth model used in [13] to obtain Zipf’s law. We
thus dare to suggest that we are dealing here with a sort of “equivalent” of a
molecular dynamics’ simulation for gases/liquids [27].
Indeed, (29) describes $N$ Brownian walkers moving in a fixed volume $\Omega$
with uniform density in $u$-space, a model used in the literature to describe
the ideal gas [27]. This scenario can also be reproduced by our free-scale
ideal gas merely by choosing to represent the system with the coordinates
$(k,v)$. In figure 1a we show the rank-plot for $k$ of a system of $N=78$
geometrical Brownian walkers with $\overline{w}=0.011$ and $\sigma_{w}=0.030$
in a volume $\Omega=4.5$, with d $k_{1}=200$ in reduced units, which
approximately describes the distribution of the population of the province of
Huelva. We also show in figure 2 the rankplot of $w$ of the same random
walkers, compared with the empirical data.
## 6 Conclusions
Our present considerations derive from the fact that, as shown in [14],
thermodynamics can be reformulated in terms of the minimization with
appropriate constraints of Fisher’s information. We have applied such
reformulation in order to discuss the thermodynamics of scale-free systems and
derived the density distribution in configuration space and the entropic
expression for the equilibrium state of what we call SFIG: the scale-free
ideal gas (in the thermodynamic limit). We have encountered convincing
empirical evidences of the SFIG actual existence in sociological scenarios.
Thus, we have dealt with city populations, electoral results and citations in
Physics journals. In such a context it is seen that Zipf’s law emerges
naturally as the equilibrium density of the non-interacting system when the
volume grows without bounds, a situation that we call the Zipf regime. Using
empirical data we have revealed that this regime can be understood as a
density-decay at the “surface” separating the bulk from an empty and very
large volume. Finally, we have shown with a simulation of city-populations
that geometrical Brownian motion can describe such systems at a microscopic
level.
## Acknowledgments
We would like to thank D. Puigdomenech, D. Villuendas, M. Barranco, R.
Frieden, and B. H. Soffer for useful discussions. This work has been partially
performed under grant FIS2008-00421/FIS from DGI, Spain (FEDER).
## References
* [1] Fractals in Physics: Essays in Honour of Benoit B Mandelbrot, edited by A. Aharony and J. Feder, Proceedings of a Conference in honor of B.B. Mandelbrot on His 65th in Vence, France, North Holland, Amsterdam, 1989.
* [2] K. Paech, W. Bauer, and S. Pratt , Phys. Rev. C 76 (2007), p. 054603; X. Campi and H. Krivine, Phys. Rev. C 72 (2005) 057602; Y. G. Ma et al., Phys. Rev. C 71 (2005) 054606.
* [3] C. Furusawa and K. Kaneko, Phys. Rev. Lett. 90 (2003) 088102.
* [4] G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley Press, Cambridge, Mass., 1949; I. Kanter and D. A. Kessler, Phys. Rev. Lett. 74 (1995) 4559.
* [5] M. E. J. Newman, Phys. Rev. E 64 (2001) 016131.
* [6] A.-L. Barabasi and R. Albert, Rev. Mod. Phys. 74 (2002) 47.
* [7] T. Maillart, D. Sornette, S. Spaeth, and G. von Krogh, Phys. Rev. Lett. 101 (2008) 218701.
* [8] R. N. Costa Filho, M. P. Almeida, J. S. Andrade, and J. E. Moreira, Phys. Rev. E 60 (1999) 1067.
* [9] A. A. Moreira, D. R. Paula, R. N. Costa Filho, and J. S. Andrade, Phys. Rev. E, 73 (2006) 065101(R).
* [10] L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, Phys. Rev. E 65 (2001) 017106.
* [11] M. Marsili and Yi-Cheng Zhang, Phys. Rev. Lett. 80 (1998) 2741.
* [12] R. L. Axtell, Science 293 (2001) 1818.
* [13] X. Gabaix, Quarterly Journal of Economics 114 (1999) 739.
* [14] B. R. Frieden, A. Plastino, A. R. Plastino, and B. H. Soffer, Phys. Rev. E 60 (1999) 48; 66 (2002) 046128.
* [15] B. R. Frieden and B. H. Soffer, Phys. Rev. E 52 (1995) 2274; B. R. Frieden, Physics from Fisher Information, Cambridge University Press, Cambridge, England, 1998; B. R. Frieden, Science from Fisher Information, Cambridge University Press, Cambridge, 2004.
* [16] M. W. Zemansky and R. H. Dittmann,Heat and Thermodynamics, McGraw-Hill, London, 1981.
* [17] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd Ed., Addison Wesley, San Francisco, 2002.
* [18] C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum Mechanics, Wiley-Interscience, New York, 2006.
* [19] W. J. Reed, and B. D. Hughes, Phys. Rev. E 66 (2002) 067103.
* [20] K. E. Kechedzhi, O. V. Usatenko, and V. A. Yampol’skii, Phys. Rev. E 72, 046138 (2005); S. Ree, Phys. Rev. E 73 (2006) 026115.
* [21] National Statistics Institute, Spain, www.ine.es.
* [22] Ministry of the Interior, Spain, www.elecciones.mir.es
* [23] Electoral Commission, Government of the UK, www.electoralcommission.org.uk.
* [24] A. Hernando, D. Villuendas, M. Abad and C. Vesperinas, arXiv:0905.3704v1, 2009.
* [25] A. Hernando, D. Puigdomenech, D. Villuendas, C. Vesperinas and A. Platino, submitted to Phys. Lett. A.
* [26] Journal Citation Reports (JCR) for 2007, Thomson Reuters.
* [27] Gould H and Tobochnik J 1996 _An Introduction to Computer Simulation Methods: Applications to Physical Systems_ , 2nd Ed. (Addison-Wesley).
|
arxiv-papers
| 2009-08-04T17:58:29 |
2024-09-04T02:49:04.432500
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Hernando, C. Vesperinas, A. Plastino",
"submitter": "Alberto Hernando",
"url": "https://arxiv.org/abs/0908.0504"
}
|
0908.0565
|
11institutetext: 1: Department of Physics, University of California, Davis,
CA 9516, USA; 11email: curro@physics.ucdavis.edu
2: Department of Electrophysics, National Chiao Tung University, Hsinchu
30010, Taiwan; 11email: blyoung@mail.nctu.edu.tw
3: National High Magnetic Field Laboratory, Florida State University,
Tallahassee, Florida 32306-4005, USA; 11email: urbano@magnet.fsu.edu
4: Theoretical Division, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA; 11email: graf@lanl.gov
# Hyperfine fields and magnetic structure in the B phase of CeCoIn5
Nicholas J. Curro1 Ben-Li Young2 Ricardo R. Urbano3 Matthias J. Graf4
###### Abstract
We re-analyze Nuclear Magnetic Resonance (NMR) spectra observed at low
temperatures and high magnetic fields in the field-induced B-phase of CeCoIn5.
The NMR spectra are consistent with incommensurate antiferromagnetic order of
the Ce magnetic moments. However, we find that the spectra of the In(2) sites
depend critically on the direction of the ordered moments, the ordering
wavevector and the symmetry of the hyperfine coupling to the Ce spins.
Assuming isotropic hyperfine coupling, the NMR spectra observed for
$\mathbf{H}~{}||~{}[100]$ are consistent with magnetic order with wavevector
$\mathbf{Q}=\pi(\frac{1+\delta}{a},\frac{1}{a},\frac{1}{c})$ and Ce moments
ordered antiferromagnetically along the [100] direction in real space. If the
hyperfine coupling has dipolar symmetry, then the NMR spectra require Ce
moments along the [001] direction. The dipolar scenario is also consistent
with recent neutron scattering measurements that find an ordered moment of
0.15$\mu_{B}$ along [001] and
$\mathbf{Q_{n}}=\pi(\frac{1+\delta}{a},\frac{1+\delta}{a},\frac{1}{c})$ with
incommensuration $\delta=0.12$ for field $\mathbf{H}~{}||~{}[1\bar{1}0]$.
Using these parameters, we find that a hyperfine field with dipolar
contribution is consistent with findings from both experiments. We speculate
that the B phase of CeCoIn5 represents an intrinsic phase of modulated
superconductivity and antiferromagnetism that can only emerge in a highly
clean system.
###### Keywords:
NMR superconductivity heavy fermion magnetism
###### pacs:
P76.60.-k 75.30.Fv 74.10.+v
††journal: Journal of Low Temperature Physics
## 1 Introduction
The heavy-fermion superconductor CeCoIn5 has attracted considerable attention
since its discovery in 2001.1 Not only does this unconventional d-wave
superconductor exhibit non-Fermi liquid behavior associated with proximity to
a proposed quantum critical point, but it is unique among the heavy-fermion
superconductors in that it also exhibits a new thermodynamic phase (B phase)
that exists only within the superconducting phase near $H_{c2}$.2, 3 Initially
this B phase was identified as the elusive Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) superconducting phase first predicted to exist in Pauli-limited
superconductors over 40 years ago.4, 5, 6, 7, 8 In fact, recent NMR work by
Young and coworkers9 identified the presence of incommensurate
antiferromagnetic order in the B phase in contrast to the standard predictions
for the FFLO phase.10, 11 Signatures of magnetism were also seen in other NMR
experiments.12, 13 The NMR spectra of the In sites in the B phase do not
reveal additional paramagnetic resonances that might be associated with
macroscopic phase separation, but rather are consistent with superconducting
order coexisting with a homogeneous modulation of antiferromagnetic order.9
Despite initial arguments to the contrary,14 recent neutron scattering results
by Kenzelmann and coworkers now provide conclusive proof for long-range static
incommensurate antiferromagnetic order.15
Figure 1: (Color online) The phase diagram of CeCoIn5 in high field as
determined by specific heat.5 Solid points represent second order phase
transitions and open points are first order transitions. The solid blue
squares are the points at which the spectra in Fig. 3 were obtained.
Figure 2: (Color online) The unit cell of CeCoIn5. The Ce atoms (yellow) sit
at the eight corners. The In(1) atoms sit in the center of the top and bottom
faces (orange). The Co atoms are grey and the In(2) atoms are green. For the
field oriented in the $ab$ plane, there are two inequivalent In(2) atoms,
depending on whether the field is parallel (In(2a) or perpendicular (In(2b))
to the unit cell face.
The antiferromagnetism in CeCoIn5 was first identified by Young et al. due to
the presence of a broad spectrum observed at the In(2) sites in this material
(see Figs. 2 and 3). The In(1) and Co sites, in contrast, showed no splitting.
Young et al. pointed out that these observations place constraints on the
possible magnetic structure, but do not uniquely identify the structure. They
proposed a minimal model where the magnetic structure consists of ordered
local Ce spins with moments $\mathbf{S}_{0}$ along the applied magnetic field
direction (along [100]), with an ordering wavevector of the form
$\mathbf{Q}=\pi(\frac{1+\delta}{a},\frac{1}{a},\frac{1}{c})$. The structure of
the NMR spectra revealed the incommensurate nature, but the value of the
modulation $\delta$ remained undetermined since the hyperfine field at the
In(2) site depends on the product of the size of the ordered moment and the
incommensuration. In the neutron diffraction experiment, Kenzelmann et al.
oriented the field along the [1$\bar{1}$0] direction, and observed
$\mathbf{Q_{n}}=\pi(\frac{1+\delta}{a},\frac{1+\delta}{a},\frac{1}{c})$ with
$\delta=0.12$ and moments along [001]. A crucial observation was that $\delta$
is independent of the applied field in the B phase, in contrast to the
predictions for the FFLO phase. By proposing a Ginzburg-Landau model for the
coupling of antiferromagnetism and superconductivity, they showed that the
superconducting order parameter in the B phase acquires a component with
finite momentum because of the strong coupling between the incommensurate
magnetism and superconductivity. The neutron data confirm the NMR observation
that this exotic state disappears immediately above $H_{c2}$, where the system
returns to a fully homogeneous normal phase, yet with strong deviations from
conventional Fermi liquid theory.3
A priori, the NMR and neutron scattering results suggest different magnetic
structures. In order to address this discrepancy, we investigate several
possible magnetic structures allowed by the NMR results. The neutron
diffraction results suggest that the applied field, $\mathbf{H}$, the moments,
$\mathbf{S}_{0}$, and the incommensuration wavevector
$\mathbf{Q}_{i}=\frac{\pi}{a}(\delta,\delta,0)$ are all mutually orthogonal.
In contrast, the proposed NMR scenario suggested
$\mathbf{S}_{0}~{}||~{}\mathbf{H}~{}||~{}\mathbf{Q}_{0}$. As we show below,
this scenario is the most likely for isotropic transferred hyperfine couplings
between the Ce spins and the In(2) nuclei. On the other hand, if the coupling
tensors are anisotropic, then other magnetic structures are possible as argued
by Koutroulakis et al.16, 17 If the hyperfine tensor has purely dipolar
symmetry, then we find that for $\mathbf{H}~{}||~{}[100]$ the most likely
magnetic structures satisfy $\mathbf{S}_{0}~{}||~{}[001]$ and
$\mathbf{Q}_{i}\perp\mathbf{S}_{0}$. The spectra of the In(1) and In(2) differ
slightly depending on the orientation of $\mathbf{Q}_{i}$ in the plane, but
the data are most consistent with $\mathbf{Q}_{i}||[010]$ or
$\mathbf{Q}_{i}||[110]$. The anisotropic coupling scenario offers a picture
that is both physically more reasonable and consistent with neutron
diffraction observations of $\mathbf{Q}_{i}\perp\mathbf{S}_{0}$ and
$\mathbf{H}_{0}\perp\mathbf{S}_{0}$, though for fields
$\mathbf{H}_{0}~{}||~{}[1\bar{1}0]$.
## 2 Analysis
Figure 3: (Color online) Fixed field NMR spectra at 11.1 T in CeCoIn5 showing
how the In(1), In(2) and Co sites evolve from the normal state (top) to the B
phase (bottom), adapted from Young et al.9
### 2.1 Spectra
In order to explain the broad double-peak structure of the In(2) spectrum in
Fig. 3, there must be a distribution of local fields both parallel and
antiparallel to the applied field $\mathbf{H}_{0}||[100]$ with values ranging
up to 1.3 kOe.9 The resonance frequency is given by
$f=\gamma|\mathbf{H_{0}}+\mathbf{H}_{hf}|+f_{Q}$, where $\gamma$ is the
gyromagnetic ratio and $f_{Q}$ is the contribution from the quadrupolar
interaction at the nucleus. Since the electric field gradient (EFG) at all
three nuclear sites is unaffected by the onset of superconductivity or
magnetism, the dramatic line broadening effects observed in the B phase can be
attributed entirely to the onset of the static hyperfine field,
$\mathbf{H}_{hf}$. $f_{Q}$ is a temperature independent constant that depends
on the particular site and we will not address it further. Experimentally, we
find no significant broadening in the B phase for the In(1) or the Co, but a
broad, double-peak spectrum for the In(2a) site (previously referred to as
In(2)||, the In(2) site on the unit cell face that lies parallel to the
field). Independent measurements show no significant broadening at the In(2b)
(previously referred to as In(2)⟂). These results put stringent constraints on
any candidate magnetic structure.
The double-peak structure observed for the In(2a) arises because there is a
distribution of local hyperfine fields that lie either parallel or
antiparallel to the applied field. If $\mathbf{H}_{hf}$ is parallel to
$\mathbf{H}_{0}$ and is modulated along a direction $\hat{r}$, then
$f(r)=f_{0}+\gamma h_{hf}^{0}\cos(qr)$, where $h_{hf}^{0}$ is the magnitude of
the modulation, $r$ is the distance along the modulation, and $q$ is the
wavevector. In this case, the spectrum will then be given by
$\mathcal{P}_{||}(f)\propto|df/dr|^{-1}=\frac{q^{-1}}{\sqrt{\gamma^{2}(h_{hf}^{0})^{2}-(f-f_{0})^{2}}}.$
(1)
On the other hand, if $\mathbf{H}_{hf}(r)~{}\perp~{}\mathbf{H}_{0}$ then
$f(r)=\sqrt{f_{0}^{2}+\gamma^{2}(h_{hf}^{0})^{2}\cos^{2}(qr)}$ and the
spectrum is given by
$\mathcal{P}_{\perp}(f)\propto\frac{q^{-1}f}{\sqrt{f^{2}-f_{0}^{2}}\sqrt{\gamma^{2}(h_{hf}^{0})^{2}-f^{2}-f_{0}^{2}}}.$
(2)
These spectra are shown in Fig. 4, and clearly show that for the parallel
case, there are two double peaks at frequencies both below and above $f_{0}$,
whereas for the perpendicular case, there is only a single peak at higher
frequency. A key result of the re-analysis of the NMR spectra is, in agreement
with our earlier analysis, that the In(2a) sites require a hyperfine field
parallel to the applied magnetic field,
$\mathbf{H}_{hf}~{}||~{}\mathbf{H}_{0}$, to account for the broadening and
double-peak spectrum. On the other side, the In(2b) sites show little or no
broadening, consistent with a vanishing or perpendicular hyperfine field,
i.e., at the In(2b) sites either $\mathbf{H}_{hf}=0$ or
$\mathbf{H}_{hf}\perp\mathbf{H}_{0}$.
Figure 4: (Color online) The calculated spectra of $\mathcal{P}_{||}(f)$
(blue) and $\mathcal{P}_{\perp}(f)$ (red) assuming $f_{0}$= 100 MHz and
$\gamma h_{hf}^{0}$ = 1 MHz. For the perpendicular case, the spectrum is only
weakly affected by the hyperfine fields, whereas for the parallel case it
broadens dramatically resulting in a double-peak structure.
### 2.2 Hyperfine couplings
The hyperfine interaction is given by the Hamiltonian
$\mathcal{H}_{hf}=\mathbf{\hat{I}}\cdot\mathbb{A}\cdot\mathbf{S}(\mathbf{r}=\mathbf{0})+\sum_{i}\mathbf{\hat{I}}\cdot\mathbb{B}_{i}\cdot\mathbf{S}(\mathbf{r}_{i}),$
(3)
where the hyperfine coupling tensor $\mathbb{A}$ represents the on-site
coupling to an electron spin, $\mathbf{S}$, at the nuclear site
$\mathbf{r}=0$, $\mathbb{B}$ represents a transferred hyperfine coupling to an
electron spin on a distant (ligand) site at $\mathbf{r}_{i}$.18 In CeCoIn5,
these sites are the nearest neighbor Ce 4$f$ electrons, and the sum is over
the nearest neighbors. For static ordering of the Ce spins, Eq. (3) can be re-
written as:
$\mathcal{H}_{hf}=\gamma\hbar\mathbf{\hat{I}}\cdot\mathbf{H}_{hf}$, where the
magnitude and direction of the hyperfine field $\mathbf{H}_{hf}$ depend
critically on the hyperfine tensors for the particular site and the magnetic
structure. The tensorial $\mathbb{A}$ term represents hyperfine coupling to
the itinerant conduction electrons, which we will ignore, since we are only
concerned with static contributions to $\mathbf{H}_{hf}$ from the static local
Ce ordering.
The transferred hyperfine tensor $\mathbb{B}$ is generally not diagonal in the
crystal axis basis,19 and may be written as the sum of isotropic and dipolar
contributions.20 To lowest order, the tensor can be approximated by a scalar
(isotropic) interaction, since the transferred hyperfine interaction is
typically at least one order of magnitude greater than the direct dipolar
interaction. However, in the CeCoIn5 compound, there is evidence that the
hyperfine interaction is not purely isotropic, and therefore we must consider
dipolar symmetries as well. Indeed, the magnitude of the dipolar portion is
found to be enhanced by the delocalized nature of the electrons in the
solid.21 Therefore, we write the hyperfine fields at the ligand sites as:
$\displaystyle\mathbf{H}_{hf}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{4}\mathbb{B}_{i}\cdot\mathbf{S}(\mathbf{r}+\mathbf{r_{i}})/\gamma\hbar\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}
at In(1) }$ (4) $\displaystyle\mathbf{H}_{hf}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\mathbb{B}_{i}\cdot\mathbf{S}(\mathbf{r}+\mathbf{r_{j}})/\gamma\hbar\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}
at Co }$ (5) $\displaystyle\mathbf{H}_{hf}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\mathbb{B}_{i}\cdot\mathbf{S}(\mathbf{r}+\mathbf{r_{k}})/\gamma\hbar\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}
at In(2a) }$ (6) $\displaystyle\mathbf{H}_{hf}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{2}\mathbb{B}_{i}\cdot\mathbf{S}(\mathbf{r}+\mathbf{r_{l}})/\gamma\hbar,\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}
at In(2b) }$ (7)
where $\mathbf{r}_{i}=(\pm\frac{a}{2},\pm\frac{a}{2},0)$ for the In(1) nearest
neighbor Ce sites, $\mathbf{r}_{j}=(0,0,\pm\frac{c}{2})$ for the Co nearest
neighbor Ce sites, and $\mathbf{r}_{l}=(\pm\frac{a}{2},0,z_{0})$ for the
In(2a) nearest neighbor Ce sites, and
$\mathbf{r}_{k}=(0,\pm\frac{a}{2},z_{0})$ for the In(2b) nearest neighbor Ce
sites. The couplings $\mathbb{B}_{i}=\mathbb{B}_{\rm iso}+\mathbb{B}_{\rm
dip}$, where:
$\mathbb{B}_{\rm iso}=B_{\rm iso}\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)$ (8)
and
$\mathbb{B}_{\rm dip}=\frac{B_{\rm
dip}}{r^{2}}\left(\begin{array}[]{ccc}2x^{2}-y^{2}-z^{2}&3xy&3xz\\\
3xy&-x^{2}+2y^{2}-z^{2}&3yz\\\ 3xz&3yz&-x^{2}-y^{2}+2z^{2}\end{array}\right)$
(9)
are evaluated for each site. Here $\mathbf{r}=(x,y,z)$ is the vector joining
the particular site to the Ce atom in question.
Table 1: The hyperfine fields at the In(1) and In(2) sites for various magnetic structures. Here we assumed that the applied magnetic field is $\mathbf{H}_{0}~{}||~{}[100]$, the moment is 0.15 $\mu_{B}$ and its modulation is $\delta=0.12$. The column at the far right indicates magnetic structures in agreement with the NMR spectra. Case | $\mathbf{Q}_{i}$ | $\mathbf{S}_{0}$ | $\mathbb{B}$ | In(1) | In(2a) | In(2b) | Agreement?
---|---|---|---|---|---|---|---
(1.1) | [100] | [100] | iso | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}=0$ | yes (a)
(1.2) | [100] | [100] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}=0$ | no
(1.3) | [100] | [001] | iso | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | no
(1.4) | [100] | [001] | dip | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | yes (b)
(2.1) | [010] | [100] | iso | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | no
(2.2) | [010] | [100] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}\approx 0$ | no
(2.3) | [010] | [001] | iso | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}=0$ | no
(2.4) | [010] | [001] | dip | $\mathbf{H}_{\rm hf}=0$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | yes (c)
(3.1) | [110] | [100] | iso | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | no
(3.2) | [110] | [100] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | no
(3.3) | [110] | [001] | iso | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | no
(3.4) | [110] | [001] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | yes (d)
(4.1) | [1$\bar{1}$0] | [100] | iso | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | no
(4.2) | [1$\bar{1}$0] | [100] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | no
(4.3) | [1$\bar{1}$0] | [001] | iso | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | $\mathbf{H}_{\rm hf}~{}||~{}[001]$ | no
(4.4) | [1$\bar{1}$0] | [001] | dip | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | $\mathbf{H}_{\rm hf}~{}||~{}[100]$ | $\mathbf{H}_{\rm hf}~{}||~{}[010]$ | yes (e)
### 2.3 Magnetic structure
The magnetic structure is given by
$\mathbf{S}=\mathbf{S_{0}}\cos[(\mathbf{Q}_{0}+\mathbf{Q}_{i})\cdot\mathbf{r}]$,
where the antiferromagnetic wavevector
$\mathbf{Q}_{0}=(\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{c})$ is commensurate
with the lattice. Neutron diffraction reports an incommensurate wavevector
$\mathbf{Q}_{i}=\frac{\pi}{a}(\delta,\delta,0)$ with spatial modulation
$\sqrt{2}a/\delta\approx 12a\approx 5.4$ nm in the $ab$ plane, and ordered
moment $\mathbf{S_{0}}$ at the Ce site. This modulation is significantly
shorter than the average inter-vortex distance of $\sim 14$ nm in a field of
10 T.13 So it does not support a picture of overlapping extended states from
the vortex cores leading to magnetic ordering. Similarly this specific
$\mathbf{Q}_{i}$ would be inconsistent with a field along [100], if it was
related to the alignment of vortices along the [100] direction. In our re-
analysis of the NMR spectra, we consider four cases: (1)
$\mathbf{Q}_{i}~{}||~{}[100]$, (2) $\mathbf{Q}_{i}~{}||~{}[010]$, and (3)
$\mathbf{Q}_{i}~{}||~{}[110]$, (4) $\mathbf{Q}_{i}~{}||~{}[1\bar{1}0]$. Case
(1) was proposed by Young et al.9 for NMR measurements under the condition
$\mathbf{H}_{0}~{}||~{}[100]$. Case (2) should be equally likely as case (1)
because of the tetragonal symmetry of the crystal structure. Case (3) was
proposed by Kenzelmann et al.15 for neutron diffraction measurements under the
condition $\mathbf{H}_{0}~{}||~{}[1\bar{1}0]$, and case (4) should be equally
likely as case (3). We have calculated the hyperfine fields for each of these
cases for both purely isotropic and purely dipolar couplings, for
$\mathbf{H}_{0}~{}||~{}[100]$ with moments along both [100] and [001], and the
results are summarized in Table (1).
The cases that are most consistent with the NMR observations are (1.1)
$\mathbf{Q}_{i}~{}||~{}[100]$, $\mathbf{S}_{0}~{}||~{}[100]$ and isotropic
coupling, (1.4) $\mathbf{Q}_{i}~{}||~{}[100]$, $\mathbf{S}_{0}~{}||~{}[001]$
and dipolar coupling, and (2.4) $\mathbf{Q}_{i}~{}||~{}[010]$,
$\mathbf{S}_{0}||[001]$ and dipolar coupling. Cases (3.4) and (4.4) are
consistent with the In(2) spectra, but give rise to an internal field at the
In(1) site that at first sight is inconsistent with experiment. We will
discuss this in more detail below in the discussion section. Figures 6 and 6
show the magnetic structure and hyperfine fields for cases (1.1) and (2.4).
Case (1.1) is identical to the one originally proposed by Young and
coworkers,9 which most likely will not minimize the magnetic contribution to
the free energy, as the moments are either parallel or antiparallel to the
applied field. Cases (1.4), (2.4), (3.4) and (4.4), in which the moments are
perpendicular to the applied field, are physically more reasonable for
antiferromagnetic ordering and agree with the neutron diffraction results of
$\mathbf{Q}_{i}\perp\mathbf{S}_{0}$ and $\mathbf{H}_{0}\perp\mathbf{S}_{0}$.15
## 3 Discussion
Figure 5: (Color online) Magnetic structure and hyperfine fields for isotropic
hyperfine couplings to the In(2) (case (1.1)). The in-plane tetragonal
structure is outlined in gray. The Ce atoms are yellow, and their moments are
indicated by red arrows pointing along [100]. The In(1) atoms are orange, and
the Co are not shown. The In(2a) are green and the In(2b) are blue. The
hyperfine fields at the In(2a) sites are indicated by blue arrows. Here the
hyperfine fields vanish at the In(1), Co and the In(2b) sites; the direction
of $\mathbf{H}_{0}$ is shown by the black arrow.
Figure 6: (Color online) Magnetic structure and hyperfine fields for dipolar
hyperfine couplings to the In(2) (case (2.4)). Same notation as in Fig. 6. The
Ce moments point along [001]. Here the hyperfine fields vanish at the In(1)
and Co sites, but not at the In(2) sites; the direction of $\mathbf{H}_{0}$ is
shown by the black arrow.
### 3.1 Hyperfine constants from the Knight shift
In CeCoIn5 measurements of the Knight shift in the normal state show that the
hyperfine couplings for the In(2a) site
($\mathbf{r}_{k}=(\pm\frac{a}{2},0,z_{0})$) for the applied field along (100),
(010) and (001) are 10.3 kOe/$\mu_{B}$, 0.0 kOe/$\mu_{B}$, and 32.4
kOe/$\mu_{B}$, respectively.22 For this site, Equations (8) and (9) yield:
$\displaystyle K_{a}$ $\displaystyle=$ $\displaystyle(2B_{iso}+B_{dip}(1-3\cos
2\theta_{z}))\chi_{a}$ (10) $\displaystyle K_{b}$ $\displaystyle=$
$\displaystyle(2B_{iso}-2B_{dip})\chi_{b}$ (11) $\displaystyle K_{c}$
$\displaystyle=$ $\displaystyle(2B_{iso}+B_{dip}(1+3\cos
2\theta_{z}))\chi_{c},$ (12)
where $\chi_{\alpha}$ is the susceptibility in the $\alpha$ direction. Using
the experimental numbers,22 we find $B_{iso}=B_{dip}=7.1$ kOe/$\mu_{B}$ and
$\theta_{z}=29^{\circ}$. The difference between this angle and that of the
crystal structure ($\theta_{z}=45^{\circ}$) probably is related to details of
the bonding of the In $4p$ orbitals, and will need further investigations.
### 3.2 Hyperfine fields in the B phase
Figure 7: (Color online) Real-space map (15 x 15 unit cells) of the hyperfine
field at the In(2a) (Eq. 6) (upper row) and the In(2b) (Eq. 7) (middle row) in
the $ab$ plane. Shown along the horizontal are the components of the hyperfine
field along the ($\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$)
directions in (red, green blue) shading, for case (2.4) with
$\mathbf{Q}_{0}~{}||~{}[010]$ and $\mathbf{S}_{0}~{}||~{}[001]$. Black
corresponds to zero hyperfine field. The lower row shows the spin density
along the [001] direction (cyan), and the histogram of resonant frequencies
for the In(2a) (orange) and the In(2b) (yellow).
Figure 8: (Color online) The same notation as in Fig. 8, but for case (3.4)
with $\mathbf{Q}_{0}~{}||~{}[110]$ and $\mathbf{S}_{0}~{}||~{}[001]$. The
corresponding histogram of resonant frequencies is in better agreement with
experiment.
For cases (1.4), (2.4), (3.4) and (4.4) the hyperfine field at the In(2a) and
In(2b) sites varies spatially
$\sim\cos[(\mathbf{Q}_{0}+\mathbf{Q}_{i})\cdot\mathbf{r}]$ with modulus:
$\displaystyle\mathbf{H}_{hf}(2a)$ $\displaystyle=$ $\displaystyle
S_{0}B_{dip}[3\sin(2\theta_{z})\cos(\frac{\pi\delta_{x}}{2})\hat{\mathbf{a}}+(1-\cos(2\theta_{z})\sin(\frac{\pi\delta_{x}}{2})\hat{\mathbf{c}}]$
(13) $\displaystyle\mathbf{H}_{hf}(2b)$ $\displaystyle=$ $\displaystyle
S_{0}B_{dip}[3\sin(2\theta_{z})\cos(\frac{\pi\delta_{y}}{2})\hat{\mathbf{b}}+(1-\cos(2\theta_{z})\sin(\frac{\pi\delta_{y}}{2})\hat{\mathbf{c}}]$
(14)
In each case, the hyperfine field at the In(2a) site oscillates along the
modulation direction with a component along [100] and the resulting spectrum
is described by Figs. 3, 8 and 8. Using the values
$\delta_{x}=\delta_{y}\approx 0.12$ and $S_{0}\approx 0.15\mu_{B}$ as reported
by Kenzelmann et al.,15 we find $h_{hf}^{0}\approx 2.6$ kOe, which is about
twice the experimental value of 1.3 kOe. The difference may be related to
uncertainties in the hyperfine coupling itself 18 or changes in the magnetic
structure for the field along [100]. Note that the modulation and ordered
moment may differ for field oriented along [100], in which case Eq. (13) will
give a different value. In fact the NMR result is consistent with the locus of
points given by $\delta_{x}=\delta_{x0}\cos^{-1}(S_{0}^{0}/S_{0}-1)$, where
$\delta_{x0}=0.3$ and $S_{0}^{0}=0.14\mu_{B}$.
For cases (3.4) and (4.4), where the modulation is along [110] or
[1$\bar{1}$0], the hyperfine field at the In(1) site does not cancel but has a
component along the [001] direction. This field can give rise to a minor shift
and/or broadening of the In(1) line. The spectra (Fig. 3) clearly show that
the In(1) line shifts and is only slightly broadened. However, the shift and
broadening may come from the onset of spin shift suppression in the
superconducting state and the presence of superconducting vortices. Therefore,
we cannot distinguish the presence of a hyperfine field from the
antiferromagnetic structure at the In(1) site within experimental error. As
seen in Figs. 8 and 8, the calculated spectra for case (3.4) is closer to the
experimental one. We speculate that the true magnetic structure for the field
$\mathbf{H}_{0}$ along [100] is best described either by case (2.4), (3.4) or
(4.4) with $\mathbf{H}_{0}\perp\mathbf{S}_{0}$, which will minimize the free
energy of the antiferromagnet. In each case the hyperfine field at the In(2a)
and In(2b) have components perpendicular to $\mathbf{H}_{0}$, but these
components only give rise to small shifts of the resonant frequency that are
difficult to distinguish from the Knight shift. The crucial element is that
the hyperfine field at the In(2a) is along $\mathbf{H}_{0}$.
### 3.3 Nature of the B phase
The fact that the antiferromagnetism exists only in field and only within the
superconducting phase indicates a strong coupling between these order
parameters. Kenzelmann and coauthors15 analyzed the symmetry of the
superconducting state for such a coupling and concluded that the
superconducting order parameter, $\Delta_{\mathbf{Q}}$ acquires the finite
momentum $\mathbf{Q}$ of the antiferromagnetic order parameter,
${\mathbf{M_{Q}}}$. This corresponds to a modulation of the order parameter in
real space that presumably is out-of-phase with the antiferromagnetism. In
other words, the antiferromagnetic order is maximum at the nodes of
$\Delta_{\mathbf{Q}}$. Since then various microscopic models have been
proposed to explain the field induced antiferromagnetic order.23, 24
Curiously, this scenario is similar to that observed in the ferropnictide
SrFe2As2 under pressure.25 In this compound, a novel hybrid state of
coexisting superconductivity and antiferromagnetism emerges above 5 GPa. We
speculate that these two novel states may in fact be the same. Although
superconductivity and antiferromagnetism are known to coexist inhomogeneously
in a number of doped high Tc, heavy fermion, and ferropnictide systems, the
highly clean undoped CeCoIn5 and SrFe2As2 materials support the emergence of
this fragile but intrinsic thermodynamic phase of modulated antiferromagnetism
and superconductivity. In CeCoIn5, this phase is quickly destroyed by doping
and is replaced by a commensurate order at zero field for sufficiently high Cd
doping.26, 27, 28, 29 Clearly many questions about this new state of matter
remain unexplained, such as the driving mechanism(s), the origin of the
incommensurate wavevector, and the nature of the excitations.
## 4 Conclusions
In summary, we have shown that both NMR and neutron diffraction measurements
in the field-induced B phase of CeCoIn5 are consistent with magnetic
structures where $\mathbf{Q}_{i}\perp\mathbf{S}_{0}$ and
$\mathbf{H}_{0}\perp\mathbf{S}_{0}$. The incommensurate modulation
$\textbf{Q}_{i}$ lies possibly along either [010] or [110] direction for
magnetic field pointing along [100] in real space. Tetragonal equivalent
directions for $\textbf{Q}_{i}$, [100] and [1$\bar{1}$0], are also possible.
Based on our analysis of the NMR spectra, we speculate that the B phase of
CeCoIn5 represents an intrinsic phase of modulated superconductivity and
antiferromagnetism that can only emerge in a highly clean system. Further NMR
and neutron diffraction measurements are necessary for the same field
orientations to unravel the origin of the field-induced antiferromagnetic
structure.
###### Acknowledgements.
We would like to thank R. Movshovich, V. Mitrović, and M. Kenzelmann for
valuable discussions and sharing their results. Work at Los Alamos National
Laboratory was performed under the auspices of the US Department of Energy
under grant no. DE-AC52-06NA25396.
## References
* 1 C. Petrovic et al., Europhys. Lett. 53, 354 (2001).
* 2 V.A. Sidorov et al., Phys. Rev. Lett. 89, 157004 (2002).
* 3 A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003).
* 4 A. Bianchi et al., Phys. Rev. Lett. 89, 137002 (2002).
* 5 A. Bianchi et al., Phys. Rev. Lett. 91, 187004 (2003).
* 6 H.A. Radovan et al., Nature 425, 51 (2003).
* 7 K. Kakuyanagi et al., Phys. Rev. Lett. 94, 047602 (2005).
* 8 K. Kumagai et al., Phys. Rev. Lett. 97, 227002 (2006).
* 9 B.-L. Young et al., Phys. Rev. Lett. 98, 036402 (2007).
* 10 A.B. Vorontsov, J.A. Sauls, and M.J. Graf, Phys. Rev. B 72, 184501 (2005).
* 11 A.B. Vorontsov and M.J. Graf, Phys. Rev. B 74, 172504 (2006).
* 12 V. F. Mitrović et al., Phys. Rev. Lett. 97, 117002 (2006).
* 13 G. Koutroulakis et al., Phys. Rev. Lett. 101, 047004 (2008).
* 14 Y. Matsuda and H. Shimahara, J. Phys. Soc. Jpn. 76, 051005 (2007).
* 15 M. Kenzelmann et al., Science 321, 1652 (2008).
* 16 G. Koutroulakis et al., 2009 APS March Meeting, BAPS.2009.MAR.B41.6.
* 17 V. Mitrović, (private communications).
* 18 N.J. Curro, B.-L. Young, J. Schmalian, and D. Pines, Phys. Rev. B 70, 235117 (2004).
* 19 M.-A. Vachon et al., J. Phys.: Condens. Matter 20, 29225 (2008).
* 20 N.J. Curro, New. J. Phys. 8, 173 (2006).
* 21 C.P. Slichter, Principles of Nuclear Magnetic Resonance, 3rd edn. Springer-Verlag (1992).
* 22 N.J. Curro et al., Phys. Rev. B 64, 180514 (2001).
* 23 Y. Yanase and M. Sigrist, JPCS 150, 052287 (2009).
* 24 H.-Y. Kee and D. Podolsky, Europhysics Letters, 86, 57005 (2009); K. Miyake, J. Phys. Soc. Jpn. 77, 123703 (2008); Y. Yanase J. Phys. Soc. Jpn. 77, 063705 (2008); A Aperis et al., J. Superconductivity and Novel Magnetism 22 115 (2009)
* 25 K. Kitagawa et al., arXiv:0906.4740 (2009).
* 26 Y. Tokiwa et al., Phys. Rev. Lett. 101, 037001 (2008).
* 27 R.R. Urbano et al., Phys. Rev. Lett. 99, 146402 (2007).
* 28 R.R. Urbano, B.-L. Young, N.J. Curro, Physica B 403, 1056 (2008).
* 29 C.F. Miclea et al., Phys. Rev. Lett. 96, 117001 (2006).
|
arxiv-papers
| 2009-08-04T23:22:35 |
2024-09-04T02:49:04.438548
|
{
"license": "Public Domain",
"authors": "Nicholas J. Curro, Ben-Li Young, Ricardo Urbano, and Matthias J. Graf",
"submitter": "Matthias J. Graf",
"url": "https://arxiv.org/abs/0908.0565"
}
|
0908.0577
|
# Form–type Calabi–Yau equations
Jixiang Fu Institute of Mathematics
Fudan University
Shanghai 200433, China majxfu@fudan.edu.cn , Zhizhang Wang Institute of
Mathematics
Fudan University
Shanghai 200433, China youxiang163wang@163.com and Damin Wu Department of
Mathematics
The Ohio State University
1179 University Drive, Newark, OH 43055, U.S.A. dwu@math.ohio-state.edu
###### Abstract.
Motivated from mathematical aspects of the superstring theory, we introduce a
new equation on a balanced, hermitian manifold, with zero first Chern class.
Solving the equation, one will obtain, in each Bott–Chern cohomology class, a
balanced metric which is hermitian Ricci–flat. T his can be viewed as a
differential form level generalization of the classical Calabi–Yau equation.
We establish the existence and uniqueness of the equation on complex tori, and
prove certain uniqueness and openness on a general Kähler manifold.
## 1\. Setting and Equations
In the superstring theory, the internal space $X^{3}$ is a complex three-
dimensional manifold with a non-vanishing holomorphic three-form $\Omega$ [15]
(cf. [1]). The $N=1$ supersymmetry requires [15, 10]
$d(\parallel\Omega\parallel_{\omega}\omega^{2})=0,$
for some hermitian metric (form) $\omega$. The above equation in mathematics
says that $\omega$ is a conformally balanced metric. (We recall that [14] a
hermitian metric $\omega$ on an $n$-dimensional complex manifold $X^{n}$ is
called _balanced_ if $\omega$ satisfies that
$d(\omega^{n-1})=0\qquad\textup{on $X^{n}$.\,)}$
Note that [8, 6] the torus bundles over $K3$ surfaces and over complex abelian
surfaces twisted by two anti-self dual $(1,1)$-forms admit a non-vanishing
holomorphic three-form $\Omega$ and a natural balanced metric $\omega_{0}$
such that
(1.1) $\parallel\Omega\parallel_{\omega_{0}}=1.$
As important examples in the superstring theory and non-Kähler complex
geometry, the complex manifolds $\\#_{k}(S^{3}\times S^{3})$ for any $k\geq 2$
[4, 12] also admit a non-vanishing holomorphic three-form [4] and a balanced
metric [5]. Moreover, we know that $\\#_{k}(S^{3}\times S^{3})$ satisfies the
$\partial\bar{\partial}$–lemma [4]. A natural question to ask is, whether
$\\#_{k}(S^{3}\times S^{3})$ admits a balanced metric $\omega_{0}$ such that
(1.1) holds. Such a metric $\omega_{0}$, if exists, will play an important
role in the superstring theory and hermitian geometry.
More generally, let $X^{n}$ $(n\geq 3)$ be a complex $n$-dimensional manifold
with a non-vanishing holomorphic $n$-form $\Omega$ and with a balanced metric
$\omega_{0}$. We want to look for a balanced metric $\omega$ such that
(1.2)
$\omega^{n-1}=\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi,$
for some real $(n-2,n-2)$–form $\varphi$, and such that
(1.3) $\textup{$\|\Omega\|_{\omega}=$ some positive constant $C_{0}$}.$
In other words, we would like to find solutions of (1.3) in the cohomology
class $[\omega_{0}^{n-1}]\in H^{n-1,n-1}_{BC}(X)$. Here $H^{p,q}_{BC}(X)$
stands for the Bott–Chern cohomology:
$H^{p,q}_{BC}(X)=\frac{(\ker\partial\cap\ker\bar{\partial})\cap\Omega^{p,q}(X)}{\textup{im}\
\partial\bar{\partial}\cap\Omega^{p,q}(X)}.$
One can certainly normalize the constant $C_{0}$ in (1.3) to be 1, as in
(1.1). However, it may be more convenient to set
$C_{0}=\left(\int_{X}\omega^{n}\right)^{-\frac{1}{2}},$
from the equation point of view. As in the Kähler case, equation (1.3) is
equivalent to the equation
(1.4)
$\frac{\det\omega}{\det\omega_{0}}=\frac{\parallel\Omega\parallel^{2}_{\omega_{0}}}{\parallel\Omega\parallel^{2}_{\omega}}=e^{f}\frac{\int_{X}\omega^{n}}{\int_{X}\omega_{0}^{n}}.$
Here we denote
$e^{f}=\|\Omega\|^{2}_{\omega_{0}}\int_{X}\omega_{0}^{n},$
and denote
$\det\omega=\det(g_{i\bar{j}}),\qquad\textup{if
\quad$\omega=\frac{\sqrt{-1}}{2}\sum_{i,j=1}^{n}g_{i\bar{j}}dz_{i}\wedge
d\bar{z}_{j}$}.$
At the moment, we write
$\begin{split}&\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi=\Bigl{(}\frac{\sqrt{-1}}{2}\Bigr{)}^{n-1}(n-1)!\\\
&\qquad\qquad\cdot\sum_{i,j=1}^{n}(\Psi_{\varphi})_{i\bar{j}}s(i,j)dz_{1}\wedge
d\bar{z}_{1}\wedge\cdots\wedge\widehat{dz_{i}}\wedge\cdots\widehat{d\bar{z}_{j}}\wedge\cdots\wedge
dz_{n}\wedge d\bar{z}_{n}.\end{split}$
Here the sign function $s(i,j)$ is equal to $1$ if $i\leq j$, and is equal to
$-1$ if $i>j$. By (1.2) and
$\begin{split}\omega^{n-1}&=\Bigl{(}\frac{\sqrt{-1}}{2}\Bigr{)}^{n-1}(n-1)!\\\
&\quad\;\cdot(\det\omega)\sum_{i,j=1}^{n}g^{i\bar{j}}s(i,j)dz_{1}\wedge
d\bar{z}_{1}\wedge\cdots\wedge\widehat{dz_{i}}\wedge\cdots\widehat{d\bar{z}_{j}}\wedge\cdots\wedge
dz_{n}\wedge d\bar{z}_{n},\end{split}$
we have
$(\det\omega)g^{i\bar{j}}=(\Psi_{\varphi})_{i\bar{j}},\qquad\textup{for all
$1\leq i,j\leq n$}.$
Hence,
$\det\omega=\bigl{\\{}\det\big{[}(\Psi_{\varphi})_{i\bar{j}}\big{]}\bigr{\\}}^{\frac{1}{n-1}}=\bigl{\\{}\det\big{[}\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\varphi\big{]}\bigr{\\}}^{\frac{1}{n-1}}.$
Here $\det[\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\varphi]$
stands for the determinant of $n\times n$ matrix of its coefficients. Thus,
equation (1.4) is equivalent to
(1.5)
$\frac{\det[\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\varphi]}{\det\omega_{0}^{n-1}}=e^{(n-1)f}\left(\frac{\int_{X}\omega^{n}}{\int_{X}\omega_{0}^{n}}\right)^{n-1}.$
We call the above equation the form-type Calabi–Yau equation. Clearly, by
integrating (1.4), we obtain a compatibility condition
(1.6) $\int_{X}e^{f}\omega_{0}^{n}=\int_{X}\omega_{0}^{n}.$
Let us denote by $\mathcal{P}(\omega_{0})$ the set of all smooth real
$(n-2,n-2)$–forms $\psi$ such that
(1.7)
$\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\psi>0\qquad\textup{on
$X$}.$
The question is therefore reduced to find, for a given $f\in C^{\infty}(X)$
with (1.6), a smooth real $(n-2,n-2)$–form $\varphi\in\mathcal{P}(\omega_{0})$
satisfying (1.5).
Here is the geometric interpretation of our equation. Let us briefly recall
some definitions related to the hermitian connection. We follow [9]. Let $R$
be the curvature of hermitian connection with respect to metric $\omega$.
Then,
$R_{i\bar{j}k\bar{l}}=-\frac{\partial^{2}g_{i\bar{j}}}{\partial
z_{k}\partial\bar{z}_{l}}+\sum_{p,q=1}^{n}g^{p\bar{q}}\frac{\partial
g_{i\bar{q}}}{\partial z_{k}}\frac{\partial
g_{p\bar{j}}}{\partial\bar{z}_{l}}.$
We set
$R_{k\bar{l}}=\sum_{i,j=1}^{n}g^{i\bar{j}}R_{i\bar{j}k\bar{l}},$
and associate with it a real $(1,1)$-form given by
$Ric^{h}=\sqrt{-1}\sum_{k,l=1}^{n}R_{k\bar{l}}dz_{k}\wedge d\bar{z}_{l}.$
We call $Ric^{h}$ the _Ricci curvature_ of hermitian connection. Clearly,
$Ric^{h}=\sqrt{-1}\bar{\partial}\partial\log(\det\omega).$
So $\parallel\Omega\parallel_{\omega}=C_{0}$ is equivalent to the Ricci
curvature $Ric^{h}=0$.
On the other hand, we can also define the Ricci form $Ric^{s}$ of the spin
connection (i.e. Bismut connection) on a hermitian manifold. The relation
between the two Ricci forms is given by [11]
$Ric^{s}=Ric^{h}-dd^{\ast}\omega.$
Here $d^{\ast}$ is the adjoint operator of $d$ with respect to the metric
$\omega$. So when $\omega$ is balanced, $Ric^{s}=Ric^{h}$, and hence,
$\parallel\Omega\parallel_{\omega}=C_{0}$ is also equivalent to the Ricci
curvature of the spin connection is zero.
In particular, if $\omega_{0}$ is Kähler and let $\varphi$ to be
$\begin{split}\mbox{either}\quad&u\sum_{i=0}^{n-2}\binom{n-1}{i}\bigl{(}\frac{\sqrt{-1}}{2}\partial\bar{\partial}u\bigr{)}^{n-i-2}\wedge\omega_{0}^{i},\quad\mbox{or}\\\
&-\frac{\sqrt{-1}}{2}\partial
u\wedge\bar{\partial}u\wedge\sum_{i=0}^{n-3}\binom{n-1}{i}\bigl{(}\frac{\sqrt{-1}}{2}\partial\bar{\partial}u\bigr{)}^{n-i-3}\wedge\omega_{0}^{i},\end{split}$
then (1.5) is reduced to
$\frac{\det(\omega_{0}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}u)}{\det{\omega_{0}}}=e^{f}.$
This is the classic equation in the Calabi Conjecture on $c_{1}(X)=0$, which
was settled by Yau [16].
It seems to us that a form-type equation such as (1.5) has not yet been
studied. To begin with, we consider the form-type Calabi–Yau equation on
$T^{n}$, the complex $n$-torus. Let $(z_{1},\ldots,z_{n})$ be the complex
coordinates on $T^{n}$ induced from $\mathbb{C}^{n}$. Then, any non-vanishing
holomorphic $n$-form $\Omega$ on $T^{n}$ is equal to
$dz_{1}\wedge\cdots\wedge dz_{n}$
up to multiplying a nonzero constant. We fix such an $n$-form $\Omega$. By a
_constant form_ or a _constant metric_ on $T^{n}$ we mean a differential form
or a metric on $T^{n}$ with constant coefficients. Let $\omega_{0}$ be a
balanced metric on $T^{n}$. As far as the Bott–Chern cohomology class of
$\omega_{0}^{n-1}$ is concerned, we can assume, without loss of generality,
that $\omega_{0}$ is a constant metric on $T^{n}$. This is due to the fact
that any closed differential form on $T^{n}$ is cohomologous to a constant
form, and the $\partial\bar{\partial}$–Lemma. Our result is as follows:
###### Theorem 1.
Let $\Omega$ be a non-vanishing holomorphic $n$-form on $T^{n}$ $(n\geq 3)$,
and $\omega_{0}$ is a constant metric on $T^{n}$ such that
$\|\Omega\|_{\omega_{0}}=1$. We denote by $C_{0}$ a positive constant.
1. (1)
If $C_{0}\leq 1$, then for any metric $\omega$ on $T^{n}$ such that
$[\omega^{n-1}]=[\omega_{0}^{n-1}]\in H^{n-1,n-1}_{BC}(T^{n})$ and that
$\|\Omega\|_{\omega}=C_{0}$, we must have $C_{0}=1$ and
$\omega=\omega_{0}.$
2. (2)
For each $C_{0}>1$, there exists a metric $\omega$ on $T^{n}$ such that
$[\omega^{n-1}]=[\omega_{0}^{n-1}]$ and that
$\|\Omega\|_{\omega}=C_{0}.$
One can see from Theorem 1 that the normalization constant $C_{0}$ plays a
role here. When $C_{0}\leq 1$, the theorem tells us that the Calabi–Yau metric
is the unique canonical balanced metric. It is the second case, $C_{0}>1$,
that marks the difference between a form-type equation and a usual function-
type equation. In this case, we establish the existence of a desired balanced
metric which is not Calabi–Yau. We further generalize the uniqueness part,
Theorem 1 (1), to an arbitrary Calabi–Yau manifold:
###### Theorem 2.
Let $X$ be a compact Kähler manifold with a non-vanishing holomorphic $n$-form
$\Omega$. Let $\omega_{0}$ be a Calabi–Yau metric such that
$\|\Omega\|_{\omega_{0}}=1$. Then, for any balanced metric $\omega$ on $X$
such that $\omega^{n-1}$ represents the Bott–Chern cohomology class of
$\omega_{0}^{n-1}$ and such that $\|\Omega\|_{\omega}=C_{0}\leq 1$, we have
$\omega=\omega_{0}.$
For a general case that $\omega_{0}$ is non-Kähler, one can use the continuity
method to solve (1.5). As an initial step we consider the openness. Here we
have to assume $X$ to be a Kähler manifold, endowed with a Kähler metric
$\eta$. For nonnegative integers $k$ and $m$, and a real number $0<\alpha<1$,
we denote by $C^{k,\alpha}(\Lambda^{m,m}(X))$ the Hölder space of real
$(m,m)$–forms on $X$, and in particular, $C^{k,\alpha}(\Lambda^{0,0}(X))\equiv
C^{k,\alpha}(X)$. Let
$\mathcal{F}^{k,\alpha}(X)=\left\\{g\in
C^{k,\alpha}(X);\int_{X}e^{g}\,\omega_{0}^{n}=\int_{X}\omega_{0}^{n}\right\\}.$
Then $\mathcal{F}^{k,\alpha}(X)$ is a hypersurface in the Banach space
$C^{k,\alpha}(X)$. Let $\omega_{0}$ be a Hermitian metric on $X$, and
$\mathcal{P}(\omega_{0})$ be the set given by (1.7). We define a map
$M:\mathcal{P}(\omega_{0})\cap
C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))\to\mathcal{F}^{k,\alpha}(X)$ by
$M(\psi)=\log\left(\frac{\omega_{\psi}^{n}}{\omega_{0}^{n}}\right)-\log\left(\frac{\int_{X}\omega_{\psi}^{n}}{\int_{X}\omega_{0}^{n}}\right),$
where, by abuse of notation, $\mathcal{P}(\omega_{0})\cap
C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$ stands for
$\left\\{\psi\in
C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X));\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\psi>0\right\\},$
and for each $\psi\in\mathcal{P}(\omega_{0})\cap
C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$, we denote by $\omega_{\psi}$ the
positive $(1,1)$–form on $X$ such that
$\omega_{\psi}^{n-1}=\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\psi.$
Note that equation (1.5) can be written as
$M(\varphi)=f.$
###### Theorem 3.
Let $X$ be an $n$-dimensional Kähler manifold $(n\geq 3)$, $\omega_{0}$ be a
Hermitian metric on $X$, $k\geq n+4$ be an integer, and $0<\alpha<1$ be a real
number. Given $f\in\mathcal{F}^{k,\alpha}(X)$, suppose that
$\varphi\in\mathcal{P}(\omega_{0})\cap C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$
satisfies
$M(\varphi)=f.$
Then, there is a positive number $\delta$, such that for any
$g\in\mathcal{F}^{k,\alpha}(X)$ with $\|g-f\|_{C^{k,\alpha}(X)}\leq\delta$,
there exists a function $\psi\in\mathcal{P}(\omega_{0})\cap
C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$ such that
$M(\psi)=g.$
The rest of the paper is organized as follows: In Section 2, we first show
Theorem 1 (1). Next, we prove Theorem 1 (2) by explicitly constructing a
smooth solution $\varphi\in\mathcal{P}(\omega_{0})$ for the form-type
equation. These arguments make use of special properties such as the flat
structure of $T^{n}$. We prove Theorem 2 at the end of Section 2. In this
respect, we essentially present two proofs for the uniqueness on $T^{n}$, as
they may have interests of their own. In Section 3, we prove Theorem 3 in full
details, where one can see the compatibility condition is crucial. Moreover,
the approach differs from the standard one in that, the special
$(n-2,n-2)$–forms $(u\eta^{n-2})$ are taken, and also in the argument of
Proposition 15 and Proposition 16.
###### Acknowledgment.
The authors would like to thank Professor S.-T. Yau and also L.-S. Tseng for
helpful discussion. Part of the work was done while the third named author was
visiting Fudan University, he would like to thank their warm hospitality. Fu
is supported in part by NSFC grants 10771037 and 10831008.
## 2\. Uniqueness and Existence
In this section, we adopt the following index convention, unless otherwise
indicated. For an $(n-1,n-1)$–form $\Theta$, we denote
$\begin{split}\Theta&=\Big{(}\frac{\sqrt{-1}}{2}\Big{)}^{n-1}(n-1)!\\\
&\quad\cdot\sum_{p,q}s(p,q)\Theta_{p\bar{q}}dz^{1}\wedge
d\bar{z}^{1}\cdots\wedge\widehat{dz^{p}}\wedge d\bar{z}^{p}\wedge\cdots\wedge
d\bar{z}^{q}\wedge\widehat{d\bar{z}^{q}}\wedge\cdots\wedge dz^{n}\wedge
d\bar{z}^{n},\end{split}$
in which
(2.1) $s(p,q)=\begin{cases}-1,&\textup{if $p>q$};\\\ 1,&\textup{if $p\leq
q$}.\end{cases}$
Here we introduce the sign function $s$ so that,
$\begin{split}&dz^{p}\wedge d\bar{z}^{q}\wedge s(p,q)dz^{1}\wedge
d\bar{z}^{1}\cdots\wedge\widehat{dz^{p}}\wedge d\bar{z}^{p}\wedge\cdots\wedge
d\bar{z}^{q}\wedge\widehat{d\bar{z}^{q}}\wedge\cdots\wedge dz^{n}\wedge
d\bar{z}^{n}\\\ &=dz^{1}\wedge d\bar{z}^{1}\wedge\cdots\wedge dz^{n}\wedge
d\bar{z}^{n},\qquad\textup{for all $1\leq p,q\leq n$}.\end{split}$
And, if the matrix $(\Theta_{p\bar{q}})$ is invertible, we denote by
$(\Theta^{p\bar{q}})$ the transposed inverse of $(\Theta_{p\bar{q}})$, i.e.,
$\sum_{l}\Theta_{i\bar{l}}\Theta^{j\bar{l}}=\delta_{ij}.$
In the following, we may also use the summation convention on repeating
indices.
### 2.1. Torus case
Throughout this subsection, we consider $X=T^{n}$, the complex $n$-torus with
$n\geq 3$. We shall prove Theorem 1. Note that the first part of Theorem 1
follows immediately from Lemma 4 below. We shall prove the second part in
Lemma 7.
###### Lemma 4.
Let $\omega_{0}$ be a constant metric on $T^{n}$. Suppose that there exists an
$(n-2,n-2)$–form $\varphi\in\mathcal{P}(\omega_{0})$ and a constant
$0<C_{0}\leq 1$ such that
(2.2)
$C_{0}\det\left(\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\right)=\det\omega_{0}^{n-1}.$
Then, we have $C_{0}=1$ and
$\sqrt{-1}\partial\bar{\partial}\varphi=0.$
We need two propositions to derive Lemma 4. Let $(z_{1},\ldots,z_{n})$ be the
complex coordinates on $T^{n}$ induced from $\mathbb{C}^{n}$. The
corresponding real coordinates are $(x_{1},\ldots,x_{2n})$. Here we denote
(2.3) $z_{i}=x_{2i-1}+\sqrt{-1}x_{2i},\quad\textup{and hence,
$\frac{\partial}{\partial z_{i}}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{2i-1}}-\sqrt{-1}\frac{\partial}{\partial x_{2i}}\right)$},$
for all $1\leq i\leq n$. We choose the following volume form on $T^{n}$:
$\displaystyle dV=(\sqrt{-1}/2)^{n}dz_{1}\wedge d\bar{z}_{1}\wedge\cdots\wedge
dz_{n}\wedge d\bar{z}_{n}.$
Here are two elementary facts:
###### Proposition 5.
For any smooth complex function $f$ defined on $T^{n}$, we have
$\displaystyle\int_{T^{n}}\frac{\partial^{2}f}{\partial z_{i}\partial
z_{j}}dV=0,\qquad\textup{for all $i,j=1\cdots,n$.}$
###### Proof.
We write
$\displaystyle f=f_{1}+\sqrt{-1}f_{2},$
where $f_{1},f_{2}$ are real functions on $T^{n}$. Then,
$\displaystyle 4\frac{\partial^{2}f_{1}}{\partial z_{i}\partial
z_{j}}=\frac{\partial^{2}f_{1}}{\partial x_{2i-1}\partial
x_{2j-1}}-\frac{\partial^{2}f_{1}}{\partial x_{2i}\partial
x_{2j}}-\sqrt{-1}\left(\frac{\partial^{2}f_{1}}{\partial x_{2i}\partial
x_{2j-1}}+\frac{\partial^{2}f_{1}}{\partial x_{2i-1}\partial x_{2j}}\right).$
We have a similar equation for $f_{2}$. And note that
$\displaystyle dV=dx_{1}\wedge\cdots\wedge dx_{2n}.$
The result then obviously follows from the fundamental theorem of calculus. ∎
###### Proposition 6.
Let $B=(b_{i\bar{j}})$ be a hermitian matrix on $T^{n}$, in which each entry
$b_{i\bar{j}}$ is a complex smooth function defined on $T^{n}$ such that
$\displaystyle\int_{T^{n}}b_{i\bar{j}}\,dV=0.$
Assume that $I+B$ is everywhere positive definite, and there is a constant
$c\geq 1$ such that
$\displaystyle\det(I+B)=c\quad\textup{on $T^{n}$, \; where
$I\equiv(\delta_{i\bar{j}})$}.$
Then, $c=1$ and $B=0$.
###### Proof.
Since $I+B$ is positive definite, we have
(2.4)
$\displaystyle\frac{\text{tr}(I+B)}{n}\geqq\sqrt[n]{\det(I+B)}=\sqrt[n]{c}\qquad\textup{on
$T^{n}$}.$
Integrating (2.4) over $T^{n}$, we obtain
$\displaystyle\int_{T^{n}}dV=\int_{T^{n}}\frac{\text{tr}(I+B)}{n}dV\geq\sqrt[n]{c}\int_{T^{n}}dV.$
Thus, $c=1$, and the inequality of (2.4) is in fact an equality. That is,
(2.5)
$\displaystyle\frac{\text{tr}(I+B)}{n}=\sqrt[n]{\det(I+B)}=1,\qquad\textup{on
$T^{n}$}.$
Now at an arbitrary point $x$ in $T^{n}$, we choose a unitary matrix $U$ such
that
$\displaystyle UB\bar{U}^{T}=\text{dial}\\{\lambda_{1},\cdots,\lambda_{n}\\}.$
Then (2.5) is equivalent to that
$\displaystyle 1+\lambda_{1}=1+\lambda_{2}=\cdots=1+\lambda_{n}=1.$
This implies that
$\displaystyle\lambda_{i}=0,\qquad\textup{for all $i=1,\ldots,n$}.$
Therefore, $B=0$ at $x$. Since $x$ is arbitrary, this finishes the proof. ∎
Let us now proceed to prove Lemma 4:
###### Proof of Lemma 4.
Let
$\begin{split}\omega_{0}^{n-1}&=\Big{(}\frac{\sqrt{-1}}{2}\Big{)}^{n-1}(n-1)!\\\
&\quad\cdot\sum_{p,q}\Psi_{p\bar{q}}s(p,q)dz^{1}\wedge
d\bar{z}^{1}\wedge\cdots\wedge\widehat{dz^{p}}\wedge\cdots\wedge\cdots\wedge\widehat{d\bar{z}^{q}}\wedge\cdots\wedge
dz^{n}\wedge d\bar{z}^{n}.\end{split}$
Here $(\Psi_{i\bar{j}})$ is a constant, positive definite, hermitian matrix,
and $s(p,q)$ is given by (2.1). We can then take a non-degenerate constant
matrix $A$ such that
(2.6) $\displaystyle A(\Psi_{i\bar{j}})\bar{A}^{T}=I.$
We define a hermitian matrix $F_{\varphi}=((F_{\varphi})_{i\bar{j}})$ on
$T^{n}$ by
$\begin{split}\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi&=\Big{(}\frac{\sqrt{-1}}{2}\Big{)}^{n-1}(n-1)!\\\
&\cdot\sum_{p,q}(F_{\varphi})_{p\bar{q}}s(p,q)dz^{1}\wedge
d\bar{z}^{1}\wedge\cdots\wedge\widehat{dz^{p}}\wedge\cdots\wedge\cdots\wedge\widehat{d\bar{z}^{q}}\wedge\cdots\wedge
dz^{n}\wedge d\bar{z}^{n}.\end{split}$
It follows from Proposition 5 that
$\displaystyle\int_{T^{n}}(F_{\varphi})_{i\bar{j}}dV=0.$
Then, by (2.2) and (2.6),
$\displaystyle\det(I+AF_{\varphi}\bar{A}^{T})=C^{-1}_{0}.$
Since $\varphi\in\mathcal{P}(\omega_{0})$, we obtain
$\displaystyle I+AF_{\varphi}\bar{A}^{T}>0\qquad\textup{on $T^{n}$}.$
Applying Proposition 6 yields that $C_{0}=1$, and
$\displaystyle AF_{\varphi}\bar{A}^{T}=0,$
and therefore,
$F_{\varphi}=0.$
∎
The following lemma establishes the second part of Theorem 1. By a linear
transformation, if necessary, we can assume the constant metric $\omega_{0}$
on $T^{n}$ to be the standard metric:
$\displaystyle\omega_{0}=\frac{\sqrt{-1}}{2}\big{(}dz_{1}\wedge
d\bar{z}_{1}+\cdots+dz_{n}\wedge d\bar{z}_{n}\big{)}.$
###### Lemma 7.
For any $0<\delta<1$, there exists a smooth $(n-2,n-2)$–form
$\varphi\in\mathcal{P}(\omega_{0})$ such that
(2.7)
$\displaystyle\det\left(\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\right)=\delta\det\omega_{0}^{n-1}.$
###### Proof of Lemma 7.
We set
(2.8)
$\begin{split}\varphi&=(n-1)!\left(\frac{\sqrt{-1}}{2}\right)^{n-2}\Big{[}u(z_{1},\bar{z}_{1})dz_{3}\wedge
d\bar{z}_{3}\wedge\cdots\wedge dz_{n}\wedge d\bar{z}_{n}\\\
&\quad+\,v(z_{1},\bar{z}_{1})dz_{2}\wedge
d\bar{z}_{2}\wedge\widehat{dz_{3}}\wedge\widehat{d\bar{z}_{3}}\wedge
dz_{4}\wedge d\bar{z}_{4}\wedge\cdots\wedge dz_{n}\wedge
d\bar{z}_{n}\Big{]}.\end{split}$
Here $u,v$ are two real, smooth, periodic functions to be determined, with
$1+\Delta u>0$ and $1+\Delta v>0$. Since $u$ and $v$ depend only on the first
variable, the equation (2.7) becomes that
(2.9) $\Big{(}1+\frac{\partial^{2}u}{\partial
z_{1}\partial\bar{z}_{1}}\Big{)}\Big{(}1+\frac{\partial^{2}v}{\partial
z_{1}\partial\bar{z}_{1}}\Big{)}=\delta.$
This reduces to an equation on $T^{1}$. Note that
$\frac{\partial^{2}u}{\partial z_{1}\partial\bar{z}_{1}}=\Delta
u,\quad\frac{\partial^{2}v}{\partial z_{1}\partial\bar{z}_{1}}=\Delta v,$
where $\Delta$ is the standard Laplacian on $T^{1}$, i.e., the Laplacian
associated with $\omega_{0}|_{T^{1}}$. We can rewrite (2.9) as
(2.10) $1+\Delta u=\frac{\delta}{1+\Delta v}.$
Our strategy is to fix a function $v$ and then solve (2.10) for a function
$u$. Note that for a fixed $v$, the necessary and sufficient condition to
solve (2.10) is that
(2.11)
$\int_{T^{1}}\omega_{0}|_{T^{1}}=\delta\int_{T^{1}}\frac{\omega_{0}|_{T^{1}}}{1+\Delta
v}.$
Now let
(2.12) $v=-4k\sin\Big{(}\frac{z_{1}+\bar{z}_{1}}{2}\Big{)}=-4k\sin x_{1},$
where $0<k<1$ is a constant to be determined, and the change of coordinates is
given by (2.3). Then, (2.11) becomes that
$\int_{T^{1}}dx_{1}\wedge dx_{2}=\int_{T^{1}}\frac{\delta}{1+k\sin
x_{1}}dx_{1}\wedge dx_{2},$
that is,
(2.13) $\displaystyle\int^{2\pi}_{0}\frac{\delta}{1+k\sin x_{1}}dx_{1}=2\pi.$
It follows from the proposition below that, for each $0<\delta<1$, there
exists a real number $0<k<1$, depending only on $\delta$, such that (2.13)
holds. Therefore, for $v$ given by (2.12), there is a smooth function $u$,
unique up to a constant, satisfies (2.10). Also, by the construction,
$1+\Delta v>0,\qquad 1+\Delta u>0.$
Thus, by (2.8) we obtain an $(n-2,n-2)$–form
$\varphi\in\mathcal{P}(\omega_{0})$ which solves (2.7). ∎
###### Proposition 8.
Let
(2.14) $\displaystyle Z(k)=\frac{1}{2\pi}\int^{2\pi}_{0}\frac{1}{1+k\sin
x}dx,\qquad\textup{for all $0\leq k<1$}.$
Then, for any $0<\delta<1$, there exists a unique number $0<k_{\delta}<1$ such
that
$Z(k_{\delta})=\delta^{-1}.$
###### Proof.
Clearly, the function $Z$ is smooth on $0\leq k<1$. Note that $Z(0)=1$, and
that
$\begin{split}Z(k)&\geq\frac{1}{2\pi}\int_{3\pi/2}^{2\pi}\frac{dx}{1+k\sin
x}\\\
&=\frac{1}{\pi\sqrt{1-k^{2}}}\arctan\sqrt{\frac{1+k}{1-k}}\to+\infty,\quad\textup{as
$k\to 1^{-}$}.\end{split}$
The existence then follows from the intermediate value theorem in calculus.
The uniqueness is due to the monotonicity of $Z$ on $[0,1)$, which is readily
seen by verifying $Z^{\prime}(0)=0$ and $Z^{\prime\prime}(k)>0$ on $[0,1)$. ∎
Lemma 7 can be easily generalized to the case of the product of a compact
hermitian manifold with $T^{k}$, $k\geq 3$. See the corollary below:
###### Corollary 9.
Let $M^{n}=N^{n-k}\times T^{k}$, $k\geq 3$, where $(N^{n-k},\omega_{N})$ is an
$(n-k)$-dimensional compact hermitian manifold. We denote
$\omega=\omega_{N}+\omega_{0}$, where $\omega_{0}$ is a constant metric on
$T^{k}$. Then, for any $1>\delta>0$, there exists a smooth $(n-2,n-2)$–form
$\psi\in\mathcal{P}(\omega)$ on $M$ such that
$\det\left(\omega^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\psi\right)=\delta\det(\omega^{n-1}).$
###### Proof.
Let
$\psi=\omega_{N}^{n-k}\wedge\varphi,$
where $\varphi$ is the $(k-2,k-2)$–form on $T^{k}$ obtained by Lemma 7, i.e.,
$\varphi\in\mathcal{P}(\omega_{0})$ satisfies that such that
$\det\left(\omega_{0}^{k-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\right)=\delta\det(\omega_{0}^{k-1}).$
Then, obviously $\psi$ satisfies the requirement. ∎
### 2.2. Kähler case
In this subsection, we shall prove Theorem 2. Observe that it is sufficient to
prove the following lemma.
###### Lemma 10.
Let $(X,\omega_{0})$ be a compact Kähler manifold. Consider
$\det\left(\omega_{0}^{n-1}+\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\right)=C_{1}\det\omega_{0}^{n-1},$
where $\varphi\in\mathcal{P}(\omega_{0})$, and $C_{1}>0$ is a constant. If
$C_{1}\geq 1$, then
$\sqrt{-1}\partial\bar{\partial}\varphi=0.$
###### Proof.
By a direct calculation, since $\omega_{0}$ is Kähler, we have
$\displaystyle\int_{X}(\omega_{0}^{n-1})^{i\bar{j}}\Big{(}\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\Big{)}_{i\bar{j}}\,\omega_{0}^{n}=n\int_{X}\omega_{0}\wedge\Big{(}\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\Big{)}=0.$
Similar to the torus case, we apply the arithmetic–geometric mean inequality
to obtain
(2.15)
$\begin{split}C_{1}^{1/n}&=\left[\frac{\det(\omega_{\varphi}^{n-1})}{\det(\omega_{0}^{n-1})}\right]^{1/n}\\\
&\leq
1+\frac{1}{n}\sum_{i,j}(\omega_{0}^{n-1})^{i\bar{j}}\Big{(}\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi\Big{)}_{i\bar{j}}.\end{split}$
Integrating over $X$ with respect to $\omega_{0}$ and using first equality
yields that
$C_{1}^{1/n}\int_{X}\omega_{0}^{n}\leq\int_{X}\omega_{0}^{n}.$
This shows that $C_{1}=1$ and we must have a pointwise equality in (2.15).
This forces that
$\frac{\sqrt{-1}}{2}\partial\bar{\partial}\varphi=0.$
∎
## 3\. Openness
Let $(X,\eta)$ be a Kähler manifold, and $\omega_{0}$ be a Hermitian metric on
$X$. Given $f\in C^{\infty}(X)$, we would like to study the solution
$\varphi\in\mathcal{P}(\omega_{0})$ of the following equation
(3.1)
$\frac{\omega_{\varphi}^{n}}{\omega_{0}^{n}}=\frac{e^{f}}{V}\int_{X}\omega_{\varphi}^{n}.$
Here $\omega_{\varphi}$ is a positive $(1,1)$–form on $X$ such that
$\omega_{\varphi}^{n-1}=\omega_{0}^{n-1}+(\sqrt{-1}/2)\partial\bar{\partial}\varphi,$
and
$V=\int_{X}\omega_{0}^{n}.$
Equation (3.1) is the same as (1.4), which is equivalent to the form-type
Calabi–Yau equation (1.5). A compatibility condition for (3.1) is
$\int_{X}e^{f}\omega_{0}^{n}=V.$
In what follows, we fix $k$ to be an integer greater than $n+3$, and fix a
real number $\alpha$ with $0<\alpha<1$. We denote by $C^{k,\alpha}(X)$ the
usual Hölder space of real-valued functions on $X$. Recall that
$\mathcal{F}^{k,\alpha}(X)=\left\\{g\in
C^{k,\alpha}(X);\int_{X}e^{g}\,\omega_{0}^{n}=V\right\\},$
which is a hypersurface in the Banach space $C^{k,\alpha}(X)$. For any $\psi$
contained in the intersection of $\mathcal{P}(\omega_{0})$ and
$C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$,
$M(\psi)\equiv\log\frac{\omega_{\psi}^{n}}{\omega_{0}^{n}}-\log\left(\frac{1}{V}\int_{X}\omega_{\psi}^{n}\right)\in\mathcal{F}^{k,\alpha}(X).$
By the map $M$, equation (3.1) can be rewritten as
$M(\varphi)=f.$
To prove Theorem 3, we first compute the linearization of $M$.
###### Proposition 11.
Let $G(\varphi)=\omega_{\varphi}^{n}$ for all
$\varphi\in\mathcal{P}(\omega_{0})$, and denote by $G_{\varphi}$ the Fréchet
derivative of $G$ at $\varphi$. Then, given
$\varphi\in\mathcal{P}(\omega_{0})$, we have
$G_{\varphi}(\psi)=\frac{n\sqrt{-1}}{2(n-1)}\partial\bar{\partial}\psi\wedge\omega_{\varphi},$
for all $\psi\in C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$.
###### Proof.
For any real $(n-2,n-2)$–form $\psi$,
$\displaystyle G_{\varphi}(\psi)$
$\displaystyle=\left.\frac{d}{ds}\left(\omega_{\varphi+s\psi}^{n}\right)\right|_{s=0}$
(3.2)
$\displaystyle=n\omega_{\varphi}^{n-1}\wedge\left.\frac{d}{ds}(\omega_{\varphi+s\psi})\right|_{s=0}$
$\displaystyle=\left.\frac{d}{ds}(\omega^{n-1}_{\varphi+s\psi})\right|_{s=0}\wedge\omega_{\varphi}+\omega_{\varphi}^{n-1}\wedge\left.\frac{d}{ds}(\omega_{\varphi+s\psi})\right|_{s=0}$
(3.3)
$\displaystyle=(\sqrt{-1}/2)\partial\bar{\partial}\psi\wedge\omega_{\varphi}+\omega_{\varphi}^{n-1}\wedge\left.\frac{d}{ds}(\omega_{\varphi+s\psi})\right|_{s=0}.$
Comparing (3.2) with (3.3), we obtain that
$G_{\varphi}(\psi)=\frac{n}{n-1}(\sqrt{-1}/2)\partial\bar{\partial}\psi\wedge\omega_{\varphi}.$
∎
###### Corollary 12.
For any $\varphi\in\mathcal{P}(\omega_{0})$, the Fréchet derivative of $M$ at
$\varphi$ is given by
$M_{\varphi}(\psi)=\frac{n(\sqrt{-1}/2)\partial\bar{\partial}\psi\wedge\omega_{\varphi}}{(n-1)\omega_{\varphi}^{n}}-\frac{n\int_{X}(\sqrt{-1}/2)\partial\bar{\partial}\psi\wedge\omega_{\varphi}}{(n-1)\int_{X}\omega_{\varphi}^{n}},$
for all $\psi\in C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$.
Next, we recall the Local Surjectivity Theorem (see [13, p. 175 and p. 108],
for example).
###### Theorem (Local Surjectivity Theorem).
Let $\mathcal{E}$ and $\mathcal{F}$ be Banach manifolds, and
$U\subset\mathcal{E}$ be an open subset. If $\mathfrak{F}:U\to\mathcal{F}$ is
a $C^{1}$ map, and $\mathfrak{F}_{\xi}\equiv D\mathfrak{F}(\xi)$ is onto from
the tangent space $T_{\xi}\mathcal{E}$ to the tangent space
$T_{\mathfrak{F}(\xi)}\mathcal{F}$, then $\mathfrak{F}$ is locally onto; that
is, there exist open neighborhoods $U_{1}$ of $\xi$ and $V_{1}$ of
$\mathfrak{F}(\xi)$ such that $\mathfrak{F}|_{U_{1}}:U_{1}\to V_{1}$ is onto.
Thus, to show Theorem 3, it suffices to show that the linearization
$M_{\varphi}$ is surjective from $C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))$ to
$T_{f}\mathcal{F}^{k,\alpha}(X)$, which denotes the tangent space of
$\mathcal{F}^{k,\alpha}(X)$ at $f$. Now let us introduce the space
$\mathcal{E}^{k,\alpha}(X)=\left\\{h\in
C^{k,\alpha}(X);\int_{X}h\,\omega_{\varphi}^{n}=0\right\\}.$
Note that $\mathcal{E}^{k,\alpha}(X)$ is itself a Banach space, as a closed
subspace in $C^{k,\alpha}(X)$. There is another point of view: We can define
an equivalence relation on the elements in $C^{k,\alpha}(X)$ by
$h\sim g\quad\mbox{if and only if $h-g\equiv$ some constant.}$
In this regard, $\mathcal{E}^{k,\alpha}(X)=C^{k,\alpha}(X)/\sim$. Observe that
$T_{f}\mathcal{F}^{k,\alpha}(X)=\mathcal{E}^{k,\alpha}(X).$
To prove the surjectivity of $M_{\varphi}$, we consider a special class of the
$(n-2,n-2)$–forms, that is,
(3.4) $\psi=u\eta^{n-2},\qquad\textup{where
$u\in\mathcal{E}^{k+2,\alpha}(X)$}.$
We recall that $\eta$ is the Kähler metric on $X$. For simplicity we denote
$L(u)=M_{\varphi}(u\eta^{n-2}).$
Then, by Corollary 12,
(3.5)
$Lu=\frac{n(\sqrt{-1}/2)\partial\bar{\partial}u\wedge\eta^{n-2}\wedge\omega_{\varphi}}{(n-1)\omega_{\varphi}^{n}}-\frac{n\int_{X}(\sqrt{-1}/2)\partial\bar{\partial}u\wedge\eta^{n-2}\wedge\omega_{\varphi}}{(n-1)\int_{X}\omega_{\varphi}^{n}}.$
We shall prove the following result:
###### Lemma 13.
Let $k\geq n+4$, and $0<\alpha<1$. For any $h\in\mathcal{E}^{k,\alpha}(X)$,
there exists a unique function $u\in\mathcal{E}^{k+2,\alpha}(X)$ satisfying
that
(3.6) $Lu=h.$
Lemma 13 implies that
$M_{\varphi}:C^{k+2,\alpha}(\Lambda^{n-2,n-2}(X))\to\mathcal{E}^{k,\alpha}(X)$
is surjective, and hence, Theorem 3 follows.
The rest of this section is devoted to prove Lemma 13. We denote by
$W^{k,p}(\Omega,\omega_{\varphi})$ the usual Sobolev space with respect to
$\omega_{\varphi}$ on a domain $\Omega$ in $X$. In the rest of this section,
we may denote $W^{k,p}(\Omega)=W^{k,p}(\Omega,\omega_{\varphi})$ for
simplicity; furthermore, when $\Omega=X$, we abbreviate
$W^{k,p}=W^{k,p}(X)=W^{k,p}(X,\omega_{\varphi})$. Notice that
$W^{0,2}(X)\equiv L^{2}(X)$.
We introduce the following spaces:
$\mathcal{H}=\left\\{v\in
W^{1,2}(X);\int_{X}v\;\omega_{\varphi}^{n}=0\right\\},$
and
$\mathcal{L}=\left\\{v\in
L^{2}(X);\int_{X}v\;\omega_{\varphi}^{n}=0\right\\}.$
Clearly, $\mathcal{H}$ and $\mathcal{L}$ are Hilbert spaces, as closed
subspaces in $W^{1,2}(X)$ and $L^{2}(X)$, respectively. We define a bilinear
map $A:\mathcal{H}\times\mathcal{H}\to\mathbb{R}$ by
$\begin{split}A(u,v)&=\frac{n\sqrt{-1}}{4(n-1)}\int_{X}\eta^{n-2}\wedge\omega_{\varphi}\wedge\big{(}\partial
u\wedge\bar{\partial}v+\partial v\wedge\bar{\partial}u\big{)}\\\
&\quad+\frac{n\sqrt{-1}}{4(n-1)}\int_{X}v\eta^{n-2}\wedge\big{(}\partial
u\wedge\bar{\partial}\omega_{\varphi}+\partial\omega_{\varphi}\wedge\bar{\partial}u\big{)}.\end{split}$
###### Definition 14.
Given $h\in\mathcal{L}$, we say that $u\in\mathcal{H}$ is a _weak_ solution of
the equation
(3.7) $-Lu=h,$
if $u$ satisfies that
(3.8) $A(u,v)=\int_{X}hv\,\omega_{\varphi}^{n}\equiv\langle
h,v\rangle_{L^{2}},\qquad\textup{for all $v\in\mathcal{H}$}.$
Let us remark that, if $u$ is a _classical_ solution of (3.7), i.e., $u\in
C^{2}(X)$, then one can obtain (3.8) by integrating (3.7) by parts with
respect to $\omega_{\varphi}^{n}$. Conversely, we have the following result:
###### Proposition 15.
If $u\in C^{3}(X)$ satisfies (3.8) for some $h\in C^{1}(X)\cap\mathcal{L}$,
then
$-Lu=h.$
###### Proof.
First, we claim the following fact: If $\chi\in C^{1}(X)$ satisfy that
(3.9) $\int_{X}\chi v\,\omega_{\varphi}^{n}=0,\qquad\textup{for all
$v\in\mathcal{H}$},$
then $\chi$ is a constant function on $X$. To see this, let
$v=\chi-\frac{\int_{X}\chi\omega_{\varphi}^{n}}{\int_{X}\omega_{\varphi}^{n}};$
then $v\in\mathcal{H}$ and (3.9) implies that
$\int_{X}|v|^{2}\omega_{\varphi}^{n}=0.$
This proves the claim. It follows that
$\frac{n(\sqrt{-1}/2)\partial\bar{\partial}u\wedge\eta^{n-2}\wedge\omega_{\varphi}}{(n-1)\omega_{\varphi}^{n}}-h=\mbox{some
constant}.$
Thus, integrating with respect to $\omega_{\varphi}^{n}$ yields the result. ∎
The following weak maximum principle is similar to that on a domain in the
Euclidean space (see, for example, Gilbarg–Trudinger [7, p. 179]). Proposition
16 is trivial, if $d\omega_{\varphi}=0$.
###### Proposition 16.
Suppose that $u\in\mathcal{H}$ satisfies
(3.10) $A(u,v)=0,\qquad\textup{for all $v\in\mathcal{H}$}.$
Then, $u=0$.
###### Proof.
It suffices to prove $\sup_{X}u\leq 0$, as one can then replace $u$ by $-u$.
(Here $\sup$ stands for the essential supremum.) Suppose the contrary. Take a
constant $\delta$ such that $0<\delta<\sup_{X}u$, and define
(3.11)
$v=(u-\delta)^{+}-\frac{\int_{X}(u-\delta)^{+}\omega_{\varphi}^{n}}{\int_{X}\omega_{\varphi}^{n}},$
in which $(u-\delta)^{+}=\max\\{u-\delta,0\\}$. Then, $v\in\mathcal{H}$ and
$dv=d(u-\delta)^{+}=\begin{cases}du,&\textup{if $u>\delta$},\\\ 0,&\textup{if
$u\leq\delta$}.\end{cases}$
Let us denote by $\Gamma$ the compact support of $dv$. Then, we obtain by
(3.10) and metric equivalence of $\eta$, $\omega_{\varphi}$, that
$\|\nabla v\|^{2}_{L^{2}}=\int_{\Gamma}|\nabla v|^{2}\omega_{\varphi}^{n}\leq
C\int_{\Gamma}|v||\nabla v|\omega_{\varphi}^{n}.$
Here and below, we denote by $C$ a generic positive constant depending only on
$\eta$, $\omega_{\varphi}$, and $n$. Apply Hölder’s inequality to get
(3.12) $\|\nabla v\|_{L^{2}}\leq C\|v\|_{L^{2}(\Gamma)}.$
On the other hand, combining the Sobolev inequality and Poincaré inequality
yields that
(3.13) $\|v\|_{L^{2n/(n-1)}}\leq C(\|\nabla v\|_{L^{2}}+\|v\|_{L^{2}})\leq
C\|\nabla v\|_{L^{2}}.$
Hence, by (3.12) and (3.13),
$\|v\|_{L^{2n/(n-1)}}\leq C\|v\|_{L^{2}(\Gamma)}\leq
C|\Gamma|^{\frac{1}{2n}}\|v\|_{L^{2n/(n-1)}},$
in which $|\Gamma|$ denotes the measure of $\Gamma$ with respect to
$\omega_{\varphi}$. It follows that
(3.14) $|\Gamma|=|\\{u>\delta,|du|>0\\}|\geq C^{-1}.$
Letting $\delta$ tend to $\sup u$ implies that $|du|>0$ on a set of positive
measure in $\\{x\in X;u(x)=\sup_{X}u\\}$, which is evidently impossible by
Lemma 7.7 in Gilbarg–Trudinger [7, p. 152]. This proves that $\sup u\leq 0$. ∎
The next two propositions are standard, for which we need the Lax–Milgram
Theorem (see Evans [2, p. 297], for example) and the Fredholm alternative (see
[2, p. 641] for example). We include them here for completeness.
###### Theorem (Lax–Milgram Theorem).
Let $H$ be a real Hilbert space, and $I:H\times H\to\mathbb{R}$ be a bilinear
mapping. Assume that, there exist positive constants $\beta$ and $\mu$ such
that
$|I(u,v)|\leq\beta\|u\|\|v\|,\qquad\textup{for all $u,v\in H$},$
and
$I(v,v)\geq\mu\|v\|^{2},\qquad\textup{for all $v\in H$}.$
Then, for any bounded linear functional $f$ on $H$, there exists a unique
element $u\in H$ satisfying that
$I(u,v)=f(v)\qquad\textup{for all $v\in H$}.$
###### Theorem (Fredholm alternative).
Let $E$ be a Banach space and $K:E\to E$ be a compact linear operator. Then,
$\textup{$\ker(I-K)=\\{0\\}$ \quad if and only if \quad$\textup{Im}(I-K)=E$},$
where $I:E\to E$ is the identity operator.
###### Proposition 17.
There exists a nonnegative constant $\gamma$, depending on $\omega_{\varphi}$
and $\eta$, such that for any $h\in\mathcal{L}$, there exists a unique weak
solution $u\in\mathcal{H}$ of
(3.15) $-L_{\gamma}u\equiv-Lu+\gamma u=h.$
That is, the function $u$ satisfies
(3.16) $A(u,v)+\gamma\langle u,v\rangle_{L^{2}}=\langle
h,v\rangle_{L^{2}},\qquad\textup{for all $v\in\mathcal{H}$}.$
###### Proof.
We have, by the metric equivalence of $\eta$ and $\omega_{\varphi}$,
$|A(u,v)|\leq\beta\|u\|_{W^{1,2}}\|v\|_{W^{1,2}},$
and
$A(u,u)+\gamma\|u\|_{L^{2}}\geq\mu\|u\|_{W^{1,2}}.$
Here $\beta>0$, $\gamma\geq 0$, and $\mu>0$ are constants depending only on
$\eta$ and $\omega_{\varphi}$. The result then follows from applying
Lax–Milgram Theorem to
$I(u,v)=A(u,v)+\gamma\langle u,v\rangle_{L^{2}},\qquad\textup{for all
$u,v\in\mathcal{H}$}.$
∎
###### Proposition 18.
For any $h\in\mathcal{L}$, there exists a unique weak solution
$u\in\mathcal{H}$ of
$-Lu=h.$
###### Proof.
By Proposition 17 we can define a map
$L_{\gamma}^{-1}:\mathcal{L}\to\mathcal{H}$ as follows: For each
$f\in\mathcal{L}$, we define $L_{\gamma}^{-1}(f)$ to be the unique function
$w\in\mathcal{H}$ satisfying
$A(w,v)+\gamma\langle w,v\rangle_{L^{2}}=\langle f,v\rangle_{L^{2}}.$
Clearly, $L_{\gamma}^{-1}$ is linear, and is a compact operator from
$\mathcal{L}$ to $\mathcal{L}$, in view of Rellich Theorem. To prove the
result, it suffices to show that, for a given $h\in\mathcal{L}$, there exists
a unique $u\in\mathcal{L}$ satisfying that
$u=L_{\gamma}^{-1}(h+\gamma u).$
Equivalently, we need to solve a unique $u\in\mathcal{L}$ for the following
equation:
$(I-\gamma L_{\gamma}^{-1})u=L^{-1}_{\gamma}h.$
To invoke the Fredholm alternative, we turn to the kernel of $(I-\gamma
L_{\gamma}^{-1})$ in $\mathcal{L}$, i.e.,
$\\{u\in\mathcal{L};\,u-\gamma L_{\gamma}^{-1}u=0\\}.$
This is equivalent to investigate the function $u\in\mathcal{H}$ such that
$A(u,v)=0\qquad\textup{for all $v\in\mathcal{H}$}.$
By Proposition 16, $u=0$. The result then follows from the Fredholm
alternative. ∎
Now we are in a position to prove Lemma 13:
###### Proof of Lemma 13.
The uniqueness of (3.6) is an immediate consequence of Proposition 18, since a
$C^{2}$ solution of (3.6) is in particular a weak solution of $-Lu=-h$.
Given $h\in C^{k,\alpha}(X)$, we have $h\in W^{k,2}(X)$, since $X$ is compact.
Then, by Proposition 18, equation (3.6) has a weak solution $u\in W^{1,2}(X)$.
Then, we obtain
$u\in W^{k+2,2}(X),$
by the local regularity theorem (see, for example, Evans [2, p. 314] or
Gilbarg–Trudinger [7, p. 186]). Since $k\geq n+4$, $k-2n/2-1\geq 3$. We apply
the Sobolev imbedding theorem to obtain that
$u\in C^{3}(X).$
By Proposition 15, $u$ is the classical solution for (3.6). It follows from
the bootstrap argument ([7, p. 109]) that
$u\in C^{k+2,\alpha}(X).$
∎
## References
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|
arxiv-papers
| 2009-08-05T02:26:20 |
2024-09-04T02:49:04.445047
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jixiang Fu, Zhizhang Wang and Damin Wu",
"submitter": "Jixiang Fu",
"url": "https://arxiv.org/abs/0908.0577"
}
|
0908.0578
|
# Dipolar Evolution in a Coronal Hole Region
Shuhong Yang, Jun Zhang Key Laboratory of Solar Activity, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
[shuhongyang;zjun]@ourstar.bao.ac.cn Juan Manuel Borrero Max Planck Institut
for Solar System Research, Max Planck Strasse 2, 37191, Katlenburg-Lindau,
Germany borrero@mps.mpg.de
###### Abstract
Using observations from the SOHO, STEREO and Hinode, we investigate magnetic
field evolution in an equatorial coronal hole region. Two dipoles emerge one
by one. The negative element of the first dipole disappears due to the
interaction with the positive element of the second dipole. During this
process, a jet and a plasma eruption are observed. The opposite polarities of
the second dipole separate at first, and then cancel with each other, which is
first reported in a coronal hole. With the reduction of unsigned magnetic flux
of the second dipole from 9.8$\times$1020 Mx to 3.0$\times$1020 Mx in two
days, 171 Å brightness decreases by 75% and coronal loops shrink obviously. At
the cancellation sites, the transverse fields are strong and point directly
from the positive elements to the negative ones, meanwhile Doppler red-shifts
with an average velocity of 0.9 km s-1 are observed, comparable to the
horizontal velocity (1.0 km s-1) derived from the cancelling island motion.
Several days later, the northeastern part of the coronal hole, where the
dipoles are located, appears as a quiet region. These observations support the
idea that the interaction between the two dipoles is caused by flux
reconnection, while the cancellation between the opposite polarities of the
second dipole is due to the submergence of original loops. These results will
help us to understand coronal hole evolution.
Sun: magnetic fields — Sun: evolution — Sun: photosphere — Sun: corona — Sun:
UV radiation — Sun: Doppler shifts
## 1 Introduction
Coronal holes (CHs) are dark and void areas on the Sun if observed with X-ray
(Underwood & Muney 1967) and EUV lines (Reeves & Parkinson 1970). They are low
density and temperature regions compared with the quiet Sun (QS) (Munro &
Withbroe 1972; Harvey 1996; Chiuderi Drago et al. 1999). In CHs, magnetic
fields are dominated by one polarity and open magnetic lines are concentrated
(Bohlin 1977). These open magnetic lines extend to the interplanetary space
along which plasma escapes, giving rise to fast solar wind (Krieger et al.
1973; Zirker 1977; Wang et al. 1996; Harvey & Recely 2002; Tu et al. 2005).
However, magnetic fields are not exclusively unipolar in CHs and there also
exist closed coronal loops besides the open flux (eg. Levine 1977; Zhang et
al. 2006). Wiegelmann & Solanki (2004) computed some properties of coronal
loops in CHs and the QS. They found that high and long closed loops in CHs are
extremely rare, whereas short and low-lying loops are almost as abundant as in
the QS. Fisk (2005) predicted that a CH is a region with a local minimum in
the rate of emerging dipoles. This prediction was supported by Zhang et al.
(2006), whose results reveal that the dipole emergence rate in the QS is 4.3
times as large as that in the CH. According to Fisk & Schwadron (2001) and
Fisk (2005), dipoles can transport open flux on the solar surface through
magnetic flux reconnection and result in the CH evolution by affecting
magnetic field distribution and characters.
Magnetic flux cancellation is an observational phenomenon of magnetic flux
disappearance in the encounter of two magnetic elements of opposite polarities
(Martin et al. 1985; Livi et al. 1985; Zhang et al. 2001). Zwaan (1978, 1987)
illustrated three modes for removal of magnetic flux with different polarities
from the photosphere: (1) If two poles are still connected by coronal loops,
then the disappearance of the magnetic flux can be caused by retraction of
flux loops to below the photosphere; (2) When two poles with no initial
connection encounter, magnetic reconnection, creating both $\Omega$$-$shaped
and U$-$shaped loops, is required to remove the magnetic flux from the
photosphere. If reconnection occurs below the photosphere, the newly formed
U$-$shaped loop is pulled out of the photosphere; (3) If reconnection takes
place above the photosphere, the $\Omega$$-$shaped loop submerges (retracts)
below the photosphere. Magnetic reconnection may not be necessary for forming
the emerging U$-$shaped loop if two poles connect below the photosphere
(Parker 1984; Lites et al. 1995).
Generally, dipoles appear to rise from below the solar surface, separate and
dissipate (Wallenhorst & Topka 1982; Liggett & Zirin 1983). However, some
dipoles do not decouple from subsurface fields (Zirin 1985). An excellent
example exhibiting the submergence of part of an active region was reported by
Rabin et al. (1984). Another example displaying the submergence of entire
region is the disappearance of a sunspot group, studied by Zirin (1985) with
videomagnetograms and H$\alpha$ observations.
When collision occurs between opposite polarities, transverse fields and
Doppler velocities at the cancellation sites have been studied by many
authors. Zhang et al. (2009) found that transverse fields connecting
cancelling magnetic elements are formed. Harvey et al. (1999) estimated
vertical velocity of magnetic flux descent ranging from about 0.3 to 1.0 km
s-1, and Yurchyshyn & Wang (2001) observed an upflow of about 0.6 km s-1 at a
cancellation site. The first direct observational evidence of flux retraction
in cancelling magnetic features was presented by Chae et al. (2004). They
found that the magnetic fields were nearly horizontal at the place where two
opposite polarities cancelled each other. In addition, they observed
significant magnetic flux submergence of about 1.0 km s-1 near the polarity
inversion line. Recently, magnetic filed properties at the cancellation sites
were investigated by Kubo & Shimizu (2007) based on more collision events.
Due to the restriction of observations, not all the properties of CHs are well
known, such as CH formation and decay. These CH properties are related to
magnetic field structures and evolution, the key characters to understand most
solar phenomena, so it is still important for us to investigate magnetic
fields in CHs with high spatial and temporal resolution data, especially the
vector fields and other plentiful information from the Hinode (Kosugi et al.
2007). In this work, we study the evolution process of two dipoles in an
equatorial CH, using observations from the Solar and Heliospheric Observatory
(SOHO; Domingo et al. 1995), the Solar Terrestrial Relations Observatory
(STEREO; Howard et al. 2008; Kaiser et al. 2008) and the Hinode. We describe
the observations in Sect. 2 and data analysis in Sect. 3. The two parts of
Sect. 4 show respectively the interaction between the two dipoles and the
cancellation between the opposite polarities of the second dipole. The
conclusions and discussion are presented in Sect. 5.
## 2 Observations
The CH shown in Fig. 1 was observed with the Michelson Doppler Imager (MDI;
Scherrer et al. 1995) and the Extreme-ultraviolet Imaging Telescope (EIT;
Delaboudinière et al. 1995) on SOHO, the Extreme Ultra Violet Imager (EUVI;
Howard et al. 2008), part of the Sun-Earth Connection Coronal and Heliospheric
Investigation (SECCHI; Howard et al. 2008) aboard STEREO, combined with the
Spectro-Polarimeter (SP; Lites et al. 2001) and the Narrowband Filter Imager
(NFI; Kosugi et al. 2007), two components of the Solar Optical Telescope (SOT;
Ichimoto et al. 2008; Shimizu et al. 2008; Suematsu et al. 2008; Tsuneta et
al. 2008) on board Hinode.
We adopt the MDI, EIT and EUVI full-Sun observations between 08:00 UT 02 March
and 12:00 UT 07 March 2008. The data sets used in this study are summarized in
Table 1. MDI provides full-disk longitudinal magnetograms with pixel sampling
of 1″.97. The cadence of MDI magnetograms was 1 minute during the periods of
11:35 UT – 14:36 UT 02 March and 00:57 UT 03 March – 12:00 UT 07 March, and 96
minutes during the other periods. EIT observed the CH with a pixel size of
2″.63 and a cadence of 12 minutes in Fe XII 195 Å. STEREO A and B
simultaneously observed full-Sun with a 1″.59 pixel resolution and a 2.5-min
cadence in Fe IX 171 Å. On the early of 05 March 2008, when the CH was located
at disk center viewed from the Earth, the STEREO A and B were 44$\arcdeg$
ahead and 48$\arcdeg$ behind the Earth. In order to maintain the CH close to
disk center in each image, we employ STEREO B observations before 00:00 UT 05
March, and STEREO A observations after that.
Hinode SP and NFI observed only part of the CH, but covered majority of our
target during 03 March and 05 March, 2008. The SP, which provides full Stokes
profiles (I, Q, U and V) of Fe 6301.5 Å ($g_{eff}=1.67$) and 6302.5 Å
($g_{eff}=2.5$) lines, scanned the target along East$-$West direction in the
fast map mode with a step of 0″.295, and the pixel size along the slit is
0″.32. The NFI obtained the Stokes I and V images at an offset wavelength of
$-$172 mÅ from the center of the Na 5896 Å line with a cadence of 3 minutes
and a pixel resolution of 0″.16.
## 3 Data analysis
At first, all the images from the EIT, EUVI and NFI are prepared by applying
the standard processing routines, including flat field correction, dark
current and pedestal subtraction, cosmic ray removal, et al.. Then we rotate
the target in the MDI, EIT and EUVI images to the central meridian. After that
we re-scale and uniform their pixel sizes. In order to remove the drift due to
the correlation tracker motion (Shimizu et al. 2008), NFI V/I (i.e. Stokes V
signals divided by Stokes I signals obtained at the same time) images are
coaligned with each other by using the image cross-correlation method. Then
the MDI magnetograms were aligned to the NFI V/I images by cross-correlating
specific features after being re-scaled. The same coalignment method is also
applied to match the EIT and EUVI images. Since the MDI and the closest EIT
images are from the same satellite and can be coalined easily, we think the
MDI, EIT, EUVI and NFI images are coaligned well now. Due to the high
conductivity, the coronal plasma is frozen into the magnetic field.
Consequently, the emitting plasma structures outline the magnetic field lines
(Wiegelmann & Solanki 2004). So in order to identify loop connections well
(see Kano & Tsuneta 1995; Kubo & Shimizu 2007), the EUVI structures are
contrast enhanced by using an unsharp mask filter (refer to Feng et al. 2007).
The spectropolarimetric data from Hinode/SP are processed with the routine _sp
-prep.pro_ available in the _Solar Software (SSW)_ package. This routine
performs standard calibrations such as flat-fielding, dark current correction,
polarimetric and instrumental calibration (Ichimoto et al. 2008; Kubo et al.
2008). The calibrated Stokes profiles are analyzed using the VFISV inversion
code (Borrero et al. 2009). This code uses the Milne$-$Eddington solution for
the radiative transfer equation to produce synthetic Stokes profiles that are
then compared with the observed ones. The free parameters of the model are:
magnetic field strength $B$, inclination ($\gamma$) and azimuth ($\phi$) of
the magnetic field vector in the observer’s reference frame, line-of-sight
(LOS) velocity $V_{los}$, continuum to core absorption coefficient $\eta_{0}$,
Doppler width $\Delta\lambda_{D}$, source function and source function
gradient $S_{0}$ and $S_{1}$, and finally the filling factor of the magnetic
component $\alpha_{mag}$. We do not consider the damping parameter and
macroturbulent velocities as free parameters since their effect can be
mimicked by the other thermodynamic parameters and they do not affect the
determination of the important quantities such as the magnetic field vector
and velocity. With this, we have a total number of 9 free parameters, which
are iteratively modified (using the Levenberg$-$Marquardt non-linear least
squares fitting technique) in other to achieve a better match between
synthetic and observed profiles.
The non-magnetic component is obtained by averaging the pixels of the map that
possess a polarization signal below the noise level (1.2$\times$10-3 _I c_,
where _I c_ is the continuum intensity). The same non-magnetic component is
used in the inversion of all pixels in the map. Our approach is therefore
slightly different from that of Orozco Suárez et al. (2007), who employed a
local (average around each inverted pixel) non-magnetic component. We have
also repeated our inversions following that approach, but our results did not
change. Given the small amount of scattered light known to exist within the SP
instrument on-board Hinode (Danilovic et al. 2008), the amount of non-magnetic
component $(1-\alpha_{mag})$ can be either interpreted as true non-magnetic
unresolved component inside the SP resolution element or as a degradation of
the polarization signal due to diffraction (Orozco Suárez et al. 2007).
The zero LOS velocity $V_{los}$ has been obtained by calculating the
convective blue-shift in the Fe I 6302.5 Å line using the Fourier Transform
Spectrometer atlas at disk center (FTS; Brault & Neckel 1987) and also by
subtracting the solar gravitational red-shift. Since the observed region was
located close to disk center no further corrections due to solar rotation were
necessary.
In the vector field measurements based on the Zeeman effect, there exists a
180° ambiguity in determining the field azimuth, $\phi$. However, it is not
unresolvable. Potential field approximation is one of the fairly acceptable
methods to resolve the ambiguity (Lites et al. 1995). We construct the
photospheric vector magnetic fields by computing the potential fields using
the SP longitudinal megnetogram, and the azimuth angles with 180° ambiguity
are disambiguated by being selected and determined to be close to that we have
constructed. Finally, the SP maps are coaligned with the NFI images by using
the SP longitudinal magnetogram.
By using the similar method introduced in Chae et al. (2007), we convert the
Na V/I signals to the LOS magnetic fields according to the temporally closest
SP data. A linear relation, $B_{los}$=$\beta$$\times$V/I, between the circular
polarization, V/I, and the line-of-sight field strength, $B_{los}$, is
applied. We determine the calibration coefficient, $\beta$, for each interval
partitioned by V/I=$-$0.07, $-$0.06, …, 0.09. The 18 values of $\beta$ range
from 4.0 kG to 12.3 kG and the mean calibration coefficient is 6.8 kG.
The CH boundary (see Fig. 1) is determined with the brightness gradient method
which was developed by Shen et al. (2006; see also Luo et al. 2008). In an
EUVI 284 Å image, the pixel value, b, varies in a range. For any value of b,
we can plot a contour and calculate the area, A, enclosed by it. The CH
boundary is at the place where _f_ =$\delta$_b_ $/$$\delta$_A_ =_f_ max.
## 4 Results
The CH in this study is dominated by the positive polarity (see the bottom
left panel in Fig. 1). Within a 90″$\times$90″ region (outlined by white
squares in Fig. 1), two dipoles emerge one by one. We focus on the interaction
between the two dipoles and the disappearance of the second dipole from 02
March to 07 March, 2008.
### 4.1 Interaction between the two dipoles
Figure 2 presents the interaction process between the two dipoles. Top panels
are time sequence of MDI magnetograms, which show the emergence and
interaction of the two dipoles. The middle panels exhibit the coronal response
in EUVI 171 Å line. Loop connections can be seen much more clearly in the
contrast enhanced 171 Å images (bottom panels). The positive element “A” of
the first dipole (denoted by arrows “1”) appeared obvious at 09:39 UT 02 March
and then grew larger, and brightening point appeared simultaneously at the
corresponding location as shown in the leftmost column. Then the negative
element “B” began to emerge and the dipole “1” reached its maximum size with
total unsigned LOS flux of 2.0$\times 10$20 Mx at 20:51 UT 02 March. At this
time, the loop connections between elements “A” and “B” are emphasized with
green curves in the contrast enhanced image at 20:47 UT 02 March. The second
dipole (indicated by arrows “2”) started to appear in the magnetogram from
20:51 UT 02 March with slightly distinguishable features and became quite
obvious at 22:27 UT 02 march. The magnetogram acquired at 00:03 UT on 03 March
shows that its positive element “C” was located in contact with element “B”.
Element “B” split into two segments during its disappearance process, as shown
in the magnetogram at 03:11 UT 03 March. Both two segments of “B” disappeared
completely at 08:22 UT 03 March. Element “A” moved toward “C” and merged with
“C” into “A$+$C”, still called element “C” by us considering the small size of
“A” compared to that of “C”. Dipole “2” continued emerging and reached its
maximum with total unsigned LOS flux of 9.8$\times 10$20 Mx at 19:01 UT 03
March (top right panel).
During the interaction process between the two dipoles, a jet and a plasma
eruption have been observed, as shown in Fig. 3. At the early emerging stage
of the dipole “2”, there was a small bright point (denoted by arrow “1”) at
the adjacent area of elements “B” and “C” (top left panel). Two and a half
minutes later, a jet (indicated by arrows “2”) rooting at the bright point was
observed. The lifetime of the jet is only 5 minutes. The second dipole
continued to emerge and another obvious brightening point (indicated by arrow
“3”) appeared at the contacted region of “B” and “C” (bottom left panel). At
23:46 UT, a cloud of plasma was disturbed (denoted by arrow “a”) and erupted
five minutes later (arrow “b”). After the eruption, a dimming area appeared
(arrow “c”).
### 4.2 Magnetic flux cancellation of the second dipole
Two elements of dipole “2” separated as flux continually emerged until 19:01
UT 03 March. Then they began to cancel with each other, as shown in Fig. 4.
Top five panels display the evolution of dipole “2” in five days. When the
dipole well developed, the flux of each polarity was almost concentrated. As
the cancellation began, each pole broke into several fragments. Then these
fragments with opposite polarities moved together and cancelled gradually.
Accompanying the cancellation process, the corresponding coronal region in 171
Å images became darker (middle panels), while the general loop connections
were not changed much (bottom panels). However, the loop systems became fewer
and fuzzier, and the length of loops became shorter. At 20:47 UT 03 March, the
amount of identifiable loops was 7 while there was no more than 4 at 06:23 UT
05 March. At 19:01 UT 03 March, when the dipole well developed, the length of
the longest loop was about 30 Mm. Two days later, most of the long coronal
loops were about 15 Mm. At 06:24 07 March, only small dispersed elements of
dipole “2” remained (bottom right panel in Fig. 1). The northeastern part of
the CH, where the dipoles were located, appeared as a quiet region and the
underlying magnetic fields evolved to mixed polarities (see the rectangle
region in Fig. 1).
Figure 5 shows the temporal variations of negative and positive magnetic flux
in MDI magnetograms and brightness in 171 Å images obtained from the ellipse
region in Fig. 4. In two days, both the negative and positive flux reduced
smoothly by 3.4$\times$1020 Mx at a steady cancellation rate of
0.7$\times$1019 Mx h-1. During this period, the brightness in 171 Å images
decreased by about 75%.
The Na V/I high spatial resolution magnetograms are used to study the
cancellation between two polarities of dipole “2” in details. Two obvious
cancellation processes are displayed in Fig. 6. In the circle region (top
panels), a negative flux island moved straightly toward the positive flux,
cancelled with it and disappeared totally at 14:29 UT 04 March. In the octagon
area (bottom panels), another flux island observed on 05 March also cancelled
with the positive flux. We measure the cancellation rates between the opposite
polarities of the second dipole in the calibrated Na V/I magnetograms (within
the rectangle area outlined in Fig. 4). It indicates that both the positive
and negative polarities decreased at an average rate of 0.8$\times$1019 Mx h-1
during the period of 11:20–16:17 on 04 March and of 0.7$\times$1019 Mx h-1
from 11:39 to 15:27 on 05 March. We also measure the disappearing rate of the
two small negative islands shown in Fig. 6. At the pre-cancellation stage, the
flux of the two islands is $-$0.4$\times$1019 Mx (outlined with circles in the
top panels) and $-$0.5$\times$1019 Mx (outlined with octagons in the bottom
panels), respectively. Both the two islands disappeared at a mean rate of
0.3$\times$1019 Mx h-1.
The negative island shown in the circle region in Fig. 6 moved toward the
positive island during its cancellation course. Its distance from the initial
site and apparent horizontal velocity at different time are presented in Fig.
7. The average horizontal velocity is 1.0 km s-1. Although the V/I
magnetograms have been coaligned with the cross-correlation method, there
still exists an uncertainty in determining the magnetic island position. A
position error (one pixel) introduces an error of the velocity of (one
pixel)/(time interval). In order to reduce the velocity error, we measure the
island position every 6 minutes. As the size of one pixel is 0″.16 and the
time interval is 6 minutes, the error becomes 0.3 km s-1.
Hinode$/$SP also observed the cancellation regions. The vector magnetic fields
and Doppler velocities derived from the SP data help us to investigate the
physical essence of magnetic field evolution at the cancellation sites. Figure
8 shows the appearance of three cancellation stages in small sub-regions
observed with SP. From left to right: longitudinal fields
($\alpha_{mag}Bcos\gamma$), transverse fields
($(\alpha_{mag})^{1/2}Bsin\gamma$), inclinations ($\gamma$) and Doppler
velocities ($V_{los}$). Top panels show the properties of four parameters at
the cancellation area at 11:24 UT 04 March. In the longitudinal magnetogram
(top left panel), the cancellation takes place at the area between the
positive and negative elements (outlined by the parallelogram), and the
transverse fields are strong and point directly from the positive island to
the negative one (second panel). The third panel shows magnetic field
inclinations where the black areas (inclination of 90$\arcdeg$) indicate
horizontal orientations and white areas (0$\arcdeg$ and 180$\arcdeg$) vertical
ones. At the polarity inversion line (dotted curve), the magnetic fields are
nearly horizontal. Within the parallelogram region in the Dopplergram (top
right panel), larger Doppler red-shifts with a mean downward velocity of 1.15
km s-1 are observed. The second and third rows exhibit other two cancellation
stages similar to the first one, and the average Doppler velocities within the
parallelogram regions are 0.84 km s-1 and 0.70 km s-1, respectively.
## 5 Conclusions and discussion
Using coordinated SOHO, STEREO and Hinode observations, we investigate the
evolution of two dipoles in a CH region. The negative element of the first
dipole disappears due to its interaction with the positive element of the
second dipole. During this process, a jet and a plasma eruption are observed.
Two opposite poles of the second dipole constantly emerge and separate at
first, and then cancel with each other. With the decrease of magnetic flux
caused by cancellation, the brightness in 171 Å images decreases and coronal
loops shrink obviously. At the cancellation sites of the second dipole, the
transverse fields are strong and point directly from the positive elements to
the negative ones. Larger Doppler red-shifts are also observed between the
cancelling elements. At last, the northeastern part of the CH, where the
dipoles are located, appears as a quiet region.
Based on the observational results in this study, we consider that the
interaction between the two dipoles is caused by flux reconnection, while that
between the opposite polarities of the second dipole is due to the submergence
of original loops. This phenomenon is first reported in a CH. To illustrate
the evolution process of the two dipoles, a series of cartoons are sketched
out (see Fig. 9). Dipole “1” appears first and dipole “2” emerges later at the
adjacent area of dipole “1” (Fig. 9a). We mark the positive and negative
elements of dipole “1” (“2”) with “A” and “B” (“C” and “D”), respectively.
Magnetic flux reconnection occurs between two groups of loops connecting the
opposite polarities. Magnetic loops are restructured to a configuration of
lower potential energy accompanied with energy release. Meanwhile, small loops
form and submerge, leading to an observational phenomenon, magnetic flux
cancellation. Element “B” totally disappears due to the cancellation with part
of element “C”. Two poles “C” and “D”, which are originally connected by flux
loops, draw back and cancel with each other due to flux submergence (Fig. 9c).
Observational evidences, e.g. the jet and the plasma eruption exhibited in
Fig. 3 indicate that the interaction between elements “B” and “C” represents a
flux reconnection process. When magnetic reconnection occurs, magnetic energy
is converted into thermal energy and kinetic energy. Then plasma jet forms and
ejects along the field line (Yokoyama & Shibata 1995). Accompanying the
reconnection, plasma eruption may also be formed due to the change of magnetic
configuration. However, we can not absolutely rule out the possibility that
the disappearance of element “B” is caused by the emergence of U$-$shaped
magnetic loops connecting “B” and “C” from below the photosphere (Parker 1984;
Lites et al. 1995), for we lack more relevant observational information.
Flux loops are always affected by magnetic buoyancy and tension of subsurface
field lines. When the tension gains the upper hand, the flux loops are pulled
back down by magnetic tension and submerge (Zirin 1985). The movies of 1-min
cadence MDI magnetograms and 3-min cadence Na V/I magnetograms and the results
revealed in Figs. 4$-$8 let us believe that the cancellation between “C” and
“D” is caused by the submergence of original loops. From Fig. 7, we obtain an
average horizontal velocity of 1.0 km s-1. When we assume flux loops
connecting the cancelling elements are approximately semicircular, the
vertical velocity of submergence is 1.0 km s-1. At the cancellation sites in
this study, we observe a mean downward velocity of 0.9 km s-1 (averaging the
values of 1.15 km s-1, 0.84 km s-1 and 0.70 km s-1 in the three stages), much
higher than that of the surrounding areas (Fig. 8), comparable to the velocity
obtained from the cancelling flux motion in Fig. 7 (1.0 km s-1) and consistent
with the velocities reported by Harvey et al. (1999) (0.3$-$1.0 km s-1) and
Chae et al. (2004) (1.0 km s-1).
Evolution of CHs refers to several aspects, such as CH formation and decay,
and the temporal variation of CH boundary. The key quantity for understanding
these aspects is magnetic field. Magnetic flux emergence, cancellation,
mergence and dispersion are the main forms of magnetic field evolution. They
are all found in the dipolar evolution process in the CH in this study. An
interesting cancelling form, i.e. submergence of initial loops after
emergence, is also observed for the first time in the CH. At the late stage of
the dipolar evolution, the area where the dipoles are located becomes mixed
polarities instead of unipolar fields, resulting in the change of the
overlying corona from a CH area to a quiet region (see the rectangle region in
Fig. 1). This confirms the result of Zhang et al. (2007) that one of the
signatures of decay of a CH is the disappearance of the magnetic flux
imbalance. These results enlighten us that, in order to understand the CH
evolution, it is important to study magnetic field evolution in CHs. In
particular, to investigate the evolution of dipoles may be an efficient
approach to understand CH decay and disappearance.
The authors are grateful to the anonymous referee for the constructive
comments and detailed suggestions to improve this manuscript. We acknowledge
the SOHO, STEREO and Hinode teams for providing the data. SOHO is a project of
international co-operation between ESA and NASA, and STEREO a NASA project.
Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as
domestic partner and NASA and STFC (UK) as international partners. It is
operated by these agencies in co-operation with ESA and NSC (Norway). This
work is supported by the National Natural Science Foundations of China
(G40674081, 40890161, 10703007, and 10733020), the CAS Project KJCX2-YW-T04,
the National Basic Research Program of China under grant G2006CB806303, and
the Young Researcher Grant of National Astronomical Observatories, Chinese
Academy of Sciences.
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Table 1: Data sets used in this study. Observation | Period | Cadence | Pixel Size | Field of View
---|---|---|---|---
| (UT) | (minutes) | (arcsec) | (arcsec2)
SOHO$/$MDI | 02 08:03 – 07 12:47 | 96 | 1.97 | full disk
| 02 11:35 – 02 14:36 | 1 | 1.97 | full disk
| 03 00:57 – 07 12:00 | 1 | 1.97 | full disk
SOHO$/$EIT(195Å) | 02 08:00 – 07 12:00 | 12 | 2.63 | full disk
STEREO-B$/$EUVI(171Å) | 02 08:00 – 05 00:00 | 2.5 | 1.59 | full disk
STEREO-A$/$EUVI(171Å) | 05 00:00 – 07 12:00 | 2.5 | 1.59 | full disk
Hinode$/$SP(Fe full Stokes) | 03 12:00 – 14:42 | 50aaScan time for one SP map is 12.5 minutes. | 0.32 | 59.04${\times}$162.30
| 04 11:20 – 15:55 | 50aaScan time for one SP map is 12.5 minutes. | 0.32 | 59.04${\times}$162.30
| 05 11:40 – 15:15 | 45aaScan time for one SP map is 12.5 minutes. | 0.32 | 59.04${\times}$162.30
Hinode$/$NFI(Na Stokes I, V) | 03 10:53 – 15:27 | 3 | 0.16 | 163.84${\times}$163.84
| 04 11:20 – 16:17 | 3 | 0.16 | 64.00${\times}$163.84
| 05 11:39 – 15:27 | 3 | 0.16 | 64.00${\times}$163.84
Figure 1: EIT 195 Å images (_top panels_) and MDI magnetograms (_bottom
panels_) showing the evolution of the CH. Black and blue curves delineate the
CH boundary derived from the EUVI 284 Å image obtained at 03:06:30 UT 02 March
2008. The squares outline the field-of-view of Figs. 2$-$4 and rectangles
enclose a QS region.
Figure 2: Interaction between two dipoles. From top to bottom: MDI
magnetograms, EUVI 171 Å images and corresponding contrast enhanced images.
Red and blue curves are contours of the positive ($+$70 G) and negative ($-$70
G) magnetic fields, while green curves emphasize and figure out loop
connections. Arrows “1” and “2” denote two dipoles emerging one after the
other. We denote the positive and negative elements of dipole “1” (“2”) with
arrows “A” and “B” (“C” and “D”), respectively.
Figure 3: EUVI images displaying a jet and a plasma eruption. “A”–“D” are
contours of two dipoles at $\pm$30 G levels from MDI magnetogram at 22:27 UT
02 March (first two panels) and at 00:03 UT 03 March (other panels). Arrows
“1” and “3” denote two brightening points, while arrows “2” indicate an EUV
jet. Arrows “a”–“c” show the different stages of a plasma eruption.
Figure 4: Similar to Fig. 2 but for magnetic flux cancellation between two
polarities of dipole “2”. Small square on the second MDI magnetogram outlines
the field-of-view of Fig. 6 and larger rectangle the area where the positive
and negative flux of the second dipole are measured in the Na V/I
magnetograms. Ellipse outlines the region where magnetic flux in MDI
magnetograms and brightness in 171 Å images are measured.
Figure 5: Temporal variations of negative and positive magnetic flux in MDI
magentograms and brightness in 171 Å images derived from the ellipse region in
Fig. 4.
Figure 6: Time sequence of Hinode$/$NFI Na V/I magnetograms displaying
magnetic flux cancellation. Circles and octagons show two cancellation events
on two days, respectively. Rectangles “I” “II” and “III” outline the locations
of SP maps from top to bottom in Fig. 8.
Figure 7: Temporal variations of distance and apparent horizontal velocity of
the negative island in the circle region shown in Fig. 6.
Figure 8: Appearance of cancellation regions observed by Hinode$/$SP. From
left to right: longitudinal fields, transverse fields, inclinations and
Doppler velocities. Inclinations of 90$\arcdeg$ correspond to magnetic fields
with horizontal orientations, and positive Doppler velocities to red-shifts.
Blue curves are contours of negative elements ($-$100 G) and other solid
curves outline the positive ones (+100 G). Dotted curves represent the
polarity inversion lines. Arrows denote the transverse fields and
parallelograms outline the areas where cancellations take place.
Figure 9: Cartoons illustrating evolution process of the two dipoles. (a) Pre-
interaction state of the dipoles. (b) Flux disappearance due to reconnection
accompanied with energy release. (c) Flux cancellation caused by submergence
of original loops connecting the dipolar elements. “A” and “B” (“C” and “D”)
represent the positive and negative elements of dipole “1” (“2”),
respectively. Asterisk marks magnetic flux reconnection and cross hatched
arrows indicate flux submergence.
|
arxiv-papers
| 2009-08-05T03:29:29 |
2024-09-04T02:49:04.451850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuhong Yang, Jun Zhang, and Juan Manuel Borrero",
"submitter": "Shuhong Yang",
"url": "https://arxiv.org/abs/0908.0578"
}
|
0908.0606
|
# Interacting new agegraphic dark energy in non-flat Brans-Dicke cosmology
Ahmad Sheykhi 111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
We construct a cosmological model of late acceleration based on the new
agegraphic dark energy model in the framework of Brans-Dicke cosmology where
the new agegraphic energy density $\rho_{D}=3n^{2}m^{2}_{p}/\eta^{2}$ is
replaced with $\rho_{D}={3n^{2}\phi^{2}}/({4\omega\eta^{2}}$). We show that
the combination of Brans-Dicke field and agegraphic dark energy can
accommodate $w_{D}=-1$ crossing for the equation of state of noninteracting
dark energy. When an interaction between dark energy and dark matter is taken
into account, the transition of $w_{D}$ to phantom regime can be more easily
accounted for than when resort to the Einstein field equations is made. In the
limiting case $\alpha=0$ $(\omega\rightarrow\infty)$, all previous results of
the new agegraphic dark energy in Einstein gravity are restored.
## I Introduction
One of the most dramatic discoveries of the modern cosmology in the past
decade is that our universe is currently accelerating Rie . A great variety of
scenarios have been proposed to explain this acceleration while most of them
cannot explain all the features of universe or they have so many parameters
that makes them difficult to fit. For a recent review on dark energy proposals
see Pad . Many theoretical studies on the dark energy problem are devoted to
understand and shed the light on it in the framework of a fundamental theory
such as string theory or quantum gravity. Although a complete theory of
quantum gravity has not established yet today, we still can make some attempts
to investigate the nature of dark energy according to some principles of
quantum gravity. The holographic dark energy and the agegraphic dark energy
(ADE) models are just such examples, which are originated from some
considerations of the features of the quantum theory of gravity. That is to
say, the holographic and ADE models possess some significant features of
quantum gravity. The former, that arose a lot of enthusiasm recently Coh ;
wang , is motivated from the holographic hypothesis Suss1 and has been tested
and constrained by various astronomical observations Xin . The later (ADE) is
based on the uncertainty relation of quantum mechanics together with the
gravitational effect in general relativity. The ADE model assumes that the
observed dark energy comes from the spacetime and matter field fluctuations in
the universe Cai1 ; Wei2 . Following the line of quantum fluctuations of
spacetime, Karolyhazy et al. Kar1 discussed that the distance $t$ in
Minkowski spacetime cannot be known to a better accuracy than $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ where $\beta$ is a dimensionless constant of order unity.
Based on Karolyhazy relation, Maziashvili Maz argued that the energy density
of spacetime fluctuations is given by
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (1)
where $t_{p}$ and $m_{p}$ are the reduced Planck time and mass, respectively.
On these basis, Cai wrote down the energy density of the original ADE as Cai1
$\rho_{D}=\frac{3n^{2}m^{2}_{p}}{T^{2}},$ (2)
where $T$ is the age of the universe and the numerical factor $3n^{2}$ is
introduced to parameterize some uncertainties, such as the species of quantum
fields in the universe. However, the original ADE model has some difficulties
Cai1 . In particular, it suffers from the difficulty to describe the matter-
dominated epoch. Therefore, a new model of ADE was proposed Wei2 , while the
time scale is chosen to be the conformal time $\eta$ instead of the age of the
universe, which is defined by $dt=ad\eta$, where $t$ is the cosmic time. It is
worth noting that the Karolyhazy relation $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$
was derived for Minkowski spacetime $ds^{2}=dt^{2}-d\mathrm{x^{2}}$ Kar1 ; Maz
. In case of the FRW universe, we have
$ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$. Thus,
it might be more reasonable to choose the time scale in Eq. (2) to be the
conformal time $\eta$ since it is the causal time in the Penrose diagram of
the FRW universe. The new ADE contains some new features different from the
original ADE and overcome some unsatisfactory points. The ADE models have been
examined and constrained by various astronomical observations age ; shey0 .
On the other front, it is quite possible that gravity is not given by the
Einstein action, at least at sufficiently high energies. In string theory,
gravity becomes scalar-tensor in nature. The low energy limit of string theory
leads to the Einstein gravity, coupled non-minimally to a scalar field Wit1 .
Although the pioneering study on scalar-tensor theories was done several
decades ago BD , it has got a new impetus recently as it arises naturally as
the low energy limit of many theories of quantum gravity such as superstring
theory or Kaluza-Klein theory. Because the agegraphic energy density belongs
to a dynamical cosmological constant, we need a dynamical frame to accommodate
it instead of Einstein gravity. Therefore the investigation on the agegraphic
models of dark energy in the framework of Brans-Dicke theory is well
motivated. In the framework of Brans-Dicke cosmology, holographic models of
dark energy have also been studied Pavon2 . Our aim in this paper is to
construct a cosmological model of late acceleration based on the Brans-Dicke
theory of gravity and on the assumption that the pressureless dark matter and
new ADE do not conserve separately but interact with each other.
## II NEW ADE in Branse-Dicke Theory
We start from the action of Brans-Dicke theory which in the canonical form can
be written Arik
$S=\int{d^{4}x\sqrt{g}\left(-\frac{1}{8\omega}\phi^{2}{R}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+L_{M}\right)},$
(3)
where ${R}$ is the scalar curvature and $\phi$ is the Brans-Dicke scalar
field. The non-minimal coupling term $\phi^{2}R$ replaces with the Einstein-
Hilbert term ${R}/{G}$ in such a way that
$G^{-1}_{\mathrm{eff}}={2\pi\phi^{2}}/{\omega}$ where $G_{\mathrm{eff}}$ is
the effective gravitational constant as long as the dynamical scalar field
$\phi$ varies slowly. The signs of the non-minimal coupling term and the
kinetic energy term are properly adopted to $(+---)$ metric signature. The new
ADE model will be accommodated in the non-flat Friedmann-Robertson-Walker
(FRW) universe which is described by the line element
$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(4)
where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . Varying action (3) with respect to metric
(4) for a universe filled with dust and ADE yields the following field
equations
$\displaystyle\frac{3}{4\omega}\phi^{2}\left(H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{2}\dot{\phi}^{2}+\frac{3}{2\omega}H\dot{\phi}\phi=\rho_{m}+\rho_{D},$
(5)
$\displaystyle\frac{-1}{4\omega}\phi^{2}\left(2\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)-\frac{1}{\omega}H\dot{\phi}\phi-\frac{1}{2\omega}\ddot{\phi}\phi-\frac{1}{2}\left(1+\frac{1}{\omega}\right)\dot{\phi}^{2}=p_{D},$
(6)
$\displaystyle\ddot{\phi}+3H\dot{\phi}-\frac{3}{2\omega}\left(\frac{{\ddot{a}}}{a}+H^{2}+\frac{k}{a^{2}}\right)\phi=0,$
(7)
where the dot is the derivative with respect to time and $H=\dot{a}/a$ is the
Hubble parameter. Here $\rho_{D}$, $p_{D}$ and $\rho_{m}$ are, respectively,
the dark energy density, dark energy pressure and energy density of dust (dark
matter). We shall assume that Brans-Dicke field can be described as a power
law of the scale factor, $\phi\propto a^{\alpha}$. A case of particular
interest is that when $\alpha$ is small whereas $\omega$ is high so that the
product $\alpha\omega$ results of order unity Pavon2 . This is interesting
because local astronomical experiments set a very high lower bound on
$\omega$; in particular, the Cassini experiment implies that $\omega>10^{4}$
Bert . Taking the derivative with respect to time of relation $\phi\propto
a^{\alpha}$, we get
$\displaystyle\dot{\phi}=\alpha H\phi,$ (8)
$\displaystyle\ddot{\phi}=\alpha^{2}H^{2}\phi+\alpha\phi\dot{H}.$ (9)
The energy density of the new ADE can be written Wei2
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (10)
where the conformal time is given by
$\eta=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (11)
In the framework of Brans-Dicke cosmology, we write down the new agegraphic
energy density of the quantum fluctuations in the universe as
$\rho_{D}=\frac{3n^{2}\phi^{2}}{4\omega\eta^{2}}.$ (12)
where $\phi^{2}={\omega}/{2\pi G_{\mathrm{eff}}}$. In the limit of Einstein
gravity, $G_{\mathrm{eff}}\rightarrow G$, expression (12) recovers the
standard new agegraphic energy density in Einstein gravity. We define the
critical energy density, $\rho_{\mathrm{cr}}$, and the energy density of the
curvature, $\rho_{k}$, as
$\displaystyle\rho_{\mathrm{cr}}=\frac{3\phi^{2}H^{2}}{4\omega},\hskip
22.76228pt\rho_{k}=\frac{3k\phi^{2}}{4\omega a^{2}}.$ (13)
We also introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{\rho_{\mathrm{cr}}}=\frac{4\omega\rho_{m}}{3\phi^{2}H^{2}},$
(14)
$\displaystyle\Omega_{k}=\frac{\rho_{k}}{\rho_{\mathrm{cr}}}=\frac{k}{H^{2}a^{2}}$
(15)
$\displaystyle\Omega_{D}=\frac{\rho_{D}}{\rho_{\mathrm{cr}}}=\frac{n^{2}}{H^{2}\eta^{2}}.$
(16)
### II.1 Noninteracting case
Let us begin with the noninteracting case, in which the dark energy and dark
matter evolves according to their conservation laws
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (17)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (18)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of the new
ADE. Differentiating Eq. (12) and using Eqs. (8) and (16) we have
$\displaystyle\dot{\rho}_{D}=2H\rho_{D}\left(\alpha-\frac{\sqrt{\Omega_{D}}}{na}\right).$
(19)
Inserting this equation in the conservation law (17), we obtain the equation
of state parameter of the new ADE
$\displaystyle w_{D}=-1-\frac{2\alpha}{3}+\frac{2}{3na}\sqrt{\Omega_{D}}.$
(20)
It is important to note that when $\alpha=0$, the Brans-Dicke scalar field
becomes trivial and Eq. (20) reduces to its respective expression in new ADE
in general relativity Wei2
$\displaystyle w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (21)
In this case ($\alpha=0$), the present accelerated expansion of our universe
can be derived only if $n>1$ Wei2 . Note that we take $a=1$ for the present
time. In addition, $w_{D}$ is always larger than $-1$ and cannot cross the
phantom divide $w_{D}=-1$. However, in the presence of the Brans-Dicke field
($\alpha>0$) the condition $n>1$ is no longer necessary to derive the present
accelerated expansion. Besides, from Eq. (20) one can easily see that $w_{D}$
can cross the phantom divide provided $na\alpha>\sqrt{\Omega_{D}}$. If we take
$\Omega_{D}=0.73$ and $a=1$ for the present time, the phantom-like equation of
state can be accounted if $n\alpha>0.85$. For instance, for $n=1$ and
$\alpha=0.9$, we get $w_{D}=-1.03$. Therefore, with the combination of new
agegraphic energy density with the Brans-Dicke field $w_{D}$ of noninteracting
new ADE can cross the phantom divide.
Let us examine the behavior of $w_{D}$ in two different stages. In the late
time where $\Omega_{D}\rightarrow 1$ and $a\rightarrow\infty$ we have
$w_{D}=-1-\frac{2\alpha}{3}$. Thus $w_{D}<-1$ for $\alpha>0$. This implies
that in the late time $w_{D}$ necessary crosses the phantom divide in the
framework of Brans-Dicke theory. In the early time where
$\Omega_{D}\rightarrow 0$ and $a\rightarrow 0$ we cannot find $w_{D}$ from Eq.
(20) directly. Let us consider the matter-dominated epoch,
$H^{2}\propto\rho_{m}\propto a^{-3}$. Therefore $\sqrt{a}da\propto dt=ad\eta$.
Thus $\eta\propto\sqrt{a}$. From Eq. (12) we have $\rho_{D}\propto
a^{2\alpha-1}$. Putting this in conservation law,
$\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0$, we obtain $w_{D}=-{2}/{3}-2\alpha/3$.
Substituting this $w_{D}$ in Eq. (20) we find that $\Omega_{D}=n^{2}a^{2}/4$
in the matter dominated epoch as expected. We will see below that this is
exactly the result one obtains for $\Omega_{D}$ from its equation of motion in
the matter-dominated epoch.
Since in our model the dynamics of the scale factor is governed not only by
the dark matter and new ADE, but also by the Brans-Dicke field, the signature
of the deceleration parameter, $q=-\ddot{a}/(aH^{2})$, has to be examined
carefully. When the deceleration parameter is combined with the Hubble
parameter and the dimensionless density parameters, form a set of useful
parameters for the description of the astrophysical observations. Dividing Eq.
(6) by $H^{2}$, and using Eqs. (8), (9) and (12)-(16) we obtain
$\displaystyle
q=\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}+3\Omega_{D}w_{D}\right].$
(22)
Substituting $w_{D}$ from Eq. (20), we reach
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+3)\Omega_{D}+\frac{2}{na}{\Omega^{3/2}_{D}}\right].$
(23)
When $\alpha=0$, Eq. (23) restores the deceleration parameter of the new ADE
in general relativity shey0
$\displaystyle
q=\frac{1}{2}(1+\Omega_{k})-\frac{3}{2}\Omega_{D}+\frac{\Omega^{3/2}_{D}}{na}.$
(24)
Finally, we obtain the equation of motion for $\Omega_{D}$. Taking the
derivative of Eq. (16) and using relation
${\dot{\Omega}_{D}}=H{\Omega^{\prime}_{D}}$, we get
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{na}\sqrt{\Omega_{D}}\right),$
(25)
where the prime denotes the derivative with respect to $x=\ln{a}$. Using
relation $q=-1-\frac{\dot{H}}{H^{2}}$, we have
$\displaystyle{\Omega^{\prime}_{D}}=2\Omega_{D}\left(1+q-\frac{\sqrt{\Omega_{D}}}{na}\right),$
(26)
where $q$ is given by Eq. (23). Let us examine the above equation for matter
dominated epoch where $a\ll 1$ and $\Omega_{D}\ll 1$. Substituting $q$ from
Eq. (23) in (26) with $\Omega_{k}\ll 1$, $\alpha\ll 1$ and
$\alpha\omega\approx 1$, this equation reads as
$\displaystyle\frac{d\Omega_{D}}{da}\simeq\frac{\Omega_{D}}{a}\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right).$
(27)
Solving this equation we find $\Omega_{D}=n^{2}a^{2}/4$, which is consistent
with our previous result. Therefore, all things are consistent. The confusion
in the original ADE is removed in this new model.
### II.2 Interacting case
Next we generalize our study to the case where the pressureless dark matter
and the new ADE do not conserve separately but interact with each other. Given
the unknown nature of both dark matter and dark energy there is nothing in
principle against their mutual interaction and it seems very special that
these two major components in the universe are entirely independent. Indeed,
this possibility is receiving growing attention in the literature Ame and
appears to be compatible with SNIa and CMB data Oli . The total energy density
satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0.$ (28)
However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (29)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (30)
where $Q=\Gamma\rho_{D}$ stands for the interaction term with $\Gamma>0$.
Using Eqs. (8) and (13), we can rewrite the first Friedmann equation (5) as
$\displaystyle\rho_{\mathrm{cr}}+\rho_{k}=\rho_{m}+\rho_{D}+\rho_{\phi},$ (31)
where we have defined
$\displaystyle\rho_{\phi}\equiv\frac{1}{2}\alpha
H^{2}\phi^{2}\left(\alpha-\frac{3}{\omega}\right).$ (32)
Dividing Eq. (31) by $\rho_{\mathrm{cr}}$, this equation can be written as
$\displaystyle\Omega_{m}+\Omega_{D}+\Omega_{\phi}=1+\Omega_{k},$ (33)
where
$\displaystyle\Omega_{\phi}=\frac{\rho_{\phi}}{\rho_{\mathrm{cr}}}=-2\alpha\left(1-\frac{\alpha\omega}{3}\right).$
(34)
We also assume $\Gamma=3b^{2}(1+r)H$ where $r={\rho_{m}}/{\rho_{D}}$ and
$b^{2}$ is a coupling constant. Therefore, the interaction term $Q$ can be
expressed as
$\displaystyle Q=3b^{2}H\rho_{D}(1+r),$ (35)
where
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\Omega_{m}}{\Omega_{D}}=-1+{\Omega^{-1}_{D}}\left[1+\Omega_{k}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(36)
Combining Eqs. (19), (35) and (36) with Eq. (30) we can obtain the equation of
state parameter
$\displaystyle w_{D}$ $\displaystyle=$
$\displaystyle-1-\frac{2\alpha}{3}+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}{\Omega^{-1}_{D}}\left[1+\Omega_{k}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right].$
(37)
When $\alpha=0$, Eq. (37) recovers its respective expression of interacting
new ADE model in general relativity shey0 . From Eq. (37) we see that with the
combination of the new ADE and Brans-Dicke field, the transition of $w_{D}$
from the phantom divide can be more easily accounted than in Einstein gravity.
For completeness we also present the deceleration parameter for the
interacting case
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha+2}\left[(2\alpha+1)^{2}+2\alpha(\alpha\omega-1)+\Omega_{k}-(2\alpha+3)\Omega_{D}+\frac{2}{na}{\Omega^{3/2}_{D}}\right.\
$ (38)
$\displaystyle\left.-3b^{2}\left(1+\Omega_{k}+2\alpha\left(1-\frac{\alpha\omega}{3}\right)\right)\right].$
In the limiting case $\alpha=0$, Eq. (38) restores the deceleration parameter
for the standard interacting new ADE in a non-flat universe shey0
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\Omega_{k})-\frac{3}{2}{\Omega_{D}}+\frac{\Omega^{3/2}_{D}}{na}-\frac{3b^{2}}{2}(1+\Omega_{k}).$
(39)
For flat universe, $\Omega_{k}=0$, and we recover exactly the result of Wei2 .
The equation of motion for $\Omega_{D}$ takes the form (26), where $q$ is now
given by Eq. (38).
## III Conclusions
An interesting attempt for probing the nature of dark energy within the
framework of quantum gravity is the so-called ADE proposal. Since ADE models
belong to a dynamical cosmological constant, it is more natural to study them
in the framework of Brans-Dicke theory than in Einstein gravity. In this
paper, we studied a cosmological model of late acceleration based on the new
ADE model in the framework of non-flat Brans-Dicke cosmology where the new
agegraphic energy density $\rho_{D}={3n^{2}m^{2}_{p}}/\eta^{2}$ is replaced
with $\rho_{D}={3n^{2}\phi^{2}}/({4\omega\eta^{2}})$. With this replacement in
Brans-Dicke theory, we found that the acceleration of the universe expansion
will be more easily achieved for than when the standard new ADE in general
relativity is employed. Interestingly enough, we found that with the
combination of Brans-Dicke field and ADE the equation of state of
noninteracting new ADE can cross the phantom divide. This is in contrast to
Einstein gravity where the equation of state of noninteracting new ADE cannot
cross the phantom divide Wei2 . When an interaction between dark energy and
dark matter is taken into account, the transition to phantom regime for the
equation of state of new ADE can be more easily accounted for than when resort
to the Einstein field equations is made.
###### Acknowledgements.
I thank the anonymous referee for constructive comments. This work has been
supported by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.
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|
arxiv-papers
| 2009-08-05T04:01:14 |
2024-09-04T02:49:04.458992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0908.0606"
}
|
0908.0615
|
††thanks: Author to whom correspondence should be addressed
# Spin relaxation in $n$-type ZnO quantum wells
C. Lü Hefei National Laboratory for Physical Sciences at Microscale and
Department of Physics, University of Science and Technology of China, Hefei,
Anhui, 230026, China J. L. Cheng jlcheng@mail.ustc.edu.cn. Hefei National
Laboratory for Physical Sciences at Microscale and Department of Physics,
University of Science and Technology of China, Hefei, Anhui, 230026, China
###### Abstract
We perform an investigation on the spin relaxation for $n$-type ZnO (0001)
quantum wells by numerically solving the kinetic spin Bloch equations with all
the relevant scattering explicitly included. We show the temperature and
electron density dependence of the spin relaxation time under various
conditions such as impurity density, well width, and external electric field.
We find a peak in the temperature dependence of the spin relaxation time at
low impurity density. This peak can survive even at 100 K, much higher than
the prediction and measurement value in GaAs. There also exhibits a peak in
the electron density dependence at low temperature. These two peaks originate
from the nonmonotonic temperature and electron density dependence of the
Coulomb scattering. The spin relaxation time can reach the order of nanosecond
at low temperature and high impurity density.
###### pacs:
72.25.Rb, 73.21.Fg, 71.10.-w
## I Introduction
Much attention has been devoted to the spin degree of the freedom of carriers
in the zinc oxide (ZnO) with wurtzite structure in the last few years,ZnO
partly because of the very long spin relaxation time (SRT)Ghosh ; Gamelin ;
Lagrade and the prediction that ZnO can become ferromagnetic with a Curie
temperature above room temperature if doped with manganese.Ferrand However,
only few works investigate on the spin dynamics properties of ZnO:
Experimentally, Ghosh _et al._Ghosh investigated the electron spin properties
in $n$-type bulk ZnO and discovered the electron spin relaxation time varying
from 20 ns to 190 ps when the temperature increased from 10 to 280 K. Liu _et
al._Gamelin measured the SRT in colloidal $n$-type ZnO quantum dots could be
as long as 25 ns at room temperature by electron paramagnetic resonance
spectroscopy. Theoretically, Harmon _et al._ calculated the SRT in bulk
material in the framework of a single-particle model.Joynt However, a
theoretical investigation on the SRT in quantum wells (QWs) is still rare,
which is our task in the present paper.
The spin relaxation can be induced by the following mechanisms: (i) D’yakonov-
Perel’(DP) spin relaxation mechanism,DPmech which is revealed to be the
dominant mechanism in $n$-doped semiconductors.Zhou ; jjh For $n$-doped ZnO
QW, the spin-orbit coupling (SOC) is at least one order of magnitude weaker
compared with that of the well studied semiconductor GaAsfu ; Voon , a simple
estimation shows that the criterion of the strong scatteringZhou is always
satisfied even the momentum scattering is also weaker than that in GaAs due to
the larger effective mass $m^{\ast}$. Therefore the DP mechanism here can be
described by the motional narrowing pictureFabian qualitatively and the
induced SRT is
$\tau\propto\frac{1}{\langle\Omega_{\mathbf{k}}^{2}\rangle\tau_{p}}\ ,$ (1)
with $\tau_{p}$ standing for the momentum scattering time and
$\langle\Omega_{\mathbf{k}}^{2}\rangle$ for the inhomogeneous broadening
induced by the SOC. As pointed out first by Wu et al.Wu1 ; Wu5 ; Weng and
then by Glazov and Ivchenko,Ivchenko the electron-electron scattering has
important contribution to the spin relaxation process. Therefore $\tau_{p}$
used in Eq. (1) should be revised as
$\tau_{p}^{\ast}=\left[\left(\tau_{p}\right)^{-1}+\left(\tau_{p}^{ee}\right)^{-1}\right]^{-1}$
to include the electron-electron momentum relaxation time $\tau_{p}^{ee}$. Wu1
; Wu5 ; Weng ; Zhou ; jjh ; Ivchenko ; Glazov ; Harley (ii) Elliott-Yafet
mechanism.EYmech The revised criterion in Eq. (22) of Refs. [jjh, ] gives
$\Theta\approx 340$ eV, which means the Elliott-Yafet mechanism is negligible
compared with the DP mechanism Joynt due to the small spin split off energy,
the large band gap, and the large $m^{\ast}$. (iii) Bir-Aronov-Pikus
mechanism, BAPmech which is always unimportant in $n$-doped semiconductor.
Zhou ; jjh Therefore, we only investigate the SRT induced by the DP mechanism
for $n$-type ZnO QWs in the following.
In this paper, we quantitatively calculate the SRT for $n$-type ZnO QWs by
using the fully microscopic spin kinetic Bloch equation (KSBE) approach, which
has been successfully used in investigating the spin relaxation in QWsWu5 ;
Weng ; schu ; Zhou ; Wu1 and in bulk semiconductors.jjh ; shen With all the
relevant scattering included, the influence of temperature, electron density,
impurity density, well width and electric field on the SRT are studied
detailedly. The temperature and density dependence of the SRT is shown to be
nomonotonic, and we find that the SRT increases with the electric field
monotonically.
This paper is organized as follows: In Sec. II we describe our model and the
KSBEs. Our numerical results are presented in Sec. III. We conclude in Sec.
IV.
## II Model and KSBEs
We start our investigation from a $n$-doped ZnO QW of well width $a$ grown in
(0001) direction, considered to be $z$ axis. Due to the confinement of QW, the
momentum states along $z$-axis is quantized by subband index $n$. With the
momentum vector $\mathbf{k}=(k_{x},k_{y})$ and the spin index $\sigma$, the
electron Hamiltonian can be written as
$H_{e}=\sum\limits_{n\mathbf{k}\atop\sigma_{1}\sigma_{2}}\left\\{{\cal
E}_{n\sigma_{1}\sigma_{2}\mathbf{k}}-e\mathbf{E}\cdot\mathbf{R}\delta_{\sigma_{1}\sigma_{2}}\right\\}a^{{\dagger}}_{n\sigma_{1}\mathbf{k}}a_{n\sigma_{2}\mathbf{k}}+H_{I}$.
Here ${\cal
E}_{n\sigma_{1}\sigma_{2}\mathbf{k}}=\varepsilon_{n\mathbf{k}}\delta_{\sigma_{1}\sigma_{2}}+[\mathbf{h}_{R}(\mathbf{k})+(\mathbf{h}_{D})_{n}(\mathbf{k})]\cdot\mbox{\boldmath$\sigma$\unboldmath}_{\sigma_{1}\sigma_{2}}$
with $\varepsilon_{n\mathbf{k}}=\frac{\mathbf{k}^{2}}{2m^{\ast}}+\frac{\langle
k_{z}^{2}\rangle_{n}}{m^{\ast}}$ is the energy spectrum, $\mathbf{R}=(x,y)$ is
the position, the effective magnetic field given by the Rashba due to the
intrinsic wurtzite structure inversion asymmetry and Dresselhaus SOC can be
written as: fu
$\displaystyle\mathbf{h}_{R}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\alpha_{e}(k_{y},-k_{x},0)\ ,$
$\displaystyle(\mathbf{h}_{D})_{n}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\gamma_{e}(b\langle
k_{z}^{2}\rangle_{n}-k_{\|}^{2})(k_{y},-k_{x},0)\ ,$ (2)
with $\alpha_{e}$, $\gamma_{e}$ and $b$ standing for the SOC coefficients.
$\langle k_{z}^{2}\rangle_{n}=\frac{\hbar^{2}\pi^{2}n^{2}}{a^{2}}$ is the
subband energy in a hard-wall confinement potential. The scattering
Hamiltonian $H_{I}$ includes all the scatterings, such as electron-nomagnetic
impurity scattering, electron-phonon scattering, and electron-electron
scattering.
We construct the KSBEs in the collinear statistics by using the non-
equilibrium Green function method as follows:Wu0 ; Wu1 ; Weng ; Wu5 ; Haug
$\partial_{t}\rho_{\mathbf{k}}-e\mathbf{E}\cdot\mbox{\boldmath$\nabla$\unboldmath}_{\mathbf{k}}\rho_{\mathbf{k}}=\partial_{t}\rho_{\mathbf{k}}|_{\rm{coh}}+\partial_{t}\rho_{\mathbf{k}}|_{\rm{scat}}.$
(3)
The density matrix $\rho_{\mathbf{k}}$ for momentum $\mathbf{k}$ is a matrix
with matrix elements $[\rho_{\mathbf{k}}]_{n_{1}\sigma_{1};n_{2}\sigma_{2}}$
which include all the coherence between different subbands and different
spins. The second terms on the left-hand side of the kinetic equations
describe the electric field $\mathbf{E}$ driven effect.
$\partial_{t}\rho_{\mathbf{k}}|_{\rm{coh}}$ is the coherent term.
$\partial_{t}\rho_{\mathbf{k}}|_{\rm{scat}}$ denotes the scattering, including
the electron-impurity, the electron-phonon, as well as the electron-electron
scattering. The expressions for these terms are given in Appendix A.
Before we give our numerical results, the qualitative analysis of the DP
mechanism due to the electron-electron scattering can be made at strong
scattering limit. The perturbation theory shows the effective electron-
electron momentum scattering time $\tau_{p}^{ee}$ in degenerate and
nondegenerate limits satisfiesVignale
$\frac{1}{\tau_{p}^{ee}}\propto\begin{cases}T^{2}N_{e}^{-1},&\mbox{for }T\ll
T_{F}\ ,\\\ T^{-1}N_{e},&\mbox{for }T\gg T_{F}\ ,\end{cases}$ (4)
which has nonmonotonic temperature $T$ and electron density $N_{e}$ dependence
as the electron gas undergoes the transition from the degenerate case to the
nondegenerate case at the Fermi temperature $T_{F}$. As the SOC in ZnO QWs
mainly depends on $\mathbf{k}$ linearly, the inhomogeneous broadening is given
by
$\langle\Omega_{\mathbf{k}}^{2}\rangle\propto\begin{cases}N_{e},&\mbox{for
}T\ll T_{F}\ ,\\\ T,&\mbox{for }T\gg T_{F}\ .\end{cases}$ (5)
Then by Eq. (1), the electron-electron scattering contributes to the SRT
$\tau$ as Harley
$\tau\propto\begin{cases}T^{2}N_{e}^{-2},&\mbox{for }T\ll T_{F}\ ,\\\
T^{-2}N_{e},&\mbox{for }T\gg T_{F}\ .\end{cases}$ (6)
The SRT is expected to reach a minimum in $T$ dependence or a maximum in
$N_{e}$ dependence, and the location of the extreme points satisfie $T\approx
T_{F}$.
## III Numerical Results
We numerically solve the KSBEs for the spin density matrix $\rho$, from which
we obtain the time evolution of the spin polarization along $z$-direction:
Weng
$\displaystyle P_{z}(t)=\sum_{n,\bf
k}\\{[\rho_{\mathbf{k}}]_{n\uparrow;n\uparrow}(t)-[\rho_{\mathbf{k}}]_{n\downarrow;n\downarrow}(t)\\}/N_{e}\
,$ (7)
where $N_{e}$ is the total electron density. The SRT $\tau$ is extracted from
the exponential decay of the envelope of $P_{z}(t)$. The initial condition at
$t=0$ is taken to be
$\displaystyle[\rho_{\mathbf{k}}]_{n_{1}\sigma_{1};n_{2}\sigma_{2}}=f_{T,\mu_{\sigma_{1}}}(\mathbf{k})\delta_{n_{1}n_{2}}\delta_{\sigma_{1}\sigma_{2}}\
.$ (8)
Here
$f_{T,\mu}(\mathbf{k})=[1+e^{(\varepsilon_{\mathbf{k}}-\mu)/(k_{B}T)}]^{-1}$
gives the Fermi-Dirac distribution. The spin dependent chemical potential
$\mu_{\sigma}$ is chosen to satisfy $P(0)=2.5\%$. The electron density and the
quantum well width are taken as $N_{e}=4\times 10^{11}$/cm2 and $a=10$ nm
respectively unless otherwise specified. All used parameters are listed in
Table 1. In the calculation, only the lowest two subbands are taken into
account.
Table 1: Material parameters used in the calculation (from Ref. parameter,
unless otherwise specified).
$\kappa_{\infty}$ | $3.7$ | | | | | | | | $\kappa_{0}$ | $7.8$
---|---|---|---|---|---|---|---|---|---|---
$m_{e}/m_{0}$ | $0.25$ | | | | | | | | $v_{1}$ (km/s) | 6.08
$v_{3}$ (km/s) | 6.09 | | | | | | | | $v_{4}$ (km/s) | 2.73
$v_{6}$ (km/s) | 2.79 | | | | | | | | b | 3.91a
$\gamma_{e}$ (eVÅ3) | 0.33a | | | | | | | | $\alpha_{e}$ (meVÅ) | 1.1b
$e_{15}$ (V/m) | $-0.35\times 10^{9}$ | | | | | | | | $e_{33}$ (V/m) | $1.56\times 10^{9}$
a Ref. [fu, ]; b Ref. [Voon, ].
### III.1 Temperature dependence
Figure 1: SRT $\tau$ vs. temperature $T$ at different impurity densities. The
dashed curve is obtained from the calculation of excluding the electron-phonon
scattering.
We now study the temperature dependence of the SRT presented in Fig. 1 for
different impurity densities. The results are similar to that in GaAs QWsZhou
; Ji and can be understood as follows: (i) The SRT always increases with the
impurity density $N_{i}$. It is because the system is in the strong scattering
regime as stated above due to the weak SOC, thus the spin relaxation can be
explained by the motional narrowing picture qualitativelyFabian ; Zhou , and
the additional scattering leads to longer SRT. (ii) The electron-phonon
scattering is shown to be negligible over the whole temperature regime by
comparing the temperature dependence of SRT with (solid curve with
$\blacktriangle$) or without (dashed curve with $\blacktriangle$) the
electron-phonon scattering for the impurity free case in the same figure.
(iii) The SRT presents a peak at very low $N_{i}$. In these cases, the
electron-electron scattering is the dominant scattering, therefore, as shown
in Eq. (6), the SRT shows a maximum and the transition temperature $T_{F}$ for
$N_{e}=4\times 10^{11}$ /cm2 is about $44$ K, which is agree with our
numerical results. (iv) When the impurity density is high enough such as
$N_{i}=N_{e}$, the SRT decreases monotonically with $T$. In this case the
total scattering is mainly determined by the impurity scattering, which
depends weakly on the temperature. However, the inhomogeneous broadening from
the DP term increases with the temperature, and results in shorter SRT.
For GaAs QWs, the temperature peak of the SRT can only be observed at low
electron density (i.e. low transition temperature) and low impurity
densityZhou ; Ji , because the electron-phonon scattering becomes strong
enough to destroy the nonmonotonic $T$ dependence of the scattering time
induced by the electron-electron scattering. Such case can be avoid in ZnO
QWs, in which the electron-phonon scattering is always pretty weak due to the
large optical phonon energies ($\sim 800$ K). Thus the temperature peak can be
found even for high electron density samples. One can easily find from the
chained curve in Fig. 1, which is calculated with parameters $N_{e}=10^{12}$
/cm2 and $N_{i}=0$, that the peak moves to $T\sim 100K$. Therefore, the high
mobility ZnO QW is a good system for studying the electron-electron
scattering.
### III.2 Doping and well width dependence
Figure 2: SRT vs. the electron density with different impurity densities and
temperatures. (a) $T=20$ K; (b) $T=300$ K. $\blacktriangle$: $N_{i}/N_{e}=0$;
$\blacksquare$: $N_{i}/N_{e}=0.1$; $\blacklozenge$: $N_{i}/N_{e}=1$.
Then we investigate the density dependence of the SRT at different
temperatures and impurity densities. In Fig. 2 (a) we plot the SRT as a
function of the electron density with $T=20$ K. One can see that for the low
impurity density case, the SRT reaches a maximum at $N_{e}\approx 2\times
10^{11}$ cm-2, which has been pointed out in $n$-type bulk III-V
semiconductorsjjh ; shen . It originates from the transition from the
nondegenerate electron gas to the degenerate electron gas and can be well
explained by Eq. (6). Our calculation gives the transition density of $2\times
10^{11}$ cm-2, corresponding $T_{F}\sim 22$ K, close to the lattice
temperature of $20$ K. For the case of $N_{i}=N_{e}$, the electron-impurity
scattering time has the same $N_{e}$ dependence as that for electron-electron
scattering: in the nondegenerate regime, $\frac{1}{\tau_{p}^{ei}}\propto
N_{i}\langle U_{q}^{2}\rangle$, in which $\langle U_{q}^{2}\rangle$ changes
little; in the degenerate regime, $\frac{1}{\tau_{p}^{ei}}\sim
N_{i}U_{k_{f}}^{2}\propto N_{e}/k_{F}^{4}\propto N_{e}^{-1}$. Consequently,
the peak still exists and is almost at the same position. In comparison, the
SRT as a function of $N_{e}$ with $T=300$ K is plotted in Fig. 2 (b). In this
case, one finds that the SRT increases monotonically with $N_{e}$. This could
be easily understood for that $T_{F}\ll T$ is satisfied and it is in the
nondegenerate regime, in which the SRT increases with density as discussed
above.
Figure 3: SRT vs. the temperature at different quantum well widths.
We further show the effect of quantum well width on the spin relaxation. In
Fig. 3 the SRTs versus temperature at well widths $a=10$ nm and $20$ nm are
plotted respectively. Both the SOC and the scatteringGlazov ; Harley depend
on the quantum well width. However, comparing to the weak well width
dependence of the scattering, the fast decrease of $\langle k_{z}^{2}\rangle$
in the DP term with $a$ dominates and so the SRT increases with well width.
Figure 4: SRT vs. the electric field at different temperature and impurity
densities. $\blacktriangle$: $N_{i}/N_{e}=0$; $\blacksquare$:
$N_{i}/N_{e}=0.1$; $\bullet$: $N_{i}/N_{e}=1$.
### III.3 Electric field dependence
Then we investigate the electric field dependence of the SRT at different
temperatures and impurity densities. In Fig. 4 we plot the SRT as a function
of the electric field for different $T$. The electric field is applied along
the $x$ axis. One can see that the SRT increases monotonically for both low
temperature and high temperature cases. According to the previous
investigation Wu5 , the electric field will enhance both the momentum
scattering due to the hot-electron effect, and the inhomogeneous broadening
due to the drift of the electron distribution to larger $\mathbf{k}$ states.
These two effects are competing effects for the SRT: the former tends to
enhance the SRT while the later tends to suppress it. Wu5 For SOC with linear
$\mathbf{k}$ dependence, the hot-electron effect dominates,jjh thus the SRT
always increases with $E$.
## IV Conclusion
In conclusion, we have investigated the spin relaxation for $n$-type ZnO
(0001) QWs by numerically solving the KSBSs with all the relevant scattering
explicitly included. It is shown that the electron-phonon scattering is pretty
weak in ZnO QWs, while the Coulomb scattering always plays an important role.
Therefore the ZnO QW is a good carrier to study the electron-electron
scattering. We find there exists a peak of SRT both in the temperature
dependence for a given electron density at low impurity density and in the
electron density dependence at low temperature. Both these two peaks originate
from the different temperature and electron density dependence of
$\tau^{ee}_{p}$ in degenerate and non-degenerate case. Compared with the same
effect in III-V semiconductor, Zhou ; Ji ; jjh ; shen this peak position can
occur at the temperature as high as 100 K and is easier to observe in
experiments due to the weak electron-phonon scattering. When the impurity
density is high, the peak in the temperature dependence disappears and the SRT
decreases with temperature monotonously. Moreover, the peak in the electron
density dependence moves to larger electron density which is beyond the scope
of our interest when the temperature is high. We also investigate the hot-
electron effect and show that the SRT always increases with the electric
field. It is also shown that the SRT reaches the order of nonosecond at low
temperature and high impurity density.
###### Acknowledgements.
The authors would like to thank M.W. Wu for proposing the topic as well as the
directions during the investigation. This work was supported by the Natural
Science Foundation of China under Grant No. 10725417, the National Basic
Research Program of China under Grant No. 2006CB922005, and the Knowledge
Innovation Project of the Chinese Academy of Sciences. J.L.C was partially
supported by China Postdoctoral Science Foundation.
## Appendix A EXPRESSIONS FOR KINETIC BLOCH EQUATIONS
Here we write the expressions for the coherent terms and the scattering terms
in the kinetic Bloch equations. The coherent terms in Eq. (3) can be written
as
$\displaystyle\partial_{t}\rho_{\mathbf{k}}\Big{|}_{\rm
coh}=-i\Big{[}E_{e}(\bf{k})+\sum_{\mathbf{Q}}V_{\mathbf{Q}}I_{q_{z}}{\rho}_{\bf{k}-\bf{q}}I_{q_{z}},\rho_{\bf{k}}\Big{]}\
,$ (9)
where $[A,B]=AB-BA$ denotes the commutator.
$[E_{e}(\mathbf{k})]_{n_{1}\sigma_{1};n_{2}\sigma_{2}}={\cal
E}_{n_{1}\sigma_{1}\sigma_{2}\mathbf{k}}\delta_{n_{1}n_{2}}$. The Coulomb
Hartree-Fock term, which is always negligible for small spin polarization,
Weng ; schu is also included. The form factor $I_{q_{z}}$ is also a matrix
with matrix elements
$\displaystyle[I_{q}]_{n_{1}\sigma_{1};n_{2}\sigma_{2}}=iaq\delta_{\sigma_{1},\sigma_{2}}[e^{iaq}\cos{\pi(n_{1}-n_{2})}-1]$
$\displaystyle\times\left[\frac{1}{\pi^{2}(n_{1}-n_{2})^{2}-a^{2}q^{2}}-\frac{1}{\pi^{2}(n_{1}+n_{2})^{2}-a^{2}q^{2}}\right].$
(10)
The statically screened Coulomb potential in the random-phase approximation
(RPA) reads Haug $V_{\bf Q}=\frac{v_{Q}}{\epsilon(\mathbf{q})}$ with the
dielectric function $\epsilon(\mathbf{q})=1-\sum\limits_{q_{z},\mathbf{k}\atop
n_{1};n_{2}}v_{Q}\big{|}[I_{q_{z}}]_{n_{1};n_{2}}\big{|}^{2}\frac{f_{\sigma}(\varepsilon_{n_{1},\mathbf{k+q}})-f_{\sigma}(\varepsilon_{n_{2},\mathbf{k}})}{\varepsilon_{n_{1},\bf
k+q}-\varepsilon_{n_{2},\bf k}}$ and $\mathbf{Q}=(q_{x},q_{y},q_{z})$. The
bare Coulomb potential is $v_{Q}=4\pi e^{2}/Q^{2}$.
$f_{\sigma}(\varepsilon_{n,\mathbf{k}})=[\rho_{\mathbf{k}}]_{n\sigma;n\sigma}$.
The scattering terms in Eq. (3) can be written as
$\partial_{t}\rho_{{\bf{k}}}|_{\rm scat}=\partial_{t}\rho_{{{\bf k}}}|_{\rm
im}+\partial_{t}\rho_{{{\bf k}}}|_{\rm ph}+\partial_{t}\rho_{{{\bf k}}}|_{\rm
ee}$ in the Markovian limit with
$\displaystyle\partial_{t}\rho_{\mathbf{k}}\Big{|}_{\rm im}$ $\displaystyle=$
$\displaystyle\pi
N_{i}\sum_{\mathbf{Q},n_{1},n_{2}}|U^{i}_{\mathbf{Q}}|^{2}\delta(\varepsilon_{n_{1},{{\bf
k}-{\bf q}}}-\varepsilon_{n_{2},{\bf k}})I_{q_{z}}[(1-{\rho}_{{{\bf k}-{\bf
q}}})T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}{\rho}_{\bf k}-\rho_{{{\bf k}-{\bf
q}}}T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}(1-\rho_{\bf k})]+h.c.\ ,$
$\displaystyle\partial_{t}\rho_{\mathbf{k}}\Big{|}_{\rm ph}$ $\displaystyle=$
$\displaystyle\pi\sum_{{\mathbf{Q}},n_{1},n_{2},\lambda}|M_{{\mathbf{Q}},\lambda}|^{2}I_{q_{z}}\\{\delta(\varepsilon_{n_{1},{{\bf
k}-{\bf q}}}-\varepsilon_{n_{2},{\bf
k}}+\omega_{{\mathbf{Q}},\lambda})[(N_{{\mathbf{Q}},\lambda}+1)(1-{\rho}_{{{\bf
k}-{\bf q}}})T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}{\rho}_{\bf k}$
$\displaystyle\mbox{}-N_{\mathbf{Q},\lambda}\rho_{{{\bf k}-{\bf
q}}}T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}(1-\rho_{\bf
k})]+\delta(\varepsilon_{n_{1},{{\bf k}-{\bf q}}}-\varepsilon_{n_{2},{\bf
k}}-\omega_{{\mathbf{Q}},\lambda})[N_{{\mathbf{Q}},\lambda}(1-{\rho}_{{{\bf
k}-{\bf q}}})T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}{\rho}_{\bf k}$
$\displaystyle\mbox{}-(N_{\mathbf{Q},\lambda}+1)\rho_{{{\bf k}-{\bf
q}}}T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}(1-\rho_{\bf k})]\\}+h.c.\ ,$
$\displaystyle\partial_{t}\rho_{\mathbf{k}}\Big{|}_{\rm ee}$ $\displaystyle=$
$\displaystyle\pi\sum_{{\mathbf{q},q_{z}q_{z}^{\prime}},{\bf
k}^{\prime}}\sum_{n_{1},n_{2},n_{3},n_{4}}V_{\mathbf{Q}}V_{\mathbf{Q}^{\prime}}\delta(\varepsilon_{n_{1},{{\bf
k}}-{\bf q}}-\varepsilon_{n_{2},{\bf k}}+\varepsilon_{n_{3},{\bf
k}^{\prime}}-\varepsilon_{n_{4},{\bf k}^{\prime}-{\bf q}})I_{q_{z}}$ (11)
$\displaystyle\mbox{}\times\\{(1-{\rho}_{{{\bf k}-{\bf
q}}})T_{n_{1}}I_{-{q_{z}}}T_{n_{2}}{\rho}_{\bf k}\mbox{Tr}[(1-\rho_{{\bf
k}^{\prime}})T_{n_{3}}I_{q_{z}^{\prime}}T_{n_{4}}\rho_{{\bf k}^{\prime}-{\bf
q}}I_{-q_{z}^{\prime}}]$ $\displaystyle\mbox{}-\rho_{\mathbf{k}-{\bf
q}}T_{n_{1}}I_{-q_{z}}T_{n_{2}}(1-\rho_{\bf k})\mbox{Tr}[\rho_{{\bf
k}^{\prime}}T_{n_{3}}I_{q_{z}^{\prime}}T_{n_{4}}(1-\rho_{{\bf k}^{\prime}-{\bf
q}})I_{-q_{z}^{\prime}}]\\}+h.c.\ \ ,$
in which
$[T_{n_{1}}]_{n,n^{\prime}}=\delta_{n_{1},n}\delta_{n_{1},n^{\prime}}$ and
$\mathbf{Q}^{\prime}=(q_{x},q_{y},q_{z}^{\prime})$. $N_{i}$ is the density of
impurities, and $|U^{i}_{\mathbf{Q}}|^{2}$ is the screened impurity potential.
$|M_{\mathbf{Q},\lambda}|^{2}$ and
$N_{\mathbf{Q},\lambda}=[\mbox{exp}(\omega_{\mathbf{Q},\lambda}/k_{B}T)-1]^{-1}$
are the matrix elements of the electron-phonon interaction and the Bose
distribution function respectively. $\omega_{\mathbf{Q},\lambda}$ is the
phonon energy spectrum. For the electron-phonon scattering, we only include
electron AC-phonon scattering, for which the explicit expressions can be found
in Refs. Lee, and Rossi, . The electron-optical phonon scattering is ignored
for that the optical phonon energies are around $70$ meV and are out of our
temperature range.parameter
## References
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|
arxiv-papers
| 2009-08-05T08:18:10 |
2024-09-04T02:49:04.464234
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. L\\\"u and J. L. Cheng",
"submitter": "J. L. Cheng",
"url": "https://arxiv.org/abs/0908.0615"
}
|
0908.0646
|
# The California Molecular Cloud
Charles J. Lada Harvard-Smithsonian Center for Astrophysics, 60 Garden Street
Cambridge, MA 02138 clada@cfa.harvard.edu Marco Lombardi European Southern
Observatory, Karl-Schwarzschild-Strasse 2, 85748, Garching Germany
mlombard@eso.org João F. Alves Calar Alto Observatory, C/Jesus Durban Remon,
2-2, 04004 Almeria, Spain jalves@caha.es
###### Abstract
We present an analysis of wide-field infrared extinction maps of a region in
Perseus just north of the Taurus-Auriga dark cloud complex. From this analysis
we have identified a massive, nearby, but previously unrecognized, giant
molecular cloud (GMC). Both a uniform foreground star density and measurements
of the cloud’s velocity field from CO observations indicate that this cloud is
likely a coherent structure at a single distance. From comparison of
foreground star counts with Galactic models we derive a distance of 450 $\pm$
23 parsecs to the cloud. At this distance the cloud extends over roughly 80 pc
and has a mass of $\approx$ 105 M⊙, rivaling the Orion (A) Molecular Cloud as
the largest and most massive GMC in the solar neighborhood. Although
surprisingly similar in mass and size to the more famous Orion Molecular Cloud
(OMC) the newly recognized cloud displays significantly less star formation
activity with more than an order of magnitude fewer Young Stellar Objects
(YSOs) than found in the OMC, suggesting that both the level of star formation
and perhaps the star formation rate in this cloud are an order of magnitude or
more lower than in the OMC. Analysis of extinction maps of both clouds shows
that the new cloud contains only 10% the amount of high extinction (A${}_{K}>$
1.0 mag) material as is found in the OMC. This, in turn, suggests that the
level of star formation activity and perhaps the star formation rate in these
two clouds may be directly proportional to the total amount of high extinction
material and presumably high density gas within them and that there might be a
density threshold for star formation on the order of n(H2) $\approx$ a few
$\times$ 104 cm-3.
stars: formation; molecular clouds
††slugcomment: to appear in the Astrophysical Journal, September 2009
## 1 Introduction
In this paper we report and analyze new observations of a little studied
molecular cloud in Perseus. This cloud attracted our attention in the course
of analyzing an extinction mapping survey of the Taurus-Auriga-Perseus region.
This survey was obtained as part of an ongoing systematic program to construct
and analyze wide-field infrared extinction maps of prominent dark clouds in
the solar vicinity. These clouds include the Pipe Nebula region (Lombardi et
al., 2006, LAL06), the Ophiuchus and Lupus cloud clouds (Lombardi et al.,
2008), the Orion/MonR2 region (Lombardi et al., 2009b, LAL09) as well as the
Taurus-Perseus-Auriga region (Lombardi et al., 2009a, LLA09). Examination of
the latter survey revealed a cloud with significantly more foreground star
contamination than either the Taurus-Auriga or Perseus clouds. Inspection of
an early CO survey of the same region (Ungerechts & Thaddeus, 1987) revealed
the cloud to also have a distinctly larger radial velocity than either the
Taurus or Perseus clouds. Using a modern version of the classical method
pioneered by Wolf (1923), we combined measurements of the foreground star
density with galactic models to derive a distance to the cloud of 450 pc. This
is considerably more distant than either the Taurus-Auriga (150 pc) or Perseus
(240 pc) clouds but comparable to the more famous Orion molecular clouds (400
pc). We determine the mass of this complex to be approximately 105 M⊙ making
it a bona fide giant molecular cloud (GMC). Interestingly this cloud is very
similar in size and shape to the Orion A GMC, but somewhat more massive. Yet
despite being perhaps the most massive GMC within 0.5 kpc of the solar system,
surprisingly little is known about it.
With this paper we begin to remedy this situation by providing measurements of
some of its basic physical properties, including distance, size, mass,
structure and star forming activity. The cloud lies entirely within the
Perseus constellation and so to avoid confusion with the closer, better known
(Perseus) cloud to its south, we designate this cloud the California Molecular
Cloud after the most prominent optical feature associated with it: the
California Nebula. We compare the properties of the California Molecular Cloud
to those of the Orion A Molecular Cloud and find the striking similarities in
masses, sizes and shapes of these two clouds to be in stark contrast to the
striking differences in their star formation activity and density structure.
## 2 Observational Data
### 2.1 Extinction Maps
Infrared extinction maps of the California Molecular Cloud (CMC) were
constructed using the NICER and NICEST extinction mapping methods (Lombardi,
2005, 2009) with infrared observations from the Two Micron All Sky Survey
(2MASS; Kleinmann et al. (1994)). The maps were constructed as part of a wide
field survey of the Taurus-Auriga-Perseus molecular clouds (LLA09) and their
surroundings. The detailed procedures used for creating the map are described
in LLA09 and in LAL04. Briefly the infrared color excesses were derived for 23
million stars in a 3500 square degree area including the Taurus-Auriga,
Perseus and California molecular clouds. The color excesses were derived by
subtracting from the observed colors of each star the corresponding intrinsic
colors derived from stars in a nearby control field with negligible
extinction. These color indices were then scaled by the appropriate extinction
law (Indebetouw et al. 2005) to derive infrared extinctions, (e.g., $A_{K}$)
to each star. The individual extinction measurements were then smoothed with a
gaussian weighting function (FWHM = 80.0 arc sec) and sampled with a spatial
frequency of 40.0 arc sec to produce a completely sampled grid from which the
map of the California Molecular Cloud was constructed. The resulting map
produced by the NICER method is presented in Figure 1.
Besides the California Nebula located at its southern edge, the previously
cataloged dark clouds L1441, L1442, L1449, L1456, L1459, L1473, L1478, L1482
and L1483 all appear to be part of the CMC complex. In addition the reflection
nebula NGC 1579 which is illuminated by LK H$\alpha$ 101 is embedded at the
eastern edge of the cloud.
### 2.2 CO Maps
CO maps were obtained from the archive of the Galactic plane survey performed
with the 1.2 meter Millimeter-wave Telescope at the Center for Astrophysics
(Dame, Hartmann & Thaddeus 2001). The data were uniformly sampled with a
spatial frequency equal to the beamwidth of 8 arc minutes and with a velocity
resolution of 0.65 km s-1. These maps represent a significant improvement over
the undersampled maps of Ungerechts & Thaddeus (1987). Figure 2 shows the
spatial map of 12CO emission from the CMC in Galactic coordinates and a
position-velocity (pv) map through the cloud complex made along galactic
longitude. The pv map has been integrated over 3 degrees in Galactic latitude
as shown in the figure and roughly parallels the primary axis of the cloud.
The CO spatial map shows basically the same overall morphology as the
extinction map despite the large (x 6) difference in angular resolutions
between the maps. (This is perhaps most clearly evident by comparison with
Figure 6 where the extinction map is plotted in Galactic Coordinates.) The pv
map shows that the cloud is continuous in velocity as well as spatially in
projection and this provides strong evidence that the emission originates in a
single contiguous cloud at the same distance. There is a significant velocity
gradient along Galactic longitude between $l_{II}=$ 156 and 163 degrees. The
magnitude of this gradient is approximately 0.9 km s-1 deg-1 or roughly 0.1 km
s-1 pc-1, which is typical, if not relatively modest, for a GMC.
## 3 Basic Physical Parameters
### 3.1 Distance
We derive the distance to this cloud using the density of foreground stars in
a manner similar to the classical method of Wolf (1923) and following the
method used by Lombardi et al. (2009a; 2009b) to determine the distances to
the Taurus-Auriga, Perseus, Orion and Mon R2 clouds. The Taurus-Auriga and
Perseus clouds are in the same general direction of the Galaxy as the
California cloud and the distances LLA09 derive for these clouds are in
excellent agreement with VLBI parallax measurements.
Briefly we first identify foreground stars in regions of high extinction. To
do this we select all the high extinction pixels ($A_{K}>$ 0.6 magnitudes) in
the map and search for stars projected on these pixels that show “no”
extinction, that is stars whose extinction is less than 3-$\sigma$ above the
background. We then calculate the density of foreground stars takinig into
account the area of the sky occupied by the high extinction pixels. In this
way we found 119 foreground stars within an area of 0.27 square degrees
yielding a foreground star density of $\rho$ $=$ 440$\pm$ 30 deg-2. The
foreground density was also found to be uniform over the extent of the cloud
indicating that the entire complex is likely at a single distance. We compared
this density to the Galactic model of Robin et al. (2003) which predicts
stellar densities as a function of distance and direction in the Galaxy.
Figure 3 shows the plot of stellar density as a function of distance predicted
from the Robin et al. model for the direction of the CMC. The observed
foreground stellar density is also indicated and the intersection of this
value with the model curve gives the distance to the obscuring dust cloud.
This comparison yields a distance to the cloud of 450 $\pm$ 23 pc.
Previous distance estimates for this region range from 125 to 700 pc. Eklöf
(1959) found two extinction layers in this direction from optical photometry
of field stars. The two layers or clouds were found to have distances of 125
and 300-380 pc, respectively. These distances are lower than our value but
since the line-of-sight to the California cloud passes near both the Taurus-
Auriga ( $\approx$ 140 pc distant) and the Perseus ($\approx$ 250 pc distant)
clouds, the layers identified by Eklöf are likely associated with these
foreground complexes. Recently Herbig et al. (2004) derived a spectroscopic
parallax distance of 700 $\pm$ 200 pc to the young stellar cluster embedded in
NGC 1579 at the east end of the California cloud. This estimate is only
marginally compatible with our star count estimate. Finally, the Hipparcos
parallax measurements (Perryman et al., 1997) of $\xi$ Per, the exciting star
of the California Nebula, NGC 1499, suggests a distance between 394 and 877 pc
for the star, consistent with both our distance and that of Herbig et al.
(2004). However, for the remainder of this paper we adopt the star count
distance of 450 pc for the cloud. As we show below, at this distance the cloud
rivals the Orion A (L1641) cloud as the largest and most massive GMC in the
solar neighborhood (i.e., D $<$ 0.5 kpc).
### 3.2 Mass, Size & Structure
We derive the cloud mass directly from the extinction map by integrating the
dust column density over the area of the cloud and assuming a gas-to-dust
ratio:
$M=D^{2}\mu\beta_{K}\int_{cloud}A_{K}(\theta)d^{2}\theta$
where $D$ is the distance to the cloud, $\mu$ is the mean molecular weight
corrected for helium and $\beta_{K}$ is the gas-to-dust ratio [N(HI) $+$
2N(H2)]/$A_{K}$ = 1.67 x 1022 cm-2mag-1 (Lilley, 1955; Bohlin et al., 1978).
The total mass of the CMC is found to be 1.12 $\times$ 105 M⊙ above an
extinction of AK = 0.1 mag. This makes the mass of the CMC, comparable to, if
not slightly greater than, the mass of the Orion, L1641 GMC, usually
considered to be the most massive molecular cloud within 0.5 kpc of the sun.
The CMC is characterized by a filamentary structure and extends over about 10
degrees on the sky which at the distance of 450 pc corresponds to a maximum
physical extent of approximately 80 pc. Above an extinction of about AK = 0.2
mag the cloud has a width of typically 1.5 degrees or 11 pc. However in
regions of high extinction (A${}_{K}>$1.0, mag) the cloud is extremely narrow,
only barely resolved and less than roughly 0.2 pc in width (see Figure 1). The
northern portion of the cloud appears to split into 3 parallel filaments
giving it the appearence of a trident (Figure 1). At the southern end the
cloud again appears to split into (two) parallel filaments which are shorter
and more closely spaced.
The distribution of mass within the CMC provides useful information about its
internal structure and physical state. Figure 4 shows the cummulative mass
fraction of material in the cloud as a function of (infrared) extinction. This
profile was generated using extinction derived by the NICEST method which more
accurately measures the highest extinction regions than using data generated
with the NICER method. The function falls very steeply from low to high
extinctions. For example, the cloud contains less than 1% of its total mass at
extinctions of A${}_{K}>$1 magnitude (i.e., approximatley AV $>$ 9 mag.) In
Table 1 we list the masses of the cloud enclosed by increasing levels of
extinction (1st column). As the extinction level increases by a factor of 10
from AK = 0.1 to 1.0 magnitudes, the enclosed mass decreases by a factor of
100. Inspection of figure 1 shows that for the eastern half of the cloud
(i.e., $\alpha_{2000}>4^{\rm hr}20^{\rm min}$) the highest extinction (AK $>$
0.4 mag) regions are confined to a very narrow spine centered on the primary
axis of the cloud. This indicates that the cloud is stratified, with an
outwardly decreasing density gradient. This in turn implies that gravity is an
important factor in determining the structure of the cloud. Such stratified
structure is common for dense, dark cloud filamentary structures (e.g., Alves
et al. (1998); Lada et al. (1999)) and suggests that such clouds are
dynamically evolved, quasi-stable objects in approximate pressure equilibrium
with their surroundings. In this situation one expects gravity to be more
important than turbulence in determining the structure of the cloud.
### 3.3 Star Formation Activity
Despite its large size and mass, the CMC appears to be very modest in its star
formation activity. Besides the California Nebula, there are no prominent HII
regions in this complex, indicating an absence of recently formed massive O
stars. The California Nebula is excited by the O star, $\xi$ Per, but this
star is a runaway O star from the Per OB2 association located more than 100 pc
closer to the solar system and is not a product of star formation in the CMC.
The best known and most prominent region of star formation within the cloud
appears to be associated with NGC 1579, a reflection nebula containing a young
embedded cluster with about 100 member stars (Andrews & Wolk, 2008). The most
massive young star in the CMC may very well be LK H$\alpha$ 101 which is a
member of the embedded cluster in NGC 1579 and is likely an early B star
(Herbig et al., 2004).
To obtain an estimate of the overall star formation activity in the cloud we
surveyed IRAS point source catalog. We selected all IRAS sources within or
near the boundaries of the cloud that had high quality fluxes in both the 25
and 60 micron bands. In all we found only 24 sources that satisfied our
criteria and can be considered candidate YSOs. In figure 5 we display an image
of IRAS emission from the CMC with the locations of the candidate YSOs
indicated by crosses. In Table 2 we list the IRAS YSO candidates we
identified. All but two of these sources fall within the boundaries of the
cloud delineated by the lowest contour in Figure 1. One of these is a late-
type giant star (IRC 40094) and not a YSO (see table 2). Of the remaining
sources, 17 are projected onto regions of highest extinction, which for the
most part make up the very narrow filamentary spine of the cloud. In
particular 11 sources line up along a dense and narrow filamentary ridge at
the southern end of the cloud. Four of these sources exhibit the IRAS colors
of Planetary Nebulae (Preite-Martinez, 1988), but their close association with
the high extinction ridge makes their status as Planetary Nebulae seem
dubious. The reflection nebula NGC 1579 and its young cluster are embedded
near the southernmost end of this ridge. The brightest IRAS source in the
cloud is associated with LK H$\alpha$ 101, confirming that this source is
likely the most luminous and massive star in the cloud. Besides LK H$\alpha$
101, five other candidates are associated with reflection nebulae and may be
very young Class I protostars. Two of these are also associated with known HH
objects and one of these may be an FU Ori star (see Table 2). The status of
the remaining stars as YSOs requires further confirmation. Overall the IRAS
observations indicate that active star formation is occurring in this cloud,
but at modest levels for a cloud this mass and size.
## 4 Analysis & Discussion: Comparison with Orion
In figure 6 we compare the NICER extinction images of the CMC and OMC at the
same angular scale which closely corresponds to the same physical scale given
the similarity of the distances to both clouds. The extinction map for the OMC
was also derived using 2MASS data in a similar fashion to the CMC map
(LAL09b). The CMC complex is easily as large as the OMC cloud. Moreover, the
two clouds show a surprising similarity in their overall morphology. Both are
filamentary in structure with a long central spine that appears to fork into
two parallel filaments at both ends of the cloud. The OMC does appear to have
many more dark (high extinction) pixels than the CMC however.
Despite the overall close similarity in size, filamentary structure,
kinematics and mass, the two clouds differ dramatically in their level of star
formation activity. For example, while LK H$\alpha$ 101 is the most massive
and only known B star in the CMC, the Orion Nebula region of the OMC alone
contains 20 OB stars, the most massive of which is an O5.5 star (Muench et
al., 2008). Moreover, the CMC appears to contain only one significant embedded
cluster (associated with NGC 1579 and LK H$\alpha$ 101). This cluster contains
about 100 members (Herbig et al., 2004; Andrews & Wolk, 2008). The OMC
contains two significant embedded clusters (ONC and L1641S) and numerous
prominent groups or aggregates of YSOs (e.g., NGC 1977, OMC2, L1641N, HBC 498,
L 1641 C; Peterson & Megeath (2008); Allen & Davis (2008)). The ONC alone
contains approximately 1700 members (Muench et al., 2008; Peterson & Megeath,
2008) so is considerably (an order of magnitude) more populous than NGC 1579,
while L 1641S is similarly rich as NGC 1579. The L1641 dark cloud is the
portion of the OMC south of the Orion Nebula and NGC 1999 regions. In this
region of the cloud Chen et al. (1993) used the co-added IRAS survey to
identify about 100 sources likely to be YSOs. Recent reports of observations
from the Spitzer Space Telescope suggest there may be as many as 750 YSOs in
the L1641 portion of the cloud (Allen & Davis, 2008) suggesting that the
entire OMC presently contains 2000 - 2400 YSOs. Clearly by any measure the
star formation activity in the OMC dwarfs that in the CMC. Indeed, it is
likely that the OMC has produced more than an order of magnitude more stars
than the CMC.
That two such similar nearby GMCs could have such drastically different levels
of star formation is interesting. This indicates that although GMCs are always
sites of star formation, the level of star formation can vary considerably and
is not necessarily sensitive to the mass and size of the cloud. This raises
the interesting question about what factors determine the amount and rate of
star formation in GMCs. Among the factors that could be important are time
(evolution), structure and external influences.
From the HR diagram for NGC 1579 constructed by Herbig et al. (2004) we
estimate that the age of the clusters members would be 1-2 Myr for a distance
of 450 pc, not all that different from the ages of the other nearby active
star forming regions, including the ONC, Taurus, Perseus and Ophiuchus. So any
possible age differences between the CMC and the OMC are relatively small
compared to the large difference in star formation activity between these two
clouds. So it is unlikely that the difference in the yield of star formation
between the two clouds is due to an age difference. Indeed, given the similar
ages of the young stellar populations in both clouds, the enormous difference
in the star formation yields indicates that star formation rates between the
two clouds also significantly differ.
Although structurally similar in most respects there is one aspect that is
significantly different between the two clouds. Specifically, as mentioned
above, the Orion cloud has more pixels at high extinction than does the
California cloud. To make a more careful comparison of this difference in
cloud structure we need to compare the NICEST extinction measurements of the
two clouds at the same physical resolution and with the same foreground and
background stellar densities. To do this we followed the prescription of
Lombardi et al. (2009a), i.e. we recomputed a new, modified map for the
California cloud using parameters that make the physical properties of the
California and Orion map equivalent. More precisely:
* •
We enlarged the pixel scale in the California cloud in order to have the same
physical resolution of the Orion map, $0.113\mbox{ pc pixel}^{-1}$;
* •
The previous operation increased the already relatively large number of
background stars per pixel used to build the California map. Therefore, in
order to have the same average number of background stars per pixel in both
maps, we randomly discarded $\sim 40\%$ of the stars in the California field.
* •
We estimated the density of foreground stars in both clouds, and we required
both clouds to have the same number of foreground stars per pixel. For this
purpose, we added to the California cloud a few ($\sim 104\mbox{ stars
deg}^{-2}$) artificial foreground stars, generated from the colors observed in
the control field.
Finally, we used this modified (lower resolution) map of California cloud to
re-compute the cloud cumulative mass fraction as a function of extinction to
enable a direct comparison with the corresponding profile derived for the
Orion cloud.
In Figure 7 we plot the cumulative mass fraction of each cloud as a function
of infrared extinction ($A_{K}$). The two profiles are strikingly different.
The CMC has a substantially lower fraction of its mass at high extinction.
This difference is also tabulated in table 1, where we have calculated the
masses for both clouds above extinction levels, $A_{K}$, of 0.1, 0.2 and 1.0
magnitudes. As can be ascertained from both the figure and table, the CMC
contains somewhat less than 1% of its mass above $A_{K}=$ 1 magnitude
($A_{V}\approx$ 9 mag) while the OMC contains slightly under 10% of its mass
above the same extinction level.
This difference may provide a significant clue concerning the physical origin
of the different levels of star formation activity between the two clouds. It
has been known for some time that stars exclusively form in dense cores within
GMCs (Lada, 1992). These cores have mean densities of typically 104 to 105
cm-3. Thus one expects that the amount of star formation in a cloud will be
directly related to the total amount of mass it contains at such high
densities. Star forming cores are also dark, characterized by visual
extinctions ($A_{V}$) typically in excess of 5-10 magnitudes. Thus regions of
relatively high density are also the regions of relatively high extinction and
high extinction can be used as a proxy for high density. For example, as
pointed out earlier, regions with visual extinctions in excess of about 10
magnitudes are charactrized by size scales between 0.1 - 0.2 pc which
corresponds to mean molecular hydrogen densities n$(H_{2})>1-2\times 10^{4}$
cm-3. It is interesting in this context to note that in the Ophiuchus cloud,
such dense cores appear visible in submillimeter dust emission only where the
mean extinctions are in excess of at least 7 visual magnitudes (Johnstone,
DiFrancesco & Kirk 2004). If we use AK = 1.0 mag as the indicator of the high
extinction star forming material in a cloud, then we see that the amount of
such material in the CMC is about an order of magnitude less than that in the
OMC. The difference in cloud masses above this extinction threshold is of the
same order as the difference in the number of YSOs in each cloud and perhaps
even the star formation rate. Interestingly the CMC has roughly the same
fraction of mass above this threshold as does the extremely quiescent Pipe
Nebula (LAL06). In the Pipe the total mass above this threshold is only about
200 M⊙ since the Pipe is overall a much smaller complex. A recent deep Spitzer
Space Telescope survey for YSOs in the Pipe uncovered only 18 such objects
over the entire cloud (Forbrich et al. 2009). The ratio of known YSOs in the
Pipe to the number of YSOs ($\sim$200) that we crudely estimate for the CMC,
is roughly equal to 0.1 and is close in value to the ratio (0.2) of the total
mass of high extinction/ high density material in the Pipe to that in the CMC.
If such trends hold with improved inventories of star formation activity in
the CMC and with comparisons between additional clouds, this would imply that
there exists a threshold extinction and presumably volume density for star
formation and once reached there is a more or less constant star formation
efficiency achieved in the dense gas component of molecular clouds. We then
expect the star formation rate to go as SFR $\sim$ $M_{dg}/\tau_{sf}$, where
$M_{dg}$ is the total amount of dense gas (above the threshold) and
$\tau_{sf}$ is the appropriate star formation time scale. If this time scale
is given by the free-fall time ( $\tau_{ff}\sim(G\rho)^{-0.5}$) at the
threshold density, then the star formation rate will be directly proportional
to the total mass at high density. It has not escaped our notice that in
external galaxies the comparison of global FIR star formation rates and
molecular emission from the HCN molecule, a dense gas tracer, suggests such a
relation may characterize the global star formation in galaxies ranging from
normal spirals to ultraluminous starbursts (Gao & Solomon, 2004). Indeed, this
relation between star formation rate and HCN luminosity appears to extend down
to Galactic GMCs (Wu et al. 2005).
It is not clear why the CMC and OMC differ so significantly in their contents
of high extinction/high density material. One possibility could be a
difference in external environments. The OMC is associated with an OB
association within which multiple supernovae are believed to have occurred
over the last 107 years. As a result of the collective action of these
supernovae the cloud may have been compressed by direct interaction with the
supernovae remnants or as a result of increased pressure in the surrounding
environment due to the hot bubble created by the supernovae. The presence of
Barnard’s Loop and the Eridanus superbubble in the region immediately
surrounding the OMC (Bally 2008) indicates the presence of a hot (105-6 K) and
possibly high pressure medium external to the OMC. Similar activity is not
observed near the CMC. However, whether this or some other factor is the cause
of the differences in the extinction profiles between the two clouds cannot be
presently ascertained with any confidence and requires further study.
## 5 Summary & Conclusions
From an analysis of wide-field infrared extinction maps we have identified a
nearby, previously unrecognized, massive molecular cloud within Perseus. Both
a uniform foreground star density and measurements of the cloud’s velocity
field from CO observations indicate that the cloud is likely a coherent
structure at a single distance. We designate this cloud the California
Molecular Cloud due to its physical association with the well known California
Nebula which is located on the cloud’s southern border. From comparison of
foreground star counts with Galactic models we derive a distance of 450 $\pm$
23 parsecs to the cloud. At this distance the cloud extends over roughly 80 pc
and has a mass (derived from the extinction measuremens) of $\approx$ 105 M⊙.
The cloud thus rivals the Orion A Molecular Cloud as the largest and most
massive GMC in the solar neighborhood.
Although surpisingly similar in mass and size to the more famous Orion
Molecular Cloud the California Molecular Cloud displays significantly less
star formation activity. There are more than an order of magnitude fewer YSOs
in the CMC than in the OMC suggesting that both the level of star formation
and perhaps the star formation rate in the CMC are correspondingly an order of
magnitude or more lower than in the OMC.
Analysis of extinction maps of both clouds shows that the CMC contains only
about 10% the amount of high extinction (A${}_{K}>$ 1.0 mag) material as is
found in the OMC. This in turn suggests that the level of star formation
activity and the perhaps star formation rate in these two clouds may be
directly proportional to the total amount of high extinction material and
presumably high density gas within the clouds and that there might be a
density threshold for star formation of order n(H2) $\approx$ a few $\times$
104 cm-3. Once this threshold is reached there would be a more or less
constant star formation efficiency achieved in the dense gas. What adds
somewhat to this surmise is that the CMC contains an order of magnitude more
YSOs than are found in the extremely quiescent Pipe Molecular Cloud and, as it
turns out, the Pipe Cloud contains only 20% the amount of high extinction
material as does the CMC (LAL06). Thus, the overall levels of star formation
activity in these three clouds, the OMC, the CMC and the Pipe appear to be
directly related to the total content of the high extinction/dense material
within them.
Finally, we find the structure of eastern half of the CMC to be well behaved
and centrally condensed with the highest extinction material confined to a
very thin spine along the primary axis of the elongated cloud. This
systematically stratified structure suggests that this portion of the cloud
may be in near pressure equilibrium with its surroundings and strongly self-
gravitating. It is also the region of most active star formation in the cloud.
We are grateful to Tom Dame for kindly providing the CO data and Figure 2.
## References
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* Alves et al. (1998) Alves, J., Lada, C. J., Lada, E. A., Kenyon, S. J., & Phelps, R. 1998, ApJ, 506, 292
* Andrews & Wolk (2008) Andrews, S. M. & Wolk, S. J. 2008, in Handbook of Star Forming Regions Volume 1: The Northern Sky, ed. B. Reipurth, (Astronomical Society of the Pacific, San Francisco), p. 390
* Bally (2008) Bally, J. 2008, in Handbook of Star Forming Regions Volume 1: The Northern Sky, ed. B. Reipurth, (Astronomical Society of the Pacific, San Francisco), p. 459
* Bohlin et al. (1978) Bohlin, R. C., Savage, B. D., & Drake, J. F. 1978, ApJ, 224, 132
* Chen et al. (1993) Chen, H., Tokunaga, A. T., & Fukui, Y. 1993, ApJ, 416, 235
* Connelley et al. (2007) Connelley, M. S., Reipurth, B., & Tokunaga, A. T. 2007, AJ, 133, 1528
* Dame et al. (2001) Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792
* Eklöf (1959) Eklöf, O. 1959 Arkiv for Astronomi, 2, 213
* Forbirch et al. (2009) Forbrich, J., Lada, C.J., Muench, A.A., Alves, J. & Lombardi, M. 2009, ApJ, in press.
* Gao & Solomon (2004) Gao, Y., & Solomon, P. M. 2004, ApJ, 606, 271
* Herbig et al. (2004) Herbig, G. H., Andrews, S. M., & Dahm, S. E. 2004, AJ, 128, 1233
* Indebetouw et al. (2005) Indebetouw, R., et al. 2005, ApJ, 619, 931
* Johnstone et al. (2004) Johnstone, D., Di Francesco, J., & Kirk, H. 2004, ApJ, 611, L45
* Kleinmann et al. (1994) Kleinmann, S. G., et al. 1994, Experimental Astronomy, 3, 65
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* Lombardi (2005) Lombardi, M. 2005, A&A, 438, 169
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* Lombardi et al. (2009a) Lombardi, M., Lada, C. J.& Alves, J. 2009, A&A, submitted.
* Lombardi et al. (2009b) Lombardi, M., Alves, J., & Lada, C. J. 2009b, A&A, in preparation
* Lombardi et al. (2006) Lombardi, M., Alves, J., & Lada, C. J. 2006, A&A, 454, 781
* Lombardi et al. (2008) Lombardi, M., Lada, C. J., & Alves, J. 2008, A&A, 489, 143
* Magakian (2003) Magakian, T. Y. 2003, A&A, 399, 141
* Muench et al. (2008) Muench, A., Getman, K., Hillenbrand, L., & Preibisch, T. 2008, in Handbook of Star Forming Regions, Volume 1: The Northern Sky, ed. B. Reipurth, (Astronomical Society of the Pacific: San Francisco), p.483
* Perryman et al. (1997) Perryman, M. A. C., et al. 1997, A&A, 323, L49
* Peterson & Megeath (2008) Peterson, D. E., & Megeath, S. T. 2008, Handbook of Star Forming Regions, Volume 1: The Northern Sky, ed. B. Reipurth, (Astronomical Society of the Pacific: San Francisco), p. 590
* Preite-Martinez (1988) Preite-Martinez, A. 1988, A&AS, 76, 317
* Robin et al. (2003) Robin, A.C., Reyle, C. Derriere, S., & Picaud, S. 2003 A&A, 409, 523
* Sandell & Aspin (1998) Sandell, G., & Aspin, C. 1998, A&A, 333, 1016
* Ungerechts & Thaddeus (1987) Ungerechts, H., & Thaddeus, P. 1987, ApJS, 63, 645
* Vogt (1973) Vogt, S. S. 1973, AJ, 78, 389
* Wolf (1923) Wolf, M. 1923, Astronomische Nachrichten, 219, 109
* Wu et al. (2005) Wu, J., Evans, N. J., Gao, Y., Solomon, P. M., Shirle, Y. L., & Vanden Bout, P. A. 2005, ApJ, 635, L173
Figure 1: The NICER Infrared extinction map of the California Molecular Cloud.
Contours start at an infrared extinction of AK = 0.2 mag and increase
uniformly in steps of 0.2 mag. The locations of prominent Lynds dark clouds
are indicated as well as the location of NGC 1579, a reflection nebula
illuminated by the early B star, LKH$\alpha$ 101\. The approximate location of
the famous California Nebula is also indicated.
Figure 2: Map of the integrated intensity of 12CO emission toward the
California cloud from the survey of Dame et al. (2001). Top panel shows the CO
spatial map in Galactic coordinates. This map covers roughly the same region
as the extinction map in figure 1. (See also fig 6). The bottom panel shows
the position-velocity map along Galactic longitude. The parallel dashed lines
in the top panel indicate the range of Galactic latitude integrated to produce
the position-velocity map in the bottom panel.
Figure 3: The plot of foreground stellar density vs distance from the Galactic
models of Robin et al. (2003). The observed foreground star density and its
uncertainties toward the highest extinction regions of the cloud are indicated
by the horizontal lines. The intersection of these lines with the model give
the distance to the cloud and the corresponding uncertainties in that
distance.
Figure 4: Plot of the cumulative mass fraction as a function of infrared
extinction, AK, for the CMC.
Figure 5: Three color image of IRAS emission from the California Molecular
Cloud. The region is the same as displayed in figure 1. Yellow crosses mark
the positions of IRAS sources that are candidate YSOs. The bright, saturated
nebulosity in the southwest is the California Nebula, the compact bright
saturated nebulosity in the southeast marks the position of NGC 1579 and the
young emission-line star LK H$\alpha$ 101\.
Figure 6: Comparison of the infrared extinction images of the CMC and OMC
clouds. Given the similar distances, the CMC cloud is comparable in physical
size and is very similar in structure compared to the better known OMC.
Figure 7: The cumulative mass fraction profile of the California Molecular
Cloud compared to that of the Orion Molecular Cloud. The measurements of the
California cloud were corrected for distance so that the measurements of the
two clouds were made at the same physical resolution and properly adjusted for
distance dependent foreground and background stellar densities. The two clouds
have strikingly different profiles, in particular the California cloud
contains substantially less of its mass at high extinction than does the Orion
cloud.
Table 1: Cloud Masses $A_{K}$(mag) | CMC Massaafor 450 pc distance (M⊙) | Adjusted MassbbTo facilitate comparison the individual map pixels of the CMC are here adjusted to have the same foreground & background stellar densities and physical resolution as the pixels in the extinction map of the OMC (see text). (M⊙) | OMC Massccfor 390 pc distance (M⊙) |
---|---|---|---|---
$>$ 0.1 | $1.12\times 10^{5}$ | $1.09\times 10^{5}$ | $7.66\times 10^{4}$ |
$>$ 0.2 | $5.34\times 10^{4}$ | $5.03\times 10^{4}$ | $5.32\times 10^{4}$ |
$>$ 1.0 | $1.09\times 10^{3}$ | $8.12\times 10^{2}$ | $6.59\times 10^{3}$ |
Table 2: Candidate IRAS YSOs
IRAS Source | $F_{12}$ | $F_{25}$ | $F_{60}$ | $F_{100}$ | Notes | ref
---|---|---|---|---|---|---
03507+3801 | 0.78 | 1.62 | 4.81 | 10.5 | IR RNe, HH 462 | 4
03530+4120 | 0.25 | 0.71 | 1.86 | 3.9 | PNe? | 3
04056+4011 | 0.34 | 1.03 | 4.07 | 4.7 | $\cdots$ | $\cdots$
04067+3954 | 0.76 | 8.25 | 38.1 | 82.8 | IR RNe | 4
04073+3800 | 6.48 | 20.1 | 47.9 | 61.5 | HH 464; FU*ddFU Ori object; IR RNe | 4,7,
04088+3834 | 0.25 | 0.33 | 2.54 | 4.8 | $\cdots$ | $\cdots$
04182+3727 | 0.40 | 0.50††Poor quality IRAS flux | 4.35 | 22.1 | $\cdots$ | $\cdots$
04223+3700 | 0.71 | 1.11 | 3.29 | 18.8 | $\cdots$ | $\cdots$
04253+3618 | 0.35 | 1.24 | 6.16 | 12.2 | $\cdots$ | $\cdots$
04256+3624 | 0.25 | 1.03 | 3.89 | 12.2 | $\cdots$ | $\cdots$
04265+3605 | 0.26 | 0.58 | 0.78 | 11.0 | $\cdots$ | $\cdots$
04269+3510 | 362.0 | 340.0 | 3120.0 | 5020.0 | LkH$\alpha$101, NGC 1579 | 1, 2
04269+3550 | 2.83 | 9.92 | 21.3 | 15.6 | PNe?aaPNe $=$ Planetary Nebula. | 3
04271+3538 | 0.34 | 1.75 | 5.74 | 5.2 | $\cdots$ | $\cdots$
04272+3529 | 0.26 | 0.63 | 1.61 | 12.8 | $\cdots$ | $\cdots$
04273+3548 | 0.25 | 0.66 | 2.64 | 15.6 | $\cdots$ | $\cdots$
04274+3553 | 0.36 | 1.58 | 2.84 | 10.6 | PNe?aaPNe $=$ Planetary Nebula. | 3
04275+3519 | 1.58 | 1.69 | 20.9 | 65.1 | RNe | 5
04275+3531 | 0.25 | 0.88 | 5.36 | 6.6 | IR RNebbRNe $=$ Reflection Nebula. | 4
04275+3452 | 0.25 | 0.69 | 1.47 | 9.2 | PNe? | 3
04276+3732 | 0.33 | 0.94 | 4.46 | 4.9 | $\cdots$ | $\cdots$
04315+3617 | 0.71 | 1.37 | 1.98 | 6.6 | $\cdots$ | $\cdots$
04316+3427 | 0.25 | 0.60 | 2.92 | 4.3 | $\cdots$ | $\cdots$
04332+3658 | 8.68 | 4.01 | 0.84 | 8.3 | IRC 40094ccField Giant, SpT: M9. | 6
References. — (1) Andrews & Wolk 2008; (2) Herbig et al. 2004; (3) Preite-
Martinez 1988; (4) Connelley et al. 2007; (5) Magakian, T.Y. 2003; (6) Vogt
1973; (7) Sandell & Aspin 1998
|
arxiv-papers
| 2009-08-05T19:45:33 |
2024-09-04T02:49:04.469786
|
{
"license": "Public Domain",
"authors": "Charles J. Lada, Marco Lombardi, Joao F. Alves",
"submitter": "Charles J. Lada",
"url": "https://arxiv.org/abs/0908.0646"
}
|
0908.0673
|
# BEM3D: a free adaptive fast multipole boundary element library
M. J. Carley
###### Abstract
The design, implementation and analysis of a free library for boundary element
calculations is presented. The library is free in the sense of the GNU General
Public Licence and is intended to allow users to solve a wide range of
problems using the boundary element method. The library incorporates a fast
multipole method which is tailored to boundary elements of order higher than
zero, taking account of the finite extent of the elements in the generation of
the domain tree. The method is tested on a sphere and on a cube, to test its
ability to handle sharp edges, and is found to be accurate, efficient and
convergent.
## 1 INTRODUCTION
The boundary element method has become accepted as a standard computational
technique for many problems and, when combined with a fast multipole method,
it has proven capable of solving very large problems, primarily in scattering
calculations for acoustic and electromagnetic applications. This paper
describes bem3d, a free library released under the GNU General Public Licence,
for the solution of general problems using the boundary element method,
incorporating a fast multipole method which is tailored to boundary element
problems using higher-order elements.
Free software has become an important part of the resources available to those
working in scientific and computational fields over the last few years.
Standard codes and libraries for many of the operations routinely carried out
in computation are now freely available and, because their source code is
openly published, they can be modified and improved by experts in the field.
Because codes are distributed as libraries for a specific purpose, it is
possible to select modules which can be combined in codes for particular
applications. This avoids the problem of reinventing the wheel, speeding code
development and improving reliability.
The library described in this paper makes use of a number of existing free
libraries [1, 2, 3] for computational applications, as well as a standard
library of portability and utility functions. The existing software is
extended by adding functions for the handling of boundary elements, for fast
multipole calculations and some support for parallel systems. The intention is
that other users will be able to make use of bem3d to solve their own problems
by adding a Green’s function for the partial differential equation in
question. This paper describes the design choices which have been made in
bem3d and the implementation of certain features including a novel adaptive
fast multipole method for boundary elements.
## 2 BASIC CODE STRUCTURE
The boundary element library, bem3d, has been written with a number of goals
in mind. The first of these is generality: users should be able to solve
different physical problems and implement different solution techniques and
elements as required. The second is that existing wheels should be used rather
than being reinvented: the codes use other, high-quality, free libraries for
various aspects of the work. These are the GNU Triangulated Surface library
(GTS) for geometry handling [1], the GNU Scientific Library [2], LAPACK [3] in
the iterative solver and the GLIB library for portability functions and
special data structures, such as the tree used in the fast multipole method.
The GMSH suite of meshing and visualization tools [4] is used to generate
geometries and examine results. Using existing free software improves code
reliability and gives users an automatic upgrade as these codes are improved.
The library and its programs have been written in C, partly because C is
nearly universally supported across the range of computing platforms, partly
because the other libraries used have C interfaces, and also because C allows
the use of programming techniques which make extension and customization
easier, for example, through the use of loadable modules, an approach which is
supported by the GLIB library. The resulting code has been written as a number
of separate libraries to facilitate code re-use and to encourage generality in
the coding.
### 2.1 Problem definition
For computational purposes, the partial differential equation for a problem is
defined by its Green’s function. Boundary integral equations have been applied
in many areas of engineering and science and here we concentrate on the
Laplace and Helmholtz potential problems which have applications in fluid
dynamics [5] and in acoustic scattering [6, 7]. The form of the integral
equation is identical in each case, with only the Green’s function being
different. For an unbounded domain containing surface(s) $S$ with outward
pointing normal $\mathbf{n}$, the potential $\phi$ is given by:
$\displaystyle C(\mathbf{x})\phi(\mathbf{x})$
$\displaystyle=\int_{S}G\frac{\partial\phi}{\partial n_{1}}-\phi\frac{\partial
G}{\partial n_{1}}\,\mathrm{d}S,\quad G=\frac{1}{4\pi R}$ (1a) $\displaystyle
C(\mathbf{x})\phi(\mathbf{x})$
$\displaystyle=\int_{S}G\frac{\partial\phi}{\partial n_{1}}-\phi\frac{\partial
G}{\partial n_{1}}\,\mathrm{d}S,\quad G=\frac{\mathrm{e}^{\mathrm{j}kR}}{4\pi
R},$ (1b)
where $k=\omega/c$ is the wavenumber of the Helmholtz problem,
$R=|\mathbf{x}-\mathbf{x}_{1}|$ and subscript $1$ denotes variables of
integration. The constant $C$ depends on the field point position
$\mathbf{x}$:
$\displaystyle C(\mathbf{x})$
$\displaystyle=\left\\{\begin{matrix}0&&\text{$\mathbf{x}$ inside $S$};\\\
1&&\text{$\mathbf{x}$ outside $S$};\\\ 1+\int_{S}\frac{\partial
G_{0}}{\partial n_{1}}\,\mathrm{d}S&&\text{$\mathbf{x}$ on
$S$}\end{matrix}\right.$ (2)
with $G_{0}=1/4\pi R$. When $\mathbf{x}$ is on $S$ with the boundary condition
$\chi=\partial\phi/\partial n$ prescribed, Equation 1 is a boundary integral
equation for $\phi$ and can be solved using the boundary element method.
The geometry $S$ is discretized into a number of elements with given nodes
$\mathbf{x}_{i}$, $i=1,\ldots,N$ and interpolation (shape) functions $L_{i}$.
This yields the system of equations:
$\displaystyle\sum_{j=1}^{N}A_{ij}\phi_{j}$
$\displaystyle=\sum_{j=1}^{N}B_{ij}\chi_{j},$ (3)
with
$\displaystyle A_{ij}$ $\displaystyle=C_{ij}+\sum_{m}\int L_{j}\frac{\partial
G}{\partial n_{1}}\mathrm{d}S_{m},$ $\displaystyle C_{ij}$
$\displaystyle=\left\\{\begin{matrix}1+\int\partial G_{0}/\partial
n_{1}\,\mathrm{d}S,\quad i=j\\\ 0,\quad i\neq j,\end{matrix}\right.$
$\displaystyle B_{ij}$ $\displaystyle=\sum_{m}\int L_{j}G\,\mathrm{d}S_{m},$
where the summation over $m$ is taken on elements which contain collocation
point $j$, the shape function $L_{j}$ is the shape function on element $m$
corresponding to point $j$ and $S_{m}$ is the surface of element $m$.
Conceptually, the boundary element method consists of discretizing the
surface, assembling the matrices $A$ and $B$, and solving the system of
Equation 3. In practice, as we shall see below, this is not necessarily a
feasible approach for large problems, but it is the default method and one
which can be used as a baseline for assessing other techniques.
In bem3d, the integrations in assembling the matrices are performed using
quadrature rules for singular integrands [8], Hayami’s transformation for
near-singular integrals [9] and the symmetric rules of Wandzura and Xiao [10].
The system is solved using a library, sisl, based on LAPACK [3], which
implements the ‘templates’ of Barrett et. al [11]. The solver library allows
for parallel solution of problems using the MPI standard and also has a
matrix-free option for use with fast-multipole methods, as described below,
§3.
### 2.2 Element types
Within the code of bem3d, elements are represented as a collection of
triangles, based on the underlying GTS triangle data type. This raises the
question of how to link computational information to geometric while retaining
the freedom to introduce new element types. Previous work using GTS surfaces
[12, 13, 14] introduced a new data type for the vertices which allowed them to
have an integer index which could be used in assembling the system matrices.
This is not a satisfactory solution, however, when more general elements are
to be implemented. The first reason is that only linear triangular elements
based on the underlying triangles can be be used; the second is that there is
no good way to introduce discontinuities at sharp edges, an essential part of
representing general geometries.
The solution which has been adopted is shown in Figure 1. Within the library,
an element is composed of a list of triangles, a list of collocation points
with their indices, a list of geometric nodes, interpolation functions for
surface data and interpolation functions for the geometry. For an
isoparametric element, the interpolation functions for the geometry are the
same as those for the surface data and the geometric nodes are the same as the
collocation points. This approach allows nodes to be indexed independently on
each element, to support discontinuities, while allowing the underlying GTS
surface to be geometrically valid.
Figure 1 shows the triangular elements which have been implemented;
quadrilateral elements have also been implemented using the same approach. The
shape functions employed are polynomials and elements of order zero to three
have been developed. Figure 1 shows each element with its underlying GTS
triangles and the collocation points. In the case of a zero order element,
there is one collocation point, placed at the centre of the triangular
element. The element is made up of three GTS triangles in order to allow for a
constant solution across the computational triangle. In the case of a linear
element, the computational triangle coincides with a GTS triangle and likewise
the collocation points. For the second and third order elements, the element
does not coincide with GTS triangles due to the element curvature, with the
underlying triangles being formed by joining the collocation points with
straight line segments.
Finally, a mesh is implemented as a GTS surface supplemented with a list of
elements, a lookup table connecting elements to triangles and a lookup table
connecting collocation points to their indices. These lookup tables are
implemented as hash tables, a well-known method for efficient referencing of
data.
|
---|---
|
Figure 1: Triangular element types: zero, first, second and third order:
circles show computational nodes, solid lines the element boundary and dashed
lines the GTS triangles.
### 2.3 Detection and indexing of sharp edges
Figure 2: Renumbering the nodes at the corner of a cube
In order to solve problems involving general geometries, a boundary element
code must have some method of dealing with sharp edges. In the case of zero-
order elements where collocation points lie on one, and only one, element,
there is no difficulty, but in the general case where collocation points lie
on more than one element, it is required that the scheme be able to support a
discontinuity in solution or boundary condition at a sharp edge. As in
previous work, this is implemented by giving points of discontinuity multiple
indices. This allows for a discontinuity as the sharp edge is approached. An
example of such indexing, at the corner of a cube, is shown in Figure 2.
Points are shown with their surface normals and their indices, while elements
are labelled with capital letters. As an example of a sharp edge, nodes on the
line between elements B and C each have two indices while those between
elements B and A have only one: the surface is smooth at these points. The
exception is the corner of the cube where the node has three indices
corresponding to the three planes which meet there.
A method has been developed to automatically detect and index sharp edges and
corners, an important practical point in applications such as aerodynamics
[15] where the imposition of an edge condition can be difficult and will
ideally be performed without user intervention. To index the nodes of a mesh,
each element is visited in turn. On each element, the currently unindexed
nodes are visited. If a node is shared with adjoining elements, the surface
normal at the node is computed on all of the elements. If the angle between
normals is less than some specified value, the surface is taken to be smooth
and the node is given one index. Otherwise, it is given a different index for
each normal which deviates from the reference normal by more than the
specified tolerance.
### 2.4 Code structure
Figure 3: Structure of bem3d codes: arrows indicate links to libraries.
Executables are shown boxed and libraries circled.
The structure of the codes in bem3d is shown in Figure 3. The sequence shown
contains all of the steps which might be needed, but in many problems they
will not all be necessary. The first is a pre-processing stage to generate the
geometry and elements and to index the nodes. In most cases, the geometry is
generated using GMSH [4] and converted to the bem3d format and indexed using a
conversion program. Boundary conditions are then supplied, either directly, or
by computing the field and its gradients from a prescribed source. If the
problem is being solved by direct solution of a matrix system, the matrices
are assembled and stored to be used by the solver; if the fast multipole
method is used, the information required for the matrix multiplications is
generated and applied directly in the same code. If required, the field
quantities can then be computed. Finally, any post-processing and
visualization which might be required are performed by converting the
solutions and mesh to the GMSH format.
The linkages between libraries and codes are shown in Figure 3 with
executables shown boxed and libraries circled. The libraries which are used
are the main bem3d library, wmpi, a set of wrapper functions for interfacing
to MPI, sisl, a simple iterative solver library and gmc, which contains the
code for fast and accelerated multipole calculations.
## 3 ACCELERATED EVALUATION OF INTEGRALS
Since its introduction, the fast multipole method has been used by a number of
workers to accelerate the boundary element and to reduce its memory
requirements [16]. The standard approach of direct solution of the matrix
system has time and memory requirements which scale as $N^{2}$, which quickly
exceed the resources available on even the most powerful computers. The method
implemented in bem3d allows problems to be solved directly on parallel systems
using the MPI standard but it was considered better to implement a fast
multipole method for solution of problems and for calculation of radiated
fields. The fast multipole method was originally developed for point sources,
rather than elements of finite extent, and is usually implemented using
spherical harmonics. In the method implemented in bem3d, the basic algorithm
is similar to the original adaptive fast multipole method [17], but with two
important differences. The first is the use of Taylor series in place of
spherical harmonics, as used by Tausch in his non-adaptive method [18, 19], in
order to cater for a wide variety of Green’s functions. The second is a
convergence radius criterion for the near field, an idea developed [20] to
allow for finite size elements which may not be completely contained within a
box of the tree decomposing the computational surface. The rest of this
section details the sequence of procedures involved in performing a fast
multipole solution of the boundary element equations.
### 3.1 Hierarchical domain decomposition
Figure 4: Decomposition of a box of points: the parent box is shown with one child box indicated in bold | |
---|---|---
a | b | c
Figure 5: Decomposition of a set of points Figure 6: Tree structure for
decomposition of Figure 5
The first step in a fast multipole method is to recursively decompose the
domain into a tree structure. The tree is made up of boxes aligned with the
coordinate axes with each box containing up to eight boxes, formed by dividing
the main box along each of the axes, Figure 4. The main box is called a
‘parent box’ and the boxes formed by subdivision are the ‘child boxes’. The
decomposition of the domain is initialized by forming a box which contains all
of the vertices of the computational surface $S$. This box is said to be at
level zero. The box is then subdivided to form the level one boxes, which are
themselves divided to form the level two boxes and so on. At each stage of
division, the number of vertices in a box is checked. If it is less than some
prescribed number $B$, division of the box is terminated and the box is called
a ‘leaf’. Recursive division in this manner gives rise to a tree structure
where the box at each node of the tree encloses all of the boxes in the nodes
below it.
The process of subdivision is shown in two dimensions in Figure 5. An initial
distribution of points is enclosed by a level zero box which is divided into
four level one child boxes, labelled A, B, C and D, Figure 5a. Following the
decomposition of box C, Figure 5b shows the level two boxes a, b, c and d. In
this case, box c has fewer than $B$ points and is subdivided no more.
Following the subdivision one more time, box a is divided into four level
three boxes, 1, 2, 3 and 4, of which boxes 2 and 3 are not divided again.
Part of the resulting tree structure is shown in Figure 6, with leaf nodes of
the tree shown boxed and parent nodes shown circled. The parent node $R$ at
level 0 has four child nodes, corresponding to the four boxes of Figure 5a and
so forth. The tree structure can be used to rapidly locate a point and as the
basis for fast integration methods.
The final step in generating the tree is to attach the elements of the mesh to
the leaf nodes. The method adopted here is to list the elements attached to
the vertices of a leaf box and link this list to the leaf node.
### 3.2 Taylor multipole expansions
A fast multipole method works by replacing a list of elements attached to a
box with a set of multipole coefficients which can be used to compute the
field due to the elements in the box far field, where ‘far field’ will be
defined more precisely below. This was originally done using an expansion of
the field in spherical harmonics, but here Taylor series expansions have been
used since they allow the unified treatment of a range of Green’s functions
[18, 19], an important point in constructing a general code.
The field integrals to be considered are of the form:
$\displaystyle I(\mathbf{x})$
$\displaystyle=\int_{S_{b}}f(\mathbf{x_{1}})G(R)\,\mathrm{d}S_{b},$ (4)
where $f(\cdot)$ is a source term and $S_{b}$ is the surface of the elements
inside some box of the tree decomposing the surface $S$. To approximate
$I(\mathbf{x})$, we expand the Green’s function in a Taylor series about some
point $\mathbf{x}_{c}$ inside the box:
$\displaystyle G(R)$
$\displaystyle\approx\sum_{h=0}^{H}\sum_{m,n,k}(-1)^{h}G_{mnk}(x_{1}-x_{c})^{m}(y_{1}-y_{c})^{n}(z_{1}-z_{c})^{k},$
(5a) $\displaystyle G_{mnk}$
$\displaystyle=\frac{1}{m!}\frac{1}{n!}\frac{1}{k!}\left.\frac{\partial^{h}G}{\partial
x^{m}\partial y^{n}\partial z^{k}}\right|_{\mathbf{x}_{1}=\mathbf{x}_{c}},$
(5b)
where the summation over $m$, $n$ and $k$ is taken over all values of
$m+n+k=h$ and the symmetry relation $\partial G/\partial x_{1}=-\partial
G/\partial x$ has been used. The integral can then be rewritten:
$\displaystyle f(\mathbf{x})$
$\displaystyle\approx\sum_{h=0}^{H}\sum_{m,n,k}(-1)^{h}G_{mnk}I_{mnk},$ (6)
$\displaystyle I_{mnk}$
$\displaystyle=\int_{S_{b}}f(\mathbf{x}_{1})(x_{1}-x_{c})^{m}(y_{1}-y_{c})^{n}(z_{1}-z_{c})^{k}\,\mathrm{d}S_{b}.$
The integrations can be performed the interpolation functions on each element
so that:
$\displaystyle I_{mnk}$ $\displaystyle=\sum_{i}f_{i}w^{(mnk)}_{i}$ (7)
where the index $i$ runs over all points on elements connected to the source
box and $w_{i}^{(mnk)}$ is a weight precomputed as:
$\displaystyle w_{i}^{(mnk)}$
$\displaystyle=\int_{S_{b}}L_{i}(x_{1}-x_{c})^{m}(y_{1}-y_{c})^{n}(z_{1}-z_{c})^{k}\,\mathrm{d}S_{b}$
(8)
which is independent of the underlying Green’s function and so can be stored
and used in multiple problems.
To evaluate the field due to the source elements contained in a box, the
Taylor series can be used in an expansion about a point
$\mathbf{x}^{\prime}_{c}$:
$\displaystyle f(\mathbf{x})$
$\displaystyle\approx\sum_{l=0}^{L}\sum_{m,n,k}F_{mnk}(x-x^{\prime}_{c})^{m}(y-y^{\prime}_{c})^{n}(z-z^{\prime}_{c})^{k}$
where the expansion coefficients $F_{mnk}$ are computed by differentiation of
Equation 6:
$\displaystyle\left.\frac{\partial^{q+r+s}f}{\partial x^{q}\partial
y^{r}\partial z^{s}}\right|_{\mathbf{x}=\mathbf{x}_{c}^{\prime}}$
$\displaystyle\approx\sum_{h=0}^{H}\sum_{m,n,k}(-1)^{h}G_{m+q,n+r,k+s}I_{mnk},$
$\displaystyle F_{qrs}$
$\displaystyle=\sum_{h=0}^{H}\sum_{m,n,k}(-1)^{h}\frac{1}{q!r!s!}G_{m+q,n+r,k+s}I_{mnk},$
(9)
which can be used to generate a local expansion about the centre of one box of
the field due to another box.
Two basic tools are now required which allow the multipole expansions of a
tree of boxes to be used to accelerate the evaluation of the field integrals.
The first is the coefficient shift operator which allows coefficients
$I_{mnk}$ evaluated about one centre to be shifted to another centre. This
allows the coefficients about the centre of a parent box to be evaluated from
the coefficients of its child boxes rather than having to be recomputed.
Writing:
$\displaystyle I_{mnk}^{\prime}$
$\displaystyle=\int_{S}f(\mathbf{x}_{1})(x_{1}-x_{c}^{\prime})^{m}(y_{1}-y_{c}^{\prime})^{n}(z_{1}-z_{c}^{\prime})^{k}\,\mathrm{d}S,$
$\displaystyle=\int_{S}f(\mathbf{x}_{1})(x_{1}-x_{c}+(x_{c}-x_{c}^{\prime}))^{m}(y_{1}-y_{c}+(y_{c}-y_{c}^{\prime}))^{n}(z_{1}-z_{c}+(z_{c}-z_{c}^{\prime}))^{k}\,\mathrm{d}S,$
and applying the binomial theorem:
$\displaystyle I_{mnk}^{\prime}$
$\displaystyle=\sum_{q=0}^{m}\sum_{r=0}^{n}\sum_{s=0}^{k}\binom{m}{q}\binom{n}{r}\binom{k}{s}(x_{c}-x_{c}^{\prime})^{q}(y_{c}-y_{c}^{\prime})^{r}(z_{c}-z_{c}^{\prime})^{s}I_{(m-q)(n-r)(k-s)},$
(10)
which allows the coefficients $I_{mnk}^{\prime}$ about centre
$\mathbf{x}_{c}^{\prime}$ to be computed in terms of coefficients $I_{mnk}$
about centre(s) $\mathbf{x}_{c}$.
The second basic tool is the expansion shift operator which shifts the
expansion about one centre $\mathbf{x}_{c}$, Equation 9, to give an expansion
about another centre $\mathbf{x}_{c}^{\prime}$. Writing:
$\displaystyle f(\mathbf{x})$
$\displaystyle=\sum_{l=0}^{L}\sum_{m,n,k}F^{\prime}_{mnk}(x-x^{\prime}_{0})^{m}(y-y^{\prime}_{0})^{n}(z-z^{\prime}_{0})^{k},$
$\displaystyle=\sum_{l=0}^{L}\sum_{m,n,k}F_{mnk}(x-x^{\prime}_{0}+(x_{0}^{\prime}-x_{0}))^{m}(y-y^{\prime}_{0}+(y_{0}^{\prime}-y_{0}))^{n}(z-z^{\prime}_{0}++(z_{0}^{\prime}-z_{0}))^{k},$
$\displaystyle=\sum_{l=0}^{L}\sum_{m,n,k}F_{mnk}\sum_{q=0}^{m}\sum_{r=0}^{n}\sum_{s=0}^{k}(x-x^{\prime}_{0})^{m-q}(x_{0}^{\prime}-x_{0})^{q}(y-y^{\prime}_{0})^{n-r}(y_{0}^{\prime}-y_{0})^{r}(z-z^{\prime}_{0})^{k-s}(z_{0}^{\prime}-z_{0})^{s},$
(11)
from which the contribution of the original coefficients $F_{mnk}$ to the
shifted coefficients $F_{mnk}^{\prime}$ can be identified. This allows a
parent box expansion to shifted to its children.
### 3.3 Derivatives of Green’s functions
In order to compute the field due to a set of multipole coefficients, we
require the derivatives of the Green’s function of a problem. Tausch [18]
gives a recursive algorithm which efficiently and stably generates the
derivatives of a Green’s function up to some required order.
Given a Green’s function $G(R)$ which is a function of source-observer
distance $R=|\mathbf{r}|$, $\mathbf{r}=(x,y,z)$, the function:
$\displaystyle G^{(h)}(R)$
$\displaystyle=\left(\frac{1}{R}\frac{\partial}{\partial R}\right)^{(h)}G(R),$
is defined. The sequence of functions $G^{(h)}$ can be computed recursively.
For the Laplace Green’s function, $G=1/4\pi R$ and:
$\displaystyle G^{(0)}(R)$ $\displaystyle=\frac{1}{4\pi R},\quad
G^{(h+1)}=-\frac{2h+1}{R^{2}}G^{(h)}(R),$
while for the Helmholtz Green’s function, $G=\exp[\mathrm{j}kR]/4\pi R$:
$\displaystyle G^{(h+1)}$
$\displaystyle=-\frac{2h+1}{R^{2}}G^{(h)}(R)-\frac{k^{2}}{R^{2}}G^{(q-1)}(R),$
with the correction of a typographical error in the original reference.
Given the sequence $G^{(h)}$, the derivatives of the functions with respect to
the components of $\mathbf{r}$ can be found from:
$\displaystyle\frac{\partial^{m+1+n+k}G^{(h)}}{\partial x^{m+1}\partial
y^{m}\partial z^{k}}$
$\displaystyle=m\frac{\partial^{m-1+n+k}G^{(h+1)}}{\partial x^{m-1}\partial
y^{n}\partial z^{k}}+x\frac{\partial^{m+n+k}G^{(h+1)}}{\partial x^{m}\partial
y^{m}\partial z^{k}},$ (12a)
$\displaystyle\frac{\partial^{m+n+1+k}G^{(h)}}{\partial x^{m}\partial
y^{n+1}\partial z^{k}}$
$\displaystyle=n\frac{\partial^{m+n-1+k}G^{(h+1)}}{\partial x^{m}\partial
y^{n-1}\partial z^{k}}+y\frac{\partial^{m+n+k}G^{(h+1)}}{\partial
x^{m}\partial y^{m}\partial z^{k}},$ (12b)
$\displaystyle\frac{\partial^{m+n+k+1}G^{(h)}}{\partial x^{m}\partial
y^{n}\partial z^{k+1}}$
$\displaystyle=k\frac{\partial^{m+n+k-1}G^{(h+1)}}{\partial x^{m}\partial
y^{n}\partial z^{k-1}}+z\frac{\partial^{m+n+k}G^{(h+1)}}{\partial
x^{m}\partial y^{n}\partial z^{k}}.$ (12c)
To compute the derivatives of the Green’s function up to a given order $H$,
Tausch gives the following scheme [18]:
1. 1.
compute $G^{(h)}$, $h=0,\ldots,H$;
2. 2.
compute $\partial^{m+n+k}G^{(h)}/\partial x^{m}\partial y^{n}\partial z^{k}$,
for $m+n+k\leq H-h$, $h=H-1,\ldots,0$, using Equations 12;
3. 3.
set $\partial^{m+n+k}G/\partial x^{m}\partial y^{n}\partial
z^{k}=\partial^{m+n+k}G^{(0)}/\partial x^{m}\partial y^{n}\partial z^{k}$, for
$m+n+k=0,\ldots,H$.
### 3.4 Box near field
Figure 7: Elements stick out of their boxes Figure 8: ‘Inheritance’ of child
convergence radius $r_{C}$ by parent box with convergence radius $r_{P}$
The tree decomposing the domain can be used to accelerate the computation of
field integrals, using the methods described in the next two sections. The
basic approach is to use the multipole expansion about a box centre to compute
the field ‘far’ from the box and to use full integration over the box elements
‘close’ to the box. This requires a definition of ‘near’ and ‘far’ which can
be used in deciding which procedure to use. In the original version of the
fast multipole method [17], the point sources used were completely contained
in a box and the near field of a box was defined as its neighbours. When
elements of finite extent are used, they will inevitably stick out of the
boxes to which they are attached and it is not clear how to define the
boundary of the ‘far field’ of the box.
A number of approaches have been discussed to deal with this problem, some of
them limited to zero order elements. Tausch [19] defines a separation ratio
which can be used in deciding when boxes lie in each others’ near field. This
has the disadvantage that it requires the boxes to be at the same level in the
tree, i.e. that they be the same size. Another approach, employed in a
spherical harmonic fast multipole code [20], is to use the convergence radius
of the expansion about a box centre define the box near field. This approach
has been used in bem3d, with the convergence radius $r_{C}$ given by the
maximum distance between the box centre and a vertex of an element attached to
the box (which may well not lie in the box proper). A point lies in the far
field of the box if its distance to the box centre is greater than $r_{C}$.
Two boxes, which need not be the same size, lie in each others’ far fields if
the distance between their centres is greater than the sum of their
convergence radii.
Finally, the convergence radius $r_{P}$ of a parent box can be derived from
the radii of its child boxes using the approach shown in Figure 8:
$\displaystyle r_{P}$ $\displaystyle=r_{C}+d_{C}/2,$ (13)
where $r_{C}$ is the maximum of the convergence radii of the child boxes and
$d_{C}$ is the length of a child box’s diagonal.
### 3.5 Accelerated field evaluation
The first principal application of the basic techniques of the previous
sections is in accelerated evaluation of the field due to a known source
distribution over a surface. Given the tree decomposition of Section 3.1, the
multipole moments computed at each leaf node and shifted upwards in the tree,
Section 3.2 and a field point $\mathbf{x}$, the field may be evaluated by
traversing the tree and computing the contribution from each box of the tree
in turn, descending the tree only far enough to evaluate the contribution of a
branch to sufficient accuracy.
The algorithm may be summarized as follows:
1. 1.
set $I(\mathbf{x})=0$;
2. 2.
set the current box to the root of the tree at level 0;
3. 3.
for the current box:
1. (a)
if the distance to the box centre $|\mathbf{x}-\mathbf{x}_{c}|>r_{C}$,
evaluate the multipole expansion of Equation 9 and add to $I(\mathbf{x})$;
terminate descent of this branch;
2. (b)
if the distance to the box centre $|\mathbf{x}-\mathbf{x}_{c}|\leq r_{C}$ and
the current box is a parent, repeat step 3 for each child box;
3. (c)
if the distance to the box centre $|\mathbf{x}-\mathbf{x}_{c}|\leq r_{C}$ and
the current box is a leaf node, evaluate the field by direct integration over
the elements of the box and add to $I(\mathbf{x})$; terminate descent of this
branch.
This algorithm can be used to evaluate the field at a small number of points
which may be far from the surface $S$: it is efficient and avoids the setup
costs involved in the fast multipole method proper. When the field must be
evaluated at a large number of points, for matrix multiplication, say, the
fast multipole method described in the next section is used.
### 3.6 Fast matrix multiplication
In order to further accelerate the computation of integrals in the boundary
element method, the technique of the previous section can be extended to a
true fast multipole method, at the expense of some extra pre-processing. This
yields a method which speeds up the matrix multiplications required for the
solution of the boundary element equations by using local expansions of the
field about the centre of each leaf node of the tree.
Figure 9: Interaction and near field lists for a tree box
The algorithm is similar to that of the previous section but with some extra
procedures related to the evaluation of the near field terms and the
separation of boxes into near and far field. The first step is, as before, to
generate the tree decomposing the surface, including the multipole
coefficients and convergence radii for the boxes. Each box now has linked to
it two lists: the near-field list and the interaction list. The near-field
list consists of boxes whose contribution to the field in the box must be
computed by direct integration over the attached elements. The interaction
list is made up of boxes whose contribution can be computed using the
multipole expansion. Figure 9 shows an example. The light grey box in the
middle of the grid is the box whose lists are shown. The darker boxes are
those in the near-field list, using the criterion of Section 3.4 which states
that boxes are well-separated iff:
$\displaystyle|\mathbf{x}_{c}^{(1)}-\mathbf{x}_{c}^{(2)}|>r_{C}^{(1)}+r_{C}^{(2)},$
(14)
where $\mathbf{x}_{c}^{(i)}$ and $r_{c}^{(i)}$ are the centre and radius of
convergence of each of the boxes. The darkest boxes are those on the
interaction list whose contribution to the field in the main box can be
computed using multipole expansions. The field in the box is computed as a sum
of contributions from the near field and interaction list boxes and from the
local expansion about the centre of the parent box which is computed in the
same way. In Figure 9, the box labelled 1 is in the near field list, even
though it is larger than the main box, because it is a leaf node and must
contribute directly to the field. The box labelled 2 is in the near field
list, even though it does not touch the main box, because it does not meet the
separation criterion of Equation 14.
The near-field and interaction lists can be generated using the following
algorithm, applied to each box $B$ descending the tree starting at level zero:
1. 1.
initialize the interaction and near-field lists of $B$ to be empty;
2. 2.
traverse the near field list of the box’s parent $P$ and for each box $N$ in
the list:
1. (a)
if $N$ is a leaf node:
1. i.
if $N$ and $B$ are well-separated, add $N$ to the interaction list of $B$;
2. ii.
if $N$ and $B$ are not well-separated, add $N$ to the near-field list of $B$;
2. (b)
if $N$ is not a leaf node, traverse the child boxes $C$ of $N$ and:
1. i.
if $C$ and $B$ are well-separated, add $C$ to the interaction list of $B$;
2. ii.
if $C$ and $B$ are not well-separated, add $C$ to the near-field list of $B$;
Given a tree with multipole coefficient weights $w^{(mnk)}$ up to order
$m+n+k=H$, and near-field and interaction lists for each box, a matrix
multiplication can be performed as follows:
1. 1.
for each leaf box, compute the multipole moments
$I_{mnk}=\sum_{i}f_{i}w^{(mnk)}_{i}$, $m+n+k\leq H$;
2. 2.
traverse tree from bottom to top, calculating box moments by accumulating the
contribution from child boxes using Equation 10;
3. 3.
traverse tree from top to bottom, computing local expansions to order $L$:
1. (a)
shift parent local expansion to box centre using Equation 11;
2. (b)
add local expansion terms due to boxes in interaction list;
each leaf box now has an expansion about its centre which gives the field in
the box due to all other boxes except those in the near-field list;
4. 4.
for each leaf box, evaluate the local expansion at each enclosed vertex and
add the contribution from boxes in the near-field list.
In practice, the near-field contribution is precomputed and stored as a sparse
matrix for re-use in the solution procedure and the expansion order $L$ is a
function of depth in the tree [19] in order to avoid excessively high order
expansions in small boxes. The order $L_{l}$ at level $l$ is set to
$L_{l}=\min(L_{\max},L_{\min}+l_{\max}-l)$.
## 4 NUMERICAL TESTS
The performance of the boundary element codes has been assessed in terms of
accuracy, speed and memory. The test uses a point source placed inside the
surface to generate a potential and a boundary condition (potential gradient).
The system is solved for the boundary condition which should recover the
original surface potential due to the point source. The error measure is the
rms difference between the original and computed potential. Two geometries
have been used, a unit sphere and a cube of edge length 2 centred on the
origin. The sphere is generated by successive refinement of an initial
surface, allowing tests to be carried out on a smooth surface with control
over the number of vertices. The cube is generated using GMSH [4] with
successively smaller edge lengths. This tests the ability of bem3d to deal
with sharp edges using the methods outlined earlier. Tests have been conducted
with first and second order elements and the Laplace and Helmholtz ($k=2$)
equations have been solved. The fast multipole method was used to solve
problems of all sizes and the direct matrix method was used for the smaller
problems which would fit in memory. Calculations were performed on a 3 G Hz
Pentium 4 with 1Gb of memory.
Data reported are the number of vertices, the maximum number of vertices $B$
in a tree box, the setup time and solution time and the maximum memory used
during solution. For matrix solution, the setup time is the time required to
assemble the system matrices. For the fast multipole method, the setup time is
that needed to generate the tree for the mesh and compute and store the near-
field matrix. In neither case is the time needed for disk storage and recovery
included. In the fast multipole case, the time reported is the total setup
time for one problem. In practice, because the multipole coefficient weights
are independent of the problem being solved, the setup time for multipole
problems could be reduced by re-using the weights. In both cases, the solution
time reported is that needed for solution using the stabilized biconjugate
gradient method [11].
As a reference case, the fast multipole solver uses a minimum expansion order
of 3 and a maximum of 8. For larger problems, the error ceases to reduce with
element size. This appears to be due to the multipole precision being larger
than the discretization error. As a check, a number of the larger problems
were solved with the minimum expansion order increased to 4. These results are
included in the tabulated data and indicated by an asterisk.
Table 1: Code performance for Laplace equation on sphere using first order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
42 | 8 | $3.03\times 10^{-2}$ | 1 | 0 | 2 | $5.31\times 10^{-3}$ | 1 | 0 | 0
162 | 16 | $1.55\times 10^{-2}$ | 6 | 3 | 5 | $1.38\times 10^{-3}$ | 5 | 0 | 2
642 | 32 | $3.48\times 10^{-4}$ | 40 | 7 | 10 | $3.46\times 10^{-4}$ | 80 | 0 | 8
2562 | 64 | $9.03\times 10^{-5}$ | 473 | 8 | 49 | $8.62\times 10^{-5}$ | 1322 | 0 | 104
2562∗ | 64 | $8.59\times 10^{-5}$ | 491 | 11 | 49 | | | |
10242 | 128 | $7.32\times 10^{-5}$ | 1758 | 54 | 203 | | | |
10242∗ | 128 | $2.96\times 10^{-5}$ | 1897 | 70 | 203 | | | |
Table 2: Code performance for Laplace equation on sphere using second order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
162 | 8 | $2.04\times 10^{-4}$ | 5 | 3 | 5 | $2.03\times 10^{-4}$ | 3 | 0 | 2
642 | 16 | $2.55\times 10^{-5}$ | 25 | 7 | 11 | $2.20\times 10^{-5}$ | 32 | 0 | 8
2562 | 32 | $3.33\times 10^{-5}$ | 102 | 47 | 37 | $2.32\times 10^{-6}$ | 474 | 0 | 104
10242 | 64 | $4.82\times 10^{-5}$ | 749 | 53 | 193 | | | |
40962 | 128 | $8.12\times 10^{-5}$ | 3611 | 264 | 842 | | | |
Table 3: Code performance for Helmholtz equation on sphere using first order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
42 | 8 | $3.46\times 10^{-2}$ | 1 | 0 | 2 | $5.14\times 10^{-3}$ | 1 | 0 | 430
162 | 16 | $1.45\times 10^{-2}$ | 10 | 20 | 6 | $1.28\times 10^{-3}$ | 6 | 0 | 3
642 | 32 | $3.30\times 10^{-4}$ | 68 | 35 | 12 | $3.15\times 10^{-4}$ | 118 | 0 | 15
2562 | 64 | $1.18\times 10^{-4}$ | 843 | 40 | 70 | $7.73\times 10^{-5}$ | 1758 | 1 | 207
2562∗ | 64 | $7.75\times 10^{-5}$ | 1398 | 25 | 122 | | | |
10242 | 128 | $1.83\times 10^{-4}$ | 3020 | 284 | 284 | | | |
10242∗ | 128 | $2.06\times 10^{-5}$ | 4721 | 231 | 475 | | | |
Table 4: Code performance for Helmholtz equation on sphere using second order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
162 | 16 | $1.71\times 10^{-2}$ | 5 | 9 | 6 | $3.62\times 10^{-4}$ | 3 | 0 | 0
642 | 32 | $8.00\times 10^{-5}$ | 31 | 23 | 14 | $4.30\times 10^{-5}$ | 39 | 0 | 15
2562 | 64 | $9.64\times 10^{-5}$ | 247 | 25 | 74 | $4.64\times 10^{-6}$ | 583 | 1 | 206
2562∗ | 64 | $7.75\times 10^{-5}$ | 1398 | 25 | 122 | | | |
10242 | 128 | $1.38\times 10^{-4}$ | 888 | 165 | 282 | | | |
10242∗ | 128 | $2.06\times 10^{-5}$ | 4721 | 231 | 475 | | | |
Table 5: Code performance for Laplace equation on cube using first order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
117 | 5 | $2.22\times 10^{-3}$ | 7 | 9 | 5 | $2.22\times 10^{-3}$ | 4 | 0 | 0
431 | 10 | $4.85\times 10^{-4}$ | 21 | 21 | 9 | $4.87\times 10^{-4}$ | 30 | 0 | 5
1763 | 20 | $1.17\times 10^{-4}$ | 114 | 81 | 29 | $1.19\times 10^{-4}$ | 552 | 0 | 50
7281 | 40 | $2.55\times 10^{-5}$ | 618 | 124 | 98 | | | |
28029 | 80 | $2.70\times 10^{-5}$ | 5003 | 227 | 512 | | | |
28029∗ | 80 | $7.85\times 10^{-6}$ | 17691 | 391 | 854 | | | |
Table 6: Code performance for Laplace equation on cube using second order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
378 | 10 | $1.25\times 10^{-4}$ | 27 | 47 | 8 | $1.24\times 10^{-4}$ | 10 | 0 | 4
1538 | 20 | $1.71\times 10^{-5}$ | 69 | 69 | 27 | $5.61\times 10^{-6}$ | 154 | 0 | 39
1538∗ | 20 | $6.90\times 10^{-6}$ | 73 | 94 | 27 | | | |
6674 | 40 | $9.10\times 10^{-6}$ | 361 | 147 | 106 | $2.78\times 10^{-7}$ | 2982 | 1 | 695
6674∗ | 40 | $5.45\times 10^{-7}$ | 614 | 279 | 147 | | | |
28362 | 80 | $2.09\times 10^{-5}$ | 2551 | 253 | 636 | | | |
Table 7: Code performance for Helmholtz equation on cube using first order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
117 | 5 | $3.46\times 10^{-3}$ | 8 | 67 | 5 | $3.47\times 10^{-3}$ | 5 | 0 | 2
431 | 10 | $7.34\times 10^{-4}$ | 46 | 158 | 34 | $7.43\times 10^{-4}$ | 37 | 0 | 8
1763 | 20 | $1.69\times 10^{-4}$ | 269 | 489 | 36 | $1.68\times 10^{-4}$ | 657 | 1 | 98
7281 | 40 | $5.11\times 10^{-5}$ | 854 | 408 | 128 | | | |
28029 | 80 | $7.18\times 10^{-5}$ | 6246 | 1142 | 713 | | | |
Table 8: Code performance for Helmholtz equation on cube using second order elements | FMM | Direct
---|---|---
$N$ | $B$ | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb | $\epsilon$ | $t_{\text{setup}}/\mathrm{s}$ | $t/\mathrm{s}$ | Mb
378 | 10 | $2.49\times 10^{-4}$ | 31 | 168 | 10 | $2.45\times 10^{-4}$ | 12 | 0 | 6
1538 | 20 | $5.73\times 10^{-5}$ | 81 | 313 | 34 | $9.12\times 10^{-6}$ | 189 | 1 | 76
1538∗ | 20 | $1.66\times 10^{-5}$ | 87 | 421 | 34 | | | |
6674 | 40 | $2.51\times 10^{-5}$ | 431 | 1149 | 147 | | | |
6674∗ | 40 | $1.35\times 10^{-6}$ | 775 | 1672 | 205 | | | |
28362 | 80 | $5.36\times 10^{-5}$ | 3009 | 901 | 892 | | | |
The first data presented are for the Laplace equation solved on a sphere,
Tables 1 and 2. The number of points increases by a factor of about four
between test cases and the computing demands increase at the expected rate,
with the memory requirement scaling roughly as $N^{2}$ for the matrix solution
and roughly linearly for the fast multipole method. This is also true for the
setup time but not for the solution time, which for the matrix method is less
than one second in all cases while it increases roughly linearly for the fast
multipole method. For the smaller problems $N\leq 2562$, the error is
essentially the same for both solution methods but beyond this point the error
from the fast multipole method stops falling. This behaviour can be cured by
increasing the minimum expansion order as shown by the test cases marked with
an asterisk in Table 1 where the reduction in error with $N$ has been
restored. It can also be noted that, even without re-using multipole weights,
the setup time for the fast multipole method is much less than that for the
direct solver, far outweighing any advantage in solution time for the direct
method, even for relatively small problems.
The solvers behave similarly when applied to the Helmholtz problem. The
decline in error with $N$ is similar, with a need to increase the minimum
expansion order at large $N\geq 2562$.
The second test case for the Laplace and Helmholtz solvers is that of a cube
of edge length two, used to check the performance of the solver on a geometry
with sharp edges, a type of problem known to be quite ill-conditioned. In the
case of the Laplace equation, Tables 5 and 6, the fast multipole solver is
comparable in accuracy to the direct method, with the caveat that the minimum
expansion order must be increased for the larger problems, but is far superior
in time and memory consumption. Likewise, the results for the Helmholtz
equation show the expected reduction in error with vertex number $N$ and a
much lower setup time for the fast multipole method, although the solution
time is rather large for small $N$ using second order elements. This appears
to be due to the inherent ill-conditioning of the problem and because the
ratio of edge vertices to non-edge vertices is greater for smaller problems.
No attempt was made to optimize the mesh to improve the handling of sharp
edges, as in other studies [19], since it is intended that bem3d be able to
handle unstructured meshes produced by a standard mesh generator, without
requiring intervention from the user.
## 5 CONCLUSIONS
A new library has been developed for the boundary element solution of general
problems. The library is free software released under the GNU General Public
Licence. It incorporates a new adaptive fast multipole method for boundary
element problems which gives comparable accuracy to direct matrix solution at
a much lower cost in terms of time and memory, making possible the solution of
problems which would otherwise be too large to solve with the available
resources. It includes higher order elements and has provision for the
addition of user-defined elements and Green’s functions without extensive
recoding.
Development of bem3d continues with the intention of adding further element
types and a parallel implementation of the fast multipole method.
## References
* [1] Stephane Popinet. GTS: GNU Triangulated Surface library. http://gts.sourceforge.net/, 2000–2004.
* [2] Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi. GNU Scientific Library Reference Manual. Network Theory Ltd, Bristol, United Kingdom, 2005.
* [3] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999.
* [4] Christophe Geuzaine and Jean-François Remacle. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, 2009\.
* [5] Luigi Morino. Boundary integral equations in aerodynamics. Applied Mechanics Reviews, 46(8):445–466, 1993.
* [6] A. D. Pierce. Acoustics: An introduction to its physical principles and applications. Acoustical Society of America, New York, 1989.
* [7] Miguel C. Junger and David Feit. Sound, structures, and their interaction. Acoustical Society of America, 1993.
* [8] Michael A. Khayat and Donald R. Wilton. Numerical evaluation of singular and near-singular potential integrals. IEEE Transactions on Antennas and Propagation, 53(10):3180–3190, October 2005.
* [9] Ken Hayami. Variable transformations for nearly singular integrals in the boundary element method. Publications of the Research Institute for Mathematical Sciences, 41:821–842, 2005.
* [10] S. Wandzura and H. Xiao. Symmetric quadrature rules on a triangle. Computers and Mathematics with Applications, 45:1829–1840, 2003\.
* [11] R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst. Templates for the solutions of linear systems: Building blocks for iterative methods. SIAM, Philadelphia, PA, USA, 1994.
* [12] Panagiota Pantazopoulou. Boundary integral methods for acoustic scattering and radiation. PhD, University of Bath, Claverton Down, Bath BA2 7AY, England, 2006\.
* [13] Panagiota Pantazopoulou and Michael Carley. Acoustic scattering from a wing in non-uniform flow. Journal of Sound and Vibration, Submitted 2006.
* [14] Panagiota Pantazopoulou, Henry Rice, and Michael Carley. Boundary integral methods for scattering in non-uniform flows. Number AIAA paper 2005-2985. AIAA, AIAA, 2005.
* [15] David J. Willis, Jaime Peraire, and Jacob K. White. A combined pFFT-multipole tree code, unsteady panel method with vortex particle wakes. International Journal for Numerical Methods in Fluids, 53:1399–1422, 2007.
* [16] N. Nishimura. Fast multipole accelerated boundary integral equation methods. Applied Mechanics Reviews, 55(4):299–324, July 2002.
* [17] H. Cheng, L. Greengard, and V. Rokhlin. A fast adaptive multipole algorithm in three dimensions. Journal of Computational Physics, 155:468–498, 1999.
* [18] Johannes Tausch. The fast multipole method for arbitrary Green’s functions. In Z. Chen, R. Glowinski, and K. Li, editors, Current Trends in Scientific Computing, pages 307–314. American Mathematical Society, 2003.
* [19] Johannes Tausch. The variable order fast multipole method for boundary integral equations of the second kind. Computing, 72:267–291, 2004.
* [20] André Buchau, Wolfgang Hafla, Friedemann Groh, and Wolfgang M. Rucker. Improved grouping scheme and meshing strategies for the fast multipole method. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 22(3):495–507, 2003.
|
arxiv-papers
| 2009-08-05T14:22:41 |
2024-09-04T02:49:04.477368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Carley",
"submitter": "Michael Carley",
"url": "https://arxiv.org/abs/0908.0673"
}
|
0908.0884
|
# Periodic and solitary wave solutions to the Fornberg-Whitham equation
Jiangbo Zhou Corresponding author. zhoujiangbo@yahoo.cn Lixin Tian Nonlinear
Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang,
Jiangsu 212013, China
###### Abstract
In this paper, new travelling wave solutions to the Fornberg-Whitham equation
$u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx}$
are investigated. They are characterized by two parameters. The expresssions
of the periodic and solitary wave solutions are obtained.
###### keywords:
Fornberg-Whitham equation , solitary wave, periodic wave
,
## 1 Introduction
Recently, Ivanov [1] investigated the integrability of a class of nonlinear
dispersive wave equations
$u_{t}-u_{xxt}+\partial_{x}(\kappa u+\alpha u^{2}+\beta u^{3})=\nu
u_{x}u_{xx}+\gamma uu_{xxx},$ (1.1)
where and $\alpha,\beta,\gamma,\kappa,\nu$ are real constants.
The important cases of Eq.(1.1) are:
The hyperelastic-rod wave equation
$u_{t}-u_{xxt}+3uu_{x}=\gamma(2u_{x}u_{xx}+uu_{xxx}),$ (1.2)
has been recently studied as a model, describing nonlinear dispersive waves in
cylindrical compressible hyperelastic rods [2]-[7]. The physical parameters of
various compressible materials put $\gamma$ in the range from -29.4760 to
3.4174 [2, 4].
The Camassa-Holm equation
$u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx},$ (1.3)
describes the unidirectional propagation of shallow water waves over a flat
bottom [8, 9]. It is completely integrable [1] and admits, in addition to
smooth waves, a multitude of travelling wave solutions with singularities:
peakons, cuspons, stumpons and composite waves [9]-[12]. The solitary waves of
Eq.(1.2) are smooth if $\kappa>0$ and peaked if $\kappa=0$ [9, 10]. Its
solitary waves are stable solitons [13, 14], retaining their shape and form
after interactions [15]. It models wave breaking [16]-[18].
The Degasperis-Procesi equation
$u_{t}-u_{xxt}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx},$ (1.4)
models nonlinear shallow water dynamics. It is completely integrable [1] and
has a variety of travelling wave solutions including solitary wave solutions,
peakon solutions and shock waves solutions [20]-[27].
The Fornberg-Whitham equation
$u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx},$ (1.5)
appeared in the study qualitative behaviors of wave-breaking [28]. It admits a
wave of greatest height, as a peaked limiting form of the travelling wave
solution [29], $u(x,t)=A\exp(-\frac{1}{2}\left|{x-\frac{4}{3}t}\right|)$,
where $A$ is an arbitrary constant. It is not completely integrable [1].
The regularized long-wave or BBM equation
$u_{t}-u_{xxt}+u_{x}+uu_{x}=0,$ (1.6)
and the modified BBM equation
$u_{t}-u_{xxt}+u_{x}+3u^{2}u_{x}=0,$ (1.7)
have also been investigated by many authors [30]-[38].
Many efforts have been devoted to study Eq.(1.2)-(1.4),(1.6) and (1.7),
however, little attention was paid to study Eq.(1.5). In [39], we constructed
two types of bounded travelling wave solutions $u(\xi)(\xi=x-ct)$ to Eq.(1.5),
which are defined on semifinal bounded domains and called kink-like and
antikink-like wave solutions. In this paper, we continue to study the
travelling wave solutions to Eq.(1.5). Following Vakhnenko and Parkes’s
strategy in [19], we obtain some periodic and solitary wave solutions $u(\xi)$
to Eq.(1.5) which are defined on $(-\infty,+\infty)$. The travelling wave
solutions obtained in this paper are obviously different from those obtained
in our previous work [39]. To the best of our knowledge, these solutions are
new for Eq.(1.5). Our work may help people to know deeply the described
physical process and possible applications of the Fornberg-Whitham equation.
The remainder of the paper is organized as follows. In Section 2, for
completeness and readability, we repeat Appendix A in [19], which discuss the
solutions to a first-order ordinary differential equaion. In Section 3, we
show that, for travelling wave solutions, Eq.(1.5) may be reduced to a first-
order ordinary differential equation involving two arbitrary integration
constants $a$ and $b$. We show that there are four distinct periodic solutions
corresponding to four different ranges of values of $a$ and restricted ranges
of values of $b$. A short conclusion is given in Section 4.
## 2 Solutions to a first-order ordinary differential equaion
This section is due to Vakhnenko and Parkes (see Appendix A in [19]). For
completeness and readability, we repeat it in the following.
Consider solutions to the following ordinary differential equation
$(\varphi\varphi_{\xi})^{2}=\varepsilon^{2}f(\varphi),$ (2.1)
where
$f(\varphi)=(\varphi-\varphi_{1})(\varphi-\varphi_{2})(\varphi_{3}-\varphi)(\varphi_{4}-\varphi),$
(2.2)
and $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\varphi_{4}$ are chosen to
be real constants with
$\varphi_{1}\leq\varphi_{2}\leq\varphi\leq\varphi_{3}\leq\varphi_{4}$.
Following [40] we introduce $\zeta$ defined by
$\frac{d\xi}{d\zeta}=\frac{\varphi}{\varepsilon},$ (2.3)
so that Eq.(2.1) becomes
$(\varphi_{\zeta})^{2}=f(\varphi).$ (2.4)
Eq.(2.4) has two possible forms of solution. The first form is found using
result 254.00 in [41]. Its parametric form is
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{1}+(\varphi_{2}-\varphi_{1})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.5)
with $w$ as the parameter, where
$m=\frac{(\varphi_{3}-\varphi_{2})(\varphi_{4}-\varphi_{1})}{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},p=\frac{1}{2}\sqrt{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},w=p\zeta,$
(2.6)
and
$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{3}-\varphi_{1}}.$ (2.7)
In (2.5) $\mathrm{sn}(w|m)$ is a Jacobian elliptic function, where the
notation is as used in Chapter 16 of [42]. $\Pi(n;w|m)$ is the elliptic
integral of the third kind and the notation is as used in Section 17.2.15 of
[42].
The solution to (2.1) is given in parametric form by (2.5) with $w$ as the
parameter. With respect to $w$, $\varphi$ in (2.5) is periodic with period
$2K(m)$, where $K(m)$ is the complete elliptic integral of the first kind. It
follows from (2.5) that the wavelength $\lambda$ of the solution to (2.1) is
$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{1}K(m)+(\varphi_{2}-\varphi_{1})\Pi(n|m)|,$ (2.8)
where $\Pi(n|m)$ is the complete elliptic integral of the third kind.
When $\varphi_{3}=\varphi_{4}$, $m=1$, (2.5) becomes
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{3}}{\textstyle
p}-2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.9)
The second form of solution of (2.5) is found using result 255.00 in [41]. Its
parametric form is
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{4}-(\varphi_{4}-\varphi_{3})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.10)
where $m,p,w$ are as in (2.6), and
$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{4}-\varphi_{2}}.$ (2.11)
The solution to (2.1) is given in parametric form by (2.10) with $w$ as the
parameter. The wavelength $\lambda$ of the solution to (2.1) is
$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{4}K(m)-(\varphi_{4}-\varphi_{3})\Pi(n|m)|.$ (2.12)
When $\varphi_{1}=\varphi_{2}$, $m=1$, (2.10) becomes
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{2}}{\textstyle
p}+2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.13)
## 3 Periodic and solitary wave solutions to Eq.(1.5)
Eq.(1.5) can also be written in the form
$(u_{t}+uu_{x})_{xx}=u_{t}+uu_{x}+u_{x}.$ (3.1)
Let $u=\varphi(\xi)+c$ with $\xi=x-ct$ be a travelling wave solution to
Eq.(3.1), then it follows that
$(\varphi\varphi_{\xi})_{\xi\xi}=\varphi\varphi_{\xi}+\varphi_{\xi}.$ (3.2)
Integrating (3.2) twice with respect to $\xi$, we have
$(\varphi\varphi_{\xi})^{2}=\frac{1}{4}(\varphi^{4}+\frac{8}{3}\varphi^{3}+a\varphi^{2}+b),$
(3.3)
where $a$ and $b$ are two arbitrary integration constants.
Eq.(3.3) is in the form of Eq.(2.1) with $\varepsilon=\frac{\textstyle
1}{\textstyle 2}$ and $f(\varphi)=(\varphi^{4}+\frac{\textstyle 8}{\textstyle
3}\varphi^{3}+a\varphi^{2}+b)$. For convenience we define $g(\varphi)$ and
$h(\varphi)$ by
$f(\varphi)=\varphi^{2}g(\varphi)+b,\ \mbox{where}\
g(\varphi)=\varphi^{2}+\frac{8}{3}\varphi+a,$ (3.4)
$f^{\prime}(\varphi)=2\varphi h(\varphi),\ \mbox{where}\
h(\varphi)=2\varphi^{2}+4\varphi+a,$ (3.5)
and define $\varphi_{L}$, $\varphi_{R}$, $b_{L}$, and $b_{R}$ by
$\varphi_{L}=-\frac{\textstyle 1}{\textstyle
2}(2+\sqrt{4-2a}),\varphi_{R}=-\frac{\textstyle 1}{\textstyle
2}(2-\sqrt{4-2a}),$ (3.6)
$b_{L}=-\varphi_{L}^{2}g(\varphi_{L})=\frac{\textstyle a^{2}}{\textstyle
4}-2a+\frac{\textstyle 8}{\textstyle 3}+\frac{\textstyle 2}{\textstyle
3}(2-a)\sqrt{4-2a},$ (3.7)
$b_{R}=-\varphi_{R}^{2}g(\varphi_{R})=\frac{\textstyle a^{2}}{\textstyle
4}-2a+\frac{\textstyle 8}{\textstyle 3}-\frac{\textstyle 2}{\textstyle
3}(2-a)\sqrt{4-2a}.$ (3.8)
Obviously, $\varphi_{L}$, $\varphi_{R}$ are the roots of $h(\varphi)=0$.
In the following, suppose that $a<2$ and $a\neq 0$ such that $f(\varphi)$ has
three distinct stationary points: $\varphi_{L}$, $\varphi_{R}$, $0$ and
comprise two minimums separated by a maximum. Under this assumption, Eq.(3.3)
has periodic and solitary wave solutions that have different analytical forms
depending on the values of $a$ and $b$ as follows:
(1) $a<0$
In this case $\varphi_{L}<0<\varphi_{R}$ and $f(\varphi_{L})<f(\varphi_{R})$.
For each value $a<0$ and $0<b<b_{R}$ (a corresponding curve of $f(\varphi)$ is
shown in Fig.1(a)), there are periodic inverted loop-like solutions to
Eq.(3.3) given by (2.5) so that $0<m<1$, and with wavelength given by (2.8);
see Fig.2(a) for an example.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 1: The curve of $f(\varphi)$. (a) $a=-50$, $b=233$; (b) $a=-50$,
$b=374.1346$; (c) $a=1.5$, $b=-0.05$; (d) $a=1.5$, $b=0$; (e)
$a=\frac{\textstyle 16}{\textstyle 9}$, $b=-0.1$; (f) $a=\frac{\textstyle
16}{\textstyle 9}$, $b=0$; (g) $a=1.9$, $b=-0.24$; (h) $a=1.9$, $b=-0.2010$.
The case $a<0$ and $b=b_{R}$ (a corresponding curve of $f(\varphi)$ is shown
in Fig.1(b)) corresponds to the limit $\varphi_{3}=\varphi_{4}=\varphi_{R}$ so
that $m=1$, and then the solution is an inverted loop-like solitary wave given
by (2.9) with $\varphi_{2}\leq\varphi<\varphi_{R}$ and
$\varphi_{1}=-\frac{\textstyle 1}{\textstyle
6}(2+3\sqrt{4-2a}+2\sqrt{4+6\sqrt{4-2a}}),$ (3.9)
$\varphi_{2}=-\frac{\textstyle 1}{\textstyle
6}(2+3\sqrt{4-2a}-2\sqrt{4+6\sqrt{4-2a}}),$ (3.10)
see Fig.3(a) for an example.
(2) $0<a<\frac{\textstyle 16}{\textstyle 9}$
In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})<f(0)$. For each
value $0<a<\frac{\textstyle 16}{\textstyle 9}$ and $b_{R}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(c)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.5) so that $0<m<1$, and with
wavelength given by (2.8); see Fig.2(b) for an example.
The case $0<a<\frac{\textstyle 16}{\textstyle 9}$ and $b=0$ (a corresponding
curve of $f(\varphi)$ is shown in Fig.1(d)) corresponds to the limit
$\varphi_{3}=\varphi_{4}=0$ so that $m=1$, and then the solution can be given
by (2.9) with $\varphi_{1}$ and $\varphi_{2}$ given by the roots of
$g(\varphi)=0$, namely
$\varphi_{1}=-\frac{\textstyle 4}{\textstyle 3}-\frac{\textstyle 1}{\textstyle
3}\sqrt{16-9a},\varphi_{2}=-\frac{\textstyle 4}{\textstyle 3}+\frac{\textstyle
1}{\textstyle 3}\sqrt{16-9a}.$ (3.11)
In this case we obtain a weak solution, namely the periodic upward-cusp wave
$\varphi=\varphi(\xi-2j\xi_{m}),(2j-1)\xi_{m}<\xi<(2j+1)\xi_{m},\ j=0,\pm
1,\pm 2,\cdots,$ (3.12)
where
$\varphi(\xi)=(\varphi_{2}-\varphi_{1}\tanh^{2}(\xi/4))\cosh^{2}(\xi/4),$
(3.13)
and
$\xi_{m}=4\tanh^{-1}\sqrt{\frac{\varphi_{2}}{\varphi_{1}}},$ (3.14)
see Fig.3(b) for an example.
(a)
(b)
(c)
(d)
Figure 2: Periodic solutions to Eq.(3.3) with $0<m<1$. (a) $a=-50$, $b=233$ so
$m=0.7885$; (b) $a=1.5$, $b=-0.05$ so $m=0.6893$; (c) $a=\frac{\textstyle
16}{\textstyle 9}$, $b=-0.1$ so $m=0.8254$; (d) $a=1.9$, $b=-0.24$ so
$m=0.6121$.
(a)
(b)
(c)
(d)
Figure 3: Solutions to Eq.(3.3) with $m=1$. (a) $a=-50$, $b=374.1346$; (b)
$a=1.5$, $b=0$; (c) $a=\frac{\textstyle 16}{\textstyle 9}$, $b=0$; (d)
$a=1.9$, $b=-0.2010$.
(3) $a=\frac{\textstyle 16}{\textstyle 9}$
In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})=f(0)$. For
$a=\frac{\textstyle 16}{\textstyle 9}$ and each value $b_{R}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(e)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.10) so that $0<m<1$, and with
wavelength given by (2.12); see Fig.2(c) for an example.
The case $a=\frac{\textstyle 16}{\textstyle 9}$ and $b=0$ (a corresponding
curve of $f(\varphi)$ is shown in Fig.1(f)) corresponds to the limit
$\varphi_{1}=\varphi_{2}=\varphi_{L}=-\frac{\textstyle 4}{\textstyle 3}$ and
$\varphi_{3}=\varphi_{4}=0$ so that $m=1$. In this case neither (2.5) nor
(2.10) is appropriate. Instead we consider Eq.(3.3) with
$f(\varphi)=\frac{\textstyle 1}{\textstyle
4}\varphi^{2}(\varphi+\frac{\textstyle 4}{\textstyle 3})^{2}$ and note that
the bound solution has $-\frac{\textstyle 4}{\textstyle 3}<\varphi\leq 0$. On
integrating Eq.(3.3) and setting $\varphi=0$ at $\xi=0$ we obtain a weak
solution
$\varphi=\frac{\textstyle 4}{\textstyle 3}\exp{(-\frac{\textstyle
1}{\textstyle 2}|\xi|)}-\frac{\textstyle 4}{\textstyle 3},$ (3.15)
i.e. a single peakon solution with amplitude $\frac{\textstyle 4}{\textstyle
3}$, see Fig.3(c).
(4) $\frac{\textstyle 16}{\textstyle 9}<a<2$
In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})>f(0)$. For each
value $\frac{\textstyle 16}{\textstyle 9}<a<2$ and $b_{R}<b<b_{L}$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(g)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.10) so that $0<m<1$, and with
wavelength given by (2.12); see Fig.2(d) for an example.
The case $\frac{\textstyle 16}{\textstyle 9}<a<2$ and $b=b_{L}$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(h)) corresponds to the
limit $\varphi_{1}=\varphi_{2}=\varphi_{L}$ so that $m=1$, and then the
solution is a hump-like solitary wave given by (2.13) with
$\varphi_{L}<\varphi\leq\varphi_{3}$ and
$\varphi_{3}=\frac{\textstyle 1}{\textstyle
6}(-2+3\sqrt{4-2a}-2\sqrt{4-6\sqrt{4-2a}}),$ (3.16)
$\varphi_{4}=\frac{\textstyle 1}{\textstyle
6}(-2+3\sqrt{4-2a}+2\sqrt{4-6\sqrt{4-2a}}),$ (3.17)
see Fig.3(d) for an example.
On the above, we have obtained expressions of parametric form for periodic and
solitary wave solutions $\varphi(\xi)$ to Eq.(3.3). So in terms of
$u=\varphi(\xi)+c$, we can get expressions for the periodic and solitary wave
solutions $u(\xi)$ to Eq.(1.5).
## 4 Conclusion
In this paper, we have found expressions for new travelling wave solutions to
the Fornberg-Whitham equation. These solutions depend, in effect, on two
parameters $a$ and $m$. For $m=1$, there are inverted loop-like ($a<0$),
single peaked ($a=\frac{\textstyle 16}{\textstyle 9}$) and hump-like
($\frac{\textstyle 16}{\textstyle 9}<a<2$) solitary wave solutions. For
$m=1,0<a<\frac{\textstyle 16}{\textstyle 9}$ or $0<m<1,a<2$ and $a\neq 0$,
there are periodic hump-like wave solutions.
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|
arxiv-papers
| 2009-08-06T14:58:32 |
2024-09-04T02:49:04.486890
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangbo Zhou, Lixin Tian",
"submitter": "Jiangbo Zhou",
"url": "https://arxiv.org/abs/0908.0884"
}
|
0908.0914
|
# New exact travelling wave solutions for the $K(2,2)$ equation with osmosis
dispersion
Jiangbo Zhou Corresponding author. Tel.: +86-511-88969336; Fax:
+86-511-88969336. zhoujiangbo@yahoo.cn Lixin Tian Xinghua Fan Nonlinear
Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang,
Jiangsu 212013, China
###### Abstract
In this paper, by using bifurcation method, we successfully find the $K(2,2)$
equation with osmosis dispersion $u_{t}+(u^{2})_{x}-(u^{2})_{xxx}=0$ possess
two new types of travelling wave solutions called kink-like wave solutions and
antikink-like wave solutions. They are defined on some semifinal bounded
domains and possess properties of kink waves and anti-kink waves. Their
implicit expressions are obtained. For some concrete data, the graphs of the
implicit functions are displayed, and the numerical simulation of travelling
wave system is made by Maple. The results show that our theoretical analysis
agrees with the numerical simulation.
###### keywords:
$K(2,2)$ equation; travelling wave solution; bifurcation method
, ,
## 1 Introduction
In recent years, many nonlinear partial differential equations (NLPDEs) have
been derived from physics, mechanics, engineering, biology, chemistry and
other fields. Since exact solutions can help people know deeply the described
process and possible applications, seeking exact solutions for NLPDEs is of
great importance.
In 1993, Rosenau and Hyman [1] introduced a genuinely nonlinear dispersive
equation, a special type of KdV equation, of the form
$u_{t}+a(u^{n})_{x}+(u^{n})_{xxx}=0,n>1,$ (1.1)
where $a$ is a constant and both the convection term $(u^{n})_{x}$ and the
dispersion effect term $(u^{n})_{xxx}$ are nonlinear. These equations arise in
the process of understanding the role of nonlinear dispersion in the formation
of structures like liquid drops. Rosenau and Hyman derived solutions called
compactons for Eq.(1.1) and showed that while compactons are the essence of
the focusing branch where $a>0$, spikes, peaks, and cusps are the hallmark of
the defocusing branch where $a<0$ which also supports the motion of kinks.
Further, the negative branch, where $a<0$, was found to give rise to solitary
patterns having cusps or infinite slopes. The focusing branch and the
defocusing branch represent two different models, each leading to a different
physical structure. Many powerful methods were applied to construct the exact
solutions for Eq.(1.1), such as Adomain method [2], homotopy perturbation
method [3], Exp-function method [4], variational iteration method [5],
variational method [6, 7]. In [8], Wazwaz studied a generalized forms of the
Eq.(1.1), that is $mK(n,n)$ equations and defined by
$u^{n-1}u_{t}+a(u^{n})_{x}+b(u^{n})_{xxx}=0,n>1,$ (1.2)
where $a,b$ are constants. He showed how to construct compact and noncompact
solutions for Eq.(1.2) and discussed it in higher dimensional spaces in [9].
Chen et al. [10] showed how to construct the general solutions and some
special exact solutions for Eq.(1.2) in higher dimensional spatial domains. He
et al. [11] considered the bifurcation behavior of travelling wave solutions
for Eq.(1.2). Under different parametric conditions, smooth and non-smooth
periodic wave solutions, solitary wave solutions and kink and anti-kink wave
solutions were obtained. Yan [12] further extended Eq.(1.2) to be a more
general form
$u^{m-1}u_{t}+a(u^{n})_{x}+b(u^{k})_{xxx}=0,nk\neq 1,$ (1.3)
And using some direct ansatze, some abundant new compacton solutions, solitary
wave solutions and periodic wave solutions of Eq.(1.3) were obtained. By using
some transformations, Yan [13] obtained some Jacobi elliptic function
solutions for Eq.(1.3). Biswas [14] obtained 1-soliton solution of equation
with the generalized evolution term
$(u^{l})_{t}+a(u^{m})u_{x}+b(u^{n})_{xxx}=0,$ (1.4)
where $a,b$ are constants, while $l,m$ and $n$ are positive integers. Zhu et
al. [15] applied the decomposition method and symbolic computation system to
develop some new exact solitary wave solutions for the $K(2,2,1)$ equation
$u_{t}+(u^{2})_{x}-(u^{2})_{xxx}+u_{xxxxx}=0,$ (1.5)
and the $K(3,3,1)$ equation
$u_{t}+(u^{3})_{x}-(u^{3})_{xxx}+u_{xxxxx}=0.$ (1.6)
In [16], Xu and Tian introduced the osmosis $K(2,2)$ equation
$u_{t}+(u^{2})_{x}-(u^{2})_{xxx}=0,$ (1.7)
where the negative coefficient of dispersive term denotes the contracting
dispersion. They obtained the peaked solitary wave solution and the periodic
cusp wave solution for Eq.(1.7). In this paper, we’ll continue their work and
using the bifurcation method of planar dynamical systems to derive two new
types of bounded travelling wave solutions for Eq.(1.7). They are defined on
some semifinal bounded domains and possess properties of kink waves and anti-
kink waves. To our knowledge, such type of travelling wave solution has never
been found for Eq.(1.7) in the former literature.
The remainder of the paper is organized as follows. In Section 2, we give the
phase portrait of the travelling wave system and use Maple to show the graphs
of the orbits connecting with the saddle points for our purpose. In Section 3,
we state the main results which are implicit expressions of the kink-like and
antikink-like wave solutions. In Section 4, we give the proof of the main
results. For some concrete data, we use Maple to display the graphs of the
implicit functions. In Section 5, the numerical simulations of travelling wave
system are made by Maple. A short conclusion is given in Section 6.
## 2 Phase portrait of the travelling wave system
Eq.(1.7) also takes the form
$u_{t}+2uu_{x}-6u_{x}u_{xx}-2uu_{xxx}=0,$ (2.1)
Let $u=\varphi(\xi)$ with $\xi=x-ct(c\neq 0)$ be the solution for Eq.(2.1),
then it follows that
$-c\varphi^{\prime}+2\varphi\varphi^{\prime}-6\varphi^{\prime}\varphi^{\prime\prime}-2\varphi\varphi^{\prime\prime\prime}=0.$
(2.2)
Integrating (2.2) once we have
$-c\varphi+(\varphi)^{2}-2(\varphi^{\prime})^{2}-2\varphi\varphi^{\prime\prime}=g,$
(2.3)
where $g$ is the integral constant.
Let $y=\varphi^{\prime}$, then we get the following planar dynamical system:
$\left\\{{\begin{array}[]{l}\frac{\textstyle d\varphi}{\textstyle d\xi}=y\\\
\frac{\textstyle dy}{\textstyle
d\xi}=\frac{\textstyle\varphi^{2}-c\varphi-g-2y^{2}}{\textstyle 2\varphi}\\\
\end{array}}\right.$ (2.4)
with a first integral
$H(\varphi,y)=\varphi^{2}(y^{2}-\frac{1}{4}\varphi^{2}+\frac{c}{3}\varphi+\frac{1}{2}g)=h,$
(2.5)
where $h$ is a constant.
Note that (2.4) has a singular line $\varphi=0$, to avoid the line temporarily
we make transformation $d\xi=2\varphi d\zeta$. Under this transformation,
Eq.(2.4) becomes
$\left\\{{\begin{array}[]{l}\frac{\textstyle d\varphi}{\textstyle
d\zeta}=2\varphi y\\\ \frac{\textstyle dy}{\textstyle
d\zeta}=\varphi^{2}-c\varphi-g-2y^{2}\\\ \end{array}}\right.$ (2.6)
Eq.(2.4) and Eq.(2.6) have the same first integral as (2.5). Consequently,
system (2.4) has the same topological phase portraits as system (2.6) except
for the straight line $\varphi=0$. Obviously, $\varphi=0$ is an invariant
straight-line solution for system (2.6).
Now we consider the singular points of system (2.6) and their properties. Note
that for a fixed $h$, (2.5) determines a set of invariant curves of (2.6). As
$h$ is varied, (2.5) determines different families of orbits of (2.6) having
different dynamical behaviors. Let $M(\varphi_{e},y_{e})$ be the coefficient
matrix of the linearized system of (2.6) at the equilibrium point
$(\varphi_{e},y_{e})$, then
$M(\varphi_{e},y_{e})=\left({{\begin{array}[]{*{20}c}{\indent
y_{e}}&&&{\indent 2\varphi_{e}}\\\ {2\varphi_{e}-c}&&&\indent{-4y_{e}}\\\
\end{array}}}\right)$
and at this equilibrium point, we have
$J(\varphi_{e},y_{e})=\det
M(\varphi_{e},y_{e})=-4y_{e}^{2}-4\varphi_{e}(\varphi_{e}-\frac{c}{2}\varphi_{e}),$
$p(\varphi_{e},y_{e})=\mathrm{trace}(M(\varphi_{e},y_{e}))=-3y_{e}.$
By the theory of planar dynamical system (see [17]), for an equilibrium point
of a planar dynamical system, if $J<0$, then this equilibrium point is a
saddle point; it is a center point if $J>0$ and $p=0$; if $J=0$ and the
Poincáre index of the equilibrium point is 0, then it is a cusp.
Although the distribution and properties of equilibrium points of system (2.4)
has been given in [16]. Here we also state it briefly for our purpose. System
(2.4) has the following properties:
(1) When $g>0$, system (2.4) has two equilibrium points $(\varphi_{0}^{-},0)$
and $(\varphi_{0}^{+},0)$. They are saddle points. (i) If $c<0$, then there is
inequality $\varphi_{0}^{-}<\frac{c}{2}<0<\varphi_{0}^{+}$; (ii) If $c>0$,
then there is inequality $\varphi_{0}^{-}<0<\frac{c}{2}<\varphi_{0}^{+}$.
(2) When $g=0$, system (2.4) has two equilibrium points $(0,0)$ and $(c,0)$.
$(0,0)$ is a cusp, and $(c,0)$ is a saddle point.
(3) When $-\frac{c^{2}}{4}<g<0$, system (2.4) has two equilibrium points
$(\varphi_{0}^{-},0)$ and $(\varphi_{0}^{+},0)$. (i) If $c<0$, then
$(\varphi_{0}^{-},0)$ is a saddle point while $(\varphi_{0}^{+},0)$ is a
center point. There is inequality
$\varphi_{0}^{-}<\frac{c}{2}<\varphi_{0}^{+}<0$; (ii) If $c>0$ , then
$(\varphi_{0}^{-},0)$ is a center point while $(\varphi_{0}^{+},0)$ is a
saddle point. There is inequality
$0<\varphi_{0}^{-}<\frac{c}{2}<\varphi_{0}^{+}$.
(4) When $g=-\frac{c^{2}}{4}$, system (2.4) has only one equilibrium point
$(\frac{c}{2},0)$. It is a cusp.
(5) When $g<-\frac{c^{2}}{4}$, system (2.4) has no equilibrium point.
###### Remark 2.1
Suppose that $\varphi(\xi)(\xi=x-ct)$ is a travelling wave solution for
Eq.(1.7) for $\xi\in(-\infty,+\infty)$, and
$\mathop{\lim}\limits_{\xi\to-\infty}\varphi(\xi)=A$,
$\mathop{\lim}\limits_{\xi\to\infty}\varphi(\xi)=B$, where $A$ and $B$ are two
constants. If $A=B$, then $\varphi(\xi)$ is called a solitary wave solution.
If $A\neq B$, then $\varphi(\xi)$ is called a kink (or an anti-kink) solution.
Usually, a solitary wave solution for Eq.(1.7) corresponds to a homoclinic
orbit of system (2.6) and a periodic orbit of system (2.6) corresponds to a
periodic travelling wave solution of Eq.(1.7). Similarly, a kink (or an anti-
kink) wave solution of Eq.(1.7) corresponds to a heteroclinic orbit (or so-
called connecting orbit) of system (2.6). In [16], Xu and Tian reported that
when $-\frac{c^{2}}{4}<g<-\frac{2c^{2}}{9}$, system (2.6) has homoclinic
orbits, and when $-\frac{c^{2}}{4}<g<0$, system (2.6) has a periodic orbit
which consist of an arc and a line segment. They obtained peakon solutions
from the limit of solitary waves and from the limit of periodic cusp waves.
We’ll obtain a new type of bounded travelling wave solutions called kink-like
and antikink-like wave solutions for Eq.(1.7) when $g>-\frac{2c^{2}}{9}$,
which correspond to the orbits of system (2.6) connecting with the saddle
points.
We show the phase portraits in each region and on the bifurcation curves in
Fig.1. From Fig.1, we can see that when $g>-\frac{2c^{2}}{9}$, system (2.6)
has orbits connecting with the saddle points.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Figure 1: Phase portrait of system (2.6). (a) $g>0$, $c<0$; (b) $g>0$, $c>0$;
(c) $g=0$, $c<0$; (d) $g=0$, $c>0$; (e) $-\frac{2c^{2}}{9}<g<0$, $c<0$; (f)
$-\frac{2c^{2}}{9}<g<0$, $c>0$; (g) $g=-\frac{2c^{2}}{9}$, $c<0$; (h)
$g=-\frac{2c^{2}}{9}$, $c>0$; (i) $-\frac{c^{2}}{4}<g<-\frac{2c^{2}}{9}$,
$c<0$; (j) $-\frac{c^{2}}{4}<g<-\frac{2c^{2}}{9}$, $c>0$; (k)
$g=-\frac{c^{2}}{4}$, $c<0$; (l) $g=-\frac{c^{2}}{4}$, $c>0$.
## 3 Main results
We state our main result as follows.
###### Theorem 3.1
For given constant $c\neq 0$, let
$\xi=x-ct,$ (3.1) $\varphi_{0}^{\pm}=\frac{c\pm\sqrt{c^{2}+4g}}{2},$ (3.2)
(1) When $g>0$ and $c<0$, Eq.(1.7) has two kink-like wave solutions
$u=\varphi_{1}(\xi)$ and $u=\varphi_{3}(\xi)$ and two antikink-like wave
solutions $u=\varphi_{2}(\xi)$ and $u=\varphi_{4}(\xi)$.
$\beta_{1}(\varphi_{1})=\beta_{1}(a)\exp(-\frac{1}{2}\xi),\\\
\quad\xi\in(-\infty,\xi_{0}^{1}),$ (3.3)
$\beta_{1}(\varphi_{2})=\beta_{1}(a)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{1},\infty),$ (3.4)
$\beta_{2}(\varphi_{3})=\beta_{2}(b)\exp(-\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{3},\infty),$ (3.5)
$\beta_{2}(\varphi_{4})=\beta_{2}(b)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\infty,\xi_{0}^{3}),$ (3.6)
(2) When $g>0$ and $c>0$, Eq.(1.7) has two kink-like wave solutions
$u=\varphi_{1}(\xi)$ and $u=\varphi_{3}(\xi)$ and two antikink-like wave
solutions $u=\varphi_{2}(\xi)$ and $u=\varphi_{4}(\xi)$.
$\beta_{1}(\varphi_{1})=\beta_{1}(a)\exp(-\frac{1}{2}\xi),\\\
\quad\xi\in(-\infty,\xi_{0}^{5}),$ (3.7)
$\beta_{1}(\varphi_{2})=\beta_{1}(a)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{5},\infty),$ (3.8)
$\beta_{2}(\varphi_{3})=\beta_{2}(b)\exp(-\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{7},\infty),$ (3.9)
$\beta_{2}(\varphi_{4})=\beta_{2}(b)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\infty,\xi_{0}^{7}),$ (3.10)
(3) When $-\frac{2c^{2}}{9}<g\leq 0$ and $c<0$, Eq.(1.7) has a kink-like wave
solution $u=\varphi_{5}(\xi)$ and an antikink-like wave solution
$u=\varphi_{6}(\xi)$.
$\beta_{1}(\varphi_{5})=\beta_{1}(d)\exp(-\frac{1}{2}\xi),\\\
\quad\xi\in(-\infty,\xi_{0}^{9}),$ (3.11)
$\beta_{1}(\varphi_{6})=\beta_{1}(d)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{9},\infty),$ (3.12)
(4) When $-\frac{2c^{2}}{9}<g\leq 0$ and $c>0$, Eq.(1.7) has a kink-like wave
solution $u=\varphi_{7}(\xi)$ and an antikink-like wave solution
$u=\varphi_{8}(\xi)$.
$\beta_{2}(\varphi_{7})=\beta_{2}(k)\exp(-\frac{1}{2}\xi),\\\
\quad\xi\in(-\infty,\xi_{0}^{11}),$ (3.13)
$\beta_{2}(\varphi_{8})=\beta_{2}(k)\exp(\frac{1}{2}\xi),\\\
\quad\quad\xi\in(-\xi_{0}^{11},\infty),$ (3.14)
where
$\varphi_{1}^{*}=\frac{c-\sqrt{c^{2}+3g}}{2},$ (3.15)
$\varphi_{2}^{*}=\frac{c+\sqrt{c^{2}+3g}}{2},$ (3.16)
$l_{1}=-\frac{1}{3}(c+3\sqrt{c^{2}+4g}),$ (3.17)
$l_{2}=\frac{1}{6}(c^{2}+6g-c\sqrt{c^{2}+4g}),$ (3.18)
$l_{3}=\frac{2}{3}(c^{2}+6g-c\sqrt{c^{2}+4g}),$ (3.19)
$m_{1}=-\frac{1}{3}(c-3\sqrt{c^{2}+4g}),$ (3.20)
$m_{2}=\frac{1}{6}(c^{2}+6g+c\sqrt{c^{2}+4g}),$ (3.21)
$m_{3}=\frac{2}{3}(c^{2}+6g+c\sqrt{c^{2}+4g}),$ (3.22)
$a_{1}=c^{2}+4g-c\sqrt{c^{2}+4g},$ (3.23) $a_{2}=c^{2}+4g+c\sqrt{c^{2}+4g},$
(3.24) $b_{1}=\frac{1}{3}(2c-6\sqrt{c^{2}+4g}),$ (3.25)
$b_{2}=\frac{1}{3}(2c+6\sqrt{c^{2}+4g}),$ (3.26)
$\alpha_{1}=\frac{c-\sqrt{c^{2}+4g}}{2\sqrt{c^{2}+4g-c\sqrt{c^{2}+4g}}},$
(3.27)
$\alpha_{2}=\frac{c+\sqrt{c^{2}+4g}}{2\sqrt{c^{2}+4g+c\sqrt{c^{2}+4g}}},$
(3.28)
$\beta_{1}(\varphi)=\frac{(2\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}+2\varphi+l_{1})(\varphi-\varphi_{0}^{-})^{\alpha_{1}}}{(2\sqrt{a_{1}}\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}+b_{1}\varphi+l_{3})^{\alpha_{1}}},$
(3.29)
$\beta_{2}(\varphi)=\frac{(2\sqrt{\varphi^{2}+m_{1}\varphi+m_{2}}+2\varphi+m_{1})(\varphi-\varphi_{0}^{+})^{\alpha_{2}}}{(2\sqrt{a_{2}}\sqrt{\varphi^{2}+m_{1}\varphi+m_{2}}+b_{2}\varphi+m_{3})^{\alpha_{2}}},$
(3.30) $\xi_{0}^{1}=2\ln(\beta_{1}(a)/\beta_{1}(\frac{c}{2})),$ (3.31)
$\xi_{0}^{3}=2\ln(\beta_{2}(\varphi_{2}^{*})/\beta_{2}(b))$ (3.32)
$\xi_{0}^{5}=2\ln(\beta_{1}(a)/\beta_{1}(\varphi_{1}^{*})),$ (3.33)
$\xi_{0}^{7}=2\ln(\beta_{2}(\frac{c}{2})/\beta_{2}(b)),$ (3.34)
$\xi_{0}^{9}=2\ln(\beta_{1}(d)/\beta_{1}(\frac{c}{2})),$ (3.35)
$\xi_{0}^{11}=2\ln(\beta_{2}(\frac{c}{2})/\beta_{2}(k)),$ (3.36)
$a$, $b$, $d$, $k$ are four constants satisfying
$\varphi_{1}(0)=\varphi_{2}(0)=a$, $\varphi_{3}(0)=\varphi_{4}(0)=b$,
$\varphi_{5}(0)=\varphi_{6}(0)=d$, $\varphi_{7}(0)=\varphi_{8}(0)=k$, and
there are inequalities
$\varphi_{0}^{-}<a<\frac{c}{2}<0<\varphi_{2}^{*}<b<\varphi_{0}^{+}$ for $c<0$,
$\varphi_{0}^{-}<a<\varphi_{1}^{*}<0<\frac{c}{2}<b<\varphi_{0}^{+}$ for $c>0$,
$\varphi_{0}^{-}<d<\frac{c}{2}<\varphi_{0}^{+}<0$ for $c<0$ and
$0<\varphi_{0}^{-}<\frac{c}{2}<k<\varphi_{0}^{+}$ for $c>0$.
We will give the proof of this theorem in Section 4. Now we take a set of data
and employ Maple to display the graphs of
$u=\varphi_{i}(\xi)(i=1,2,3,4,5,6,7,8)$.
###### Example 3.1
Taking $g=5$, $c=-1$ (corresponding to (1) in Theorem (3.1)), it follows that
$\varphi_{0}^{-}=-2.79129$ , $\varphi_{0}^{+}=1.79129$, $l_{1}=-4.24924$,
$l_{2}=5.93043$, $l_{3}=23.7217$, $a_{1}=25.5826$, $b_{1}=-9.83182$ ,
$\alpha_{1}=-0.551865$. Further, choosing
$a=-0.75\in(\varphi_{0}^{-},\frac{c}{2})$, we obtain $\xi_{0}^{1}=0.0482492$.
We present the graphs of the solutions $\varphi_{1}(\xi)$ and
$\varphi_{2}(\xi)$ in Fig.2 (a) and (b), respectively. Meanwhile, we get
$m_{1}=4.51591$, $m_{2}=4.4029$, $m_{3}=17.6116$, $a_{2}=16.4174$,
$b_{2}=8.49848$, $\alpha_{2}=0.442092$, $\varphi_{2}^{*}=1.5$. Further,
choosing $b=1.6\in(\varphi_{2}^{*},\varphi_{0}^{+})$, we get
$\xi_{0}^{3}=0.343656$. The graphs of the solutions $\varphi_{3}(\xi)$ and
$\varphi_{4}(\xi)$ are presented in Fig.2(c) and (d), respentively. The graphs
in Fig.2 show that $\varphi_{1}(\xi)$ and $\varphi_{3}(\xi)$ are two kink-like
wave solutions and $\varphi_{2}(\xi)$ and $\varphi_{4}(\xi)$ are two antikink-
like wave solutions.
(a)
(b)
(c)
(d)
Figure 2: The graphs of $\varphi_{i}(\xi)(i=1,2,3,4)$ when $g=5$, $c=-1$,
$a=-0.75$, $b=1.6$.
###### Example 3.2
Taking $g=5$, $c=1$ (corresponding to (2) in Theorem (3.1)), it follows that
$\varphi_{0}^{-}=-1.79129$ , $\varphi_{0}^{+}=2.79129$, $l_{1}=-4.51591$,
$l_{2}=4.4029$, $l_{3}=17.6116$, $a_{1}=16.4174$, $b_{1}=-8.49848$ ,
$\alpha_{1}=-0.442092$, $\varphi_{1}^{*}=-1.5$. Further, choosing
$a=-1.6\in(\varphi_{0}^{-},\varphi_{1}^{*})$, we obtain
$\xi_{0}^{5}=0.343656$. We present the graphs of the solutions
$\varphi_{1}(\xi)$ and $\varphi_{2}(\xi)$ in Fig.3 (a) and (b), respectively.
Meanwhile, we get $m_{1}=4.249241$, $m_{2}=5.93043$, $m_{3}=23.7214$,
$a_{2}=25.58264$, $b_{2}=9.93182$, $\alpha_{2}=0.551865$. Further, choosing
$b=2\in(\frac{c}{2},\varphi_{0}^{+})$, we get $\xi_{0}^{7}=0.773847$. The
graphs of the solutions $\varphi_{3}(\xi)$ and $\varphi_{4}(\xi)$ are
presented in Fig.3(c) and (d), respentively. The graphs in Fig.3 show that
$\varphi_{1}(\xi)$ and $\varphi_{3}(\xi)$ are two kink-like wave solutions and
$\varphi_{2}(\xi)$ and $\varphi_{4}(\xi)$ are two antikink-like wave
solutions.
(a)
(b)
(c)
(d)
Figure 3: The graphs of $\varphi_{i}(\xi)(i=1,2,3,4)$ when $g=5$, $c=1$,
$a=-1.6$, $b=2$.
###### Example 3.3
Taking $g=-0.5$, $c=-2$ (corresponding to (3) in Theorem (3.1)), it follows
that $\varphi_{0}^{-}=-1.70711$ , $\varphi_{0}^{+}=-0.292893$,
$l_{1}=-0.747547$, $l_{2}=0.638071$, $l_{3}=2.5528$, $a_{1}=4.82843$,
$b_{1}=-4.16176$ , $\alpha_{1}=-0.776887$. Further, choosing
$d=-1.2\in(\varphi_{0}^{-},\frac{c}{2})$, we obtain $\xi_{0}^{9}=0.448123$. We
present the graphs of the solutions $\varphi_{5}(\xi)$ and $\varphi_{6}(\xi)$
in Fig.4 (a) and (b), respectively. The graphs in Fig.4 show that
$\varphi_{5}(\xi)$ is a kink-like wave solution and $\varphi_{6}(\xi)$ is an
antikink-like wave solution.
(a)
(b)
Figure 4: The graphs of $\varphi_{i}(\xi)(i=5,6)$ when $g=-0.5$, $c=-2$,
$d=-1.2$.
###### Example 3.4
Taking $g=-0.5$, $c=2$ (corresponding to (4) in Theorem (3.1)), it follows
that $\varphi_{0}^{-}=0.292893$ , $\varphi_{0}^{+}=1.70711$, $m_{1}=0.747547$,
$m_{2}=0638071$, $m_{3}=2.55228$, $a_{2}=4.82843$, $b_{2}=4.1676$,
$\alpha_{2}=0.776887$. Further, choosing
$k=1.2\in(\frac{c}{2},\varphi_{0}^{+})$, we get $\xi_{0}^{11}=0.448123$. The
graphs of the solutions $\varphi_{7}(\xi)$ and $\varphi_{8}(\xi)$ are
presented in Fig.5 (a) and (b), respentively. The graphs in Fig.5 show that
$\varphi_{7}(\xi)$ is a kink-like wave solutions and $\varphi_{8}(\xi)$ is an
antikink-like wave solutions.
(a)
(b)
Figure 5: The graphs of $\varphi_{i}(\xi)(i=7,8)$ when $g=-0.5$, $c=2$,
$k=1.2$.
## 4 Proof of main results
Suppose $g>0$ and $c<0$, then system (2.4) has two saddle points
$(\varphi_{0}^{-},0)$ and $(\varphi_{0}^{+},0)$. There are four orbits
connecting with $(\varphi_{0}^{-},0)$. We use $l_{\varphi_{0}^{-}}^{2}$ to
denote the two orbits lying on the right side of $(\varphi_{0}^{-},0)$ (see
Fig.6(a)). Meanwhile, there are four orbits connecting with
$(\varphi_{0}^{+},0)$. We employ $l_{\varphi_{0}^{+}}^{1}$ and
$l_{\varphi_{0}^{+}}^{2}$ to denote the two orbits lying on the left side of
$(\varphi_{0}^{+},0)$ (see Fig.6(a)).
(a)
(b)
(c)
(d)
Figure 6: The sketches of orbits connecting with saddle points. (a)$g>0$,
$c<0$; (b)$g>0$, $c>0$; (c) $-\frac{2c^{2}}{9}<g\leq 0$, $c<0$;
(d)$-\frac{2c^{2}}{9}<g\leq 0$, $c>0$.
On the $\varphi-y$ plane, the orbits $l_{\varphi_{0}^{-}}^{1}$,
$l_{\varphi_{0}^{-}}^{2}$, $l_{\varphi_{0}^{+}}^{1}$ and
$l_{\varphi_{0}^{+}}^{2}$ have the following expressions, respectively,
$l_{\varphi_{0}^{-}}^{1}:\quad
y=\frac{(\varphi_{0}^{-}-\varphi)\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}}{2\varphi}$
(4.1) $l_{\varphi_{0}^{-}}^{2}:\quad
y=\frac{(\varphi-\varphi_{0}^{-})\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}}{2\varphi}$
(4.2) $l_{\varphi_{0}^{+}}^{1}:\quad
y=\frac{(\varphi_{0}^{+}-\varphi)\sqrt{\varphi^{2}+m_{1}\varphi+m_{2}}}{2\varphi}$
(4.3) $l_{\varphi_{0}^{+}}^{2}:\quad
y=\frac{(\varphi-\varphi_{0}^{+})\sqrt{\varphi^{2}+m_{1}\varphi+m_{2}}}{2\varphi}$
(4.4)
where $\varphi_{0}^{-}$ and $\varphi_{0}^{+}$ are in (3.2) , $l_{1}$ and
$l_{2}$ are in (3.17) and (3.18), $m_{1}$ and $m_{2}$ are in (3.20) and
(3.21), respectively.
Assume that $\varphi=\varphi_{1}(\xi)$, $\varphi=\varphi_{2}(\xi)$,
$\varphi=\varphi_{3}(\xi)$ and $\varphi=\varphi_{4}(\xi)$ on
$l_{\varphi_{0}^{-}}^{1}$, $l_{\varphi_{0}^{-}}^{2}$,
$l_{\varphi_{0}^{+}}^{1}$ and $l_{\varphi_{0}^{+}}^{2}$, respectively and
$\varphi_{1}(0)=\varphi_{2}(0)=a$, $\varphi_{3}(0)=\varphi_{4}(0)=b$,
$\varphi_{2}^{*}=\frac{c+\sqrt{c^{2}+3g}}{2}$, where $a$ and $b$ are two
constants satisfying $\varphi_{0}^{-}<a<\frac{c}{2}$ and
$\varphi_{2}^{*}<b<\varphi_{0}^{+}$. Substituting (4.1)-(4.4) into the first
equation of (2.4) and integrating along the corresponding orbits,
respectively, we have
$\int_{a}^{\varphi_{1}}{\frac{-s}{(s-\varphi_{0}^{-})\sqrt{s^{2}+l_{1}s+l_{2}}}ds=\frac{1}{2}\int_{0}^{\xi}{ds}}\quad\quad(\textmd{along}\quad
l_{\varphi_{0}^{-}}^{1}),$ (4.5)
$\int_{\varphi_{2}}^{a}{\frac{s}{(s-\varphi_{0}^{-})\sqrt{s^{2}+l_{1}s+l_{2}}}ds=\frac{1}{2}\int_{\xi}^{0}{ds}}\quad\quad(\textmd{along}\quad
l_{\varphi_{0}^{-}}^{2}),$ (4.6)
$\int_{\varphi_{3}}^{b}{\frac{-s}{(s-\varphi_{0}^{+})\sqrt{s^{2}+m_{1}s+m_{2}}}ds=\frac{1}{2}\int_{\xi}^{0}{ds}}\quad(\textmd{along}\quad
l_{\varphi_{0}^{+}}^{1}),$ (4.7)
$\int_{b}^{\varphi_{4}}{\frac{s}{(s-\varphi_{0}^{+})\sqrt{s^{2}+m_{1}s+m_{2}}}ds=\frac{1}{2}\int_{0}^{\xi}{ds}}\quad(\textmd{along}\quad
l_{\varphi_{0}^{+}}^{2}).$ (4.8)
With the aim of Maple, we obtain the implicit expressions of
$\varphi_{i}(\xi)$ as in (3.3)-(3.6).
Meanwhile, suppose that $\varphi_{1}(\xi)\to\frac{c}{2}$ as
$\xi\to\xi_{0}^{1}$, $\varphi_{2}(\xi)\to\frac{c}{2}$ as $\xi\to-\xi_{0}^{2}$,
$\varphi_{3}(\xi)\to\varphi_{2}^{*}$ as $\xi\to-\xi_{0}^{3}$ and
$\varphi_{4}(\xi)\to\varphi_{2}^{*}$ as $\xi\to\xi_{0}^{4}$, then it follow
from (4.5) -(4.8) that
$\xi_{0}^{1}=\xi_{0}^{2}=\int_{a}^{\frac{c}{2}}{\frac{-s}{(s-\varphi_{0}^{-})\sqrt{s^{2}+l_{1}s+l_{2}}}ds}\quad\quad(\textmd{along}\quad
l_{\varphi_{0}^{-}}^{1}),$ (4.9)
$\xi_{0}^{3}=\xi_{0}^{4}=\int_{b}^{\varphi_{2}^{*}}{\frac{s}{(s-\varphi_{0}^{+})\sqrt{s^{2}+m_{1}s+m_{2}}}ds}\quad(\textmd{along}\quad
l_{\varphi_{0}^{+}}^{2}).$ (4.10)
With the aim of Maple, we get the expressions of $\xi_{0}^{1}$ and
$\xi_{0}^{3}$ as in (3.31) and (3.32). The proof of (1) in Theorem (3.1) is
finished.
Similarly, we can prove (2)-(4) in Theorem (3.1). Here we omit the details.
## 5 Numerical simulations
In this section, we will simulate the planar graphs of the kink-like and
antikink-like wave solutions.
From Section 2, we see that in the parameter expressions
$\varphi=\varphi(\xi)$ and $y=y(\xi)$ of the orbits of system (2.6), the graph
of $\varphi(\xi)$ and the integral curve of Eq.(2.3) are the same. In other
words, the integral curves of Eq.(2.3) are the planar graphs of the traveling
waves of Eq.(1.7). Therefore, we can see the planar graphs of the kink-like
and the antikink-like waves through the simulation of the integral curves of
Eq.(2.3).
###### Example 5.1
Take the same data as Example (3.1), that is $g=5$, $c=-1$, $a=-0.75$,
$b=1.6$. Let $\varphi=a=-0.75$ in (4.1) and (4.2), then we can get $y\approx
4.23397$ or $y\approx-4.23397$. And let $\varphi=b=1.6$ in (4.3) and (4.4),
then we obtain $y\approx 0.23198$ or $y\approx-0.23198$. Thus we take the
initial conditions of Eq.(2.3) as follows: (a) Corresponding to
$l_{\varphi_{0}^{-}}^{1}$ we take $\varphi(0)=-0.75$ and
$\varphi^{\prime}(0)=4.23397$; (b) Corresponding to $l_{\varphi_{0}^{-}}^{2}$
we take $\varphi(0)=-0.75$ and $\varphi^{\prime}(0)=-4.23397$; (c)
Corresponding to $l_{\varphi_{0}^{+}}^{1}$ we take $\varphi(0)=1.6$ and
$\varphi^{\prime}(0)=0.23198$; (d) Corresponding to $l_{\varphi_{0}^{+}}^{2}$
we take $\varphi(0)=1.6$ and $\varphi^{\prime}(0)=-0.23198$. Under each set of
initial conditions we use Maple to simulate the integrals curve of Eq.(2.3) in
Fig.7.
(a)
(b)
(c)
(d)
Figure 7: The numerical simulations of integral curves of Eq.(2.3) when $g=5$
and $c=-1$. (a) $\varphi(0)=-0.75$, $\varphi^{\prime}(0)=4.23397$; (b)
$\varphi(0)=-0.75$, $\varphi^{\prime}(0)=-4.23397$; (c) $\varphi(0)=1.6$,
$\varphi^{\prime}(0)=0.230189$; (d) $\varphi(0)=1.6$,
$\varphi^{\prime}(0)=-0.230189$.
###### Example 5.2
Take the same data as Example (3.2), that is $g=5$, $c=1$, $a=-1.6$, $b=2$.
Let $\varphi=a=-1.6$ in (4.1) and (4.2) , then we can get $y\approx 0.230189$
or $y\approx-0.230189$. And let $\varphi=b=2$ in (4.3) and (4.4), then we
obtain $y\approx 0.849228$ or $y\approx-0.849228$. Thus we take the initial
conditions of Eq.(2.3) as follows: (a) Corresponding to
$l_{\varphi_{0}^{-}}^{1}$ we take $\varphi(0)=-1.6$ and
$\varphi^{\prime}(0)=0.230189$; (b) Corresponding to $l_{\varphi_{0}^{-}}^{2}$
we take $\varphi(0)=-1.6$ and $\varphi^{\prime}(0)=-0.230189$; (c)
Corresponding to $l_{\varphi_{0}^{+}}^{1}$ we take $\varphi(0)=2$ and
$\varphi^{\prime}(0)=0.849228$; (d) Corresponding to $l_{\varphi_{0}^{+}}^{2}$
we take $\varphi(0)=2$ and $\varphi^{\prime}(0)=-0.849228$. Under each set of
initial conditions we use Maple to simulate the integrals curve of Eq.(2.3) in
Fig.8.
(a)
(b)
(c)
(d)
Figure 8: The numerical simulations of integral curves of Eq.(2.3) when $g=5$
and $c=1$. (a) $\varphi(0)=-1.6$, $\varphi^{\prime}(0)=0.230189$; (b)
$\varphi(0)=-1.6$, $\varphi^{\prime}(0)=-0.230189$; (c) $\varphi(0)=2$,
$\varphi^{\prime}(0)=0.849228$; (d) $\varphi(0)=2$,
$\varphi^{\prime}(0)=-0.849228$.
###### Example 5.3
Take the same data as Example (3.3), that is $g=-0.5$, $c=-2$, $d=-1.2$. Let
$\varphi=d=-1.2$ in (4.1) and (4.2), then we can get $y\approx 0.364453$ or
$y\approx-0.364453$. Thus we take the initial conditions of Eq.(2.3) as
follows: (a) Corresponding to $l_{\varphi_{0}^{-}}^{1}$ we take
$\varphi(0)=-1.2$ and $\varphi^{\prime}(0)=0.364453$; (b) Corresponding to
$l_{\varphi_{0}^{-}}^{2}$ we take $\varphi(0)=-1.2$ and
$\varphi^{\prime}(0)=-0.364453$. Under each set of initial conditions we use
Maple to simulate the integrals curve of Eq.(2.3) in Fig.9.
(a)
(b)
Figure 9: The numerical simulations of integral curves of Eq.(2.3) when
$g=-0.5$ and $c=-2$. (a) $\varphi(0)=-1.6$, $\varphi^{\prime}(0)=0.230189$;
(b) $\varphi(0)=-1.6$, $\varphi^{\prime}(0)=-0.230189$.
###### Example 5.4
Take the same data as Example (3.4), that is $g=-0.5$, $c=2$, $k=1.2$. Let
$\varphi=k=-1.6$ in (4.3) and (4.4) , then we can get $y\approx 0.364453$ or
$y\approx-0.364453$. Thus we take the initial conditions of Eq.(2.3) as
follows: (a) Corresponding to $l_{\varphi_{0}^{+}}^{1}$ we take
$\varphi(0)=1.2$ and $\varphi^{\prime}(0)=0.364453$; (b) Corresponding to
$l_{\varphi_{0}^{+}}^{2}$ we take $\varphi(0)=1.2$ and
$\varphi^{\prime}(0)=-0.364453$. Under each set of initial conditions we use
Maple to simulate the integrals curve of Eq.(2.3) in Fig.10.
(a)
(b)
Figure 10: The numerical simulations of integral curves of Eq.(2.3) when
$g=-0.5$ and $c=2$. (a) $\varphi(0)=1.2$, $\varphi^{\prime}(0)=0.364453$; (b)
$\varphi(0)=1.2$, $\varphi^{\prime}(0)=-0.364453$.
Comparing Fig.2 with Fig.7, Fig.3 with Fig.8, Fig.4 with Fig.9, and Fig.5 with
Fig.10, we can see that the graphs of $\varphi_{i}(\xi)(i=1,2,3,4,5,6,7,8)$
are the same as the corresponding integral curves of Eq.(2.3). This implies
that our theoretical results agree with the numerical simulations.
## 6 Conclusion
In this paper, we find a new type of bounded travelling wave solutions for the
$K(2,2)$ equation with osmosis dispersion. Their implicit expressions are
obtained in (3.3)-(3.14). From the graphs (see Fig.2-Fig.5) of the implicit
functions and the numerical simulations (see Fig.7-Fig.10) we see that these
new bounded solutions are defined on some semifinal bounded domains and
possess properties of kink and anti-kink wave solutions.
## References
* [1] P. Rosenau, J. M. Hyman, Compactons: solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564-567.
* [2] A. M. Wazwaz, Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl. Math. Comput. 138 (2/3) (2003) 309-319.
* [3] J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J Nonlinear Sci. Numer. Simulat. 6 (2) (2005) 207-208.
* [4] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (3) (2006) 700-708.
* [5] J. H. He, X. H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals 29 (1) (2006) 108-113.
* [6] J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J Modern Phys. B 20 (10) (2006) 1141-1199.
* [7] L. Xu, Variational approach to solitons of nonlinear dispersive equations, Chaos, Solitons and Fractals 37 (1) (2008) 137-143.
* [8] A. M. Wazwaz, General compacts solitary patterns solutions for modified nonlinear dispersive equation in higher dimensional spaces, Math. Comput. Simulat. 59 (6) (2002) 519-531.
* [9] A. M. Wazwaz, Compact and noncompact structures for a variant of KdV equation in higher dimensions, Appl. Math. Comput. 132 (1) (2002) 29-45.
* [10] Y. Chen, B. Li, H. Q. Zhang, New exact solutions for modified nonlinear dispersive equations in higher dimensions spaces, Math. Comput. Simul. 64 (5) (2004) 549-559.
* [11] B. He, Q. Meng, W. Rui , Y. Long, Bifurcations of travelling wave solutions for the equation, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 2114-2123.
* [12] Z. Y. Yan, Modified nonlinearly dispersive $mK(m,n,k)$ equations: I. New compacton solutions and solitary pattern solutions, Comput. Phys. Commun. 152 (1) (2003) 25-33.
* [13] Z. Y. Yan, Modified nonlinearly dispersive equations: II. Jacobi elliptic function solutions, Comput. Phys. Commun. 153 (1) (2003) 1-16.
* [14] A. Biswas, 1-soliton solution of the $K(m,n)$ equation with generalized evolution, Phys. Lett. A 372 (25) (2008) 4601-4602.
* [15] Y. G. Zhu, K. Tong, T. C. Lu, New exact solitary-wave solutions for the $K(2,2,1)$ and $K(3,3,1)$ equations, Chaos, Solitons and Fractals 33 (4) (2007) 1411-1416.
* [16] C. H. Xu, L. X. Tian, The bifurcation and peakon for $K(2,2)$ equation with osmosis dispersion, Chaos, Solitons and Fractals 40 (2) (2009) 893-901.
* [17] D. Luo et al., Bifurcation Theory and Methods of Dynamical Systems, World Scientific Publishing Co., London, 1997.
|
arxiv-papers
| 2009-08-06T17:21:33 |
2024-09-04T02:49:04.492996
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangbo Zhou, Lixin Tian, Xinghua Fan",
"submitter": "Jiangbo Zhou",
"url": "https://arxiv.org/abs/0908.0914"
}
|
0908.0921
|
# Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham
equation
Jiangbo Zhou Corresponding author. Tel.: +86-511-88969336; Fax:
+86-511-88969336. zhoujiangbo@yahoo.cn Lixin Tian Nonlinear Scientific
Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu
212013, China
###### Abstract
In this paper, we employ the bifurcation method of dynamical systems to
investigate the exact travelling wave solutions for the Fornberg-Whitham
equation $u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx}$. The implicit
expression for solitons is given. The explicit expressions for peakons and
periodic cusp wave solutions are also obtained. Further, we show that the
limits of soliton solutions and periodic cusp wave solutions are peakons.
###### keywords:
Fornberg-Whitham equation , soliton , peakon , periodic cusp wave solution
,
## 1 Introduction
The Fornberg-Whitham equation
$u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx},$ (1.1)
has appeared in the study of qualitative behaviors of wave breaking [1, 2]. It
is a nonlinear dispersive wave equation. Since Eq.(1.1) was derived, little
attention has been paid to studying it. In [3], Fornberg and Whitham obtained
a peaked solution of the form
$u(x,t)=A\exp{(-\frac{1}{2}\left|{x-\frac{4}{3}t}\right|)}$, where $A$ is an
arbitrary constant. In [4], we constructed a type of bounded travelling wave
solutions for Eq.(1.1), which are called kink-like and antikink-like wave
solutions. Unfortunately, the results in [3, 4] are not complete. In the
present paper, we continue to derive more travelling wave solutions for
Eq.(1.1), so that we can supplement the results of [3, 4].
The remainder of the paper is organized as follows. In Section 2, we discuss
the bifurcation curves and phase portraits of travelling wave system. In
Section 3, we obtain the implicit expression for solitons and the explicit
expressions for peakons and periodic cusp wave solutions. At the same time, we
show that the limits of solitons and periodic cusp wave solutions are peakons.
A short conclusion is given in Section 4.
## 2 Bifurcation and phase portraits of travelling wave system
Let $u=\varphi(\xi)$ with $\xi=x-ct$ be the solution for Eq.(1.1); then it
follows that
$-c\varphi^{\prime}+c\varphi^{\prime\prime\prime}+\varphi^{\prime}+\varphi\varphi^{\prime}=\varphi\varphi^{\prime\prime\prime}+3\varphi^{\prime}\varphi^{\prime\prime}.$
(2.1)
Integrating Eq. (2.1) once we have
$\varphi^{\prime\prime}(\varphi-c)=g-c\varphi+\varphi+\frac{1}{2}\varphi^{2}-(\varphi^{\prime})^{2},$
(2.2)
where $g$ is the integral constant.
Let $y=\varphi^{\prime}$; then we get the following planar dynamical system:
$\left\\{{\begin{array}[]{l}\frac{\textstyle d\varphi}{\textstyle d\xi}=y,\\\
\frac{\textstyle dy}{\textstyle d\xi}=\frac{\textstyle
g-c\varphi+\varphi+\frac{1}{2}\varphi^{2}-y^{2}}{\textstyle\varphi-c},\end{array}}\right.$
(2.3)
with a first integral
$H(\varphi,y)=(\varphi-c)^{2}(y^{2}-\frac{1}{4}\varphi^{2}+(\frac{1}{2}c-\frac{2}{3})\varphi+\frac{1}{4}c^{2}-\frac{1}{3}c-g)=h,$
(2.4)
where $h$ is a constant.
Note that (2.3) has a singular line $\varphi=c$. To avoid the line temporarily
we make transformation $d\xi=(\varphi-c)d\zeta$. Under this transformation,
Eq.(2.3) becomes
$\left\\{{\begin{array}[]{l}\frac{\textstyle d\varphi}{\textstyle
d\zeta}=(\varphi-c)y,\\\ \frac{\textstyle dy}{\textstyle
d\zeta}=g-c\varphi+\varphi+\frac{1}{2}\varphi^{2}-y^{2}.\\\
\end{array}}\right.$ (2.5)
System (2.3) and system (2.5) have the same first integral as (2.4).
Consequently, system (2.5) has the same topological phase portraits as system
(2.3) except for the straight line $\varphi=c$. Obviously, $\varphi=c$ is an
invariant straight-line solution for system (2.5).
For a fixed $h$, (2.4) determines a set of invariant curves of system (2.5).
As $h$ is varied, (2.4) determines different families of orbits of system
(2.5) having different dynamical behaviors. Let $M(\varphi_{e},y_{e})$ be the
coefficient matrix of the linearized version of (2.5) at the equilibrium point
$(\varphi_{e},y_{e})$; then
$M(\varphi_{e},y_{e})=\left({{\begin{array}[]{*{20}c}{\quad\quad
y_{e}\hfill}&&&{\varphi_{e}-c\hfill}\\\
{\varphi_{e}-(c-1)\hfill}&&&{-2y_{e}\hfill}\\\ \end{array}}}\right)$ (2.6)
and at this equilibrium point, we have
$J(\varphi_{e},y_{e})=\det
M(\varphi_{e},y_{e})=-2y_{e}^{2}-(\varphi_{e}-c)[\varphi_{e}-(c-1)],$ (2.7)
$p(\varphi_{e},y_{e})=\mathrm{trace}(M(\varphi_{e},y_{e}))=-y_{e}.$ (2.8)
By the theory of planar dynamical systems (see [5]), for an equilibrium point
of a planar dynamical system, if $J<0$, then this equilibrium point is a
saddle point; it is a center point if $J>0$ and $p=0$; if $J=0$ and the
Poincaré index of the equilibrium point is 0, then it is a cusp.
By using the first-integral value and properties of equilibrium points, we
obtain the bifurcation curves as follows:
$g_{1}(c)=\frac{1}{2}(c-1)^{2},$ (2.9)
$g_{2}(c)=\frac{1}{2}(c-1)^{2}-\frac{1}{18},$ (2.10)
$g_{3}(c)=\frac{1}{2}(c-1)^{2}-\frac{1}{2}.$ (2.11)
Obviously, the three curves have no intersection point and
$g_{3}(c)<g_{2}(c)<g_{1}(c)$ for arbitrary constant $c$.
Using the bifurcation method for vector fields (e.g., [5]), we have the
following result which describes the locations and properties of the singular
points of system (2.5).
###### Theorem 2.1
For given any constant wave speed $c\neq 0$, let
$\varphi_{1\pm}=c-1\pm\sqrt{(c-1)^{2}-2g}\quad for\quad g\leq g_{1}(c),$
(2.12) $y_{1\pm}=\pm\sqrt{g-\frac{1}{2}c^{2}+c}\quad for\quad g\geq g_{3}(c).$
(2.13)
Then we have
(1)If $g<g_{3}(c)$, then system (2.5) has two equilibrium points
$(\varphi_{1-},0)$ and $(\varphi_{1+},0)$, which are saddle points.
(2)If $g=g_{3}(c)$, then system (2.5) has two equilibrium points $(c-2,0)$ and
$(c,0)$. $(c-2,0)$ is a saddle point and $(c,0)$ is a cusp.
(3)If $g_{3}(c)<g<g_{2}(c)$, then system (2.5) has four equilibrium points
$(\varphi_{1-},0)$, $(\varphi_{1+},0)$, $(c,y_{1-})$ and $(c,y_{1+})$.
$(\varphi_{1-},0)$ is a saddle point and $(\varphi_{1+},0)$ is a center point
enclosing the orbit which connects the saddle points $(c,y_{1-})$ and
$(c,y_{1+})$.
(4)If $g=g_{2}(c)$, then system (2.5) has four equilibrium points
$(c-\frac{4}{3},0)$, $(c-\frac{2}{3},0)$, $(c,-\frac{2}{3})$ and
$(c,\frac{2}{3})$, which satisfy
$H(c-\frac{4}{3},0)=H(c,-\frac{2}{3})=H(c,\frac{2}{3})$ and form a triangular
orbit which encloses the center point $(c-\frac{2}{3},0)$.
(5)If $g_{2}(c)<g<g_{1}(c)$, then system (2.5) has four equilibrium points
$(\varphi_{1-},0)$, $(\varphi_{1+},0)$, $(c,y_{1-})$ and $(c,y_{1+})$.
$(\varphi_{1+},0)$ is a center point enclosing the orbit which is homoclinic
for the saddle point $(\varphi_{1-},0)$.
(6)If $g=g_{1}(c)$, then system (2.5) has three equilibrium points $(c-1,0)$,
$(c,-\frac{\sqrt{2}}{2})$ and $(c,\frac{\sqrt{2}}{2})$. $(c-1,0)$ is a cusp.
$(c,-\frac{\sqrt{2}}{2})$ and $(c,\frac{\sqrt{2}}{2})$ are two saddle points.
(7)If $g>g_{1}(c)$, then system (2.5) has two equilibrium points $(c,y_{1-})$
and $(c,y_{1+})$. They are saddle points.
The phase portraits of system (2.5) are given in Fig.1.
(a) (b)
(c) (d)
(e) (f)
(g)
Figure 1: The phase portraits of system (2.5). (a) $g<g_{3}(c)$; (b)
$g=g_{3}(c)$; (c) $g_{3}(c)<g<g_{2}(c)$; (d) $g=g_{2}(c)$; (e)
$g_{2}(c)<g<g_{1}(c)$; (f) $g=g_{1}(c)$; (g) $g>g_{1}(c)$.
## 3 Solitons, peakons and periodic cusp wave solutions
Suppose that $\varphi(\xi)(\xi=x-ct)$ is a travelling wave solution for
Eq.(1.1) for $\xi\in(-\infty,+\infty)$, and
$\mathop{\lim}\limits_{\xi\to-\infty}\varphi(\xi)=A$,
$\mathop{\lim}\limits_{\xi\to\infty}\varphi(\xi)=B$, where $A$ and $B$ are two
constants. If $A=B$, then $\varphi(\xi)$ is called a soliton solution. If
$A\neq B$, then $\varphi(\xi)$ is called a kink (or an antikink) solution.
Usually, a soliton solution for Eq.(1.1) corresponds to a homoclinic orbit of
system (2.3) and a periodic travelling wave solution for Eq.(1.1) corresponds
to a periodic orbit of system (2.3). Similarly, a kink (or an antikink) wave
solution of Eq.(1.1) corresponds to a heteroclinic orbit (or the so-called
connecting orbit) of system (2.3). The graphs of the homoclinic orbit,
periodic orbit and their limit cure are shown in Fig.2.
The following lemma gives the relationship of soliton solutions of Eq.(1.1)
and homoclinic orbits of system (2.3).
###### Lemma 3.1
Assume that $\Gamma$ is a homoclinic orbit of system (2.3) and its parameter
expression is $\varphi=\varphi(\xi)$ and $y=y(\xi)$; then $u=\varphi(\xi)$
with $\xi=x-ct$ is a soliton solution for Eq.(1.1).
_Proof._ From Fig.1(e), we can see that the homoclinic orbit $\Gamma$ encloses
$(\varphi_{1+},0)$ and connects $(\varphi_{1-},0)$. Therefore,
$\lim_{|\xi|\rightarrow\infty}\varphi(\xi)=\varphi_{1-}$.
On the other hand, $u=\varphi(\xi)$ is the solution for system (2.3). This
implies that $u=\varphi(\xi)$ is the solution for Eq.(2.1). Thus,
$u=\varphi(x-ct)$ is the soliton solution for Eq.(1.1).
(a) (b)
(c)
Figure 2: The orbits of system (2.3). (a) The homoclinic orbit (corresponding
to $g_{2}(c)<g<g_{1}(c)$). (b) The limit curve of homoclinic orbit and
periodic orbit (corresponding to $g=g_{2}(c)$). (c) The periodic orbit
(corresponding to $g_{3}(c)<g<g_{2}(c)$).
In Fig.2(a), the homoclinic orbit of system (2.3) can be expressed as
$y=\pm\frac{(\varphi-\varphi_{1-})\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}}{2(\varphi-c)}\quad
for\quad\varphi_{1-}<\varphi<\varphi_{2+},$ (3.1)
where
$l_{1}=\frac{2}{3}(1-3c-3\sqrt{(c-1)^{2}-2g}),$ (3.2)
$l_{2}=\frac{2}{3}(1-4c+3c^{2}-3g+(3c+1)\sqrt{(c-1)^{2}-2g}),$ (3.3)
$\varphi_{2+}=-\frac{1}{3}(1-3c-3\sqrt{(c-1)^{2}-2g}+2\sqrt{1-3\sqrt{(c-1)^{2}-2g}}).$
(3.4)
Substituting Eq.(3.1) into the first equation of system (2.3) and integrating
along the homoclinic orbits, we have
${\int_{\varphi}^{\varphi_{2+}}{\frac{s-c}{(s-\varphi_{1-})\sqrt{s^{2}+l_{1}s+l_{2}}}ds=-\frac{1}{2}|\xi|}}.$
(3.5)
It follows from (3.5) that
$\beta(\varphi_{2+})=\beta(\varphi)\exp(-\frac{1}{2}|\xi|),$ (3.6)
where
$\beta_{(}\varphi)=\frac{(2\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}+2\varphi+l_{1})(\varphi-\varphi_{1-})^{\alpha_{1}}}{(2\sqrt{a_{1}}\sqrt{\varphi^{2}+l_{1}\varphi+l_{2}}+b_{1}\varphi+l_{3})^{\alpha_{1}}},$
(3.7) $l_{1}=\frac{2}{3}(1-3c-3\sqrt{(c-1)^{2}-2g}),$ (3.8)
$l_{2}=\frac{2}{3}(1-4c+3c^{2}-3g+(3c+1)\sqrt{(c-1)^{2}-2g}),$ (3.9)
$l_{3}=\frac{4}{3}(2-5c+3c^{2}-6g+(3c+2)\sqrt{(c-1)^{2}-2g}),$ (3.10)
$a_{1}=4(1-2c+c^{2}-2g+\sqrt{(c-1)^{2}-2g}),$ (3.11)
$b_{1}=-\frac{4}{3}-4\sqrt{(c-1)^{2}-2g},$ (3.12)
$\alpha_{1}=-\frac{1+\sqrt{(c-1)^{2}-2g},}{2\sqrt{(c-1)^{2}-2g+\sqrt{(c-1)^{2}-2g}}}.$
(3.13)
(3.6) is the implicit expression for solitons for Eq.(1.1). We show the graphs
of the solitons in Fig.3 under some parameter conditions. From Fig.3, we can
see that when $g_{2}(c)<g<g_{1}(c)$ and $g$ tends to $g_{2}(c)$, the solitons
lose their smoothness and tend to peakons.
(a) (b)
(c) (d)
Figure 3: The solitons for Eq.(1.1). (a) $c=2$, $g=0.499999$; (b) $c=2$,
$g=0.49$; (c) $c=2$, $g=0.46$ ; (d) $c=2$, $g=0.45$.
Note the following facts: when $g_{2}(c)<g<g_{1}(c)$ and $g$ tends to
$g_{2}(c)$, the limit curve of such homoclinic orbit of system (2.3) is a
triangle with the following three line segments (see Fig.2(b)):
$y=\pm\frac{1}{2}(\varphi-c+\frac{4}{3})\quad
for\quad\varphi_{1-}\leq\varphi\leq\varphi_{2},$ (3.14)
and
$\varphi=c\quad for\quad-\frac{2}{3}\leq y\leq\frac{2}{3}.$ (3.15)
Let us have $g_{2}(c)<g<g_{1}(c)$ and $g$ tends to $g_{2}(c)$; then we obtain
that
$\varphi(\xi)=\frac{4}{3}\exp({-\frac{1}{2}|\xi|})+c-\frac{4}{3},$ (3.16)
which implies that for arbitrary constant $c\neq 0$, Eq.(1.1) has peakons
$u(x,t)=\frac{4}{3}\exp({-\frac{1}{2}|x-ct|})+(c-\frac{4}{3}).$ (3.17)
Obviously, $u$ has peaks at $x-ct=0$. We show graphs of the peakons in Fig.4
under some parameter conditions.
(a) (b)
Figure 4: The peakons for Eq.(1.1). (a) $c=-1$ ; (b) $c=2$.
###### Remark 3.1
(1) If we take $c=\frac{4}{3}$ in (3.17), then we can see that (3.17) agrees
with the result in [3].
(2) In the phase portaits, the triangle curve corresponds to a peakon
solution.
We have the following lemma, similar to Lemma 3.1, which indicates the
relationship of periodic wave solutions for Eq.(1.1) and periodic orbits of
system (2.3).
###### Lemma 3.2
Assume that $\Gamma$ is a periodic orbit of system (2.3) and that its
parameter expression is $\varphi=\varphi(\xi)$ and $y=y(\xi)$; then
$u=\varphi(\xi)$ with $\xi=x-ct$ is a periodic wave solution for Eq.(1.1).
In Fig.2(c), the periodic orbit can be expressed as
$y=\pm\sqrt{\frac{1}{4}\varphi^{2}-(\frac{1}{2}c-\frac{2}{3})\varphi-\frac{1}{4}c^{2}+\frac{1}{3}c+g}\quad
for\quad\varphi_{2-}\leq\varphi\leq c,$ (3.18)
and
$\varphi=c\quad for\quad y_{1-}\leq y\leq y_{1+},$ (3.19)
where
$\varphi_{2-}=\frac{1}{3}(-4+3c+\sqrt{2(9c^{2}-18c+8-18g)}).$ (3.20)
Substituting (3.18) into the first equation of system (2.3) and integrating
along the periodic orbit, we have
${\int_{\varphi}^{c}{\frac{1}{\sqrt{\varphi^{2}-(2c-\frac{8}{3})\varphi-c^{2}+\frac{4}{3}c+4g}}}ds=-\frac{1}{2}\xi}\quad
for\quad\xi<0,$ (3.21)
and
${\int_{\varphi}^{c}{\frac{1}{\sqrt{\varphi^{2}-(2c-\frac{8}{3})\varphi-c^{2}+\frac{4}{3}c+4g}}}ds=\frac{1}{2}\xi}\quad
for\quad\xi>0.$ (3.22)
It follows from (3.21) and (3.22) that
$\varphi(\xi)=l_{+}\exp({-\frac{1}{2}|\xi|})+l_{-}\exp({\frac{1}{2}|\xi|})+(c-\frac{4}{3})\quad
for\quad\varphi_{2-}\leq\varphi\leq c,$ (3.23)
where
$l_{\pm}=\frac{1}{6}(4\pm 3\sqrt{4g+4c-2c^{2}}).$ (3.24)
Let
$T=2|\ln(\varphi_{2-}-c+\frac{4}{3})-\ln(2l_{-})|.$ (3.25)
Then
$u(x,t)=\varphi(x-ct-2nT)\quad for\quad(2n-1)T<x-ct<(2n+1)T,$ (3.26)
are periodic cusp wave solutions for Eq.(1.1) with $2T$ period. Clearly, when
$g_{3}(c)<g<g_{2}(c)$ and $g\rightarrow g_{2}(c)$, $T\rightarrow\infty$,
$l_{+}\rightarrow\frac{3}{4}$, $l_{-}\rightarrow 0$, and $u(x,t)$ in (3.26)
tends to
$u(x,t)=\frac{4}{3}\exp({-\frac{1}{2}|x-ct|})+(c-\frac{4}{3}).$ (3.27)
(3.27) is identical with (3.17). The graphs of some periodic waves for
Eq.(1.1) are shown in Fig.5 under some parameter conditions. From Fig.5 we can
see that $g_{3}(c)<g<g_{2}(c)$ and $g$ tends to $g_{2}(c)$, the periodic cusp
wave solutions tend to peakons.
(a) (b)
(c) (d)
Figure 5: The periodic cusp wave solutions for Eq.(1.1). (a) $c=2$, $g=0.3$;
(b) $c=2$, $g=0.4$; (c) $c=2$, $g=0.4444444$ ; (d) $c=2$, $g=0.444444444444$.
###### Remark 3.2
(1) In the phase portaits, the semi-elliptic closed curve with one side on the
singular line $\varphi=c$ corresponds to a periodic cusp wave solution.
(2) In [4], we dealt with the case $g\leq g_{1}(c)$, and obtained the kink-
like and antikink-like wave solutions for Eq.(1.1).
## 4 Conclusion
In this work, by using the bifurcation method, we obtain the analytic
expressions for solitons, peakons and periodic wave solutions for the
Fornberg-Whitham equation, given as (3.6), (3.17) and (3.26), respectively. We
also show the relationships among the solitons, peakons and periodic cusp wave
solutions.
## References
* [1] G. B. Whitham, Variational methods and applications to water wave, Proc. R. Soc. Lond. Ser. A 299 (1967) 6-25.
* [2] R. Ivanov, On the integrability of a class of nonlinear dispersive wave equations, J. Nonlinear Math. Phys. 1294 (2005) 462-468.
* [3] B. Fornberg, G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. Lond. Ser. A 289 (1978) 373-404.
* [4] J. Zhou, L. Tian, A type of bounded traveling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261.
* [5] D. Luo, et al., Bifurcation Theory and Methods of Dynamical Systems, World Scientific Publishing Co., London, 1997.
|
arxiv-papers
| 2009-08-06T17:53:25 |
2024-09-04T02:49:04.498332
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangbo Zhou, Lixin Tian",
"submitter": "Jiangbo Zhou",
"url": "https://arxiv.org/abs/0908.0921"
}
|
0908.1001
|
# Does Unruh radiation accelerate the universe? A novel approach to the cosmic
acceleration
Hongsheng Zhang111Electronic address: hongsheng@kasi.re.kr Korea Astronomy and
Space Science Institute, Daejeon 305-348, Korea Department of Astronomy,
Beijing Normal University, Beijing 100875, China Hyerim Noh Korea Astronomy
and Space Science Institute, Daejeon 305-348, Korea Zong-Hong Zhu Department
of Astronomy, Beijing Normal University, Beijing 100875, China Hongwei Yu
Department of Physics and Institute of Physics, Hunan Normal University,
Changsha, Hunan 410081, China
###### Abstract
We present a novel mechanism for the present acceleration of the universe. We
find that the temperature of the Unruh radiation perceived by the brane is not
equal to the inherent temperature (Hawking temperature at the apparent
horizon) of the brane universe in the frame of Dvali-Gabadadze-Porrati (DGP)
braneworld model. The Unruh radiation perceived by a dust dominated brane is
always warmer than the brane measured by the geometric temperature, which
naturally induces an energy flow between bulk and brane based on the most
sound thermodynamics principles. Through a thorough investigation to the
microscopic mechanism of interaction between bulk Unruh radiation and brane
matter, we put forward that an energy influx from bulk Unruh radiation to the
dust matter on the brane accelerates the universe.
Unruh effect Cosmic acceleration
###### pacs:
98.80.-k, 95.36.+x
††preprint: arXiv: 0908.1001
## I Introduction
An unexpected discovery appeared in 1998, that is, our universe is
accelerating rather than decelerating acce . This discovery has significant
and far-reaching influence on both fundamental physics and astronomy. The
invisible sector of the universe (or the modified terms to Einstein gravity)
dominates the present evolution and determines the final destiny of the
universe. To understand its physical nature, we have to go beyond the standard
model. Various explanations of this acceleration have been proposed; see
review for recent reviews with fairly complete lists of references of
different models. However, the physical nature of the cosmic acceleration
remains as a most profound problem in sciences.
In different approaches to the cosmic acceleration problem, braneworld models
draw much attention in recent years. Inspired by string/M theory the
braneworld gravity is introduced for several problems in standard model,
firstly for the hierarchy problem. The braneworld gravity has been extensively
studied, for a review, seerev . In the braneworld scenario the standard model
particles are confined to a brane, while gravity can propagate in the whole
space time. At the low energy region general relativity recovers, while at the
high energy region the behavior of gravity is strongly modified. This may
yield remarkable changes in gravity dynamics with several different
implications for cosmology, black holes and high energy physics. Among various
braneworld models, the model proposed by Dvali, Gabadadze and Porrati (DGP)
dgpmodel is impressive and a leading one in cosmology. In the DGP model, the
bulk is a flat Minkowski spacetime, but a reduced gravity term appears on a
tensionless brane. In this model, gravity appears 4-dimensional at short
distances but is altered at distance large compared to some freely adjustable
crossover scale $r_{0}$ through the slow evaporation of the graviton off our
4-dimensional brane world universe into an unseen, yet large, 5th dimension.
At short distances the 4-dimensional curvature scalar $R$ dominates and
ensures that gravity looks 4-dimensional. At large distances the 5-dimensional
curvature scalar ${}^{(5)}R$ takes over and gravity spreads into extra
dimension. The late-time acceleration is driven by the manifestation of the
excruciatingly slow leakage of gravity off our four-dimensional world into an
extra dimension. This offers an alternative explanation for the current
acceleration of the universe dgpcosmology . However, just as LCDM model, it is
also suffered from fine-tune problem. In LCDM model, $\Lambda$ is an
unimaginatively tiny constant compared to the expectation of the vacuum if our
quantum field theory (QFT) is valid till to Planck scale. We know QFT is at
least valid at eletro-weak scale, whose expectation of zero-point energy is
still $10^{60}$ higher than the scale of the cosmological constant. To solve
the dark energy problem in frame of DGP model, people require $r_{c}\sim
H_{0}^{-1}$, where $r_{c}$ is a constant, and $H_{0}$ is the present Hubble
parameter. Similarly, we can ask why $r_{c}$ is so large that it is
approximately equals the Hubble radius.
Interaction is a universal phenomenon in the physics world. An interaction
term between bulk and brane has been invoked as a possible mechanism for the
cosmic acceleration in frame of braneworld model interbrane . However, in
previous works little attention has been devoted to the physical mechanism of
such a term. There is a natural physical origin for this interaction which has
not been adverted before, that is, Unruh effect. In the braneworld scenario a
brane is moving in the bulk, and generally speaking, its proper acceleration
does not vanish. Thus, a brane, qua an observer in the bulk space, should
perceive the Unruh radiation. The Unruh effect of RS II braneworld is
investigated in jenn .
Even for a Minkowski bulk, the brane will sense a thermal bath filled with
radiations. If the temperature is different from the inherent temperature of
the brane, energy flow will appear based on the most sound principles of
thermodynamics. There are several different particles are confined to the
brane, which are not in thermal equilibrium. Then, what is the inherent
temperature of the brane in the frame of gravity theory? It will be useful to
give a brief account of the previous approaches, especially on black hole
thermodynamics.
The relation between gravity theory and thermodynamics is an interesting and
profound issue. The key quantities bridging the gravity and thermodynamics are
temperature and entropy. Temperature of an ordinary system denotes the the
average kinetic energy of microscopic motions of a single particle. To gravity
the temperature becomes subtle. Since we do not have a complete quantum theory
of gravity, for general case we can not use the usual way to get the
temperature of the gravity field. Under this situation people set up some
thermodynamics and statistical quantities of gravity by using semi-quantum
(matter field is quantized, but gravity remains classical) theory, though the
concept of gravitational particle is not clear. The black hole thermodynamics
(in fact, spacetime thermodynamics, because the physical quantities in black
hole thermodynamics should be treated as the quantities of the globally
asymptomatic flat manifold) is set up in 4law and confirmed by Hawking
radiation hawk .
In the initial work 4law , the temperature of the spacetime is suspected by an
analogy to the ordinary thermodynamics. Shortly, it is recognized that the
temperature is just the temperature of the radiation emitted from the black
hole by a semi-quantum treatment. The method in hawk depends on the global
structure of the spacetime. The vacuum state is treated as nonvacuum state by
observers at spacelike infinity, which roots in the fact that vacuum state
around the black hole is different from the vacuum at spacelike infinity.
Along this clue Unruh found that a uniformly accelerating particle detector
with proper acceleration $A$ in Minkowski vacuum perceives different excited
modes exactly in Bose-Einstein distribution with a temperature $T=A/2\pi$
unruh . Unruh effect helps us to derive the local temperature of a
gravitational system. Unruh effect has inherent relations to Hawking effect.
After a conformal transformation, the thermal particles detected by an
accelerated detector becomes thermal particles seen by an inertial observer in
curved spacetime, which is just the Hawking effect in the case of black hole
bril .
The laws of black hole mechanics are results of classical Einstein gravity,
for qualification of laws of thermal dynamics the quantum theory is required.
The mathematical form keeps invariant. This implies that the classical gravity
theory may hide information from quantum theory. This possibility is
investigated in jaco . The Einstein equation is reproduced from the
proportionality of entropy and horizon area together with the first law of
thermal dynamics, $\delta Q=TdS$, jointing to heat, entropy, and temperature,
where the temperature is the Unruh temperature to an observer just behind a
causal Rindler horizon. The entropy is supposed to be proportional to the area
of this horizon and the heat flow is measured by this observer. The
significance of this derivation is that the Unruh effect is a result of
quantum field theory but the derivation of classical Einstein theory depends
on it.
More directly, for a dynamical spacetime, a similar procedure reproduces the
Friedmann equation. In this case one should apply the first law of
thermodynamics to the trapped surface (apparent horizon) of an FRW universe
and assume the geometric entropy given by a quarter of the apparent horizon
area and the temperature given by the inverse of the apparent horizon cai .
There are several arguments that the apparent horizon should be a causal
horizon and is associated with the gravitational entropy and Hawking
temperature bak . So, for an expanding universe, the Hawking temperature at
the apparent horizon should be treated as the inherent temperature of the
universe. For a brane universe its inherent temperature may be higher or lower
than the temperature of the Unruh radiation in the bulk, which thus triggers
an energy flow between brane and bulk. We will study this possibility in frame
of DGP braneworld model.
This paper is organized as follows: In the next section we give the basic
construction of DGP model and present the Unruh temperature perceived by an
inertial observer (inertial respective to the brane, which we will explain in
detail) on the brane. In section III, we investigate the condition for the
temperature to be valid. In section IV, we study geometric temperature of a
DGP brane in detail and find the thermal equilibrium condition for the Unruh
radiation and the brane (evaluated by its geometric temperature). In section
V, we present a detailed study of the energy exchange between bulk Unruh
radiation and the brane matter by statistical mechanics. Our conclusion and
discussion appear in section VI.
## II Unruh effect for a DGP brane
In this section we discuss the Unruh effect for a DGP brane. Before studying
the Unruh effect of a brane, we shall first give a brief review of the
gravitational thermodynamics and Unruh effect in 4 dimensional theory,
especially the reality of this significant effect.
Unruh effect states that the concept of particle depends on observer. This
amazing effect generates some puzzles, sometimes even treated as paradoxes.
First, Unruh’s original construction is not consistent because his
quantization is not unitarily equivalent to the standard construction
associated with Minkowski vacuum. Hence some authors used mathematically more
rigorous methods to solve this problem math .
Second, the temperature of an Unruh particle detector, which is in thermal
equilibrium with Unruh radiation it experiences in Minkowski vacuum, is higher
than the temperature of the vacuum to an inertial observer. Hence, does it
$really$ emit radiation for an inertial observer, just like an accelerated
charged particle? It was argued that there was no radiation flux from an Unruh
observer no . Unruh made an almost the same calculation as in no , but he
found extra terms in the two point correlation function of the field which
would contribute to the excitation of a detector 1992 . It was pointed out
that the extra terms were missing in no , which shape a polarization cloud
about an Unruh detector cloud . The above analysis and detailed analysis
(including non-uniformly accelerated observers) in non showed that in a
2-dimensional toy model, there is no radiation flux from the detector.
Recently it was found that there exists a positive radiated power of quantum
nature emitted by the detector in 4 dimensional space lin .
It should be noted that the response of an Unruh observer to the Minkowski
vacuum is independent of its inner structure: The distribution of the
different excited modes perceived by the observer depends only on the
acceleration of the observer. This implies that the Unruh temperature is an
inherent property of the quantum field. The detector only plays the role of
test particles. This effect still exists even without a detector, which can be
seen clearly from the derivation by Bogolubov transformation. The Unruh effect
can be also derived in terms of the spontaneous excitation of accelerating
atoms Audretsch94 ; ZYL06 ; ZY07 .
Several contemporary research themes, for example, the black hole
thermodynamics wald , the quantum entanglement state yuhan , the Lorentz
symmetry breaking kkrs . For extensive references of the Unruh effect, see a
recent review article ureview .
Before studying the Unruh effect for a DGP brane we warm up by a short review
the DGP brane. The basic construction of DGP model can be written as follows
dgpmodel ,
$S=S_{\rm bulk}+S_{\rm brane},$ (1)
where
$S_{\rm bulk}=\int d^{5}x\sqrt{-{}^{(5)}g}\left[{1\over
2\kappa^{2}}{}^{(5)}R\right],$ (2)
and
$S_{\rm brane}=\int d^{4}x\sqrt{-g}\left[{1\over\kappa^{2}}K^{\pm}+L_{\rm
brane}(g_{\alpha\beta},\psi)\right].$ (3)
Here $\kappa^{2}$ is the 5-dimensional gravitational constant, ${}^{(5)}R$ is
the 5-dimensional scalar curvature. The induced metric $g_{\mu\nu}$ is defined
as
$g_{\mu\nu}={}^{(5)}g_{\mu\nu}+n_{\mu}n_{\nu},$ (4)
where the lower case of Greeks run from $0\sim 3$, and $n_{\mu}$ is the normal
to the brane. $x^{\mu}~{}(\mu=0,1,2,3)$ are the induced 4-dimensional
coordinates on the brane, $K^{\pm}$ is the trace of extrinsic curvature on
either side of the brane and $L_{\rm brane}(g_{\alpha\beta},\psi)$ is the
effective 4-dimensional Lagrangian, which is given by a generic functional of
the brane metric $g_{\alpha\beta}$ and matter fields $\psi$ on the brane. In
this article we adopt a mostly negative signature. For DGP model, an induced 4
dimensional Ricci scalar term appears in the brane Lagrangian $L_{\rm brane}$,
$\displaystyle L_{\rm brane}={\mu^{2}\over 2}R+L_{\rm m},$ (5)
where $\mu$ is the reduced 4 -dimensional Planck mass, $R$ denotes the scalar
curvature on the brane, and $L_{\rm m}$ stands for the Lagrangian of matters
on the brane.
Assuming a Friedmann-Robertson-Walker (FRW) metric on the brane, we can derive
the Friedmann equation on the brane dgpcosmology (for a more extensive study
of the DGP cosmology in which a Gauss-Bonnet term a Weyl term appear in the
bulk, see gbzhang ),
$H^{2}+\frac{k}{a^{2}}=\frac{\rho}{3\mu^{2}}+\frac{2}{r_{c}^{2}}+\frac{2\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{1/2},$
(6)
where $\rho$ denotes matter energy density on the brane,
$r_{c}=\kappa^{2}\mu^{2}$, denotes the cross radius of DGP brane,
$\epsilon=\pm 1$, represents the two branches of DGP model, $H$ is Hubble
parameter, $k$ is spatial curvature, and $a$ is the scale factor of the brane.
The branch $\epsilon=1$ was treated as an unstable branch based on the linear
order perturbations. However, a recent study shows that ghosts do not appear
under small fluctuations of an empty background, and conformal sources do not
yield instabilities either stable .
Now we begin to investigate the Unruh effect for a DGP brane in a Minkowski
bulk. We work in a bulk-based coordinate system,
$dS^{2}=d\hat{T}^{2}-dR^{2}-R^{2}d\Omega_{3}^{2},$ (7)
where $dS^{2}$ is the 5 dimensional line-element, $d\Omega_{3}^{2}$ denotes a
3 dimensional sphere. A free falling observer on the brane, which is just the
comoving observer in the FRW universe, can be described in the FRW
coordinates,
$x^{\mu}=(t,x^{1}_{0},x^{2}_{0},x^{3}_{0}),$ (8)
where $t=x_{0}$ is the standard world time function, and
$x^{1}_{0},x^{2}_{0},x^{3}_{0}$ are three coordinates of the unit 3-sphere
$\Omega_{3}$. For a comoving observer, they are constants. Hence, clearly its
acceleration is zero in the brane-viewpoint. However, it accelerates in the
cross direction of the brane, or the direction of extradimension. The observer
moves along a non-geodesic orbit measured by the full 5-dimensional metric.
This is the exact meaning of “A brane accelerates in the bulk”, and the
acceleration of the brane indicates the acceleration of such comoving
observers.
We assume a free-falling particle on the brane, as a particle detector, moves
along the worldline
$x^{B}(t)=(\hat{T},R,x^{1}_{0},x^{2}_{0},x^{3}_{0}),$ (9)
where the capital Latin letters run from $0\sim 4$. For a moving brane in the
$R$ direction, the velocity of a free falling observer on the brane reads
$U^{A}=(\dot{\hat{T}},\dot{R},0,0,0),$ (10)
which satisfies the normalization condition,
$U_{A}U^{A}=1.$ (11)
The induced metric on the brane reads,
${ds}^{2}=dS^{2}-U\otimes U=dt^{2}-a(t)^{2}d\Omega_{3}^{2},$ (12)
where we have labeled the radius coordinate $R$ by a new symbol $a$, which is
more frequently used in cosmology.
The unit normal of the brane is vertical to the velocity and spacelike,
$n_{A}U^{A}=0,~{}n_{A}n^{A}=-1.$ (13)
The acceleration of the brane is defined as,
$A^{C}=U^{B}\nabla_{B}U^{C}.$ (14)
We consider the component of the extrinsic curvature along the velocity,
$K_{tt}=K_{AB}U^{A}U^{B}=-n_{C}A^{C}.$ (15)
By using the junction condition across the brane
$[K_{\mu\nu}]=-\kappa^{2}(s_{\mu\nu}-\frac{1}{3}g_{\mu\nu}s),$ (16)
where $[K_{\mu\nu}]$ denotes the difference of the extrinsic curvatures of the
two sides of the brane, $s_{\mu\nu}$ represents the energy-momentum confined
to the brane, we derive the amplitude of the acceleration of brane lang ,
$A=\left|{}^{(5)}g_{CD}A^{C}A^{D}\right|^{1/2}=\frac{\kappa^{2}}{6}\left|2\rho_{e}+3p_{e}\right|.$
(17)
Here $\rho_{e}$ and $p_{e}$ are the effective energy density and pressure of
the brane, which are defined as
$\displaystyle\rho_{e}=s_{0}^{0},$ (18) $\displaystyle p_{e}=-s_{i}^{i},$ (19)
where $i$ is an arbitrary spatial index and we need not sum over the index
$i$. And $s$ is the effective energy momentum confined to the brane,
$s_{\mu\nu}\triangleq\frac{2}{\sqrt{-g}}\frac{\delta({\sqrt{-g}L_{\rm
brane}})}{\delta g_{\mu\nu}},$ (20)
where $L_{\rm brane}$ is given by (5). Due to the above equation, the energy
momentum $s_{\mu\nu}$ includes the contribution of the matter term $L_{m}$, as
well as the induced curvature term $R$ on the brane. That is the reason why we
call the density and pressure in (18) and (19) effective density and pressure.
We stress that the effective density $\rho_{e}$ is different from the local
density $\rho$ of the matter confined to the brane in (6), which does not
include the geometric effect of the induced Ricci term $R$.
We see that, generally speaking, $A$ does not vanish. So the brane may
perceive Unruh-type radiation in the bulk. To discuss Unruh radiations seen by
the observer on the brane, we introduce a 5-dimensional scalar field in the
bulk, which is in its Minkowski vacuum state,
${\cal
L}_{\phi}=\frac{1}{2}\partial_{A}\phi\partial^{A}\phi-\frac{1}{2}m^{2}\phi^{2},$
(21)
where $m$ denotes $\phi$’s mass parameter. Since the scalar is in its vacuum
state, classically it does no work on the brane and bulk dynamics.
Several different methods have been proposed to derive Unruh effect since
Unruh’s original work bril . Here we use the Green function method. We
consider a detector minimally coupled to a 5-dimensional scalar field $\phi$
in the bulk 222One often adds a conformal coupling term to gravitational field
in the action $\L_{{\rm con}}=-\frac{1}{2}\xi R\phi^{2}$. But there are some
subtles in this couple, because there exists two Ricci scalars in the DGP
model and two types of conformal transformations. Wether the Lagrangian is
conformally invariant depends on definition. Here we just consider the minimal
coupling case..
The Lagrangian $cm(t)\phi[x(t)]$ describes the interaction between detector
and field, where $c$ is the coupling constant and $m$ denotes the moment
operator of the detector. It is shown in bril that, for a small number $c$,
the probability amplitude of the transition from the ground state $|E_{0}>$ of
a particle detector coupled to a scalar field in its vacuum state $|0_{M}>$ to
an excited state $|E>$, where $M$ stands for Minkowski, reads
$c^{2}\sum_{E}|<E|m|0_{M},E_{0}>|^{2}{\cal F}(E-E_{0}),$ (22)
where the detector response function ${\cal F}(E)$ is defined as
${\cal
F}(E-E_{0})=\int^{\infty}_{-\infty}dt\int^{\infty}_{-\infty}dt^{\prime}G^{+}(x(t),x(t^{\prime}))e^{-i(E-E_{0})(t-t^{\prime})},$
(23)
and the Wightman Green function $G^{+}(x,x^{\prime})$ is defined as
$G^{+}=<0_{M}|\phi(x)\phi(x^{\prime})|0_{M}>.$ (24)
We note that the term $|<E|m|0_{M},E_{0}>|^{2}$ depends on the structure of
the particle detector, but ${\cal F}(E-E_{0})$ does not, which reflects the
inherent properties of quantum fields. It is clear that for a general
trajectory the response function is not zero.
For a uniformly accelerated trajectory the Wightman Green function becomes a
function of $\Delta t=t-t^{\prime}$. Since the detector will detect infinite
particles in its whole history, the response function (23) is not well-
defined. Under this situation it will make sense to consider the unit response
function, which describes in unit time interval,
${\cal U}(E-E_{0})=\int^{\infty}_{-\infty}d\Delta te^{-i(E-E_{0})\Delta
t}G^{+}(\Delta t).$ (25)
Since $x_{0}^{1},x_{0}^{2},x_{0}^{3}$ are constant, we just set
$x_{0}^{1}=x_{0}^{2}=x_{0}^{3}=0$ after a coordinates transformation. For a
uniformly accelerated observer in the cross-brane direction, the coordinates
in (9) read,
$x_{0}^{1}=x_{0}^{2}=x_{0}^{3}=0,~{}\hat{T}=\frac{\sinh(tA)}{A},R=\frac{\cosh(tA)}{A},$
(26)
where $A$ is a constant, denoting the magnitude of acceleration of the
detector. Then, integrating (24) directly with proper boundary condition
(corresponding to proper contour bril ), we obtain its concrete form in
McDonald function (Bessel function with imaginary arguments),
$G^{+}(\Delta t)=\frac{e^{i5\pi/4}(mA)^{3/2}}{16\pi^{5/2}(\sinh(A\Delta
t/2-i\varepsilon))^{3/2}}K_{3/2}\left(\frac{i2m}{A}\sinh(A\Delta
t/2-i\varepsilon)\right),$ (27)
where, as usual, $\varepsilon$ is a small positive number. The Wightman Green
function for massless mode reads,
$D^{+}({\Delta t})=\lim_{m\to 0}G^{+}(\Delta
t)=-\frac{i}{64\pi^{2}}\frac{A^{3}}{(\sinh(A\Delta t/2-i\varepsilon))^{3}}.$
(28)
Generally speaking, Unruh effect is a very weak effect. For example, in 4
dimensional Minkowski space, it needs about an acceleration $10^{21}$ meter
per second2 to increase temperature 1 K, hence various massive modes are
difficult to be excited. The mode with zero mass is the easiest mode to be
excited. Therefore, we consider excitations of massless modes. Substituting
$D^{+}({\Delta t})$ into (25), one obtains the unit response function with a
contour closed at lower half plane of complex $\Delta t$ by using the Jordan’s
lemma,
${\cal
U}(E-E_{0})=\frac{1}{32\sqrt{\pi}}\frac{A^{2}+4(E-E_{0})^{2}}{e^{2\pi(E-E_{0})/A}+1}.$
(29)
This is the response function for a particle detector confined to the FRW
brane, and comoving to the FRW brane, which describes photon gas system at
temperature $A/2\pi$. We see that a Fermi-Dirac factor appears, which implies
that the particles of Rindler radiation behaves as Fermions, though all the
excited modes we integrated to derive the response function are Bosonic. It is
not a phenomenon completely new. In 1986, Unruh pointed out that the Fermi-
Dirac factor would appear in the response function for an accelerated monopole
of a massless field in an odd number of space dimension, arising from
integration over all modes for a scalar field unruh .
In the above text of this section, only a brane with positive spatial
curvature is discussed. A spatially flat brane can be treated as a limit when
$R$ is large enough. To the case of a negative spatial curvature, we need to
replace the 3-sphere by a 3-hyperbola $H_{3}$ in (7). A 5-dimensional
Minkowski space sliced with space-like 3-hyperbola can be written as,
$dS^{2}=-d\hat{T}^{2}+dR^{2}-R^{2}dH_{3}^{2}.$ (30)
Then the following discussions exactly follow the case of positive spatial
curvature. The resulting distribution function (29) is still valid.
## III Quasi-stationary acceleration stage
Up to now our analysis is limited to uniformly accelerated detectors. But from
(17) we see that the acceleration of the brane is not a constant. Physically,
the formula (29) remains valid when the acceleration varies slowly.
Mathematically we can estimate the time over which the constant acceleration
approximation is valid by imposing that the variations in the acceleration are
small, e.g., expanding the acceleration around some time $t_{0}$ to the first
order $t-t_{0}$:
$A(t)=A(t_{0})+(t-t_{0})\frac{dA}{dt}\left|{}_{t=t_{0}}\right.$ (31)
Here, we adopt the cosmic time (proper time) $t$ rather than the proper time
coordinate time $\hat{T}$ of the detector since the cosmic time is directly
related to our measurement of cosmological parameters. For evaluating the
variation of the acceleration, define characteristic time
$t_{c}\triangleq\left|\frac{A(t_{0})}{dA/dt|_{t=t_{0}}}\right|,$ (32)
which is a function of $t_{0}$, i.e., it varies with the evolution of the
universe. Our condition for “acceleration varies slowly” requires that the
time scale of a physical process we concerned is much less than the
characteristic time,
$t_{i}\ll t_{c}.$ (33)
The time scale of a physical process $t_{i}$ must be shorter than the age of
the universe. So if the characteristic time $t_{c}$ is much larger than the
time scale of the universe, we can safely treat the acceleration as a constant
for any physical processes.
Here we present some examples to explain this condition. We work in a frame of
“$r_{c}$CDM” model, that is, a DGP brane universe filled with dust, whose
density function $\rho$ in Friedmann equation (6) only includes pressureless
matter, with about $30\%$ of the critical density.
First, we consider the inflationary phase at the early universe. At such a
high energy scale, the infrared correction of DGP model to the standard
general relativity, which becomes important at late time universe, can be
omitted safely. So the effective energy density and pressure in (17) are just
the ordinary density and pressure of the universe. At the whole inflationary
phase, the density of the universe is approximately constant, and the pressure
$p=-\rho$. Hence,
$\frac{dA}{dt}=-\frac{\kappa^{2}}{6}\frac{d\rho}{dt}\sim 0,$ (34)
that is, from (32), the characteristic time is very long. In this case, the
Unruh temperature is well defined.
Second, the universe enters a radiation dominated phase after the inflation.
At this stage, the DGP-correction to the standard model is still tiny. Thus,
we adopt the the same approximation as above, $\rho_{e}=\rho,~{}p_{e}=p$. For
a radiation dominated universe, we have
$p=\frac{1}{3}\rho.$ (35)
The continuity equation reads,
$\frac{d\rho}{dt}+3H(\rho+p)=0.$ (36)
Substitute (35) and (36) into (32), we reach,
$t_{c}=\frac{\sqrt{3}}{4}\frac{\mu}{\sqrt{\rho}},$ (37)
from which we see that the characteristic time becomes longer when the energy
density becomes lower. For instance, when $\rho=(0.1$Mev)4, we get $t_{c}\sim
1$s, which is much shorter than the age of the universe at that time.
And then the universe is diluted to be thinner and thinner. The DGP correction
becomes more and more important. Under this situation we have to introduce the
correction terms. By using (20) (see also eff ), we derive
$\displaystyle\rho_{e}=\rho-\mu^{2}(3X),$ (38) $\displaystyle
p_{e}=p+\mu^{2}(\frac{2\ddot{a}}{a}+X),$ (39)
where $X$ is defined as
$X\triangleq H^{2}+k/a^{2}.$ (40)
Substituting to the Friedmann equation (6), we obtain
$X=\frac{\rho}{3\mu^{2}}+\frac{2}{r_{c}^{2}}+\frac{2\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{1/2}.$
(41)
As for standard cosmology, $\frac{2\ddot{a}}{a}$ can be obtained from the
Friedmann equation and the continuity equation . Here, similarly, from
Friedmann equation (6) and the continuity equation (75), we derive
$\frac{\ddot{a}}{a}=X-\frac{1}{2\mu^{2}}\frac{\rho+p}{1+\frac{\epsilon}{\sqrt{X}}\frac{1}{r_{c}}}.$
(42)
Then substituting (38) , (39) and (17) into (32) , we arrive at the following
rather complicated from,
$t_{c}=\frac{1}{3}\left[2+\frac{3\mu^{2}X}{\rho}-{3}{(1+\frac{\epsilon}{r_{c}\sqrt{X}})^{-1}}\right](X-\frac{k}{a^{2}})^{-1/2}V^{-1},$
(43)
where
$V=3-\frac{1}{3}(1+\frac{\epsilon}{r_{c}\sqrt{X}})^{-1}+\frac{\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{-1/2}-\frac{1}{2}\frac{\rho}{r_{c}^{2}\mu^{2}X^{3/2}}\left[1+\frac{\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{-1/2}\right].$
(44)
Substituting the current parameter of the universe $\rho=(1.8\times
10^{-3}$ev$)^{4}$ and we suppose that the universe is spatially flat, then we
obtain, either for the negative or the positive branch, $t_{c}\sim 10^{17}$s,
which is at the same scale of the age of the universe. So the constant
acceleration is a perfect approximation under this condition.
Then, at least of late time universe, we can safely use (29), which is a
thermal distribution at a temperature of
$T=\frac{A}{2\pi}.$ (45)
Associating with (17), we get
$T=\frac{\kappa^{2}}{12\pi}|2\rho_{e}+3p_{e}|.$ (46)
## IV compared to geometric temperature of the brane
We see that the temperature of Unruh radiation the brane perceives, as
displayed in (46), only relates to the energy density and pressure of the
brane for a given DGP model. Then, a natural question emerges: Is the Unruh
radiation hotter, colder or at the same temperature to the brane? If there is
energy exchange between bulk and brane, we also need the temperature of the
brane to decide the direction of the energy flux. But there are several
different particles on the brane. They had gone out of thermal equilibrium
long before. Hence we’d better to find the characteristic temperature of the
braneworld independent of its detailed microscopic construction. As we have
pointed out Unruh radiation is a geometric feature of the space, which is
unrelated to the construction of the detector. Just as well we need the
characteristic temperature of the brane.
We know that the formulae of black hole entropy and temperature have a certain
universality in the sense that the horizon area and surface gravity are purely
geometric quantities determined by the space geometry, once Einstein equation
determines the space geometry. As we have mentioned above, Einstein equation
can be reproduced by thermal dynamics considerations. As for the case of
cosmology, applying the first law of thermodynamics to the apparent horizon of
an FRW universe and assuming the geometric entropy given by a quarter of the
apparent horizon area and the temperature given by the inverse of the apparent
horizon, the Friedmann equation can be derived. This celebrated result implies
that the inverse number of the apparent horizon is the geometric temperature
of the universe, which is independent of the microscopic structures of the
particles confined to the brane. The apparent horizon in the dynamical
universe is a marginally trapped surface with vanishing expansion.
Straightforward calculation yields the radius of the apparent horizon
$R_{A}=X^{-1/2},$ (47)
where $X$ is defined in (40). And then the geometric temperature of the brane
reads
$T^{\prime}=\frac{R_{A}^{-1}}{2\pi}=\frac{X^{1/2}}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{\rho}{3\mu^{2}}+\frac{2}{r_{c}^{2}}+\frac{2\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{1/2}}~{}~{}.$
(48)
Substituting (42) into (39) and (39) and (38) into (46), we arrive at
$T=\left|\frac{r_{c}}{12\pi\mu^{2}}\left(2\rho+3p\right)+\frac{r_{c}}{4\pi}\left(X-\frac{1}{\mu^{2}}\frac{\rho+p}{1+\frac{\epsilon}{\sqrt{X}}\frac{1}{r_{c}}}\right)\right|,$
(49)
where $X$ is given by (41). Therefore we obtain the explicit form temperature
of the brane, which is determined by its energy density and pressure. By
contrast the geometric temperature only depends on the energy density, which
has no relation to the pressure. We see that both Unruh temperature (49) and
geometric temperature (48) are unrelated to the spatial curvature of the
brane.
To investigate further the evolution of the Unruh temperature and the
geometric temperature during the history of the universe, we write them in
dimensionless form,
$\frac{T^{\prime}}{H_{0}}=\frac{X^{1/2}}{2\pi
H_{0}}=\frac{1}{2\pi}\left[x\Omega_{m}+2\Omega_{r_{c}}+2\epsilon\sqrt{\Omega_{r_{c}}}(x\Omega_{m}+\Omega_{r_{c}})^{1/2}\right]^{1/2},$
(50) $\displaystyle\frac{T}{H_{0}}=\frac{A}{2\pi H_{0}}$ $\displaystyle=$
$\displaystyle\left|\frac{x\Omega_{m}}{2\pi\sqrt{\Omega_{r_{c}}}}(1+3w/2)+\frac{1}{4\pi\sqrt{\Omega_{r_{c}}}}\left[\frac{X}{H_{0}^{2}}-3\Omega_{m}x(1+w)\frac{1}{1+\frac{\epsilon}{\sqrt{X/H_{0}^{2}}}\sqrt{\Omega_{r_{c}}}}\right]\right|,$
(51) $\displaystyle=$
$\displaystyle\left|\frac{x\Omega_{m}}{2\pi\sqrt{\Omega_{r_{c}}}}(1+3w/2)+\frac{1}{4\pi\sqrt{\Omega_{r_{c}}}}\left[\frac{4\pi^{2}T^{\prime
2}}{H_{0}^{2}}-3\Omega_{m}x(1+w)\frac{1}{1+\frac{\epsilon}{\sqrt{T^{\prime}/H_{0}}}\sqrt{\Omega_{r_{c}}}}\right]\right|$
where $H_{0}$ is the present value of the Hubble parameter, $w$ is the
parameter of state equation of the matter confined to the brane,
$x\triangleq\rho/\rho_{0}$, and $\Omega_{m},~{}\Omega_{r_{c}}$ are defined as
$\displaystyle\Omega_{m}=\frac{\rho_{0}}{3\mu^{2}H_{0}^{2}},$ (52)
$\displaystyle\Omega_{r_{c}}=\frac{1}{r_{c}^{2}H_{0}^{2}}.~{}~{}$ (53)
Here $\rho_{0}$ denotes the present density. First, we study various limits of
the two types of the temperatures. When
$r_{c}\to\infty,~{}\Omega_{r_{c}}\rightarrow 0$, we expect that DGP brane
theory reduces to standard 4-dimensional cosmology. It is right in the case
for geometric temperature of the brane,
$\lim_{r_{c}\to\infty}T^{\prime}=\frac{\sqrt{\rho}}{2\pi\sqrt{3}\mu}.$ (54)
But for Unruh temperature, it is a different situation. Though all dynamical
effect of the 5th dimension vanishes when $r_{c}\to\infty$, the Unruh
temperature does not vanish. One can check
$\lim_{r_{c}\to\infty}T=\frac{1}{2}\left|5+3w\right|\frac{\sqrt{\rho}}{2\pi\sqrt{3}\mu}.$
(55)
This result implies that even when the gravitational effect of the extra
dimension vanishes, the quantum effect is saved. The other important limit is
the limit when the matter on the brane is infinitely diluted, i.e., $\rho\to
0$,
$\displaystyle\lim_{\rho\to
0}T^{\prime}=\frac{\sqrt{2}}{2\pi}\frac{\sqrt{1+\epsilon}}{r_{c}},$ (56)
$\displaystyle\lim_{\rho\to
0}T=\frac{1}{2\pi}\frac{1+\epsilon}{r_{c}}.~{}~{}~{}~{}$ (57)
One may conclude
$\lim_{\rho\to 0}T^{\prime}=\lim_{\rho\to 0}T,$ (58)
for either $\epsilon=1$ or $\epsilon=-1$. However
$\lim_{\rho\to 0}\frac{T^{\prime}}{T}=1,$ (59)
for $\epsilon=1$, and
$\lim_{\rho\to 0}\frac{T^{\prime}}{T}=\frac{1}{|2+3w|},$ (60)
for $\epsilon=-1$, since the “speeds” are different when $T$ and $T^{\prime}$
go to zero.
An important case for cosmology is a universe filled with dust matter, i.e.,
$w=0$. In this case, (48) and (49) become,
$T^{\prime}=Z+\epsilon Y,$ (61)
and
$T=\left|Y+\epsilon Z+\frac{3}{2}\frac{Z^{2}-Y^{2}}{2Y+\epsilon Z}\right|,$
(62)
where we set
$Y=\frac{1}{2\pi}H_{0}\sqrt{\Omega_{r_{c}}}~{},$ (63)
and
$Z=\frac{1}{2\pi}H_{0}\sqrt{\Omega_{r_{c}}+x\Omega_{m}}~{}.$ (64)
For a reasonable cosmological model, we have $Y>0,~{}Z>0$ and $Z>Y$. The
calculation to take absolute value is difficult to deal with. Hence we
consider different cases in which we can determine the sign in the absolute
calculation.
First, in the branch $\epsilon=1$, $T$ becomes
$T=Y+Z+\frac{3}{2}\frac{Z^{2}-Y^{2}}{2Y+Z},$ (65)
because every term is larger than zero. Then
$T-T^{\prime}=\frac{3}{2}\frac{Z^{2}-Y^{2}}{2Y+Z}>0.$ (66)
Therefore, the Unruh radiation perceived by a brane is warmer than the brane
measured by the geometric temperature.
Second, we consider the branch $\epsilon=-1$. The Unruh temperature in this
case is much more complicated. First, at the point $2Y=Z$, or equally
$3\Omega_{r_{c}}=x\Omega_{m}$, there is a singularity $T\to\infty$ in (62),
which means the particle numbers are equal at every energy level. This is
valid only for a system endowed with finite energy levels, which indicates our
theory must be cut off at some energy level. On the right hand of this
singularity $2Y-Z>0$, $T$ can be decomposed in the form
$T=\left|\frac{(Y-Z)(Y-5Z)}{2(2Y-Z)}\right|,$ (67)
where $Y-Z<0$, $Y-5Z<0$ and $2Y-Z>0$, hence $\frac{(Y-Z)(Y-5Z)}{2(2Y-Z)}>0$.
So we remove the absolute sign directly and compare $T$ and $T^{\prime}$ by
$T-T^{\prime}=\frac{(Y-Z)(5Y-7Z)}{2(2Y-Z)}>0,$ (68)
since $Y-Z<0$, $5Y-7Z<0$ and $2Y-Z>0$. On the left hand of this singularity
$2Y-Z<0$, similar to the case $2Y-Z>0$, we can prove
$\frac{(Y-Z)(Y-5Z)}{2(2Y-Z)}<0$. Hence we remove the absolute sign by
inserting a minus sign and, $T-T^{\prime}$ becomes
$T-T^{\prime}=-\frac{3}{2}\frac{Z^{2}-Y^{2}}{2Y-Z}>0.$ (69)
Thus we prove that the Unruh radiation perceived by a dust dominated brane is
warmer than the brane measured by the geometric temperature all the time.
The other point deserving to be noted is that $T^{\prime}$ will be always
equal to $T$ if the parameter of state equation of the matter is confined to
brane $w=-1$. In this case,
$T=T^{\prime}=\frac{1}{2\pi}\left(\epsilon\frac{1}{r_{c}}+\sqrt{\frac{1}{r_{c}^{2}}+\frac{\rho}{3\mu^{2}}}\right).$
(70)
Figure 1: $\eta\triangleq T^{\prime}/T$ as a function of $x$ and
$\Omega_{r_{c}}$. In this figure we consider quintessence-like matter
dominated universe, in which $w=-1/2$ and we set $\Omega_{m}=0.3$. (a) The
branch $\epsilon=-1$. (b) The branch $\epsilon=+1$.
Figure 2: $u\triangleq 2\pi T/H_{0}$ as a function of $v\triangleq 2\pi
T^{\prime}/H_{0}$. In this figure we also consider quintessence-like, in which
$w=-1/2$ and we set $\Omega_{m}=0.3$. (a) The branch $\epsilon=1$. (b) The
branch $\epsilon=-1$.
For general case the expressions of $T$ and $T^{\prime}$ are rather
complicated. Hence we plot two figures to give their visual profiles. Fig. 1
illustrates $T^{\prime}/T$ as a function of $x$ and $\Omega_{r_{c}}$. As we
have pointed out, when $\rho\to 0$, $T^{\prime}/T\to 1$ for the branch
$\epsilon=1$; while when $\rho\to 0$, $T^{\prime}/T\to 1/|2+3w|=1/2$ for the
branch $\epsilon=1$. Fig. 2 directly displays $T$ as a function of
$T^{\prime}$, in which the interval of the argument $x$ is $x\in(0,3)$.
Recalling $x=\rho/\rho_{0}$, we see that we consider the temperatures in some
low redshift region and in the future in figure 2. It is clear that $T\to 0$
when $T^{\prime}\to 0$, as shown by (56) and (57) for the branch $\epsilon=1$.
Our numerical result also shows an interesting property of the branch
$\epsilon=-1$: $T$ is almost a linear function of $T^{\prime}$.
## V Unruh radiation accelerates the universe
With the detailed studies in the last sections, we see that the Unruh
temperature does not vanish even at the limit $r_{c}\to\infty$, which is the
condition for the vanishing of the gravitational effect of the 5th dimension.
Furthermore, the Unruh temperature is always higher than geometric temperature
for a dust dominated brane in the whole history of the universe for both of
the two branches and for all three types of spatial curvature. So an influx
from bulk to brane will appear if brane interacts with bulk based on the most
sound principle of thermodynamics.
Although this difference in temperature indicates an energy flux, it offers no
hint of the form of the interaction term. Under this situation, we explore the
microscopic mechanism based on statistical mechanics and particle physics to
derive the interaction term between Unruh radiation and dark matter confined
to the brane.
In section II, we have considered a detector coupling to a scalar field
$L_{\phi}=\frac{1}{2}\partial_{A}\phi\partial^{A}\phi-\frac{1}{2}m^{2}\phi^{2}$,
which satisfies 5 dimensional Klein-Gordon equation. At the massless limit, it
obeys
$\square^{(5)}\phi=0.$ (71)
By using Green function method, we derive the temperature of the Unruh
radiation,
$T=\frac{A}{2\pi},$ (72)
where $A$ is given by (17).
To investigate the microscopic mechanism of interaction between brane and
bulk, one should decompose the 5-dimensional modes, which satisfies (71), into
modes along the brane and modes transverse to the brane. Following yling , we
call them type I modes and type II modes, respectively. In a pseudo-Euclidean
coordinates system, type I mode reads, $\phi^{(I)}=e^{-ik_{\mu}x^{\mu}}$,
where $k_{\mu}$ denotes the momentum of the mode in different directions, and
$\mu=0,1,2,3$. Type II modes reads,
$\phi^{(II)}=e^{-ik_{\mu}x^{\mu}-ik_{y}y}$, where $y$ is the coordinate of the
extra dimension, and we require $k_{y}\neq 0$. An arbitrary 5 dimensional mode
must be either type I or type II. It was shown in yling that Type II modes of
the bulk fluctuations do not interact with the brane, for which the brane is
effectively a mirror. Hence only type I mode can exchange energy with matter
confined to the brane. We show a schematic plan of type I and type II modes in
Fig. 3.
Figure 3: A sketch of type I and type II modes. Only type I mode can exchange
energy with particles confined to the brane, while the brane behaves as a
perfect mirror for type II mode (for a color version on line).
By a standard process of canonical quantization, type I modes become photons
on the brane. Here photons mean quanta without mass, which do not necessarily
satisfy Maxwell equations. In fact we do not even need to study its
commutativity in detail. Whether they obey Fermi-Dirac statistics or Bose-
Einstein statistics the number density of the photon gas $n_{\gamma}$ is
proportional to the cubic of its temperature,
$n_{\gamma}\propto T_{\gamma}^{3},$ (73)
where $T_{\gamma}$ is just the temperature of the Unruh radiation (45). The
particles confined to the brane are immerged in the thermal bath of these
photons. Therefore, based on statistical mechanics the reaction rate $\Gamma$
between dark matter particles and the type I photons is proportional to the
number densities of photons and dark matter particles, the relative velocity
of the photons and matter particles, and the scattering cross section
$\Gamma\propto n_{\rm dm}n_{\gamma}\sigma v_{\rm pm}$. Here, $n_{\rm dm}$
denotes the number density of dark matter particle, $v_{\rm pm}$ stands for
the relative velocity, which is a constant, and $\sigma$ represents the
scattering cross section. In a low energy region, the internal freedoms can
not be excited. This paper concentrates on the late time universe, hence the
cross section is effectively constant. Therefore, the reaction rate can be
written as an equation by inserting a constant $b$,
$\Gamma=b\rho A^{3},$ (74)
where we have used (73), (45) and $n_{dm}\propto\rho$. Here, we assume there
is only pressureless dark matter on the brane, which interacts with the Unruh
photons. Therefore, the continuity equation of the brane becomes,
$\dot{\rho}+3H\rho=\Gamma.$ (75)
As we explained before, the DGP braneworld model also suffers from fine-tune
problem, which says why the three RHS terms of (6) are at the same scale. To
overcome this hurdle, we consider a limiting DGP model, that is, $r_{c}\gg
H_{0}^{-1}$. In such a model, the fine-tuned problem is evaded, and at the
same time the gravitational effect of the 5th dimension is no longer
responsible for the present acceleration. We have seen that the Unruh effect
does not vanish even when $r_{c}\to\infty$. We will prove that under the
situation $r_{c}\gg H_{0}^{-1}$ the interaction between dark matter on the
brane and Unruh radiation can drive the observed acceleration of the universe.
In a limiting DGP model , substituting the Friedmann equation (6) into (75),
we derive
$\frac{\ddot{a}}{{a}}=-\frac{\rho}{6\mu^{2}}\left[1-bA^{3}\left(\frac{\rho}{3\mu^{2}}-\frac{k}{a^{2}}\right)^{-1/2}\right].$
(76)
The acceleration $A$ of the brane in the bulk does not directly depend on
energy density of the matter on the brane $\rho$, but through the effective
density $\rho_{e}$ and and effective pressure $p_{e}$ in (17), which are
presented in (38) and (39), respectively.
Associating (17), (38), (39), and (76) we derive
$A^{-2}=\frac{\kappa^{2}}{6}b\rho\left(\frac{\rho}{3\mu^{2}}-\frac{k}{a^{2}}\right)^{-1/2}.$
(77)
The formula (77) is general for all the three cases of curvatures. In the case
of a spatially flat brane, it takes a simple form,
$A^{-2}=C\sqrt{\rho},$ (78)
where $C=\kappa^{2}\mu b$. Interestingly, we see that the acceleration of the
brane $A$ inversely correlates to the energy density of the brane, which is
completely different from the first sight at equation (17). Therefore, we
expect the effect of the bulk Unruh radiation is negligible in the early time
for a dust dominated limiting DGP braneworld. Only in some low energy region
the bulk Unruh radiation becomes important.
Before presenting the exact cosmic solution, we study some qualitative side of
the Firedmann equation (6) and continuity equation (75) to see whether our
model is stable. We consider a spatially flat universe, in which the
stagnation point dwells at $\delta H=0$, or equivalently $\delta\rho=0$. From
the continuity equation (75), we derive the energy density $\rho_{s}$ at the
stagnation point,
$\rho_{s}^{5/4}={b\mu C^{-3/2}}/{\sqrt{3}}.$ (79)
To investigate the stability of the cosmic fluid in the neighbourhood of the
stagnation point, we impose a perturbation to the continuity equation,
$(\delta\rho)^{.}=\delta\rho\left(\frac{1}{4}b\rho^{-3/4}C^{-3/2}-\frac{3\sqrt{3}}{2\mu}\rho^{1/2}\right).$
(80)
At the stagnation point, $\rho=\rho_{s}$, hence
$\displaystyle(\delta\rho)^{.}|_{\rho=\rho_{s}}=-\frac{5\sqrt{3}}{4\mu}\rho_{s}^{1/2}\delta\rho,$
(81)
which means it is a stable point. Now we consider the deceleration parameter,
which is one the most significant parameters from the viewpoint of
observations. Here the deceleration parameter reads,
$\displaystyle
q=\frac{1}{2}\left(1-\frac{bA^{3}}{\sqrt{\frac{\rho}{3\mu^{2}}-\frac{k}{a^{2}}}}\right)\left(1-\frac{3\mu^{2}}{\rho}\frac{k}{a^{2}}\right)^{-1},$
(82)
where $A$ is given by (77). In the case of a spatially flat universe, it
degenerates to
$\displaystyle q=\frac{1}{2}\left(1-\sqrt{3}\mu bC^{-3/2}\rho^{-5/4}\right).$
(83)
This equation clearly shows that in the early universe $q\to 1/2$, hence the
universe behaves as dust dominated one, and with the decreasing of energy
density the deceleration parameter becomes smaller. Finally at the stagnation
point the deceleration parameter ceases at $q=\frac{1}{2}\left(1-\sqrt{3}\mu
bC^{-3/2}\rho_{s}^{-5/4}\right)=-1$, which implies that the universe enters a
de Sitter phase.
Though, for an arbitrary spatial curvature the analytical solution does not
exist, we find an exact solution for a spatially flat universe driven by bulk
Unruh radiation. For a spatially flat universe, associating Friedmann equation
(6) with the continuity equation (75) we obtain,
$\rho=\left(c_{1}a^{-15/4}+\frac{L}{3}\right)^{4/{5}},$ (84)
where $c_{1}$ is an integration constant, and $L$ is defined by $L=\sqrt{3}\mu
bC^{-3/2}$. When $c_{1}>0$, from this exact solution, the universe behaves as
dust dominated one in a high energy region ($a$ small enough), and becomes de
Sitter universe in a low energy region ($a$ large enough), which is exactly
the same as we concluded before from behaviors of the deceleration parameter
in the history of the universe. When $c_{1}<0$, there is a bounce in the early
universe, and the universe also enters a de Sitter phase in a low energy
region. Since our investigations for the acceleration driven by Unruh
radiation concentrate on the low energy region, it is only a toy model in the
early universe.
With the exact solution (84) and (6), we obtain an analytical expression of
cosmic time $t$ as function of the scale factor $a$ by using a hypergeometric
function,
$\displaystyle\frac{t}{\sqrt{3}\mu}+c_{2}=\frac{2}{3}\frac{a^{3/2}}{c_{1}^{2/{5}}}F(\frac{2}{5},\frac{2}{5},\frac{7}{5},-a^{15/4}\frac{L}{3c_{1}}),$
(85)
where $c_{2}$ is an integration constant, and $F$ denotes Gauss hypergeometric
function. The requirement
$\lim_{a\to 0}t=0,$ (86)
yields $c_{2}=0$.
To visualize the physical meaning of this solution, we demand the various
limits of it. In the high energy limit (small $a$), expanding (85) in series
around $a=0$,
$\frac{t}{\sqrt{3}\mu}=\frac{2}{3c_{1}^{2/5}}a^{3/2}-\frac{8L}{315c_{1}^{7/5}}a^{21/4}+{\cal{O}}(a^{17/2}),$
(87)
where if we only keep the dominated term we just obtain $a\sim t^{2/3}$, as we
expected. In a low energy region when $a$ is large enough, the series becomes,
$\displaystyle\frac{t}{\sqrt{3}\mu}=\frac{4}{5}\frac{1}{3^{3/5}L^{2/5}}\left[-\gamma+\ln(\frac{L}{3c_{1}})+\frac{15}{4}\ln
a-\psi(\frac{2}{5})\right]$
$\displaystyle+\frac{3^{2/5}}{25}\frac{8c_{1}a^{-15/4}}{L^{7/5}}+{\cal
O}(a^{-29/4}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(88)
where $\gamma$ is the Euler constant, and $\psi$ is the digamma function. If
we only take the leading term, we obtain $a=Me^{Nt}$, where
$M=\exp\\{\frac{4}{15}[\gamma-\ln(L/3c)+\psi(2/5)]\\}$,
$N=L^{2/5}3^{3/5}(3\sqrt{3}\mu)^{-1}$, which is just a de Sitter space.
Thus, we complete a cosmic solution of the limiting DGP braneworld model,
where the universe is self-accelerated through the bulk Unruh radiation
perceived by the brane. From the viewpoint of observations our model belongs
to the class of unified dark energy model, that is, there is only one
component (usually dust) in the universe, but due to different reasons it
evolves in a non-standard way. Therefore, it can drive the present
acceleration of the universe. Such models also have been phenomenologically
investigated in, for example, chap .
Finally, we would like present some preliminary discussions to confront the
observations. First, the parameter $b$ is critical to our model, which
encloses all the undetermined information of interaction between dark matter
and Unruh radiation. Here we present a preliminary estimation of its value
from the deceleration parameter (83) in a spatially flat universe. From
various observations the present value of deceleration parameter $q\sim-0.5$,
therefore,
$\frac{\sqrt{2}}{\sqrt{bH_{0}^{2}}}(r_{c}H_{0})^{-3/2}\left(\frac{\rho}{3\mu^{2}H_{0}^{2}}\right)^{-5/4}\sim
1,$ (89)
where $\frac{\rho}{3\mu^{2}H_{0}^{2}}=1$ in our model. We see that a larger
cross radius $r_{c}$ needs a smaller coupling constant $b$. And we have set
$r_{c}H_{0}\gg 1$ in previous constructions. Hence the dimensionless coupling
constant $bH_{0}^{2}$ is a tiny number, which eludes our laboratory
experiments even if the dark matter particles were found. Alternatively, in
future work we expect astronomical observations to fit our model and hence to
determine the value of $b$, which is also helpful to constrain the cross
radius $r_{c}$ in DGP model.
An exotic matter with negative pressure, call dark energy, is frequently
introduced to explain the cosmic acceleration in frame of general relativity.
To explain observed accelerated expansion, we calculate the equation of state
$w$ of the effective “dark energy” caused by the induced Ricci term and energy
influx from the bulk Unruh radiation by comparing the modified Friedmann
equation in the brane world scenario and the standard Friedmann equation in
general relativity, since almost all observed properties of dark energy are
“derived” in general relativity. The Friedmann equation in the four
dimensional general relativity can be written as
$H^{2}+\frac{k}{a^{2}}=\frac{1}{3\mu^{2}}(\rho+\rho_{de}),$ (90)
where the first term of RHS of the above equation represents the dust matter
and the second term stands for the dark energy. Generally speaking the Bianchi
identity requires,
$\frac{d\rho_{de}}{dt}+3H(\rho_{de}+p_{de})=0,$ (91)
we can then express the equation of state for the dark energy as
$w_{de}=\frac{p_{de}}{\rho_{de}}=-1-\frac{1}{3}\frac{d\ln\rho_{de}}{dlna}.$
(92)
Comparing (90) and (6), we derive
$\rho_{de}=\frac{2}{r_{c}^{2}}+\frac{2\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{1/2}.$
(93)
Note that in our model there is no exotic matter confined to the brane.
$\rho_{de}$ is in fact geometric and interaction effect. We call $\rho_{de}$
equivalent or virtual density of dark energy. Various evidences, which are
independent to cosmological models, implies the existence of dark matter with
present density about $\Omega_{m}=0.2\sim 0.4$ darkmatter . Maybe more or
less, but the density of dark matter does reach the density to flat the space.
In our model, the geometric contribution of $r_{c}$ can be very small. So
generally speaking, we need a curvature term. Substituting (93) into (92), and
recalling that the matter confined to the brane is pressureless, we obtain,
$w_{de}=-1+\frac{\epsilon}{3}\frac{\dot{\rho}}{3\mu^{2}H}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)^{-1/2}\left[\frac{2\epsilon}{r_{c}}\left(\frac{\rho}{3\mu^{2}}+\frac{1}{r_{c}^{2}}\right)+\frac{2}{r_{c}^{2}}\right]^{-1},$
(94)
where $\dot{\rho}$ is given by (75), and $\Gamma$ can be calculated by (74).
Far a large $r_{c}$, after complicated but straightforward algebraic
calculating, we deduce,
$w_{de}=-\frac{3}{2}+Q\Omega_{m}^{-3/2}(1+z)^{-9/2}\left(\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}\right)^{1/4},$
(95)
where $Q$ is defined as,
$Q=\frac{18^{3/2}}{6}(r_{c}H_{0})^{-3/2}(bH_{0}^{2})^{-1/2},$ (96)
$\Omega_{k}=\frac{-k}{H_{0}^{2}a^{2}},$ (97)
and $z$ is the redshift. It is clear that when
$Q\Omega_{m}^{-3/2}(\Omega_{m}+\Omega_{k})^{1/4}>0.5$, the virtual dark energy
behaves as quintessence; while when
$Q\Omega_{m}^{-3/2}(\Omega_{m}+\Omega_{k})^{1/4}<0.5$, it behaves as phantom.
We stress for the second time that the dark energy in this model is only some
virtual, not actual stuff. The most sensible quantity in observation is the
deceleration parameter $q$, which is given in (82). Substituting (77) into
(82), we reach,
$q=\frac{1}{2}-3^{-1/2}Q\Omega_{m}^{-3/2}(1+z)^{-9/2}\left(\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}\right)^{1/4},$
(98)
where we also supposed a large $r_{c}$. Note that for both $q$ and $w_{de}$,
$\epsilon$ vanishes at the large $r_{c}$ limit. This reasonable, since the two
branches have no difference at such a limi.t
As numerical examples, we plot a figure for $q$ for two sets of parameters in
fig 4. This figure illuminates that the universe is accelerating and $q$ goes
to $0.5$ quickly at high redshift region, ie, the universe becomes a dust one.
Figure 4: The evolution of deceleration parameter with respect to $z$. For
both of the two curves, $\Omega_{m}=0.3$, $\Omega_{k}=0.7$. $Q=0.4$ on the
solid curve, and $Q=0.3$ on the dashed curve.
## VI conclusion
Our scenario is that the brane is accelerating in the bulk, such that it is
bathed in the Unruh radiation in the bulk. The temperature of the Unruh
radiation is higher than the brane, which yields energy flux from the Unruh
radiation to the brane matter, which accelerates the brane universe.
In this article we study the cosmology case of a DGP braneworld model, that
is, an FRW brane imbedded in the bulk spacetime. We first investigate the
Unruh temperature that the brane perceives and its “own” temperature.
Generally speaking the brane accelerates in the bulk. Hence the particles
confined to the brane and inertial on the brane should also perceive Unruh
radiation in the bulk. In this paper we show that for a DGP brane in a
Minkowski bulk it is just the case. We investigate the case of the brane, as a
particle detector, coupled to a massless scalar field in the bulk. As for a
point detector, the temperature of the radiation perceived is proportional to
the acceleration of the brane.
As for the brane’s own temperature, we should seek a characteristic
temperature of the brane. The temperature can reveal the inherent property of
gravity. The Friedmann equation can be reproduced by applying the first law of
thermal dynamics to the apparent horizon. Thus we take the temperature of the
apparent horizon as the characteristic temperature of the brane, which we
called geometric temperature.
We find that generally speaking the Unruh temperature the brane perceived and
geometric temperature of the brane are not equal. We compare these two
temperatures in various cases. We find that for a dust dominated brane, the
temperature of the Unruh radiation perceived by a brane is always higher than
the geometric temperature of the brane, either in the branch $\epsilon=1$ or
$\epsilon=-1$, no matter what the value of the cross radius $r_{c}$ and the
spatial curvature of the brane. So, generally speaking the brane and the Unruh
radiation it perceived can not reach to thermal equilibrium.
As we pointed out before, the validity of Unruh radiation has been confirmed
in detail review ,lin . Therefore if energy exchange is allowed between bulk
and matter brane, an energy flux between the bulk radiation and the matter
confined to the brane will come forth, which may accelerate our universe. In
all cases $\rho>0$, the Unruh temperature of the bulk radiation $T$ is higher
than than the geometric temperature $T^{\prime}$ for a dust dominated brane,
which means an energy influx to the brane can appear. So we study this
possibility in section V.
The DGP braneworld model seems to be a hopeful candidate to explain the
cosmological acceleration. But, as we pointed out before, it also suffers from
fine-tune problem. To evade this problem, we consider the limiting DGP model,
that is, $r_{c}>H_{0}^{-1}$, which means the gravitational effect of the 5th
dimension is negligible. Under this condition, the cosmological acceleration
does not happen in original DGP model under this situation if there is only
dust on the brane. By considering the possible energy influx from the bulk
Unruh radiation to the brane induced by the temperature grads, we find the
universe can accelerate through interacting the bulk Unruh radiation in a
limiting DGP model, even the brane is dust dominated. Differently from
previous works on the brane-bulk interaction, we find the interaction form
through careful studies on the microscopic mechanism of interaction between
brane and bulk. It is shown that the interaction term can be settled up to a
constant factor $b$. Based on these constructions we find the acceleration of
the brane $A$ is inversely correlated with the energy density of the brane for
a dust dominated limiting DGP brane. Finally we derive an exact solution for a
spatially flat model. This solution shows clearly that the universe behaves as
a dust dominated one at early time and enters a de Sitter phase at late time,
which is consistent with observations. We also show the de Sitter phase is
stable.
In this paper only massless Unruh mode is considered. Although it is the most
important mode in low energy region, the massive mode also deserves to study
further for a full Unruh effect in a high energy region. Also, in the enough
high energy region, the universe is radiation dominated, and at the same time
the scattering cross section between dark matter particles and Unruh photons
becomes temperature-dependent, since the internal freedoms can be excited.
Under this situation the Unruh effect may be important again, which deserves
to investigate further.
At the observation side, we should not only take a special set of parameters
to show the property of this model, but constrain the parameters, especially
$b$ by various observations in the future.
Acknowledgments. HS Zhang thanks Prof. W. Unruh and Prof. D. Jennings for
helpful discussions . H. Noh was supported by grant No. C00022 from the Korea
Research Foundation. ZH Zhu was supported by the National Science Foundation
of China under the Distinguished Young Scholar Grant 10825313, the Key Project
Grant 10533010, and by the Ministry of Science and Technology national basic
science Program (Project 973) under grant No. 2007CB815401. HW Yu was
supported by the National Natural Science Foundation of China under Grants No.
10575035 and No. 10775050, the SRFDP under Grant No. 20070542002, and the
Program for the key discipline in Hunan Province.
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|
arxiv-papers
| 2009-08-07T08:32:45 |
2024-09-04T02:49:04.504157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hongsheng Zhang, Hyerim Noh, Zong-Hong Zhu, Hongwei Yu",
"submitter": "Hongsheng Zhang",
"url": "https://arxiv.org/abs/0908.1001"
}
|
0908.1046
|
# Pairing and Quantum Double of Finite Hopf C*-Algebras 111Supported by
National Natural Science Foundation of China (No.10301004)and Excellent Young
Scholars Research Fund of Beijing Institute of Technology(000Y07-25)
Ming LIU, Li Ning JIANG, Guo Sheng ZHANG
Department of Mathematics,
Beijing Institute of Technology, Beijing, 100081, P.R. China
E-mail: sunny-sunabc@yahoo.com.cn
Abstract This paper defines a pairing of two finite Hopf C*-algebras $A$ and
$B$, and investigates the interactions between them. If the pairing is non-
degenerate, then the quantum double construction is given. This construction
yields a new finite Hopf C*-algebra $D(A,B)$. The canonical embedding maps of
$A$ and $B$ into the double are both isometric.
Key words Hopf C*-algebra, paring, quantum double, GNS representation
MR (2000) Subject Classification 46K70;16W30;81R05
Abbreviated title: Pairing of Hopf C*-algebras
## 1 Introduction
The Yang-Baxter equation firstly came up in a paper ([1]) as a factorization
condition of the scattering $S$ matrix in the many-body problems in one
dimension and in Baxter’s work on exactly solvable models in statistical
mechanics. The equation also plays an important role in the quantum inverses
scattering method created by Feddeev, Sklyamin and Takhtadjian for the
construction of quantum integrable systems. Since braided Hopf algebras ([2])
can provide solutions for the Yang-Baxter equation, attempts to find its
solutions in a systematic way have led to the construction of braided Hopf
algebras, and moreover, led to the theory of quantum group. Based on ([3]),
Woronowicz ([4]) exhibited C*-algebra structures of quantum groups in the
framework of C*-algebra. From then on, the research on Hopf algebra has always
been going with that on C*-algebra. This leads to the concept of Hopf
C*-algebra ([5]). Indeed, by the Gelfand-Naimark Theorem, an abelian
C*-algebra can be understood as the space of all complex continuous functions
vanishing at infinity on a locally compact space, and for this reason a
C*-algebra can be considered as a noncommutative locally compact quantum
space. Henceforth, a Hopf C*-algebra can be regarded as a noncommutative
locally compact quantum group.
In this paper, we are interested in finite Hopf C*-algebras. Firstly, we
propose the definition of Hopf *-algebra.
Definition 1.1 ([6, 7]) Let $A$ be a *-algebra with a unit 1. Suppose that
$\Delta:A\longrightarrow A\otimes A$ is a *-homomorphism such that
$(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta,$
and $\varepsilon:A\longrightarrow\mathbb{C}$ is a *-homomorphism such that
$(\varepsilon\otimes\iota)\Delta=(\iota\otimes\varepsilon)\Delta=\iota,$
where $\iota$ denotes the identity map. Finally, assume that
$S:A\longrightarrow A$ is a linear, anti-multiplicative map so that for all
$a\in A$,
$S(S(a)^{*})^{*}=a,$ $m(S\otimes\iota)\Delta(a)=m(\iota\otimes
S)\Delta(a)=\varepsilon(a)1,$
where $m:A\otimes A\longrightarrow A$ is the multiplication defined by
$m(a\otimes b)=ab$. Then $A$ is called a Hopf *-algebra, and
$\Delta,\varepsilon,S$ are called comultiplication, counit and antipode
respectively.
Besides the assumption that $A$ is a finite dimensional Hopf *-algebra, if $A$
is also a C*-algebra, $A$ is called a finite Hopf C*-algebra, which satisfies
the convolution property, i.e., $S^{2}=\iota$ naturally. And by ([8]), there
exists an invariant functional $\varphi$ on $A$ so that $\forall a\in A$,
$\varphi=\varphi\circ S$ and
$(\varphi\otimes\iota)\Delta(a)=(\iota\otimes\varphi)\Delta(a)=\varphi(a)1.$
After posing the notion of pairing of finite Hopf C*-algebras and exhibiting
the actions of dually paired Hopf C*-algebras on each other, this paper gives
the quantum double construction ([9]) out of the underlying two Hopf
C*-algebras and shows that the consequence of this construction is again a
finite Hopf C*-algebra with an invariant integral.
Much of our work is inspired by the work of ([10, 11]). All algebras in this
paper will be algebras over the complex field $\mathbb{C}$. Please refer to
([12]) for general results on Hopf algebra. In many of our calculations, we
use the standard Sweedler notation ([13]). For instance, formula like
$m(S\otimes\iota)\Delta(a)=\varepsilon(a)1$ can be written as
$\sum\limits_{(a)}S(a_{(1)})a_{(2)}=\varepsilon(a)1.$
## 2 Pairing of Finite Hopf C*-Algebras
In this section, we consider a bilinear form between finite Hopf C*-algebras.
Definition 2.1 Let $A$ and $B$ be two finite Hopf C*-algebras, and
$<\cdot,\cdot>:A\otimes B\longrightarrow\mathbb{C}$ be a bilinear form. Assume
that they satisfy: $\forall a_{1},a_{2},a\in A,b_{1},b_{2},b\in B$,
$<\Delta(a),b_{1}\otimes b_{2}>=<a,b_{1}b_{2}>,$ $<a_{1}\otimes
a_{2},\Delta(b)>=<a_{1}a_{2},b>,$ $<a^{*},b>=\overline{<a,S_{B}(b)^{*}>},$
$<a,1_{B}>=\varepsilon_{A}(a),$ $<1_{A},b>=\varepsilon_{B}(b),$
$<S_{A}(a),b>=<a,S_{B}(b)>,$
where $\varepsilon_{A},S_{A}$ (resp. $\varepsilon_{B},S_{B}$) denote the
counit and antipode on $A$ (resp. $B$) respectively. Then
$(A,B,<\cdot,\cdot>)$ is called a pairing of finite Hopf C*-algebras.
Definition 2.2 ([14]) Suppose that $(A,B,<\cdot,\cdot>)$ is a pairing of
finite Hopf C*-algebras. If $B$ (resp. $A$) can separate the points of $A$
(resp. $B$) (i.e., if $a_{0}\in A$ (resp. $b_{0}\in B$) such that $\forall
b\in B$ (resp. $a\in A$) $<a_{0},b>=0$ (resp.$<a,b_{0}>=0$), then $a_{0}=0$
(resp. $b_{0}=0$)) and we call the pairing is non-degenerate.
Similar to the discussions in ([10]), we have the following results. Firstly
for any pairing of finite Hopf C*-algebras $(A,B,<\cdot,\cdot>)$, we can
define the linear mappings by using the standard Sweedler notation:
$\mu_{A,B}^{l}:A\otimes B\longrightarrow B,a\otimes
b\mapsto\sum\limits_{(b)}b_{(1)}<a,b_{(2)}>,$ $\mu_{A,B}^{r}:B\otimes
A\longrightarrow B,b\otimes a\mapsto\sum\limits_{(b)}<a,b_{(1)}>b_{(2)},$
$\mu_{B,A}^{l}:B\otimes A\longrightarrow A,b\otimes
a\mapsto\sum\limits_{(a)}a_{(1)}<a_{(2)},b>,$ $\mu_{B,A}^{r}:A\otimes
B\longrightarrow B,a\otimes b\mapsto\sum\limits_{(a)}<a_{(1)},b>a_{(2)}.$
Proposition 2.3 Let $(A,B,<\cdot,\cdot>)$ be a pairing of finite Hopf
C*-algebras. Then the maps $\mu_{A,B}^{l}$ and $\mu_{A,B}^{r}$ are left and
right actions of $A$ on $B$, i.e., ($B$, $\mu_{A,B}^{l}$) is a left $A$-module
and ($B$, $\mu_{A,B}^{r}$) is a right $A$-module, respectively. Analogously,
$\mu_{B,A}^{l}$ and $\mu_{B,A}^{r}$ are left and right actions of $B$ on $A$,
respectively.
Proof For all $a,a^{\prime}\in A,b,b^{\prime}\in B$, we can obtain
$\begin{array}[]{llll}\mu_{A,B}^{l}(aa^{\prime}\otimes
b)&=&\sum\limits_{(b)}b_{(1)}<aa^{\prime},b_{(2)}>\\\
&=&\sum\limits_{(b)}b_{(1)}<a,b_{(2)}><a^{\prime},b_{(3)}>\\\
&=&\sum\limits_{(b)}b_{(1)}<a,b_{(2)}<a^{\prime},b_{(3)}>>\\\
&=&\mu_{A,B}^{l}(a\otimes\mu_{A,B}^{l}(a^{\prime}\otimes b)),\end{array}$
which shows that ($B$, $\mu_{A,B}^{l}$) is a left $A$-module. In a similar
way, we can check other relations and we omit them here. $\blacksquare$
For convenience, the previous actions will be denoted by $\triangleright$ and
$\triangleleft$ :
$\mu_{A,B}^{l}(a\otimes b):=a\triangleright b,\ \ \mu_{A,B}^{r}(b\otimes
a):=b\triangleleft a,$ $\mu_{B,A}^{l}(b\otimes a):=b\triangleright a,\ \
\mu_{B,A}^{r}(a\otimes b):=a\triangleleft b,$
which mean “$a$ acts from the left or right on $b$” and “$b$ acts from the
left or right on $a$” respectively, according to the directions of the arrows
$\triangleright$ and $\triangleleft$.
Lemma 2.4 Let $(A,B,<\cdot,\cdot>)$ be a pairing of finite Hopf C*-algebras.
Then for all $a,a^{\prime}\in A,b,b^{\prime}\in B$,
$<b\triangleright a,b^{\prime}>=<a,b^{\prime}b>,\ \ \ <a\triangleleft
b,b^{\prime}>=<a,bb^{\prime}>,$ $<a,a^{\prime}\triangleright
b>=<aa^{\prime},b>,\ \ \ <a,b\triangleleft a^{\prime}>=<a^{\prime}a,b>.$
Proof From the implications of the notations “$\triangleright$” and
“$\triangleleft$”, the proof is obvious. $\blacksquare$
From Lemma 2.4, we can get the following proposition at once.
Proposition 2.5 Suppose that $(A,B,<\cdot,\cdot>)$ is a non-degenerate paring
of finite Hopf C*-algebras. Then ($A$, $\mu_{B,A}^{l}$, $\mu_{B,A}^{r}$) is a
$B$-bimodule and ($B$, $\mu_{A,B}^{l}$, $\mu_{A,B}^{r}$) is an $A$-bimodule.
Proof Let $a\in A$ and $b_{1},b_{2},b_{3}\in B$, and check
$(b_{1}\triangleright a)\triangleleft b_{2}$ and
$b_{1}\triangleright(a\triangleleft b_{2})$ paired with $b_{3}$. Using Lemma
2.4, the associativity of $B$ implies
$<(b_{1}\triangleright a)\triangleleft
b_{2},b_{3}>=<a,b_{2}b_{3}b_{1}>=<b_{1}\triangleright(a\triangleleft
b_{2}),b_{3}>,$
which proves the proposition for the non-degeneracy of the pairing.
$\blacksquare$
From Proposition 2.5, we will write $(b_{1}\triangleright a)\triangleleft
b_{2}$ as $b_{1}\triangleright a\triangleleft b_{2}$ briefly in sequence.
Remark 2.6 (1) The third axiom in Definition 2.1 is also symmetric in $A$ and
$B$:
$\begin{array}[]{lll}<a,b^{*}>&=&\overline{<a^{*},S_{B}(b^{*})^{*}>}\\\
&=&\overline{<S_{A}^{-1}(a^{*}),S_{B}(S_{B}(b)^{*})^{*}>}\\\
&=&\overline{<S_{A}(a)^{*},b>}.\end{array}$
(2) The last three axioms in Definition 2.1 are redundant if the pairing
$<\cdot,\cdot>$ is non-degenerate. From the first three axioms of Definition
2.1 and Lemma 2.4, one can obtain for all $a,a^{\prime}\in A,b,b^{\prime}\in
B$, $<T_{2}^{A}(a\otimes a^{\prime}),b\otimes b^{\prime}>=<a\otimes
a^{\prime},T_{1}^{B}(b\otimes b^{\prime})>$, which also holds for the inverse
mappings of $T_{2}^{A}$ and $T_{1}^{B}$. Put $a=1_{A}$, $b=1_{B}$, by the non-
degeneracy of $<\cdot,\cdot>$,
$<a^{\prime}_{(1)},1_{B}>a^{\prime}_{(2)}=a^{\prime}$. Applying $\varepsilon$
to the two sides of this equation yields
$<a^{\prime},1_{B}>=\varepsilon_{A}(a^{\prime})$. Similarly,
$<1_{A},b^{\prime}>=\varepsilon_{B}(b^{\prime})$. Using $<T_{2}^{A-1}(a\otimes
a^{\prime}),b\otimes b^{\prime}>=<a\otimes a^{\prime},T_{1}^{B-1}(b\otimes
b^{\prime})>$ and Lemma 2.4, one can get $<S_{A}(b^{\prime}\triangleright
a^{\prime}),b\triangleleft a>=<b^{\prime}\triangleright
a^{\prime},S_{B}(b\triangleleft a)>,$ which implies the last axiom.
## 3 The Quantum Double
In what follows, we will only consider the action of $B$ on $A$, where $A$ and
$B$ are two dually paired finite Hopf C*-algebras. It is easy to see that
$A\otimes B$ can be made into a linear space of finite dimension in a natural
way ([15]). Furthermore, we can turn the linear space $A\otimes B$ into an
associative algebra which has an analogous algebra structure to the classical
Drinfeld’s quantum double.
Definition 3.1 The quantum double $D(A,B)$ of a non-degenerate paring of
finite Hopf C*-algebras $(A,B,<\cdot,\cdot>)$ is the algebra ($A\otimes B$,
$m_{D}$) with the multiplication map defined through
$\begin{array}[]{llll}m_{D}((a,b)(a^{\prime},b^{\prime}))=\sum\limits_{(b)}(a(b_{(3)})\triangleright
a^{\prime}\triangleleft(S^{-1}_{B}b_{(1)}),b_{(2)}b^{\prime})\\\
=\sum\limits_{(a^{\prime})(b)}(aa^{\prime}_{(2)},b_{(2)}b^{\prime})<a^{\prime}_{(1)},S^{-1}_{B}(b_{(3)})><a^{\prime}_{(3)},b_{(1)}>,\end{array}$
where $(a,b),(a^{\prime},b^{\prime})$ are in the linear basis
$B_{D}:=\\{(a,b)\mid a\in A,b\in B\\}$ of $D(A,B)$.
Following, we will write $m_{D}((a,b)(a^{\prime},b^{\prime}))$ as
$(a,b)(a^{\prime},b^{\prime})$ directly. It is easy to see
$(a,1_{B})(a^{\prime},b)=(aa^{\prime},b)$ and
$(a,b)(1_{A},b^{\prime})=(a,bb^{\prime})$. In particular, $(1_{A},1_{B})$ is
the unit of $D(A,B)$. Under the canonical embedding maps
$i_{A}:a\mapsto(a,1_{B})$ and $i_{B}:b\mapsto(1_{A},b)$, $A$ and $B$ become
subalgebras of $D(A,B)$.
Proposition 3.2 The multiplication $m_{D}$ of the quantum double $D(A,B)$ is
non-degenerate.
Proof For a fixed element $(a,b)\in D(A,B)$, suppose
$(a,b)(a^{\prime},b^{\prime})=0$ for all $(a^{\prime},b^{\prime})\in D(A,B)$.
Particularly pick $a^{\prime}=1_{A}$. Then
$(a,b)(a^{\prime},b^{\prime})=(a,bb^{\prime})=0$. If $a\neq 0$, then
$bb^{\prime}=0$ for all $b^{\prime}\in B$, which implies $b=0$ for the non-
degeneracy of the product on $B$. Thus we have $a=0$ or $b=0$, i.e.,
$(a,b)=0$. Similarly one can prove that $(a,b)(a^{\prime},b^{\prime})=0$ for
all $(a,b)\in D(A,B)$ if and only if $(a^{\prime},b^{\prime})=0$.
$\blacksquare$
In order to avoid using too many brackets, we will use $Sa$ for $S(a)$. On the
basis $B_{D}$, set
$*_{D}(a,b)=(a,b)^{*}:=\sum\limits_{(a)(b)}(a^{*}_{(2)},b^{*}_{(2)})<a^{*}_{(3)},b_{(1)^{*}}><a^{*}_{(1)},S^{*}_{B}b_{(3)}>,$
and extend it anti-linearly to the whole space of $D(A,B)$. Then
$(a,1_{B})^{*}=(a^{*},1_{B})$, $(1_{A},b)^{*}=(1_{A},b^{*})$. To describe the
*-structure of $D(A,B)$ exactly, we firstly do some preparing work.
Lemma 3.3 $\forall(a,b)\in D(A,B)$, $(a,b)^{**}=(a,b)$.
Proof
$\begin{array}[]{llll}(a,b)^{**}\\\
=\sum\limits_{(a)(b)}(a^{*}_{(2)},b^{*}_{(2)})^{*}<a_{(3)},S_{B}b_{(1)}><a_{(1)},b_{(3)}>\\\
=\sum\limits_{(a)(b)}(a_{(3)},b_{(3)})\overline{<a^{*}_{(4)},S^{*}_{B}b_{(2)}><a^{*}_{(2)},b^{*}_{(4)}>}<a_{(5)},S_{B}b_{(1)}><a_{(1)},b_{(5)}>\\\
=\sum\limits_{(a)(b)}(a_{(3)},b_{(3)})\overline{<a^{*}_{(4)},S^{*}_{B}b_{(2)}>}<a_{(2)},S_{B}b_{(4)}><a_{(5)},S_{B}b_{(1)}><a_{(1)},b_{(5)}>\\\
=\sum\limits_{(a)(b)}(a_{(3)},b_{(3)})[<a_{(1)},b_{(5)}><a_{(2)},S_{B}b_{(4)}>][<a_{(4)},b_{(2)}><a_{(5)},S_{B}b_{(1)}>]\\\
=\sum\limits_{(a)(b)}(a_{(2)},b_{(3)})<a_{(1)},b_{(5)}S_{B}b_{(4)}><a_{(3)},b_{(2)}S_{B}b_{(1)}>\\\
=\sum\limits_{(a)(b)}(a_{(2)},b_{(2)})[\varepsilon_{B}(b_{(3)})<a_{(1)},1_{B}>][\varepsilon_{B}(b_{(1)})<a_{(3)},1_{B}>]\\\
=\sum\limits_{(a)(b)}(a_{(2)},b_{(1)})\varepsilon_{B}(b_{(2)})\varepsilon_{A}(a_{(1)})\varepsilon_{A}(a_{(3)})\\\
=\sum\limits_{(a)}(a_{(2)},b)\varepsilon_{A}(a_{(1)})\varepsilon_{A}(a_{(3)})\\\
=(a,b),\end{array}$
where we use relations $S_{A}((S_{A}a)^{*})^{*}=a$ and
$<a^{*},b>=\overline{<a,S_{B}b^{*}>}$ in the third and forth equations.
$\blacksquare$
Lemma 3.4 $\forall(a,b),(a^{\prime},b^{\prime})\in D(A,B)$,
$[(a,b)(a^{\prime},b^{\prime})]^{*}=(a^{\prime},b^{\prime})^{*}(a,b)^{*}$.
Proof We firstly prove the relation
$[(1_{A},b)(a^{\prime},b^{\prime})]^{*}=(a^{\prime},b^{\prime})^{*}(1_{A},b)^{*}$.
$\begin{array}[]{llll}[(1_{A},b)(a^{\prime},b^{\prime})]^{*}\\\
=[\sum\limits_{(a^{\prime})(b)}(a^{\prime}_{(2)},b_{(2)}b^{\prime})<a^{\prime}_{(1)},S_{B}b_{(3)}><a^{\prime}_{(3)},b_{(1)}>]^{*}\\\
=\sum\limits_{(a^{\prime})(b)}(a^{\prime}_{(2)},b_{(2)}b^{\prime})^{*}\overline{<a^{\prime}_{(1)},S_{B}b_{(3)}><a^{\prime}_{(3)},b_{(1)}>}\\\
=\sum\limits_{(a^{\prime})(b)(b^{\prime})}(a^{\prime*}_{(3)},b^{\prime*}_{(2)}b^{*}_{(3)})<a^{\prime*}_{(4)},b^{\prime*}_{(1)}b^{*}_{(2)}><a^{\prime*}_{(2)},S_{B}b^{*}_{(4)}S_{B}b^{\prime*}_{(3)}>\times\\\
\overline{<a^{\prime}_{(1)},S_{B}b_{(5)}><a^{\prime}_{(5)},b_{(1)}>}\\\
=\sum\limits_{(a^{\prime})(b)(b^{\prime})}(a^{\prime*}_{(3)},b^{\prime*}_{(2)}b^{*}_{(3)})\times\\\
\overline{<a^{\prime}_{(1)},S_{B}b_{(5)}><a^{\prime}_{(2)},b_{(4)}b^{\prime}_{(3)}><a^{\prime}_{(4)},S_{B}b^{\prime}_{(1)}S_{B}b_{(2)}><a^{\prime}_{(5)},b_{(1)}>}\\\
=\sum\limits_{(a^{\prime})(b)(b^{\prime})}(a^{\prime*}_{(3)},b^{\prime*}_{(2)}b^{*}_{(3)})\times\\\
\overline{<a^{\prime}_{(1)},1_{B})><a^{\prime}_{(2)},b^{\prime}_{(3)}><a^{\prime}_{(4)},S_{B}b^{\prime}_{(1)}S_{B}b_{(2)}><a^{\prime}_{(5)},b_{(1)}>}\\\
=\sum\limits_{(a^{\prime})(b)(b^{\prime})}(a^{\prime*}_{(3)},b^{\prime*}_{(2)}b^{*}_{(3)})\times\\\
\overline{<a^{\prime}_{(1)},1_{B})><a^{\prime}_{(2)},b^{\prime}_{(3)}><a^{\prime}_{(4)},S_{B}b^{\prime}_{(1)}><a^{\prime}_{(5)},S_{B}b_{(2)}><a^{\prime}_{(6)},b_{(1)}>}\\\
=\sum\limits_{(a^{\prime})(b^{\prime})}(a^{\prime*}_{(3)},b^{\prime*}_{(2)}b^{*})\overline{<a^{\prime}_{(1)},1_{B}><a^{\prime}_{(2)},b^{\prime}_{(3)}><a^{\prime}_{(4)},S_{B}b^{\prime}_{(1)}><a^{\prime}_{(5)},1_{B}>}\\\
=\sum\limits_{(a^{\prime})(b^{\prime})}(a^{\prime*}_{(2)},b^{\prime*}_{(2)})(1_{A},b)^{*}\overline{<a^{\prime}_{(1)},b^{\prime}_{(3)}><a^{\prime}_{(3)},S_{B}b^{\prime}_{(1)}><a^{\prime}_{(4)},1_{B}>}\\\
=\sum\limits_{(a^{\prime})(b^{\prime})}(a^{\prime*}_{(2)},b^{\prime*}_{(2)})(1_{A},b^{*})\overline{<a^{\prime}_{(1)},b^{\prime}_{(3)}><a^{\prime}_{(3)},S_{B}b^{\prime}_{(1)}>}\\\
=(a^{\prime},b^{\prime})^{*}(1_{A},b)^{*}.\end{array}$
Similarly, $(a,b)^{*}=[(a,1_{B})(1_{A},b)]^{*}=(1_{A},b)^{*}(a,1_{B})^{*}$ and
then
$\begin{array}[]{llll}[(a,b)(a^{\prime},b^{\prime})]^{*}&=&[(a,1_{B})(1_{A},b)(a^{\prime},b^{\prime})]^{*}\\\
&=&[(1_{A},b)(a^{\prime},b^{\prime})]^{*}(a,1_{B})^{*}\\\
&=&(a^{\prime},b^{\prime})^{*}(1_{A},b)^{*}(a,1_{B})^{*}\\\
&=&(a^{\prime},b^{\prime})^{*}(a,b)^{*},\end{array}$
which completes the proof. $\blacksquare$
Using Lemma 3.3 and Lemma 3.4, one can immediately get the following result.
Proposition 3.5 The involution $*_{D}$ renders $D(A,B)$ into a non-degenerate
*-algebra.
Furthermore, one can show that $D(A,B)$ has a Hopf *-algebra structure.
Indeed, under the following structure maps, $D(A,B)$ becomes a finite
dimensional Hopf algebra naturally ([16]): $\forall(a,b)\in D(A,B)$,
$\Delta_{D}(a,b)=\sum\limits_{(a)(b)}(a_{(1)},b_{(1)})\otimes(a_{(2)},b_{(2)}),$
$\varepsilon_{D}(a,b)=\varepsilon_{A}(a)\varepsilon_{B}(b),$
$S_{D}(a,b)=\sum\limits_{(a)(b)}(S_{A}a_{(2)},S_{B}b_{(2)})<a_{(1)},S_{B}b_{(3)}><a_{(3)},b_{(1)}>.$
Theorem 3.6 $D(A,B)$ is a Hopf *-algebra.
Proof It suffices to show that $\Delta_{D}$ and $\varepsilon_{D}$ are
*-homomorphisms and $\forall(a,b)\in D(A,B)$,
$S_{D}(S_{D}(a,b)^{*})^{*}=(a,b)$.
(1) $\Delta_{D}$ is a *-homomorphism.
$\begin{array}[]{llll}\Delta_{D}((a,b)^{*})&=&\Delta_{D}(((a,1_{B})(1_{A},b))^{*})\\\
&=&\Delta_{D}((1_{A},b^{*})(a^{*},1_{B}))\\\
&=&\Delta_{D}(1_{A},b^{*})\Delta_{D}(a^{*},1_{B})\\\
&=&\sum\limits_{(b)}(1_{A},b^{*}_{(1)})\otimes(1_{A},b^{*}_{(2)})\sum\limits_{(a)}(a^{*}_{(1)},1_{B})\otimes(a^{*}_{(2)},1_{B})\\\
&=&\sum\limits_{(a)(b)}(1_{A},b_{(1)})^{*}(a_{(1)},1_{B})^{*}\otimes(1_{A},b_{(2)})^{*}(a_{(2)},1_{B})^{*}\\\
&=&\sum\limits_{(a)(b)}[(a_{(1)},1_{B})(1_{A},b_{(1)})]^{*}\otimes[(a_{(2)},1_{B})(1_{A},b_{(2)})]^{*}\\\
&=&\sum\limits_{(a)(b)}(a_{(1)},b_{(1)})^{*}\otimes(a_{(2)},b_{(2)})^{*}\\\
&=&\sum\limits_{(a)(b)}[(a_{(1)},b_{(1)})\otimes(a_{(2)},b_{(2)})]^{*}\\\
&=&(\Delta_{D}(a,b))^{*}.\end{array}$
Similarly, $\varepsilon_{D}$ is a *-homomorphism.
(2) It is easy to see $S_{D}(a,1_{B})=(S_{A}a,1_{B})$ and
$S_{D}(1_{A},b)=(1_{A},S_{B}b)$. Thus
$S_{D}(a,b)=S_{D}[(a,1_{B})(1_{A},b)]=S_{D}(1_{A},b)S_{D}(a,1_{B})=(1_{A},S_{B}b)(S_{A}a,1_{B}),$
and therefore,
$(S_{D}(a,b))^{*}=(S_{A}a,1_{B})^{*}(1_{A},S_{B}b)^{*}=(S^{*}_{A}a,S^{*}_{B}b).$
Using these two relations, we have
$\begin{array}[]{llll}S_{D}(S_{D}(a,b)^{*})\\\
=S_{D}(S^{*}_{A}a,S^{*}_{B}b)\\\
=\sum\limits_{(S^{*}_{A}a)(S^{*}_{B}b)}(S_{A}(S^{*}_{A}a)_{(2)},S_{B}(S^{*}_{B}b)_{(2)})\times\\\
<S^{*}_{A}a_{(1)},S_{B}(S^{*}_{B}b)_{(3)}><S^{*}_{A}a_{(3)},S^{*}_{B}b_{(1)}>\\\
=\sum\limits_{(a)(b)}(a^{*}_{(2)},b^{*}_{(2)})<a^{*}_{(3)},b^{*}_{(1)}><S^{*}_{A}a_{(1)},S_{B}(S^{*}_{B}b)_{(3)}>\\\
=\sum\limits_{(a)(b)}(a^{*}_{(2)},b^{*}_{(2)})<a^{*}_{(3)},b^{*}_{(1)}><a^{*}_{(1)},S_{B}b^{*}_{(3)}>\\\
=(a,b)^{*}.\end{array}$
$\blacksquare$
Now it is time to consider the C*-algebra structure of $D(A,B)$.
Lemma 3.7 Let $\varphi_{A}$ and $\varphi_{B}$ be invariant integrals on $A$
and $B$, respectively. $\forall(a,b)\in D(A,B)$, set
$\theta((a,b)):=\varphi_{A}(a)\varphi_{B}(b).$
Then $\theta$ is a faithful positive linear functional on $D(A,B)$.
Proof
$(a,b)(a,b)^{*}=(a,b)(1_{A},b^{*})(a^{*},1_{B})=(a,bb^{*})(a^{*},1_{B})$. In
the following, we denote $bb^{*}$ by $c$ briefly.
$\begin{array}[]{llll}\theta((a,b)(a,b)^{*})&=&\theta((a,c)(a^{*},1_{B}))\\\
&=&\sum\limits_{(a)(c)}\theta((aa^{*}_{(2)},c_{(2)}))<a^{*}_{(1)},S_{B}c_{(3)}><a^{*}_{(3)},c_{(1)}>\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(aa^{*}_{(2)})\varphi_{B}(c_{(2)})<a^{*}_{(1)},S_{B}c_{(3)}><a^{*}_{(3)},c_{(1)}>\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(aa^{*}_{(2)})<a^{*}_{(1)},S_{B}c_{(3)}><a^{*}_{(3)},\varphi_{B}(c_{(2)})c_{(1)}>\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(aa^{*}_{(2)})<a^{*}_{(1)},S_{B}c_{(2)}><a^{*}_{(3)},1_{B}>\varphi_{B}(c_{(1)})\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(a\varepsilon_{A}(a^{*}_{(3)})a^{*}_{(2)})<a^{*}_{(1)},S_{B}c_{(2)}>\varphi_{B}(c_{(1)})\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(aa^{*}_{(2)})<a^{*}_{(1)},S_{B}c_{(2)}>\varphi_{B}(c_{(1)})\\\
&=&\sum\limits_{(a)(c)}\varphi_{A}(aa^{*}_{(2)})<a^{*}_{(1)},\varphi_{B}\circ
S_{B}c_{(1)}S_{B}c_{(2)}>\\\
&=&\sum\limits_{(a)}\varphi_{A}(aa^{*}_{(2)})<a^{*}_{(1)},1_{B}>\varphi_{B}(c)\\\
&=&\sum\limits_{(a)}\varphi_{A}(aa^{*}_{(2)})\varepsilon_{A}(a^{*}_{(1)})\varphi_{B}(c)\\\
&=&\varphi_{A}(aa^{*})\varphi_{B}(c)\geq 0,\end{array}$
where we use the relation $\varphi_{B}\circ S_{B}=\varphi_{B}$ for the last
third and forth equations.
It is clear that $\theta((a,b)(a,b)^{*})=0$ if and only if $a=0$ or $b=0$,
which implies $(a,b)=0$. Thus $\theta$ is a faithful positive linear
functional on $D(A,B)$. $\blacksquare$
Theorem 3.8 $D(A,B)$ is a finite Hopf C*-algebra.
Proof Using the result in Lemma 3.7, one can construct the associated GNS
representation of $D(A,B)$ ([17]): $\forall x,y\in D(A,B)$, set
$<x,y>_{{}_{\theta}}=\theta(y^{*}x),$
where $<x,y>_{{}_{\theta}}$ denotes the inner product of $x$ and $y$. Thus
$D(A,B)$ turns into a Hilbert space $K$. For $d\in D(A,B)$, define
$\pi(d):K\longrightarrow K,\ \ x\mapsto dx.$
Using ([17]), $(\pi,K)$ is a faithful *-representation of $D(A,B)$, and hence
$D(A,B)$ can embeds into $B(K)$ isometrically through
$\pi:D(A,B)\longrightarrow B(K),\ \ d\mapsto\pi(d).$
Again $D(A,B)$ is finite dimensional, therefore, it is a C*-algebra with
C*-norm $\|(a,b)\|=(\theta((a,b)(a,b)^{*}))^{1/2}$. $\blacksquare$
Remark 3.9 A short calculation shows that $\theta$ coincides with
$\varphi_{A}\otimes\varphi_{B}$, which is indeed an integral on $D(A,B)$.
Using the relation
$\theta((a,b)(a,b)^{*})=\varphi_{A}(aa^{*})\varphi_{B}(bb^{*})$, one can get
$\|(a,b)\|=\|a\|\|b\|$. In particular, $\|(a,1_{B})\|=\|a\|$ (resp.
$\|(1_{A},b)\|=\|b\|$), which implies that the canonical embedding map $i_{A}$
(resp. $i_{B}$) is isometric.
Example 3.10 Let $H$ be a finite Hopf C*-algebra and $H^{\prime}$ be its dual,
which is also a finite Hopf C*-algebra by ([8]). They are naturally dually
pairing and have invariant integrals, denoted by $h$ and $h^{\prime}$
respectively. Drinfeld’s quantum double $D(H)$ of $H$, which is defined as the
bicrossed product of $H$ and $H^{\prime}$, is a special case of our
construction. One ([11]) can prove that it is also a finite Hopf C*-algebra
and has an invariant integral $h\otimes h^{\prime}$.
## References
* [1] Yang, C.N.: Some exact results for the many-body problems in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19:1312-1315 (1967)
* [2] Kassel, C.: Quantum groups, Springer-Verlag, NewYork, 1995
* [3] Drinfeld, V.G.: Quantum groups, International Congress of Mathematicians, Berkeley, 1986
* [4] Woronowicz, S.L.: Compact matrix psedogroups, Comm. Math. Phys., 111:613-665 (1987)
* [5] Vaes, S., Daele, A.V.: Hopf C*-algebras, Proc.Land.Math.Soc., 82(3):337-384 (2001)
* [6] Daele, A.V.: Dual pairs of Hopf *-algebras, Bull London Math. Soc., 25:209-220 (1993)
* [7] Jiang, L.N., Li, Z.Y.: Representation and duality of finite Hopf C*-algebra, Acta Mathematica Sinica, 47(6): 1155-1160 (in Chinese)
* [8] Daele, A.V.: The Haar measure on finite quantum groups, Proc. Amer. Math. Soc., 125:3489-3500 (1997)
* [9] Delvaux, L., Daele, A.V.: The Drinfel’d double for group-cograded multiplier Hopf algebras, arXiv:math.QA/0404029
* [10] Drabant, B., Daele, A.V.: Pairing and quantum double of multiplier Hopf algebras, Algebras Represet. Theory, 4:109-132 (2001)
* [11] Jiang, L.N.: C*-structure of quantum double of finite Hopf C*-algebras, J. Beijing Institute of Technology, 14(3):328-331 (2005)
* [12] Abe, E.: Hopf algebras, Cambridge University Press, Cambridge, 1977
* [13] Daele, A.V., Wang, S.: The Larson-Sweedler theorem for multiplier Hopf algebras, arXiv:math.QA/0408218
* [14] Guo, M.Z., Jiang, L.N., Zhao, Y.W.: A paring theorem between a braided bialgebra and its dual bialgebra, J. Algebra, 245:532-551 (2001)
* [15] Mueger, M.: From subfactors to categories and topology II. the quantum double of tensor categories and subfactors, J. Pure Appl. Alg., 180:159-219 (2003)
* [16] Majid, S.: Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebras, 130:17-64 (1990)
* [17] Li, B.R.: Operate algebra, Science Press, Beijing, 1992 (in Chinese)
|
arxiv-papers
| 2009-08-07T13:07:50 |
2024-09-04T02:49:04.513077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ming Liu, Li Ning Jiang, Guo Sheng Zhang",
"submitter": "Liu Ming",
"url": "https://arxiv.org/abs/0908.1046"
}
|
0908.1214
|
# Brane-Bulk energy exchange and agegraphic dark energy
Ahmad Sheykhi 111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
We consider the agegraphic models of dark energy in a braneworld scenario with
brane-bulk energy exchange. We assume that the adiabatic equation for the dark
matter is satisfied while it is violated for the agegraphic dark energy due to
the energy exchange between the brane and the bulk. Our study shows that with
the brane-bulk interaction, the equation of state parameter of agegraphic dark
energy on the brane, $w_{D}$, can have a transition from normal state where
$w_{D}>-1$ to the phantom regime where $w_{D}<-1$, while the effective
equation of state for dark energy always satisfies
$w^{\mathrm{eff}}_{D}\geq-1$.
## I Introduction
The observed acceleration in the universe expansion rate is usually attributed
to the presence of an exotic kind of energy, called “dark energy” Rie . A
great variety of dark energy models have been proposed, but most of them are
not able to explain all features of the universe, or are artificially
constructed in the sense that it introduces too many free parameters to be
able to fit with the experimental data. For a recent review on dark energy
candidates see Pad . Many theoretical attempts toward understanding the dark
energy problem are focused to shed light on it in the framework of a
fundamental theory such as string theory or quantum gravity. Although a
complete theory of quantum gravity has not established until now, we still can
make some attempts to investigate the nature of dark energy according to some
principles of quantum gravity. An interesting attempt for probing the nature
of dark energy within the framework of quantum gravity (and thus compute it
from first principles) is the so-called “Agegraphic Dark Energy” (ADE)
proposal. This model is based on the uncertainty relation of quantum mechanics
together with the gravitational effect in general relativity. Following the
line of quantum fluctuations of spacetime, Karolyhazy et al. Kar1 argued that
the distance $t$ in Minkowski spacetime cannot be known to a better accuracy
than $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$ where $\beta$ is a dimensionless
constant of order unity. Based on Karolyhazy relation, Maziashvili discussed
that the energy density of the metric fluctuations of Minkowski spacetime is
given by Maz
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (1)
where $t_{p}$ is the reduced Planck time. Throughout this paper we use the
units $c=\hbar=k_{b}=1$. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$
and $m_{p}$ are the reduced Planck length and mass, respectively. The ADE
model assumes that the observed dark energy comes from the spacetime and
matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1 . The agegraphic
models of dark energy have been examined and constrained by various
astronomical observations age ; shey1 ; shey2 ; setare ; age2 .
Independent of the challenge we deal with the dark energy puzzle, in recent
years, theories of large extra dimensions, in which the observed universe is
realized as a brane embedded in a higher dimensional spacetime, have received
a lot of interest. According to the braneworld scenario the standard model of
particle fields are confined to the brane while, in contrast, the gravity is
free to propagate in the whole spacetime (RSII, ). In this theory the
cosmological evolution on the brane is described by an effective Friedmann
equation that incorporates non-trivially with the effects of the bulk into the
brane (Bin, ). An interesting consequence of the braneworld scenario is that
it allows the presence of five-dimensional matter which can propagate in the
bulk space and may interact with the matter content in the braneworld. It has
been shown that such interaction can alter the profile of the cosmic expansion
and lead to a behavior that would resemble the dark energy. The cosmic
evolution of the braneworld models with energy exchange between the brane and
bulk has been studied in the different setups (Kirit, ; Cai, ; Bog, ; Sahni, ;
Sheykhi, ; Sheykhi2, ). In these models, due to the energy exchange between
the bulk and the brane, the usual energy conservation law on the brane is
broken down and consequently it was found that the equation of state of the
dark energy may experience the transition behavior. In the context of
holographic dark energy braneworld model with bulk-brane interaction has also
been studied Setare0 . Other studies on the dark energy models in the context
of braneworld scenarios have been carried out in Setare1 .
The purpose of the present work is to disclose the effect of the energy
exchange between the brane and the bulk in RSII braneworld scenario on the
evolution of the universe by considering the flow of energy onto or away from
the brane. Employing the agegraphic model of dark energy in a non-flat
universe, we obtain the equation of state parameter for ADE density. We shall
assume that the adiabatic equation for the dark matter is satisfied while it
is violated for the ADE due to the energy exchange between the brane and the
bulk. We will show that by suitably choosing model parameters, our model can
exhibit accelerated expansion of the universe. In addition, we will present a
profile of the $w_{D}$ crossing $-1$ phenomenon which is in good agreement
with observations.
This paper is organized as follows. In section II, we review the formalism of
bulk-brane energy exchange. In section III, we study the original ADE in
braneworld where the time scale is chosen to be the age of the universe. In
section IV, we consider the new model of ADE while the time scale is chosen to
be the conformal time instead of the age of the universe. The last section is
devoted to conclusions and discussions.
## II Braneworld With Brane-Bulk Interaction
The theory we are considering is five-dimensional and has an action of the
form
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2{\kappa}^{2}}\int{d^{5}x\sqrt{-{g}}\left({R}-2\Lambda\right)}+\int{d^{5}x\sqrt{-{g}}{L}_{\mathrm{bulk}}^{m}}$
(2)
$\displaystyle+\int{d^{4}x\sqrt{-\tilde{g}}({L}_{\mathrm{brane}}^{m}-\sigma)},$
where $R$ is the 5D scalar curvature and $\Lambda<0$ is the bulk cosmological
constant. $g$ and $\tilde{g}$ are the bulk and the brane metrics,
respectively. We have also included arbitrary matter content both in the bulk
and on the brane through ${L}_{\mathrm{bulk}}^{m}$ and
${L}_{\mathrm{brane}}^{m}$, respectively, and $\sigma$ is the positive brane
tension. The field equations can be obtained by varying action (2) with
respect to the bulk metric $g_{AB}$. The result is
$\displaystyle G_{AB}+\Lambda g_{AB}=\kappa^{2}T_{AB}.$ (3)
For convenience and without loss of generality, we can choose the extra-
dimensional coordinate $y$ such that the brane is located at $y=0$ and bulk
has $\mathbb{Z}_{2}$ symmetry. We are interested in the cosmological solution
with a metric
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-n^{2}(t,y)dt^{2}+a^{2}(t,y)\gamma_{ij}dx^{i}dx^{j}+b^{2}(t,y)dy^{2},$
(4)
where $\gamma_{ij}$ is a maximally symmetric three-dimensional metric for the
surface ($t$=const., $y$=const.), whose spatial curvature is parameterized by
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . The metric coefficients are chosen so that,
$n(t,0)=1$ and $b(t,0)=1$, where $t$ is cosmic time on the brane. The total
energy-momentum tensor has bulk and brane components and can be written as
${T}_{AB}={T}_{AB}\mid_{\mathrm{brane}}+{T}_{AB}\mid_{\sigma}+{T}_{AB}\mid_{\mathrm{bulk}}.$
(5)
The first and the second terms are the contribution from the energy-momentum
tensor of the matter field confined to the brane and the brane tension
$\displaystyle T^{A}_{\,\,B}\mid_{\mathrm{brane}}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{diag}(-\rho,p,p,p,0)\frac{\delta(y)}{b},{}$ (6)
$\displaystyle T^{A}_{\,\,B}\mid_{\sigma}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{diag}(-\sigma,-\sigma,-\sigma,-\sigma,0)\frac{\delta(y)}{b},{}$
(7)
where $\rho$ and $p$, being the energy density and pressure on the brane,
respectively. In addition we assume an energy-momentum tensor for the bulk
content of the form
$T^{A}_{\ B}\mid_{\mathrm{bulk}}\,=\,\left(\begin{array}[]{ccc}T^{0}_{\
0}&\,0&\,T^{0}_{\ 5}\\\ \,0&\,T^{i}_{\ j}\delta^{i}_{\ j}&\,0\\\
-\frac{n^{2}}{b^{2}}T^{0}_{\ 5}&\,0&\,T^{5}_{\
5}\end{array}\right)\,\,.\,\,\,$ (8)
The quantities which are of interest here are $T^{5}_{\ 5}$ and $T^{0}_{\ 5}$,
as these two enter the cosmological equations of motion. In fact, $T^{0}_{\
5}$ is the term responsible for energy exchange between the brane and the
bulk. Inserting the ansatz (4) for the metric, the non-vanishing components of
the Einstein tensor ${G}_{AB}$ are found to be
$\displaystyle{G}_{00}$ $\displaystyle=$ $\displaystyle
3\left\\{\frac{\dot{a}}{a}\left(\frac{\dot{a}}{a}+\frac{\dot{b}}{b}\right)-\frac{n^{2}}{b^{2}}\left(\frac{a^{\prime\prime}}{a}+\frac{a^{\prime}}{a}\left(\frac{a^{\prime}}{a}-\frac{b^{\prime}}{b}\right)\right)+k\frac{n^{2}}{b^{2}}\right\\},$
(9) $\displaystyle{G}_{ij}$ $\displaystyle=$
$\displaystyle\frac{a^{2}}{b^{2}}\gamma_{ij}\left\\{\frac{a^{\prime}}{a}\left(\frac{a^{\prime}}{a}+2\frac{n^{\prime}}{n}\right)-\frac{b^{\prime}}{b}\left(\frac{n^{\prime}}{n}+2\frac{a^{\prime}}{a}\right)+2\frac{a^{\prime\prime}}{a}+\frac{n^{\prime\prime}}{n}\right\\}$
(10)
$\displaystyle+\frac{a^{2}}{n^{2}}\gamma_{ij}\left\\{\frac{\dot{a}}{a}\left(-\frac{\dot{a}}{a}+2\frac{\dot{n}}{n}\right)-2\frac{\ddot{a}}{a}+\frac{\dot{b}}{b}\left(-2\frac{\dot{a}}{a}+\frac{\dot{n}}{n}\right)-\frac{\ddot{b}}{b}\right\\}-k\gamma_{ij},$
$\displaystyle{G}_{05}$ $\displaystyle=$ $\displaystyle
3\left(\frac{n^{\prime}}{n}\frac{\dot{a}}{a}+\frac{a^{\prime}}{a}\frac{\dot{b}}{b}-\frac{\dot{a}^{\prime}}{a}\right),$
(11) $\displaystyle{G}_{55}$ $\displaystyle=$ $\displaystyle
3\left\\{\frac{a^{\prime}}{a}\left(\frac{a^{\prime}}{a}+\frac{n^{\prime}}{n}\right)-\frac{b^{2}}{n^{2}}\left(\frac{\dot{a}}{a}\left(\frac{\dot{a}}{a}-\frac{\dot{n}}{n}\right)+\frac{\ddot{a}}{a}\right)-k\frac{b^{2}}{a^{2}}\right\\}.$
(12)
In the above expressions, primes and dots stand for derivatives with respect
to $y$ and $t$, respectively. Integrating Eqs. (9) and (10) across the brane
and imposing $\mathbb{Z}_{2}$ symmetry, we obtain the jumps across the brane
$\displaystyle\frac{a^{\prime}_{+}}{a_{0}}=-\frac{\kappa^{2}}{6}(\rho+\sigma),$
(13)
$\displaystyle\frac{n^{\prime}_{+}}{n_{0}}=\frac{\kappa^{2}}{6}(2\rho+3p-\sigma),$
(14)
where $2a^{\prime}_{+}=-2a^{\prime}_{-}$ and
$2n^{\prime}_{+}=-2n^{\prime}_{-}$ are the discontinuities of the first
derivative, and the subscript “ 0” denotes quantities are evaluated at $y=0$.
Substituting the junction conditions $(\ref{jun1})$ and $(\ref{jun2})$ into
the $(05)$ and $(55)$ components of the field equations (3), we obtain the
modified Friedmann equation and the semi-conservation law on the brane
$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle-2T^{0}_{\
5},$ (15) $\displaystyle 2H^{2}+\dot{H}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{4}}{36}\left[\sigma\left(3p-\rho\right)+\rho\left(\rho+3p\right)\right],$
(16)
$\displaystyle+\frac{\kappa^{2}}{3}\left(\Lambda+\frac{\kappa^{2}\sigma^{2}}{6}\right)-\frac{\kappa^{2}}{3}T^{5}_{\
5},$
where $a=a_{0}=a(t,0)$ and $H=\dot{a}/a$ is the Hubble parameter on the brane.
We shall assume an equation of state $p=w\rho$ which represents a relation
between the energy density and pressure of the matter on the brane. We also
neglect $T^{5}_{\ 5}$ term by assuming that the bulk matter relative to the
bulk vacuum energy is much less than the ratio of the brane matter to the
brane vacuum energy (Kirit, ). Considering this we get
$\displaystyle 2H^{2}+\dot{H}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\gamma\rho\left(1-3w\right)-\beta\rho^{2}\left(1+3w\right)+\frac{\lambda}{3},$
(17) $\displaystyle\dot{\rho}+3H\rho(1+w)$ $\displaystyle=$
$\displaystyle-2T^{0}_{\ 5},$ (18)
where we have used the usual definition $\beta=\kappa^{4}/{36}$,
$\gamma=\beta\sigma$ and
$\lambda=\kappa^{2}(\Lambda+{\kappa^{2}\sigma^{2}}/{6})$. Assuming the
Randall-Sundrum fine-tuning
$\lambda=\kappa^{2}(\Lambda+{\kappa^{2}\sigma^{2}}/{6})=0$ holds on the brane,
one can easily check that the Friedmann equation (17) is equivalent to the
following equations
$\displaystyle{H}^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\beta\rho^{2}+2\gamma(\rho+\chi),$ (19)
$\displaystyle\dot{\chi}+4\,H\chi$ $\displaystyle=$ $\displaystyle 2T^{0}_{\
5}\left(\frac{\rho}{\sigma}+1\right).$ (20)
Equation (19) is the modified Friedmann equation describing cosmological
evolution on the brane. The auxiliary field $\chi$ incorporates non-trivial
contributions of dark energy which differ from the standard matter fields
confined to the brane. The bulk matter contributes to the energy conservation
equation (18) through $T^{0}_{\ 5}$ which is responsible for the energy
exchange between the brane and bulk. We are interested in the scenarios where
the energy density of the brane is much lower than the brane tension, namely
$\rho\ll\sigma$, therefore our system of equations can be simplified in the
following form
$\displaystyle{H}^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{3m^{2}_{p}}(\rho+\chi),$ (21)
$\displaystyle\dot{\chi}+4\,H\chi$ $\displaystyle\approx$ $\displaystyle 2\
{T^{0}_{\ 5}}=Q,$ (22) $\displaystyle\dot{\rho}+3H\rho(1+w)$ $\displaystyle=$
$\displaystyle-2T^{0}_{\ 5}=-Q.$ (23)
Here $m^{2}_{p}=(8\pi G_{4})^{-1}$ is the reduced Planck mass, where
$G_{4}=3\gamma/4\pi$ is the $4$D Newtonian constant. We assume that there are
two dark components in the universe, dark matter and dark energy, and thus the
total energy density is $\rho=\rho_{m}+\rho_{D}$, where $\rho_{m}$ and
$\rho_{D}$ are the energy density of dark matter and dark energy,
respectively. With the energy exchange between the bulk and brane, the usual
energy conservation is broken down. Here we assume that the adiabatic equation
for the dark matter is satisfied while it is violated for the dark energy due
to the energy exchange between the brane and the bulk
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (24)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-2T^{0}_{\ 5}=-Q.$ (25)
Here $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of ADE and
$Q=\Gamma\rho_{D}$ stands for the interaction term between the bulk and the
brane with interaction rate $\Gamma$. Therefore, until now we have obtained
the set of equations describing the dynamics of our universe in braneworld
with bulk-brane interaction.
## III THE ORIGINAL ADE and Bulk-brane interaction
The original ADE density has the form (1) where $t$ is chosen to be the age of
the universe
$T=\int{dt}=\int_{0}^{a}{\frac{da}{Ha}}.$ (26)
Thus, the energy density of the original ADE is given by Cai1
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (27)
where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time and so on. The Friedmann equation (21) can be
reexpressed as
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}+\chi\right).$
(28)
If we introduce, as usual, the fractional energy densities such as Setare0
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},\hskip
14.22636pt\Omega_{\chi}=\frac{\chi}{3m_{p}^{2}H^{2}},$ (29)
then, the Friedmann equation (28) can be written as
$\displaystyle\Omega_{m}+\Omega_{D}+\Omega_{\chi}=1+\Omega_{k}.$ (30)
Using Eq. (27), we have
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (31)
We choose the following ansatz for the interaction rate wang
$\displaystyle\Gamma=3b^{2}(1+r)H,$ (32)
where $b^{2}$ is a coupling constant and $r=\chi/\rho_{D}$ is the ratio of two
energy densities Setare0 ,
$\displaystyle
r=\frac{\Omega_{\chi}}{\Omega_{D}}=-1+\frac{1}{\Omega_{D}}\left[1+\Omega_{k}-\Omega_{m}\right].$
(33)
Using the continuity equation (24), it is easy to show that
$\displaystyle\Omega_{m}=\Omega_{m0}a^{-3}=\Omega_{m0}(1+z)^{3},$ (34)
where $\Omega_{m0}=0.28\pm 0.02$ is the present value of all part of the
matter confined to the brane. Taking the derivative of Eq. (27) with respect
to the cosmic time and using Eq. (31) we reach
$\displaystyle\dot{\rho}_{D}=-2H\rho_{D}\frac{\sqrt{\Omega_{D}}}{n}.$ (35)
Inserting this equation in the conservation law (25) and using Eqs. (32)-(34)
we find the equation of state parameter of the original ADE on the brane
$\displaystyle
w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}-b^{2}{\Omega^{-1}_{D}}\left\\{1+\Omega_{k}-\Omega_{m0}(1+z)^{3}\right\\}.$
(36)
One can easily check that $w_{D}$ can cross the phantom divide if
$3nb^{2}(1+\Omega_{k}-\Omega_{m})>2{\Omega^{3/2}_{D}}$. If we take
$\Omega_{D}\approx 0.72$, $\Omega_{m0}\approx 0.28$ and $\Omega_{k}\approx
0.01$ for the present time, the phantom-like equation of state for $w_{D}$ can
be achieved provided $nb^{2}>0.56$. The joint analysis of the astronomical
data for the new agegraphic dark energy gives the best-fit value (with
$1\sigma$ uncertainty) $n=2.7$ age2 . Thus, the condition $w_{D}<-1$ leads to
$b^{2}>0.2$ for the coupling between dark energy and dark matter. For
instance, if we take $b^{2}=0.25$ we get $w_{D}=-1.04$. If we define,
following kim ; Setare2 , the effective equation of state parameter as
$\displaystyle w^{\mathrm{eff}}_{D}=w_{D}+\frac{\Gamma}{3H},$ (37)
then, the continuity equation (25) for dark energy can be written in the
standard form
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w^{\mathrm{eff}}_{D})=0.$ (38)
Substituting Eqs. (32), (33) and (36) into Eq. (37), we find
$\displaystyle w^{\mathrm{eff}}_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}.$ (39)
From Eq. (39) we see that $w^{\mathrm{eff}}_{D}$ is always larger than $-1$
and cannot cross the phantom divide $w^{\mathrm{eff}}_{D}=-1$. Let us study
the behavior of $w^{\mathrm{eff}}_{D}$ in two different stages. In the early
time (matter-dominated epoch) where $\Omega_{D}\rightarrow 0$ we have
$w^{\mathrm{eff}}_{D}=-1$. Namely, the effective equation of state mimics a
cosmological constant in the matter-dominated epoch. In the late time where
$\Omega_{D}\rightarrow 1$ we have $w^{\mathrm{eff}}_{D}=-1+{2}/{3n}$. Thus we
have $w^{\mathrm{eff}}_{D}<-2/3$ provided $n>2$ which is consistent with
recent cosmological data age2 . Next, we obtain the equation of motion of
$\Omega_{D}$. Differentiating Eq. (31) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(40)
where the dot is the derivative with respect to the cosmic time and the prime
denotes the derivative with respect to $x=\ln{a}$. Taking the derivative of
both side of the Friedmann equation (28) with respect to the cosmic time, and
using Eqs. (22), (24), (30), (31) and (35), it is easy to find that
$\displaystyle\frac{\dot{H}}{H^{2}}=-2+\frac{3b^{2}}{2}-\frac{\Omega_{k}}{2}(2-3b^{2})+\frac{\Omega_{m}}{2}(1-3b^{2})+\Omega_{D}\left(2-\frac{\sqrt{\Omega_{D}}}{n}\right).$
(41)
Substituting this relation into Eq. (40), we obtain the equation of motion of
the original ADE
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left\\{4(1-\Omega_{D})\left(1-\frac{\sqrt{\Omega_{D}}}{2n}\right)+2\Omega_{k}-\Omega_{m}-3b^{2}(1+\Omega_{k}-\Omega_{m})\right\\}.$
(42)
This equation describes the evolution behavior of the original ADE in
braneworld cosmology with brane-bulk energy exchange. For completeness, we
give the deceleration parameter
$\displaystyle q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (43)
which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. Substituting Eq. (41) into (43) we get
$\displaystyle q$ $\displaystyle=$ $\displaystyle
3+\Omega_{k}-\frac{\Omega_{m}}{2}-\Omega_{D}\left(2-\frac{\sqrt{\Omega_{D}}}{n}\right)-\frac{3b^{2}}{2}\left(1+\Omega_{k}-\Omega_{m}\right).$
(44)
If we take $\Omega_{D}=0.72$, $\Omega_{m0}\approx 0.28$ and $\Omega_{k}\approx
0.01$ for the present time and choosing $n=2.4$, $b^{2}=2$ we obtain
$q\approx-0.5$ for the present value of the deceleration parameter which is in
good agreement with recent observational results Daly .
## IV THE NEW ADE and Bulk-brane interaction
Soon after the original ADE model was introduced by Cai Cai1 , an alternative
model dubbed “ new agegraphic dark energy” was proposed by Wei and Cai Wei2 ,
while the time scale is chosen to be the conformal time $\eta$ instead of the
age of the universe, which is defined by $dt=ad\eta$, where $t$ is the cosmic
time. It is important to note that the Karolyhazy relation $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ was derived for Minkowski spacetime
$ds^{2}=dt^{2}-d\mathrm{x^{2}}$ Kar1 ; Maz . In case of the FRW universe, we
have $ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$.
Thus, it might be more reasonable to choose the time scale in Eq. (27) to be
the conformal time $\eta$ since it is the causal time in the Penrose diagram
of the FRW universe. The new ADE contains some new features different from the
original ADE and overcome some unsatisfactory points. For instance, the
original ADE suffers from the difficulty to describe the matter-dominated
epoch while the new ADE resolved this issue Wei2 . The energy density of the
new ADE can be written
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (45)
where the conformal time is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (46)
The fractional energy density of the new ADE is given by
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (47)
Taking the derivative of Eq. (45) with respect to time and using Eq. (47) we
reach ($\dot{\eta}=1/a$)
$\displaystyle\dot{\rho}_{D}=-2H\rho_{D}\frac{\sqrt{\Omega_{D}}}{na}.$ (48)
Inserting this equation in the conservation law (25) and using Eqs. (32)-(34)
we can find the equation of state parameter
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}{\Omega^{-1}_{D}}\left\\{1+\Omega_{k}-\Omega_{m0}(1+z)^{3}\right\\}.$
(49)
Again we see that $w_{D}$ can cross the phantom divide provided
$3nab^{2}(1+\Omega_{k}-\Omega_{m})>2{\Omega^{3/2}_{D}}$. The effective
equation of state $w^{\mathrm{eff}}_{D}$ reads as
$\displaystyle w^{\mathrm{eff}}_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (50)
In the late time where $a\rightarrow\infty$ and $\Omega_{D}\rightarrow 1$,
from Eq. (50) we have $w^{\mathrm{eff}}_{D}=-1$, while from Eq. (49) it is
necessary to have $w_{D}<-1$. Thus the effective equation of state
$w^{\mathrm{eff}}_{D}$ behaves like a cosmological constant in the late time,
while $w_{D}$ crosses the phantom divide $w_{D}=-1$. We can also find the
equation of motion for $\Omega_{D}$ by differentiating Eq. (47). The result is
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{na}\sqrt{\Omega_{D}}\right).$
(51)
Taking the derivative of both side of the Friedman equation (28) with respect
to the cosmic time, and using Eqs. (22), (24), (30), (47) and (48), it is easy
to find that
$\displaystyle\frac{\dot{H}}{H^{2}}=-2+\frac{3b^{2}}{2}-\frac{\Omega_{k}}{2}(2-3b^{2})+\frac{\Omega_{m}}{2}(1-3b^{2})+\Omega_{D}\left(2-\frac{\sqrt{\Omega_{D}}}{na}\right).$
(52)
Substituting this relation into Eq. (51), we obtain the equation of motion of
the new ADE
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left\\{4(1-\Omega_{D})\left(1-\frac{\sqrt{\Omega_{D}}}{2na}\right)+2\Omega_{k}-\Omega_{m}-3b^{2}(1+\Omega_{k}-\Omega_{m})\right\\}.$
(53)
The deceleration parameter is now given by
$\displaystyle q$ $\displaystyle=$ $\displaystyle
3+\Omega_{k}-\frac{\Omega_{m}}{2}-\Omega_{D}\left(2-\frac{\sqrt{\Omega_{D}}}{na}\right)-\frac{3b^{2}}{2}\left(1+\Omega_{k}-\Omega_{m}\right).$
(54)
Comparing Eqs. (48)-(54) with their respective equations obtained in the
previous section, we see that the scale factor $a$ enters Eqs. (48)-(54)
explicitly. Besides, comparing the results obtained in this work with those
presented in Cai1 ; Wei2 ; shey1 for ADE models in standard cosmology we find
that the energy exchange between the brane and bulk seriously modifies our
basic equations.
## V Conclusions and Discussions
Among different candidates for probing the nature of dark energy, the
holographic dark energy model arose a lot of enthusiasm recently Coh ; Li ;
Huang ; Hsu ; Setare3 . However, there are some difficulties in holographic
dark energy model. Choosing the event horizon of the universe as the length
scale, the holographic dark energy gives the observation value of dark energy
in the universe and can drive the universe to an accelerated expansion phase.
But an obvious drawback concerning causality appears in this proposal. Event
horizon is a global concept of spacetime; existence of event horizon of the
universe depends on future evolution of the universe; and event horizon exists
only for universe with forever accelerated expansion. In addition, more
recently, it has been argued that this proposal might be in contradiction to
the age of some old high redshift objects, unless a lower Hubble parameter is
considered Wei0 .
In this work we have studied the agegraphic dark energy in the framework of
RSII braneworld scenario with bulk-brane energy exchange. Considering the
effects of the interaction between the brane and the bulk we have obtained the
equation of state for the ADE in a non-flat universe on the brane. We found
that although the equation of state parameter of ADE on the brane, $w_{D}$,
can cross the phantom divide, the effective equation of state parameter
$w^{\mathrm{eff}}_{D}=w_{D}+\frac{\Gamma}{3H}$ is always larger than $-1$ and
cannot cross the phantom divide $w^{\mathrm{eff}}_{D}=-1$, where $\Gamma$ is
the rate of the bulk-brane interaction. For instance, taking $n=2.7$ age2 and
$\Omega_{D}=0.72$ for the present time, we found $w^{\mathrm{eff}}_{D}=-0.8$.
This indicates that one cannot generate phantom-like effective equation of
state from an ADE in a braneworld model with bulk-brane interaction. For new
ADE, in the late time where $a\rightarrow\infty$ and $\Omega_{D}\rightarrow
1$, we found $w^{\mathrm{eff}}_{D}=-1$ while $w_{D}<-1$. Thus in the new model
of ADE the effective equation of state $w^{\mathrm{eff}}_{D}$ mimics a
cosmological constant in the late time, while $w_{D}$ necessary have a
transition to the phantom regime in the presence of bulk-brane interaction.
In agegraphic models of dark energy with bulk-brane interaction, the
properties of ADE is determined by the parameters $n$ and $b$ together. These
parameters would be obtained by confronting with cosmic observational data. In
this work we just restricted our numerical fitting to limited observational
data. Giving the wide range of cosmological data available, in the future we
expect to further constrain our model parameter space and test the viability
of this model.
###### Acknowledgements.
I thank the anonymous referee for constructive comments. This work has been
supported by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.
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|
arxiv-papers
| 2009-08-09T04:55:03 |
2024-09-04T02:49:04.519861
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0908.1214"
}
|
0908.1473
|
# Proof of a Universal Lower Bound on the Shear Viscosity to Entropy Density
Ratio
Ram Brustein (1), A.J.M. Medved (2)
(1) Department of Physics, Ben-Gurion University,
Beer-Sheva 84105, Israel
(2) Physics Department, University of Seoul
Seoul 130-743 Korea
E-mail: ramyb@bgu.ac.il, allan@physics.uos.ac.kr
###### Abstract
It has been conjectured, on the basis of the gauge-gravity duality, that the
ratio of the shear viscosity to the entropy density should be universally
bounded from below by $1/4\pi$ in units of the Planck constant divided by the
Boltzmann constant. Here, we prove the bound for any ghost-free extension of
Einstein gravity and the field-theory dual thereof. Our proof is based on the
fact that, for such an extension, any gravitational coupling can only increase
from its Einstein value. Therefore, since the shear viscosity is a particular
gravitational coupling, it is minimal for Einstein gravity. Meanwhile, we show
that the entropy density can always be calibrated to its Einstein value. Our
general principles are demonstrated for a pair of specific models, one with
ghosts and one without.
Kovtun, Son and Starinets (KSS) have proposed that the ratio of the shear
viscosity $\eta$ to the entropy density $s$ should be universally bounded from
below by $1/4\pi$ in units of the Planck constant divided by the Boltzmann
constant $\hbar/k_{B}$ [1]. In kinetic theory, the shear viscosity of a fluid
is directly proportional to the mean free path of the quasiparticles,
suggesting that $\eta/s$ is much larger than $\hbar/k_{B}$ in weakly coupled
fluids for which the mean free path of the quasiparticles is always large.
Thus, a bound on $\eta/s$ is mostly relevant to strongly interacting fluids.
The uncertainty principle can be used in this context to argue that the ratio
$\eta/s$ in units of $\hbar/k_{B}$ is bounded from below by a constant of
order unity [1]. So far, two classes of quantum fluids are known to have
values of $\eta/s$ that approach $\frac{1}{4\pi}\frac{\hbar}{k_{B}}$: Strongly
correlated ultracold Fermi gases and the quark–gluon plasma. (See [2] for a
recent review and many references.) Experiments in both systems [3, 4] have
now reached the necessary precision to probe the KSS bound.
The gauge–gravity duality [5, 6] provides a hydrodynamic description of
strongly coupled field theories in terms of the hydrodynamics of a black brane
in an asymptotically anti-de Sitter (AdS) spacetime [7]. It was shown in [8]
that, for strongly coupled field theories, $\eta/s=1/4\pi$ when the bulk
gravitational theory is Einstein’s (from here on, $\hbar,\;k_{B},\;c=1$). This
value has since been confirmed in many examples. (For references and further
discussion, see [9].) However, recent findings have cast doubts over the
universal nature of the bound. For instance, when the gravitational Lagrangian
includes the square of the $4$-index Riemann tensor, the ratio $\eta/s$ can
either be smaller or larger than its Einstein value [10, 11]. These
modifications can be understood from the observation [12] that $\eta/s$ is
equivalently a ratio of two different gravitational couplings, each associated
with a differently polarized graviton [13]. If the gravitational theory is
Einstein’s or related to Einstein’s by a field redefinition, then the
couplings will be independent of the polarization and $\eta/s=1/4\pi$. In
general, however, the couplings for differently polarized gravitons are
distinct, and there is no longer any reason to expect that $\eta/s=1/4\pi$.
As will be explained, if we impose the physical requirement that extensions of
Einstein gravity must be ghost free, then any gravitational coupling can only
increase from its Einstein value. We will show, in particular, how this
outcome applies to the shear viscosity. It will then be demonstrated how the
entropy density for any extension can always be calibrated to its Einstein
value. Combining both facts will allow us to establish that, if the ratio
$\eta/s$ does differ from the Einstein result, then it must necessarily be
larger than $1/4\pi$.
We will consider a general theory of gravity in AdS whose action depends on
the metric $g_{\mu\nu}$, the Riemann tensor ${\cal R}_{\rho\mu\lambda\nu}$,
matter fields $\phi$ and their covariant derivatives:
$I=\int\\!\\!d^{d+1}x\sqrt{-g}\ \mathscr{L}\left({\cal
R}_{\rho\mu\lambda\nu},g_{\mu\nu},\nabla_{\sigma}R_{\rho\mu\lambda\nu},\phi,\nabla\phi,\ldots\right)$
with $d\geq 3$. We will further assume the existence of stationary
$(p+2)$-black brane solutions with a bifurcate Killing horizon and described
by the following metric:
$ds^{2}=-g_{tt}(r)\,dt^{2}+g_{rr}(r)\,dr^{2}+g_{xx}(r)\,dx^{i}\,dx_{i}\;,\
i=1,\dots,p.$ The black brane event horizon is at $r=r_{h}$, where $g_{tt}$
has a first-order zero, $g_{rr}$ has a first-order pole and all other metric
components are finite. All the metric components are taken to depend only on
$r$ and, therefore, the metric is Poincare invariant in the $(t,x_{i})$
subspace. Also, the AdS boundary is taken to be at $r\to\infty$, where the
metric asymptotically approaches its AdS form.
Let us discuss small perturbations of the brane metric: $g_{\mu\nu}\rightarrow
g_{\mu\nu}+h_{\mu\nu}$ with $h_{\mu\nu}\ll 1$. We will take $z$ as their
propagating direction on the brane. It is well known that, for a suitable
choice of gauge [14], the highest-helicity polarization of the $h_{xy}$
gravitons decouples from all others. (Obviously, $x,y$ can be replaced by any
other orthogonal-to-$z$ transverse dimensions.) A standard procedure [15, 9]
that involves taking the hydrodynamic limit and using the Kubo formula allows
one to extract the shear viscosity from the correlation function of the
dissipative energy-momentum tensor $T_{xy}\sim\eta\partial_{t}h_{xy}$. As
explained in [12], this procedure is valid for extensions of Einstein gravity
and amounts to extracting the gravitational coupling for the $h_{xy}$
gravitons. The very same procedure can also be implemented at any radial
distance from the brane [16, 17]. Applying it at the AdS boundary, one then
learns about the shear viscosity in the dual field theory.
We can determine the $h_{xy}$ coupling by calculating the propagator in the
one-particle exchange approximation, which is valid because $h_{xy}\ll 1$. For
Einstein gravity, only massless spin-$2$ gravitons are exchanged, but
gravitons can, for a general theory, be either massless or massive and of
either spin-$0$ or spin-$2$. Particles of any other spin, in particular
vectors, can not couple linearly to a conserved source and so can safely be
neglected when evaluating the propagator. Accordingly, the graviton propagator
$[{\cal D}(q^{2})]^{\ \nu\;\beta}_{\mu\;\alpha}\equiv\langle h_{\mu}^{\
\nu}(q)h_{\alpha}^{\ \beta}(-q)\rangle$ must be of the following irreducibly
decomposed form [18, 19, 20, 21]:
$\displaystyle[{\cal D}(q^{2})]^{\ \nu\;\beta}_{\mu\;\alpha}\;$
$\displaystyle=$
$\displaystyle\;\left(\rho_{E}(q^{2})+\rho_{NE}(q^{2})\right)\left[\delta^{\
\beta}_{\mu}\delta^{\ \nu}_{\alpha}-\frac{1}{2}\delta^{\ \nu}_{\mu}\delta^{\
\beta}_{\alpha}\right]\frac{G_{E}}{q^{2}}$ (1) $\displaystyle+$
$\displaystyle\\!\\!\sum_{i}\rho^{i}_{NE}(q^{2})\left(\delta^{\
\beta}_{\mu}\delta^{\ \nu}_{\alpha}-\frac{1}{3}\delta^{\ \nu}_{\mu}\delta^{\
\beta}_{\alpha}\right)\frac{G_{E}}{q^{2}-m_{i}^{2}}$ $\displaystyle+$
$\displaystyle\\!\\!\sum_{j}{\widetilde{\rho}}^{\;j}_{NE}(q^{2})\;\delta^{\
\nu}_{\mu}\delta^{\
\beta}_{\alpha}\;\frac{G_{E}}{q^{2}-{\widetilde{m}}_{j}^{2}}\;.$
Here, $G_{E}$ is Newton’s constant and we have denoted the Einstein and “Non-
Einstein” parts of the gravitational couplings $\rho$ by the subscripts $E$
and $NE$. We have separated the contribution of the massless spin-$2$
particles, massive spin-$2$ particles with mass $m_{i}$ and scalar particles
with mass ${\widetilde{m}}_{j}$. Some of the masses $m_{i}$,
${\widetilde{m}}_{j}$ may vanish or be parametrically small in certain cases.
The couplings or $\rho$’s are dimensionless quantities and can depend on the
momentum scale $q$; in particular, $\rho_{E}(0)=1$.
For a ghost-free theory, all of the couplings must remain positive at all
energy scales [20]; meaning that the propagator can only increase relative to
its Einstein value.
As discussed above, the shear viscosity for any theory of gravity can be
determined directly from the propagator $\langle h_{xy}(q)h_{xy}(-q)\rangle$
when taken to the hydrodynamic limit. In this limit, the temperature $T$ is
the largest relevant scale, so that both the energy $\omega$ and momentum
$\vec{q}\;$ have to satisfy $\omega/T$, $|\vec{q}|/T\ll 1$ (with $\omega$ and
$|\vec{q}|$ not necessarily of the same magnitude). For Einstein’s theory, the
above procedure yields the well-known answer $\eta_{E}=1/(16\pi G_{E})$. For a
general theory, the corrections can be read off the propagator in Eq. (1). As
we only have to consider mode contributions such that $m/T\to 0$, the shear
viscosity $\eta_{X}$ for a generic theory $X$ can be expressed as follows:
$\frac{\eta_{X}}{\eta_{E}}\;=\;\left[\frac{\langle h_{x}^{\ y}h_{y}^{\
x}\rangle_{X}}{\langle h_{x}^{\ y}h_{y}^{\
x}\rangle_{E}}\right]\;=\;1+\frac{1}{\rho_{E}(0)}\sum\limits_{i}\rho_{NE}^{\;i}(0)\;.$
(2)
The sum represents the non-Einstein contribution of spin-2 particles to the
$xy$-polarization channel in Eq. (1). The scalars and the trace parts of the
massless and massive gravitons do not contribute to the sum in Eq. (2).
Irrespective of the precise nature of the corrections, we know that
$\rho_{E}(0)=1$ and that, for any ghost-free theory of gravity, the $\rho$’s
must be positive. It follows that their effect can only be to increase $\eta$
relative to its Einstein value:
$\frac{\eta_{X}}{\eta_{E}}\geq 1\;.$ (3)
This lower bound must be true for any coordinate system or choice of field
definitions, as the absence of ghosts is an invariant statement that is
insensitive to these choices. Further, as previously discussed, this bound is
valid at any radial distance from the brane and, in particular, near the AdS
boundary at $r\to\infty$. The gauge–gravity duality then implies that the
shear viscosity of any field-theory dual also satisfies Eq. (3). That is, the
smallest $\eta$ for a field theory must be for the theory dual to Einstein
gravity.
To bring the discussion back to the ratio of interest $\eta/s$, let us next
consider the entropy density. The gauge–gravity duality tells us that the
entropy density for a given field theory is the same as that of its black
brane dual. Also, the temperature $T$ of the field theory can be identified
with the Hawking temperature of the black brane. The latter is fixed by the
horizon radius $r_{h}$ (along with any charges) in units of the AdS curvature
scale and depends explicitly on the geometry but not on the underlying
Lagrangian. Working at a fixed value of temperature or fixed $r_{h}$, we will
demonstrate that, although the black brane entropy $S_{X}$ for a generic
gravity theory can be different from the Einstein value $S_{E}$, it is always
possible to find a field redefinition that calibrates the entropy density
$s_{X}$ to the Einstein density $s_{E}$.
For a $(p+2)$-black brane with $p\geq 2$ transverse dimensions, the entropy
for Einstein’s theory is $S_{E}\;=\;V_{\bot}{r_{h}^{p}}/{(4G_{E})}\;,$ where
$V_{\bot}$ is the transverse volume of the brane which includes all other
numerical factors and inverse factors of the AdS curvature radius. For a
general theory $X$ that extends Einstein gravity, the entropy is given by
Wald’s formula [22, 23, 24] and can be expressed as a correction to the
Einstein value:
$S_{X}\;=\;\frac{V_{\bot}r_{h}^{p}}{4G_{X}}\;=\;\frac{V_{\bot}r_{h}^{p}}{4G_{E}+\lambda\delta
G}\;+\;{\cal O}[\lambda^{2}]\;.$ (4)
Here, $G_{X}$ is the gravitational coupling for $X$ and we have fixed the
horizon radius of the brane since our interest is to compare the theories at a
fixed temperature. $\lambda$ is a parameter that controls the strength of the
correction to Einstein’s theory and $\delta G$ indicates the first-order (in
$\lambda$) shift in the coupling. One can determine $G_{X}$ and, hence,
$\delta G$ by calculating the Wald entropy as prescribed in [13] and then
Taylor expanding to any desired order of accuracy. Results are presented at
the lowest non-trivial order for simplicity and clarity.
Next, let us consider a conformal field redefinition of the metric such that
$g_{\mu\nu}\rightarrow{\tilde{g}}_{\mu\nu}=e^{\Omega}g_{\mu\nu}\;$ with
$\Omega=-\frac{2}{p}\lambda\delta G/G_{E}$. This field redefinition should be
accompanied by an appropriate rescaling $G_{X}\to G_{\widetilde{X}}$ to
preserve the form of the leading-order Lagrangian $(16\pi
G)^{-1}\sqrt{-g}{\cal R}$. The position of the horizon can be determined as
the largest root of the ratio $|g_{tt}/g_{rr}|$ and so will be left unchanged.
The entropy for the transformed theory is given by
$S_{\widetilde{X}}\;=\;\frac{{\widetilde{V}}_{\bot}\;r_{h}^{p}}{4G_{\widetilde{X}}}\;=\;\frac{\left(V_{\bot}-\lambda\frac{\delta
G}{G_{E}}\right)r_{h}^{p}}{4G_{E}}\;+\;{\cal O}[\lambda^{2}]\;,$ (5)
which, again, can be extended to any desired order in $\lambda$. The entropy
is invariant under the field redefinition, as it must be. It is also clear
from Eq. (5) that $S_{\widetilde{X}}/S_{E}={\widetilde{V}}_{\bot}/{V_{\bot}}$,
and so
$s_{\widetilde{X}}=s_{E}.$ (6)
This last result and the gauge–gravity duality implies that the entropy
densities of the dual field theories are also equal, while Eq. (3) tells us
that their shear viscosities satisfy $\eta_{\widetilde{X}}\geq\eta_{E}$.
Hence,
$\frac{\eta_{X}}{s_{X}}\;\geq\;\frac{\eta_{E}}{s_{E}}\;=\;\frac{1}{4\pi}\;,$
(7)
where the tildes have been dropped for what must be a field-redefinition-
invariant statement. We have thus proved the KSS bound for any consistent
extension of Einstein gravity and its field-theory dual.
Let us now discuss some examples. Obviously, in any theory which is equivalent
to Einstein gravity with simple enough matter interactions, $\eta/s=1/4\pi$.
This can occur for theories that contain only topological corrections such as
Gauss-Bonnet gravity in $4D$ and Lovelock gravity in higher even-numbered
dimensions, or for theories that can be brought into Einstein’s by a field
redefinition such as $f({\cal R})$ gravity.
To obtain a gravity theory without ghosts that extends Einstein gravity in a
non-trivial way, one can start with a ghost-free theory and then integrate out
some of the matter or gravity degrees of freedom in a consistent way. We have
chosen to discuss a simple model in this class: Einstein’s theory in $4+n$
dimensions, with the $n$ extra dimensions compactified on a torus of radius
$R$. From a $4$-dimensional point of view, everything is trivial in the far
infrared when the horizon radius is larger than the compactification scale or
$r_{h}\gg R$. But, when the energy scales are high enough to probe the
compactified dimensions or $r_{h}\ll R$, the correct description is the
higher-derivative theory that is obtained after the Kaluza–Klein (KK) modes of
the torus have consistently been integrated out.
From a $4D$ point of view, each extra dimension $i={1,2,\ldots n}$ induces an
infinite tower of massive KK modes with uniformly spaced masses $m_{k_{i}}\sim
k_{i}/R$ ($k_{i}=1,2,\ldots\infty$) for particles of spin-$0$, $1$ and $2$.
Only the spin-$2$ particles will be relevant to the shear ($xy$) channel of
the two-graviton propagator (1), and this contribution can be expressed as the
following sum:
$\left[{\cal D}_{KK}\right]^{\ y\ x}_{x\
y}\;\sim\;R^{2}G_{E}\sum_{i=1}^{n}\sum_{k_{i}=1}^{\infty}\frac{\rho_{k_{i}}(q^{2})}{R^{2}q^{2}-k_{i}^{2}}\;.$
(8)
The coefficients $\rho_{k_{i}}$ must be non-negative for this ghost-free
theory at all energy scales. Then, using the translational symmetry of the
torus, one finds that ${\cal D}_{KK}$ is vanishingly small for $q\ll 1/R$ (as
anticipated), whereas ${\cal D}_{KK}\sim R/q$ for $q\gtrsim 1/R$. In this
latter case, the KK contribution clearly dominates over the standard $1/q^{2}$
contributions. Recall that, for the hydrodynamic limit, the masses
contributing to Eq. (8) need to be small or $m\sim 1/R\ll T$, which is indeed
the case when $r_{h}\ll R$ since $r_{h}\sim 1/T$.
It is evident that, for scales $q\gtrsim 1/R$, the shear viscosity must
increase from its Einstein value. On the other hand, the entropy density can
always be computed in either $4+n$ dimensions or just $4$, with the same
result guaranteed. This is true by virtue of the compactified dimensions
making the same $R^{n}$ contribution to both the area density in the numerator
and the gravitational coupling in the denominator. Consequently, from a $4D$
point of view, $\eta/s$ saturates the bound (7) in the IR where the theory is
Einstein’s and then increases towards the UV due to the increase in $\eta$.
Our assertion is that the bound (7) has to hold for a ghost-free theory.
However, the converse is not true. It is possible, as demonstrated below, that
the bound holds for some theories with ghosts but not for others; apparently,
some ghosts are “friendlier” than others. For concreteness, let us discuss
$5D$ Gauss–Bonnet gravity. The Lagrangian density of this theory is ${\cal
L}\;=\;{\cal R}+12\;+\;\lambda G_{E}\left[{\cal R}_{abcd}{\cal
R}^{abcd}-4{\cal R}_{ab}{\cal R}^{ab}+{\cal R}^{2}\right]$ where we have set
the AdS curvature radius equal to unity and $\lambda$ is a dimensionless
constant. It is well known that this theory has ghosts for any value of
$\lambda$, as one can readily see by performing a field redefinition that
retains only the $4$-index (squared) correction $\sim{\cal R}_{abcd}{\cal
R}^{abcd}$ and then computing the complete propagator.
The graviton propagator is calculated as follows: We expand the metric
$g_{\mu\nu}\rightarrow g_{\mu\nu}^{(0)}+h_{\mu\nu}$ (with a superscript of
$(0)$ always denoting the $\lambda=0$ solution) and then calculate the
graviton kinetic terms which contain exactly two $h$’s and two derivatives.
From the formalism of [13] (in particular, Eq. (24)), these are ${\cal
L}_{kin}\;=\;\frac{1}{2}\left(\frac{\delta{\cal L}}{\delta{\cal
R}_{abcd}}\right)^{(0)}\nabla^{(0)}_{e}h_{ad}\nabla^{e(0)}h_{bc}\;.$ We have
neglected any inessential factors, as well as a term that makes no
contribution when coupled to a conserved source. The background quantities
have to be evaluated on the $\lambda=0$ solution, since each $h$ already
represents one order of $\lambda$.
The $\lambda=0$ solution is the well-known AdS $3$-brane, for which
$-g_{tt}=g_{rr}=r^{2}\left[r^{2}-\frac{r_{h}^{2}}{r^{2}}\right]$ (with
$r_{h}=1/(\pi T)$) and $g_{xx}=g_{yy}=g_{zz}=r^{2}$. It is also useful to note
that $({\cal R})^{(0)}=-20$ and $({\cal R}^{\ b}_{a})^{(0)}=-4\delta^{\
b}_{a}$. The calculation proceeds in a straightforward manner. Specializing to
the $h_{xy}$ gravitons of interest, we obtain ${\cal
L}_{kin}=-\frac{1}{4}\sum_{a\neq b}^{\\{x,y,z\\}}\left[\left(1-8\lambda
G_{E}+4\lambda G_{E}{\cal R}^{ab}_{\;\;\;ab}\right)h_{ab}\Box h^{ab}\right],$
where some boundary terms have been dropped, as well as a bulk term that is
not of the kinetic form, and the usual summation conventions should be
ignored. The leading-order term is just the standard graviton propagator prior
to any gauge fixing, and so we need to compare the sign of the higher-order
terms with this one. To proceed, let us evaluate the four-index Riemann tensor
on the horizon, although any other radial surface would serve just as well. In
this case, ${\cal R}^{xy}_{\;\;\;xy}=0$, with all other relevant possibilities
being redundant. So that the extremely simple end result is (with the
summation suppressed) ${\cal L}_{kin}=-\frac{1}{4}\left(1-8\lambda
G_{E}\right)h_{ab}\Box h^{ab}\;.$ Evidently, $\lambda>0$ induces a negative or
ghost contribution to the propagator. This agrees perfectly with the
observation that $\lambda>0$ Gauss–Bonnet models violate the KSS bound [10,
11]. The reason is clear: For $\lambda>0$, $\eta$ picks up a correction of
$-8\lambda$ while $s$ at the horizon is found to receive no such correction.
There is an important lesson that can be learned from the two examples. Had we
first integrated out the KK modes and then truncated the resulting theory, the
gravitational action would end up looking very similar to the just-discussed
Gauss–Bonnet model. In string theory, which is ghost free, when one integrates
out the massive modes and truncates the ensuing expansion, the leading-order
result is indeed a Gauss–Bonnet theory. It may well be that, for the
calculation of scattering amplitudes and some other physical quantities, such
a truncation is perfectly fine. However, since the ratio $\eta/s$ is
especially sensitive to the presence of ghosts, extra care must now be taken.
Acknowledgments: The research of RB was supported by The Israel Science
Foundation grant no 470/06. The research of AJMM is supported by the
University of Seoul. AJMM thanks Ben-Gurion University for their hospitality
during his visit.
## References
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* [4] CERES Collaboration, “Viscosity of the matter created in nucleus-nucleus collisions at the SPS measured via two-pion interferometry,” arXiv:0907.2799 [nucl-ex].
* [5] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200].
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* [9] D. T. Son and A. O. Starinets, “Viscosity, Black Holes, and Quantum Field Theory,” Ann. Rev. Nucl. Part. Sci. 57, 95 (2007) [arXiv:0704.0240 [hep-th]]
* [10] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity Bound Violation in Higher Derivative Gravity,” Phys. Rev. D 77, 126006 (2008) [arXiv:0712.0805 [hep-th]].
* [11] Y. Kats and P. Petrov, “Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory,” JHEP 0901, 044 (2009) [arXiv:0712.0743 [hep-th]].
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* [15] P. K. Kovtun and A. O. Starinets, “Quasinormal modes and holography,” Phys. Rev. D 72, 086009 (2005) [arXiv:hep-th/0506184].
* [16] N. Iqbal and H. Liu, “Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm,” Phys. Rev. D 79, 025023 (2009) [arXiv:0809.3808 [hep-th]].
* [17] R. Brustein and A. J. M. Medved, “The shear diffusion coefficient for generalized theories of gravity,” Phys. Lett. B 671, 119 (2009) [arXiv:0810.2193 [hep-th]].
* [18] V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz. 12, 447 (1970)].
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|
arxiv-papers
| 2009-08-11T07:48:55 |
2024-09-04T02:49:04.529908
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ram Brustein, A.J.M. Medved",
"submitter": "Ram Brustein",
"url": "https://arxiv.org/abs/0908.1473"
}
|
0908.1474
|
# A finite excluded volume bond-fluctuation model:
Static properties of dense polymer melts revisited
J.P. Wittmer jwittmer@ics.u-strasbg.fr Institut Charles Sadron, 23 rue du
Loess, 67037 Strasbourg Cédex, France A. Cavallo Institut Charles Sadron, 23
rue du Loess, 67037 Strasbourg Cédex, France T. Kreer Institut Charles
Sadron, 23 rue du Loess, 67037 Strasbourg Cédex, France J. Baschnagel
Institut Charles Sadron, 23 rue du Loess, 67037 Strasbourg Cédex, France A.
Johner http://www.ics-u.strasbg.fr/~etsp/welcome.php Institut Charles Sadron,
23 rue du Loess, 67037 Strasbourg Cédex, France
###### Abstract
The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo
algorithm for coarse-grained polymer chains where each monomer occupies
exclusively a certain number of lattice sites. In this paper we propose a
generalization of the BFM where we relax this constraint and allow the overlap
of monomers subject to a finite energy penalty $\varepsilon$. This is done to
vary systematically the dimensionless compressibility $g$ of the solution in
order to investigate the influence of density fluctuations in dense polymer
melts on various static properties at constant overall monomer density. The
compressibility is obtained directly from the low-wavevector limit of the
static structure factor. We consider, e.g., the intrachain bond-bond
correlation function, $P(s)$, of two bonds separated by $s$ monomers along the
chain. It is shown that the excluded volume interactions are never fully
screened for very long chains. If distances smaller than the thermal blob size
are probed ($s\ll g$) the chains are swollen according to the classical Fixman
expansion where, e.g., $P(s)\sim g^{-1}s^{-1/2}$. More importantly, the
polymers behave on larger distances ($s\gg g$) like swollen chains of
incompressible blobs with $P(s)\sim g^{0}s^{-3/2}$.
###### pacs:
05.10.Ln, 61.25.hk, 61.25.hp
## I Introduction
#### The bond-fluctuation model.
The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo
(MC) algorithm for coarse-grained polymer chains where each monomer occupies
exclusively a certain number of lattice sites on a simple cubic lattice
Carmesin and Kremer (1988); Deutsch and Dickman (1990); Deutsch and Binder
(1991). It was proposed in 1988 by Carmesin and Kremer Carmesin and Kremer
(1988) as an alternative to single-site self-avoiding walk models, which
retains the computational efficiency of the lattice without being plagued by
severe ergodicity problems. The key idea is to increase the size of a monomer
which now occupies a whole unit cell of the lattice, as illustrated in Fig. 1.
The multitude of possible bond lengths and angles allows a better
representation of the continuous-space behavior of real polymer solutions and
melts. The BFM algorithm has been used for a huge range of problems addressing
the generic behavior of long polymer chains of very different molecular
architectures and geometries: statics Deutsch and Dickman (1990); Deutsch and
Binder (1991); Paul et al. (1991a); Müller and Paul (1994); Wilding and Müller
(1994); Müller et al. (2000a); Stukan et al. (2002); Wittmer et al. (2004);
Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al. (2007b); Meyer
et al. (2008) and dynamics Paul et al. (1991b); Wittmer et al. (1992); Kreer
et al. (2001); Mattioni et al. (2003); Azuma and Takayama (1999) of linear
chains and rings Müller et al. (1996); Müller et al. (2000b), polymer blends
and interfaces Müller (1995, 1999); Cavallo et al. (2003), gels and networks
Sommer and Saalwächter (2005), glass transition Binder et al. (2003),
(co-)polymers at surfaces Werner et al. (1999), brushes Wittmer et al. (1994);
Kopf et al. (1996); Wittmer et al. (1996), thin films Khalatur et al. (2000);
Mischler et al. (2001); Cavallo et al. (2005), equilibrium polymers Wittmer et
al. (1998); Wittmer et al. (2007c); Beckrich et al. (2007) and general self-
assembly Cavallo et al. (2006, 2008), …. For recent reviews see Refs.
Baschnagel et al. (2004); Müller (2005).
#### A BFM version allowing a systematic compressibility variation.
As sketched in Fig. 1, we propose here a generalization of the BFM where we
relax the no-overlap constraint and allow the overlap of monomers subject to a
finite energy penalty $\varepsilon$. This is done to vary systematically the
strength of density fluctuations in dense solutions and melts to study their
influence on static and dynamical properties. More specifically, we want to
test the standard perturbation theory of weakly interacting three-dimensional
polymer melts Doi and Edwards (1986) and to verify whether certain long-range
correlations Wittmer et al. (2004); Wittmer et al. (2007a); Beckrich et al.
(2007); Wittmer et al. (2007b); Meyer et al. (2008), which have been found
recently for incompressible melts, are also present in melts with finite
compressibility. We will see that this is indeed the case if one considers
properties on scales larger than the screening length $\xi\approx bg^{1/2}$ of
the density fluctuations where $b$ indicates the effective bond length of the
chain and $g$ the number of monomers spanning the “thermal blob” de Gennes
(1979); Doi and Edwards (1986); foo (a). Interestingly, $g$ is related to the
isothermal compressibility of the solution $\kappa_{\text{T}}$ Semenov and
Johner (2003); Semenov and Obukhov (2005); Beckrich et al. (2007), i.e. the
thermodynamic property which measures the strength of the density
fluctuations. It can thus be determined directly experimentally or in a
computer simulation from the low-wavevector limit of the total monomer
structure factor McQuarrie (2000); Frenkel and Smit (2002); foo (b)
$G(q)\equiv\frac{1}{n_{\text{mon}}}\sum_{n,m=1}^{n_{\text{mon}}}\exp\left(-i\bm{q}\cdot(\bm{r}_{n}-\bm{r}_{m})\right)\stackrel{{\scriptstyle
q\to 0}}{{\Longrightarrow}}g\equiv T\ \kappa_{\text{T}}\rho,$ (1)
with $\bm{q}$ being the wavevector, $\bm{r}_{i}$ the position of monomer $i$,
$n_{\text{mon}}=\rho V$ the total monomer number, $\rho$ the monomer number
density, $V=L^{3}$ the volume of the system and $T\equiv 1/\beta$ the
temperature. (Boltzmann’s constant $k_{\text{B}}$ is set to unity throughout
this paper.) Due to the definition given in Eq. (1), $g$ is also called
“dimensionless compressibility” foo (c). We will show that the variation of
the operational parameter $x=\varepsilon/T$ foo (d), characterizing the
strength of the overlap penalty, allows us to scan $g(x)$ over four orders of
magnitude foo (a). This puts us in the position to test various theoretical
predictions which are sketched below.
#### Thermodynamic properties for $x\ll 1$.
To characterize the proposed soft BFM model we will first investigate
thermodynamic properties such as the mean overlap energy per monomer $e$, the
excess chemical potential $\mu_{\text{ex}}$ or the (already mentioned)
dimensionless compressibility $g$ as functions of $x$. In the limit of weak
overlap penalties these properties have been calculated long ago by Edwards
theory Doi and Edwards (1986) and we will compare our numerical results with
his predictions. For later reference we postulate here the contributions to
the free energy per monomer $f(\beta)$ relevant for this comparison
$\displaystyle\beta f(\beta)$ $\displaystyle=$ $\displaystyle\beta
e_{\text{self}}$ (2) $\displaystyle+$
$\displaystyle\frac{1}{N}\left(\log(\rho/N)-1\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2}v(x)\rho\
\underline{-\frac{1}{12\pi}\frac{1}{\xi(x)^{3}\rho}}\ +\ldots.$
The first term is due to the constant intrachain self-energy discussed in Sec.
III.1. It is due to the reference energy chosen in this study and it is not
accessible experimentally foo (e). The second and third line of Eq. (2) can be
readily obtained from the literature, e.g., by integrating the osmotic
pressure given by Eq. (5.45) or Eq. (5.II.5) of Doi and Edwards (1986) with
respect to the density $\rho$. (The first line can be considered as the
integration constant with respect to this integration.) The second line
represents the translational invariance of monodisperse chains of length $N$
(van’t Hoff’s law). Due to this contribution the compressibilities depend in
general on $N$ as will be discussed in Sec. III.4. The (bare) excluded volume
interaction between the monomers is accounted for by the first term in the
third line with $v(x)$ being the second virial coefficient of a solution of
unconnected monomers. The underlined term represents the leading correction to
the previous term due to the fact that the monomers are connected by bonds
summing over the density fluctuations to quadratic order. As one expects de
Gennes (1979), the corresponding correlations of the density fluctuations
reduce the free energy by about one $k_{\text{B}}T$ per thermal blob of volume
$\xi^{3}$. Consistently with Refs. Doi and Edwards (1986); Semenov and Obukhov
(2005); Meyer et al. (2008) the correlation length $\xi$ of the density
fluctuations has been defined here as
$\xi^{2}(x)\equiv b^{2}(x)g(x)/12\approx b^{2}(x)/(12v(x)\rho).$ (3)
Note that $g(x)$ and $b(x)$ refer respectively to the dimensionless
compressibility and the effective bond length of asymptotically long chains
foo (a, f). The density fluctuation contribution to the free energy will be
demonstrated numerically from the scaling of the specific heat $c_{\text{V}}$
with respect to overlap strength $x$ and density $\rho$ (Sec. III.2). We note
finally that Eqs. (2) and (3) are supposed to apply as long as
$G_{\text{z}}(x)\equiv\frac{1}{\sqrt{g(x)}\ b^{3}(x)\rho}\ll 1,$ (4)
with the Ginzburg parameter $G_{\text{z}}$ being the small parameter of the
perturbation theory foo (g). This restricts — as we shall see — the related
predictions to overlap penalties with $x\ll 1$.
#### Conformational properties of asymptotically long chains for all $x$.
This paper aims ultimately to characterize an important intrachain
conformational property, the size of chain segments of arc-length $s$ in very
long chains where finite-$N$ effects may be neglected foo (h). Specifically,
we will investigate the mean-square end-to-end distance
$R^{2}(s)=\left<\left(\bm{r}_{m=n+s}-\bm{r}_{n}\right)^{2}\right>$ where the
average is performed over all possible pairs of monomers $(n,m=n+s)$. The
total chain end-to-end distance is $R_{\text{e}}(N)\equiv R(s=N-1)$. If
appropriately plotted Wittmer et al. (2007b), the segment size $R(s)$ will
allow us to obtain an estimation of the effective bond length $b(x)$ by
extrapolation foo (a). Our numerical results will be compared with an
analytical prediction obtained by a standard one-loop perturbation calculation
very similar to the analytic results already presented elsewhere Wittmer et
al. (2004); Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al.
(2007b); Meyer et al. (2008). Details of the calculation for general soft
melts will be presented in a future paper Wittmer et al. (2009). Focusing in
this paper mainly on computational issues, we only quote here the “key
relation” put to the test
$1-\frac{R^{2}(s)}{b^{2}s}=\frac{c_{\text{s}}}{g^{1/2}}\left(\frac{1}{\sqrt{u}}-\frac{\sqrt{\pi/8}}{u}\left(1-e^{2u}\mbox{erfc}(\sqrt{2u})\right)\right),$
(5)
where we have used $u=s/g(x)$ for the reduced curvilinear arc-length and
$\mbox{erfc}(x)$ for the complementary error function Abramowitz and Stegun
(1964). The correction to Gaussianity expressed by the r.h.s. terms of the key
prediction is always positive and corresponds to a weak swelling of the chain.
The “swelling coefficient” $c_{\text{s}}=\sqrt{24/\pi^{3}}/\rho b^{3}$ Wittmer
et al. (2007b) measures the strength of the effect. While Eq. (2) fails for
large $x$ where the Ginzburg parameter $G_{\text{z}}$ becomes of order unity,
Eq. (5) is supposed to hold for all $x$ provided that the considered segment
length $s$ is large enough. As explained in Sec. II of Ref. Wittmer et al.
(2007b), the relevant small parameter of the perturbation theory is here the
so-called “correlation hole potential” of the chains
$u(s)\approx G_{\text{z}}\sqrt{g/s}\approx 1/\sqrt{s}\ll 1\mbox{ for }s\gg 1$
(6)
rather than $G_{\text{z}}$. For a discussion of chain end effects in
incompressible melts see Ref. Wittmer et al. (2007b).
#### Check of limiting behavior.
Although no formal derivation of Eq. (5) is given, we verify briefly whether
the suggested key prediction is reasonable by discussing the limits of large
and small reduced arc-length. We remind first that according to Flory’s
“ideality hypothesis” Flory (1949); de Gennes (1979), polymer chains in the
melt are thought to display Gaussian statistics for segment sizes somewhat
larger than the persistence length de Gennes (1979); Doi and Edwards (1986);
Flory (1949) which implies that the r.h.s. of Eq. (5) is traditionally assumed
to vanish (exponentially) for finite $s$. At variance to this, the expansion
of the complementary error function Abramowitz and Stegun (1964) for $u=s/g\gg
1$ yields
$1-\frac{R^{2}(s)}{b^{2}s}\approx\frac{c_{\text{s}}}{\sqrt{s}},$ (7)
suggesting in fact that corrections to Gaussianity must be taken into account
for all finite $s$. As one expects, however, Gaussianity is still recovered in
three dimensions if $s\to\infty$. (Incidentally, this does not hold for
effectively two-dimensional melts as may be seen from the discussion of
ultrathin polymer films in Refs. Semenov and Johner (2003); Cavallo et al.
(2005); Meyer et al. (2007).) Most remarkably, the explicit compressibility
dependence drops out for large $u$ for all $g(x)$ provided that the chains are
long enough, such that $g\ll s\ll N$. Eq. (7) is precisely the relation which
has been discussed in detail both theoretically and numerically Wittmer et al.
(2004); Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al. (2007b)
for highly incompressible melts with full excluded volume interactions
($x=\infty$). Hence, this is the expected limiting behavior if the polymer
chains are renormalized in terms of an incompressible packing of thermal blobs
with $g$ monomers. In the opposite limit of small reduced arc-lengths,
$u=s/g\ll 1$, the expansion of Eq. (5) yields
$1-\frac{R^{2}(s)}{b^{2}s}\approx\frac{c_{\text{s}}}{\sqrt{g}}\left(\sqrt{\frac{\pi}{2}}-\frac{4}{3}\sqrt{u}\right).$
(8)
This is consistent with the classical expansion result of the chain size in
terms of the “Fixman parameter” $z\approx
v\sqrt{s}/b^{3}\approx\sqrt{u}/(\sqrt{g}b^{3}\rho)$ Doi and Edwards (1986);
foo (c) since the deviations from ideality expressed by the last term of Eq.
(8) become then proportional to $-c_{\text{s}}\sqrt{u/g}\approx-z$, in
agreement with the leading correction term for the total chain size
$R_{\text{e}}(N)$ presented in textbooks foo (i).
#### Outline.
In Section II the algorithm is introduced and some technical details are
discussed. First we summarize the classical BFM without monomer overlap (Sec.
II.1), which has been used in previous work Wittmer et al. (2007b) providing
the start configuration for the present study, and introduce then its
generalization with finite overlap penalty (Sec. II.2). The central Section
III presents our numerical results starting with the thermodynamic properties
(Secs. III.1-III.4), in particular the dimensionless compressibility $g(x)$.
We characterize then (Secs. III.5-III.7) various intrachain properties, as for
instance the effective bond length $b(x)$ or the bond-bond correlation
function $P(s)$, as functions of $g(x)$. A synopsis of our results is
presented in Section IV.
## II Algorithm and technical details
### II.1 Bond fluctuation model without monomer overlap
#### A lattice Monte Carlo scheme for topology conserving polymers.
We have used the three-dimensional version of the bond fluctuation model
Wittmann and Kremer (1990); Deutsch and Dickman (1990); Deutsch and Binder
(1991); Paul et al. (1991a); Wittmer et al. (1992) where each effective
coarse-grained monomer is represented by a cube of eight adjacent sites on a
simple cubic lattice, as illustrated in Fig. 1. Even the partial overlap of
monomers is forbidden. The lattice constant $a$ is naturally chosen as the
unit length. Polymers of length $N$ consist of $N$ cubes connected by $N-1$
bonds, as shown in the sketch for $N=3$. These bonds are taken from the set
$\left\\{P(2,0,0),P(2,1,0),P(2,1,1),P(2,2,1),P(3,0,0),P(3,1,0)\right\\}$ (9)
of allowed bond vectors, where $P$ stands for all permutations and sign
combinations of coordinates. This corresponds to 108 different bond vectors of
5 possible bond lengths ($2$, $\sqrt{5}$, $\sqrt{6}$, $3$, $\sqrt{10}$) and
$100$ possible angles between consecutive bonds. The smallest $13$ angles do
not appear for the classical BFM without monomer overlap, because excluded
volume forbids sharp backfolding of bonds. If only local Monte Carlo (MC)
moves of the monomers to the six nearest neighbor sites are performed — called
“L06” moves Wittmer et al. (2007b) — this set of vectors ensures automatically
that polymer chains cannot cross. Topological constraints, e.g. in ring
polymers Müller et al. (1996), hence are conserved and the polymer dynamics
may be expected to be of reptation type Paul et al. (1991a); Müller et al.
(2000b); Kreer et al. (2001); Wittmer et al. (2007b). It is this fact which
has originally motivated the choice of allowed bonds. We keep it for
consistency with previous work although the non-crossing constraint is
irrelevant for the present study. Note that the classical BFM algorithm with
L06 moves is strictly speaking not ergodic, since some (thermodynamically
irrelevant) configurations may be easily constructed which are not accessible
starting from an initial configuration of stretched linear chains Baschnagel
et al. (2004).
#### Obtaining athermal start configurations with topology violating moves.
The algorithm is up to now athermal and the only control parameters are the
monomer density $\rho$ and the chain size $N$. Melt conditions are realized
for $\rho=0.5/8$, where half of the lattice sites are occupied Paul et al.
(1991a). We use (if not stated otherwise) periodic simulation boxes of linear
dimension $L=256$ which contain $n_{\text{mon}}=\rho L^{3}=2^{20}\approx
10^{6}$ monomers. This large system size allows to eliminate finite-size
effects even for the longest chain lengths studied. Our simulations have been
carried out by a mixture of local, slithering snake Kron (1965); Wall and
Mandel (1975); Mattioni et al. (2003), and double-bridging MC moves
Karayiannis et al. (2002); Banaszak and de Pablo (2003); Auhl et al. (2003);
Baschnagel et al. (2004) which allow us to equilibrate polymer melts with
chain lengths up to $N=8192$. Instead of the more realistic but very slow L06
dynamical scheme we make so-called “L26” jump attempts to the 26 sites of the
cube surrounding the current monomer position. This permits the crossing of
chains which dramatically speeds up the dynamics, expecially for long chains
$(N>512)$. Details of the equilibration procedure and possible caveats are
discussed in Ref. Wittmer et al. (2007b). We stress that if these topology
violating MC moves are included all possible configurations become accessible,
i.e. the BFM becomes fully ergodic. Concerning the static properties ergodic
and non-ergodic BFM versions are, however, practically equivalent. This has
been confirmed by comparing various static properties and by counting the
number of monomers which become “blocked” once one returns to the original
local L06 moves.
### II.2 Bond fluctuation model with finite excluded volume penalty
#### The model Hamiltonian.
Fig. 1 shows how finite energy penalties are introduced. The overlap of two
cube corners on one lattice site ($N_{\text{ov}}=1$) corresponds to an energy
cost of $\varepsilon/8$, the full overlap of two monomers ($N_{\text{ov}}=8$)
to an energy $\varepsilon$. More generally, with $N_{\text{ov}}$ being the
total number of interacting cube corners the total interaction energy of a
configuration is
$E=\frac{\varepsilon}{8}N_{\text{ov}}.$ (10)
With the energies of the final ($E_{\text{f}}$) and the initial configurations
($E_{\text{i}}$) we accept the MC move according to the Metropolis criterion
Baschnagel et al. (2004); Landau and Binder (2000) with probability
$\mbox{min}(1,\exp[-(E_{\text{f}}-E_{\text{i}})/T])$. We set arbitrarily
$\varepsilon=1$ and vary the ratio $x=\varepsilon/T$ foo (d) starting from
$x=\infty$ corresponding to the athermal classical BFM and systematically
increase the temperature $T$ as shown in Table 1 for unconnected monomers
($N=1$).
#### Second virial coefficient.
To illustrate this interaction we indicate already here the second virial of
an imperfect gas of unconnected monomers, $v=\int
d{\bm{\delta}}(1-e^{-E({\bm{\delta}})/T})$, which is shown below to be useful
for roughly characterizing the effective strength of the potential.
$\bm{\delta}$ stands for a possible lattice vector between the centers of two
interacting cubes. It is easy to see that there are 8 vectors corresponding to
$N_{\text{ov}}=1$ (as in Fig. 1), 12 to $N_{\text{ov}}=2$ (overlap of two cube
corners), 6 to $N_{\text{ov}}=4$ (overlap of two faces), and 1 to
$N_{\text{ov}}=8$ (full overlap). This leads to a second virial
$v=8\times(1-\exp(-x/8))+12\times(1-\exp(-x/4))+6\times(1-\exp(-x/2))+1\times(1-\exp(-x))$
(11)
given in units of the lattice cube volume $a^{3}$. We note that the second
virial becomes constant, $v=27$, in the low temperature limit ($x\gg 1$)
corresponding to the classical athermal BFM result Deutsch and Dickman (1990).
In the opposite high temperature limit ($x\ll 1$) it decays as
$v\approx 8x-\frac{27}{16}x^{2}.$ (12)
#### Implementation of the algorithm.
We briefly explain the implementation of BFM chains with the soft overlap
Hamiltonian, Eq. (10). Following Paul et al. (1991a, b); Wittmer et al.
(1992); Kreer et al. (2001); Mattioni et al. (2003) it is convenient to keep
one list of the monomer positions in absolute space (since we are also
interested in dynamical properties) and one for the corresponding lattice
positions. We identify each of the 108 allowed bonds of the set [Eq. (9)] with
a unique bond index and keep a list of these indices. This allows to verify
rapidly whether an attempted bond vector is acceptable. Since the bond index
(being less than 128) can be encoded as a byte, this compressed list is stored
when we write down the configuration ensemble for further analysis. Additional
lists allow to handle efficiently the periodic boundary conditions, the change
of a bond index for a given monomer move and the local interactions relevant
for local L06 or L26 moves.
Since the soft overlap Hamiltonian allows the occupation of a lattice site by
more than one monomer, it is not possible to use a compact boolean occupation
lattice (corresponding to a spin $\sigma=0$ and $\sigma=1$ for an empty or
filled lattice site) or an integer lattice filled with the monomer indices as
in previous implementations. Instead we have mapped Eq. (10) onto a Potts spin
model Landau and Binder (2000)
$E=\frac{1}{2}\sum_{\bm{r}}\sigma(\bm{r})\sum_{\bm{\delta}}J(\bm{\delta})\sigma(\bm{r}+\bm{\delta})-\frac{1}{2}\varepsilon
n_{\text{mon}}$ (13)
with constant monomer number
$n_{\text{mon}}=\sum_{\bm{r}}\sigma(\bm{r})\stackrel{{\scriptstyle!}}{{=}}L^{3}\rho$
using a Wigner-Seitz representation of the cubic lattice following Müller
Müller (1995, 2005). In this representation an integer spin variable
$\sigma(\bm{r})$ counts the number of BFM monomers ($\sigma=0,1,2,\ldots$)
with cubes centered at a Wigner-Seitz lattice position $\bm{r}$. In other
words, each cube is not represented by 8 lattice entries for the cube corners,
but just by one for its center. Since we have now to compute the interaction
between cube centers instead of cube corners, the coupling constant $J$
characterizing the interaction between two spins depends only on the relative
distance $\bm{\delta}$:
$J(\bm{\delta})=\varepsilon\left\\{\begin{array}[]{ll}1/8&\mbox{if
$\bm{\delta}=P(1,1,1)$ (cube corners),}\\\ 1/4&\mbox{if $\bm{\delta}=P(1,1,0)$
(cube edges),}\\\ 1/2&\mbox{if $\bm{\delta}=P(1,0,0)$ (cube faces),}\\\
1&\mbox{if $\bm{\delta}=P(0,0,0)$ (full overlap),}\\\
0&\mbox{otherwise.}\end{array}\right.$ (14)
Since the interaction is still short-ranged and the values of $J$ are readily
tabulated, this remains an efficient rendering of the monomer interactions.
Note that the first term on the r.h.s. of Eq. (13) contains a constant self-
interaction contribution of the $n_{\text{mon}}$ monomers with themselves for
$\bm{\delta}=\bm{0}$, which is substracted by the second term foo (j).
#### Equilibration and system properties of high-molecular melts.
As already stated we have used as start configurations the equilibrated BFM
configurations without monomer overlap from our computationally much more
expensive previous studies obtained with topology violating moves Wittmer et
al. (2007b). Since the soft BFM simulations are also ergodic, these are the
relevant reference configurations. Starting with these configurations we
increase the temperature. As one may expect, the configurational properties
essentially are found unchanged for low temperatures ($x\gg 5$). Local L26
moves need to be added to global slithering-snakes moves for $x\geq 1$.
Otherwise the pure snake motion will become ineffective as it is well known
from a previous study of the snake dynamics without overlap Mattioni et al.
(2003). Simple slithering-snake moves without additional local moves are
sufficient for smaller $x$. We have crosschecked our results in this regime
for $N=2048$ and $N=8192$ using boxes of linear size $L=512$ by starting our
simulations with Gaussian chains at $x=0$ and increasing then the penalty.
Tables 2 and 3 present some system properties obtained for our reference
density $\rho=0.5/8$. Averages are performed over all chains and at least 100
configurations. Table 2 summarizes the properties extrapolated for
asymptotically long chains. Similar information is given in Table 3 for a
constant penalty $x=1$ as a function of chain length $N$. Density effects have
been studied only briefly for chains of length $N=8192$ and weak overlap
penalties ($x\ll 1$). This has been done to investigate the intrachain
contributions to the mean energy. We begin the discussion of our numerical
results by addressing this issue.
## III Computational results
### III.1 The mean overlap energy
#### Qualitative behavior.
From the numerical point of view the simplest thermodynamic property to be
investigated here is the mean interaction energy per monomer,
$e=\left<E\right>/n_{\text{mon}}$, due to the Hamiltonian, Eq. (10). Fig. 2
presents the dimensionless energy $y=e/\varepsilon$ as a function of the
reduced overlap penalty $x=\varepsilon/T=\varepsilon\beta$ foo (d) for melts
at monomer number density $\rho=0.5/8$ and various chain lengths $N$ as
indicated. We increase the temperature $T$ from the right to the left starting
with configurations obtained recently Wittmer et al. (2007b) for the classical
athermal BFM. As one expects, the interaction energy increases exponentially
for small $T$ and levels off for large $T$ where chains and their monomers
freely overlap ($x\ll 1$). The data for unconnected beads ($N=1$) represented
by the filled spheres and polymer chains ($N\gg 1$) are broadly speaking
similar, especially for large overlap penalties, $x>1$. Interestingly, the
mean energy of polymer melts increases more strongly for $x\ll 1$ as can be
seen better from the log-linear data representation chosen in the inset of
Fig. 2. Also shown in the inset is the mean intrachain self-energy per chain
monomer $e_{\text{self}}$ (filled triangles) obtained for the largest chain
length $N$ available for a given $x$. As can be seen also from Table 2, about
half of the energy of polymer melts for all $x$ is due to these intrachain
interactions. For $x\ll 1$ the self-energy becomes
$e_{\text{self}}/\varepsilon\approx 0.18$ which is exactly the observed energy
difference between polymer and bead systems.
#### Second virial contribution.
Before addressing this point let us first consider the energy of unconnected
soft BFM beads for which the second virial coefficient $v(x)$ has been given
in Eq. (11). Since $e=\partial_{\beta}(\beta f(\beta))$ the mean energy per
bead becomes to leading order McQuarrie (2000)
$y\approx\frac{1}{2}\rho\frac{\partial v(x)}{\partial
x}=\frac{\rho}{2}\left(\exp(-x/8)+3\exp(-x/4)+3\exp(-x/2)+\exp(-x)\right)$
(15)
corresponding to the first term in the third line of Eq. (2). Eq. (15) is
represented by the dashed line in Fig. 2. It corresponds to a Arrhenius
behavior with $y\approx\rho\exp(-x/8)/2$ (dash-dotted line) in the low
temperature region and, as expected, to $y\rightarrow\frac{1}{2}8\rho=4\rho$
for large temperatures. This simple formula predicts well the bead data over
the entire range of $x$ (underestimating slightly the mean energy at $x\approx
10$) and yields also a remarkable fit for polymer chains with larger overlap
penalties.
#### Self-energy in the high temperature limit.
The above-mentioned energy difference between polymer chains and beads for
$x\ll 1$ has been accounted for by the first free energy contribution
indicated in Eq. (2). This contribution is further investigated in Fig. 3
presenting data for such a high temperature ($x=0.001$) that the entropy
dominates essentially all conformational properties. The self-energy of a
chain is thus given by the probability $p(s,\bm{\delta})$ that a random walk
of $s$ BFM bonds returns to a relative position $\bm{\delta}$ with respect to
a reference monomer at $\bm{r}$. The self-energy per monomer is then
$e_{\text{self}}=\frac{2}{N}\sum_{\bm{\delta}}\sum_{s=2}^{N-1}(N-s)J(\bm{\delta})p(s,\bm{\delta})$
(16)
where the first sum runs over all positions with non-vanishing coupling
constant $J(\bm{\delta})$ as defined in Eq. (14). The probability
$p(s,\bm{\delta})$ and the weights $J(\bm{\delta})p(s,\bm{\delta})$ can be
tabulated in principle for small $s$. Since the return probability decreases
strongly with $s$, these model-specific small-$s$ values dominate the
integral, Eq. (16). As can be seen from the inset of Fig. 3 for single chains
(corresponding to an overall density $\rho=0$) the self-energy per monomer
becomes $e_{\text{self}}\approx 0.18\varepsilon$ for large $N$. The weak chain
length dependence visible in the panel stems from the upper integration
boundary over the Gaussian return probability which leads to a chain length
correction linear in $t\equiv 1/\sqrt{N-1}$. This is indicated by the bold
line presented in the panel. Also shown in the panel are energies for our
reference density $\rho=0.5/8$. They are shifted vertically by the mean field
energy $4\rho$ assuming that densities fluctuations of different chains do not
couple. The main panel presents the mean energy $e$ and the mean self-energy
$e_{\text{self}}$ as functions of the density $\rho$ for chains of length
$N=8192$. The self-energy (triangles) stays essentially density-independent.
The total interaction energy sums over the self-energy and mean-field energy
contributions as shown by the dashed line. The self-energy contribution can
only be neglected for very large densities corresponding to volume fractions
larger than unity.
#### Temperature dependence in the high temperature limit.
Summarizing Eqs. (2), (3) and (12) the energy per bead should scale to leading
order in $x$ as
$y\approx 0.18+4\rho-\
\underline{\frac{24^{3/2}}{\pi}\frac{\sqrt{x\rho}}{l^{3}\rho}}+\ldots\mbox{
for }x\ll 1$ (17)
where the two $x$-independent contributions have already been discussed above.
The underlined term stems from the density fluctuation contribution for long
polymer chains predicted by Edwards Doi and Edwards (1986) indicated in Eq.
(2). Here we have approximated the effective bond length $b(x)$ by the mean-
squared bond length $l\sim x^{0}$. As can be seen from Table 2 this
approximation (further discussed in Sec. III.6) is justified for $x\ll 1$. Eq.
(17) is indicated by the bold line in the inset of Fig. 2. It yields a
reasonable description of the temperature dependence of the mean energy for
small $x$. Since the energy is dominated by the two constant contributions to
Eq. (17) for $x\leq 0.001$ and since higher expansion terms become relevant
for $x>0.1$, the predicted $\sqrt{x}$-decay corresponds unfortunately only to
a small $x-$regime. In order to show that it is indeed the density fluctuation
term which dominates the temperature dependence for $x\ll 1$ we will consider
in the next paragraph the specific heat $c_{\text{V}}$, i.e. the second
derivative of the free energy with respect to $\beta$.
### III.2 Energy fluctuations
#### Specific heat.
The fluctuations of the interaction energy are addressed in Fig. 4 displaying
the enthalpic contribution to the specific heat per monomer,
$c_{\text{V}}=-\beta^{2}\partial^{2}_{\beta}(\beta
f(\beta))=\frac{1}{T^{2}}\left(\left<E^{2}\right>-\left<E\right>^{2}\right)/n_{\text{mon}}$
McQuarrie (2000). Using again the second virial of soft BFM beads, Eq. (11),
one obtains the simple estimate for the specific heat
$c_{\text{V}}=\frac{\rho}{2}x^{2}\left(\frac{1}{8}\exp(-x/8)+\frac{3}{4}\exp(-x/4)+\frac{3}{2}\exp(-x/2)+\exp(-x)\right)$
(18)
represented by the dashed line. In the large-$x$ limit, this corresponds to
the exponential decay, $c_{\text{V}}\approx\rho x^{2}\exp(-x/8)/16$, indicated
by the dash-dotted line. For barely interacting beads ($x\ll 1$), Eq. (18)
yields a power-law limiting behavior, $c_{\text{V}}\approx\frac{27}{16}\rho
x^{2}\sim 1/T^{2}$. As can be seen from the plot, Eq. (18) predicts the energy
fluctuations of BFM beads for essentially all $x$, slightly underestimating
again the maximum of $c_{\text{V}}$ at $x\approx 10$. Since the specific heat
for beads and polymer chains is similar for $x>1$, the virial formula is also
good for polymer chains in this limit.
#### High temperature limit for polymer melts.
Strong chain length effects are, however, visible for high temperatures ($x\ll
1$) where the specific heat is found to increase monotonously with $N$. This
can better be seen from the inset where the specific heat is plotted as a
function of the reduced chain length $u=N/g$ with $g$ being the dimensionless
compressibility determined below in Sec. III.4. (Since $e$ and $c_{\text{V}}$
correspond to different derivatives of the free energy $f$ with respect to
$\beta$, there is obviously no thermodynamic inconsistency in the finding that
$c_{\text{V}}$ reveals much larger chain length effects than $e$.) For large
chains with $u\gg 1$ this increase levels off at a chain length independent
envelope
$c_{\text{V}}\approx\frac{24\sqrt{6}}{\pi}\frac{\rho^{1/2}}{l^{3}}x^{3/2}N^{0}+\ldots$
(19)
as anticipated by the density fluctuation contribution predicted in Eq. (2).
In contrast to Eq. (17) for the mean energy the density fluctuation term does
now correspond to the leading contribution to the numerically measured
property. This increases the range where the density fluctuation contribution
can be demonstrated to over three decades in $x$. Eq. (19) is indicated by the
bold lines in the main panel and the inset of Fig. 4.
#### Scaling with chain length $N$.
We have still to clarify the scaling observed for $u=N/g(x)\ll 1$ in the inset
of Fig. 4. Chains which are smaller than the thermal blob ($u\ll 1$) behave
obviously as random walks and the density fluctuations decouple from the
interaction strength $x$. Due to this factorization the specific heat for
these short chains must scale as $x^{2}$, just as for beads. This is shown in
the main figure for $N=16$ (thin solid line). Consistency with Eq. (19)
implies the scaling $c_{\text{V}}(x,N)\approx x^{3/2}\rho^{1/2}h(u)$ with
$h(u)$ being a universal function scaling as $h(u)\sim u^{0}$ in the large-$u$
limit. Since $c_{\text{V}}\sim x^{2}$ and $g(x)\sim 1/(x\rho)$ (as shown in
Sec. III.4) for $u\ll 1$, it follows that $h(u)\approx u^{1/2}$ as confirmed
by the dashed line indicated in the inset. Hence, $c_{\text{V}}\sim\rho
x^{2}N^{1/2}$ for $u\ll 1$.
### III.3 Chemical potential
#### Scaling of the chemical potential.
Fig. 5 presents the excess chemical potential per monomer,
$y\equiv\mu_{\text{ex}}/TN$, obtained using thermodynamic integration (as
explained below) for three chain lengths $N=1$, $64$, and $2048$ as functions
of the overlap penalty $x=\varepsilon/T$. As one expects, the chemical
potential increases first linearly with $x$ and then levels off. Chain length
effects are again small on the logarithmic scale chosen in the plot foo (k).
For large $x$ the chemical potential becomes slightly larger for beads
($y\approx 2.64$) than for long chains where $y\approx 2.1$ (dash-dotted
line). That the chemical potential of polymer chains is reduced compared to
melts of unconnected beads is of course expected for all $x$ due to the
(effectively) attractive bond potential. For weak interactions this reduction
should be described by the density fluctuation contribution to the free energy
[Eq. (2)] which corresponds to an excess chemical potential
$y=\frac{\partial(\beta f(\beta)\rho)}{\partial\rho}\approx v(x)\rho\left(1-\
\underline{\frac{3\sqrt{3}}{\pi}\frac{(v(x)\rho)^{1/2}}{b(x)^{3}\rho}}+\
\ldots\right)\mbox{ for }x\ll 1$ (20)
with $v(x)$ being the second virial of unconnected beads. The dashed line in
Fig. 5 presents the leading contribution $v(x)\rho$ for unconnected beads, the
bold line in addition the underlined connectivity contribution given in Eq.
(20). It turns out that the simple second virial approximation provides a much
better fit of the data over the entire $x$-range than the full prediction.
(The weak underestimation of the chemical potential for $x>10$ must be
attributed to higher order correlations relevant in this limit.) That the
density fluctuation contribution overestimates the reduction of the chemical
potential for $x>1$ is in agreement with Eq. (4) and the Ginzburg parameters
indicated in Table 2. Hence, Eq. (20) in principle can be tested only for
$x\ll 1$. Unfortunately, in this limit the relative correction, scaling as
$\sqrt{x/\rho}$, becomes too small to allow a fair test of the theory. A
systematic variation of the density and an improved numerical accuracy of the
chemical potentials measured are warranted to achieve this goal.
#### Thermodynamic integration.
We now explain how the data of Fig. 5 have been obtained numerically. The
simple insertion method due to Widom Frenkel and Smit (2002) obviously becomes
rapidly inefficient with increasing $x$, especially for longer chains.
Slightly generalizing the method suggested in Müller and Paul (1994); Wilding
and Müller (1994) we therefore have performed a thermodynamic integration
Frenkel and Smit (2002)
$\beta\mu_{\text{ex}}=\int_{\lambda(\varepsilon)}^{1}d\lambda\frac{\left<N_{\text{sg}}\right>}{\lambda}$
(21)
over discrete values of the interaction affinity
$\lambda=\exp(-\varepsilon_{\text{sg}}\beta/8)$ characterizing the excluded
volume interaction of a ghost (g) chain that is gradually inserted into an
equilibrated system (s). $\left<N_{\text{sg}}\right>$ refers to the mean
number of lattice sites where system and ghost monomer cube corners overlap at
a given interaction $\lambda$. Generalizing the Potts spin mapping, Eq. (13),
of the excluded volume interactions for homopolymers presented above, we use
now two spin lattices, $\sigma_{\text{s}}(\bm{r})$ describing (as before) the
interaction of the system monomers and $\sigma_{\text{g}}(\bm{r})$ the ghost
chain. The spin lattices are kept at the same temperature and are both
characterized by the same (arbitrary) energy $\varepsilon=1$ which has to be
paid for a complete overlap of two system monomers or two ghost monomers. The
interaction of both spins is described by
$\Delta
E_{sg}=\sum_{\bm{r}}\sigma_{\text{s}}(\bm{r})\sum_{\bm{\delta}}J_{\text{sg}}(\bm{\delta})\sigma_{\text{g}}(\bm{r}+\bm{\delta})$
(22)
with coupling constants
$J_{\text{sg}}(\bm{\delta})\sim\varepsilon_{\text{sg}}$ defined as in Eq. (14)
taken apart the energy parameter $\varepsilon$ which is replaced by the
tuneable interaction energy $\varepsilon_{\text{sg}}$. Starting with decoupled
system and ghost configurations at $\varepsilon_{\text{sg}}=0$, i.e.
$\lambda=1$, we gradually increase the interaction parameter up to
$\varepsilon_{\text{sg}}=\varepsilon$, i.e.
$\lambda(\varepsilon)=\exp(-\varepsilon\beta/8)$, always keeping the coupled
system at equilibrium. Monitoring the distribution of the number
$N_{\text{sg}}$ of overlaps between system and ghost cube corners we use
multihistogram methods as described in Müller and Paul (1994); Wilding and
Müller (1994) to improve the precision of the integral.
The mean overlap number $\left<N_{\text{sg}}\right>$ (devided by $8N$) is
shown in the inset of Fig. 5 as a function of $\lambda$ for $N=2048$ and two
inverse temperatures $x=3$ and $x=100$. Starting from $\lambda=1$ the overlap
number decreases monotonously with increasing coupling between system and
ghost monomers, i.e. decreasing $\lambda$. Interestingly, a power law behavior
$\left<N_{\text{sg}}\right>/N\approx\lambda^{1/4}$ is found empirically for
large $x\gg 10$ (dashed line). Fitting this power law and integrating then
analytically Eq. (21) provides a useful crosscheck of the numerical
integration using the multihistogram analysis. This is a technically important
finding, since the multihistogram analysis requires overlapping distributions
of $N_{\text{sg}}$ and hence much more equilibrated intermediate values
$\lambda$ as indicated for $x=100$. A detailed explanation for the observed
power law still is missing, but it is presumably due to the systematic
screening of the long range correlations of the ghost chain which is swollen
at $\lambda=1$ becoming more and more Gaussian as it feels the compression due
to the surrounding host chains de Gennes (1979).
### III.4 The compressibility
#### Compressibility $g(x,N)$ and excess compressibility $g_{\text{ex}}(x)$.
To test the key relation Eq. (5) announced in the Introduction we need
accurate values for the dimensionless compressibilities
$g(x)\equiv\lim_{N\to\infty}g(x,N)$ of asymptotically long chains. As
suggested by Eq. (1), we compute first the dimensionless compressibility
$g(x,N)=\lim_{q\to 0}G(q)$ from the low-$q$ limit of the total monomer
structure factor for different overlap penalties $x$ and chain lengths $N$
(see below for details). These raw data are presented in Fig. 6 as a function
of $x$. As one expects, $g(x,N)$ decreases monotonously with overlap strength
$x$. In contrast to the thermodynamic integration performed for the chemical
potential [Eq. (21)], the structure factor measures the complete
compressibility, not just the excess contribution. The strong $N$-dependence
visible in the plot thus is expected from the translational entropy of the
chains. As can be seen, e.g., from Eq. (2) or from the virial expansion of
polymer solutions de Gennes (1979), the compressibility can be written in
general as
$\frac{1}{g(x,N)}=\rho\ \frac{\partial^{2}(\beta
f(\beta)\rho)}{\partial\rho^{2}}=\frac{1}{N}+\frac{1}{g_{\text{ex}}(x,N)}$
(23)
for all $x$ with $g_{\text{ex}}(x,N)$ being the excess contribution to the
compressibility which may, at least in principle, depend on $N$ foo (k). As
can be seen from the inset of Fig. 6, all compressibilities collapse, however,
on one $N$-independent master curve if one plots $1/g(x,N)-1/N$ as a function
of $x$, even the compressibilities obtained for unconnected beads ($N=1$).
Within numerical accuracy the $N$-dependence observed for $g(x,N)$ can
therefore be attributed to the trivial osmotic contribution and the excess
compressibility $g_{\text{ex}}\sim N^{0}$ is thus identical to the
compressibility $g(x)$ of asymptotically long chains. The bold line indicated
in the inset presents the best values of $g(x)$ summarized in Table 2. These
values have been obtained from the excess compressibilities for the largest
chain length available for $x\geq 0.001$. A precise numerical determination of
$g_{\text{ex}}(x)$ becomes impossible for even smaller overlap penalties. We
thus have used for the smallest $x$-values sampled the theoretical prediction
$\frac{1}{g(x)}\approx v(x)\rho\left(1-\
\underline{\frac{3\sqrt{3}}{2\pi}\frac{(v(x)\rho)^{1/2}}{b(x)^{3}\rho}}\ldots\right)\mbox{
for }x\ll 1$ (24)
due to the free energy [Eq. (2)] postulated in the Introduction. The prefactor
$v(x)\rho$ representing the bare monomer interaction is indicated by the
dashed line in the main panel of Fig. 6. Hence, $g(x)\approx 1/(8x\rho)$ for
weak interactions, i.e. the compressibility increases linearly with
temperature. The underlined term is the leading correction due to the density
fluctuation contribution to the free energy. It implies that the excess
compressibilities for polymer melts and unconnected beads cannot be completely
identical. However, as before for the chemical potential, the difference is
far too small to be measurable in the limit where Eq. (24) applies. Although
this result is unfortunate from the theoretical point of view, the data
collapse observed in the inset suggests that it is acceptable to numerically
estimate the long chain compressibility $g(x)$ by computing the structure
factors of rather short chains.
#### Total monomer structure factor.
We now turn to the total structure factor $G(q)$ shown in Fig. 7 to explain
how the compressibilites $g(x,N)$ have been obtained. Only chains of length
$N=2048$ are presented for clarity. The total monomer structure factor is
obtained by computing
$G(q)=\frac{1}{n_{\text{mon}}}\left<[\sum_{n}\cos(\bm{q}\cdot\bm{r}_{n})]^{2}+[\sum_{n}\sin(\bm{q}\cdot\bm{r}_{n})]^{2}\right>$
where the sums run over all the $n_{\text{mon}}$ monomers of the box and the
wavevectors are commensurate with the cubic box of linear length $L$. Since
the smallest possible wavevector is $2\pi/L$, it thus is important to have
large box sizes to scan over a sufficiently important $q$-range allowing a
reasonable determination of $g(x,N)$. Note that around and above $q\approx 2$
monomer structure and lattice effects become important. Since only smaller
wavevectors are of interest if one is interested in universal physical
behavior, we will focus below on wavevectors $q<1$. For comparison, we have
also included the single chain form factor
$F(q)=\frac{1}{N}\left<[\sum_{n=1}^{N}\cos(\bm{q}\cdot\bm{r}_{n})]^{2}+[\sum_{n=1}^{N}\sin(\bm{q}\cdot\bm{r}_{n})]^{2}\right>$
Doi and Edwards (1986) for $x=0.001$ (bold line). Note that the qualitative
shape of $F(q)$ — decaying monotonously with $q$ from its maximum value
$F(q=0)=N$ — depends very little on the temperature (not shown). We remind
that the “random phase approximation” (RPA) formula de Gennes (1979); Doi and
Edwards (1986)
$\frac{1}{G(q)}=\frac{1}{F(q)}+\frac{1}{g(x)}$ (25)
relates the total structure factor to the measured form factor. Eq. (25) is of
course consistent with Eq. (23) in the $q\to 0$ limit. It allows to directly
fit for the excess compressibility $g_{\text{ex}}(x)$ using the measured
structure factor $G(q)$ and form factor $F(q)$, at least in the $x$-range
where the RPA approximation applies. As may be seen from the figure, $G(q)$
indeed decreases systematically with $x$, i.e. with decreasing $g(x)$. For
large temperatures ($x\leq 3$) it also decays monotonously with $q$, again in
agreement with Eq. (25). Interestingly, this becomes qualitatively different
for larger excluded volume interactions ($x>3$) where the total structure
factor is essentially constant (in double-logarithmic coordinates), very
weakly increasing monotonously with $q$. The RPA formula apparently does not
apply in this limit in agreement with Eq. (4). Fortunately, this is of no
concern for our main purpose — to compute $g(x)$ — since in precisely this
limit the compressibility is readily obtained from a broad plateau (even for
much smaller boxes) which in addition becomes chain length independent, as we
have already seen from the inset of Fiq. (6). Using boxes with $L=256$ it is
possible to directly measure the plateau values for $x\leq 0.3$. For smaller
$x$ we have simulated boxes with $L=512$ containing $n_{\text{mon}}\approx
8.4\cdot 10^{6}$ monomers and corresponding to a smallest wavevector $q\approx
0.01$. This box size becomes again insufficient for the largest temperatures
we have simulated, as shown in Fig. 7 for $x=0.001$ (dashed line). It is for
these values where the RPA formula, Eq. (25), allowing to fit the deviation
from the (barely visible) plateau, has been particulary useful.
#### Approximated RPA formula.
We finally note that in the intermediate wavevector regime (where $q$
corresponds to distances much smaller than the radius of gyration and much
larger than the monomer size) the general RPA Eq. (25) may be written as
$\frac{1}{G(q)}=\frac{1}{g(x)}+\frac{1}{12}b^{2}(x)q^{2}=\frac{1}{g(x)}\left(1+(q\xi)^{2}\right)$
(26)
which justifies the definition given in Eq. (3) for the correlation length of
the density fluctuations $\xi$. Here we have used that the form factor becomes
$F(q)\approx 12/b^{2}q^{2}$ Doi and Edwards (1986). This assumes that
corrections to Gaussian chain statistics may be ignored Wittmer et al.
(2007a); Beckrich et al. (2007) and that finite chain size effects are
negligible. From the numerical point of view the approximated RPA Eq. (26) has
the disadvantage that the effective bond length $b(x)$ needs to be determined
first. As shown in Fig. 8, it has the advantage that it allows for an
additional test of the values of $g(x)$ and $b(x)$ indicated in Table 2. The
main panel presents the rescaled structure factor $G(q)/g(x)$ for chains of
length $N=8192$ as a function of $Q\equiv q\xi$ with $\xi$ being obtained from
$g(x)$ using Eq. (3). All data collapse on the master curve $1/(1+Q^{2})$
indicated by the bold line provided that the wavevector $q$ remains
sufficiently small and no local physics is probed. That the used
compressibilities are accurate is emphasized further in the inset where
$g(x)/G(q)-1$ is plotted as a function of $Q^{2}$ using only sufficiently
small wavevectors $q$. According to Eq. (26) all data should collapse on the
bisection line in double-logarithmic coordinates if the correct
compressibilities are used. This is indeed the case. Unfortunately, even this
rather precise method still does not allow to demonstrate the density
fluctuation contribution in Eq. (24) for $x\ll 1$ since the same scaling
collapse is obtained for the simple choice $1/g(x)\equiv v(x)\rho$. Please
note the weak deviations visible for $x=1$ which is due to the breakdown of
the RPA formula for large $x$ mentioned above.
### III.5 Bond properties
Up to now we focused on some thermodynamic features of the soft BFM model,
i.e. on large-scale properties. Turning to configurational issues we begin by
characterizing local-scale features of the algorithm. (Readers only interested
in universal properties may wish to skip this paragraph.)
#### Mean bond length.
By definition of our version of the BFM algorithm the bond length is allowed
to fluctuate strongly between $2$ and $\sqrt{10}$. One expects that switching
on the excluded volume interaction, i.e. decreasing the temperature, will
suppress large bonds due to the increasing pressure. The mean bond length
commonly is characterized by the root-mean-square length
$l=\left<\bm{l}^{2}\right>^{1/2}$. (Other moments would yield similar
results.) The mean bond length rapidly becomes ($N>20$) chain length
independent Paul et al. (1991a). As a function of overlap penalty $x$ the bond
length shows a monotonous decay between $x\approx 3$ and $x\approx 20$ as can
be seen from Fig. 9. As other local properties, $l$ becomes constant in the
small-$x$ and large-$x$ limits (dashed horizontal lines). The value
$l(x=0)=2.718$ gives the lower bound for the effective bond length $b(x)$ of
asymptotically long chains (stars) obtained below.
#### Mean bond angle and local chain rigidity.
Defining the bond angle $\theta$ between two subsequent bonds by the scalar
product $\cos(\theta)=\bm{e}_{n}\cdot\bm{e}_{n+1}$ of the normalized bond
vectors $\bm{e}_{i}=\bm{l}_{i}/|\bm{l}_{i}|$, the local chain rigidity may be
characterized by $\left<\theta\right>$ and $\left<\cos(\theta)\right>$. Note
that $\left<\theta\right>$ and $\left<\cos(\theta)\right>$ can be regarded as
chain length independent, just as the mean bond length. As can be seen from
Table 2, the local rigidity is negligible for $x\ll 1$, i.e.
$\left<\theta\right>\approx 90^{\circ}$ and $\left<\cos(\theta))\right>\approx
0$ due to the symmetry of the distribution $p(\theta)$ with respect to
$90^{\circ}$. The rigidity then increases around $x\approx 1$ and becomes
constant again for large $x$ where $\left<\theta\right>\approx 82.2^{\circ}$
and $\left<\cos(\theta)\right>\approx 0.106$. The increase of the local
rigidity for larger excluded volume interactions is of course expected due to
the suppression of immediate backfoldings corresponding to bond angles
$\theta>143^{\circ}$ Wittmer et al. (1992). The distribution $p(\theta)$
therefore becomes lopsided towards smaller $\theta$ (not shown). It is well
known Doi and Edwards (1986) that for chains characterized by the “freely
rotating” (FR) chain model such a local rigidity would lead to an effective
bond length $b(x)=l(x)\sqrt{c_{\text{FR}}}$ with
$c_{\text{FR}}=(1+\left<\cos(\theta)\right>)/(1-\left<\cos(\theta)\right>)$.
This simple model, indicated by the crosses in Fig. 9, yields a qualitatively
reasonable trend (monotonous increase of the effective bond length at
$x\approx 1$) but fails to fit the directly measured effective bond lengths
quantitatively.
### III.6 Chain and segment size
#### Total chain size $R_{\text{e}}(N)$.
One way to characterize the total chain size is to measure the second moment
of the chain end-to-end distance
$R_{\text{e}}^{2}(x,N)=\left<(\bm{r}_{N}-\bm{r}_{1})^{2}\right>$. (Other
moments yield similar behavior Wittmer et al. (2007b).) We consider the
effective bond length $b(x,N)\equiv R_{\text{e}}(x,N)/\sqrt{N-1}$ to compare
the measured chain size with the ideal chain behavior which is commonly taken
as granted de Gennes (1979); Doi and Edwards (1986); Flory (1949) and which is
the basis of our perturbation calculation. We use the notation
$b(x)\equiv\lim_{N\to\infty}b(x,N)$ for the effective bond length of
asymptotically long chains foo (a). The effective bond lengths $b(x,N)$ for
$N=64$ and $N=2048$ and the asymptotic limit $b(x)$ — obtained by
extrapolation as described below — are presented in Fig. 9 as functions of
$x$. Obviously, $b(x,N)\to l(x=0)$ for all $N$ in the small-$x$ limit.
$b(x,N)$ increases then in the intermediate $x$-window before it levels off at
$x\approx 10$. The swelling due to the excluded volume interaction is the
stronger the larger the chain length, i.e. $b(x,N)$ increases monotonously
with $N$. This swelling therefore cannot be attributed to a local persistence
length as described, e.g., by the freely-rotating chain model.
The chain length effects can be seen better in Fig. 10 where we have plotted
$b(x,N)$ for several penalties $x$ as a function of $t=1/\sqrt{N-1}$. The
choice of the horizontal axis is motivated by Eq. (7) suggesting the linear
relation
$b^{2}(x,N)\approx b^{2}(x)\ (1-c(x)c_{\text{s}}(x)t)$ (27)
for $u=N/g\gg 1$ with $c_{\text{s}}(x)\equiv\sqrt{24/\pi^{3}}/\rho b(x)^{3}$
being the swelling coefficient defined in Sec. I. $c(x)$ is an additional
numerical prefactor of order unity which has been introduced in agreement with
Eq. (19) of Ref. Wittmer et al. (2007b). The reason for this coefficient is
that the corrections to Gaussian behavior differ slightly for internal chain
segments [as described by Eq. (7)] and the total chain size which is
characterized in Fig. 10. It has been shown that $c\to 1.59$ for large $N$
Wittmer et al. (2007b). However, since this value corresponds to the limit of
a very slowly converging integral Wittmer et al. (2007b) it is better to use
Eq. (27) as a two-parameter fit for $b(x)$ and $c(x)$ and to crosscheck then
whether the fitted $c$ is of order unity. As shown in the figure for three
overlap penalties, this method can be used reasonably for overlap penalties as
low as $x\approx 0.1$, albeit with decreasing $x$ systematically
underestimating the “true” $b$-values indicated in Table 2. Please note that
$N/g\approx 400$ for $x=0.1$ and $N=8192$. Chains with $N\gg 8192$ would be
required to use this method for even smaller $x$. In this limit it is better
to use as a simple first step the value $b(x,N=8192)$ of the largest chain
length simulated as a (rather reasonable) lower bound for $b(x)$.
#### Segment size $R(s)$.
As we have already stressed in Ref. Wittmer et al. (2007b), it is technically
better to extrapolate for the effective bond length $b$ using the distribution
$R(s)$ of the mean-squared size of segments of curvilinear arc-length $s$
defined in Sec. I. It follows from $c>1$ that the total chain ratio
$b^{2}(x,N)$ converges less rapidly to the asymptotic Gaussian behavior as
$R(s)^{2}/s$. (See Fig. 4 of Ref. Wittmer et al. (2007b).) The ratio of
$R^{2}(s)/s$ as a function of $s$ is plotted in the inset of Fig. 11 for
$N=2048$ and for several $x$. As can be seen, it increases systematically with
segment length $s$. The swelling levels off for large $s$, but rather
gradually. Therefore it would not be appropriate to identify the maximum
around $s\approx N$ as the asymptotic plateau. This again would yield an
underestimation of $b(x)$. A more precise method to obtain $b(x)$ uses the
predicted correction, Eq. (7), to the Gaussian limit. We recommend to plot,
using double-logarithmic coordinates, $1-R(s)^{2}/b^{2}s$ as a function of
$c_{\text{s}}/\sqrt{s}$ and to tune $b(x)$ until the data for intermediate
chain segments with $g\ll s\ll N$ collapses on the bisection line. This one-
parameter fit yields good estimates down to $x=0.01$ where $N/g\approx 40$.
Since the corresponding plot is very similar to Fig. 5 of Ref. Wittmer et al.
(2007b), it is not reproduced here. We rather show in Fig. 11 a scaling plot
motivated by the key relation, Eq. (5), which uses our best values of $g(x)$
and $b(x)$ for asymptotically long chains (Table 2). The data collapse on the
theoretical prediction (bold line) is remarkable, especially considering that
$u$ covers seven orders in magnitude. The “Fixman limit”, Eq. (8), for $u\ll
1$ (dashed line) fits the data with the smallest overlap penalty $x\approx
0.001$ confirming the chosen value of $b$. The limiting behavior for $u\gg 1$
[Eq. (7)], characterizing an incompressible melt of thermal blobs of length
$g$, is indicated by the dash-dotted line.
#### Predicting the effective bond length.
Up to now, we have used theory to improve the fit of $b(x)$, rather than to
predict it from the thermodynamic properties and local model features such as
the bond length $l(x)$. The increase of the effective bond length for weakly
interacting and asymptotically long polymer melts has been calculated long ago
by Edwards [see Eq. (5.55) of Ref. Doi and Edwards (1986) or Eq. (11) of Ref.
Wittmer et al. (2007b)]. Reformulated using our notations and substituting the
bare excluded volume parameter $v(x)$ by $1/g\rho$ Semenov and Obukhov (2005);
Wittmer et al. (2007b); foo (c) his result reads
$b^{2}=l^{2}\left(1+\frac{\sqrt{12}}{\pi}G_{\text{z}}\right)\mbox{ with
}G_{\text{z}}=\frac{1}{\sqrt{g}b_{\text{r}}^{3}\rho}$ (28)
where $b_{\text{r}}$ is the bond length of the unperturbed reference chain of
the calculation and $G_{\text{z}}(b_{\text{r}},g)$ the relevant Ginzburg
parameter quantifying the strength of the interaction acting on a chain
segment of length $s=g$ (see Eq. (6) of Ref. Wittmer et al. (2007b)). Since
$G_{\text{z}}$ becomes small for large compressibilities $g(x)$, one expects
good agreement with our data for small $x$. The question is now what actually
might be the best reference bond length to allow a prediction over the
broadest possible $x$-range. The simplest choice to associate $b_{\text{r}}$
with the bond length $l(x)$ yields the dash-dotted line indicated in Fig. 9.
As can be seen, this choice of $b_{\text{r}}$ allows a reasonable prediction
only up to $x\approx 0.01$. The predictive power of Eq. (28) can be
considerably improved over nearly two decades up to $x\approx 1$ if one
applies the formula iteratively starting with $b_{\text{r}}=l$ and using the
effective bond length obtained as input for the Ginzburg parameter ($b\to
b_{\text{r}}$) in the next step. This recursion relation converges rapidly as
shown by the bold line indicated in Fig. 9 obtained after 20 iterations. This
iterative renormalization of the bond length of the reference chain and the
associated Ginzburg parameter has been suggested by Muthukumar and Edwards
Muthukumar and Edwards (1982). Essentially the same result is obtained up to
$x\approx 1$ if one sets directly $b_{\text{r}}=b$ using the measured
effective bond length (not shown), i.e. these values correspond to the fix-
point solution of Eq. (28). The Ginzburg parameters $G_{\text{z}}$ obtained
using the measured $b(x)$ are listed in Table 2. Note that $G_{\text{z}}<0.34$
for $x<1$ where Eq. (28) fits our data nicely. The fix-point solution of Eq.
(28) does not capture correctly the leveling off of $b(x)$ setting in above
$x\approx 1$. Since the Ginzburg parameter becomes there of order one (Table
2), this is to be expected. In summary, we have shown that the iteration of
Eq. (28) allows a good prediction for $b(x)$ for $x<1$ such that
$G_{\text{z}}\ll 1$. If reliable values for the compressibility $g(x)$ are
available [by means of the extrapolation method implied by Eq. (23)], this is
the method of choice if one cannot afford to simulate very long chains.
### III.7 Bond-bond correlation function
#### Motivation and theoretical prediction.
We return now to the deviations from Flory’s ideality hypothesis predicted in
Eq. (5) for the mean-square segment size $R^{2}(s)$. As we have seen above
(Fig. 11), this property requires to substract a large Gaussian contribution
$b^{2}s$ from the measured $R^{2}(s)$ to demonstrate the existence and the
scaling of the deviations. Unfortunately, this requires as a first step the
precise determination of the effective bond length $b(x)$ for asymptotically
long chains which might not always be available. Indeed we have used in the
preceeding Sec. III.6 the fact that the scaling of Eq. (5) critically depends
on this accurate value to improve the estimation of the effective bond length
$b(x)$ for asymptotically long chains. Hence, it would be nice to demonstrate
directly the scaling implied by our key prediction without any tuneable
parameter. The trick to achieve this is similar to our demonstration of the
density fluctuation contributions to the free energy, Eq. (2), presented in
Sec. III.2: We consider the curvature of $R^{2}(s)$, i.e. its second
derivative with respect to $s$, to eliminate the large Gaussian contribution.
In principle this can be achieved by fitting $R^{2}(s)$ by a sufficiently high
polynomial whose second derivative with respect to $s$ then is compared to the
theory. Following Wittmer et al. (2004); Wittmer et al. (2007b) we use a more
direct numerical route where we compute the well-known bond-bond correlation
function $P(s)\equiv\left<\bm{l}_{m=n+s}\cdot\bm{l}_{n}\right>/l^{2}$ with
$\bm{l}_{i}=\bm{r}_{i+1}-\bm{r}_{i}$ denoting the bond vector between two
adjacent monomers $i$ and $i+1$ and $l^{2}$ the mean-square bond length (Sec.
III.5). The average is performed as before over all chains and all pairs of
monomers $(n,m+s)$ possible in a chain of length $N$. We use this definition
rather than the more common first Legendre polynomial
$\left<\bm{e}_{n}\cdot\bm{e}_{m}\right>$ since it allows to relate the bond-
bond correlation function to the segment size by
$P(s)=\frac{1}{2l^{2}}\frac{d^{2}}{ds^{2}}R^{2}(s).$ (29)
This formula is obtained from
$\left<\bm{l}_{n}\cdot\bm{l}_{m}\right>\approx\left<\partial_{n}\bm{r}_{n}\cdot\partial_{m}\bm{r}_{m}\right>=-\partial_{n}\partial_{m}\left<(\bm{r}_{n}-\bm{r}_{m})^{2}\right>/2$.
Using Eq. (29) the key prediction, Eq. (5), implies for the bond-bond
correlation function
$P(s)=\frac{c_{\text{P}}}{g(x)^{3/2}}\left(\frac{4}{\sqrt{u}}-4\sqrt{2\pi}e^{2u}\mbox{erfc}(\sqrt{2u})\right)$
(30)
where we have introduced the coefficient
$c_{\text{P}}=c_{\text{s}}(b/l)^{2}/8$. Eq. (30) corresponds to the limiting
behavior
$P(s)\approx\frac{c_{\text{P}}}{g^{3/2}}\frac{4}{\sqrt{u}}$ (31)
for small reduced arc-lengths $u\ll 1$. The explicit compressibility
dependence drops out in the opposite limit ($u\gg 1$) where the bond-bond
correlation function becomes
$P(s)\approx c_{\text{P}}/s^{3/2},$ (32)
in agreement with Eq. (7). Please note that $c_{\text{P}}$ depends implicitly
on the compressibility. (Obviously, both asymptotic behaviors could have been
obtained directly from the corresponding limits for $R^{2}(s)$, Eqs. (8) and
(7).)
#### Numerical confirmation.
The bond-bond correlation function $P(s)$ for different overlap penalties $x$
is presented in Fig. 12 for chains of length $N=2048$. As can be seen from the
unscaled data shown in the inset, $P(s)$ approaches a power law with exponent
$\omega=1/2$ (dashed line) in the limit of weak overlap penalties in agreement
with Eq. (31). For $x\geq 1$ our data is compatible with an exponent
$\omega=3/2$ (dash-dotted line) as suggested by Eq. (32). Hence, we have
demonstrated without any tunable parameter that Flory’s ideality hypothesis is
systematically violated for all segment lengths $s$ and all overlap penalties
$x$.
We consider now the prefactors and the scaling with $x$. As suggested by Eq.
(30), the main figure presents $P(s)/(c_{\text{P}}/g(x)^{3/2})$ as a function
of the reduced arc-length $u=s/g(x)$ using the dimensionless compressibilities
$g(x)$ and effective bond lengths $b(x)$ from Table 2. The data collapse is
remarkable as long as $1\ll s\ll N$. The relation Eq. (30) is indicated by the
bold line; it is in perfect agreement with the simulation data foo (l). The
asymptotic power law behavior with exponents $\omega=1/2$ for $u\ll 1$ and
$\omega=3/2$ for $u\gg 1$ is shown by the dashed and dash-dotted lines,
respectively. As predicted by Eq. (32), one recovers the power law
$P(s)=c_{\text{P}}/s^{3/2}$ — already observed for incompressible melts
Wittmer et al. (2004); Wittmer et al. (2007b) — for scales larger than the
thermal blob irrespective of the blob size $g$. This demonstrates that the
exponent $\omega=3/2$ is not due to local physics on the monomer scale, since
for $s\gg g\gg 1$ distances much larger than the monomer or even the thermal
blob are probed.
## IV Conclusion
#### Thermodynamic properties of a BFM version with finite overlap penalty.
In this paper we have discussed a generalization of the standard bond-
fluctuation model (BFM) where the monomers may overlap subject to a finite
energy penalty $\varepsilon$ (Fig. 1). This allows us to switch on
systematically the excluded volume interaction between the monomers as
suggested by perturbation theory Doi and Edwards (1986) and to tune the
density fluctuations of the solution at constant monomer density. In this
study we have focused on dense polymer melts containing flexible linear chains
which are athermal apart from the finite overlap penalty. The central
thermodynamic parameter characterizing these systems is the excess part of the
dimensionless compressibility $g=T\kappa_{\text{T}}\rho$ of the solution which
has been obtained directly from the low-wavevector limit of the static
structure factor (Figs. 7 and 8). Scanning the overlap penalty (or,
equivalently, the temperature $T$ foo (d)) from $x=\varepsilon/T=\infty$ (no
overlap) down to $x=0.0001$ leads to a variation of $g(x)$ over four orders of
magnitude (Fig. 6). This allows for a systematic study of the thermodynamic
properties (Figs. 2-6) and the intrachain configurational statistics (Figs.
9-12). Particular attention has been paid to the thermodynamic properties of
weakly interacting melts ($x\ll 1$). We have verified that our results are
consistent with the free energy, Eq. (2), postulated in agreement with Edwards
Doi and Edwards (1986). The main result of this part of our study is that we
have been able to demonstrate the density fluctuation contribution to the free
energy induced by the chain connectivity from the scaling of the specific heat
$c_{\text{V}}$ with respect to overlap penalty $x$ (Fig. 4).
#### Intrachain conformational properties: Violation of Flory’s ideality
hypothesis.
The broad variation of $g(x)$ puts us into a position to test the recently
proposed Eq. (5) predicted by perturbation theory Meyer et al. (2008); Wittmer
et al. (2009) describing the systematic swelling of chain segments as function
of the segment size $s$ and the compressibility $g(x)$. As outlined in the
Introduction, this key relation suggests that the repulsive interactions
between chain segments in the same chain are not fully screened at variance to
Flory’s ideality hypothesis for polymers in dense melts de Gennes (1979). The
violation of the ideality hypothesis is demonstrated in Fig. 11 for the mean-
square segment size $R^{2}(s)$ and in Fig. 12 for the bond-bond-correlation
function $P(s)$. We show that data obtained for systems with very different
compressibilities $g(x)$ can be superimposed on the predicted master curves if
plotted as a function of the reduced arc-length $u=s/g$. The scaling of $R(s)$
allows a precise determination of an important intrachain property, the
effective bond length $b(g)$ for asymptotically long chains (Fig. 9). These
values compare well for $x<1$ with the fix-point solution of the recursion
relation, Eq. (28) Doi and Edwards (1986); Muthukumar and Edwards (1982);
Wittmer et al. (2007b). The bond-bond correlation function $P(s)$ being the
second derivative of $R^{2}(s)$ with respect to $s$ allows an even more direct
test of the predicted deviations. The reason is that the large Gaussian
contribution $b^{2}s$, which must be subtracted from $R^{2}(s)$ (see the
vertical axis of Fig. 11), drops out due to the differentiation. In contrast
to Flory’s hypothesis, $P(s)$ does not vanish rapidly on scales corresponding
to the local persistence length (Fig. 12). In perfect agreement with theory
foo (l), the scaling plot shows two power law regimes characterized by
exponents $\omega=1/2$ for small ($u\ll 1$) and $\omega=3/2$ for large ($u\gg
1$) reduced arc-length.
The central result of this study is that even for polymer melts with finite
overlap penalty excluded volume interactions are not fully screened. If
distances smaller than the thermal blob size are probed the chains are swollen
according to the standard Fixman parameter expansion. More importantly, even
on distances larger than the thermal blob size ($s/g\gg 1$) deviations from
ideal chain behavior are found. Interestingly, in this limit the explicit
compressibility dependence drops out and the relations established for
incompressible melts Wittmer et al. (2004); Wittmer et al. (2007b) are
recovered. This shows that soft melts behave on large scales as incompressible
packings of blobs.
#### Outlook.
Since the presented soft BFM is fully ergodic (in contrast to the classical
BFM) and very efficient, it may be an interesting alternative to various
popular coarse-grained simulation approaches using soft effective interaction
parameters Likos (2001); Eurich and Maass (2001); Yatsenko et al. (2004);
Müller and Smith (2005); Daoulas and Müller (2006). The presented model is
part of a broader attempt to describe systematically the effects of correlated
density fluctuations in dense polymer systems, both for static Semenov (1996);
Semenov and Johner (2003); Semenov and Obukhov (2005); Obukhov and Semenov
(2005) and dynamical Semenov (1997); Semenov and Rubinstein (1998); Mattioni
et al. (2003) properties. This also involves the comparison with (off-lattice)
molecular dynamics simulation using a standard bead-spring model which is
discussed elsewhere Meyer and Müller-Plathe (2001); Wittmer et al. (2007b);
Meyer et al. (2008); foo (m). An important unresolved question is for instance
whether recently predicted long-range repulsive forces of van der Waals type
(“Anti-Casimir effect”) Semenov and Obukhov (2005); Obukhov and Semenov (2005)
can be demonstrated numerically from specific non-analytic deviations from the
RPA formula, Eq. (25), at intermediate overlap strengths ($x\approx 1)$. In
order to do this, we are currently improving the statistics of our data.
In this paper we have discussed only static properties of the soft BFM.
Similar scaling behavior also has been obtained for the static Rouse mode
correlation function which displays systematic deviations from the scaling
expected for ideal chains Meyer et al. (2008). We currently are working out
how these deviations may influence the dynamics for polymer chains without
topological constraints. (These constraints can be switched off even for
$x=\infty$ by using the “L26” local moves described in Sec. II.) Conceptually
important issues can be addressed if the (artifical) slithering-snake dynamics
is analysed and compared to predictions of the “activated-reptation dynamics”
hypothesis suggested by Semenov Semenov (1997); Semenov and Rubinstein (1998)
for real, although extremely long polymer chains. If no overlap is allowed,
the slithering-snake dynamics is known to show anomalous curvilinear diffusion
and correlated motion of neighboring snakes Mattioni et al. (2003). Since for
$x=\infty$ the lattice might influence the results, it is important to verify
if qualitative similar behavior is also found for soft BFM melts with thermal
blobs much larger than the local monomer scale and how the anomalous
curvilinear diffusion changes with compressibility.
###### Acknowledgements.
We thank H. Meyer and A.N. Semenov (both ICS, Strasbourg, France), S.P.
Obukhov (Gainesville, Florida) and M. Müller (Göttingen) for helpful
discussions. A generous grant of computer time by the IDRIS (Orsay) is also
gratefully acknowledged. We are indebted to the Université de Strasbourg and
the ESF STIPOMAT programme for financial support. J.B. acknowledges financial
support by the IUF.
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* foo (b) For the computational definition of $G(q)$ see Sec. III.4. Note that our normalization of the structure factor is consistent with Eq. (5.39) of Doi and Edwards (1986), i.e. $G(q)$ is dimensionless. Quite generally, we present in this paper properties normalized by the total number of monomers $n_{\text{mon}}$, i.e. essentially by the degree of freedom of the system, rather than by its volume $V$.
* foo (c) Following Semenov and Obukhov (2005); Beckrich et al. (2007); Wittmer et al. (2007b) the “bulk modulus” $\tilde{v}\equiv 1/g\rho$ is sometimes called “effective excluded volume” which refers to the fact that convergence and self-consistency of the perturbation theory Doi and Edwards (1986) suggests to substitute (renormalize) the bare excluded volume $v$ of the monomers by the explicitly measured $\tilde{v}$ summing also over higher virial contributions, not only pair interactions. However, since higher virial contributions become negligible for weak excluded volume interactions one expects that to leading order $v(x)\approx\tilde{v}(x)$ for $x\ll 1$. This will be crosschecked in Sec. III.4 (Fig. 6). Since the perturbation calculation summarized by Eq. (2) is anyway restricted to the same $x$-regime [Eq. (4)] we do not use $\tilde{v}$ in the main text.
* foo (d) Since the overlap penalty is the only energy scale in this study one may either vary the overlap parameter $\varepsilon$ or the temperature $T$. Since the presentation of thermodynamic properties (especially Figs. 2-5) becomes, however, slightly simpler if $T$ is the control parameter we fix arbitrarily $\varepsilon=1$. The inverse temperature $\beta=1/T$ and the dimensionless overlap strength $x=\varepsilon/T$ are thus numerically equal. We keep both notations for dimensional reasons and for future generalization to models with more than one energy scale.
* foo (e) A similar intrachain energy contribution to the free energy arises also from Eq. (5.43) of Doi and Edwards (1986) if an upper cut-off $q_{\text{max}}$ is introduced for the wavevectors $q$ to avoid the ultra-violet divergence. Such an upper cut-off is justified by the discreteness of the monomers of real polymers. This leads necessarily to a non-universal free energy contribution.
* foo (f) Interestingly, according to Edwards Doi and Edwards (1986) the chain connectivity, i.e. the presence of attractive forces between bonded monomers, does not change the excluded volume $v(x)$ — as one would expect naively — but rather gives rise to an additional term scaling differently with density.
* foo (g) The Ginzburg parameter is defined here in terms of the measured compressibility $g(x)$ and the measured effective bond length $b(x)$. Eq. (4) is consistent with Eq. (5.46) of Ref. Doi and Edwards (1986).
* foo (h) Since the intrachain form factor is essentially the Fourier transform of the segmental size distribution Wittmer et al. (2007b); Beckrich et al. (2007), $R(s)$ is related to an experimentally relevant quantity.
* Wittmer et al. (2009) J. P. Wittmer, A. Johner, J. Baschnagel, and S. P. Obukhov (2009), in preparation.
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* foo (i) See, e.g., Eq. (2.100) of Ref. Doi and Edwards (1986), which presents the excluded volume effect for the total chain size $R_{\text{e}}(N)$ as obtained by expansion with respect to the bare excluded volume coefficient $v$. As already mentioned Semenov and Obukhov (2005); Wittmer et al. (2007b); foo (c), $v$ must be replaced by the effective modulus $\tilde{v}=1/g\rho$. Note that the leading correction term $\frac{4}{3}z\sim\sqrt{u}$ differs by a numerical coefficent of order one from the corresponding deviation $\frac{c_{\text{s}}}{\sqrt{g}}\frac{4}{3}\sqrt{u}$ we have calculated. This is due to the fact that we consider the size of inner chain segments which involves the two additional graphs in the perturbation calculation illustrated on the right of Fig. 14 given in Ref. Wittmer et al. (2007b). Only the first diagram shown in this figure is taken into account for the total chain size.
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* foo (j) Attractive interactions similar to the ones used in Müller (1995, 1999); Cavallo et al. (2003) may be easily added to the Potts spin formulation of the soft BFM. The simulation of polymer blends requires additional Potts spin lattices interacting as indicated in Eq. (22) for the two lattices used to obtain the chemical potential in Sec. III.3.
* foo (k) Small corrections to $g(x)\sim N^{0}$ and $\mu_{\text{ex}}\sim N$ may arise due to the long-range correlations which lead to the swelling described by Eq. (5). This effect is, however, too weak to be visible in our data. It can be demonstrated from the number distribution of equilibrium polymers Wittmer et al. (1998); Wittmer et al. (2007c) which differs slightly from the exponential behavior generally admitted Semenov and Johner (2003); Beckrich et al. (2007).
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$x$ | $e$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/T$ | $g$
---|---|---|---|---
0.0001 | 0.2499 | 1.14E-09 | 5.0E-05 | $\approx 1$
0.001 | 0.2499 | 1.08E-07 | 5.0E-04 | $\approx 1$
0.01 | 0.2489 | 1.05E-05 | 0.0049 | $\approx 1$
0.1 | 0.2400 | 0.00093 | 0.049 | 0.95
0.3 | 0.2228 | 0.00709 | 0.14 | 0.88
1.0 | 0.1799 | 0.04854 | 0.43 | 0.69
3.0 | 0.1147 | 0.21538 | 1.04 | 0.47
10 | 0.0321 | 0.57795 | 2.15 | 0.25
20 | 0.0223 | 0.36671 | 2.55 | 0.20
30 | 0.0012 | 0.16318 | 2.62 | 0.20
50 | 8.9E-05 | 0.02874 | 2.63 | 0.20
100 | 2.2E-07 | 0.00039 | 2.635 | 0.20
$\infty$ | 0 | 0 | 2.635 | 0.20
Table 1: Various properties for soft BFM beads ($N=1$) at monomer number
density $\rho=0.5/8$ (corresponding to a volume fraction $0.5$) and linear box
size $L=256$ as a function of the reduced overlap strength $x=\varepsilon/T$.
The limit $x=\infty$ corresponds to the classical BFM without monomer overlap,
the limit $x=0$ to non-interacting monomers. Indicated are the mean energy per
bead $e$, the specific heat $c_{\text{V}}$ per bead, the excess part of the
chemical potential $\mu_{\text{ex}}/T$, and the dimensionless compressibility
$g(x,N=1)$. Within statistical accuracy we obtain below $x\approx 0.1$ the
ideal gas compressibility, $g\approx 1$, and above $x\approx 10$ the
compressibility for a melt without monomer overlap.
$x$ | $e$ | $e_{\text{self}}$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/(TN)$ | $g$ | $l$ | $b$ | $\left<\theta\right>$ | $\left<\cos(\theta)\right>$ | $c_{\text{s}}$ | $G_{\text{z}}$
---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | $\infty$ | 2.718 | 2.72 | $90^{\circ}$ | 0 | - | 0
0.0001 | 0.42 | 0.18 | 2.5E-07 | 4.9E-05 | 20094 | 2.718 | 2.75 | $90^{\circ}$ | 0 | 0.68 | 0
0.001 | 0.42 | 0.18 | 8.8E-06 | 4.9E-04 | 2029 | 2.718 | 2.75 | $89.99^{\circ}$ | 1.7E-04 | 0.68 | 0.017
0.01 | 0.39 | 0.17 | 2.2E-04 | 4.5E-03 | 209 | 2.718 | 2.80 | $89.9^{\circ}$ | 1.1E-03 | 0.65 | 0.052
0.1 | 0.32 | 0.15 | 4.5E-03 | 0.05 | 22 | 2.719 | 2.92 | $89.4^{\circ}$ | 9.2E-03 | 0.57 | 0.14
0.3 | 0.26 | 0.12 | 0.015 | 0.1 | 7.1 | 2.720 | 3.01 | $88.5^{\circ}$ | 0.021 | 0.52 | 0.22
1 | 0.18 | 0.08 | 0.06 | 0.4 | 2.4 | 2.721 | 3.13 | $86.9^{\circ}$ | 0.043 | 0.46 | 0.34
3 | 0.11 | 0.05 | 0.3 | 0.9 | 0.85 | 2.721 | 3.21 | $84.9^{\circ}$ | 0.069 | 0.42 | 0.52
10 | 0.03 | 0.01 | 0.5 | 1.8 | 0.32 | 2.670 | 3.24 | $82.9^{\circ}$ | 0.096 | 0.41 | 0.83
20 | 0.004 | 0.002 | 0.26 | 2.0 | 0.25 | 2.643 | 3.24 | $82.4^{\circ}$ | 0.104 | 0.41 | 0.94
30 | 9.7E-04 | 4.0E-04 | 0.11 | 2.0 | 0.25 | 2.638 | 3.24 | $82.3^{\circ}$ | 0.105 | 0.41 | 0.94
50 | 7.1E-05 | 2.9E-05 | 0.019 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
100 | 2.2E-07 | - | 4.5E-04 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
$\infty$ | 0 | 0 | 0 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
Table 2: Various properties for asymptotically long BFM chains at number
density $\rho=0.5/8$ as a function of $x=\varepsilon/T$. Apart from the
properties already presented in Table 1 for beads we indicate here the
intrachain self-energy $e_{\text{self}}$, the root-mean-square bond length
$l=\left<\bm{l}_{n}^{2}\right>^{1/2}$, the effective bond length $b$, the mean
angle $\left<\theta\right>$ and the mean cosine
$\left<\cos(\theta)\right>=\left<\bm{e}_{n}\cdot\bm{e}_{n+1}\right>$ of two
subsequent bonds with $\bm{e}_{n}=\bm{l}_{n}/|\bm{l}_{n}|$ being the
normalized bond vector, the swelling coefficient
$c_{\text{s}}\equiv\sqrt{24/\pi^{3}}/\rho b^{3}$, and the Ginzburg parameter
$G_{\text{z}}=1/\sqrt{g}b^{3}\rho$. The excess part of the chemical potential
of a chain is given in units of the chain length $N$ (column 4). The effective
bond length $b(x)$ has been obtained using an extrapolation scheme implied by
Eq. (7) and discussed in Sec. III.6.
$N$ | $e$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/(TN)$ | $g$ | $l$ | $b$
---|---|---|---|---|---|---
1 | 0.1799 | 0.0485 | 0.43 | 0.69 | - | -
4 | 0.1767 | 0.0482 | 0.40 | 1.5 | 2.717 | 2.77
16 | 0.1811 | 0.0691 | 0.37 | 2.0 | 2.720 | 2.89
64 | 0.1819 | 0.0562 | 0.35 | 2.3 | 2.721 | 2.99
256 | 0.1820 | 0.0877 | 0.35 | 2.4 | 2.721 | 3.05
1024 | 0.1820 | 0.0871 | 0.34 | 2.4 | 2.721 | 3.08
2048 | 0.1820 | 0.0639 | 0.34 | 2.4 | 2.721 | 3.09
4096 | 0.1820 | 0.0697 | 0.34 | 2.4 | 2.721 | 3.10
8192 | 0.1820 | 0.0795 | 0.34 | 2.4 | 2.721 | 3.11
Table 3: Various properties for BFM melts of number density $\rho=0.5/8$ at
overlap strength $x=\varepsilon/T=1$ as a function of chain length $N$. For
small chains the overlap energy $e$ and its fluctuation $c_{\text{V}}$
increase slightly while the chemical potential per bead decreases. The
compressibility $g(x=1,N)$ becomes chain length independent for $N>64$. The
chain length dependence visible for small $N$ is described by Eq. (23), i.e.
the data is consistent with an excess compressibility $g_{\text{ex}}(x)\approx
2.4N^{0}$ for all $N$. The last column indicates the rescaled end-to-end
distance $b(x=1,N)\equiv R_{\text{e}}(N)/(N-1)^{1/2}$ which approaches the
effective bond length $b(x)\approx 3.13$ of asymptotically long chains
monotonously from below, just as for classical BFM melts Wittmer et al.
(2007b). Interestingly, $b(x=1,N)$ has not yet reached the asymptotic limit
$b(x)$ even for $N=8192$ albeit all other quantities indicated can be regarded
(within statistical accuracy) as independent of chain length above $N\approx
256$.
Figure 1: Sketch of the bond-fluctuation model (BFM) with finite excluded
volume penalty. The BFM algorithm represents monomers by cubes of length $a$
on a simple cubic lattice (of lattice constant $a$) which are connected by a
set of allowed bond vectors given by Eq. (9). Two short chains of length $N=3$
are shown. The classical BFM model Carmesin and Kremer (1988); Deutsch and
Binder (1991); Baschnagel et al. (2004) assumes that all lattice sites are at
most occupied once. We relax this constraint and penalize double occupation by
a finite interaction energy $\varepsilon$ which has to be paid if two cubes
totally overlap. A corresponding fraction of the energy penality is associated
with a partial monomer overlap, as sketched in the figure for two cube corners
occupying the same lattice site. Varying systematically the ratio
$x=\varepsilon/T$ allows us to put to a test various theoretical results
obtained by perturbation calculation Wittmer et al. (2004); Beckrich et al.
(2007); Wittmer et al. (2007b); Wittmer et al. (2009) for flexible polymer
chains in the melt.
Figure 2: Reduced mean overlap energy per monomer $y=e/\varepsilon$ as a
function of the overlap penalty $x=\varepsilon/T$ for several chain lengths
$N$ as indicated. The energy decreases monotonously with increasing $x$. The
decay becomes Arrhenius-like for $x\gg 10$ (dash-dotted line). The dashed line
indicates the energy predicted from the second virial of soft BFM beads, Eq.
(15). The main figure demonstrates the weak chain length dependence on
logarithmic scales, especially for strong excluded volume interactions
($x>1$). Inset: Same data plotted with linear vertical axis emphasizing the
higher mean energy for long polymers ($N>64$) for $x\ll 1$ caused by a self-
energy contribution $e_{\text{self}}/\varepsilon\approx 0.18$. The self-
energies are indicated by the triangles. The bold line shows the temperature
dependence predicted by Eq. (17).
Figure 3: Reduced mean energy $e/\varepsilon$ (spheres) and self-energy
$e_{\text{self}}/\varepsilon$ (triangles) as functions of the number density
$\rho$ for $N=8192$, $L=512$ and $x=0.001$. As shown by the dashed line,
$e(\rho)$ is a superposition of the mean field energy $4\rho$ and the
(essentially) constant self-energy $e_{\text{self}}/\varepsilon\approx
0.18N^{0}x^{0}\rho^{0}$. Inset: $e/\varepsilon-4\rho$ as a function of chain
length $1/\sqrt{N-1}$ for our reference density $\rho=0.5/8$ and for a single
chain ($\rho=0$). The linear slope (bold line) is expected from the return
probability of Gaussian chains.
Figure 4: Specific heat per bead $c_{\text{V}}$ vs. $x$ for chain length $N$
as indicated. The dashed line indicates the energy fluctuations predicted from
the second virial, Eq. (18), which fits nicely the data of soft BFM beads
($N=1)$ over six decades. While the chain length appears not to matter for
strong excluded volume interactions, the energy fluctuations are found to
increase strongly with $N$ for $x\ll 1$. For short chains we observe
$c_{\text{V}}\sim\rho N^{1/2}x^{2}$ as can be seen for $N=16$ (thin solid
line). The chain length effect saturates for long chains where
$c_{\text{V}}\approx\rho^{1/2}x^{3/2}N^{0}$ (bold line) in agreement with Eq.
(19). Inset: $c_{\text{V}}/(\rho^{1/2}x^{3/2})$ as a function of the reduced
chain length $u=N/g(x)$ with $g(x)$ being the dimensionless compressibility
(Table 2).
Figure 5: Excess chemical potential $y=\mu_{\text{ex}}/TN$ as a function of
the inverse temperature $x=\varepsilon/T$. Increasing linearly (dashed line)
for small $x$ it levels off for large $x\gg 1$ (dash-dotted line). The dashed
line indicates the simple second virial approximation $y\approx v(x)\rho$ for
unconnected beads, fitting successfully the data below $x\approx 1$. The bold
line corresponds to the high temperature prediction Eq. (20) taking into
account the density fluctuation contribution induced by the chain
connectivity. Inset: The chemical potential has been obtained by thermodynamic
integration over the excluded volume interaction of an inserted ghost chain
generalizing the method suggested in Ref. Müller and Paul (1994). The mean
number of lattice sites where monomers and ghost monomers overlap,
$\left<N_{\text{sg}}\right>$, is presented for $N=2048$ as a function of
$\lambda=\exp(-\varepsilon_{\text{sg}}/8T)$ for $x=3$ and $x=100$. A power law
increase of $\left<N_{\text{sg}}\right>$ is found for large $x$ (dashed line).
Figure 6: Dimensionless compressibility $g(x,N)$ as a function of $x$ for
different chain lengths $N$ using the same symbols as in Fig. 4. Main panel:
Raw data as obtained from the low-wavevector limit of the structure factor.
Chain length effects become irrelevant for $x\geq 0.1$ if $N\geq 64$ and for
$x>0.001$ if $N\geq 2048$. The data are compared to the simple second virial
approximation $1/v(x)\rho$ (dashed line) which reduces to $1/(8x\rho)$ for
$x\ll 1$. As one expects, the compressibility levels off for large $x$ and
becomes identical to the value $g\approx 0.25$, known for the classical BFM
Wittmer et al. (2007b) (dash-dotted line). Inset: As suggested by Eq. (23) the
excess part of the inverse compressibility $1/g(x,N)-1/N$ becomes chain length
independent, i.e. the data points for all $N$ collapse. The master curve
indicated by the bold line corresponds to the long chain limit
$g(x)=\lim_{N\to\infty}g(x,N)$ indicated in Table 2.
Figure 7: Total structure factor $G(q)$ as a function of wavevector $q$ for
$N=2048$ for different overlap penalties $x=\varepsilon/T$ as indicated. For
comparison, we have also included the single chain form factor $F(q)$ for
$x=0.001$. The low-wavevector limit of the structure factor is used to
determine the dimensionless compressibility $g(x,N)$ [Eq. (1)]. Only for
$x\leq 3$ does the structure factor decay monotoneously with $q$ as suggested
by the RPA formula, Eq. (25). $G(q)$ becomes essentially constant for smaller
temperatures except for wavevectors corresponding to the first sharp
diffraction peak (called here “Bragg peak”). The box size $L=256$ allows only
a direct and fair determination of $g(x,N)$ for $x>0.1$. We have been forced
to increase the box size to $L=512$ for smaller $x$ as may be seen for an
example with $x=0.1$ (dash-dotted line). As shown by the bold dashed line, the
RPA formula is used to improve the estimation of $g(x,N)$ for small $x$.
Figure 8: Rescaled total structure factor $G(q)/g(x)$ as a function of the
reduced wavevector $Q\equiv q\xi$ for chain length $N=8192$ and several $x\leq
1$ as indicated. The screening length $\xi$ of the thermal blob is obtained
according to Eq. (3) using the dimensionless compressibility $g(x)$ and the
effective bond length $b(x)$ from Table 2. The bold line compares the data
with the approximated RPA, Eq. (26). If replotted as indicated in the inset
the data collapse on the bisection line. Deviations from the RPA formula
become visible for larger $x$ as shown for $x=1$ (crosses).
Figure 9: The (effective) bond length as a function of the reduced overlap
penalty $x=\varepsilon/T$. The data for the root-mean-square bond length
$l(x)$ and the effective bond length $b(x)$ for asymptotically long chains are
listed in Tab. 2. The dash-dotted line indicates the effective bond length as
predicted by Eq. (28) assuming $b_{\text{r}}=l(x)$ for the bond length of the
reference chain. The bold line shows the fix points obtained by iteration of
Eq. (28) using as an input for the Ginzburg parameter the effective bond
length of the previous iteration step: $b\rightarrow b_{\text{r}}$. See the
main text for details.
Figure 10: Rescaled mean-square chain end-to-end distance $b(x,N)^{2}\equiv
R_{\text{e}}^{2}(N)/(N-1)$ as a function of $t=1/\sqrt{N-1}$ for different $x$
as indicated. The chains only remain Gaussian on all scales and all $N$ for
extremely small $x$. For $x\geq 0.1$ one observes $b(x,N)^{2}$ to decay
linearly in agreement with Eq. (27). This can be used for a simple two-
parameter fit for $b(x)$ as indicated for $x=0.1$, $1.0$ and $\infty$. Note
that the coefficient $c$ is slightly above unity as expected from Eq. (19) of
Ref. Wittmer et al. (2007b).
Figure 11: Segment size $R(s)$ for overlap penalty $x$ as indicated for chain
length $N=2048$. Inset: $R(s)^{2}/s$ as a function of segment length $s$
increases monotonously approaching from below the asymptotic limit for large
$s$, i.e. the chains are swollen. Main figure: As suggested by Eq. (5), the
rescaled data
$\left(1-R^{2}(s)/b^{2}(x)s\right)/\left(c_{\text{s}}(x)/g(x)^{1/2}\right)$ is
plotted as a function of the reduced arc-length $u=s/g$. The data collapse is
successful for $1\ll s\ll N$ which confirms the values $g(x)$ and $b(x)$ for
asymtotically long chains (Table 2). The bold line shows the full prediction
from Eq. (5). We indicate the limiting behavior for small and large $u$ by the
dashed and dash-dotted lines, representing respectively Eq. (8) and Eq. (7).
Figure 12: Bond-bond correlation function $P(s)$ for different overlap
penalties $x$ as indicated in the figure. Inset: $P(s)$ as a function of
segment length $s$ in log-log coordinates. The data approaches a power law
behavior, $P(s)\sim 1/s^{\omega}$, with exponent $\omega=1/2$ for small $x$
(dashed line) and $\omega=3/2$ for $x\geq 1$ (dash-dotted line). Main panel:
Rescaled bond-bond correlation function
$P(s)/\left[c_{\text{P}}(g)/g^{3/2}\right]$ plotted as a function of $u=s/g$
as suggested by Eq. (30). For large $u$, where an incompressible packing of
thermal blobs is probed, all data collapse onto the dash-dotted line as
predicted by Eq. (32), i.e. $P(s)$ becomes independent of the compressibility
$g$. That this holds not only for the classical BFM with $x=\infty$ (stars)
but also for finite $x$ is the central result of this study.
|
arxiv-papers
| 2009-08-11T07:59:38 |
2024-09-04T02:49:04.535938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.P. Wittmer, A. Cavallo, T. Kreer, J. Baschnagel, A. Johner",
"submitter": "J. P. Wittmer",
"url": "https://arxiv.org/abs/0908.1474"
}
|
0908.1488
|
# Stability on Kähler-Ricci flow, I
Xiaohua Zhu Xiaohua Zhu
Department of Mathematics, Peking University, Beijing, 100871, China
xhzhu@math.pku.edu.cn
###### Abstract.
In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein
metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as
an initial Kähler metric is very closed to $g_{KE}$ (or $g_{KS}$) if a compact
Kähler manifold with $c_{1}(M)>0$ admits a Kähler Einstein metric $g_{KE}$ (or
a Kähler-Ricci soliton $g_{KS}$). The result improves Main Theorem in [TZ3] in
the sense of stability of Kähler-Ricci flow.
###### Key words and phrases:
Kähler-Ricci flow, Kähler-Einstein metric, Kähler-Ricci soliton
###### 1991 Mathematics Subject Classification:
Primary: 53C25; Secondary: 53C55, 58E11
Partially supported by NSF10425102 in China.
## 0\. Introduction
The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying
manifold $M$ is Kähler with positive first Chern class $c_{1}(M)>0$, it is
more natural to study the following Kähler-Ricci flow (normalized),
$\displaystyle\frac{\partial g(t,\cdot)}{\partial
t}=-\text{Ric}(g(t,\cdot))+g(t,\cdot),$ (0.1) $\displaystyle g(0,\cdot)=g,$
where $g$ is an initial Kähler metric with its Kähler form $\omega_{g}\in 2\pi
c_{1}(M)>0.$ It can be shown that (0.1) preserves the Kähler class. Moreover,
(0.1) has a global solution $g_{t}=g(t,\cdot)$ for any $t>0$ ([Ca]). So, the
main interest and difficulty of (0.1) is to study the limiting behavior of
$g_{t}$ as $t$ tends to $\infty$ (cf. [CT1], [CT2], [TZ3], etc.).
In this paper, we study a stability problem of Kähler-Ricci flow (0.1),
namely, we assume that $M$ admits a Kähler-Einstein metric or a Kähler-Ricci
soliton, and then we analysis the behavior of evolved Kähler metrics $g_{t}$
of (0.1). We shall prove
###### Theorem 0.1 (Main Theorem).
Let $M$ be a compact Kähler manifold with $c_{1}(M)>0$ which admits a Kähler
Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $(g_{KS,X_{0}})$ with
respect some holomorphic vector field $X_{0}$ on $M$) with its Kähler form in
$2\pi c_{1}(M)$. Let $\psi$ be a Kähler potential of an initial metric $g$ of
(0.1) and $\varphi=\varphi_{t}$ be a family of Kähler potentials of evolved
metrics $g_{t}$ of (0.1), i.e.,
$\omega_{g}=\omega_{KE}+\sqrt{-1}\partial\bar{\partial}\psi$ (or
$\omega_{g}=\omega_{KS}+\sqrt{-1}\partial\bar{\partial}\psi$) and
$\omega_{\varphi}=\omega_{KE}+\sqrt{-1}\partial\bar{\partial}\varphi$ (or
$\omega_{g}=\omega_{KS}+\sqrt{-1}\partial\bar{\partial}\varphi$), where
$\omega_{g}$, $\omega_{\varphi}=\omega_{g_{t}}$ and $\omega_{KE}$ (or
$\omega_{KS}$) denote Kähler forms of $g$, $g_{t}$ and $g_{KE}$ (or $g_{KS}$),
respectively. Then there exists a small $\epsilon$ such that if
$\|\psi-\underline{\psi}\|_{C^{2,\alpha}}\leq\epsilon,$
where
$\underline{\psi}=\frac{1}{\int_{M}\omega_{KE}^{n}}\int_{M}\psi\omega_{KE}^{n}$
(or
$\underline{\psi}=\frac{1}{\int_{M}\omega_{KS}^{n}}\int_{M}\psi\omega_{KS}^{n}$),
then there exist a family of holomorphisms $\sigma=\sigma_{t}$ on $M$ such
that Kähler potentials $(\varphi_{\sigma}-\underline{\varphi_{\sigma}})$ are
$C^{k}$-norm uniformly bounded, where
$\varphi_{\sigma}=\sigma^{*}\varphi+\rho$ and $\rho=\rho_{t}$ are Kähler
potentials defined by
$\rho^{*}(\omega_{KE})=\omega_{KE}+\sqrt{-1}\partial\overline{\partial}\rho$
and $\int_{M}e^{-\rho}\omega_{KE}^{n}=\int_{M}\omega_{KE}^{n}$ (or
$\rho^{*}(\omega_{KS})=\omega_{KS}+\sqrt{-1}\partial\overline{\partial}\rho$
and $\int_{M}e^{-\rho-X_{0}(\rho)}\omega_{KS}^{n}=\int_{M}\omega_{KS}^{n}$).
As a consequence, $g_{t}$ converge to $g_{KE}$ (or $g_{KS}$) smoothly in the
sense of Cheeger-Gromov.
The main step in the proof of Theorem 0.1 is to obtain a decay estimate for
$\dot{\varphi}$ and $\varphi$ both when one studies the convergence of Kähler-
Ricci flow as in [CT2], [PS], [TZ3] etc. In case that $M$ admits a Kähler
Einstein metric or $M$ admits a Kähler-Ricci soliton and an initial potential
$\psi$ is $K_{X_{0}}$-invariant, we can obtain an exponential decay estimate
for both $\dot{\varphi}$ and $\varphi$, so we can improve that Kähler
potentials $(\sigma^{*}\varphi+\rho)$ in the theorem exponentially converge to
zero as long as $\|\psi-\underline{\psi}\|_{C^{2,\alpha}}$ is small, where
$K_{X_{0}}$ is an one-parameter compact subgroup generated by the imaginary
part $X^{\prime}$ of $X_{0}$ ([TZ1],[TZ2]). This result is also obtained in
[TZ3] where a crucial step is to use the monotonicity and the properness of
the Mabuchi’s K-energy on a Kähler-Einstein manifold with $c_{1}(M)>0$ (or the
monotonicity and the properness of the generalized K-energy on a compact
Kähler manifold which admits a Kähler-Ricci soliton , cf. [CTZ]). But at the
present paper, we avoid to use these energies in our case of the stability
problem. This advantage allows us to remove the $K_{X_{0}}$-invariant
condition for the initial potential $\psi$ in case of Kähler-Ricci soliton in
Theorem 0.1, although we need more careful computations than the case of
Kähler-Einstein metric. Basically, we shall use the generalized Futaki-
invariant and the Gauge Transformation induced by the reductive subgroup
$\text{Aut}_{r}(M)$ of holomorphisms transformation group $\text{Aut}(M)$ on
$M$ to control the modified Kähler potentials $(\sigma^{*}\varphi+\rho)$ along
the Kähler-Ricci flow. We note that the definition of generalized Futaki-
invariant is independent of the choice of Kähler metric, which needs no
$K_{X_{0}}$-invariant condition ([TZ2]). Unfortunately, we could not improve
the convergence of $(\sigma^{*}\varphi+\rho)$ exponentially without the
assumption of $K_{X_{0}}$-invariant condition. But we believe that it is still
true if one can extend the Gauge Transformation $\text{Aut}_{r}(M)$ to
$\text{Aut}(M)$ (cf. Proposition 2.10).
Theorem 0.1 will be proved in Section 1 and Section 2 while in Section 1 we
consider the case of Kähler-Einstein metric and in Section 2, we consider the
case of Kähler-Ricci soliton. The rest of paper is as follows: In Section 3,
we prove a uniqueness result for the limit of Kähler-Ricci flow as an
application of Theorem 0.1; Section 4 and Section 5 are two appendixes, one is
a lemma about a $W^{k,2}$-estimate for $\dot{\varphi}_{t}$ and another is a
lemma about the existence of almost orthonormality of a Kähler potential to
the space of first eigenvalue-functions of operator $(P,\omega_{KS})$ defined
in Lemma 2.2 in Section 2.
The author would like to thank professor Gang Tian and professor Xiuxiong Chen
for valuable discussions.
## 1\. In case of Kähler-Einstein metric
In this section, we assume that $M$ admits a Kähler Einstein metric $g_{KE}$
with its Kähler form $\omega_{KE}\in 2\pi c_{1}(M)$. For simplicity, we set a
class of Kähler potentials by
$\displaystyle\mathcal{M}(\omega_{KE})=\\{\phi\in
C^{\infty}(M,R)|~{}~{}\omega_{KE}+\sqrt{-1}\partial\bar{\partial}\phi>0\\}.$
Then for any Kähler metric $g$ with its Kähler form $\omega_{g}\in 2\pi
c_{1}(M)$, we have
$\omega_{g}=\omega_{KE}+\sqrt{-1}\partial\bar{\partial}\psi$ for some
$\psi\in\mathcal{M}(\omega_{KE})$ and Kähler-Ricci flow (0.1) is equivalent to
a parabolic equation of complex Monge-Ampère type for Kähler potentials
$\varphi_{t}=\varphi(t,\cdot)$ with
$\omega_{g_{t}}=\omega_{KE}+\sqrt{-1}\partial\bar{\partial}\varphi_{t}$,
$\displaystyle\frac{\partial\varphi}{\partial
t}=\log\frac{\omega^{n}_{\varphi}}{\omega_{KE}^{n}}+\varphi,$ (1.1)
$\displaystyle\varphi(0)=\psi-\underline{\psi},$
where $\underline{\psi}=\frac{1}{V}\int_{M}\psi\omega_{KE}^{n}$ and
$V=\int_{M}\omega_{KE}^{n}$.
Set a Hölder space by
$\mathcal{K}(\epsilon_{0})=\\{\phi\in\mathcal{M}(\omega_{KE})|~{}~{}\|\phi-\underline{\phi}\|_{C^{2,\alpha}}\leq\epsilon_{0}\\}.$
Let $\text{Aut}_{0}(M)$ be the connected component of holomorphisms
transformation group of $M$ which contains the identity map of $M$. Then we
shall prove
###### Theorem 1.1.
There exists a small $\epsilon$ such that for any initial data
$\psi\in\mathcal{K}(\epsilon)$ in equation (1.1),
$\|\varphi-\underline{\varphi}\|_{C^{2,\alpha}}$ are uniformly bounded, where
$\varphi=\varphi_{t}=\varphi(t,\cdot)$ are evolved Kähler potentials of (1.1).
Moreover, there exist a family of $\sigma=\sigma_{t}\in\text{Aut}_{0}(M)$ such
that Kähler potentials $(\varphi_{\sigma}-\underline{\varphi_{\sigma}})$
converge exponentially to $0$ as $t\to\infty$, where
$\varphi_{\sigma}=(\sigma^{*}\varphi+\rho)$, and $\rho=\rho_{t}$ are Kähler
potentials defined by
$\displaystyle\sigma^{*}(\omega_{KE})=\omega_{KE}+\sqrt{-1}\partial\overline{\partial}\rho,$
(1.2) $\displaystyle\int_{M}e^{-\rho}\omega_{KE}^{n}=\int_{M}\omega_{KE}^{n}.$
As a consequence, Kähler metrics $\sigma^{*}(\omega_{\varphi})$ converge
exponentially to $\omega_{KE}$.
We need several lemmas to prove Theorem 1.1. Let $\Lambda_{1}(M,\omega_{KE})$
be a finite dimensional linear space of the first eigenvalue-functions of
Lapalace operator $\triangle_{\omega_{KE}}$ associated to the metric
$\omega_{KE}$. Then by using the Bochner formula, it is well-known that the
first non-zero eigenvalue is $1$ and
$\Lambda_{1}(M,\omega_{KE})=\text{span}\\{\theta_{X}|~{}X\in\eta(M)\\}$, where
$\eta(M)$ is a linear space consisting of holomorphic vector fields on $M$
which is isomorphic to the Lie algebra of $\text{Aut}_{0}(M)$ and $\theta_{X}$
is a potential of $X$ defined by
$\displaystyle\sqrt{-1}\overline{\partial}\theta_{X}=i_{X}(\omega_{KE}),$
(1.3) $\displaystyle\int_{M}\theta_{X}\omega_{KE}^{n}=0.$
By using the continuity of eigenvalues of Lapalcian operators, one sees
###### Lemma 1.2.
Let $\lambda_{1}(\omega_{\phi})$ and $\lambda_{2}(\omega_{\phi})$ be the first
and the second eigenvalues of Lapalcian operator associated to Kähler metric
$\omega_{\phi}$, respectively. Then there exists a $\delta_{0}$ such that for
any $\phi\in\mathcal{K}(\epsilon_{0})$, we have
$\displaystyle\lambda_{1}(\omega_{\phi})\geq
1+\delta_{0},~{}~{}\text{if}~{}~{}\eta(M)=0,$
$\displaystyle\lambda_{2}(\omega_{\phi})\geq
1+\delta_{0},~{}~{}\text{if}~{}~{}\eta(M)\neq 0,$
where $\epsilon_{0}$ is a small positive number.
Fix a large number $T$ and $N$, we can choose a sufficient small $\epsilon$
depends on $T$, $\epsilon_{0}$ and $N$ such that for any $t\leq T$, evolved
Kähler potentials $\varphi_{t}$ of (1.1) lie in
$\mathcal{K}(\frac{\epsilon_{0}}{2})$ and satisfy
(1.4)
$\displaystyle|\dot{\varphi}_{t}-c(t)|_{C^{0}}\leq(\frac{\epsilon_{0}}{2N})^{2},\text{
and }osc(\varphi_{t})\leq\frac{\epsilon_{0}}{4N},$
whenever $\|\psi-\underline{\psi}\|_{C^{2,\alpha}}\leq\epsilon$. Here
$c(t)=\frac{1}{V}\int_{M}\dot{\varphi}_{t}\omega_{\varphi_{t}}^{n}.$ Choose a
maximal $\delta(T)$ such that $\varphi_{t}\in\mathcal{K}(\epsilon_{0})$ for
any $t<T+\delta(T)$. We shall show that $\delta(T)$ must be the infinity
whenever $T$ and $N$ are large enough. First we prove
###### Lemma 1.3.
Let
$H(t)=\frac{1}{V}\int_{M}|\dot{\varphi}_{t}-c(t)|^{2}\omega_{\varphi_{t}}^{n}$.
Then for any $t\in[0,T+\delta(T))$, there exists a $\theta>0$ such that
(1.5) $\displaystyle H(t)\leq H(0)e^{-\theta t}.$
###### Proof.
For simplicity, we let $\varphi=\varphi_{t}$. By (1.1), one sees
$|\dot{\varphi}|\leq 3\epsilon_{0},~{}~{}\forall t\in[0,T+\delta(T)).$
Since $\varphi$ satisfies,
(1.6) $\displaystyle\ddot{\varphi}=\triangle\dot{\varphi}+\dot{\varphi},$
then by a direct computation, we have
$\displaystyle\frac{d}{dt}H_{0}(t)$
$\displaystyle=2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))(\ddot{\varphi}-\dot{c}(t))\omega^{n}_{\varphi}+\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}\triangle_{\varphi}\dot{\varphi}\omega_{\varphi}^{n}$
$\displaystyle=2\frac{1}{V}\int_{M}(\dot{\varphi}_{t}-c(t))(\triangle_{\varphi}\dot{\varphi}+\dot{\varphi})\omega^{n}_{\varphi}+\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}\triangle_{\varphi}\dot{\varphi}\omega_{\varphi}^{n}$
$\displaystyle=-2\frac{1}{V}\int_{M}|\nabla(\dot{\varphi}-c(t))|^{2}\omega^{n}_{\varphi}+2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}\omega^{n}_{\varphi}$
$\displaystyle-2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))\|\nabla(\dot{\varphi}-c(t))\|^{2}\omega_{\varphi}^{n}$
$\displaystyle=2H_{0}(t)-2\frac{1}{V}\int_{M}(1+\dot{\varphi}-c(t))\|\nabla(\dot{\varphi}-c(t))\|^{2}\omega_{\varphi}^{n}$
(1.7) $\displaystyle\leq
2H_{0}(t)-2(1-6\epsilon_{0})\frac{1}{V}\int_{M}\|\nabla(\dot{\varphi}-c(t))\|^{2}\omega_{\varphi}^{n}.$
Case 1, $\eta(M)=0$. Then by (1.7) and Lemma 1.1, we have
$\frac{d}{dt}H_{0}(t)\leq-[-2+2(1-6\epsilon_{0})(1+\delta_{0})]H_{0}(t).$
By choosing $\theta=-2+2(1-4\epsilon_{0})(1+\delta_{0})\geq\delta_{0}$, we
will get
(1.8) $\displaystyle H_{0}(t)\leq H_{0}(0)e^{-\theta t}.$
Case 2, $\eta(M)\neq 0$. Since the Futaki-invariant vanishes, for any
$X\in\eta(M)$, we have
$\int_{M}\triangle(\theta_{X}+X(\varphi))(\dot{\varphi}-c(t))\omega^{n}_{\varphi}=0,$
where $\theta_{X}$ is the potential of $X$ defined by (1.3) and $X(\phi)$ is
the derivative of $\phi$ along $X$. It follows
$|\int_{M}\theta_{X}(\dot{\varphi}-c(t))\omega^{n}_{\varphi}|\leq
C\epsilon_{0}\int_{M}|\dot{\varphi}-c(t)|\omega^{n}_{\varphi},$
for any $X\in\eta(M)$ with satisfying
$\int_{M}\|X\|^{2}_{\omega_{KE}}\omega_{KE}^{n}=1.$ Here we used an estimate
$\|\varphi-\underline{\varphi}\|_{C^{3}}=O(\epsilon_{0})$
by the regularity of Kähler potentials $\varphi\in\mathcal{K}(\epsilon_{0})$,
which can be obtained by the Implicity Functional Theorem for equation (1.22)
(cf. an argument at the last paragraph of this section) with the help of
$W^{k,2}$-estimate
$\|\dot{\varphi}-c(t)\|_{W^{k,2}}=O(\epsilon_{0})$
for $\dot{\varphi}$ (cf. the argument in Appendix 4.1). Thus by using the
continuity of the eigenvalue functions, one sees
(1.9)
$\displaystyle|\int_{M}\psi^{i}(\dot{\varphi}-c(t))\omega^{n}_{\varphi}|\leq
C^{\prime}\epsilon_{0}\int_{M}|\dot{\varphi}-c(t)|\omega^{n}_{\varphi},$
where $\psi^{i}$ are the first eigenvalue functions of the Lapalacian operator
associated to the metric $\omega_{\varphi}$, which satisfy
$\int_{M}|\psi^{i}|^{2}\omega_{\varphi}^{n}=1.$
Let $\Lambda_{1}(M,\omega_{\varphi})$ be a linear space spanned by a basis
$\\{\psi^{i}\\}$ and $\Lambda_{1}^{\bot}(M,\omega_{\varphi})$ be a subspace of
$L^{2}$-integral functions which are orthogonal to
$\Lambda_{1}(M,\omega_{\varphi})\cup\mathbb{R}$. Then we can decompose
$(\dot{\varphi}_{t}-c(t))$ as
$\dot{\varphi}_{t}-c(t)=\phi+\phi^{\bot},$
with $\phi\in\Lambda_{1}(M,\omega_{\varphi})$ and
$\phi^{\bot}\in\Lambda_{1}^{\bot}(M,\omega_{\varphi})$. Thus by (1.9), we get
$\int_{M}|\phi|^{2}\omega^{n}_{\varphi}\leq
A\epsilon_{0}^{2}\int_{M}(\dot{\varphi}_{t}-c(t))^{2}\omega^{n}_{\varphi},$
for some uniform constant $A$. It follows
(1.10)
$\displaystyle\int_{M}|\phi^{\bot}|^{2}\omega^{2}_{\varphi}\geq(1-A\epsilon_{0}^{2})\int_{M}(\dot{\varphi}_{t}-c(t))^{2}\omega^{2}_{\varphi}.$
Hence by Lemma 1.1, we get
$\displaystyle\int_{M}\|\nabla(\dot{\varphi}_{t}-c(t)\|^{2}\omega^{n}_{\varphi}$
$\displaystyle\geq\int_{M}\|\nabla\psi^{\bot}\|^{2}\omega^{n}_{\varphi}$
(1.11)
$\displaystyle\geq(1+\sigma_{0})(1-A\epsilon_{0}^{2})\int_{M}(\dot{\varphi}_{t}-c(t))^{2}\omega^{n}_{\varphi}.$
By choosing
$\theta=2(1-6\epsilon_{0})(1+\sigma_{0})(1-C\epsilon_{0}^{2})-2\geq\sigma_{0}$,
we obtain from (1.7),
$\frac{dH_{0}(t)}{dt}\leq-\theta H_{0}(t).$
As a consequence, we have
$H_{0}(t)\leq H_{0}(0)e^{-\theta t}.$
∎
Next we want to use Perelman’s deep estimates for the gradient of
$\dot{\varphi}_{t}$ and the non-collapsing result for metric
$\omega_{\varphi_{t}}$ to get a $C^{0}$-estimate of $\dot{\varphi}_{t}$ with
help of Lemma 1.3. Let’s state the Perelman’s result (a detailed proof can be
found in [ST]).
###### Lemma 1.4.
(Perelman) Let $g_{t}$ be the evolved Kähler metrics of (0.1) and
$\varphi=\varphi_{t}$ be Kähler potentials of $g_{t}$. Then there exists a
uniform constant $C$ independent of $t$ (just depending on the initial metric
$g$) such that the following two facts hold,
i) $\|\nabla\dot{\varphi}\|_{\omega_{\varphi}}\leq C$;
ii) for $x\in M$ and $0<r\leq\text{diam}(M,g(t))$,
$\int_{B_{r}(x)}\omega^{n}>C^{-1}r^{2n}$, where $\text{diam}(M,g(t))$ denote
the diameters of $(M,g(t))$ which are uniformly bounded.
###### Lemma 1.5.
For any $t\in[0,T+\delta(T))$, we have
(1.12)
$\displaystyle|\dot{\varphi}_{t}-c(t)|\leq\min\\{(\frac{\epsilon_{0}}{2N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}$
and
(1.13) $\displaystyle||\dot{\varphi}_{t}-c(t)||_{C^{\alpha}}\leq
C\\{\min\\{(\frac{\epsilon_{0}}{2N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}\\}^{\frac{1}{2}}.$
Here $\alpha\leq\frac{1}{4}$ and $C$ depends only on the constant in Lemma
1.4.
###### Proof.
We suffice to consider the case of $\alpha=\frac{1}{4}$. Let $x_{0}$ be the
point where $|\tilde{h}|=|\tilde{h}_{t}|=|\dot{\varphi}_{t}-c(t)|$ achieves
its maximum. Choose a small ball $B_{r}(x_{0})$ for
$r=\min\\{\text{diam}(M,g(t)),e^{-\frac{\theta}{2(n+1)}t}\\}$. So we have for
any $x\in B_{r}(x_{0})$,
(1.14) $\displaystyle
0\leq|\tilde{h}(x_{0})|\leq|\tilde{h}(x)|+\|\nabla\tilde{h}\|r=|\tilde{h}(x)|+\|\nabla
h\|r.$
Case 1), $e^{-\frac{\theta}{2(n+1)}t}\geq\text{diam}(M,g(t))$. Then by Lemma
1.4, one sees
$\displaystyle\int_{M}|\tilde{h}(x_{0})|^{2}\omega^{n}_{\varphi}$
$\displaystyle\leq
2\int_{M}|\tilde{h}(x)|^{2}\omega^{n}_{\varphi}+2V\text{diam}(M,g(t))^{2}\|\nabla\tilde{h}\|^{2}$
$\displaystyle\leq Ce^{-\frac{\theta}{n+1}t}.$
Thus
(1.15) $\displaystyle|\tilde{h}(x_{0})|\leq Ce^{-\frac{\theta}{2(n+1)}t}.$
Case 2), $e^{-\frac{\theta}{2(n+1)}t}\leq\text{diam}(M,g(t))$. Then
$\displaystyle\frac{1}{V(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}|\tilde{h}(x_{0})|^{2}\omega^{n}_{\varphi}$
$\displaystyle\leq\frac{2}{V(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}|\tilde{h}(x)|^{2}\omega^{n}_{\varphi}$
$\displaystyle+\frac{2}{V(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}\|\nabla\tilde{h}\|^{2}r^{2}\omega^{n}_{\varphi}.$
Thus by Lemma 1.4, we get
$\displaystyle|\tilde{h}(x_{0})|^{2}$ $\displaystyle\leq
Ce^{\frac{2n\theta}{2(n+1)}t}\int_{M}|\tilde{h}(x)|^{2}\omega^{n}_{\varphi}+Cr^{2}$
$\displaystyle\leq Ce^{-\frac{\theta}{n+1}t}.$
It follows
(1.16) $\displaystyle|\tilde{h}(x_{0})|\leq
C^{\prime}e^{-\frac{\theta}{2(n+1)}t}.$
Therefore, both (1.15) and (1.16) give the estimate (1.12).
For any $x,y\in M$, by (1.12), we have: if
$\text{dist}(x,y)=\|x-y\|_{\omega_{\varphi}}\geq e^{-\frac{\theta}{2(n+1)}t}$,
$\displaystyle\frac{|\tilde{h}(x)-\tilde{h}(y)|}{\|x-y\|^{\frac{1}{4}}_{\omega_{KE}}}\leq
2\frac{|\tilde{h}(x)-\tilde{h}(y)|}{\|x-y\|^{\frac{1}{4}}_{\omega_{\varphi}}}$
$\displaystyle\leq
2|\tilde{h}(x)-\tilde{h}(y)|^{\frac{1}{2}}\frac{|\tilde{h}(x)-\tilde{h}(y)|^{\frac{1}{4}}_{\omega_{\varphi}}}{\|x-y\|^{\frac{1}{4}}_{\omega_{\varphi}}}$
(1.17) $\displaystyle\leq
C\\{\min\\{(\frac{\epsilon_{0}}{2N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}\\}^{\frac{1}{2}};$
if $\text{dist}(x,y)\leq e^{-\frac{\theta}{2(n+1)}t}$,
$\displaystyle\frac{|\tilde{h}(x)-\tilde{h}(y)|}{\|x-y\|^{\frac{1}{4}}_{\omega_{KE}}}\leq
2\frac{|\tilde{h}(x)-\tilde{h}(y)|}{\|x-y\|^{\frac{1}{4}}_{\omega_{\varphi}}}$
$\displaystyle=2\frac{|\tilde{h}(x)-\tilde{h}(y)|^{\frac{1}{2}}|\tilde{h}(x)-\tilde{h}(y)|^{\frac{1}{2}}}{\|x-y\|^{\frac{1}{2}}_{\omega_{\varphi}}}\|x-y\|^{\frac{1}{4}}_{\omega_{\varphi}}$
$\displaystyle\leq
C|\tilde{h}|^{\frac{1}{2}}_{C^{0}}(\text{diam}(M,g(t)))^{\frac{1}{4}}$ (1.18)
$\displaystyle\leq
C^{\prime}\\{\min\\{(\frac{\epsilon_{0}}{2N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}\\}^{\frac{1}{2}}.$
Here we used Perelman’s estimates again. (1.17) and (1.18) give the estimate
(1.13). ∎
###### Remark 1.6.
We can avoid to use Perelman’s estimates to prove Lemma 1.5 by replacing to
estimate the $W^{k,2}$-norm of $\dot{\varphi}_{t}$. See Appendix 1 in this
paper.
###### Proposition 1.7.
Choose some large $T$ such that
$C\frac{4(n+1)}{\theta}e^{-\frac{\theta}{2(n+1)}T}\leq\frac{\epsilon_{0}}{4N},$
where $C$ is the constant chosen in Lemma 1.5. Then
(1.19)
$\displaystyle|\tilde{\varphi}|\leq\frac{3\epsilon_{0}}{4N},~{}~{}\forall~{}t\in[0,T+\delta(T)),$
where
$\tilde{\varphi}=\tilde{\varphi}_{t}=\varphi(t)-\frac{1}{V}\int_{M}\varphi\omega_{\varphi}^{n}.$
###### Proof.
Notice that
$\frac{d}{dt}\tilde{\varphi}=\tilde{h}-\frac{1}{V}\int_{M}\tilde{h}\triangle_{\varphi}\varphi\omega_{\varphi}^{n}.$
Then by Lemma 1.5, we have
$\displaystyle\tilde{\varphi}$
$\displaystyle=\tilde{\varphi}_{T}+\int_{T}^{T+\delta(T)}\tilde{h}dt-\int_{T}^{T+\delta(T)}\frac{1}{V}\int_{M}\tilde{h}\triangle_{\varphi}\varphi\omega_{\varphi}^{n}dt$
$\displaystyle\leq\frac{\epsilon_{0}}{2N}+C\int_{T}^{T+\delta(T)}e^{-\frac{\theta}{2(n+1)}t}dt+2C\epsilon_{0}\int_{T}^{T+\delta(T)}e^{-\frac{\theta}{2(n+1)}t}dt$
$\displaystyle\leq\frac{\epsilon_{0}}{2N}+2C\frac{2(n+1)}{\theta}e^{-\frac{\theta}{2(n+1)}T}.$
∎
###### Proof of Theorem 1.1.
First we want to show that $\varphi_{t}\in\mathcal{K}(\epsilon_{0})$ for any
$t>0$. By the contradiction, we may assume that there exists a number
$\delta(T)<\infty$ such that $\varphi_{t}\in\mathcal{K}(\epsilon_{0})$ for any
$t<T+\delta(T)$ and there exists a sequence of $t_{i}\to T+\delta(T)$ such
that
(1.20)
$\displaystyle\|\overline{\varphi_{t_{i}}}\|_{C^{2,\alpha}}=\|\varphi_{t_{i}}-\underline{\varphi_{t_{i}}}\|_{C^{2,\alpha}}\rightarrow\epsilon_{0}.$
Let $b_{t}$ be a constant so that $\overline{\varphi}=\tilde{\varphi}+b_{t}$.
Then by Proposition 1.7, it is easy to see $b_{t}\leq\frac{2\epsilon_{0}}{N}$.
Decompose $\overline{\varphi}$ by $\overline{\varphi}=\phi+\phi^{\bot}$, where
$\phi\in\Lambda_{1}(M,\omega_{KE})$ and
$\phi^{\bot}\in\Lambda_{1}^{\bot}(M,\omega_{KE})$, where
$\Lambda_{1}^{\bot}(M,\omega_{KE})$ is a subspace of $L^{2}$-integral
functions which are orthogonal to $\Lambda_{1}(M,\omega_{KE})\cup\mathbb{R}$.
Thus $\phi=\sum_{i}a_{i}\theta_{i}$ for some constants $a_{i}$, where
$\theta_{i}$ is a basis of the space $\Lambda_{1}(M,\omega_{KE})$. As a
consequence, by Proposition 1.7, we have $|a_{i}|\leq\frac{2\epsilon_{0}}{N}$,
so
(1.21) $\displaystyle\|\phi\|_{C^{2,\alpha}}\leq\frac{A_{0}\epsilon_{0}}{N},$
for some uniform constant $A_{0}$.
By equation (1.1), we have
(1.22)
$\displaystyle\omega_{\varphi}^{n}=\omega_{KE}^{n}e^{\tilde{h}+\overline{\varphi}+a},$
where $\tilde{h}=\dot{\phi}_{t}-c_{t}$ and $a=a_{t}$ are constants. By Lemma
1.5 and Proposition 1.7, it is easy to see that
$|a|\leq\frac{4A_{0}\epsilon_{0}}{N}$. Let $P$ be a projection from Banach
space $H^{2,\alpha}(M)$ to Banach space
$H^{\alpha}(M)\cap\Lambda_{1}^{\bot}(M,\omega_{KE})$. Then $\phi^{\bot}$ is a
solution of equation
$P[\log(\frac{[\omega_{\phi+\phi^{\bot}}]^{n}}{\omega_{KE}^{n}})]-\phi^{\bot}=P(\tilde{h}+a),$
where $\phi$ and $\tilde{h}+a$ are regarded as two perturbation functions. On
the other hand, by Lemma 1.5, we have
$\|P(\tilde{h}+a)\|_{C^{\alpha}}=\|P(\tilde{h})\|_{C^{\alpha}}\leq
C\min\\{(\frac{2\epsilon_{0}}{N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}^{\frac{1}{2}}.$
Thus by using the Implicity Functional Theorem, we get
(1.23) $\displaystyle\|\phi^{\bot}\|_{C^{2,\alpha}}\leq
c=O(\frac{\epsilon_{0}}{N}),$
where constant $c$ is independent of $t$ and $\epsilon_{0}$ and goes to zero
as $N\to\infty$. Consequently, $c\leq\frac{\epsilon_{0}}{4}$ by choosing a
large number $N$. Hence by combining (1.21) and (1.23), we obtain
(1.24)
$\displaystyle\|\overline{\varphi}\|_{C^{2,\alpha}}\leq\frac{\epsilon_{0}}{2}.\forall~{}~{}t\in[T,T+\delta(T)).$
But this is impossible according to (1.20). Therefore we prove that
$\varphi_{t}\in\mathcal{K}(\epsilon_{0})$ for any $t>0$.
By the above argument and lemma 1.5 and Proposition 1.7, we conclude that
there exists an $\epsilon$ such that if
$\|\psi-\underline{\psi}\|_{C^{2,\alpha}}\leq\epsilon$, then for any $t>0$, we
have
(1.25) $\displaystyle\text{a)}~{}~{}\varphi_{t}\in\mathcal{K}(\epsilon_{0}),$
(1.26)
$\displaystyle\text{b)}~{}~{}|\tilde{\varphi}|\leq\frac{3\epsilon_{0}}{4N},$
(1.27) $\displaystyle\text{c)}~{}~{}\|\tilde{h}\|_{C^{\alpha}}\leq
C\\{\min\\{(\frac{\epsilon_{0}}{2N})^{2},Ce^{-\frac{\theta}{2(n+1)}t}\\}\\}^{\frac{1}{2}}.$
On the other hand, according to [BM], one can choose an element
$\sigma_{t}\in\text{Aut}_{0}(M)$ for each $\varphi$ such that potential
$(\varphi_{\sigma}-\underline{\varphi_{\sigma}})$ lies in
$\Lambda_{1}^{\bot}(M,\omega_{KE})$, where
$\varphi_{\sigma}=\varphi_{\sigma_{t}}=\varphi_{t}(\sigma_{t}(.))+\rho_{t}(.)$
and $\rho_{t}$ is Kähler potential defined by (1.2). Furthermore, by the fact
$\varphi\in\mathcal{K}(\epsilon_{0})$, one can prove easily
$\text{dist}(\sigma,Id)\leq 1.$
Consequently, by (1.27), we have
$\|\tilde{h}(\sigma_{t}(.))\|_{C^{\alpha}}\leq Ce^{-\frac{\theta}{(n+1)}t}.$
Thus by applying the Implicity Functional Theorem to the modified equation of
(1.22),
$\omega_{\varphi_{\sigma}}^{n}=\omega_{KE}^{n}e^{\tilde{h}(\sigma_{t}(.))-\varphi_{\sigma}+a},$
we have
$\|\varphi_{\sigma}-\underline{\varphi_{\sigma}}\|_{C^{2,\alpha}}\leq
C(\|\tilde{h}(\sigma_{t}(.))\|_{C^{\alpha}}).$
Furthermore, one can get an explicit estimate
$\|\varphi_{\sigma}-\underline{\varphi_{\sigma}}\|_{C^{2,\alpha}}\leq
2\|\tilde{h}(\sigma_{t}(.))\|_{C^{\alpha}}\leq
C^{\prime}e^{-\frac{\theta}{(n+1)}t}.$
To get higher-order estimates for the modified Kähler potentials
$\varphi_{\rho}$, one can use Lemma 4.1 in Appendix 1 and the embedding theory
of Sobolev spaces to obtain
$\|\tilde{\dot{\varphi}}\|_{C^{k,\alpha}}\leq
C_{k}e^{-\frac{\theta}{n+1}t},~{}~{}\forall~{}t>0,$
where constants $C_{k}$ depends only on $k,\epsilon_{0}$ and higher-order
derivatives of the initial Kähler potential $\psi$ (we may assume that $\psi$
is smooth since we can replace it by an evolved Kähler metric
$\varphi_{t=1}$). Then by the Implicity Functional Theorem as the above, we
derive
$\|\varphi_{\sigma}-\underline{\varphi_{\sigma}}\|_{C^{k+2,\alpha}}\leq
2\|\tilde{\dot{\varphi}}\|_{C^{k,\alpha}}\leq
C_{k}^{\prime}e^{-\frac{\theta}{n+1}t}.$
Therefore we prove that Kähler metrics $\sigma^{*}(\omega_{\varphi})$ converge
exponentially to $\omega_{KE}$.
∎
## 2\. In case of Kähler-Ricci soliton
In this section, we assume that $M$ admits a Kähler Ricci soliton
$(\omega_{KS},X_{0})$ with some holomorphic vector field $X_{0}$ on $M$, i.e.,
$(\omega_{KS},X_{0})$ satisfies equation,
$\text{Ric}(\omega_{KS})-\omega_{KS}=L_{X_{0}}\omega_{KS},$
where $L_{X_{0}}$ is a Lie derivative along the vector field $X_{0}$. By the
Hodge theorem, one can define a real-valued potential $\theta_{X}$ of $X_{0}$
by
$\displaystyle
L_{X_{0}}\omega_{KS}=\sqrt{-1}\partial\overline{\partial}\theta_{X},$
$\displaystyle\int_{M}e^{\theta_{X}}\omega_{KS}^{n}=\int_{M}\omega_{KS}^{n}.$
So if we let $X$ and $X^{\prime}$ be a real part and imaginary part of
$X_{0}$, respectively, then for any $\phi\in\mathcal{M}(\omega_{KS})$, we have
(2.1) $\displaystyle
L_{X_{0}}\omega_{\phi}=\sqrt{-1}\partial\overline{\partial}(\theta_{X}+X_{0}(\phi)),$
and so
$L_{X}\omega_{\phi}=\sqrt{-1}\partial\overline{\partial}(\theta_{X}+X(\phi))$
and
$L_{X^{\prime}}\omega_{\phi}=\sqrt{-1}\partial\overline{\partial}(X^{\prime}(\phi)).$
(2.1) also implies that for any $\psi\in C^{\infty}(M)$ it holds
$<\overline{\partial}(\theta_{X}+X_{0}(\phi)),\overline{\partial}\psi>_{\omega_{\phi}}=X_{0}(\psi)=X(\psi)+\sqrt{-1}X^{\prime}(\psi).$
Thus
(2.2)
$\displaystyle|<\nabla(\theta_{X}+X(\phi)),\nabla\psi>_{\omega_{\phi}}-X(\psi)|\leq|X^{\prime}(\phi)|\|\nabla\psi\|_{\omega_{\phi}}.$
We now consider a modified equation of (0.1),
$\displaystyle\frac{\partial g(t,\cdot)}{\partial t}=-\text{Ric}(g)+g+L_{X}g,$
(2.3) $\displaystyle g(0)=g.$
Then (2.3) is equivalent to a parabolic equation of complex Monge-Ampère type,
$\displaystyle\frac{\partial\varphi}{\partial
t}=\log\frac{\omega^{n}_{\varphi}}{\omega_{KS}^{n}}+\varphi+X(\varphi),$ (2.4)
$\displaystyle\varphi(0)=\psi-\underline{\psi},$
where $\underline{\psi}=\frac{1}{V}\int_{M}\psi\omega_{KS}^{n}$,
$V=\int_{M}\omega_{KE}^{n}$, and $\varphi=\varphi_{t}$ are potentials of
evolved Kähler metrics $g_{t}$ of (2.3). Let $K_{X_{0}}$ be an one-parameter
compact subgroup of $\text{Aut}_{0}(M)$ generated by the imaginary part
$X^{\prime}$ of $X_{0}$. By choosing a reductive subgroup $\text{Aut}_{r}(M)$
of $\text{Aut}_{0}(M)$ such that $\text{Aut}_{r}(M)$ contains $K_{X_{0}}$, we
can prove
###### Theorem 2.1.
Let $M$ be a compact Kähler manifold $M$ with $c_{1}(M)>0$ which admits a
Kähler-Ricci soliton $\omega_{KS}$. Then there exists a small $\epsilon$ such
that for any initial data, potential $\psi\in\mathcal{M}(\omega_{KS})$ in
equation (2.4) with $\psi\in\mathcal{K}(\epsilon)$, there exist a family of
$\sigma=\sigma_{t}\in\text{Aut}_{r}(M)$ for evolved Kähler potentials
$\varphi=\varphi_{t}$ of (2.4) at $t$ such that Kähler potentials
$(\varphi_{\sigma}-\underline{\varphi_{\sigma}})$ are $C^{k}$-norm uniformly
bounded, where $\varphi_{\sigma}=\sigma^{*}\varphi+\rho$ and $\rho=\rho_{t}$
are Kähler potentials defined by
$\displaystyle\rho^{*}(\omega_{KS})=\omega_{KS}+\sqrt{-1}\partial\overline{\partial}\rho,$
(2.5) $\displaystyle\int_{M}e^{-\rho-
X_{0}(\rho)}\omega_{KS}^{n}=\int_{M}\omega_{KS}^{n}.$
As a consequence, evolved Kähler metrics $g_{t}$ of (2.3) converge to $g_{KS}$
smoothly in the sense of Cheeger-Gromov. Furthermore, if in addition that
$\psi$ is $K_{X_{0}}$-invariant, then there exist a family of
$\sigma=\sigma_{t}\in\text{Aut}_{r}(M)$ such that
$(\varphi_{\sigma}-\underline{\varphi_{\sigma}})$ converge exponentially to
$0$ as $t\to\infty$, and consequently Kähler metrics
$\sigma^{*}(\omega_{\varphi})$ converge exponentially to $\omega_{KS}$.
As in Section 1, to prove Theorem 2.1, we need to estimate $\dot{\varphi}$ of
Kähler potentials $\varphi=\varphi_{t}$ of (2.4). We introduce a modified
functional of $H_{0}(t)$ by
$\tilde{H}_{0}(t)=\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}e^{\tilde{h}}\omega^{n}_{\varphi},$
where $c(t)=\int_{M}\dot{\varphi}e^{\tilde{h}}\omega_{\varphi}^{n}$ is a
constant, $\tilde{h}=\tilde{h}_{t}=\theta_{X}+X(\varphi)-\dot{\varphi}$ and
$V=\int_{M}\omega_{KS}^{n}$. By a direct computation, one shows
$\displaystyle\frac{d}{dt}\tilde{H}_{0}(t)=2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))(\ddot{\varphi}-\dot{c}(t))e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle+\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}(\triangle_{\varphi}\dot{\varphi}+X(\dot{\varphi})-\ddot{\varphi})e^{\tilde{h}}\omega_{\varphi}^{n}$
$\displaystyle=2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))(\triangle_{\varphi}\dot{\varphi}+\dot{\varphi}+X(\dot{\varphi})-c(t))e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle+\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}(\triangle_{\varphi}\dot{\varphi}+X(\dot{\varphi})-\ddot{\varphi})e^{\tilde{h}}\omega_{\varphi}^{n}$
$\displaystyle=2\frac{1}{V}\int_{M}[\dot{\varphi}-c(t)](\triangle_{\varphi}\dot{\varphi}+X(\dot{\varphi}))e^{\tilde{h}}\omega^{n}_{\varphi}$
(2.6)
$\displaystyle+2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}(1+\dot{\varphi})e^{\tilde{h}}\omega^{n}_{\varphi}.$
On the other hand, by (2.2), we see
$\displaystyle\int_{M}[\dot{\varphi}-c(t)](\triangle_{\varphi}\dot{\varphi}+X(\dot{\varphi}))e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle=-\int_{M}\|\nabla(\dot{\varphi}-c(t))\|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle+\int_{M}[\dot{\varphi}-c(t)][X(\dot{\varphi})-<\nabla(\theta_{X}+X(\phi)-\dot{\varphi}),\nabla\dot{\varphi}>]e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle\leq-\int_{M}\|\nabla(\dot{\varphi}-c(t))\|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}+V|\dot{\varphi}-c(t)||X^{\prime}(\dot{\varphi})|\|\nabla{\dot{\varphi}}\|.$
Thus inserting the above inequality into (2.6), we get
$\displaystyle\frac{d}{dt}\tilde{H}_{0}(t)\leq
2\frac{1}{V}\int_{M}(\dot{\varphi}-c(t))^{2}(1+\dot{\varphi})e^{\tilde{h}}\omega^{n}_{\varphi}$
$\displaystyle-2\frac{1}{V}\int_{M}\|\nabla(\dot{\varphi}-c(t))\|^{2}e^{\tilde{h}}\omega_{\varphi}^{n}$
(2.7)
$\displaystyle+2|\dot{\varphi}-c(t)||X^{\prime}(\dot{\varphi})|\|\nabla{\dot{\varphi}}\|.$
We shall estimate the $L^{2}$-integral of $\nabla\dot{\varphi}$ and need the
following lemma,
###### Lemma 2.2.
Let $P=(P,\omega_{\phi})$ be an elliptic operator on $C^{k,\alpha}(M)$ defined
by
$P\psi=\triangle\psi+\psi+\text{Re}<\overline{\partial}{h},\overline{\partial}\psi>_{\omega_{\phi}},$
where $\triangle$ is the Lapalace operator with respect to a Kähler metric
$\omega_{\phi}$ and $h$ is a Ricci potential of $\omega_{\phi}$. Then
$\text{ker}(P,\omega_{\phi})\subset\eta_{r}(M),$ where $\eta_{r}(M)$ is a
reductive part of Lie algebraic $\eta(M)$ consisting of all holomorphic vector
fields on $M$. Moreover, if $\omega_{\phi}=\omega_{KS}$, then
$\text{ker}(P,\omega_{KS})\cong\eta_{r}(M)$.
###### Proof.
Let
$L\psi=\triangle\psi+\psi+<\overline{\partial}{h},\overline{\partial}\psi>_{\omega_{\phi}}$
and
$\overline{L}\psi=\triangle\psi+\psi+\overline{<\overline{\partial}{h},\overline{\partial}\psi>_{\omega_{\phi}}}$,
where $h$ is a Ricci potential of the metric $\omega_{\phi}$. Then by the
Bochner formula, one can show (cf. Lemma 3.1 in [TZ3]),
(2.8) $\displaystyle\int_{M}-(L\psi)\psi
e^{h}\omega_{\phi}^{n}=\int_{M}(\|\nabla\psi\|^{2}-\psi^{2})e^{h}\omega_{\phi}^{n}\geq
0,$
and
(2.9) $\displaystyle\int_{M}-(\overline{L}\psi)\psi
e^{h}\omega_{\phi}^{n}=\int_{M}(\|\nabla\psi\|^{2}-\psi^{2})e^{h}\omega_{\phi}^{n}\geq
0.$
Moreover, the equality (2.8) or (2.9) holds if and only if the corresponding
vector field of $(0,1)$-form $\overline{\partial}\phi$ is holomorphic. Thus
$\displaystyle-2\int_{M}(P\psi)\psi
e^{h}\omega_{\phi}^{n}=-\int_{M}(L\psi+\overline{L}\psi)\psi
e^{h}\omega_{\phi}^{n}$ (2.10)
$\displaystyle=2\int_{M}(\|\nabla\psi\|^{2}-\psi^{2})e^{h}\omega_{\phi}^{n}\geq
0,$
and the equality holds if and only if the corresponding vector field of
$(0,1)$-form $\overline{\partial}\psi$ is holomorphic. Since $\psi$ is a real-
valued function the corresponding vector field must lie in $\eta_{r}(M)$.
Furthermore, if one defines a potential $\theta_{Y}^{\prime}$ by
$L_{Y}\omega_{KS}=\sqrt{-1}\partial\overline{\partial}\theta_{Y}^{\prime}~{}~{}\text{and}~{}~{}\int_{M}\theta_{Y}^{\prime}e^{\theta_{X}}\omega_{KS}^{n}=0,$
for an element $Y$ in $\eta_{r}(M)$, then in case of
$\omega_{\phi}=\omega_{KS}$, by using the fact that $X_{0}$ is an element of
center of $\eta_{r}(M)$ [TZ1] and $h=\theta_{X_{0}}$, one can show
$\theta_{Y}^{\prime}$ must be in $\text{ker}(P,\omega_{KS})$. ∎
Set a Banach space by
$\overline{\mathcal{K}}(\epsilon_{0})=\\{\phi\in\mathcal{M}(\omega_{KS})|~{}~{}\|\phi-\underline{\phi}\|_{C^{2,\alpha}}\leq\epsilon_{0}\\}.$
Then we have
###### Lemma 2.3.
Let $\varphi=\varphi_{t}$ be an evolved Kähler potential of (2.4) at $t$ and
$\theta_{Y}^{\prime}\in\text{ker}(P,\omega_{\varphi})$ be a potential of
$Y\in\eta_{r}(M)$ with $\int_{M}\|Y\|^{2}\omega_{KS}^{n}=1$. If
$\varphi\in\overline{\mathcal{K}}(\epsilon_{0})$, then there exist two uniform
constants $C_{1}$ and $C_{2}$ such that
$\displaystyle|\int_{M}\theta_{Y}^{\prime}(\dot{\varphi}_{t}-c(t))e^{\theta_{X}+X(\varphi)}\omega_{\varphi}^{n}|$
(2.11) $\displaystyle\leq
C_{1}\epsilon_{0}\int_{M}|\dot{\varphi}_{t}-c(t)|e^{\theta_{X}+X(\varphi)}\omega_{\varphi}^{n}+C_{2}\epsilon_{0}^{2}.$
###### Proof.
Recall a generalized Futaki-invariant defined in [TZ2] by
$F_{X_{0}}(Y)=\int_{M}Y[h_{\omega_{\phi}}-(\theta_{X}+X_{0}(\phi))]e^{\theta_{X}+X_{0}(\phi))}\omega_{\phi}^{n},~{}~{}\forall~{}Y\in\eta(M).$
It was proved that the invariant is independent of the choice of Kähler metric
$\omega_{\phi}$ on $M$ and the invariant vanishes if $M$ admits the Kähler-
Ricci soliton $(\omega_{KS},X_{0})$. So we have
$F_{X_{0}}(Y)\equiv 0,~{}~{}\forall~{}Y\in\eta(M).$
By applying the metrics $\omega_{\varphi}$ to the above identity, one sees
$\int_{M}Y[\dot{\varphi}-c(t)-\sqrt{-1}X^{\prime}(\varphi)]e^{\theta_{X}+X_{0}(\varphi))}\omega_{\varphi}^{n}=0,~{}~{}\forall~{}Y\in\eta_{r}(M).$
It follows
$\displaystyle|\text{Re}(\int_{M}Y[\dot{\varphi}-c(t)-\sqrt{-1}X^{\prime}(\varphi)]e^{\theta_{X}+X(\varphi)+\cos(X^{\prime}(\varphi))}\omega_{\varphi}^{n})|$
(2.12) $\displaystyle\leq A_{0}\|\varphi\|_{C^{2}}|X^{\prime}(\varphi)|.$
On the other hand, by using the Stoke’s formula, we have
$\displaystyle\int_{M}Y[\dot{\varphi}-c(t)-\sqrt{-1}X^{\prime}(\varphi)]e^{\theta_{X}+X(\varphi)+\ln\cos(X^{\prime}(\varphi))}\omega_{\varphi}^{n}$
$\displaystyle=-\int_{M}(\dot{\varphi}-c(t))-\sqrt{-1}X^{\prime}(\varphi))$
$\displaystyle\times[\triangle(\theta_{Y}^{\prime}+Y(\varphi))+<\overline{\partial}(\theta_{Y}^{\prime}+Y(\varphi)),\overline{\partial}(\theta_{X}+X(\varphi)+\ln\cos(X^{\prime}(\varphi)))>]$
$\displaystyle\times
e^{\theta_{X}+X(\varphi)+\ln\cos(X^{\prime}(\varphi))}\omega_{\varphi}^{n}$
$\displaystyle=\int_{M}(\dot{\varphi}-c(t)-\sqrt{-1}X^{\prime}(\varphi))(\triangle\theta_{Y}^{\prime}+<\overline{\partial}\theta_{Y}^{\prime},\overline{\partial}\theta_{X}>)$
$\displaystyle\times
e^{\theta_{X}+X(\varphi)+\ln\cos(X^{\prime}(\varphi))}\omega_{\varphi}^{n}+O(\epsilon_{0}^{2})$
$\displaystyle=\int_{M}(\dot{\varphi}-c(t)-\sqrt{-1}X^{\prime}(\varphi))(\triangle_{\omega_{KS}}\theta_{Y}^{\prime}+<\overline{\partial}\theta_{Y}^{\prime},\overline{\partial}\theta_{X}>_{\omega_{KS}})$
(2.13) $\displaystyle\times
e^{\theta_{X}+X(\varphi)+\ln\cos(X^{\prime}(\varphi))}\omega_{\varphi}^{n}+O(\epsilon_{0}^{2}).$
Note that
$<\overline{\partial}\theta_{Y}^{\prime},\overline{\partial}\theta_{X}>_{\omega_{KS}}=Y(\theta_{X})=X(\theta_{Y}^{\prime})=<\overline{\partial}\theta_{X},\overline{\partial}\theta_{Y}^{\prime}>_{\omega_{KS}}$
is a real-valued function ([TZ1]). Thus
$\triangle_{\omega_{KS}}\theta_{Y}^{\prime}+<\overline{\partial}\theta_{Y}^{\prime},\overline{\partial}\theta_{X}>_{\omega_{KS}}=\triangle_{\omega_{KS}}\theta_{Y}^{\prime}+<\overline{\partial}\theta_{X},\overline{\partial}\theta_{Y}^{\prime}>_{\omega_{KS}}=-\theta_{Y}^{\prime}.$
Therefore, inserting (2.13) into (2.12), one will get (2.11). ∎
By using Lemma 2.2 and Lemma 2.3, we can complete the $L^{2}$-estimate of
$\dot{\varphi}$.
###### Lemma 2.4.
Let $\epsilon_{0}<<1$. Then
(2.14) $\displaystyle\tilde{H}_{0}(t)\leq\tilde{H}_{0}(0)e^{-\theta
t}+\frac{B_{0}}{\theta}\epsilon_{0}^{3},~{}~{}\forall~{}~{}t\in[0,T),$
if $\varphi_{t}$ lies in $\overline{\mathcal{K}}(\epsilon_{0})$ and
$\tilde{H}_{0}(t)\geq\frac{B_{0}}{\theta}\epsilon_{0}^{3}$ for any $t$ in
$[0,T)$, where the constant $B_{0}=B_{0}(\|X^{\prime}\|_{C^{0}})$ depends only
on $\|X^{\prime}\|_{C^{0}}$ and the constant $\theta>0$ depends only on the
gap of the first two eigenvalues of the operator $P$ associated to the metric
$\omega_{KS}$ in Lemma 2.2.
###### Proof.
Let $\psi^{i}$ be the first eigenvalue functions of the operator
$(P,\omega_{\varphi})$ with respect to the metric $\omega_{\varphi}$ with
satisfying $\int_{M}|\psi^{i}|^{2}e^{\tilde{h}}\omega_{\varphi}^{n}=1.$ Then
by the continuity of eigenvalue functions and (2.11), one sees that there
exists two constants $C$ and $A_{0}$ such that
$\displaystyle|\int_{M}\psi^{i}(\dot{\varphi}_{t}-c(t))e^{\tilde{h}}\omega_{\varphi}^{n}$
(2.15) $\displaystyle\leq
C\epsilon_{0}\int_{M}|\dot{\varphi}_{t}-c(t)|e^{\tilde{h}}\omega_{\varphi}^{n}+A_{0}\epsilon_{0}^{2}.$
Now as same as in the proof of Lemma 1.3, we decompose
$\dot{\varphi}_{t}-c(t)$ as $\psi+\psi^{\bot}$ with
$\psi\in\Lambda_{1}(M,\omega_{\varphi})$ and
$\psi^{\bot}\in\Lambda_{1}^{\bot}(M,\omega_{\varphi})$, where
$\Lambda_{1}(M,\omega_{\varphi})$ is a linear space spanned by a basis
$\\{\psi^{i}\\}$ and $\Lambda_{1}^{\bot}(M,\omega_{\varphi})$ be a subspace of
$L^{2}$-weighted integral functions which are orthogonal to
$\Lambda_{1}(M,\omega_{\varphi})\cap\mathbb{R}$ in the sense of
$\int_{M}\psi\psi^{\prime}e^{\tilde{h}}\omega_{\varphi}^{n}=0,~{}~{}\forall~{}\psi\in\Lambda_{1}(M,\omega_{\varphi}),\psi^{\prime}\in\Lambda_{1}^{\bot}(M,\omega_{\varphi}).$
Then we get
$\int_{M}|\psi|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}\leq
C^{\prime}\epsilon_{0}^{2}\int_{M}(\dot{\varphi}-c(t))^{2}e^{\tilde{h}}\omega^{n}_{\varphi}+nA_{0}^{2}\epsilon_{0}^{4},$
and so
$\displaystyle\int_{M}|\psi^{\bot}|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}$
(2.16)
$\displaystyle\geq(1-C^{\prime}\epsilon_{0}^{2})\int_{M}(\dot{\varphi}-c(t))^{2}e^{\tilde{h}}\omega^{n}_{\varphi}-nA_{0}^{2}\epsilon_{0}^{4}.$
On the other hand, by using the continuity of eigenvalues and Lemma 2.2, there
exists a number $\delta_{0}>0$ (compared to Lemma 1.2), which depends only on
the gap of the first two eigenvalues of the operator $(P,\omega_{KS})$ with
respect to the metric $\omega_{KS}$ in Lemma 2.2, such that for any
$\varphi\in\overline{\mathcal{K}}(\epsilon_{0})$, we have
$\int_{M}\|\nabla\psi^{\bot}\|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}\\\
\geq(1+\delta_{0})\int_{M}(\psi^{\bot})^{2}e^{\tilde{h}}\omega^{n}_{\varphi}.$
Thus by (2.16), we get
$\displaystyle\int_{M}\|\nabla(\dot{\varphi}_{t}-c(t))\|^{2}e^{\tilde{h}}\omega^{n}_{\varphi}$
(2.17)
$\displaystyle\geq(1+\delta_{0})(1-C^{\prime}\epsilon_{0}^{2})\int_{M}(\dot{\varphi}_{t}-c(t))^{2}e^{\tilde{h}}\omega^{n}_{\varphi}-nA_{0}^{2}\epsilon_{0}^{4}.$
By inserting (2.17) into (2.7), we obtain
$\displaystyle\frac{d\tilde{H}_{0}(t)}{dt}$
$\displaystyle\leq-2[(1-2\epsilon_{0})(1+\delta_{0})(1-C^{\prime}\epsilon_{0}^{2})-(1+\epsilon_{0})]\tilde{H}_{0}(t)+B_{0}\epsilon_{0}^{3}$
(2.18) $\displaystyle\leq-\theta\tilde{H}_{0}(t)+B_{0}\epsilon_{0}^{3},$
where the constant $B_{0}=B_{0}(\|X^{\prime}\|_{C^{0}})$ depends only on
$\|X^{\prime}\|_{C^{0}}$ and
$\theta=2[(1-2\epsilon_{0})(1+\delta_{0})(1-C^{\prime}\epsilon_{0}^{2})-(1+\epsilon_{0})]\geq\delta_{0}$
as $\epsilon_{0}$ is small enough.
By (2.18), we have
$\frac{d(\tilde{H}_{0}(t)-\frac{B_{0}\epsilon_{0}^{3}}{\theta})}{dt}\leq-\theta(\tilde{H}_{0}(t)-\frac{B_{0}\epsilon_{0}^{3}}{\theta}).$
Since $\tilde{H}_{0}(t)\geq\frac{B_{0}\epsilon_{0}^{3}}{\theta}$, we get
$\displaystyle\tilde{H}_{0}(t)\leq$ $\displaystyle e^{-\theta
t}(\tilde{H}_{0}(0)-\frac{B_{0}\epsilon_{0}^{3}}{\theta})+\frac{B_{0}\epsilon_{0}^{3}}{\theta}$
$\displaystyle\leq e^{-\theta
t}\tilde{H}_{0}(0)+\frac{B_{0}\epsilon_{0}^{3}}{\theta}.$
∎
###### Remark 2.5.
From (2.7) and (2.12), we see that if in addition that the initial Kähler
potential $\psi$ is $K_{X_{0}}$-invariant, then (2.11) can be improved as
$\tilde{H}_{0}(t)\leq\tilde{H}_{0}(0)e^{-\theta
t},~{}~{}\forall~{}~{}t\in[0,T)$
whenever $\varphi_{t}$ lies in $\overline{\mathcal{K}}(\epsilon_{0})$.
To get a $C^{0}$-estimate and $C^{\alpha}$-estimate for $\dot{\varphi}$, we
use a method as in Appendix to estimate $W^{k,2}$-estimates ($k\geq 1$) for
$\dot{\varphi}$. Let
$\tilde{H}_{k}(t)=\int_{M}\|\nabla^{k}\dot{\varphi}\|^{2}e^{\tilde{h}}\omega_{\varphi}^{n}.$
Then we have
###### Proposition 2.6.
Let $\epsilon_{0}<<1$. Then under the same condition in Lemma 2.4, there exist
two uniform constants $\theta^{\prime},B>0$ which depend only on the metric
$\omega_{KS}$ and integer number $k$ such that
(2.19) $\displaystyle\tilde{H}_{k}(t)\leq
e^{-\theta^{\prime}t}(\tilde{H}_{k}(0)+B\tilde{H}_{0}(0))+\frac{B_{0}B}{\theta^{\prime}}\epsilon_{0}^{3},~{}~{}\forall~{}t\in[0,T),$
if
$\tilde{H}_{k}(t)+B\tilde{H}_{0}(t))\geq\frac{B_{0}B}{\theta^{\prime}}\epsilon_{0}^{3}$
for any $t\leq T$, where $B_{0}$ is the constant determined in Lemma 2.4.
###### Proof.
First note that similarly to (4.1) in Appendix 1, we can obtain
$\displaystyle\frac{d\|\nabla^{k}\dot{\varphi}\|^{2}}{dt}$
$\displaystyle\leq-2\|\nabla^{k+1}\dot{\varphi}\|^{2}+C_{1}\|\nabla^{k}\dot{\varphi}\|^{2}+C_{2}\|\dot{\varphi}-c(t)\|^{2}.$
It follows
$\displaystyle\frac{d\tilde{H}_{k}(t)}{dt}$
$\displaystyle=\int_{M}\frac{d\|\nabla^{k}\dot{\varphi}\|^{2}}{dt}e^{\tilde{h}}\omega_{\phi}^{n}+\int_{M}\|\nabla^{k}\dot{\varphi}\|^{2}(\triangle\dot{\varphi}+X(\dot{\varphi})-\ddot{\varphi})e^{\tilde{h}}\omega_{\varphi}^{n}$
$\displaystyle\leq-2\tilde{H}_{k+1}(t)+C_{1}^{\prime}\tilde{H}_{k}(t)+C_{2}^{\prime}\|\dot{\varphi}-c(t)\|^{2}$
$\displaystyle\leq-\theta^{\prime}\tilde{H}_{k}(t)+C_{3}\tilde{H}_{0}(t).$
On the other hand, by (2.18), we have
$\frac{d\tilde{H}_{0}(t)}{dt}\leq-\theta\tilde{H}_{0}(t)+B_{0}\epsilon_{0}^{3},~{}~{}\forall~{}t\in[0,T),$
since we may also assume that
$\tilde{H}_{0}(t)\geq\frac{B_{0}}{\theta}\epsilon_{0}^{3}$ for any $t$ in
$[0,T)$, Thus combining the above two inequalities, we get
$\displaystyle\frac{d(\tilde{H}_{k}(t)+B\tilde{H}_{0}(t))}{dt}$
$\displaystyle\leq-\theta^{\prime}[\tilde{H}_{k}(t)+\frac{(B\theta-
C_{3})}{\theta^{\prime}}\tilde{H}_{0}(t)]+B_{0}B\epsilon_{0}^{3}$ (2.20)
$\displaystyle\leq-\theta^{\prime}(\tilde{H}_{k}(t)+B\tilde{H}_{0}(t))+B_{0}B\epsilon_{0}^{3},$
where $B$ is a sufficiently large number independent of $\epsilon_{0}$. From
(2.20), one can easily get
$(\tilde{H}_{k}(t)+B\tilde{H}_{0}(t))\leq
e^{-\theta^{\prime}t}(\tilde{H}_{k}(0)+B\tilde{H}_{0}(0))+\frac{B_{0}B}{\theta^{\prime}}\epsilon_{0}^{3},~{}~{}\forall~{}t\in[0,T),$
and so (2.19) is true.
∎
By the embedding theory of Sobolev spaces, we get
###### Corollary 2.7.
Let $\epsilon_{0}<<1$. Then under the same condition in Lemma 2.4, there exist
two uniform constants $\theta_{0},C_{0}>0$ which depend only on the metric
$\omega_{KS}$ such that
(2.21) $\displaystyle\|\tilde{\dot{\varphi_{t}}}\|_{C^{\alpha}}\leq
C_{0}[e^{-\theta_{0}t}\|\psi-\underline{\psi}\|_{C^{2,\alpha}}+\epsilon_{0}^{\frac{3}{2}}],~{}~{}\forall~{}t\in[0,T),$
if $\|\tilde{\dot{\varphi_{t}}}\|_{C^{\alpha}}\geq
C_{0}\epsilon_{0}^{\frac{3}{2}}$ for any $t\leq T$.
###### Remark 2.8.
By Remark 2.5, according to the proof of Proposition 2.6, we see that if in
addition that the initial Kähler potential $\psi$ is $K_{X_{0}}$-invariant,
then (2.19) can be improved as
$\tilde{H}_{k}(t)\leq(\tilde{H}_{0}(0)+B\tilde{H}_{k}(0))e^{-\theta^{\prime}t},~{}~{}\forall~{}~{}t\in[0,T)$
whenever $\varphi_{t}$ lies in $\overline{\mathcal{K}}(\epsilon_{0})$. Thus
(2.21) can be improved as
$\|\tilde{\dot{\varphi_{t}}}\|_{C^{\alpha}}\leq
C_{0}e^{-\theta_{0}t}\|\psi-\underline{\psi}\|_{C^{2,\alpha}}.$
The following lemma can be easily proved by using apriori estimates for
solution $\varphi(t,\cdot)$ of (2.4) at finite time (cf [TZ3]).
###### Lemma 2.9.
Let $\psi\in\mathcal{K}(\frac{\epsilon_{0}}{N})$. Then there exists $T=T_{N}$
such that evolved Kähler potentials $\varphi_{t}$ of (2.4) with $\psi$ as an
initial potential lies in $\mathcal{K}(\epsilon_{0})$ for any $t<T$.
We are now going to do a key estimate for the proof of Theorem 2.1.
###### Proposition 2.10.
There exist a small $\epsilon_{0}$ and a large number $N$ such that if the
initial data $\psi\in\mathcal{M}(\omega_{KS})$ in (2.4) satisfies
$\|\psi-\underline{\psi}\|_{C^{2,\alpha}}\leq\frac{\epsilon_{0}}{N}$, then
there there exist a family of $\sigma_{t}\in\text{Aut}_{r}(M)$ such that
(2.22)
$\displaystyle\|\varphi_{\sigma_{t}}-\underline{\varphi_{\sigma_{t}}}\|_{C^{2,\alpha}}\leq\epsilon_{0},~{}~{}\forall~{}t>0,$
where $\varphi_{\sigma_{t}}=(\sigma_{t})^{*}\varphi_{t}+\rho_{t}$ and
$\rho_{t}$ are Kähler potentials defined by (2.5) in Theorem 2.1.
###### Proof.
The proof is a modification of one of Theorem 1.1. Let $N_{0}$ be a very big
number and choose another big number $N$ with $N_{0}\leq
N\leq\frac{1}{\epsilon_{0}^{1/4}}$ such that
$C_{0}e^{-\theta_{0}T_{N}}\leq\frac{1}{N_{0}}$, where $C_{0}$ and $T_{N}$ are
two uniform numbers determined in Corollary 2.7 and Lemma 2.9, respectively.
Now we consider the solution $\varphi=\varphi_{T_{N}}$ of (2.4) at time
$T_{N}$. By Lemma 5.1 in Appendix 2, we see that there exists
$\sigma=\sigma_{T_{N}}$ such that for any $Y\in\eta_{r}(M)$ with
$\int_{M}\|Y\|^{2}\omega_{KS}^{n}=1$, it holds
(2.23)
$\displaystyle|\int_{M}\theta_{Y}^{\prime}\varphi_{\sigma}e^{\theta_{X}}\omega_{KS}^{n}|\leq
O(\epsilon_{0}^{2}),$
where $\varphi_{\sigma}=\sigma^{*}\varphi+\rho_{\sigma}$. By adding a constant
to $\varphi_{\sigma}$ so that
$\widetilde{\varphi_{\sigma}}=\varphi_{\sigma}+const.$ satisfies
$\int_{M}\widetilde{\varphi_{\sigma}}e^{\theta_{X}}\omega_{KS}^{n}=0$, then we
can decompose $\widetilde{\varphi_{\sigma}}$ into
$\widetilde{\varphi_{\sigma}}=\phi+\phi^{\bot}$ with
$\phi\in\Lambda_{1}(M,\omega_{KE})$ and
$\phi^{\bot}\in\Lambda_{1}^{\bot}(M,\omega_{KE})$, where
$\Lambda_{1}^{\bot}(M,\omega_{KS})$ is a subspace of weighted $L^{2}$-integral
functions which are orthogonal to $\Lambda_{1}(M,\omega_{KS})\cup\mathbb{R}$.
Then $\phi=\sum_{i}a_{i}\theta_{i}$ for some constants $a_{i}$, where
$\theta_{i}$ is a basis of the space $\Lambda_{1}(M,\omega_{KS})$. As a
consequence, by (2.23), we see that $a_{i}=O(\epsilon_{0}^{2})$ and so
(2.24) $\displaystyle\|\phi\|_{C^{2,\alpha}}\leq O(\epsilon_{0}^{2}).$
Since $\rho=\rho_{\sigma}$ satisfies equation,
$\omega_{\rho}^{n}=\omega_{KS}^{n}e^{-\rho-X(\rho)},$
by equation (2.4), we have
(2.25)
$\displaystyle\omega_{\widetilde{\varphi_{\sigma}}}^{n}=\omega_{KS}^{n}e^{\sigma^{*}(\tilde{\dot{\varphi}})-\widetilde{\varphi_{\sigma}}-X(\widetilde{\varphi_{\sigma})}+b},$
where $b$ is a constant. Then $\phi^{\bot}$ is a solutions of equation
$P[\log(\frac{[\omega_{\phi+\phi^{\bot}}]^{n}}{\omega_{KS}^{n}})]+\varphi^{\bot}+X(\phi^{\bot})=P[\sigma^{*}(\tilde{\dot{\varphi}})+b-X(\phi)],$
where $P$ is a projection from Banach space $H^{2,\alpha}(M)$ to Banach space
$H^{\alpha}(M)\cap\Lambda_{1}^{\bot}(M,\omega_{KS})$, and $\phi$ and
$\sigma^{*}(\tilde{\dot{\varphi}})$ are regarded as two peturbation functions.
Without of the generality, we may assume that
$\|\sigma^{*}(\tilde{\dot{\varphi}})\|_{C^{\alpha}}\geq
C_{0}\epsilon_{0}^{\frac{3}{2}},$
where $C_{0}$ is the constant in Corollary 2.7. Then according to Corollary
2.6 and (2.24), we have
$\|P[\sigma^{*}(\tilde{\dot{\varphi}})+b-X(\phi)]\|_{C^{\alpha}}=\|P[\sigma^{*}(\tilde{\dot{\varphi}})-X(\phi)]\|_{C^{\alpha}}\leq\frac{2\epsilon_{0}}{NN_{0}}.$
Thus we can use the Implicity Functional Theorem to get
(2.26) $\displaystyle\|\phi^{\bot}\|_{C^{2,\alpha}}\leq
2(\|P[\sigma^{*}(\tilde{\dot{\varphi}})+b-X(\phi)]\|_{C^{\alpha}}+\|\phi\|_{C^{2,\alpha}})\leq\frac{8\epsilon_{0}}{NN_{0}}.$
(2.24) and (2.26) implies
(2.27)
$\displaystyle\|\widetilde{\varphi_{\rho}}\|_{C^{2,\alpha}}\leq\frac{16\epsilon_{0}}{NN_{0}},$
so $\widetilde{\varphi_{\rho}}\in\mathcal{K}(\frac{\epsilon_{0}}{N})$.
At the next step (Step 2) we consider equation (2.4) with
$\widetilde{\varphi_{\rho}}$ as an initial potential to replace $\psi$. By
Lemma 2.9, one sees that equation is solvable for any $t\in T_{N}$ with
evolved Kähler potentials $\varphi_{t}^{(2)}\in\mathcal{K}(\epsilon_{0})$ for
any $t\leq T_{N}$. So by the argument at the last step (Step 1), we can also
show that there exists $\sigma^{(2)}=\sigma_{T_{N}}^{(2)}\in\text{Aut}_{r}(M)$
such that
$\displaystyle\|\widetilde{\varphi_{\rho}^{(2)}}\|_{C^{2,\alpha}}$
$\displaystyle=\|\widetilde{(\sigma^{(2)})^{*}\varphi_{T_{N}}^{(2)}+\rho_{\sigma^{(2)}}}\|_{C^{2,\alpha}}$
(2.28)
$\displaystyle\leq(\frac{16}{N_{0}})^{2}\frac{\epsilon_{0}}{N}<\frac{\epsilon_{0}}{N}.$
Repeating to use the above step for finite times, we can obtain
$\|\widetilde{\dot{\varphi^{(k)}}}\|_{C^{\alpha}}\leq
C_{0}\epsilon_{0}^{\frac{3}{2}}$
for some integer $k$. Then also by using the argument in Step 1, we can find
$\sigma^{(k+1)}=\sigma_{T_{N}}^{(k+1)}\in\text{Aut}_{r}(M)$ such that
$\|\widetilde{\varphi_{\rho}^{(k)}}\|_{C^{2,\alpha}}=O(\epsilon_{0}^{\frac{3}{2}}).$
Now (Step 3) we considering equation (2.4) with
$\widetilde{\varphi_{\rho}^{(k)}}$ as an initial potential. Then we conclude
that either evolved Kähler potentials $\varphi^{(k+1)}_{t}$ lies in
$\mathcal{K}(\frac{\epsilon_{0}}{N})$ for any $t$ or there exists some time
$T$ such that $\|\varphi^{(k+1)}_{T}\|_{C^{2,\alpha}}=\frac{\epsilon_{0}}{N}$
for an evolved Kähler potential $\varphi^{(k+1)}_{T}$ at time $T$. If the
first case happens, then we will finish all steps. If the second case happens,
then we can repeat the Step1-3 and we can finally prove that there exist a
family of $\sigma_{t}\in\text{Aut}_{r}(M)$ such that (2.22) satisfies for any
evolved Kähler potential $\varphi_{t}$ of (2.4) at $t$ as long as the initial
potential $\psi$ lies in $\mathcal{K}(\frac{\epsilon_{0}}{N})$.
∎
###### Proof of Theorem 2.1.
We suffice to do higher-order estimates for the modified evolved Kähler
potentials $((\sigma_{t})^{*}\varphi_{t}+\rho_{t})$ of equation (2.4) in
Proposition 2.10. Here we use a trick in [CT2] to choose a modified family of
holomorphic transformations $\overline{\sigma}_{t}\in\text{Aut}_{r}(M)$
($0<t<\infty$, $\overline{\sigma}_{0}=\text{Id}$) to replace $\sigma_{t}$ such
that for any $t\in(0,\infty)$ (cf. [TZ3]),
$\|\sigma_{t}^{-1}\overline{\sigma}_{t}-Id\|\leq C,$
and
$\|(\overline{\sigma}_{t}^{-1})_{*}\frac{\partial\overline{\sigma}_{t}}{\partial
t}\|_{g}\leq C,$
where
$(\overline{\sigma}_{t}^{-1})_{*}\frac{\partial\overline{\sigma}_{t}}{\partial
t}=\overline{X}_{t}\in\eta_{r}(M)$ is a family of holomorphic vector fields on
$M$. Furthermore, for any $k\geq 0$, we may assume that there is a constant
$C_{k}$ such that
$\|\frac{\partial^{k}\overline{X}_{t}}{\partial t^{k}}\|_{g}\leq C_{k}.$
Note that the choice of such $\overline{\sigma}_{t}$ just depends on the
$C^{0}$-estimate of
$\widetilde{\varphi_{\sigma}}=\widetilde{((\sigma_{t})^{*}\varphi_{t}+\rho_{t})}$.
Thus by Proposition 2.10, we also have
$(\overline{\varphi}-\frac{1}{V}\int_{M}\overline{\varphi}\omega_{KS}^{n})\in\mathcal{K}(\epsilon_{0})$.
On the other hand, by equation (2.4), the new modified potential
$\overline{\varphi}=\varphi_{\overline{\sigma}_{t}}=(\overline{\sigma}_{t})^{*}(\varphi_{t}+\overline{\rho}_{t})$
will satisfy equation,
$\displaystyle\frac{\partial\overline{\varphi}}{\partial
t}=\log\frac{\omega^{n}_{\overline{\varphi}}}{\omega_{KS}^{n}}+\overline{\varphi}+\overline{X}(\overline{\varphi}),$
(2.29) $\displaystyle\overline{\varphi}(0)=\psi-\underline{\psi}.$
Now for each $t$, we can consider solution $\varphi^{\prime}$ of equation
(2.29) on the interval $[t-1,t+1]$ with
$(\overline{\varphi}_{t-1}-\frac{1}{V}\int_{M}\overline{\varphi}_{t-1}\omega_{KS}^{n})$
as an initial data. By the Maximal Principle, it is easy to see that both
$\varphi_{s}^{\prime}$ and $\dot{\varphi}_{s}^{\prime}$ are uniformly bounded
in $[t-1,t+1]$. Since
$\|\overline{\varphi}_{s}^{\prime}-\frac{1}{V}\int_{M}\overline{\varphi}_{s}^{\prime}\omega_{KS}^{n}\|_{C^{2,\alpha}}=\|\overline{\varphi}_{s}-\frac{1}{V}\int_{M}\overline{\varphi}_{s}\omega_{KS}^{n}\|_{C^{2,\alpha}},$
by the regularity theory of parabolic equation, we get all bounded
$C^{k}$-estimates for $\overline{\varphi}_{t}^{\prime}$. This implies that all
$C^{k}$-norms of
$(\overline{\varphi}_{t}-\frac{1}{V}\int_{M}\overline{\varphi}_{t}\omega_{KS}^{n})$
are uniformly bounded, and so are $\widetilde{\varphi_{\sigma}}$.
From the above estimates, we see that for any sequence of Kähler metrics
$\omega_{\varphi_{\sigma_{i}}}$, there exists a limit Kähler metric
$\omega_{\infty}$ of subsequence of $\omega_{\varphi_{\sigma_{i}}}$ in the
sense of $C^{k}$-convergence. By applying Perelman’s $W$-function in [Pe] to
the normalized Ricci equation (0.1), one conculdes that $\omega_{\infty}$ must
be a Kähler-Ricci soliton (cf. [Se]). Since the Kähler-Ricci solition is
unique, we see that there exists an element
$\tau_{\infty}\in\text{Aut}_{0}(M)$ such that
$\omega_{\infty}=\tau_{\infty}^{*}\omega_{KS}$. By using the fact that the
convergent sequence is arbitary, the above implies that there exists a family
of $\tau=\tau_{t}\in\text{Aut}_{0}(M)$ such that evolved Kähler metrics
$\tau^{*}g$ converge to $g_{KS}$ smoothly.
If in addition that the initial Kähler potential $\psi$ is
$K_{X_{0}}$-invariant, by Remark 2.7, we can follow the argument in the proof
of Theorem 1.1 to apply the Implicity Functional Theorem to equation (2.25) in
the proof of Proposition 2.10 to show that there exists a family of
$\sigma=\sigma_{t}\in\text{Aut}_{r}(M)$ such that the modified solution
$\varphi_{\sigma}=((\sigma_{t})^{*}\varphi_{t}+\rho_{t})$ of equation (2.25)
satisfy
$\displaystyle\|\widetilde{\varphi_{\sigma}}\|_{C^{2,\alpha}}$
$\displaystyle=\|((\sigma_{t})^{*}\varphi_{t}+\rho_{t}-\frac{1}{V}\int_{M}(\sigma_{t})^{*}\varphi_{t}+\rho_{t}))\omega_{KS}^{n})\|_{C^{2,\alpha}}$
(2.30) $\displaystyle\leq
2\|P(\sigma^{*}(\tilde{\dot{\varphi}})\|_{C^{\alpha}}\leq
Ce^{-\theta^{\prime}t},~{}~{}\forall~{}t>0.$
Similarly, we can also prove that for any $k$ it holds
$\|\widetilde{\varphi_{\sigma}}\|_{C^{k+2,\alpha}}\leq
C_{k}e^{-\theta^{\prime}t},~{}~{}\forall~{}t>0,$
since by Remark 2.8 and the embedding theory of Sobolev spaces we have
$\|\tilde{\dot{\varphi}}\|_{C^{k,\alpha}}\leq
C_{k}^{\prime}e^{-\theta^{\prime}t},~{}~{}\forall~{}t>0,$
where $C_{k}$ and $C_{k}^{\prime}$ are uniform constants which depends only on
$k,\epsilon_{0}$ and higher-order derivatives of the initial Kähler potential
$\psi$. Therefore we prove that Kähler metrics $\sigma^{*}(\omega_{\varphi})$
converge exponentially to the Kähler-Ricci soliton $\omega_{KS}$. ∎
## 3\. Uniqueness of the limit of Kähler Ricci flow
By Theorem 1.1 and Theorem 2.2 in Section 1 and 2, we complete the proof of
Theorem 0.1. As an application of Theorem 0.1, we have the following
uniqueness result about the limit of Kähler-Ricci flow.
###### Theorem 3.1.
Let $g_{t}$ be the evolved Kähler metrics of Kähler-Ricci flow (0.1) on $M$.
Suppose that there exists a sequence $g_{i}$ of $g_{t}$ and a sequence of
holomorphic transformations $\sigma_{i}\in\text{Aut}(M)$ such that
$\sigma_{i}^{*}g_{i}$ converge to a limit Kähler metric $g_{\infty}$ in the
sense of $C^{2,\alpha}$-norm of Kähler potentials. Then the Kähler-Ricci flow
converges to $g_{\infty}$ smoothly in the sense of Cheeger-Gromov.
###### Proof.
First we note that by applying Perelman’s $W$-function in [Pe], the limit
Kähler metric $g_{\infty}$ must be a Kähler-Ricci soliton $g_{KS}$ on $M$. On
the other hand, by the convergence of $g_{i}$, one sees that for any
$\epsilon<<1$ there exists a big index $i$ such that the potential
$\psi=\psi_{i}$ of $g_{i}$ satisfies
$\|\psi-\underline{\psi}\|_{C^{2,\alpha}}\leq\epsilon,$
where $\omega_{\psi}=\omega_{KS}+\sqrt{-1}\partial\bar{\partial}\psi$. Now we
consider the Kähler-Ricci flow (0.1) with $\omega_{g}=\omega_{\psi}$ as an
initial Kähler metric. Then by Theorem 0.1, this flow converges to $g_{KS}$
smoothly in the sense of Cheeger-Gromov, so the theorem is proved.
∎
###### Remark 3.2.
In a subsequence paper, we will prove the uniqueness of the limit of Kähler-
Ricci flow in more general. Namely, Theorem 3.1 is still true if we assume
that there exists a sequence $g_{i}$ of $g_{t}$ of equation (0.1) which
converge to a limit Riemanian metric $g_{\infty}$ in $C^{2,\alpha}$-norm in
the sense of Cheeger-Gromov.
## 4\. Appendix 1
In this appendix, we prove a lemma about $W^{k,2}$-estimates of
$\dot{\varphi}$ for evolved Kähler metrics $\varphi$ of flow (1.1) under the
assumption $\varphi\in\mathcal{K}(\epsilon_{0})$. Recall that a $k-norm$
$\|\nabla^{k}{\dot{\varphi}}\|^{2}$ is defined by
$\|\nabla^{k}\dot{\varphi}\|^{2}=\sum
g^{i_{1}j_{1}}...g^{i_{k}j_{k}}{\dot{\varphi}}_{i_{1}...i_{k}}{\dot{\varphi}}_{j_{1}...j_{k}},$
where ${\dot{\varphi}}_{i_{1}...i_{k}}$ are components of the $k$-covariant
derivative of $\dot{\varphi}$ with respect to $g=\omega_{\varphi}$ as a
Riemannian metric.
Since
${\dot{\varphi}}_{i_{1}...i_{k}}=\frac{\partial^{k}\dot{\varphi}}{\partial
x^{i_{1}}...\partial
x^{i_{k}}}+\Phi_{1}(\dot{\varphi},...,{\dot{\varphi}}_{i_{1}...i_{k-1}}),$
we have
$\displaystyle\frac{d{\dot{\varphi}}_{i_{1}...i_{k}}}{dt}$
$\displaystyle=\frac{\partial^{k}\ddot{\varphi}}{\partial x^{i_{1}}...\partial
x^{i_{k}}}+\frac{d\Phi_{1}}{dt}$
$\displaystyle={\ddot{\varphi}}_{i_{1}...i_{k}}+\Phi_{2}(\dot{\varphi}_{i},...,{\dot{\varphi}}_{i_{1}...i_{k-1}})+\frac{d\Phi_{1}}{dt},$
where $\Phi_{1}$ and $\Phi_{2}$ are two polynomials with variables
$\dot{\varphi}_{i},...,{\dot{\varphi}}_{i_{1}...i_{k-1}}$ and coefficients
$g_{ij},\partial^{l}g_{ij},l=1,...,k$. Note that $\frac{d\Phi_{1}}{dt}$ is
uniformly bounded. Then by equations (0.1) and (1.1), one can estimate
$\displaystyle\frac{d\|\nabla^{k}\dot{\varphi}\|^{2}}{dt}$
$\displaystyle=\sum_{i_{1},...,i_{k}}\sum_{\alpha}(R_{i_{\alpha}i_{\alpha}}-g_{i_{\alpha}i_{\alpha}}){\dot{\varphi}}_{i_{1},...,i_{\alpha},...,i_{k}}{\dot{\varphi}}_{i_{1},...,i_{\alpha},...,i_{k}}$
$\displaystyle+2\sum
g^{i_{1}j_{1}}...g^{i_{k}j_{k}}\frac{d{\dot{\varphi}}_{i_{1}...i_{k}}}{dt}{\dot{\varphi}}_{j_{1}...j_{k}}$
$\displaystyle\leq
C_{1}\|\nabla^{k}\dot{\varphi}\|^{2}+C_{2}\|\nabla\dot{\varphi}\|^{2}+2(\ddot{\varphi})_{i_{1}...i_{k}}{\dot{\varphi}}_{j_{1}...j_{k}}$
(4.1)
$\displaystyle\leq-2\|\nabla^{k+1}\dot{\varphi}\|^{2}+C_{1}^{\prime}\|\nabla^{k}\dot{\varphi}\|^{2}+C_{2}^{\prime}\|\dot{\varphi}-c(t)\|^{2}.$
Let
$H_{k}(t)=\int_{M}\|\nabla^{k}\dot{\varphi}\|^{2}\omega_{\varphi}^{n}.$
Then by (4.1), we have
###### Lemma 4.1.
Let $T$ be any positive number. Suppose that $\varphi_{t}$ lies
$\mathcal{K}(\epsilon_{0})$ for any $t\in[0,T)$. Then
(4.2) $\displaystyle H_{k}(t)\leq
Ce^{-\theta^{\prime}t},~{}~{}\forall~{}t\in[0,T).$
###### Proof.
By (4.1), we have
$\displaystyle\frac{dH_{k}(t)}{dt}$
$\displaystyle=\int_{M}\frac{d\|\nabla^{k}\dot{\varphi}\|^{2}}{dt}\omega_{\varphi}^{n}+\int_{M}\|\nabla^{k}\dot{\varphi}\|^{2}\triangle\dot{\varphi}\omega_{\varphi}^{n}$
$\displaystyle\leq-2H_{k+1}(t)+C_{3}H_{k}(t)+C_{2}^{\prime}\|\dot{\varphi}-c(t)\|^{2}$
(4.3) $\displaystyle\leq-\theta^{\prime}H_{k}(t)+C_{4}H_{0}(t).$
On the other hand, from the proof of Lemma 1.3, we in fact prove that
$\frac{dH_{0}(t)}{dt}\leq-\theta H_{0}(t),~{}~{}\forall~{}t\in[0,T),$
if $\varphi\in\mathcal{K}(\epsilon_{0}),~{}~{}\forall~{}t\in[0,T)$. Thus
Combining the above inequality with (4.3), we get
$\frac{d(H_{k}(t)+AH_{0}(t))}{dt}\leq-\theta^{\prime}[H_{k}(t)+\frac{(A\theta-
C_{4})}{\theta^{\prime}}H_{0}(t)],$
where $A$ is a sufficiently large number. It follows
$\frac{d\ln(H_{k}(t)+AH_{0}(t))}{dt}\leq-\theta^{\prime}\frac{H_{k}(t)+\frac{(A\theta-
C_{4})}{\theta^{\prime}}H_{0}(t)}{H_{k}(t)+AH_{0}(t)}\leq-\theta^{\prime}.$
Thus
$H_{k}(t)+AH_{0}(t)\leq(H_{k}(0)+AH_{0})e^{-\theta^{\prime}t}$
and so (4.2) follows. ∎
## 5\. Appendix 2
The following lemma is about the existence of almost orthonormality of a
Kähler potential to the space of first eigenvalue-functions of operator
$(P,\omega_{KS})$ defined in Lemma 2.2 in Section 2. The lemma is crucial in
the proof of Proposition 2.10.
###### Lemma 5.1.
Let $M$ be a compact Kähler manifold $M$ with $c_{1}(M)>0$ which admits a
Kähler-Ricci soliton $(\omega_{KS},X_{0})$. Then for any Kähler potential
$\phi\in\mathcal{K}(\epsilon_{0})$ there exists a $\sigma\in\text{Aut}_{r}(M)$
with bounded $\text{dist}(\sigma,Id)$ such that for any $Y\in\eta_{r}(M)$ with
$\int_{M}\|Y\|^{2}\omega_{KS}^{n}=1$, it holds
$|\int_{M}\theta_{Y}^{\prime}(\sigma^{*}\phi+\rho_{\sigma})e^{\theta_{X}}\omega_{KS}^{n}|\leq
C\|X^{\prime}(\phi)\|_{C^{0}}^{2}=O(\epsilon_{0}^{2}),$
where $\theta_{Y}\in\text{ker}(P,\omega_{KS})$ and $\rho_{\sigma}$ is a Kähler
potential defined by (2.5) in Section 2.
###### Proof.
This lemma was proved in [TZ1] if $\phi$ is $K_{0}$-invariant. The key point
in the proof is to use a functional defined on a space of Kähler-Ricci
solitons
$\\{\omega_{KS}^{\prime}=\sigma^{*}(\omega_{KE})=\omega_{KS}+\sqrt{-1}\partial\bar{\partial}\rho_{\sigma}|~{}~{}\sigma\in\text{Aut}_{r}(M)\\},$
which was introduced in [Zh] by
$\displaystyle(I-J)(\omega_{\phi},\omega_{KS}^{\prime})$
$\displaystyle=\int_{0}^{1}dt\int_{M}\dot{\phi}_{t}e^{\theta_{X_{0}}(\phi_{t})}\omega_{\phi_{t}}^{n}-\int_{M}(-\phi+\rho)e^{\theta_{X_{0}}+X(\rho)}(\omega_{KS}^{\prime})^{n},$
where $\phi_{t}$ is a $K_{X_{0}}$-invariant path in $\mathcal{M}(\omega_{KS})$
which connects $0$ and $-\phi+\rho$, and $\theta_{X_{0}}(\phi_{t})$ are
potentials of $X_{0}$ associated to metric $\omega_{\phi_{t}}$ defined by
(2.1). It is proved in [Zh] that this well-defined for a $K_{0}$-invariant
$\phi$, i.e., the functional is independent of the choice of a
$K_{0}$-invariant path. But for a general Kähler potential $\phi$, one can
also show that $(I-J)(\omega_{\phi},\omega_{KS}^{\prime})$ is not well-defined
(to see (5.4) below), so we shall introduce another functional defined on
whole space $\mathcal{M}(\omega_{KS})$ to replace it. In fact, we consider the
following functional
$\displaystyle\mathcal{F}(\omega_{\phi},\omega_{KS}^{\prime})$
$\displaystyle=\text{Re}[\int_{0}^{1}dt\int_{M}(-\phi+\rho_{\sigma})e^{\theta_{X_{0}}(t(-\phi+\rho_{\sigma}))}\omega_{t(-\phi+\rho_{\sigma})}^{n}$
(5.1)
$\displaystyle-\int_{M}(-\phi+\rho)e^{\theta_{X_{0}}^{\prime}}(\omega_{KS}^{\prime})^{n}].$
Clearly, the definition of $\mathcal{F}$ just uses a real part of
$(I-J)(\omega_{\phi},\omega_{KS}^{\prime})$ while a Kähler potentials path is
chosen by $\phi_{t}=t(-\phi+\rho_{\sigma})$. We now consider a Kähler
potentials path $\rho_{t}$ induced by an one-parameter subgroup $\sigma_{t}$
generated by the real part of $Y\in\eta_{r}(M)$, i.e. $\rho_{t}$ are defined
by
$\omega_{t}=\sigma_{t}^{*}\omega_{KS}^{\prime}=\omega_{KS}^{\prime}+\sqrt{-1}\partial\bar{\partial}\rho_{t}$.
Let
(5.2) $\displaystyle
f_{Y}(t)=\text{Re}[\int_{0}^{1+t}ds\int_{M}(\dot{\phi}_{s})e^{\theta_{X_{0}}(\phi_{s})}\omega_{\phi_{s}}^{n}-\int_{M}(-\phi+\rho_{t})e^{\theta_{X_{0}}(\omega_{t})}\omega_{t}^{n}],$
where $\phi_{s}$ is a path in $\mathcal{M}(\omega_{KS})$ defined by
$\phi_{s}=s(-\phi+\rho_{\sigma}),~{}\forall~{}0\leq s\leq 1$ and
$\phi_{s}=-\phi+\rho_{\sigma}+\rho_{t}$, $1\leq s\leq 1+t$. It is easy to see
$\frac{d}{dt}f_{Y}(t)|_{t=0}=\int_{M}\theta_{Y}^{\prime}(-\varphi+\rho_{\sigma})e^{\theta_{X}^{\prime}}(\omega_{KS}^{\prime})^{n}.$
This implies
(5.3)
$\displaystyle\frac{d}{dt}f_{Y}(t)|_{t=0}=-\int_{M}\theta_{Y}^{\prime}((\sigma^{-1})^{*}\varphi+\rho_{\sigma^{-1}})e^{\theta_{X}}\omega_{KS}^{n}$
The gap between $f_{Y}(t)$ and $\mathcal{F}(\omega_{\phi},\omega_{t})$ can be
computed as follows. Let $\Delta=\\{(\tau,s)|~{}0\leq\tau\leq 1,~{}0\leq
s\leq\tau+(1-\tau)(1+t)$} be a domain in $\mathbb{R}^{2}$. Let
$\Phi=\Phi(\tau,s;\cdot)$ be Kähler potentials with two parameters
$(\tau,s)\in\Delta$ which satisfy:
$\displaystyle\Phi=s(-\phi+\rho_{\sigma}+\rho_{t}),~{}~{}0\leq s\leq
1,~{}~{}~{}\text{as}~{}\tau=1;$ $\displaystyle\Phi=\phi_{s},~{}0\leq s\leq
1+t,~{}\text{as}~{}\tau=0;$
$\displaystyle\Phi=0,~{}\text{as}~{}s=0;\Phi=-\phi+\rho_{\sigma}+\rho_{t},~{}s=\tau+(1-\tau)(1+t).$
Then by using the Stoke’s formula, we have
$\displaystyle|f_{Y}(t)-\mathcal{F}(\omega_{\phi},\omega_{t})|$
$\displaystyle=|\text{Re}\\{\int_{\partial\Delta}\int_{M}d_{\tau,s}\Phi(\tau,s;\cdot)e^{\theta_{X_{0}}(\phi_{s})}\omega_{\phi_{s}}^{n}\\}|$
$\displaystyle=|\text{Re}\\{\int_{\Delta}d\tau
ds\int_{M}\dot{\Phi}_{\tau}(<\overline{\partial}\dot{\Phi}_{s},\overline{\partial}\theta_{X_{0}}(\Phi)>-$
$\displaystyle<\overline{\partial}\theta_{X_{0}}(\Phi),\overline{\partial}\dot{\Phi}_{s}>)e^{\theta_{X_{0}}(\Phi)}\omega_{\Phi}^{n}\\}|$
$\displaystyle=2|\text{Re}\\{\int_{\Delta}d\tau
ds\int_{M}\dot{\Phi}_{\tau}\text{Im}(X_{0}(\Phi_{s}))e^{\theta_{X_{0}}(\Phi)}\omega_{\Phi}^{n}\\}|$
(5.4) $\displaystyle\leq C\|X^{\prime}(\phi)\|_{C^{0}}^{2}.$
At the last inequality, we used a fact that $X_{0}(\rho_{\sigma})$ and
$X_{0}(\rho_{t})$ are both real-valued. Similarly, we can get
(5.5)
$\displaystyle|\frac{d}{dt}(f_{Y}(t)-\mathcal{F}(\omega_{\phi},\omega_{t}))|_{t=0}\leq
C\|X^{\prime}(\phi)\|_{C^{0}}^{2}.$
Next we claim
(5.6)
$\displaystyle\mathcal{F}(\sigma)=\mathcal{F}(\omega_{\phi},\omega_{KS}^{\prime})\geq
0.$
To prove the claim, we let
$\displaystyle g(t)$
$\displaystyle=\text{Re}[\int_{0}^{t}ds\int_{M}(-\phi+\rho_{\sigma})e^{\theta_{X_{0}}(s(-\phi+\rho_{\sigma}))}\omega_{s(-\phi+\rho_{\sigma})}^{n}$
$\displaystyle-\int_{M}(-\phi+\rho)e^{\theta_{X_{0}}(t(-\phi+\rho_{\sigma}))}\omega_{t(-\phi+\rho_{\sigma})}^{n}].$
Then
$\mathcal{F}(\sigma)=g(1)=\int_{0}^{1}g(t)^{\prime}dt.$
On the other hand, we have
$\displaystyle g(t)^{\prime}$
$\displaystyle=\text{Re}[n\sqrt{-1}\int_{M}\partial(-\phi+\rho_{\sigma})\wedge\overline{\partial}(-\phi+\rho_{\sigma})e^{\theta_{X_{0}}(t(-\phi+\rho_{\sigma}))}\omega_{t(-\phi+\rho_{\sigma})}^{n}]$
$\displaystyle\geq 0.$
Thus we get $g(1)\geq 0$ and prove the claim.
By the above claim, we can take a minimizing sequence of $\mathcal{F}(\sigma)$
in $\text{Aut}_{r}(M)$ and we see that for any small
$\epsilon\leq\epsilon_{0}$, there exists a $\sigma\in\text{Aut}_{r}(M)$ with
bounded $\text{dist}(\sigma,Id)$ such that for any $Y\in\eta_{r}(M)$ with
$\int_{M}\|Y\|^{2}\omega_{KS}^{n}=1$, we have
(5.7) $\displaystyle|D\mathcal{F}(\sigma)(Y)|\leq\epsilon.$
Therefore combining (5.3), (5.5) and (5.7), we prove the lemma while $\sigma$
is replaced by $\sigma^{-1}$.
∎
## References
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* [22] [ST] Sesum, N. and Tian, G., Bounding scalar curvature and diameter along the Kähler-Ricci flow (after Perelman), J. Inst. Math., Jussieu, 7 (2008), 575-587.
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|
arxiv-papers
| 2009-08-11T09:33:15 |
2024-09-04T02:49:04.547242
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaohua Zhu",
"submitter": "Xiaohua Zhu",
"url": "https://arxiv.org/abs/0908.1488"
}
|
0908.1495
|
# A Cosmological Model without Singularity and its Explanation for Evolution
of the Universe and the natures of Huge Voids
Shi-Hao Chen Institute of Theoretical Physics, Northeast Normal University,
Changchun 130024, China.
shchen@nenu.edu.cn
###### Abstract
The new conjectures are proposed that there are s-matter and v-matter which
are symmetric and whose gravitational masses are opposite to each other. There
are two sorts of symmetry breaking modes called V-breaking and S-breaking,
respectively. In the V-breaking, v-particles get their masses and form
v-galaxies etc., while s-fermions and s-gauge bosons are still massless and
form s-SU(5) color-single states which loosely distribute in space and cause
space to expand with an acceleration. The curvature factor K in the RW metric
is regarded as a function of gravitational mass density in the comoving
coordinates. In the S-breaking, space can contract and causes temperature to
rise. When it reaches the critical temperature, masses of all particles are
zero so that s-particles and v-particles transform from one to another so that
the gravitational mass density is zero. Consequently space inflation occurs.
After the reheating process, the state with the highest symmetry transits into
the V-breaking. In the V-breaking, space first expands with a deceleration;
then comes to static, and finally expands with an acceleration up to now. The
cosmological constant is determined to be zero, although the energy density of
the vacuum state is still large. There is no space-time singularity in the
present model. There are the critical temperature, the highest temperature and
the least scale in the universe. A formula is obtained which well describes
the relation between a luminous distance and its redshift. The equations of
structure formation have been derived, on which, galaxies can form earlier
than the conventional theory. The universe is composed of infinite s-cosmic
islands and v-cosmic islands. A huge v-voids is not empty, in which there must
be s-matter with its bigger density, and which is equivalent to a huge concave
lens. The densities of hydrogen and helium in the huge voids must be more less
than that predicted by the conventional theory. The gravitation between two
galaxies distant enough will be less than that predicted by the conventional
theory. It is possible that a v-black hole with its big enough mass and
density can transform into a huge white hole by its self-gravitation.
Nucleosynthesis and CMBR are explained. Space-time is open or K$<$0 according
to the present model. w can change from w $>$ 0 to w ~-1.
Cosmology: theory; Inflation; early Universe; large-scale structure of
Universe; Primordial Nucleosynthesis; Mosmic Microwave Backgrond.
###### pacs:
98.65.Dx; 98.80.Bp; 95.36.+x; 98.80.Cq; 98.62.Sb; 26.35.+C; 98.70.Vc.
LABEL:FirstPage1 LABEL:LastPage#146
###### Contents
1. I Introduction
2. II Action, energy-momentum tensor and field equations
1. II.1 Conjectures
2. II.2 Explanation for the conjectures
1. II.2.1 There is no contradiction between conjecture 1 and experiments and observations up to now.
2. II.2.2 The conjectures 2-4 are consistent with the conventional theory
3. II.2.3 The inferences of the conjectures are consistent with the cosmological observations
3. II.3 Action
4. II.4 Equations of motion and energy-momentum tensors
5. II.5 The difference of motion equations of a v-particle and a s-particle in the same gravitational field
3. III Spontaneous breaking of symmetry and the gravitational mass density of the vacuum state
1. III.1 Spontaneous breaking of symmetry
2. III.2 The characters of the vacuum state
4. IV Evolving equations of space
1. IV.1 Evolving equations when curvature factor $k$ is regarded as a constant
2. IV.2 The evolving equations when $K=K(\underline{\rho}_{g}\left(t,R\left(t\right)\right))$
5. V Temperature effect
1. V.1 Two sorts of temperature
2. V.2 The influences of finite temperature on the Higgs potential
1. V.2.1 Effective potentials
2. V.2.2 Critical temperatures and masses of the Higgs particles
3. V.3 $\rho_{s}$ and $\rho_{v}$ transform from one into another when $T_{s}\longrightarrow T_{cr}$ so that $\rho_{s}-\rho_{v}\longrightarrow 0$
6. VI Simplification of the Higgs potential
1. VI.1 Change of $w$ from $>0$ to $<-1$ and $q$
7. VII Contraction of space, the highest temperature and inflation of space
1. VII.1 Contraction of space, proof of non-singularity, the highest temperature and inflation of space
2. VII.2 The result above is not contradictory to the singularity theorems
3. VII.3 The process of space inflation
8. VIII Expansion of space after inflation
1. VIII.1 The reheating process
2. VIII.2 Change of gravitational mass density in comoving coordinates
3. VIII.3 Space expands with a deceleration
4. VIII.4 Space expands with an acceleration
5. VIII.5 To determine $a\left(t\right)$
6. VIII.6 The relation between redshift and luminosity distance
7. VIII.7 Repulsive potential energy chiefly transforms into the kinetic energy of SU(5) color single states
9. IX Existing and distributive forms of s-SU(5) color single states in the V-breaking
1. IX.1 Sorts of s-SU(5) color single states in the V-breaking
2. IX.2 The characters of the s-SU(5) color single states in V-breaking
10. X Dynamics of v-structure formation and the distributive form of the s-SU(5) color single states
11. XI Some guesses, new predictions and an inference
1. XI.1 Some guesses
1. XI.1.1 The universe is composed of infinite s-cosmic islands and v-cosmic islands
2. XI.1.2 Mass redshifts
2. XI.2 New predictions
1. XI.2.1 It is possible that huge voids are equivalent to huge concave lenses. The densities of hydrogen and helium in the huge voids predicted by the present model must be more less than that predicted by the conventional theory.
2. XI.2.2 The gravitation between two galaxies distant enough will be less than that predicted by the conventional theory.
3. XI.2.3 A black hole with its mass and density big enough will transform into a white hole
4. XI.2.4 The transformation of the cosmic ultimate
3. XI.3 An inference :$\lambda_{eff}=\lambda=0,$ although $\rho_{0}\neq 0$
12. XII Primordial nucleosynthesis
1. XII.1 F-W dark matter model
2. XII.2 Primordial nucleosynthesis
13. XIII Cosmic microwave background radiation ($CMBR)$
1. XIII.1 The recombination temperature $T_{rec}$
2. XIII.2 The temperature $T_{eq}$ of matter-radiation equality
3. XIII.3 Decoupling temperature
4. XIII.4 Space-time is open, i. e. $K<0$.
14. XIV Conclusions
## I Introduction
We consider that it is impossible to solve the singularity issue of space-time
and the issue of the cosmological constant in the frame of the conventional
theory, so that new conjectures are necessary. On the basis of a new essential
conjecture, we can solve the two issues, well explain evolution of the
universe and the characters of huge voids, and give new predictions and some
guesses.
As is now well known, there is space-time singularity under certain
conditions[1]. These conditions fall into three categories. First, there is
the requirement that gravity shall be attractive. Secondly, there is the
requirement that there is enough matter present in some region to prevent
anything escaping from that region. The third requirement is that there should
be no causality violations. Because of the theorems, there must be space-time
singularity in the conventional theory. On the other hand, there should be no
space-time singularity in physics. Hence this problem must be solved.
Recent astronomical observations show that the universe expanded with a
deceleration earlier while is expanding with an acceleration now. This implies
that there is dark energy. $0.73$ of the total energy density of the universe
is dark energy density${}^{[2]}.$What is dark energy? Many possible answers
have been given. One possible interpretation is in terms of the effective
cosmological constant $\lambda_{eff}=$ $\lambda+\rho_{g0},$ here $\lambda$ and
$\rho_{g0}$ are respectively the Einstein’s cosmological constant and the
gravitational mass density of the vacuum state. According to the equivalent
principle, $\rho_{g0}=\rho_{0}$, $\rho_{0}$ is the mass density of the vacuum
state, hence $\lambda_{eff}$ may written as $\lambda+\rho_{0}.$
$\lambda_{eff}$ cannot be derived from basic theories[3] and
$\rho_{0}\ggg\lambda_{eff}$. Hence the interpretation is unsatisfactory.
Alternatively, dark energy is associated with the dynamics of scalar field
$\phi\left(t\right)$ that is uniform in space[4]. This is a seesaw
cosmology[5]. Thus, discussion about the universe expansion with an
acceleration is still open to the public.
$\rho_{g0}=\rho_{0}\ggg 0$ originates from the conventional quantum field
theory and the equivalent principle. $\rho_{0}\ggg\lambda_{eff}$ and the
singularity issue imply that the conventional theory is not self-consistent.
$\rho_{0}=0$ is a necessary result of our quantum field theory without
divergence${}^{[6]}.$ In this theory, there is no divergence of loop
corrections, and dumpling dark matter is predicted${}^{[7]}.$ It is different
from the supersymmetric quantum field theory in which $\rho_{0}=0$ can be
obtained in only some models but is not necessary. Thus, issue of the
cosmological constant is open as well.
Huge voids in the cosmos have been observed${}^{[8]}.$ The model in which the
hot dark matter (e.g. neutrinos) is major can explain the phenomenon, however,
it cannot explain the structure with middle and small scales. Hence this is an
open problem as well.
We consider that all important existing forms of matter, dark matter and dark
energy, to be presented. Hence the basic problems, e.g. divergence problem in
quantum field theory, cosmological constant and space-time singularity
problems in cosmology, should be solved based on the important existing forms
of matter. We have constructed a quantum field theory without divergence by
dark matter. We construct a model of the universe without singularity in the
paper by dark energy.
We consider the following conditions 1 and 2 to be necessary in order to solve
the singularity and the cosmological constant problems in the basis of the
classical cosmology and the frame of the conventional quantum field theory.
The third condition is necessary to cause the space inflation.
Condition 1
There are two sorts of matter which are symmetric, whose gravitational masses
are opposite to each other and whose energies are all positive.
The two sorts of matter are called $s-matter$ and $v-matter$, respectively.
The condition implies that if $\rho_{s}=\rho_{v},$ $\rho_{gs}=-\rho_{gv}.$
Condition 2
When temperature is high enough, the thermal equilibrium between the two sorts
of matter can come into being, $\rho_{s}$ and $\rho_{v}$ can transform from
one into another so that $\rho_{g}=\rho_{gs}+\rho_{gv}=\rho_{s}-\rho_{v}=0$.
Condition 3
$\rho_{g}=\rho_{s}-\rho_{v}=0$ and $V=V_{0}=V_{\max}>0$ when $T\geq T_{cr},$
here $T_{cr}$ is the critical temperature.
The conditions 1 and 2 cannot be realized in the conventional theory. In order
to uniformly solve the above four problems on one basis, we present new
conjectures (see section 2) and construct a cosmological model on the basis of
the conjectures[9].
The basic idea of the present model is the conjecture 1, which realizes the
conditions Consequently there is no singularity of space-time and
$\rho_{g0}=0$ is proven, although $\rho_{0}$ is still very large. Thus, there
is no the fine tuning problem, even if $\lambda_{eff}\neq 0$. There are two
sorts of breaking modes which are called $S-breaking$ and $V-breaking$ due to
the conjecture 1. $\rho_{g}>0,$ $=0$ or $<0$ are all possible due to the
conjecture 1. Hence the curvature factor $k$ in the $RW$ metric should be
changeable. We consider $k$ to be a function of $\underline{\rho}_{g},$
$\underline{\rho}_{g}$ is the gravitational mass density in comoving
coordinates, i.e.,
$K=K\left(\underline{\rho}_{g}\left(t,R\left(t\right)\right)\right)$. The
evolving equations corresponding to $K$ have been derived[9].
An important basis of the present model is the temperature effect on
expectation values of Higgs fields.
The present model have the following results.
According to the present model, the evolving process of space is as follows.
In the $S-breaking,$ space contracts $\longrightarrow$temperature $T$ rises to
the highest temperature $T_{\max}$ and the highest symmetry comes into being
$\longrightarrow$ space inflation $\longrightarrow$ reheating process
$\longrightarrow$ the state with the highest symmetry transits to the state
with the $V-breaking$ and space expands with a deceleration $\longrightarrow$
then comes to static $\longrightarrow$ finally expands with an acceleration up
to now.
$w=p/\rho$ can change from $w>0$ to $w<-1$.
There are the critical temperature $T_{cr}$, the highest temperature
$T_{\max}$, the least scale $R_{\min}$ and the largest energy density
$\rho_{\max}$ in the universe. $R_{\min}$ and $T_{cr}$ are two new important
constants, $T_{\max}$ and $\rho_{\max}$ are determined by
$R\left(T_{cr}\right)$.
A formula comes out which well describes the relation between a luminosity
distance and its redshift.
The new predictions of the present model are as follows.
$V-huge$ voids in the $V-breaking$ are not empty, but there is $s-matter$ with
its density $\rho_{s}\gg\rho_{v}$ in them. Their effects are similar to those
of huge concave lenses for $v-photons$. In huge voids, both matter and dark
matter are even shorter in the huge void. Thus the density of hydrogen in the
huge voids must be more less that that predicted by the conventional theory.
Right or mistake of the predict can be confirmed by the observation of
distribution of hydrogen.
When the distance between two $v-galaxies$ is very large, the gravitation
between both will be less than that predicted by the conventional theory,
because there must be $s-matter$ between both.
We generalize equations governing nonrelativistic fluid motion to present
model. The equations of structure formation have been derived. According to
the equations, galaxies can form earlier and easier than that in the
conventional theory.
On the basis of this model, we have three guesses.
The universe is composed of infinite $s-universal$ islands and $v-universal$
islands.
Some huge redshifts (e.g. the big redshifts of quasi-stellar objects) are
explained as the mass redshifts which is caused by less mass $m\prime_{e}$ of
an electron than given $m_{e}$.
It is possible that a $v-black$ hole with its big enough mass and density can
transform into a huge white hole by its self-gravitation. Of course, the
effects of quantum mechanics, e.g. Hawking radiation, must be considered. But
there is no contradiction between the transformation and quantum mechanics.
It seems that there are some difficulties in the present model, e.g., it seems
that this model is not consistent with primordial nucleosynthesis and the
$CMBR$ data. It is a misunderstanding. In fact, in a broad scope of
parameters, the primordial nucleosynthesis and $CMBR$ can be explained based
on the $F-W$ dark matter model (or the mirror dark matter model) and this
cosmological model.
The first peak of the $CBMR$ power spectra is the evidence of existence of the
elementary wave. The elementary wave began at reheating and ended at
recombination after $\bigtriangleup t_{hc}^{\prime}\equiv 3.8\times 10^{5}$
years according to the conventional theory. But according to the present
model, it is necessary that $\bigtriangleup t_{hc}>\bigtriangleup
t_{hc}^{\prime},$ because the sound speed $c_{s}\sim c_{s}^{\prime}$ and
$H=\eta\rho_{g}<H^{\prime}=\eta\rho_{g}^{\prime}=\eta\rho^{\prime},$ here
$\bigtriangleup t_{hc}$ is the period in which $T_{reh}$ descends into
$T_{rec},$ $c_{s}^{\prime},$ $\bigtriangleup t_{hc}^{\prime}$ and $H^{\prime}$
are the physical quantities in the conventional theory. Consequently, space-
time is open or $K<0$ according to the present model.
In section 2, action, energy-momentum tensor and field equations are
presented; In section 3, spontaneous breaking of symmetry and the
gravitational mass density of the vacuum state are discussed; In section 4,
evolving equations of space are derived out; In section 5, temperature effects
are considered; In section 6, the inflation process and change of $w$ and $q$
are discussed; In section 7, contraction of space, the highest temperature and
inflation of space are considered; In section 8, expansion of space after
inflation is discussed; In section 9, existing and distributive forms of
$s-SU(5)$ color single states in the $V-breaking$ are discussed; In section
10, dynamics of $v-structure$ formation and the distributive form of the
$s-SU(5)$ color single states are got; In section 11, some guesses, new
predictions and an inference are given; In section 12, the primordial
nucleosynthesis is discussed; In section 14, cosmic microwave background
radiation is explained; Section 14 is the conclusions; Section 15 is
discussion about conjecture 1.
## II Action, energy-momentum tensor and field equations
### II.1 Conjectures
In order to solve the above problems, we propose the following conjectures.
###### Conjecture 1
There are two sorts of matter which are called $solid-matter$ ($s-matter$) and
$void-matter$ ($v-matter$), respectively. Both are symmetric and their
contributions to the Einstein tensor are opposite each other. There is no
other interaction between both except the interaction described by $(10)$
between $s-Higgs$ $fields$ and $v-Higgs$ $fields$.
###### Conjecture 2
$\lambda_{eff}=0,$ where $\lambda_{eff}$ is the effective cosmological
constant.
###### Conjecture 3
The curvature factor
$K=K\left(\underline{\rho}_{g}\left(t,R\left(t\right)\right)\right)$ in the
Robertson -Walker metric is a monotone and finite function of
$\underline{\rho}_{g},$ $dK/d\underline{\rho}_{g}>0$ and $K=0$ for
$\underline{\rho}_{g}=0,$ here $\underline{\rho}_{g}$ is the gravitational
mass density in the comoving coordinates.
###### Conjecture 4
When $SU(5)$ symmetry holds water and temperature is low, all particles in
free states must exist in $SU(5)$ color single states.
The other premise of the present model is the conventional $SU(5)$ grand
unified theory $\left(GUT\right).$ But it is easily seen that the present
model does not rely on the special $GUT$. Provided the conjecture 1 and such a
coupling as $\left(10\right)$ are kept in a $GUT,$ the $GUT$ can be accepted.
In fact, only the conjecture 1 is essential. The other conjectures are
obviously consistent with the conventional theory.
All the following inferences hold water when $S\rightleftarrows V$ and
$s\leftrightarrows v$ due to the conjecture 1.
### II.2 Explanation for the conjectures
#### II.2.1 There is no contradiction between conjecture 1 and experiments
and observations up to now.
$\mathbf{1.}$ $S-matter$ and $v-matter$ are asymmetric because of the symmetry
spontaneously breaking.
Matter determines properties of space-time. Different breaking modes of Higgs
fields correspond to different ground states. There are two sorts of breaking
modes which are called $S-breaking$ and $V-breaking$. In the $S-breaking,$ the
expectation values of $s-Higgs$ fields are not zero and the expectation values
of all $v-Higgs$ fields are zero. Consequently, the $s-SU(5)$ symmetry finally
breaks into $s-SU(3)\times U(1),$ $s-particles$ can get their masses and form
$s-atoms,$ $s-observers$ and $s-galaxies$, and the $v-SU(5)$ symmetry is still
strictly kept, all $v-fermions$ and $v-gauge$ bosons are massless and must
form $v-SU(5)$ color-single states when temperature is low. There is no
electroweak gauge interaction among the $v-SU(5)$ color-single states because
the $v-SU(5)$ is a simple group. Consequently the $v-SU(5)$ color-single
states cannot form $v-atoms,$ $v-observers$ and $v-galaxies$, and must loosely
distribute in space as the so-called dark energy. Thus, in the $S-breaking$
$s-matter$ is identified with the conventional matter forming the given world,
and $v-matter$ can cause space to expand with an acceleration as dark energy
and cannot be observed except by the repulsion. In contrast with the dark
energy, the gravitational masses of $v-matter$ is negative in the
$S-breaking.$
$\mathbf{2}$. There are only the repulsion between $s-matter$ and $v-matter$
when temperature is low.
The interaction $\left(10\right)$ between the $v-Higgs$ fields and the
$s-Higgs$ fields is repulsive, the masses of Higgs particles are all very
large and the Higgs particles must decay fast at low temperature. Hence the
interaction may be ignored when temperature is low. Thus There are only the
repulsion between $s-matter$ and $v-matter$ when $T\ll T_{cr}.$ Consequently,
any bound state is composed of only $s-particles$ or only $v-particles$, there
is no the transformation of $s-particles$ and $v-particles$ from one into
another when $T\ll T_{cr}$, and if $\rho_{v}$ is very large, $\rho_{s}$ must
be very little in the same region. Thus, in the $V-breaking,$ there must be
$\rho_{s}\ll\rho_{v}$ in a $v-galaxy$ so that $\rho_{s}$ may be ignored.
$\mathbf{3.}$ The equivalence principle still holds for the ordinary
particles.
In the $V-breaking,$ $v-particles$ are identified as the ordinary particles to
form the given world and there are only the $v-observers,$ and there is no
$s-observer,$ hence the gravitational masses of $v-matter$ must be positive,
i.e. $m_{vg}=m_{v},$ and the gravitational masses of $s-matter$ must be
negative relatively to the $v-observers,$ i.e. $m_{sg}=-m_{s},$ because of the
conjecture 1. Thus the equivalence principle still holds for $v-matter$ (given
matter), but is violated for $s-matter$. Thus a $v-photon$ falling in a
gravitational field must have redshift, but a $s-particle$ (there is no
$s-photon$ and there are only $s-SU(5)$ color single states) will have purple
shift. This result does not contradict the experiments and observations up to
now, because of the above reasons. In fact, it is too difficult that a
$v-observor$ observes $s-particles$, because $\rho_{s}\ll\rho_{v}$ in a
$v-galaxy$, there is only the repulsion between $s-matter$ and $v-matter$ and
the $s-SU(5)$ color single states can only loosely distribute in space. In the
other hand, there is no reason to demands demand unknown matter to satisfy the
equivalence principle.
$\mathbf{4.}$ $\rho_{s}$ and $\rho_{v}$ can transform from one into another
when temperature is high enough, i.e., $T\sim T_{cr}$ (see later)..
When $T\sim T_{cr},$ the expectation values of all Higgs fields and the masses
of all particles are zero and the interaction $\left(10\right)$ between the
$s-Higgs$ fields and the $v-Higgs$ fields is important. Consequently,
$\rho_{s}$ and $\rho_{v}$ can transform from one into another by
$\left(10\right)$ so that $\rho_{s}=\rho_{v}$, $T_{s}=T_{v}\sim T_{cr}$ and
the symmetry $v-SU(5)\times s-SU(5)$ holds in this case. This is a new case
which is different from any given experiment and observation.
In addition, conjecture 1 is necessary in order to solve the following issues.
$\mathbf{1.}$ The cosmological constant issue.
$\mathbf{2.}$ The space-time singularity issue.
#### II.2.2 The conjectures 2-4 are consistent with the conventional theory
$\left(1\right)$. $\lambda_{eff}=0$ is a necessary inference because we can
explain evolution of space without $\lambda_{eff}$. On the basis, the
cosmological constant problem is easily solved.
$\left(2\right).$ In contrast with the conventional theory, all
$\underline{\rho}_{g}>0,$ $=0$ and $<0$ are possible, hence the curvature
factor $K>0,$ $=0$ and $<0$ are all possible as well. Consequently $K$ is
regarded as a function of the gravitational mass density
$\underline{\rho}_{g}$ in the comoving coordinates, i.e.
$K=K\left(\underline{\rho}_{g}\left(t,R\left(t\right)\right)\right)$. The
evolving equations corresponding to $K$ have been derived[9].
$\left(3\right).$As is well known, the $SU(3)$ theory has proven that there
can be the $SU(3)$ glue-balls whose masses are not zero. The $SU(5)$ color
single states can be regarded as generalization of the $SU(3)$ glue-balls. In
contrast with the $SU(3)$ glue-balls, there is no the interaction similar with
$U(1)$ gauge interaction among the $SU(5)$ color single states because $SU(5)$
is a single group.
Sum up, in the $v-breaking,$ the$\ s-particles$ have only the cosmological
effects and cannot be observed. Consequently there is no contradiction between
conjectures and experiments and observations up to now.
#### II.2.3 The inferences of the conjectures are consistent with the
cosmological observations
Based on the conjecture and the $F-W$ dark matter model (or mirror dark matter
model), the following inferences are consistent with the cosmological
observations.
$\mathbf{1.}$ The evolution of the universe and the relation between distance
and redshift;
$\mathbf{2.}$ Large-scale structure formation;
$\mathbf{3.}$ The features of huge voids;
$\mathbf{4.}$ Primordial nucleosynthesis (it is necessary to consider
simultaneously the conjecture and the $F-W$ dark matter model);
$\mathbf{5.}$ Cosmic microwave background radiation (it is necessary to
consider simultaneously the conjecture and the $F-W$ dark matter model).
$\mathbf{6.}$ $w=p/\rho$ can change from $w>0$ to $w<-1.$
Based on the conjectures, there are predicts to be confirmed.
### II.3 Action
The breaking mode of the symmetry is only one of the $S-breaking$ and the
$V-breaking$ due to $\left(10\right)$. In the $S-breaking,$ there are only the
$s-observators.$ In the $V-breaking,$ there are only the $v-observators$.
Hence the actions should be written as two sorts of form, $I_{S}$ in the
$S-breaking$ and $I_{V}$ in the $V-breaking$. Because of the conjecture 1, the
structures of $I_{S}$ and $I_{V}$ are the same, i.e. $I_{V}\rightleftarrows
I_{S}$ when $S\rightleftarrows V$ and $s\rightleftarrows v$. Thus, at the
zero-temperature we have
$I_{S}=I_{g}+I_{SM}=I_{g}+I_{VM}=I_{V},$ (1) $I_{g}=\frac{1}{16\pi
G}\left(\mathop{\displaystyle\int}_{\Sigma}R\sqrt{-g}d^{4}x+2\mathop{\displaystyle\int}\nolimits_{\partial\Sigma}K\sqrt{\pm
h}d^{3}x\right),$ (2) $I_{SM}=\int d^{4}x\sqrt{-g}\mathcal{L}_{SM},\text{ \ \
\
}\mathcal{L}_{SM}=\alpha\mathcal{L}_{s}+\beta\mathcal{L}_{v}+V_{0}+\frac{1}{2}\left(\alpha+\beta\right)V_{sv},$
(3) $I_{VM}=\int d^{4}x\sqrt{-g}\mathcal{L}_{VM},\text{ \ \ \
}\mathcal{L}_{VM}=\alpha\mathcal{L}_{v}+\beta\mathcal{L}_{s}+V_{0}+\frac{1}{2}\left(\alpha+\beta\right)V_{vs},$
(4)
$\mathcal{L}_{s}=\mathcal{L}_{sM}\left(\Psi_{s},g\left(x\right),g\left(x\right),_{\mu}\right)+V_{s}\left(\omega_{s}\right),$
(5)
$\mathcal{L}_{v}=\mathcal{L}_{vM}\left(\Psi_{v},g\left(x\right),g\left(x\right),_{\mu}\right)+V_{v}\left(\omega_{v}\right),$
(6)
$V_{sv}\left(\omega_{s},\omega_{v}\right)=V_{vs}\left(\omega_{s},\omega_{v}\right);$
(7) $\omega_{s}\equiv\Omega_{s},\;\Phi_{s},\;\chi_{s};\;\ \ \ \ \
\omega_{v}\equiv\Omega_{v},\;\Phi_{v},\;\chi_{v},$
where the meanings of the symbols are as follows. $g=\det(g_{\mu\nu})$,
$g_{\mu\nu}=diag(-1,1,1,1)$ in flat space. $R$ is the scalar curvature. Here
$\alpha$ and $\beta$ are two parameters and we finally take $\alpha=-\beta=1$.
$V_{0}$ is a parameter which is so taken that
$V_{s\min}\left(\varpi_{s}\right)+V_{0}=0$ in the $S-breaking$ or
$V_{v\min}\left(\varpi_{v}\right)+V_{0}=0$ in the V-breaking at the $zero-
temperature,$ $\varpi=\langle\omega\rangle$. $\mathcal{L}_{sM}$
($\mathcal{L}_{vM}$) is the Lagrangian density of all $s-fields$ ($v-fields$)
and their couplings of the $SU(5)$ $GUT$ except the Higgs potentials $V_{s},$
$V_{v}$ and $V_{sv}$. $\Psi_{s}$ and $\Psi_{v}$ represents all $s-fields$ and
all $v-fields$, respectively. $\mathcal{L}_{s}$ and $\mathcal{L}_{s}$ do not
contain the contribution of the gravitational and repulsive fields. It is seen
that the set of equation $(1)-(7)$ is unchanged when the subscripts
$s\rightleftarrows v$ and $S\rightleftarrows V$. This shows the symmetry
between $s-matter$ and $v-matter.$
Gibbons and Hawking pointed out[10] that in order to get the Einstein field
equations, it is necessary
$I_{g}^{\prime}=\frac{1}{16\pi
G}\mathop{\displaystyle\int}\nolimits_{\Sigma}R\sqrt{-g}d^{4}x\longrightarrow
I_{g}=\frac{1}{16\pi
G}\left(\mathop{\displaystyle\int}_{\Sigma}R\sqrt{-g}d^{4}x+2\mathop{\displaystyle\int}\nolimits_{\partial\Sigma}K\sqrt{\pm
h}d^{3}x\right).$
This is because it is not necessary that $\delta\Gamma_{\mu\nu}^{\alpha}=0$ in
$\delta I_{g}^{\prime}$ on the boundary $\partial\Sigma.$ Hence
$I_{g}^{\prime}$ is replaced by $I_{g}$ in $\left(2\right).$ $\Sigma$ is a
manifold with four dimensions. $\partial\Sigma$ is the boundary of $\Sigma.$
$K=trK_{j}^{i}.$ $K_{ij}=-\nabla_{i}n_{j}$ is the outer curvature on
$\partial\Sigma.$ $n_{j}$ is the vertical vector on $\partial\Sigma.$ $h=\mid
h_{ij}\mid,$ and $h_{ij}$ is the induced outer metric on $\partial\Sigma$.
When $\partial\Sigma$ is space-like, $\sqrt{\pm h}$ takes positive sign. When
$\partial\Sigma$ is time-like, $\sqrt{\pm h}$ takes negative sign.
The Higgs potentials in $\left(5\right)-\left(7\right)$ is the following.
$\displaystyle V_{s}$
$\displaystyle=-\frac{1}{2}\mu^{2}\Omega_{s}^{2}+\frac{1}{4}\lambda\Omega_{s}^{4}$
$\displaystyle-\frac{1}{2}w\Omega_{s}^{2}Tr\Phi_{s}^{2}+\frac{1}{4}a\left(Tr\Phi_{s}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{s}^{4}\right)$
$\displaystyle-\frac{1}{2}\varsigma\Omega_{s}^{2}\chi_{s}^{+}\chi_{s}+\frac{1}{4}\xi\left(\chi_{s}^{+}\chi_{s}\right)^{2},$
(8)
$\displaystyle V_{v}$
$\displaystyle=-\frac{1}{2}\mu^{2}\Omega_{v}^{2}+\frac{1}{4}\lambda\Omega_{v}^{4}$
$\displaystyle-\frac{1}{2}w\Omega_{v}^{2}Tr\Phi_{v}^{2}+\frac{1}{4}a\left(Tr\Phi_{v}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{v}^{4}\right)$
$\displaystyle-\frac{1}{2}\varsigma\Omega_{v}^{2}\chi_{v}^{+}\chi_{v}+\frac{1}{4}\xi\left(\chi_{v}^{+}\chi_{v}\right)^{2},$
(9) $\displaystyle V_{sv}$
$\displaystyle=\frac{1}{2}\Lambda\Omega_{s}^{2}\Omega_{v}^{2}+\frac{1}{2}\alpha\Omega_{s}^{2}Tr\Phi_{v}^{2}+\frac{1}{2}\beta\Omega_{s}^{2}\chi_{v}^{+}\chi_{v}$
$\displaystyle+\frac{1}{2}\alpha\Omega_{v}^{2}Tr\Phi_{s}^{2}+\frac{1}{2}\beta\Omega_{v}^{2}\chi_{s}^{+}\chi_{s},$
(10)
where $\Omega_{a}$,
$\Phi_{a}=\overset{24}{\underset{i=1}{\sum}}\left(T_{i}/\sqrt{2}\right)\varphi_{ai}$
and $\chi_{a}$ are respectively $\underline{1}$ , $\underline{24}$ and
$\underline{5}$ dimensional representations of the $SU(5)$ group, $a=s,v$,
here the couplings of $\Phi_{a}$ and $\chi_{a}$ are ignored for
short${}^{[11]}.$ We do not consider the terms coupling to curvature scalar,
e.g. $\xi R\Omega^{2}$, for a time. In fact, $\xi
R\left(\langle\Omega_{s}^{2}\rangle-\langle\Omega_{v}^{2}\rangle\right)\sim 0$
when temperature $T$ is high enough due to the symmetry between $s-matter$ and
$v-matter$.
For short, we take $\alpha=w$ in the paper. We will see in the following paper
that when $\alpha>w,$ the duration $\tau$ of inflation can be long enough
without the slow approximation.
### II.4 Equations of motion and energy-momentum tensors
By the conventional method, from $\left(2\right)$ we can get
$\delta I_{g}=\frac{1}{16\pi
G}\mathop{\displaystyle\int}\nolimits_{\Sigma}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)\delta
g^{\mu\nu}\sqrt{-g}d^{4}x.$ (11a)
Considering $\alpha=-\beta=1$, from $\left(3\right)$ we have
$\displaystyle\delta I_{SM}$
$\displaystyle=\mathop{\displaystyle\int}\frac{1}{\sqrt{-g}}\left[\frac{\partial\mathcal{L}_{SM}\sqrt{-g}}{\partial
g^{\mu\nu}}-\left(\frac{\partial\mathcal{L}_{SM}\sqrt{-g}}{\partial
g_{,\sigma}^{\mu\nu}}\right)_{,\sigma}\right]\delta g^{\mu\nu}\sqrt{-g}d^{4}x$
$\displaystyle=\mathop{\displaystyle\int}\left\\{\frac{1}{\sqrt{-g}}\left[\frac{\partial\mathcal{L}_{sM}\sqrt{-g}}{\partial
g^{\mu\nu}}-\left(\frac{\partial\mathcal{L}_{sM}\sqrt{-g}}{\partial
g_{,\sigma}^{\mu\nu}}\right)_{,\sigma}\right]-\frac{1}{2}g_{\mu\nu}\left(V_{s}+V_{0}\right)\right\\}\delta
g^{\mu\nu}\sqrt{-g}d^{4}x$
$\displaystyle-\mathop{\displaystyle\int}\left\\{\frac{1}{\sqrt{-g}}\left[\frac{\partial\mathcal{L}_{vM}\sqrt{-g}}{\partial
g^{\mu\nu}}-\left(\frac{\partial\mathcal{L}_{vM}\sqrt{-g}}{\partial
g_{,\sigma}^{\mu\nu}}\right)_{,\sigma}\right]-\frac{1}{2}g_{\mu\nu}V_{v}\right\\}\delta
g^{\mu\nu}\sqrt{-g}d^{4}x$
$\displaystyle=\mathop{\displaystyle\int}\frac{1}{2}\left(T_{s\mu\nu}-g_{\mu\nu}V_{0}-T_{v\mu\nu}\right)\delta
g^{\mu\nu}\sqrt{-g}d^{4}x,$ (11b)
where
$\displaystyle T_{s\mu\nu}$ $\displaystyle=T_{sM\mu\nu}-g_{\mu\nu}V_{s}$
$\displaystyle=2\frac{1}{\sqrt{-g}}\left[\frac{\partial\left(\sqrt{-g}\mathcal{L}_{sM}\right)}{\partial
g^{\mu\nu}}-\left(\frac{\partial\left(\sqrt{-g}\mathcal{L}_{sM}\right)}{\partial
g^{\mu\nu},_{\sigma}}\right),_{\sigma}\right]-g_{\mu\nu}V_{s},$ (12a)
$\displaystyle T_{v\mu\nu}$ $\displaystyle=T_{vM\mu\nu}-g_{\mu\nu}V_{v}$
$\displaystyle=2\frac{1}{\sqrt{-g}}\left[\frac{\partial\left(\sqrt{-g}\mathcal{L}_{vM}\right)}{\partial
g^{\mu\nu}}-\left(\frac{\partial\left(\sqrt{-g}\mathcal{L}_{vM}\right)}{\partial
g^{\mu\nu},_{\sigma}}\right),_{\sigma}\right]-g_{\mu\nu}V_{v}.$ (12b)
From $\left(11\right)$ we obtain
$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=-8\pi
G\left(T_{s\mu\nu}-g_{\mu\nu}V_{0}-T_{v\mu\nu}\right)\equiv-8\pi
GT_{Sg\mu\nu},$ (13a) $\displaystyle T_{Sg\mu\nu}$ $\displaystyle\equiv
T_{s\mu\nu}-g_{\mu\nu}V_{0}-T_{v\mu\nu}=T_{SMg\mu\nu}-g_{\mu\nu}V_{Sg}$ (13b)
$\displaystyle T_{SMg\mu\nu}$ $\displaystyle\equiv
T_{sM\mu\nu}-T_{vM\mu\nu},\text{ \ \ \ }V_{Sg}=V_{s}+V_{0}-V_{v},$ (13c)
in the $S-breaking.$ It is seen from $\left(13c\right)$ that $V_{Sg}$ is
independent of $V_{sv}.$ In fact, $V_{v\min}\left(\varpi_{v}\right)=0$ due
$\langle\omega_{v}\rangle=0$ in the $S-breaking$, hence
$V_{Sg\min}\left(\varpi_{s},\varpi_{v}\right)=V_{s\min}+V_{0}-V_{v\min}=V_{s\min}+V_{0}.$
(13d)
$T_{Sg\mu\nu},$ $T_{SMg\mu\nu}$ and $V_{Sg}$ are the gravitational energy-
momentum tensor density, the gravitational energy-momentum tensor density
without the Higgs potential and the gravitational potential density of the
Higgs fields in the $S-breaking$, respectively. Analogously, from
$\left(2\right)$ and $\left(4\right)$ we obtain
$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=-8\pi
G\left(T_{v\mu\nu}-g_{\mu\nu}V_{0}-T_{s\mu\nu}\right)\equiv-8\pi
GT_{Vg\mu\nu},$ (14a) $\displaystyle T_{Vg\mu\nu}$ $\displaystyle\equiv
T_{v\mu\nu}-g_{\mu\nu}V_{0}-T_{s\mu\nu}=T_{VMg\mu\nu}-g_{\mu\nu}V_{Vg},$ (14b)
$\displaystyle T_{VMg\mu\nu}$ $\displaystyle\equiv
T_{vM\mu\nu}-T_{sM\mu\nu},\text{ \ \ \ }V_{Vg}=V_{v}+V_{0}-V_{s},$ (14c)
in the $V-breaking.$
From $(1)$ the energy-momentum tensor density which does not contain the
energy-momentum tensor of gravitational and repulsive fields can be defined as
$\displaystyle T_{S\mu\nu}$
$\displaystyle=\frac{2}{\sqrt{-g}}\left(\frac{\partial}{\partial\alpha}+\frac{\partial}{\partial\beta}\right)\left[\frac{\partial\left(\sqrt{-g}\mathcal{L}_{S}\right)}{\partial
g^{\mu\nu}}-\left(\frac{\partial\left(\sqrt{-g}\mathcal{L}_{S}\right)}{\partial
g^{\mu\nu},_{\sigma}}\right),_{\sigma}\right]$ $\displaystyle\equiv
T_{s\mu\nu}+T_{v\mu\nu}-g_{\mu\nu}\left(V_{sv}+V_{0}\right)=T_{SM\mu\nu}-g_{\mu\nu}V_{S}=T_{V\mu\nu}\equiv
T_{\mu\nu},$ (15a) $\displaystyle T_{SM\mu\nu}$
$\displaystyle=T_{sM\mu\nu}+T_{vM\mu\nu}=T_{VM\mu\nu}\equiv T_{M\mu\nu},$
$\displaystyle V_{S}$ $\displaystyle=V_{s}+V_{v}+V_{sv}+V_{0}=V_{V}\equiv V.$
(15b)
In fact, $V_{v\min}=V_{sv\min}=0$ due $\langle\omega_{v}\rangle=0$ in the
$S-breaking$, hence $V_{S\min}=V_{s\min}+V_{0}.$
It should be pointed out that only $\left(13\right)$ is applicable in the
$S-breaking,$ and only $\left(14\right)$ applicable in the $V-breaking.$
It should be noticed from $(13)-(15)$ that the potential energy $V_{sv}$ is
different from other energies in essence. There is no contribution of $V_{sv}$
to $R_{\mu\nu},$ i.e., there is no gravitation and repulsion of the potential
energy $V_{sv}.$ This does not satisfy the equivalence principle. But this
does not cause any contradiction with all given experiments and astronomical
observations because $V_{sv}=0$ in either of breaking mode.
It is proved that the necessary and sufficient condition of
$T_{;\nu}^{\mu\nu}=0$ is $I_{M}$ to be a scalar quantity[12]. $I_{S}$ and
$I_{V}$ are all scalar quantities, hence
$T_{S;\nu}^{\mu\nu}=T_{V;\nu}^{\mu\nu}=0.$ (16)
### II.5 The difference of motion equations of a v-particle and a s-particle
in the same gravitational field
The geodesic equations of the present model are the same as the conventional
equations, i.e.,
$\frac{d^{2}x^{\mu}}{d\sigma^{2}}+\Gamma_{\alpha\beta}^{\mu}\frac{dx^{\alpha}}{d\sigma}\frac{dx^{\beta}}{d\sigma}=0.$
(17)
The field equation can be rewritten as
$R_{\mu\nu}=8\pi
G\left[\left(T_{s\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{s}\right)-\left(T_{v\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{v}\right)\right],$
(18)
in the $S-breaking$.
In order to compare the equation of motion of a $s-particle$ and the equation
of motion of a v-particle in $S-breaking$. We take the Newtonian
approximation. Under the Newtonian limit, from $\left(17\right)$ we have[13].
$\Gamma_{00}^{k}\simeq\frac{1}{2}\frac{\partial g_{00}}{\partial x^{k}},$ (19)
$\overset{\cdot\cdot}{x}^{k}\simeq-\Gamma_{00}^{k}\simeq-\frac{1}{2}\frac{\partial
g_{00}}{\partial x^{k}}.$ (20)
From $\left(18\right)$ we have
$R_{00}=4\pi G\left(\rho_{s}-\rho_{v}\right).$ (21)
On the other hand, the approximate value of $R_{00}$ can be found from
expression of the Ricci tensor, After neglecting the nonlinear terms and the
terms that are time derivative, one finds
$\displaystyle R_{00}$
$\displaystyle=\frac{\partial\Gamma_{00}^{\alpha}}{\partial
x^{\alpha}}-\frac{\partial\Gamma_{0\alpha}^{\alpha}}{\partial
x^{0}}+\Gamma_{00}^{\alpha}\Gamma_{\beta\alpha}^{\beta}-\Gamma_{0\beta}^{\alpha}\Gamma_{0\alpha}^{\beta}$
$\displaystyle\simeq\frac{\partial\Gamma_{00}^{k}}{\partial
x^{k}}.\simeq\frac{1}{2}\nabla^{2}g_{00}.$ (22)
From $\left(21\right)$ and $\left(22\right)$ we have
$\nabla^{2}g_{00}=8\pi G\left(\rho_{s}-\rho_{v}\right),$ (23a)
$\nabla^{2}g_{00}=8\pi G\rho_{s},\ \text{\ \ when }\rho_{v}=0,$ (23b)
$\nabla^{2}\left(-g_{00}\right)=8\pi G\rho_{v},\text{ \ \ \ when }\rho_{s}=0.$
(23c)
Consequently, from $\left(20\right)$ and $\left(23\right)$ we have
$\overset{\cdot\cdot}{x}^{k}\simeq-\Gamma_{00}^{k}\simeq-\frac{1}{2}\frac{\partial
g_{00}}{\partial x^{k}},\text{ \ \ for }\rho_{s},$ (24a)
$\overset{\cdot\cdot}{x}^{k}\simeq\Gamma_{00}^{k}\simeq\frac{1}{2}\frac{\partial
g_{00}}{\partial x^{k}},\text{ \ \ for }\rho_{v}.$ (24b)
$\left(24a\right)$ is the same as $\left(3.2.9\right)$ in Ref.
$\left[13\right],$ and the equation $\left(24b\right)$ of motion of a
$s-particle$ is different from that of a $v-particle$ in the same
gravitational field.
Eq. $\left(24a\right)$ and $\left(24b\right)$ are consistent with observed
data. The reasons are as follows.
$A$. In the $s-breaking,$ in a $s-galaxy$ $\rho_{v}$ must be very small and
$\rho_{s}$ must be large. Hence the equivalent principle still holds for
$s-particles,$ and the gravitational field, the equation $\left(24a\right)$
and trajectories of motion of $s-particles$ are still the same as those of the
conventional theory in observed precision.
$B.$ In the $s-breaking,$ the equation $\left(24b\right)$ and trajectories of
motion of $v-particles$ must be different from those of the conventional
theory. But it is impossible to observe the $v-SU(5)$ color single states by a
$s-observor$ in practice, because $v-SU(5)$ color single states cannot form
dumpling and must loosely distribute in space, and there is only the repulsion
between $s-matter$ and $v-matter$. In fact, $\left(24b\right)$ has only
theoretical meanings.
$C$. In fact, only the cosmological effects of $v-matter$ are important and
are consistent with the observed data up to now.
## III Spontaneous breaking of symmetry and the gravitational mass density of
the vacuum state
### III.1 Spontaneous breaking of symmetry
Ignoring the couplings of $\Phi_{s}$ and $\chi_{s\text{ }}$and suitably
choosing the parameters of the Higgs potential, analogously to Ref.[11]. we
can prove from $(8)-(10)$ that there are the following vacuum expectation
values at the zero-temperature and the tree-level approximation,
$\left\langle
0\left|\Omega_{v}\right|0\right\rangle=\overline{\Omega}_{v0}=\left\langle
0\left|\Phi_{v}\right|0\right\rangle=\overline{\Phi}_{v0}=\left\langle
0\left|\chi_{v}\right|0\right\rangle=\overline{\chi}_{v0}=0,$ (25)
$\left\langle
0\left|\Omega_{s}\right|0\right\rangle=\overline{\Omega}_{s0}\equiv\upsilon_{\Omega
0},$ (26)
$\left\langle
0\left|\Phi_{s}\right|0\right\rangle=\overline{\Phi}_{s0}=Diagonal\left(1,1,1,-\frac{3}{2},-\frac{3}{2}\right)\upsilon_{\varphi
0},$ (27)
$\left\langle
0\left|\chi_{s}\right|0\right\rangle^{+}=\overline{\chi}_{0}=\frac{\upsilon_{\chi
0}}{\sqrt{2}}\left(0,0,0,0,1\right),$ (28)
$\upsilon_{\Omega 0}^{2}=\frac{\mu^{2}}{f},\;\ \ \ \ \ \
f\equiv\lambda-\frac{15w^{2}}{\left(15a+7b\right)}-\frac{\varsigma^{2}}{\xi}.$
(29a)
Ignoring the contributions of $\Phi_{s}$ and $\chi_{s}$ to
$\overline{\Omega}_{s0},$ at the zero-temperature we get
$\upsilon_{\Omega 0}^{2}=\frac{\mu^{2}}{\lambda},$ (29b)
$\upsilon_{\varphi 0}^{2}=\frac{2w}{15a+7b}\upsilon_{\Omega 0}^{2},$ (30)
$\upsilon_{\chi 0}^{2}=\frac{2\zeta}{\xi}\upsilon_{\Omega 0}^{2}.$ (31)
We take $\Lambda>\lambda>15w^{2}/\left(15a+7b\right)+\zeta^{2}/\xi.$ From
$\left(9\right)$-$\left(10\right)$ and $\left(25\right)$-$\left(29a\right)$ it
can be proved that all $v-Higgs$ bosons can get their masses big enough. The
masses of the Higgs particles exclusive of the $\Phi_{s}-particles$ and the
$\chi_{s}-particles\ $are respectively in the $S-breaking$
$m^{2}\left(\Omega_{s}\right)=2\mu^{2},$ (32)
$m^{2}\left(\Omega_{v}\right)=\Lambda\upsilon_{\Omega 0}^{2}-\mu^{2},$ (33)
$m^{2}\left(\Phi_{v}\right)=\frac{1}{2}\alpha\upsilon_{\Omega 0}^{2},$ (34)
$m^{2}\left(\chi_{v}\right)=\beta\upsilon_{\Omega 0}^{2}.$ (35)
We can choose such parameters so that
$m\left(\Omega_{s}\right)\simeq m\left(\Omega_{v}\right)\gg
m\left(\varphi_{v}\right)\sim m\left(\varphi_{s}\right)\gg
m\left(\chi_{v}\right)\sim m\left(\chi_{s}\right),$ (36)
e.g., $m\left(\Omega_{s}\right)\sim 10^{16}Gev,$
$m\left(\varphi_{s}\right)\sim 10^{14}Gev$ and $m\left(\chi_{s}\right)\sim
10^{2}Gev$. It is easily seen from $(32)-(35)$ that all real components of
$\Phi_{v}$ have the same mass $m\left(\Phi_{v}\right)$, all real components of
$\chi_{v}$ have the same mass $m\left(\chi_{v}\right)$ in the $S-breaking.$
The $S-breaking$ and the $V-breaking$ are symmetric because $s-matter$ and
$v-matter$ are symmetric. Hence when $s\rightleftarrows v$ and
$S\rightleftarrows V$ in $(25)-(35),$ the formulas are still kept.
### III.2 The characters of the vacuum state
The characters of the vacuum state are as follows. Taking $V_{0}=-V_{s\min}$
at the zero-temperature, considering $\left(13d\right)$ and $V_{sv}=V_{svg}=0$
in the $S-breaking$ and the symmetry between $s-matter$ and $v-matter,$ in the
vacuum state we have
$V_{Sg\min}\equiv
V_{s\min}\left(\varpi_{s}\right)+V_{0}-V_{v\min}\left(\varpi_{v}\right)=0,$
(37) $V_{S\min}=\left(V_{s}+V_{sv}\right)_{\min}+V_{0}=0.$
Applying the conventional quantum field theory to the present model, we have
$\rho_{0}=\rho_{s0}+\rho_{v0}\gg 0.$ Because of the conjecture 1,
$\rho_{s0}=\rho_{v0}$ and $\rho_{sg0}=-\rho_{vg0}.$ Consequently, we have
$\rho_{sg0}=-\rho_{vg0},\text{ \ \ }\rho_{g0}=\rho_{sg0}+\rho_{vg0}=0.$ (39a)
here $\rho_{0}$ and $\rho_{g0}$ are the mass density and the gravitational
mass density of the vacuum state, respectively.
According to the our quantum field theory[6],
$\rho_{s0}=\rho_{v0}=\rho_{0}=0,\text{ \ \ }\rho_{g0}=0.$ (39b)
It is seen that in any case, $\rho_{g0}=0$ is necessary provided the
conjecture 1 is valid.
## IV Evolving equations of space
### IV.1 Evolving equations when curvature factor $k$ is regarded as a
constant
Provided the cosmological principle is valid, the metric tensor is the
Robertson-Walker metric which can be written
$(ds)^{2}=-\left(dt\right)^{2}+R^{2}\left(t\right)\left\\{\frac{\left(dr\right)^{2}}{1-kr^{2}}+\left(rd\theta\right)^{2}+\left(r\sin\theta
d\varphi\right)^{2}\right\\},$ (40)
where $k$ is the curvature factor. $k$ is regarded an arbitrary real constant
in the Friedmann model. When $r\longrightarrow\alpha r,$ $R\longrightarrow
R/\alpha$ and $k\longrightarrow k/\alpha^{2}$, $\left(40\right)$ is unchanged.
Thus, without losing generality, $k$ may be taken as $1$, $0$ or $-1$. Here
$\alpha$ must be a positive number, hence $k$ cannot change from $1$ into $0$
or $-1$ by altering $a.$
Matter in the universe may approximately be regarded as ideal gas evenly
distributed in the whole space when temperature is not very high. The energy-
momentum tensor densities of the ideal gas are
$T_{sM\mu\nu}=\left(\rho_{s}+p_{s}\right)U_{s\mu}U_{s\nu}+p_{s}g_{\mu\nu},$
(41)
$T_{vM\mu\nu}=\left(\rho_{v}+p_{v}\right)U_{v\mu}U_{v\nu}+p_{v}g_{\mu\nu},$
(42)
where $U_{a\mu}$ is a 4-velocity, $a=s,v$. Considering the potential energy
densities in $(12)$, we can rewrite $\left(41\right)-\left(42\right)$ and
$\left(-g_{\mu\nu}V_{0}\right)$ as
$T_{s\mu\nu}=\left[\widetilde{\rho}_{s}+\widetilde{p}_{s}\right]U_{\mu}U_{\nu}+\widetilde{p}_{s}g_{\mu\nu},$
(43) $\widetilde{\rho}_{s}=\rho_{s}+V_{s}\left(\varpi_{s}\right),\;\ \ \
\widetilde{p}_{s}=p_{s}-V_{s}(\varpi_{s}),$ (44)
$T_{v\mu\nu}=\left[\widetilde{\rho}_{v}+\widetilde{p}_{v}\right]U_{\mu}U_{\nu}+\widetilde{p}_{v}g_{\mu\nu},$
(45) $\widetilde{\rho}_{v}=\rho_{v}+V_{v}\left(\varpi_{v}\right),\;\ \ \
\widetilde{p}_{v}=p_{v}-V_{v}(\varpi_{v}),$ (46)
$-g_{\mu\nu}V_{0}=\left(\widetilde{\rho}\left(V_{0}\right)+\widetilde{p}\left(V_{0}\right)\right)U_{\mu}U_{\nu}+g_{\mu\nu}\widetilde{p}\left(V_{0}\right),\text{\
\ }\widetilde{\rho}\left(V_{0}\right)=V_{0},\text{ \
}\widetilde{p}\left(V_{0}\right)=-V_{0.}$ (47)
In fact, $V_{v}=0$ due $\langle\omega_{v}\rangle=0$ in the $S-breaking$, hence
$\widetilde{\rho}_{v}=\rho_{v},\ \widetilde{p}_{v}=p_{v}.$
Substituting $(43)-(47)$ and the RW metric into $(13)$ and considering
$\left(37\right)$ and $U_{\mu}=(1,0,0,0)$ which implies that a medium moves
only as expanding of space, we get the Friedmann equations in the $S-breaking$
$\displaystyle\overset{\cdot}{R}^{2}+k$
$\displaystyle=\eta\left[\left(\rho_{s}+V_{s}+V_{0}\right)-\left(\rho_{v}+V_{v}\right)\right]R^{2}=\eta\left[\rho_{Sg}+V_{Sg}\right]R^{2}\equiv\eta\widetilde{\rho}_{Sg}R^{2},$
(48a) $\displaystyle\rho_{Sg}$ $\displaystyle\equiv\rho_{s}-\rho_{v},\text{\
}V_{Sg}\equiv V_{s}+V_{0}-V_{v},\text{ \
}\widetilde{\rho}_{Sg}=\rho_{Sg}+V_{Sg},\text{ \ }\eta\equiv 8\pi G/3,$ (48b)
$\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=-\frac{1}{2}\eta\left[\left(\rho_{s}+3p_{s}\right)-2\left(V_{s}+V_{0}-V_{v}\right)-\left(\rho_{v}+3p_{v}\right)\right]R.\equiv-\frac{1}{2}\eta\left(\widetilde{\rho}_{Sg}+3\widetilde{p}_{Sg}\right)R$
$\displaystyle=-\frac{1}{2}\eta\left[\rho_{Sg}+3p_{Sg}-2V_{Sg}\right],\text{ \
\ }p_{Sg}\equiv p_{s}-p_{v},\text{ \ \ }\widetilde{p}_{Sg}=p_{Sg}-V_{Sg}.$
(49)
Analogously, from $(14)$, $(44)$ and $(46)$, we get
$\overset{\cdot}{R}^{2}+k=\eta\left[\left(\rho_{v}+V_{v}+V_{0}\right)-\left(\rho_{s}+V_{s}\right)\right]R^{2}=\eta\widetilde{\rho}_{Vg}R^{2},$
(50)
$\overset{\cdot\cdot}{R}=-\frac{1}{2}\eta\left[\left(\rho_{v}+3p_{v}\right)-2\left(V_{v}+V_{0}-V_{s}\right)-\left(\rho_{s}+3p_{s}\right)\right]R=-\frac{1}{2}\eta\left(\widetilde{\rho}_{Vg}+3\widetilde{p}_{Vg}\right)R,$
(51)
in the $V-breaking.$
In the $S-breaking,$ only $(48)-(49)$ is applicable, and in the $V-breaking,$
only $(50)-(51)$ is applicable.
It is impossible when temperature is low that the $S-breaking$ transforms into
the $V-breaking$ because of the coupling $\left(10\right)$. But it is possible
that $\langle\omega_{s}\rangle=$ $\langle\omega_{v}\rangle=0$ when temperature
is high enough. Thus the transformation
$\langle\omega_{s}\rangle\neq 0\text{ and
}\langle\omega_{v}\rangle=0\longrightarrow\langle\omega_{s}\rangle=\langle\omega_{v}\rangle=0\longrightarrow\langle\omega_{s}\rangle=0\text{
and }\langle\omega_{v}\rangle\neq 0$ (52)
is possible, i.e., the $S-breaking$ can transform into the $V-breaking$ via
the highest temperature (see the following).
From $(48)-(49)$ we obtain
$d\left[\left(\rho_{s}-\rho_{v}\right)R^{3}\right]/dt+R^{3}\left(dV_{Sg}/dt\right)=-\left(p_{s}-p_{v}\right)dR^{3}/dt+R\left(dk/\eta
dt\right).$ (53)
Let $\rho_{m}$ and $\rho_{\gamma}$ denote the mass density of particles whose
static masses are not zero and the mass density of photon-like particles,
respectively, we have $\rho=\rho_{m}+\rho_{\gamma}$ and $p=p_{m}+p_{\gamma}.$
$p_{\gamma}=\rho_{\gamma}/3$ and $p_{m}$ may be ignored when temperature is
low. When $\rho_{m}$ is major and $\left(p_{s}-p_{v}\right),$ $dk/\eta dt$
(see the following section) and $\left(dV_{Sg}/dt\right)$ may be ignored, from
$\left(53\right)$ we have
$\left(\rho_{sm}-\rho_{vm}\right)R^{3}=C_{S},$ (54a)
in the $S-breaking$, here $C_{S}$ is a constant. In contrast to the
conventional theory, it is possible $C_{S}>0,$ $C_{S}=0$ or $C_{S}<0.$ When
$T_{s}\ll T_{cr}\equiv 2\mu/\sqrt{\lambda},$ both $m\left(\Omega_{s}\right)$
and $m\left(\Omega_{v}\right)$ are all very big, hence $\rho_{sm}$ cannot
transform into $\rho_{vm}$ by $\left(10\right).$ Consequently one has
$\rho_{sm}R^{3}=C_{s},\text{ \ \ }\rho_{vm}R^{3}=C_{v},\text{ \
}C_{S}=C_{s}-C_{v}.$ (54b)
Analogously, we have
$\displaystyle\left(\rho_{vm}-\rho_{sm}\right)R^{3}$ $\displaystyle=C_{V}$
(55a) $\displaystyle\rho_{vm}R^{3}$ $\displaystyle=C_{v},\text{ \ \
}\rho_{sm}R^{3}=C_{s},\text{ \ }C_{V}=C_{v}-C_{s},$ (55b)
in the $V-breaking$.
When photon-like gases are major and $dk/\eta dt$ and $dV_{Sg}/dt$ may be
ignored, after thermal equilibrium, $p_{a\gamma}\sim\rho_{a\gamma}/3,$ we have
$\displaystyle
d\left[\left(\rho_{s\gamma}-\rho_{v\gamma}\right)R^{4}\right]dt$
$\displaystyle=-\overset{\cdot}{V}_{Sg}R^{4}\sim 0,$
$\displaystyle\left(\rho_{s\gamma}-\rho_{v\gamma}\right)R^{4}$
$\displaystyle=D_{S},$ (56a)
in the $S-breaking.$ When the transformation $\rho_{s\gamma}$ into
$\rho_{v\gamma}$ may be ignored, we have
$\rho_{s\gamma}R^{4}=D_{s},\text{ \ \ \ \ }\rho_{v\gamma}R^{4}=D_{v},\text{ \
\ }D_{S}=D_{s}-D_{v}.$ (56b)
Similarly, one has
$\left(\rho_{Vv\gamma}-\rho_{Vs\gamma}\right)R^{4}=D_{V}=D_{s}-D_{v},$ (57a)
$\rho_{v\gamma}R_{V}^{4}=D_{v},\text{ \ \ }\rho_{s\gamma}R^{4}=D_{s},\text{ \
}D_{V}=D_{v}-D_{s}.$ (57b)
in the $V-breaking$.
### IV.2 The evolving equations when
$K=K(\underline{\rho}_{g}\left(t,R\left(t\right)\right))$
Provided the cosmological principle is valid, the metric tensor is the
Robertson-Walker metric. $k$ in $\left(40\right)$ is regarded an arbitrary
real constant in the Friedmann model. When $r\longrightarrow\alpha r,$
$R\longrightarrow R/\alpha$ and $k\longrightarrow k/\alpha^{2}$,
$\left(40\right)$ is unchanged. Thus, without losing generality, $k$ may be
taken as $1$, $0$ or $-1$. Here $\alpha$ must be a positive number, hence $k$
cannot change from $1$ into $0$ or $-1$ by altering $a.$ The R-W metric is
undoubtedly right when all gravitation masses are positive, i.e. $m_{g}=m.$ In
contrast with the conventional theory, according to the present model, all
$\rho_{g}>0,$ $=0$ and $<0$ are possible. Hence $k$ in $\left(40\right)$
should be changeable from $k=1$ to $-1$ corresponding to change of $\rho_{g}$
from $\rho_{g}>0$ to $\rho_{g}<0$. We consider $k$ to be a function of
$\underline{\rho}_{g},$ $\underline{\rho}_{g}$ is the gravitational mass
density in comoving coordinates.
$\underline{\rho}_{g}=\rho_{gm}+\rho_{g\gamma}$ can change as $R$ because
$\rho_{gm}=\rho_{vm}-\rho_{sm}\propto R^{-3}$ and
$\rho_{g\gamma}=\rho_{v\gamma}\propto R^{-4}.$ On the other hand, although
$\overset{\cdot}{R}=0,$ $\underline{\rho}_{g}$ can also change (see section
8.7). This is because the repulsive potential energy is chiefly transformed
into the kinetic energy of color single states. Hence we have
$k\longrightarrow
K\left(t\right)=K(\underline{\rho}_{g}\left(t,R\left(t\right)\right)).$
$\overset{\cdot}{R}^{2}+3K=8\pi
G\rho_{g}R^{2}+\frac{2}{3}\frac{R\overset{\cdot}{R}\overset{\cdot\cdot}{K}}{\overset{\cdot}{K}},$
(58) $\overset{\cdot\cdot}{R}=\left[-4\pi
G\left(\rho_{g}+p_{g}\right)+\frac{K}{R^{2}}\right]R-\frac{1}{3}\frac{\overset{\cdot}{R}\overset{\cdot\cdot}{K}}{\overset{\cdot}{K}}.$
(59)
$\overset{\cdot\cdot}{K}+\frac{3r^{2}\overset{\cdot}{K}^{2}}{2\left(1-Kr^{2}\right)}+3\frac{\overset{\cdot}{R}\overset{\cdot}{K}}{R}=0.$
(60)
We discuss the evolving equations as follows[9].
$1\mathbf{.}$ When $\overset{\cdot}{K}\sim 0,$ from $\left(60\right)$ we have
$\frac{\overset{\cdot\cdot}{K}}{\overset{\cdot}{K}}=-3\frac{\overset{\cdot}{R}}{R}.$
(61)
Substituting $\left(61\right)$ into $\left(58\right)-\left(59\right),$ we get
the Friedmann equations $\left(48\right)-\left(49\right)$ or
$\left(50\right)-\left(51\right)$ anew. Hence the equations are self-
consistent. Thus, when $\overset{\cdot}{K}\sim 0,$ we can still determine
$R\left(t\right)$ by $\left(48\right)-\left(49\right)$ or
$\left(50\right)-\left(51\right).$
$\overset{\cdot}{K}\sim 0$ is possible. Because
$K\left(\underline{\rho}_{g}\right)$ is a monotone and finite function of
$\underline{\rho}_{g}$ and $dK/d\underline{\rho}_{g}>0,$ it is necessary when
$\underline{\rho}_{g}\gg 0$ or $\underline{\rho}_{g}\ll 0,$ $K\left(t\right)$
slowly changes so that $\overset{\cdot}{K}\sim 0$. In fact, considering
$\underline{\rho}_{g}$ to be the gravitational mass density in the comoving
coordinates, we have
$\underline{\rho}_{g}\left(t\right)=\frac{\left(\rho_{gm}+\rho_{g\gamma}\right)R^{3}}{R_{1}^{3}}=\underline{\rho}_{gm1}+\underline{\rho}_{g\gamma
1}\frac{R_{1}}{R}.$ (62)
When $\mid\underline{\rho}_{g\gamma 1}/\underline{\rho}_{gm1}R_{1}/R\mid\ll
1,$
$\underline{\rho}_{g}\left(t\right)\sim\underline{\rho}_{gm}\left(t_{1}\right)$
so that $\overset{\cdot}{K}\sim 0.$ From the conjecture 3, we can also
determine only when $\underline{\rho}_{g}\left(t\right)\sim 0$ so that
$K\left(\underline{\rho}_{g}\right)\sim 0,$ $\overset{\cdot}{K}$ is important.
Because $K\left(\underline{\rho}_{g}\right)$ is a monotone and finite function
of $\underline{\rho}_{g},$ $K\left(\underline{\rho}_{g}=0\right)=0$ and
$dK/d\underline{\rho}_{g}>0,$ it is necessary
$K\left(\underline{\rho}_{g}\right)>0$ when $\underline{\rho}_{g}>0$ and
$K\left(\underline{\rho}_{g}\right)<0$ when $\underline{\rho}_{g}<0.$ When
$\underline{\rho}_{g}\gg 0$ or $\underline{\rho}_{g}\ll 0,$ we will regard
$K\left(\underline{\rho}_{g}\right)$ as a constant, e.g. $K=1$ for
$\underline{\rho}_{g}\gg 0$ and $K=-1$ for $\underline{\rho}_{g}\ll 0.$
$2.$ When $\overset{\cdot}{R}\longrightarrow 0$, from
$\left(58\right)-\left(59\right)$ we have
$\displaystyle\overset{\cdot}{R}$ $\displaystyle=0,\text{ \ \ \
}K=\eta\rho_{g}R^{2},$ (63a) $\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=-\frac{\eta}{2}\left(\rho_{g}+3p_{g}\right)R$
$\displaystyle\simeq-\frac{\eta}{2}\left(\rho_{g}+\rho_{g\gamma}\right)R=-\frac{\eta
R_{1}^{3}}{2R^{2}}\left(\rho_{gm1}+2\rho_{g\gamma 1}\frac{R_{1}}{R}\right).$
(63b)
This is similar to the conventional theory only when $\rho_{g}>0.$ According
to present model, $\rho_{g}$ and $\rho_{g}+\rho_{g\gamma}=0$ or $<0$ are
possible as well. Thus, the present model is different from the conventional
theory.
$3.$ When
$\rho_{g}=\rho_{gm}+\rho_{g\gamma}=\frac{R_{1}^{3}}{R^{3}}\left(\rho_{gm1}+\rho_{g\gamma
1}\frac{R_{1}}{R}\right)=\frac{R_{1}^{3}}{R^{3}}\underline{\rho}_{g}=0\text{ \
so that }K\left(\underline{\rho}_{g}\right)=0,$ (64)
$\left(58\right)-\left(59\right)$ becomes
$\overset{\cdot}{R}^{2}=\frac{2}{3}\frac{R\overset{\cdot}{R}\overset{\cdot\cdot}{K}}{\overset{\cdot}{K}},$
(65a)
$\overset{\cdot\cdot}{R}=-\frac{\eta}{2}\rho_{g\gamma}R-\frac{1}{3}\frac{\overset{\cdot}{R}\overset{\cdot\cdot}{K}}{\overset{\cdot}{K}}=-\frac{\eta}{2}\rho_{g\gamma}R-\frac{1}{2}\frac{\overset{\cdot}{R}^{2}}{R}<0.$
(65b)
This is because only when $\rho_{gm}<0$, it is possible that
$\rho_{gm}+\rho_{g\gamma}=0.$ Hence $\rho_{g\gamma}>0$. It is seen that
although $\rho_{g}=K=0,$ it is still possible $\overset{\cdot}{R}>0$ and
$\overset{\cdot\cdot}{R}<0.$ In the case space expands with a deceleration.
This is different from the conventional theory in which when $\rho_{g}=k=0,$
$\overset{\cdot}{R}=\overset{\cdot\cdot}{R}=0$ is necessary. It is seen that
when $\rho_{g}\sim 0$ and $K\sim 0,$ $\overset{\cdot}{K}$ must be considered.
$4$. Although $K$ is a function of $r$ when $K\sim 0,$ $\rho_{g}$ and $R$ are
still independent of $r^{[9]}.$
$5$. In the $S-breaking$, $\rho_{s}$ can transform into $\rho_{v}$ because of
$\partial K/\partial t<0$ when $\rho_{s}\sim\rho_{v}.$ After reheating, in
fact, $\partial V_{g}/\partial t\sim 0.$ It is necessary that
$\rho_{s}=\rho_{sm}+\rho_{s\gamma}\sim\rho_{v}$ at some a time, because
$\rho_{sm}\varpropto R^{-3},$ $\rho_{s\gamma}\varpropto R^{-4}$, and
$\rho_{sm}<\rho_{v}=\rho_{vm}\varpropto R^{-3}.$ When $\rho_{s}\sim\rho_{v},$
$K\sim 0,$ $\partial K/\partial t<0$ is marked and the universe is matter-
dominated so that $p_{s}-p_{v}$ may be ignored. Consequently from
$\left(53\right)$ we see
$d\left[\left(\rho_{s}-\rho_{v}\right)R^{3}\right]/dt=R\left(dK/\eta
dt\right)<0.$ (65c)
This implies $\rho_{s}$ can transform into $\rho_{v}$ because of $\partial
K/\partial t<0$ when $\rho_{s}\sim\rho_{v}.$ In fact, in this stage,
$s-galaxies$ can be fast formed and $\left(65c\right)$ holds.
## V Temperature effect
### V.1 Two sorts of temperature
The thermal equilibrium between the $v-particles$ and the $s-particles$ can be
realized by only $\left(10\right).$ The Higgs bosons are hardly produced
because the masses of the Higgs particles are all very big at low temperature.
Consequently, the interaction between the $v-particles$ and the $s-particles$
may be ignored so that there is no thermal equilibrium between the
$v-particles$ and the $s-particles.$ Thus, when temperature is low, we should
use two sorts of temperature $T_{v}$ and $T_{s}$ respectively to describe the
thermal equilibrium of $v-matter$ and the thermal equilibrium of $s-matter.$
Generally speaking, $T_{v}\neq T_{s}$. When temperature $T_{s}$ is so high
that $\langle\Omega_{s}\rangle\longrightarrow 0$ in $S-breaking,$ the masses
of $s-Higgs$ bosons will tend to zero (see below). On the other hand, because
$\langle\Omega_{v}\rangle=0$ in the $S-breaking,$ $m\left(\Omega_{v}\right),$
$m\left(\Phi_{v}\right)$ and $m\left(\chi_{v}\right)$ will tend to zero as
$\langle\Omega_{s}\rangle$ tends to zero as well. Consequently $\Omega_{s},$
$\Phi_{s}$ and $\chi_{s}$ can be enormously produced and easily transformed
into $\Omega_{v},$ $\Phi_{v}$ and $\chi_{v}$ by the couplings in $(10).$ Other
$v-particles$ can be easily produced by the couplings of $v-SU(5)$ as well.
Consequently thermal equilibrium between the $v-particles$ and the
$s-particles$ will appear provided $T_{s}$ is high enough.. In the case,
contraction of space will stop and inflation must occur. Thus there must be
the highest temperature $T_{\max}$.
### V.2 The influences of finite temperature on the Higgs potential
#### V.2.1 Effective potentials
The influence of finite temperature on the Higgs potential in the present
model are consistent with the conventional theory. For short, we consider only
$\Omega_{a}$ and $\varphi_{a},$ $a=s,v$. When $\chi_{a}$ is considered as
well, the following inferences are still qualitatively valid. For
$V(\Omega_{s})=-\frac{\mu^{2}}{2}\Omega_{s}^{2}+\frac{\lambda}{4}\Omega_{s}^{4},$
to ignore the terms proportional $\lambda^{n},$ $n>1$, the finite-temperature
effective potential approximate to 1-loop in flat space is[14,15]
$V_{eff}^{(1)T}\left(\overline{\Omega}_{s},T_{s}\right)=-\frac{1}{2}\left(\mu^{2}-\frac{\lambda}{4}T_{s}^{2}\right)\overline{\Omega}_{s}^{2}+\frac{\lambda}{4}\overline{\Omega}_{s}^{4}-\frac{\pi^{2}}{90}T_{s}^{4}+\frac{\mu^{2}}{24}T_{s}^{2}.$
(66a)
Considering the influence of the expectation values $v_{\Omega
v}\left(T_{s},T_{v}\right),$ $v_{\varphi s}\left(T_{s},T_{v}\right)$ and
$v_{\varphi v}\left(T_{s},T_{v}\right)$ and ignoring the terms irrelevant to
$\Omega_{s},$ we have
$V_{eff}^{(1)T}\left(\overline{\Omega}_{s},T_{s},T_{v}\right)=-\frac{1}{2}\mu_{s}^{2}\left(T_{s}\right)\overline{\Omega}_{s}^{2}+\frac{\lambda}{4}\overline{\Omega}_{s}^{4},$
(66b)
$\mu_{s}^{2}\left(T_{s},T_{v}\right)\equiv\mu^{2}-\frac{\lambda}{4}T_{s}^{2}-\Lambda
v_{\Omega v}^{2}\left(T_{s},T_{v}\right)-\frac{15}{2}\left(\alpha v_{\varphi
v}^{2}\left(T_{s},T_{v}\right)-wv_{\varphi
s}^{2}\left(T_{s},T_{v}\right)\right).$ (66c)
In the $S-breaking$, $v_{\Omega s}\left(T_{s}\right)\neq 0,$ $v_{\varphi
s}\left(T_{s},T_{v}\right)\neq 0,$ and $v_{\Omega
v}\left(T_{s},T_{v}\right)=v_{\varphi v}\left(T_{s},T_{v}\right)=0.$ From
$\left(66b,c\right)$ we find
$\displaystyle v_{\Omega s}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\mu_{s}^{2}\left(T_{s},T_{v}\right)/\lambda,\text{ \ when
}\mu_{s}^{2}\left(T_{s},T_{v}\right)>0,$ $\displaystyle v_{\Omega
s}\left(T_{s},T_{v}\right)$ $\displaystyle=0,\text{ \ when
}\mu_{s}^{2}\left(T_{s},T_{v}\right)\leq 0.$ (66d)
Similarly, for $\Omega_{v}$ we have
$V_{eff}^{(1)T}\left(\overline{\Omega}_{v},T_{s},T_{v}\right)=-\frac{1}{2}\mu_{v}^{2}\left(T_{s},T_{v}\right)\overline{\Omega}_{v}^{2}+\frac{\lambda}{4}\overline{\Omega}_{v}^{4},$
(67a) $\mu_{v}^{2}\left(T_{s},T_{v}\right)\equiv\mu^{2}-\Lambda v_{\Omega
s}^{2}\left(T_{s},T_{v}\right)-\frac{\lambda}{4}T_{v}^{2}-\frac{15}{2}\left(\alpha
v_{\varphi s}^{2}\left(T_{s},T_{v}\right)-wv_{\varphi
v}^{2}\left(T_{s},T_{v}\right)\right).$ (67b) $\displaystyle v_{\Omega
v}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\mu_{v}^{2}\left(T_{s},T_{v}\right)/\lambda,\text{ \ when
}\mu_{v}^{2}\left(T_{s},T_{v}\right)>0,$ $\displaystyle v_{\Omega
v}\left(T_{s},T_{v}\right)$ $\displaystyle=0,\text{ \ when
}\mu_{v}^{2}\left(T_{s},T_{v}\right)\leq 0.$ (67c)
In the $S-breaking$, because of $\left(10\right),$ it will be proved (see
section 5.3)
$\mu_{v}^{2}\left(T_{v}\right)\leq 0,\text{ and\ }v_{\Omega
v}\left(T_{s},T_{v}\right)=0.$ (67d)
For
$V(\Phi_{s})=\left(\frac{1}{2}\alpha\Omega_{v}^{2}-w\Omega_{s}^{2}\right)Tr\Phi_{s}^{2}+\frac{1}{4}a(Tr\Phi_{s}^{2})^{2}+\frac{1}{2}bTr\Phi_{s}^{4},$
(68)
ignoring the contributions of the Higgs fields and the fermion fields to one
loop correction and only considering the contribution of the gauge fields to
one-loop correction, when $\overline{\varphi}_{s}\ll kT$, $k$ is the Boltzmann
constant, the finite-temperature effective potential approximate to 1-loop in
flat space is[14]
$V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s}\right)=V(\overline{\varphi}_{s})+B\overline{\varphi}_{s}^{4}\left(\ln\frac{\overline{\varphi}_{s}^{2}}{\sigma^{2}}-\frac{25}{6}\right)+CT_{s}^{2}\overline{\varphi}_{s}^{2}-\frac{\pi^{2}}{15}\left(kT_{s}\right)^{4},$
(69)
where $B=\left(5625/1024\pi^{2}\right)g^{4},$
$\Phi_{s}=Diagonal\left(1,1,1,-\frac{3}{2},-\frac{3}{2}\right)\overline{\varphi}_{s}.$
In general, $w$ and $\alpha<\lambda\sim
g^{4}<C=\left(75/16\right)\left(kg\right)^{2}.$ We take $w\simeq\alpha$ for
simplicity. $\sigma$ is a parameter at which the renormalized coupling-
constant $\lambda$ is defined,
$\frac{d^{4}V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)}{d\overline{\varphi}_{s}}\mid_{\overline{\varphi}_{s}=\sigma}=\lambda.$
Only considering the contribution of the expectation values of $\Omega_{s}$
and $\Omega_{v}$ to $V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s}\right),$
taking $\left(15/16\right)\left(15a+7b\right)=\left(11/3\right)B$, and
ignoring the term $\left(kT_{s}\right)^{4}$ unconnected with
$\overline{\varphi}_{v},$ from $\left(8\right),$
$\left(68\right)-\left(69\right)$ we have
$\displaystyle V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)$
$\displaystyle=A_{s}^{2}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)\overline{\varphi}_{s}^{2}+B\overline{\varphi}_{s}^{4}\left(\ln\frac{\overline{\varphi}_{s}^{2}}{\sigma^{2}}-\frac{1}{2}\right),$
(70a) $\displaystyle A_{s}^{2}\left(\varphi_{s},T_{s},T_{v}\right)$
$\displaystyle\equiv\frac{15}{4}\alpha\left(v_{\Omega
v}^{2}\left(T_{s}\right)-v_{\Omega s}^{2}\left(T_{s}\right)\right)+CT_{s}^{2}$
(70b)
Similarly, we have
$\displaystyle V_{eff}^{(1)T}\left(\overline{\varphi}_{v},T_{v}\right)$
$\displaystyle=A_{v}^{2}\left(\overline{\varphi}_{v},T_{s},T_{v}\right)\overline{\varphi}_{v}^{2}+B\overline{\varphi}_{v}^{4}\left(\ln\frac{\overline{\varphi}_{v}^{2}}{\sigma^{2}}-\frac{1}{2}\right),$
(71a) $\displaystyle A_{v}^{2}\left(\varphi_{v},T_{s},T_{v}\right)$
$\displaystyle=\frac{15}{4}\alpha\left(v_{\Omega
s}^{2}\left(T_{s}\right)-v_{\Omega
v}^{2}\left(T_{s}\right)\right)+CT_{v}^{2}.$ (71b)
#### V.2.2 Critical temperatures and masses of the Higgs particles
The critical temperature $T_{s,\varphi cr}$ is such a temperature at which the
minima are degenerate, i.e.,
$V_{eff\min}^{(1)T}\left(\overline{\varphi}_{s},T_{s,\varphi
cr}\right)=V_{eff}^{(1)T}\left(v_{\varphi scr},T_{s,\varphi
cr}\right)=V_{eff}^{(1)T}\left(0,T_{s,\varphi cr}\right)=0$. In other words,
$\langle\varphi_{s}\rangle=v_{\varphi s}\neq 0$ when $T_{s}<T_{s,\varphi cr},\
$and $v_{\varphi s}=0$ when $T_{s}\geq T_{s,\varphi cr}.$ $T_{s,\varphi cr}$
and $A_{v}^{2}\left(\varphi_{v},T_{s},T_{v}\right)$ can be determined from
$\left(70\right)$ by that when $\overline{\varphi}_{s}=v_{\varphi scr}$
$V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s,\varphi cr}\right)=0,\text{ \
}\frac{\partial}{\partial\overline{\varphi}_{s}}V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s,\varphi
cr}\right)=0.$ (72) $\displaystyle v_{\varphi scr}^{2}$
$\displaystyle=\sigma^{2}e^{-1/2},\text{ \
}A_{scr}^{2}\left(\varphi_{s},T_{s},T_{v}\right)=B\sigma^{2}e^{-1/2},$ (73a)
$\displaystyle T_{s,\varphi cr}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\left[B\sigma^{2}e^{-1/2}+\left(15/4\right)\left(wv_{\Omega
s}^{2}\left(T_{s},T_{v}\right)-\alpha v_{\Omega
v}^{2}\left(T_{s},T_{v}\right)\right)\right]/C.$ (73b)
In the $S-breaking$, $\varpi_{v}=0=0.$ Considering $v_{\varphi s}=0$ when
$T_{s}=T_{s,\varphi cr}$ and $C>\lambda,$ and taking
$B\sigma^{2}e^{-1/2}<\mu^{2},$ we find
$T_{s,\varphi
cr}^{2}\left(T_{s},T_{v}\right)=\left[B\sigma^{2}e^{-1/2}+\frac{15}{4}\frac{\alpha}{\lambda}\mu^{2}\right]/\left(C+\frac{15}{4}\alpha\right)<\frac{\mu^{2}}{\lambda}.$
(73c)
When $T_{s}\geq T_{s,\varphi cr},$
$\displaystyle v_{\varphi s}$ $\displaystyle=0,\text{
}V_{eff\min}^{(1)T}\left(\overline{\varphi}_{s},T_{s}\right)=V_{eff}^{(1)T}\left(0,T_{s}\right)=0,$
$\displaystyle m^{2}\left(\varphi_{s},T_{s}\right)$
$\displaystyle=2\left[-\frac{15}{4}\alpha v_{\Omega
s}^{2}\left(T_{s}\right)+CT_{s}^{2}\right],$ (74a)
where $m\left(\varphi_{s},T_{s}\right)$ is called the effective mass of
$\varphi_{s}$ which implies that the temperature effect is considered. When
$T_{s}<T_{s,\varphi cr}$
$\displaystyle v_{\varphi s}\left(T_{s},T_{v}\right)$ $\displaystyle\neq
0,\text{ \
}V_{eff\min}^{(1)T}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)=V_{eff}^{(1)T}\left(v_{\varphi
s},T_{s},T_{v}\right)<0,$ $\displaystyle m^{2}\left(\varphi_{s},T_{s}\right)$
$\displaystyle=15wv_{\Omega
s}^{2}\left(T_{s},T_{v}\right)-4CT_{s}^{2}+8Bv_{\varphi
s}^{2}\left(T_{s},T_{v}\right).$ (74b)
Similarly, from $\left(71\right)$ we have
$\displaystyle v_{\varphi v}^{2}$ $\displaystyle=\sigma^{2}e^{-1/2},\text{ \
}A_{vcr}^{2}\left(\overline{\varphi}_{v},T_{s},T_{v}\right)=B\sigma^{2}e^{-1/2},$
(75a) $\displaystyle T_{v,\varphi cr}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\left[B\sigma^{2}e^{-1/2}+\left(15/4\right)\alpha\left(v_{\Omega
v}^{2}\left(T_{s},T_{v}\right)-v_{\Omega
s}^{2}\left(T_{s},T_{v}\right)\right)\right]/C.$ (75b)
In the $S-breaking$, $v_{\Omega v}\left(T_{s},T_{v}\right)=v_{\varphi
v}\left(T_{s},T_{v}\right)=0.$ Considering $C>\lambda,$ and taking
$B\sigma^{2}e^{-1/2}<\mu^{2},$ we find
$\displaystyle T_{v,\varphi cr}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\left[B\sigma^{2}e^{-1/2}-\frac{15}{4}\frac{\alpha}{\lambda}\left(\mu^{2}-\frac{\lambda}{4}T_{s}^{2}+\frac{15}{2}\alpha
v_{\varphi s}^{2}\left(T_{s},T_{v}\right)\right)\right]/C,$ (75c)
$\displaystyle T_{v,\varphi cr\max}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=B\sigma^{2}e^{-1/2}/C<B\sigma^{2}e^{-1/2}/\lambda<\mu^{2}/\lambda.$
(75d)
$T_{v,\varphi cr}^{2}\leq 0$ implies $v_{\varphi
v}\left(T_{s},T_{v}\right)=0.$ $T_{v,\varphi cr}^{2}$ will increase from $0$
to $B\sigma^{2}e^{-1/2}$ as $v_{\Omega s}\left(T_{s}\right)$ decreases from
$v_{\Omega s}\left(0\right)$ to $0$ due to $\left(75b\right).$ In the case, it
seems $T_{v}^{2}<B\sigma^{2}e^{-1/2}$ so that $v_{\varphi
v}\left(T_{s},T_{v}\right)\neq 0$ to be possible. In fact, this is impossible
for the space-contraction process. In the case, it is necessary
$T_{s}^{2}\sim\mu^{2}/\lambda$ due to $v_{\Omega s}\left(T_{s}\right)\sim 0.$
Hence $\Omega_{s}$ and $\Omega_{v}$ or $\varphi_{v},$ and $\varphi_{s}$ and
$\Omega_{v}$ can transform from one into another by $\left(10\right)$ and
$T_{v}/T_{v1}=R_{1}/R=T_{s}/T_{s1}$. Consequently $\rho_{s}\sim\rho_{v}$ and
$T_{v}\longrightarrow T_{s}>T_{\varphi cr}$ so that $v_{\varphi
v}\left(T_{s},T_{v}\right)=0$ (see below). Hence in the $S-breaking$,
$v_{\varphi v}\left(T_{s},T_{v}\right)=0.$
When $T_{v}\geq T_{v,\varphi cr},$
$\displaystyle v_{\varphi v}\left(T_{s},T_{v}\right)$ $\displaystyle=0,\text{
}V_{eff\min}^{(1)T}\left(\overline{\varphi}_{v},T_{s},T_{v}\right)=V_{eff}^{(1)T}\left(0,T_{s},T_{v}\right)=0,$
(76a) $\displaystyle m^{2}\left(\varphi_{v},T_{v}\right)$
$\displaystyle=2\left[\frac{15}{4}\alpha v_{\Omega
s}^{2}\left(T_{s}\right)+CT_{v}^{2}\right].$ (76b)
In fact, only when $v_{\Omega s}\left(T_{s},T_{v}\right)\sim 0,$ $v_{\varphi
v}\left(T_{s},T_{v}\right)\neq 0$ due to $\left(10\right).$ Hence when
$T_{v}<T_{v,\varphi cr},$
$\displaystyle m^{2}\left(\varphi_{v},T_{v}\right)$ $\displaystyle=-15\alpha
v_{\Omega s}^{2}\left(T_{s}\right)-4CT_{v}^{2}+8B\sigma_{v}^{2}$
$\displaystyle=-4CT_{v}^{2}+8B\sigma_{v}^{2}\text{.}$ (77)
In fact, in general, the case cannot emerge, because when $v_{\Omega
s}\left(T_{s},T_{v}\right)\sim 0,$ $T_{s}$ is very large. Hence $\rho_{s}$ and
$\rho_{v}$ can transform from one into another so that $T_{v}\sim T_{s}.$ Thus
$T_{v}<T_{v,\varphi cr}$ cannot emerge.
If $v_{\Omega s}\left(T_{s}\right)=0=v_{\Omega v}\left(T_{v}\right),$
$T_{s,\varphi cr}^{2}=T_{v,\varphi cr}^{2}\equiv T_{\varphi
cr}^{2}=B\sigma^{2}e^{-1/2}/C.$ (78a)
It is obvious
$T_{s,\varphi cr}>T_{\varphi cr}>T_{v,\varphi cr},\text{ \ \ when \ }v_{\Omega
s}\left(T_{s}\right)\neq 0.$ (78b)
In the $S-breaking$, when $T_{s}\geq\mu/\sqrt{\lambda}>T_{s,\varphi cr},$
$v_{\varphi v}\left(T_{s},T_{v}\right)=v_{\varphi
s}\left(T_{s},T_{v}\right)=v_{\Omega v}=0.$ Thus the critical temperature
$T_{\Omega cr}\equiv T_{cr}$ is determined by $\left(66c\right)$
$T_{cr}=\frac{2\mu}{\sqrt{\lambda}}>T_{s,\varphi cr},.$ (79)
Thus, when $T_{s}\geq T_{cr},$
$\displaystyle v_{\Omega s}\left(T_{s}\right)$ $\displaystyle=0,\text{
}V_{eff\min}^{(1)T}\left(\overline{\Omega}_{s},T_{s}\right)=V_{eff}^{(1)T}\left(0,T_{s}\right)=0,$
(80a) $\displaystyle m^{2}\left(\overline{\Omega}_{s},T_{s}\right)$
$\displaystyle=\frac{1}{2}\lambda T_{s}^{2}-2\mu^{2},$ (80b)
when $T_{s}<T_{cr}$
$\displaystyle v_{\Omega s}^{2}\left(T_{s},T_{v}\right)$
$\displaystyle=\mu_{s}^{2}\left(T_{s},T_{v}\right)/\lambda>0,\text{ \
}V_{eff\min}^{(1)}\left(\overline{\Omega}_{s},T_{s}\right)=-\frac{\mu_{s}^{4}\left(T_{s},T_{v}\right)}{4\lambda}$
(80c) $\displaystyle m^{2}\left(\Omega_{s}\right)$
$\displaystyle=2\mu_{s}^{2}\left(T_{s},T_{v}\right)\text{.}$ (80d)
When $\mu_{v}^{2}\left(T_{s},T_{v}\right)\leq 0,$ $v_{\Omega
v}\left(T_{s},T_{v}\right)=0,$ and when
$\mu_{v}^{2}\left(T_{s},T_{v}\right)>0,$ $v_{\Omega
v}\left(T_{s},T_{v}\right)\neq 0.$ Considering $v_{\Omega
s}\left(T_{s}\right)=0$ when $T_{s}=T_{cr},$ from $\left(67a,b\right)$ we find
$T_{vcr}=T_{cr}.$ (81a)
When $T_{v}\gtrsim T_{cr}$,
$\displaystyle v_{\Omega v}\left(T_{s}\right)$ $\displaystyle=0,\text{
}V_{eff\min}^{(1)T}\left(\overline{\Omega}_{v},T_{v}\right)=V_{eff}^{(1)T}\left(0,T_{v}\right)=0,$
$\displaystyle m^{2}\left(\overline{\Omega}_{v},T_{s}\right)$
$\displaystyle=\frac{1}{2}\lambda T_{v}^{2}-2\mu^{2}.$ (81b)
In the $S-breaking$, when $v_{\Omega s}\left(T_{s},T_{v}\right)\longrightarrow
0,$ it is necessary $T_{s}^{2}\longrightarrow 4\mu^{2}/\lambda.$ In the case,
$\rho_{s}$ and $\rho_{v}$ can transform from one into another and
$T_{v}/T_{v1}=R_{1}/R=T_{s}/T_{s1},$
hence $T_{v}^{2}\longrightarrow T_{s}^{2}\sim 4\mu^{2}/\lambda.$ Consequently
there still is $\mu_{v}^{2}\left(T_{s},T_{v}\right)\leq 0.$ Hence in the
$S-breaking$, when $T_{v}<T_{cr},$ there still are
$\displaystyle v_{\Omega v}\left(T_{s},T_{v}\right)$ $\displaystyle=0,\text{
}V_{eff\min}^{(1)T}\left(\overline{\Omega}_{v},T_{s},T_{v}\right)=V_{eff}^{(1)T}\left(0,T_{s},T_{v}\right)=0,$
$\displaystyle m^{2}\left(\overline{\Omega}_{v},T_{s},T_{v}\right)$
$\displaystyle=-2\mu_{v}^{2}\left(T_{s},T_{v}\right).$ (81c)
### V.3 $\rho_{s}$ and $\rho_{v}$ transform from one into another when
$T_{s}\longrightarrow T_{cr}$ so that $\rho_{s}-\rho_{v}\longrightarrow 0$
In the $s-breakng,$ we consider the space contraction process. In low
temperatures, $V_{Sg}\sim 0$, $\rho_{sm}\gg\rho_{s\gamma},$ and the masses of
all $v-colour$ single states are not zero, i.e., $\rho_{v\gamma}=0.$ Thus
$\rho_{sm}$ and $\rho_{vm}$ are major.
$\underline{\widetilde{\rho}}_{Sg}=\widetilde{\underline{\rho}}_{s}-\widetilde{\underline{\rho}}_{v}=\rho_{s}-\rho_{vm}>0$
so that space can contract. According to the conjecture 3 and the discussion
about $\left(61\right)$, $K>0$ and $\overset{\cdot}{K}\sim 0$ when$\
\underline{\widetilde{\rho}}_{Sg}=\underline{\widetilde{\rho}}_{s}-\widetilde{\underline{\rho}}_{v}\gg
0$ so that $\left(48\right)$-$\left(49\right)$ are still applicable.
When $T_{s}\ll T_{cr},$ the transformation $\rho_{s}$ into $\rho_{v}$ may be
ignored. From $\left(54\right)$ one can rewrite
$\left(48\right)$-$\left(49\right)$ as
$\displaystyle\overset{\cdot}{R}^{2}$ $\displaystyle=-K+\eta\frac{C_{S}}{R},$
$\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=-\frac{\eta}{2}\frac{C_{S}}{R^{2}},$
where $C_{S}=C_{s}-C_{v}>0,$
$C_{a}=\rho_{a}\left(T_{a}\right)R^{3}\left(T_{a}\right),$ $a=s,v$ and
$T_{a}\ll T_{cr}.$ It is seen that space will monotonously contract faster and
faster.
$T_{s}$ must go up high as $R$ decreases, because the non-zero momentum of a
free particle $p\propto 1/R(t),$ $\bigtriangleup p\bigtriangleup x\gtrsim 1$
and $\rho_{m}\propto 1/R^{3}(t).$
When $T_{s}$ and $T_{v}$ are high enough so that masses of particles may be
ignored, if the transformation $\rho_{s}$ into $\rho_{v}$ may still be
ignored, from $\left(56b\right)$ and $\left(57b\right)$ we have
$\rho_{a}\left(T_{a}\right)R^{4}\left(T_{a}\right)=D_{a},\text{ \
}\rho_{a}\left(T_{a}\right)=\frac{\pi^{2}}{30}g_{a}^{\ast}T_{a}^{4},\text{ \
}T_{a}R\left(T_{a}\right)=\left(30D_{a}/\pi^{2}g_{a}^{\ast}\right)^{1/4},\text{
\ }a=s,v,$ (82)
where $g_{a}^{\ast}=g_{aB}+7g_{aF}/8,$ $g_{aB}$ $\left(g_{aF}\right)$ is the
total number of the spin states of $a-bosons$ $\left(a-fermions\right).$
Considering $s-matter$ and $v-matter$ are symmetric, we have
$g_{s}^{\ast}=g_{sB}+7g_{sF}/8=g_{v}^{\ast}=g_{vB}+7g_{vF}/8\equiv g^{\ast}.$
(83)
It is seen that space contraction will cause $T_{s}$ and $T_{v}$ to rise. If
$T_{s}$ and $T_{v}$ are high enough and the transformation $\rho_{s}$ into
$\rho_{v}$ cannot be ignored, there will be $\rho_{s}\sim\rho_{v}$ and
$T_{s}\sim T_{v}.$ This is a expected result.
If there is only one sort of matter as the conventional theory or $\rho_{s}$
and $\rho_{v}$ cannot transform from one into other, space will continue to
contract and $T_{s}$ and $T_{v}$ will continue to rise provided
$\widetilde{\rho}_{Sg}R^{2}-K>0.$ In fact, in this case, space will contract
to a singular point and $T_{s}$ and $T_{v}$ tend to infinite.
In contrast with the conventional theory, when $T_{s}$ and $T_{v}$ are high
enough, $\rho_{s}$ and $\rho_{v}$ can transform from one into another by
$\left(10\right)$ i.e.,
$\Omega_{s}+\Omega_{s}\rightleftarrows\Omega_{v}+\Omega_{v},\text{ \
}\Omega_{s}+\Omega_{s}\rightleftarrows\varphi_{v}+\varphi_{v},\text{ \
}\varphi_{s}+\varphi_{s}\rightleftarrows\Omega_{v}+\Omega_{v},$
so that
$\displaystyle T_{s}$ $\displaystyle=T_{v}=T,\text{ \ \
}\rho_{s}\left(T_{s}\right)=\rho_{v}\left(T_{v}\right)\equiv\rho\left(T\right)=\frac{\pi^{2}}{30}g^{\ast}T^{4},$
$\displaystyle\widetilde{\rho}_{Sg}$
$\displaystyle=\rho_{s}\left(T_{s}\right)-\rho_{v}\left(T_{v}\right)+V_{s}+V_{0}-V_{v}=V_{0}.$
(84)
The expectation values $v_{s\Omega}$, $v_{v\Omega}$, $v_{\varphi s}$ and
$v_{\varphi v}$ will tend to zero when $T_{s}\sim T_{v}\gtrsim T_{cr}$ because
of $\left(79\right),$ $\left(73c\right),$ $\left(75\right)$ and
$\left(78\right).$ Consequently, the masses of all particles which originating
from the couplings with $v_{s\Omega}$, $v_{v\Omega}$, $v_{\varphi s}$ and
$v_{\varphi v}$ are zero. Thus, the $a-Higgs$ particles and the $a-fermions$
or the $a-gauge$ particles can transform from one into another by the
$a-SU(5)$ couplings so that the $a-Higgs$ particles can enormously emerge
$(a=s,v)$. The $s-Higgs$ particles and the $v-Higgs$ particles can transform
from one into another by the coupling $\left(10\right)$ so that the thermal
equilibrium between $s-matter$ and $v-matter$ comes into being. After thermal
equilibrium, the number and the energy of every sort of particles will satisfy
statistical distribution determined by their spins. Thus we prove that
$\left(84\right)$ is a necessary result of space contraction because of
$\left(83\right).$ It is seen that when $T\geq T_{cr},$ i.e.,
$v_{s\Omega}=v_{v\Omega}=v_{\varphi s}=v_{\varphi v}=0,$ $s-matter$ and
$v-matter$ are completely symmetric, and both $s-SU(5)$ and $v-SU(5)$ strictly
hold water.
It is seen the end of space contraction is in the most symmetric state in
which $\langle\omega_{v}\rangle=\langle\omega_{s}\rangle=0.$.
$\langle\omega_{v}\rangle=\langle\omega_{s}\rangle=0$ is the sufficient but is
not the necessary condition for $\left(84\right)$. In fact, provided the
following conditions can be realized, $\left(84\right)$ can come into being
when $T_{s}$ and $T_{v}<T_{cr}$ as well.
$A$. $m\left(\omega_{a},T_{v},T_{s}\right)\sim
m\left(f_{a},T_{v},T_{s}\right)$ or $m\left(g_{a},T_{v},T_{s}\right),$ or
temperature is high enough so that $\omega_{a}$ can enormously emerge. Here
$f_{a}$ and $g_{a}$ denote fermions and gauge bosons, respectively.
$m\left(f_{a},T_{v},T_{s}\right)$ and $m\left(g_{a},T_{v},T_{s}\right)$ are
small. Thus $\Omega_{a}$ or $\varphi_{a}$ and $f_{a}$ or $g_{a}$ can transform
from one into other by the $SU(5)$ couplings.
$B$ $m\left(\omega_{s},T_{v},T_{s}\right)\sim
m\left(\omega_{v},T_{v},T_{s}\right).$ Thus, $\omega_{s}$ and $\omega_{v}$ can
transform from one into other by $\left(10\right)$
The two conditions can be realized as well when $T_{s}\sim T_{v}\lesssim
T_{cr}.$
From $\left(80d\right)$ and $\left(81c\right)$, in the $S-breaking$ we can
rewritten $m^{2}\left(\Omega_{v},T_{v},T_{s}\right)$ as
$\displaystyle m^{2}\left(\Omega_{v},T_{v},T_{s}\right)$
$\displaystyle=-2\mu_{v}^{2}\left(T_{v},T_{s}\right)$
$\displaystyle=\left(\frac{\Lambda}{\lambda}-1\right)m^{2}\left(\Omega_{s},T_{s},T_{v}\right)-\frac{\lambda}{2}\left(T_{s}^{2}-T_{v}^{2}\right)+30\alpha\left(v_{\varphi
s}^{2}-v_{\varphi v}^{2}\right).$ (85a)
When $T_{s\varphi cr}<T<T_{cr},$ $v_{\varphi s}=v_{\varphi v}=0,$
$m^{2}\left(\Omega_{s},T_{v},T_{s}\right)\longrightarrow 0$ as
$T_{s}\longrightarrow T_{cr},$ and
$m^{2}\left(\Omega_{v},T_{v},T_{s}\right)=\left(\frac{\Lambda}{\lambda}-1\right)m^{2}\left(\Omega_{s},T_{s},T_{v}\right)-\frac{\lambda}{2}\left(T_{s}^{2}-T_{v}^{2}\right).$
(85b)
In the $S-breaking,$ $m^{2}\left(\Omega_{s},T_{s}\right)\gtrsim 0$ and
$T_{s}\geq T_{v}$ when $T_{s}\lesssim T_{cr}$ and space contracts. Hence when
$m\left(\Omega_{v},T_{v},T_{s}\right)\sim 0,$ it is necessary
$m\left(\Omega_{s}\right)\gtrsim m\left(\Omega_{v},T_{v},T_{s}\right)\sim 0.$
In fact, if $\lambda\left(T_{s}^{2}-T_{v}^{2}\right)/2$ in $\left(85b\right)$
is very large, $\lambda T_{s}^{2}/2$ is definitely very large, e.g.
$T_{s}^{2}\lesssim 4\mu^{2}/\lambda.$ Consequently, because of
$\left(79c\right)$ and $\left(66c\right),$ $m\left(\Omega_{s},T_{s}\right)$ is
definitely very small and $\Lambda/\lambda\gg 1$. This case satisfies the two
conditions $A$ and $B$ above. If $\lambda\left(T_{s}^{2}-T_{v}^{2}\right)/2$
is very small, $m^{2}\left(\Omega_{s},T_{s}\right)$ must be less than
$\lambda\left(T_{s}^{2}-T_{v}^{2}\right)/2$ provided
$\left(\Lambda/\lambda-1\right)>1.$ When
$m^{2}\left(\Omega_{v},T_{v},T_{s}\right)=0,$
$m^{2}\left(\Omega_{s},T_{s}\right)>0$ because $T_{s}^{2}-T_{v}^{2}>0.$ Hence
$m\left(\Omega_{s},T_{s}\right)\geq m\left(\Omega_{v},T_{v},T_{s}\right)$ when
$m\left(\Omega_{v},T_{v},T_{s}\right)\longrightarrow 0.$ It is seen provided
$m^{2}\left(\Omega_{v},T_{v},T_{s}\right)\sim 0,$
$m\left(\Omega_{s},T_{s}\right)$ is definitely very small and the conditions
$A$ and $B$ above are definitely satisfied. Thus, $\Omega_{s}$ and
$\varphi_{s}$ can transform into $\Omega_{v}$ by $\left(10\right)$ so that
$T_{v}$ and $\rho_{v}$ increase. Hence $\left(84\right)$ can come into being,
and $\mu_{s}^{2}\left(T_{v},T_{s}\right)\leq 0$ in the $S-breaking$ when
$T_{s}$ and $T_{v}\lesssim T_{cr}$ as well. Thus we prove $\left(67d\right)$
to be valid. When $v_{\Omega s}^{2}\left(T_{s}\right)$ continues to decrease
to zero as $T_{s}$ rises to $T_{cr}$ due to space contraction, $v_{\Omega
s}\left(T_{cr}\right)=v_{\Omega v}\left(T_{cr}\right)=0$ and
$m\left(\Omega_{v},T_{v},T_{s}\right)=m\left(\Omega_{s},T_{s}\right)=0.$
Even $\chi_{s}$ and $\chi_{v}$ are considered, the above conclusions still
hold water qualitatively.
## VI Simplification of the Higgs potential
If we only discuss space inflation of the present model, the Higgs potentials
$\left(8\right)-\left(10\right)$ can be simplified as follows.
$V_{s}=-\frac{\mu^{2}}{15}\left(Tr\Phi_{s}^{2}\right)+\frac{1}{4}a\left(Tr\Phi_{s}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{s}^{4}\right)-\frac{1}{2}\varsigma\left(Tr\Phi_{s}^{2}\right)\chi_{s}^{+}\chi_{s}+\frac{1}{4}\xi\left(\chi_{s}^{+}\chi_{s}\right)^{2},$
(86a)
$V_{v}=-\frac{\mu^{2}}{15}\left(Tr\Phi_{v}^{2}\right)+\frac{1}{4}a\left(Tr\Phi_{v}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{v}^{4}\right)-\frac{1}{2}\varsigma\left(Tr\Phi_{v}^{2}\right)\chi_{v}^{+}\chi_{v}+\frac{1}{4}\xi\left(\chi_{v}^{+}\chi_{v}\right)^{2},$
(86b)
$V_{sv}=\frac{2}{225}\alpha\left(Tr\Phi_{v}^{2}\right)Tr\Phi_{v}^{2}+\frac{1}{2}\beta\left(Tr\Phi_{s}^{2}\right)\chi_{v}^{+}\chi_{v}+\frac{1}{2}\beta\left(Tr\Phi_{v}^{2}\right)\chi_{s}^{+}\chi_{s}.$
(86c)
Because of $\left(86c\right),$ the breaking mode can only be the $S-breaking$
$\left(\varpi_{s}=v_{s}\left(T_{s},T_{v}\right),\text{ }\varpi_{v}=0\right)$
or the $V-breaking$ $\left(\varpi_{v}=v_{v}\left(T_{s},T_{v}\right),\text{
}\varpi_{s}=0\right).$ For simplicity, we only consider the $V-breaking$ and
ignore $\chi_{v}$ and $\chi_{s}$ for a time.
$\displaystyle V_{eff}^{(1)T}\left(\overline{\varphi}_{v},T_{s},T_{v}\right)$
$\displaystyle=\frac{1}{2}\left(-\mu^{2}+\alpha v_{\varphi
s}^{2}+2CT_{v}^{2}\right)\overline{\varphi}_{v}^{2}+B\overline{\varphi}_{v}^{4}\left(\ln\frac{\overline{\varphi}_{v}^{2}}{\sigma^{2}}-\frac{1}{2}\right),$
(87a) $\displaystyle
V_{eff}^{(1)T}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)$
$\displaystyle=\frac{1}{2}\left(-\mu^{2}+\alpha v_{\varphi
v}^{2}+2CT_{s}^{2}\right)\overline{\varphi}_{s}^{2}+B\overline{\varphi}_{s}^{4}\left(\ln\frac{\overline{\varphi}_{s}^{2}}{\sigma^{2}}-\frac{1}{2}\right),$
(87b)
The critical temperatures and the effective masses are determined by
$\displaystyle T_{\varphi vcr}^{2}$
$\displaystyle=\left(B\sigma^{2}e^{-1/2}+\mu^{2}/2-\alpha v_{\varphi
s}^{2}/2\right)/C,$ (88a) $\displaystyle T_{\varphi scr}^{2}$
$\displaystyle=\left(B\sigma^{2}e^{-1/2}+\mu^{2}/2-\alpha v_{\varphi
v}^{2}/2\right)/C.$ (88b) $\displaystyle
m^{2}\left(\varphi_{v},T_{s},T_{v}\right)$
$\displaystyle=2\left(\mu^{2}-\alpha v_{\varphi
s}^{2}-2CT_{v}^{2}\right)+8Bv_{\varphi v}^{2},\text{ \ when }T_{\varphi
v}<T_{\varphi vcr},$ (88c) $\displaystyle
m^{2}\left(\varphi_{v},T_{s},T_{v}\right)$
$\displaystyle=\left(-\mu^{2}+\alpha v_{\varphi
s}^{2}+2CT_{v}^{2}\right),\text{ \ when }T_{\varphi v}\geq T_{\varphi vcr},$
(88d) $\displaystyle m^{2}\left(\varphi_{s},T_{s},T_{v}\right)$
$\displaystyle=2\left(\mu^{2}-\alpha v_{\varphi
v}^{2}-2CT_{s}^{2}\right)+8Bv_{\varphi s}^{2},\text{ \ when }T_{\varphi
s}<T_{\varphi scr},$ (88e) $\displaystyle
m^{2}\left(\varphi_{s},T_{s},T_{v}\right)$
$\displaystyle=\left(-\mu^{2}+\alpha v_{\varphi
v}^{2}+2CT_{s}^{2}\right),\text{ \ when }T_{\varphi s}\geq T_{\varphi scr}.$
(88f)
Substituting $\left(v_{\varphi v}\left(T_{s},T_{v}\right)\neq 0,v_{\varphi
s}\left(T_{s},T_{v}\right)=0\right)$ or $\left(v_{\varphi
v}\left(T_{s},T_{v}\right)=0,v_{\varphi s}\left(T_{s},T_{v}\right)\neq
0\right)$ into $\left(88\right),$ we obtain the critical temperatures and the
effective masses in the $V-breaking$ or in the $S-breaking.$ There is no
influence of $\varphi_{s}$ to
$V_{eff}^{\left(1\right)T}\left(\varphi_{v},T_{s},T_{v}\right)$ because
$\varpi_{s}=0$ in the $V-breaking.$ Hence the potential $\left(86\right)$, in
fact, is equivalent to the $SU(5)$ potential discussed by Ref. $[14,15].$
When temperature descends due to space inflation so that $T_{v}<T_{\varphi
cr},$
$V_{eff\min}^{\left(1\right)T}\left(\overline{\varphi}_{v},T_{s},T_{v}\right)=V_{eff}^{\left(1\right)T}\left(v_{v}\left(T_{s},T_{v}\right)\right)<0,$
$V_{eff\min}^{\left(1\right)T}\left(\overline{\varphi}_{s},T_{s},T_{v}\right)=0.$
In the case, there is space inflation in the slow roll approximation. The
duration $\tau\sim 1/T_{b}$ ($T_{b}\sim 10^{8}Gev$) of inflation has been
estimated by the Ref. $[15]$.
The difference between $\left(86\right)$ and $\left(8\right)-\left(10\right)$
is the following.
The minimum
$V_{v,eff\min}^{\left(1\right)T}\left(\overline{\varphi}_{v}\right)$ is the
absolute minimum at the zero-temperature in the $V-breaking$ according to
$\left(86\right)$. In contrast to $\left(86\right),$ according to
$\left(8\right)-\left(10\right),$
$V_{v,eff\min}^{\left(1\right)T}\left(\overline{\varphi}_{v}\right)$ is not
absolute minimum at the zero-temperature in the $V-breaking$, but
$V_{s,eff\min}^{\left(1\right)T}\left(\overline{\Omega}_{v},\overline{\varphi}_{v}\right)$
$\ll V_{v,eff\min}^{\left(1\right)T}\left(\overline{\varphi}_{v}\right)$ is
the absolute minimum at the zero-temperature. We will see
$\left(8\right)-\left(10\right)$ has more important significance.
### VI.1 Change of $w$ from $>0$ to $<-1$ and $q$
In the inflation period (see Section 7.1, $\left(103\right)$), $\rho_{g}=0$
and $V_{g}=V_{0}.$ $\left(51\right)$ can be rewritten as
$\frac{\overset{\cdot\cdot}{R}}{R}=-\frac{\eta}{2}\left(V_{0}-3V_{0}\right).$
(89)
After reheating, $V_{g}=0.$ In the $V-breaking$,
$\displaystyle\frac{\overset{\cdot\cdot}{R}}{R}$
$\displaystyle=-\frac{\eta}{2}\left[\left(\rho_{vm}+\rho_{v\gamma}+3p_{vm}+3p_{v\gamma}\right)-\left(\rho_{sm}+\rho_{s\gamma}+3p_{sm}+3p_{s\gamma}\right)\right]$
$\displaystyle=-\frac{\eta}{2}\left[\rho+3p\right],$ (90)
where
$\displaystyle\rho$ $\displaystyle\equiv\rho_{vm}+\rho_{v\gamma},$
$\displaystyle p$ $\displaystyle\equiv
p_{vm}+p_{v\gamma}-\rho_{sm}/3-\rho_{s\gamma}/3-p_{sm}-p_{s\gamma},$ (91)
$\rho$ is the conventional positive mass density, and $p$ is the effective
pressure density relative to $\rho$. When the evolution equation is in the
form $\left(90\right),$ $w$ is defined as
$w=p/\rho.$ (92)
Thus, in the inflation period, from $\left(89\right)$ we find
$w=-1.$ (93)
For $\left(90\right)$ we have
$w=\frac{p_{vm}+p_{v\gamma}-\rho_{sm}/3-\rho_{s\gamma}/3-p_{sm}-p_{s\gamma}}{\rho_{vm}+\rho_{v\gamma}}$
(94)
The static masses of all color-single states are non-zero, hence
$\rho_{s\gamma}=0.$ In the early period after reheating, temperature is very
high so that the masses of particles may be neglected,
$p_{vm}\sim\rho_{vm}/3,$ $p_{v\gamma}\sim\rho_{v\gamma}/3$ and
$p_{sm}\sim\rho_{sm}/3$. Consequently,
$w\sim\frac{\rho_{vm}+\rho_{v\gamma}-2\rho_{sm}}{3\left(\rho_{vm}+\rho_{v\gamma}\right)}>0.$
(95)
From $\left(65c\right)$ and section 8.7, we know $\rho_{sm}/\rho_{vm}$ is
changeable. If after some a time,
$\rho_{sm}\sim 3\rho_{vm},$ (96)
considering $\rho_{\gamma}\propto R^{-4}$ and $\rho_{m}\propto R^{-3},$ when
$R$ is large enough, $\rho_{v\gamma}\sim 0$ and temperature is low so that
$p_{vm}\sim 0$ and $p_{sm}\sim 0,$ from $\left(94\right)$ and
$\left(96\right)$ we have
$w\sim\frac{-3\rho_{sm}}{3\rho_{vm}}\sim-1.$ (97)
It is seen that $w$ can change from $w>0$ to $w\sim-1$ according to the
present model..
According to the simplified Higgs potential $\left(86\right),$ in the slow
rolling approximation, we can get the results similar to $\left(93\right)$ and
$\left(96\right)-\left(97\right).$
In fact, space inflation or expansion with an acceleration is independent of
the slow approximation, and $w$ is not important for the present model.
Although there is no slow approximation, space inflation can be explained
because $V_{g}=V_{0}$ in the inflation process and the parameters $\alpha$ and
$w$ in $\left(8\right)-\left(10\right)$ are so chosen that $\alpha>w$.
Although $V_{g}=0$ after reheating, space expansion with a deceleration or an
acceleration can still be explained by the present model.
$q$ is defined as
$q=-\frac{\overset{\cdot\cdot}{R}R}{\overset{\cdot}{R}^{2}}=\frac{-K/R^{2}+\eta\left(\rho_{g}+V_{g}\right)}{\eta\left(\rho_{g}/2+3p_{g}/2-V_{g}\right)}.$
(98)
When $\overline{\Omega}_{s}=\overline{\Omega}_{v}=0$ (or
$\overline{\varphi}_{s}=\overline{\varphi}_{v}=0$ according to the simplified
Higgs potential $\left(86\right)$), $\rho_{v}=\rho_{s},$ $p_{v}=p_{s},$
$V_{s}=V_{v}$, $K\sim 1,$ $R\geq R_{\min}$ to be finite and
$V_{g}=V_{0}\gg-K/R^{2}.$ Consequently,
$q=\frac{-K/R^{2}+\eta V_{g}}{-\eta V_{g}}\lesssim-1..$ (99)
After the reheating, $V_{v}\sim-V_{0},$ $V_{s}=0,$ and
$V_{g}=V_{v}+V_{0}-V_{s}=0.$ When $\rho_{g}=\rho_{v}-\rho_{s}>0,$
$p_{v}>p_{s}$ and $-K/R^{2}+\eta\rho_{g}>0,$
$q=\frac{-K/R^{2}+\eta\rho_{g}}{\eta\rho_{g}/2+3p_{g}/2}>0,$ (100)
where $K\sim 1$ due to $\rho_{g}>0$.
When $\rho_{g}=0,$ $K=0,$ from $\left(65\right)$ we have
$q=-\frac{\overset{\cdot\cdot}{R}R}{\overset{\cdot}{R}^{2}}=\frac{3}{4}\frac{\eta\rho_{g\gamma}R\overset{\cdot}{K}}{\overset{\cdot}{R}\overset{\cdot\cdot}{K}}+\frac{1}{2}.$
(101)
When $\rho_{g}<0,$ $K\sim-1$. Considering $\rho_{g}=\rho_{mg}+\rho_{\gamma
g},$ $\rho_{mg}\varpropto R^{-3},$ $\rho_{\gamma g}\varpropto R^{-4},$ $p_{g}$
may be neglected and $-K/R^{2}+\eta\rho_{g}>0$ when $R$ is large enough, we
have
$q=\frac{-K/R^{2}+\eta\rho_{g}}{\eta\rho_{g}/2}<0.$ (102)
In the case, space expands with an acceleration, although $V_{g}=0.$
## VII Contraction of space, the highest temperature and inflation of space
On the basis of the cosmological principle, if there is the space-time
singularity, it is a result of space contraction. Thus, we discuss the
contracting process and find the condition of space inflation. From the
contracting process we will see that there is no space-time singularity in
present model
We do not consider the couplings of the Higgs fields with the Ricci scalar $R$
for a time. We will see in the following paper that the following conclusions
still hold water when such couplings as $\xi R\Omega_{s}^{2}$ are considered.
In fact, $\xi
R\left(\overline{\Omega}_{s}^{2}-\overline{\Omega}_{v}^{2}\right)=0$ because
there is the strict symmetry between $s-matter$ and $v-matter$ when $T\gtrsim
T_{cr}$.
We chiefly discuss change of $\langle\Omega_{a}\left(T_{a}\right)\rangle$ and
$\langle\varphi_{a}\left(T_{a}\right)\rangle$ as temperature, $a=s,v,$ in the
contracting process of space for short. When
$\langle\chi_{a}\left(T_{a}\right)\rangle$ is considered as well, the
inferences are still valid qualitatively.
### VII.1 Contraction of space, proof of non-singularity, the highest
temperature and inflation of space
Proof. There no singularity of space-time in the present model.
$1.$ The end of space contraction is in the most symmetric state in which
$\langle\omega_{v}\rangle=\langle\omega_{s}\rangle=0,$ $T_{s}=T_{v}$ and
$\rho_{s}=\rho_{v}$ so that $\widetilde{\rho}_{Sg}=V_{0}$
This has been proved in section
$2.$ There is no singularity in the present model on the basis of the
cosmological principle.
If space does not contract because $\overset{\cdot}{R}>0$ or
$\widetilde{\rho}_{Sg}=0,$ it is necessary that there is no space-time
singularity. Provided space contract because $\widetilde{\rho}_{Sg}=0,$
$T_{s}$ and $T_{v}$ must rise. When $T_{s}\sim T_{v}\geq
T_{cr}\equiv\mu/\sqrt{\lambda},$
$\langle\omega_{v}\rangle=\langle\omega_{s}\rangle=0,$ the masses of all
particles which originate from the couplings with $\varpi_{s}$ and
$\varpi_{v}$ are zero. Consequently $\rho_{s}$ and $\rho_{v}$ can transform
from one into another. Thus $\rho_{s}=\rho_{v}$, $T_{s}=T_{v}$ and
$\widetilde{\rho}_{Sg}=V_{0},$ i.e., the most symmetric state comes into
being. In the state, both the $s-SU(5)$ and the $v-SU(5)$ are strictly kept.
In the case, from the conjecture 3 and the discussion about $\left(61\right),$
we may take $K=1$. Thus $\left(48\right)-\left(49\right)$ is reduced to
$\overset{\cdot}{R}^{2}=-1+\eta V_{0}R^{2},$ (103a)
$\overset{\cdot\cdot}{R}=\eta V_{0}R.$ (103b)
Consequently space inflation must occur and temperature will fast descend.
Let $R_{cr}=R\left(T_{cr}\right).$ If
$\overset{\cdot}{R}_{cr}^{2}=-1+\eta V_{0}R_{cr}^{2}\geq 0,\text{ \ i.e. \
}R_{cr}\geq\sqrt{1/\eta V_{0}},$ (104)
$R$ can continue to decrease with a deceleration or stop contracting. Hence
there must be the least scale $R_{\min}\leq R_{cr},$ the critical temperature
$T_{cr}$, the highest temperature $T_{\max}$ and the largest energy density
$\rho_{\max},$
$\displaystyle R_{\min}$ $\displaystyle=\sqrt{1/\eta V_{0}}\leq R_{cr},\text{
\ }T_{cr}\equiv 2\mu/\sqrt{\lambda},$ (105a) $\displaystyle T_{\max}$
$\displaystyle=T\left(R_{\min}\right)=T_{cr}R_{cr}/R_{\min}\geq T_{cr},$
(105b) $\displaystyle\rho_{\max}$
$\displaystyle=\rho_{s\max}+\rho_{v\max}=2\frac{\pi^{2}}{30}g^{\ast}T_{\max}^{4},\text{
\ and \ }\widetilde{\rho}_{\max}=\rho_{\max}+V_{0}.$ (105c)
Thus, when $R$ decreases to $R_{\min},$ and space inflation must occur,
$\displaystyle R$ $\displaystyle=\sqrt{\frac{1}{\eta V_{0}}}\cosh\sqrt{\eta
V_{0}}\left(t-t_{FI}\right)=\sqrt{\frac{1}{\eta V_{0}}}\cosh
H\left(t-t_{FI}\right),\text{ \ }\sqrt{\eta V_{0}}\equiv H,$ (106a)
$\displaystyle=\sqrt{\frac{1}{\eta V_{0}}}=R_{\min},\text{ \ \ when }t=t_{FI}$
(106b) $\displaystyle\sim\frac{1}{2}\sqrt{\frac{1}{\eta V_{0}}}\exp
H\left(t-t_{FI}\right),\text{ \ when }H\left(t-t_{FI}\right)>>1,$ (106c)
where $\sqrt{\eta V_{0}}\equiv H$ is the Hubble constant at $t\gtrsim t_{FI}$.
$t_{FI}$ is just the last moment of the world in the $S-breaking$ and the
initial moment of the world in the $V-breaking.$ $R_{\min}$ and $T_{cr}$ are
two new important constants, and $T_{\max}$ and $\rho_{\max}$ are determined
by $R_{cr}$. It is seen that all $R$, $T$ and $\rho$ must be finite in the
case. $\left(56b\right)$ is considered in $\left(105b\right).$ The meanings of
the parameters are that when $T=T_{cr},$
$\langle\omega_{s}\rangle=\langle\omega_{v}\rangle=0$ and $R=R_{cr},$ and when
$R=R_{\min}$, $T=T_{\max}$ or $\rho=\rho_{\max}$ and $\overset{\cdot}{R}=0.$
We know that the duration of inflation $\tau$ may be long enough for inflation
(see below $\left(88\right)$). After $\tau,$ $R$ has a large enough increase.
$\left(104\right)$ is the condition of space inflation. Because the masses of
all particles which originate from the couplings with
$\langle\omega_{s}\rangle$ or $\langle\omega_{v}\rangle$ are zero and
$\rho_{s}=\rho_{v}$ when $T\gtrsim T_{cr}$, considering $\left(56b\right),$ we
have
$\displaystyle\rho_{s}R^{4}$
$\displaystyle=\rho_{scr}R^{4}\left(T_{cr}\right)\equiv
D_{s}=D_{v}=\rho_{v}R^{4}=\rho_{vcr}R^{4}\left(T_{cr}\right),\text{ \
}\rho_{a}=\frac{\pi^{2}}{30}g^{\ast}T_{a}^{4},$ $\displaystyle\rho_{scr}$
$\displaystyle=\rho_{vcr}=\rho_{cr}\equiv\frac{\pi^{2}}{30}g^{\ast}T_{cr}^{4}=\frac{\pi^{2}}{30}g^{\ast}\frac{16\mu^{4}}{\lambda^{2}},\text{
\ }T_{a}^{4}R^{4}=T_{cr}^{4}R^{4}\left(T_{cr}\right).$ (107)
Thus, when $T\sim T_{cr}$, from $\left(118\right)$ we can rewrite the
condition of inflation $\left(104\right)$ as
$D_{v}=D_{s}\geq\left(\frac{K}{\eta}\right)^{2}\frac{16\mu^{4}g^{\ast}}{V_{0}^{2}\lambda^{2}}=g^{\ast}\frac{1}{\eta^{2}}\left(\frac{4}{\mu}\right)^{4}\equiv
D_{cr},$ (108)
here $V_{0}=\mu^{4}/4\lambda$ and $K=1$ is considered. Thus, when $D_{s}\geq
D_{cr},$ there must be space inflation.
If $R\left(T_{cr}\right)<\sqrt{1/\eta V_{0}}$ or $D_{s}<D_{cr},$ this implies
that $\overset{\cdot}{R}=0$ already occurs before $R$ contracts to
$R\left(T_{cr}\right)$ or $T_{s}$ rises to $T_{cr},$ i.e.,
$R_{\min}>R\left(T_{cr}\right)$ and $T\left(R_{\min}\right)=T_{\max}<T_{cr}.$
Consequently $T_{cr}$ and $R\left(T_{cr}\right)$ cannot be arrived and there
still are $\langle\omega_{s}\left(T_{s}\right)\rangle\neq 0$ and
$\langle\omega_{v}\rangle=0$. In the case, all $R_{\min}$, $T_{\max},$
$\rho_{g},$ $\rho_{s}$ and $\rho_{v}$ must still be finite because of the
cosmological principle, i.e. there is no space-time singularity. In the case,
it is necessary
$\overset{\cdot}{R}=0,\text{ \ \ }\overset{\cdot\cdot}{R}>0,\text{ \ when
}R=R_{\min},$
because $R_{\min}$ is the end of contracting $R$. In the case, when $R\geq
R_{\min},$ the evolving equations are still $\left(48\right)-\left(49\right)$
and space will expand still in the $S-breaking$ mode, but space inflation
cannot occur.
To sum up, we see that in any case of the contracting process, there must be
$R_{\min}>0$ and the finite $T_{\max}.$ Because of the cosmological principle,
all $\rho_{s},$ $\rho_{v},$
$\widetilde{\rho}_{Sg}=\widetilde{\rho}_{s}-\widetilde{\rho}_{v}$ and
$p\leqslant\rho/3$ are finite because of the cosmological principle. Hence
$T_{Ss\mu\nu},$ $T_{Sv\mu\nu}$ and $T_{Ss\mu\nu}-T_{Sv\mu\nu}$ must be finite
due to $(43)-(47).$ Substituting the finite $T_{Ss\mu\nu}-T_{Sv\mu\nu}$ into
the Einstein field equation $\left(13\right),$ we see that $R_{\mu\nu}$ and
$g_{\mu\nu}$ must be finite.
Thus, we have proved that there is no singularity in present model.
In fact, when
$\widetilde{\rho}_{g}=\widetilde{\rho}_{v}-\widetilde{\rho}_{s}=V_{0},$
$\left(103\right)$ is consistent with the Lemaitre model without singularity
in which $\rho_{g}=0$, $k=1$ and the cosmological constant
$\lambda_{eff}>0^{[16]}.$
### VII.2 The result above is not contradictory to the singularity theorems
We first intuitively explain the reasons that there is no space-time
singularity. It has been proved that there is space-time singularity under
certain conditions[1]. These conditions fall into three categories. First,
there is the requirement that gravity shall be attractive. Secondly, there is
the requirement that there is enough matter present in some region to prevent
anything escaping from that region. The third requirement is that there should
be no causality violations.
Hawking considers it is a reasonable conjecture that $\rho_{g}>0$ and
$p_{g}\geq 0^{[1]}.$ But this conjecture is not valid. The gravitational mass
density $\rho_{g}=\rho_{s}-\rho_{v}>0,$ $=0$ or $<0$ are all possible in the
present model.
It is necessary $\rho_{g}=\rho_{s}-\rho_{v}=0$ because $\rho_{s}$ and
$\rho_{v}$ can transform from one to another when $T\gtrsim T_{cr}$. It is
seen that $\rho_{g}$ does not increase not only, but also decreases to zero
when $T\gtrsim T_{cr}.$ Hence the second condition is violated.
The key of non-singularity is that there are $s-matter$ and $v-matter$ with
opposite gravitational masses and both can transform from one to another when
$T\gtrsim T_{cr}$
We explain the reasons that there is no space-time singularity from the
Hawking theorem as follows. S.W. Hawking has proven the following theorem[1].
The following three conditions cannot all hold:
$(a)$ every inextendible non-spacelike geodesic contains a pair of conjugate
point;
$(b)$ the chronology condition holds on ${\Huge\mu;}$
$(c)$ there is an achronal set $\mathfrak{T}$ such that $E^{+}(\mathfrak{T})$
or $E^{-}(\mathfrak{T})$ is compact.
The alternative version of the theorem can obtained by the following two
propositions.
Proposition $1^{[1]}$:
If $R_{ab}V^{a}V^{b}\geq 0$ and if at some point $p=\gamma(s_{1})$ the tidal
force $R_{abcd}V^{c}V^{d}$ is non-zero, there will be values $s_{0}$ and
$s_{2}$ such that $q=\gamma\left(s_{0}\right)$ and
$r=\gamma\left(s_{2}\right)$ will be conjugate along $\gamma\left(s\right)$,
providing that $\gamma\left(s\right)$ can be extended to these values.
Proposition $2^{[1]}$:
If $R_{ab}V^{a}V^{b}\geq 0$ everywhere and if at $p=\gamma(v_{1}),$
$K^{a}K^{b}K_{[a}R_{b]cd[e}K_{f]}$ is non-zero, there will be $v_{0}$ and
$v_{2}$ such that $q=\gamma\left(v_{0}\right)$ and
$r=\gamma\left(v_{2}\right)$ will be conjugate along $\gamma\left(v\right)$
provided that $\gamma\left(v\right)$ can be extended to these values.
An alternative version of the above theorem is the following.
Space-time (${\Huge\mu,g}$) is not timelike and null geodesically complete if:
$(1)$ $R^{ab}K_{a}K_{b}\geq 0$ for every non-spacelike vector
$\boldsymbol{K}\mathbf{.}$
$(2)$ The generic condition is satisfied, i.e. every non-spacelike geodesic
contains a point at which $K_{[a}R_{b]cd[e}K_{f]}K^{c}K^{d}\neq 0$, where
$\boldsymbol{K}\mathcal{\ }$is the tangent vector to the geodesic.
$(3)$ The chronology condition holds on ${\Huge\mu}$ (i.e. there are no closed
timelike curves).
$(4)$ There exists at least one of the following:
$(A)$ a compact achronal set without edge,
$(B)$ a closed trapped surface,
$(C)$ a point $p$ such that on every past (or every future) null geodesic from
$p$ the divergence $\widehat{\vartheta}$ of the null geodesics from $p$
becomes negative (i.e. the null geodesics from $p$ are focussed by the matter
or curvature and start to reconverge).
This theorem is an alternative version of the above theorem. This is because
if ${\Huge\mu}$ is timelike and null geodesically complete, $(1)$ and $(2)$
would imply $\left(a\right)$ by above propositions $1$ and $2,$ $(1)$ and
$(4)$ would imply $\left(c\right)$, and $\left(3\right)$ is the same as
$\left(b\right).$
In fact, $R_{ab}$ is determined by the gravitational energy-momentum tensor
$T_{gab}$. According to the conventional theory, $T_{gab}=T_{ab}$ so that the
above theorem holds.
In contrast with the conventional theory, according to conjecture $1$, in the
$s-breaking$,
$\displaystyle R_{\mu\nu}$ $\displaystyle=-8\pi
G\left(T_{g\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{g}\right)$ $\displaystyle=-8\pi
G\left[\left(T_{s\mu\nu}-T_{v\mu\nu}\right)-\frac{1}{2}g_{\mu\nu}\left(T_{s}-T_{v}\right)\right].$
(109a)
Consequently, $R_{00}>0,$ $=0$ and $<0$ are all possible. Thus, although the
strong energy condition still holds, i.e.
$\left[\left(T_{s}^{ab}+T_{v}^{ab}\right)-\frac{1}{2}g^{ab}\left(T_{s}+T_{v}\right)\right]K_{a}K_{b}\geq
0,$ (109b)
the conditions of propositions $1$ and $2$ and condition $\left(1\right)$ do
no longer hold, because the gravitational mass density $\rho_{g}$ determines
$R_{\mu\nu}$ and $\rho_{g}=\rho_{v}-\rho_{s}\neq\rho_{v}+\rho_{s}=\rho.$ Hence
$\left(a\right)$ and $\left(c\right)$ do not hold, but $\left(b\right)$ still
holds, and ${\Huge\mu}$ is timelike and null geodesically complete..
### VII.3 The process of space inflation
As mentioned before, the duration of space inflation is finite (see below
$\left(88\right)$). Supposing $\lambda\sim g^{4}$, $g^{2}\sim 4\pi/45$ for
$SU(5),$ and considering $m(\Omega_{s})=\sqrt{2}\mu$ (see $\left(32\right)$),
from $(66)$ we can estimate $T_{\max}$ ,
$T_{\max}\gtrsim\frac{2\mu}{\sqrt{\lambda}}\sim\frac{2\mu}{g^{2}}\sim\frac{\sqrt{2}m(\Omega_{s})}{4\pi/45}=5m(\Omega_{s}).$
(110)
The temperature will strikingly decrease in the process of inflation, but the
potential energy $V\left(\varpi_{s}\sim\varpi_{v}\sim 0\right)\sim V_{0}$
cannot decrease to $V_{\min}\left(T_{v}\right)$ at once, because this is a
super-cooling process.
We can get the expecting results by suitably choosing the parameters in
$(8)-(10).$ In order to estimate $H=\sqrt{\eta V_{0}},$ taking
$V_{0}\sim\mu^{4}/4\lambda,$ from $(110)$ we have
$H=aT_{\max}^{2},\text{ \ \ }a\equiv\sqrt{\eta\lambda}/8\sim
g^{2}\sqrt{\eta}/8.$ (111)
We can take $T_{\max}$ to be the temperature corresponding to $GUT$ because
the $SU(5)$ symmetry strictly holds water at $T_{\max}$.
Taking $T_{\max}\sim 5m\left(\Omega_{s}\right)\sim 5\times 10^{15}Gev$ and
$\sqrt{\lambda}/8\sim g^{2}\sim 0.035,$ we have $H^{-1}=10^{-35}s.$ If the
duration of the super-cooling state is
$10^{-33}s\sim\left(10^{8}Gev\right)^{-1},$ $R_{\min}$ will increase
$exp100\sim 10^{43}$ times. As mentioned before (see below $\left(88\right)$),
the duration $\tau$ of inflation may be long enough, $\tau\sim
1/T_{b}^{[15]}.$ Taking $T_{\max}\sim 10^{15}Gev$ and $T_{b}\sim 10^{8}Gev,$
we have $H^{-1}=10^{-35}s$ and $\tau\sim 10^{-33}s$. The result is consistent
with the Guth’s inflation model[17].
Before inflation, the world in the $S-breaking$ is in thermal equilibrating
state. If there is no $v-matter,$ because of contraction by gravitation, the
world would become a thermal-equilibrating singular point, i.e., the world
would be in the hot death state. As seen, it is necessary that there are both
$s-matter$ and $v-matter$ and both the $S-breaking$ and the $V-breaking$.
The parameters in the Higgs potential can be so suitalbly chosen that $\tau$
is suitable. We discuss the problem in the following paper.
## VIII Expansion of space after inflation
### VIII.1 The reheating process
After inflation, the temperature must sharply descend. In the case, it is
easily seen that the state with
$\langle\omega_{s}\rangle=\langle\omega_{v}\rangle=0$ is no longer stable and
must decay into such a state with $V_{\min}.$ Either of the $S-breaking$ and
the $V-breaking$ can come into being, because $s-matter$ and $v-matter$ are
completely symmetric at $T_{\max}\gtrsim T_{cr}$. Let the $V-breaking$ comes
into being, then $v-SU(5)\longrightarrow v-SU(3)\times U(1)$ and $s-SU(5)$
symmetry is kept. In this case, $t_{FI}$ can be regarded as the initial moment
for the world in the $V-breaking.$ Thus we take $t_{FI}=0$. Ignoring the
effect of temperature $\left(T\sim 0\right)$, from
$\left(8\right)-\left(10\right)$ we see the phase transition of the vacuum is
as follows,
$\displaystyle\varpi_{v}(T_{\max})$
$\displaystyle=0\longrightarrow\varpi_{v}(T_{v}\sim 0)=\varpi_{v0},\text{ }$
$\displaystyle\varpi_{s}(T_{\max})$
$\displaystyle=0\longrightarrow\varpi_{s}(T_{s}\sim 0)=\varpi_{s0}=0,$
$\displaystyle V_{v}(\varpi_{v},T_{\max})$ $\displaystyle=0\longrightarrow
V_{v}(\varpi_{v0},0)=-V_{0},$ $\displaystyle V_{s}(\varpi_{s},T_{\max})$
$\displaystyle=0\longrightarrow V_{s}(\varpi_{s0},0)=0,\text{ \
}V_{sv}=0\longrightarrow 0,$ $\displaystyle V_{Vg}$
$\displaystyle=V_{v}(T_{\max})+V_{0}-V_{s}\left(T_{\max}\right)$
$\displaystyle=V_{0}\longrightarrow
V_{v}(\varpi_{v0},0)+V_{0}-V_{s}(\varpi_{s0},0)=0.$ (112)
After transition, $V_{v}\left(T_{\max}\right)-V_{v}\left(T_{v}\sim
0\right)=V_{0}$ must firstly transform into a lot of the $v-Higgs$ particles.
The $v-Higgs$ particles can decay fast into the $v-gauge$ bosons and the
$v-fermions$ by the $v-SU(5)$ couplings. On the other hand, because of the
coupling $(10),$ the $v-Higgs$ particles can transform into the $s-Higgs$
particles as well. The $s-Higgs$ particles $\Omega_{s},$ $\varphi_{s}$ and
$\chi_{s}$ can fast decay into the $s-gauge$ bosons and the $s-fermions$ by
$s-SU(5)$ couplings. Let $\alpha V_{0}$ transform the $v-energy$, then
$(1-\alpha)V_{0}$ transforms the $s-energy.$ From the decaying process we see
it is necessary $\alpha>(1-\alpha)$. Let $\rho_{v}^{\prime}=\rho_{s}^{\prime}$
before the transition, it is necessary after transition that
$\rho_{v}=\rho_{v}^{\prime}+\alpha
V_{0}>\rho_{s}=\rho_{s}^{\prime}+(1-\alpha)V_{0}.$ (113)
This is the reheating process, after which, the $v-particles$ get their
masses, but the $s-gauge$ bosons and the elementary $s-fermions$ are still
massless since $s-SU(5)$ is not broken. Both $v-matter$ and $s-matter$ can
exist in the form of plasma for a short period, because $T_{s}$ and $T_{v}$
must be very high in the initial period after reheating.
In the $V-breaking,$ the space evolving equations are
$\left(50\right)-\left(51\right).$ After the reheating process, $V\sim 0.$
Thus, $\left(50\right)-\left(51\right)$ reduce
$\overset{\cdot}{R}^{2}=-k+\eta\left(\rho_{v}-\rho_{s}\right)R^{2}=-k+\eta\rho_{Vg}R^{2}.$
(114a)
$\overset{\cdot\cdot}{R}=-\frac{\eta}{2}\left[\left(\rho_{v}-\rho_{s}\right)+3\left(p_{v}-p_{s}\right)\right]R=-\frac{\eta}{2}\left(\rho_{Vg}+3p_{Vg}\right)R.$
(114b)
After $T_{s}$ and $T_{v}$ decline further, the $v-particles$ exist in forms of
nucleons, leptons and photons, and the $s-particles$ will form $S-SU(5)$ color
single states whose masses are non-zero.
### VIII.2 Change of gravitational mass density in comoving coordinates
We take the order of time to be $t_{0}>t_{1}\geq t_{2}>t_{3}>t_{FI}=0.$
After reheating process, all Higgs particles have got their very big masses
due to $T_{v}\sim T_{s}\sim 0$ , hence $s-matter$ and $v-matter$ cannot
transform into each other. $\rho_{v}=\rho_{vm}+\rho_{v\gamma}$ because
$v-SU(5)\longrightarrow v-SU\left(3\right)\times U\left(1\right).$ All
$s-particles$ must be the $s-colour$ single states and there is no
$s-photonlike$ particle so that $\rho_{s\gamma}=0$ so that
$\rho_{s}=\rho_{sm}$, because $s-SU(5)$ is not broken in the $V-breaking.$
Hence $\rho_{v}/\rho_{s}$ must decrease as $R$ increases because
$\rho_{m}\propto R^{-3}$ and $\rho_{\gamma}\propto R^{-4}$. In the case,
$V_{Vg}=0$ and $\left(114\right)$ reduces to
$\overset{\cdot}{R}^{2}+k=\eta\left(\rho_{vm}+\rho_{v\gamma}-\rho_{sm}\right)R^{2}=\eta\rho_{Vg}R^{2},$
(115) $\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=-\frac{\eta}{2}\left[\left(\rho_{vm}+\rho_{v\gamma}-\rho_{sm}\right)+3\left(p_{vm}+p_{v\gamma}-p_{sm}\right)\right]R,$
(116a)
$\displaystyle\simeq-\frac{\eta}{2}\left(\rho_{vm}+2\rho_{v\gamma}-\rho_{sm}\right)R=-\frac{\eta}{2}\left(\rho_{Vmg}+2\rho_{v\gamma
g}\right)R,$ (116b)
where $p_{vm}$ and $p_{sm}$ are neglected and $p_{v\gamma}=\rho_{v\gamma}/3$
is considered.
Suppose when $t=t_{3}$ (e.g. $t_{3}\sim 10^{4}\sim 10^{5}a$), $v-atoms$ have
formed, $v-photons$ have decoupled, $s-SU(5)$ color single states have formed,
$\rho_{sm}\left(t_{3}\right)=x\rho_{vm}\left(t_{3}\right),$
$\rho_{v\gamma}\left(t_{3}\right)=y_{3}\rho_{vm}\left(t_{3}\right)$ and
$\displaystyle\rho_{v}$
$\displaystyle=\rho_{v\gamma}\left(t_{3}\right)+\rho_{vm}\left(t_{3}\right)>\rho_{sm}\left(t_{3}\right)>3\rho_{vm}\left(t_{3}\right),$
$\displaystyle\text{or \ }y_{3}+1$ $\displaystyle>x>3.$ (117)
After the photons decoupled, the number $n_{vN}$ of the $v-nucleons$ and the
number $n_{v\gamma}$ of the $v-photons$ are invariant and $n_{vN0}/n_{v\gamma
0}\sim 5\times 10^{-10}$ when space expansion. The number $n_{s}$ of the
$s-colour$ single states is invariant as well. Let
$\underline{\rho}_{Vg}\left(t,R\left(t\right)\right)=\underline{\rho}_{Vg}\left(R\left(t\right)\right),$
then $\rho_{sm}\varpropto R^{-3},$ $\rho_{vm}\varpropto R^{-3}$ and
$\rho_{v\gamma}\varpropto R^{-4}.$ Thus the gravitational mass density
$\underline{\rho}_{Vg}\left(t\right)$ in comoving coordinates is changeable,
$\displaystyle\underline{\rho}_{Vg}\left(t\right)$
$\displaystyle=\underline{\rho}_{vm}+\underline{\rho}_{v\gamma}-\underline{\rho}_{sm}=\underline{\rho}_{vm}\left(t_{3}\right)\left(1-x+yR\left(t_{3}\right)/R\left(t\right)\right)$
$\displaystyle=\underline{\rho}_{vm}\left(t_{0}\right)\left(1-x\right)+\underline{\rho}_{v\gamma
0}R\left(t_{0}\right)/R\left(t\right)$
$\displaystyle=\underline{\rho}_{vm}\left(t_{0}\right)\left(1-x+y_{0}R\left(t_{0}\right)/R\left(t\right)\right),$
(118)
where $y_{0}=\rho_{v\gamma 0}/\rho_{vm0}$ and
$\underline{\rho}_{vm}\left(t_{3}\right)=\underline{\rho}_{vm}\left(t_{0}\right)$
is considered. Let $\rho_{Vg}\left(t_{2}\right)=0,$ then considering the
conjecture 3, we have
$\rho_{Vg}\left(t_{2}\right)=K\left(t_{2}\right)=0,\text{\ \
}\frac{R\left(t_{3}\right)}{R\left(t_{2}\right)}=\left(x-1\right)/y_{3},\text{
\ or \
}\frac{R\left(t_{0}\right)}{R\left(t_{2}\right)}=\left(x-1\right)/y_{0}.$
(119)
Thus when $t>t_{2},$ $\rho_{Vg}\left(t\right)<0$ and $K\left(t\right)<0$. $x$
is invariant and $y$ is changeable. For example, we may take $y_{3}=1$ and
$y_{0}=1/6000$.
$\underline{\rho}_{vm}-\underline{\rho}_{sm}$ will also change because of
$\left(65c\right)$ and the reason in section 8.7.
### VIII.3 Space expands with a deceleration
From $\left(118\right)-\left(119\right)$ we see when $0<t<t_{2},$
$\rho_{Vg}\left(t\right)>0$ so that $K\left(\underline{\rho}_{g}\right)>0.$
From $\left(115\right),$ $\left(116b\right)$ and $\left(118\right)$ we have
$\displaystyle\overset{\cdot}{R}^{2}+K$
$\displaystyle=\eta\rho_{Vg}R^{2}=\eta\frac{R^{3}\left(t_{3}\right)}{R}\rho_{vm}\left(t_{3}\right)\left[\left(1-x\right)+\frac{y_{3}R\left(t_{3}\right)}{R}\right],$
(120a) $\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=-\frac{\eta}{2}\left(\rho_{Vg}+\rho_{v\gamma}\right)$
$\displaystyle=-\frac{\eta}{2}\frac{R^{3}\left(t_{3}\right)}{R^{2}}\rho_{vm}\left(t_{3}\right)\left[\left(1-x\right)+2\frac{y_{3}R\left(t_{3}\right)}{R}\right].$
(120b)
In the case, $\left(1-x\right)+y_{3}R\left(t_{3}\right)/R>0.$ Hence
$\left(1-x\right)+2y_{3}R\left(t_{3}\right)/R>0$ so that
$\overset{\cdot\cdot}{R}<0,$ i.e., space must expand with a deceleration in
the period. We analyze the process in detail as follows.
$1.$ It is possible that there is $t_{1}$ to satisfy
$\overset{\cdot}{R}\left(t_{1}\right)=0$ because $\overset{\cdot}{R}>0$ and
$\overset{\cdot\cdot}{R}<0$ when $t>t_{3}.$
When $\overset{\cdot}{R}\left(t\right)=0,$ $\left(58\right)-\left(59\right)$
reduces to $\left(63\right)$ in which $\rho_{g}=\rho_{Vg}$ and
$\rho_{g\gamma}=\rho_{vg\gamma}$ (because in the $V-breaking$ now). There are
two sorts of possibility when $t>t_{1}$.
A. $\rho_{Vg}\left(t\right)>0$ when $t\leq t_{1}$ and
$\overset{\cdot}{R}\left(t_{1}\right)=0,$ hence
$\overset{\cdot\cdot}{R}\left(t_{1}\right)\leq 0$ ($\rho_{g\gamma}>0$ as
well). Thus $R\left(t>t_{1}\right)$ will decrease as the conventional theory
(space contracts).
B. $\rho_{Vg}$ decreases because space expands. It is possible that there is
$t_{2}\leq t_{1\text{ }}$to satisfy $\rho_{Vg}\left(t_{2}\right)=0$ and
$\overset{\cdot}{R}\left(t_{2}\right)\geq 0.$ $K\left(t_{2}\right)=0$ because
of the conjecture 3. In the case, $\overset{\cdot\cdot}{K}/\overset{\cdot}{K}$
must be considered and $\left(58\right)-\left(59\right)$ reduces to
$\left(65\right)$ in which $\rho_{g\gamma}=\rho_{vg\gamma}.$ From
$\left(65\right)$ we see that there are two sorts of possibility when
$t>t_{2}.$
a. If $\overset{\cdot\cdot}{K}/\overset{\cdot}{K}=0,$ $\overset{\cdot}{R}=0$
and $\overset{\cdot\cdot}{R}=-\left(\eta/2\right)\rho_{v\gamma}<0.$ Hence
space will contract with an acceleration. In the case, $t_{2}=t_{1.\text{ }}$
b. If $\overset{\cdot\cdot}{K}/\overset{\cdot}{K}>0,$ $\overset{\cdot}{R}>0$
and $\overset{\cdot\cdot}{R}<0$ so that space continues to expands with a
deceleration.
In the case, $t_{2}<t_{1},$ $\overset{\cdot}{R}\left(t_{1}\right)=0,$
$\rho_{Vg}\left(t_{1}\right)<0$ and $K\left(t_{1}\right)<0$ because of
$\left(118\right)-\left(119\right)$ and
$R\left(t_{1}\right)>R\left(t_{2}\right).$ There are the following three sorts
of possibility, because of $\rho_{Vg}\left(t_{1}\right)<0$.
$\left(a\right).$
$\rho_{Vg}\left(t_{1}\right)+\rho_{vg\gamma}\left(t_{1}\right)>0,$ so
$\overset{\cdot\cdot}{R}\left(t_{1}\right)<0.$ Thus $R\left(t>t_{1}\right)$
will decrease due to $\overset{\cdot}{R}\left(t_{1}\right)=0$, i.e. space will
contract with an acceleration.
$\left(b\right).$
$\rho_{Vg}\left(t_{1}\right)+\rho_{vg\gamma}\left(t_{1}\right)<0,$ so
$\overset{\cdot\cdot}{R}\left(t_{1}\right)>0.$Thus $R\left(t>t_{1}\right)$
will continue to increase due to $\overset{\cdot}{R}\left(t_{1}\right)=0$,
i.e. space will expand with an acceleration.
$\left(c\right).$
$\rho_{Vg}\left(t_{1}\right)+\rho_{vg\gamma}\left(t_{1}\right)=0,$ so
$\overset{\cdot\cdot}{R}\left(t_{1}\right)=0.$Thus
$R\left(t>t_{1}\right)=R\left(t_{1}\right),$ i.e., space will be static.
In the case $\left(c\right)$, in fact,
$\rho_{Vg}=\rho_{Vg}\left(t,R\left(t\right)\right)$ which can still change
although $\overset{\cdot}{R}=0$. This is because the major part of the
repulsive potential energy between $s-matter$ and $v-matter$ will transform
into $s-energy$ when $v-galaxies$ form (see below). Consequently, after long
time, $\rho_{Vg}=\rho_{v}-\rho_{sm}$ will strikingly decrease so that space
will continue to expand with an acceleration.
The cases are different from the conventional theory. According to the
conventional theory, when $\rho_{Vg}\left(t\right)=K\left(t\right)=0,$ there
must be $\overset{\cdot}{R}=\overset{\cdot\cdot}{R}=0.$
There is no the case $\overset{\cdot\cdot}{K}/\overset{\cdot}{K}<0.$ This is
because if $\overset{\cdot\cdot}{K}/\overset{\cdot}{K}<0,$ there must be
$\overset{\cdot}{R}\left(t\right)<0.$ This is contradictory to the premise
$\overset{\cdot}{R}\left(t_{2}\right)<0$
$2\mathbf{.}$ The other possibility is when
$\overset{\cdot}{R}\left(t\right)>0,$ $\overset{\cdot\cdot}{R}$ have changed
from $\overset{\cdot\cdot}{R}<0$ in to $\overset{\cdot\cdot}{R}>0.$ In the
case, space will continue to expand with an acceleration, and there is no
stage in which $\overset{\cdot}{R}\left(t\right)=0.$
To sum up, we see that after inflation, space first expands with a
deceleration, then it is possible that space is static for a period and
finally expands with an acceleration up to now.
### VIII.4 Space expands with an acceleration
We consider the expanding process of space. After reheating, $V_{Vg}\sim 0.$
Based on the discussion about $\left(61\right)$ we may conclude when $t\gg
t_{3},$ say $t\geq t_{1},$ so that $R\left(t\right)\gg R\left(t_{3}\right),$
$\rho_{Vg}\left(t\right)\sim\rho_{vm}\left(t_{3}\right)\left(-x+1+yR\left(t_{3}\right)/R\left(t\right)\right)\ll
0,$
$\overset{\cdot}{K}\sim 0,$ $K<0$ and $K$ can be taken as $-1$. Thus, in the
$V-breaking$, from $\left(50\right)-\left(51\right)$ we have
$\displaystyle\overset{\cdot}{R}^{2}$
$\displaystyle=1+\eta\rho_{g}R^{2}=1-\eta\frac{C_{0}}{R}\left(x-1-\frac{y_{0}R_{0}}{R}\right)$
$\displaystyle\overset{\cdot}{a}^{2}$
$\displaystyle=H_{0}^{2}\left(1+\Omega_{g0}\right)\left[1-\frac{1}{\left(1+\Omega_{g0}\right)}\left(\frac{\Omega_{gm0}}{a}-\frac{\Omega_{v\gamma
0}}{a^{2}}\right)\right],$ (121a) $\displaystyle\overset{\cdot\cdot}{R}$
$\displaystyle=\frac{\eta}{2}\frac{C_{0}}{R^{2}}\left(x-1-2\frac{y_{0}R_{0}}{R}\right),\text{
\
}\overset{\cdot\cdot}{a}=\frac{H_{0}^{2}}{2}\left(\frac{\Omega_{gm0}}{a^{2}}-2\frac{\Omega_{v\gamma
0}}{a^{3}}\right),$ (121b)
where $\rho_{g}=\rho_{Vg}$ for simplicity, $C_{0}=\rho_{m0}R_{0}^{3},$
$a=R/R_{0},$
$\overset{\cdot}{a}_{0}^{2}=H_{0}^{2}=\eta\rho_{c}=\eta\rho_{gc},$
$\Omega_{gm0}=\left(\rho_{sm0}-\rho_{vm0}\right)/\rho_{c}=\left(x-1\right)\rho_{vm0}/\rho_{c},$
$\Omega_{v\gamma 0}=\rho_{v\gamma 0}/\rho_{c}=y_{0}\rho_{vm0}/\rho_{c},$
$\Omega_{g0}=\left(\rho_{sm0}-\rho_{vm0}-\rho_{v\gamma 0}\right)/\rho_{c},$
$H_{0}^{2}R_{0}^{2}=1/\left(1+\Omega g_{0}\right).$ It is obvious that when
$R_{0}/R<\left(x-1\right)/2y_{0},$ $\overset{\cdot}{R}>0$ and
$\overset{\cdot\cdot}{R}>0$, i.e., space will expand with an acceleration.
### VIII.5 To determine $a\left(t\right)$
It is difficult to uniformly describe the three evolving stages space since
$K\left(t\right)$ is difficultly determined. Considering $K=-1$ is applicable
to all cases $\rho_{g}>0,$ $\rho_{g}=0$, and $\rho_{g}<0,$ as a crude
approximation for $\rho_{g}>0$ $($thereby $K>0)$ and $\rho_{g}=0$ $($thereby
$K=0),$ we describe expansion of space by $\left(121a\right)$ in which $K=-1.$
In fact, when $\eta\rho_{g}R^{2}\gg 1,$ $K$ may be ignored. Because when
$\rho_{g}>0$ and $K>0,$ space expands with a deceleration, and when
$\rho_{g}<0$ and $K<0,$ space expands with an acceleration, the period of
$\rho_{g}=K=0$ can be approximately solved. The result is qualitatively
consistent with $\left(121a\right).$ As mentioned before, when $t\leq t_{1},$
$\overset{\cdot\cdot}{R}\left(t\right)<0$ so that space expands with a
deceleration. From $(121a)$ we have
$\displaystyle t$
$\displaystyle=t_{0}-\frac{1}{H_{0}\sqrt{1+\Omega_{g0}}}\\{\sqrt{1-M+\Gamma}-\sqrt{a^{2}-Ma+\Gamma}$
$\displaystyle+\frac{M}{2}\ln\frac{2-M+2\sqrt{1-M+\Gamma}}{2a-M+2\sqrt{a^{2}-Ma+\Gamma}}\\},$
(122)
where $M=\Omega_{gm0}/\left(1+\Omega_{g0}\right),$ $\Gamma=\Omega_{v\gamma
0}/\left(1+\Omega_{g0}\right).$
Taking $\Omega_{v\gamma 0}=0.001,$ $\Omega_{gm0}=0.3\Omega_{v\gamma
0}+2\sqrt{\Omega_{v\gamma 0}},$ $H_{0}^{-1}=9.7776\times 10^{9}h^{-1}yr^{[8]}$
and $h=0.8,$ we get $a\left(t\right).$ $a\left(t\right)$ is shown by the curve
$B$ in the figure 1 and describes evolution of the cosmos from $14\times
10^{9}yr$ ago to now. Taking $\Omega_{v\gamma 0}=0.05,$
$\Omega_{gm0}=2\sqrt{\Omega_{v\gamma 0}},$ we get the $a\left(t\right)$ which
is shown by the curve $A$ in the figure 1 and describes evolution of the
cosmos from $13.7\times 10^{9}yr$ ago to now. Provided $\Omega_{gm0}\leqslant
2\Omega_{v\gamma 0}+2\sqrt{\Omega_{v\gamma 0}}$ (the condition
$\overset{\cdot}{a}^{2}>0),$ we can get a curve of $a\left(t\right)$ which
describes evolution of the cosmological scale.
From the two curves we see that the cosmos must undergo a period in which the
cosmos expands with a deceleration in the past and present period in which the
cosmos expands with an acceleration.
Ignoring $\Omega_{v\gamma 0},$ $\Omega_{gm0}\longrightarrow-\Omega_{gm0}$ and
taking $a\sim 0$, we can reduce $(122)$ to the corresponding formula
$\left(3.44\right)$ in Ref. $[8]$
### VIII.6 The relation between redshift and luminosity distance
Considering $k=-1,$ from $(40)$ and $(120a)$ we have
$\mathop{\displaystyle\int}\nolimits_{a}^{1}\frac{cda}{R\overset{\cdot}{a}}=-\mathop{\displaystyle\int}\nolimits_{r}^{0}\frac{dr}{\sqrt{1+r^{2}}},$
(123) $\displaystyle H_{0}d_{L}$
$\displaystyle=H_{0}R_{0}r(1+z)=\frac{2c}{\left(\Omega_{gm0}-2\Omega_{v\gamma
0}\right)^{2}-4\Omega_{v\gamma 0}}\cdot$
$\displaystyle\\{2\left(1+\Omega_{g0}\right)-\left(1+z\right)\Omega_{gm0}$
$\displaystyle-\left[2\left(1+\Omega_{g0}\right)-\Omega_{gm0}\right]\sqrt{1-\left(\Omega_{gm0}-2\Omega_{v\gamma
0}\right)z+\Omega_{v\gamma 0}^{2}z^{2}}\\},$ (124)
where $z=\left(1/a\right)-1$ is the redshift caused by $R$ increasing.
Provided $\Omega_{gm0}\longrightarrow-\Omega_{gm0},$ $(124)$ is consistent
with the corresponding formula $\left(3.81\right)$ in Ref. $[8].$ Ignoring
$\Omega_{v\gamma 0},$ $\Omega_{gm0}\longrightarrow-\Omega_{gm0}$ we reduce
$(124)$ to
$H_{0}d_{L}=\frac{2c}{\Omega_{gm0}^{2}}\left\\{2+\Omega_{gm0}\left(1-z\right)-\left[2+\Omega_{gm0}\right]\sqrt{1-\Omega_{gm0}z}\right\\},$
(125a)
which is consistent with $\left(3.78\right)$ in Ref. $[8]$. Approximating to
$\Omega_{m0}^{1}$ and $z^{2}$, we obtain
$H_{0}d_{L}=z+\frac{1}{2}z^{2}\left(1+\frac{1}{2}\Omega_{gm0}\right).$ (125b)
Taking $\Omega_{v\gamma 0}=0.001,$ $\Omega_{gm0}=0.3\Omega_{v\gamma
0}+2\sqrt{\Omega_{v\gamma 0}}$ and $H_{0}^{-1}=9.7776\times
10^{9}h^{-1}yr^{[8]}$ and $h=0.8,$ from $\left(124\right)$ we get the
$d_{L}-z$ relation which is shown by the curve $A$ in the figure 2; Taking
$\Omega_{v\gamma 0}=0.05,$ $\Omega_{gm0}=2\sqrt{\Omega_{v\gamma 0}}$ we get
the $d_{L}-z$ relation which is shown by the curve $B$ in the figure 2.
Figure 1: The curve A describes evolution of $a(t)$ from $14\times 10^{9}yr$
ago to now; The curve B describes evolution of $a(t)$ from $13.7\times
10^{9}yr$ ago to now.
Figure 2: The curve $A$ describes the $d_{L}-z$ relation when $\Omega_{v\gamma
0}=0.001$and $\Omega_{m0}=0.3\Omega_{v\gamma 0}+2\sqrt{\Omega_{v\gamma 0}};$
The curve $B$ describes the $d_{L}-z$ relation when $\Omega_{v\gamma
0}=0.05$and $\Omega_{m0}=2\sqrt{\Omega_{v\gamma 0}}$
### VIII.7 Repulsive potential energy chiefly transforms into the kinetic
energy of SU(5) color single states
The repulsion potential energy between $v-matter$ and $s-matter$ is determined
by the distributing mode of $s-matter$ and $v-matter.$ In $V-breaking$
$v-particles$ with their masses can form $v-celestial$ bodies, but $s-matter$
which is $s-SU(5)$ color single states cannot form any dumpling and must
loosely distribute in space. Consequently, the huge repulsion potential energy
must chiefly transform into the kinetic energy of $s-SU(5)$ color single
states when the $v-celestial$ bodies form or space expands. In fact, when flat
space expands $N$ times, i.e., $R\longrightarrow NR,$ the repulsive-potential
energy density $V_{r}$ becomes $V_{r}/N$ and
$\bigtriangleup V_{r}=\left(1-1/N\right)V_{r}.$ (126)
Consider a system in flat space which is composed of a $v-body$ with its mass
$M$ and a $s-colour$ single state with its mass $m.$ It is easy to get the
rate $\bigtriangleup E_{m}/\bigtriangleup E_{M}$ for static $M$ and $m$ at the
initial moment..
$\frac{\bigtriangleup E_{m}}{\bigtriangleup E_{M}}=\frac{2M+\bigtriangleup
V_{r}}{2m+\bigtriangleup V_{r}}$ (127)
Because $M\gg m$, $\bigtriangleup E_{m}>\bigtriangleup E_{M}.$
Space expansion is not the necessary condition to transform repulsive
potential energy into kinetic energy. Supposing $\overset{\cdot}{R}=0$ and
some $v-matter$ gather to a region and forms a galaxy, $s-matter$ which is
initially in the region must be repulsed away from the region and increases
its kinetic energy. The repulsive potential energy chiefly transforms into the
kinetic energy of the $s-matter$ in this case as well.
## IX Existing and distributive forms of s-SU(5) color single states in the
V-breaking
When $T_{s}$ is low, the $s-SU(5)$ color single states must loosely distribute
in space or form $s-superclusterings$, i.e., huge voids relative to
$v-matter.$
### IX.1 Sorts of s-SU(5) color single states in the V-breaking
$\varpi_{s}=0$ and $\varpi_{v}\neq 0$ in the $V-breaking$. Thus the $s-SU(5)$
symmetry still holds water and $v-SU(5)\longrightarrow v-SU(3)\times U(1).$
Hence all $s-gauge$ particles and $s-fermions$ are massless (if the masses
originate from only the couplings of the $s-fermions$ with the $s-Higgs$
fields). When the temperature $T_{s}$ is high enough, all $s-gauge$ particles
and $s-fermions$ must exist in plasma form. When $T_{s}$ is low, all
$s-particles$ will exist in $s-SU(5)$ color-single state form (conjecture 4).
Let $A$, $B$, $C$, $D$, $E$ denote the 5 sorts of colors. A component of $10$
representation carries color $\alpha\beta$, $\alpha,$ $\beta=A$, $B$, $C$,
$D$, $E,$ $\alpha\neq\beta.$ A component of $5$ representation carries color
$\alpha.$ A gauge boson carries colors $\alpha\beta^{\ast}.$ There are the
following sorts of the $s-SU(5)$ color-single states.
2-fermion states: $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$,
$\alpha\neq\beta.$ When the spin of $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
is zero, we denote $\underline{\alpha}\underline{\alpha}^{\ast}$ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
by $s-\pi$. When the spin of $\alpha$$\alpha$∗ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
is 1, we denote $\underline{\alpha}\underline{\alpha}^{\ast}$ or
$\left(\underline{\alpha\beta}\right)\left(\underline{\alpha\beta}\right)^{\ast}$
by $s-\rho$. Analogously to $QCD,$ we have
$m\left(s-\pi\right)<m\left(s-\rho\right)$ (because of color-magnetic
interaction) and can suppose $m(s-\pi)\sim m(\pi)$ to be minimum in all
$s-SU(5)$ color-single states. In contrast with a given $\pi-meson,$ $s-\pi$
must be stable, because there is no the electroweak interaction in the case
$\varpi_{s}=0$.
3-fermion states: $\left(\underline{AB}\right)(\underline{CD})$$E$ or
$\left(\underline{AB}\right)\underline{A}^{\ast}\underline{B}^{\ast}.$ When
the spin of $3-fermion$ states is $1/2$, we denote them by $s-N$. $s-N$ is
stable, and we can suppose $m(s-N)\sim m(N).$ $N$ denotes a given nucleon.
4-fermion states: $\left(\underline{AB}\right)\underline{C}\underline{D}$$E$.
5-fermion states: $A$$B$$\underline{C}\underline{D}$$E$ or
$\left(\underline{AB}\right)(\underline{BC})\left(\underline{CD}\right)(\underline{DE})\left(\underline{EA}\right).$
Gauge boson single-states:
$\left(\underline{\alpha\beta^{\ast}}\right)\left(\underline{\alpha^{\ast}\beta}\right)$
or
$\left(\underline{\alpha\beta^{\ast}}\right)\left(\underline{\beta\gamma^{\ast}}\right)\left(\underline{\gamma\alpha^{\ast}}\right)$
etc., $\alpha\neq\beta\neq\gamma.$
Fermion-gauge boson single-states.
$\alpha^{\ast}$$\left(\underline{\alpha\beta^{\ast}}\right)$$\beta$,
$\alpha^{\ast}$$\left(\underline{\alpha\gamma^{\ast})(\gamma\beta^{\ast}}\right)$$\beta$
etc..
Of course, there are the $s-antiparticles$ corresponding to the $s-colour$
single states above as well.
### IX.2 The characters of the s-SU(5) color single states in V-breaking
We can only qualitatively discuss the $SU(5)$-color single states by comparing
the $SU(5)$-color single states with the $SU(3)$-color single states, because
there is no mature theory of strict $SU(5)$ symmetry up till now. The
following inferences are obtained on the basis of the sameness of the $SU(5)$
and $SU(3)$ groups, hence they are reliable.
The characters of the $s-SU(5)$ color single states in the $V-breaking$ are as
follows.
1\. If the masses of the $s-SU(5)$ fermions only originate from the coupling
of the $s-SU(5)$ fermions with the $s-SU(5)$ Higgs fields, they all must be
zero because $\langle\omega\rangle_{s0}=0$ in the $V-breaking$. If such mass
terms $m\overline{\psi}\psi+MTr\overline{\Psi}\Psi$ are added to $\mathcal{L}$
($\mathcal{L}_{V}$ and $\mathcal{L}_{S}$), the masses of the $s-SU(5)$
fermions are non-zero. Here $\psi$ and $\overline{\Psi}$ are respectively the
representations $\underline{5}$ and $10$. The masses of all $SU(3)$ color
single states (include gluon balls ) are non-zero. Consequently, we can
determine that all $SU(5)$-color single states can get their suitable masses.
2\. In contrast with $SU(3)$ color single states among which there are the
electromagnetic and weak interactions due to the gauge group is $SU(3)\times
U(1)$, there is no electroweak interaction among the $s-SU(5)$ color single
states because $s-SU(5)$ is a simple group. Hence the $s-SU(5)$ color single
states with the minimum masses must be stable.
3\. The interaction radius of the $s-SU(5)$ color single states must be finite
and much smaller ($\sim 1\times 10^{-15}m$) than the radius of hydrogen atoms
($\sim 1\times 10^{-10}m$). This is because the interactions can come into
being only by exchanging the $s-SU(5)$ color single states, and the masses of
all $s-SU(5)$ color single states are non-zero.
Thus we can approximately regard the $s-SU(5)$ color single states as ideal
gas without collision.
In a word, the $s-SU(5)$ color single states are analogous to $v-neutrinos$,
but in contrast with the $v-neutrinos$, there is the repulsion between the
$s-SU(5)$ color single states and $v-matter.$
From the characters of the $s-SU(5)$ color single states mentioned above, we
will see that the $s-SU(5)$ color single states in the $V-breaking$ cannot
form clusters and must loosely distribute in space, but it is possible they
form $s-superclusterings$ as the neutrinos.
## X Dynamics of v-structure formation and the distributive form of the
s-SU(5) color single states
Generalizing equations governing nonrelativistic fluid motion[8] to present
model, we have
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v}\mathbf{\cdot\nabla}\right)\mathbf{v}_{v}=-\frac{\nabla
p_{v}}{\rho_{v}}-\nabla\Phi,$ (128) $\frac{\partial}{\partial
t}\rho_{v}+\nabla\cdot\left(\rho_{v}\mathbf{v}_{v}\right)=0$ (129)
$\nabla^{2}\Phi=4\pi G\left(\rho_{v}-\rho_{s}\right),$ (130)
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s}\mathbf{\cdot\nabla}\right)\mathbf{v}_{s}=-\frac{\nabla
p_{s}}{\rho_{s}}+\nabla\Phi,$ (131) $\frac{\partial}{\partial
t}\rho_{s}+\nabla\cdot\left(\rho_{s}\mathbf{v}_{s}\right)=0,$ (132)
in the $V-breaking,$ where $\partial/\partial
t+\mathbf{v}_{v}\mathbf{\cdot\nabla}$ is call the convective derivative. We
can produce the linearized equations of motion by collecting terms of first
order in perturbations about a homogeneous background
$\rho_{v}=\rho_{v0}+\delta\rho_{v}$ etc. Let
$\displaystyle\mathbf{v}_{v}$
$\displaystyle=\mathbf{v}_{v0}+\delta\mathbf{v}_{v},\quad\ \ \ \
\mathbf{v}_{s}=\mathbf{v}_{s0}+\delta\mathbf{v}_{s}^{\prime}$ (133a)
$\displaystyle\rho_{v}$ $\displaystyle=\rho_{v0}+\delta\rho_{v},\quad\ \ \ \
\rho_{s}=\rho_{s0}+\delta\rho_{s},$ (133b) $\displaystyle\delta_{v}$
$\displaystyle=\frac{\delta\rho_{v}}{\rho_{v0}},\quad\ \ \ \
\delta_{s}=\frac{\delta\rho_{s}}{\rho_{s0}},$ (133c)
considering
$\mathbf{v}_{v0}=H\mathbf{x=}\left(\overset{\cdot}{a}/a\right)\mathbf{x,}$ we
get
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\mathbf{\cdot\nabla}\right)\delta\mathbf{v}_{v}=-\frac{\nabla\delta
p_{v}}{\rho_{v0}}-\nabla\delta\Phi-\left(\delta\mathbf{v}_{v}\cdot\nabla\right)\mathbf{v}_{v0},$
(134) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\cdot\nabla\right)\delta_{v}=-\nabla\delta\mathbf{v}_{v},$
(135) $\nabla^{2}\Phi=4\pi
G\left(\rho_{v0}\delta_{v}-\rho_{s0}\delta_{s}\right),$ (136)
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\mathbf{\cdot\nabla}\right)\delta\mathbf{v}_{s}=-\frac{\nabla\delta
p_{s}}{\rho_{s0}}+\nabla\delta\Phi-\left(\delta\mathbf{v}_{s}\cdot\nabla\right)\mathbf{v}_{s0},$
(137) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\cdot\nabla\right)\delta_{s}=-\nabla\delta\mathbf{v}_{s},$
(138)
Defining the comoving spatial coordinates
$\mathbf{x}(t)=a\left(t\right)\mathbf{r}\left(t\right),\quad\ \ \ \
\delta\mathbf{v}\left(t\right)=a\left(t\right)\mathbf{u}\left(t\right),$ (139)
we have $\nabla_{x}=\nabla_{r}/a.$ Let
$\displaystyle\delta_{vk}$
$\displaystyle=\delta_{vk}\left(t\right)\exp\left(-i\mathbf{k}_{v}\cdot\mathbf{r}\right),\quad\
\ \ \ \
\delta_{sk}=\delta_{sk}\left(t\right)\exp\left(-i\mathbf{k}_{s}\cdot\mathbf{r}\right),$
(140a) $\displaystyle c_{v}^{2}$ $\displaystyle=\frac{\partial
p_{v}}{\partial\rho_{v}},\quad\ \ \ \ \ \ c_{s}^{2}=\frac{\partial
p_{s}}{\partial\rho_{s}},$ (140b)
from $\left(134\right)-\left(140\right)$ can get
$\left(\frac{\partial}{\partial
t}+\mathbf{v}_{v0}\mathbf{\cdot\nabla}\right)^{2}\delta_{vk}+2\frac{\overset{\cdot}{a}}{a}\overset{\cdot}{\delta}_{vk}=4\pi
G\left(\rho_{v0}\delta_{vk}-\rho_{s0}\delta_{sk}\right)-\frac{c_{v}^{2}k_{v}^{2}}{a^{2}}\delta_{vk}.$
(141) $\left(\frac{\partial}{\partial
t}+\mathbf{v}_{s0}\mathbf{\cdot\nabla}\right)^{2}\delta_{sk}+2\frac{\overset{\cdot}{a}}{a}\overset{\cdot}{\delta}_{sk}=4\pi
G\left(\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}\right)-\frac{c_{s}^{2}k_{s}^{2}}{a^{2}}\delta_{sk}.$
(142)
It is necessary that $\delta_{sk}<0$ when $\delta_{vk}>0$ because there is
only repulsion between $s-matter$ and $v-matter.$ Consequently,
$\displaystyle\rho_{v0}\delta_{vk}-\rho_{s0}\delta_{sk}$
$\displaystyle=\rho_{v0}\delta_{vk}+\rho_{s0}\mid\delta_{sk}\mid,$ (143a)
$\displaystyle\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}$
$\displaystyle=\rho_{s0}\delta_{sk}+\rho_{v0}\mid\delta_{vk}\mid.$ (143b)
According to the present model, there must be
$\overset{\cdot}{a}/a=\left(-K/a^{2}+\eta\rho_{g}\right)^{1/2}\sim 0,$ because
$\rho_{v}=\rho_{vm}+\rho_{v\gamma},$ $\rho_{vm}\varpropto R^{-3},$
$\rho_{v\gamma}\varpropto R^{-4},$ $\rho_{s}=\rho_{sm}>\rho_{vm},$ and
$\rho_{sm}\varpropto R^{-3}$ so that $\rho_{g}=\rho_{v}-\rho_{s}=0$ and $K=0.$
There possibly is $\delta_{vk}\left(t\right)>\delta_{sk}\left(t\right)$ when
$\rho_{v}=\rho_{s}$, because $s-SU(5)$ color single states in the $V-breaking$
can be regarded as ideal gas without collision. The ideal gas has the effect
of free flux damping for clustering. Ignoring $\delta_{sk}$ in
$\left(141\right),$ for $\overset{\cdot}{a}/a\sim 0$ and $\left(4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}\right)>0,$ from $\left(141\right)$ we get
$\overset{\cdot\cdot}{\delta}_{vk}\left(t\right)=\left(4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}\right)\delta_{vk},\quad\ \ \
\delta_{vk}\left(t\right)=\exp\left(t/\tau\right)\ ,$ (144)
where $k_{v}^{\prime}=k_{v}/a,$ $\tau=1/\sqrt{4\pi
G\rho_{v0}-c_{v}^{2}k_{v}^{\prime 2}}.$ We see that
$\delta_{vk}\left(t\right)$ will exponentially grow for long-wavelength. We
cannot get the result as $\left(144\right)$ for $\delta_{sk}$ from
$\left(142\right)$, because the velocity $v_{s0}$ of a $s-SU(5)$ color single
state is invariant because there is no collision and is very big. Let the
duration in which $\overset{\cdot}{R}\sim 0$ be $\triangle t,$ the distance
$l$ to be damped out is $l=v_{s0}\triangle t.$ The perturbation whose size is
less than $l$ cannot form. Thus $s-SU(5)$ color single states can only form
superclusterings. When $\rho_{s0}>\rho_{v0},$
$\rho_{s0}\delta_{sk}-\rho_{v0}\delta_{vk}>0$ is possible. Consequently, the
$s-SU(5)$ color single states possibly form superclusterings, because when
$\left(\overset{\cdot}{a}/a\right)$ is large, the perturbation will slowly
grow in power rules.
From the above mentioned, we see that the $s-SU(5)$ color single states must
loosely distribute in space or form $s-superclustering$, i.e., huge $v-voids$,
in which $\rho_{s}\gg\rho_{v}.$
To sum up, because of the following two reasons, the perturbation
$\delta_{vk}$ will grow faster or earlier than that determined by the
conventional theory.
1\. There is such a stage in which $\rho_{s}-\rho_{v}\sim 0$ and
$K\left(\rho\right)\sim 0$ so that
$\overset{\cdot}{R}/R=\overset{\cdot}{a}\sim 0,$ because the gravitational
masses of $s-matter$ and $v-matter$ are opposite and $K\left(\rho\right)$ is
changeable. Consequently, from $\left(141\right)$ and $\left(144\right)$ we
see that $\delta_{vk}\left(t\right)$ will exponentially grow for long-
wavelength in the stage.
2\. From $\left(141\right),$ $\left(143\right)$ and $\left(144\right)$ we see
that $\delta_{vk}\left(t\right)$ can grow faster than that determined by
$\left(144\right)$ because
$\rho_{v0}\delta_{vk}+\rho_{s0}\mid\delta_{sk}\mid>\rho_{v0}\delta_{vk}.$ The
origin is the repulsion between $s-matter$ and $v-matter$.
## XI Some guesses, new predictions and an inference
### XI.1 Some guesses
#### XI.1.1 The universe is composed of infinite s-cosmic islands and
v-cosmic islands
If the whole universe is the world in the $S-breaking$ or the the world the
-$V-breaking$, the analysis above is correct. But according to the present
model, as mentioned before, there are the two sorts of symmetry breaking which
are different in essence. Thus it is possible that there are different
breaking forms in different regions of the universe.
As mentioned above, $v-matter$ and $s-matter$ are symmetric and mutually
repulsive, $v-matter$ in the $V-breaking$ can form $v-galaxies$ and $s-matter$
in the $S-breaking$ can form $s-galaxies$.
From this we present a new cosmic model as follows.
The universe is composed of infinite $s-cosmic$ islands and and $v-cosmic$
islands. $\langle\omega_{v}\rangle=0$ and
$\langle\omega_{s}\rangle=\langle\omega_{s}\rangle_{0}$ in $s-cosmic$ islands,
and $\langle\omega_{s}\rangle=0$ and
$\langle\omega_{v}\rangle=\langle\omega_{v}\rangle_{0}$ in the the $v-cosmic$
islands.
A $s-cosmic$ island or a $v-cosmic$ island must be finite. There must be a
transitional region ($T-region$) between a $s-cosmic$ island and a $v-cosmic$
island. In the $T-region,$ it is necessary $\langle\omega_{s}\rangle$ and
$\langle\omega_{v}\rangle$ change from
$\langle\omega_{s}\rangle=\langle\omega_{s}\rangle_{0}$ and
$\langle\omega_{v}\rangle=0$ into $\langle\omega_{s}\rangle=0$ and
$\langle\omega_{v}\rangle=\langle\omega_{v}\rangle_{0},$ respectively.
Consequently, the expectation values $\langle\omega_{s}\rangle_{T}$ and
$\langle\omega_{v}\rangle_{T}$ inside the $T-region$ must satisfy
$0<\mid\langle\omega_{s}\rangle_{T}\mid<\mid\langle\omega_{s}\rangle_{0}\mid,\text{
\ \
}0<\mid\langle\omega_{v}\rangle_{T}\mid<\mid\langle\omega_{v}\rangle_{0}\mid.$
(145)
There must be only $v-cosmic$ islands neighboring a $s-cosmic$ island. This is
because that if two $s-cosmic$ islands are neighboring, they must form one new
larger $s-cosmic$ island.
It is obvious that if there is only one sort of breaking, it is impossible
that such cosmic islands exist.
Based on the following reasons, the probability is very little that a
$v-observer$ accepts messages from a $s-cosmic$ island.
1. The probability must be very small that a $s-particle$ (a quark, a lepton or a photon) in the $s-cosmic$ island comes into the $v-cosmic$ island, because a $s-particle$ in a $s-cosmic$ island is $s-SU(5)$ non-color single state. If a $s-particle$ comes into the $v-cosmic$ island, it would still be non-color single state and would get very big mass. This is impossible due to color confinement. But a bound state of the $s-particles,$ e.g. $\left(u\overline{u}\mp d\overline{d}\right)/\sqrt{2}$ which is a color single state in both $V-breaking$ and $S-breaking,$ possibly comes into the $v-cosmic$ island. It is hardly funded by a $v-observer$ because the boson $\left(u\overline{u}\mp d\overline{d}\right)/\sqrt{2}$ is a $s-colour$ single state in the $v-cosmic$ island which is a particle of dark energy. .
2\. The probability must be very small that a $v-particle$ in the $s-cosmic$
island come into the $v-cosmic$ island as well. The $v-particle$ (a fermion or
a gauge boson) in the $s-cosmic$ island must be massless. If the massless
$v-particle$ comes into the $v-cosmic$ island, it would get its mass. Thus its
static mass will change from $m_{0}=0$ to $m_{0}>0$ so that it must suffer a
strong-repulsive interaction, hence it hardly comes into the $v-cosmic$
island.
3\. Higgs particles in the $s-cosmic$ island must decay fast into fermions or
gauge bosons, hence they cannot come to the $v-cosmic$ island.
4\. $T-regions$ is so big that the probability through which a particle passes
is very small..
The probability must be very small that particles leave the $v-cosmic$ island
because of the same reasons.
A $v-cosmic$ islands and a $s-cosmic$ island can influence on each other by
the Higgs potential in the $T-region$ between both.
As a consequence a $v-observer$ in the $v-cosmic$ island can regard the
$v-cosmic$ island as the whole cosmos. It is possible that Some cosmic islands
are forming or expanding, and the other cosmic islands are contracting.
Thus, according to the present model the cosmos as a whole is infinite and its
properties are always unchanging, and there is no starting point or end of
time.
#### XI.1.2 Mass redshifts
Hydrogen spectrum is
$\omega_{nk}=(E_{n}-E_{k})/\hbar=-\frac{\mu
e^{4}}{2\hbar^{3}}(\frac{1}{n^{2}}-\frac{1}{k^{2}}),\text{ \ \
}\mu=\frac{mM}{m+M},$ (146)
where $m$ is the mass of an electron, and $M$ is the mass of a proton.
According the unified model, $m\propto\upsilon_{e}$, the mass of a quark
$m_{q}\propto\upsilon_{q},$ where $\upsilon_{e}$ and $\upsilon_{q}$ are the
expectation values of the Higgs fields coupling with the electron and the
quark, respectively. $M\propto m_{q}.$
If there are some galaxies inside a $T-region,$ from $\left(145\right)$ we see
that the mass $m_{T}$ of an electron and the mass $M_{T}$ of a proton inside
the $T-region$ must be
$m_{T}<m,\text{ \ \ }M_{T}<M.$
Thus we have
$\mu_{T}<\mu,\text{ \ \
}\bigtriangleup\omega_{nk}=\omega_{nk}-\omega_{nkT}=-\frac{(\mu-\mu_{T})e^{4}}{2\hbar^{3}}(\frac{1}{n^{2}}-\frac{1}{k^{2}})<0.$
(147)
The sort of red-shifts is called $mass$ $redshift$. The mass redshift is
essentially different from the cosmological red-shift mentioned before. Thus,
the photons coming from the star in a $T-region$ must have larger red-shift
than that determined by the Hubble formula at the same distance. Thereby we
guess that some quasars are just the galaxies in the $T-region$ of our cosmos
island and they have the mass redshifts. The fine-structure constant is
considered to change based on the redshifts of some quasars according to the
conventional theory. In contrast with the conventional, we consider that the
mass of electrons possibly changes, but is not the fine-structure constant to
change.
An ordinary $s-galaxy$ and a $s-quasar$ can be neighboring, because a
$T-region$ must be neighboring to an ordinary region.
### XI.2 New predictions
#### XI.2.1 It is possible that huge voids are equivalent to huge concave
lenses. The densities of hydrogen and helium in the huge voids predicted by
the present model must be more less than that predicted by the conventional
theory.
Hot dark matter, e.g. the neutrinos, can form huge voids, but cannot explain
evolution of structure with middle and small scales. Cold dark matter can
explain evolution of structure with the middle and the small scales, but
cannot explain the huge voids. The problem of huge voids remains unsettled.
According the present model, huge $v-voids$ in the $V-breaking$ are, in fact,
superclusterings of $s-particles$. The huge $v-voids$ not empty, and in which
there is $s-matter$ with a bigger density,
$\rho_{s}^{\prime}\gg\rho_{v}^{\prime},$ $\rho_{s}^{\prime}>\rho_{s}$ and
$\rho_{v}^{\prime}<\rho_{v},$ here $\rho_{s}^{\prime}$ and $\rho_{v}^{\prime}$
denote the densities in the huge $v-voids$. Because there is the repulsion
between $s-matter$ and $v-matter$ and there is the gravity among
$s-particles,$ a huge $v-void$ can form. The forming process is analogous to
that of the neutrino-superclusterings. The characters of such a huge $v-void$
are as follows.
$A$. A $v-void$ must be huge, because there is no other interaction among the
$s-SU(5)$ color single states except the gravity and the masses of the
$s-SU(5)$ color single states are very small.
$B$. When $v-photons$ pass through such a huge $v-void$, the $v-photons$ must
suffer repulsion from $\rho_{s}^{\prime}$ and are scattered by
$\rho_{s}^{\prime}$ as they pass through a huge concave lens. Consequently,
the galaxies behind the huge $v-void$ seem to be darker and more remote.
$C$. Both density of matter and density of dark matter in huge voids must be
more lower than those predicted by the conventional theory. Specifically, the
density of hydrogen in the huge voids must be more less than that predicted by
the conventional theory. Right or mistake of the predict can be confirmed by
the observation of distribution of hydrogen.
It is seen that the present model can well explain the characters of some huge
voids. This is a decisive prediction which distinguishes the present model
from other models.
#### XI.2.2 The gravitation between two galaxies distant enough will be less
than that predicted by the conventional theory.
There must be $s-matter$ between two $v-galaxies$ distant enough, hence the
gravitation between the two $v-galaxies$ must be less than that predicted by
the conventional theory due to the repulsion between $s-matter$ and
$v-matter$. When the distance between the two $v-galaxies$ is small, the
gravitation is not influenced by $s-matter,$ because $\rho_{s}$ must be small
when $\rho_{v}$ is big.
#### XI.2.3 A black hole with its mass and density big enough will transform
into a white hole
Let there be a $v-black$ hole with its mass and density to be big enough so
that the critical temperature $T_{cr}$ can be reached in the $V-breaking.$ If
its mass is so big that its temperature $T_{v}\gtrsim 2\mu/\sqrt{\lambda}$
since the black hole contracts by its self-gravitation, then the expectation
values of the Higgs fields inside the $v-black$ hole will change from
$\varpi_{v}=\varpi_{v0}$ and $\varpi_{s}=0$ into $\varpi_{v}=\varpi_{s}=0$.
Consequently, inflation must occur. After inflation, the highest symmetry will
transit into the $V-breaking$ or the $S-breaking$. No matter which breaking
appears, the energy of the black hole must transform into both $v-energy$ and
$s-energy$. Thus, a $v-observer$ will find that the black hole disappears and
a white hole appears.
In the process, part of $v-energy$ transforms into $s-energy.$ A $v-observer$
will consider the energy not to be conservational because he cannot detect
$s-matter$ except by repulsion. The transformation of black holes is different
from the Hawking radiation. This is the transformation of the vacuum
expectation values of the Higgs fields. There is no contradiction between the
transformation and the Hawking radiation or another quantum effect, because
both describe different processes and based on different conditions (the
density and mass of the black hole must be large enough so that its
temperature $T_{v}\gtrsim 2\mu/\sqrt{\lambda}$). According to the present
model, there still are the Hawking radiation or other quantum effects of black
holes. In fact, the universe is just a huge black hold. The universe can
transform from the $S-breaking$ into the $V-breaking$ because of its
contraction or expansion. This transformation is not quantum effects.
Let there be a $v-cosmic$ island neighboring on a $s-cosmic$ island. It is
possible that the $v-cosmic$ island transforms into a $s-cosmic$ island after
it contracted. In this case, a $s-observer$ in the $s-cosmic$ island must
observe a very huge white hole.
#### XI.2.4 The transformation of the cosmic ultimate
As mentioned before, in the $V-breaking$, when $\rho_{s}>\rho_{v}$, space will
expand with an acceleration. It seems that the universe will always expand
with an acceleration. This is impossible. In the expanding process of the
universe, galaxies can be continue to exist and $v-matter$ will gather so that
a huge $v-black$ hole with its mass and density big enough can be formed after
long enough time, because the repulsion between s-matter and v-matter and the
gravitation among $v-particles$. After the temperature of the black hole
reaches the critical temperature $T_{cr}$, the expectation values of all Higgs
fields will tend to zero and the local space will expand. In the case, both
$V-breaking$ and $s-breaking$ are possibly realized. If the density of
$v-matter$ around the huge $v-black$ hole is little enough, the $S-breaking$
will is realized. Consequently the $V-breaking$ transforms into the
$S-breaking$ and the $v-world$ transforms the $s-world$. If the density of
$v-matter$ around the huge $v-black$ hole is very large, the $V-breaking$ will
is realized. This is because the transformation of symmetry breaking must
cause the transformation of existing form of matter. $V-matter$ whose
temperature is low around the huge $v-black$ hole will prevent the
transformation of the $V-breaking$ into the $S-breaking.$ After the average
density of v-matter is little because space expands and a huge $v-black$ hole
with its mass and density big enough formed, the transformation of the
$V-breaking$ into the $S-breaking$ can occur.
It is seen that in the both cases that the universe is little enough because
space contracts or is large enough because space expands, it will occur that
one sort of breaking transforms into the other and the world transforms the
other world.
### XI.3 An inference :$\lambda_{eff}=\lambda=0,$ although $\rho_{0}\neq 0$
The effective cosmological constant $\lambda_{eff}=\lambda+\rho_{g0}.$ The
conventional theory can explain evolution with a small $\lambda_{eff}.$ Such a
small $\lambda_{eff}$ cannot be derived from a elementary theory. According to
the conventional quantum field theory, $\lambda_{eff}=\lambda+\rho_{0},$
$\rho_{0}\ggg\lambda_{eff},$ In fact, $\rho_{0}$ is divergence. According to
the conventional gravitational theory, $\rho=\rho_{g}=\rho$ ($c=1$).
Consequently, the issue of the cosmological constant appears.
$\rho_{0}=0$ can be obtained by some supersymmetric model, but it is not a
necessary result. On the other hand, the particles predicted by the
supersymmetric theory have not been found, although their masses are not
large.
$\rho_{0}=0$ is a necessary result of our quantum field theory without
divergence${}^{[6]}.$ In this theory, $\rho_{0}=0$ is naturally obtained
without normal order of operators, there is no divergence of loop corrections,
and dumpling dark matter is predicted${}^{[7]}.$
According to the present model, $\lambda_{eff}=\lambda=0,$ although $\rho_{0}$
is still very big according to the conventional quantum field theory.
Proof: $\lambda_{eff}=\lambda=0,$ although $\rho_{0}\neq 0.$
Applying the conventional quantum field theory to the present model, we have
$\rho_{0}=\rho_{s0}+\rho_{v0}.$ Both $\rho_{s0}$ and $\rho_{v0}$ must be two
constants. According to the conjecture 1, $s-particles$ and $v-particles$ are
strictly symmetric in essence. Hence
$\rho_{s0}=\rho_{v0}=\rho_{0}/2.$
According to the conjecture 1, the gravitational mass of $s-matter$ is
opposite to that of $v-matter$. i.e., $\rho_{gs}=-\rho_{gv}$ when
$\rho_{s}=\rho_{v}$. Hence we have
$\displaystyle\rho_{0}$ $\displaystyle=\rho_{s0}+\rho_{v0}=2\rho_{s0}\neq 0,$
(148a) $\displaystyle\rho_{g0}$ $\displaystyle=\rho_{sg0}+\rho_{vg0}=0.$
(148b)
Thus, there is no the fine tuning problem, even if $\lambda_{eff}\neq 0$.
$\lambda_{eff}=0$ is a necessary inference, because evolution of the cosmos
can be explained by the present model without $\lambda_{eff}$. Consequently,
although $\rho_{0}\neq 0,$ we have still
$\lambda_{eff}=0=\lambda+\rho_{g0}=\lambda.$ (149)
This is an direct inference of the present model, and independent of a quantum
field theory. Thus, the cosmological constant issue has been solved.
For the vacuum state in the $S-breaking$ or the $V-breaking$, the Einstein
field equation is reduced to
$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-8\pi
G\left(T_{s0\mu\nu}-T_{v0\mu\nu}\right)=-8\pi
G\left(T_{v0\mu\nu}-T_{s0\mu\nu}\right)=0.$ (150)
This is a reasonable result.
## XII Primordial nucleosynthesis
### XII.1 F-W dark matter model
The $F-W$ dark matter model[7] is a necessary inference of the quantum field
theory without divergence. The $F-W$ dark matter model is similar with the
mirror dark matter model.
According to the mirror dark matter model, it is impossible that the density
of matter is equal to that of dark matter in order to explain the primordial
nucleosynthesis and $CMBR$. This is too difficultly understood, because matter
and mirror matter are symmetric and both can transform from one into another
when temperature is high enough.
In contrast with the mirror dark matter model, according to the $F-W$ matter
model, $F-matter$ (ordinary matter) and $W-matter$ (dark matter) are not only
symmetric, but also $\rho_{vF}=\rho_{vW}.$ If the total density of matter and
dark energy is $\rho_{t}$ and the ratio of the density of dark energy
$\rho_{de}$ to $\rho_{t\text{ }}$ is $0.73$,
$\left(\rho_{vF}+\rho_{vW}\right)/\rho_{t}=2\rho_{vF}/\rho_{t}=0.27.$
Considering only $F-baryon$ matter is visible and the radio of the density
$\rho_{vFB}$ to $\rho_{t}$ is $\rho_{vFB}/\rho_{t}=0.04,$ dark matter can be
classified into the following three sorts: invisible $F-matter$ for a time
whose density is $\rho_{vFu}=\left(0.27/2-0.4\right)\rho_{t}=0.095\rho_{t},$
invisible $W-baryon$ matter whose density is $\rho_{vWB}=0.04\rho_{t},$ and
invisible $W-non-baryon$ matter whose density is $\rho_{vWu}=0.095\rho_{t}.$
$\rho_{vWB}$ can form dark galaxies and can be observed, $\rho_{vWu}$ and
$\rho_{vFu}$ cannot form any dumpling. The $vFu-particles$ is possibly
observed in future. $\rho_{vF}$ and $\rho_{vW}$ can transform from one into
another when temperature is high enough.
According to the present cosmological model, there are $s-matter$ and
$v-matter$ which are symmetric in principle. After symmetry spontaneously
breaking, $s-matter$ and $v-matter$ are no longer symmetric. In the
$V-breaking$, $v-matter$ corresponds visible matter and dark matter,
$s-matter$ corresponds to so-called dark energy and must exist in the form of
the $s-SU(5)$ color single states. The masses of all color single states are
non-zero. But the masses are different from each other. Some masses of color
single states (their density are denoted by $\rho_{sl}$) are possibly less
than $1Mev$ so that they may be ignored when temperature $T\gtrsim 1Mev,$ and
the others (their density are denoted by $\rho_{sm}$) are larger than $1Mev$.
$\rho_{sm}$ and $\rho_{sl}$ can be determined based on observation.
When the $F-W$ dark matter model and the present cosmological model are
simultaneously considered, the primordial nucleosynthesis and CMBR can be
explained.
### XII.2 Primordial nucleosynthesis
According to the $F-W$ matter model[7] which is similar with the mirror dark
matter model, the mechanism of primordial nucleosynthesis is the same as the
conventional theory. But the mechanism of space expansion of the present model
is different from that of the conventional theory. For short, we consider only
influence of space expansion on the primordial nucleosynthesis and $CMBR.$
The primordial helium abundance $Y_{4}$ is determined by $n_{n}/n_{p}^{[8]},$
$\displaystyle Y_{4}$
$\displaystyle=2/\left[1+\left(n_{n}/n_{p}\right)^{-1}\right],$ (151a)
$\displaystyle n_{n}/n_{p}$ $\displaystyle=exp\left(-\bigtriangleup
m/kT_{1}\right),\text{ }\bigtriangleup m=m_{n}-m_{p},$ (151b)
where $n_{n}/n_{p}$ is the neutron-proton ratio in the unit comoving volume at
the freeze-out temperature $T_{1}.$ $T_{1}$ is determined by
$\Gamma_{1}=\Gamma\left(T_{1}\right)$ and
$H_{1}=\eta\rho_{g}\left(T_{1}\right),$ here .$\Gamma$ is the interaction rate
experienced by a particle.
As mentioned above,
$\rho_{vFm}=\rho_{vWm},\text{ \ }\rho_{vF\gamma}=\rho_{vW\gamma},\text{\
}\rho_{v}=2\rho_{vFm}+2\rho_{vF\gamma},\text{ \
}\rho_{s}=\rho_{sm}=\rho_{sm}^{\prime}+\rho_{sl},$
and in the $V-breaking,$ $T_{1}=T_{v1}$ and in general $T_{v1}\neq T_{s1}.$
Since the masses of all $s-color$ single states are non-zero, it is possible
there are some $s-color$ single states with their masses $m_{sm}\gtrsim 1Mev$
and the others with their masses $m_{sl}$ and $1Mev>m_{sl}>1ev.$ Thus, when
$T_{s}\gtrsim 1Mev,$ $m_{sl}$ may be ignored so that $\rho_{sl}\varpropto
T_{s}^{4},$ and when $T_{s}\lesssim 1ev$,
$\rho_{s}=\rho_{sm}^{\prime}+\rho_{sl}\varpropto T_{s}^{3}.$
Considering $m_{p}\sim m_{n}\sim 1Gev,$ $m_{e}=0.511Mev,$ $m_{\gamma}=0$ and
$m_{\nu i},$ $i=e,$ $\mu$ and $\tau$, are regarded as zero, $g_{e}=7/2,$
$g_{\gamma}=2$ and $g_{\nu i}=7/4$ so that $g^{\ast}=10.75.$ When $T\sim
T_{v1}\sim T_{s1}\sim 1Mev,$ $m_{e}$ may be ignored and the universe is
dominated by $\rho_{sl}$ and $\rho_{s\gamma},$ we have
$\displaystyle H_{1}^{2}$
$\displaystyle=\frac{\overset{\cdot}{R}^{2}\left(T_{1}\right)}{R^{2}\left(T_{1}\right)}=\eta\rho_{g}\left(T_{1}\right)=\eta\left[\rho_{v}\left(T_{v1}\right)-\rho_{s}\left(T_{s1}\right)\right]$
$\displaystyle=\eta\left[\left(\rho_{vFm}+\rho_{vWm}+\rho_{vF\gamma}+\rho_{vW\gamma}\left(T_{v1}\right)\right)-\left(\rho_{sm}^{\prime}+\rho_{sl}\left(T_{s1}\right)\right)\right]$
$\displaystyle\simeq\eta\left[\rho_{vF\gamma}+\left(\rho_{vW\gamma}\left(T_{v1}\right)-\rho_{sl}\left(T_{s1}\right)\right)\right]$
$\displaystyle=\eta\left[\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}+\left(\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}-\rho_{sl}\left(T_{s1}\right)\right)\right].$
(152) $\rho_{sl}\left(T_{s}\right)/\rho_{sl}\left(T_{s1}\right)\varpropto
T_{s}^{4}/T_{s1}^{4}=\left(R_{1}/R\right)^{4}=T_{v}^{4}/T_{v1}^{4},\text{ \
when }T_{v}\gg T_{vdec},$ (153)
where $T_{vdec}$ is the $v-photon$ decoupling temperature (see the following).
$\rho_{sl}$ and $\rho_{sm}^{\prime}$ are two parameters which should be
determined by observations. $\rho_{sl}\left(T_{s1}\right)$ can be so chosen
that
$\displaystyle\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}-\rho_{sl}\left(T_{s1}\right)$
$\displaystyle\sim 0,$ (154a) $\displaystyle H_{1}^{2}$
$\displaystyle\sim\eta\frac{\pi^{2}}{30}g^{\ast}T_{v1}^{4}.$ (154b)
For the freeze-out temperature $T_{v1}$ is determined by $\Gamma_{1}$ and
$H_{1}$. $\Gamma_{1}$ in the present model is the same as that in the
conventional theory. $\left(154b\right)$ is the same as that in the
conventional theory as well. Hence the present model can explain the
primordial nucleosynthesis and $Y_{4}$ as the conventional theory. For
example, taking the rough approximation $\Gamma_{1}=H_{1},$ considering
$\left(154b\right),$ we get the equation to determine $T_{v1}$
$\Gamma_{1}\left(T_{v1}\right)\sim
G_{F}^{2}T_{v1}^{5}=H_{1}=\left[\eta\left(\rho_{v}\left(T_{v1}\right)-\rho_{s}\left(T_{s1}\right)\right)\right]^{1/2}\simeq\left(\eta\frac{\pi^{2}}{30}g_{v}^{\ast}T_{v1}^{4}\right)^{1/2}.$
(155)
This result and $n_{n}/n_{p}$ and $Y_{4}$ corresponding to this are the same
as those of the conventional theory[19].
It is seen that although ordinary matter and dark matter are completely
symmetric so that $\rho_{vF}=\rho_{vW}$, we can still obtained the result of
conventional theory, provided the F-W dark matter model and the present model
are simultaneously considered. This is different from the mirror dark matter
model.
## XIII Cosmic microwave background radiation ($CMBR)$
### XIII.1 The recombination temperature $T_{rec}$
It is the same as the conventional theory that there are the inflation and big
bang processes in the present model. Hence there must be the cosmic microwave
background radiation $(CMBR).$
The recombination temperature of the present model is the same as that of the
conventional theory, because it is independent of s-matter in the V-breaking.
From the following formulas[19] we can determine the recombination temperature
$T_{rec},$
$\frac{1-\chi}{\chi^{2}}=1.1\times 10^{-8}\xi
T_{v}^{3/2}\exp\left(B/T_{v}\right),\text{ \ }A\equiv T_{v}/Tv0.$
where $\chi=n_{e}/n=n_{p}/n,$ $n=n_{p}+n_{H}$ to be number density and
$B=13.6ev$ is the ionization potential of hydrogen. Taking $\xi\sim 5\times
10^{-10}$ and $\chi=0.1,$ considering $n_{e}=n_{p}$,
$n_{\gamma}=\left(\zeta\left(3\right)/\pi^{2}\right)g_{\gamma}T^{3}=\left(2.4/\pi^{2}\right)T^{3},$
$n=\xi n_{\gamma},$ $T_{v}=Tv0\left(T_{v}/Tv0\right)$ and $Tv0=2.35\times
10^{-4}ev,$ we have[19]
$\displaystyle T_{rec}$ $\displaystyle=3423.5K=0.295ev.$ (156a)
$\displaystyle\left(1+z_{rec}\right)$ $\displaystyle=T_{rec}/Tv0=1255.$ (156b)
### XIII.2 The temperature $T_{eq}$ of matter-radiation equality
In contrast with the conventional theory, according to the $F-W$ dark matter
model, $F-matter$ and $W-matter$ are completely symmetric so that not only
there are $F-photon$ (ordinary photons), but also $W-photons$ (dark-matter
photons). $\rho_{v\gamma}=\rho_{vF\gamma}+\rho_{vW\gamma}=2\rho_{vF\gamma},$
$\rho_{vm}=\rho_{vFm}+\rho_{vWm}=2\rho_{vFm}.$ From this we can estimate the
temperature $T_{eq}$ of matter-radiation equality as follows.
When only the photons and the three species of neutrinos are considered (here
the three species of neutrinos are regarded as massless)., we have[18]
$\rho_{vF\gamma}=\rho_{vW\gamma}=\frac{\pi^{2}}{30}g_{\gamma}^{\ast}\left(\frac{kT_{v}}{\hbar
c}\right)^{4}.$ (157a) $g_{\gamma}^{\ast}=2+\frac{7}{8}\times
6\times\left(\frac{4}{11}\right)^{4/3}=3.36.$ (157b)
Thus, considering $T_{v0}=T_{0}=2.728K=2.35\times 10^{-4}ev,$ we get
$\displaystyle\rho_{vr0}$
$\displaystyle=2\rho_{vFr0}=2\times\left[\frac{3.36}{2}\times\frac{\pi^{2}}{30}\times
2\times\left(2.35\times 10^{-13}Gev\right)^{4}\right]$
$\displaystyle=6.7425\times 10^{-51}Gev^{4}.$ (158)
Observation shows that the total density of matter and dark matter is
$\rho_{0}=\Omega_{0}\rho_{c}=0.27\rho_{c}.$ According to the $F-W$ model, this
implies
$\displaystyle\rho_{v0}$
$\displaystyle=2\rho_{vFm0}+2\rho_{vFr0}\simeq\rho_{vm0}=\Omega_{0}\rho_{c}$
$\displaystyle=1.8789\times 10^{-26}h^{2}\Omega_{0}\cdot kg\cdot
m^{-3}=9.238\times 10^{-48}Gev^{4},\text{ \ when }h=0.65.$ (159)
where $h=0.5-0.8,$ $\rho_{vm0}=\rho_{vFm0}+\rho_{vWm0},$ and$\
\rho_{vr0}=\rho_{vF\gamma 0}+\rho_{vWr0}=2\rho_{vF\gamma 0}=2\rho_{r0}.$
From $\left(158\right)-\left(159\right)$ $T_{veq}$ can be determined as
follows
$\displaystyle\rho_{vmeq}\left(T_{veq}\right)$
$\displaystyle=\rho_{vm0}\left[R_{0}/R_{eq}\left(T_{veq}\right)\right]^{3}=\rho_{v\gamma
0}\left[R_{0}/R_{eq}\left(T_{veq}\right)\right]^{4}=\rho_{v\gamma
eq}\left(T_{veq}\right),\text{ }$ (160a) $\displaystyle\frac{R_{0}}{R_{eq}}$
$\displaystyle=\frac{T_{veq}}{T_{v0}}=\frac{\rho_{vm0}}{\rho_{v\gamma
0}}=\frac{\Omega_{0}\rho_{c}}{2\rho_{vFr0}}=\left(1+z_{eq}\right)=1370,\text{
}$ $\displaystyle T_{veq}$ $\displaystyle=0.32ev,\text{ when }h=0.65,$ (160b)
$\displaystyle\frac{R_{0}}{R_{eq}}$
$\displaystyle=\frac{T_{veq}}{T_{v0}}=\frac{\rho_{vm0}}{\rho_{v\gamma
0}}=1272\text{, }$ $\displaystyle T_{veq}$
$\displaystyle=0.295ev=T_{rec},\text{ when }h=0.624.$ (160c)
According to the present model, $\left(160\right)$ is a crude approximation,
$\rho_{v}-\rho_{s}$ is changeable not only because of $\rho_{vm}\varpropto
R^{-3},$ $\rho_{v\gamma}\varpropto R^{-4}$ and $\rho_{sm}\varpropto R^{-3},$
but also because of $\left(65c\right).$
According to the conventional theory, $\rho_{\gamma 0}=\rho_{vr0}/2$,
$\rho_{0}=\rho_{v0},$ hence if $h=0.65$ and $\Omega_{0}=0.27$,
$T_{eq}^{\prime}=T_{0}R_{0}/R_{eq}^{\prime}=\left(1+z_{eq}^{\prime}\right)T_{0}=\left(\rho_{m0}/\rho_{r0}\right)T_{0}=2\left(\rho_{m0}/\rho_{vr0}\right)T_{0}=0.64ev=2T_{veq}.$
(161)
It is seen that $T_{eq}^{\prime}$ is remarkably different from $T_{veq}.$ When
$T_{v}\sim T_{veq},$ the universe is not matter-dominated.
### XIII.3 Decoupling temperature
Let when $T_{v}=T_{veq},$ $T_{s}=T_{sq},$
$\rho_{sm}\left(T_{sq}\right)=t_{mq}\rho_{vm}\left(T_{veq}\right)$ and the
masses of all s-color single states cannot be ignored, i.e.
$\rho_{sm}\left(T_{sq}\right)=\rho_{s}\left(T_{sq}\right).$ Considering
$\frac{T_{sq}}{T_{s}}=\frac{R}{R_{eq}}=\frac{T_{veq}}{T_{v}},$ (162)
ignoring the transformation $\rho_{v}$ into $\rho_{s},$ we can rewrite
$\rho_{vm},$ $\rho_{v\gamma}$ and $\rho_{sm}$ as follows.
$\displaystyle\rho_{vm}$
$\displaystyle=\frac{\rho_{vm}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=\frac{R_{0}^{3}}{R^{3}}\Omega_{m0}\rho_{c}=A^{3}\Omega_{m0}\rho_{c},\text{
}A\equiv\frac{Tv}{Tv0}=1+z,$ (163a) $\displaystyle\rho_{v\gamma}$
$\displaystyle=\frac{\rho_{v\gamma}}{\rho_{v\gamma eq}}\frac{\rho_{v\gamma
eq}}{\rho_{vmeq}}\frac{\rho_{vmeq}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=\left(\frac{T_{v0}}{Tveq}\right)A^{4}\Omega_{m0}\rho_{c},$
(163b) $\displaystyle\rho_{sm}$
$\displaystyle=\frac{\rho_{sm}}{\rho_{sm}q}\frac{\rho_{sm}q}{\rho_{vme}q}\frac{\rho_{vmeq}}{\rho_{vm0}}\frac{\rho_{vm0}}{\rho_{c}}\rho_{c}=t_{mq}A^{3}\Omega_{m0}\rho_{c},$
(163c)
Ignoring $K,$ considering $H_{0}=\sqrt{\eta\rho_{c}}=65km\cdot\left(s\cdot
Mpc\right)^{-1}=1.4\times 10^{-42}Gev,$ $\Omega_{vm0}=0.27,$ and
$T_{v0}/Tveq=\left(2.35\times 10^{-4}\right)/0.32=0.734\times 10^{-3},$we get
$\displaystyle H^{2}$
$\displaystyle=\eta\rho_{g}=\eta\left(\rho_{vm}+\rho_{v\gamma}-\rho_{sm}\right)$
$\displaystyle=0.27\times 1.4^{2}\times
10^{-82}A^{3}\left(1-t_{mq}+0.734\times 10^{-3}A\right)Gev.$ (164)
This is the only deference between the present model and the conventional
theory in order to determine the decoupling temperature.
We have the same interaction rate experienced by one photon and Saha equation
as the conventional theory[19]
$\Gamma=n_{vFe}\sigma_{th}=\chi\xi
n_{vF\gamma}\sigma_{th}=\frac{2.4}{\pi^{2}}\xi T_{v0}^{3}\sigma_{th}\chi
A^{3}=5.4\times 10^{-36}\xi\chi A^{3}Gev,$ (165)
$\frac{1-\chi}{\chi^{2}}=1.1\times 10^{-8}\xi
T_{v}^{3/2}\exp\frac{13.6}{T_{v}}=3.96\times 10^{-14}\xi
A^{3/2}\exp\frac{57872}{A},$ (166)
where $T_{v}=T_{v0}\left(T_{v}/T_{v0}\right)=2.35\times 10^{-4}A$ is
considered. Only for comparison of the present model with the conventional
theory, we use the same equation $\left(166\right)$ and the same crude
approximation $\Gamma=H$ to evaluate the decoupling temperature. Taking
$\Gamma=H$, and $t_{mq}=1.5,$ from $\left(164\right)-\left(165\right)$ we have
$\displaystyle\chi A^{3/2}$ $\displaystyle=\xi^{-1}1.347\times
10^{-7}\left(1-t_{mq}+0.734\times 10^{-3}A\right)^{1/2}$
$\displaystyle=269\left(-0.5+0.734\times 10^{-3}A\right)^{1/2}.,\text{ \ when
\ }\xi=5\times 10^{-10},$ (167)
Substituting $\left(167\right)$ into $\left(166\right)$, we get
$\displaystyle A^{3/2}$ $\displaystyle=\xi^{-1}1.347\times
10^{-7}\left(1-t_{mq}+0.734\times 10^{-3}A\right)^{1/2}$
$\displaystyle+\xi^{-1}7.185\times 10^{-28}\times\left(1-t_{mq}+0.734\times
10^{-3}A\right)\times\exp\frac{57872}{A}$ (168)
Taking $t_{mq}=1.5$ and $\xi=5\times 10^{-10}$, we get
$A=1+z_{dec}=1097,\text{ \ \ }T_{vdec}=0.258ev,\text{ \ }\chi=0.004.$ (169)
Taking $t_{mq}=1.5$ and $\xi=3\times 10^{-10}$, we get
$A=1+z_{dec}=1108,\text{ \ }T_{vdec}=0.260ev,\text{ \ }\chi=0.0068,$ (170)
Taking place of $\left(164\right)$ by the equation in the conventional theory
$H^{2}=\eta\rho_{g}=\eta\rho_{vm}=0.27\times 1.4^{2}\times 10^{-82}A^{3}Gev,$
(171)
we have
$\displaystyle A$ $\displaystyle=1+z_{dec}=1121,\text{ \
}T_{vdec}=0.263ev,\text{ \ }\chi=0.007,$ $\displaystyle\text{when }t_{mq}$
$\displaystyle=1.5\text{, \ }\xi=5\times 10^{-10},$ (172) $\displaystyle A$
$\displaystyle=1+z_{dec}=1132,\text{ \ }T_{vdec}=0.266ev,\text{ \
}\chi=0.012,$ $\displaystyle\text{when }t_{mq}$ $\displaystyle=1.5\text{, \
}\xi=3\times 10^{-10}.$ (173)
$z_{dec}$ is not susceptible for change of $t_{mq}$ in the scope $1.1-1.7.$
Considering
$\frac{\rho_{vmdec}}{\rho_{vmeq}}=\left(\frac{T_{vdec}}{T_{veq}}\right)^{3},\text{
\ \ }\frac{\rho_{v\gamma dec}}{\rho_{v\gamma
eq}}=\left(\frac{T_{vdec}}{T_{veq}}\right)^{4},\text{ \ \
}\rho_{vmeq}=\rho_{v\gamma eq},$ (174)
we have
$\frac{\rho_{v\gamma
dec}}{\rho_{vmdec}}=\frac{T_{vdec}}{T_{veq}}=\frac{0.258}{0.32}=0.81.$ (175)
It is seen from $\left(175\right)$ that in the decoupling stage,
$\rho_{vmdec}\sim\rho_{v\gamma dec}$ and the universe is not matter-dominated.
This is different from the conventional theory.
### XIII.4 Space-time is open, i. e. $K<0$.
The first peak of the $CBMR$ power spectra is the evidence of existence of the
elementary wave. The elementary wave began at reheating ($T=T_{reh}$) and
ended at recombination after $3.8\times 10^{5}$ years ($T=T_{rec}$) according
to the conventional theory. Let the temperature of reheating is $T_{reh}$. In
the period $T_{reh}$ descends into $T_{rec},$ baryons exist in plasma. The
sound speed of plasma is $c_{s}=\partial p/\partial\rho=\sqrt{5T_{b}/3m_{p}}.$
Let the period in which $T_{reh}$ descends into $T_{rec}$ is $\bigtriangleup
t_{hc}$ according to the present model and that according to the conventional
theory is $\bigtriangleup t_{hc}^{\prime}=3.8\times 10^{5}a,$ there must be
$\bigtriangleup t_{hc}>\bigtriangleup t_{hc}^{\prime}.$ (176)
The reasons are as follows.
That $\left(152\right)$ holds implies
$\displaystyle\rho_{g}\left(T_{v1}\right)$
$\displaystyle=\rho_{vFm}\left(T_{v1}\right)+\rho_{vWm}\left(T_{v1}\right)+\rho_{vFr}\left(T_{v1}\right)-\rho_{sm}\left(T_{s1}\right)+\left(\rho_{vWr}\left(T_{v1}\right)-\rho_{sl}\left(T_{s1}\right)\right)$
$\displaystyle\sim\rho_{vFm}\left(T_{v1}\right)+\rho_{vWm}\left(T_{v1}\right)+\rho_{vFr}\left(T_{v1}\right)-\rho_{sm}\left(T_{s1}\right)\sim\rho_{vF\gamma}\left(T_{1}\right)$
$\displaystyle\sim\rho_{m}\left(T_{1}\right)+\rho_{\gamma}\left(T_{1}\right)\sim\rho_{\gamma}\left(T_{1}\right)=\rho_{g}\left(T_{1}\right),$
(177)
where $\rho_{m}\left(T\right)$ and $\rho_{\gamma}\left(T\right)$ are the mass
density in the conventional theory $(T=T_{v}),$ and
$\rho_{m}\left(T_{1}\right)\ll\rho_{\gamma}\left(T_{1}\right)$ etc are
considered. It is necessary that when $T_{v}\geq T_{v1},$ $\left(152\right)$
or $\left(177\right)$ still holds. This is because
$\rho_{sl}\left(T_{s}\right)\varpropto R^{-4},$
$\rho_{vF\gamma}\left(T_{v}\right)=\rho_{vW\gamma}\left(T_{v}\right)\varpropto
R^{-4}$ and $\rho_{v\gamma}\left(T_{v}\right)\gg\rho_{vm}\left(T_{v}\right)$
and $\rho_{sl}\left(T_{s}\right)\gg\rho_{sm}\left(T_{s}\right)$ when
$T_{v}\geq T_{v1}.$ When $T_{v}<T_{v1},$ it is necessary that
$\rho_{vW\gamma}\left(T_{v}\right)<\rho_{sl}\left(T_{s}\right)$. This is
because
$\rho_{vF\gamma}\left(T_{v}\right)=\rho_{vW\gamma}\left(T_{v}\right)\varpropto
R^{-4}$ still holds, but $\rho_{sl}\left(T_{s}\right)\varpropto R^{-3}$ (due
to $1Mev>m_{sl}\gtrsim 1ev$) when $T_{v}<T_{v1}.$ Hence when $T_{v}<T_{v1},$
$H^{2}\left(T_{v}\right)=\eta\rho_{g}\left(T_{v}\right)<\eta\left(\rho_{m}\left(T\right)+\rho_{\gamma}\left(T\right)\right)=\eta\rho_{g}^{\prime}\left(T\right)=H^{\prime
2}\left(T\right).$ (178)
This implies $\left(176\right)$ to hold. On the other hand, the sound speed
$c_{vs}=\partial p_{v}/\partial\rho_{v}$ is determined by only $p_{v}$ and
$\rho_{v}$ or $T_{v}$ and $m_{p},$ and is independent of $\rho_{s}.$ Hence
$c_{vs}\sim c_{s}^{\prime}$ when $T_{reh}$ descends into $T_{rec}$, here
$c_{s}^{\prime}$ is the sound speed in the conventional theory when
$T_{reh}\geq T\geq T_{rec}$. Thus when temperature descends from $T_{reh}$
descends into $T_{rec},$ the propagating distance of the elementary sound wave
must be longer according to the present model than that according to the
conventional theory.
Based on $\bigtriangleup t_{hc}^{\prime}=3.8\times 10^{5}a,$ space-time is
flat or $K=0$ as the conventional theory, but based on $\bigtriangleup
t_{hc}>\bigtriangleup t_{hc}^{\prime}$ and $c_{s}=c_{s}^{\prime}=\partial
p/\partial\rho,$ space-time is open or $K<0$ as the the present model. This is
consistent with the present model according which $K<0$ when
$\rho_{g}=\rho_{v}-\rho_{s}<0$ in present stage. We will discuss the issue in
detail in the following paper.
## XIV Conclusions
The new conjectures are proposed that there are $s-matter$ and $v-matter$
which are symmetric, whose gravitational mass densities are opposite to each
other and whose energies are all positive. Both can transform from one to
another when temperature $T\gtrsim T_{cr}$. Consequently there is no
singularity in the model and the cosmological constant
$\lambda=\lambda_{eff}=0$ is determined although the energy density of the
vacuum state is still large and there is no the fine tuning problem, even if
$\lambda_{eff}\neq 0$.
There are two sorts of breaking modes, i.e. the $S-breaking$ and the
$V-breaking$. In the $V-breaking$ $v-SU(5)$ symmetry is broken into
$v-SU(3)\times U\left(1\right)$ for $v-particles$ and $s-SU(5)$ symmetry is
still strictly kept for $s-particles.$ Consequently $v-particles$ get their
masses determined by the $SU(5)$ $GUT$ and form the $v-galaxies$ etc., while
$s-particles$ are massless and form $s-SU(5)$ color-single states which
loosely distribute in space and can cause space to expand with an
acceleration. In contrast with the dark energy, the gravitational masses of
$s-matter$ is negative in the $V-breaking$.
The conjectures are not in contradiction with all given experiments and
astronomical observations up to now, although the conjecture 1 violates the
equivalence principle.
The curvature factor $K$ in the RW metric is regarded as a function of the
gravitational mass density in the comoving coordinates.
Based on the present model, the space evolving process is as follows. Firstly,
in the $S-breaking$, $\rho_{Sg}=\rho_{s}-\rho_{v}>0$ and $K>0,$ hence space
contracts and $T_{s}$ rises. When $T_{v}\sim T_{s}=T_{cr}$,
$\langle\omega_{s}\rangle=\langle\omega_{v}\rangle=0,$ both $v-SU(5)$ and
$s-SU(5)$ symmetries is strictly kept (this state have the highest symmetry),
and the masses of all particles originating from the couplings with the Higgs
fields are zero so that $\rho_{s}$ and $\rho_{v}$ can transform from one into
another. As a consequence $\rho_{s}=\rho_{v},$ $T_{v}=T_{s}$ and
$\widetilde{\rho}_{Sg}=\rho_{s}-\rho_{v}+V_{0}=V_{0}$ so that inflation must
occur. After the inflation, the phase transition of the vacuum, i.e. the
reheating process, occurs. After the reheating process, this state with the
highest symmetry transits to the state with the $V-breaking$. Space in the
$V-breaking$ have three evolving stages. Space firstly expands with a
deceleration because $\rho_{Vg}=\rho_{v}-\rho_{s}>0$ and $K>0$; then comes to
static because $\rho_{Vg}=0$ and $K=0$; and finally expands with an
acceleration up to now because $\rho_{Vg}<0$ and $K<0$. The results above is
still valid when $V\rightleftarrows S$ and $v\rightleftarrows s.$ It is seen
that the world in the $S-breaking$ and the world in the $V-breaking$ can
transform from one into another. The evolving process of the cosmos is
different from that determined by the conventional theory. $w=p/\rho$ changes
from $w>0$ into $w<-1$ is obtained.
There are the critical temperature $T_{cr}$, the highest temperature
$T_{\max}$, the least scale $R_{\min}$ and the largest energy density
$\rho_{\max}$ in the universe. $R_{\min}$ and $T_{cr}$ are two new important
constants, $T_{\max}$ and $\rho_{\max}$ are determined by
$R\left(T_{cr}\right)$.
A formula is derived which well describes the relation between a luminosity
distance and its redshift.
Generalizing equations governing nonrelativistic fluid motion to the present
model, the equations of $v-structure$ formation have been derived. According
to the equations, galaxies formed earlier and more easily than the
conventional theory.
Two guesses have been presented. The universe is composed of infinite
$s-cosmic$ islands and $v-cosmic$ islands. Some huge redshifts (e.g. the big
redshifts of quasi-stellar objects) are explained as the mass redshifts which
is caused by $m_{eT}<m_{e},$ here $m_{eT}$ is the mass of an electron in a
transiting region and $m_{e}$ is the given mass of an electron in the normal
region.
Three new predicts have been given. Huge $v-voids$ in the $V-breaking$ are not
empty, but are superclusterings of $s-particles$, there must be $s-matter$
with its density $\rho_{s}\gg\rho_{v}$ in the $v-voids$. It is possible that
huge voids is equivalent to a huge concave lens. The density of hydrogen and
the density of helium in the huge voids predicted by the present model must be
more less than that predicted by the conventional theory. It is possible that
a $v-black$ hole with its big enough mass and density can transform into a
huge white hole by its self-gravitation.
The primordial nucleosynthesis and $CMBR$ are explained based on the $F-W$
dark matter model (or the mirror dark matter model) and this cosmological
model.
The first peak of the $CBMR$ power spectra is the evidence of existence of the
elementary wave. The elementary wave began at reheating and ended at
recombination after $\bigtriangleup t_{hc}^{\prime}\equiv 3.8\times 10^{5}$
years according to the conventional theory. But according to the present
model, it is necessary that $\bigtriangleup t_{hc}>\bigtriangleup
t_{hc}^{\prime},$ because the sound speed $c_{s}\sim c_{s}^{\prime}$ and
$H=\eta\rho_{g}<H^{\prime}=\eta\rho_{g}^{\prime}=\eta\rho^{\prime}.$
Consequently, space-time is open or $K<0$ according to the present model
###### Acknowledgement 5
I am very grateful to professor Zhao Zhan-yue and professor Wu Zhao-yan for
their helpful discussions and best support.
## References
* (1) S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambidge University Press, (1999) p7, p98, p101, p137, p256-298.
* (2) R.R., Caldwell, Phys. World 17, (2004) 37; T, Padmanabhan, Phys. Rep. 380 (2003) 325; P.J.E Peebles and B. Ratra, Rev. Mod. Phys. 75 (2003) 559.
* (3) S. Weinberg, Phys. Rev. Lett. 59 (1987) 2607; H. Martel, P.R. Shapiro and S. Weinberg, Astrophys.J 492 (1998) 29.
* (4) P.J.E. Peebles and B. Ratra, Astrophys. J.325 (1988) L17; B Ratra and P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; P.J.E., Peebles and B. Ratra, Rev. Mod. Phys. 75 (2003) 559.
* (5) L.J, Hall, Y. Nomura, and S.J. Oliver, Phys. Rev. Lett. 95 (2005) 141302.
* (6) S-H. Chen, ‘Quantum Field Theory Without Divergence A’, (2002a) hep-th/0203220; S-H, Chen, ‘Significance of Negative Energy State in Quantum Field Theory A’ (2002b) hep-th/0203230; S-H, Chen, ‘Quantum Field Theory :New Research’, O. Kovras Editor, Nova Science Publishers, Inc. (2005a) p103-170.
* (7) S-H, Chen, ‘A Possible Candidate for Dark Matter’, (2001) hep-th/0103234; S-H, Chen, ‘Progress in Dark Matter Research’ Editor: J. Val Blain, (2005b) pp.65-72. Nova Science Publishers, Inc. arXiv: 1001.4221.
* (8) J. A, Peacock, Cosmological Physics, Cambridge University Press, (1999) p579, p458, (3.44) in p78, (3.81) in p90, (3.78) in p89, p460-464, p664, 296 (9.81).
* (9) S-H Chen, ‘Discussion of a Possible Universal Model without Singularity’, (2009) arXiv. 0908.1495; S-H, Chen, ‘A Possible Universal Model without Singularity and its Explanation for Evolution of the Universe’, (2006) hep-th/0611283.
* (10) G. W. Gibbons, and S. W. Hawking, Phys. Rev D, 15, (1977) 2752\.
* (11) M Chaichian and N.F.Nelipa, ‘Introduction to Gauge Field Theories’ Springer-Verlag Berlin Heidelberg, (1984) p269; G. G. Ross, ‘Grand Unified Theories’, The Benjamin/Cummings Publishing Company, INC, (1984) p177-183.
* (12) S. Weinberg, Gravitation and Cosmology, New York, Wiley (1972) Chanter 12 section 3.
* (13) M. Carmeli, ‘Classical Fields: General Relativity and Gauge Theory’, World Scientific Publishing, (1982) p89-92, (3.2.1), (3.2.7), (3.2.8), (3.2.9), (3.2.1), (3.2.15), (3.2.16)-(3.2.18).
* (14) S. Coleman and E.J.Weiberg Phys. Rev.D7. (1973) 1888; R.H. Brandenberg, Rev. of Mod. Phys. 57. (1985)1.
* (15) Liu Liao, Jiang Yuanfang and Qian Zhenhua, Progress in Physics, V.9, No. 2, (1989) p159.
* (16) H.C. Ohanian, and R. Ruffini, Gravitation and Spacetime (2nd ed.), W.W. Norton and Company, Inc. (1994) Section 9.9.
* (17) A. H. Guth, Phys. Rev. D 23, (1981) 347.
* (18) A.R. Liddle and D.H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press 2000, p20 (2.27).
* (19) Yu Yunqiang, Lectures in Cosmological Physics, Peking University Press, 2002, p151, (6.21), p170-172..
|
arxiv-papers
| 2009-08-11T11:23:57 |
2024-09-04T02:49:04.558238
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shi-Hao Chen",
"submitter": "Shihao Chen",
"url": "https://arxiv.org/abs/0908.1495"
}
|
0908.1644
|
# On the gravitational origin of the Pioneer Anomaly
Siutsou I. A siutsou@icranet.org ICRANet, Pescara, Italy
65122, p-le della Repubblica, 10 ICRA and Dip. di Fisica, Universita di Roma
“Sapienza”, Roma, Italy
00185, p-le A. Moro 5 Stepanov Institute of Physics, NASB, Minsk, Belarus
220072, Nezalezhnastsi Av., 68 Tomilchik L. M Stepanov Institute of Physics,
NASB, Minsk, Belarus
220072, Nezalezhnastsi Av., 68, Tel.: +375-29-4040838
###### Abstract
From Doppler tracking data and data on circular motion of astronomical objects
we obtain a metric of the Pioneer Anomaly. The metric resolves the issue of
manifest absence of anomaly acceleration in orbits of the outer planets and
extra-Pluto objects of the Solar system. However, it turns out that the
energy-momentum tensor of matter, which generates such a gravitational field
in GR, violates energy dominance conditions. At the same time the equation of
state derived from the energy-momentum tensor is that of dark energy with
$w=-1/3$. So the model proposed must be carefully studied by ”Grand-Fit”
investigations.
###### pacs:
04.25.Nx and 04.80.Cc and 04.50.Kd
## I Introduction
Spacecrafts Pioneer 10 and 11 were launched in the early 1970’s for the
exploration of outer planets of the Solar system (see the special issue of
Science 183, No. 4122, 25 January 1974, especially Soberman1974 ; Anderson1974
). After the encounters with Jupiter and Saturn they followed hyperbolic
trajectories on leaving the Solar system. Because of rotational spin-
stabilization of these spacecrafts, which reduces the need for manoeuvres,
they represent unique experiments for testing our understanding of celestial
mechanics. The accuracy of acceleration measurements for the Pioneer
spacecrafts is about $10^{-10}$ m/s2 PioneerMissionPlan ; Scherer1997 .
During the flight spacecrafts were continuously tracked by Doppler effect on
retransmitted radio signals. Then data were fitted to theoretical ones
obtained in PPN approximation initially by ODP program of JPL (JPL’s Export
Planetary Ephemeris DE405 was used for planet motion).
But surprisingly above 10 a.u. of heliocentric distance the systematical
deviation of experimental and theoretical data was found Pioneer1998 . This
deviation can be described simply as a constant acceleration towards the Sun
with magnitude of about $8\cdot 10^{-10}$ m/s2. This value is the same —
within error limits — for all the spaceships Pioneer 10 and 11, Galileo and
Ulysses and for all distances from the Sun Pioneer2001 .
This coincidence has been interpreted as a hint of the gravitational — metric
— origin of the acceleration. But at the same time there are no signatures of
such an acceleration in the orbits of outer planets and other objects in the
Solar system. Inclusion of such acceleration leads to unavoidable deviations
from the observed planet positions Pioneer2006 ; Iorio2007 .
Many attempts to explain the anomaly were made during last 10 years. Some of
the recent work on this subject includes analyses of: the thermal radiation of
the Pioneers ISI:000263816800017 ; ISI:000261214100012 , the gravitational
attraction by the Kuiper Belt ISI:000238999400005 ; ISI:000239223200009 ;
ISI:000232936700007 , the cosmological origin of the Anomaly
ISI:000255093800006 ; ISI:000242327900022 ; ISI:000255524300012 ;
ISI:000258636700088 ; ISI:000236229700005 ; ISI:000246464000017 , the
influence of multipole moments of the Sun 2005AIPC..758..129Q , the clocks
acceleration ISI:000227551100006 and many proposes of modified gravity
ISI:000238120300014 ; ISI:000254557000003 ; ISI:000259935000020 ;
ISI:000257329300072 including even laboratory investigations on very small
acceleration dynamics ISI:000245691400011 and interesting endeavors to
constrain some parameters of modified gravity theories by the known value of
the Pioneer Anomaly ISI:000248810500006 ; ISI:000262356900001 .
In the frame of metric theories of gravitation there is an attractive
possibility to explain the Pioneer Anomaly by metric perturbation, preserving
at the same time the character of planet motion. It is possible because the
Pioneers’ trajectories are very different from planet orbits: spacecrafts
leave the Solar system almost in a radial direction, while planets orbit the
Sun almost circularly. The potential possibility of such an approach was noted
independently in a series of manuscripts 2005CQGra..22.2135J ;
2006CQGra..23..777J ; 2006CQGra..23.7561J , but authors of these works didn’t
analyze the origin of the metric. Our approach is closer to that of Kjell
Tangen Tangen2007 , but with more rigor because we don’t neglect perturbation
of the space components of metric.
The goal of paper is to find the static space-time metric close to
Schwarzschild one, in which radial motion of test bodies shows the Pioneer
Anomaly, but circular motion doesn’t. For this purpose in section II we
develop and discuss an algorithm of metric determination from data on radial
and circular motions. Metric determination does not make use of Einstein
equations so it is applicable to any pure metric theory of gravity in
terminology of Will et. al. Will1993 ; Will2006 . Then we apply the method in
the case of the Pioneer Anomaly, starting from the Schwarzschild space-time
and recover the properties of matter forming such a metric within GR (section
III). Finally some concluding remarks are made.
## II Space-time determination from radial and circular motions
### II.1 Metric choice and time definition
We begin with the interval with 3 metric functions $\tau$, $\rho$ and $\sigma$
(for simplicity it was taken $c=1$)
$\displaystyle
ds^{2}=e^{\tau(r)}dt^{2}-e^{\rho(r)}dr^{2}-e^{\sigma(r)}r^{2}(d\theta^{2}+\cos^{2}\theta
d\varphi^{2}),$ (1)
and then find the gauge relation between them for maximal simplification. The
metric functions $\tau$, $\rho$ and $\sigma$ will be referred as time, radial
and transverse metric function or coefficient, respectively. We consider
radial and circular motions in such a space-time separately and find
connections between metric functions following from the known properties of
the motion. But first of all we must recall some time convention.
The metric (1) is written in the form that is consistent with global clock
synchronization, in fact $\partial/\partial t$ being timelike Killing vector.
So in the approach proposed when the cosmological effects is totally neglected
and Solar system is supposed to be placed in a space-time, that is Minkowskian
at spatial infinity, the coordinate time is the astronomical ephemerides time
ET (up to a multiplier). This is the usual approximation used for PPN-
ephemeride calculations.
There is a difficulty, because the perturbations required by the Pioneer
Anomaly grow with the radial distance, so the perturbed space-time is not
asymptotically flat. But it is not a big problem because before the metric
perturbations grow significantly the space-time has a wide nearly-flat region
in which we can choose almost Minkowskian observers and coordinates. So
further if we talk about ”spatial infinity”, we mean this wide region, in the
Solar system scale the effects of difference between this approximation and
rigorous treatment are negligible. The same problem with the same solution is
arising in the PPN-approximation then we must place the system not in
Minkowskian background but in the cosmological one. Additionally, as it can be
shown, in PPN-approach the cosmological effects, such as mutual acceleration
of geodetically moving bodies, have the second order in $H$. Therefore even
while the Pioneer Anomaly acceleration is nearly equal to $cH$, in the
framework of pure metric theories of gravitation there is no possibility to
link it to the cosmological expansion (early but almost exhaustive analysis of
the problem was made by R. C. Tolman (Tolman, , §§153–156), for the recent
work on subject see the articles mentioned in the Introduction and
additionally FahrSiewert2006 ; MizonyLachiezeRey2005 ; Licht2001 and
references within).
The radial coordinate rescaling $r\rightarrow f(r)$ changes all metric
functions, giving a possibility to imply gauge conditions on the metric
coefficients. But there are two invariants, i.e. physically measurable
quantities, which characterize the distance from a given point to the center
of space symmetry. Firstly it is an atomic time rate in comparison with the
coordinate time rate (or atomic time rate on ”spatial infinity”) $e^{\tau/2}$,
and secondly it is an area of a sphere of points equidistant to the center of
the space $4\pi r^{2}e^{\sigma}$. So the numerical values of time and
transverse metric coefficients have clear physical meaning for a given space-
time point and only $\rho(r)$ can take arbitrary values. Usual choices include
$\rho\equiv\sigma$ corresponding to isotropic coordinates of PPN-approximation
and implicit on $\rho(r)$ relation $\sigma\equiv 0$ corresponding to
Schwarzschild coordinates.
### II.2 Radial motion and its description by Doppler radio tracking
Radial motion in the space-time is fully described by the energy
$g_{tt}\frac{dt}{ds}=e^{\tau(r)}u^{0}=k=const$ and the 4-velocity length
conservation $e^{\tau(r)}u^{0^{2}}-e^{\rho(r)}u^{1^{2}}=\varepsilon,\
\varepsilon=0$ or 1 for electromagnetic waves and test bodies, respectively:
$\displaystyle\frac{dt}{dr}=\frac{e^{\frac{\rho(r)-\tau(r)}{2}}}{\sqrt{1-\varepsilon
e^{\tau(r)}/k^{2}}},$ (2)
the constant $k$ being connected with the velocity $v$ of the spacecraft on
”space infinity”
$\displaystyle k^{2}=\frac{1}{1-v^{2}}.$ (3)
The relation (2) can be integrated to give us $t(r)$ dependence which in turn
can be inverted giving $r(t)$. But there is a gauge freedom in the result
because we can choose $\rho(r)$ freely. Moreover, this relation essentially
involve both physical $\tau$ and unphysical $\rho$.
Our goal is to determine the metric functions from observations. Direct
results of experimentation in astronomy and cosmology are the measurements of
externally originated signals received on the world line of an observer. These
signals have mostly electromagnetic character and can originate from some
external source (e. g. the light emitted by the Sun and reflected by a planet)
or be emitted by the observer him/herself and then returned to him/her after
some interaction with outer bodies (as in case of radar measurements). The
latter case is considered here as corresponding to the real situation
Pioneer2001 .
The scheme of Doppler tracking used in the Pioneer experiments is very simple
conceptually: an electromagnetic signal, emitted from the world line of the
observer, is reflected back by the mirror on the world line of the spacecraft
and then compared with the initial one again on the observer world line. To be
more precise, the monochromatic electromagnetic signal, obtained from the high
precision hydrogen maser (Pioneer2001, , subsection III.A), is emitted by the
antennae on Earth in the direction of the spacecraft. This signal is detected
by the spacecraft, amplified and reemitted back to Earth, where the measured
waveform of arrived signal is compared with the emitted one to obtain the red
shift of the signal and the time of signal travel (the very detailed
description of the process can be found in Pioneer2001 ).
Now we must describe the Doppler tracking and the signal time arrival analysis
in this space-time. In the geometric optics approximation (which is applicable
for the case considered) the Doppler shift is governed simply by the ratio
between scalar products of the 4-velocity on world lines and the null wave
vector of the signal, parallel transported along the null geodesic line
between emitter and receiver:
$\displaystyle\frac{\nu_{r}}{\nu_{e}}=\frac{s_{e}}{s_{r}}=\frac{\vec{\mathstrut
u}_{r}\cdot\vec{k}_{r}}{\vec{\mathstrut u}_{e}\cdot\vec{k}_{e}},$ (4)
where $\nu_{r}$ and $\nu_{e}$ are received and emitted frequencies, measured
by the standard atomic clocks,
$s_{r}$ and $s_{e}$ are proper times of one cycle of oscillation,
$u_{r}$ and $u_{e}$ are 4-velocities of receiver and emitter,
$k_{r}$ and $k_{e}$ are tangential null vectors (wave vector), parallel
transported along the path of the signal.
Atomic clock time deviations from ephemerides time along with all known
effects of Earth motion were taken into account during the data processing
(see Pioneer2001 ), so for the description of such a small deviation like the
Pioneer Anomaly it is sufficient to use the simple model, in which the emitter
of the initial signal and the receiver of the retranslated signal are fixed at
constant distances from the Sun on the line from the Sun to apparatus. The
signal is emitted from this ”fixed” Earth at $r_{0}$ and $t-t_{p}$, received
by the spaceship at $r$ and $t$, amplified, exactly retransmitted back to
Earth and finally compared with the initial frequency on the ”fixed” Earth
again at $r_{0}$ and $t+t_{p}$ ($t_{p}$ is the time of signal propagation, the
same for forward and backward directions).
As it can be shown easily, the frequency $\nu_{r}$ received on Earth is
connected to the initially emitted $\nu_{e}$ as
$\displaystyle\nu_{r}=\nu_{e}\frac{1-\sqrt{1-e^{\tau(r)}/k^{2}}}{1+\sqrt{1-e^{\tau(r)}/k^{2}}}.$
(5)
This expression can be readily reduced to special relativistic one in the case
of $e^{\tau}\equiv 1$.
As a relation between physical quantities only, this equation does not involve
arbitrary unphysical $\rho$ function. The problem of gauge choice comes with
the definition of $r(t)$: from (5) one can determine $e^{\tau(r(t))}$, but
because the radial coordinate $r$ (and consequently $r(t)$) is arbitrary, the
radial dependence of the time metric coefficient remains gauge dependent. It
is interesting also that this relation cannot be represented by a power series
in terms of small deviations of $e^{\tau(r)}$ and $k$ from 1. The transverse
space metric coefficient $\sigma(r)$ naturally cannot be determined from the
radial motion only.
So we come to an unavoidable alternative of a-priory $r(t)$ definition or
a-priory imposing some gauge condition on $\rho(r)$. In each case the
remaining function is defined by the experimental data, and our goal now is to
find in which case the process of metric restoration can be done without
unnecessary complications. In the next subsection we consider these
possibilities in some details and then show that the best choice is the latter
case, i.e. imposing a gauge.
### II.3 General formulae and coordinate choice
While the time of signal arrival to the spaceship $t$ can be easily determined
as a half-sum of observed times of emitting $t_{e}$ and receiving $t_{r}$ of
the signal at the ”fixed” Earth
$\displaystyle t=\frac{t_{r}+t_{e}}{2},$ (6)
the corresponding $r$ determination is not so trivial task. In general we can
measure only the time of signal travel from the ”fixed” Earth to the spaceship
as a half-difference between observed times of sending and receiving of the
signal
$\displaystyle
t_{p}=\int_{r_{0}}^{r}{e^{\frac{\rho(r)-\tau(r)}{2}}}\,dr=\frac{t_{r}-t_{e}}{2}.$
(7)
These are results of a different method of tracking — signal time arrival
analysis, which in essence represents integration of the Doppler data.
So to recover $\rho(r)$ from a given $r(t)$ we must firstly find $\tau(r)$ by
(5) from the observed redshift, and then solve an integral equation above. On
the other hand, to find $r(t)$ from a given $\rho(r)$ it is sufficient to
solve a non-integral equation following from relation (2) for the spacecraft
motion
$\displaystyle\int_{t_{0}}^{t}e^{\frac{\tau(r(t))}{2}}\sqrt{1-e^{\tau(r(t))}/k^{2}}\,dt=\int_{r_{0}}^{r}{e^{\frac{\rho(r)}{2}}}\,dr,$
(8)
where $t_{0}$ is the time when the spacecraft leaves ”fixed” Earth. The left-
hand side of the relation can be found totally from the observed frequency
shifts by (5), and the right-hand side is a known function of $r$ with given
$\rho(r)$.
Consequently maximal simplification of the problem is reached in the case of
a-priory given radial metric function $\rho(r)$. It is nonsense to define it
dependent on still unknown time and transverse metric functions, with one
interesting exclusion: if $\tau(r)\equiv\rho(r)$ then $r(t)$ can be recovered
from (7) simply as
$\displaystyle r=r_{0}+\frac{t_{r}-t_{e}}{2}.$ (9)
This choice of coordinates known as light or null coordinates is not so usual
as Schwarzschild ($\sigma\equiv 0$) or isotropic ($\rho\equiv\sigma$)
coordinates but it is the most suitable one for the situation. Both
abovementioned choices are especially inappropriate here because they rely on
transversal metric function that does not reveal itself in pure radial
motions.
### II.4 Circular motion
We know that the near-circular motion of outer Solar system objects (i.e.
Neptune or Pluto) is unperturbed by the Pioneer Anomaly acceleration
Pioneer2006 ; Iorio2007 . This gives us a way to determine the transversal
metric coefficient and therefore to find metric completely.
The angular velocity of circular motion ($\theta=0,\ \phi=\omega t$) in the
considered space-time is defined by the ratio of derivatives of time and
transversal space metric coefficients
$\displaystyle\omega^{2}(r)=\frac{(e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\frac{(e^{\tau(r(t))}){}\dot{}}{(r^{2}e^{\sigma(r(t))}){}\dot{}},$
(10)
where as usual prime denotes differentiation with respect to radial
coordinate, and dot denotes differentiation in time. So there is no dependence
on the radial metric part at all. Moreover as $\omega(r(t))$ and
$(e^{\tau(r(t))}){}\dot{}$ is directly observable, so the transverse metric
coefficient $r^{2}e^{\sigma(r(t))}$ can be obtained by a simple integration
without any notion of the radial metric function.
So again recalling the radial motion equation (2) we conclude that all gauge
conditions for $\rho(r)$ involving transverse metric coefficient are not
convenient for treatment of the Pioneer Anomaly, because all such conditions
lead to coupling of equations (5) and (10) which can be solved independently
otherwise.
### II.5 Final list of relations and concluding remarks
So we work in light or null coordinates $\tau(r)\equiv\rho(r)$, then:
$\displaystyle\frac{dt}{dr}=\frac{1}{\sqrt{1-\varepsilon e^{\tau(r)}/k^{2}}}.$
(11) The trajectory of the spacecraft $r(t)$ is recovered simply from time of
sending $t_{e}$ and arrival $t_{r}$ of signal $\displaystyle
t_{p}=r-r_{0}\quad\Rightarrow\quad t=\frac{t_{r}+t_{e}}{2},\qquad
r=r_{0}+\frac{t_{r}-t_{e}}{2},$ (12)
and the observed redshift of the signal
$\displaystyle
z(t)=\frac{\nu_{e}-\nu_{r}}{\nu_{e}}=\frac{\Delta\nu}{\nu_{e}}=\frac{2}{(1-e^{\tau(r(t))}/k^{2})^{-1/2}+1},$
(13)
can be immediately transformed into time metric coefficient
$\displaystyle
e^{\tau(r(t))}=k^{2}\left[1-\left(\frac{z(t)}{2-z(t)}\right)^{2}\right]=4k^{2}\frac{(1-z)}{(2-z)^{2}}.$
(14)
The transverse space metric coefficient is defined by the dependence of
angular velocity $\omega$ on radial coordinate $r$
$\displaystyle
r^{2}e^{\sigma(r)}=r_{0}^{2}e^{\sigma(r_{0})}-\int_{r_{0}}^{r}\frac{4k^{2}z(r)z^{\prime}(r)}{(2-z(r))^{3}\omega^{2}(r)}dr,$
(15)
which for the space-time of the Pioneer Anomaly should be the same as in the
Schwarzshild field (or very close to it). So basing on the Schwarzshild metric
one can find such perturbations of metric coefficients, that the circular
motion remains unperturbed, but the radial one shows small deviation — the
Pioneer Anomaly.
It is worth noting that the sentence ”the circular motion remains unperturbed”
denotes exactly the following: if, neglecting all mutual planet disturbances,
the period of circularly orbiting planet will be measured in the units of
ephemerides time and the distance of planet from the baricenter of the Solar
system will be found by analyzing light propagation times on straight lines
from the Sun (null coordinates!) then the values of the period and the
distance will exactly be the same as needed for the 3rd Kepler law to hold —
exactly as in the Schwarzshild field.
## III The Pioneer Anomaly and its source in GR
### III.1 Schwarzschild space-time in light coordinates
Radius $r$ in light coordinates of the Schwarzschild field is related to the
usual Schwarzschild radial coordinate $r_{s}$ as
$\displaystyle r=r_{s}+r_{g}\ln\left(\frac{r_{s}}{r_{g}}-1\right),\quad
r_{s}=r_{g}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right).$ (16)
The interval in null coordinates is as follows
$\displaystyle\begin{split}ds^{2}=&\frac{\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}{1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}(dt^{2}-dr^{2})\\\
&-r_{g}^{2}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{2}(d\theta^{2}+\cos^{2}\theta
d\varphi^{2}),\end{split}$ (17) so that $\displaystyle
e^{\tau(r)}=e^{\rho(r)}=\frac{\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}{1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}=r_{g}\mathop{W}\nolimits^{\prime}_{r}\left(e^{\frac{r}{r_{g}}-1}\right),$
(18) $\displaystyle
e^{\sigma(r)}=\frac{r_{g}^{2}}{r^{2}}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{2},$
(19) where $W(x)$ is the so called multiplicative logarithm or Lambert $W$
function $\displaystyle W(x)e^{W(x)}=x.$ (20)
### III.2 The Pioneer Anomaly. Radial perturbation
The Pioneer Anomaly is the linear in ephemerides time ET deviation of the
experimentally obtained frequency of received signal $\nu_{r}$ from the
”modelled” one $\nu_{m}$
$\frac{d}{d\text{{ET}}}(\nu_{r}-\nu_{m})=-\nu_{e}\frac{2a_{P}}{c},$ (21)
where $a_{P}\sim 8\cdot 10^{-10}\ \mbox{m/s}^{2}$ is the ”unmodelled”
acceleration Pioneer2001 . The ephemerides time coincides with time coordinate
$t$ of the metric considered (as described earlier). For the most part of
range of the Pioneer Anomaly found (from $\sim 15$ to $\sim 80$ a.u.) the
deviation of the Pioneers’ orbits from pure radial motion is comparable to or
even below experimental uncertainty in the acceleration: it can be estimated
roughly as a ratio of the semi-major axis $a$ absolute value to heliocentric
distance $r$
$\frac{|a|}{r}\lesssim\frac{10^{9}\text{km}}{40\cdot 150\cdot
10^{6}\text{km}}\simeq 17\%,$ (22)
while experimental error in the acceleration is $1.33/8.74\simeq 15\%$ (see
Appendix of Pioneer2001 ). It should be noted that this uncertainty prevents
Anderson et. al. from determining the direction of the acceleration: to Earth
or to the Sun (see beginning of section VII and especially note 73 of
Pioneer2001 ).
The modelled frequency and velocity of the spacecraft are
$\displaystyle\nu_{m}=\nu_{0}\;\frac{1+\frac{1}{W(e^{\frac{r}{r_{g}}-1})}}{1-v^{2}}\left(1-\sqrt{1-\frac{1-v^{2}}{1+\frac{1}{W(e^{\frac{r}{r_{g}}-1})}}}\right)^{2},$
(23)
$\displaystyle\left(\frac{dr}{dt}\right)_{m}=v_{m}(r)=\sqrt{\frac{v^{2}+1/W(e^{\frac{r}{r_{g}}-1})}{1+1/W(e^{\frac{r}{r_{g}}-1})}},$
(24)
so accurately expanding the expression (14) for the time metric coefficient
from the red shift $z(r(t))$, one can find that given the accuracy of the
Pioneer Anomaly measurements one can simply use the relation
$\displaystyle\delta e^{\tau(r)}\simeq-\frac{z(r)\;\delta
z(r)}{2}=a_{P}\;z(r)\Delta t(r),$ (25)
where $\Delta t=t(r)-t_{0}$ is the time from the start of the Pioneer Anomaly
(we suppose that for $r<r_{0}$ the metric coincides with the Schwarzschild
one, so before $t_{0}=t(r_{0})$ there is no anomalous acceleration), $\delta
z(r)$ is a deviation of observed red shift from the modelled one.
In the first approximation $t(r)$ dependence can be replaced with the
”modelled” time, which to the accuracy of the measurements is the same as in
the Newtonian case
$t_{m}(r)=t(r_{0})+\int_{r_{0}}^{r}\frac{dr}{\dot{r}}\simeq\\\ \simeq
t_{0}+\frac{r}{v^{2}}\sqrt{v^{2}+\frac{r_{g}}{r}}-\frac{r_{g}}{v^{3}}\sinh^{-1}\left(\sqrt{\frac{r}{r_{g}}}v\right).$
(26)
Finally inserting this and modelled $z(r)$ into the equation (25) we arrive to
the perturbation of the time metric coefficient
$\delta e^{\tau(r)}=2a_{P}\Biggl{(}r-C\sqrt{v^{2}+\frac{r_{g}}{r}}+\\\
+\frac{r_{g}}{v^{2}}\left[1-\sqrt{1+\frac{r_{g}}{r\;v^{2}}}\sinh^{-1}\left(\sqrt{\frac{r}{r_{g}}}v\right)\right]\Biggr{)},$
(27)
where $C$ is a constant which can be determined from $r_{0}$ and $v$. The
perturbation appears to be non-linear in $r$, but for the Pioneer 10/11
parameters the deviation from linearity is buried deep in the experimental
errors. It is illustrated by the figure 1, which shows the deviation of the
time metric coefficient compared to the ”naive” post-Newtonian approach, where
one simply adds to the gravitational potential $\Phi(r)$ a term linear in
radius and use $e^{\tau(r)}\simeq 1+\Phi(r)$. As we can see, the difference is
mainly in the slope, all the graphs are nearly linear. Moreover, the relative
value of the deviation from linearity is decreasing with radial distance. So
the linear approximation $\delta e^{\tau(r)}\simeq 2\eta a_{P}(r-r_{0})$ is
sufficient for the Pioneer Anomaly explanation. The only difference between
this more accurate result and the ”naive” post-Newtonian one is the presence
of $\eta$, which is always less then 1 (see table 1).
Figure 1: Metric perturbation $\delta e^{\tau(r)}$ of the Pioneer Anomaly for $v$ from 5 km/s to 50 km/s in 5 km/s steps (solid lines from bottom to top) for the metric matching the Schwarzschild metric at 12 a. u., compared to the ”naive” post-Newtonian one (dashed line) Table 1: The quantity $1-\eta$ of the best linear approximations $\delta e^{\tau(r)}\simeq 2\eta a_{P}(r-r_{0})$ on the interval $r_{0}\leq r\leq 70$ a. u. for different velocities $v$ and metric matching distances $r_{0}{}^{*}$ $r_{0}$, a. u. | $v$, km/s
---|---
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50
10 | 0.175 | 0.098 | 0.057 | 0.036 | 0.025 | 0.018 | 0.013 | 0.010 | 0.008 | 0.007
15 | 0.146 | 0.077 | 0.044 | 0.027 | 0.018 | 0.013 | 0.010 | 0.008 | 0.006 | 0.005
20 | 0.122 | 0.062 | 0.034 | 0.021 | 0.014 | 0.010 | 0.007 | 0.006 | 0.005 | 0.004
∗ $r_{0}$ is the radial coordinate at which the metric coefficients coincides
with the Schwarzschild ones.
### III.3 Transversal perturbation leaving planet orbits unchanged. Necessity
of the central source
Now we assume that in the perturbed space-time planets orbit with the same
periods as in the unperturbed Schwarzschild solution, so that there is no
signatures of the Pioneer acceleration in the orbits of planets. So the
angular velocity dependence $\omega(r)$ must be the same as in the
Schwarzschild case (see, e. g., eq. (25.40) of MTW )
$\displaystyle\omega_{s}^{2}(r)=\frac{{e^{\tau(r)}}^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\frac{1}{2r_{g}^{2}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{3}}=\frac{r_{g}}{2r_{s}^{3}},$
(28)
and the perturbation in $e^{\tau}$ must lead to such a perturbation in
$r^{2}e^{\sigma}$ that $\omega^{2}(r)$ remains invariant. In the perturbed
space-time by the general formula (10) we have from simple mathematics
$\omega_{s}^{2}(r)=\frac{{e^{\tau(r)}}^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\omega^{2}(r)=\frac{(e^{\tau(r)}+\delta
e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)}+\delta(r^{2}e^{\sigma(r)}))^{\prime}}=\\\
=\frac{(e^{\tau(r)})^{\prime}+(\delta
e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}+(\delta(r^{2}e^{\sigma(r)}))^{\prime}}=\frac{(\delta
e^{\tau(r)})^{\prime}}{(\delta(r^{2}e^{\sigma(r)}))^{\prime}}.$ (29)
So the perturbation of the transverse metric coefficient is
$\displaystyle\delta(r^{2}e^{\sigma(r)})=r^{2}\delta
e^{\sigma(r)}=\int_{r_{0}}^{r}\omega^{-2}(r)\delta({e^{\tau(r)}})^{\prime}\,dr,$
(30)
and finally
$\delta
e^{\sigma(r)}=\frac{4a_{P}\eta\,r_{g}^{2}}{r^{2}}\int_{r_{0}}^{r}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{3}\,dr\simeq\\\
\simeq\frac{4a_{P}\eta\,r_{g}^{2}}{r^{2}}\int_{r_{0}}^{r}\left(\frac{r}{r_{g}}\right)^{3}dr=\frac{4a_{P}\eta(r^{4}-r_{0}^{4})}{r^{2}r_{g}}.$
(31)
In the first approximation the perturbation of $r^{2}e^{\sigma(r)}$ grows
quartically in radius.
Note the gravitational radius of the source $r_{g}$ in the answer. So _the
effect of the Pioneer Anomaly can be reproduced only by perturbations of
Schwarzschild space-time and not Minkowski one_. So the gravitational
explanation of the Pioneer Anomaly can be obtained without the equivalence
principle violation required by various authors Pioneer2006 ; Iorio2007 ;
Tangen2007 .
### III.4 Matter corresponding to the obtained metric in General Relativity
One can try to find the gravitational field theory, that gives equations of
the gravitational field allowing the solution found for the weak field of a
point mass. But we think that it is not very promising because there are no
reliable experimental evidence in favor of any gravitational theory other than
General Relativity.
Instead we find the properties of matter surrounding the point mass in GR that
can generate the obtained metric. Because in the scale of Solar system
experiments the influence of cosmological constant is negligible, for the
determination of matter one can use the Einstein equations of the form
$G_{ij}=R_{ij}-\frac{1}{2}Rg_{ij}=\kappa T_{ij},\qquad\kappa=\frac{8\pi
G}{c^{4}}.$ (32)
Using for simplicity the metric in the form
$ds^{2}=e^{\tau(r)}dt^{2}-e^{\rho(r)}dr^{2}-e^{\sigma(r)}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (33)
one arrives at the Einstein tensor
$\displaystyle G_{ij}=\frac{e^{-\rho}}{4}\Bigl{(}\lambda_{t}T_{i}\otimes
T_{j}-\lambda_{s}S_{i}\otimes S_{j}-\lambda g_{ij}\Bigr{)},$ (34)
$\displaystyle S_{i}=\Bigl{\\{}0,e^{\frac{\rho}{2}},0,0\Bigr{\\}},\qquad
T_{i}=\Bigl{\\{}e^{\frac{\tau}{2}},0,0,0\Bigr{\\}},$ (35) $\displaystyle-
S_{i}S^{i}=T_{i}T^{i}=1,\qquad S^{i}T_{i}=0,$ (36)
$\displaystyle\lambda_{t}=4e^{\rho-\sigma}+\left(\rho^{\prime}-2\sigma^{\prime}-\tau^{\prime}\right)\left(\sigma^{\prime}-\tau^{\prime}\right)+2\left(\tau^{\prime\prime}-\sigma^{\prime\prime}\right),$
(37)
$\displaystyle\lambda_{s}=4e^{\rho-\sigma}-\tau^{\prime}\left(\sigma^{\prime}-\tau^{\prime}\right)-\rho^{\prime}\left(\sigma^{\prime}+\tau^{\prime}\right)+2\left(\tau^{\prime\prime}+\sigma^{\prime\prime}\right),$
(38) $\displaystyle\lambda=\sigma^{\prime
2}+\sigma^{\prime}\tau^{\prime}+\tau^{\prime
2}-\rho^{\prime}\left(\sigma^{\prime}+\tau^{\prime}\right)+2\left(\tau^{\prime\prime}+\sigma^{\prime\prime}\right).$
(39)
Expanding to the first power of $a_{P}$ one obtains
$\displaystyle\lambda_{t}=-96\,\frac{a_{P}\eta}{r_{g}},\quad\lambda_{s}=-32\,\frac{a_{P}\eta}{r_{g}}\,\frac{r_{0}^{4}}{r^{4}},$
$\displaystyle\lambda=16\,\frac{a_{P}\eta}{r_{g}}\left(3-\frac{r_{0}^{4}}{r^{4}}\right).$
(40)
The algebraic type of the energy-momentum tensor at spatial infinity is that
of an ideal fluid (by $\lambda_{s}\rightarrow 0$) with constant positive
pressure
$\displaystyle p=\frac{e^{-\tau}}{4\kappa}\lambda\rightarrow
12\,\frac{a_{P}\eta}{\kappa r_{g}}>0,$ (41) but _negative energy density_
$\displaystyle{\rho=\frac{e^{-\tau}}{4\kappa}(\lambda_{t}-\lambda)\rightarrow-36\,\frac{a_{P}\eta}{\kappa
r_{g}}<0.}$ (42)
It is worth noting that relation between $p$ and $\rho$ is as for an
ultrarelativistic fluid except for the sign: instead of $p=\rho/3$ one has
asymptotically $p=-\rho/3$. This is a typical equation of state of dark energy
with parameter $w=-1/3$. It is interesting that such a fluid does not change
the cosmological dynamics of Friedman-Lemaître universe (see, e.g.,
(PeeblesRatra2003, , III.E)).
## IV Conclusions
In this work we show that it is possible to perturb time metric coefficient of
the Schwarzschild space-time in such a way that the Pioneer Anomaly is
reproduced. Moreover, because planet motion is governed also by anther
component of the metric, we can tune it so that circular orbits is not
disturbed. This result applies in any pure metric theory of gravitation where
test bodies follow geodetics of the metric. It is deduced that the
perturbation of the time metric coefficient can be taken linear in $r$ without
contradiction with the accuracy of experimental data obtained up to now.
Assuming the validity of the General Relativity, we find out the energy-
momentum tensor generating the metric obtained. The tensor corresponds to an
ideal fluid, however having negative energy density. It is interesting to note
that an exact static solution with spherical symmetry is known for the ”fluid”
with $3p+\rho=0$ (StanjukovichMel'nikov1983, , §8.5). This ”fluid” does not
interact with the ordinary matter besides its gravitational influence on the
metric, so it much like WIMPs or scalar field of gravitational theories of
Brans-Dicke type. Thus the ordinary matter including spacecrafts and planets
is moving geodesically.
Naturally the found ”fluid” does change the planet orbits (if they are not
strictly circular) and light rays paths. The model proposed must be carefully
studied in view of the ”Grand-Fit” investigations Pitjeva2005 ;
2008AIPC..977..254S , but direct measurements from the planned missions for
testing General Relativity in space are preferable 2007arXiv0711.0304W ;
2005gr.qc…..6104L ; 2005gr.qc…..6139T . The absolute value of effects for the
perturbation found as well as for exact solution of Stanjukovich and Ivanov
will be studied in the forthcoming paper.
The analysis presented in this paper can encourage someone to find out which
of the known alternatives to GR can reproduce the metric found or to invent
some new theory which can do it. It is possible, but in our opinion the
Pioneer Anomaly has some simple explanation by conventional and non-
gravitational physics, which is not found yet. So the question of deep
theoretical grounds for the existence of the ”fluid” in GR or of development
of some new theory of gravitation based on the metric obtained is not in the
scope of our article. Instead we point out that negative energy density of the
”fluid” is in a direct contradiction with the properties of conventional
matter. One interesting possibility for the ”fluid” is dark energy, which has
the right equation of state. Therefore we suppose that at present the metric
(gravitational) origin of the Pioneer Anomaly cannot be ruled out.
###### Acknowledgements.
The authors thank participants of the Theoretical Physics Laboratory seminar
for helpful discussions and fruitful remarks, and also acknowledge the creator
of the RGTC package — S. Bonanos. One of us (SIA) thanks G. Vereshchagin for
the help in preparation of the present version of manuscript.
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|
arxiv-papers
| 2009-08-12T08:54:42 |
2024-09-04T02:49:04.578637
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. A. Siutsou, L. M. Tomilchik",
"submitter": "Ivan Siutsou",
"url": "https://arxiv.org/abs/0908.1644"
}
|
0908.1745
|
# Interplay between the Fulde-Ferrell phase and Larkin-Ovchinnikov phase in
the superconducting ring pierced by an Aharonov-Bohm flux
H. T. Quan and Jian-Xin Zhu Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, U.S.A.
###### Abstract
We study the phase diagram of a superconducting ring threaded by an Aharonov-
Bohm flux and an in-plane magnetic Zeeman field. The simultaneous presence of
both the external flux and the in-plane magnetic field leads to the
competition between the Fulde-Ferrell (FF) phase and the Larkin-Ovchinnikov
(LO) phase. Using the Bogoliubov-de Gennes equation, we investigate the
spacial profile of the order parameter. Both the FF phase and the LO phase are
found to exist stably in this system. The phase boundary is determined by
comparing the free energy. The distortion of the phase diagrams due to the
mesoscopic effect is also studied.
###### pacs:
74.81.-g, 74.20.Fg, 74.25.Dw, 74.78.-w
## I introduction
In recent years, one of the inhomogeneous superconducting states, known as
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, has received a lot of interest.
This superconducting state with periodical spacial variation of order
parameter (OP) was first proposed independently by Fulde and Ferrell FF64 and
by Larkin and Ovchinnikov LO64 in 1960s. The possible evidence of its
existence has been reported in certain unconventional superconductors
HARadovan03 ; ABianchi03 and the possibility of its realization in trapped
cold atoms. MWZwierlein06 ; GBPartridge06 ; XJLiu07 ; RSharma08 ; WLLu09 In
literature, the state is collectively known as the FFLO state. YMatsuda07
Actually, they are two kinds of states with slight difference: the order
parameter of the LO state is real and spatially inhomogeneous, which breaks
the translational symmetry, while the order parameter of the FF state has a
uniform magnitude, but an inhomogeneous phase similar to that of a plane wave,
breaks the time-reversal symmetry. According to previous studies, the FF state
is usually unstable, and unfavorable in comparison with the LO state. Although
the LO to FF phase transition was predicted in Ref. KYang01, , a more recent
study QWang07 shows that there is no stable FF phase in such a system and
there is no LO to FF phase transition either. The authors in Ref. QCui06, ,
mention a possible FF state in a momentum space study, but as to the best of
our knowledge, a realization of stable FF state in the presence of a Zeeman
field has not been reported yet in a real space calculation.
As is well known when a Zeeman field is added to a superconductor, the LO
state becomes favorable in comparison with the BCS state, irrespective of the
geometry of the superconductor. Meanwhile, we notice that in a superconducting
ring, which is threaded by a magnetic flux, the Aharonov-Bohm (AB) flux breaks
the time reversal symmetry in much the same spirit as that in the FF phase.
JXZhu94 As a result the FF state comes out. An interesting question is then
if we add both the magnetic flux and an in-plane magnetic field, how will the
two phases compete with each other? Motivated by this observation, we study in
this paper the interplay between this AB flux-driven FF phase and the Zeeman
field-induced LO phase. It is of great interest to study the phase transitions
and phase diagram in such a system. The investigation is carried out in a
tight-binding model for a superconducting ring pierced by an AB magnetic flux,
and in the presence of a Zeeman magnetic field. We solve self-consistently the
Bogliubov de Gennes equation for the superconducting order parameter and
determine the phase diagram by comparing the total energy. We find that for
this system, there are four different phases when we vary the two parameters,
magnetic flux $\Phi$ and the Zeeman field $h$. More interestingly, we also
study the mescscopic effect.
The paper is organized as follows: in Sec. II, we introduce the tight-binding
model and present the mean-field treatment. In Sec. III, we numerically carry
out the calculation of superconducting order parameter as a function of the
magnetic flux and Zeeman field, and determine the phase diagram by comparing
the free energies. Section IV is the discussion and conclusion.
## II Model and mean-field treatment
Figure 1: (Color online) Schematic illustration of the setup. A
superconducting ring is threaded by an external magnetic flux, denoted by
$\Phi$. A magnetic field $B$ is applied in the plane of the ring. The ring is
connected to the ground to ensure that the chemical potential is fix, but the
electron number may fluctuate
We consider a one-dimensional superconducting ring threaded by an external
magnetic flux $\Phi$ (see Fig. 1). Meanwhile, there is an in-plane magnetic
field $B$, which generates the Zeeman spliting and gives rise to the
inhomogeneous pairing. The system is described by the following Hamiltonian
$\begin{split}H=&-\sum_{i,j,\sigma}\tilde{t}_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+h\sum_{i,\sigma}\sigma
c_{i\sigma}^{\dagger}c_{i\sigma}\\\
&-V\sum_{i}n_{i\uparrow}n_{i\downarrow}-\mu\sum_{i,\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}.\end{split}$
(1)
Here $\tilde{t}_{ij}=t_{ij}e^{i2\pi\Phi/N\Phi_{0}}$, where $t_{ij}$ is the
bare hopping coefficient and $\Phi_{0}=hc/e$ is the normal-state flux quantum,
and $N$ is number of lattice sites for the ring; $c_{i\sigma}^{\dagger}$
($c_{i\sigma}$) is the creation (annihilation) operator on the $i$-th lattice
site with spin $\sigma=\pm 1$ for spin up and down electrons, arising from the
interaction between the magnetic field and the spin of the electrons;
$n_{i,\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the particle number on the
$i$-th site with spin $\sigma$; $g$ is equal to 2; $\mu_{B}$ is the Bohr
magneton and $B$ is the strength of the in-plane magnetic field; $V$ is the
strength of the on-site pairing interaction; $\mu$ is the chemical potential.
For simplicity, we define $h=g\mu_{B}B$ as the strength of the Zeeman field.
In the present work, we take $t_{ij}$ to be $t$ between nearest neighboring
sites and zero otherwise. Within the mean-field approximation, the Hamiltonian
(1) is reduced to
$\begin{split}H=&-\sum_{i,j,\sigma}\tilde{t}_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+h\sum_{i,\sigma}\sigma
c_{i\sigma}^{\dagger}c_{i\sigma}-\mu\sum_{i,\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}\\\
&+V\sum_{i}(\Delta_{i}c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}+h.c.)+\sum_{i}\frac{\left|\Delta_{i}\right|^{2}}{V},\end{split}$
(2)
where $\Delta_{i}\equiv V\left\langle
c_{i\uparrow}c_{i\downarrow}\right\rangle$ is the pair potential. To
diagonalize this Hamiltonian, we employ the following Bogoliubov
transformation
$\begin{split}c_{i\sigma}=&\sum_{\nu}\left[u_{i\sigma}^{\nu}\gamma_{\nu}-\sigma(v_{i\sigma}^{\nu})^{\ast}\gamma_{\nu}^{\dagger}\right]\\\
c_{i\sigma}^{\dagger}=&\sum_{\nu}\left[(u_{i\sigma}^{\nu})^{\ast}\gamma_{\nu}^{\dagger}-\sigma
v_{i\sigma}^{\nu}\gamma_{\nu}\right],\end{split}$ (3)
corresponding to the eigenvalues $E_{\nu}^{\sigma}$ where $\gamma_{\nu}$ and
$\gamma_{\nu}^{\dagger}$ are the quasi-particle operators. The coefficients
($u_{i\sigma}^{\nu},v_{i\sigma}^{\nu}$) satisfy the Bogoliubov-de Gennes (BdG)
equation: PGdeGennes65
$\sum_{j}\left[\begin{array}[]{cc}H_{ij\sigma}&\Delta_{i}\delta_{ij}\\\
(\Delta_{i})^{\ast}\delta_{ij},&-\overline{H}_{ij\sigma}\\\
\end{array}\right]\left[\begin{array}[]{c}u_{j\sigma}^{\nu}\\\
v_{j\overline{\sigma}}^{\nu}\\\
\end{array}\right]=E_{\nu}^{\sigma}\left[\begin{array}[]{c}u_{i\sigma}^{\nu}\\\
v_{i\overline{\sigma}}^{\nu}\end{array}\right]$ (4)
where $H_{ij\sigma}=-\tilde{t}_{ij}-\mu\delta_{ij}+\sigma h\delta_{ij}$, and
$\overline{H}_{ij\sigma}=[-\tilde{t}_{ij}-\mu\delta_{ij}]^{\ast}+\overline{\sigma}h^{\ast}\delta_{ij}$.
The self consistent equation of the pair potential
$\Delta_{i}=\frac{V}{2}\sum_{\nu=1}^{2N}u_{i\uparrow}^{\nu}(v_{i\downarrow}^{\nu})^{\ast}\tanh{\frac{E_{\nu}^{\uparrow}}{2T}}$
(5)
is solved by iteration. Here $T$ is the temperature the Boltzmann constant
$k_{B}=1$ has been taken.) . Notice that the quasiparticle energy is measured
with respect to the chemical potential.
## III Numerical results
In our numerical calculation, we take the energy unit $t=1$, and the chemical
potential $\mu=-0.5$, the interaction strength $V=2$, and the ring size
$N=50$. Though the system size is far from the thermodynamic limit, it already
gives the same phase boundary as that of infinite $N$. The order parameter
structure depends not only on the Zeeman field $h$, but also the magnetic flux
$\Phi$. We note JXZhu94 ; NByers62 that all physical quantities have already
been a function of $\Phi$ with a period of $\Phi_{0}$ even in the normal
state. Therefore, it is sufficient for us to consider the magnetic flux in the
range $\Phi\in[0,\Phi_{0}]$. In the absence of the magnetic flux $\Phi=0$, the
Bardeen-Cooper-Schrieffer (BCS) order parameter $\Delta=0.351$ for $h=0$, and
the LO state is stable for $h_{c1}<h<h_{c2}$ with $h_{c1}=0.23$ and
$h_{c2}=1.56$. The system becomes normal ($\Delta=0$) for $h>h_{c2}$. In the
presence of the magnetic flux, the magnetic flux can induce a change in the
structure of the BCS state in an $s$-wave superconductor, namely a crossover
from the BCS state in the absence of a magnetic flux to a FF state with a
magnetic flux when the Zeeman field is low $h<h_{c1}$. When the Zeeman field
increases, the LO becomes favorable and both BCS and FF states give in. If we
continue to increase the Zeeman field, the amplitude of the pairing potential
of the LO phase will be suppressed by the Zeeman field until it disappears
finally, and the system enters the normal state. In the following, we will
numerically construct the phase diagrams.
### III.1 Phase boundary in h-$\Phi$ plane
We first focus on the low temperature case $\beta=1/T=200$ (corresponding to
$T=0.005$). In the absence of the magnetic flux, there are three different
phases: BCS, LO, and normal. In the presence of the magnetic flux, there are
also three phases, FF, LO, and normal state. In the following, we study the
phase transitions and the phase boundaries for fixed temperature when varying
$h$ and $\Phi$.
In order to check if the FF state becomes the ground state, we assign a
periodic phase to the order parameter at each site as an initial condition.
Similarly, we assign a constant phase to see if BCS state becomes the ground
state. For a set of fixed parameters ($h$, $\Phi$, $N$, $T$), different stable
solutions (with different order parameter textures) could be obtained from
different initial configurations. For example, one may find both stable LO-
type OP and FF-type OP for the same set of parameters
($N=50,\;\beta=200,\;\Phi=0.25\Phi_{0},\;h=0.25$). Even there are more than
one stable LO type solutions for the same set of parameters, which means
different net momentum of the Cooper pair. To distinguish one state from other
competing states (including BCS state and FF state), we choose the
energetically most favored one by comparing their free energies. For the model
(1), the free energy is given by
$\begin{split}F=&-\frac{1}{\beta}\sum_{\nu}\ln\left(1+e^{-\beta
E_{\nu}^{\uparrow}}\right)+\sum_{i}\frac{|\Delta_{i}|^{2}}{V}-\sum_{i}(\mu+h)\end{split}$
(6)
Here, we just compare the summation of the first two terms, because the third
term is a constant for all solutions of different phases. In the following we
determine the phase boundary between FF state, LO state, and normal state.
#### III.1.1 First order transition between the FF and LO phases
When determining the pair potential self-consistently by iteration, we find
that in certain range of the strength of the in-plane Zeeman field $h$,
different initial configurations of the pair potential lead to different
stable solutions. In another word, there are more than one stable solutions
through iteration. For example, when we fix $\Phi=\Phi_{0}/4$, and vary the
magnetic field in the range $0.08<h<0.29$, stable solutions of both the FF
type and the LO type can be arrived at through iteration. The free energies of
these two types of stable solutions are listed in Table I. It can be seen that
the LO state becomes energetically favorable when the magnetic field is equal
to or greater than $h_{c1}=0.21$. In addition, the free energy at $h_{c1}$ is
continuous, but its first order derivative is not continuous. Hence, we
conclude that for a fixed magnetic flux $\Phi=\Phi_{0}/4$, there is a first-
order phase transition between the FF and LO states at $h_{c1}$. Similarly, we
fix magnetic flux $\Phi$ at different values and we can find the threshold
value of $h$ at which the system changes from the FF state to the LO state or
vise versa. Thus for a fixed temperature $\beta=200$ and fixed system size
$N=50$, the phase transition line between FF and LO state is determined by
comparing the free energy of the FF phase and the LO phase, and we plot it in
Fig. 2. To ensure that the phase boundary given by $N=50$ is close to that of
the thermodynamic limit, we change the system size to $N=200$, and we find the
phase boundary does not change. For $N=50$ and $N=200$, the magnitude of the
OP $\Delta_{i}$ in BCS phase is the same. Hence the result based on $N=50$ can
be regarded as in thermodynamic limit. It can be seen that the first-order
transition line is not parallel to the $\Phi$ axis, so we can turn the flux to
make the system change from the LO phase to the FF phase or vise versa. We
call this phase transition AB effect induced phase transition. We can also see
that the phase boundary between the LO and FF states is symmetric around
$\Phi=\Phi_{0}/4$, and the period of FF phase is $\Phi_{0}/2$.
Table 1: Free energies (up to a constant $-\sum_{i=1}^{N}(\mu+h)$) for stable solutions of FF state and LO state. Here the ring size is $N=50$, the magnetic flux $\Phi=\Phi_{0}/4$, and the temperature $\beta=200$. It can be seen that there is a first-order phase transition from the FF state to LO state when the in-plane magnetic field is tune across $h=0.21$ | $h=0.20$ | $h=0.21$ | $h=0.22$ | $h=0.23$ | $h=0.24$ | $h=0.25$
---|---|---|---|---|---|---
FF | -56.1487 | -55.6487 | -55.1487 | -54.6487 | -54.1487 | -53.6487
LO | -56.1019 | -55.6319 | -55.1619 | -54.6919 | -54.2219 | -53.7519
Figure 2: (Color online) Phase diagram of the superconducting ring in the
$h$-$\Phi$ plane. Here the ring size is $N=50$, and the temperature is
$T=0.005$ ($\beta=200$). Notice that the boundary line between the FF and LO
phases has the periodicity in $\Phi$ with a period of $\Phi_{0}/2$ while that
between the LO and normal state phases has the periodicity in $\Phi$ with a
period of $\Phi_{0}$. Specifically, the LO state exists in the range
$[0.23,1.56]$, $[0.21,1.52]$, and $[0.23,1.51]$ for $\Phi=0$, $\Phi_{0}/4$,
and $\Phi_{0}/2$, respectively.
#### III.1.2 Second-order transition between the LO and normal state phases
If we continue to increase the in-plane magnetic field above the value
$h=0.29$ for $\Phi$ fixed at $\Phi_{0}/4$, all initial configurations of the
pair potential will lead to the LO state, or only the LO state becomes stable.
Meanwhile the amplitude of the pair potential decreases and the period of the
modulation of the pair potential is shortened continuously. Further increase
of the Zeeman field leads to the reduction of the pair potential until it
vanishes gradually. When the magnetic field reaches $h_{c2}=1.52$, the
amplitude of the pairing potential vanishes, or the LO state is completely
depressed by the in-plane magnetic field, and the system changes from the LO
state to the normal state. If we do the iteration from zero pair potential, we
will find that when $h>h_{c2}$ the stable solution for the pair potential is
zero (normal state). When $h\leqslant h_{c2}$, the stable solution is an LO
state. There is no coexistence area of the LO and the normal states in the $h$
axis. Hence we conclude that the phase transition at $h_{c2}$ for a fixed
$\Phi$ is a second-order phase transitoin. Our result is consistent with
previous studies. YMatsuda07 ; HShimahara94 ; KYang98
### III.2 Phase boundary in h-T plane
Figure 3: (Color online) Phase diagram of the superconducting ring in the
$h$-$T$ plane. Here the ring size is $N=50$, and the magnetic flux are
$\Phi=0$ (a), and $\Phi=\Phi_{0}/4$ (b), respectively. For $\Phi=0$, the LO
state emerges at low temperature $T<0.12$ and relatively high magnetic field
$0.23<h<1.56$. For $\Phi=\Phi_{0}/4$, the LO state emerges at low temperature
$T<0.11$ and relatively high magnetic field $0.21<h<1.52$. The LO state to
normal state transition (Black with open circles) is of second order, while
the BCS (or FF) to the normal state transition (Red with open squares) is of
first order. The zero-field transition temperature is around $T_{c}=0.21$ and
$0.20$ for $\Phi=0$ and $\Phi_{0}/4$.
In the preceding subsection, we study the phase transitions when we vary the
magnetic flux $\Phi$ or the in-plane magnetic field $h$. The temperature is
fixed at a very low value. Hence these phase transitions can be regarded as
quantum phase transitions. In this subsection, we will study the phase
transitions induced by thermal fluctuations, and determine their phase
boundaries. We will fix the magnetic flux $\Phi$ and vary the temperature
$\beta$ or the in-plane magnetic field $h$. First we consider the case in the
absence of the magnetic flux $\Phi=0$. We fix the temperature at $\beta$=20
($T$=0.05), $\beta$=10 ($T$=0.10), $\beta$=6.67 ($T$=0.15), $\beta$=5
($T$=0.20), and $\beta$=4 ($T$=0.25) respectively, and do the iteration
separately. The phase boundary between the BCS and LO states is determined in
a similar way to that in Sec. III.A. It can be seen that when we fix the
magnetic flux to be zero, and tune the in-plane magnetic field or the
temperature, the system will change between the BCS, LO, and normal states. As
can be seen from Fig. 3(a). the LO phase emerges below the critical
temperature $T\approx 0.12$. We note that the BCS to the LO state is first
order, and the LO to normal is second order. When the magnetic flux is
nonzero, the BCS state will be replaced by FF state with the phase diagram, as
shown in Fig. 3(b), very similar to the zero-flux case.
### III.3 Mesoscopic effect
Figure 4: Phase diagram in the $h$-$\Phi$ plane. All the parameters are the
same as that in Fig. 2 except that the ring size is $N=20$ (a) and $N=10$ (b).
The empty area represents the normal state. The gray area represents the FF
state, and the area covered by the thin black lines represents the LO state.
The thick black lines represent the BCS state. It can be seen that with the
decrease of the ring size, the FF phase expand a lot and the LO phase shrink
dramatically. The period of FF phase also changes from $\Phi_{0}/2$ to
$\Phi_{0}$
Another interesting question is the mesoscopic effect. In this subsection we
will study the mesoscopic effect by fixing the temperature and decreasing the
ring size. As mentioned in the above discussion, the ring size $N=50$ already
gives the same phase boundary as that of $N\rightarrow\infty$. A simple check
is that when we increase the ring size to $N=100$, and $N=200$, we find that
the phase boundaries do not change in comparison with that for $N=50$. This
means that for the current model, $N=50$ can be treated as in the
thermodynamic limit. However, if we decrease the ring size, for example to
$N=20$, the mesoscopic effect will occur. First, in the $h$-$\Phi$ plane, the
LO phase will shrink dramatically and the FF phase will expand (see Fig.
4(a)). This is because (1) the influence of the magnetic flux on the system
will increase and the influence of the in-plane Zeeman field will decrease
relatively, and (2) having a finite size restricts the periodicity of the LO
order parameter. At a given Zeeman field, if the period is not commensurate
with the corresponding ring, solutions of the LO state will have to be
modified to be commensurate with system size, which results in some energy
cost. Therefore, the LO state will shrink. Second, the periodicity of the
magnetic flux changes from $\Phi_{0}/2$ to $\Phi_{0}$. This is because the
system size is so small that the Cooper pair can no longer be treated as a
whole, and can only be treated as two separate electrons. We show in Fig. 4(b)
the phase boundary for $N=10$ and the re-entrant behavior of various phases
can be seen in the phase diagram.
## IV Discussion and conclusion
Based upon a tight-binding model, we study a one-dimensional $s$-wave
superconducting ring subject to an in-plane Zeeman field and a magnetic flux
by solving the BdG equation in real space. In the presence of a magnetic flux,
a crossover from the BCS state to the FF state is obtained when the in-plane
magnetic field is not very strong. If we increase the strength of the in-plane
magnetic field, the LO state becomes favorable, and a FF to LO phase
transition occurs. With the further increase of the in-plane magnetic field
strength, the magnitude of the pair potential of the LO state is suppressed,
and disappears finally with the system entering the normal state. In the
absence of the magnetic flux, there is no FF phase, and the Zeeman field
induces the transitions between the BCS, LO, and normal states, which has been
studied extensively. HShimahara94 ; KYang98 ; YTanuma98 ; JXZhu00 ;
DFAgterberg01 ; QWang06a ; QWang06b ; QCui08 ; YYanase07 ; TDatta09 Our
results agree well with the previous studies in a two-dimensional system that
the energetically favorable state for $s$-wave superconductor is a one-
dimensional stripe-like LO state. This suggests the first-order transition
between the BCS and LO states while a second-order transition between the LO
to normal states. Our study goes beyond that and indicates a stable FF state
due to the magnetic flux. The mesoscopic effects are also studied. When the
system size decreases, two mesoscopic effects arise: (1) the LO phase in the
$h$-$\Phi$ plane shrinks, and the FF state expands due to the enhancement of
the Aharanov-Bohm effect; (2) the periodicity of the external magnetic flux
will change from $\Phi_{0}/2$ to $\Phi_{0}$.
The following remarks are in order: (1) Though we study a one-dimensional
model, the system should not be regarded as a mathematically one-dimensional.
The current study can be easily extended to the two-dimensional and other
geometry, such as a torus configuration threaded by a magnetic flux. It can be
expected that a similar phase transition between LO state and FF state will
occur. (2) For the one-dimensional case, the LO state exists in a broader
range of parameters space ($h-T$ space, see Fig. 2) than that of two-
dimensional and three-dimensional cases, YMatsuda07 ; YSuzumura83 ; KMachida84
which makes it easier to access experimentally. (3) In Ref. KHenderson09, , it
is reported that an trap potential with arbitrary configuration can be
achieved. Hence, we expect that the result presented in this paper should be
able to be observed experimentally in cold Fermions under current experiment
technique. (4) In the thermodynamic limit, $N\rightarrow\infty$, the FF state
reproduces the BCS state, because the phase gradient of the order parameter is
vanishingly small. This result agrees with our intuition that when the ring
size becomes infinity, the influence of the magnetic flux can be neglected.
(5) The effect of the impurity is not included in the current study, and will
be given in our future studies.
###### Acknowledgements.
One of us (H.T.Q.) thanks Rishi Sharma for stimulating discussions. This work
was supported by U.S. DOE at LANL under Contract No. DE-AC52-06NA25396, the
U.S. DOE Office of Science, and the LANL LDRD Program.
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|
arxiv-papers
| 2009-08-12T16:16:46 |
2024-09-04T02:49:04.586404
|
{
"license": "Public Domain",
"authors": "H. T. Quan and Jian-Xin Zhu",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/0908.1745"
}
|
0908.1769
|
11institutetext: Computer Science Department
Columbia University
New York, NY 10027
{bert,jebara}@cs.columbia.edu
%the␣affiliations␣are␣given␣next;␣don’t␣give␣your␣e-mail␣address%unless␣you␣accept␣that␣it␣will␣be␣publishedhttp://www.cs.columbia.edu/learning
Bert Huang and Tony Jebara
# Approximating the Permanent with Belief Propagation
Bert Huang Tony Jebara
###### Abstract
This work describes a method of approximating matrix permanents efficiently
using belief propagation. We formulate a probability distribution whose
partition function is exactly the permanent, then use Bethe free energy to
approximate this partition function. After deriving some speedups to standard
belief propagation, the resulting algorithm requires $(n^{2})$ time per
iteration. Finally, we demonstrate the advantages of using this approximation.
00footnotetext: This work was done in late 2007 and early 2008.
## 1 Introduction
The permanent is a scalar quantity computed from a matrix and has been an
active topic of research for well over a century. It plays a role in
cryptography and statistical physics where it is fundamental to Ising and
dimer models. While the determinant of an $n\times n$ matrix can be evaluated
exactly in sub-cubic time, efficient methods for computing the permanent have
remained elusive. Since the permanent is $\\#$P-complete, efficient exact
evaluations cannot be found in general. The best exact methods improve over
brute force (${\cal O}(n!)$) and include Ryser’s algorithm [13, 14] which
requires as many as $\Theta(n2^{n})$ arithmetic operations. Recently,
promising fully-polynomial randomized approximate schemes (FPRAS) have emerged
which provide arbitrarily close approximations. Many of these methods build on
initial results by Broder [3] who applied Markov chain Monte Carlo (a popular
tool in machine learning and statistics) for sampling perfect matchings to
approximate the permanent. Recently, significant progress has produced an
FPRAS that can handle arbitrary $n\times n$ matrices with non-negative entries
[10]. The method uses Markov chain Monte Carlo and only requires a polynomial
order of samples.
However, while these methods have tight theoretical guarantees, they carry
expensive constant factors, not to mention relatively high polynomial running
times that discourage their usage in practical applications. In particular, we
have experienced that the prominent algorithm in [10] is slower than Ryser’s
exact algorithm for any feasible matrix size, and project that it only becomes
faster around $n>30$.
It remains to be seen if other approximate inference methods can be brought to
bear on the permanent. For instance, loopy belief propagation has also
recently gained prominence in the machine learning community. The method is
exact for singly-connected networks such as trees. In certain special loopy
graph cases, including graphs with a single loop, bipartite matching graphs
[1] and bipartite multi-matching graphs [9], the convergence of BP has been
proven. In more general loopy graphs, loopy BP still maintains some surprising
empirical success. Theoretical understanding of the convergence of loopy BP
has recently been improved by noting certain general conditions for its fixed
points and relating them to minima of Bethe free energy. This article proposes
belief propagation for computing the permanent and investigates some
theoretical and experimental properties.
In Section 2, we describe a probability distribution parameterized by a matrix
similar to those described in [1, 9] for which the partition function is
exactly the permanent. In Section 3, we discuss Bethe free energy and
introduce belief propagation as a method of finding a suitable set of pseudo-
marginals for the Bethe approximation. In Section 4, we report results from
experiments. We then conclude with a brief discussion.
## 2 The Permanent as a Partition Function
Given an $n\times n$ non-negative matrix $W$, the matrix permanent is
$\sum_{\pi\in S_{n}}\prod_{i=1}^{n}W_{i\pi(i)}.$ (1)
Here $S_{n}$ refers to the symmetric group on the set $\\{1,\ldots,n\\}$, and
can be thought of as the set of all permutations of the columns of $W$. To
gain some intuition toward the upcoming analysis, we can think of the matrix
$W$ as defining some function $f(\pi;W)$ over $S_{n}$. In particular, the
permanent can be rewritten as
$\displaystyle\textrm{per}(W)=\sum_{\pi\in S_{n}}f(\pi;W),\hskip 14.45377pt$
$\displaystyle\textrm{ where }\hskip
14.45377ptf(\pi;W)=\prod_{i=1}^{n}W_{i\pi(i)}.$
The output of $f$ is non-negative, so we consider $f$ a density function over
the space of all permutations.
If we think of a permutation as a perfect matching or assignment between a set
of $n$ objects $A$ and another set of $n$ object $B$, we relax the domain by
considering all possible assignments of imperfect matchings for each item in
the sets.
Consider the set of assignment variables $X=\\{x_{1},\ldots,x_{n}\\}$, and the
set of assignment variables $Y=\\{y_{1},\ldots,y_{n}\\}$, such that
$x_{i},y_{j}\in\\{1,\ldots,n\\},\forall i,j$. The value of the variable
$x_{i}$ is the assignment for the $i$’th object in $A$, in other words the
value of $x_{i}$ is the object in $B$ being selected (and vice versa for the
variables $y_{j}$).
$\displaystyle\phi(x_{i})$ $\displaystyle=$
$\displaystyle\sqrt{W_{ix_{i}}},\hskip
14.45377pt\phi(y_{j})=\sqrt{W_{y_{j}j}},\hskip 14.45377pt$
$\displaystyle\psi(x_{i},y_{j})$ $\displaystyle=$ $\displaystyle
I(\neg(j=x_{i}\oplus i=y_{j})).$
We square-root the matrix entries because the factor formula multiplies both
potentials for the $x$ and $y$ variables. We use $I()$ to refer to an
indicator function such that $I(\textrm{true})=1$ and $I(\textrm{false})=0$.
Then the $\psi$ function outputs zero whenever any pair $(x_{i},y_{j})$ have
settings that cannot come from a true permutation (a perfect matching).
Specifically, if the $i$’th object in $A$ is assigned to the $j$’th object in
$B$, the $j$’th object in $B$ must be assigned to the $i$’th object in $A$
(and vice versa) or else the density function goes to zero. Given these
definitions, we can define the equivalent density function that subsumes
$f(\pi)$ as follows:
$\hat{f}(X,Y)=\prod_{i,j}\psi(x_{i},y_{j})\prod_{k}\phi(x_{k})\phi(y_{k}).$
This permits us to write the following equivalent formulation of the
permanent: $\textrm{per}(W)=\sum_{X,Y}f(X,Y)$. Finally, if we convert density
function $\hat{f}$ into a valid probability, simply add a normalization
constant to it, producing:
$p(X,Y)=\frac{1}{Z(W)}\prod_{i,j}\psi(x_{i},y_{j})\prod_{k}\phi(x_{k})\phi(y_{k}).$
(2)
The normalizer or partition function $Z(W)$ is the sum of $f(X,Y)$ for all
possible inputs $X,Y$. Therefore, the partition function of this distribution
is the matrix permanent of $W$.
## 3 Bethe Free Energy
To approximate the partition function, we use the Bethe free energy
approximation. The Bethe free energy of our distribution given a belief state
$b$ is
$\displaystyle F_{Bethe}$ $\displaystyle=$
$\displaystyle-\sum_{ij}\sum_{x_{i},y_{j}}b(x_{i},y_{j})\ln\psi(x_{i},y_{j})\phi(x_{i})\phi(y_{j})$
(3) $\displaystyle+\sum_{ij}\sum_{x_{i},y_{j}}b(x_{i},y_{j})\ln
b(x_{i},y_{j})$ $\displaystyle-(n-1)\sum_{i}\sum_{x_{i}}b(x_{i})\ln b(x_{i})$
$\displaystyle-(n-1)\sum_{j}\sum_{y_{j}}b(y_{j})\ln b(y_{j})$
The belief state $b$ is a set of pseudo-marginals that are only locally
consistent with each other, but need not necessarily achieve global
consistency and do not have to be true marginals of a single global
distribution. Thus, unlike the distributions evaluated by the exact Gibbs free
energy, the Bethe free energy involves pseudo-marginals that do not
necessarily agree with a true joint distribution over the whole state-space.
The only constraints pseudo-marginals of our bipartite distribution obey (in
addition to non-negativity) are the linear local constraints:
$\displaystyle\sum_{y_{j}}b(x_{i},y_{j})$ $\displaystyle=$ $\displaystyle
b(x_{i}),\hskip 14.45377pt\sum_{x_{i}}b(x_{i},y_{j})=b(y_{j}),\>\forall i,j,$
$\displaystyle\sum_{x_{i},y_{j}}b(x_{i},y_{j})$ $\displaystyle=$
$\displaystyle 1.$
The class of true marginals is a subset of the class of pseudo-marginals. In
particular, true marginals also obey the constraint $\sum_{X\setminus
x}p(X)=p(x)$, which pseudo-marginals are free to violate.
We will use the approximation
$\textrm{per}(W)\approx\exp\left(-\min_{b}F_{\textrm{Bethe}}(b)\right)$ (4)
### 3.1 Belief Propagation
The canonical algorithm for (locally) minimizing the Bethe free energy is
Belief Propagation. We use the dampened belief propagation described in [6],
which the author derives as a (not necessarily convex) minimization of Bethe
free energy. Belief Propagation is a message passing algorithm that
iteratively updates messages between variables that define the local beliefs.
Let $m_{x_{i}}(y_{j})$ be the message from $x_{i}$ to $y_{j}$. Then the
beliefs are defined by the messages as follows:
$b(x_{i},y_{j})\propto\psi(x_{i},y_{j})\phi(x_{i})\phi(y_{j})\prod_{k\neq
j}m_{y_{k}}(x_{i})\prod_{\ell\neq i}m_{x_{\ell}}(y_{j})\\\ $ $\displaystyle
b(x_{i})$ $\displaystyle\propto$
$\displaystyle\phi(x_{i})\prod_{k}m_{y_{k}}(x_{i}),\>\>\>\>b(y_{j})\propto\phi(y_{j})\prod_{k}m_{x_{k}}(y_{j})$
(5)
In each iteration, the messages are updated according to the following update
formula:
$m_{x_{i}}^{\textrm{new}}(y_{j})=\sum_{x_{i}}\left[\phi(x_{i})\psi(x_{i},y_{j})\prod_{k\neq
j}m_{y_{k}}(x_{i})\right]$ (6)
Finally, we dampen the messages to encourage a smoother optimization in log-
space.
$\ln m_{x_{i}}(y_{j})\leftarrow\ln m_{x_{i}}(y_{j})+\epsilon\left[\ln
m_{x_{i}}^{\textrm{new}}(y_{j})-\ln m_{x_{i}}(y_{j})\right]$ (7)
We use $\epsilon$ as a dampening rate as in [6] and dampen in log space
because the messages of BP are exponentiated Lagrange multipliers of Bethe
optimization [6, 18, 19]. We next derive faster updates of the messages (6)
and the Bethe free energy (3) with some careful algebraic tricks.
### 3.2 Algorithmic Speedups
Computing sum-product belief propagation quickly for our distribution is
challenging since any one variable sends a message vector of length $n$ to
each of its $n$ neighbors, so there are $2n^{3}$ values to update each
iteration. One way to ease the computational load is to avoid redundant
computation. In Equation (6), the only factor affected by the value of $y_{j}$
is the $\psi$ function. Therefore, we can explicitly define the update rules
based on the $\psi$ function, which will allow us to take advantage of the
fact that the computation for each setting of $y_{j}$ is similar. When
$y_{j}\neq i$, we have
$\displaystyle m_{x_{i}y_{j}}^{\textrm{not}}$ $\displaystyle=$
$\displaystyle\left(\sum_{x_{i}\neq j}\phi(x_{i})\prod_{k\neq
j}m_{y_{k}}(x_{i})\right)$ (8) $\displaystyle=$
$\displaystyle\left(\sum_{x_{i}\neq
j}\phi(x_{i})m_{y_{x_{i}}x_{i}}^{\textrm{match}}\prod_{k\neq j,k\neq
x_{i}}m_{y_{k}x_{i}}^{\textrm{not}}\right).$
When $y_{j}=i$,
$\displaystyle m_{x_{i}y_{j}}^{\textrm{match}}$ $\displaystyle=$
$\displaystyle\left(\phi(x_{i}=j)\prod_{k\neq j}m_{y_{k}}(x_{i}=j)\right)$ (9)
$\displaystyle=$ $\displaystyle\left(\phi(x_{i}=j)\prod_{k\neq
j}m_{y_{k}x_{i}}^{\textrm{not}}\right).$
We have reduced the full message vectors to only two possible values:
$m^{\textrm{not}}$ is the message for when the variables are not matched and
$m^{\textrm{match}}$ is for when they are matched. We further simplify the
messages by dividing both values by $m_{x_{i}y_{j}}^{\textrm{not}}$. This
gives us
$\displaystyle m_{x_{i}y_{j}}^{\textrm{not}}$ $\displaystyle=$ $\displaystyle
1$ $\displaystyle m_{x_{i}y_{j}}^{\textrm{match}}$ $\displaystyle=$
$\displaystyle\frac{\phi(x_{i}=j)\prod_{k\neq
j}m_{y_{k}x_{i}}^{\textrm{not}}}{\sum_{x_{i}\neq
j}\phi(x_{i})m_{y_{x_{i}}x_{i}}^{\textrm{match}}\prod_{k\neq j,k\neq
x_{i}}m_{y_{k}x_{i}}^{\textrm{not}}}$ (10) $\displaystyle=$
$\displaystyle\frac{\phi(x_{i}=j)}{\sum_{k\neq
j}\phi(x_{i}=k)m_{y_{k}x_{i}}^{\textrm{match}}}$
We can now define a fast message update rule that only needs to update one
value between each variable.
$m_{x_{i}y_{j}}\leftarrow\frac{1}{Z}\phi(x_{i}=j)/\sum_{k\neq
j}\phi(x_{i}=k)m_{y_{k}x_{i}}$ (11)
We can rewrite the belief update formulas using these new messages.
$\displaystyle b(x_{i}=j,y_{j}=i)$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{ij}}\phi(x_{i})\phi(y_{j})$ $\displaystyle
b(x_{i}\neq j,y_{j}\neq i)$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{ij}}\phi(x_{i})\phi(y_{j})m_{y_{x_{i}}x_{i}}m_{x_{y_{j}}y_{j}}$
$\displaystyle b(x_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\phi(x_{i})m_{y_{x_{i}}x_{i}},$ $\displaystyle
b(y_{j})$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\phi(y_{j})m_{x_{y_{j}}y_{j}}$ (12)
With the simplified message updates, each iteration takes ${\cal O}(n)$
operations per node, resulting in an algorithm that takes ${\cal O}(n^{2})$
operations per iteration. We demonstrate experimentally that the algorithm
converges to within a certain tolerance in a constant number of iterations
with respect to $n$, so in practice the ${\cal O}(n^{3})$ operations it takes
to compute Bethe free energy is the asymptotic bottleneck of our algorithm.
### 3.3 Convergence
One important open question about this work is whether or not we can guarantee
convergence. Empirically, by initializing belief propagation to various random
points in the feasible space, we found BP still converged to the same
solution. The max-product algorithm is guaranteed to converge to the correct
maximum matching [1, 9] via arguments on the unwrapped computation tree of
belief propagation. The matching graphical model does not not meet the
sufficient conditions provided in [7] nor does our distribution fit the
criteria for non-convex convergence provided in [16] and [8].
In our analysis, we have found that the Bethe free energy is certainly non-
convex near the vertices of the distribution. That is, if we evaluate the
Bethe free energy on pseudomarginals corresponding to exactly one matching,
and take a tiny step in the direction of a non-adjacent matching vertex, Bethe
free energy increases. On the other hand, when we initialize belief
propagation such that the beliefs are at a vertex, BP moves away from the
apparent local minimum and converges to the same solution as other
initializations. This behavior implies that, while the Bethe free energy
within the matching constraints is non-convex, it may still have a unique
zero-gradient point despite not fitting the criteria in [8], which exploit the
strength of potentials.
Since all our empirical evidence implies that BP always converges, we suspect
that we have not yet correctly analyzed the true space traversed during
optimization. In particular, the distribution described by Equation 2 is
defined over the set of all $n^{n}$ possible $X,Y$ states, while it is only
nonzero in $n!$ states. Any beliefs derived from belief propagation obey
similar constraints, so it is reasonable to suspect that careful analysis of
the optimization with special attention to the oddities of the distribution
could yield more promising theoretical guarantees.
However, without being rigorous, we can note that the matching constraints
created by the $\psi$ functions enforce that the singleton beliefs are exactly
the matched pairwise beliefs. This means we can think of these as entries in a
doubly-stochastic matrix $B$.
$\displaystyle b(x_{i}=j,y_{j}=i)=b(x_{i}=j)=b(y_{j}=i)\equiv B_{ij}$ (13)
Therefore it becomes clear that there is a strong connection to the Sinkhorn
operation [11], which iteratively scales rows and columns of a matrix until it
converges to a doubly-stochastic matrix. It has been shown that the Sinkhorn
operation effectively minimizes the pseudo-KL divergence between some matrix
and the doubly-stochastic matrix[12].
$\displaystyle\min_{B}$
$\displaystyle\sum_{ij}B_{ij}\log\frac{B_{ij}}{A_{ij}}$ s.t.
$\displaystyle\sum_{i}B_{ij}=1,\forall j,\>\>\>\sum_{j}B_{ij}=1,\forall i$
Here the pseudo-KL divergence can be interpreted as the KL for each row and
each column, each of which is an assignment distribution like in our matching
setting. The Sinkhorn procedure is guaranteed to converge for indecomposable
input matrices [11], so the fact that the the procedure is reminiscent of ours
is encouraging. However the two procedures differ enough that the guarantee
does not directly translate.
(a) Running time (b) Iterations
Figure 1: (a) Average running time until convergence of BP for $5\leq n\leq
50$. (b) Number of iterations.
## 4 Experiments
In this section we evaluate the performance of this algorithm in terms of
running time and accuracy, and finally we exemplify a possible usage of the
approximate permanent as a kernel function.
### 4.1 Running Time
We ran belief propagation to approximate the permanents of random matrices of
sizes $n=[5,50]$, recording the total running time and the number of
iterations to convergence. Surprisingly, the number of iterations to
convergence initially _decreased_ as $n$ grew, but appears to remain constant
beyond $n>10$ or so. Running time still increased because the cost of updating
each iteration well subsumes the decrease in iterations to convergence.
In our implementation, we checked for convergence by computing the absolute
change in all the messages from the previous iteration, and considered the
algorithm converged if the sum of all the changes of all $n^{3}$ messages was
less than $1e-10$. In all cases, the resulting beliefs were consistent with
each other within comparable precision to our convergence threshold. These
experiments were run on a a 2.4 Ghz Intel Core 2 Duo Apple Macintosh running
Mac OS X 10.5. The code is in $C$ and compiled using gcc version 4.0.1.
Table 1: Normalized Kendall distances between the rankings of random matrices based on their true permanents and the rankings based on approximate permanents. See Figure 2 for plots of the approximations. n | Bethe | Sampling | Det. | Diag.
---|---|---|---|---
10 | 0.00023 | 0.0248 | 0.3340 | 0.0724
8 | 0.0028 | 0.1285 | 0.4995 | 0.4057
5 | 0.0115 | 0.0914 | 0.4941 | 0.3834
Figure 2: Plots of the approximated permanent versus the true permanent using
four different methods. It is important to note that the scale of the y-axis
varies from plot to plot. The diagonal is extremely erratic and the
determinant underestimates so much that it is barely visible on the log scale.
Sampling approximates values much closer in absolute distance to the true
permanent but does not provide monotonicity in its approximations. Typically,
this is more important than absolute accuracy. Here we illustrate the results
from the $n=8$ case. We report results for $n=5$ and $10$ in Table 1.
### 4.2 Accuracy of Approximation
We evaluate the accuracy of our algorithm by creating 1000 random matrices of
sizes 5, 8 and 200 matrices of size 10. The entries of each of these matrices
were randomly drawn from a uniform distribution in the interval $[0,50]$. We
computed the true permanents of these matrices, then computed approximate
permanents using our Bethe approximation. We also computed an approximate
using a naive sampling method, where we sample by choosing random permutations
and storing a cumulative sum of each permutation’s corresponding product. We
sampled for the same amount of actual time our belief propagation algorithm
took to converge. Finally we also computed two weak approximations: the
determinant and the scaled product of the diagonal entries.
In order to be able to compare to the true permanent, we had to limit this
analysis to small matrices. However, since MCMC sampling methods such as in
[10] take ${\cal O}(n^{10})$ time to reach less than some $\epsilon$ error, as
matrix size increases, the precision achievable in comparable time to our
algorithm would decrease. We scale the cumulative sum by $\frac{n!}{s}$, where
$s$ is the number of samples. This is the ratio of the total possible
permutations and the number of samples.
In our experiments, determinants and the products of diagonals are neither
accurate nor consistent approximations of the permanent. Sampling, however, is
accurate with respect to absolute distance to the permanent, so for
applications where that is most important, it may be best to apply some sort
of sampling method. Our Bethe approximation seems the most consistent. While
the approximations of the permanent are off by a large amount, they seem to be
consistently off by some monotonic function of the true permanent. In many
cases, this virtue is more important than the absolute accuracy, since most
applications requiring a matrix permanent likely compare the permanents of
various matrices. These results are visualized for $n=8$ in Figure 2.
To measure the monotonicity and consistency of these approximations, we
consider the Kendall distance [5] between the ranking of the random matrices
according to the true permanent and their rankings according to the
approximations. Kendall distance is a popular way of measuring the distance
between two permutations. The Kendall distance between two permutations
$\pi_{1}$ and $\pi_{2}$ is
$\displaystyle D_{\textrm{Kendall}}(\pi_{1},\pi_{2})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{j=i+1}^{n}I\left((\pi_{1}(i)<\pi_{1}(j))\wedge(\pi_{2}(i)>\pi_{2}(j))\right).$
In other words, it is the total number of pairs where $\pi_{1}$ and $\pi_{2}$
disagree on the ordering. We normalize the Kendall distance by dividing by
$\frac{n(n-1)}{2}$, the maximum possible distance between permutations, so the
distance will always be in the range $[0,1]$. Table 1 lists the Kendall
distances between the true permanent ranking and the four approximations. The
Kendall distance of the Bethe approximation is consistently less than that of
our sampler.
Table 2: Left: Error rates of running SVM using various kernels on the
original three UCI datasets and data where the features are shuffled randomly
for each datum. Right: UCI resampled pendigits data with order of points
removed. Error rates of 1-versus-all multi-class SVM using various kernels.
Kernel | Heart | Pima | Ion.
---|---|---|---
Original Linear | 0.1600 | 0.2600 | 0.1640
Orig. RBF $\sigma=0.3$ | 0.2908 | 0.3160 | 0.1240
Orig. RBF $\sigma=0.5$ | 0.2158 | 0.3220 | 0.0760
Orig. RBF $\sigma=0.7$ | 0.1912 | 0.2760 | 0.0960
Shuffled Linear | 0.2456 | 0.3080 | 0.2640
Shuff. RBF $\sigma=0.3$ | 0.4742 | 0.3620 | 0.4840
Shuff. RBF $\sigma=0.5$ | 0.3294 | 0.3140 | 0.3580
Shuff. RBF $\sigma=0.7$ | 0.2964 | 0.3280 | 0.2700
Bethe $\sigma=0.3$ | 0.2192 | 0.2900 | 0.1000
Bethe $\sigma=0.5$ | 0.2140 | 0.2900 | 0.1380
Bethe $\sigma=0.7$ | 0.2164 | 0.2920 | 0.1380
Kernel | PenDigits
---|---
Sorted Linear | 0.3960
Sorted RBF $\sigma=0.2$ | 0.4223
Sorted RBF $\sigma=0.3$ | 0.3407
Sorted RBF $\sigma=0.5$ | 0.3277
Shuffled Linear | 0.7987
Shuff. RBF $\sigma=0.2$ | 0.9183
Shuff. RBF $\sigma=0.3$ | 0.9120
Shuff. RBF $\sigma=0.5$ | 0.8657
Bethe $\sigma=0.2$ | 0.1463
Bethe $\sigma=0.3$ | 0.1190
Bethe $\sigma=0.5$ | 0.1707
### 4.3 Approximate Permanent Kernel
To illustrate a possible usage of an efficient permanent approximation, we use
a recent result [2] proving that the permanent of a valid kernel matrix
between two sets of points is also a valid kernel between point sets. Since
the permanent is invariant to permutation, we decided to try a few
classification tasks using an approximate permanent kernel. The permanent
kernel is computed by first computing a valid subkernel between a pairs of
elements in two sets, then the permanent of those subkernel evaluations is
taken as the kernel value between the data. Surprisingly, in experiments the
kernel matrix produced by our algorithm was a valid positive definite matrix.
This discovery opens up some intriguing questions to be explored later.
We ran a similar experiment to [15] where we took a the first 200 examples of
each of the Cleveland Heart Disease, Pima Diabetes, and Ionosphere datasets
from the UCI repository [4], and randomly permuted the features of each
example, then compare the result of training an SVM on this shuffled data. We
also provide the performance of the kernels on the unshuffled data for
comparison. After normalizing the features of the data to the $[0,1]^{D}$ box,
we chose three reasonable settings of $\sigma$ for the RBF kernels and cross
validated over various settings of the regularization parameter $C$. We used
RBF kernels between pairs of data as the permanent subkernel. Finally, we
report the average error over 50 random splits of 150 training points and 50
testing points. Not surprisingly, the permanent kernel is robust to the
shuffling and outperforms the standard kernels.
We also tested the Bethe kernel on the pendigits dataset, also from the UCI
repository. The original pendigits data consists of stylus coordinates of test
subjects writing digits. We used the preprocessed version that has been
resampled spatially and temporally. However, we omit the order information and
treat the input as a cloud of unordered points. Since there is a natural
spatial interpretation of this data, so we compare to sorting by distance from
origin, a simple method of handling unordered data. We chose slightly
different $\sigma$ values for the RBF kernels. For this dataset, there are 10
classes, one for each digit, so we used a one-versus-all strategy for multi-
class classification. Here we averaged over only 10 random splits of 300
training points and 300 testing points (see Table 2).
Based on our experiments, the permanent kernel typically does not outperform
standard kernels when permutation invariance is not inherently necessary in
the data, but when we induce the necessity of such invariance, its utility
becomes clear.
## 5 Discussion and Future Directions
We have described an algorithm based on BP over a specific distribution that
allows an efficient approximation of the $\\#P$ matrix permanent operation. We
write a probability distribution over matchings and use Bethe free energy to
approximate the partition function of this distribution. The algorithm is
significantly faster than sampling methods, but attempts to minimize a
function that approximates the permanent. Therefore it is limited by the
quality of the Bethe approximation so it cannot be run longer to obtain a
better approximation like sampling methods can. However, we have shown that
even on small matrices where sampling methods can achieve extremely high
accuracy of approximation, our method is more well behaved than sampling,
which can approach the exact value from above or below.
In the future, we can try other methods of approximating the partition
function such as generalized belief propagation [18], which takes advantage of
higher order Kikuchi approximations of free energy. Unfortunately the
structure of our graphical model causes higher order interactions to become
expensive quickly, since each variable has exactly $N$ neighbors. Similarly,
the bounds on the partition function in [17] are based on spanning subtrees in
the graph, and again the fully connected bipartite structure makes it
difficult to capture the true behavior of the distribution with trees.
Finally, the positive definiteness of the kernels we computed is surprising,
and requires further analysis. The exact permanent of a valid kernel forms a
valid Mercer kernel [2] because it is a sum of positive products, but since
our algorithm outputs the results of an iterative approximation of the
permanent, it is not obvious why the resulting output would obey the positive
definite constraints.
#### Acknowledgments
## References
* [1] M. Bayati, D. Shah, and M. Sharma. Maximum weight matching via max-product belief propagation. In Proc. of the IEEE International Symposium on Information Theory, 2005.
* [2] M. Cuturi. Permanents, transportation polytopes and positive definite kernels on histograms. In International Joint Conference on Artificial Intelligence, IJCAI, 2007.
* [3] P. Dagum and M. Luby. Approximating the permanent of graphs with large factors. Theoretical Computer Science, 102(2):283–305, 1992.
* [4] C.L. Blake D.J. Newman, S. Hettich and C.J. Merz. UCI repository of machine learning databases, 1998.
* [5] R. Fagin, R. Kumar, and D. Sivakumar. Comparing top k lists, 2003.
* [6] T. Heskes. Stable fixed points of loopy belief propagation are local minima of the bethe free energy. In S. Thrun S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 343–350. MIT Press, Cambridge, MA, 2003.
* [7] T. Heskes. Convexity arguments for efficient minimization of the bethe and kikuchi free energies. Journal of Artificial Intelligence Research, 26, 2006.
* [8] Tom Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Comput., 16(11):2379–2413, 2004.
* [9] B. Huang and T. Jebara. Loopy belief propagation for bipartite maximum weight b-matching. In Artificial Intelligence and Statistics (AISTATS), 2007.
* [10] M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM, 51(4):671–697, 2004.
* [11] Knopp and Sinkhorn. Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics, 1967.
* [12] Anand Rangarajan, Alan Yuille, and Eric Mjolsness. Convergence properties of the softassign quadratic assignment algorithm. Neural Comput., 11(6):1455–1474, 1999.
* [13] H. J. Ryser. Combinatorial mathematics. The Carus Mathematical Monographs, (14), 1963.
* [14] R. A. Servedio and A. Wan. Computing sparse permanents faster. Inf. Process. Lett., 96(3):89–92, 2005.
* [15] P. Shivaswamy and T. Jebara. Permutation invariant svms. In International Conference on Machine Learning, ICML, 2006.
* [16] S. Tatikonda and M. Jordan. Loopy belief propagation and Gibbs measures. In Proc. Uncertainty in Artificial Intell., vol. 18, 2002.
* [17] M. Wainwright, T. Jaakkola, and A. Willsky. A new class of upper bounds on the log partition function, 2002.
* [18] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory, 51(7), 2005.
* [19] A. L. Yuille. Cccp algorithms to minimize the bethe and kikuchi free energies: Convergent alternatives to belief propagation. Neural Computation, 14(7):1691–1722, 2002.
|
arxiv-papers
| 2009-08-12T18:27:54 |
2024-09-04T02:49:04.592081
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bert Huang and Tony Jebara",
"submitter": "Bert Huang",
"url": "https://arxiv.org/abs/0908.1769"
}
|
0908.1821
|
# NOTAS SOBRE ANÁLISIS FUNCIONAL
Jaime Chica
(Date: 7 de Agosto de 2007)
###### Abstract.
Estas son las notas de clase no publicadas, sobre el curso de Ánalisis
Funcional, impartido por el Prof. Jaime Chica, en la Facultad de Matemáticas
de la U de A.
###### Contents
1. 1 Isometrías entre E.L.N
1. 1.1 El teorema de Hahn-Banach. (caso real)
2. 2 Algunas consecuencias del T.H.B
3. 3 Los espacios $\mathfrak{L}^{p}$
4. 4 Mapeos Bilineales
5. 5 La teor a algebra ca del producto tensorial.
Sean $E,F\in\text{Norm}.$
Llamaremos $\mathcal{L}(E,F)=\bigl{\\{}T:E\longrightarrow F\diagup\text{T es
Apli. Lineal}\bigr{\\}}\\\ \\\
\mathcal{L}_{c}(E,F)=\bigl{\\{}T:E\longrightarrow F\diagup\text{T es A.L
Continua}\bigr{\\}}$
La proposición que sigue es importantísima en todo el escrito.
###### Proposición 1 (Continuidad de una A.L. entre Espacios Normados).
Sean E,F esp. Normados y $T:E\longrightarrow F$ una A.L, i.e,
$T\in\mathcal{L}(E,F).$
Las siguientes afirmaciones son equivalentes:
1. (1)
T es continua, o sea, $T\in\mathcal{L}_{c}(E,F)$
2. (2)
T es continua en 0.
3. (3)
$\exists M>0$ tal que $\forall x\in E:\|T(x)\|\leqslant M\|x\|.$
4. (4)
T es Uniformemente continua.
###### Proof.
$(1)\Rightarrow(2):$ trivial.
$(2)\Rightarrow(3).$ Supongamos que T es una A.L. continua en 0. Veamos que:
$\exists M>0$ tal que
(1) $\displaystyle\forall x\in E:\|T(x)\|\leqslant M\|x\|$
Tomemos $\epsilon=1/2$
Como T es continua en 0 y $\epsilon=1/2>0,\exists\delta>0\,\,\text{tal que
$\forall x\in E:\|x\|<\delta\Rightarrow\|T(x)\|<1/2\hskip 14.22636pt\star$}$
Tomemos $\underset{fijo}{\underbrace{x}}\in E,x\neq 0.$
Entonces
$\begin{Vmatrix}\frac{\delta}{2}\frac{x}{\|x\|}\end{Vmatrix}=\frac{\delta}{2}\frac{\|x\|}{\|x\|}=\frac{\delta}{2}<\delta$
y por $\hskip
14.22636pt\star,\begin{Vmatrix}T\Biggl{(}\frac{\delta}{2}\frac{x}{\|x\|}\Biggr{)}\end{Vmatrix}<1/2$
O sea $\frac{\delta}{2}\frac{1}{\|x\|}\|T(x)\|<1/2,$ i.e:
$\|T(x)\|<\frac{1}{\delta}\|x\|.$
Luego si llamamos $M=\frac{1}{\delta}$ hemos demostrado que $\exists
M>0\,\,\text{tal que $\forall x\in E,x\neq 0:\|T(x)\|\leqslant M\|x\|$}$
desigualdad que es obvia para el caso $x=0.$
$(3)\Rightarrow(4).$ Supongamos ahora que $T:E\longrightarrow F$ es A.L. y que
$\exists M>0$ tal que $\forall x\in E:\|T(x)\|\leqslant M\|x\|\hskip
14.22636pt\star\star$
Veamos que, T es Unif. continua.
Sea $\epsilon>0.$ Tomemos $\delta=\frac{\epsilon}{M}.$
Entonces, $\forall x,y\in E$ con
$\|x-y\|<\delta=\frac{\epsilon}{M},\|T(x)-T(y)\|=\|T(x-y)\|\underset{\overset{\uparrow}{\star\star}}{\leqslant}M\|x-y\|<M\frac{\epsilon}{M}=\epsilon$
Esto demuestra que T es Unif. continua.
$(4)\Rightarrow(1)$ Supongamos ahora que $T:E\longrightarrow F$ es unif.
continua. Veamos que T es continua.
Sea $\epsilon>0.$ Como T es unif. continua, $\exists\delta>0$ tal que $\forall
x,y\in E$ con
(2) $\displaystyle\|x-y\|<\delta:\|T(x)-T(y)\|<\epsilon$
Tomemos $\underset{fijo}{\underbrace{x_{0}}}\in E$ y veamos que T es continua
en $x_{0}.$
Como $\epsilon>0,$ debemos demostrar que $\exists\delta^{\prime}>0$ tal que
$\forall x\in E,\|x-x_{0}\|<\delta^{\prime},\|T(x)-T(x_{0})\|<\epsilon.$
Tomemos $\delta^{\prime}=\frac{\delta}{2}>0.$
Entonces, $\forall x\in E$ con $\|x-x_{0}\|<\dfrac{\delta}{2}<\delta,$ se
tiene por [2] que $\|T(x)-T(y)\|<\epsilon.$
Así que dado $\epsilon>0,\exists\delta>0$ tal que $\forall x\in E,$ si
$\|x-x_{0}\|<\delta$ entonces $\|T(x)-T(x_{0})\|<\epsilon,$ lo que demuestra
que T es continua en $x_{0}$ y como $x_{0}$ es cualquier punto de E, T es
continua en E.
∎
Sabemos que $\mathcal{L}(E,F)$ es un $\mathbb{K}$ esp. vectorial.
Podemos ahora probar que $\mathcal{L}_{c}(E,F)$ es un subespacio de
$\mathcal{L}(E,F).$
###### Proposición 2.
Sean E,F esp. normados. Entonces
$\mathcal{L}_{c}(E,F)\subset\mathcal{L}(E,F).$
###### Proof.
1. i)
Consideremos el caso 0.
$\begin{diagram}$
$\begin{diagram}$
Veamos que $0\in\mathcal{L}_{c}(E,F).$
Tomemos $\underset{fijo}{\underbrace{M}}>0.$ Entonces,$\forall x\in
E:\|0_{x}\|=\|0_{F}\|=0\leqslant M\|x\|.$
Esto demuestra que $0\in\mathcal{L}_{c}(E,F)$ y por tanto,
$\mathcal{L}_{c}(E,F)\neq\emptyset.$
2. ii)
Sean $T_{1},T_{2}\in\mathcal{L}_{c}(E,F).$ Veamos que
$(T_{1}+T_{2})\subset\mathcal{L}_{c}(E,F)$
Es claro que $T_{1}+T_{2}\in\mathcal{L}(E,F)$
Resta demostrar que $\exists M>0$ tal que
(3) $\displaystyle\forall x\in E:\|(T_{1}+T_{2})_{x}\|\leqslant M\|x\|$
Como $T_{1}\in\mathcal{L}_{c}(E,F),\exists M_{1}>0$ tal que
(4) $\displaystyle\forall x\in E:\|(T_{1})_{x}\|\leqslant M_{1}\|x\|$
Como $T_{2}\in\mathcal{L}_{c}(E,F),\exists M_{2}>0$ tal que
(5) $\displaystyle\forall x\in E:\|(T_{2})_{x}\|\leqslant M_{2}\|x\|$
As que:
$\|(T_{1}+T_{2})_{x}\|=\|T_{1}(x)+T_{2}(x)\|\underset{\overset{\nearrow}{(4),(5)}}{\leqslant}\|T_{1}(x)\|+\|T_{2}(x)\|\leqslant
M_{1}\|x\|+M_{2}\|x\|=(M_{1}+M_{2})\|x\|$
y se tiene (3)
3. iii)
Se tiene $\alpha\in\mathbb{K}$ y $T\in\mathcal{L}_{c}(E,F).$ Veamos que
(6) $\displaystyle(\alpha T)\in\mathcal{L}_{c}(E,F)$
Si $\alpha=0,\alpha T=0\in\mathcal{L}_{c}(E,F)$
Supongamos $\alpha\neq 0.$
Como
(7) $\displaystyle T\in\mathcal{L}_{c}(E,F),\exists M>0\,\,\text{tal que
$\forall x\in E:\|T(x)\|\leqslant M\|x\|$}$
Ahora, $\forall x\in E:\|\bigl{(}\alpha
T\bigr{)}(x)\|=|\alpha|\|T(x)\|\leqslant|\alpha|M\|x\|,$
lo que nos demuestra (6)
∎
###### Proposición 3.
Sean E,F:ELN. Si $dimE=n,$ toda AL. $T:E\longrightarrow F$ es continua.
###### Proof.
Sea
$\begin{diagram}$
$\begin{diagram}$
T; AL, donde E,F:ELN.
La funci n
$\begin{diagram}$
$\begin{diagram}$
es una norma en E.
En efecto,
* •
${\|x\|}^{*}\geqslant 0$
Si ${\|x\|}^{*}=0,\text{m
x$\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}=0$}\Rightarrow\|x\|_{E}=0,\|T(x)\|_{F}=0$
* •
${\|\alpha x\|}^{*}=mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}\\\ \\\
=|\alpha|mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}\\\ \\\
=|\alpha|{\|x\|}^{*}$
* •
Veamos la desigualdad triangular:
${\|x+y\|}^{*}=mx\bigl{\\{}\|x+y\|_{E}\|T(x+y)\|_{F}\bigr{\\}}\\\ \\\
=\|x+y\|_{E}\leqslant\|x\|_{E}+\|y\|_{E}\\\ \\\ \leqslant
mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}+mx\bigl{\\{}\|y\|_{E}\|T(y)\|_{F}\bigr{\\}}\\\
\\\ ={\|x\|}^{*}+{\|y\|}^{*}\\\ \\\
\|T(x+y)\|_{F}=\|T(x)+T(y)\|_{F}\leqslant\|T(x)\|_{F}+\|T(y)\|_{F}\\\ \\\
\leqslant
mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}+mx\bigl{\\{}\|y\|_{E}\|T(y)\|_{F}\bigr{\\}}\\\
\\\ ={\|x\|}^{*}+{\|y\|}^{*}$
Como ${\|\|}^{*}$ es una norma en E y $dimE=n,{\|\|}^{*}\sim\|\|_{E}$ y por
tanto,
$\exists\beta>0\,\,\text{tal que $\forall x\in
E:{\|x\|}^{*}\leqslant\beta\|x\|_{E}.$}$
As que $\exists\beta>0\,\,\text{tal que $\forall x\in E:$}\\\ \\\
\|T(x)\|_{F}\leqslant
mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}={\|x\|}^{*}\leqslant\beta\|x\|_{E}\\\
\\\ {\|x\|}^{*}=mx\bigl{\\{}\|x\|_{E}\|T(x)\|_{F}\bigr{\\}}.$
Lo que nos demuestra que T es continua.
∎
Esto demuestra que M es cota superior del conjunto de números reales
$\bigl{\\{}\|T(x)\|,\|x\|\leqslant 1\bigr{\\}}.$ Luego
$\exists\,\,\underset{\|x\|\leqslant 1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}.$
###### Definición 1.
Al número real $\underset{\|x\|\leqslant 1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$
se le llama la $\underline{\text{Norma de T}}$ y se indica
$\|T\|=\underset{\|x\|\leqslant 1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$
###### Proposición 4.
Sean E y F: ELN. Sabemos que $\mathcal{L}_{c}(E,F):$ K esp. vect.
La funci n $\begin{diagram}$
$\begin{diagram}$
es una norma en $\mathcal{L}_{c}(E,F).$
###### Proof.
1. (1)
Es claro que $\|T\|\geqslant 0$
2. (2)
Supongamos que $\|T\|=0.$ Veamos que $T=0$. Entonces $\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}=0.$
O sea que $\forall x\in E$ con $\|x\|\leqslant 1:\|T(x)\|=0\Rightarrow
T(x)=0.$
Esto demuestra que
(8) $\displaystyle\forall x\in E\,\,con\|x\|\leqslant 1:\|T(x)\|=0\Rightarrow
T(x)=0$
Tomemos ahora $y\in E$ con $y\neq 0.$
$\begin{Vmatrix}\dfrac{y}{\|y\|}\end{Vmatrix}=\dfrac{1}{\|y\|}\|y\|=1.$ Luego
por (8), $\begin{Vmatrix}T\Biggl{(}\dfrac{y}{\|y\|}\Biggr{)}\end{Vmatrix}=0$
O sea que
$\dfrac{1}{\|y\|}\begin{Vmatrix}T(y)\end{Vmatrix}=0\Rightarrow\|T(y)\|=0$ i.e,
$T(y)=0.$
Luego $\forall y\in E$ con $y\neq 0,T(y)=0.$
Hemos demostrado así que si $\|T\|=0,T$ es la aplicación cero.
3. (3)
Sea $\alpha\in\mathbb{K}$ y $T\in\mathcal{L}_{c}(E,F).$ Entonces $(\alpha
T)\in\mathcal{L}_{c}(E,F).$
Ahora, $\|\alpha T\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|\alpha T(x)\|\bigr{\\}}}\\\ \\\
=|\alpha|\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}=|\alpha|\|T\|.$
4. (4)
Sean $T_{1},T_{2}\in\mathcal{L}_{c}(E,F).$
Entonces $T_{1}+T_{2}\in\mathcal{L}_{c}(E,F)$ y por tanto,
(9) $\displaystyle\|T_{1}+T_{2}\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|\bigl{(}T_{1}+T_{2}\bigr{)}(x)\|\bigr{\\}}}=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T_{1}(x)+T_{2}(x)\|\bigr{\\}}}$
Ahora, $\|T_{1}\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T_{1}(x)\|\bigr{\\}}}.$ Luego $\forall x\in E\,\,\text{con
$\|x\|\leqslant 1:\|T_{1}(x)\|\leqslant\|T_{1}\|$}$
Como $\|T_{2}\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T_{2}(x)\|\bigr{\\}}}.$ Luego $\forall x\in E\,\,\text{con
$\|x\|\leqslant 1:\|T_{2}(x)\|\leqslant\|T_{2}\|$}$
As que $\forall x\in
E:\|T_{1}(x)+T_{2}(x)\|\leqslant\|T_{1}(x)\|+\|T_{2}(x)\|\leqslant\|T_{1}\|+\|T_{2}\|$
desigualdad que nos muestra que $\|T_{1}(x)\|+\|T_{2}(x)\|$ es cota superior
del conjunto de lo números reales
$\bigl{\\{}\|T_{1}(x)+T_{2}(x)\|,\|x\|\leqslant 1\bigr{\\}}$
Luego $\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T_{1}(x)+T_{2}(x)\|\bigr{\\}}}\leqslant\|T_{1}(x)\|+\|T_{2}(x)\|$
y regresando a (9) se tiene que
$\|T_{1}\|+\|T_{2}\|\leqslant\|T_{1}\|+\|T_{2}\|.$
∎
###### Proposición 5.
Sean $E,F\in\text{Norm y $T\in\mathcal{L}_{c}(E,F).$}$
Entonces, $\forall x\in E:\|T(x)\|\leqslant\|T\|\|x\|.$
###### Proof.
Sabemos que $\|T\|_{\mathcal{L}_{c}(E,F)}=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$
Luego,
(10) $\displaystyle\forall x\in E\,\,\text{con $\|x\|_{E}\leqslant
1:\|T(x)\|\leqslant\|T\|$}$
Ahora, $\forall x\in E\,\,\text{con $x\neq
0,\begin{Vmatrix}\dfrac{x}{\|x\|}\end{Vmatrix}=1$}$ y al tener en cuenta (10),
$\begin{Vmatrix}T\Biggl{(}\dfrac{x}{\|x\|}\Biggr{)}\end{Vmatrix}\leqslant\|T\|$
o sea que $\|T(x)\|\leqslant\|T\|\|x\|.$
∎
###### Ejercicio 1.
Sea $E\in\text{Norm}.$
Demostrar que si una S. de Cauchy en
$E=\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}$ tiene una subsucesión convergente
a x, la sucesión $x_{n}\longrightarrow x.$
###### Soluci n 1.
Sea $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset E\in\text{Norm}.$ y
supongamos que
$\bigl{\\{}x_{n_{1}},x_{n_{2}},\ldots,x_{n_{j}},\ldots\bigr{\\}}\subset\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}$
tal que $\lim\limits_{n_{j}\rightarrow\infty}x_{n_{j}}=x.$ Veamos que
$x_{n}\longrightarrow x.$
Sea $\epsilon>0.$ Entonces $\dfrac{\epsilon}{2}>0$ y con
$\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}$ es una S. Cauchy en E,
(11) $\displaystyle\exists N_{1}\in\mathbb{N}\text{tal que $\forall
m,n>N_{1}:\|x_{n}-x_{m}\|<\dfrac{\epsilon}{2}$}$
Como
$\bigl{\\{}x_{n_{1}},x_{n_{2}},\ldots,x_{n_{j}},\ldots\bigr{\\}}\longrightarrow
x$ y $\dfrac{\epsilon}{2}>0,$
(12) $\displaystyle\exists N_{2}\in\mathbb{N}\text{tal que $\forall
n_{j}>N_{2}:\|x_{n_{j}}-x\|<\dfrac{\epsilon}{2}$}$
Tomemos $n>m\'{a}x\bigl{\\{}N_{1},N_{2}\bigr{\\}}.$
Entonces $\exists_{j}>N_{2}\text{tal que $n<_{j}$}$ y se tiene por (13) que
$\|x_{n}-x_{{}_{j}}\|<\dfrac{\epsilon}{2}$
Como $n,_{j}>N_{1},\|x_{n}-x_{{}_{j}}\|<\dfrac{\epsilon}{2}$
Luego
$\|x_{n}-x\|=\|\bigl{(}x_{n}-x_{{}_{j}}\bigr{)}+\bigl{(}x_{{}_{j}}-x\bigr{)}\|\leqslant\|x_{n}-x_{{}_{j}}\|+\|x_{{}_{j}}-x\|<\dfrac{\epsilon}{2}+\dfrac{\epsilon}{2}=\epsilon$
Esto prueba que $\forall
n>m\'{a}x\bigl{\\{}N_{1},N_{2}\bigr{\\}}:\|x_{n}-x\|<\epsilon,$ i.e,
$x_{n}\longrightarrow x.$
###### Ejercicio 2.
Sean $E,F\in\text{Norm}$ y $T\in\mathcal{L}_{c}(E,F).$ Hemos definido
$\|T\|=\underset{\|x\|\leqslant 1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}.$
Probar que $\|T\|=\underset{\|x\|=1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}.$
###### Soluci n 2.
Es claro que
$\bigl{\\{}\|T(x)\|\diagup\|x\|=1\bigr{\\}}\subset\bigl{\\{}\|T(x)\|\diagup\|x\|\leqslant
1\bigr{\\}}.$
Como $\|T\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}},\|T\|$ es cota superior del $2^{do}$
cjto.
Luego $\|T\|$ es cota superior del $1^{er}$ cjto y
$\exists\,\,\underset{\|x\|=1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$ teniéndose
que
$\underset{\|x\|=1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}\leqslant\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}=\underset{\|u\|\leqslant
1}{\sup\Biggl{(}\|u\|\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix}\Biggr{)}}\underset{\overset{\nearrow}{\star}}{\leqslant}\underset{\|x\|=1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$
Justifiquemos $\star.$
Llamemos
$S=\bigl{\\{}\|u\|\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix},\|u\|\leqslant
1\bigr{\\}}$
y $W=\bigl{\\{}\|T(y)\|,\|y\|=1\bigr{\\}}.$ Veamos que $\forall s\in S,\exists
w\in W$ tal que $s<w$ con lo que se tendría que $\sup S\leqslant\sup W,$ i.e,
se tendría $\star.$
Sea $s\in S.$ Entonces
(13) $\displaystyle
s=\|u\|\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix}\underset{\overset{\nearrow}{\|u\|\leqslant
1}}{\leqslant}\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix}$
Tomemos $w=\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix}.$
Es claro que $w\in W$ y que
$s\underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{d}}}{\leqslant}\begin{Vmatrix}T\Biggl{(}\dfrac{u}{\|u\|}\Biggr{)}\end{Vmatrix}=w$
###### Observación 1.
Con ligeras variantes podemos demostrar que
$\|T\|=\underset{\|x\|<1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}$
## 1\. Isometrías entre E.L.N
###### Definición 2.
Sean $E,F\in Norm.$ y $T:E\longrightarrow F,T\in\mathcal{L}_{c}(E,F).$
1. (1)
T preserva la norma si $\forall x\in E:\|T(x)\|=\|x\|$
2. (2)
T preserva la distancia si $\forall x,y\in
E:d\bigl{(}T(x),T(y)\bigr{)}=\|T(x)-T(y)\|=\|x-y\|=d\bigl{(}x,y\bigr{)}.$
###### Proposición 6.
1. (1)
Si T preserva la norma, T preseva la distancia.
2. (2)
Si T preserva la distancia, T preserva la norma.
###### Proof.
1. (1)
$\forall x,y\in
E:d\bigl{(}T(x),T(y)\bigr{)}=\|T(x)-T(y)\|=\|T(x-y)\|=\|x-y\|=d\bigl{(}x,y\bigr{)}.$
2. (2)
Sea $x\in E.$ Veamos que $\|T(x)\|=\|x\|.\\\ \\\
\|T(x)\|=\|T(x)-0_{F}\|=\|T(x)-T(0_{E})\|=d\bigl{(}T(x),T(0_{E})\bigl{)}\\\
\\\ =d\bigl{(}x,0_{E}\bigr{)}=\|x-0_{E}\|=\|x\|.$
∎
###### Definición 3.
Sean $E,F\in Norm.$ Una función $T:E\longrightarrow F$ tal que
1. (1)
$T\in\mathcal{L}(E,F).$
2. (2)
T preserva la norma (ó la distancia) se llama una
$\underline{\text{isometr\'{i}a}}$ entre E,F.
###### Proposición 7.
1. (1)
Toda isometría es 1-1.
2. (2)
La composición de isometrías es una isometría.
3. (3)
Toda isometría es Unif. continua y por tanto, toda isometría es una función
continua.
4. (4)
Si T es una isometría entre E y F y T es sobre, $T^{-1}$ es también una
isometría de F en E.
###### Proof.
1. (1)
Sean $E,F\in Norm$ y $T:E\longrightarrow F,$ una A.L. tal que $\forall x\in
E:\|T(x)\|=\|x\|$ i.e, T es una isometría entre E,F.
Veamos que $T$ es 1-1 que $KerT=\bigl{\\{}0_{E}\bigr{\\}}$
Sea $x\in KerT\Rightarrow T(x)=0_{F}\Rightarrow\|x\|=\|T(x)\|=\|0_{F}\|=0;$
i.e, $\|x\|=0\Rightarrow x=0_{E}.$
2. 2 y 3
son inmediatas.
3. 4
Sean $E,F\in Norm$ y $T:E\longrightarrow F,T\in\mathcal{L}(E,F)$ T: isometría
(y por tanto 1-1 y sobre.)
Entonces
$\begin{diagram}$
$\begin{diagram}$
donde
(14) $\displaystyle T(x)=y$
es A.L. i.e, ${T^{-1}}\in\mathcal{L}(F,E).$
Veamos que $\forall y\in F:\|T^{-1}(y)\|=\|y\|.$
Sea $y\in F.$ Entonces $\exists!x\in E$ tal que
$T^{-1}(y)=x:\|T^{-1}(y)\|=\|x\|=\|T(x)\|\underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{e}}}{=}\|y\|$
∎
###### Observación 2.
Si T es una isometría entre E,F; $T\in\mathcal{L}_{c}(E,F).$ Luego
$\|T\|=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|T(x)\|\bigr{\\}}}=\underset{\|x\|\leqslant
1}{\sup\bigl{\\{}\|x\|\bigr{\\}}}=1$ lo que demuestra que $\underline{\text{la
norma de toda isometr\'{i}a es 1.}}$
###### Definición 4 (Isomorfismo isométrico entre Espacios Normados).
Sean $E,F\in Norm$ y $T:E\longrightarrow F,$ una A.L. biyectiva. Si tanto T
como $T^{-1}$ son continuas diremos que T es un isomorfismo (topológico) entre
E y F. Y si además $\forall x\in E:\|T(x)\|=\|x\|$ se dirá que T es un
isomorfismo isométrico entre E y F.
La siguiente Prop. da un criterio para establecer cuando una A.L. biyectiva
entre E.N. es un Iso. Topológico.
###### Proposición 8.
Sea $E\in Norm,dimE=n.$ Entonces $E\cong\mathbb{K}^{n}.$
###### Proof.
Sea $\bigl{\\{}e_{1},e_{2},\ldots,e_{n}\bigr{\\}}:$ base de E, y consideremos
la A.L.
$\begin{diagram}$
$\begin{diagram}$
Entonces:
1. (1)
T es A.L. Biyectiva y $T^{-1}$ también es A.L.
2. (2)
Veamos que T es continua. Bastará con demostrar que $\exists M>0$ tal que
$\forall x\in\mathbb{K}^{n}:\|T(x)\|\leqslant M\|x\|.$
Tomemos $x=(x_{1},\ldots,x_{n})\in\mathbb{K}^{n}$ y consideremos en
$\mathbb{K}^{n}$ la norma $\|x\|=|x_{1}|+\ldots+|x_{n}|.$
Como
(15) $\begin{split}|x_{1}|\leqslant\|x_{1}\|\\\ \vdots\\\
\underline{|x_{n}|\leqslant\|x_{n}\|}\\\ \sum\limits_{i=1}^{n}|x_{i}|\leqslant
n\|x\|\end{split}$
Sea $K=mx\bigl{\\{}\|e_{1}\|,\ldots,\|e_{n}\|\bigr{\\}}.$ Entonces
$\|T(x)\|=\|T(x_{1},\ldots,x_{n})\|\\\ \\\
=\|x_{1}e_{1}+\ldots+x_{n}e_{n}\|\\\ \\\
\leqslant|x_{1}|\|e_{1}\|+\ldots+|x_{n}|\|e_{n}\|\\\ \\\
\leqslant\bigl{(}|x_{1}|+\ldots+|x_{n}|\bigl{)}K\\\ \\\
=\bigl{(}\sum\limits_{i=1}^{n}|x_{i}|\bigr{)}K\underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{f}}}{\leqslant}nK\|x\|$
Así que $\exists M=nK>0$ tal que:
$K=m\'{a}x\bigl{\\{}\|e_{1}\|,\ldots,\|e_{n}\|\bigr{\\}}.$
$\forall x\in\mathbb{K}^{n}=\|T(x)\|\leqslant(nK)\|x\|$ lo que demuestra que
$T\in\mathcal{L}_{c}(\mathbb{K}^{n},E).$
3. (3)
Veamos ahora que $T^{-1}:E\longrightarrow\mathbb{K}^{n}$ es continua.
Bastará con demostrar que $\exists m>0$ tal que $\forall
x\in\mathbb{K}^{n}:m\|x\|\leqslant\|T(x)\|.$
Consideremos la esfera unidad en $\mathbb{K}^{n}$ de centro 0 y radio 1:
$\underset{\text{cerrado y acotado
}}{\underbrace{S(0,1)}}=\bigl{\\{}x\in\mathbb{K}^{n}\diagup\|x\|=1\bigr{\\}}\subset\mathbb{K}^{n}.$
Luego por el Teorema de Heine-Borel- Lebesgue, $S(0,1)$ es compacto.
Consideremos a su vez la función compuesta definida en el diagrama siguiente:
$\begin{diagram}$
$\|\|_{E}\circ T\diagup S(0,1):\\\ \begin{diagram}$
$\begin{diagram}$
Como $S(0,1)$ es compacto, la función $\|\|_{E}\circ T\diagup S(0,1)$ alcanza
un valor mínimo absoluto en $S(0,1).$
Así que $\exists a\in S(0,1)$ tal que
(16) $\displaystyle\forall x\in S(0,1):\|T(a)\|\leqslant\|T(x)\|$
$a\in S(0,1)\Rightarrow T(a)\neq 0\,\,\text{(ya que T es
1-1)$\Rightarrow\|T(a)\|>0$}$
Tomemos $x\in\mathbb{K}^{n},x\neq 0.$
Entonces $\dfrac{x}{\|x\|}\in S(0,1)$ y por tanto, teniéndose en cuenta (16):
$0<\|T(a)\|\leqslant\|T\bigl{(}\dfrac{x}{\|x\|}\|\bigr{)}\|x\|=\dfrac{1}{\|x\|}\|T(x)\|$
O sea que $\forall x\in\mathbb{K}^{n}:\|T(a)\|\|x\|\leqslant\|T(x)\|.$
Esto demuestra que $\exists m=\|T(a)\|>0$ tal que $\forall
x\in\mathbb{K}^{n}:m\|x\|\leqslant\|T(x)\|$ y por la Prop. anterior,
$T^{-1}:E\longrightarrow\mathbb{K}^{n}$ es continua.
T es as un $\mathbb{K}^{n}\cong E$.
∎
Antes de continuar conviene tener presente ciertas propiedades topológicas de
los E.N.
Sea $E\in Norm.$ Entonces $\bigl{(}E,d\bigr{)}$ es un Esp. métrico, donde
$d(x,y)=\|x-y\|.$ A su vez la distancia $d$ induce una topología sobre E en la
que la familia de vecindades $\mathcal{N}_{x}$ de un punto $x\in E$ se define
así:
$E\supset N\in\mathcal{N}_{x}\Leftrightarrow\exists\epsilon>0$ tal que
$B(0,\epsilon)\subseteq N.$
O sea que un conjunto $N\subset E$ es vecindad del punto $x\in E$ si el
conjunto contiene una bola de centro en el punto;
$B(x,\epsilon)=\bigl{\\{}u\in E\diagup d(u,x)=\|u-x\|<\epsilon\bigr{\\}}.$
Si consideramos a $E$ dotado de la topolog a inducida por la norma, se tienen
los siguientes resultados:
###### Proposición 9.
Sea $E\in Norm$ y $A\subset E.$
Entonces:
1. (1)
$\overline{A}=\mathcal{\text{cl}}(A)=\bigl{\\{}x\in E\diagup\forall
n\in\mathbb{N}:B(x,\frac{1}{n})\bigcap A\neq\emptyset\bigr{\\}}$
2. (2)
${A}^{\circ}=\mathcal{\text{int}}(A)=\bigl{\\{}x\in E\diagup\exists
n\in\mathbb{N}:B\bigl{(}x,\frac{1}{x}\bigr{)}\subset A\bigr{\\}}$
De esta manera,
$x\in\overline{A}\Leftrightarrow\forall n\in\mathbb{N}:B(x,\frac{1}{x})\bigcap
A\neq 0$
$x\in{A}^{\circ}\Leftrightarrow\exists n\in\mathbb{N}:B(x,\frac{1}{x})\subset
A$
###### Proof.
1. (1)
Sea $x\in\mathcal{\text{cl}}(A)=\overline{A}.$
Entonces
(17) $\displaystyle\forall N\in\mathcal{N}_{x}:N\cap A\neq\emptyset$
Recordemos que $E\supset N\in\mathcal{N}_{x}\Leftrightarrow\exists
B(x,\epsilon)\subseteq N.$
As que la familia
$\bigl{\\{}B(0,\epsilon),\epsilon>0\bigr{\\}}\subset\mathcal{N}_{x}.$
Luego al tener en cuenta (17), se tiene que
$\forall n\in\mathbb{N}:B\bigl{(}x,\frac{1}{n}\bigr{)}\bigcap A\neq\emptyset.$
Sea ahora $x\in E$ con la siguiente propiedad:
$\forall n\in\mathbb{N}:B\bigl{(}x,\frac{1}{n}\bigr{)}\bigcap A\neq\emptyset.$
Veamos que: $x\in\mathcal{\text{cl}}(A)$ o que $\forall
N\in\mathcal{N}_{x}:N\cap A\neq\emptyset.$
Tomemos $\underset{fijo}{\underbrace{N}}\in\mathcal{N}_{x}$
$\exists G:\text{abierto, tal que $x\in G\subset
N$}\Rightarrow\exists\epsilon>0$ tal que $B(x,\epsilon)\subset G\subset N.$
Pero si $\epsilon>0,\exists n\in\mathbb{N}$ tal que
$\frac{1}{n}<\epsilon\Rightarrow B(x,\frac{1}{n})\subset B(x,\epsilon)\subset
N.$
Ahora, por hip tesis, $B(x,\frac{1}{n})\bigcap A\neq 0.$ i.e, A encuentra a la
$B(x,\frac{1}{n})\subset N.$
Luego A encuentra a N; i.e, $N\cap A\neq\emptyset,$ cualquiera sea
$N\in\mathcal{N}_{x}.$
2. (2)
Se omite.
∎
###### Proposición 10.
Sea $E\in Norm,$ $A\subset E$ y $x\in E.$
Entonces
$x\in\overline{A}\Longleftrightarrow\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset
A$ tal que $\lim\limits_{n\rightarrow\infty}x_{n}=x.$
###### Proof.
$"\Longrightarrow"$ Sea $x\in\overline{A}.$
Entonces, por la Prop. anterior, $\forall
n\in\mathbb{N}:B(x,\frac{1}{n})\bigcap A\neq\emptyset.$ O sea que $\forall
n\in\mathbb{N}:\exists x_{n}\in A$ tal que $x_{n}\in B(x,\frac{1}{n}).$
Esto a su vez quiere decir que
$\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset A$ tal que
$x_{1}\in B(x,1)\\\ \\\ x_{2}\in B(x,\frac{1}{2})\\\ \\\ x_{3}\in
B(x,\frac{1}{3})\\\ \\\ \ldots\ldots\\\ $
Es claro que fijado $\in\mathbb{N},\forall n>:x_{n}\in B(x,\frac{1}{}).$
Resta demostrar que $x_{n}\longrightarrow x.$
Sea $\epsilon>0.$ Entonces $\in\mathbb{N}$ tal que
$\frac{1}{}<\epsilon\Longrightarrow B(x,\frac{1}{})\subset B(x,\epsilon).$
Pero $\forall n>:x_{n}\in B(x,\frac{1}{})\subset B(x,\epsilon)$, i.e, $\forall
n>:x_{n}\in B(x,\epsilon)$,
lo que demuestra que $x_{n}\longrightarrow x.$
$"\Longleftarrow"$ Supongamos ahora que $x\in E$ tiene esta propiedad:
(18) $\displaystyle\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset
A\,\,\text{tal que $\lim\limits_{n\rightarrow\infty}x_{n}=x$}$
Veamos $x\in\overline{A}.$
Por la Prop. anterior bastar con dm. que $\forall
n\in\mathbb{N}:B\bigl{(}x,\frac{1}{n}\bigr{)}\bigcap A\neq\emptyset.$
Tomemos $n\in\mathbb{N}.$ Entonces $\frac{1}{n}>0$ y como
$x_{n}\longrightarrow x,\exists N\in\mathbb{N}$ tal que $\forall p>N:x_{p}\in
B(x,\frac{1}{n}).$
Pero por (18), $x_{p}\in A,\forall p>N.$
As que $\exists N\in\mathbb{N}$ tal que $p>N:x_{p}\in B(x,\frac{1}{n})\bigcap
A.$
Esto demuestra que $\forall n\in\mathbb{N}:B(x,\frac{1}{n})\bigcap
A\neq\emptyset.$
∎
Recordemos la definici n de sucesi n convergente en un E. Normado.
Sea $E\in Norm,\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}$ sucesi n en E y $x\in
E.$
$x=\lim\limits_{n\rightarrow\infty}x_{n}\Longleftrightarrow\forall\epsilon>0\\\
\exists N\in\mathbb{N}\text{tal que $\forall n>N,\|x_{n}-x\|<\epsilon$}.$
La prueba de la unicidad del l mite es trivial.
Los siguientes hechos se establecen tambi n de manera trivial.
Sea $E\in Norm,\forall x,y\in E:\left|\|x\|-\|y\|\right|\leqslant\|x\pm
y\|\leqslant\|x\|+\|y\|.$
Si $x_{n}\longrightarrow x,\alpha x_{n}\longrightarrow\alpha x.$
Si $x_{n}\longrightarrow x,y_{n}\longrightarrow y,x_{n}+y_{n}\longrightarrow
x+y.$
###### Proposición 11.
La clausura de todo subespacio de un E.L.N. es un subespacio.
###### Proof.
Sea $E\in Norm$ y $E\supset S:$ subespacio de E. Veamos que $\overline{S}$ es
un subespacio de E.
1. (1)
Como $E\supset S$ y S: subespacio de E, $S\neq\emptyset.$ Ahora
$S\subset\overline{S}.$ Luego $\overline{S}\neq\emptyset.$
2. (2)
Sean $x,y\in\overline{S}.$ Veamos que $(x+y)\in\overline{S}.$
Batar con demostrar por la Prop. anterior, que
$\exists\bigl{\\{}u_{n}\bigr{\\}}_{n=1}^{\infty}\subset S.$ tal que
$\lim\limits_{n\rightarrow\infty}u_{n}=x+y.$
Como $x\in\overline{S},\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset
S$ tal que $\lim\limits_{n\rightarrow\infty}x_{n}=x$
Como $y\in\overline{S},\exists\bigl{\\{}y_{n}\bigr{\\}}\subset S$ tal que
$\lim\limits_{n\rightarrow\infty}y_{n}=y.$
Como S es subespacio de E,
$\bigl{\\{}u_{n}\bigr{\\}}=\bigl{\\{}x_{n}+y_{n}\bigr{\\}}_{n=1}^{\infty}\subset
S.$
Luego $\bigl{\\{}u_{n}\bigr{\\}}_{n=1}^{\infty}\subset$ y adem s,
$\lim\limits_{n\rightarrow\infty}u_{n}=\lim\limits_{n\rightarrow\infty}(x_{n}+y_{n})=\lim\limits_{n\rightarrow\infty}x_{n}+\lim\limits_{n\rightarrow\infty}y_{n}=x+y$
3. (3)
Sea $x\in\overline{S}$ y $\alpha\in\mathbb{K}.$ Veamos que $(\alpha
x)\in\overline{S}.$
Como $x\in\overline{S},\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset
S$ tal que
$\lim\limits_{n\rightarrow\infty}x_{n}=x\Longrightarrow\bigl{\\{}\alpha
x\bigr{\\}}_{n=1}^{\infty}\subset$ y adem s,
$\lim\limits_{n\rightarrow\infty}(\alpha
x_{n})=\alpha\lim\limits_{n\rightarrow\infty}x_{n}=\alpha x.$
Esto demuestra que $\overline{S}$ es subespacio de E.
∎
###### Definición 5.
1. (1)
Sea $E\in Norm$ y $\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset E.$
Decimos que $\bigl{\\{}x_{n}\bigr{\\}}$ es una S. de Cauchy en E si la sucesi
n $\bigl{\\{}x_{n}\bigr{\\}}$ tiene la siguiente propiedad:
$\forall\epsilon>0,\exists N\in\mathbb{N}$ tal que $\forall
m,n>N:\|x_{m}-x_{n}\|<\epsilon.$
2. (2)
Sea $E\in Norm.$
Decimos que E es un E. de Banach si toda sucesi n de Cauchy en E converge.
###### Proposición 12.
Todo subespacio cerrado de un E. de Banach es tambi n de Banach.
###### Proof.
Es claro que S: Norm. Veamos que S:Banach.
Sea $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset S.$ Veamos que
$x_{n}\longrightarrow x\in S.$
Como $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset S\subset
E,\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset E.$
Luego, $x_{n}\longrightarrow x\in E.$ Resta demostrar que $x\in S.$
Como S es cerrado, $S=\overline{S},$ bastar con dm. que $x\in\overline{S},$ lo
cual resulta claro ya que si $\bigl{\\{}x_{n}\bigr{\\}}\subset S$ y
$x_{n}\longrightarrow x,x\in\overline{S}.$
∎
Vamos a establecer otras propiedades topol gicas de los E.N.
Sea $E\in Norm,a\in E$ y $\gamma>0.$
Consideremos la $B(a,\gamma)=\bigl{\\{}x\in E\diagup\|x-a\|<\gamma\bigr{\\}}$
y $S(a,\gamma)=\bigl{\\{}x\in E\diagup\|x-a\|=\gamma\bigr{\\}}.$
Vamos a demostrar que si E es infinito, estos conjuntos tienen infinitos
puntos.
* •
Tomemos $x\in E,x\neq 0.$
Entonces $\left(a+\dfrac{\gamma}{2\|x\|}x\right)\in B(a,\gamma).$
En
efecto:$\left\|a+\dfrac{\gamma}{2\|x\|}x-a\right\|=\dfrac{r}{2}\left\|\dfrac{x}{\|x\|}\right\|=\dfrac{\gamma}{2}<\gamma$
lo que demuestra que $\forall x\in E;\left(a+\dfrac{\gamma}{2\|x\|}x\right)\in
B(a,\gamma)$ y por tanto, si E es infinito, la $B(a,\gamma)$ contiene $\infty$
puntos.
* •
Adem s,
$\left\|a+\dfrac{\gamma}{2\|x\|}x-a\right\|=\gamma\left\|\dfrac{x}{\|x\|}\right\|=\gamma$
lo que prueba que $\forall x\in E,\\\ \left(a+\dfrac{\gamma}{\|x\|}x\right)\in
S(a,\gamma)$ y de nuevo, si E es infinito, $S(a,\gamma)$ es un conjunto
infinito.
###### Proposición 13.
Sea $E\in Norm.$
La clausura de una bola abierta es la bola cerrada con el mismo radio y
centro. O sea:
(19) $\displaystyle B^{*}(a,\gamma)=\overline{B(a,\gamma)}$
###### Proof.
Es claro que $B(a,\gamma)\subset B^{*}(a,\gamma)\hskip
14.22636pt\overline{B(a,\gamma)}\subset\overline{B^{*}{a,\gamma}}=B^{*}(a,\gamma)$
O sea que $\overline{B(a,\gamma)}\subset B^{*}(a,\gamma).$
Para tener (16) resta demostrar que
$B^{*}(a,\gamma)\subset\overline{B(a,\gamma)}.$
Es claro que $B^{*}(a,\gamma)=B(a,\gamma)\cup S(a,\gamma).$ Sea $y\in
B^{*}(a,\gamma).$ Veamos que $y\in\overline{B(a,\gamma)}.$
1. (1)
Supongamos que $y\in\overline{B(a,\gamma)}.$ Como
$B(a,\gamma)\subset\overline{B(a,\gamma)},y\in\overline{B(a,\gamma)}.$
2. (2)
Supongamos que $y\in S(a,\gamma),$ i.e:
(20) $\displaystyle\|y-a\|=r$
Debemos probar que $y\in\overline{B(a,\gamma)}\Longrightarrow\forall
N\in\mathcal{N}_{y}:N\cap B(a,\gamma)\neq\emptyset.$
Sea $N\in\mathcal{N}_{y}.$ Entonces $\exists G:\text{abierto en E tal que
$y\in G\subset N\Longrightarrow\exists\gamma>0$}$
tal que $B(y,\gamma)\subset G\subset N.$
Si logramos probar que $B(y,\gamma)\cap B(a,\gamma)\neq\emptyset,$ y puesto
que $B(y\gamma)\subset N$ se tendr a que $N\cap B(a,\gamma)\neq\emptyset$ como
se quiere.
Tomemos $\underset{fijo}{\underbrace{\epsilon}}<2\gamma.$
Sea $z=a+\left(1-\dfrac{\epsilon}{2\gamma}\right)(y-a)$ y veamos que $z\in
B(y,\gamma),z\in B(a,\gamma).$
* •
$\|z-y\|=\left\|a+\left(1-\dfrac{\epsilon}{2\gamma}\right)(y-a)-y\right\|\\\
\\\
=\left\|\dfrac{\epsilon}{2\gamma}a-\dfrac{\epsilon}{2\gamma}y\right\|=\dfrac{\epsilon}{2\gamma}\|y-a\|\underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{j}}}{=}\dfrac{\epsilon}{2\gamma}\gamma=\dfrac{\epsilon}{2}<\gamma$
Lo que prueba que $z\in B(y,\gamma).$
* •
$\|z-a\|=\left\|a+\left(1-\dfrac{\epsilon}{2\gamma}\right)(y-a)-y\right\|=\dfrac{(2\gamma-\epsilon)}{2\gamma}\|y-a\|\\\
\\\ \underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{j}}}{=}\dfrac{\gamma}{2\gamma}(2\gamma-\epsilon)\\\ \\\
=\dfrac{1}{2}(2\gamma-\epsilon)<\dfrac{2\gamma}{2}=\gamma$
Lo que demuestra que $z\in B(a,\gamma).$
∎
###### Definición 6 (Conjuntos Convexos en un E.L.N).
Sea $E\in Norm$ y $A\subset E.$
Decimos que A es convexo si $\forall x,y\in A$ y $\forall\mathsf{t}\in[0,1]$
el vector $\left((1-\mathsf{t}x+\mathsf{t}y)\right)\in A.$
###### Proposición 14.
Sea $E\in Norm$ y $\underset{convexo}{\underbrace{A}}\subset E.$ Entonces
$\overline{A}$ y $A^{\circ}$ son convexos.
###### Proof.
1. (1)
Por Hip, A es convexo. Veamos que $\overline{A}$ es convexo.
Tomemos $x,y\in\overline{A}$ y $\mathsf{t}\in[0,1].$
Bastar con demostrar que
$\left[(1-\mathsf{t}x+\mathsf{t}y)\right]\in\overline{A},$ o que
$\exists\bigl{\\{}w_{n}\bigr{\\}}_{n=1}^{\infty}\subset A$ tal que
(21)
$\displaystyle\lim\limits_{n\rightarrow\infty}w_{n}=(1-\mathbf{t})x+\mathsf{t}y$
Como
$x\in\overline{A},\exists\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}\subset\overline{A}$
tal que $\lim\limits_{n\rightarrow\infty}x_{n}=x.$
Como
$y\in\overline{A},\exists\bigl{\\{}y_{n}\bigr{\\}}_{n=1}^{\infty}\subset\overline{A}$
tal que $\lim\limits_{n\rightarrow\infty}y_{n}=y$
Fijemos $x\in\mathbb{N}.$ Entonces $x_{n},y_{n}\in A$ y como A es convexo por
Hip, $\left[(1-\mathsf{t}x+\mathsf{t}y)\right]\in A$ cualquiera sea
$n\in\mathbb{N}.$
Consiremos ahora la sucesi n $\bigl{\\{}w_{n}\bigr{\\}}_{n=1}^{\infty}\subset
A.$
Ahora,
$\lim\limits_{n\rightarrow\infty}w_{n}=\lim\limits_{n\rightarrow\infty}\left[(1-\mathsf{t}x+\mathsf{t}y)\right]=(1-\mathsf{t})x+\mathsf{t}y$
y se tiene (21).
2. (2)
Por Hip. A es convexo. Veamos que $A^{\circ}$ es convexo.
De nuevo se toman $x,y\in A^{\circ}$ y $\mathsf{t}\in[0,1].$ Veamos que
$\left[(1-\mathsf{t}x+\mathsf{t}y)\right]\in
A^{\circ}\Longrightarrow\exists\gamma>0$ tal que
$B\left((1-\mathsf{t})x+\mathsf{t}y,\gamma\right)\subset A.$
Como $x\in A^{\circ},\exists\gamma_{1}>0$ tal que $B(x,\gamma_{1})\subset A.$
Como $y\in A^{\circ},\exists\gamma_{2}>0$ tal que $B(y,\gamma_{2})\subset A.$
Tomando $\gamma<mn\bigl{\\{}\gamma_{1},\gamma_{2}\bigr{\\}}$ se tiene que
$B(x,\gamma)\subset A,B(y,\gamma)\subset A.$
Consideremos la $B\left((1-\mathsf{t})x+\mathsf{t}y,\gamma\right).$ Veamos que
$B\left((1-\mathsf{t})x+\mathsf{t}y,\gamma\right)\subset A.$
Sea $w\in\left((1-\mathsf{t})x+\mathsf{t}y,\gamma\right).$
Entonces
(22)
$\displaystyle\left\|w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\right\|<\gamma$
Definamos $x_{1}=x+w-\left[(1-\mathsf{t})x+\mathsf{t}y\right].$ Entonces
$x_{1}-x=w-\left[(1-\mathsf{t})x+\mathsf{t}y,\gamma\right]$ y por tanto,
$\|x_{1}-x\|=\left\|w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\right\|<\gamma$,
lo que nos demuestra que $x_{1}\in B(x,\gamma)\subset A,$ i.e,
$\boxed{x_{1}\in A}.$
Definamos $y_{1}=y+w-\left[(1-\mathsf{t})x+\mathsf{t}y\right].$ Entonces
$y_{1}-y=w-\left[(1-\mathsf{t})x+\mathsf{t}y,\right]$ y por tanto,
$\|y_{1}-y\|=\left\|w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\right\|<\gamma$,
lo que nos demuestra que $y_{1}\in B(y,\gamma)\subset A,$ i.e,
$\boxed{y_{1}\in A}.$
Como $\mathsf{t}\in[0,1]$ y $x_{1},y_{1}\in A;$ convexo se tiene que
$(1-\mathsf{t})x_{1}+\mathsf{t}y_{1}\in A.$
O sea que
$(1-\mathsf{t})\left\\{x+w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\right\\}+\mathsf{t}\left\\{y+w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\right\\}\in
A$
i.e, $(1-\mathsf{t})+\mathsf{t}y+w-\left[(1-\mathsf{t})x+\mathsf{t}y\right]\in
A.$
Lo que nos demuestra que $w\in A.$
∎
###### Proposición 15.
Toda bola abierta cerrada de un E.N. es un conjunto convexo.
###### Proof.
$E\in Norm,a\in E$ y $\gamma>0.$
Consideremos la $B(a,\gamma)=\bigl{\\{}x\in\diagup\|x-a\|<\gamma\bigr{\\}}.$
Veamos que $B(a,\gamma):\text{convexo}.$
Tomemos $x,y\in B(a,\gamma)$ y
$\underset{fijo}{\underbrace{\mathsf{t}}}\in[0,1].$
Entonces
(23) $\displaystyle\|x-a\|<\gamma,\|y-a\|<\gamma$
Entonces $(1-\mathsf{t})x+\mathsf{t}y\in
B(a,\gamma)\Longrightarrow\left\|(1-\mathsf{t})x+\mathsf{t}y-a\right\|<\gamma.$
$\left\|(1-\mathsf{t})x+\mathsf{t}y-a\right\|<\gamma=\left\|(1-\mathsf{t})x+\mathsf{t}a+\mathsf{t}y-a-\mathsf{t}a\right\|\\\
\\\ =\left\|(1-\mathsf{t})(x-a)+\mathsf{t}(y-a)\right\|\\\ \\\
\leqslant(1-\mathsf{t})\|x-a\|+\mathsf{t}\|y-a\|\underset{\overset{\nearrow}{\leavevmode\nobreak\
\eqref{l}}}{<}(1-\mathsf{t})\gamma+\mathsf{t}\gamma=\gamma$
Ahora, $B^{*}(a,\gamma)=\overline{B(a,\gamma)}.$ Como $B(a,\gamma)$ es
convexo, $\overline{B(a,\gamma)}$ es convexo y por lo tanto, $B^{*}(a,\gamma)$
es convexo.
∎
El siguente lema ser til al $\underline{\text{completar}}$ un E.N. $(E,\|\|).$
###### Lema 1 (Un criterio para establecer cuando un E.L.N. es de Banach).
Sea $E\in Norm.$ Si $\exists A\subset E$ tal que:
1. (1)
$\overline{A}=E$ i.e. A es denso E.
2. (2)
$\forall\underset{\text{S. de
Cauchy}}{\underbrace{\bigl{\\{}x_{n}\bigr{\\}}}}\subset
A,\bigl{\\{}x_{n}\bigr{\\}}$ converge en E.
Entonces $E\in Ban.$
###### Proof.
Sea $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}:$ S. de Cauchy en E. Veamos que
$\bigl{\\{}x_{n}\bigr{\\}}$ converge en E.
Tomemos un t rmino $\underset{fijo}{\underbrace{x_{N}}}$ de la sucesi n.
Entonces $x_{N}\in E=\overline{A}$ (Hip.) Luego $x_{N}\in\overline{A}$ y por
tanto, $\forall n\in\mathbb{N}:B(x_{N},\frac{1}{n})\bigcap A\neq\emptyset.$
Si $n=1,\exists\beta_{1}^{N}\in A\,\,\text{tal que
$\|\beta_{1}^{N}-x_{N}\|<1$}$
Si $n=2,\exists\beta_{2}^{N}\in A\,\,\text{tal que
$\|\beta_{2}^{N}-x_{N}\|<\frac{1}{2}$}$
Si $n=3,\exists\beta_{3}^{N}\in A\,\,\text{tal que
$\|\beta_{3}^{N}-x_{N}\|<\frac{1}{3}$}$
$\text{etc}\ldots\ldots\ldots$
De este modo, si variamos a N:
para $x_{1},\exists\bigl{\\{}\beta_{n}^{1}\bigr{\\}}_{n=1}^{\infty}\subset
A\,\,\text{tal que $\|\beta_{1}^{1}-x_{1}\|<1$}$
$\|\beta_{2}^{1}-x_{1}\|<\frac{1}{2}$
$\|\beta_{3}^{1}-x_{1}\|<\frac{1}{3}$
$\vdots$
para $x_{2},\exists\bigl{\\{}\beta_{n}^{2}\bigr{\\}}_{n=1}^{\infty}\subset
A\,\,\text{tal que $\|\beta_{1}^{2}-x_{2}\|<1$}$
$\|\beta_{2}^{2}-x_{2}\|<\frac{1}{2}$
$\|\beta_{3}^{2}-x_{2}\|<\frac{1}{3}$
$\vdots$
para $x_{3},\exists\bigl{\\{}\beta_{n}^{3}\bigr{\\}}_{n=1}^{\infty}\subset
A\,\,\text{tal que $\|\beta_{1}^{3}-x_{3}\|<1$}$
$\|\beta_{2}^{3}-x_{3}\|<\frac{1}{2}$
$\|\beta_{3}^{3}-x_{3}\|<\frac{1}{3}$
$\vdots$
Consideremos la sucesi n
$\bigl{\\{}\beta_{n}^{n}\bigr{\\}}_{n\in\mathbb{N}}\subset A.$
Se tiene que $\forall n\in\mathbb{N}:\|\beta_{n}^{n}-x_{n}\|<\frac{1}{n}.$
Se deja como ejercicio al lector probar que la sucesi n
$\bigl{\\{}\beta_{n}^{n}\bigr{\\}}$ es una S. de Cauchy en A (y por la Hip.
converger en E).
∎
Continuando entonces con las propiedades de los E.L.N; presentamos ahora, uno
de los resultamos m s impotantes del An lisis Funcional.
###### Proposición 16.
En un espacio vectorial de dimensi n finita, todas las normas son
equivalentes.
###### Proof.
(La prueba que haremos es tomada del texto, Funtional Analysis, by Bachman-
Narici)
Sea X;K esp. vec., $dimX=n$ y sea $\|\|$ una norma cualquiera en X.
1. Paso 1.
Vamos a construir una cierta norma $\|\|_{0}$ en X.
2. Paso 2.
Enseguida demostraremos que $\|\|\sim\|\|_{0}.$
Esto demostrar que todas las normas en X son equivalentes.
3. Paso 1.
Tomemos
$\underset{fija}{\underbrace{\bigl{\\{}X_{1},X_{2},\ldots,X_{n}\bigr{\\}}}}:$
Base de X.
Sea $x\in X.$ Entonces $\exists!\alpha_{1},\ldots,\alpha_{n}\in K$ tal que
$x=\alpha_{1}X_{1}+\ldots+\alpha_{n}X_{n}.$
Esto permite que podamos definir la funci n
$\begin{diagram}$
$\begin{diagram}$
Es f cil probar que $\|\|_{0}$ es una norma en X. La llamaremos
$\underline{\text{norma cero en X asociada a la Base
$\bigl{\\{}X_{\alpha}\bigr{\\}}$}}.$ (Si cambiamos de base, cambia la
representaci n del vector y por lo tanto, cambia la norma.)
4. Paso 2.
Sea $\|\|$ una norma cualquiera en X. Nuestra tarea es demostrar que $\exists
a,b>0$ tal que:
(24) $\displaystyle\forall x\in X:a\|x\|_{0}\leqslant\|x\|\leqslant
b\|x\|_{0}$
En efecto:
$\left\|x\|=\|\alpha_{1}X_{1}+\ldots+\alpha_{n}X_{n}\right\|\leqslant\|x\|_{0}\|X_{1}\|+\ldots\ldots+\|X\|_{0}\|X_{n}\|\leqslant\\\
\underset{b}{\underbrace{\left(\|X_{1}\|+\ldots\ldots+\|X_{n}\|\right)}}\|X\|_{0}$
$\begin{cases}\|X\|_{0}=\underset{i=1,\ldots,n}{mx|\alpha_{i}|}\\\ \\\
x=\alpha_{1}X_{1}+\ldots+\alpha_{n}X_{n}\\\ \\\ \therefore\hskip
14.22636pt|\alpha_{1}|\leqslant\underset{i=1,\ldots,n}{mx|\alpha_{i}|}=\|X\|_{0}\\\
\\\ \hskip 42.67912pt\vdots\\\ \\\
|\alpha_{n}|\leqslant\underset{i=1,\ldots,n}{mx|\alpha_{i}|}=\|X\|_{0}\end{cases}$
Esto demuestra la desigualdad de la derecha en (24).
Resta demostrar que $\exists a>0$ tal que $\forall x\in
X:a\|x\|_{0}\leqslant\|x\|\hskip 14.22636pt\star$
Esto ya no es tan simple!!
Procederemos por inducci n sobre la dimensi n de X.
5. (1)
Supongamos que $dimX=1.$
Sea $\bigl{\\{}x_{1}\bigr{\\}}:$ Base de X. Tomemos $x\in X.$ Entonces
$x=\alpha_{1}X_{1}.$
$\therefore\|X\|_{0}=|\alpha_{1}|.$ Sea ahora $\|\|$ una norma cualquiera en
X.
$\|X\|=\|\alpha_{1}X_{1}\|=|\alpha|_{1}\|X_{1}\|=\|X_{1}\|\|X\|_{0}$ y por
tanto, $\underset{a}{\underbrace{\|X_{1}\|}}\|X\|_{0}\leqslant\|X\|$.
6. (2)
Hip. de Inducci n:
Asumimos que la propiedad es cierta para espacios de demensi n $p-1.$
Sea X: K esp. vectorial con $dimX=p$ y sea $\|\|$ una norma cualquiera en X.
El plan es demostrar que
(25) $\displaystyle\exists a>0\text{ tal que $\forall x\in
X:a\underset{i=1,\ldots,n}{mx|\alpha_{i}|}=a\|X\|_{0}\leqslant\|X\|.$}$
Tomemos $M=Sg\bigl{\\{}x_{1},\ldots,x_{n-1}\bigr{\\}}.$
Es claro que siendo $M\subset(X,\|\|),\|\|$ es una norma en M,
$\bigl{\\{}x_{1},\ldots,x_{n-1}\bigr{\\}}:$ Base de M y $dimM=p-1.$
Como $dimM=p-1,$ se tiene, por la Hip. de Inducci n que toda norma en M es
$\sim$ a la norma $\|\|_{0}$ en M asociada a la Base de
M:$\bigl{\\{}x_{1},\ldots,x_{n-1}\bigr{\\}}.$
Ahora, $\|\|$ es una norma en M. Por tanto, $\|\|\sim\|\|_{0}$ y se tiene que
(26) $\displaystyle\exists a>0\,\,\text{tal que $\forall y\in
M:a\|y\|_{0}\leqslant\|y\|\Longrightarrow\|y\|_{0}=\underset{i=1,\ldots,n}{mx|\alpha_{i}|}$}$
Compare (25) y (26): Observe los cuantificadores.
Vamos ahora a demostrar que $(M,\|\|):$ Banach.
Sea $\bigl{\\{}y_{n}\bigr{\\}}\subset(M,\|\|).$ Veamos que
(27) $\displaystyle y_{n}\overset{\|\|}{\longrightarrow}y\in M$
Como $\|\|\sim\|\|_{0}$ en M, $\bigl{\\{}y_{n}\bigr{\\}}$ es una S. de Cauchy
en $(M,\|\|_{0})$
Denotemos
$\bigl{\\{}y_{n}\bigr{\\}}=\bigl{\\{}y_{1},y_{1},\ldots,y_{n}\bigr{\\}}\subset
M.$
Ahora, como $\bigl{\\{}x_{1},\ldots,x_{n-1}\bigr{\\}}:$ Base de M, podemos
escribir:
$y_{1}=\alpha_{1}^{1}x_{1}+\alpha_{1}^{2}x_{2}+\ldots\ldots+\alpha_{1}^{n-1}x_{n-1}\\\
\\\
y_{2}=\alpha_{2}^{1}x_{1}+\alpha_{2}^{2}x_{2}+\ldots\ldots+\alpha_{2}^{n-1}x_{n-1}\\\
\\\
y_{3}=\alpha_{3}^{1}x_{1}+\alpha_{3}^{2}x_{2}+\ldots\ldots+\alpha_{3}^{n-1}x_{n-1}\\\
\\\ \centerline{\hbox{\vdots}}$
Considremos ahora las suces. columna en K $(\mathbb{R}\text{ $\mathbb{C}$})$:
$\bigl{\\{}\alpha_{j}^{1}\bigr{\\}}_{j=1}^{\infty}$
$\bigl{\\{}\alpha_{j}^{2}\bigr{\\}}_{j=1}^{\infty}\ldots\ldots$
$\bigl{\\{}\alpha_{j}^{n-1}\bigr{\\}}_{j=1}^{\infty}$ $\alpha_{1}^{1}$
$\alpha_{1}^{2}\ldots\ldots$ $\alpha_{1}^{n-1}$ $\alpha_{2}^{1}$
$\alpha_{2}^{2}\ldots\ldots$ $\alpha_{2}^{n-1}$ $\alpha_{3}^{1}$
$\alpha_{3}^{2}\ldots\ldots$ $\alpha_{3}^{n-1}$ $\downarrow$
$\downarrow\ldots\ldots$ $\downarrow$
Vamos a demostrar que c/u de estas $(n-1)$ Sucesiones converge en K.
Sea $\epsilon>0.$
Como $\bigl{\\{}y_{n}\bigr{\\}}:$ S. de Cauchy en $(M,\|\|_{0}),\exists
N\in\mathbb{N}\text{tal que $\forall
p,q>N:\underset{i=1,\ldots,n-1}{mx|\alpha_{p}^{i}-\alpha_{q}^{i}|=\|y_{p}-y_{q}\|_{0}<\epsilon}$}$
$y_{p}=\alpha_{p}^{1}x_{1}+\ldots\ldots+\alpha_{p}^{n-1}x_{n-1}\\\ \\\
y_{q}=\alpha_{q}^{1}x_{1}+\ldots\ldots+\alpha_{q}^{n-1}x_{n-1}$
Fijemos $p,q>N.$
Entonces:
$\begin{cases}|\alpha_{p}^{1}-\alpha_{q}^{1}|\leqslant\underset{i=1,\ldots,n-1}{mx|\alpha_{p}^{i}-\alpha_{q}^{i}|}<\epsilon,&\text{siempre
que $p,q>N$}\\\ \\\ \vdots\\\ \\\
|\alpha_{p}^{n-1}-\alpha_{q}^{n-1}|\leqslant\underset{i=1,\ldots,n-1}{mx|\alpha_{p}^{i}-\alpha_{q}^{i}|}<\epsilon\end{cases}$
Esto significa que
$\begin{cases}\text{la sucesi n
$\bigl{\\{}\alpha_{j}^{1}\bigr{\\}}_{j=1}^{\infty}$ es una S de Cauchy en K y
como K es completo,
$\alpha_{j}^{1}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{1}$}\\\
\\\ \ldots\ldots\\\ \\\ \text{la sucesi n
$\bigl{\\{}\alpha_{j}^{n-1}\bigr{\\}}_{j=1}^{\infty}$ es una S de Cauchy en K
y como K es completo,
$\alpha_{j}^{n-1}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{n-1}$}\end{cases}$
Definamos $y=\alpha_{1}x_{1}+\ldots\ldots+\alpha_{n-1}x_{n_{1}}.$
Es claro que $y\in M.$
Para tener (27), veamos que $y_{n}\overset{\|\|}{\longrightarrow}y.$
Bastar con demostrar que $y_{n}\overset{\|\|_{0}}{\longrightarrow}y$ ya que
siendo $\|\|\sim\|\|_{0}$ en M, $y_{n}\overset{\|\|}{\longrightarrow}y.$
Veamos pues, que $y_{n}\overset{\|\|_{0}}{\longrightarrow}y.$
Sea $\epsilon>0.$
Debemos demostrar que $\exists N\in\mathbb{N}$ tal que $\forall
q>N:\underset{i=1,\ldots,n}{mx|\alpha_{q}^{i}-\alpha_{i}|}=\|y_{q}-y\|_{0}<\epsilon$
$\begin{cases}\text{Dado que
$\alpha_{j}^{1}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{1}$ y que
$\epsilon>0,$ se tiene que}\\\ \\\ \exists N_{1}\in\mathbb{N}\,\,\text{tal que
$\forall q>N_{1}:|\alpha_{q}^{i}-\alpha_{i}|<\epsilon$}\\\ \\\ \ldots\ldots\\\
\\\ \text{Como
$\alpha_{j}^{n-1}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{n-1}$ y
que $\epsilon>0,$ se tiene que}\\\ \\\ \exists
N_{n-1}\in\mathbb{N}\,\,\text{tal que $\forall
q>N_{n-1}:|\alpha_{q}^{n-1}-\alpha_{n-1}|<\epsilon$}\end{cases}$
Luego si escogemos $N>mx\bigl{\\{}N_{1},\ldots,N_{n-1}\bigr{\\}},$ se tiene
que $\forall q>N$:$\begin{cases}|\alpha_{q}^{1}-\alpha_{1}|<\epsilon\\\ \\\
\ldots\ldots\\\ \\\ |\alpha_{q}^{n-1}-\alpha_{n-1}|<\epsilon.\end{cases}$
y por lo tanto,
$\underset{i=1,\ldots,n-1}{mx|\alpha_{q}^{i}-\alpha_{i}|}<\epsilon$ siempre
que $q>N.$
Pero
$\underset{i=1,\ldots,n-1}{mx|\alpha_{q}^{i}-\alpha_{i}|}=\|y_{q}-y\|_{0}.$
Luego $\|y_{q}-y\|_{0}<\epsilon$ siempre que $q>N.$
Esto demuestra que $y_{n}\overset{\|\|_{0}}{\longrightarrow}y$ y por tanto,
$y_{n}\overset{\|\|}{\longrightarrow}y.$
Hemos demostrado as que $(M,\|\|):$ Esp. Banach.
Vamos ahora a probar que $M\subset X$ es un cjto cerrado en X.
Debemos probar que $\overline{M}=M.$
Es claro que $M\subset\overline{M}.$ Resta demostrar que $\overline{M}\subset
M.$
Tomemos $x\in\overline{M}$ y veamos que $x\in M.$
$x\in\overline{M}\Longrightarrow\exists\bigl{\\{}x_{n}\bigr{\\}}\subset
M\text{tal que $\lim_{n\rightarrow\infty}x_{n}=x\in X.$}$
Pero $(M,\|\|)\in Banach$ y como $\bigl{\\{}x_{n}\bigr{\\}}\subset M,$
entonces $\bigl{\\{}x_{n}\bigr{\\}}$ es una S. de Cauchy en
$(M,\|\|)\Longrightarrow\lim_{n\rightarrow\infty}x_{n}\in M.$
O sea que $x\in M.$ Esto prueba que M es un cjto cerrado en X.
Consideremos ahora el conjunto $x_{n}+M=\bigl{\\{}x_{n}+z,z\in
M\bigr{\\}}\subset X.$
$x_{n}+M,$ no es subespacio de X ya que $0\notin x_{n}+M.$ En efecto, si $0\in
x_{n}+M,\exists\bigl{(}\beta_{1}x_{1}+\ldots+\beta_{n_{1}}x_{n-1}\bigr{)}\in
M$ tal que $0_{x}=x_{n}+\beta_{1}x_{1}+\ldots,\beta_{n-1}x_{n-1}$ y se tendr a
que $x_{n}$ es una C.L. de
$\bigl{\\{}x_{1},\ldots,x_{n-1}\bigr{\\}},\rightarrow\leftarrow$ ya que el
cjto. $\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}$ es L.I.
Considremos ahora en X la traslaci n definida por el vector $-x_{n}:$
$\begin{diagram}$
$\begin{diagram}$
Es f cil demostrar que T es continua. Ahora, como M es cerrado en X,
$T_{-x_{n}}^{-1}(M)$ es cerrado en X.
Pero $T_{-x_{n}}^{-1}(M)=\bigl{\\{}x\in\text{tal que $T_{-x_{n}}(x)=x-x_{n}\in
M$}\bigr{\\}}\\\ \\\ =\bigl{\\{}x\in\text{tal que
$x-x_{n}=\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1},\alpha_{i}\in
K$}\bigr{\\}}\\\ \\\ =\bigl{\\{}x\in\text{tal que $\underset{\in
M}{\underbrace{x=\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1},\alpha_{i}+x_{n}}}\in
K$}\bigr{\\}}\\\ \\\ =x_{n}+M.$
Por lo tanto,
$\underset{cerrado}{\underbrace{x_{n}+M}}\subset(X,\|\|)\Longrightarrow$
$C_{x}(x_{n}+M):$ abto en X.
Como $0\in C_{x}(x_{n}+M),\exists c_{n}>0$ tal que
$B_{(x,\|\|)}(0;c_{n})\subset C_{x}(x_{n}+M).$
i.e, $\exists c_{n}>0$ tal que $\forall x\in X:\text{si $\|x\|<c_{n},$
entonces, $x\in C_{x}(x_{n}+M)$};$ i.e $x\notin x_{n}+M.$
Tomando contrarec proco, tendremos que $\forall x\in X;$ si $x\in x_{n}+M$
entonces $\|x\|\geqslant c_{n}.$
Tomemos ahora $\underset{fijo}{\underbrace{x}}\in x_{n}+M.$ Entonces
$x=\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1}+x_{n},\alpha_{i}\in K.$ Y se
tiene que:
$\|\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1}+x_{n}\|\geqslant c_{n},$
cualquiera sean $\alpha_{1},\alpha_{n-1}\in K.$
Hemos demostrado as que $\exists c_{n}>0$ tal que
$\forall\alpha_{1},\alpha_{n-1}\in
K:c_{n}\leqslant\|\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1}+x_{n}\|.$ De aqu
resulta claro que $\forall\underset{\neq 0}{\alpha_{n}}\in
K,c_{n}\leqslant\left\|\dfrac{\alpha_{1}}{\alpha_{n}}x_{1}+\ldots+\dfrac{\alpha_{n-1}}{\alpha_{n}}x_{n-1}+x_{n}\right\|,$
o lo que es lo mismo,
$|\alpha_{n}|c_{n}\leqslant\|\alpha_{1}x_{1}+\ldots+\alpha_{n-1}x_{n-1}+\alpha_{n}x_{n}\|$
lo que nos dice que $\forall x\in X:|\alpha_{n}|c_{n}\leqslant\|x\|,$ donde
$\alpha$ es la cta $n^{a}$ del vector x en la base
$\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}.$
Resumiendo, hemos demostrado que si $M=Sg(x_{1},\ldots,x_{n-1}),$ se tiene que
$\exists c_{n}>0$ tal que $\forall x\in X:|\alpha_{n}|c_{n}\leqslant\|x\|$
donde $\alpha$ es la cta $n^{a}$ del vector x en la base
$\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}$.
Si ahora tomamos como $M=Sg(x_{1},x_{2},\ldots,x_{n-2},x_{n})$ y desarrollamos
el mismo an lisis que se hizo para el caso en que M era el
$Sg(x_{1},\ldots,x_{n-1}),$ podr amos demostrar que $\exists c_{n-1}>0$ tal
que $\forall x\in X:|\alpha_{n_{1}}|c_{n_{1}}\|x\|$ donde $\alpha_{n-1}$ es la
cta $(n-1)$ del vector x en la base $\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}.$
etc$\ldots,\ldots$
As que en el fondo de todo lo que se tiene es que $\exists
c_{1},c_{2},\ldots,c_{n}>0$ tal que $\forall x\in X:$
$\begin{cases}|\alpha_{1}|c_{1}\leqslant\|x\|;\alpha_{1}:\text{cte $1^{a}$ de
X en la base $\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}$}\\\ \\\
|\alpha_{2}|c_{2}\leqslant\|x\|;\alpha_{1}:\text{cte $2^{a}$ de X en la base
$\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}$}\\\ \\\ $\vdots$\\\ \\\
|\alpha_{n}|c_{n}\leqslant\|x\|;\alpha_{n}:\text{cte $n^{a}$ de X en la base
$\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}$}\end{cases}$
Fijemos x en X. Entonces
$x=\beta_{1}x_{1}+\ldots+\beta_{n-1}x_{n-1}+\beta_{n}x_{n}$ y se tiene que:
$\begin{cases}|\beta_{1}|\underset{j=1,\ldots,n}{mnC_{j}}\leqslant\|x\|\\\ \\\
|\beta_{2}|\underset{j=1,\ldots,n}{mnC_{j}}\leqslant\|x\|\\\ \\\ \vdots\\\ \\\
|\beta_{n}|\underset{j=1,\ldots,n}{mnC_{j}}\leqslant\|x\|\end{cases}$
Y si llamamos $a=\underset{j=1,\ldots,n}{mnC_{j}}$ se tiene que $\forall
i=1,\ldots,n:a|\beta_{i}|\leqslant\|x\|\Longrightarrow
a\underset{i=1,\ldots,n}{mx|\beta_{i}|}\leqslant\|x\|.$ Pero
$\underset{i=1,\ldots,n}{mx|\beta_{i}|}=\|x\|_{0}$
As que $a\|x\|_{0}\leqslant\|x\|,$ y como x es cualquier vector en X, hemos
conseguido demostrar que $\exists a>0$ tal que $\forall x\in
X:a\|x\|_{0}\leqslant\|x\|$ que era plan que nos hab amos propuesto en 16.
∎
###### Corolario 1.
Sea $(X,\|\|):$ E.L.N. Si $E\subset X$ y $dimE=n,$ entonces $(E,\|\|):$
Banach.
###### Proof.
Es claro que siendo $E\subset X,$ $(X,\|\|):$ E.L.N.
Sea $\bigl{\\{}y_{n}\bigr{\\}}_{n=1}^{\infty}$ una S. de Cauchy en $(E,\|\|).$
Veamos que $y_{n}\overset{\|\|}{\longrightarrow}y\in E.$
Como $dimE=n,$ sea $\underset{fija}{\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}}:$
base de E.
Tomemos $\mathsf{v}\in E.$ Entonces $\exists!\alpha_{1},\ldots,\alpha_{n}\in
K$ tal que $\mathsf{v}=\alpha_{1}x_{1},\ldots,\alpha_{n}x_{n}$ y podemos
definir
$\begin{diagram}$
$\begin{diagram}$
Es f cil probar que $\|v\|_{0}$ es una norma en E. As que $(E,\|\|_{0}):$
E.L.N
Dado que $\|\|$ y $\|\|_{0}$ son normas en E y como $dimE=n,$ se tiene, por la
Prop. anterior que $\|\|\sin\|\|_{0}.$
Como $\bigl{\\{}y_{n}\bigr{\\}}_{n=1}^{\infty}\subset(E,\|\|),$ entonces,
$\bigl{\\{}y_{n}\bigr{\\}}$ es una S. Cauchy en $(E,\|\|_{0}).$
Si logramos demostrar que $y_{n}\overset{\|\|_{0}}{\longrightarrow}y\in E,$
entonces, $y_{n}\overset{\|\|}{\longrightarrow}y\in E$ que es lo que se quiere
probar.
As que todo consiste en probar que
$y_{n}\overset{\|\|_{0}}{\longrightarrow}y\in E.$
Sea $\varepsilon>0.$ Como $\bigl{\\{}y_{n}\bigr{\\}}:$ S. de Cauchy en
$(E,\|\|_{0}),$
(28) $\displaystyle\exists N\in\mathbb{N}\,\,\text{tal que $\forall
p,q>N:\|y_{p}-y_{q}\|_{0}<\varepsilon$}$
Como $\bigl{\\{}x_{1},\ldots,x_{n}\bigr{\\}}:$ Base de E y
$\bigl{\\{}y_{n}\bigr{\\}}_{n=1}^{\infty}\subset E,$ los t rminos de la sucesi
n $y_{n}$ se pueden escribir as :
$y_{1}=\alpha_{1}^{1}x_{1}+\alpha_{1}^{2}x_{2}+\ldots\ldots+\alpha_{1}^{n}x_{n}\\\
\\\
y_{2}=\alpha_{2}^{1}x_{1}+\alpha_{2}^{2}x_{2}+\ldots\ldots+\alpha_{2}^{n}x_{n}\\\
\\\ \ldots\ldots\\\ \\\
y_{N}=\alpha_{N}^{1}x_{1}+\alpha_{N}^{2}x_{2}+\ldots\ldots+\alpha_{N}^{n}x_{n}\\\
\\\ \ldots\ldots\\\ \\\
y_{p}=\alpha_{p}^{1}x_{1}+\alpha_{p}^{2}x_{2}+\ldots\ldots+\alpha_{p}^{n}x_{n}\\\
\\\
y_{q}=\underset{\downarrow}{\alpha_{q}^{1}x_{1}}+\underset{\downarrow}{\alpha_{q}^{2}x_{2}}+\ldots\ldots+\underset{\downarrow}{\alpha_{q}^{n}x_{n}}$
Regresemos a (28).
Tomemos $\underset{fijos}{\underbrace{p,q}}>N.$ Entonces
$\|y_{p}-y_{q}\|_{0}<\varepsilon.$
$\left\|\alpha_{p}^{1}x_{1}+\alpha_{p}^{2}x_{2}+\ldots\ldots+\alpha_{p}^{n}x_{n}-\bigl{(}\alpha_{q}^{1}x_{1}+\alpha_{q}^{2}x_{2}+\ldots\ldots+\alpha_{q}^{n}x_{n}\bigr{)}\right\|_{0}\\\
\\\
=mx\bigl{\\{}|\alpha_{p}^{1}-\alpha_{q}^{1}|,\ldots\ldots,|\alpha_{p}^{n}-\alpha_{q}^{n}|\bigr{\\}}$
As que
$\begin{cases}\left|\alpha_{p}^{1}-\alpha_{q}^{1}\right|<\varepsilon,\text{siempre
que $p,q>N$}\\\ \\\ \ldots\ldots\\\ \\\
\left|\alpha_{p}^{n}-\alpha_{q}^{n}\right|<\varepsilon,\text{siempre que
$p,q>N$}\end{cases}$
$\begin{cases}\text{la sucesi n de columnas
$\bigl{\\{}\alpha_{j}^{1}\bigr{\\}}_{j=1}^{\infty}$ es una S. de Cauchy en
$K(\mathbb{R}\,\,\text{ }\mathbb{C})$}\\\ \\\ \text{y como K es completo,
$\alpha_{1}^{j}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{1}$}\\\
\\\ \ldots\ldots\\\ \\\ \text{la sucesi n de columnas
$\bigl{\\{}\alpha_{j}^{n}\bigr{\\}}_{j=1}^{\infty}$ es una S. de Cauchy en
$K(\mathbb{R}\,\,\text{ }\mathbb{C})$}\\\ \\\ \text{y como K es completo,
$\alpha_{j}^{n}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{n}$}\end{cases}$
Definamos ahora el vector $y=\alpha_{1}x_{1}+\ldots\ldots+\alpha_{n}x_{n}$
Resta demostrar que $y\in E$ (lo cual es obvio) y que
$y_{n}\overset{\|\|_{0}}{\longrightarrow}y$
Sea $\varepsilon>0.$ Debemos demostrar que $\exists N\in\mathbb{N}$ tal que
(29) $\displaystyle\forall
j>N:mx\bigl{\\{}|\alpha_{j}^{1}-\alpha_{n}|,\ldots\ldots,|\alpha_{j}^{n}-\alpha_{n}|\bigr{\\}}=\left\|y_{j}-y\right\|_{0}<\varepsilon$
Como
$\alpha_{j}^{1}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{1},\exists
N_{1}\in\mathbb{N}$ tal que $\forall
j>N_{1}:\left|\alpha_{j}^{1}-\alpha_{1}\right|<\varepsilon$
Como
$\alpha_{j}^{2}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{2},\exists
N_{2}\in\mathbb{N}$ tal que $\forall
j>N_{2}:\left|\alpha_{j}^{2}-\alpha_{2}\right|<\varepsilon$
$\ldots\ldots$
Como
$\alpha_{j}^{n}\overset{j\rightarrow\infty}{\longrightarrow}\alpha_{n},\exists
N_{n}\in\mathbb{N}$ tal que $\forall
j>N_{n}:\left|\alpha_{j}^{n}-\alpha_{n}\right|<\varepsilon$
Luego si escogemos $N>mx\bigl{\\{}N_{1},\ldots,N_{n}\bigr{\\}}$ se tendr que
$\displaystyle\forall j>N:\left|\alpha_{j}^{1}-\alpha_{1}\right|<\varepsilon$
$\displaystyle\left|\alpha_{j}^{2}-\alpha_{2}\right|<\varepsilon$
$\displaystyle\vdots$
$\displaystyle\underline{\left|\alpha_{j}^{n}-\alpha_{n}\right|<\varepsilon}$
$\displaystyle\therefore\hskip
14.22636ptmx\bigl{\\{}\left|\alpha_{j}^{i}-\alpha_{1}\right|,\ldots\ldots,\left|\alpha_{j}^{n}-\alpha_{n}\right|<\varepsilon,\text{siempre
que $j>N$ y se tiene \leavevmode\nobreak\ \eqref{s}.}$
∎
###### Corolario 2.
Todo E.L.N. de dimensi n finita es un E. de Banach.
###### Corolario 3.
Sea $(X,\|\|):$ E.L.N. $E\subset X.$ Entonces:
1. (1)
$(E,\|\|):$ Banach.
2. (2)
E es cerrado en X.
O sea que todo subespacio vectorial de dimensi n finita de un E.L.N. es un
cjto cerrado.
###### Proof.
1. (1)
$\checkmark$
2. (2)
Debemos demostrar que $\overline{E}=E.$ $E\subset\overline{E}\checkmark.$
Resta probar que $\overline{E}\subset E.$
Tomemos $x\in\overline{E}$ y veamos que $x\in E.$
$x\in\overline{E}\Longrightarrow\exists\bigl{\\{}x_{n}\bigr{\\}}\subset E$ tal
que $\lim\limits_{n\rightarrow\infty}x_{n}=x(x\in X).$
Pero $(E,\|\|):$Banach y como
$\bigl{\\{}x_{n}\bigr{\\}}\subset(E,\|\|),\bigl{\\{}x_{n}\bigr{\\}}$ es una S.
de Cauchy en $(E,\|\|)\hskip
14.22636pt\therefore\left(\lim\limits_{n\rightarrow\infty}x_{n}\right)\in E.$
O sea que $x\in E.$
∎
###### Ejemplo 1.
Sabemo que $\mathbb{C}:$Esp. vectorial.
Vamos a demostrar que $(\mathbb{C},\|\|):$ E.L.N. donde $\forall
z\in\mathbb{C}:\|z\|=\sqrt{x^{2}+y^{2}}=|z|.$
Una vez demostremos que $(\mathbb{C,\|\|}):$ E.L.N, como $dim\mathbb{C}=1,$ se
tendr que $(\mathbb{C,\|\|}):$ E.L.N. de dimensi n finita. Luego
$(\mathbb{C,\|\|}):$ Banach.
1. (1)
Es claro que $|z|\geqslant 0.$ Si $z=0,|z|=|0|=0.$
Supongamos $|z|=0.$ Entonces $\sqrt{x^{2}+y^{2}}=0\Rightarrow x=y=0;z=0.$
2. (2)
$|\alpha z|^{2}=(\alpha z)\overline{(\alpha z)}=\alpha
z\overline{\alpha}\overline{z}=(\alpha\overline{\alpha})(z\overline{z})=|\alpha|^{2}|z|^{2}.$
3. (3)
$|z_{1}+z_{2}|\leqslant|z_{1}|+|z_{2}|\\\ \\\
|z_{1}+z_{2}|^{2}=(z_{1}+z_{2})(\overline{z_{1}+z_{2}})\\\ \\\
=(z_{1}+z_{2})(\overline{z_{1}}+\overline{z_{2}})\\\ \\\
=z_{1}\overline{z_{1}}+z_{2}\overline{z_{2}}+(z_{1}\overline{z_{2}}+\overline{z_{1}}z_{2})\\\
\\\
=|z_{1}|^{2}+|z_{2}|^{2}+2\mathsf{Re}(z_{1}\overline{z_{2}})\leqslant|z_{1}|^{2}+|z_{2}|^{2}+2|z_{1}\overline{z_{2}}|\\\
\\\ =|z_{1}|^{2}+|z_{2}|^{2}+2|z_{1}||z_{2}|\\\ \\\
=\left(|z_{1}|+|z_{2}|\right)^{2}\Longrightarrow|z_{1}+z_{2}|\leqslant|z_{1}|+|z_{2}|$
###### Ejemplo 2.
Es posible dm. de una manera directa (i.e, sin utilizar el Tma. ”todo E.V. de
dim. finita es Banach”.) que $(\mathbb{C},\|\|)$ es Banach.
$|z|=\sqrt{x^{2}+y^{2}}$ siendo $z=x+iy$
### 1.1. El teorema de Hahn-Banach. (caso real)
Este importante Tma. del An. Funcional establece que todo funcional lineal
continuo definido sobre un subespacio de un E.L.N. siempre se puede extender a
todo el espacio conserv ndose la norma del funcional.
Comenzamos con un
###### Lema 2.
Sea $(X,\|\|):\mathbb{R}$ esp.vectorial normado, $M\subset X$ y $f\in
M^{\prime}=\mathcal{L}_{c}(M,\mathbb{R})\diagup\text{dual topol gico de M}.$
Sea $x_{0}\in X,x_{0}\notin M.$
Entonces $\exists g\in S_{g}(M\cup\bigl{\\{}x_{0}\bigr{\\}}),$ o sea
$g:S_{g}M\cup\bigl{\\{}x_{0}\bigr{\\}}\longrightarrow\mathbb{R}$
tal que:
1. (1)
$g\diagup M=f$
2. (2)
$\|g\|=\|f\|.$
###### Proof.
Tomemos $\underset{fijo}{\underbrace{y_{1}}}\in M.$
Entonces, $\forall y\in M:\\\ \\\
\left||f(y_{1})|-|f(y)|\right|\leqslant\left|f(y_{1})-f(y)\right|=\left|f(y_{1}-y)\right|\leqslant\|f\|\|y_{1}-y\|\\\
\\\
=\|f\|\|(y_{1}+x_{0})+(-y-x_{0})\|\leqslant\|f\|\|y_{1}+x_{0}\|+\|f\|\|y+x_{0}\|\\\
\\\ \therefore\\\ \\\ |f(y)|-\|f\|\|y_{1}+x_{0}\|-\|f\|\|y+x_{0}\|\leqslant
f(y_{1})\leqslant|f(y)|+\|f\|\|y_{1}+x_{0}\|+\|f\|\|y+x_{0}\|\\\ \\\
-|f(y)|-\|f\|\|y+x_{0}\|\leqslant\|f\|\|y_{1}+x_{0}\|-\|f(y_{1})\|$ lo que dm.
que el real es cota superior del cjto $\left\\{-|f(y)|-\|f\|\|y+x_{0}\|,y\in
M\right\\}\\\ \\\ \therefore\hskip 14.22636pt\exists
Sup\left\\{-|f(y)|-\|f\|\|y+x_{0}\|,y\in M\right\\}=a\\\ \\\
-\|f(y_{1})\|-\|f\|\|y_{1}+x_{0}\|\leqslant|f(y)|+\|f\|\|y+x_{0}\|$ lo que dm.
que el real es cota inferior del cjto $\left\\{-|f(y)|-\|f\|\|y+x_{0}\|,y\in
M\right\\}.$
Luego $\exists\text{ nf}\left\\{-|f(y)|-\|f\|\|y+x_{0}\|,y\in M\right\\}=b.$
Veamos que $a\leqslant b.$
Recordemos que el Sup de un cjto de n meros reales es la m nima cota superior
del cjto.
Si logramos dm. que b es cota superior del cjto
$\left\\{-|f(y)|-\|f\|\|y+x_{0}\|,y\in M\right\\},$ tendremos que $a\leqslant
b.$
As que vamos a dm. que b es cota superior del cjto citado, o lo que es lo
mismo, que
(30) $\displaystyle\forall y\in M:-|f(y)|-\|f\|\|y+x_{0}\|\leqslant b$
Razonemos por R. abs.
O sea, supongamos que (30) no es cierta.
Entonces $\exists\overset{\sim}{y}\in M$ tal que
$b<-|f(\overset{\sim}{y})|-\|f\|\|\overset{\sim}{y}+x_{0}\|\\\ \\\
\therefore\hskip
14.22636pt0<\varepsilon=-|f(\overset{\sim}{y})|-\|f\|\|\overset{\sim}{y}+x_{0}\|-b$
Pero $b=\text{ nf}\left\\{-|f(y)|-\|f\|\|y+x_{0}\|\right\\}$ y como
$\varepsilon>0,$ se tiene, por la propiedad de aproximaci n del nf, que
$\exists\overset{\sim}{\overset{\sim}{y}}\in M$ tal que
$-|f(\overset{\sim}{\overset{\sim}{y}})|-\|f\|\|\overset{\sim}{\overset{\sim}{y}}+x_{0}\|<b+\varepsilon\\\
\\\ =-|f(\overset{\sim}{y})|-\|f\|\|\overset{\sim}{y}+x_{0}\|\\\ \\\
\therefore\\\ \\\
\|f\|\left(\|\overset{\sim}{\overset{\sim}{y}}+x_{0}\|+\|\overset{\sim}{y}+x_{0}\|\right)=\|f\|\|\overset{\sim}{y}+x_{0}\|+\|f\|\|\overset{\sim}{\overset{\sim}{y}}+x_{0}\|<\left|f(\overset{\sim}{\overset{\sim}{y}})-f(\overset{\sim}{y})\right|\\\
\\\
\|f\|\|\overset{\sim}{\overset{\sim}{y}}-\overset{\sim}{y}\|\geqslant\left|f(\overset{\sim}{\overset{\sim}{y}}-\overset{\sim}{y})\right|=\left|f(\overset{\sim}{\overset{\sim}{y}})-f(\overset{\sim}{y})\right|\geqslant\left||f(\overset{\sim}{\overset{\sim}{y}})|-|f(\overset{\sim}{y})|\right|$
O sea que
$\left||f(\overset{\sim}{\overset{\sim}{y}})|-|f(\overset{\sim}{y})|\right|<|f(\overset{\sim}{\overset{\sim}{y}})|-|f(\overset{\sim}{y})|,(\rightarrow\leftarrow)$
Luego $a\leqslant b.$
Supongamos que $a<b$ y tomemos $a<\underset{fijo}{\underbrace{c}}<b.$
Como $\text{Sup}\left\\{-|f(y)|-\|f\|\|y+x_{0}\|\right\\}=a<c<b=\text{
nf}\left\\{-|f(y)|+\|f\|\|y+x_{0}\|\right\\}$
se tiene que $\forall m\in M:\\\ \\\
-|f(m)|-\|f\|\|m+x_{0}\|<c<-|f(m)|+\|f\|\|m+x_{0}\|$
O sea que $-\|f\|\|m+x_{0}\|<f(m)+c<\|f\|\|m+x_{0}\|,$ i.e
(31) $\displaystyle\left|f(m)+c\right|\leqslant\|f\|\|m+x_{0}\|$
cualquiera sea $m\in M.$
Ahora, es f cil dm. que
$S_{g}\left(M\cup\bigl{\\{}x_{0}\bigr{\\}}\right)=\left\\{m+\lambda x_{0},m\in
M,\lambda\in\mathbb{R}\right\\}.$
Definamos
$\begin{diagram}$
$\begin{diagram}$
Es f cil dm. que $g$ es A.L. H galo!
Veamos que $g\diagup M=f.$
Tomemos $m\in M$ y veamos que $g(m)=f(m).$
$m=m+0.x_{0}$ y $g(m)=g(m+0.x_{0})=f(m)+0.c=f(m).$
Veamos que $g$ es continua.
Tomemos $x\in S_{g}\left(M\cup\bigl{\\{}x_{0}\bigr{\\}}\right).$ Entonces
$x=m+\lambda x_{0}.$ Asumamos $\lambda\neq 0.$
$|g(x)|=|g(m+\lambda x_{0})|=|f(m)+\lambda c|=|\lambda
f({\lambda}^{-1}m)+\lambda c|\\\ \\\
=|\lambda||f({\lambda}^{-1}m)+c|\underset{\leavevmode\nobreak\
\eqref{u}}{\leqslant}|\lambda|\|f\|\|{\lambda}^{-1}m+x_{0}\|\\\ \\\
=\|f\|\|m+\lambda x_{0}\|\\\ \\\ =\|f\|\|x\|\hskip 14.22636pt\star.$
(Si $\lambda=0\text{se llega a lo mismo}$)
As que $\forall x\in M:|g(x)|\leqslant\|f\|\|x\|$ lo que prueba que $g$ es
continua. Resta dm que $\|g\|=\|f\|.$
Veamos que
(32) $\displaystyle\|g\|\leqslant\|f\|$
Como $\|g\|=\underset{\|x\|\leqslant 1}{\text{Sup}|g(x)|},$ para obtener (32)
basta con dm. que $\|f\|$ es cota superior del cjto
$S=\left\\{g(x),\|x\|\leqslant 1\diagup x\in
S_{g}(M\cup\bigl{\\{}x_{0}\bigr{\\}})\right\\}.$
Sea $\xi\in S.$ Veamos que $\xi\leqslant\|f\|.\xi\in S\Longrightarrow\exists
x=m+\lambda x_{0}\in S_{g}(M\cup\bigl{\\{}x_{0}\bigr{\\}})$ con
$\|x\|\leqslant 1$ tal que $\xi=|g(x)|.$
Si tenemos en cuenta $\star$ se tiene que:
$\xi=|g(x)|\leqslant\|f\|\|x\|\leqslant\|f\|\,\,\text{l.q.q.d}$
Veamos finalmente que
(33) $\displaystyle\|f\|\leqslant\|g\|$
$\|f\|=\underset{\|x\|\leqslant 1}{\text{Sup}|f(x)|}.$ Para obtener (33) basta
dm. que $\|g\|$ es cota superior del cjto $T=\left\\{|f(x)|,\|x\|\leqslant
1,x\in M\right\\}.$
Sea $\xi\in T.$ Veamos que $\xi\leqslant\|g\|.$ $\xi\in
T\Longrightarrow\exists x\in M$ con $\|x\|\leqslant 1$ tal que $g=|f(x)|.$
Por tanto, $g(x)=g(x+0.x_{0})=f(x)+0.c=f(x).\\\ \\\ \therefore\hskip
14.22636pt|g(x)|=|f(x)|=g,$ i.e,
(34) $|g(x)|=\xi$
Como $x\in S_{g}(M\cup\bigl{\\{}x_{0}\bigr{\\}})$ y
$\|g\|=\underset{\|y\|\leqslant 1}{\text{Sup}|g(y)|}$
$|g(x)|\leqslant\|g\|$ y regresando a (34) se tiene que $\xi\leqslant\|g\|.$
∎
###### Proposición 17 (El Teorema de Hahn-Banach).
Sea $(X,\|\|):\mathbb{R}$ esp. vectorial normado, $M\subset X$ y sea $f\in
M^{\prime}=\mathcal{L}_{c}(M,\mathbb{R}).$ O sea,
$f:M\longrightarrow\mathbb{R}\diagup\text{A.L. continua.}$
Entonces $\exists F\in X^{\prime}=\mathcal{L}_{c}(X,\mathbb{R}),$ i.e,
$F:X\longrightarrow\mathbb{R}$ tal que:
1. (1)
$F\diagup M=f$
2. (2)
$\|F\|=\|f\|$
Y en palabras, ”todo funcional lineal continuo definido sobre un subespacio de
un E.L.N se puede extender (prolongar) a todo el espacio preservando la
norma.”
###### Proof.
Sea S=$\begin{cases}(1).\overline{f}:M\subset
D_{\overline{f}}\longrightarrow\mathbb{R},\overline{f}\in(D_{\overline{f}})^{\prime}\\\
\\\ (2).\overline{f}\,\,\text{es una extesi n de $f$ a
$D_{\overline{f}}$},i.e,\overline{f}\diagup M=f\\\ \\\
(3).\|\overline{f}\|=\|f\|\end{cases}$
O sea que $\forall\overline{f}\in S,D_{\overline{f}}$ es un subespacio de X m
s grande que M y que contiene a M.
Las $\overline{f}\in S$ son A.L.
$\begin{diagram}$
$\begin{diagram}$
Por el lema anterior, $S\neq\emptyset.$
Vamos a definir en S una $\mathcal{R.O.P}$ as :
Sean $\overline{f_{1}},\overline{f_{2}}\in S.$ Entonces
$\overline{f_{1}}\preceq\overline{f_{2}}\Longleftrightarrow
D_{\overline{f_{1}}}\subseteq D_{\overline{f_{2}}}$ y $\overline{f_{2}}$ es
una extensi n de $\overline{f_{1}}$ i.e, $\overline{f_{2}}\diagup
D_{\overline{f_{1}}}=\overline{f_{1}}.$
O sea que $\forall x\in
D_{\overline{f_{1}}},\overline{f_{2}}(x)=\overline{f_{1}}(x)$
Vamos a dm. que $(S,\preceq):$ cjto P.O. o que $\preceq$ es reflexiva,antisim
trica y transitiva.
1. (1)
$D_{\overline{f_{1}}}\subset D_{\overline{f_{1}}}$ y $\overline{f_{1}}\diagup
D_{\overline{f_{1}}}=\overline{f_{1}},\overline{f_{1}}\preceq\overline{f_{1}}$
2. (2)
Supongamos que $\overline{f_{1}}\preceq\overline{f_{2}}$ y
$\overline{f_{2}}\preceq\overline{f_{1}}.$ Veamos que:
$\overline{f_{1}}=\overline{f_{2}}$ o que
$D_{\overline{f_{1}}}=D_{\overline{f_{2}}}$ y que $\forall x\in
D_{\overline{f_{1}}}=D_{\overline{f_{2}}},\overline{f_{1}}(x)=\overline{f_{2}}(x).$
Como
(35)
$\displaystyle\overline{f_{1}}\preceq\overline{f_{2}},D_{\overline{f_{1}}}\subseteq
D_{\overline{f_{2}}}$
y
(36) $\displaystyle\overline{f_{2}}\diagup
D_{\overline{f_{1}}}=\overline{f_{1}}\,\,\text{i.e,}\forall x\in
D_{\overline{f_{1}}}:\overline{f_{2}}(x)=\overline{f_{1}}(x)$
Como
(37)
$\displaystyle\overline{f_{2}}\preceq\overline{f_{1}},D_{\overline{f_{2}}}\subseteq
D_{\overline{f_{1}}}$
y $\overline{f_{1}}\diagup D_{\overline{f_{2}}}=\overline{f_{2}}.$
De (35) y (37): $D_{\overline{f_{1}}}=D_{\overline{f_{2}}}.$
Seg n (36), $\forall x\in
D_{\overline{f_{1}}}=D_{\overline{f_{2}}}:\overline{f_{2}}(x)=\overline{f_{1}}(x).$
Esto dm. que $\overline{f_{1}}=\overline{f_{2}}.$
3. (3)
Supongamos ahora que $\overline{f_{1}}\preceq\overline{f_{2}}$ y
$\overline{f_{2}}\preceq\overline{f_{2}}.$ Veamos que
$\overline{f_{1}}\preceq\overline{f_{3}}.$
Dos cosas se deben probar:
1. i)
$D_{\overline{f_{1}}}\subset\overline{f_{3}}$
2. ii)
$\overline{f_{3}}\diagup D_{\overline{f_{1}}}=\overline{f_{1}}.$ o que
$\forall x\in D_{\overline{f_{1}}},\overline{f_{3}}(x)=\overline{f_{1}}(x).$
Como $\overline{f_{1}}\preceq\overline{f_{2}},D_{\overline{f_{1}}}\subseteq
D_{\overline{f_{2}}}\\\
\text{como}\,\,\overline{f_{2}}\preceq\overline{f_{3}},D_{\overline{f_{2}}}\subseteq
D_{\overline{f_{3}}}\hskip 14.22636pt\therefore\hskip
14.22636ptD_{\overline{f_{1}}}\subseteq D_{\overline{f_{3}}}$ y se tiene i).
O sea que
(38) $\displaystyle\forall x\in
D_{\overline{f_{1}}}:\overline{f_{2}}(x)=\overline{f_{1}}(x)$
Como $\overline{f_{2}}\preceq\overline{f_{3}},\overline{f_{3}}\diagup
D_{\overline{f_{2}}}=\overline{f_{2}}$
i.e que $\forall y\in
D_{\overline{f_{2}}}:\overline{f_{3}}(y)=\overline{f_{2}}(y)\hskip
14.22636pt\star$
Sea $x\in D_{\overline{f_{1}}}\Longrightarrow x\in
D_{\overline{f_{2}}}\underset{\star}{\Longrightarrow}\overline{f_{3}}(x)=\overline{f_{2}}(x).$
As que
(39) $\displaystyle\forall x\in
D_{\overline{f_{1}}}:\overline{f_{3}}(x)=\overline{f_{2}}(x)$
De (38) y (39) se concluye que $\forall x\in
D_{\overline{f_{1}}}:\overline{f_{3}}(x)=\overline{f_{1}}(x)$ y se tiene ii).
Hemos dm. que $(S,\preceq)$ es un cjto. P.O.
V a aplicar el lema de Zorn, sea $\underset{\text{cjto. totalmente ordenado
}}{\bigl{\\{}\overline{f_{\alpha}}\bigr{\\}}_{\alpha\in I}}\subset S.$
O sea que $\forall\alpha\in I,\overline{f_{\alpha}}$ es una extensi n de f.
Siendo $\bigl{\\{}\overline{f_{\alpha}}\bigr{\\}}_{\alpha\in I}$ totalmente
ordenado, se tiene que $\forall\alpha_{1},\alpha_{2}\in
I,\overline{f_{\alpha_{1}}}\preceq\overline{f_{\alpha_{2}}}$
$\overline{f_{\alpha_{2}}}\preceq\overline{f_{\alpha_{1}}}$.
Vamos a demostrar que $\bigl{\\{}\overline{f_{\alpha}}\bigr{\\}}_{\alpha\in
I}$ tiene cota superior en S, o que $\exists\overline{f}\in S$ tal que
$\forall\alpha\in I:\overline{f_{\alpha}}\preceq f.$
Consideremos el cjto. $\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\subset X.$
Veamos que $\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$ es un
subespacio de X.
1. i)
Como $\forall\alpha\in I:M\subset
D_{\overline{f_{\alpha}}},M\subset\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}$ y por lo tanto $\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\neq\emptyset.$
2. ii)
Sea $x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}},\beta\in\mathbb{R}.$ Veamos que: $\beta
x\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}.\\\ \\\
x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\Rightarrow\exists\alpha_{1}\in I$ tal que $x\in
D_{\overline{f_{\alpha_{1}}}}\hskip 14.22636pt\therefore\hskip
14.22636pt(\beta x)\in D_{\overline{f_{\alpha_{1}}}}$.
$(\beta x)\in D_{\overline{f_{\alpha_{1}}}}\Rightarrow(\beta
x)\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}.$
3. iii)
Sean $x,y\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$. Veamos
que $(x+y)\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}.$
$x,y\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\Rightarrow\exists\alpha_{1},\alpha_{2}\in I$ tal
que $x\in D_{\overline{f_{\alpha_{1}}}},y\in D_{\overline{f_{\alpha_{2}}}}.$
Pero $\bigl{\\{}\overline{f_{\alpha}}\bigr{\\}}_{\alpha\in I}\subset S.$
Entonces $\overline{f_{\alpha_{1}}}\preceq\overline{f_{\alpha_{2}}}$
$\overline{f_{\alpha_{2}}}\preceq\overline{f_{\alpha_{1}}},$ i.e,
$D_{\overline{f_{\alpha_{1}}}}\subseteq D_{\overline{f_{\alpha_{2}}}}$
$D_{\overline{f_{\alpha_{2}}}}\subseteq D_{\overline{f_{\alpha_{1}}}}$
Supongamos que se da $D_{\overline{f_{\alpha_{1}}}}\subseteq
D_{\overline{f_{\alpha_{2}}}}.$ Como $x\in D_{\overline{f_{\alpha_{1}}}},x\in
D_{\overline{f_{\alpha_{2}}}},y\in
D_{\overline{f_{\alpha_{2}}}}\Longrightarrow(x+y)\in
D_{\overline{f_{\alpha_{2}}}}\hskip 14.22636pt\therefore\hskip
14.22636pt(x+y)\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}.$
Si $D_{\overline{f_{\alpha_{2}}}}\subseteq
D_{\overline{f_{\alpha_{1}}}},\cdots\cdots$
Esto demuestra que $\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$
es un subespacio de X. Pasemos ahora a definir a $\overline{f}.$
Tomemos $x\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}.$
Entonces $\exists\beta\in I$ tal que $x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\beta}}}\Longrightarrow\overline{f_{\beta}}(x)\in\mathbb{R}.$
Definamos
$\begin{diagram}$
$\begin{diagram}$
Debemos dm. que $\overline{f}$ est bi n definida.
Supongamos que $\exists\alpha,\beta\in I$ tal que $x\in
D_{\overline{f_{\alpha}}}$ y $x\in D_{\overline{f_{\beta}}}.$
Para establecer que $\overline{f}$ est bien definida debemos dm. que
(40) $\displaystyle\overline{f_{\alpha}}(x)=\overline{f_{\beta}}(x)$
Como $\alpha,\beta\in I,\overline{f_{\alpha}}\preceq\overline{f_{\beta}}$
$\overline{f_{\beta}}\preceq\overline{f_{\alpha}}.$
Veamos que en cualquier caso,
$\overline{f_{\alpha}}(x)=\overline{f_{\beta}}(x).$
Suponagmos que $\overline{f_{\alpha}}\preceq\overline{f_{\beta}}.$
Entonces $D_{\overline{f_{\alpha}}}\subseteq D_{\overline{f_{\beta}}}$ y
$\overline{f_{\beta}}\diagup D_{\overline{f_{\alpha}}}=\overline{f_{\alpha}}.$
O sea que $\forall y\in
D_{\overline{f_{\alpha}}}:\overline{f_{\beta}}(y)=\overline{f_{\alpha}}(y).$
Como $x\in
D_{\overline{f_{\alpha}}},\overline{f_{\beta}}(x)=\overline{f_{\alpha}}(x)$ y
se tiene (41).
Si se dice que $\overline{f_{\beta}}\preceq\overline{f_{\alpha}}$ se procede
de manera an loga. Esto dm. $\overline{f}$ est bien definida.
Veamos ahora que $\overline{f}\in S$ o que
4. i)
$\overline{f}\diagup M=f$
5. ii)
$\overline{f}\in\left(\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\right)^{\prime}$
6. iii)
$\|\overline{f}\|=\|f\|.$
7. (a)
Sea $x\in M.$ Veamos que: $\overline{f}=f(x).$
Como $x\in M\subset\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}},\exists\beta\in I$ tal que $x\in
D_{\overline{f_{\beta}}}\Longrightarrow\overline{f}(x)=\overline{f_{\beta}}(x)=f(x)\\\
\\\ \begin{cases}S\in\overline{f_{\beta}}\,\,\text{y por lo tanto},\\\ \\\
\overline{f_{\beta}}\diagup M=f.\,\,\text{Como}x\in
M,\overline{f_{\beta}}(x)=f(x).\end{cases}$
8. ii)
Veamos $\overline{f}\in\left(\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\right)^{\prime}.$ Primero veamos que
${\overline{f}\in\left(\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\right)}^{*},$ i.e, que f es A.L.
Sean $x,y\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\Longrightarrow\exists\beta,\varpi\in I$ tal que
$x\in D_{\overline{f_{\beta}}}$ y $y\in D_{\overline{f_{\varpi}}}.$ Entonces
$x+y\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$ ya que es un
subespacio de X. Veamos que
$\overline{f}(x+y)=\overline{f}(x)+\overline{f}(y)$.
Pero $\overline{f_{\beta}}\text{y
$\overline{f_{\varpi}}$}\in\bigl{\\{}\overline{f_{\alpha}}\bigr{\\}}_{\alpha\in
I}\subset S.$
Luego $D_{\overline{f_{\beta}}}\subseteq D_{\overline{f_{\varpi}}}$
$D_{\overline{f_{\varpi}}}\subseteq D_{\overline{f_{\beta}}}.$
Suponganos que $D_{\overline{f_{\beta}}}\subseteq D_{\overline{f_{\varpi}}}.$
Como $x\in D_{\overline{f_{\beta}}},x\in D_{\overline{f_{\varpi}}};y\in
D_{\overline{f_{\varpi}}}\Longrightarrow(x+y)\in D_{\overline{f_{\varpi}}}.$
$\overline{f}(x+y)=\overline{f_{\varpi}}(x+y).\,\,{\overline{f_{\varpi}}\in\left(D_{\overline{f_{\varpi}}}\right)}^{*}=\overline{f_{\varpi}}(x)+\overline{f_{\varpi}}(y)=f(x)+f(y).$
Sea $x\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$ y
$\lambda\in\mathbb{R}.\,\,x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\Longrightarrow\exists\beta\in I$ tal que $x\in
D_{\overline{f_{\beta}}}:$ subespacio de X.$\hskip 14.22636pt\therefore\hskip
14.22636pt\Longrightarrow\lambda x\in
D_{\overline{f_{\beta}}}\Longrightarrow\overline{f}(\lambda
x)=\overline{f_{\beta}}(\lambda
x)=\lambda\overline{f_{\beta}}(x)=\lambda\overline{f}(x).$
Veamos $\overline{f}$ es continua. Sea $x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}.$
Entonces $\exists\beta\in I$ tal que $x\in
D_{\overline{f_{\beta}}}\Longrightarrow\overline{f}(x)\in\mathbb{R}$ y por la
def. de $\overline{f},f(x)=\overline{f_{\beta}}(x).\hskip
14.22636pt\therefore\hskip
14.22636pt\left|\overline{f}(x)\right|=\left|\overline{f_{\beta}}(x)\right|\leqslant\|\overline{f_{\beta}}\|\|x\|=\|f\|\|x\|$
Como $\overline{f_{\beta}}\in S,\|\overline{f_{\beta}}\|=\|f\|.$
As que $\forall x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}},\left|\overline{f}(x)\right|\leqslant\|f\|\|x\|$ lo
que dm. que $\overline{f}$ es continua.
Veamos ahora que $\|\overline{f}\|=\|f\|.$
$\|\overline{f}\|=\underset{\|x\|\leqslant
1}{\sup\left|\overline{f}(x)\right|}.$
Sea $x\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$ con
$\|x\|\leqslant 1.$ Entonces, por lo establecido antes, $\exists\beta\in I$
tal que $\left|\overline{f}(x)\right|\leqslant\|f\|\|x\|\leqslant\|f\|.$
Esto dm. que el real $\|f\|$ es cota superior del cjto.
$\left\\{\left|\overline{f}(x)\right|,x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}},\|x\|\leqslant 1\right\\}.$
Luego $\underset{\|x\|\leqslant
1}{\sup\left|\overline{f}(x)\right|}\leqslant\|f\|$, i.e,
$\|\overline{f}|\leqslant\|f\|.$
Veamos que $\|f\|\leqslant\|\overline{f}\|.$
Batar con dm. que
(41) $\displaystyle\left\\{\left|\overline{f}(x)\right|,x\in M,\|x\|\leqslant
1\right\\}\subset\left\\{\left|\overline{f}(x)\right|,x\in\underset{\alpha\in
I}{\bigcup D}_{\overline{f_{\alpha}}},\|x\|\leqslant 1\right\\}$
ya que el tomar el supremo se tendr que $\underset{x\in M,\|x\|\leqslant
1}{\sup\left|f(x)\right|}\leqslant\underset{x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}}{\sup\left|\overline{f}(x)\right|}$
i.e, $\|f\|\leqslant\|\overline{f}\|.$
Establezcamos pues, (41).
Sea $y$ un vector en el primer cjto.
Entonces $\exists x\in M\Longrightarrow x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}$ con $\|x\|\leqslant 1$ y tal que $y=\|f(x)\|$.
$x\in\underset{\alpha\in I}{\bigcup
D}_{\overline{f_{\alpha}}}\Longrightarrow\exists\beta\in I$ tal que $x\in
D_{\overline{f_{\beta}}}\Longrightarrow\overline{f}(x)=\overline{f_{\beta}}(x)\Longrightarrow
f(x)=\overline{f}(x).$ Pero $y=\left|f(x)\right|.$ Entonces
$y=\left|\overline{f}(x)\right|.$
$\exists\beta\in I\Longrightarrow\overline{f_{\beta}}\in S.\hskip
14.22636pt\therefore\hskip 14.22636pt\overline{f_{\beta}}\diagup M=f.$ Como
$x\in M,\overline{f_{\beta}}=f(x).$
As que $\exists x\in\underset{\alpha\in I}{\bigcup D}_{\overline{f_{\alpha}}}$
con $\|x\|\leqslant 1$ tal que $y=\left|\overline{f}(x)\right|,$ lo que dm.
que $y$ est en el $2^{do}$ cjto.
Veamos que $\forall\alpha\in I:\overline{f_{\alpha}}\preceq\overline{f}$ con
lo que habremos dm. que $\left\\{\overline{f_{\alpha}}\right\\}_{\alpha\in I}$
tiene una cota superior en S.
Fijemos $\alpha\in I.$ Veamos que $\overline{f_{\alpha}}\preceq\overline{f}.$
Se debe probar que $D_{\overline{f_{\beta}}}\subseteq D_{\overline{f}}$ (esto
es claro). y que $\overline{f}\diagup D_{\overline{f}}=\overline{f_{\alpha}},$
o que $\forall x\in
D_{\overline{f_{\alpha}}},\overline{f}=\overline{f_{\alpha}}.$
Tomemos $x\in D_{\overline{f_{\beta}}}.$ Entonces, por la def. de
$\overline{f},\overline{f}(x)=\overline{f_{\alpha}}(x).$
Hemos dm. hasta aqu que
$\left\\{(S,\preceq):\text{cjto. P.O. y
que}\forall\left(\overline{f_{\alpha}}\right)_{\alpha\in I},\\\ \\\
\left\\{\overline{f_{\alpha}}\right\\}_{\alpha\in I}\,\,\text{tiene cota
superior en S}.\right\\}$
Luego, por el Lema de Zorn, S tiene elemento maximal, i.e,$\exists D_{F}:$
sub.vect. de X con $M\subset D_{F}$ y $\exists F\in(D_{F})^{\prime}$ i.e,
$F:D_{F}\longrightarrow\mathbb{R}\diagup\text{A.L. Continua}$ tal que
$F\diagup M=f\\\ \|F\|=\|f\|$
y $\nexists\overline{f}\in S$ tal que
(42) $\displaystyle F\preceq\overline{f}$
o lo que es lo mismo, $\nexists\overline{f}\in S$ tal que $D_{F}\subset D_{f}$
y $\overline{f}\diagup D_{F}=F.$
Para completar la prueba del T.H.B. solo resta dm. que $D_{F}=X.$
Razonemos por R.Abs.
Supongamos que no es as . Entonces $\exists x_{0}\in X,x_{0}\notin D_{F}$ y
por el
$\text{Lema},\exists\overline{f}\in\left(Sg(D_{F})\cup\bigl{\\{}x_{0}\bigr{\\}}\right)^{\prime}=(D_{F})^{\prime}$
tal que $\overline{f}\diagup D_{F}=F$ y $\|\overline{f}\|=\|F\|.$
Veamos que $\overline{f}\in S$ y que $F\preceq\overline{f}$
lo que constituye una $(\rightarrow\leftarrow)$ con (42)
De esta manera $X=D_{F}$ y termina la dm.
Establezcamos pues (42). Es claro que
$\displaystyle\overline{f}\in(D_{F})^{\prime}$ $\displaystyle M\subset D_{F}$
$\displaystyle\|\overline{f}\|=\|F\|=\|f\|$
Veamos que $\overline{f}\diagup M=f$ o que $\forall x\in
M:\overline{f}(x)=f(x).$
$\forall x\in M\Longrightarrow F\diagup M=f.F(x)=f(x).$ Como $x\in M\subset
D_{F}$ y $\overline{f}\diagup D_{F}=F,\overline{f}(x)=F(x).$
Esto dm. que $\overline{f}\in S.$
Finalmente, como $D_{F}\subset D_{\overline{f}}$ y $\overline{f}\diagup
D_{F}=F$ se tiene que $F\preceq f$ y se tiene (42).
∎
## 2\. Algunas consecuencias del T.H.B
Recordemos el enunciado del T.H.B.
”todo funcional continuo definido sobre un subespacio vectorial de un E.L.N se
puede extender (prolongar) a todo el espacio conservando la norma”:
$(X,\|\|):\mathbb{R}$ Esp. vec. normado. $\exists F\in
X^{\prime}=\mathcal{L}_{c}(X,\mathbb{R})$ tal que
1. (1)
$F\diagup M=f$
2. (2)
$\|F\|=\|f\|$
lo que es lo mismo, $\underset{x\in X;\|x\|\leqslant
1}{\sup\left|F(x)\right|}=\underset{x\in X;\|x\|\leqslant
1}{\sup\left|f(x)\right|}$
###### Consecuencia 1.
Sea $(X,\|\|):\mathbb{R}$ esp. vect. normado y sea
$\underset{fijo}{\underbrace{x}}\in X,x\neq 0.$ Entonces $\exists
x^{\prime}\in X$ tal que
1. (1)
$\|x^{\prime}\|=1$
2. (2)
$\langle x,x^{\prime}\rangle=\|x\|$
###### Proof.
Definamos
$\begin{diagram}$
$\begin{diagram}$
Veamos que
$f\in\left(Sg\bigl{\\{}x\bigr{\\}}\right)^{\prime}=\mathcal{L}_{c}(Sg\bigl{\\{}x\bigr{\\}},\mathbb{R}).$
Tomemos $\alpha_{1}x,\alpha_{2}x\in Sg\bigl{\\{}x\bigr{\\}}.$
Entonces
$f\left(\alpha_{1}x+\alpha_{2}x\right)=f\left((\alpha_{1}+\alpha_{2})\right)x\\\
\\\
=(\alpha_{1}+\alpha_{2})\|x\|=\alpha_{1}\|x\|+\alpha_{2}\|x\|=f(\alpha_{1},x)+f(\alpha_{2},x).$
Sea $\lambda\in\mathbb{R},(\alpha x)\in Sg\bigl{\\{}x\bigr{\\}}.$
$f\left(\lambda(\alpha
x)\right)=f\left((\lambda\alpha)x\right)=\lambda\alpha\|x\|=\lambda f(\alpha
x).$
Esto dm. que f es A.L.
Ahora, $\forall(\alpha x)\in Sg\bigl{\\{}x\bigr{\\}}:\\\ \\\ \left|f(\alpha
x)\right|=\left|\alpha\|x\|\right|=|\alpha|\|x\|=\|\alpha x\|=1.\|\alpha x\|,$
lo que dm. que f es continua y de este modo se tiene que
$f\in\left(Sg\bigl{\\{}x\bigr{\\}}\right)^{\prime}.$
Luego, por el T.H.B, $\exists x^{\prime}\in X^{\prime}$ i.e
$\begin{diagram}$
$\begin{diagram}$
tal que
1. (1)
$x^{\prime}\diagup Sg\bigl{\\{}x\bigr{\\}}=f$
2. (2)
$\|x^{\prime}\|=\|f\|$
Seg n (1), $\forall\alpha x\in Sg\bigl{\\{}x\bigr{\\}}:\langle\alpha
x,x^{\prime}\rangle=f(\alpha x).$
Luego si $\alpha=1,\langle x,x^{\prime}\rangle=f(1.x)=f(x)=\langle
x,f\rangle=\|x\|$
De esta manera hemos dm que $\exists x^{\prime}\in X^{\prime}$ tal que
$\langle x,x^{\prime}\rangle=\|x\|.$
Veamos ahora que $\|x^{\prime}\|=1\hskip 14.22636pt\star.$
Seg n (2), $\|x^{\prime}\|=\|f\|.$ Luego para tener $\star$ veamos que
$\|f\|=1.$
$\|f\|=\underset{\|\alpha x\|\leqslant 1}{\sup\left|\langle\alpha
x,f\rangle\right|}=\sup|\alpha|\left|\langle
x,f\rangle\right|=\|x\|\sup|\alpha|=\|x\|\dfrac{1}{\|x\|}=1$
∎
###### Consecuencia 2.
Sea $(X,\|\|):\mathbb{R}$ esp. vect. Normado, $F\subset X,F$ cerrado y $x\in
X;x\notin F.$
$\exists x^{\prime}\in X^{\prime}$ tal que
1. (1)
$\langle x,x^{\prime}\rangle=1$
2. (2)
$\forall\in F:\langle y,x^{\prime}\rangle=0.$
O sea que ”si $F\subset(X,\|\|)$ y $x\notin F,$ hay un funcional continuo en X
que vale 1 en x y se anula en F”.
###### Proof.
Sea $M=F+Sg\bigl{\\{}x\bigr{\\}}=\left\\{y+\alpha x,y\in
F,\alpha\in\mathbb{R}\right\\}$
Es claro que M es un subespacio vect. de X.
Sea
$\begin{diagram}$
$\begin{diagram}$
f es A.L. ya que
$f\left((y_{1}+\alpha_{1}x)+(y_{2}+\alpha_{2}x)\right)=f\left((y_{1}+y_{2})+(\alpha_{1}+\alpha_{2})x\right)\\\
=\alpha_{1}+\alpha_{2}=f(y_{1}+\alpha_{1}x)+f(y_{1}+\alpha_{2}x).\\\ \\\
f\left(\lambda(y+\alpha x)\right)=f\left(\lambda
y+(\lambda\alpha)x\right)=\lambda\alpha=\lambda f(y+\alpha x)$
Esto dm. que f es A.L.
N tese a dem s que de la def. de f,
(43) $\begin{split}\forall y\in F:f(y)=f(y+0.x)=0\\\ \\\ \text{Adem
s},f(x)=f(0+1.x)=1\end{split}$
Recordemos ahora el sgte. resultado de la topolog a:
$\begin{cases}SeaA\subset(X,\|\|)\,\,\text{y$x\in X$}.\\\
x\in\overline{A}\Longleftrightarrow d(x,A)=\underset{y\in
A}{\inf\|x-y\|=0}\end{cases}$
En nuestro caso $F\subset(X,\|\|)$ y $x\notin F.$ Luego $x\notin\overline{F}$
y por lo tanto $d(x,F)=\underset{y\in F}{\inf\|x-y\|}>0.$
O sea que
(44) $\displaystyle\forall y\in F:\|x-y\|\geqslant d(x,F)>0$
Tomemos $(y-\alpha x)\in M$ con $\alpha\neq 0.$ Entones $\|y-\alpha
x\|=\left\|\alpha\left(\frac{1}{\alpha}y-x\right)\right\|\\\ \\\
=|\alpha|\left\|\frac{1}{\alpha}y-x\right\|\underset{\leavevmode\nobreak\
\eqref{8}}{\geqslant}|\alpha|\alpha(x,F)$
O sea que $|\alpha|\leqslant\dfrac{\|y-\alpha x\|}{d(x,F)}.$
∎
## 3\. Los espacios $\mathfrak{L}^{p}$
###### Definición 7.
Sea $\underset{fijo}{p}\geqslant 1.$ El espacio $\mathfrak{L}^{p}$ se define
como
$\mathfrak{L}^{p}=\left\\{x=(\xi_{j})_{j=1}^{\infty}\in
S(\mathbb{K})\,\,\text{tal que la serie de n meros
reales}\,\,|\xi_{1}|^{p}+|\xi_{2}|^{p}+\ldots\ldots+\text{es
convergente}\right\\}$
$=\left\\{x=(\xi_{j})_{j=1}^{\infty}\in S(\mathbb{K})\,\,\text{tal
que}\sum\limits_{j=1}^{\infty}|\xi_{j}|^{p}<\infty\right\\}$
###### Proposición 18.
$\mathfrak{L}^{p}$ es un subespacio de $S(\mathbb{K})$.
###### Proof.
1. (1)
Veamos que $\mathfrak{L}^{p}\neq\emptyset.$
Consideremos una sucesi n cualquiera que tenga una cola de infinitos ceros:
$x=\left\\{\xi_{1},\xi_{2},\ldots,\ldots,\xi_{n},0,0,\ldots\ldots\right\\}$
Es claro que $|\xi_{1}|^{p}+|\xi_{2}|^{p}+\ldots\ldots+$ converge a
$\sum\limits_{j=1}^{n}|\xi_{j}|^{p}$ y por lo tanto
$\mathfrak{L}^{p}\neq\emptyset.$
Otra forma de verlo es la siguiente:$\forall p>1,$ si
$x=\left\\{\dfrac{1}{j}\right\\}_{j=1}^{\infty}\in S(\mathbb{R}),$ la serie
$1+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\ldots\ldots+$ converge. Luego
$x=\left\\{\dfrac{1}{j}\right\\}_{j=1}^{\infty}\in\mathfrak{L}^{p}$ lo que dm.
una vez m s que $\mathfrak{L}^{p}\neq\emptyset.$
2. (2)
Sea
$x=\\{\xi_{j}\\}_{j=1}^{\infty},y=\\{\eta_{j}\\}_{j=1}^{\infty}\in\mathfrak{L}^{p},$
o sea que las sucesiones
$|\xi_{1}|^{p}+|\xi_{2}|^{p}+\ldots\ldots+\,\,\text{converge
a}\,\,\sum\limits_{i=1}^{\infty}|\xi_{i}|^{p}\\\
|\eta_{1}|^{p}+|\eta_{2}|^{p}+\ldots\ldots+\,\,\text{converge
a}\,\,\sum\limits_{i=1}^{\infty}|\eta_{i}|^{p}$
Veamos que $(x+y)\in\mathfrak{L}^{p}$ o que la serie
$|\xi_{1}+\eta_{1}|^{p}+|\xi_{2}+\eta_{2}|^{p}+\ldots\ldots+\sum\limits_{i=1}^{\infty}|\xi_{i}+\eta_{i}|^{p}$
Fijemos n en $\mathbb{N}$.
Por la Desigualdad de Minkowski en el
ELN$(\mathbb{K}^{n},\|\|_{p})\diagup\|x+y\|_{p}\leqslant\|x\|_{p}+\|y\|_{p}$
(45)
$\displaystyle\left(\sum\limits_{i=1}^{n}|\xi_{i}+\eta_{i}|^{p}\right)^{1/p}\leqslant\left(\sum\limits_{i=1}^{n}|\xi_{i}|^{p}\right)^{1/p}+\left(\sum\limits_{i=1}^{n}|\eta_{i}|^{p}\right)^{1/p}$
Pero como
$x\in\mathfrak{L}^{p},\sum\limits_{i=1}^{n}|\xi_{i}|^{p}\leqslant\sum\limits_{i=1}^{\infty}|\xi_{i}|^{p}$
y como
$y\in\mathfrak{L}^{p},\sum\limits_{i=1}^{n}|\eta_{i}|^{p}\leqslant\sum\limits_{i=1}^{\infty}|\eta_{i}|^{p}$
$\Longrightarrow\left(\sum\limits_{i=1}^{n}|\xi_{i}|^{p}\right)^{1/p}+\left(\sum\limits_{i=1}^{n}|\eta_{i}|^{p}\right)^{1/p}\leqslant\left(\sum\limits_{i=1}^{\infty}|\xi_{i}|^{p}\right)^{1/p}+\left(\sum\limits_{i=1}^{\infty}|\eta_{i}|^{p}\right)^{1/p}$
que en (45)
$\Longrightarrow\left(\sum\limits_{i=1}^{n}|\xi_{i}+\eta_{i}|^{p}\right)^{1/p}\leqslant\left(\sum\limits_{i=1}^{n}|\xi_{i}|^{p}\right)^{1/p}+\left(\sum\limits_{i=1}^{n}|\eta_{i}|^{p}\right)^{1/p}\\\
\\\ \\\
\Longrightarrow\sum\limits_{i=1}^{n}|\xi_{1}+\eta_{i}|^{p}\leqslant{\left[\left(\sum\limits_{i=1}^{\infty}|\xi_{i}|^{p}\right)^{1/p}+\left(\sum\limits_{i=1}^{\infty}|\eta_{i}|^{p}\right)^{1/p}\right]}^{p}$
O sea que la sucesi n de S. parciales de la de t rminos positivos
$|\xi_{1}+\eta_{1}|^{p}+|\xi_{2}+\eta_{2}|^{p}+\ldots\ldots+$ tiene una cota
superior y por tanto converge.
3. (3)
tomemos $\lambda\in\mathbb{K}$ y
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p}.$ Veamos que
$(\lambda x)=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p}.$
Debemos dm. que
$|\lambda\xi_{1}|^{p}+|\lambda\xi_{2}|^{p}+\ldots\ldots+\sum\limits_{j=1}^{\infty}|\lambda\xi_{i}|^{p}$
Como
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},|\xi_{1}|^{p}+|\xi_{2}|^{p}+\ldots\ldots+\sum\limits_{j=1}^{\infty}|\xi_{i}|^{p}\\\
\Longrightarrow|\lambda\xi_{1}|^{p}+|\lambda\xi_{2}|^{p}+\ldots\ldots+|\lambda|^{p}\sum\limits_{j=1}^{\infty}|\xi_{i}|^{p}$
Luego, $\mathfrak{L}^{p}$ es un subespacio de $S(\mathbb{K})$ y por lo tanto
$\mathfrak{L}^{p}$ es un $\mathbb{K}$ Esp. Vect.
∎
###### Proposición 19.
Sea $p\geqslant 1.$ La funci n
$\begin{diagram}$
$\begin{diagram}$
es una norma en $\mathfrak{L}^{p}$ y por tanto $(\mathfrak{L}^{p},\|\|_{p}):$
E.L.N
###### Proof.
1. (1)
Es claro que $\|x\|_{p}\geqslant 0.$ Supongamos que $\|x\|_{p}=0$ entonces
${\left(\sum\limits_{j=1}^{\infty}|\xi_{j}|^{p}\right)}^{1/p}=0\Longrightarrow\bigl{\\{}\xi_{j}\bigr{\\}}=0\,\,\forall
j,$ i.e, x es la sucesi n cero.
2. (2)
Sea
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},\alpha\in\mathbb{K}.$
Entonces $\alpha
x=\bigl{\\{}\alpha\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p}\Longrightarrow\sum\limits_{j=1}^{\infty}|\alpha\xi_{j}|^{p}$
converge a $|\alpha|\sum\limits_{j=1}^{\infty}|\xi_{i}|^{p}.\\\ \\\ \|\alpha
x\|={\left(\sum\limits_{j=1}^{\infty}|\alpha\xi_{j}|^{p}\right)}^{1/2}=|\alpha|{\left(\sum\limits_{j=1}^{\infty}|\xi_{j}|^{p}\right)}^{1/2}=|\alpha|\|x\|_{p}$
3. (3)
Supongamos $p=1$ y sean $x,y\in\mathfrak{L}^{1}.$ Veamos que
$\|x+y\|_{p}\leqslant\|x\|_{p}+\|y\|_{p}.$
Entonces $|\xi_{1}|+|\xi_{2}|+\ldots+\ldots+$ converge a
$\bigl{\\{}x_{j}\bigr{\\}}_{j=1}^{\infty}\\\ \\\
|\eta_{1}|+|\eta_{2}|+\ldots\ldots+$ converge a $\bigl{\\{}\eta_{j}\bigr{\\}}$
Por la desigualdad triangular,
$|\xi_{j}+\eta_{j}|\leqslant|\xi_{j}|+|\eta_{j}|$
Como $|\xi_{1}|+|\xi_{2}|+\ldots+\ldots$ converge y
$|\eta_{1}|+|\eta_{2}|+\ldots\ldots+$ converge, la serie
$(|\xi_{1}|+|\eta_{1}|)+(|\xi_{2}|+|\eta_{2}|)+\ldots\ldots+$ converge a
$\left(\sum\limits_{j=1}^{\infty}|\xi_{j}|+\sum\limits_{j=1}^{\infty}|\eta_{j}|\right)$
As que $(|\xi_{1}|+|\eta_{1}|)+(|\xi_{2}|+|\eta_{2}|)+\ldots\ldots+$ converge
y $|\xi_{j}+\eta_{j}|\leqslant|\xi_{j}|+|\eta_{j}|$ Luego por el criterio de
comparaci n, $|\xi_{1}+\eta_{1}|+|\xi_{2}+\eta_{2}|+\ldots\ldots$ converge y
su suma es tal
que:$\sum\limits_{j=i}^{\infty}|\xi_{j}+\eta_{j}|\leqslant\sum\limits_{j=1}^{\infty}|\xi_{j}|+\sum\limits_{j=1}^{\infty}|\eta_{j}|$
y se tiene as la desigualdad $\Delta_{r}$ para $p=1.$
Sea ahora $p>1$ y consideremos las sucesiones $x,y\in\mathfrak{L}^{p}.$
Entonces $\bigl{\\{}x+y\bigr{\\}}\in\mathfrak{L}^{p}.$ Veamos que
$\|x+y\|_{p}\leqslant\|x\|_{p}+\|y\|_{p}.$
Fijemos $n\in\mathbb{N}.$
Como
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},\sum\limits_{j=1}^{n}|\xi|^{p}\leqslant\sum\limits_{j=1}^{\infty}|\xi|^{p}\Longrightarrow{\left(\sum\limits_{j=1}^{n}|\xi|^{p}\right)}^{1/2}\leqslant{\left(\sum\limits_{j=1}^{n}|\xi|^{p}\right)}^{1/2}=\|x\|_{p}$
Como
$y=\bigl{\\{}\eta_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},\sum\limits_{j=1}^{n}|\eta|^{p}\leqslant\sum\limits_{j=1}^{\infty}|\eta|^{p}\Longrightarrow{\left(\sum\limits_{j=1}^{n}|\eta|^{p}\right)}^{1/2}\leqslant{\left(\sum\limits_{j=1}^{n}|\eta|^{p}\right)}^{1/2}=\|y\|_{p}$
Ahora por la Des. de Minkowski:
${\left(\sum\limits_{j=1}^{n}|\xi_{i}+\eta_{j}|^{p}\right)}^{1/2}\leqslant{\left(\sum\limits_{j=1}^{\infty}|\xi_{j}|\right)}^{1/2}+{\left(\sum\limits_{j=1}^{\infty}|\eta_{j}|\right)}^{1/2}\leqslant\|x\|_{p}+\|y\|_{p}$
e.i,
$\sum\limits_{j=1}^{n}|\xi_{i}+\eta_{i}|^{p}\leqslant{\left(\|x\|_{p}+\|y\|_{p}\right)}^{p}$
i.e, la suc. de sumas parciales de la serie
$|\xi_{1}+\eta_{1}|^{p}+|\xi_{2}+\eta_{2}|^{p}+\ldots\ldots+$ est acotada
superiormente. La serie
$|\xi_{1}+\eta_{1}|^{p}+|\xi_{2}+\eta_{2}|^{p}+\ldots\ldots+$ converge y
$\sum\limits_{j=1}^{\infty}|\xi_{i}+\eta_{i}|^{p}\leqslant{\left(\|x\|_{p}+\|y\|_{p}\right)}^{1/2}$
i.e $\|x+y\|_{p}\leqslant\|x\|_{p}+\|y\|_{p}$
∎
###### Proposición 20 (Desigualdad de Hölder en los $\mathfrak{L}^{p}$ ).
Sean p,q exponentes conjungados, y sean
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},y=\bigl{\\{}\eta_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{q}.$
Entonces la serie $|\xi_{1}\eta_{1}|+|\xi_{2}\eta_{2}|+\ldots\ldots$ converge
y $\sum\limits_{j=1}^{\infty}|\xi_{l}\eta_{j}|\leqslant\|x\|_{p}\|y\|_{q}.$
###### Proof.
Como
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{p},y=\bigl{\\{}\eta_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{q},\sum\limits_{j=1}^{\infty}|\xi_{j}|^{p}<\infty,\sum\limits_{j=1}^{\infty}|\eta_{j}|^{q}<\infty.$
Definamos las sucesiones:
(46)
$\displaystyle\overline{\xi_{j}}=\dfrac{\xi_{j}}{{\left(\sum\limits_{k=1}^{\infty}|\xi_{k}|^{p}\right)}^{1/p}}\hskip
14.22636pt,\hskip
14.22636pt\overline{\eta_{j}}=\dfrac{\eta_{j}}{{\left(\sum\limits_{k=1}^{\infty}|\eta_{k}|^{q}\right)}^{1/q}}$
Entonces
$|\overline{\xi_{j}}|^{p}=\dfrac{|\xi_{j}|^{p}}{\sum\limits_{k=1}^{\infty}|\xi_{k}|^{p}}\hskip
14.22636pt,\hskip
14.22636pt|\overline{\eta_{j}}|^{q}=\dfrac{|\eta_{j}|^{q}}{\sum\limits_{k=1}^{\infty}|\eta_{k}|^{q}}$
Y las series
$|\overline{\xi_{1}}|^{p}+|\overline{\xi_{2}}|^{p}+\ldots\ldots\\\ \\\
|\overline{\eta_{1}}|^{p}+|\overline{\eta_{2}}|^{p}+\ldots\ldots$ convergen a
1. En efecto,
$|\overline{\xi_{1}}|^{p}+|\overline{\xi_{2}}|^{p}=\dfrac{1}{\sum\limits_{k=1}^{\infty}|\xi_{k}|^{p}}\left(|\xi_{1}|^{p}+|\xi_{2}|^{p}+\ldots\ldots\right)=\dfrac{\sum\limits_{k=1}^{\infty}|\xi_{k}|^{p}}{\sum\limits_{k=1}^{\infty}|\xi_{k}|^{p}}=1\\\
\\\
|\overline{\eta_{1}}|^{q}+|\overline{\eta_{2}}|^{q}\dfrac{1}{\sum\limits_{k=1}^{\infty}|\eta_{k}|^{q}}\left(|\eta_{1}|^{q}+|\eta_{2}|^{p}+\ldots\ldots\right)=\dfrac{\sum\limits_{k=1}^{\infty}|\eta_{k}|^{q}}{\sum\limits_{k=1}^{\infty}|\eta_{k}|^{q}}=1$
Ahora
$|\overline{\xi_{j}}\overline{\eta_{j}}|\leqslant\dfrac{|\overline{\eta_{j}}|^{p}}{p}+\dfrac{|\overline{\eta_{j}}|^{q}}{q}$
Como la serie $\dfrac{1}{p}\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}|^{p}$
converge a $\dfrac{1}{p};$ y como la serie
$\dfrac{1}{q}\sum\limits_{j=1}^{\infty}|\overline{\eta_{j}}|^{q},$ converge a
$\dfrac{1}{q}$ entonces por el criterio de comparaci n, la serie
$\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}\overline{\eta_{j}}|$ converge,
y se tiene que su suma est acotada
$\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}\overline{\eta_{j}}|\leqslant\dfrac{1}{p}\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}|^{p}+\dfrac{1}{q}\sum\limits_{j=1}^{\infty}|\overline{\eta_{j}}|^{q}=\dfrac{1}{p}+\dfrac{1}{q}=1$
y si se tiene encuenta (46),
$\dfrac{\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}\overline{\eta_{j}}|}{\|x\|_{p}.\|y\|_{q}}\leqslant
1\Longrightarrow\sum\limits_{j=1}^{\infty}|\overline{\xi_{j}}\overline{\eta_{j}}|\leqslant\|x\|_{p}\|y\|_{q}$
∎
###### Proposición 21.
$\mathfrak{L}^{2}$ es completo y por lo tanto $\mathfrak{L}^{2}:$ E. Banach.
###### Proof.
Sea $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}:x_{1},x_{2},x_{3},\ldots\ldots,$
una S. de Cauchy en $\mathfrak{L}^{2}$
i.e,
$x_{1}=\left(\xi_{1}^{1},\xi_{2}^{1},\xi_{3}^{1},\ldots\ldots,\xi_{j}^{i},\ldots\right)$
S. de Cauchy en $\sum\limits_{j=1}^{\infty}|\xi_{j}^{1}|^{2}<\infty\\\ \\\
x_{2}=\left(\xi_{1}^{2},\xi_{2}^{2},\xi_{3}^{2},\ldots\ldots,\xi_{j}^{2},\ldots\right)$
S. de Cauchy en $\sum\limits_{j=1}^{\infty}|\xi_{j}^{2}|^{2}<\infty\\\ \\\
x_{3}=\left(\xi_{1}^{3},\xi_{2}^{3},\xi_{3}^{3},\ldots\ldots,\xi_{j}^{3},\ldots\right)$
S. de Cauchy en $\sum\limits_{j=1}^{\infty}|\xi_{j}^{3}|^{2}<\infty\\\ \\\
\ldots\ldots$
Sea $\varepsilon>0.$ Como $\bigl{\\{}x_{n}\bigr{\\}}_{n=1}^{\infty}$ es una S.
de Cauchy en $\mathfrak{L}^{2},\exists N\in\mathbb{N}$ tal que $\forall
m,n>N:{\left(\sum\limits_{j=1}^{\infty}|\xi_{j}^{m}-\eta_{j}^{n}|^{2}\right)}^{1/2}=\|x_{m}-x_{n}\|_{2}<\varepsilon$
i.e, $\forall m,n>N,\forall
j=1,2,\ldots\ldots:|\xi_{j}^{m}-\eta_{j}^{n}|<\varepsilon^{2}$
o tambi n $\forall m,n>N,\forall
j=1,2,\ldots\ldots:|\xi_{j}^{m}-\eta_{j}^{n}|<\varepsilon.$
As que si fijamos $j,j=1,2,3,\ldots\ldots$ la columna $\xi_{j}^{1}$
$\xi_{j}^{2}$ $\vdots$ $\xi_{j}^{m}$ $\xi_{j}^{n}$ $\downarrow$ $\xi_{j}$ es
una S. de Cauchy en K y como K es Banach, las sucesi n columna $j^{a}$
converge. Llamemos $\xi_{j}=\lim\limits_{m\longrightarrow\infty}\xi_{j}^{m}$
Esto permite que podamos definir la sucesi n
$x=\left(\xi_{1},\xi_{2},\ldots\ldots,\right)$
$x=\left(\lim\limits_{m\longrightarrow\infty}\xi_{1}^{m},\lim\limits_{m\longrightarrow\infty}\xi_{2}^{m},\ldots\ldots,\right)$
Vamos ahora a dm. que
1. (1)
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{2}$
2. (2)
$\lim\limits_{m\longrightarrow\infty}=x$ con lo que quedar establecido que
$\mathfrak{L}^{2}$ es Banach.
3. 1
Fijemos $p\in\mathbb{N}.$
Sea $\varepsilon>0.$ Entonces $\dfrac{\varepsilon}{2\sqrt{p}}>0.$
Como $\xi_{1}^{m}\overset{m\rightarrow\infty}{\longrightarrow\xi_{1}},\exists
N\in\mathbb{N}^{1}$ tal que $\forall
m>N^{1}:|\xi_{1}^{m}-\xi_{1}|<\dfrac{\varepsilon}{2\sqrt{p}}\\\ \\\
\xi_{2}^{m}\overset{m\rightarrow\infty}{\longrightarrow\xi_{1}},\exists
N\in\mathbb{N}^{2}$ tal que $\forall
m>N^{2}:|\xi_{2}^{m}-\xi_{2}|<\dfrac{\varepsilon}{2\sqrt{p}}\\\ \\\ \vdots\\\
\\\ \xi_{p}^{m}\overset{m\rightarrow\infty}{\longrightarrow\xi_{p}},\exists
N\in\mathbb{N}^{p}$ tal que $\forall
m>N^{p}:|\xi_{p}^{m}-\xi_{p}|<\dfrac{\varepsilon}{2\sqrt{p}}$
As que si escogemos
$N_{2}>\max\bigl{\\{}N^{1},N^{2},\ldots\ldots,N^{p}\bigr{\\}}$ se tendr que
$\forall
m>N_{2}:|\xi_{j}^{m}-\xi_{j}|<\dfrac{\varepsilon}{2\sqrt{p}},j=1,2,\ldots\ldots,p$
y al colacar y sumar sobre j de a p,
$\left(\sum\limits_{j=1}^{p}|\xi_{m}^{j}-\xi_{j}|^{2}\right)<\dfrac{\varepsilon^{2}}{4}$
siempre que $m>N_{2}$ i.e, $\exists N_{2}\in\mathbb{N}$ tal que $\forall
n>N_{2}:{\left(\sum\limits_{j=1}^{p}|\xi_{m}^{j}-\xi_{j}|^{2}\right)}^{1/2}<\dfrac{\varepsilon}{2}$
Por lo tanto, si $\underset{fijo}{m}>N_{2}\\\ \\\
\left\|(\xi_{1},\xi_{2},\ldots,\xi_{p})\right\|_{2}=\left\|(\xi_{1},\xi_{2},\ldots,\xi_{p})-(\xi_{1}^{m},\xi_{2}^{m},\ldots,\xi_{p}^{m})+(\xi_{1}^{m},\xi_{2}^{m},\ldots,\xi_{p}^{m})\right\|_{2}\\\
\\\
\leqslant\left\|(\xi_{1}^{m}-\xi_{1},\ldots\ldots,\xi_{p}^{m}-\xi_{p})\right\|_{2}+\left\|(\xi_{1}^{m},\ldots\ldots,\xi_{p}^{m})\right\|_{2}$
i.e,
${\left(\sum\limits_{j=1}^{p}|\xi_{j}|^{2}\right)}^{1/2}\leqslant{\left(\sum\limits_{j=1}^{p}|\xi_{j}^{m}-\xi_{j}|^{2}\right)}^{1/2}+{\left(\sum\limits_{j=1}^{p}|\xi_{j}^{m}|^{2}\right)}^{1/2}<\dfrac{\varepsilon}{2}+\|x_{m}\|_{2}\\\
\\\
\Longrightarrow\sum\limits_{j=1}^{2}|\xi_{j}|^{2}\leqslant\left(\dfrac{\varepsilon}{2}+\|x_{m}\|_{2}\right)^{2},\forall
p\in\mathbb{N}.$
lo cual significa que es cota superior de la S. Sumas parciales de la serie
$|\xi_{1}|^{2}+|\xi_{2}|^{2}+\ldots\ldots$ O sea que la serie
$|\xi_{1}|^{2}+|\xi_{2}|^{2}+\ldots\ldots$ converge, i.e,
$x=\bigl{\\{}\xi_{j}\bigr{\\}}_{j=1}^{\infty}\in\mathfrak{L}^{2}.$
4. 2.
Veamos finalmente que $\lim\limits_{m\rightarrow\infty}x_{m}=x,$ o que
$\forall\epsilon>0\,\,\exists N\in\mathbb{N}$ tal que $\forall
n>N:\|x_{m}-x\|_{2}<\epsilon,$ i.e,
${\left(\sum\limits_{j=1}^{\infty}|\xi_{j}^{m}-\xi_{j}|^{2}\right)}^{1/2}<\epsilon$
siempre que $m>N.$
Seg n (1), dado $\epsilon>0,\exists N\in\mathbb{N}$ tal que $\forall
m>N,p\in\mathbb{N},{\left(\sum\limits_{j=1}^{\infty}|\xi_{j}^{m}-\xi_{j}|^{2}\right)}^{1/2}<\epsilon\Longrightarrow\left(\sum\limits_{j=1}^{\infty}|\xi_{j}^{m}-\xi_{j}|^{2}\right)<\epsilon^{2},\forall
p\in\mathbb{N}\Longrightarrow{\left(\sum\limits_{j=1}^{\infty}|\xi_{j}^{m}-\xi_{j}|^{2}\right)}^{1/2}<\epsilon$
siempre que $m>N.$
∎
###### Proposición 22.
El dual de $\mathfrak{L}^{p}$ es $\mathfrak{L}^{q},$ o sea que
$\mathfrak{L}^{p^{\prime}}:$ el dual de $\mathfrak{L}^{p}$ es isom tricamente
isomorfo a $\mathfrak{L}^{q}.$
###### Proof.
Sea
$\begin{diagram}$
$\begin{diagram}$
$\begin{diagram}$
Sea $f=\bigl{\\{}\alpha_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{q}.$
Veamos que $\phi(f)$ est bien definida. En primer lugar, por la Desigualdad de
Hölder:$\sum\limits_{j=1}^{n}|\alpha_{j}\xi_{j}|\leqslant{\left(\sum\limits_{j=1}^{n}|\alpha_{j}|^{q}\right)}^{1/q}{\left(\sum\limits_{j=1}^{n}|\xi_{j}|^{p}\right)}^{1/p}\leqslant\|f\|_{q}\|x\|_{p}$
Lo que nos dm. que la serie $\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}$ es
ABS. Convergente, y por tanto la serie
$\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}$ converge. Esto prueba que
$\phi(f)$ est bien definida, teni ndose adem s que
(47)
$\displaystyle\left|\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}\right|\leqslant\sum\limits_{n=1}^{\infty}|\alpha_{n}\xi_{n}|\leqslant\|f\|_{q}\|x\|_{p}$
Sea
$x=\bigl{\\{}\xi_{n}\bigr{\\}}_{n=1}^{\infty},y=\bigl{\\{}\beta_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}.$
Veamos que $\phi_{f}(x+y)=\phi_{f}(x)+\phi_{f}(y).$
$x+y=\bigl{\\{}\xi_{n}+\beta_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}$
y por tanto
$\phi_{f}(x+y)=\sum\limits_{n=1}^{\infty}\alpha_{n}(\xi_{n}+\beta_{n})=\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}+\sum\limits_{n=1}^{\infty}\alpha_{n}\beta_{n}=\phi_{f}(x)+\phi_{f}(y).$
Esto dm. que $\phi_{f}\in\mathfrak{L}^{p*}:\text{dual algebraico de
$\mathfrak{L}^{p}$}.$
Seg n (47)
(48) $\displaystyle\forall
x\in\mathfrak{L}^{p}:\left|\phi_{f}(x)\right|=\left|\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}\right|\leqslant\sum\limits_{n=1}^{\infty}|\alpha_{n}\xi_{n}|\leqslant\|f\|_{q}\|x\|_{p}$
lo que dm. que $\forall f\in\mathfrak{L}^{q}:\phi_{f}$ es continua, i.e,
$\forall f\in\mathfrak{L}^{q}:\phi_{f}\in\mathfrak{L}^{p^{\prime}}$
As que
$\begin{diagram}$
$\begin{diagram}$
est bien definida.
Veamos ahora que $\phi$ es A.L.
Sean
$f=\bigl{\\{}\alpha_{n}\bigr{\\}}_{n=1}^{\infty},g=\bigl{\\{}\theta_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{q}.$
Veamos que
(49) $\displaystyle\phi_{(f+g)}=\phi_{f}+\phi_{g}$
$f+g=\bigl{\\{}\alpha_{n}+\theta_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{q}$
y por tanto $\phi_{f+g}\in\mathfrak{L}^{p^{\prime}}.$
Tomemos $x=\bigl{\\{}\xi_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}.$
Para dm, (49) bastar con probar que $\phi_{f+g}(x)=\phi_{f}(x)+\phi_{g}(x)$
$\phi_{f+g}(x)=\sum\limits_{n=1}^{\infty}(\alpha_{n}+\theta_{n})\xi_{n}=\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}+\sum\limits_{n=1}^{\infty}\theta_{n}\xi_{n}=\phi_{f}(x)+\phi_{g}(x).$
Esto dm. que
$\phi\in\mathcal{L}\left(\mathfrak{L}^{q},\mathfrak{L}^{p^{\prime}}\right)$
Veamos ahora que $\phi$ es continua.
$\forall f\in\mathfrak{L}^{q},\phi_{f}\mathfrak{L}^{p^{\prime}},\\\
\begin{diagram}$
$\begin{diagram}$
Debemos dm. que $\exists M>0$ tal que $\forall
f\in\mathfrak{L}^{q}:\|\phi_{f}\|\leqslant M\|f\|_{q}$
Sea $f\in\mathfrak{L}^{q}.$ Entonces $\phi_{f}\in\mathfrak{L}^{p^{\prime}}$ y
(50) $\displaystyle\|\phi_{f}\|=\sup\limits_{\|x\|_{p}\leqslant
1}\left|\phi_{f}(x)\right|=\sup\limits_{\|x\|\leqslant
1}\|f\|_{q}\|x\|_{q}=\|f\|_{q}$
$\left|\phi_{f}(x)\right|\leqslant\|f\|_{q}\|x\|_{p}.$
As que
(51) $\displaystyle\forall f\in\mathfrak{L}^{q}:\|\phi_{f}\|\leqslant
1\|f\|_{q}$
lo que nos dm. que
$\phi\in\mathcal{L}_{c}\left(\mathfrak{L}^{q},\mathfrak{L}^{p^{\prime}}\right).$
Veamos que $\phi$ es sobre y que
(52) $\displaystyle\|\phi_{f}\|\geqslant\|f\|_{q}$
Una vez establezcamos lo anterior, de (51) y (52) se concluye que
$\|\phi_{f}\|=\|f\|$ y por tanto $\phi$ es isometr a. Luego $\phi$ es A.L.
biyectiva y continua y por tanto
$\mathfrak{L}^{q}=\mathfrak{L}^{p^{\prime}}$(isomorfismo isom trico)
Sea $\overline{f}\in\mathfrak{L}^{p^{\prime}}.$ Veamos que: $\exists
f\in\mathfrak{L}^{p}$ tal que $\phi(f)=\overline{f}.$
$\begin{diagram}$
$\begin{diagram}$
es A.L. continua.
Como $\forall
n\in\mathbb{N}:e_{n}(0,0,\ldots,1,0,\ldots)\in\mathfrak{L}^{p},\overline{f}(e_{n})\in\mathbb{K}.$
(53)
$\displaystyle\overline{f}(e_{n})=\alpha_{n}\in\mathbb{K}=|\alpha_{n}|e^{i\theta_{n}}\diagup
0\leqslant\theta\leqslant{360}^{\circ}$
los $|\alpha_{n}|$ y $\theta_{n}$ conocidos.
Fijemos $m\in\mathbb{N}$ y definamos
$\beta_{k}=|\alpha_{k}|^{q-1}e^{-i\theta_{k}},k=1,\ldots,m\Longrightarrow|\beta_{k}|=|\alpha_{k}|^{q-1}$
Consideremos ahora la sucesi n:
$\bigl{\\{}w_{n}\bigr{\\}}_{n=1}^{\infty}=(\beta_{1},\beta_{2},\ldots\ldots,\beta_{m},0,0,\ldots\ldots)=\left(|\alpha_{1}|^{q-1}e^{-i\theta_{1}},|\alpha_{2}|^{q-1}e^{-i\theta_{2}},\ldots\ldots,|\alpha_{m}|^{q-1}e^{-i\theta_{m}},0,0,\ldots\ldots\right)\\\
\\\
=\left(|\alpha_{1}|^{q-1}e^{-i\theta_{1}}\right)_{e_{1}}+\left(|\alpha_{2}|^{q-1}e^{-i\theta_{2}}\right)_{e_{2}}+\ldots\ldots+\left(|\alpha_{m}|^{q-1}e^{-i\theta_{m}}\right)_{e_{m}}$
Es claro que $\bigl{\\{}w_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}$ ya
que tiene una cola de ceros.
Luego
$\overline{f}_{(w_{n})}=|\alpha_{1}|^{q-1}e^{-i\theta_{1}}\overline{f}_{e_{1}}+|\alpha_{2}|^{q-1}e^{-i\theta_{2}}\overline{f}_{e_{2}}+\ldots\ldots,|\alpha_{m}|^{q-1}e^{-i\theta_{m}}\overline{f}_{e_{m}}\\\
\\\ \leavevmode\nobreak\
\eqref{16}=|\alpha_{1}|^{q-1}e^{-i\theta_{1}}|\alpha_{1}|e^{i\theta_{1}}+|\alpha_{2}|^{q-1}e^{-i\theta_{2}}|\alpha_{2}|e^{i\theta_{2}}+\ldots\ldots+|\alpha_{m}|^{q-1}e^{-i\theta_{m}}|\alpha_{m}|e^{i\theta_{m}}\\\
\\\
=|\alpha_{1}|^{q}+\ldots\ldots+|\alpha_{m}|^{q}=\sum\limits_{k=1}^{\infty}|\alpha_{k}|^{q}$
Como $\overline{f}\in\mathfrak{L}^{p^{\prime}},\text{y
$\bigl{\\{}w_{n}\bigr{\\}}\in\mathfrak{L}^{p}$};\left|\overline{f}_{(w_{n})}\right|\leqslant\left\|\overline{f}\right\|\left\|\bigl{\\{}w_{n}\bigr{\\}}\right\|_{p}$
O sea que
$\sum\limits_{k=1}^{m}|\alpha_{k}|^{q}\leqslant\|\overline{f}\|{\left(\sum\limits_{k=1}^{m}|\beta_{k}|^{p}\right)}^{1/p}\\\
\\\
=\|\overline{f}\|{\left(\sum\limits_{k=1}^{m}{\left(|\alpha_{k}|^{q-1}\right)}^{p}\right)}^{1/p}=\|\overline{f}\|{\left(\sum\limits_{k=1}^{m}|\alpha_{k}|^{q}\right)}^{1/q}\\\
\\\ \\\ \frac{1}{p}+\frac{1}{q}=1\\\ \\\ p+q=pq\\\ \\\
(q-1)p=pq-p=p+q-p=q\hskip 14.22636pt\therefore\hskip
14.22636pt{\left(\sum\limits_{k=1}^{m}|\alpha_{k}|^{q}\right)}^{1-\frac{1}{p}}\leqslant\|\overline{f}\|$
i.e
${\left(\sum\limits_{k=1}^{m}|\alpha_{k}|^{q}\right)}^{\frac{1}{p}}\leqslant\|\overline{f}\|,$
cualqiuera sea $m\in\mathbb{N}$ lo cual significa que
$f=\bigl{\\{}\alpha_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{q}$ y que
(54) $\displaystyle\|f\|_{q}\leqslant\|\overline{f}\|$
Veamos ahora que
(55) $\displaystyle\phi(f)=\overline{f}$
Como
$f=\bigl{\\{}\alpha_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{q},{\left(\sum\limits_{n=1}^{\infty}|\alpha_{n}|^{q}\right)}^{1/q}=\|f\|_{q}$
$\begin{diagram}$
$\begin{diagram}$
$\begin{diagram}$
$\begin{diagram}$
Para obtener (55) debemos dm. que $\forall
x=\bigl{\\{}\xi_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}:\phi_{f}(x)=\overline{f}(x).$
Tomemos $x=\bigl{\\{}\xi_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p}$
Entonces $\phi_{f}(x)=\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}$
Como
$x=\bigl{\\{}\xi_{n}\bigr{\\}}_{n=1}^{\infty}\in\mathfrak{L}^{p},S_{m}=\sum\limits_{n=1}^{m}|\xi_{n}|^{p}\overset{m\rightarrow\infty}{\longrightarrow}\sum\limits_{n=1}^{\infty}|\xi_{n}|^{p}$
Sea $\epsilon>0.$ Entonces $\exists M\in\mathbb{N}$ tal que $\forall
m>M:\left|S_{m}-\sum\limits_{n=1}^{\infty}|\xi_{n}|^{p}\right|<{\epsilon}^{p},$
i.e, $\sum\limits_{n=m+1}^{\infty}|\xi_{n}|^{p}<{\epsilon}^{p}$ siempre que
$m>M.$ O sea que
(56)
$\displaystyle\|x-x_{m}\|_{p}={\left(\sum\limits_{n=m+1}^{\infty}|\xi_{n}|^{p}\right)}^{1/p}<\epsilon$
siempre que $m>M.$
$x_{m}=(\xi_{1},\xi_{2},\ldots\ldots,\xi_{m},0,0,\ldots\ldots,)\in\mathfrak{L}^{p};(x-x_{m})\in\mathfrak{L}^{p}$
De otra
parte,$x_{m}=(\xi_{1},\xi_{2},\ldots\ldots,\xi_{m},0,0,\ldots\ldots,)\in\mathfrak{L}^{p}=\xi_{1}e_{1}+\xi_{2}e_{2}+\ldots\ldots+\xi_{m}e_{m}\\\
\\\ \therefore\hskip
14.22636pt\overline{f}(x_{m})=\xi_{1}\overline{f}(e_{1})+\xi_{2}\overline{f}(e_{2})+\ldots\ldots+\xi_{m}\overline{f}(e_{m})=\alpha_{1}\xi_{1}+\alpha_{2}\xi_{2}+\ldots\ldots+\alpha_{m}\xi_{m}=\sum\limits_{n=1}^{m}\alpha_{n}\xi_{n}\\\
\\\
\|\overline{f}(x)-\overline{f}{x_{m}}\|=\|\overline{f}{x-x_{m}}\|\leqslant\|\overline{f}\|\|x-x_{m}\|_{p}\underset{\leavevmode\nobreak\
\eqref{19}}{\leqslant}\|f\|\epsilon,\\\ \\\
\left|\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}-\sum\limits_{n=1}^{m}\alpha_{n}\xi_{n}\right|=\left|\sum\limits_{n=m+1}^{\infty}\right|\leqslant\sum\limits_{n=m+1}^{\infty}|\alpha_{n}\xi_{n}|\leqslant\|f\|_{q}\|x-x_{m}\|_{p}<\|\overline{f}\|\epsilon$
siempre que $m>M$
$\left|\overline{f}(x)-\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}\right|\leqslant\left|\overline{f}(x)-\sum\limits_{n=1}^{m}\alpha_{n}\xi_{n}\right|+\left|\sum\limits_{n=1}^{m}\alpha_{n}\xi_{n}-\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}\right|\\\
\\\
\|\overline{f}\|\epsilon+\epsilon\|\overline{f}\|=2\|\overline{f}\|\epsilon,\hskip
14.22636pt\forall\epsilon$
Luego, $\overline{f}(x)=\sum\limits_{n=1}^{\infty}\alpha_{n}\xi_{n}.$
∎
## 4\. Mapeos Bilineales
En esta secci n se tratan los Mapeos Bilineales y se realiza un especial
nfasis al respecto sobre las diferencias entre stos y los Mapeos Lineales;
aunque estos est n relacionados ntimamente con el tema.
Sean $E,F,G$ esp.vectoriales sobre alg n campo escalar $\mathbb{K}=\mathbb{R}$
$\mathbb{C}$ de n meros reales o complejos. Un mapeo
$\begin{diagram}$
es denominado bilineal si los mapeos
$\begin{diagram}\hskip 14.22636pt\text{y}\hskip 14.22636pt\begin{diagram}$
$\begin{diagram}$
son lineales $\forall x\in E,F,$ s mbolos: $\phi\in\mathsf{Bil}(E,F,G)$ si
$\phi_{x}\in L(F,G)\,\,\text{y $\phi_{y}\in L(E,G)$}$
$\forall x\in E\,\,\text{y}\,\,y\in F.$ Por simplicidad:
$\mathsf{Bil}(E,F)=\mathsf{Bil}(E,F,\mathbb{K}).$ Si $E,F,G$ son espacios
normados (m s generalmente espacios vectoriales topol gicos), el conjunto de
mapeos bilineales continuos $E\times F\longrightarrow G$ se puede denotar por
$\mathsf{Bil}(E,F,G)$ y $\mathsf{Bil}(E,F)$ si $G=\mathbb{K}.$
Para
$\phi(x,y)-\phi(x_{0},y_{0})=\phi(x-x_{0},y-y_{0})+\phi(x_{0},y-y_{0})$
Los siguientes desarrollos son sencillos de deducir.
###### Proposición 23.
Para $\phi\in\mathsf{Bil}(E,F;G)$ las siguientes afirmaciones son equivalentes
entre s :
1. (a)
$\phi$ es continua
2. (b)
$\phi$ es continua (0,0)
3. (c)
Sea C una constante, tal que $C\geqslant 0,$ se tiene
$\|\phi(x,y)\|_{G}\leqslant C\|x\|_{E}\|y\|_{F},\forall(x,y)\in E\times F$
Se observa claramente que $\|\phi\|=\min\bigl{\\{}C\geqslant
0\bigr{\\}}=\sup\bigl{\\{}\|\phi(x,y)\|_{G}\diagup x\in B_{E},y\in
B_{F}\bigr{\\}}$
define una norma sobre $\mathsf{Bil}(E,F;G)$ que es uniforme y una norma
completa si $\|.\|_{G}$ lo es.
N tese que el mapeo Bilineal continuo no es uniformemente continuo puesto que,
la resticci n $\mathbb{R}^{2}\longrightarrow\mathbb{R},(x,y)\rightsquigarrow
xy$ sobre la diagonal es la funci n $\mathbb{R}\ni x\rightsquigarrow
x^{2}\in\mathbb{R}.$
Un mapeo Bilineal $\phi\in\mathsf{Bil}(E,F;G)$ es separable continuo si para
todo $\phi_{x}:F\longrightarrow G$ y $\phi_{y}:E\longrightarrow G$ son
continuos.
###### Teorema 1.
Sean $E,F,G$ espacios normados y E es completo. Para todo mapeo Bilineal
continuo separable $\phi\in\mathsf{Bil}(E,F;G)$ es continuo.
###### Proof.
El conjunto $D=\bigl{\\{}z^{\prime}\circ\phi_{y}\diagup z^{\prime}\in
B_{G}^{\prime},y\in B_{F}\bigr{\\}}\subset E^{\prime}$ es
$\sigma(E^{\prime},E)-$ continuo puesto que para todo $x\in E$
$\left|\langle z^{\prime}\circ_{y},x\rangle\right|=\left|\langle
z^{\prime},\phi(x,y)\rangle\right|\leqslant\|z^{\prime}\|\|\phi(x,y)\|\leqslant\|x\phi\|$
Por teorema de MacKey’s $\diagup$ la uniformidad principal continua muestra
que D es $\textit{uniformemente continuo},$ i.e all la constante $c\geqslant
0$ tal que $\forall z^{\prime}\in B_{G}^{\prime}$ y $y\in B_{F}$
$\left|\langle z^{\prime},\phi(x,y)\rangle\right|=\left|\langle
z^{\prime}\circ\phi_{y},x\rangle\right|\leqslant c\|x\|_{E}$
para todo $x\in E.$ Esto prueba que $\|\phi\|\leqslant c.$
∎
Algunos ejemplos de mapeos Bilineales:
1. (1)
Para $x^{\prime}\in E^{\prime}$ y $y^{\prime}\in F^{\prime}$
$[x^{\prime}\underline{\otimes}y^{\prime}](x,y)=\langle
x^{\prime},x\rangle\langle y^{\prime},y\rangle$
define una forma Bilineal continua y
$\|x^{\prime}\underline{\otimes}y^{\prime}\|=\|x^{\prime}\|\|y^{\prime}\|.$ Si
$x^{\prime}_{n}$ y $y^{\prime}_{n}$ pertenecen a una bola unitaria y
$(\lambda_{n})\in\mathcal{L}_{1}$, entonces
$\varphi(x,y)=\sum\limits_{n=1}^{\infty}\lambda_{n}[x^{\prime}\underline{\otimes}y^{\prime}](x,y)$
es una funci n bien definida y
$\|\varphi\|\leqslant\sum\limits_{n=1}^{\infty}|\lambda_{n}|;$ pertenecen a la
clase de las formas bilineales denominadas $\textit{nucleares}.$
2. (2)
La condici n del mapeo sobre el espacio $\mathcal{L}(E,F)$ de operadores
lineales continuos
$\begin{diagram}$
tiene norma 1 (si E y F son no triviales).
3. (3)
Si E y F son esp. vectoriales de dimensi n finita, entonces todo mapeo
bilineal $E\times F\longrightarrow G$ es continuo (empleando bases).
4. (4)
El mapeo de convoluci n
$\begin{diagram}$
es bilineal.
5. (5)
Tomemos las funciones continuas sobre un espacio compacto K y sea E un espacio
normado. Entonces
$\begin{diagram}$
es bilineal.
Los mapeos
$\begin{diagram}$
$\begin{diagram}$
son espacios vectoriales isomorfos y son inversos a cualquier otro. Puesto que
$\|\varphi\|=\sup\bigl{\\{}\left|\varphi(x,y)\right|\diagup x\in B_{E},y\in
B_{F}\bigr{\\}}=\sup\bigl{\\{}\|L\varphi x\|\diagup x\in
B_{F}\bigr{\\}}=\|L\varphi\|\in[0,\infty];$ este isomorfismo reduce las formas
bilineales continuas a espacios isom tricos normados
$\mathsf{Bil}(E,F)=\mathcal{L}(E,F^{\prime})$
$\|l\varphi\|=\|\varphi\|\,\,\text{y $\|\beta T\|=\|T\|$}$
Esta relaci n es b sica para la compresi n de las ideas que desarrollaremos
acontinuaci n: La forma bilineal continua sobre $E\times F$ son ex ctamente
los operadores lineales continuos $E\longrightarrow F^{\prime}.$
El Teorema de Hahn-Banach para operadores, no es completo para formas
bilineales continuas en el siguiente sentido: Sea $G\subset E$ un subespacio y
$\varphi\in\mathsf{Bil}(G,F)$;
¿No existe all una extensi n $\overline{\varphi}\in\mathsf{L}(E,F^{\prime})$
de $\varphi$? Esto puede pensarse, por la identificaci n de las formas
bilineales y los operadores; $\forall T\in\mathcal{L}(G,F^{\prime})$ puede
tener una extensi n $\overline{T}\in\mathcal{L}(E,F^{\prime}).$
As se pueden observar algunos ejemplos de operadores que no son extensi n del
caso especial dado en que $G=F^{\prime}$ y $T=idG$. La identidad del mapeo: La
extensi n $\overline{T}$ puede ser una proyecci n de E sobre G.
* •
Debido a un resultado famoso de Lindenstrauss-Tzafriri [Sobre el problema del
complemento de los subespacios; Israel J. Math. 9(1971) 263-69] todos los
espacios de Banach de dimensi n infinita que no son isomorfos al espacio de
Hilbert no son complemento de subespacios cerrados.
* •
Puede observarse m s en concreto en el ejemplo dado por la funci n de
Rademacher definida sobre $[0,1]$
$r_{n}(t)=(-1)^{k}\text{s
$t\in\left[\dfrac{k}{2^{n}},\dfrac{k+1}{2^{n}}\right[$}$
(La forma es ortonormal al sistema en $L_{2}[0,1],$ medida de Lebesgue) y
consid rese la inyecci n
$\begin{diagram}$
La desigualdad de $\textit{Khintchine}:$ ”Para $1\leqslant p<\infty$ all son
constantes $a_{p}$ y $b_{p}\geqslant 1$ tal que
$a_{p}^{-1}{\left(\sum\limits_{k=1}^{n}|\alpha_{k}|^{2}\right)}^{1/2}\leqslant{\left(\int\limits_{D_{n}}{|\sum\limits_{k=1}^{n}\alpha_{k}\xi_{k}(w)|}^{p}\mu_{n}(dw)\right)}^{1/2}\leqslant
b_{p}{\left(\sum\limits_{k=1}^{n}|\alpha_{k}|^{2}\right)}^{1/2}$
para todo $n\in\mathbb{N}$ y $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{C}.$”
puede mostrarnos que $L_{1}$ induce una norma equivalente sobre
$\mathcal{L}_{2}$
La extensi n de las complexiones, afortunadamente, no es un problema. bservese
que all no son uniformemente continuas!
###### Proposición 24.
Sean E,F,G espacios normados y G completo. Para todo
$\phi\in\mathsf{Bil}(E,F;G)$ existe una extensi n nica
$\overline{\phi}\in\mathsf{Bil}(\overline{E},\overline{F};G).$ Adem s,
$\|\phi\|=\|\overline{\phi}\|.$
Este desarrollo es sencillo para la relaci n isom trica
$\mathsf{Bil}(E,F;G)=\mathcal{L}\left(E,\mathcal{L}(F,G)\right)$
y la extensi n de los operadores lineales continuos.
###### Observación 3.
Si E es un espacio normado de dimensi n menor que 2, entonces la multiplicaci
n por el escalar $k,\mathbb{K}\times E\longrightarrow E$ es bilineal,
continua, sobreyectiva s lo s E es cerrado.
###### Proof.
Tomemos un conjuto abierto no vac o $V\subset E$ y un funcional $y^{\prime}\in
E^{\prime}$ con
$\inf|\langle y^{\prime},x\rangle|>0$
Si U es la bola unitaria abierta en $\mathbb{K},$ entonces $0\in U.V,$ s lo si
0 no es un punto interior de $U.V$ puesto que $U.V\cap\ker
y^{\prime}=\bigl{\\{}0\bigr{\\}}$
∎
Esto tambi n es posible para los ejemplos $\phi\in\mathsf{Bil}(E,F;G)$ que son
sobreyectivos y conjuntos cerrados en cero, i.e, cero pertenece al interior de
$\phi(B_{E},B_{F}).$
Otro propiedad negativa de los mapeos bilineales continuos es que ellos no
permanecen continuos para las topolog as d viles: los vectores unitarios
$e_{n}$ en $\mathcal{L}_{2}$ (en $\mathbb{R}$) convergen d vilmente a cero
exceptuando $(e_{n}|e_{n})_{\mathcal{L}_{2}}=1$
Para espacios normados E y F se tiene el siguiente desarrollo isom trico:
$\phi:\mathsf{Bil}(E,F)=\mathcal{L}(E,F^{\prime})\hookrightarrow\mathcal{L}(F^{\prime\prime},E^{\prime})=\mathsf{Bil}(F^{\prime\prime},E)=\mathsf{Bil}(E,F^{\prime\prime}),$
$T\rightsquigarrow T^{\prime}$
donde la ltima igualdad es la ”transposici n” obvia $U^{t}(x,y)=U(y,x).$ Para
todo $\phi\in\mathsf{Bil}(E,F)$ se define
${\varphi}^{\wedge}=\phi(\varphi)\in\mathsf{Bil}(E,F^{\prime\prime});$ si
satisface $\|{\varphi}^{\wedge}\|=\|\varphi\|=\|L_{\varphi}\|$ y
${\varphi}^{\wedge}(x,y^{\prime\prime})=\left\langle
L^{\prime}_{\varphi}(y^{\prime\prime}),x\right\rangle_{E^{\prime},E}=\left\langle
y^{\prime\prime},L_{\varphi}(x)\right\rangle_{F^{\prime\prime},F}=\left\langle
y^{\prime\prime},\varphi(x,.)\right\rangle_{F^{\prime\prime},F}$
para todo $(x,y^{\prime\prime})\in E\times F^{\prime\prime}.$ Puesto que, por
definici n $\varphi(x,y)=\langle y,L_{\varphi}(x)\rangle_{F,F^{\prime\prime}}$
para todo $x\in E$ y $y\in F,$ el mapeo ${\varphi}^{\wedge}$ extensi n de
$\varphi$ para $E\times F$ a $E\times F^{\prime\prime}$ con igual normal.
${\varphi}^{\wedge}$ es denominada la extensi n can nica derecha de $\varphi$.
###### Proposición 25.
Sean E y F espacios normados y $\varphi\in\mathsf{Bil}(E,F).$ Entonces
${\varphi}^{\wedge}$ es la nica forma bilineal separada
$\sigma(E,E^{\prime})-\sigma(F^{\prime\prime},F^{\prime})$-mapeo continua
$\psi:E\times F^{\prime\prime}\longrightarrow\mathbb{K}$ que extiende a
$\varphi.$
###### Proof.
Que ${\varphi}^{\wedge}$ es una extensi n es claro por el desarrolllo de la
definici n para la ecuaci n; se obtiene de las desigualdades para los
funcionales $\sigma(F^{\prime\prime},F^{\prime})$\- densidad de F en F”.
Claro, all se tiene la extensi n can nica izquierda ${}^{\wedge}{\varphi}$
sobre $E^{\prime\prime}\times F$ definida por
${}^{\wedge}{\varphi}={\left({(\varphi^{t})}^{\wedge}\right)}^{t}$ dado por
${}^{\wedge}{\varphi}(x^{\prime\prime},y)=\langle
x^{\prime\prime},(L_{\varphi}\circ
k_{f})y\rangle_{E^{\prime\prime},E^{\prime}}=\langle
x^{\prime\prime},\varphi(.,y)\rangle_{E^{\prime\prime},E^{\prime}}$
∎
De que manera ¿Son los funcionales ${(^{\wedge}{\varphi}})^{\wedge}$ y
${}^{\wedge}{({\varphi}}^{\wedge})$ sobre $E^{\prime\prime}\times
F^{\prime\prime}$ relativos? Bastante sorprendente, el siguiente desarrollo
exacto:
###### Corolario 4.
Para $\varphi\in\mathsf{Bil}(E,F)$ el desarrollo de los tres estamentos
siguientes son equivalentes:
1. (1)
Las dos extensiones ”can nicas” ${(^{\wedge}{\varphi}})^{\wedge}$ y
${}^{\wedge}{({\varphi}}^{\wedge})$ de $\varphi$ en $E^{\prime\prime}\times
F^{\prime\prime}$ coinciden.
2. (2)
All $\psi\in\mathsf{Bil}(E^{\prime\prime},F^{\prime\prime})$ que es separable
$\sigma(E^{\prime\prime},E^{\prime})-\sigma(F^{\prime\prime},F^{\prime})-$ son
continuos y extensiones de $\varphi$.
3. (3)
$L_{\varphi}:E\longrightarrow F$ es compacto-d vil.
En este caso el funcional $\psi$ en [2] aes igual a
${(^{\wedge}{\varphi}})^{\wedge}=^{\wedge}{({\varphi}}^{\wedge})$
###### Proof.
La proposici n implica sencillamente que $(a)\Leftrightarrow(b)$. Se observa
la equivalencia de (a) y (c), por
$L{(^{\wedge}{\varphi}})^{\wedge}=K_{F^{\prime}}\circ P_{F^{\prime}}\circ
L^{\prime\prime}_{\varphi}:E^{\prime\prime}\longrightarrow F^{\prime\prime}$
$L^{\wedge}{({\varphi}}^{\wedge})=P_{F^{\prime\prime\prime}}\circ
L^{\prime\prime}_{{\varphi}^{\wedge}}=P_{F^{\prime\prime\prime}}(K_{F}^{\prime}\circ
L_{\varphi})^{\prime\prime}=$
$P_{F^{\prime\prime\prime}}\circ K^{\prime\prime}_{F^{\prime}}\circ
L^{\prime\prime}_{\varphi}=L^{\prime\prime}_{\varphi}$
Ahora afirmamos que $L_{\varphi}$ es compacto-d vil sii
$L^{\prime\prime}_{\varphi}(E^{\prime\prime})\subset F^{\prime}.$
∎
## 5\. La teor a algebra ca del producto tensorial.
El objeto de estudio de los mapeos bilineales puede reducirse al estudio de
los mapeos lineales. La construcci n de estos nuevos espacios vectoriales
$E\otimes F$ es, dada por el an lisis, de una manera simple.
Para un conjunto arbitrario A definamos $\mathcal{F}(A)$ como el conjunto de
todos las funciones $f:A\longrightarrow\mathbb{K}$ dentro de un soporte
finito, i.e. $f(\alpha)=0$ excepto sobre un subconjunto finito de A. Para
$\alpha\in A$ el ${}^{\prime\prime}\alpha-\text{vector
unitario}^{\prime\prime}$ es la funci n $e_{\alpha}\in\mathcal{F}(A)$ definida
por
$e_{\alpha}(\beta)=\delta_{\alpha\beta}$
del Delta de Kronecker. Es claro que para cada $f\in\mathcal{F}(A)$ se tiene
la nica representaci n
$f=\sum\limits_{\alpha}\in Af(\alpha)e_{\alpha},$
en otras palabras:$(e_{\alpha})_{\alpha\in A}$ es un conjunto algebraico b
sico de $\mathcal{F}(A).$ Ahora tomemos dos conjunto A,B y consideremos, el
mapeo bilineal
$\begin{diagram}$
Entonces $e(\alpha,\beta)=\Psi_{0}(e_{\alpha},e_{\beta}),$ es claro que
$\text{ext im$\Psi_{0}$}=\mathcal{F}(A\times B)$
Para $\psi\in\mathsf{Bil}(\mathcal{F}(A),\mathcal{F}(B),G)$ definamos un
funcional $T\in L(\mathcal{F}(A\times B),G)$ por
$T(e(\alpha,\beta))=\psi(e_{\alpha},e_{\beta})$
(una extensi n lineal). Es obvio que T es el nico mapeo lineal
$\mathcal{F}(A\times B)\longrightarrow G$ en el interior de
$T\circ\Psi_{0}=\psi$
Esto demuestra que
$L(\mathcal{F}(A\times B),G)=\mathsf{Bil}(\mathcal{F}(A),\mathcal{F}(B);G)$
$T\rightsquigarrow T\circ\Psi_{0}$
es un isomorfismo lineal de espacios vectoriales.
|
arxiv-papers
| 2009-08-13T02:10:47 |
2024-09-04T02:49:04.600846
|
{
"license": "Public Domain",
"authors": "Jaime Chica",
"submitter": "Mauricio Velasquez",
"url": "https://arxiv.org/abs/0908.1821"
}
|
0908.1823
|
Originally published: ]October 1825; transcribed: August 13, 2009; revised:
February 24, 2010
# ”Uber die Berechnung der geographischen L”angen und Breiten aus
geod”atischen Vermessungen111This transcription of Astronomische Nachrichten
4(86), 241–254 (1826), doi:10.1002/asna.18260041601, has been edited by
Charles F. F. Karney $\langle$ckarney@sarnoff.com$\rangle$ and Rodney E.
Deakin $\langle$rod.deakin@rmit.edu.au$\rangle$. The paper also appears in
Abhandlungen von Friedrich Wilhelm Bessel, Vol. 3 (W. Engelmann, Leipzig,
1876). The text follows the original; however the mathematical notation has
been updated to conform to current conventions. Several errors have been
corrected and the tables have been recomputed. An English translation of this
paper is available at arXiv:0908.1824.
F. W. Bessel K”onigsberger Sternwarte
([)
## 1
Die Aufgabe: aus der gegebenen Polh”ohe eines Punkts $A$, aus der, auf der
geod”atischen Linie gemessenen Entfernung eines anderen Punkts $B$ von dem
ersteren, und aus dem Winkel dieser Linie mit dem Meridiane von $A$, die
Polh”ohe und den Mittagsunterschied des zweiten Punkts, so wie den Winkel der
geod”atischen Linie mit dem Meridiane desselben zu finden, ist von so
h”aufiger Anwendung, da”s ich die Mittheilung der von mir, zur Erleichterung
der Rechnung entworfenen Tafeln, nicht f”ur unn”utz halte. Wie die
Vermessungen so berechnet werden k”onnen, da”s man die Entfernungen aller
Punkte derselben, von dem Hauptpunkte, auf geod”atischen Linien gemessen, und
die Azimuthe dieser Linien erh”alt, habe ich in Nr. 3 und 6 der Astronomischen
Nachrichten gezeigt.
Meine Tafeln weichen, in ihrer Einrichtung, von anderen, desselben Zwecks
wegen bekannt gemachten Tafeln ab, und k”onnen auch im Resultate, mit der oft
angewandten Dusejour schen Berechnungsart nicht ganz ”ubereinstimmen, indem
diese die Entfernungen und Azimuthe nicht auf die geod”atischen Linien
bezieht. Um dieselben deutlich erkl”aren zu k”onnen, werde ich die
Entwickelungen, worauf sie beruhen, ganz mittheilen, und dabei mit der
Ableitung der Eigenschaften der geod”atischen Linien auf dem
Rotationssph”aroid anfangen, selbst wenn Bekanntes dadurch wiederholt wird. Da
das hier nothwendige sich kurz genug fassen l”a”st, so darf der Vortheil,
alles hierhergeh”orige beisammen zu haben, nicht theuer erkauft werden.
## 2
Wenn man zwei Punkte $A$ und $B$, auf der Oberfl”ache eines Rotations-
Sph”aroids, durch eine nach irgend einem Gesetze gezogene Curve verbindet, und
in derselben zwei unendlich nahe Punkte annimmt, welchen die Polh”ohen $\phi$
und $\phi+d\phi$, und die vom Meridiane von $A$ gez”ahlten geographischen
L”angen $w$ und $w+dw$ (”ostlich positiv, westlich negativ genommen)
zugeh”oren; wenn man ferner die Entfernung derselben durch $ds$, den Winkel,
in welchem der von $A$ kommende Theil der Curve den Meridian durchschneidet
(von Norden rechts herum, von $0$ bis $360^{\circ}$ gez”ahlt) durch $\alpha$,
den Halbmesser des Parallelkreises durch $r$, den Kr”ummungshalbmesser durch
$R$ bezeichnet, so hat man:
$\begin{split}ds\cos\alpha&=-R\,d\phi=\frac{dr}{\sin\phi}\\\
ds\sin\alpha&=-r\,dw\end{split}$ (1)
woraus folgt
$ds=\sqrt{R^{2}\,d\phi^{2}+r^{2}\,dw^{2}}$
oder, wenn man, um abzuk”urzen, $p$ f”ur $d\phi/dw$ und $U$ f”ur
$\sqrt{R^{2}p^{2}+r^{2}}$ schreibt,
$ds=U\,dw.$
Die Entfernung der beiden Punkte $A$ und $B$, auf der Curve gemessen, ist
daher
$s=\int U\,dw$
das Integral von $A$ bis $B$ genommen. Soll die Curve die geod”atische oder
k”urzeste Linie sein, so mu”s ihr Gesetz, oder die Relation zwischen $\phi$
und $w$, so angenommen werden, da”s dieses Integral ein Minimum wird; oder
wenn man, statt derjenigen Relation zwischen $\phi$ und $w$, welche dieses
leistet, eine andere setzt, in welcher demselben $w$ nicht $\phi$, sondern
$\phi+z$ zugeh”ort, wo $z$ eine willk”urliche Function von $w$ ist, welche an
den Punkten $A$ und $B$ verschwindet, indem diese Punkte beiden Curven
gemeinschaftlich sind, so mu”s
$s^{\prime}=\int U^{\prime}\,dw$
gro”ser sein als $s$ und zwar f”ur ein unbestimmtes $z$.
Man erh”alt aber, nach dem Taylor schen Lehrsatze,
$U^{\prime}=U+\biggl{(}\frac{dU}{d\phi}\biggr{)}z+\biggl{(}\frac{dU}{dp}\biggr{)}\frac{dz}{dw}+\ldots$
und daher
$s^{\prime}=s+\int\biggl{(}\frac{dU}{d\phi}\biggr{)}z\,dw+\int\biggl{(}\frac{dU}{dp}\biggr{)}dz+\ldots$
wo die geschriebenen Glieder von der Ordnung von $z$, die nur angedeuteten
aber von h”oheren Ordnungen sind; es mu”s also
$\int\biggl{(}\frac{dU}{d\phi}\biggr{)}z\,dw+\int\biggl{(}\frac{dU}{dp}\biggr{)}dz+\ldots$
keinen negativen Werth haben, welche Function man auch f”ur $z$ annehmen mag.
Da dieses auch f”ur entgegengesetzte Annahmen, $z$ und $-z$ gelten mu”s, und
man die Freiheit hat, $z$ so klein anzunehmen, da”s die Glieder der ersten
Ordnung gr”o”ser werden, als die Summe der ”ubrigen, au”ser wenn jene
verschwinden, so folgt, da”s das Minimum nur stattfinden kann, wenn die
Glieder der ersten Ordnung verschwinden: man hat also die Bedingung des
Minimums,
$0=\int\biggl{(}\frac{dU}{d\phi}\biggr{)}z\,dw+\int\biggl{(}\frac{dU}{dp}\biggr{)}dz$
und wenn man f”ur das zweite Glied $z({dU}/{dp})-\int z\,d({dU}/{dp})$ setzt
und sich erinnert, da”s $z$ f”ur beide Grenzen des Integrals verschwindet,
$0=\int
z\biggl{\\{}\biggl{(}\frac{dU}{d\phi}\biggr{)}dw-d\biggl{(}\frac{dU}{dp}\biggr{)}\biggr{\\}}.$
Da dieses Integral f”ur ein unbestimmtes $z$ verschwinden mu”s, so ist
$0=\biggl{(}\frac{dU}{d\phi}\biggr{)}dw-d\biggl{(}\frac{dU}{dp}\biggr{)}$
oder, durch Multiplication mit $d\phi/dw=p$
$0=\biggl{(}\frac{dU}{d\phi}\biggr{)}d\phi-p\,d\biggl{(}\frac{dU}{dp}\biggr{)}$
wovon das Integral
$\mathrm{const.}=U-p\biggl{(}\frac{dU}{dp}\biggr{)}$
ist. Setzt man f”ur $U$ seinen Ausdruck, n”amlich $\sqrt{r^{2}+R^{2}p^{2}}$,
so erh”alt man
$\mathrm{const.}=\frac{r}{\sqrt{1+(R^{2}/r^{2})p^{2}}}=-r\sin\alpha$
welches die bekannte characteristische Eigenschaft der geod”atischen Linie
ist.
Man hat also, wenn das Azimuth der geod”atischen Linie am Punkte $A$, durch
$\alpha^{\prime}$, und die Entfernung dieses Punkts von der Drehungsaxe durch
$r^{\prime}$ bezeichnet werden,
$r^{\prime}\sin(\alpha^{\prime}+180^{\circ})=r\sin\alpha$
oder
$r^{\prime}\sin\alpha^{\prime}=-r\sin\alpha$ (2)
## 3
Bezeichnet man die gr”o”ste Entfernung des Sph”aroids von der Rotationsaxe
durch $a$, so sind $r$ und $r^{\prime}$ kleiner, oder wenigstens nicht
gr”o”ser als $a$, und man kann setzen
$r^{\prime}=a\cos u^{\prime};\qquad r=a\cos u$
wodurch die Gleichung (2) die Form
$\cos u^{\prime}\sin\alpha^{\prime}=-\cos u\sin\alpha$ (3)
annimmt. Diese Gleichung enth”alt die Relation zwischen zwei Seiten eines
sph”arischen Dreiecks $90^{\circ}-u^{\prime}$ und $90^{\circ}-u$ und den ihnen
gegen”uberstehenden Winkeln $360^{\circ}-\alpha$ und $\alpha^{\prime}$. Die
dritte Seite desselben sph”arischen Dreiecks und der ihr gegen”uberstehende
Winkel, welche ich durch $\sigma$ und $\omega$ bezeichnen werde, geben, wenn
man sie in die Rechnung einf”uhrt, elegante Ausdr”ucke f”ur die
zusammengeh”origen Ver”anderungen von $s$, $u$ und $w$. Man hat n”amlich,
durch die bekannten Differentialformeln der sph”arischen Trigonometrie
$\displaystyle du$ $\displaystyle=-\cos\alpha\,d\sigma$ $\displaystyle\cos
u\,d\omega$ $\displaystyle=-\sin\alpha\,d\sigma$
und wenn man dieses in die Gleichungen (1) setzt, nachdem man in denselben $r$
durch $u$ ausgedr”uckt hat,
$\begin{split}ds&=a\frac{\sin u}{\sin\phi}d\sigma\\\ dw&=\frac{\sin
u}{\sin\phi}d\omega\end{split}$ (4)
## 4
Die Erdmeridiane werde ich jetzt als elliptisch annehmen, und ihre halbe
gro”se Axe durch $a$, die halbe kleine Axe durch $b$, die Excentricit”at durch
$e$ bezeichnen. Differentiirt man die Gleichung der Ellipse f”ur Coordinaten
aus dem Mittelpunkte
$1=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
und setzt man $-\cot\phi$ f”ur $dy/dx$, so erh”alt man
$0=\frac{x\sin\phi}{a^{2}}-\frac{y\cos\phi}{b^{2}}$
und aus der Verbindung beider Gleichungen
$x=\frac{a\cos\phi}{\sqrt{1-e^{2}\sin^{2}\phi}}.$
Dieses $x$ ist unser $r$, also $=a\cos u$, woraus man erh”alt
$\displaystyle\cos u$
$\displaystyle=\frac{\cos\phi}{\sqrt{1-e^{2}\sin^{2}\phi}};$
$\displaystyle\cos\phi$ $\displaystyle=\frac{\cos
u\sqrt{1-e^{2}}}{\sqrt{1-e^{2}\cos^{2}u}}$ $\displaystyle\sin u$
$\displaystyle=\frac{\sin\phi\sqrt{1-e^{2}}}{\sqrt{1-e^{2}\sin^{2}\phi}};$
$\displaystyle\sin\phi$ $\displaystyle=\frac{\sin u}{\sqrt{1-e^{2}\cos^{2}u}}$
$\displaystyle\tan u$ $\displaystyle=\tan\phi\sqrt{1-e^{2}};$
$\displaystyle\tan\phi$ $\displaystyle=\frac{\tan u}{\sqrt{1-e^{2}}}$
und
$\frac{\sin u}{\sin\phi}=\sqrt{1-e^{2}\cos^{2}u}$
Substituirt man dieses in den Differentialen (4), so erh”alt man f”ur das
elliptische Rotationssph”aroid
$\begin{split}ds&=a\sqrt{1-e^{2}\cos^{2}u}\,d\sigma\\\
dw&=\sqrt{1-e^{2}\cos^{2}u}\,d\omega\end{split}$ (5)
## 5
Um das erste dieser Differentiale zu integriren, werde ich den drei
Gleichungen zwischen $u^{\prime}$, $u$, $\alpha^{\prime}$, $\alpha$ und
$\sigma$,
$\begin{split}\sin u&=\sin u^{\prime}\cos\sigma+\cos
u^{\prime}\sin\sigma\cos\alpha^{\prime}\\\ \cos u\cos\alpha&=\sin
u^{\prime}\sin\sigma-\cos u^{\prime}\cos\sigma\cos\alpha^{\prime}\\\ \cos
u\sin\alpha&=-\cos u^{\prime}\sin\alpha^{\prime}\end{split}$ (6)
durch Einf”uhrung der H”ulfswinkel $m$ und $M$, welche ich nach den Formeln
$\begin{split}\sin u^{\prime}&=\cos m\sin M\\\ \cos
u^{\prime}\cos\alpha^{\prime}&=\cos m\cos M\\\ \cos
u^{\prime}\sin\alpha^{\prime}&=\sin m\end{split}$ (7)
bestimme, die Form
$\begin{split}\sin u&=\cos m\sin(M+\sigma)\\\ \cos u\cos\alpha&=-\cos
m\cos(M+\sigma)\\\ \cos u\sin\alpha&=-\sin m\end{split}$ (8)
geben. Dadurch erh”alt man
$\cos^{2}u=1-\cos^{2}m\sin^{2}(M+\sigma)$
und
$ds=a\sqrt{1-e^{2}+e^{2}\cos^{2}m\sin^{2}(M+\sigma)}\,d\sigma$ (9)
Die Integration dieses Differentials h”angt von den elliptischen
Transcendenten ab und ist von Legendre, in den Exercices de calcul intégral,
gegeben. Allein die H”ulfsmittel zur Berechnung dieser Transcendenten,
scheinen noch nicht eine solche Vollst”andigkeit oder Geschmeidigkeit erlangt
zu haben, da”s die Entwickelung in eine Reihe, welche, bei der Kleinheit von
$e^{2}$ sehr schnell convergirt, nicht bequemer sein sollte. Man erh”alt
dieselbe am leichtesten durch Zerf”allung der Gr”o”se unter dem Wurzelzeichen
in zwei imagin”are Factoren, n”amlich
$ds={\textstyle\frac{1}{2}}a\,d\sigma\bigl{(}\sqrt{1-e^{2}\sin^{2}m}+\sqrt{1-e^{2}}\bigr{)}\times\\\
\bigl{(}1-\epsilon c^{2i(M+\sigma)}\bigr{)}^{1/2}\bigl{(}1-\epsilon
c^{-2i(M+\sigma)}\bigr{)}^{1/2}$
wo $c$ die Grundzahl der nat”urlichen Logarithmen und $i$ die Quadratwurzel
aus $-1$ bezeichnen, und $\epsilon$ f”ur
$\frac{\sqrt{1-e^{2}\sin^{2}m}-\sqrt{1-e^{2}}}{\sqrt{1-e^{2}\sin^{2}m}+\sqrt{1-e^{2}}}$
geschrieben ist. Setzt man hier
$\frac{e\cos m}{\sqrt{1-e^{2}}}=\tan E.$
so wird $\epsilon=\tan^{2}\frac{1}{2}E$ und
$ds=a\sqrt{1-e^{2}}\,\frac{\cos^{2}\frac{1}{2}E}{\cos E}\,d\sigma\times\\\
\sqrt{1-\epsilon c^{2i(M+\sigma)}}\sqrt{1-\epsilon c^{-2i(M+\sigma)}}$
L”oset man die beiden Factoren unter dem Wurzelzeichen in unendliche Reihen
auf, und multiplicirt man diese in einander, so erh”alt man
$ds=a\sqrt{1-e^{2}}\,\frac{\cos^{2}\frac{1}{2}E}{\cos
E}\,d\sigma\bigl{[}A-2B\cos 2(M+\sigma)\\\ -2C\cos 4(M+\sigma)-2D\cos
6(M+\sigma)-\ldots\bigr{]}$
wo $A$, $B$, $C$ … folgende unendliche Reihen bezeichnen:
$\displaystyle A$
$\displaystyle=1+\biggl{(}\frac{1}{2}\biggr{)}^{2}\epsilon^{2}+\biggl{(}\frac{1\\!\cdot\\!1}{2\\!\cdot\\!4}\biggr{)}^{2}\epsilon^{4}+\biggl{(}\frac{1\\!\cdot\\!1\\!\cdot\\!3}{2\\!\cdot\\!4\\!\cdot\\!6}\biggr{)}^{2}\epsilon^{6}+\ldots$
$\displaystyle B$
$\displaystyle=\frac{1}{2}\epsilon-\frac{1\\!\cdot\\!1}{2\\!\cdot\\!4}\,\frac{1}{2}\epsilon^{3}-\frac{1\\!\cdot\\!1\\!\cdot\\!3}{2\\!\cdot\\!4\\!\cdot\\!6}\,\frac{1\\!\cdot\\!1}{2\\!\cdot\\!4}\epsilon^{5}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{1\\!\cdot\\!1\\!\cdot\\!3\\!\cdot\\!5}{2\\!\cdot\\!4\\!\cdot\\!6\\!\cdot\\!8}\,\frac{1\\!\cdot\\!1\\!\cdot\\!3}{2\\!\cdot\\!4\\!\cdot\\!6}\epsilon^{7}-\ldots$
$\displaystyle C$
$\displaystyle=\frac{1\\!\cdot\\!1}{2\\!\cdot\\!4}\epsilon^{2}-\frac{1\\!\cdot\\!1\\!\cdot\\!3}{2\\!\cdot\\!4\\!\cdot\\!6}\,\frac{1}{2}\epsilon^{4}-\frac{1\\!\cdot\\!1\\!\cdot\\!3\\!\cdot\\!5}{2\\!\cdot\\!4\\!\cdot\\!6\\!\cdot\\!8}\,\frac{1\\!\cdot\\!1}{2\\!\cdot\\!4}\epsilon^{6}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{1\\!\cdot\\!1\\!\cdot\\!3\\!\cdot\\!5\\!\cdot\\!7}{2\\!\cdot\\!4\\!\cdot\\!6\\!\cdot\\!8\\!\cdot\\!10}\,\frac{1\\!\cdot\\!1\\!\cdot\\!3}{2\\!\cdot\\!4\\!\cdot\\!6}\epsilon^{8}-\ldots$
$\displaystyle\mathrm{etc.}$
Das Integral dieses Differentials, von $\sigma=0$ angerechnet, ist daher
$\displaystyle s=b\frac{\cos^{2}\frac{1}{2}E}{\cos E}\bigl{[}A\sigma$
$\displaystyle-2B\cos(2M+\sigma)\sin\sigma$
$\displaystyle-{\textstyle\frac{2}{2}}C\cos(4M+2\sigma)\sin 2\sigma$
$\displaystyle-{\textstyle\frac{2}{3}}D\cos(6M+3\sigma)\sin 3\sigma$
$\displaystyle-\ldots\bigr{]}$ (10)
## 6
Diese Reihe giebt die Entfernung $s$ der Punkte $A$ und $B$, durch
$u^{\prime}$, $\alpha^{\prime}$ und $\sigma$ ausgedr”uckt; sind dagegen $s$
und $\alpha^{\prime}$ durch die Vermessung und $u^{\prime}$ durch die Polh”ohe
des Punkts $A$, bekannt, so findet man $\sigma$ durch Aufl”osung der eben
gegebenen transcendenten Gleichung; die Polh”ohe des Punkts $B$ und das
Azimuth der geod”atischen Linie an demselben, finden sich dann nach (8). Die
Aufl”osung der transcendenten Gleichung kann man, entweder durch Umkehrung der
Reihe (10), oder durch successive Ann”aherungen erhalten; — der letzte Weg ist
aber der bequemste, wenn man ihn durch die Tafeln erleichtert, welche ich hier
mittheile.
Ich setze n”amlich
$\sigma=\frac{\alpha}{b}s+\beta\cos(2M+\sigma)\sin\sigma\\\
+\gamma\cos(4M+2\sigma)\sin 2\sigma+\ldots$ (11)
wo
$\begin{split}\alpha&=\frac{648\,000}{\pi}\,\frac{\cos
E}{\cos^{2}\frac{1}{2}E}\,\frac{1}{A}\\\
\beta&=\frac{648\,000}{\pi}\,\frac{2B}{A}\\\
\gamma&=\frac{648\,000}{\pi}\,\frac{C}{A}\\\
\delta&=\frac{648\,000}{\pi}\,\frac{2D}{3A}\\\ &\mathrm{etc.}\end{split}$
Die Tafeln enthalten die Logarithmen von $\alpha$, $\beta$, $\gamma$ und sind
so eingerichtet, da”s ihr Argument
$=\log\frac{e\cos m}{\sqrt{1-e^{2}}}$
ist. Durch diese Einrichtung erlangt man den Vortheil, da”s die Zahlen in der
Tafel f”ur $\log\beta$, immer sehr nahe um die doppelte Differenz des
Arguments, und in der Tafel f”ur $\log\gamma$, um die vierfache Differenz des
Arguments wachsen, wodurch der Gebrauch der Tafeln sehr erleichtert wird.
Man nimmt $\alpha s/b$ als den ersten N”aherungswerth von $\sigma$ an,
substituirt denselben im zweiten Gliede und erh”alt dadurch einen zweiten
N”aherungswerth, mit welchem man das zweite Glied neu berechnet und das dritte
hinzuf”ugt. Die Convergenz der Reihe ist so gro”s, da”s, wenn man das Argument
auch $9,\\!1$ annimmt (welchen Werth es nur bei einer Abplattung
$>\frac{1}{128}$ erlangen kann), die N”aherung nie weiter getrieben zu werden
braucht, ohne $\sigma$ um $0,\\!001^{\prime\prime}$ fehlerhaft zu geben. Das
von $\delta$ abh”angige Glied betr”agt, f”ur diesen Werth des Arguments, nur
$0,\\!0005^{\prime\prime}$.
## 7
Die Tafel f”ur $\log\alpha$ hat 8 Decimalen; ein Fehler einer halben Einheit
der letzten Decimale erzeugt erst f”ur $\sigma=12^{\circ}4^{\prime}$, oder
f”ur etwa 700 000 Toisen Entfernung, einen Fehler von
$0,\\!0005^{\prime\prime}$, welcher $0,\\!008$ Toisen entspricht. F”ur
denselben Werth von $\sigma$ rechnet man, mit der Tafel f”ur $\log\beta$, wenn
man alle Decimalen benutzt, welche sie enth”alt, bis auf eine noch kleinere
Gr”o”se genau; da eine gr”o”sere Genauigkeit kein Interesse zu haben scheint,
indem die Genauigkeit der Tafeln die Sicherheit der Vermessungen schon weit
”uberschreitet, so da”s es unn”utz sein w”urde, mit mehr als 8 Decimalen zu
rechnen, so habe ich der Tafel f”ur $\log\beta$ zwar am Ende 6 Decimalen
gegeben, allein fr”uher davon so viele weggelassen, als geschehen konnte, ohne
das Resultat $0,\\!0005^{\prime\prime}$ zweifelhaft zu machen. Das dritte
Glied ist, f”ur den angegebenen Werth von $\sigma$, selbst am Ende der Tafel,
nie gr”o”ser als $0,\\!17^{\prime\prime}$, weshalb ich der Tafel f”ur
$\log\gamma$ nur 3 Decimalen gegeben habe. — Die Tafeln werden also, wenn die
Entfernungen nicht gr”o”ser sind als 700 000 Toisen, die Ann”aherung eines
Tausendtheils einer Secunde geben, und selbst wenn die Entfernung einen
Erdquadranten betr”uge, so w”urden die Tafeln ein Hunderttheil einer Secunde
nicht zweifelhaft lassen.
## 8
Um den Gebrauch der Tafeln durch ein Beispiel zu erl”autern, werde ich die
Lage von D”unkirchen, gegen Seeberg, so annehmen, wie sie Herr General-
Lieutenant von M”uffling im $27^{\mathrm{sten}}$ St”ucke der Astronomischen
Nachrichten, aus seiner gro”sen Vermessung gefolgert hat; n”amlich
$\begin{split}\log s&=5,\\!478\,303\,14\\\
\alpha^{\prime}&=274^{\circ}\,21^{\prime}\,3,\\!18^{\prime\prime}\end{split}$
ferner nehme ich die Polh”ohe der Seeberger Sternwarte
$\phi^{\prime}=50^{\circ}\,56^{\prime}\,6,\\!7^{\prime\prime}$; $\log
b=6,\\!513\,354\,64$, $\log e=8,\\!905\,4355$.
Aus der Formel $\tan u^{\prime}=\sqrt{1-e^{2}}\tan\phi^{\prime}$ findet man,
$\displaystyle\log\tan\phi^{\prime}$ $\displaystyle=0,\\!090\,626\,65$
$\displaystyle\log\sqrt{1-e^{2}}$ $\displaystyle=9,\\!998\,590\,60$
$\displaystyle\log\tan u^{\prime}$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}0,\\!089\,217\,25};$
$\displaystyle u^{\prime}$
$\displaystyle=50^{\circ}\,50^{\prime}\,39,\\!057^{\prime\prime}$
Aus $u^{\prime}$ and $\alpha^{\prime}$ erh”alt man $M$, $\cos m$ und $\sin m$:
$\displaystyle\log\sin u^{\prime}$ $\displaystyle=9,\\!889\,543\,51$
$\displaystyle\log\cos u^{\prime}$ $\displaystyle=9,\\!800\,326\,27$
$\displaystyle\log\cos\alpha^{\prime}$ $\displaystyle=8,\\!880\,037\,33$
$\displaystyle\log\sin\alpha^{\prime}$ $\displaystyle=9,\\!998\,746\,62(-)$
$\displaystyle\log(\cos m\sin M)$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}9,\\!889\,543\,51}$
$\displaystyle\log(\cos m\cos M)$ $\displaystyle=8,\\!680\,363\,60$
$\displaystyle\log\,\sin m$ $\displaystyle=9,\\!799\,072\,89(-)$
$\displaystyle M$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}86^{\circ}\,27^{\prime}\,53,\\!949^{\prime\prime}};$
$\displaystyle\\!\\!2M$
$\displaystyle=172^{\circ}\,55^{\prime}\,47,\\!9^{\prime\prime}$
$\displaystyle\log\cos m$ $\displaystyle=9,\\!890\,370\,63$
$\displaystyle\\!\\!4M$
$\displaystyle=345^{\circ}\,51^{\prime}\,36^{\prime\prime}$
Das Argument der Tafeln ist $\log\bigl{(}(e/\sqrt{1-e^{2}})\cos m\bigr{)}$:
$\displaystyle\log\frac{e}{\sqrt{1-e^{2}}}$ $\displaystyle=8,\\!906\,845$
$\displaystyle\log\cos m$ $\displaystyle=9,\\!890\,371$
$\displaystyle\mathrm{Argument}$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}8,\\!797\,216}$
hiermit erh”alt man aus den Tafeln $\log\alpha$, wodurch $\alpha s/b$
berechnet wird:
$\displaystyle\log\alpha$ $\displaystyle=5,\\!313\,998\,92$
$\displaystyle\mathop{\mathrm{colog}}\nolimits b$
$\displaystyle=3,\\!486\,645\,36$ $\displaystyle\log s$
$\displaystyle=5,\\!478\,303\,14$ $\displaystyle\log\frac{\alpha s}{b}$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}4,\\!278\,947\,42};$
$\displaystyle\frac{\alpha}{b}s$
$\displaystyle=5^{\circ}\,16^{\prime}\,48,\\!481^{\prime\prime}$
Dieses als erste Ann”aherung an den Werth von $\sigma$ angenommen, findet sich
die zweite, durch Hinzuf”ugung des ersten Gliedes der Reihe (11):
$\displaystyle\log\beta$ $\displaystyle=2,\\!305\,94$
$\displaystyle\log\cos(2M+\sigma)$ $\displaystyle=9,\\!999\,79(-)$
$\displaystyle\log\sin\sigma$ $\displaystyle=8,\\!963\,91$
$\displaystyle\hphantom{=\;\;}\overline{\vphantom{\rule{0.0pt}{10.76385pt}}1,\\!269\,64(-)}=-18,\\!61^{\prime\prime}$
Die genauere Berechnung dieses Gliedes, mit der zweiten Ann”aherung von
$\sigma=5^{\circ}\,16^{\prime}\,29,\\!9^{\prime\prime}$, so wie die des
dritten, ergiebt:
$\displaystyle\log\beta$ $\displaystyle=2,\\!305\,94$
$\displaystyle\log\cos(2M+\sigma)$ $\displaystyle=9,\\!999\,79(-)$
$\displaystyle\log\sin\sigma$ $\displaystyle=8,\\!963\,48$
$\displaystyle\hphantom{=\;\;}\overline{\vphantom{\rule{0.0pt}{10.76385pt}}1,\\!269\,21(-)}=-18,\\!587^{\prime\prime}$
$\displaystyle\log\gamma$ $\displaystyle=8,\\!394$
$\displaystyle\log\cos(4M+2\sigma)$ $\displaystyle=9,\\!999$
$\displaystyle\log\sin 2\sigma$ $\displaystyle=9,\\!263$
$\displaystyle\hphantom{=\;\;}\overline{\vphantom{\rule{0.0pt}{10.76385pt}}7,\\!656}=+0,\\!005^{\prime\prime}$
Man hat also $\sigma=5^{\circ}\,16^{\prime}\,29,\\!899^{\prime\prime}$, und
endlich $\alpha$, $u$ und $\phi$ aus den Formeln (8)
$\displaystyle M+\sigma$
$\displaystyle=91^{\circ}\,44^{\prime}\,23,\\!848^{\prime\prime}$
$\displaystyle\log\sin(M+\sigma)$ $\displaystyle=9,\\!999\,799\,71$
$\displaystyle\log\bigl{(}-\cos(M+\sigma)\bigr{)}$
$\displaystyle=8,\\!482\,349\,32$ $\displaystyle\log\cos m$
$\displaystyle=9,\\!890\,370\,63$ $\displaystyle\log(-\sin m)$
$\displaystyle=9,\\!799\,072\,89$ $\displaystyle\log\,\sin u$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}9,\\!890\,170\,34}$
$\displaystyle\log(\cos u\cos\alpha)$ $\displaystyle=8,\\!372\,719\,95$
$\displaystyle\log(\cos u\sin\alpha)$ $\displaystyle=9,\\!799\,072\,89$
$\displaystyle\alpha$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}87^{\circ}\,51^{\prime}\,15,\\!523^{\prime\prime}}$
$\displaystyle\log\cos u$ $\displaystyle=9,\\!799\,377\,50$
$\displaystyle\log\tan u$ $\displaystyle=0,\\!090\,792\,84$
$\displaystyle\mathop{\mathrm{colog}}\nolimits\sqrt{1-e^{2}}$
$\displaystyle=0,\\!001\,409\,40$ $\displaystyle\log\tan\phi$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}0,\\!092\,202\,24};$
$\displaystyle\\!\\!\\!\\!\phi$
$\displaystyle=51^{\circ}\,2^{\prime}\,12,\\!719^{\prime\prime}$
In diesem Beispiele habe ich die trigonometrische Rechnung mit 8 Decimalen
gef”uhrt, weil selbst bei dieser Ann”aherung, $\alpha$ und $\phi$ noch nicht
die Sicherheit erhalten, welche die Tafeln f”ur $\log\alpha$, $\log\beta$,
$\log\gamma$, gew”ahren. Will man nur die gew”ohnlichen Logarithmentafeln mit
7 Decimalstellen anwenden, so kann man auch in den Tafeln f”ur $\log\alpha$,
$\log\beta$, $\log\gamma$ die letzte Decimale vernachl”assigen.
## 9
Es ist nun noch der Mittagsunterschied $w$, durch die Integration des
Differentials (5)
$dw=\sqrt{1-e^{2}\cos^{2}u}\,d\omega$
zu finden. Dieses Integral enth”alt aber zwei getrennte Constanten $m$ und
$e$, welche sich nicht vereinigen lassen, so da”s man die streng richtige
Aufl”osung dieser Aufgabe, nicht auf Tafeln zur”uckf”uhren kann, welche f”ur
alle Werthe von $e$ g”ultig sind. Man darf aber, um dieses zu erreichen, von
der strengen Richtigkeit nur so wenig aufopfern, da”s es f”ur die Anwendung
von keinem Belange ist.
Setzt man
$dw=d\omega-\bigl{(}1-\sqrt{1-e^{2}\cos^{2}u}\bigr{)}d\omega$
und schreibt man f”ur $d\omega$, im zweiten Gliede
$\frac{\sin\alpha^{\prime}\cos u^{\prime}}{\cos^{2}u}\,d\sigma$
so erh”alt man, nach der Integration,
$w=\omega-\sin\alpha^{\prime}\cos
u^{\prime}\int\frac{1-\sqrt{1-e^{2}\cos^{2}u}}{\cos^{2}u}\,d\sigma$
Setzt man ferner
$\frac{1-\sqrt{1-e^{2}\cos^{2}u}}{\cos^{2}u}=\frac{e^{2}}{2}(1+e^{2}p\cos^{2}u)^{q}(1+y)$
so erh”alt man
$1+y=\frac{2(1-\sqrt{1-e^{2}\cos^{2}u})}{e^{2}\cos^{2}u(1+e^{2}p\cos^{2}u)^{q}}\\\
=\frac{1+\frac{1}{4}e^{2}\cos^{2}u+\frac{1}{8}e^{4}\cos^{4}u+\frac{5}{64}e^{6}\cos^{6}u+\ldots}{\Biggl{(}\begin{aligned}
\textstyle 1+qpe^{2}\cos^{2}u+\frac{q(q-1)}{1\cdot
2}p^{2}e^{4}\cos^{4}u\qquad\qquad\\\
\textstyle\qquad\qquad+\frac{q(q-1)(q-2)}{1\cdot 2\cdot
3}p^{3}e^{6}\cos^{6}u+\ldots\end{aligned}\Biggr{)}};$
im Nenner dieses Ausdrucks werden die 3 ersten Glieder den drei ersten
Gliedern im Z”ahler gleich, wenn man
$p=-{\textstyle\frac{3}{4}};\qquad q=-{\textstyle\frac{1}{3}}$
annimmt; man erh”alt dadurch
$\displaystyle 1+y$
$\displaystyle=\frac{1+\frac{1}{4}e^{2}\cos^{2}u+\frac{1}{8}e^{4}\cos^{4}u+\frac{5}{64}e^{6}\cos^{6}u+\ldots}{1+\frac{1}{4}e^{2}\cos^{2}u+\frac{1}{8}e^{4}\cos^{4}u+\frac{7}{96}e^{6}\cos^{6}u+\ldots}$
$\displaystyle=1+{\textstyle\frac{1}{192}}e^{6}\cos^{6}u+\ldots$
woraus also hervorgeht, da”s man durch die Vernachl”assigung von $y$, nur
einen Fehler von der Ordnung von $e^{8}$ begeht; das Maximum des Einflusses
dieses Fehlers auf $w$ ist $=\frac{1}{384}e^{8}\sigma$, und daher selbst dann
unmerklich, wenn man auch sehr weit ausgedehnte Vermessungen mit
Logarithmentafeln von 10 Decimalen berechnen wollte.
Man kann also, f”ur die Anwendung, $y=0$ setzen und dann das Integral auf
Tafeln reduciren, welche f”ur jeden Werth von $e$ gelten.
## 10
Dieser Bemerkung zufolge ist
$\displaystyle w$ $\displaystyle=\omega-\frac{e^{2}}{2}\sin
m\int\frac{d\sigma}{\sqrt[3]{1-\frac{3}{4}e^{2}\cos^{2}u}}$
$\displaystyle=\omega-\frac{e^{2}}{2}\sin
m\\!\int\\!\\!\frac{d\sigma}{\sqrt[3]{1-\frac{3}{4}e^{2}+\frac{3}{4}e^{2}\cos^{2}m\sin^{2}(M+\sigma)}}$
Wenn man
$\tan E^{\prime}=\frac{e\sqrt{{\scriptstyle\frac{3}{4}}}\cos
m}{\sqrt{1-{\scriptstyle\frac{3}{4}}e^{2}}}$
und $\tan^{2}\frac{1}{2}E^{\prime}=\epsilon^{\prime}$ setzt, so verwandelt
sich das Integral im zweiten Gliede, in
$\int\frac{d\sigma}{\sqrt[3]{1-\frac{3}{4}e^{2}}\sqrt[3]{1+\tan^{2}E^{\prime}\sin^{2}(M+\sigma)}}$
und durch Zerlegung in zwei imagin”are Factoren, in
$\int\frac{\sqrt[3]{(1-\epsilon^{\prime})^{2}}}{\sqrt[3]{1-{\scriptstyle\frac{3}{4}}e^{2}}}\frac{1}{(1-\epsilon^{\prime}c^{2i(M+\sigma)})^{1/3}(1-\epsilon^{\prime}c^{-2i(M+\sigma)})^{1/3}}\,d\sigma$
L”oset man die imagin”aren Factoren in unendliche Reihen auf, so giebt das
Product derselben
$\frac{2}{\sqrt[3]{1-{\scriptstyle\frac{3}{4}}e^{2}}}\int\biggl{(}\begin{aligned}
\alpha^{\prime}+\beta^{\prime}\cos 2(M+\sigma)\qquad\quad\\\
\quad{}+2\gamma^{\prime}\cos 4(M+\sigma)+\ldots\end{aligned}\biggr{)}d\sigma$
wo
$\displaystyle\alpha^{\prime}$
$\displaystyle={\textstyle\frac{1}{2}}\sqrt[3]{(1-\epsilon^{\prime})^{2}}\biggl{[}1+\biggl{(}\frac{1}{3}\biggr{)}^{2}\epsilon^{\prime
2}+\biggl{(}\frac{1\\!\cdot\\!4}{3\\!\cdot\\!6}\biggr{)}^{2}\epsilon^{\prime
4}+\ldots\biggr{]}$ $\displaystyle\beta^{\prime}$
$\displaystyle=\sqrt[3]{(1-\epsilon^{\prime})^{2}}\biggl{[}\frac{1}{3}\epsilon^{\prime}+\frac{1\\!\cdot\\!4}{3\\!\cdot\\!6}\,\frac{1}{3}\epsilon^{\prime
3}+\frac{1\\!\cdot\\!4\\!\cdot\\!7}{3\\!\cdot\\!6\\!\cdot\\!9}\,\frac{1\\!\cdot\\!4}{3\\!\cdot\\!6}\epsilon^{\prime
5}+\ldots\biggr{]}$ $\displaystyle\gamma^{\prime}$
$\displaystyle={\textstyle\frac{1}{2}}\sqrt[3]{(1-\epsilon^{\prime})^{2}}\biggl{[}\frac{1\\!\cdot\\!4}{3\\!\cdot\\!6}\epsilon^{\prime
2}+\frac{1\\!\cdot\\!4\\!\cdot\\!7}{3\\!\cdot\\!6\\!\cdot\\!9}\,\frac{1}{3}\epsilon^{\prime
4}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\frac{1\\!\cdot\\!4\\!\cdot\\!7\\!\cdot\\!10}{3\\!\cdot\\!6\\!\cdot\\!9\\!\cdot\\!12}\,\frac{1\\!\cdot\\!4}{3\\!\cdot\\!6}\epsilon^{\prime
6}+\ldots\biggr{]}$ $\displaystyle\mathrm{etc.}$
sind. Das Integral, von $\sigma=0$ angerechnet, giebt daher
$\displaystyle w$ $\displaystyle=\omega-\frac{e^{2}\sin
m}{\sqrt[3]{1-{\scriptstyle\frac{3}{4}}e^{2}}}\Bigl{(}\alpha^{\prime}\sigma+\beta^{\prime}\cos(2M+\sigma)\sin\sigma$
$\displaystyle\qquad\qquad\qquad+\gamma^{\prime}\cos(4M+2\sigma)\sin
2\sigma+\ldots\Bigr{)}$ (12)
## 11
Die beiden ersten Coefficienten dieser Reihe sind in der $4^{\mathrm{sten}}$
und $5^{\mathrm{sten}}$ Columne der Tafel enthalten; das Argument derselben
ist
$\log\biggl{(}\frac{e\sqrt{{\scriptstyle\frac{3}{4}}}}{\sqrt{1-{\scriptstyle\frac{3}{4}}e^{2}}}\cos
m\biggr{)}.$
Die Ann”aherung ist der der 3 fr”uheren Columnen der Tafel angemessen. Man
berechnet daher $\omega$ nach einer dem sph”arischen Dreiecke (§3)
zugeh”origen Formel, entweder
$\sin\omega=\frac{\sin\sigma\sin\alpha^{\prime}}{\cos
u}=\frac{-\sin\sigma\sin\alpha}{\cos u^{\prime}}=\frac{\sin\sigma\sin m}{\cos
u\cos u^{\prime}}$
oder
$\displaystyle\tan{\textstyle\frac{1}{2}}\omega$
$\displaystyle=\frac{\sin\frac{1}{2}(u^{\prime}-u)}{\cos\frac{1}{2}(u^{\prime}+u)}\cot{\textstyle\frac{1}{2}}(\alpha^{\prime}+\alpha)$
$\displaystyle=\frac{\cos\frac{1}{2}(u^{\prime}-u)}{\sin\frac{1}{2}(u^{\prime}+u)}\cot{\textstyle\frac{1}{2}}(\alpha^{\prime}-\alpha)$
und die Reduction auf $w$ mit H”ulfe der Tafel.
Nach dieser Vorschrift werde ich das im 8 Art. berechnete Beispiel fortsetzen,
um auch den Mittagsunterschied zwischen D”unkirchen und Seeberg zu bestimmen.
$\displaystyle\log\sin\sigma$ $\displaystyle=8,\\!963\,483\,83$
$\displaystyle\log(-\sin\alpha)$ $\displaystyle=9,\\!999\,695\,39(-)$
$\displaystyle\mathop{\mathrm{colog}}\nolimits\cos u^{\prime}$
$\displaystyle=0,\\!199\,673\,73$ $\displaystyle\log\sin\omega$
$\displaystyle=\overline{\vphantom{\rule{0.0pt}{10.76385pt}}9,\\!162\,852\,95}(-);$
$\displaystyle\omega$
$\displaystyle=-8^{\circ}\,21^{\prime}\,57,\\!741^{\prime\prime}$
Das Argument der beiden letzten Columnen der Tafel ist
$\log\bigl{(}(e\sqrt{\scriptstyle\frac{3}{4}}/\sqrt{1-{\scriptstyle\frac{3}{4}}e^{2}})\cos
m\bigr{)}$
$\displaystyle\log\frac{e\sqrt{{\scriptstyle\frac{3}{4}}}}{\sqrt{1-{\scriptstyle\frac{3}{4}}e^{2}}}$
$\displaystyle=8,\\!844\,022$ $\displaystyle\log\cos m$
$\displaystyle=9,\\!890\,371$ $\displaystyle\mathrm{Argument}$
$\displaystyle=8,\\!734\,393$
$\displaystyle\log\alpha^{\prime}$ $\displaystyle=9,\\!698\,758$
$\displaystyle\log(-\sin m)$ $\displaystyle=9,\\!799\,073$
$\displaystyle\log\frac{e^{2}}{\sqrt[3]{1-{\scriptstyle\frac{3}{4}}e^{2}}}$
$\displaystyle=7,\\!811\,575$ $\displaystyle\log\sigma$
$\displaystyle=4,\\!278\,523$
$\displaystyle\hphantom{=\;\;}\overline{\vphantom{\rule{0.0pt}{10.76385pt}}1,\\!587\,929}=+38,\\!719^{\prime\prime}$
$\displaystyle\log\beta^{\prime}$ $\displaystyle=1,\\!703$
$\displaystyle\log(-\sin m)$ $\displaystyle=9,\\!799$
$\displaystyle\log\frac{e^{2}}{\sqrt[3]{1-{\scriptstyle\frac{3}{4}}e^{2}}}$
$\displaystyle=7,\\!812$
$\displaystyle\log\bigl{(}\cos(2M+\sigma)\sin\sigma\bigr{)}$
$\displaystyle=8,\\!963(-)$
$\displaystyle\hphantom{=\;\;}\overline{\vphantom{\rule{0.0pt}{10.76385pt}}8,\\!277}(-)=-0,\\!019^{\prime\prime}$
also die Summe beider Glieder $=+38,\\!700^{\prime\prime}$, und der gesuchte
L”angenunterschied
$w=-8^{\circ}\,21^{\prime}\,19,\\!041^{\prime\prime}$
## 12
Die Erkl”arung, welche ich von diesen Tafeln gegeben habe, zeigt, da”s bei dem
Gebrauche derselben nicht etwa h”ohere Potenzen der Excentricit”at
vernachl”assigt werden, sondern da”s sie das Resultat so genau geben, als die
Anzahl ihrer Decimalstellen erlaubt. Die Rechnung, welche dieses leistet, ist
meistentheils dieselbe, welche man f”uhren mu”s, wenn man die Erde als
sph”arisch annimmt, und es ist dieser sph”arischen Rechnung, durch die
Ber”ucksichtigung der Ellipticit”at der Erde, nur die Aufl”osung der Gleichung
(11) und die Berechnung der Reihe (10) hinzugekommen. — Da diese Rechnung,
selbst f”ur den h”aufigen Gebrauch, bequem genug ist, so scheint es mir
unn”othig, N”aherungen anzuwenden, welche auf der Bedingung beruhen, da”s die
Vermessung eine geringe Ausdehnung besitzt.
(Die Tafeln werden einem der n”achsten St”ucke beigelegt.)
Tables for computing geodesics 1, $\vphantom{\bigr{|}^{2^{2}}}\mathrm{Arg}$
$\log\alpha$ $\hskip 9.00002pt-\Delta$ $\log\beta$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\log\gamma$ $\hskip 9.00002pt\hphantom{-}\Delta$
$\log\alpha^{\prime}$ $\hskip 9.00002pt-\Delta$ $\log\beta^{\prime}$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\vphantom{\bigr{|}^{2^{2}}}6,\\!4\hphantom{0}$
$5,\\!314\,425\,13$ $\hskip 9.00002pt1$ $7,\\!5124$ $\hskip 9.00002pt2000$
$9,\\!698\,970$ $\hskip 9.00002pt0$ $7,\\!035$ $\hskip 9.00002pt200$
$6,\\!5\hphantom{0}$ $5,\\!314\,425\,12$ $\hskip 9.00002pt0$ $7,\\!7124$
$\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt0$ $7,\\!235$ $\hskip
9.00002pt200$ $6,\\!6\hphantom{0}$ $5,\\!314\,425\,12$ $\hskip 9.00002pt1$
$7,\\!9124$ $\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt0$
$7,\\!435$ $\hskip 9.00002pt200$ $6,\\!7\hphantom{0}$ $5,\\!314\,425\,11$
$\hskip 9.00002pt2$ $8,\\!1124$ $\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip
9.00002pt0$ $7,\\!635$ $\hskip 9.00002pt200$ $6,\\!8\hphantom{0}$
$5,\\!314\,425\,09$ $\hskip 9.00002pt3$ $8,\\!3124$ $\hskip 9.00002pt2000$
$9,\\!698\,970$ $\hskip 9.00002pt0$ $7,\\!835$ $\hskip 9.00002pt200$
$6,\\!9\hphantom{0}$ $5,\\!314\,425\,06$ $\hskip 9.00002pt4$ $8,\\!5124$
$\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt0$ $8,\\!035$ $\hskip
9.00002pt200$ $7,\\!0\hphantom{0}$ $5,\\!314\,425\,02$ $\hskip 9.00002pt6$
$8,\\!7124$ $\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt0$
$8,\\!235$ $\hskip 9.00002pt200$ $7,\\!1\hphantom{0}$ $5,\\!314\,424\,96$
$\hskip 9.00002pt10$ $8,\\!9124$ $\hskip 9.00002pt2000$ $9,\\!698\,970$
$\hskip 9.00002pt0$ $8,\\!435$ $\hskip 9.00002pt200$ $7,\\!2\hphantom{0}$
$5,\\!314\,424\,86$ $\hskip 9.00002pt16$ $9,\\!1124$ $\hskip 9.00002pt2000$
$9,\\!698\,970$ $\hskip 9.00002pt0$ $8,\\!635$ $\hskip 9.00002pt200$
$7,\\!3\hphantom{0}$ $5,\\!314\,424\,70$ $\hskip 9.00002pt25$ $9,\\!3124$
$\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt0$ $8,\\!835$ $\hskip
9.00002pt200$ $7,\\!4\hphantom{0}$ $5,\\!314\,424\,45$ $\hskip 9.00002pt40$
$9,\\!5124$ $\hskip 9.00002pt2000$ $9,\\!698\,970$ $\hskip 9.00002pt1$
$9,\\!035$ $\hskip 9.00002pt200$ $7,\\!50$ $5,\\!314\,424\,05$ $\hskip
9.00002pt5$ $9,\\!7124$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!235$ $\hskip 9.00002pt20$ $7,\\!51$ $5,\\!314\,424\,00$
$\hskip 9.00002pt6$ $9,\\!7324$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!255$ $\hskip 9.00002pt20$ $7,\\!52$ $5,\\!314\,423\,94$
$\hskip 9.00002pt5$ $9,\\!7524$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!275$ $\hskip 9.00002pt20$ $7,\\!53$ $5,\\!314\,423\,89$
$\hskip 9.00002pt6$ $9,\\!7724$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!295$ $\hskip 9.00002pt20$ $7,\\!54$ $5,\\!314\,423\,83$
$\hskip 9.00002pt6$ $9,\\!7924$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!315$ $\hskip 9.00002pt20$ $7,\\!55$ $5,\\!314\,423\,77$
$\hskip 9.00002pt7$ $9,\\!8124$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!335$ $\hskip 9.00002pt20$ $7,\\!56$ $5,\\!314\,423\,70$
$\hskip 9.00002pt7$ $9,\\!8324$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!355$ $\hskip 9.00002pt20$ $7,\\!57$ $5,\\!314\,423\,63$
$\hskip 9.00002pt7$ $9,\\!8524$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!375$ $\hskip 9.00002pt20$ $7,\\!58$ $5,\\!314\,423\,56$
$\hskip 9.00002pt7$ $9,\\!8724$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!395$ $\hskip 9.00002pt20$ $7,\\!59$ $5,\\!314\,423\,49$
$\hskip 9.00002pt8$ $9,\\!8924$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!415$ $\hskip 9.00002pt20$ $7,\\!60$ $5,\\!314\,423\,41$
$\hskip 9.00002pt8$ $9,\\!9124$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!435$ $\hskip 9.00002pt20$ $7,\\!61$ $5,\\!314\,423\,33$
$\hskip 9.00002pt8$ $9,\\!9324$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!455$ $\hskip 9.00002pt20$ $7,\\!62$ $5,\\!314\,423\,25$
$\hskip 9.00002pt9$ $9,\\!9524$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!475$ $\hskip 9.00002pt20$ $7,\\!63$ $5,\\!314\,423\,16$
$\hskip 9.00002pt10$ $9,\\!9724$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!495$ $\hskip 9.00002pt20$ $7,\\!64$ $5,\\!314\,423\,06$
$\hskip 9.00002pt9$ $9,\\!9924$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt0$ $9,\\!515$ $\hskip 9.00002pt20$ $7,\\!65$ $5,\\!314\,422\,97$
$\hskip 9.00002pt11$ $0,\\!0124$ $\hskip 9.00002pt200$ $9,\\!698\,969$ $\hskip
9.00002pt1$ $9,\\!535$ $\hskip 9.00002pt20$ $7,\\!66$ $5,\\!314\,422\,86$
$\hskip 9.00002pt10$ $0,\\!0324$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!555$ $\hskip 9.00002pt20$ $7,\\!67$ $5,\\!314\,422\,76$
$\hskip 9.00002pt11$ $0,\\!0524$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!575$ $\hskip 9.00002pt20$ $7,\\!68$ $5,\\!314\,422\,65$
$\hskip 9.00002pt12$ $0,\\!0724$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!595$ $\hskip 9.00002pt20$ $7,\\!69$ $5,\\!314\,422\,53$
$\hskip 9.00002pt12$ $0,\\!0924$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!615$ $\hskip 9.00002pt20$ $7,\\!70$ $5,\\!314\,422\,41$
$\hskip 9.00002pt13$ $0,\\!1124$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!635$ $\hskip 9.00002pt20$ $7,\\!71$ $5,\\!314\,422\,28$
$\hskip 9.00002pt14$ $0,\\!1324$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!655$ $\hskip 9.00002pt20$ $7,\\!72$ $5,\\!314\,422\,14$
$\hskip 9.00002pt14$ $0,\\!1524$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!675$ $\hskip 9.00002pt20$ $7,\\!73$ $5,\\!314\,422\,00$
$\hskip 9.00002pt15$ $0,\\!1724$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!695$ $\hskip 9.00002pt20$ $7,\\!74$ $5,\\!314\,421\,85$
$\hskip 9.00002pt15$ $0,\\!1924$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!715$ $\hskip 9.00002pt20$ $7,\\!75$ $5,\\!314\,421\,70$
$\hskip 9.00002pt16$ $0,\\!2124$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt0$ $9,\\!735$ $\hskip 9.00002pt20$ $7,\\!76$ $5,\\!314\,421\,54$
$\hskip 9.00002pt17$ $0,\\!2324$ $\hskip 9.00002pt200$ $9,\\!698\,968$ $\hskip
9.00002pt1$ $9,\\!755$ $\hskip 9.00002pt20$ $7,\\!77$ $5,\\!314\,421\,37$
$\hskip 9.00002pt18$ $0,\\!2524$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!775$ $\hskip 9.00002pt20$ $7,\\!78$ $5,\\!314\,421\,19$
$\hskip 9.00002pt18$ $0,\\!2724$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!795$ $\hskip 9.00002pt20$ $7,\\!79$ $5,\\!314\,421\,01$
$\hskip 9.00002pt20$ $0,\\!2924$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!815$ $\hskip 9.00002pt20$ $7,\\!80$ $5,\\!314\,420\,81$
$\hskip 9.00002pt20$ $0,\\!3124$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!835$ $\hskip 9.00002pt20$ $7,\\!81$ $5,\\!314\,420\,61$
$\hskip 9.00002pt22$ $0,\\!3324$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!855$ $\hskip 9.00002pt20$ $7,\\!82$ $5,\\!314\,420\,39$
$\hskip 9.00002pt22$ $0,\\!3524$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!875$ $\hskip 9.00002pt20$ $7,\\!83$ $5,\\!314\,420\,17$
$\hskip 9.00002pt23$ $0,\\!3724$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt0$ $9,\\!895$ $\hskip 9.00002pt20$ $7,\\!84$ $5,\\!314\,419\,94$
$\hskip 9.00002pt25$ $0,\\!3924$ $\hskip 9.00002pt200$ $9,\\!698\,967$ $\hskip
9.00002pt1$ $9,\\!915$ $\hskip 9.00002pt20$ $7,\\!85$ $5,\\!314\,419\,69$
$\hskip 9.00002pt25$ $0,\\!4124$ $\hskip 9.00002pt200$ $9,\\!698\,966$ $\hskip
9.00002pt0$ $9,\\!935$ $\hskip 9.00002pt20$ $7,\\!86$ $5,\\!314\,419\,44$
$\hskip 9.00002pt27$ $0,\\!4324$ $\hskip 9.00002pt200$ $9,\\!698\,966$ $\hskip
9.00002pt0$ $9,\\!955$ $\hskip 9.00002pt20$ $7,\\!87$ $5,\\!314\,419\,17$
$\hskip 9.00002pt28$ $0,\\!4524$ $\hskip 9.00002pt200$ $9,\\!698\,966$ $\hskip
9.00002pt0$ $9,\\!975$ $\hskip 9.00002pt20$ $7,\\!88$ $5,\\!314\,418\,89$
$\hskip 9.00002pt30$ $0,\\!4724$ $\hskip 9.00002pt200$ $9,\\!698\,966$ $\hskip
9.00002pt0$ $9,\\!995$ $\hskip 9.00002pt20$ $7,\\!89$ $5,\\!314\,418\,59$
$\hskip 9.00002pt31$ $0,\\!4924$ $\hskip 9.00002pt200$ $9,\\!698\,966$ $\hskip
9.00002pt1$ $0,\\!015$ $\hskip 9.00002pt20$ $7,\\!90$ $5,\\!314\,418\,28$
$0,\\!5124$ $9,\\!698\,965$ $0,\\!035$
Tables for computing geodesics 2, $\vphantom{\bigr{|}^{2^{2}}}\mathrm{Arg}$
$\log\alpha$ $\hskip 9.00002pt-\Delta$ $\log\beta$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\log\gamma$ $\hskip 9.00002pt\hphantom{-}\Delta$
$\log\alpha^{\prime}$ $\hskip 9.00002pt-\Delta$ $\log\beta^{\prime}$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\vphantom{\bigr{|}^{2^{2}}}7,\\!90$
$5,\\!314\,418\,28$ $\hskip 9.00002pt32$ $0,\\!512\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,965$ $\hskip 9.00002pt0$ $0,\\!035$ $\hskip 9.00002pt20$ $7,\\!91$
$5,\\!314\,417\,96$ $\hskip 9.00002pt34$ $0,\\!532\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,965$ $\hskip 9.00002pt0$ $0,\\!055$ $\hskip 9.00002pt20$ $7,\\!92$
$5,\\!314\,417\,62$ $\hskip 9.00002pt35$ $0,\\!552\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,965$ $\hskip 9.00002pt0$ $0,\\!075$ $\hskip 9.00002pt20$ $7,\\!93$
$5,\\!314\,417\,27$ $\hskip 9.00002pt37$ $0,\\!572\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,965$ $\hskip 9.00002pt0$ $0,\\!095$ $\hskip 9.00002pt20$ $7,\\!94$
$5,\\!314\,416\,90$ $\hskip 9.00002pt39$ $0,\\!592\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,965$ $\hskip 9.00002pt1$ $0,\\!115$ $\hskip 9.00002pt20$ $7,\\!95$
$5,\\!314\,416\,51$ $\hskip 9.00002pt41$ $0,\\!612\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,964$ $\hskip 9.00002pt0$ $0,\\!135$ $\hskip 9.00002pt20$ $7,\\!96$
$5,\\!314\,416\,10$ $\hskip 9.00002pt42$ $0,\\!632\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,964$ $\hskip 9.00002pt0$ $0,\\!155$ $\hskip 9.00002pt20$ $7,\\!97$
$5,\\!314\,415\,68$ $\hskip 9.00002pt45$ $0,\\!652\,35$ $\hskip 9.00002pt2000$
$9,\\!698\,964$ $\hskip 9.00002pt1$ $0,\\!175$ $\hskip 9.00002pt20$ $7,\\!98$
$5,\\!314\,415\,23$ $\hskip 9.00002pt47$ $0,\\!672\,35$ $\hskip 9.00002pt1999$
$9,\\!698\,963$ $\hskip 9.00002pt0$ $0,\\!195$ $\hskip 9.00002pt20$ $7,\\!99$
$5,\\!314\,414\,76$ $\hskip 9.00002pt48$ $0,\\!692\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,963$ $\hskip 9.00002pt0$ $0,\\!215$ $\hskip 9.00002pt20$ $8,\\!00$
$5,\\!314\,414\,28$ $\hskip 9.00002pt52$ $0,\\!712\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,963$ $\hskip 9.00002pt1$ $0,\\!235$ $\hskip 9.00002pt20$ $8,\\!01$
$5,\\!314\,413\,76$ $\hskip 9.00002pt53$ $0,\\!732\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,962$ $\hskip 9.00002pt0$ $0,\\!255$ $\hskip 9.00002pt20$ $8,\\!02$
$5,\\!314\,413\,23$ $\hskip 9.00002pt56$ $0,\\!752\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,962$ $\hskip 9.00002pt0$ $0,\\!275$ $\hskip 9.00002pt20$ $8,\\!03$
$5,\\!314\,412\,67$ $\hskip 9.00002pt59$ $0,\\!772\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,962$ $\hskip 9.00002pt1$ $0,\\!295$ $\hskip 9.00002pt20$ $8,\\!04$
$5,\\!314\,412\,08$ $\hskip 9.00002pt61$ $0,\\!792\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,961$ $\hskip 9.00002pt0$ $0,\\!315$ $\hskip 9.00002pt20$ $8,\\!05$
$5,\\!314\,411\,47$ $\hskip 9.00002pt65$ $0,\\!812\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,961$ $\hskip 9.00002pt1$ $0,\\!335$ $\hskip 9.00002pt20$ $8,\\!06$
$5,\\!314\,410\,82$ $\hskip 9.00002pt67$ $0,\\!832\,34$ $\hskip 9.00002pt2000$
$9,\\!698\,960$ $\hskip 9.00002pt0$ $0,\\!355$ $\hskip 9.00002pt20$ $8,\\!07$
$5,\\!314\,410\,15$ $\hskip 9.00002pt71$ $0,\\!852\,34$ $\hskip 9.00002pt1999$
$9,\\!698\,960$ $\hskip 9.00002pt0$ $0,\\!375$ $\hskip 9.00002pt20$ $8,\\!08$
$5,\\!314\,409\,44$ $\hskip 9.00002pt74$ $0,\\!872\,33$ $\hskip 9.00002pt2000$
$9,\\!698\,960$ $\hskip 9.00002pt1$ $0,\\!395$ $\hskip 9.00002pt20$ $8,\\!09$
$5,\\!314\,408\,70$ $\hskip 9.00002pt77$ $0,\\!892\,33$ $\hskip 9.00002pt2000$
$9,\\!698\,959$ $\hskip 9.00002pt0$ $0,\\!415$ $\hskip 9.00002pt20$ $8,\\!10$
$5,\\!314\,407\,93$ $\hskip 9.00002pt81$ $0,\\!912\,33$ $\hskip 9.00002pt2000$
$9,\\!698\,959$ $\hskip 9.00002pt1$ $0,\\!435$ $\hskip 9.00002pt20$ $8,\\!11$
$5,\\!314\,407\,12$ $\hskip 9.00002pt85$ $0,\\!932\,33$ $\hskip 9.00002pt2000$
$9,\\!698\,958$ $\hskip 9.00002pt1$ $0,\\!455$ $\hskip 9.00002pt20$ $8,\\!12$
$5,\\!314\,406\,27$ $\hskip 9.00002pt89$ $0,\\!952\,33$ $\hskip 9.00002pt2000$
$9,\\!698\,957$ $\hskip 9.00002pt0$ $0,\\!475$ $\hskip 9.00002pt20$ $8,\\!13$
$5,\\!314\,405\,38$ $\hskip 9.00002pt93$ $0,\\!972\,33$ $\hskip 9.00002pt1999$
$9,\\!698\,957$ $\hskip 9.00002pt1$ $0,\\!495$ $\hskip 9.00002pt20$ $8,\\!14$
$5,\\!314\,404\,45$ $\hskip 9.00002pt98$ $0,\\!992\,32$ $\hskip 9.00002pt2000$
$9,\\!698\,956$ $\hskip 9.00002pt0$ $0,\\!515$ $\hskip 9.00002pt20$ $8,\\!15$
$5,\\!314\,403\,47$ $\hskip 9.00002pt102$ $1,\\!012\,32$ $\hskip
9.00002pt2000$ $9,\\!698\,956$ $\hskip 9.00002pt1$ $0,\\!535$ $\hskip
9.00002pt20$ $8,\\!16$ $5,\\!314\,402\,45$ $\hskip 9.00002pt107$
$1,\\!032\,32$ $\hskip 9.00002pt2000$ $9,\\!698\,955$ $\hskip 9.00002pt1$
$0,\\!555$ $\hskip 9.00002pt20$ $8,\\!17$ $5,\\!314\,401\,38$ $\hskip
9.00002pt112$ $1,\\!052\,32$ $\hskip 9.00002pt2000$ $9,\\!698\,954$ $\hskip
9.00002pt1$ $0,\\!575$ $\hskip 9.00002pt20$ $8,\\!18$ $5,\\!314\,400\,26$
$\hskip 9.00002pt117$ $1,\\!072\,32$ $\hskip 9.00002pt1999$ $9,\\!698\,953$
$\hskip 9.00002pt0$ $0,\\!595$ $\hskip 9.00002pt20$ $8,\\!19$
$5,\\!314\,399\,09$ $\hskip 9.00002pt123$ $1,\\!092\,31$ $\hskip
9.00002pt2000$ $9,\\!698\,953$ $\hskip 9.00002pt1$ $0,\\!615$ $\hskip
9.00002pt20$ $8,\\!20$ $5,\\!314\,397\,86$ $\hskip 9.00002pt128$
$1,\\!112\,31$ $\hskip 9.00002pt2000$ $9,\\!698\,952$ $\hskip 9.00002pt1$
$0,\\!635$ $\hskip 9.00002pt20$ $8,\\!21$ $5,\\!314\,396\,58$ $\hskip
9.00002pt135$ $1,\\!132\,31$ $\hskip 9.00002pt2000$ $9,\\!698\,951$ $\hskip
9.00002pt1$ $0,\\!655$ $\hskip 9.00002pt20$ $8,\\!22$ $5,\\!314\,395\,23$
$\hskip 9.00002pt141$ $1,\\!152\,31$ $\hskip 9.00002pt1999$ $9,\\!698\,950$
$\hskip 9.00002pt1$ $0,\\!675$ $\hskip 9.00002pt20$ $8,\\!23$
$5,\\!314\,393\,82$ $\hskip 9.00002pt147$ $1,\\!172\,30$ $\hskip
9.00002pt2000$ $9,\\!698\,949$ $\hskip 9.00002pt1$ $0,\\!695$ $\hskip
9.00002pt20$ $8,\\!24$ $5,\\!314\,392\,35$ $\hskip 9.00002pt155$
$1,\\!192\,30$ $\hskip 9.00002pt2000$ $9,\\!698\,948$ $\hskip 9.00002pt1$
$0,\\!715$ $\hskip 9.00002pt20$ $8,\\!25$ $5,\\!314\,390\,80$ $\hskip
9.00002pt162$ $1,\\!212\,30$ $\hskip 9.00002pt1999$ $6,\\!207$ $\hskip
9.00002pt40$ $9,\\!698\,947$ $\hskip 9.00002pt1$ $0,\\!735$ $\hskip
9.00002pt20$ $8,\\!26$ $5,\\!314\,389\,18$ $\hskip 9.00002pt169$
$1,\\!232\,29$ $\hskip 9.00002pt2000$ $6,\\!247$ $\hskip 9.00002pt40$
$9,\\!698\,946$ $\hskip 9.00002pt1$ $0,\\!755$ $\hskip 9.00002pt20$ $8,\\!27$
$5,\\!314\,387\,49$ $\hskip 9.00002pt177$ $1,\\!252\,29$ $\hskip
9.00002pt2000$ $6,\\!287$ $\hskip 9.00002pt40$ $9,\\!698\,945$ $\hskip
9.00002pt1$ $0,\\!775$ $\hskip 9.00002pt20$ $8,\\!28$ $5,\\!314\,385\,72$
$\hskip 9.00002pt186$ $1,\\!272\,29$ $\hskip 9.00002pt1999$ $6,\\!327$ $\hskip
9.00002pt40$ $9,\\!698\,944$ $\hskip 9.00002pt2$ $0,\\!795$ $\hskip
9.00002pt20$ $8,\\!29$ $5,\\!314\,383\,86$ $\hskip 9.00002pt195$
$1,\\!292\,28$ $\hskip 9.00002pt2000$ $6,\\!367$ $\hskip 9.00002pt40$
$9,\\!698\,942$ $\hskip 9.00002pt1$ $0,\\!815$ $\hskip 9.00002pt20$ $8,\\!30$
$5,\\!314\,381\,91$ $\hskip 9.00002pt203$ $1,\\!312\,28$ $\hskip
9.00002pt1999$ $6,\\!407$ $\hskip 9.00002pt40$ $9,\\!698\,941$ $\hskip
9.00002pt1$ $0,\\!835$ $\hskip 9.00002pt20$ $8,\\!31$ $5,\\!314\,379\,88$
$\hskip 9.00002pt213$ $1,\\!332\,27$ $\hskip 9.00002pt2000$ $6,\\!447$ $\hskip
9.00002pt40$ $9,\\!698\,940$ $\hskip 9.00002pt2$ $0,\\!855$ $\hskip
9.00002pt20$ $8,\\!32$ $5,\\!314\,377\,75$ $\hskip 9.00002pt224$
$1,\\!352\,27$ $\hskip 9.00002pt2000$ $6,\\!487$ $\hskip 9.00002pt40$
$9,\\!698\,938$ $\hskip 9.00002pt1$ $0,\\!875$ $\hskip 9.00002pt20$ $8,\\!33$
$5,\\!314\,375\,51$ $\hskip 9.00002pt234$ $1,\\!372\,27$ $\hskip
9.00002pt1999$ $6,\\!527$ $\hskip 9.00002pt40$ $9,\\!698\,937$ $\hskip
9.00002pt2$ $0,\\!895$ $\hskip 9.00002pt20$ $8,\\!34$ $5,\\!314\,373\,17$
$\hskip 9.00002pt244$ $1,\\!392\,26$ $\hskip 9.00002pt2000$ $6,\\!567$ $\hskip
9.00002pt40$ $9,\\!698\,935$ $\hskip 9.00002pt1$ $0,\\!915$ $\hskip
9.00002pt20$ $8,\\!35$ $5,\\!314\,370\,73$ $\hskip 9.00002pt257$
$1,\\!412\,26$ $\hskip 9.00002pt1999$ $6,\\!607$ $\hskip 9.00002pt40$
$9,\\!698\,934$ $\hskip 9.00002pt2$ $0,\\!935$ $\hskip 9.00002pt20$ $8,\\!36$
$5,\\!314\,368\,16$ $\hskip 9.00002pt268$ $1,\\!432\,25$ $\hskip
9.00002pt2000$ $6,\\!647$ $\hskip 9.00002pt40$ $9,\\!698\,932$ $\hskip
9.00002pt2$ $0,\\!955$ $\hskip 9.00002pt20$ $8,\\!37$ $5,\\!314\,365\,48$
$\hskip 9.00002pt281$ $1,\\!452\,25$ $\hskip 9.00002pt1999$ $6,\\!687$ $\hskip
9.00002pt40$ $9,\\!698\,930$ $\hskip 9.00002pt2$ $0,\\!975$ $\hskip
9.00002pt20$ $8,\\!38$ $5,\\!314\,362\,67$ $\hskip 9.00002pt295$
$1,\\!472\,24$ $\hskip 9.00002pt1999$ $6,\\!727$ $\hskip 9.00002pt40$
$9,\\!698\,928$ $\hskip 9.00002pt2$ $0,\\!995$ $\hskip 9.00002pt20$ $8,\\!39$
$5,\\!314\,359\,72$ $\hskip 9.00002pt308$ $1,\\!492\,23$ $\hskip
9.00002pt2000$ $6,\\!767$ $\hskip 9.00002pt40$ $9,\\!698\,926$ $\hskip
9.00002pt2$ $1,\\!015$ $\hskip 9.00002pt20$ $8,\\!40$ $5,\\!314\,356\,64$
$1,\\!512\,23$ $6,\\!807$ $9,\\!698\,924$ $1,\\!035$
Tables for computing geodesics 3, $\vphantom{\bigr{|}^{2^{2}}}\mathrm{Arg}$
$\log\alpha$ $\hskip 9.00002pt-\Delta$ $\log\beta$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\log\gamma$ $\hskip 9.00002pt\hphantom{-}\Delta$
$\log\alpha^{\prime}$ $\hskip 9.00002pt-\Delta$ $\log\beta^{\prime}$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\vphantom{\bigr{|}^{2^{2}}}8,\\!40\hphantom{0}$
$5,\\!314\,356\,64$ $\hskip 9.00002pt323$ $1,\\!512\,23$ $\hskip
9.00002pt1999$ $6,\\!807$ $\hskip 9.00002pt40$ $9,\\!698\,924$ $\hskip
9.00002pt2$ $1,\\!035$ $\hskip 9.00002pt20$ $8,\\!41\hphantom{0}$
$5,\\!314\,353\,41$ $\hskip 9.00002pt338$ $1,\\!532\,22$ $\hskip
9.00002pt1999$ $6,\\!847$ $\hskip 9.00002pt40$ $9,\\!698\,922$ $\hskip
9.00002pt2$ $1,\\!055$ $\hskip 9.00002pt20$ $8,\\!42\hphantom{0}$
$5,\\!314\,350\,03$ $\hskip 9.00002pt353$ $1,\\!552\,21$ $\hskip
9.00002pt2000$ $6,\\!887$ $\hskip 9.00002pt40$ $9,\\!698\,920$ $\hskip
9.00002pt2$ $1,\\!075$ $\hskip 9.00002pt20$ $8,\\!43\hphantom{0}$
$5,\\!314\,346\,50$ $\hskip 9.00002pt371$ $1,\\!572\,21$ $\hskip
9.00002pt1999$ $6,\\!927$ $\hskip 9.00002pt40$ $9,\\!698\,918$ $\hskip
9.00002pt3$ $1,\\!095$ $\hskip 9.00002pt20$ $8,\\!44\hphantom{0}$
$5,\\!314\,342\,79$ $\hskip 9.00002pt388$ $1,\\!592\,20$ $\hskip
9.00002pt1999$ $6,\\!967$ $\hskip 9.00002pt40$ $9,\\!698\,915$ $\hskip
9.00002pt2$ $1,\\!115$ $\hskip 9.00002pt20$ $8,\\!45\hphantom{0}$
$5,\\!314\,338\,91$ $\hskip 9.00002pt406$ $1,\\!612\,19$ $\hskip
9.00002pt1999$ $7,\\!007$ $\hskip 9.00002pt40$ $9,\\!698\,913$ $\hskip
9.00002pt3$ $1,\\!135$ $\hskip 9.00002pt20$ $8,\\!46\hphantom{0}$
$5,\\!314\,334\,85$ $\hskip 9.00002pt425$ $1,\\!632\,18$ $\hskip
9.00002pt2000$ $7,\\!047$ $\hskip 9.00002pt40$ $9,\\!698\,910$ $\hskip
9.00002pt3$ $1,\\!155$ $\hskip 9.00002pt20$ $8,\\!47\hphantom{0}$
$5,\\!314\,330\,60$ $\hskip 9.00002pt446$ $1,\\!652\,18$ $\hskip
9.00002pt1999$ $7,\\!087$ $\hskip 9.00002pt40$ $9,\\!698\,907$ $\hskip
9.00002pt3$ $1,\\!175$ $\hskip 9.00002pt20$ $8,\\!48\hphantom{0}$
$5,\\!314\,326\,14$ $\hskip 9.00002pt466$ $1,\\!672\,17$ $\hskip
9.00002pt1999$ $7,\\!127$ $\hskip 9.00002pt40$ $9,\\!698\,904$ $\hskip
9.00002pt3$ $1,\\!195$ $\hskip 9.00002pt20$ $8,\\!49\hphantom{0}$
$5,\\!314\,321\,48$ $\hskip 9.00002pt489$ $1,\\!692\,16$ $\hskip
9.00002pt1999$ $7,\\!167$ $\hskip 9.00002pt40$ $9,\\!698\,901$ $\hskip
9.00002pt3$ $1,\\!215$ $\hskip 9.00002pt20$ $8,\\!50\hphantom{0}$
$5,\\!314\,316\,59$ $\hskip 9.00002pt511$ $1,\\!712\,15$ $\hskip
9.00002pt1999$ $7,\\!207$ $\hskip 9.00002pt40$ $9,\\!698\,898$ $\hskip
9.00002pt4$ $1,\\!235$ $\hskip 9.00002pt20$ $8,\\!51\hphantom{0}$
$5,\\!314\,311\,48$ $\hskip 9.00002pt535$ $1,\\!732\,14$ $\hskip
9.00002pt1999$ $7,\\!247$ $\hskip 9.00002pt40$ $9,\\!698\,894$ $\hskip
9.00002pt3$ $1,\\!255$ $\hskip 9.00002pt20$ $8,\\!52\hphantom{0}$
$5,\\!314\,306\,13$ $\hskip 9.00002pt561$ $1,\\!752\,13$ $\hskip
9.00002pt1999$ $7,\\!287$ $\hskip 9.00002pt40$ $9,\\!698\,891$ $\hskip
9.00002pt4$ $1,\\!275$ $\hskip 9.00002pt20$ $8,\\!53\hphantom{0}$
$5,\\!314\,300\,52$ $\hskip 9.00002pt587$ $1,\\!772\,12$ $\hskip
9.00002pt1998$ $7,\\!327$ $\hskip 9.00002pt40$ $9,\\!698\,887$ $\hskip
9.00002pt4$ $1,\\!295$ $\hskip 9.00002pt20$ $8,\\!54\hphantom{0}$
$5,\\!314\,294\,65$ $\hskip 9.00002pt615$ $1,\\!792\,10$ $\hskip
9.00002pt1999$ $7,\\!367$ $\hskip 9.00002pt40$ $9,\\!698\,883$ $\hskip
9.00002pt4$ $1,\\!315$ $\hskip 9.00002pt20$ $8,\\!55\hphantom{0}$
$5,\\!314\,288\,50$ $\hskip 9.00002pt644$ $1,\\!812\,09$ $\hskip
9.00002pt1999$ $7,\\!407$ $\hskip 9.00002pt40$ $9,\\!698\,879$ $\hskip
9.00002pt4$ $1,\\!335$ $\hskip 9.00002pt20$ $8,\\!56\hphantom{0}$
$5,\\!314\,282\,06$ $\hskip 9.00002pt674$ $1,\\!832\,08$ $\hskip
9.00002pt1999$ $7,\\!447$ $\hskip 9.00002pt40$ $9,\\!698\,875$ $\hskip
9.00002pt5$ $1,\\!355$ $\hskip 9.00002pt20$ $8,\\!57\hphantom{0}$
$5,\\!314\,275\,32$ $\hskip 9.00002pt705$ $1,\\!852\,07$ $\hskip
9.00002pt1998$ $7,\\!487$ $\hskip 9.00002pt40$ $9,\\!698\,870$ $\hskip
9.00002pt5$ $1,\\!375$ $\hskip 9.00002pt20$ $8,\\!58\hphantom{0}$
$5,\\!314\,268\,27$ $\hskip 9.00002pt739$ $1,\\!872\,05$ $\hskip
9.00002pt1999$ $7,\\!527$ $\hskip 9.00002pt40$ $9,\\!698\,865$ $\hskip
9.00002pt4$ $1,\\!395$ $\hskip 9.00002pt20$ $8,\\!59\hphantom{0}$
$5,\\!314\,260\,88$ $\hskip 9.00002pt774$ $1,\\!892\,04$ $\hskip
9.00002pt1998$ $7,\\!567$ $\hskip 9.00002pt40$ $9,\\!698\,861$ $\hskip
9.00002pt6$ $1,\\!415$ $\hskip 9.00002pt20$ $8,\\!60\hphantom{0}$
$5,\\!314\,253\,14$ $\hskip 9.00002pt810$ $1,\\!912\,02$ $\hskip
9.00002pt1998$ $7,\\!607$ $\hskip 9.00002pt39$ $9,\\!698\,855$ $\hskip
9.00002pt5$ $1,\\!435$ $\hskip 9.00002pt20$ $8,\\!61\hphantom{0}$
$5,\\!314\,245\,04$ $\hskip 9.00002pt848$ $1,\\!932\,00$ $\hskip
9.00002pt1999$ $7,\\!646$ $\hskip 9.00002pt40$ $9,\\!698\,850$ $\hskip
9.00002pt6$ $1,\\!455$ $\hskip 9.00002pt20$ $8,\\!62\hphantom{0}$
$5,\\!314\,236\,56$ $\hskip 9.00002pt889$ $1,\\!951\,99$ $\hskip
9.00002pt1998$ $7,\\!686$ $\hskip 9.00002pt40$ $9,\\!698\,844$ $\hskip
9.00002pt6$ $1,\\!475$ $\hskip 9.00002pt20$ $8,\\!63\hphantom{0}$
$5,\\!314\,227\,67$ $\hskip 9.00002pt930$ $1,\\!971\,97$ $\hskip
9.00002pt1998$ $7,\\!726$ $\hskip 9.00002pt40$ $9,\\!698\,838$ $\hskip
9.00002pt6$ $1,\\!495$ $\hskip 9.00002pt20$ $8,\\!64\hphantom{0}$
$5,\\!314\,218\,37$ $\hskip 9.00002pt973$ $1,\\!991\,95$ $\hskip
9.00002pt1998$ $7,\\!766$ $\hskip 9.00002pt40$ $9,\\!698\,832$ $\hskip
9.00002pt6$ $1,\\!515$ $\hskip 9.00002pt20$ $8,\\!65\hphantom{0}$
$5,\\!314\,208\,64$ $\hskip 9.00002pt1020$ $2,\\!011\,93$ $\hskip
9.00002pt1998$ $7,\\!806$ $\hskip 9.00002pt40$ $9,\\!698\,826$ $\hskip
9.00002pt7$ $1,\\!535$ $\hskip 9.00002pt20$ $8,\\!66\hphantom{0}$
$5,\\!314\,198\,44$ $\hskip 9.00002pt1068$ $2,\\!031\,91$ $\hskip
9.00002pt1998$ $7,\\!846$ $\hskip 9.00002pt40$ $9,\\!698\,819$ $\hskip
9.00002pt7$ $1,\\!555$ $\hskip 9.00002pt20$ $8,\\!67\hphantom{0}$
$5,\\!314\,187\,76$ $\hskip 9.00002pt1118$ $2,\\!051\,89$ $\hskip
9.00002pt1998$ $7,\\!886$ $\hskip 9.00002pt40$ $9,\\!698\,812$ $\hskip
9.00002pt8$ $1,\\!575$ $\hskip 9.00002pt20$ $8,\\!68\hphantom{0}$
$5,\\!314\,176\,58$ $\hskip 9.00002pt1170$ $2,\\!071\,87$ $\hskip
9.00002pt1997$ $7,\\!926$ $\hskip 9.00002pt40$ $9,\\!698\,804$ $\hskip
9.00002pt7$ $1,\\!595$ $\hskip 9.00002pt20$ $8,\\!69\hphantom{0}$
$5,\\!314\,164\,88$ $\hskip 9.00002pt1226$ $2,\\!091\,84$ $\hskip
9.00002pt1998$ $7,\\!966$ $\hskip 9.00002pt40$ $9,\\!698\,797$ $\hskip
9.00002pt9$ $1,\\!615$ $\hskip 9.00002pt20$ $8,\\!70\hphantom{0}$
$5,\\!314\,152\,62$ $\hskip 9.00002pt1283$ $2,\\!111\,82$ $\hskip
9.00002pt1997$ $8,\\!006$ $\hskip 9.00002pt40$ $9,\\!698\,788$ $\hskip
9.00002pt8$ $1,\\!635$ $\hskip 9.00002pt19$ $8,\\!71\hphantom{0}$
$5,\\!314\,139\,79$ $\hskip 9.00002pt1344$ $2,\\!131\,79$ $\hskip
9.00002pt1998$ $8,\\!046$ $\hskip 9.00002pt40$ $9,\\!698\,780$ $\hskip
9.00002pt9$ $1,\\!654$ $\hskip 9.00002pt20$ $8,\\!72\hphantom{0}$
$5,\\!314\,126\,35$ $\hskip 9.00002pt1406$ $2,\\!151\,77$ $\hskip
9.00002pt1997$ $8,\\!086$ $\hskip 9.00002pt40$ $9,\\!698\,771$ $\hskip
9.00002pt9$ $1,\\!674$ $\hskip 9.00002pt20$ $8,\\!73\hphantom{0}$
$5,\\!314\,112\,29$ $\hskip 9.00002pt1473$ $2,\\!171\,74$ $\hskip
9.00002pt1997$ $8,\\!126$ $\hskip 9.00002pt40$ $9,\\!698\,762$ $\hskip
9.00002pt10$ $1,\\!694$ $\hskip 9.00002pt20$ $8,\\!74\hphantom{0}$
$5,\\!314\,097\,56$ $\hskip 9.00002pt1543$ $2,\\!191\,71$ $\hskip
9.00002pt1997$ $8,\\!166$ $\hskip 9.00002pt40$ $9,\\!698\,752$ $\hskip
9.00002pt11$ $1,\\!714$ $\hskip 9.00002pt20$ $8,\\!75\hphantom{0}$
$5,\\!314\,082\,13$ $\hskip 9.00002pt1615$ $2,\\!211\,68$ $\hskip
9.00002pt1997$ $8,\\!206$ $\hskip 9.00002pt40$ $9,\\!698\,741$ $\hskip
9.00002pt10$ $1,\\!734$ $\hskip 9.00002pt20$ $8,\\!76\hphantom{0}$
$5,\\!314\,065\,98$ $\hskip 9.00002pt1690$ $2,\\!231\,65$ $\hskip
9.00002pt1996$ $8,\\!246$ $\hskip 9.00002pt40$ $9,\\!698\,731$ $\hskip
9.00002pt12$ $1,\\!754$ $\hskip 9.00002pt20$ $8,\\!77\hphantom{0}$
$5,\\!314\,049\,08$ $\hskip 9.00002pt1771$ $2,\\!251\,61$ $\hskip
9.00002pt1997$ $8,\\!286$ $\hskip 9.00002pt40$ $9,\\!698\,719$ $\hskip
9.00002pt11$ $1,\\!774$ $\hskip 9.00002pt20$ $8,\\!78\hphantom{0}$
$5,\\!314\,031\,37$ $\hskip 9.00002pt1853$ $2,\\!271\,58$ $\hskip
9.00002pt1996$ $8,\\!326$ $\hskip 9.00002pt40$ $9,\\!698\,708$ $\hskip
9.00002pt13$ $1,\\!794$ $\hskip 9.00002pt20$ $8,\\!79\hphantom{0}$
$5,\\!314\,012\,84$ $\hskip 9.00002pt1941$ $2,\\!291\,54$ $\hskip
9.00002pt1996$ $8,\\!366$ $\hskip 9.00002pt39$ $9,\\!698\,695$ $\hskip
9.00002pt13$ $1,\\!814$ $\hskip 9.00002pt20$ $8,\\!800$ $5,\\!313\,993\,43$
$\hskip 9.00002pt1004$ $2,\\!311\,50$ $\hskip 9.00002pt998$ $8,\\!405$ $\hskip
9.00002pt20$ $9,\\!698\,682$ $\hskip 9.00002pt6$ $1,\\!834$ $\hskip
9.00002pt10$ $8,\\!805$ $5,\\!313\,983\,39$ $\hskip 9.00002pt1028$
$2,\\!321\,48$ $\hskip 9.00002pt998$ $8,\\!425$ $\hskip 9.00002pt20$
$9,\\!698\,676$ $\hskip 9.00002pt7$ $1,\\!844$ $\hskip 9.00002pt10$ $8,\\!810$
$5,\\!313\,973\,11$ $\hskip 9.00002pt1051$ $2,\\!331\,46$ $\hskip
9.00002pt998$ $8,\\!445$ $\hskip 9.00002pt20$ $9,\\!698\,669$ $\hskip
9.00002pt7$ $1,\\!854$ $\hskip 9.00002pt10$ $8,\\!815$ $5,\\!313\,962\,60$
$\hskip 9.00002pt1076$ $2,\\!341\,44$ $\hskip 9.00002pt998$ $8,\\!465$ $\hskip
9.00002pt20$ $9,\\!698\,662$ $\hskip 9.00002pt7$ $1,\\!864$ $\hskip
9.00002pt10$ $8,\\!820$ $5,\\!313\,951\,84$ $\hskip 9.00002pt1101$
$2,\\!351\,42$ $\hskip 9.00002pt998$ $8,\\!485$ $\hskip 9.00002pt20$
$9,\\!698\,655$ $\hskip 9.00002pt8$ $1,\\!874$ $\hskip 9.00002pt10$ $8,\\!825$
$5,\\!313\,940\,83$ $\hskip 9.00002pt1127$ $2,\\!361\,40$ $\hskip
9.00002pt997$ $8,\\!505$ $\hskip 9.00002pt20$ $9,\\!698\,647$ $\hskip
9.00002pt7$ $1,\\!884$ $\hskip 9.00002pt10$ $8,\\!830$ $5,\\!313\,929\,56$
$\hskip 9.00002pt1152$ $2,\\!371\,37$ $\hskip 9.00002pt998$ $8,\\!525$ $\hskip
9.00002pt20$ $9,\\!698\,640$ $\hskip 9.00002pt8$ $1,\\!894$ $\hskip
9.00002pt10$ $8,\\!835$ $5,\\!313\,918\,04$ $\hskip 9.00002pt1180$
$2,\\!381\,35$ $\hskip 9.00002pt998$ $8,\\!545$ $\hskip 9.00002pt20$
$9,\\!698\,632$ $\hskip 9.00002pt8$ $1,\\!904$ $\hskip 9.00002pt10$ $8,\\!840$
$5,\\!313\,906\,24$ $\hskip 9.00002pt1207$ $2,\\!391\,33$ $\hskip
9.00002pt997$ $8,\\!565$ $\hskip 9.00002pt20$ $9,\\!698\,624$ $\hskip
9.00002pt8$ $1,\\!914$ $\hskip 9.00002pt10$ $8,\\!845$ $5,\\!313\,894\,17$
$\hskip 9.00002pt1234$ $2,\\!401\,30$ $\hskip 9.00002pt998$ $8,\\!585$ $\hskip
9.00002pt20$ $9,\\!698\,616$ $\hskip 9.00002pt8$ $1,\\!924$ $\hskip
9.00002pt10$ $8,\\!850$ $5,\\!313\,881\,83$ $2,\\!411\,28$ $8,\\!605$
$9,\\!698\,608$ $1,\\!934$
Tables for computing geodesics 4, $\vphantom{\bigr{|}^{2^{2}}}\mathrm{Arg}$
$\log\alpha$ $\hskip 9.00002pt-\Delta$ $\log\beta$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\log\gamma$ $\hskip 9.00002pt\hphantom{-}\Delta$
$\log\alpha^{\prime}$ $\hskip 9.00002pt-\Delta$ $\log\beta^{\prime}$ $\hskip
9.00002pt\hphantom{-}\Delta$ $\vphantom{\bigr{|}^{2^{2}}}8,\\!850$
$5,\\!313\,881\,83$ $\hskip 9.00002pt1264$ $2,\\!411\,279$ $\hskip
9.00002pt9974$ $8,\\!605$ $\hskip 9.00002pt20$ $9,\\!698\,608$ $\hskip
9.00002pt8$ $1,\\!934$ $\hskip 9.00002pt10$ $8,\\!855$ $5,\\!313\,869\,19$
$\hskip 9.00002pt1293$ $2,\\!421\,253$ $\hskip 9.00002pt9974$ $8,\\!625$
$\hskip 9.00002pt20$ $9,\\!698\,600$ $\hskip 9.00002pt9$ $1,\\!944$ $\hskip
9.00002pt10$ $8,\\!860$ $5,\\!313\,856\,26$ $\hskip 9.00002pt1323$
$2,\\!431\,227$ $\hskip 9.00002pt9974$ $8,\\!645$ $\hskip 9.00002pt20$
$9,\\!698\,591$ $\hskip 9.00002pt9$ $1,\\!954$ $\hskip 9.00002pt10$ $8,\\!865$
$5,\\!313\,843\,03$ $\hskip 9.00002pt1353$ $2,\\!441\,201$ $\hskip
9.00002pt9973$ $8,\\!665$ $\hskip 9.00002pt20$ $9,\\!698\,582$ $\hskip
9.00002pt9$ $1,\\!964$ $\hskip 9.00002pt10$ $8,\\!870$ $5,\\!313\,829\,50$
$\hskip 9.00002pt1385$ $2,\\!451\,174$ $\hskip 9.00002pt9972$ $8,\\!685$
$\hskip 9.00002pt20$ $9,\\!698\,573$ $\hskip 9.00002pt9$ $1,\\!974$ $\hskip
9.00002pt10$ $8,\\!875$ $5,\\!313\,815\,65$ $\hskip 9.00002pt1417$
$2,\\!461\,146$ $\hskip 9.00002pt9972$ $8,\\!705$ $\hskip 9.00002pt20$
$9,\\!698\,564$ $\hskip 9.00002pt10$ $1,\\!984$ $\hskip 9.00002pt10$
$8,\\!880$ $5,\\!313\,801\,48$ $\hskip 9.00002pt1450$ $2,\\!471\,118$ $\hskip
9.00002pt9971$ $8,\\!725$ $\hskip 9.00002pt20$ $9,\\!698\,554$ $\hskip
9.00002pt9$ $1,\\!994$ $\hskip 9.00002pt10$ $8,\\!885$ $5,\\!313\,786\,98$
$\hskip 9.00002pt1484$ $2,\\!481\,089$ $\hskip 9.00002pt9970$ $8,\\!745$
$\hskip 9.00002pt20$ $9,\\!698\,545$ $\hskip 9.00002pt10$ $2,\\!004$ $\hskip
9.00002pt10$ $8,\\!890$ $5,\\!313\,772\,14$ $\hskip 9.00002pt1518$
$2,\\!491\,059$ $\hskip 9.00002pt9970$ $8,\\!765$ $\hskip 9.00002pt20$
$9,\\!698\,535$ $\hskip 9.00002pt10$ $2,\\!014$ $\hskip 9.00002pt9$ $8,\\!895$
$5,\\!313\,756\,96$ $\hskip 9.00002pt1553$ $2,\\!501\,029$ $\hskip
9.00002pt9969$ $8,\\!785$ $\hskip 9.00002pt19$ $9,\\!698\,525$ $\hskip
9.00002pt11$ $2,\\!023$ $\hskip 9.00002pt10$ $8,\\!900$ $5,\\!313\,741\,43$
$\hskip 9.00002pt1590$ $2,\\!510\,998$ $\hskip 9.00002pt9968$ $8,\\!804$
$\hskip 9.00002pt20$ $9,\\!698\,514$ $\hskip 9.00002pt10$ $2,\\!033$ $\hskip
9.00002pt10$ $8,\\!905$ $5,\\!313\,725\,53$ $\hskip 9.00002pt1626$
$2,\\!520\,966$ $\hskip 9.00002pt9968$ $8,\\!824$ $\hskip 9.00002pt20$
$9,\\!698\,504$ $\hskip 9.00002pt11$ $2,\\!043$ $\hskip 9.00002pt10$
$8,\\!910$ $5,\\!313\,709\,27$ $\hskip 9.00002pt1664$ $2,\\!530\,934$ $\hskip
9.00002pt9966$ $8,\\!844$ $\hskip 9.00002pt20$ $9,\\!698\,493$ $\hskip
9.00002pt11$ $2,\\!053$ $\hskip 9.00002pt10$ $8,\\!915$ $5,\\!313\,692\,63$
$\hskip 9.00002pt1702$ $2,\\!540\,900$ $\hskip 9.00002pt9966$ $8,\\!864$
$\hskip 9.00002pt20$ $9,\\!698\,482$ $\hskip 9.00002pt11$ $2,\\!063$ $\hskip
9.00002pt10$ $8,\\!920$ $5,\\!313\,675\,61$ $\hskip 9.00002pt1742$
$2,\\!550\,866$ $\hskip 9.00002pt9965$ $8,\\!884$ $\hskip 9.00002pt20$
$9,\\!698\,471$ $\hskip 9.00002pt12$ $2,\\!073$ $\hskip 9.00002pt10$
$8,\\!925$ $5,\\!313\,658\,19$ $\hskip 9.00002pt1783$ $2,\\!560\,831$ $\hskip
9.00002pt9965$ $8,\\!904$ $\hskip 9.00002pt20$ $9,\\!698\,459$ $\hskip
9.00002pt12$ $2,\\!083$ $\hskip 9.00002pt10$ $8,\\!930$ $5,\\!313\,640\,36$
$\hskip 9.00002pt1824$ $2,\\!570\,796$ $\hskip 9.00002pt9963$ $8,\\!924$
$\hskip 9.00002pt20$ $9,\\!698\,447$ $\hskip 9.00002pt12$ $2,\\!093$ $\hskip
9.00002pt10$ $8,\\!935$ $5,\\!313\,622\,12$ $\hskip 9.00002pt1866$
$2,\\!580\,759$ $\hskip 9.00002pt9963$ $8,\\!944$ $\hskip 9.00002pt20$
$9,\\!698\,435$ $\hskip 9.00002pt12$ $2,\\!103$ $\hskip 9.00002pt10$
$8,\\!940$ $5,\\!313\,603\,46$ $\hskip 9.00002pt1909$ $2,\\!590\,722$ $\hskip
9.00002pt9962$ $8,\\!964$ $\hskip 9.00002pt20$ $9,\\!698\,423$ $\hskip
9.00002pt13$ $2,\\!113$ $\hskip 9.00002pt10$ $8,\\!945$ $5,\\!313\,584\,37$
$\hskip 9.00002pt1953$ $2,\\!600\,684$ $\hskip 9.00002pt9961$ $8,\\!984$
$\hskip 9.00002pt20$ $9,\\!698\,410$ $\hskip 9.00002pt13$ $2,\\!123$ $\hskip
9.00002pt10$ $8,\\!950$ $5,\\!313\,564\,84$ $\hskip 9.00002pt1999$
$2,\\!610\,645$ $\hskip 9.00002pt9960$ $9,\\!004$ $\hskip 9.00002pt20$
$9,\\!698\,397$ $\hskip 9.00002pt13$ $2,\\!133$ $\hskip 9.00002pt10$
$8,\\!955$ $5,\\!313\,544\,85$ $\hskip 9.00002pt2045$ $2,\\!620\,605$ $\hskip
9.00002pt9959$ $9,\\!024$ $\hskip 9.00002pt20$ $9,\\!698\,384$ $\hskip
9.00002pt14$ $2,\\!143$ $\hskip 9.00002pt10$ $8,\\!960$ $5,\\!313\,524\,40$
$\hskip 9.00002pt2093$ $2,\\!630\,564$ $\hskip 9.00002pt9958$ $9,\\!044$
$\hskip 9.00002pt20$ $9,\\!698\,370$ $\hskip 9.00002pt14$ $2,\\!153$ $\hskip
9.00002pt10$ $8,\\!965$ $5,\\!313\,503\,47$ $\hskip 9.00002pt2141$
$2,\\!640\,522$ $\hskip 9.00002pt9957$ $9,\\!064$ $\hskip 9.00002pt19$
$9,\\!698\,356$ $\hskip 9.00002pt14$ $2,\\!163$ $\hskip 9.00002pt10$
$8,\\!970$ $5,\\!313\,482\,06$ $\hskip 9.00002pt2191$ $2,\\!650\,479$ $\hskip
9.00002pt9956$ $9,\\!083$ $\hskip 9.00002pt20$ $9,\\!698\,342$ $\hskip
9.00002pt15$ $2,\\!173$ $\hskip 9.00002pt10$ $8,\\!975$ $5,\\!313\,460\,15$
$\hskip 9.00002pt2241$ $2,\\!660\,435$ $\hskip 9.00002pt9956$ $9,\\!103$
$\hskip 9.00002pt20$ $9,\\!698\,327$ $\hskip 9.00002pt15$ $2,\\!183$ $\hskip
9.00002pt10$ $8,\\!980$ $5,\\!313\,437\,74$ $\hskip 9.00002pt2293$
$2,\\!670\,391$ $\hskip 9.00002pt9954$ $9,\\!123$ $\hskip 9.00002pt20$
$9,\\!698\,312$ $\hskip 9.00002pt15$ $2,\\!193$ $\hskip 9.00002pt10$
$8,\\!985$ $5,\\!313\,414\,81$ $\hskip 9.00002pt2347$ $2,\\!680\,345$ $\hskip
9.00002pt9953$ $9,\\!143$ $\hskip 9.00002pt20$ $9,\\!698\,297$ $\hskip
9.00002pt16$ $2,\\!203$ $\hskip 9.00002pt9$ $8,\\!990$ $5,\\!313\,391\,34$
$\hskip 9.00002pt2400$ $2,\\!690\,298$ $\hskip 9.00002pt9952$ $9,\\!163$
$\hskip 9.00002pt20$ $9,\\!698\,281$ $\hskip 9.00002pt15$ $2,\\!212$ $\hskip
9.00002pt10$ $8,\\!995$ $5,\\!313\,367\,34$ $\hskip 9.00002pt2457$
$2,\\!700\,250$ $\hskip 9.00002pt9951$ $9,\\!183$ $\hskip 9.00002pt20$
$9,\\!698\,266$ $\hskip 9.00002pt17$ $2,\\!222$ $\hskip 9.00002pt10$
$9,\\!000$ $5,\\!313\,342\,77$ $\hskip 9.00002pt2513$ $2,\\!710\,201$ $\hskip
9.00002pt9950$ $9,\\!203$ $\hskip 9.00002pt20$ $9,\\!698\,249$ $\hskip
9.00002pt17$ $2,\\!232$ $\hskip 9.00002pt10$ $9,\\!005$ $5,\\!313\,317\,64$
$\hskip 9.00002pt2571$ $2,\\!720\,151$ $\hskip 9.00002pt9948$ $9,\\!223$
$\hskip 9.00002pt20$ $9,\\!698\,232$ $\hskip 9.00002pt17$ $2,\\!242$ $\hskip
9.00002pt10$ $9,\\!010$ $5,\\!313\,291\,93$ $\hskip 9.00002pt2631$
$2,\\!730\,099$ $\hskip 9.00002pt9948$ $9,\\!243$ $\hskip 9.00002pt20$
$9,\\!698\,215$ $\hskip 9.00002pt17$ $2,\\!252$ $\hskip 9.00002pt10$
$9,\\!015$ $5,\\!313\,265\,62$ $\hskip 9.00002pt2691$ $2,\\!740\,047$ $\hskip
9.00002pt9946$ $9,\\!263$ $\hskip 9.00002pt19$ $9,\\!698\,198$ $\hskip
9.00002pt18$ $2,\\!262$ $\hskip 9.00002pt10$ $9,\\!020$ $5,\\!313\,238\,71$
$\hskip 9.00002pt2754$ $2,\\!749\,993$ $\hskip 9.00002pt9945$ $9,\\!282$
$\hskip 9.00002pt20$ $9,\\!698\,180$ $\hskip 9.00002pt18$ $2,\\!272$ $\hskip
9.00002pt10$ $9,\\!025$ $5,\\!313\,211\,17$ $\hskip 9.00002pt2818$
$2,\\!759\,938$ $\hskip 9.00002pt9943$ $9,\\!302$ $\hskip 9.00002pt20$
$9,\\!698\,162$ $\hskip 9.00002pt19$ $2,\\!282$ $\hskip 9.00002pt10$
$9,\\!030$ $5,\\!313\,182\,99$ $\hskip 9.00002pt2883$ $2,\\!769\,881$ $\hskip
9.00002pt9943$ $9,\\!322$ $\hskip 9.00002pt20$ $9,\\!698\,143$ $\hskip
9.00002pt19$ $2,\\!292$ $\hskip 9.00002pt10$ $9,\\!035$ $5,\\!313\,154\,16$
$\hskip 9.00002pt2949$ $2,\\!779\,824$ $\hskip 9.00002pt9941$ $9,\\!342$
$\hskip 9.00002pt20$ $9,\\!698\,124$ $\hskip 9.00002pt20$ $2,\\!302$ $\hskip
9.00002pt10$ $9,\\!040$ $5,\\!313\,124\,67$ $\hskip 9.00002pt3018$
$2,\\!789\,765$ $\hskip 9.00002pt9939$ $9,\\!362$ $\hskip 9.00002pt20$
$9,\\!698\,104$ $\hskip 9.00002pt20$ $2,\\!312$ $\hskip 9.00002pt10$
$9,\\!045$ $5,\\!313\,094\,49$ $\hskip 9.00002pt3087$ $2,\\!799\,704$ $\hskip
9.00002pt9939$ $9,\\!382$ $\hskip 9.00002pt20$ $9,\\!698\,084$ $\hskip
9.00002pt20$ $2,\\!322$ $\hskip 9.00002pt10$ $9,\\!050$ $5,\\!313\,063\,62$
$\hskip 9.00002pt3159$ $2,\\!809\,643$ $\hskip 9.00002pt9936$ $9,\\!402$
$\hskip 9.00002pt20$ $9,\\!698\,064$ $\hskip 9.00002pt21$ $2,\\!332$ $\hskip
9.00002pt10$ $9,\\!055$ $5,\\!313\,032\,03$ $\hskip 9.00002pt3232$
$2,\\!819\,579$ $\hskip 9.00002pt9936$ $9,\\!422$ $\hskip 9.00002pt20$
$9,\\!698\,043$ $\hskip 9.00002pt22$ $2,\\!342$ $\hskip 9.00002pt9$ $9,\\!060$
$5,\\!312\,999\,71$ $\hskip 9.00002pt3306$ $2,\\!829\,515$ $\hskip
9.00002pt9934$ $9,\\!442$ $\hskip 9.00002pt19$ $9,\\!698\,021$ $\hskip
9.00002pt22$ $2,\\!351$ $\hskip 9.00002pt10$ $9,\\!065$ $5,\\!312\,966\,65$
$\hskip 9.00002pt3383$ $2,\\!839\,449$ $\hskip 9.00002pt9932$ $9,\\!461$
$\hskip 9.00002pt20$ $9,\\!697\,999$ $\hskip 9.00002pt22$ $2,\\!361$ $\hskip
9.00002pt10$ $9,\\!070$ $5,\\!312\,932\,82$ $\hskip 9.00002pt3460$
$2,\\!849\,381$ $\hskip 9.00002pt9931$ $9,\\!481$ $\hskip 9.00002pt20$
$9,\\!697\,977$ $\hskip 9.00002pt23$ $2,\\!371$ $\hskip 9.00002pt10$
$9,\\!075$ $5,\\!312\,898\,22$ $\hskip 9.00002pt3541$ $2,\\!859\,312$ $\hskip
9.00002pt9929$ $9,\\!501$ $\hskip 9.00002pt20$ $9,\\!697\,954$ $\hskip
9.00002pt24$ $2,\\!381$ $\hskip 9.00002pt10$ $9,\\!080$ $5,\\!312\,862\,81$
$\hskip 9.00002pt3623$ $2,\\!869\,241$ $\hskip 9.00002pt9928$ $9,\\!521$
$\hskip 9.00002pt20$ $9,\\!697\,930$ $\hskip 9.00002pt24$ $2,\\!391$ $\hskip
9.00002pt10$ $9,\\!085$ $5,\\!312\,826\,58$ $\hskip 9.00002pt3706$
$2,\\!879\,169$ $\hskip 9.00002pt9926$ $9,\\!541$ $\hskip 9.00002pt20$
$9,\\!697\,906$ $\hskip 9.00002pt25$ $2,\\!401$ $\hskip 9.00002pt10$
$9,\\!090$ $5,\\!312\,789\,52$ $\hskip 9.00002pt3791$ $2,\\!889\,095$ $\hskip
9.00002pt9924$ $9,\\!561$ $\hskip 9.00002pt20$ $9,\\!697\,881$ $\hskip
9.00002pt25$ $2,\\!411$ $\hskip 9.00002pt10$ $9,\\!095$ $5,\\!312\,751\,61$
$\hskip 9.00002pt3879$ $2,\\!899\,019$ $\hskip 9.00002pt9922$ $9,\\!581$
$\hskip 9.00002pt19$ $9,\\!697\,856$ $\hskip 9.00002pt26$ $2,\\!421$ $\hskip
9.00002pt10$ $9,\\!100$ $5,\\!312\,712\,82$ $2,\\!908\,941$ $9,\\!600$
$9,\\!697\,830$ $2,\\!431$
|
arxiv-papers
| 2009-08-13T18:21:40 |
2024-09-04T02:49:04.615988
|
{
"license": "Public Domain",
"authors": "F. W. Bessel (K\\\"onigsberg Observatory), Charles F. F. Karney (Sarnoff\n Corp.), Rodney E. Deakin (RMIT)",
"submitter": "Charles Karney",
"url": "https://arxiv.org/abs/0908.1823"
}
|
0908.1863
|
11institutetext: Institut für Astronomie & Astrophysik, Universität Tübingen,
Auf der Morgenstelle 10, D-72076 Tübingen, Germany 22institutetext: Max-
Planck-Institut für Astronomie, Heidelberg
# Planet migration in three-dimensional radiative discs
Wilhelm Kley 11 Bertram Bitsch 11 Hubert Klahr 22 wilhelm.kley@uni-
tuebingen.de
(February 25, 2009, revised May 27, 2009)
###### Abstract
Context. The migration of growing protoplanets depends on the thermodynamics
of the ambient disc. Standard modelling, using locally isothermal discs,
indicate in the low planet mass regime an inward (type-I) migration. Taking
into account non-isothermal effects, recent studies have shown that the
direction of the type-I migration can change from inward to outward.
Aims. In this paper we extend previous two-dimensional studies, and
investigate the planet-disc interaction in viscous, radiative discs using
fully three-dimensional radiation hydrodynamical simulations of protoplanetary
accretion discs with embedded planets, for a range of planetary masses.
Methods. We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA
that includes full tensor viscosity. We have added implicit radiation
transport in the flux-limited diffusion approximation, and to speed up the
simulations significantly we have newly adapted and implemented the FARGO-
algorithm in a 3D context.
Results. First, we present results of test simulations that demonstrate the
accuracy of the newly implemented FARGO-method in 3D. For a planet mass of 20
$M_{\rm earth}$ we then show that the inclusion of radiative effects yields a
torque reversal also in full 3D. For the same opacity law used the effect is
even stronger in 3D than in the corresponding 2D simulations, due to a
slightly thinner disc. Finally, we demonstrate the extent of the torque
reversal by calculating a sequence of planet masses.
Conclusions. Through full 3D simulations of embedded planets in viscous,
radiative discs we confirm that the migration can be directed outwards up to
planet masses of about 33 $M_{\rm earth}$. Hence, the effect may help to
resolve the problem of too rapid inward migration of planets during their
type-I phase.
###### Key Words.:
accretion discs – planet formation – hydrodynamics – radiative transport
††offprints: W. Kley,
## 1 Introduction
The process of migration in protoplanetary discs allows forming planets to
move away from their location of creation and finally end up at a different
position. The cause of this change in distance from the star are the tidal
torques acting from the disturbed disc back on the protoplanet. These can be
separated into two parts: i) the so-called Lindblad torques that are created
by the two spiral arms in the disc and ii) the corotation torques that are
caused by the co-orbital material as it periodically exchanges angular
momentum with the planet on its horseshoe orbit. For comprehensive
introductions to the field see for example Papaloizou et al. (2007) or Masset
(2008), and references therein. The Lindblad torques, caused by density waves
launched at Lindblad resonances, quite generally lead to an inward motion of
the planet explaining quite nicely the existence of the observed hot planets
(Ward 1997). The corotation torques are caused mainly by two effects, first by
a gradient in the vortensity (Tanaka et al. 2002), and second by a gradient in
the entropy (Baruteau & Masset 2008). For typical protoplanetary discs both
contributions can be positive, possibly counterbalancing the negative Lindblad
torques (Paardekooper & Mellema 2006; Baruteau & Masset 2008). For the
typically considered locally isothermal discs where the temperature depends
only on the radial distance from the star the net torque is negative and
migration directed inwards for typical disc parameter (Tanaka et al. 2002).
Through planetary synthesis models the inferred rapid inward migration of
planetary cores has been found to be inconsistent with the observed mass-
distance distribution of exoplanets (Alibert et al. 2004; Ida & Lin 2008).
Possible remedies are the retention of icy cores at the snow line or a strong
reduction in the speed of type-I migration (of embedded small mass planets).
Here we focus of the latter process.
Different mechanism for slowing down the too rapid inward migration have been
discussed (Masset et al. 2006; Li et al. 2009), but the inclusion of more
realistic physics seems to be particularly appealing. As has been pointed out
first by Paardekooper & Mellema (2006) the inclusion of radiative transfer can
cause a strong reduction in the migration speed. The process has been
subsequently investigated by several groups (Baruteau & Masset 2008;
Paardekooper & Mellema 2008; Paardekooper & Papaloizou 2008; Kley & Crida
2008), who show that indeed the migration process can be slowed down or even
reversed for sufficiently low mass planets. This new effect occurs in non-
isothermal discs and scales with the gradient of the entropy (Baruteau &
Masset 2008), hence entropy-related torque. However, in a strictly adiabatic
situation after a few libration time scales the entropy gradient will flatten
within the corotation region due to phase mixing. This will lead to a
saturation (and subsequent disappearance) of the part of the corotation torque
that is caused by the entropy gradient in the horseshoe region (Baruteau &
Masset 2008; Paardekooper & Papaloizou 2009). To prevent saturation of this
entropy-related torque some radiative diffusion (or local radiative cooling)
is required (Kley & Crida 2008). Since for genuinely inviscid flows, the
streamlines in the horseshoe region will be closed and symmetric with respect
to the planet’s location, some level of viscosity is always necessary to avoid
torque saturation (Masset 2001; Ogilvie & Lubow 2003; Paardekooper &
Papaloizou 2008, 2009). This applies to both, the vortensity- and entropy-
related corotation torques. The maximum planet mass for which a change of
migration may occur due to this effect lies for typical disc masses in the
range of about 40 earth masses, beyond which the migration rate follows the
standard (isothermal) case, as gap formation sets in which reduces the
corotation effects (Kley & Crida 2008). Most of the above simulations have
studied only the two-dimensional case, while three dimensional models
including radiative effects have been presented only for very small masses
(Paardekooper & Mellema 2006), or for Jupiter type planets (Klahr & Kley
2006). A range of planet masses has not yet been studied systematically in
full 3D.
In this paper we investigate the planet-disc interaction in radiative discs
using fully three-dimensional radiation hydrodynamical simulations of
protoplanetary accretion discs with embedded planets for a variety of
planetary masses. For that purpose we modified and substantially extended an
existing multi-dimensional hydrodynamical code Nirvana (Ziegler & Yorke 1997a;
Kley et al. 2001) by incorporating the FARGO-algorithm (Masset 2000a) and
radiative transport in the flux-limited diffusion approximation (Levermore &
Pomraning 1981; Kley 1989). The code Nirvana can in principle handle nested
grids which allows to zoom-in on the detailed structure in the vicinity of the
planet (D’Angelo et al. 2002, 2003), however in the present context we limit
ourselves to single grid simulations. We present several test cases to
demonstrate first the accuracy of the FARGO-method in 3D. We then proceed to
analyse the effects of radiative transport on the disc structure and torque
balance. For our standard planet of $20M_{\rm earth}$, we find that the effect
of torque reversal appears to be even stronger in 3D than in 2D for an
otherwise identical physical setup. We have a more detailed look at the
implementation of the planet potential and show that it has definitely an
influence on the strength of the effect. Finally, we perform simulations for a
sequence of different planet masses to evaluate the mass range over which the
migration may be reversed. The consequence for the migration process and the
overall evolution of planets in discs is discussed.
## 2 Physical modelling
The protoplanetary disc is treated as a three-dimensional (3D), non-self-
gravitating gas whose motion is described by the Navier-Stokes equations. The
turbulence in discs is thought to be driven by magneto-hydrodynamical
instabilities (Balbus & Hawley 1998). Since we are interested in this study
primarily on the average effect the disc has on the planet, we prefer in this
work to simplify and treat the disc as a viscous medium. The dissipative
effects can then be described via the standard viscous stress-tensor approach
(eg. Mihalas & Weibel Mihalas 1984). We assume that the heating of the disc
occurs solely through internal viscous dissipation and ignore in the present
study the influence of additional energy sources such as irradiation from the
central star or other external sources. The internally produced energy is then
radiatively diffused through the disc and eventually emitted from its
surfaces. To describe this process we utilise the flux-limited diffusion
approximation (FLD, Levermore & Pomraning 1981) which allows to treat
approximately the transition from optically thick to thin regions near the
disc’s surface.
### 2.1 Basic equations
Discs with embedded planets have mostly been modelled through 2D simulations
in which the disc is assumed to be infinitesimal thin, and vertical integrated
quantities are used to describe the time evolution of the disc with the
embedded planet. This procedure saves considerable computational effort but is
naturally not as accurate as truly 3D simulations, in particular the radiation
transport is difficult to model in a 2D context.
In this work we present an efficient method for 3D disc simulations based on
the FARGO algorithm (Masset 2000a). For accretion discs where material is
orbiting a central object the best suited coordinates are spherical polar
coordinates $(r,\theta,\varphi)$ where $r$ denotes the radial distance from
the origin, $\theta$ the polar angle measured from the $z$-axis, and $\varphi$
denotes the azimuthal coordinate starting from the $x$-axis.
In this coordinate system, the mid-plane of the disc coincides with the
equator ($\theta=\pi/2$), and the origin of the coordinate system is centred
on the star. Sometimes we will need the radial distance from the polar axis
which we denote by a lower case $s$, which is the radial coordinate in
cylindrical coordinates.
For a better resolution of the flow in the vicinity of the planet, we work in
a rotating coordinate system which rotates with the orbital angular velocity
$\Omega$, which is identical to the orbital angular velocity of the planet
$\Omega_{P}=\left[\frac{G\,(M_{*}+m_{p})}{a^{3}}\right]^{1/2}$ (1)
where $M_{*}$ is the mass of the star, $m_{p}$ the mass of the planet, and $a$
the semi-major axis of the planet. Only for testing purposes for our
implementation of the FARGO-method we let the planet move under the action of
the disc.
The Navier-Stokes equations in a rotating coordinate system in spherical
coordinates read explicitly:
a) Continuity Equation
$\frac{\partial\,\rho}{\partial t}+\nabla\cdot(\rho\,{\bf u})=0$ (2)
Here $\rho$ denotes the density of the gas and ${\bf
u}=(u_{r},u_{\theta},u_{\varphi})$ its velocity.
b) Radial Momentum
$\displaystyle\frac{\partial\,\rho u_{r}}{\partial
t}+\nabla\cdot(\rho\,u_{r}\,{\bf u})$ $\displaystyle=$
$\displaystyle\rho\frac{u_{\theta}^{2}}{r}+\rho
r\sin^{2}\theta\,(\omega+\Omega)^{2}$ (3) $\displaystyle+$ $\displaystyle\rho
a_{r}-\frac{\partial p}{\partial r}-\rho\,\frac{\partial\Phi}{\partial
r}+f_{\mathrm{r}}.$
Here $\omega$ is the azimuthal angular velocity as measured in the rotating
frame, $p$ is the gas pressure, and $\Phi$ denotes the gravitational potential
due to the star and the planet. The vector ${\bf
a}=(a_{r},a_{\theta},a_{\varphi})$ represents inertial forces generated by the
accelerated origin of the coordinate system. Specifically, ${\bf a}$ equals
the negative acceleration acting on the star due to the planet(s).
c) Meridional Momentum
$\displaystyle\frac{\partial\,\rho ru_{\theta}}{\partial t}+\nabla\cdot(\rho
ru_{\theta}{\bf u})$ $\displaystyle=$ $\displaystyle\rho
r^{2}\sin\theta\cos\theta\,(\omega+\Omega)^{2}$ (4) $\displaystyle+$
$\displaystyle\rho r\,a_{\theta}-\frac{\partial
p}{\partial\theta}-\rho\,\frac{\partial\Phi}{\partial\theta}+r\
f_{\mathrm{\theta}}$
d) Angular Momentum
$\frac{\partial\,\rho h_{t}}{\partial t}+\nabla\cdot\left(\rho\,h_{t}\,{\bf
u}\right)=\rho r\sin\theta\ a_{\varphi}-\frac{\partial
p}{\partial\varphi}-\rho\,\frac{\partial\Phi}{\partial\varphi}+r\sin\theta\
f_{\mathrm{\varphi}};$ (5)
where we defined the total specific angular momentum (in the inertial frame)
$h_{t}=r^{2}\,\sin^{2}\theta\left(\omega+\Omega\right);$ (6)
i.e. the azimuthal velocity in the rotating frame is given by
$u_{\varphi}=\omega\,r\,\sin\theta$.
The Coriolis force in Eq. (5) for $u_{\varphi}$ (or $h_{t}$) has been
incorporated into the left hand side. Thus, it is written in such a way as to
conserve total angular momentum best. This conservative treatment is necessary
to obtain an accurate solution of the embedded planet problem (Kley 1998).
The function ${\bf f}=(f_{r},f_{\theta},f_{\varphi})$ in the momentum
equations denotes the viscous forces which are stated explicitly for the
three-dimensional case in spherical polar coordinates in Tassoul (1978). For
the description of the viscosity we use a constant kinematic viscosity
coefficient $\nu$.
e) Energy equation (internal energy)
$\frac{\partial\rho c_{v}T}{\partial t}+\nabla\cdot\left(\rho\,c_{v}T{\bf
u}\right)=-p\nabla\cdot{\bf u}+Q^{+}-\nabla\cdot\@vec{F}$ (7)
Here $T$ denotes the gas temperature in the disc and $c_{\mathrm{v}}$ is the
specific heat at constant volume. On the right hand side, the first term
describes compressional heating, $Q^{+}$ the viscous dissipation function, and
$\@vec{F}$ denotes the radiative flux. In writing eq. (7) we have assumed that
the radiation energy density $E=a_{R}T^{4}$ is always small in comparison to
the thermal energy density $e=\rho c_{v}T$. Here, $a_{R}$ denotes the
radiation constant. Furthermore, we utilise the one-temperature approach and
write for the radiative flux, using flux-limited diffusion (FLD)
$\qquad\@vec{F}=-\frac{\lambda c\,4a_{R}T^{3}}{\rho(\kappa+\sigma)}\,\nabla T\
,$ (8)
where $c$ is the speed of light, $\kappa$ the Rosseland mean opacity, $\sigma$
the scattering coefficient, and $\lambda$ the flux-limiter. Using FLD allows
us to perform stable accretion disc models that cover several vertical scale
heights. We use here the FLD approach described in Levermore & Pomraning
(1981) with the flux-limiter of Kley (1989). Its suitability for protostellar
discs has been shown in Kley & Lin (1996, 1999), and for embedded planets in
Klahr & Kley (2006). In this work we use for the Rosseland mean opacity
$\kappa(\rho,T)$ the analytical formulae by Lin & Papaloizou (1985) and set
the scattering coefficient $\sigma$ to zero. To close the basic system of
equations we use an ideal gas equation of state where the gas pressure is
given by $p=R_{gas}\rho T/\mu$, with the mean molecular weight $\mu$ and gas
constant $R_{gas}$. For a standard solar mixture we assume here $\mu=2.35$.
The speed of sound is calculated from $c_{\mathrm{s}}=\sqrt{\gamma p/\rho}$
with the adiabatic index $\gamma=1.43$.
Figure 1: The gravitational potential of a $20M_{\rm earth}$ planet with
different smoothings applied, see Eqs. (9) and (10). The distance $d$ from the
planet is given in units of $a_{p}$ (here $a_{Jup}$), and the smoothing length
$r_{sm}$ in units of the Hill radius of the planet which refers here to
$R_{H}=0.0271a_{p}$. Note that the data points are drawn directly from the
used 3D computational grid and indicate our standard numerical resolution (see
section 2.4).
### 2.2 Planetary potential
The total potential $\Phi$ acting on the disc consists of two contributions,
one from the star $\Phi_{*}$, the other from the planet $\Phi_{p}$
$\Phi=\,\Phi_{*}+\Phi_{p}\,=-\,\frac{GM_{*}}{r}-\frac{Gm_{\mathrm{p}}}{\sqrt{({\bf
r}-{\bf r}_{p})^{2}}},$
where ${\bf r}_{p}$ denotes the radius vector of the planet location. The
embedded planet is modelled as a point mass that orbits the central star on a
fixed circular orbit. In numerical simulations the planetary potential has to
be smoothed over a few gridcells to avoid divergences.
Typically in 2D simulations the planetary potential is modelled by an
$\epsilon$-potential
$\Phi_{p}^{\epsilon}=-\frac{Gm_{p}}{\sqrt{d^{2}+\epsilon^{2}}}\,,$ (9)
where we denote the distance of the disc element to the planet with $d=|{\bf
r}-{\bf r}_{p}|$ and $\epsilon$ is the smoothing length. In a 2D configuration
a potential of this form is indeed very convenient, as the smoothing takes
effects of the otherwise neglected vertical extent of the disc into account.
To account for the finite disc thickness $H$ which depends on the temperature
in the disc, an often used value for the smoothing length in 2D-simulations is
$\epsilon=r_{sm}=0.6H$. The obtained 2D torques are then found to be in
reasonable agreement with three-dimensional analytical estimates of the
torques, that do not include a softening length for the planet potential.
However in a 3D configuration the same approach is not necessary and would
lead to an unphysical ’spreading’ of the potential over a large region. Hence,
we apply a different type of smoothing, follow Klahr & Kley (2006) and use a
cubic-potential of the form
$\Phi_{p}^{cub}=\left\\{\begin{array}[]{cc}-\frac{m_{p}G}{d}\,\left[\left(\frac{d}{r_{\mathrm{sm}}}\right)^{4}-2\left(\frac{d}{r_{\mathrm{sm}}}\right)^{3}+2\frac{d}{r_{\mathrm{sm}}}\right]&\mbox{for}\quad
d\leq r_{\mathrm{sm}}\\\ -\frac{m_{p}G}{d}&\mbox{for}\quad
d>r_{\mathrm{sm}}\end{array}\right.$ (10)
This potential is constructed in such a way as to yield for distances $d$
larger than $r_{\mathrm{sm}}$ the correct $1/r$ potential of the planet, and
inside that radius ($d<r_{\mathrm{sm}}$) the potential is smoothed with a
cubic polynomial such that at the transition radius $r_{\mathrm{sm}}$ the
potential and its first and second derivative agree with the analytic outside
$1/r$-potential. To illustrate the various types of smoothing, we display in
Fig. 1 the behaviour of the different forms of the planetary potential.
Clearly the $\epsilon$-potential leads for the same values of
$r_{\mathrm{sm}}$ to a much wider and shallower potential than our cubic-
approach. For the often used value $\epsilon=0.6H$ the $\epsilon$-potential is
felt way outside the Hill radius
$R_{H}=a_{p}\left(\frac{m_{p}}{3M_{*}}\right)^{1/3}\,,$
and leads to a significant underestimate of the potential depth already at
$r_{\mathrm{sm}}$. The cubic-potential (Eq. 10) will always be accurate down
to $d=r_{\mathrm{sm}}$ and is inside much deeper than the
$\epsilon$-potential, and hence more accurate. In the simulations presented
below we study in detail the influence that the potential description has on
the value of the torques acting on the planet.
We calculate the gravitational torques acting on the planet by integrating
over the whole disc, where we apply a tapering function to exclude the inner
parts of the Hill sphere of the planet. Specifically, we use the smooth
(Fermi-type) function
$f_{b}(d)=\left[\exp\left(-\frac{d/R_{H}-b}{b/10}\right)+1\right]^{-1}$ (11)
which increases from 0 at the planet location ($d=0$) to 1 outside $d\geq
R_{H}$ with a midpoint $f_{b}=1/2$ at $d=bR_{H}$, i.e. the quantity $b$
denotes the torque-cutoff radius in units of the Hill radius. This torque
cutoff is necessary to avoid first a large, possibly very noisy contribution
from the inner parts of the Roche lobe, and second to disregard material that
is gravitationally bound to the planet. The question of torque cutoff and
exclusion of Roche lobe material becomes very important when (i) the disc
self-gravity is neglected, and (ii) there exists material bound to the planet
(e.g. a circumplanetary disc). This issue should definitely be addressed in
the future. Here, we assume a transition radius of $b=0.8$ Hill radii (see
Crida et al. 2008, Fig. 2). For reference we quote the width of the horseshoe
region which is given for an isothermal disc approximately by (Masset et al.
2006)
$x_{s}=1.16\,a_{p}\,\sqrt{\frac{q}{(H/r)}}$ (12)
with the mass ratio $q=m_{p}/M_{*}$ and the local relative disc thickness
$H/r$. For an adiabatic disc $H$ has to be replaced by $\gamma H$ (Baruteau &
Masset 2008).
Figure 2: Evolution of semi-major axis, eccentricity and inclination as a
function of time for a 20 $M_{earth}$ planet in a three-dimensional disc.
Results are displayed for different numerical resolutions
($N_{r},N_{\theta},N_{\varphi}$), and using Fargo and Non-Fargo runs.
### 2.3 Setup
The three-dimensional ($r,\theta,\varphi$) computational domain consists of a
complete annulus of the protoplanetary disc centred on the star, extending
from $r_{\mathrm{min}}=0.4$ to $r_{\mathrm{max}}=2.5$ in units of
$r_{0}=a_{Jup}=5.2$AU. In the vertical direction the annulus extends from the
disc’s midplane (at $\theta=90^{\circ}$) to about $7^{\circ}$ (or
$\theta=83^{\circ}$) above the midplane. In case of an inclined planet the
domain has to be extended and cover the upper and lower half of the disc. The
mass of the central star is one solar mass $M_{*}=M_{\odot}$, and the total
disc mass inside $[r_{\mathrm{min}},r_{\mathrm{max}}]$ is
$M_{disc}=0.01M_{\odot}$. For the present study, we use a constant kinematic
viscosity coefficient with a value of $\nu=10^{15}$ cm2/s, a value that
relates to an equivalent $\alpha=0.004$ at $r_{0}$ for a disc aspect ratio of
$H/r=0.05$, where $\nu=\alpha H^{2}\Omega_{K}$. In standard dimensionless
units we have $\nu=10^{-5}$.
The models are initialised with a locally isothermal configuration where the
temperature is constant on cylinders and has the profile $T(s)\propto s^{-1}$,
where $s$ is related to $r$ through $s=r\sin\theta$. This yields a constant
ratio of the disc’s vertical height $H$ to the radius $s$. The initial
vertical density stratification is approximately given by a Gaussian:
$\rho(r,\theta)=\rho_{0}(r)\,\exp\left[-\frac{(\pi/2-\theta)^{2}\,r^{2}}{H^{2}}\right].$
(13)
Here, the density in the midplane is $\rho_{0}(r)\propto r^{-1.5}$ which leads
to a $\Sigma(r)\propto\,r^{-1/2}$ profile of the vertically integrated surface
density. The vertical and radial velocities, $u_{\theta}$ and $u_{r}$, are
initialised to zero. The initial azimuthal velocity $u_{\varphi}$ is given by
the equilibrium of gravity, centrifugal acceleration and the radial pressure
gradient. In case of purely isothermal calculations this setup is equal to the
equilibrium configuration (in the case of closed radial boundaries). For fully
radiative simulations the model is first run in a 2D axisymmetric mode to
obtain a new self-consistent equilibrium where viscous heating balances
radiative transport/cooling from the surfaces (see Sect. 4.1 below). This
initialisation through an axisymmetric 2D phase (in the $r-\theta$ plane)
reduces the required computational effort substantially, as the evolution from
the initial isothermal state towards the radiative equilibrium takes about 100
orbits, if the disc is started with an isothermal equilibrium having constant
$H/r$. After reaching the equilibrium between viscous heating and radiative
transport/cooling, we extend this model to a full 3D simulation, by expanding
the grid into the $\phi$-direction, and the planet is embedded.
### 2.4 Numerics
We adopt a coordinate system, which rotates at the orbital frequency of the
planet. For our standard cases, we use an equidistant grid in
$r,\theta,\varphi$ with a resolution of
$(N_{r},N_{\theta},N_{\varphi})=(266,32,768)$. To minimise disturbances (wave
reflections) from the radial boundaries, we impose, at $r_{\mathrm{min}}$ and
$r_{\mathrm{max}}$, damping boundary conditions where all three velocity
components are relaxed towards their initial state on a timescale of
approximately the local orbital period. The radial velocities at the inner and
outer radius vanish. The angular velocity is relaxed towards the Keplerian
values. For the density and temperature, we apply reflective radial boundary
conditions. In the azimuthal direction, periodic boundary conditions are
imposed for all variables. In the vertical direction we apply outflow boundary
conditions. The boundary conditions do not allow for mass accretion through
the disc, such that the total disc mass remains nearly constant during the
time evolution, despite a possible small change due to little outflow through
the vertical boundaries and the used density floor (see below).
The numerical details of the used finite volume code (NIRVANA) relevant for
these planet disc simulations were described in Kley et al. (2001) and
D’Angelo et al. (2003). In the latter paper the usage of the nested grid-
technique is described in more detail as well. The original version of the
NIRVANA code, on which our programme is based upon, has been developed by
Ziegler & Yorke (1997b). The empowerment with FARGO is based on the original
work by Masset (2000a). Our implementation appears to be the first inclusion
of the FARGO-algorithm in a 3D spherical coordinate system. More details about
the implementation are given in the appendix. The basic algorithm of the newly
implemented radiation part in the energy equation (7) is presented in the
appendix as well. To avoid possible time step limitations this part is always
solved implicitly.
## 3 Test calculations
### 3.1 The FARGO-algorithm
To test the 3D implementation of the FARGO-algorithm in our NIRVANA-code we
have run several models with planets on circular, elliptic and inclined orbits
with and without the FARGO-method applied. Here, we follow closely the models
presented in Cresswell et al. (2007) and consider moving planets in 3D discs.
As the tests are dynamically already complicated we use here only the
isothermal setup. The different setups gave very similar results in all cases,
and we present results for one combined case of a 20 $M_{\rm earth}$ planet
embedded in a locally isothermal disc with an initial non-zero eccentricity
($e=0.2$) and non-zero inclination ($i=5^{\circ}$). All physical parameters of
this run are identical to those described in Cresswell et al. (2007), and we
compare our results to the last models presented in that paper (their Fig.
16). The outcome of this comparison is shown in Fig. 2, where we display the
results of a standard non-Fargo run with the resolution
$(N_{r},N_{\theta},N_{\phi})=(264,80,800)$ with the data taken from Cresswell
et al. (2007) (where a different code has been used) to two runs having a
lower resolution of $(N_{r},N_{\theta},N_{\varphi})=(128,34,384)$, one with
FARGO and the other one without. We can see that all 3 models (obtained with
two different codes, methods and resolutions) yield very similar results. The
scatter of the data points is slightly smaller in the FARGO-run.
Figure 3: Radial stratification of surface density (left) and the midplane
temperature (right) in the disc. This dashed lines represent simple
approximations to the 3D stratified results. The solid (green) curves labelled
’flat’ refer to corresponding results for a vertically integrated flat 2D disc
using the same input physics.
Figure 4: Vertical stratification of density (left) and temperature (right) at
a radius of $r=1.44$. Thin dashed lines just represent simple functional
relations.
### 3.2 The radiation algorithm
To obtain an independent test of the newly implemented radiation transport
module in our NIRVANA-code we performed a run with the standard setup as
described above but with no embedded planet. Hence, this setup refers to an
axisymmetric disc with internal heating and radiative cooling. For a fixed,
closed computational domain it is only the total mass enclosed that determines
the final equilibrium state of the system, once the physics (viscosity,
opacity, and equation of state) have been prescribed. The radial dependence of
the vertically integrated surface density and the midplane temperature are
displayed in Fig. 3, and the corresponding vertical profiles at a radius of
$r=1.44$ in Fig. 4. First of all, these new results obtained with NIRVANA
agree very well with those obtained with the completely independent 2D-code
RH2D used in the $r-\theta$ mode, such as presented for example in Kley et al.
(1993), which are not shown in the figures, however. As the final
configuration of the system is given by the equilibrium of internal (viscous)
dissipation and radiative transport, this test demonstrates the consistency of
our implementations.
To relate our 3D results to previous radiative 2D runs which use vertically
integrated quantities, and hence can only use an approximative energy
transport and cooling (Kley & Crida 2008), we compare in Fig. 3 the results
obtained with the two methods. Both models are constructed for the same disc
mass and identical physics. The label ’flat’ in the Figure refers to the flat
2D case (obtained with RH2D, see Kley & Crida (2008)) and the ’stratified’
label to our new 3D implementation presented here. The left graph displays the
vertically integrated surface density distribution, here
$\Sigma(r)=f_{\Sigma}\int_{\theta_{max}}^{\theta_{min}}\rho
r\sin(\theta)d\theta$, with $f_{\Sigma}=1$ for two-sided and 2 for one-sided
discs. Our result is well represented by a $\Sigma\propto r^{-1/2}$ profile as
expected for a closed domain and constant viscosity. Interesting is the
irregular structure at radii smaller than $r\approx 1.0$ in the full 3D
stratified case, and we point out that these refer to the onset of convection
inside that radius. To model convection is of course not possible in a flat 2D
approach. The temperature distribution for the full 3D case follows
approximately a $T\propto r^{-1.7}$ profile. Here, the approximate flat-disc
model leads to midplane temperatures that are about 40% higher for the bulk
part of the domain than in the true 3D case. Possibly a refined modelling of
the vertical averaging procedure and the radiative losses in the flat 2D case
could improve the agreement here, but in the presence of convection we may
expect differences in any case.
In Fig. 4 we display the vertical stratification of the disc at a specific
radius in the middle of the computational domain at $r=1.44$. Two simple
approximations are over-plotted as dashed lines. Note, that in these plots the
stratification is plotted along the $r=const.$ lines which deviates for thin
discs only slightly from $z=r\cos\theta$. Taking $z_{0}=0.08$, the Gaussian
curve for the density refers to $\rho_{0}\exp\left[-(z/z_{0})^{2}\right]$ and
the temperature fit to $T(z)=T_{0}[1-0.4\,(z/z_{0})^{2}]$. These simple
formulae are intended to guide the eye rather than meant to model exactly the
structure at this radius which depends on the used opacities. Given the
simplicity of these, it is interesting that they approximate the true solution
reasonably well within one scale height.
### 3.3 Density Floor
By expanding the computed area in the $\theta$-direction beyond the
$90^{\circ}$ to $83^{\circ}$ region of our standard model, the code would have
to cover several orders of magnitude in the density. Thus, many more grid
cells would be required to resolve the physical quantities. In order to avoid
this and save computation time, we apply a minimum density function (floor)
for the low-density regions high above the equatorial plane of the disc. It
reads
$\rho=\left\\{\begin{array}[]{cc}\rho&\quad\mbox{for}\quad\rho>\rho_{min}\\\
\rho_{min}&\quad\mbox{for}\quad\rho\leq\rho_{min}\end{array}\right.\ .$ (14)
Of course, applying a density floor like this will create mass inside the
computed domain. The density floor $\rho_{min}$ has now to be chosen such
that: firstly the computation is not handicapped by too small values and
secondly the inner (optically thick) parts of the disc are not influenced. To
test the sensitivity of the disc structure against the density floor we
performed a series of test calculations, and show the results of simulations
with different $\rho_{min}$ in Fig. 5. These runs cover $\theta=90^{\circ}$ to
$70^{\circ}$, a range about 3 times as large as before. The density and
temperature profiles in these simulations do not differ for the regions near
the equatorial plane, because the density is too high for the minimum density
to take effect. Indeed, all curves are nearly indistinguishable in the region
for optical depths larger than $\tau=1.0$, with
$\tau(z)=\int_{z}^{\infty}\rho\kappa dz.$ (15)
Please note, that in the plot we do display the results along lines of
constant (spherical) radius. Moving further away from the equatorial plane,
one can see in the density profile the different minimum densities, but in the
temperature profile there is hardly any difference at all. Above a certain
distance from the equatorial plane the temperature remains constant. The
little fluctuations visible in the profile are due to oscillations in the
temperature for the low mass regions.
Figure 5: Vertical stratification of density in logarithmic scale (top) and
temperature (bottom) at a radius of $r=1.00$ for a simulation covering the
$\theta=90^{\circ}$ to $70^{\circ}$ region. The optical depth $\tau=1.0$ is
reached at about $z=0.055$.
By applying a minimum density the code is capable of resolving large distances
above the equatorial plane with a reasonable number of grid cells. Also note
that it is not necessary to use a minimum density for calculations covering
only the $\theta=90^{\circ}$ to $83^{\circ}$ regions, as the density is always
high enough.
## 4 Models with an embedded planet
For all the models with embedded planets we use our standard disc setup as
described in section 2.3 with the corresponding boundary conditions in section
2.4. Here, we briefly summarise some important parameter of the setup. The
three-dimensional ($r,\theta,\varphi$) structure of the disc extends form
$r_{\mathrm{min}}=0.4$ to $r_{\mathrm{max}}=2.5$ in units of
$r_{0}=a_{Jup}=5.2$AU. In the vertical direction the annulus extends from the
disc’s midplane (at $\theta=90^{\circ}$) to about $7^{\circ}$ (or
$\theta=83^{\circ}$) above the midplane. For our chosen grid size of
$(N_{r},N_{\varphi},N_{\theta})=266,32,768)$ this refers to linear grid
resolution of $\Delta\approx 0.008$ at the location of the planet, which
corresponds to $3.3$ gridcells per Hill radius, and to about 5 gridcells per
horseshoe half-width for a $20M_{earth}$ planet in a disc with $H/r=0.05$. In
this configuration the planet is located exactly at the corner of a gridcell.
In the fully radiative disc, the temperature at the disc surface is kept at
the fixed ambient temperature of 10 K. This simple ’low-temperature’ boundary
condition ensures that all the internally generated energy is liberated freely
at the disc’s surface. It is only suitable for optically thin boundaries and
does not influence the inner parts of the optically thick disc (see Fig. 5).
The disc has a mass of $0.01M_{\odot}$, and an aspect ratio $H/r=0.05$ in the
beginning.
### 4.1 Initial setup
Before placing the planet into the 3D disc we have to bring it first into a
radiative equilibrium state such that our results are not corrupted by initial
transients. As described above this initial equilibration is performed in an
axisymmetric 2D setup that is then expanded to full 3D. Tests with our code
have shown that we reach the 3D equilibrium state (a constant torque) in a
calculation with embedded planets about $50\%$ faster when starting first with
the 2D radiative equilibrium disc.
In Fig. 6 the 2D density and temperature distributions for such an
equilibrated disc are displayed.
Figure 6: Density (top) and Temperature (bottom) for a fully radiative model
in a 2D axisymmetric simulation
In the equilibrium state of the fully radiative model the disc is much thinner
in comparison to the isothermal starting case, see Fig. 7. Consequently, the
density is increased in the equatorial plane, leaving the areas high above and
below the disc with less material. Apparently, for this disc mass and the
chosen values of viscosity and opacity, the balance of viscous heating and
radiative cooling reduces the aspect ratio of the disc from initially $0.05$
to about $0.037$ in the radiative case. Had we started with an initially
thinner disc, the difference would of course not be that pronounced.
Figure 7: Vertical density distribution at $r=1.4$ for a fully radiative model
in a 2D axisymmetric simulation. ’+’: isothermal starting configuration, ’x’:
relaxed radiation equilibrium configuration
After successfully completing the equilibration we now embed a 20 $M_{\rm
earth}$ planet into the disc. The planet is held on a fixed orbit and we
calculate the torques acting on it through integrating over the whole disc
taking into account the above tapering function with a cutoff
$r_{torq}=0.8R_{H}$, which refers to $b=0.8$ in Eq. 11. In addition to this
value of the torque cutoff he have tested how the obtained total torque
changes when using $b=0.6$. For our standard $20M_{\rm earth}$ planet
presented in the following we found that for the isothermal cases the results
change by about 10% and in the radiative case by about 30%, which can be
considered as a rough estimate of the numerical uncertainties of the results.
The deviation is larger in the radiative situation because in this case
important (corotation) contributions to the total torque originate from a
region very close to the planet, which is influenced stronger by the applied
torque cutoff. Here, cancellation effects caused by adding the negative
Lindblad and the positive corotation torque may explain part of the larger
relative uncertainty in the radiative case. We note, that our applied torque
cutoff is not hard but refers to the smooth function (11). Keeping in mind
that there are only 3.3 gridcells per Hill radius, smaller values for $b$ are
not useful.
### 4.2 Isothermal discs
Due to the applied smoothing, we expect the planetary potential to modify the
density structure of the disc near the planet and subsequently change the
torques acting on the planet. First, we investigate the influence of the
planetary potentials (see Fig. 1) on the disc and torques in the isothermal
regime. The 2D surface density distribution in the disc’s midplane at 100
planetary orbits corresponding to our two extreme planetary potentials (the
shallowest and the deepest) is displayed in Fig. 8, where we used a cutoff for
the maximum displayed density to make both cases comparable. As expected, a
deeper planetary potential results in a higher density concentration inside
the planetary Roche lobe and to a slightly reduced density in the immediate
surroundings. This accumulation of mass near the planet for deeper potentials
is illustrated in more detail Fig. 9. For our deepest $r_{sm}=0.5$ cubic
potential the maximum density inside the planet’s Roche-lobe is over an order
of magnitude larger than in the shallowest $r_{sm}=0.8$ $\epsilon$-potential.
Figure 8: Surface density distribution for isothermal simulations with
$H/r=0.05$ at 100 planetary orbits. Displayed are results for the shallowest
and deepest potential. Top: $\epsilon$-potential with $r_{sm}=0.8$, Bottom:
cubic with $r_{sm}=0.5$. Figure 9: Radial density distribution in the equator
along ($\rho(r,\varphi=\pi,\theta=\pi/2)$), i.e. along a ray through the
location of the planet for all 4 planetary potentials used.
In Fig. 10 we show the specific torques acting onto the planet using different
potentials for the case of $H/r=0.05$. The total torque is continuously
monitored and plotted versus time in the upper panel. The radial torque
density $\Gamma(r)$ for the same models is displayed in the lower panel. Here,
$\Gamma(r)$ is defined such that the total torque $T^{tot}$ acting on the
planet is given by
$T^{tot}=\int_{r_{min}}^{r_{max}}\,\Gamma(r)\,dr.$ (16)
The time evolution of the total torque displays a characteristic behaviour.
Starting from the axisymmetric case, a first intermediate plateau is reached
at early times between $t\approx 5-10$, after which the torques settle on
longer timescales towards their final equilibria. The initial plateaus
correspond to the values of the torques shortly after the disc material has
started its horseshoe-type motion in the co-orbital region. The level of this
so-called unsaturated torque depends on the local disc properties and on the
applied smoothing of the potential, as indicated clearly in the top panel of
Fig. 10. In the following evolution, the material in this horseshoe region
will be mixed thoroughly, the torques decline and settle eventually to their
final equilibria, here reached after about 40 orbits. This process of phase
mixing inside the horseshoe region is called torque saturation, and it occurs
on timescales of the order of the libration time, which is given by
$\tau_{lib}=\frac{4\,a_{p}}{3\,x_{s}}\,{P_{p}}$ (17)
where $P_{p}$ is the orbital period of the planet, and $x_{s}$ the half-width
of the horseshoe region (see eq. 12). In our case (for $q=6\times 10^{-5}$ and
$H/r=0.05$) the libration time is about $30P$. The different initial values of
the unsaturated torques depend on the form of the potential (eg. smoothing
length), but note that the timescale to reach equilibrium is similar in all
cases. This particular time behaviour of the torques and the process of
saturation has been described recently for isothermal discs by Paardekooper &
Papaloizou (2009), see also Masset et al. (2006).
The two runs with the $\epsilon$-potential result in the most negative torque
values, i.e. the fastest inward migration (lower two curves in the upper
panel). While the total torques of the two $\epsilon$-potentials are nearly
identical, in the corresponding radial torque distribution the cases are
clearly separated, a fact which is due to cancellation effects when adding the
inner (positive) and outer (negative) contribution. The slightly deeper cubic
$r_{sm}=0.8$ potential leads to a marginally decreased (in magnitude) torque
compared to the simulations with $\epsilon$-potential. For the cubic
$r_{sm}=0.5$ potential we obtain an even less negative equilibrium torque
compared to all the other isothermal simulations. As most of the corotation
torque is generated in the vicinity of the planet, a change in the density
structure there (by deepening the potential) may have a significant impact on
the torque values. We can compare our values of the torque with the well known
formulae for the specific torque in a 3D strictly isothermal disc as presented
by Tanaka et al. (2002)
$T^{tot}_{0}=-\,f_{\Gamma}\,q\,\left(\frac{H}{r}\right)^{-2}\,\left(\frac{\Sigma
a_{p}^{2}}{M_{*}}\right)\,a_{p}^{2}\Omega_{p}^{2}$ (18)
with
$f_{\Gamma}=(1.364+0.541\,\alpha_{\Sigma})$ (19)
where $\alpha_{\Sigma}$ denotes the radial gradient of the surface density
through $\Sigma\propto r^{-\alpha_{\Sigma}}$. For our standard parameter this
formula gives about $T^{tot}_{0}=-2.5\,10^{-5}a_{p}^{2}\Omega_{p}^{2}$, which
is, in absolute value, about a factor $1.4-2.2$ times larger than our results.
We note however, that eq. (18) has been derived for constant temperature,
inviscid disc. The influence of viscosity on the torque has been studied by
Masset (2002), who found that a reduction of the viscosity from our used value
of $10^{-5}$ to zero will fully saturate the (vortensity-related) corotation
torque, leading easily to a reduction of the total torque by a factor of two.
Additional simulations with much smaller viscosity (not shown here) indicate
indeed that then the equilibrium torque is in good agreement with eq. (18).
Figure 10: Specific torque (in units of $a_{p}^{2}\Omega_{p}^{2}$) acting on
the planet using 4 different smoothings for the planetary potential in the
isothermal case with $H/r=0.05$. Top: Evolution of total torque with time.
Bottom: Radial variation of the specific torque density $\Gamma(r)$ at $t=80$
orbits.
It seems at first surprising and unpleasing that the torques depend so much on
the treatment of the planetary potential. However, an $\epsilon$-potential has
an influence far beyond the Roche-radius of the planet and certainly will
change the torques acting on the planet. Here the corotation torques are
affected most prominently and become more and more positive as the smoothing
length is lowered (see also Paardekooper & Papaloizou 2009). Nevertheless, in
two-dimensional simulations it has become customary to rely on
$\epsilon$-potentials for the purpose to take into account the finite
thickness of the disc. In a three-dimensional context, the more localised
cubic-potential with its finite region of influence may be more realistic. But
for the isothermal case the increased potential depth leads to a very large
accumulation of mass, as seen in Fig. 9. In such a case it will be very
difficult to achieve convergence. In the more realistic radiative case the
situation is eased somewhat through a temperature increase near the planet, as
outlined below.
To check numerical convergence we performed additional runs using a larger
number of gridcells. In Fig. 11 we display the total torque versus time and
the radial torque density $\Gamma(r)$ for different grid resolutions, again
for the isothermal disc case. In contrast to the previous plot we use here a
slightly cooler disc with $H/r=0.037$, as this matches more closely the
results from the fully radiative calculations presented below. In this case
the initial unsaturated torques reach even positive values due to the smaller
thickness of the disc. The grid resolution seems to be sufficient for
resolving the structures near the Roche-lobe. For the displayed $\Gamma(r)$
distribution in the lower panel both cases are very similar and the higher
resolution case is a bit smoother.
Figure 11: Specific torque acting on the planet using different grid
resolutions for the isothermal case with $H/r=0.037$. In all cases the cubic
potential with $r_{sm}=0.5$ has been used. Top: Evolution of total torque with
time. Bottom: Radial variation of the specific torque density, at $t=80$
orbits for the standard and medium resolution, and at $60$ orbits for the high
resolution.
### 4.3 Fully radiative discs
The simulations are started from the radiative disc in equilibrium as
described above, and are continued with an embedded planet of $20M_{\rm
earth}$. The obtained equilibrium configuration for the surface density and
midplane-temperature is displayed in Fig. 12 after an evolutionary time of 100
orbits. As in the isothermal case the density within the Roche lobe of the
planet is strongly enhanced for the deeper potentials, displayed are the two
extreme cases of our different potentials. Comparing with the corresponding
density maps of the isothermal case 8, one can also observe slightly smaller
opening angles of the spiral arms in the radiative case. For identical $H/r$
the sound speed would be $\sqrt{\gamma}$ times larger in the radiative case
leading to a larger opening angle. Here, the effect is overcompensated by the
reduced temperature (lower thickness) in the radiative case. A different
opening angle of the spiral arms will affect the corresponding Lindblad
torques acting on the planet.
At the same time, a slight density enhancement is visible ’ahead’ of the
planet ($\varphi>180^{\circ}$) at a slightly smaller radius
($r\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle{\displaystyle<}}}{{\sim}}$}1$).
This feature that is not visible in the isothermal case is caused by including
the thermodynamics of the disc. Let us consider an adiabatic situation just
after the planet has been inserted into the disc, and follow material on its
horseshoe orbit (in the co-rotating frame) as it makes a turn from the outer
disc ($r>1,\varphi>180^{\circ}$) to the inner ($r<1$). The radial temperature
and density gradient imply for our ideal gas law a gradient in the entropy
function $S$ in the disc through
$S\propto\frac{p}{\rho^{\gamma}}.$
As shown above, in our simulations we find for the surface density
$\Sigma\propto r^{-1/2}$ as due to the assumption of a constant viscosity, and
the midplane temperature follows $T\propto r^{-1.7}$. Due to this gradient in
$S$ a parcel (coming from outside) has in our case a smaller entropy than the
inner disc which it is entering. Now the entropy remains constant on its path,
due to the adiabatic assumption. Additionally, dynamical equilibrium requires
that the pressure of the parcel does not change significantly upon its turn,
and entropy conservation then implies that the density has to increase. At the
same time the density ‘behind’ the planet ($\varphi<180^{\circ}$ and
$r\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle{\displaystyle>}}}{{\sim}}$}1$)
will be lowered by similar reasoning. Both components produce a positive
contribution to this entropy-related corotation torque that acts on the
planet, and which adds to the negative Lindblad torque and the positive
vortensity-related corotation torque. In truly adiabatic discs this effect
will disappear after a few libration times (Baruteau & Masset 2008; Kley &
Crida 2008) because the material, being within the horseshoe region, will
start interacting with itself, and the density and entropy will be smeared out
due to the mixing, leading to the described torque saturation process. Adding
radiative diffusion will prevent this and keep the entropy-related torques
unsaturated, and a non-zero viscosity is also required.
Figure 12: Surface density (upper two panels) and temperature in the
equatorial plane (lower two) for fully radiative simulations at 100 planetary
orbits. Displayed are results for the shallowest and deepest potential. Upper
panels refer to the $\epsilon$-potential with $r_{sm}=0.8$, and lower to the
cubic-potential with $r_{sm}=0.5$, respectively.
In this fully radiative case the temperature within the Roche radius of the
planet has also increased substantially due to compressional heating of the
gas (lower two panels in Fig. 12). In addition, the temperature in the spiral
arms is increased as well due to shock heating.
Figure 13: Radial density (top) and temperature (bottom) distribution in the
equator along a ray through the location of the planet for all 4 planetary
potentials used, for the fully radiative case.
The density and temperature runs in the disc midplane along a radial line at
$\varphi=180^{\circ}$ cutting through the planet are displayed in Fig. 13. As
in the isothermal case, deeper potentials lead to higher densities within the
Roche lobe. The increase is somewhat lower because now the temperature is
higher as well due to the compression of the material. The larger pressure
lowers the density in comparison to the isothermal case. Interesting is that
the maximum temperature is substantially higher than in the ambient disc even
for this very low mass planet of $20M_{\rm earth}$. Considering accretion onto
the planet the increase in temperature might be even stronger due to the
expected accretion luminosity.
Figure 14: Specific torques acting on a 20 $M_{\rm earth}$ planet for
different numerical potentials in the fully radiative case. Top: Evolution of
total torque with time. Bottom: Radial variation of the specific torque for
$t=80$ orbits.
Figure 15: Specific torque acting on the planet using different grid
resolutions for the fully radiative case. In all cases the cubic potential
with $r_{sm}=0.5$ has been used. Top: Evolution of total torque with time.
Bottom: Radial variation of the specific torque density at $t=50$ orbits.
### 4.4 Torque analysis for the radiative case
In the upper panel of Fig. 14 we display the time evolution of the total
specific torque acting on a 20 $M_{\rm earth}$ planet for the full radiative
case. In contrast to the isothermal situation, all four potentials result now
in a positive total torque acting on the planet. As in the previous isothermal
runs, the torques reach their maximum shortly after the onset of the
simulations (between $t\approx 10-20$) and then settle toward their final
value. In the corresponding isothermal case with $H/r=0.037$ the difference
between the initial positive unsaturated torque and the final saturated value
has been very pronounced (see Fig. 11). In contrast, in this fully radiative
case the inclusion of energy diffusion and the subsequent radiative cooling of
the disc will prevent saturation of the entropy-related corotation torque,
resulting in a positive equilibrium torque. Very similar results have been
found previously in the fully radiative regime in 2D simulations (Kley & Crida
2008). It is important to notice, that the two cubic-potentials (which are
more realistic in the 3D case) yield very similar results. The more
unrealistic $\epsilon$-potentials show rather strong deviations because, due
to their extended smoothing of the potential, they tend to weaken in
particular the corotation torques which originate in the close vicinity of the
planet. In the lower panel of Fig. 14 the radial torque distribution is
displayed for the same 4 potentials. In comparison to the corresponding plot
for the isothermal $H/r=0.037$ (see Fig. 11) we notice that the regular
Lindblad part is slightly reduced in the radiative case due to the larger
sound speed.
Figure 16: Radial variation of the specific torque acting on the planet using
different thermodynamical disc models. In all cases the cubic potential with
$r_{sm}=0.5$ and standard resolution have been used. The adiabatic model at
$t=10$ corresponds to the time where the corresponding total torque has its
maximum. The models at $t=80$ have all reached their equilibria.
Additionally, clearly seen is the additional positive contribution just inside
$r=1$ which appears to be responsible for the torque reversal. This feature is
caused by an asymmetric distribution of the density in the very vicinity of
the planet, see also Fig. 17 below. It pulls the planet gravitationally ahead,
increasing its angular momentum, leading to a positive torque. Above we argued
that this effect may be due to the entropy-related corotation torque of
material moving on horseshoe orbits (see also Baruteau & Masset 2008). Due to
the symmetry of the problem one might expect a similar feature caused by the
material moving from inside out. However, there is no sign of this present in
the lower panel of Fig. 14. To analyse this asymmetry, we performed additional
simulations varying the grid resolution and disc thermodynamics. That the
feature is not caused by lack of numerical resolution is demonstrated in Fig.
15, where results obtained with two different grids are displayed. Both models
show the same characteristic torque enhancement just inside the planet. In
Fig. 16 we compare the radial torque density of the isothermal and a new
adiabatic model for $H/r=0.037$ with the fully radiative model, all for the
cubic potential with $r_{sm}=0.5$ at intermediate resolution. For the
adiabatic case we show $\Gamma(r)$ at two different times. The first at $t=10$
when the torques are unsaturated, and the second at $t=80$ after saturation
has occurred. Please note, that the isothermal and adiabatic models start from
the same initial conditions (locally isothermal), while the radiative model
starts from the radiative equilibrium without the planet. While the adiabatic
model at $t=10$ shows signs of the enhanced torque just inside the planet,
there is no sign of a similar feature at a radius just outside of the planet.
Hence, this asymmetry of the entropy-related corotation torque is visible in
both the adiabatic and radiative case. Outside of the planet the adiabatic
model and the radiative agree very well as $H/r$ is similar, while the
isothermal model deviates due to the different sound speed. Whether the
location of the maximum in the torque density is identical in the radiative
and adiabatic case is hard to say from these simulations because in 3D
adiabatic runs the peak appears to be substantially broader with respect to
corresponding 2D cases. It has been argued by Baruteau & Masset (2008) that it
should occur exactly at the corotation radius, which is shifted (very
slightly) from the planet’s location due to the pressure gradient in the disc.
In our radiative simulations it seems that the maximum is slightly shifted
inwards, an effect which may be caused by adding radiative diffusion to the
models and consider discs in equilibrium. An issue that certainly needs
further investigation.
Figure 17: Results of a 2D fully radiative model using a resolution of
$512\times 1536$ gridcells. The gray dot indicates the location of the planet,
and the curve its Roche lobe. The solid black lines show the streamlines. Top:
Perturbed surface density with respect to the case without an embedded planet.
The values are scaled as $\Sigma^{1/2}$. Bottom: The net torque acting on the
planet caused by the mass in each individual gridcell using the prescribed
smoothed torque cutoff function with $b=0.8$, see explanation in text. The
values are scaled as $(\tilde{\Gamma}^{\pm})^{1/2}$.
In Fig. 17 we present additional results of supporting 2D simulations for the
fully radiative case, as shown for lower resolution in Kley & Crida (2008).
All physical parameter are identical to our 3D fully radiative case. The top
panel shows the surface density distribution next to the planet. Seen is the
density enhancement just inside and ahead of the planet, and some indication
for a lowering outside and behind. Please note, that the planet moves counter-
clockwise into the positive $\phi$-direction. i.e. upward in Fig. 17. In the
lower panel we display for each gridcell the net torque
($\tilde{\Gamma}^{\pm}$) acting on the planet. It is constructed by adding
each cell’s individual contribution to the torque and that of the symmetric
cell with respect to the planet location, i.e.
$\tilde{\Gamma}^{\pm}_{i,j}=\pm\left[\Gamma(r_{i},\phi_{planet}+\phi_{j})+\Gamma(r_{i},\phi_{planet}-\phi_{j})\right].$
(20)
Hence, in absolute values the bottom half of the plot (with
$\phi<\phi_{planet}$) resembles exactly the top half ($\phi>\phi_{planet}$).
The colours are chosen such that blue refers to negative
$\tilde{\Gamma}^{\pm}_{i,j}$ and yellow/red to positive values. The signs of
$\tilde{\Gamma}^{\pm}_{i,j}$ are chosen such that the top left and lower right
quadrant have the correct sign ($+$) and the other two are just reversed. Due
to this redundancy in the plot, only the upper left and lower right quadrant
should be taken into account to estimate global effects. The mirroring process
at the $\phi=\pi$-line (with the mirroring of the colour scale) allows an easy
evaluation and comparison of the individual contributions. One can notice that
the net torque will be positive due to excess material just ahead and inside
of the planet. From the plot it is also clear that there exists indeed an
asymmetry of the torques induced by horseshoe material coming from outside-in
versus material turning inside-out. In the figure, there is only a weak
indication of a marginal positive contribution just below the planet. In
additional simulations for purely adiabatic discs with different (positive and
negative) entropy gradients which have either constant density or temperature
it has become apparent that the asymmetry is caused by the entropy gradient.
In the case of a negative entropy gradient (as in our fully radiative model)
the positive excess torque comes from inside/ahead the planet, while for a
positive gradient the negative excess torque comes from outside/behind the
planet. Whether the maximum of $\Gamma(r)$ lies at corotation (Baruteau &
Masset 2008) or is slightly shifted when radiative effects are considered may
deserve further studies.
Figure 18: Perturbed entropy (Top) and perturbed density (Bottom) for the 2D
fully radiative equilibrium model using a resolution of $512\times 1536$
gridcells. The values are scaled as $\Sigma^{1/2}$ and $S^{1/2}$,
respectively.
In Fig. 18 we show the perturbed entropy and density in the 2D fully radiative
model in equilibrium, for a larger domain. Caused by the flow in the horseshoe
region, there is an entropy minimum for larger $\phi$ inside of the planet,
and a maximum for lower $\phi$ outside, for $r+r_{p}$. Both lie inside the
horseshoe region and close to the separatrix. The overall entropy distribution
is very similar to that found by Baruteau & Masset (2008) for adiabatic discs
shortly after the insertion of the planet. Due to the included radiative
diffusion this effect does not saturate in our case, and we clearly support
their findings even for the long term evolution. The disturbed entropy shows
in fact a slight asymmetry (in amplitude) with respect to the planet, that
reflects back on to to the density distribution (bottom panel).
### 4.5 Planets with different planetary masses
Following the results obtained in the previous sections we adopt now the cubic
$r_{sm}=0.5$ planetary potential assuming that it is closest to reality, and
study the effects of planets with various masses in fully radiative discs.
Starting from the 2D radiative equilibrium state (see section 4.1) we now
place planets with masses ranging from $5$ up to $100$ $M_{\rm earth}$ in the
initially axisymmetric 3D disc. The numerical parameters for these simulations
are identical to those discussed above.
In recent 2D simulations of radiative discs with embedded low mass planets the
torque acting on the planet depends on the planetary mass in such a way that
for planets with a size smaller than about $40$ earth masses the total torque
is positive implying outward migration (Kley & Crida 2008). For larger masses
the forming gap reduces the contribution of the corotation torques, and the
results of the radiative simulations approach those of the fixed temperature
(locally isothermal) runs. Our 3D simulations show indeed very similar results
for planets in this mass regime, see Fig. 19. Planets in the isothermal regime
migrate inward with a torque proportional to the planet mass squared, as
predicted for low mass planets undergoing Type-I-Migration. Note, that we use
in these models the temperature distribution for a fixed $H/r=0.037$. The
values for the three lowest mass planets (with $5,10,15M_{\rm earth}$) are not
as accurate due to the insufficient grid resolution, remember the ’kink’ in
Fig. 11 which refers to $20M_{\rm earth}$ at standard resolution. For the
fully radiative disc the planets up to about $33M_{\rm earth}$ experience a
positive torque, while larger mass planets migrate inward, due to the negative
torque acting on them.
When comparing the 3D torques to the corresponding 2D values as obtained by
Kley & Crida (2008) for the same disc mass and opacity law, we note two
differences: i) The absolute magnitudes of the torques in the radiative case
are enhanced in the 3D simulations with respect to the corresponding 2D
results, resulting in even faster outward migration of the planets. This
result can be explained by the reduced temperature (i.e. vertical thickness)
of the 3D disc with respect to the 2D counterpart (cf. Fig. 3), as a reduction
in $H$ typically increases the torques (Tanaka et al. 2002). ii) The turnover
mass from positive to negative torques is reduced in the 3D simulations. This
effect is caused again by the reduced disc thickness, as now the onset of gap
formation ($R_{H}\approx H$) occurs for smaller planetary masses. The
different form of the potential and the softening length may also play a role
in explaining some of the differences observed between the 3D and 2D results.
Figure 19: Specific torques acting on planets of different masses in the fully
radiative (blue crosses) and isothermal (red plus signs) regime. Note, that
the isothermal models are run for a fixed $H/r=0.037$. All torque values are
displayed at a time when the equilibrium has been reached. Figure 20: Radial
torque distribution in equilibrium for various planet masses. The vertical
dotted line indicates the location of the maximum.
Finally, in Fig. 20 we show that the position of the maximum of the radial
torque density is independent of the planet mass, and is therefore a result of
the underlying disc physics.
## 5 Summary
We have investigated the migration of planets in discs using fully three-
dimensional numerical simulations including radiative transport using the code
NIRVANA. For this purpose we have presented and described our implementation
of implicit radiative transport in the flux-limited diffusion approximation,
and secondly our new FARGO-implementation in full 3D.
Before embedding the planets we studied the evolution of axisymmetric,
radiative accretion discs in 2D. Starting with an isothermal disc model having
a fixed $H/r=0.05$, we find that for our physical disc parameter the inclusion
of radiative transport yields discs that are thinner ($H/r=0.037$ at $r=1$).
We note that in the isothermal case the disc thickness is a chosen input
parameter, while in the fully radiative situation it depends on the local
surface density and the chosen viscosity and opacity. Interesting is here the
direct comparison to the equivalent 2D models using the same viscosity,
opacity and disc mass (Kley & Crida 2008), as displayed in Fig. 3. Here, our
new 3D disc yields smaller temperatures (by a factor of 0.6-0.7) than the 2D
runs. Since the 2D simulations have to work with vertically averaged
quantities, it will be interesting whether it might be possible to adjust
those as to yield results in better agreement to our 3D results.
Concerning planetary migration we have confirmed the occurrence of outward
migration for planetary cores in radiative discs. As noticed in previous
research, the effect is driven by a radial entropy gradient across the
horseshoe region in the disc, that is maintained by radiative diffusion. Our
results show that planets below the turnover mass of about $m\approx 33M_{\rm
earth}$ migrate outward while larger masses drift inward. The reduced
temperature in the 3D versus 2D runs has direct influence on the magnitude of
the resulting torques acting on the planet. As the disc is thinner in 3D the
resulting torques, corotation as well as Lindblad, are also stronger. The
turnover mass from outward to inward migration is slightly reduced as well for
the 3D disc since the smaller vertical thickness allows for gap opening at
lower planet masses. Due to the reduced temperature in the 3D case, the spiral
waves have a slightly smaller opening angle compared to the isothermal case.
Another interesting, partly numerical, issue that we have addressed concerns
the influence that the smoothing of the planetary potential has on the density
structure in the vicinity of the planet and the speed of migration. We have
compared the standard $\epsilon$-potentials in contrast with so-called cubic-
potentials. The results indicate, that a deeper (cubic) potential results in a
higher density inside the planet’s Roche lobe for isothermal, as well as
radiative discs. Since the potential depth influences the density in the
immediate vicinity of the planet the resulting torques show some dependence on
the chosen smoothing length. We note that for the more realistic cubic-
potential, changing the smoothing from $0.8$ to $0.5$ does not alter the
results for the fully radiative simulations considerably, and runs at
different numerical resolutions have indicated numerical convergence. Hence,
we believe that this value is suitable for performing planet disc simulations
in 3D. The usage of an $\epsilon$-potential cannot be recommended in 3D.
Outside of the planet’s Roche lobe one can hardly notice a difference in the
density structure. Since the cubic-potential agrees with the true planetary
potential outside of the smoothing-length $r_{sm}$, it is of course desirable
to choose this transition radius as small as possible, but the achievable
numerical resolution always will set a lower limit. We found $r_{sm}=0.5$ a
suitable value for our grid resolution and used that in our parameter studies
for different planet masses. We note, that independent of the chosen form of
the potential the outward migration of planetary cores seems to be a robust
result for radiative discs. As the magnitude (and direction) of the effect
depends on the viscosity and opacity, further studies, investigating different
radial locations in the disc, will be very interesting.
The outward migration of planet embryos with several earth masses certainly
represents a solution to the too rapid inward migration found in this mass
regime of classical type-I migration. Growing planets can spend more time in
the outer disc regions and move then later via type-II migration towards the
star. On the other hand it may be difficult to reconcile this finding with the
presence of the discovered Neptune-mass planets that reside closer to the
central star ($a\approx 0.1$AU), but still too far away to be ablated by
stellar irradiation.
As seen in our simulations, parts of the disc can display convection due to
the form of the opacity law used. It will be certainly interesting in the
future to analyse what influence the convective motions have on the migration
properties of the embedded protoplanets. Additionally, fully radiative 3D-MHD
simulations are definitely required to judge the efficiency of this process in
turbulent discs.
###### Acknowledgements.
Very fruitful discussions with Aurélien Crida and Fredéric Masset are
gratefully acknowledged. H. Klahr and W. Kley acknowledge the support through
the German Research Foundation (DFG) through grant KL 650/11 within the
Collaborative Research Group FOR 759: The formation of Planets: The Critical
First Growth Phase. B. Bitsch has been sponsored through the German D-grid
initiative. The calculations were performed on systems of the Computer centre
of the University of Tübingen (ZDV) and systems operated by the ZDV on behalf
of bwGRiD, the grid of the Baden Württemberg state. Finally, we gratefully
acknowledge the very helpful and constructive comments of an anonymous referee
that inspired us to perform more detailed analysis.
## Appendix A Fargo algorithm
Multi-dimensional simulations of accretion discs that include the
$\varphi$-direction typically suffer from severe timestep limitations. This is
due to the fact that the azimuthal velocity $u_{\varphi}$ is Keplerian and
falls off with radius. Hence, the innermost rings determine the maximum
timestep allowed even though the region of interest lies much further out. One
suggestion to resolve this issue is given by the FARGO algorithm which stands
for “Fast Advection in Rotating Gaseous Objects” (Masset 2000a). It has
originally been developed for 2D disc simulations in a cylindrical coordinate
system, for details of the implementation see Masset (2000a, b). Here, we
briefly describe our extension to three spatial dimensions in spherical polar
coordinates.
The basic method relies on a directional splitting of the advection part,
where first the radial and meridional (in $\theta$ direction) advection are
performed in the standard way. To calculate the azimuthal part we follow
Masset (2000a) and split the angular velocity into three parts: From the
angular velocity of each grid cell
$\omega_{i,j,k}=(u_{\varphi})_{i,j,k}/r_{i}$ first an average angular velocity
$\bar{\omega}_{i}$ is calculated for each radial ring $i$, which is obtained
here by averaging over the azimuthal (index $k$) and vertical (index $j$)
direction
$\bar{\omega}_{i}=\frac{1}{N_{\varphi}\,N_{\theta}}\,\sum_{j,k}\omega_{i,j,k}$
(21)
where the summation runs of all azimuthal and meridional gridcells, and
$N_{\varphi},N_{\theta}$ denote the number of these gridcells, respectively.
We note, that the summation over the vertical direction with index $j$ is not
required at this point. In our case, for a thin disc where the angular
velocity does not vary much with height, the vertical averaging simplifies
matters somewhat. From this, one calculates an integer-valued shift quantity
$n_{i}={\tt Nint}\left(\bar{\omega}_{i}\Delta t/\Delta\varphi\right),$ (22)
where Nint denotes the nearest integer function. This corresponds to a
transport by the angular ’shift velocity’
$\omega_{i}^{SH}=n_{i}\frac{\Delta\varphi}{\Delta t}.$ (23)
Then we calculate the constant residual velocity of each ring
$\omega_{i}^{cr}=\bar{\omega}_{i}-\omega_{i}^{SH},$ (24)
and finally the residual velocity for each individual gridcell
$\omega_{i,j,k}^{res}=\omega_{i,j,k}-\bar{\omega}_{i}.$ (25)
Rewritten, we find for the angular velocity the following expression
$\omega_{i,j,k}\,=\,\omega_{i,j,k}^{res}+\omega_{i}^{cr}+\omega_{i}^{SH}$ (26)
The advection algorithm in the $\varphi$-direction proceeds now in three
steps. In the first two steps all quantities are advected using the standard
advection routine with the transport velocities $\omega_{i}^{cr}$ and
$\omega_{i,j,k}^{res}$ and then all quantities are shifted by the integer
values $n_{i}$ in each ring $i$ which corresponds to a transport velocity
$\omega_{i}^{SH}$. Using this splitting, the transport velocities in the
advection part are given by the two residual velocities $\omega_{i}^{cr}$ and
$\omega_{i,j,k}^{res}$, which are typically much smaller than
$\omega_{i,j,k}$. Hence, the time step limitation for the azimuthal direction
is determined by the local variation from the mean azimuthal flow in the disc
which is typically much smaller than the Keplerian value. In our case of a 3D
disc the time step criterion is first given by the normal CFL-criterion as
presented for example in Stone & Norman (1992) where the angular velocity
$\omega_{i,j,k}$ is just replaced by the residual cell values
$\omega_{i,j,k}^{res}$ and $\omega_{i}^{cr}$. This change provides the major
reduction in the transport velocity and a substantial increase in the time
step size. An additional time step limitation is given by the requirement that
the shift should not disconnect two neighbouring grid cells in the radial and
in the meridional direction (Masset 2000a). Here, this additional limit on the
time step reads
$\Delta
t_{shear}=0.5\,\min_{i,j,k}\left\\{\frac{\Delta\varphi}{\left|\omega_{i,j,k}-\omega_{i-1,j,k}\right|},\,\frac{\Delta\varphi}{\left|\omega_{i,j,k}-\omega_{i,j-1,k}\right|}\right\\}$
(27)
The second restriction is only necessary in the case, where the above vertical
averaging in Eq. 21 has not been performed. The sequencing of the advection
sweeps in the FARGO algorithm has to be such that the azimuthal sweep comes at
the end, hence in our simulations we use always: radial, meridional, and
finally azimuthal.
In a staggered mesh code such as our NIRVANA code, that is essentially based
on the ZEUS method, an additional complication arises in the straightforward
application of the FARGO method, due to the fact that the velocity variables
are located at the cell interfaces and not at the centres. Hence, the
corresponding ’momentum cells’ for the radial and meridional momenta ($\rho
u_{r}$ and $\rho ru_{\theta}$) are shifted with respect to the standard
density cells by half a gridcell in the radial or meridional direction,
respectively. To apply the FARGO method one has first to split all the
momentum cells in two halves, use the algorithm outlined above on each of the
halves, and then combine them again afterwards to calculate from the updated
momenta the new velocities on the interfaces. This leads of course to an
overhead in the simulation cost which is counterbalanced however by the much
larger time step.
## Appendix B Radiative transport
Here, we outline briefly the method to solve the flux-limited diffusion
equation in 3D. Radiative transport is treated as a sub-step of the
integration procedure. In equilibrium viscous heating is balancing radiative
diffusion and to ensure this also numerically we incorporate the dissipation
into this sub-step. Using the appropriate parts of the energy equation (7) and
the flux (8) we obtain a diffusion equation for the gas temperature.
$\frac{\partial T}{\partial
t}=\frac{1}{c_{\mathrm{v}}\rho}\,\left[\nabla\cdot\,D\nabla T+Q^{+}\right]$
(28)
where the diffusion coefficient is given by
$D=\frac{\lambda c4a_{R}T^{3}}{\rho(\kappa+\sigma)},$ (29)
and $Q^{+}$ denotes the viscous dissipation that is added to the system. The
flux-limiter $\lambda$ depends on the local physical state of the gas and
approaches $\lambda=1/3$ in the optically thick parts and reduces the flux to
$F=ca_{R}T^{4}$ in the optically thin parts. Here we use an expressions for
$\lambda$ as given in Kley (1989).
A straight forward finite difference form of Equation (28) in Cartesian
Coordinates is given by
$\displaystyle\frac{T^{n+1}_{i,j,k}-T^{n}_{i,j,k}}{\Delta t}$ $\displaystyle=$
$\displaystyle\frac{1}{(c_{\mathrm{v}}\rho)_{i,j,k}}\left[\right.$ (30)
$\displaystyle\frac{1}{\Delta
x}\left(\bar{D}^{x}_{i+1,j,k}\frac{T_{i+1,j,k}-T_{i,j,k}}{\Delta
x}-\bar{D}^{x}_{i,j,k}\frac{T_{i,j,k}-T_{i-1,j,k}}{\Delta x}\right)$
$\displaystyle+$ $\displaystyle\frac{1}{\Delta
y}\left(\bar{D}^{y}_{i,j+1,k}\frac{T_{i,j+1,k}-T_{i,j,k}}{\Delta
y}-\bar{D}^{y}_{i,j,k}\frac{T_{i,j,k}-T_{i,j-1,k}}{\Delta y}\right)$
$\displaystyle+$ $\displaystyle\frac{1}{\Delta
z}\left(\bar{D}^{z}_{i,j,k+1}\frac{T_{i,j,k+1}-T_{i,j,k}}{\Delta
z}-\bar{D}^{z}_{i,j,k}\frac{T_{i,j,k}-T_{i,j,k-1}}{\Delta
z}\right)\left.\right]$
In orthogonal curvilinear coordinates additional geometry terms have to be
added in the above equation. Here $\bar{D}^{x}_{i,j,k}$ denotes
$\bar{D}^{x}_{i,j,k}=\frac{1}{2}\left(D_{i,j,k}+D_{i-1,j,k}\right)$ (31)
and so forth.
The grid structure from which Eq. (30) follows is outlined in the two-
dimensional case in Fig.21. One has to keep in mind that the temperature
(being a scalar) is defined in the cell centre, while the values of
$\bar{D}^{x}_{i,j,k}$ are defined at the cell interfaces.
Figure 21: The grid structure used in a staggered grid code for a two-
dimensional example. Shown is the cell $i,j$ where the coordinates of the cell
centre [$x^{c}_{i},y^{c}_{j}$] is given by
[$1/2(x_{i,j}+x_{i+1,j}),1/2(y_{i,j}+y_{i,j+1})$]. The temperature $T_{i,j}$
is located at the cell centre where also the diffusion coefficient $D$ is
defined. The averaged diffusion coefficients $\bar{D}$ are defined at the cell
interfaces.
In Eq. (30) no time levels are specified on the right hand side. For explicit
differencing the time level should be $t^{n}$ such that the new temperature on
the left $T^{n+1}$ is entirely given by the old values $T^{n}$ at time
$t^{n}$. This might lead to very small timesteps since the timestep limitation
is approximately given by
$\Delta t\leq\min_{i,j,k}\left(\frac{\Delta x^{2},\Delta y^{2},\Delta
z^{2}}{\tilde{D}_{i,j,k}}\right)$ (32)
where $\tilde{D}$ is given by $D/(\rho c_{v})$.
Hence, often an implicit version of the equation has to be used, where all the
temperature values $T_{i,j,k}$ on the r.h.s. are evaluated at the new time
$t^{n+1}$ or an arithmetic mean between new and old times. Even though the
diffusion coefficients may depend on temperature, we always evaluate those at
the old time $t^{n}$. Otherwise this would lead to a non-linear matrix
equation.
Collection all the terms in eq. (30) this leads to a linear system of
equations with the form
$\displaystyle
A^{x}_{i,j,k}T_{i-1,j,k}+C^{x}_{i,j,k}T_{i+1,j,k}+A^{y}_{i,j,k}T_{i,j-1,k}$
$\displaystyle+$ $\displaystyle C^{y}_{i,j,k}T_{i,j+1,k}$ $\displaystyle
A^{z}_{i,j,k}T_{i,j,k-1}+C^{z}_{i,j,k}T_{i,j,k+1}+B_{i,j,k}T_{i,j,k}$
$\displaystyle=$ $\displaystyle R_{i,j,k}$ (33)
where the superscript $n+1$ has been omitted on the left hand side. The
coefficients $A^{x}_{i,j,k}$ to $C^{z}_{i,j,k}$ can be obtained
straightforwardly from Eq. (30). The right hand side is given by
$R_{i,j,k}=T^{n}_{i,j,k}+\frac{1}{(c_{v}\rho)_{i,j,k}}\,Q^{+}_{i,j,k}$
Written in matrix notation Eq. (33) reads
${\sf M}\@vec{T}^{n+1}=\@vec{R}$ (34)
Obviously the matrix ${\sf M}$ is a sparse matrix with a banded structure.
Usually ${\sf M}$ is diagonally dominant but in situations with extended
optically thin regions this property will be lost.
Equation (34) can in principle be solved by any linear equation package. For
simplicity and testing purposes we work presently with a standard SOR solver.
Using an optimised relaxation parameter $\tilde{\omega}$ we need about 130
iterations per timestep in the initial phase which is far from equilibrium and
only 80 iterations at later times near equilibrium.
The radiation module of the code has been tested extensively in different
coordinate systems in Bitsch (2008), and we have compared our new results on
radiative viscous discs obtained with the 3D version of NIRVANA in detail with
those of the existing 2D code RH2D.
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|
arxiv-papers
| 2009-08-13T10:19:39 |
2024-09-04T02:49:04.628036
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Willy Kley (1), Bertram Bitsch (1), Hubert Klahr (2) ((1) University\n of Tuebingen, (2) Max-Planck Institue of Astronomy)",
"submitter": "Willy Kley",
"url": "https://arxiv.org/abs/0908.1863"
}
|
0908.1899
|
# Relativistic coupled-cluster calculations of 20Ne, 40Ar, 84Kr and 129Xe:
correlation energies and dipole polarizabilities
B. K. Mani and D. Angom Physical Research Laboratory, Navarangpura-380009,
Gujarat, India
###### Abstract
We have carried out a detailed and systematic study of the correlation
energies of inert gas atoms Ne, Ar, Kr and Xe using relativistic many-body
perturbation theory and relativistic coupled-cluster theory. In the
relativistic coupled-cluster calculations, we implement perturbative triples
and include these in the correlation energy calculations. We then calculate
the dipole polarizability of the ground states using perturbed coupled-cluster
theory.
###### pacs:
31.15.bw, 31.15.ve, 31.15.ap, 31.15.am
## I Introduction
High precision atomic experiments are at the core of several investigations
into fundamental physics and high end technology developments. Selected
examples are search for electric dipole moment (EDM) griffith-09 and
observation of parity nonconservation tsigutkin-09 . These endeavours, in
general, require precision atomic theory calculations to analyse the results
and understand systematics. The challenging part of precision atomic structure
and properties calculations is obtaining accurate wave functions. In the case
of high $Z$ atoms, the need to incorporate relativity adds to the difficulty.
A systematic study of the correlation energy is one of the possible methods to
test the accuracy of the atomic wave function. In this paper, we report the
results of correlation energy calculations of inert gas atoms Ne, Ar, Kr and
Xe. For this we employ many body perturbation theory (MBPT) and calculate the
second order correlation energy. A comparative study reveals the changing
nature of electron correlations in the group. Our interest in particular is
Xe, which is a candidate for EDM experiments rosenberry-01 and theoretical
calculations dzuba-02 .
For completeness, in the presentation of the paper, we give an overview of
MBPT. It is a powerful theory and forms the basis of other more sophisticated
and elaborate many-body methods. However, one drawback of MBPT is the
complexity of expressions at higher orders. This renders the theory
inappropriate to incorporate strong correlation effects in heavy atoms. Yet,
at lower orders its simplicity makes it an ideal choice to test and optimize
basis sets. We use this insight to generate basis sets for coupled-cluster
calculations.
The coupled-cluster theory, first developed in nuclear many body physics
coester-58 ; coester-60 , is considered the most accurate many body theory. In
recent times, it has been used with great success in nuclear hagen-08 , atomic
nataraj-08 ; pal-07 , molecular isaev-04 and condensed matter bishop-09
calculations. It is equivalent to incorporating electron correlation effects
to all orders in perturbation. The theory has been used in performing high
precision calculations to study the atomic structure and properties. These
include atomic electric dipole moments nataraj-08 ; latha-09 , parity
nonconservation wansbeek-08 , hyperfine structure constants pal-07 ; sahoo-09
and electromagnetic transition properties thierfelder-09 ; sahoo-09a . In the
present work we use the relativistic coupled-cluster singles and doubles
(CCSD) approximation to calculate correlation energy and dipole polarizability
of inert gas atoms Ne, Ar, Kr and Xe. In the dipole polarizability
calculations, the dipole interaction Hamiltonian is introduced as a
perturbation. A modified theory, recently developed latha-08 , incorporates
the perturbation within the coupled-cluster theory. This theory has the
advantage of subsuming correlation effects more accurately. The results
provide a stringent test on the quality of the wave functions as the dipole
polarizability of inert gas atoms are known to high accuracy hohm-90 . Based
on the CCSD method, we also estimate the third order correlation energy.
Further more, perturbative triples are incorporated in the coupled-cluster
calculations.
In the paper we give a brief description of MBPT in Section.II and discuss the
method to calculate electron correlation energy to the second and third order
in residual Coulomb interaction. The coupled-cluster theory is described in
Section.III, where we also discuss linearized coupled-cluster theory and
correlation energy calculation using coupled-cluster theory. Then the
inclusion of approximate triples to the correlation energy is explained and
illustrated. Section.IV is a condensed description of the perturbed coupled-
cluster theory and provide details of how to incorporate the effects of an
additional perturbation to the residual Coulomb interaction in atomic systems.
Results are presented and discussed in Section.V. In the paper, all the
calculations and mathematical expressions are in atomic units
($e=\hbar=m_{e}=1$).
## II Correlation Energy from MBPT
In this section, to illustrate the stages of our calculations and compare with
coupled-cluster theory, we provide a brief description of many-body
perturbation theory. Detailed and complete exposition of the method, in the
context of atomic many-body theory, can be found in ref lindgren-85 .
The Dirac-Coulomb Hamiltonian $H^{\rm DC}$ is an appropriate choice to
incorporate relativistic effects in atoms. This is particularly true for heavy
atoms, where the relativistic effects are large for the inner core electrons
due to the high nuclear charge. As the name indicates, $H^{\rm DC}$ is fully
relativistic for the one-body terms only. For an $N$ electron atom
$H^{\rm
DC}=\sum_{i=1}^{N}\left[c\bm{\alpha}_{i}\cdot\bm{p}_{i}+(\beta-1)c^{2}-V_{N}(r_{i})\right]+\sum_{i<j}\frac{1}{r_{ij}},$
(1)
where $\alpha_{i}$ and $\beta$ are the Dirac matrices. For the nuclear
potential $V_{N}(r)$, we consider the nucleus as finite size and the volume
effects are taken into account by modeling the nuclear charge distribution as
a Fermi two-parameter distribution. Then the nuclear density is
$\rho_{\rm nuc}(r)=\frac{\rho_{0}}{1+e^{\frac{(r-c)}{a}}},$ (2)
here, $a=t4\ln 3$. The parameter $c$ is the half-charge radius, that is
$\rho_{\rm nuc}(c)=\rho_{0}/2$ and $t$ is the skin thickness. The eigen states
of $H^{\rm DC}$ are $|\Psi_{i}\rangle$, the correlated many-particle states
with eigenvalues $E_{i}$. The eigenvalue equation is
$H^{\rm DC}|\Psi_{i}\rangle=E_{i}|\Psi_{i}\rangle.$ (3)
It is however impossible to solve this equation exactly due to the relative
coordinates in the electron-electron Coulomb interaction. Many-body
perturbation theory is one approach which, starting from a mean field
approximation, incorporates the electron correlation effects systematically.
The starting point of perturbative scheme in many-body theory is to split the
Hamiltonian as
$H^{\rm DC}=H_{0}+V,$ (4)
where
$H_{0}=\sum_{i}[c\alpha_{i}.p_{i}+(\beta_{i}-1)c^{2}-V_{N}{r_{i}}+u(\bm{r}_{i})]$,
is the unperturbed or zeroth order Hamiltonian. It is the exactly solvable
part of the total Hamiltonian and correspond to independent particle model. In
this model, each electron is assumed to move independently of the others in an
average field arising from the nucleus and other electrons. The average field
of the other electrons is the Dirac-Fock central potential $u(\bm{r}_{i})$.
The remaining part of the electron-electron Coulomb interaction
$V=\sum_{i<j}^{N}\frac{1}{\bm{r}_{ij}}-\sum_{i}u(\bm{r}_{i})$, is the residual
Coulomb interaction. The purpose of any atomic many-body theory is to account
for this part as accurately as possible. The Hamiltonian $H_{0}$ satisfies the
eigenvalue equation
$H_{0}|\Phi_{i}\rangle=E_{i}^{0}|\Phi_{i}\rangle,$ (5)
where $|\Phi_{i}\rangle$ is a many-particle state and $E_{i}^{0}$ is the
eigenvalue. The eigenstates are generally Slater determinants, antysymmetrised
direct product of single particle states and $E_{i}^{0}$ is the sum of the
single particle energies. The difference between the exact and mean field
energy, $\Delta E_{i}=E_{i}-E_{i}^{0}$, is the correlation energy of the
$i^{\rm Th}$ state. At the single particle level, the relativistic spin
orbitals are of the form
$\psi_{n\kappa
m}(\bm{r})=\frac{1}{r}\left(\begin{array}[]{r}P_{n\kappa}(r)\chi_{\kappa
m}(\bm{r}/r)\\\ IQ_{n\kappa}(r)\chi_{-\kappa m}(\bm{r}/r)\end{array}\right),$
(6)
where $P_{n\kappa}(r)$ and $Q_{n\kappa}(r)$ are the large and small component
radial wave functions, $\kappa$ is the relativistic total angular momentum
quantum number and $\chi_{\kappa m}(\bm{r}/r)$ are the spin or spherical
harmonics. One representation of the radial components is to define these as
linear combination of Gaussian like functions and are referred to as Gaussian
type orbitals (GTOs). Then, the large and small components mohanty-89 ;
chaudhuri-99 are
$\displaystyle P_{n\kappa}(r)=\sum_{p}C^{L}_{\kappa p}g^{L}_{\kappa p}(r),$
$\displaystyle Q_{n\kappa}(r)=\sum_{p}C^{S}_{\kappa p}g^{S}_{\kappa p}(r).$
(7)
The index $p$ varies over the number of the basis functions. For large
component we choose
$g^{L}_{\kappa p}(r)=C^{L}_{\kappa i}r^{n_{\kappa}}e^{-\alpha_{p}r^{2}},$ (8)
here $n_{\kappa}$ is an integer. Similarly, the small component are derived
from the large components using kinetic balance condition. The exponents in
the above expression follow the general relation
$\alpha_{p}=\alpha_{0}\beta^{p-1}.$ (9)
The parameters $\alpha_{0}$ and $\beta$ are optimized for an atom to provide
good description of the atomic properties. In our case the optimization is to
reproduce the numerical result of the total and orbital energies. Besides GTO,
B-splines is another class of basis functions widely used in relativistic
atomic many body calculations johnson-88 . A description of B-splines with
details of implementation and examples are given in ref johnson-07 .
The next step in perturbative calculations is to divide the entire Hilbert
space of $H_{0}$ into two manifolds: model and complementary spaces $P$ and
$Q$ respectively. The model space has, in single reference calculation, the
eigen state $|\Phi_{i}\rangle$ of $H_{0}$ which is a good approximation of the
exact eigenstate $|\Psi_{i}\rangle$ to be calculated. The other eigenstates
constitute the complementary space. The corresponding projection operators are
defined as
$P=|\Phi_{i}\rangle\langle\Phi_{i}|\;\;\;\;{\rm
and}\;\;\;\;Q=\sum_{|\Phi_{j}\rangle\notin
P}|\Phi_{j}\rangle\langle\Phi_{j}|.$ (10)
The operator $P$ projects out the component of the exact eigenstate which lies
in the model space, $P|\Psi_{i}\rangle=|\Phi_{i}\rangle$ and $Q$ projects out
the component in the orthogonal space and $P+Q=1$. In the present paper, we
restrict to calculating the ground state $|\Psi_{0}\rangle$ of the closed
shell inert gas atoms. From here on, for a consistent description, the model
space consist of $|\Phi_{0}\rangle$.
The most crucial part of perturbation theory is to define a wave operator
$\Omega$ which operates on $|\Phi_{0}\rangle$ and transform it to
$|\Psi_{0}\rangle$ as
$|\Psi_{0}\rangle=\Omega|\Phi_{0}\rangle.$ (11)
Then, with the intermediate normalization approximation
$\langle\Psi_{0}|\Phi_{0}\rangle=1$, the wave operator is evaluated in orders
of the perturbation as $\Omega=\sum_{i=0}^{\infty}\Omega^{(i)}$ with
$\Omega^{(0)}=1$. It is possible to evaluate $\Omega^{(i)}$ iteratively or
recursively from the Bloch equation
$[\Omega,H_{0}]P=QV\Omega P-\chi PV\Omega P,$ (12)
where $\chi=\sum_{i=1}^{\infty}\Omega^{(i)}$ is the correlation operator. In
the second quantization notations, the wave operator and perturbation can be
expressed in terms of particle excitations. Then, the effect of correlation is
incorporated as linear combination of excited states. For simplification, in
the normal form the perturbation is separated as lindgren-85
$V=V_{0}+V_{1}+V_{2}$. These zero-, one- and two-body operators and are
defined as
$\displaystyle V_{0}$ $\displaystyle=\sum^{core}_{a}\langle
a|-u|a\rangle+\frac{1}{2}\sum^{core}_{ab}(\langle
ab|r^{-1}_{ab}|ab\rangle-\langle ba|r^{-1}_{ab}|ab\rangle),$ $\displaystyle
V_{1}$ $\displaystyle=\sum_{ij}\\{a_{i}^{\dagger}a_{j}\\}\langle
i|v|j\rangle,$ $\displaystyle V_{2}$
$\displaystyle=\frac{1}{2}\sum_{ijkl}\\{a_{i}^{\dagger}a_{j}^{\dagger}a_{l}a_{k}\\}\langle
ij|r_{12}^{-1}|kl\rangle.$ (13)
The operators $a^{\dagger}$ ($a$) create (annihilate) electrons in virtual
($p,q,r,s,....$ etc) and core ($a,b,c,d,....$ etc) shells. The indexes
$i,j,k,l$ etc are general representations of orbitals, it could either be core
or virtual. The operator $V_{0}$ acting on $|\Phi_{0}\rangle$ leaves it
unchanged, while $V_{1}$ and $V_{2}$ produce single and double excitations.
From these definition, the first-order wave operator can be separated as
$\Omega^{(1)}=\Omega^{(1)}_{1}+\Omega^{(1)}_{2}.$ (14)
Here, $\Omega^{(1)}_{1}$ and $\Omega^{(1)}_{2}$ are one- and two-body
components of the first order wave operator. The corresponding algebraic
expressions are
$\displaystyle\Omega^{(1)}_{1}$ $\displaystyle=$
$\displaystyle\sum_{ap}a^{\dagger}_{p}a_{a}\frac{\langle
p|v|a\rangle}{\epsilon_{a}-\epsilon_{p}},$ (15)
$\displaystyle\Omega^{(1)}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{abpq}a^{\dagger}_{p}a^{\dagger}_{q}a_{b}a_{a}\frac{\langle
pq|v|ab\rangle}{\epsilon_{a}+\epsilon_{b}-\epsilon_{p}-\epsilon_{q}}.$ (16)
We get singly (doubly) excited states $|\Phi_{a}^{p}\rangle$
($|\Phi{ab}^{pq}\rangle$) when $\Omega^{(1)}_{1}$ ($\Omega^{(1)}_{1}$ )
operates on the reference state $|\Phi_{0}$. The complexity of the expressions
increases with order of perturbation and is hard to manage. One powerful tool
in many-body perturbation theory is the diagrammatic evaluation of the
perturbation expansion. Then, the tedious algebraic evaluations are reduced to
a sequence of diagrams and equivalent algebraic expressions are derived with
simple rules. Even with this approach, it is computationally not practical to
go beyond fourth order.
Figure 1: The diagrammatic representations of the one- and two-body wave
operator. Lines with downward (upward) arrows represent core (virtual) single
particle states.
### II.1 Second-Order Correlation Energy
The ground state correlation energy $\Delta E_{0}$, as described earlier, is
the difference between the exact energy and the mean field energy. It is the
sum total of the energy corrections from all orders in perturbation. At the
$n^{\rm th}$ order, the energy correction $E_{\rm
corr}^{(n)}=\langle\Phi_{0}|V\Omega^{(n-1)}|\Phi_{0}\rangle$ and $\Delta
E_{0}=\sum_{n}E_{\rm corr}^{(n)}$. Then the second order correlation energy is
$E^{(2)}_{\rm
corr}=\langle\Phi_{0}|(V_{1}+V_{2})(\Omega^{(1)}_{1}+\Omega^{(1)}_{2})|\Phi_{0}\rangle.$
(17)
When Dirac-Fock orbitals are used, the diagonal matrix elements of $V_{1}$ are
the orbital energies and off diagonal matrix elements are zero. For this
reason, it does not contribute to the second-order energy. Then, the second
order correlation energy is
$E^{(2)}_{\rm corr}=\langle\Phi_{0}|V_{2}\Omega^{(1)}_{2}|\Phi_{0}\rangle.$
(18)
There are two diagrams arising from the above expression and these are as
shown in Fig.2.
Figure 2: MBPT diagrams arising from the second-order correlation energy.
In terms of algebraic expressions
$E^{(2)}_{\rm corr}=\sum_{abpq}\left[\frac{\langle ab|V_{2}|pq\rangle\langle
pq|V_{2}|ab\rangle}{\epsilon_{a}+\epsilon_{b}-\epsilon_{p}-\epsilon_{q}}-\frac{\langle
ba|V_{2}|pq\rangle\langle
pq|V_{2}|ab\rangle}{\epsilon_{a}+\epsilon_{b}-\epsilon_{p}-\epsilon_{q}}\right].$
(19)
In the above expression, the first and second terms on the right hand side are
the direct and exchange. Though the expression is fairly straight forward to
derive, we have given explicitly for easy reference while analysing the
results.
### II.2 Third-Order Correlation Energy
The diagrammatic representation of the second order wave operator
$\Omega^{(2)}$ consists of single, double, triple and quadruple excitations.
The singles are non-zero starting from the second order when Dirac-Fock
orbitals are used. And, the triples and quadruples begin to contribute from
this order. The triples consist of connected diagrams, whereas all the
quadruples are disconnected. The third order correlation energy is
$E^{(3)}_{\rm
corr}=\langle\Phi_{0}|(V_{1}+V_{2})(\Omega^{(2)}_{1}+\Omega^{(2)}_{2}+\Omega^{(2)}_{3}+\Omega^{(2)}_{4})|\Phi_{0}\rangle.$
(20)
The triple and quadruple excitations do not contribute as $V$ at the most can
contract with double excitations. For the same reason mentioned earlier, in
second order, $V_{1}$ also does not contribute. Then the third order
correlation energy is simplified to
$E^{(3)}_{\rm corr}=\langle\Phi_{0}|V_{2}\Omega^{(2)}_{2})|\Phi_{0}\rangle.$
(21)
This is similar in form to the second order correlation energy. In general,
the $n^{\rm th}$ order correlation energy has non-zero contribution from the
term $V_{2}\Omega^{(n-1)}_{2}$ only. It must be mentioned that the connected
triples begin to contribute from the fourth order energy. This is utilized in
perturbative inclusion of triples, in later sections of the paper, while
discussing coupled-cluster calculations.
## III Coupled-Cluster Theory
The coupled-cluster theory is a non-perturbative many-body theory and
considered as one of the best. A recent review bartlett-07 provides an
excellent overview of recent developments and different variations. In the
context of diagrammatic analysis of many-body perturbation theory, coupled-
cluster theory is equivalent to a selective evaluation of the connected
diagrams to all orders. Then casting the disconnected but linked diagrams as
products of connected diagrams. In coupled-cluster theory, for a closed-shell
atom, the exact ground state is
$|\Psi_{0}\rangle=e^{T^{(0)}}|\Phi_{0}\rangle,$ (22)
where $T^{(0)}$ is the cluster operator. The superscript is a tag to identify
cluster operators arising from different perturbations. For the case of $N$
electron atoms, the cluster operator is
$T^{(0)}=\sum_{i=1}^{N}T^{(0)}_{i}.$ (23)
Figure 3: Diagrammatic representation of unperturbed single and double cluster
operators.
In closed shell atoms, the single and doubles provide a good approximation of
the exact ground state. Then, the cluster operator
$T^{(0)}=T_{1}^{(0)}+T_{2}^{(0)}$ and is referred to as the coupled-cluster
single and doubles (CCSD). The cluster operators in the second quantized
notations are
$\displaystyle T^{(0)}_{1}$ $\displaystyle=$
$\displaystyle\sum_{a,p}t_{a}^{p}a_{p}^{\dagger}a_{a},$ (24) $\displaystyle
T^{(0)}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\sum_{a,b,p,q}t_{ab}^{pq}.a_{p}^{\dagger}a_{q}^{\dagger}a_{b}a_{a}.$
(25)
Here, $t_{a}^{p}$ and $t_{ab}^{pq}$ are the single and double cluster
amplitudes respectively. Subtracting $\langle\Phi_{0}|H|\Phi_{0}\rangle$ from
both sides of Eq.(3) and using the normal form of an operator,
$O_{N}=O-\langle\Phi_{0}|O|\Phi_{0}\rangle$, we get
$H_{\rm N}|\Psi_{0}\rangle=\Delta E|\Psi_{0}\rangle,$ (26)
where $\Delta E=E-\langle\Phi_{0}|H|\Phi_{0}\rangle$, as defined earlier, is
the correlation energy. Operating with $e^{-T^{(0)}}$ and projecting the above
equation on excited states we get the cluster amplitude equations
$\displaystyle\langle\Phi^{p}_{a}|\overline{H}_{\rm N}|\Phi_{0}\rangle=0,$
(27) $\displaystyle\langle\Phi^{pq}_{ab}|\overline{H}_{\rm
N}|\Phi_{0}\rangle=0,$ (28)
where $\overline{H}_{\rm N}=e^{-T^{(0)}}H_{\rm N}e^{T^{(0)}}$ is the
similarity transformed or dressed Hamiltonian. Following Wick’s theorem and
structure of $H_{\rm N}$, in general
$\displaystyle\overline{H}_{\rm N}=$ $\displaystyle H_{\rm
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width=0.50003pt,height=0.0pt,depth=8.61108pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=8.61108pt\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=8.61108pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm
N}T^{(0)}T^{(0)}T^{(0)}\\}+\frac{1}{4!}\\{\mathchoice{\vbox{\hbox
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3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=10.76385pt\vrule
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width=0.50003pt,height=0.0pt,depth=10.76385pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt\vrule
width=55.54344pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt\vrule
width=39.24706pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt\vrule
width=33.2717pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=10.76385pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}T^{(0)}T^{(0)}T^{(0)}\\},$
Here $\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.75pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 1.875pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}A\ldots B$ denote contraction between two operators $A$ and $B$.
The single and double cluster amplitudes are solutions of Eq.(27) and (28)
respectively. These are set of coupled nonlinear equations and iterative
methods are the ideal choice to solve these equations.
Figure 4: Diagrams which contribute to the singles, unperturbed cluster
operator ($T^{(0)}_{1}$), in the linearised coupled-cluster.
Figure 5: Diagrams which contribute to the doubles, unperturbed cluster
operator ($T^{(0)}_{2}$), in the linearised coupled-cluster.
### III.1 Linearized Coupled-Cluster
The nonlinearity in the cluster amplitude equation arises from the two and
higher contractions in the dressed Hamiltonian. An approximation often used as
a starting point of coupled-cluster calculations is to retain only the first
two terms in $\overline{H}_{N}$, then
$\overline{H}_{\rm N}=H_{\rm N}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}\\}.$ (30)
The cluster equations are then a pair of linear equations
$\displaystyle\langle\Phi^{p}_{a}|H_{\rm N}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}\\}|\Phi_{0}\rangle=0,$ (31)
$\displaystyle\langle\Phi^{pq}_{ab}|H_{\rm N}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}\\}|\Phi_{0}\rangle=0.$ (32)
In the CCSD approximation $T^{(0)}=T^{(0)}_{1}+T^{(0)}_{2}$, these equations
are then
$\displaystyle\langle\Phi^{p}_{a}|\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}_{1}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}_{2}\\}|\Phi_{0}\rangle=-\langle\Phi^{p}_{a}|H_{\rm
N}|\Phi_{0}\rangle$
$\displaystyle\langle\Phi^{pq}_{ab}|\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T^{(0)}_{1}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm
N}T^{(0)}_{2}\\}|\Phi_{0}\rangle=-\langle\Phi^{pq}_{ab}|H_{\rm
N}|\Phi_{0}\rangle.$
These are the linearized coupled-cluster equations of single and double
cluster amplitudes. This can be combined as the matrix equation
$\displaystyle\left(\begin{array}[]{cc}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{array}\right)\left(\begin{array}[]{c}t_{1}\\\
t_{2}\end{array}\right)=-\left(\begin{array}[]{c}H_{10}\\\
H_{20}\end{array}\right),$ (40)
where $H_{11}=\langle\Phi^{p}_{a}|H_{\rm N}|\Phi^{s}_{b}\rangle$,
$H_{12}=\langle\Phi^{p}_{a}|H_{\rm N}|\Phi^{st}_{bc}\rangle$ and so on. The
equations are set of coupled linear equations and solved using standard or
specialized linear algebra solvers. In the literature several authors refer to
linearized coupled-cluster as all-order method. A description of the all-order
method and applications are given in ref safronova-07 . In a recent work, the
authors report the combination of all-order method and configuration
interaction safronova-09 .
### III.2 Correlation Energy and approximate triples
From Eq.(26) the ground state correlation energy, in coupled-cluster theory,
is the ground state expectation value of $\overline{H}_{N}$. That is
$\Delta E=\langle\Phi_{0}|\overline{H}_{\rm N}|\Phi_{0}\rangle.$ (41)
The diagrams arising from the above expression are shown in the Fig.6. The
dominant contributions are from the diagrams (a) and (b), which is natural as
the doubles cluster amplitudes are larger in value than the singles. Diagram
(e) does not contribute to the correlation energy when Dirac-Fock orbitals are
used.
Figure 6: Coupled-cluster correlation energy diagrams. The diagram (e) is
equal to zero when Dirac-Fock orbitals are used.
To go beyond the CCSD approximation, we incorporate selected correlation
energy diagrams arising from approximate triples. The approximate triples are
perturbative contraction of $V_{2}$ with the $T^{(0)}$ cluster amplitudes
krishnan-89 ; porsev-06 . Example diagrams of the approximate triples and
correlation energies are shown in Fig.7. There are two categories of triples,
first is $V_{2}$ contracted with $T^{(0)}$ through a hole line, and second
contraction through a particle line (Fig.7a). To calculate the correlation
energy from the triples, these are contracted perturbatively with $V_{2}$ and
reduced to a double excitation diagram. Then the correlation energy is
obtained after another contraction with $V_{2}$. These two contractions
generate several diagrams. The triples correlation energy diagrams are
separated into three categories based on the number of internal lines. These
are: two particle and two hole internal lines (2p-2h), three particle one hole
internal lines (3p-1h), and one particle three hole internal lines (1p-3h). In
the present calculations eight diagrams from the first category and two each
from other remaining two categories are considered.
Figure 7: Diagrams of approximate triples calculated perturbatively:(a)
approximate triples cluster operator and (b) correlation energy arising from
approximate triples.
## IV Perturbed Coupled-Cluster
The atomic properties of interest are, in general, associated with additional
interactions. The interaction are either internal like hyperfine interaction
or external like static electric field. These are treated as perturbations
which modify the wave function and energy of the atom. This section briefly
describes a method to incorporate an additional perturbation within the frame
work of relativistic coupled-cluster. The scheme is referred to as perturbed
coupled-cluster theory. It has been tried and tested in precision atomic
properties and structure calculations. In the presence of a perturbation
$H_{1}$, the eigen value equation is
$\left(H^{\rm DC}+\lambda
H_{1}\right)|\widetilde{\Psi}_{0}\rangle=\widetilde{E}|\widetilde{\Psi}_{0}\rangle.$
(42)
Here $|\widetilde{\Psi}_{0}\rangle$ is the perturbed wave function,
$\widetilde{E}$ is the corresponding eigenvalue and $\lambda$ is the
perturbation parameter. The perturbed wave function is the sum of the
unperturbed wave function and a correction $|\overline{\Psi}^{1}_{0}\rangle$
arising from $H_{1}$. That is
$|\widetilde{\Psi}_{0}\rangle=|\Psi_{0}\rangle+\lambda|\overline{\Psi}^{1}_{0}\rangle.$
(43)
Figure 8: Diagrams of single and double perturbed cluster operators.
Following the earlier description of coupled-cluster wave function, the
perturbed wave function is
$|\widetilde{\Psi}_{0}\rangle=e^{T^{(0)}+\lambda T^{(1)}}|\Phi_{0}\rangle.$
(44)
The cluster operators $T^{(0)}$, as defined earlier, incorporate the effects
of residual Coulomb interaction. For clarity these are referred as unperturbed
cluster operator. The $T^{(1)}$ cluster operators arise from $H_{1}$ and are
referred to as the perturbed cluster operators. It acts on the reference state
$|\Phi_{0}\rangle$ to generate the correction. Consider the perturbation
expansion to first order in $\lambda$, we get
$|\widetilde{\Psi}_{0}\rangle=e^{T^{(0)}}(1+\lambda T^{(1)})|\Phi_{0}\rangle.$
(45)
To derive the cluster equations use this in Eq.(42), then operate with
$e^{-T^{(0)}}$ and project on excited states. We get the equations for singles
and doubles perturbed cluster amplitudes
$\displaystyle\langle\Phi^{p}_{a}|\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\overline{H}_{\rm
N}T^{(1)}\\}|\Phi_{0}\rangle=-\langle\Phi^{p}_{a}|\bar{H}_{1}|\Phi_{0}\rangle,$
(46) $\displaystyle\langle\Phi^{pq}_{ab}|\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=12.3788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=7.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\overline{H}_{\rm
N}T^{(1)}\\}|\Phi_{0}\rangle=-\langle\Phi^{pq}_{ab}|\overline{H}_{1}|\Phi_{0}\rangle.$
(47)
The dressed Hamiltonian $\overline{H}_{\rm N}$ is same as in Eq.(29). Like in
linearized coupled-cluster, these form a set of linear algebraic equations.
### IV.1 Approximate triples
Like in $T^{(0)}$, a perturbed triple cluster Fig.9(a) is a perturbative
contraction between $V_{2}$ and $T^{(1)}_{2}$. As in the case of unperturbed
approximate triples discussed earlier, there are two types of diagrams in the
present case as well. One arises from particle line contraction and the other
from hole line contraction between $V_{2}$ and $T^{(1)}$ diagrams. In this
work we implement approximate triples while calculating properties. In
particular, to calculate dipole polarizability and an example diagram is shown
in Fig. 9(b). The algebraic expression of this diagram is
$\sum_{a,b,c,p,q,r,s}\frac{\langle ab|{T^{(0)}_{2}}^{\dagger}|pq\rangle\langle
c|\bm{d}|s\rangle\langle qs|V_{2}|rc\rangle\langle
pr|T^{(1)}_{2}|ab\rangle}{\epsilon_{a}+\epsilon_{b}+\epsilon_{c}-\epsilon_{p}-\epsilon_{q}-\epsilon_{s}},$
(48)
here, $\bm{d}$ is the dipole operator. In total there are twentyfour
properties diagrams arising from the perturbative triples and we include all
of these diagrams in the calculations.
### IV.2 Dipole Polarizability
When an atom is placed in an external electric field $\cal\bm{E}$, the charge
distribution of electron cloud is distorted and an electric dipole moment
$\bm{D}_{\rm ind}$ is induced. The dipole polarizability of the atom $\alpha$
is then the ratio of the induced dipole moment to the applied electric field,
that is
$\bm{D}_{\rm ind}=\alpha\cal\bm{E}.$ (49)
By definition, the dipole polarizability of the ground state is
$\alpha=-2\sum_{I}\frac{|\langle\Psi_{0}|\bm{D}|\Psi_{I}\rangle|^{2}}{E_{0}-E_{I}},$
(50)
where $|\Psi_{I}\rangle$ are intermediate atomic states. These are opposite in
parity to the ground state $|\Psi_{0}\rangle$. The expression of $\alpha$ can
be rewritten as
$\alpha=-2\langle\Psi_{0}|\bm{D}|\overline{\Psi}_{0}^{1}\rangle.$ (51)
Here
$|\overline{\Psi}_{0}^{1}\rangle=\sum_{I}(|\Psi_{I}\rangle\langle\Psi_{I}|\bm{D}|\Psi_{0}\rangle)/(E_{0}-E_{I})$,
which follows from the first order time independent perturbation theory. The
perturbation Hamiltonian is $H_{1}=-\bm{D}.\cal\bm{E}$ and external field
$\cal\bm{E}$ is the perturbation parameter. One short coming of calculating
$\alpha$ from Eq.(50) is, for practical reasons, the summation over $I$ is
limited to the most dominant intermediate states. However, the summation is
avoided altogether when the perturbed coupled-cluster wave functions are used
in the calculations. From Eq.(44) the perturbed wave function
$|\overline{\Psi}_{0}^{1}\rangle=e^{T^{(0)}}T^{(1)}|\Phi_{0}\rangle.$ (52)
In a more compact form, the dipole polarizability in terms of the perturbed
coupled-cluster wave function is
$\alpha=\langle\widetilde{\Psi}_{0}|\bm{D}|\widetilde{\Psi}_{0}\rangle.$ (53)
After simplification, using the perturbed wave function in Eq.(44), we get
$\alpha=\langle\overline{\Psi}_{0}^{1}|\bm{D}|\Psi_{0}\rangle+\langle\Psi_{0}|\bm{D}|\overline{\Psi}_{0}^{1}\rangle.$
(54)
The correction $|\overline{\Psi}_{0}^{1}\rangle$, as described earlier, is
opposite in parity to $|\Phi_{0}\rangle$. Hence the matrix elements
$\langle\Psi_{0}|\bm{D}|\Psi_{0}\rangle$ and
$\langle\overline{\Psi}_{0}^{1}|\bm{D}|\overline{\Psi}_{0}^{1}\rangle$ are
zero. As $D$ is hermitian, the two terms on the left hand side are identical
and the above expression is same as Eq.(51). Considering the leading terms
$\alpha=\langle\Phi_{0}|{T^{(1)}}^{\dagger}\overline{D}^{(0)}+\overline{D}^{(0)}T^{(1)}|\Phi_{0}\rangle.$
(55)
Here, the operator $\overline{D}^{(0)}=e^{{T^{(0)}}^{\dagger}}De^{T^{(0)}}$ is
the unitary transformed electric dipole operator. It is explicitly evident
that the dipole polarizability, in terms of perturbed cluster operator, does
not have a sum over states. In this scheme, contributions from all
intermediate states within the chosen configuration space are included. For
precision calculations, this is a very important advantage.
Figure 9: Diagrams of approximate triples calculated perturbatively:(a) (a)
Representation of approximate perturbed triples. (b) Contribution of
approximate perturbed triples to the dipole polarizability.
## V Results
In order to get reliable results, in atomic structure and properties
calculations, one prerequisite is good quality orbital basis set. In all
calculations described in the paper, we employ GTOs as orbital functions. In
particular, we use even tempered basis in which the parameters $\alpha_{0}$
and $\beta$, in Eq. (9), are different for each symmetries. We use the basis
parameters of Tatewaki and Watanabe tatewaki-04 as starting values and
optimized further to obtain $E_{\rm DC}^{(0)}$ (ground state Dirac-Fock
energy) and $\epsilon_{a}$ (single particle energies of core orbitals) in
agreement with the numerical results. The numerical results are obtained from
the GRASP92 parpia-96 code. In order to obtain converged $E^{(2)}_{\rm
corr}$, we consider orbital basis set consisting of all the core orbitals and
virtual orbitals up to 10,000–11,000 in single particle energies.
The working equations of coupled-cluster theory are coupled nonlinear
equations. Solving these equations is a computational challenge. The number of
unknowns, cluster amplitudes, are in the order of millions. In addition,
implementing fast and efficient algorithms demand huge memory to tabulate and
store two-electron integrals. This is essential as the two-electron integrals
are needed repeatedly and are compute intensive. For the larger basis sets in
the present work, the number of two-electron integrals is more than $2\times
10^{8}$. In order to utilize memory efficiently, we have developed a scheme
which parallelize the tabulation and storage of two-electron integrals. To
improve performance further, we also tabulate and store $6j$ symbols. This is
desirable as the angular part of the perturbed cluster diagrams involve large
number of $6j$ symbols. Quantitatively, we observe a performance gain of 30%
or more with the $6j$ symbols tabulation. We shall describe and discuss the
various computational schemes developed in a future publication.
The unperturbed and perturbed cluster amplitude equations are solved
iteratively using Jacobi method. We chose the method as it is relatively
simple to parallelize. One drawback of the method is, it converges slowly. To
obtain faster convergence, we employ direct inversion in iterated subspace
(DIIS) pulay-80 convergence acceleration.
### V.1 Second-Order Correlation Energy
Table 1: The SCF $E^{(0)}_{DC}$, the second-order correlation $E^{(2)}_{\rm corr}$ and the total energies $E$ of Ne, Ar, Kr and Xe. All the values listed are in atomic units (Hartrees). Atom | Z | Atomic mass | This work | Other work
---|---|---|---|---
| | | $E^{(0)}_{DC}$ | $E^{(2)}_{\rm corr}$ | $E$ | $E^{(0)}_{DC}$ | $E^{(2)}_{\rm corr}$ | $E$
Ne | 10 | 20.18 | $-128.6932$ | $-0.3830$ | $-129.0762$ | $-128.6919$ | $-0.3834$111ReferenceIshikawa-94 . | $-129.0753$
| | | | | | | $-0.3836$222Referencelindgren-80 . |
| | | | | | | $-0.3822$333ReferenceNesbet-68 . |
| | | | | | | $-0.3697$444ReferenceSasaki-74 . |
| | | | | | | $-0.3804$555ReferenceIshikawa-90 . |
Ar | 18 | 39.95 | $-528.6882$ | $-0.6938$ | $-529.3820$ | $-528.6838$ | $-0.6981$111ReferenceIshikawa-94 . | $-529.3819$
| | | | | | | $-0.6822$555ReferenceIshikawa-90 . |
| | | | | | | $-0.685$666ReferenceCooper-73 . |
| | | | | | | $-0.790$777ReferenceClementi-65 . |
Kr | 36 | 83.80 | $-2788.8659$ | $-1.8426$ | $-2790.7085$ | $-2788.8615$ | $-1.8468$111ReferenceIshikawa-94 . | $-2790.7083$
Xe | 54 | 131.29 | $-7446.8887$ | $-2.9767$ | $-7449.8654$ | $-7446.8880$ | $-2.9587$111ReferenceIshikawa-94 . | $-7449.8467$
The SCF energy $E^{(0)}_{\rm DC}$, second-order correlation energy
$E^{(2)}_{\rm corr}$ and the total energy $E$ from our calculations are listed
in Table.1. For comparison the results of previous calculations are also
listed. It is evident that our second-order correlation energy and total
energy, sum of the SCF and second order correlation energy, are in agreement
with the results of Ishikawa et al Ishikawa-94 for all the atoms studied.
Among the previous works, we select the results of Ishikawa et al Ishikawa-94
for detailed comparison as their calculations are relativistic. The other
results listed in the table are from non-relativistic calculations. For all
the atoms, our SCF energy $E_{\rm DC}^{0}$ are lower and there are no
discernible trends, as a function of nuclear charge, in the difference.
Interestingly, except for Xe, our second order correlation energies are
higher. This compensates the lower $E^{(0)}_{\rm DC}$ and subsequently, the
total energies $E$ of the two calculations are in excellent agreement. The
lack of trend indicates the choice of the basis set could be the source of the
observed differences of $E^{(0)}_{\rm DC}$ and $E^{(2)}_{\rm corr}$ between
the two calculations.
Figure 10: The second-order correlation energy when orbitals upto a particular symmetry are included in the virtual space. Table 2: Cumulative second-order correlation energy when orbitals upto a particular symmetry are included in the virtual space. All the values are in atomic units. Symmetry | Ne | Ar | Kr | Xe
---|---|---|---|---
$s$ | -0.0194 | -0.0210 | -0.0236 | -0.0247
$p$ | -0.1920 | -0.2043 | -0.2479 | -0.2687
$d$ | -0.3216 | -0.5401 | -0.9512 | -1.0419
$f$ | -0.3589 | -0.6330 | -1.5213 | -2.2972
$g$ | -0.3732 | -0.6695 | -1.7077 | -2.6879
$h$ | -0.3786 | -0.6830 | -1.7843 | -2.8520
$i$ | -0.3811 | -0.6891 | -1.8179 | -2.9238
$j$ | -0.3823 | -0.6921 | -1.8343 | -2.9591
$k$ | -0.3830 | -0.6938 | -1.8426 | -2.9767
The Table.2 lists the cumulative contributions from various symmetries to
$E^{(2)}_{\rm corr}$. Among the previous works, Lindgren and collaborators
lindgren-80 provide cumulative $E^{(2)}_{\rm corr}$ for Ne up to $i$
symmetry. Their converged result is -0.3836, this compares well with our
result of -0.3811 calculated with orbitals up to $i$ symmetry. However, in our
calculation, we get converged results of -0.3830 after including $j$ and $k$
symmetry orbitals. For Ne, $E^{(2)}_{\rm corr}$ is considered converged when
the change with additional symmetry is below millihartree. However, for Ar, Kr
and Xe orbitals of $l$ and higher symmetries are essential to obtain
$E^{(2)}_{\rm corr}$ converged up to millihartree. Since the magnitude of
$E^{(2)}_{\rm corr}$ increases with $Z$, a representative measure of
convergence is the percentage change. The contribution from $k$ symmetry to
$E^{(2)}_{\rm corr}$ are -0.0017, -0.0083 and -0.0176 for Ar, Kr and Xe
respectively. These are larger than that of Ne, which is -0.0007. However,
these correspond to 0.24%, 0.45% and 0.59% for Ar, Kr and Xe respectively,
these compare very well with that of Ne 0.20%. For Ar there is a variation in
the previous values of $E^{(2)}_{\rm corr}$, these range from the lowest value
of Clementi Clementi-65 -0.790 to that of Ishikawa Ishikawa-94 -0.6981. Our
value of -0.6938 is closer to that of Ishikawa.
There is a pattern in the change of the correlation energy with symmetry wise
augmentation of the virtual orbital set. There is an initial increase, reaches
a maximum and then decreases. The maximum change occurs with the addition of
$p$, $d$, $d$ and $f$ symmetry for Ne, Ar, Kr and Xe respectively. The pattern
is evident in Fig.10, which plots the change in $E^{(2)}_{\rm corr}$ with
symmetry wise augmentation of the virtual space. The pattern arise from the
distribution of the contributions from each of the core orbitals. From
Eq.(19), depending on the core orbital combination $ab$, there are two types
of correlation effects. These are inter and intra core shell correlations
corresponding to $a=b$ and $a\neq b$ respectively. Among the various
combinations, the $2p_{3/2}2p_{3/2}$, $3p_{3/2}3p_{3/2}$, $3d_{5/2}3d_{5/2}$
and $4d_{5/2}4d_{5/2}$ core orbital pairings have leading contributions in Ne,
Ar, Kr and Xe respectively. Here for Ne and Ar the leading pairs correspond to
the valence shell but it is the last $d$ shell for Kr and Xe. This correlates
with the pattern observed in the symmetry wise augmentation.
### V.2 Third-Order Correlation Energy
We calculate the third order correlation energy $E^{(3)}_{\rm corr}$ from the
linearized CCSD equations. This is possible when the first order MBPT wave
operator $\Omega^{(1)}$ is chosen as the initial guess and iterate the
linearized coupled-cluster equations once. The two-body wave operator so
obtained is $\Omega_{2}^{(2)}$ and from Eq.(21) $E^{(3)}_{\rm
corr}=\langle\Phi_{0}|V_{2}\Omega_{2}^{(2)}|\Phi_{0}\rangle$. This approach,
however, is not applicable beyond third order. The reason is, starting from
the fourth order correlation energy the triples contribute to $\Delta E$ and
triples are not part of the linearized CCSD equations. The results of
$E^{(3)}_{\rm corr}$, obtained from our calculations, are listed in Table.3.
For comparison, results from previous works are also listed. For Ne, Jankowski
and Malinowski Jankowski reported a value of 0.0024. Their calculations were
done with a limited basis set and hence, could leave out less significant
contributions. The results of Lindgren and collaborators lindgren-80 0.0035
is perhaps more accurate and reliable on account of larger basis set. They
include virtuals up to $i$ symmetry and then extrapolate. Similarly, in our
calculations we include virtual orbitals up to $i$ symmetry, then extrapolate
up to $k$ symmetry based on $E^{(2)}_{\rm corr}$ results. We obtain 0.0019,
which is in better agreement with the result of Jankowski and
MalinowskiJankowski . As expected, $E^{(3)}_{\rm corr}$ increases with $Z$ and
to our knowledge, our results of Ar, Kr and Xe are the first reported
calculations in literature. Interestingly, $E^{(3)}_{\rm corr}$ is positive
for Ne, Kr and Xe but it is negative for Ar.
Table 3: Third-order correlation energy in atomic units. Atom | $E_{3}$
---|---
| This work | Other work
Ne | 0.0019 | 0.0035111Referencelindgren-80 .
| | 0.0024222ReferenceJankowski .
Ar | -0.0127 | -
| | -
Kr | 0.0789 | -
| | -
Xe | 0.1526 | -
| | -
### V.3 Coupled-Cluster correlation energy
The MBPT correlation energies $E_{\rm corr}^{(i)}$ converges with relatively
large basis set. For example, the $E_{\rm corr}^{(2)}$ of Ne converge when
virtual orbitals up to $k$ symmetry are included in the calculations. This
correspond to a total of 224 virtual orbitals. Similar or larger number of
virtual orbitals are required to obtain converged $E_{\rm corr}^{(2)}$ of Ar,
Kr and Xe as well. However, it is not practical to have such large basis sets
in relativistic coupled-cluster calculations. The $n_{v}^{4}n_{c}^{3}$, where
$n_{v}$ and $n_{c}$ are the number of the virtual and core orbitals
respectively, scaling of arithmetic operations in CCSD render computations
with large $n_{v}$ beyond the scope of detailed studies. Hence, in the CCSD
calculations, the size of the virtual orbital set is reduced to a manageable
level and restrict up to the $h$ symmetry. To choose the optimal set, after
considering the most dominant ones, the virtual orbitals are augmented in
layers. Where one layer consists of one virtual orbital each from all the
symmetries considered.
The CCSD correlation energies for two basis sets are listed in Table.4. The
first is with a basis set considered optimal and manageable size for CCSD
calculations after a series of calculations. Then the next is with an
additional layer of virtual orbitals. The change in the linearized CCSD
$\Delta E$ with the additional layer of virtual orbitals are 0.2%, 1.7%, 6.0%
and 5.0% for Ne, Ar, Kr and Xe respectively. Changes of similar order are
observed in the corresponding $\Delta E$ of the non-linear CCSD calculations.
It must be mentioned that, though the difference in $\Delta E$ is small, the
computational cost of non-linear CCSD is much higher than the linearized CCSD
calculations. The percentage changes indicate the basis size of Kr and Xe are
not large enough. The orbital basis of Xe, with the additional layer, consists
of 17 core and 129 virtual orbitals. This translates to $\sim 5.0\times
10^{6}$ cluster amplitudes, which follows from the $n_{v}^{2}n_{c}^{2}$
scaling of the number of cluster amplitudes. At this stage, the computational
efforts and costs far out weight the gain in accuracy. To account for the
correlation energy from the other virtual orbitals, not included in the CCSD
calculations, we resort to the second order correlation energy. For this we
calculate $E_{\rm corr}^{(2)}$ with the basis set chosen in CCSD calculations
and subtract from the converged $E_{\rm corr}^{(2)}$. The estimated $\Delta E$
in Table.4 is the sum of this difference and CCSD $\Delta E$. This includes
the correlation effects from $i$, $j$ and $k$ symmetries as well. For Ne, the
estimated experimental value of correlation energy lies between the range
0.385 and 0.390 Sasaki-74 ; Bunge-70 . Our coupled-cluster result, estimated
value, is in excellent agreement.
Table 4: Correlation energy from coupled-cluster. All the values are in atomic units. Atom | Active Orbitals | $\Delta E$(ccsd)
---|---|---
| | Linear | Nonlinear
Ne | 17$s$10$p$10$d$9$f$9$g$8$h$ | -0.3783 | -0.3760
| 18$s$11$p$11$d$10$f$10$g$9$h$ | -0.3805 | -0.3782
| Estimated | -0.3905 | -0.3882
Ar | 17$s$11$p$11$d$9$f$9$g$9$h$ | -0.6884 | -0.6829
| 18$s$12$p$12$d$10$f$10$g$10$h$ | -0.7001 | -0.6945
| Estimated | -0.7258 | -0.7202
Kr | 22$s$13$p$11$d$9$f$9$g$9$h$ | -1.5700 | -1.5688
| 23$s$14$p$12$d$10$f$10$g$10$h$ | -1.6730 | -1.6716
| Estimated | -1.8480 | -1.8466
Xe | 23$s$14$p$12$d$10$f$10$g$10$h$ | -2.5500 | -2.5509
| 24$s$15$p$13$d$11$f$11$g$11$h$ | -2.6874 | -2.6881
| Estimated | -2.9973 | -2.9979
The contributions to the correlation energy arising from the approximate
triples are listed in Table.5. As discussed in Section.III.2, the correlation
energy diagrams corresponding to the approximate triples are grouped into
three classes. Out of these we have selected a few: eight from 2p-2h and two
each from 3p-1h and 1p-3h. In Table.5, $\Delta E$ arising from these are
listed. It is evident from the table, the contribution from 1p-3h and 3p-1h
are negative and adds to the magnitude of $\Delta E$. Whereas, the
contribution from 2p-2h is positive and reduces the magnitude of $\Delta E$.
Table 5: Correlation energy arising from the approximate triples in the coupled-cluster theory. All the values are in atomic units. Atom | Basis size | $\Delta E$
---|---|---
| | 2p-2h | 1p-3h | 3p-1h
Ne | 18$s$11$p$11$d$10$f$10$g$9$h$ | 0.00672 | -0.00145 | -0.00164
Ar | 18$s$12$p$12$d$10$f$10$g$10$h$ | 0.00805 | -0.00066 | -0.00192
Kr | 22$s$13$p$11$d$9$f$9$g$9$h$ | 0.01546 | -0.00171 | -0.00305
Xe | 19$s$15$p$10$d$9$f$5$g$2$h$ | 0.02011 | -0.00148 | -0.00260
### V.4 Dipole Polarizability
One constraint while using perturbed coupled-cluster theory to calculate
dipole polarizability is the form of $\overline{D}$. It is a unitary
transformation of the dipole operator and expands to a non terminating series.
For the present calculations we consider the leading terms in
${T^{(1)}}^{\dagger}\overline{D}$. That is, we use the approximation
$\displaystyle{T^{(1)}}^{\dagger}\overline{D}$ $\displaystyle\approx$
$\displaystyle{T^{(1)}_{1}}^{\dagger}\left[D+DT^{(0)}_{1}+DT^{(0)}_{2}\right]+$
(56)
$\displaystyle{T^{(1)}_{2}}^{\dagger}\left[DT^{(0)}_{2}+DT^{(0)}_{1}\right].$
The ground state dipole polarizabilities of Ne, Ar, Kr and Xe calculated with
this approximation are given in Table.6. Among the various terms, the first
term ${T^{(1)}}^{\dagger}D$ subsumes contributions arising from Dirac-Fock and
random phase approximation. We can thus expect this term to have the most
dominant contribution. This is evident in Table.6, which shows that the
contribution from ${T^{(1)}}^{\dagger}D$ is far larger than the others. The
next leading term is ${T^{(1)}_{1}}^{\dagger}DT^{(0)}_{2}$. This is attributed
to the larger values, compared to $T_{1}^{(0)}$, of $T_{2}^{(0)}$ cluster
amplitudes. The dipole polarizability calculations with relativistic coupled-
cluster theory involve two sets of cluster amplitudes. These are the $T^{(0)}$
and $T^{(1)}$ cluster amplitudes. As mentioned earlier, solving coupled-
cluster equations is compute intensive. To minimize the computational costs,
we optimize the basis parameters $\alpha_{0}$ and $\beta$ to contract the size
of the orbital basis set.
One pattern discernible in the results is the better agreement between the
${T^{(1)}}^{\dagger}D$ results and experimental data. The deviations from the
experimental data are large when we consider the total (CCSD) result. For Ne,
the deviation from experimental data is 2%, where as it is 9% in the case of
Xe atom. We attribute this to the approximation in Eq.(56) and partly to the
basis set. To confirm this, however, requires detailed computations with
higher order terms in the unitary transformation. This poses considerable
computational challenges and shall be addressed in future publications. We
also implement the approximate triples excitation of the perturbed cluster
amplitudes and contribution to $\alpha$ are listed in the table.
Table 6: Dipole polarizability of the ground state of neutral rare-gas atoms (in a. u.). Contributions | Ne | Ar | Kr | Xe
---|---|---|---|---
${T^{(1)}_{1}}^{\dagger}D$ | 2.7108 | 11.3330 | 17.2115 | 27.7427
${T^{(1)}_{1}}^{\dagger}DT^{(0)}_{1}$ | 0.0771 | 0.0486 | 0.0429 | -0.1495
${T^{(1)}_{1}}^{\dagger}DT^{(0)}_{2}$ | -0.0703 | -0.8264 | -1.2721 | -2.3286
${T^{(1)}_{2}}^{\dagger}DT^{(0)}_{1}$ | -0.0004 | -0.0001 | 0.0002 | 0.0027
${T^{(1)}_{2}}^{\dagger}DT^{(0)}_{2}$ | 0.0053 | 0.2490 | 0.0439 | 0.0786
Total(CCSD) | 2.7225 | 10.8041 | 16.0264 | 25.3459
Approx. triples | 2.7281 | 10.7360 | 16.0115 | 25.2974
Exp. values111hohm-90 . | 2.670$\pm$0.005 | 11.070(7) | 17.075 | 27.815
## VI Conclusion
We have done a systematic study of the electron correlation energy of neutral
inert gas atoms using relativistic MBPT and coupled-cluster theory. Our MBPT
results are based on larger basis sets consisting of higher symmetries than
the previous works. Hence more reliable and accurate. Our study shows that in
heavier atoms Kr and Xe, the inner core electrons in $d$ shells dominates the
electron correlation effects. This ought to be considered in high precision
properties calculations. For example, the EDM calculations of Xe arising from
nuclear Schiff moment. The dipole polarizability calculated with the perturbed
coupled-cluster show systematic deviation from the experimental data. However,
the leading term is in good agreement. The deviations might decrease when
higher order terms are incorporated in the calculations. From these results,
it is evident that the basis set chosen is of good quality and appropriate for
precision calculations.
###### Acknowledgements.
We wish to thank S. Chattopadhyay, S. Gautam, K. V. P. Latha, B. Sahoo and S.
A. Silotri for useful discussions. The results presented in the paper are
based on computations using the HPC cluster of the Center for computational
Material Science, JNCASR, Bangalore.
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|
arxiv-papers
| 2009-08-13T13:43:19 |
2024-09-04T02:49:04.639419
|
{
"license": "Public Domain",
"authors": "B. K. Mani and D. Angom",
"submitter": "Brajesh Mani",
"url": "https://arxiv.org/abs/0908.1899"
}
|
0908.1959
|
# Nonlinear dielectric effect of dipolar fluids
I. Szalai szalai@almos.vein.hu Institute of Physics, University of Pannonia,
H-8201 Veszprém, POBox 158, Hungary S. Nagy Institute of Physics, University
of Pannonia, H-8201 Veszprém, POBox 158, Hungary S. Dietrich
dietrich@mf.mpg.de Max-Planck-Institut für Metallforschung, Heisenbergstr. 3,
D-70569 Stuttgart, Germany Institut für Theoretische und Angewandte Physik,
Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
###### Abstract
The nonlinear dielectric effect for dipolar fluids is studied within the
framework of the mean spherical approximation (MSA) of hard core dipolar
Yukawa fluids. Based on earlier results for the electric field dependence of
the polarization our analytical results show so-called normal saturation
effects which are in good agreement with corresponding NVT ensemble Monte
Carlo simulation data. The linear and the nonlinear dielectric permittivities
obtained from MC simulations are determined from the fluctuations of the total
dipole moment of the system in the absence of an applied electric field. We
compare the MSA based theoretical results with the corresponding Langevin and
Debye-Weiss behaviors.
## I Introduction
The studies of dielectric polarization in fluids bo1 are based on the
following relation between the polarization $\mathbf{P}$ and the electric
field strength $\mathbf{E}$ inside the dielectric:
$4\pi\mathbf{P}=(\epsilon_{E}-1)\mathbf{E},$ (1)
where $\epsilon_{E}$ is the field dependent dielectric permittivity. The
internal field is often called Maxwell field bo1 ; di1 which differs from the
external field $\mathbf{E}_{0}$ applied to the dielectric medium. In the low-
field limit the linear dielectric permittivity $\epsilon_{0}$ of an isotropic
fluid is given by the ratio of the polarization to the internal field
strength:
$\epsilon_{0}=1+4\pi\lim_{E\rightarrow{0}}\frac{P}{E}=1+4\pi\left(\frac{\partial{P}}{\partial{E}}\right)_{E=0}.$
(2)
In strong electric fields the polarization response acquires in addition
nonlinear contributions, so that the electric permittivity turns into a
nonlinear function of the Maxwell field:
$\epsilon_{E}=\epsilon_{0}+\epsilon_{2}{E}^{2}+\epsilon_{4}{E}^{4}+...\,\,\,\,\,\,\,.$
(3)
For simple molecular liquids the coefficients of the power series decrease
rapidly rz1 . Thus in general dielectric experiments can be well described in
terms of the linear dielectric permittivity. For liquids consisting of small
molecules this series can be limited to the second term. The corresponding
nonlinear dielectric effect (NDE) is determined by the coefficient
$\epsilon_{2}$ of the contribution $\sim{E^{2}}$ in Eq. (3):
$\epsilon_{2}\equiv 4\pi\lambda=\lim_{E\rightarrow
0}\frac{\epsilon_{E}-\epsilon_{0}}{E^{2}}=\lim_{E\rightarrow
0}\frac{\Delta\epsilon_{E}}{E^{2}},$ (4)
where $\lambda$ is the nonlinear dielectric permittivity which, upon
substituting Eq. (3) into Eq. (1), is given by
$\lambda=\frac{1}{3!}\left(\frac{\partial^{3}{P}}{\partial{E^{3}}}\right)_{E=0}.$
(5)
Although the nonlinear dielectric behavior of liquids has enjoyed a long
lasting scientific interest he1 ; pi1 accurate measurements of NDEs have
become possible only recently due to the development of new techniques rz1 ,
which in an applied electric field are capable to separate from the NDE, e.g.,
the Joule effect due to heating of the sample induced by its finite
conductivity. Experimentally, $\lambda$ is determined from the small change of
the dielectric permittivity ($\Delta\epsilon_{E}$) induced by static he1 ; pi1
or pulsed ma1 ; sr1 ; rz1 ; rz2 strong electric fields and is detected by a
weak radio-frequency probing field. In nonlinear optics the NDE can also
provide useful information for laser induced molecular reorientations in
isotropic and liquid crystalline phases kh1 . The first NDE measurements were
carried out by Herweg in diethyl ether he1 yielding $\epsilon_{2}<0$. If
$\epsilon_{2}$ has a sign opposite to that of $\epsilon_{0}>0$ one often
speaks of normal saturation because in this case the first correction term to
the linear behavior $P{\sim}E$ for $E{\rightarrow}0$ is in line with the
levelling off at large $E$. This effect is mainly connected to the ordering of
the orientation of dipoles in strong electric fields. The negative value of
$\epsilon_{2}$ corresponds to the negative value of the third order term of
the power expansion of the Langevin function for small $E$. Strongly dipolar
liquids, such as nitrobenzene, show anomalous (positive) dielectric saturation
because in such systems the external electric field influences the formation
of antiparallel pairs of dipolar nitrobenzene molecules pi1 . Nonlinear
dielectric effects have become also very useful for analyzing intermolecular
association ma2 , conformational equilibria no1 , and critical phenomena in
liquids and liquid mixtures rz3 ; rz4 . Recently the NDE has also been used to
study isotropic - mesophase transitions of various liquid crystals dr1 .
The theoretical models of nonlinear dielectric phenomena are based on
classical electrostatics and statistical mechanics of liquids. There have been
early attempts by Debye de1 , Onsager on1 , and Kirkwood ki1 to calculate the
nonlinear dielectric permittivity on the basis of phenomenological theories of
dielectric continua. Nice summaries of these nonlinear theories can be found
in Refs. bo1 and co1 . From a microscopic point of view Rasaiah et al. ra1
and Martina and Stell st1 have proposed a statistical mechanics description
for NDE and electrostriction on the basis of a quadratic hypernetted chain
approximation. Using molecular dynamics (MD) simulations the dielectric
saturation of water has been studied by Alper and Levy al1 , but they have not
published any numerical value for the nonlinear dielectric permittivity. Yeh
and Berkowitz ye1 have obtained new MD simulation data for the electric field
dependence of the dielectric permittivity of water and they found that
$\epsilon_{E}$ decreases with increasing applied field strength in accordance
with the expectation of a normal saturation effect. Their external field
simulation data show good agreement with a phenomenological equation proposed
by Booth bh1 . This equation has been used in the calculation of dielectric
saturation of water in membrane protein channels ag1 . Recently Fulton fu1
has compared the nonlinear dielectric permittivity of water obtained from
simulation, theory, and experiment. He concluded that an upgraded approach by
Booth bh2 renders the best agreement between the simulation and experimental
data; but the extent of the agreement depends on the water model used for
determining the corresponding correlation functions. It is also shown that the
calculated nonlinear dielectric permittivity strongly depends on the
application of phenomenological cavity and reaction field corrections. These
findings underscore the ongoing interest in statistical mechanics analyses of
nonlinear dielectric effects of dipolar liquids.
Within the framework of density functional theory (DFT) and the mean spherical
approximation (MSA) we have proposed an equation sz1 for the magnetic field
dependence of the magnetization of ferrofluids, which turned out to be
successful in comparison with corresponding Monte Carlo (MC) simulation data.
Translated into the synonymous electric language this means that we have
obtained an analytical (implicit) equation for the electric field dependence
of the polarization. Here, from our previous results sz1 , we deduce a formula
for the nonlinear dielectric permittivity of dipolar fluids. Moreover, we
compare our theoretical findings with canonical MC simulation data using the
corresponding fluctuation formulae in the absence of external fields. The
motivation for our study is to deduce an analytical theory of a realistic
model for nonlinear phenomena in dipolar liquids. This theory has to be
quantitatively correct in the sense that it withstands the comparison with MC
simulation data and it has to be easily applicable for interpreting
corresponding experimental data. We expect that our model calculations can be
used to shed light on the role of nonlinear dielectric saturation for
solvation effects of ions in dipolar solvents ou1 ; ou2 .
## II Theory
### II.1 Microscopic model
We study hard core dipolar Yukawa fluids which consist of spherical particles
interacting via a hard core Yukawa (Y) potential characterized by parameters
$\sigma$, $\varepsilon_{Y}$, and $\kappa$:
$u_{Y}(r_{12})=\left\\{\begin{array}[]{lll}\infty&,&r_{12}<\sigma\\\
-\varepsilon_{Y}\sigma{(r_{12})^{-1}}\exp[-\kappa(r_{12}-\sigma)]&,&r_{12}\geq\sigma.\\\
\end{array}\right.$ (6)
In addition there is a dipolar interaction due to point dipoles embedded at
the centers of the particles:
$u_{D}(\mathbf{r}_{12},\omega_{1},\omega_{2})=-\frac{m^{2}}{r_{12}^{3}}D(\omega_{12},\omega_{1},\omega_{2}),$
(7)
with the rotationally invariant function
$D(\omega_{12},\omega_{1},\omega_{2})=3(\mathbf{\widehat{m}}_{1}\cdot\mathbf{\widehat{r}}_{12})(\mathbf{\widehat{m}}_{2}\cdot\mathbf{\widehat{r}}_{12})-(\mathbf{\widehat{m}}_{1}\cdot\mathbf{\widehat{m}}_{2}),$
(8)
where particle 1 (2) is located at $\mathbf{r}_{1}$ ($\mathbf{r}_{2}$) and
carries a dipole moment of strength $m$ with an orientation given by the unit
vector $\mathbf{\widehat{m}}_{1}(\omega_{1})$
($\mathbf{\widehat{m}}_{2}(\omega_{2})$) with polar angles
$\omega_{1}=(\theta_{1},\phi_{1})$ ($\omega_{2}=(\theta_{2},\phi_{2})$);
$\mathbf{r}_{12}=\mathbf{r}_{1}-\mathbf{r}_{2}$ is the difference vector
between the centers of particle 1 and 2, $r_{12}=|\mathbf{r}_{12}|$, and
$\mathbf{\widehat{r}}_{12}=\mathbf{r}_{12}/r_{12}$ is a unit vector with
orientation $\omega_{12}=(\theta_{12},\phi_{12})$. The hard core dipolar
Yukawa interaction potential is defined by the sum of the aforementioned
potentials as
$u_{DY}(\mathbf{r}_{12},\omega_{1},\omega_{2})=u_{Y}(r_{12})+u_{D}(\mathbf{r}_{12},\omega_{1},\omega_{2}).$
(9)
### II.2 MSA for dipolar Yukawa fluids
The MSA is defined by three equations relating the total correlation function
$h(\mathbf{r}_{12},\omega_{1},\omega_{2})$ and the direct correlation function
$c(\mathbf{r}_{12},\omega_{1},\omega_{2})$ as follows:
$\displaystyle
h(\mathbf{r}_{12},\omega_{1},\omega_{2})=c(\mathbf{r}_{12},\omega_{1},\omega_{2})+$
$\displaystyle\frac{\rho}{4\pi}\int{d\omega_{3}}\int{d^{3}r_{3}}h(\mathbf{r}_{13},\omega_{1},\omega_{3})c(\mathbf{r}_{23},\omega_{2},\omega_{3}),$
(10) $h(\mathbf{r}_{12},\omega_{1},\omega_{2})=-1,\,\,\,\,\,r_{12}<\sigma,$
(11)
and
$c(\mathbf{r}_{12},\omega_{1},\omega_{2})=-\beta{u(\mathbf{r}_{12},\omega_{1},\omega_{2})},\,\,\,\,\,r_{12}\geq\sigma.$
(12)
Here and in the following $\beta=1/k_{B}T$, where $k_{B}$ is the Boltzmann
constant and $T$ is the temperature, and
$u(\mathbf{r}_{12},\omega_{1},\omega_{2})$ is the pair potential
characterizing the system. The number density $\rho=N/V$ is given by the
number $N$ of molecules in the system of volume $V$. Equation (10) is the
Ornstein-Zernike (OZ) relation, Eq. (11) is an exact relation for hard
spheres, and Eq. (12) is the closure relation for the MSA (for details see
Ref. ha1 ). For dipolar hard sphere (DHS) fluids an analytical solution of the
MSA was reported by Wertheim we1 . Later the MSA for hard core Yukawa fluids
has also been solved analytically by Waisman wa1 . Following the ideas of
Wertheim, for hard core dipolar Yukawa fluids Henderson et al. sz2 ; sz3
found an analytical solution in the framework of MSA. Within this theory the
direct correlation function of the DY fluid can be expressed as
$\displaystyle c_{DY}(\mathbf{r}_{12},\omega_{1},\omega_{2})=c_{Y}(r_{12})+$
$\displaystyle
c_{D}(r_{12})D(\omega_{12},\omega_{1},\omega_{2})+c_{\Delta}(r_{12})\Delta(\omega_{1},\omega_{2}),$
(13)
where
$\Delta(\omega_{1},\omega_{2})=\mathbf{\widehat{m}}_{1}\cdot\mathbf{\widehat{m}}_{2}$
(14)
is a rotationally invariant function. (We note that a similar equation is
valid for the total correlation function
$h_{DY}(\mathbf{r}_{12},\omega_{1},\omega_{2})$ of the DY fluid.) The radially
symmetric function $c_{Y}(r_{12})$ is the solution of the OZ equation with a
simple hard-core Yukawa MSA closure and is given in Refs. wa1 ; hh1 while the
functions $c_{D}(r_{12})$ and $c_{\Delta}(r_{12})$ are the solutions of two
MSA integral equations (derived from the OZ equations) with the corresponding
dipolar hard sphere MSA closure, depending therefore on the dipole moment of
the molecules we1 . In Eq. (13) $D$ and $\Delta$ are given by Eqs. (8) and
(14) respectively. The coefficients $c_{D}(r_{12})$ and $c_{\Delta}(r_{12})$
(together with $h_{D}(r_{12})$ and $h_{\Delta}(r_{12})$ ) are independent of
the Yukawa potential parameters $\varepsilon_{Y}$ and $\kappa$ as well as of
$\omega_{12}$, $\omega_{1}$, $\omega_{2}$ and are given in Ref. we1 . The main
feature of this solution is that it decomposes into contributions from the
Yukawa potential and from the dipolar hard sphere potential with the latter
ones factorizing into radial and angular dependences. In Ref. sz2 it has been
shown that, within the framework of MSA, the free energy $F_{DY}$ of the DY
fluids can be written as
$F_{DY}=F_{ID}+F_{DY}^{ex}=F_{ID}+F_{HS}^{ex}+F_{Y}^{ex}+F_{DHS}^{ex},$ (15)
where $F_{ID}$ is the ideal gas free energy and $F_{HS}^{ex}$, $F_{Y}^{ex}$,
and $F_{DHS}^{ex}$ are the excess free energies of hard sphere ha1 , hard core
Yukawa wa1 , and dipolar hard sphere fluids we1 , respectively. Within this
approximation the dielectric constant of the DY fluid is given by the formula
due to Wertheim we1 and Henderson et al. sz2 :
$\epsilon_{0}=\frac{q(2\xi(y))}{q(-\xi(y))},$ (16)
where
$q(x)=\frac{(1+2x)^{2}}{(1-x)^{4}}$ (17)
is the reduced inverse compressibility function of the hard sphere fluid. The
parameter $\xi(y)$ stems from the DHS MSA and is given by the implicit
equation
$3y=q(2\xi)-q(-\xi),$ (18)
where
$y=\frac{4\pi}{9}\frac{{m^{2}}\rho}{k_{B}T}$ (19)
measures the reduced dipole strength.
### II.3 Field dependence of polarization
The disadvantage of the MSA is that it can predict the polarization only for
weak electric fields because in essence it is a linear response theory. In
order to overcome this shortcoming we resort to density functional theory.
Within this framework we first have to determine the orientational
distribution function in an applied external field. For an inhomogeneous and
anisotropic dipolar fluid the one-particle distribution function
${\rho}(\mathbf{r},\omega)$ depends on the position $\mathbf{r}$ and the
orientation $\omega$ of the particles. Accordingly the number density is given
by ${\rho}({\bf{r}})=\int{d\omega}{\rho}(\mathbf{r},\omega)$ and
$\alpha(\mathbf{r},\omega)={\rho}(\mathbf{r},\omega)/\rho(\bf{r})$ is the
orientational distribution function. In a homogeneously polarized bulk phase
(concerning influences from the sample shape see below) the number density
$\rho$ is spatially constant and the orientational distribution function
$\alpha({\bf{r}},\omega)=\alpha(\theta)$ depends only on the angle $\theta$,
which measures the orientation of the dipole of a single particle relative to
the field direction. (Here we do not consider electrostriction, i.e., the
dependence of $\rho$ on $E$, because this contribution to $\Delta\epsilon$ is
one order of magnitude smaller than the corresponding contribution of the
orientational ordering of dipoles in the presence of an applied electric field
kr1 ). The orientational distribution function can be obtained by minimizing
the following grand canonical variational functional:
$\displaystyle\Omega[\rho,\\{\alpha(\theta)\\},T,\mu]=F_{ID}[\rho,\\{\alpha(\theta)\\},T]+$
$\displaystyle F^{ex}_{DY}[\rho,\\{\alpha(\theta)\\},T]-$
$\displaystyle\rho\int{d^{3}r}{d\omega}\alpha(\theta)(\mu+mE\cos{\theta}),$
(20)
where $F_{ID}$ and $F_{DY}^{ex}$ are the ideal gas and the excess dipolar
Yukawa free energy density functionals, respectively, $\mu$ is the chemical
potential, and $E$ is the internal electric field in the sample. In this
theory we assume that the volume of the fluid $V$ has the shape of a
macroscopic prolate rotational ellipsoid (elongated around the electric field
direction) so that the internal field strength $E$ is equal to the strength of
the externally applied electric field $E_{0}$ (see Ref. di1 ). The excess DY
free energy functional for an anisotropic system is not known. However, it can
be approximated by a functional Taylor series, expanded around a homogeneous
isotropic reference system with bulk number density $\rho$. Neglecting all
terms beyond second order, one has
$\displaystyle{\beta}F^{ex}_{DY}[\rho,\\{\alpha(\theta)\\},T]={\beta}F^{ex}_{DY}(\rho,T)-$
$\displaystyle\frac{\rho^{2}}{2}\int{d^{3}r_{1}}{d\omega_{1}}\int{d^{3}r_{2}}{d\omega_{2}}\Delta\alpha(\theta_{1})\Delta\alpha(\theta_{2})\times$
$\displaystyle c_{DY}(\mathbf{r}_{12},\omega_{1},\omega_{2}),$ (21)
where $\Delta\alpha(\theta)=\alpha(\theta)-1/(4\pi)$ is the difference between
the actual anisotropic and the isotropic orientational distribution function
and $F_{DY}^{ex}(\rho,T)$ is the excess dipolar Yukawa free energy for an
isotropic distribution given by the last three terms in Eq. (15). Using the
MSA direct correlation function of the DY fluid (Eq. (13)) within this
approximation the anisotropic excess free energy functional
$F^{ex}_{DY}[\rho,\\{\alpha(\theta)\\},T]$ can be calculated analytically,
which allows one to obtain $\alpha(\theta)$ by minimizing the grand canonical
functional in Eq. (20). From the orientational distribution function the
polarization $P$ (the direction of which coincides with the direction of the
external electric field) can be obtained as
$P=\rho\int{d\omega}\alpha(\theta)m\cos\theta.$ (22)
Based on the orientational distribution function $\alpha(\theta)$ Eq. (22)
leads to the following polarization function:
$P=m\rho{L}\left(\beta{m}E+3P\frac{(1-q(-\xi(y)))}{m\rho}\right),$ (23)
where $L(x)=\coth x-1/x$ is the well known Langevin function bo1 . This is an
implicit equation for the external field dependence of the polarization. The
details of this calculation can be found in Ref. sz1 (where the equivalent
magnetic language was adopted, i.e., the particles carry magnetic dipole
moments and interact with an applied magnetic field).
### II.4 Nonlinear dielectric effect
In order to obtain the nonlinear dielectric permittivity $\lambda$ the third
order derivative of the polarization with respect to the Maxwell field has to
be calculated (see Eq. (5)). To this end we introduce the following
dimensionless form of Eq. (23):
$p(e)=L\left(e+{\Gamma}p(e)\right),$ (24)
where
$p=P/(m\rho),\,\,\,\,\,e={\beta}mE$ (25)
are the dimensionless polarization and electric field strength, respectively,
and $\Gamma=3(1-q(-\xi(y)))$ is a field independent parameter. From Eq. (24)
one obtains
$\frac{dp}{de}=\frac{dL(x)}{dx}{\Bigg{|}}_{x=a}\left(1+\Gamma\frac{dp}{de}\right),$
(26)
where $a=e+\Gamma{p(e)}$. The corresponding second order derivative is
$\
\frac{d^{2}p}{de^{2}}=\frac{d^{2}L(x)}{dx^{2}}{\Bigg{|}}_{x=a}\left(1+\Gamma\frac{dp}{de}\right)^{2}+\Gamma\frac{dL(x)}{dx}{\Bigg{|}}_{x=a}\frac{d^{2}p}{de^{2}}.$
(27)
For the third order derivative one has
$\displaystyle\frac{d^{3}p}{de^{3}}=\frac{d^{3}L(x)}{dx^{3}}{\Bigg{|}}_{x=a}\left(1+\Gamma\frac{dp}{de}\right)^{3}+$
$\displaystyle\Gamma\frac{dL(x)}{dx}{\Bigg{|}}_{x=a}\frac{d^{3}p}{de^{3}}+$
$\displaystyle
3\Gamma\frac{d^{2}L(x)}{dx^{2}}{\Bigg{|}}_{x=a}\left(1+\Gamma\frac{dp}{de}\right)\frac{d^{2}p}{de^{2}}.$
(28)
Since we consider thermodynamic states without spontaneous polarization (see
Ref. sz1 ) one has $L(x=0)=0$ so that $p(e=0)=0$. With
$\displaystyle\frac{dL(x)}{dx}{\Bigg{|}}_{x=0}=\frac{1}{3},\,\,\,\,\,\,\,\,\,\frac{d^{2}L(x)}{dx^{2}}{\Bigg{|}}_{x=0}=0,$
$\displaystyle\frac{d^{3}L(x)}{dx^{3}}{\Bigg{|}}_{x=0}=-\frac{2}{15}$ (29)
one finds
$\frac{dp}{de}{\Bigg{|}}_{e=0}=\frac{1}{3-\Gamma}=\frac{1}{3q(-\xi(y))},\ $
(30) $\frac{d^{2}p}{de^{2}}{\Bigg{|}}_{e=0}=0,$ (31)
and
$\frac{d^{3}p}{de^{3}}{\Bigg{|}}_{e=0}=-\frac{54}{5}\frac{1}{(3-\Gamma)^{4}}=-\frac{2}{15}\frac{1}{q^{4}(-\xi(y))}\,\,.$
(32)
Equations (2), (18), and (30) render back the expression in Eq. (16) for the
linear dielectric permittivity. For the nonlinear dielectric permittivity Eqs.
(5) and (32) yield
$\displaystyle\lambda=-\frac{m^{4}\rho}{45(k_{B}T)^{3}}\frac{1}{q^{4}(-\xi(y))}=$
$\displaystyle-\frac{m^{2}}{20\pi(k_{B}T)^{2}}\frac{y}{q^{4}(-\xi(y))}.$ (33)
We note that for $\Gamma=0$ Eq. (24) reduces to the Langevin equation for non-
interacting dipoles,
$p(e)=L(e),$ (34)
while for $\Gamma=3y$ one obtains the well known mean field Debye-Weiss
polarization equation
$p(e)=L(e+3yp(e)).$ (35)
Accordingly, for non-interacting dipoles (i.e., $\Gamma=0$ and $q(-\xi)=1$)
one has for the linear and nonlinear dielectric permittivities
$\displaystyle\epsilon_{0}=1+3y,$
$\displaystyle\lambda=-\frac{m^{4}\rho}{45(k_{B}T)^{3}}=-\frac{m^{2}}{20\pi(k_{B}T)^{2}}y,$
(36)
respectively, and within the framework of the Debye-Weiss theory (i.e.,
$\Gamma=3y$ and $q(-\xi)=1-y$)
$\displaystyle\epsilon_{0}=1+3\frac{y}{1-y},$
$\displaystyle\lambda=-\frac{m^{4}\rho}{45(k_{B}T)^{3}}\frac{1}{(1-y)^{4}}=$
$\displaystyle-\frac{m^{2}}{20\pi(k_{B}T)^{2}}\frac{y}{(1-y)^{4}}.$ (37)
The latter equations show that not only the linear but also the nonlinear
dielectric permittivity diverge for
$y=4\pi\rho{m^{2}}/(9k_{B}T)\rightarrow{1}$, which for a fixed value $m$ of
the dipole moment provides the Debye-Weiss critical temperature
$k_{B}T_{c}=4\pi\rho{m^{2}}/9$ of the isotropic liquid - ferroelectric liquid
second-order phase transition. In the following these theoretical predictions
will be compared with corresponding Monte Carlo simulation results.
## III Monte Carlo simulations
We have performed Monte Carlo simulations for DY fluids using the canonical
NVT ensemble and applying Boltzmann sampling, periodic boundary conditions,
and the minimum image convention al2 . In order to take into account the long-
ranged character of the dipolar interaction the so-called reaction field (RF)
method is used. According to this method we consider our system to be a
macroscopic spherical sample composed of a number of replicas of the basic
simulation cell embedded in a dielectric continuum with dielectric constant
$\epsilon_{RF}$. In this case in the spherical sample the internal (Maxwell)
field $\mathbf{E}$ is not equal to the external applied field
$\mathbf{E}_{0}$, but ne1
$\mathbf{E}=\left(\frac{2\epsilon_{RF}+1}{2\epsilon_{RF}+\epsilon_{E}}\right)\mathbf{E}_{0}.$
(38)
For such a system, using the third order Taylor series expansion of the
polarization with respect to the external field, Kusalik ku1 has shown that
$\displaystyle\epsilon_{E}\simeq\epsilon_{0}+$
$\displaystyle\left(\frac{\epsilon_{0}+2\epsilon_{RF}}{2\epsilon_{RF}+1}\right)^{2}\left[\frac{4\pi\beta^{3}}{90V}\left(3\langle{\mathbf{M}^{4}}\rangle_{0}-5\langle{\mathbf{M}^{2}}\rangle^{2}_{0}\right)\right]E_{0}^{2},$
(39)
where
$\epsilon_{0}=\frac{1+2\epsilon_{RF}+\frac{8\pi\beta\epsilon_{RF}}{3V}\langle{\mathbf{M}^{2}}\rangle_{0}}{1+2\epsilon_{RF}-\frac{4\pi\beta}{3V}\langle{\mathbf{M}^{2}}\rangle_{0}}.$
(40)
In Eqs. (39) and (40) $\mathbf{M}$ is the total dipole moment of the system of
volume $V$,
$\mathbf{M}=\sum_{i=1}^{N}{\mathbf{m}_{i}},$ (41)
and $\langle{\mathbf{M}^{2}}\rangle_{0}$, $\langle{\mathbf{M}^{4}}\rangle_{0}$
are the ensemble averages of the corresponding moments in zero external field.
In our simulations we apply a conducting boundary condition which means
$\epsilon_{RF}\rightarrow\infty$. In this limit Eqs. (38), (39), and (40) for
the internal field dependent dielectric permittivity lead to
$\epsilon_{E}=\epsilon_{0}+\frac{4\pi\beta^{3}}{90V}\left(3\langle{\mathbf{M}^{4}}\rangle_{0}-5\langle{\mathbf{M}^{2}}\rangle_{0}^{2}\right)E^{2},$
(42)
with
$\epsilon_{0}=1+\frac{4\pi\beta}{3V}\langle{\mathbf{M}^{2}}\rangle_{0}.\ $
(43)
Comparing Eq. (42) with Eq. (4) for the nonlinear dielectric permittivity we
obtain
$\lambda=\frac{\beta^{3}}{90V}\left(3\langle{\mathbf{M}^{4}}\rangle_{0}-5\langle{\mathbf{M}^{2}}\rangle_{0}^{2}\right).$
(44)
In our NVT ensemble MC simulations $\epsilon_{0}$ and $\lambda$ are calculated
from Eqs. (43) and (44), respectively.
A spherical cutoff of the hard core DY pair potentials at half of the cubic
box has been applied and long-ranged corrections (LRC) were taken into account
al2 . In our simulations $N=256$ particles have been used. We have not
resorted to any finite-size scaling analysis for detecting the occurrence of
the isotropic - anisotropic phase transitions. The simulations were started
from a hcp lattice configuration with randomly oriented dipoles. After 20.000
equilibration cycles, $2\times 10^{6}$ \- $4\times 10^{6}$ production cycles
were used. Statistical errors were calculated from the standard deviations of
sub-averages containing $2\times 10^{5}$ cycles.
## IV Results and discussion
In the following we shall use reduced quantities:
$T^{*}=k_{B}T/\varepsilon_{Y}$ as the reduced temperature,
$\rho^{*}=\rho\sigma^{3}$ as the reduced density,
$m^{*}=m/\sqrt{\varepsilon_{Y}\sigma^{3}}$ as the reduced dipole moment, and
$\lambda^{*}=\lambda\varepsilon_{Y}/\sigma^{3}$ as the reduced nonlinear
dielectric permittivity. Concerning the range of the Yukawa potential, all our
results correspond to $\kappa=1.8/\sigma$.
Figure 1 shows the linear and nonlinear dielectric permittivity as function of
the reduced number density $\rho^{*}$. For $(m^{*})^{2}=0.5$ and $T^{*}=1$ the
predictions of our MSA theory (see Eqs. (16)-(18) and (33))
Figure 1: Linear (a) and nonlinear (b) MSA dielectric properties of hard core
dipolar Yukawa fluids for $(m^{*})^{2}=0.5$ and $T^{*}=1$ in comparison with
the corresponding Langevin and Debye-Weiss approximations. The symbols in both
figures represent our Monte Carlo simulation data whereas the lines are
obtained from the three corresponding theories. In (a) the size of the error
bars is that of the symbols. Within the Debye-Weiss theory both $\epsilon_{0}$
and $\lambda^{*}$ diverge at $\rho_{c}^{*}=1.432$. Note that according to Eqs.
(47) and (49) $\lambda^{*}$ vanishes linearly for $\rho^{*}\rightarrow{0}$.
are compared with the Langevin (Eq. (36)) and Debye-Weiss (Eq. (37))
approximations and our MC data. The Langevin theory predicts, in general, a
linear dependence of $\epsilon_{0}$ and $\lambda$ on $\rho$. Figure 1 shows
that the interparticle interaction enhances $\epsilon_{0}$ and
$\rvert\lambda^{*}\rvert$ relative to the corresponding values of the Langevin
theory. The Debye-Weiss theory overestimates the MSA results for
$\epsilon_{0}$ and $\rvert\lambda^{*}\rvert$. The critical density, at which
within the Debye-Weiss theory $\epsilon_{0}$ and $\lambda$ diverge, is
$\rho_{c}=1.432$. The behavior of $\epsilon_{0}$ and $\lambda$ shown in Fig. 1
for various approximations can be understood in terms of a low density
expansion which amounts to an expansion in terms of $y<1$ which is
proportional to $\rho$ (see Eq. (19)). Within MSA the linear dielectric
permittivity of the hard core dipolar Yukawa fluid has the expansion sz3
$(\epsilon_{0})_{MSA}=1+3y+3y^{2}+\frac{3}{16}y^{3}+O(y^{4}).$ (45)
From Eqs. (16), (18), and (45) one finds
$\frac{1}{q(-\xi(y))}=\frac{\epsilon_{0}-1}{3y}=1+y+\frac{1}{16}y^{2}+O(y^{3}),$
(46)
so that
$\displaystyle\lambda_{MSA}=-\frac{m^{2}}{20\pi(k_{B}T)^{2}}\frac{y}{q^{4}(-\xi(y))}=$
$\displaystyle-\frac{m^{2}}{20\pi(k_{B}T)^{2}}(y+4y^{2}+\frac{25}{4}y^{3})+O(y^{4}).$
(47)
The corresponding expansions within the Debye-Weiss approximation are
$\displaystyle(\epsilon_{0})_{DW}=1+3\frac{y}{1-y}=$ $\displaystyle
1+3y+3y^{2}+3y^{3}+O(y^{4})=$
$\displaystyle(\epsilon_{0})_{MSA}+\frac{45}{16}y^{3}+O(y^{4})$ (48)
and
$\displaystyle\lambda_{DW}=-\frac{m^{2}}{20\pi(k_{B}T)^{2}}\frac{y}{(1-y)^{4}}=$
$\displaystyle-\frac{m^{2}}{20\pi(k_{B}T)^{2}}(y+4y^{2}+10y^{3})+O(y^{4})=$
$\displaystyle\lambda_{MSA}-\frac{3m^{2}}{16\pi(k_{B}T)^{2}}y^{3}+O(y^{4}).$
(49)
Equations (48) and (49) explain the trends of the various curves shown in Fig.
1. For the present choice of $m^{*}$ and $T^{*}$ the MC data and the MSA
results for the linear dielectric permittivity agree very well. This agreement
remains rather good also for the nonlinear dielectric permittivity. We note
that the very good agreement between the MSA and the MC simulation data for
the linear dielectric permittivity has also been reported in Ref. sz3 .
Figure 2: Same as in Fig. 1 for $T^{*}=2$. Within the Debye-Weiss theory both
$\epsilon_{0}$ and $\lambda^{*}$ diverge at $\rho_{c}^{*}=2.865$.
As shown in Fig. 2, the increase of the reduced temperature from $T^{*}=1$ to
$T^{*}=2$ does not change the character of the curves. The MSA results are in
good agreement with the simulation data. With increasing temperature the
discrepancy between the Debye-Weiss theory and the simulation data is reduced
because for higher temperatures the critical density, at which within the
Debye-Weiss theory $\epsilon_{0}$ and $\lambda$ diverge, is shifted to higher
densities (here $\rho^{*}_{c}=2.865$; we note that this value of the density
is physically not accessible, because the corresponding packing fraction
$\eta=\pi\rho^{*}/6$ would be larger than 1). The absolute values of
$\epsilon_{0}$ and $\lambda$ are considerably reduced upon raising the
temperature.
Figure 3: Same as in Fig. 1 for $(m^{*})^{2}=1$. Within the Debye-Weiss theory
both $\epsilon_{0}$ and $\lambda^{*}$ diverge at $\rho_{c}^{*}=0.716$.
Figure 3 shows that upon increasing the dipole moment from $(m^{*})^{2}=0.5$
to $(m^{*})^{2}=1$ the agreement between the MSA and the Monte Carlo
simulation data does not change significantly. However, the discrepancy
between the Debye-Weiss theory and the simulation data has widened because due
to the increase of the dipole moment the critical density, at which within the
Debye-Weiss theory $\epsilon_{0}$ and $\lambda$ diverge, is shifted to a lower
value ($\rho^{*}_{c}=0.716$). The Langevin theory underestimates both
$\epsilon_{0}$ and $\rvert\lambda^{*}\rvert$. The increase of the dipole
moment significantly increases $\epsilon_{0}$ and $\rvert\lambda^{*}\rvert$.
Figure 4 shows that upon increasing the temperature from $T^{*}=1$ to
$T^{*}=2$ at a fixed value $(m^{*})^{2}=1$ of the dipole moment the agreement
between the MSA and the simulation data is improved and the values of
$\epsilon_{0}$ and $\rvert\lambda^{*}\rvert$ are significantly reduced. The
discrepancy between the Debye-Weiss theory and the simulation data is
decreased because the critical density $\rho^{*}_{c}=1.432$, at which
$\epsilon_{0}$ and $\lambda^{*}$ diverge according to the Debye-Weiss theory,
is increased.
Figure 4: Same as in Fig. 3 for $T^{*}=2$. Within the Debye-Weiss theory both
$\epsilon_{0}$ and $\lambda^{*}$ diverge at $\rho_{c}^{*}=1.432$.
From Figs. 1-4 we conclude that for hard core dipolar Yukawa fluids the MSA
describes the linear and nonlinear dielectric permittivities with an adequate
accuracy up to liquid number densities $\rho^{*}{\lesssim}\,0.8$ (note that
freezing occurs at $\rho^{*}\simeq{0.8}$; see Ref. kl1 for the related
Stockmayer system), while the Langevin and the Debye-Weiss theory can be used
only for $\rho^{*}{\lesssim}\,0.1$ and $\rho^{*}{\lesssim}\,0.2$,
respectively.
Within the framework of our MSA theory and in agreement with our MC simulation
data we have obtained a negative nonlinear dielectric permittivity for hard
core dipolar Yukawa fluids. We note that using a cluster expansion and the
quadratic hypernetted chain (QHNC) approximation Martina and Stell st1
obtained positive nonlinear dielectric permittivities for dipolar hard sphere
fluids. They tried to explain the positive sign in terms of electrostriction
(i.e., the dependence of $\rho$ on $E$) which is included in their theory.
However, from an experimental point of view this explanation is unlikely
because the effect of electrostriction is so weak kr1 that it cannot
overcompensate the contribution to $\Delta\epsilon$ of molecular orientations
in the presence of an applied external field (see Ref. kr1 ). Note that Eq.
(33) as well as Eq. (16) are also valid for dipolar hard spheres, which
implies that within MSA the dispersion forces do not influence the dielectric
properties.
## V Summary
In the present study of the nonlinear dielectric effect for dipolar fluids the
following main results have been obtained:
1) We have applied an extension of the mean spherical approximation (MSA)
theory to determine the internal electric field dependence of the polarization
for hard core dipolar Yukawa fluids.
2) From the electric field dependence of the polarization analytical equations
have been obtained for the linear and the nonlinear dielectric permittivity of
these dipolar fluids. The predicted nonlinear dielectric permittivity is
negative, which corresponds to the so-called normal saturation effect of
dielectric media.
3) Canonical Monte Carlo simulations have been carried out for the
determination of the linear and nonlinear dielectric permittivity of dipolar
Yukawa fluids. There is good agreement between the results from the MSA and
the Monte Carlo simulation data for the reduced dipole moments
$(m^{*})^{2}\leq 1$ studied here (see Figs. 1-4).
4) We have compared our theoretical results with the corresponding Langevin
and Debye-Weiss approximations. The observed trends are in agreement with the
low density behavior of the various approximations (see Eqs. (48) and (49)).
Our new theoretical approach provides a quantitatively reliable description
for the nonlinear dielectric permittivity. This raises the expectation that
actual experimental systems can be analyzed and understood along these lines.
It will be also interesting to detect model systems which exhibit anomalous
dielectric saturation, i.e., a positive nonlinear dielectric permittivity.
###### Acknowledgements.
I. Szalai would like to thank the Hungarian Scientific Research Fund (Grant
No. OTKA K61314) for financial support.
## References
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* (28) H. Kitamura and A. Onuki, J. Chem. Phys. 123, 124513 (2005).
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* (30) J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids (Elsevier, Amsterdam, 2005).
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|
arxiv-papers
| 2009-08-13T19:18:08 |
2024-09-04T02:49:04.648484
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Szalai, S. Nagy, and S. Dietrich",
"submitter": "Istvan Szalai",
"url": "https://arxiv.org/abs/0908.1959"
}
|
0908.2271
|
This paper has been withdrawn by the author due to the paper is far from
complishment.
|
arxiv-papers
| 2009-08-17T14:13:47 |
2024-09-04T02:49:04.658005
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin-zhang Tang, Bin Chen, Shi Pi",
"submitter": "Jinzhang Tang",
"url": "https://arxiv.org/abs/0908.2271"
}
|
0908.2289
|
Spherical means in annular regions in the
$n$-dimensional real hyperbolic spaces
Rama Rawat and R. K. Srivastava
###### Abstract
Let $Z(Ann(r,R))$ be the class of all continuous functions $f$ on the annulus
$Ann(r,R)$ in the real hyperbolic space $\mathbb{B}^{n}$ with spherical means
$M_{s}f(x)=0$, whenever $s>0$ and $x\in\mathbb{B}^{n}$ are such that the
sphere $S_{s}(x)\subset\mbox{Ann}(r,R)$ and $B_{r}(0)\subseteq B_{s}(x).$ In
this article, we give a characterization for functions in $Z(Ann(r,R))$. In
the case $R=\infty$, this result gives a new proof of Helgason’s support
theorem for spherical means in the real hyperbolic spaces.
AMS Classification: 30F45, 33C55, 43A85.
## 1 Introduction
Let $g$ be a continuous function on the open annulus
$\\{x\in\mathbb{R}^{d}:r<|x|<R)\\},~{}0\leq r<R\leq\infty,d\geq 2.$ We say
that $g$ satisfies the Vanishing Spherical Means Condition if
$\int_{|x-y|=s}g(y)d\sigma_{s}(y)=0$
for every sphere $\\{y\in\mathbb{R}^{d}:|x-y|=s\\}$ which is contained in the
annulus and is such that the closed ball $\\{y\in\mathbb{R}^{d}:|y|\leq r\\}$
is contained in the closed ball $\\{y\in\mathbb{R}^{d}:|x-y|\leq s\\}.$ Here
$d\sigma_{s}$ is the surface measure on the sphere
$\\{y\in\mathbb{R}^{d}:|x-y|=s\\}.$
For a continuous function $g$ on $\mathbb{R}^{d},$ let
$g(x)=\sum_{k=0}^{\infty}\sum_{j=1}^{d_{k}}~{}a_{kj}(\rho)~{}Y_{kj}(\omega)$ (
1.1 )
be the spherical harmonic expansion, where $x=\rho\omega,\rho=|x|,\omega\in
S^{d-1}$ and $\\{Y_{kj}(\omega):~{}j=1,2,\cdots\cdots d_{k}\\}$ is an
orthonormal basis for the space $V_{k}$ of homogeneous harmonic polynomials in
$d$ variables of degree $k$ restricted to the unit sphere $S^{d-1}.$ Then the
following interesting results has been proved in [EK] by Epstein and Kleiner:
###### Theorem 1.1.
Let $g$ be a continuous function on the annulus
$\\{x\in\mathbb{R}^{d}:r<|x|<R\\},~{}0\leq r<R\leq\infty.$ Then $g$ satisfies
the Vanishing Spherical Means Condition if and only if
$a_{kj}(\rho)=\sum_{i=0}^{k-1}~{}\alpha_{kj}^{i}~{}\rho^{k-d-2i},~{}\quad\alpha_{kj}^{i}\in~{}\mathbb{C},$
for all $k>0,$ $1\leq j\leq d_{k},$ and $a_{0}(\rho)=0$ whenever $r<\rho<R.$
In this paper, we investigate the following analogous problem for spherical
means in real hyperbolic spaces. Let
$\mathbb{B}^{n}=\\{x\in\mathbb{R}^{n}:~{}|x|^{2}=\sum x_{i}^{2}<1\\}$ be the
open unit ball in $\mathbb{R}^{n},~{}n\geq 2$, endowed with the Poincare
metric $ds^{2}=\lambda^{2}(dx_{1}^{2}+\cdots+dx_{n}^{2})$, where
$\lambda=2(1-|x|^{2})^{-1}$. Let $B_{s}(0)=\\{x\in\mathbb{B}^{n}:d(x,0)\leq
s\\}$ be the closed geodesic ball of radius $s$ with centre at origin and
$\mbox{Ann}(r,R)=\\{x\in\mathbb{B}^{n}:r<d(x,0)<R\\},~{}0\leq r<R\leq\infty,$
be an open annulus in $\mathbb{B}^{n}.$
For $s>0,$ let $\mu_{s}$ denote the surface measure on the geodesic sphere
$S_{s}(x)=\\{y\in\mathbb{B}^{n}:d(x,y)=s\\}.$ Let $f$ be a continuous function
on $\mathbb{B}^{n}$. Define the spherical means of $f$ by
$M_{s}f(x)=\dfrac{1}{A(s)}\int_{S_{s}(x)}f(y)d\mu(y),~{}~{}x\in\mathbb{B}^{n},$
( 1.2 )
where $A(s)=(\Omega_{n})^{-1}(\sinh s)^{-n+1}.$
Let $Z(\mbox{Ann}(r,R))$ be the class of all continuous functions on
$\mbox{Ann}(r,R)$ with the spherical means $M_{s}f(x)=0,$ whenever $s>0,$ and
$x\in\mathbb{B}^{n}$ are such that the sphere $S_{s}(x)\subset\mbox{Ann}(r,R)$
and ball $B_{r}(0)\subseteq B_{s}(x).$
Our main result is the following characterization theorem.
###### Theorem 1.2.
Let $f$ be a continuous function on $\mbox{Ann}(r,R)$. The a necessary and
sufficient condition for $f$ to be in $Z(\mbox{Ann}(r,R))$ is that its
spherical harmonic coefficients $a_{kj}(\rho)$ satisfy
$a_{kj}(\rho)=\sum_{i=1}^{k}C_{kj}^{i}\frac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-2}},~{}\forall~{}j,1\leq
j\leq d_{k}(n)~{}\mbox{and}~{}k\geq 1,~{}C_{kj}^{i}\in\mathbb{C}$
and $a_{0}(\rho)\equiv 0$ whenever $\tanh\frac{r}{2}<\rho<\tanh\frac{R}{2}$.
As the authors in [EK] have observed, their result for Euclidean spherical
means, can be used to derive result for some cases, real hyperbolic spaces
being one of them. We would however like to give a direct proof of Theorem 1.2
using the underline geometry of the real hyperbolic spaces. The case of other
real rank one symmetric spaces can be dealt with in a similar way.
## 2 Notation and Preliminaries
We begin with the realization of real hyperbolic spaces ( see [M], [Re]). Let
$O(1,n+1)$ be the group of all linear transformations which preserve the
quadratic form
$\langle y,y\rangle=y_{0}^{2}-\sum_{i=1}^{n+1}y_{i}^{2},y=(y_{0},y_{1},\ldots
y_{n+1})$
on $\mathbb{R}^{n+2}.$ This group is known as the Lorentz group and is equal
to
$\\{g\in M_{n+2}(\mathbb{R}):g^{t}Jg=J,~{}J=\text{diag}(1,-1,\ldots,-1)\\}.$
In particular, $O(1,n+1)$ leaves invariant the cone
$C=\left\\{y\in\mathbb{R}^{n+2}:\langle y,y\rangle=0\right\\}.$
With the inhomogeneous coordinates $\eta_{i}=y_{i}/y_{0}~{},i=1,\ldots,n+1,$
the relation $\langle y,y\rangle=0$ would imply that $\eta$ is in
$S^{n}=\\{\eta\in\mathbb{R}^{n+1}:~{}|\eta|=1\\}.$ Thus a point on $C$ gets
identified with a point on the sphere $S^{n}.$ Conversely for $\eta\in S^{n},$
$\eta^{*}=(1,\eta_{1},\ldots,\eta_{n+1})$ gives a point on the cone $C.$ As
$g\in O(1,n+1)$ acts on $\eta^{*}$ and $g\eta^{*}\in C,$ $g$ acts on $S^{n}$
via the above identification. More explicitly, $g\eta^{*}$ can be identified
with the point
$\left(\dfrac{(g\eta^{*})_{1}}{(g\eta^{*})_{0}},\ldots,\dfrac{(g\eta^{*})_{n+1}}{(g\eta^{*})_{0}}\right)$
in $S^{n}.$ ($(g\eta^{*})_{0}$ is nonzero, as $\eta^{*}$ is nonzero and
$g\eta^{*}$ is in $C.$ )
Let $O_{\pm}(1,n+1)\cong O(1,n+1)/\\{\pm I\\}$ be the subgroup of $O(1,n+1)$
which leaves invariant the positive cone
$C^{+}=\left\\{y=(y_{0},y_{1}\ldots,y_{n+1})\in\mathbb{R}^{n+2}:\langle
y,y\rangle=y_{0}^{2}-\sum_{i=1}^{n+1}y_{i}^{2}>0,~{}y_{0}>0\right\\}.$
Equivalently, $O_{\pm}(1,n+1)$ is equal to
$\\{g\in M_{n+2}(\mathbb{R}):g^{t}Jg=J,~{}J=\text{diag}(1,-1,\ldots,-1)\mbox{
with }g_{00}>0\\},$
where $g_{00}$ is the top left entry in the matrix of $g.$ In particular,
$O_{\pm}(1,n+1)$ leaves the cone $C^{0}=\left\\{y\in\mathbb{R}^{n+2}:\langle
y,y\rangle=0,~{}y_{0}>0\right\\}$ invariant. Moreover, as the action of $g$
and $-g$ in $O(1,n+1)$ on the sphere $S^{n}$ coincides, $O_{\pm}(1,n+1)$ also
acts on $S^{n}.$ In fact, this is the group of Mobius transforms on $S^{n}$.
The real hyperbolic space $\mathbb{B}^{n}$ is then isomorphic to the quotient
space $SO_{\pm}(1,n)/SO(n).$ This isomorphism is established as follows.
We identify $S^{n}\setminus\\{e_{n+1}\\}$ with $\mathbb{R}^{n}$ under the
stereographic projection from the point
$e_{n+1}=(0,\ldots,0,1)\in\mathbb{R}^{n+1}$ onto the plane $\eta_{n+1}=0.$
Then the $O_{\pm}(1,n+1)$ action on $S^{n}$ induces an action on
$\mathbb{R}^{n}\cup\\{\infty\\}$ and vice versa. It turns out that the
subgroup of $O_{\pm}(1,n+1)$ which stabilizes $\mathbb{B}^{n}$ is isomorphic
to $O_{\pm}(1,n)$. This can be seen as follows.
Let $x=(x_{1},\ldots,x_{n})\in\mathbb{B}^{n}$. Then the inverse stereographic
projection of $\eta\in S^{n}$ of $x$ is given by
$\eta_{i}=\dfrac{2x_{i}}{1+|x|^{2}},~{}i=1\ldots,n~{}\mbox{and}~{}\eta_{n+1}=\dfrac{|x|^{2}-1}{|x|^{2}+1}.$
( 2.3 )
Therefore, $x\in\mathbb{B}^{n}$ if and only if $\eta_{n+1}<0.$ Thus a subgroup
of $O_{\pm}(1,n+1)$ stabilizes the open unit ball $\mathbb{B}^{n}$ if and only
if it stabilizes the lower hemisphere $\\{\eta\in S^{n}:~{}\eta_{n+1}<0\\}$.
This subgroup in turn is isomorphic to $O_{\pm}(1,n)$, (see [M]). The elements
of this subgroup realized as elements of $O_{\pm}(1,n+1)$ look like
$\left(\begin{array}[]{cccc}g&0\\\ 0&1\\\ \end{array}\right),$
with $g\in O_{\pm}(1,n).$ Moreover, this action of $O_{\pm}(1,n)$ on
$\mathbb{B}^{n}$ is transitive and the orthogonal group $O(n)$ thought of as
$\left(\begin{array}[]{cccc}1&0\\\ 0&g\\\ \end{array}\right)$
inside $O_{\pm}(1,n)$ is the isotropy subgroup of the point origin in the ball
$\mathbb{B}^{n}.$ Thus $\mathbb{B}^{n}$ is isomorphic to the quotient space
$O_{\pm}(1,n)/O(n).$ Likewise, $\mathbb{B}^{n}\cong SO_{\pm}(1,n)/SO(n).$ Let
$G=SO_{\pm}(1,n)$ and $K=SO(n)$. Hence onwards, we will work with the
representation $G/K$ of $\mathbb{B}^{n}.$ Using the $G$-invariant metric
$dy_{0}^{2}-dy_{1}^{2}-\cdots-dy_{n}^{2}$ on the positive cone
$y_{0}^{2}-y_{1}^{2}-\cdots-y_{n}^{2}=1,~{}y_{0}>0$, $\mathbb{B}^{n}$ can be
endowed with a $G$-invariant Riemannian metric given by
$ds^{2}=\lambda^{2}|dx|^{2}.$ The distance $d(x,y)$ between points
$x,y\in\mathbb{B}^{n}$, in this metric, is then given by the formula
$\tanh\frac{1}{2}d(x,y)=\frac{|x-y|}{\sqrt{1-2x.y+|x|^{2}|y|^{2}}}.$
This makes $(\mathbb{B}^{n},d)$ into a Riemannian symmetric space. Group
theoretically, $\mathbb{B}^{n}=G/K$ is a real rank one symmetric space.
Further, let $G=K\overline{A_{+}}K$ be the Cartan decomposition of $G$, where
$A=\left\\{\left(\begin{array}[]{ccc}\cosh\frac{t}{2}&0&\sinh\frac{t}{2}\\\
0&I_{n-1}&0\\\ \sinh\frac{t}{2}&0&\cosh\frac{t}{2}\\\
\end{array}\right):~{}t\in\mathbb{R}\right\\},$
is the maximal abelian subgroup of $G$ and $A_{+}$ is the Weyl chamber
$\\{a_{t}:t>0\\}.$ Let M be the centralizer $\\{k\in K:ka=ak,\forall a\in
A\\}$ of $A$ in $K$. Therefore, $M$ is given by
$M=\left\\{\left(\begin{array}[]{ccc}1&0&0\\\ 0&m&0\\\ 0&0&1\\\
\end{array}\right):~{}m\in SO(n-1)\right\\}.$
Thus the boundary $S^{n-1}$ of $\mathbb{B}^{n}$ gets identified with $K/M$
under the map $\sigma M\rightarrow\sigma.e_{n},$ $\sigma\in K$ where
$e_{n}=(0,0,\ldots,1)\in\mathbb{R}^{n}$ and the elements of $G/K$ can be
thought as pairs $(a_{t},\omega),~{}t\geq 0,~{}\omega\in S_{n-1}.$ The point
$(a_{t},\omega)$ then is identified with the point
$(\cosh\frac{t}{2},~{}\sinh\frac{t}{2}.\omega)$ on the positive cone in
$\mathbb{R}^{n+1}$ and this point in turn, is identified with the point
$\tanh\frac{t}{2}\omega$ in $\mathbb{B}^{n}$.
Next we recall certain standard facts about spherical harmonics, for more
details see [T], p. 12.
Let $\hat{K}_{M}$ denote the set of all the equivalence classes of irreducible
unitary representations of $K$ which have a nonzero $M$-fixed vector. It is
well known that each representation in $\hat{K}_{M}$ has in fact a unique
nonzero $M$-fixed vector, up to a scalar multiple.
For a $\delta\in\hat{K}_{M},$ which is realized on $V_{\delta},$ let
$\\{e_{1},\ldots,e_{d(\delta)}\\}$ be an orthonormal basis of $V_{\delta},$
with $e_{1}$ as the $M$-fixed vector. Let $t^{ji}_{\delta}(\sigma)=\langle
e_{i},\delta(\sigma)e_{j}\rangle,$ $\sigma\in K$ and $\langle,\rangle$ stand
for the innerproduct on $V_{\delta}.$ By Peter-Weyl theorem, it follows that
$\\{\sqrt{d(\delta)}t^{j1}_{\delta}:1\leq j\leq
d(\delta),\delta\in\hat{K}_{M}\\}$ is an orthonormal basis of $L^{2}(K/M).$
We would further need a concrete realization of the representations in
$\hat{K}_{M},$ which can be done in the following way.
Let $\mathbb{Z}^{+}$ denote the set of all non-negative integers. For
$k\in\mathbb{Z}^{+}$, let $P_{k}$ denote the space of all homogeneous
polynomials $P$ in $n$ variables of degree $k.$ Let $H_{k}=\\{P\in
P_{k}~{}:~{}\Delta P=0\\}$ where $\Delta$ is the standard Laplacian on
$\mathbb{R}^{n}.$ The elements of $H_{k}$ are called the solid spherical
harmonics of degree $k.$ It is easy to see that the natural action of $K$
leaves the space $H_{k}$ invariant. In fact the corresponding unitary
representation $\pi_{k}$ is in $\hat{K}_{M}.$ Moreover, $\hat{K}_{M}$ can be
identified, up to unitary equivalence, with the collection
$\\{\pi_{k}:k\in\mathbb{Z}^{+}.\\}$
Define the spherical harmonics on the sphere $S^{n-1}$ by
$Y_{kj}(\omega)=\sqrt{d_{k}}t^{j1}_{\pi_{k}}(\sigma),$ where
$\omega=\sigma.e_{n}\in S^{n-1},$ $\sigma\in K$ and $d_{k}$ is the dimension
of $H_{k}.$ Then $\\{Y_{kj}:1\leq j\leq d_{k},k\in\mathbb{Z}^{+}\\}$ forms an
orthonormal basis for $L^{2}(S^{n-1}).$ Therefore, for a continuous function
$f$ on $\mathbb{B}^{n},$ writing $y=\rho\,\omega,$ where $0<\rho<1$ and
$\omega\in S^{n-1},$ we can expand the function $f$ in terms of spherical
harmonics as in the (1.1) For each non negative integer $k$, the $k^{th}$
spherical harmonic projection, $\Pi_{k}(f)$ of the function $f$ is defined by
$\Pi_{k}(f)(y)=~{}\sum_{j=1}^{d_{k}}~{}a_{kj}(\rho)~{}Y_{kj}(\omega).$ ( 2.4 )
## 3 Auxiliary results
We begin with the observation that the $K$-invariance of the annulus and the
measure $\mu_{s}$ implies that for any $f$ in $Z(Ann(r,R))$ and
$k\in\mathbb{Z}^{+}$, $\Pi_{k}(f),$ as defined in equation (2.4), also belongs
to $Z(Ann(r,R)).$ In fact the following stronger result is true.
###### Lemma 3.1.
Let $f\in Z(\mbox{Ann}(r,R))$. Then each spherical harmonic projection
$\Pi_{k}(f)$ belongs to $Z(\mbox{Ann}(r,R))$ and
$a_{kj}(\rho)Y_{kj}(\omega)\in Z(\mbox{Ann}(r,R))~{}\forall~{}j,~{}1\leq j\leq
d_{k}$ and for all $k\geq 0.$
###### Proof.
Since the measure $\mu_{s}$ and space $\mbox{Ann}(r,R)$ both are rotation
invariant. Therefore, it is easy to verify that, if $f\in Z(\mbox{Ann}(r,R))$,
then the function $f(\tau.y)\in Z(\mbox{Ann}(r,R))$ for each $\tau\in K$.
Since space $H_{k}$ is $K$-invariant, therefore for $\tau\in K$ and a
spherical harmonic $Y_{kj}$, we have
$Y_{kj}(\tau^{-1}\omega)=\sum_{m=1}^{d_{k}}\overline{t^{mj}_{\pi_{k}}(\tau)}Y_{km}(\omega).$
Hence from the equation (1.1), the function $f(\tau^{-1}.)$ can be decomposed
as
$f(\tau^{-1}\rho\omega)=\sum_{k\geq
0}\sum_{j,m=1}^{d_{k}}a_{kj}(\rho)\overline{t^{mj}_{\pi_{k}}(\tau)}Y_{km}(\omega).$
Since, the set $\\{t^{mj}_{\pi_{k}}:1\leq j,m\leq d_{k},k\geq 0\\}$ form an
orthonomal basis for $L^{2}(K)$. Therefore,
$a_{kj}(\rho)Y_{km}(\omega)=d_{k}\int_{K}f(\tau^{-1}\rho\omega)t^{mj}_{\pi_{k}}(\tau)d\tau\in
Z(\mbox{Ann}(r,R)).$
Subsequently, each projection $\Pi_{k}(f)$ belongs to $Z(\mbox{Ann}(r,R))$. ∎
Next we need the following explicit expression for action of $G$ on
$\mathbb{B}^{n},$ which has been derived in [J].
###### Lemma 3.2.
Let $g\in G$ and $\;x\in\mathbb{B}^{n}$. Then $g.(x_{1},\ldots
x_{n})=(y_{1},\ldots,y_{n})$, where
$y_{j}=\dfrac{\frac{(1+|x|^{2})}{2}g_{j0}+\sum_{l=1}^{n}g_{jl}x_{l}}{\frac{1-|x|^{2}}{2}+\frac{(1+|x|^{2})}{2}g_{00}+\sum_{l=1}^{n}g_{0l}x_{l}},~{}j=1,\ldots,n.$
( 3.5 )
###### Proof.
By equation (2.3), a point $x\in\mathbb{B}^{n}$ is mapped to the point
$\eta\in S^{n}$ via the the inverse stereographic projection. By definition,
for $g\in G$,
$g\cdot\eta=\left(\begin{array}[]{cccc}g&0\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{c}1\\\ \eta\\\
\end{array}\right)=\alpha,$
where $\alpha=(\alpha_{0},\ldots,\alpha_{n},\eta_{n+1})$ and
$\alpha_{j}=g_{j0}+\sum_{l=1}^{n}g_{jl}\eta_{l},~{}l=0,1,\ldots,n$. Since the
cone $C^{0}$ is $G$-invariant, it follows that $\alpha_{0}>0$. In the
inhomogeneous coordinates, introduced earlier, the point $\alpha$ gets
identified with the point
$\left(\dfrac{\alpha_{1}}{\alpha_{0}},\ldots,\dfrac{\alpha_{n}}{\alpha_{0}},\dfrac{\eta_{n+1}}{\alpha_{0}}\right)$
on the sphere $S^{n}$. The image of this point, under the stereographic
projection is the point $y=(y_{1},\ldots y_{n})\in\mathbb{B}^{n}$, where
$y_{j}=\dfrac{\alpha_{j}/\alpha_{0}}{1-\eta_{n+1}/\alpha_{0}},~{}j=1,\ldots,n.$
That is
$y_{j}=\dfrac{g_{j0}+\sum_{l=1}^{n}g_{jl}\eta_{l}}{g_{00}+\sum_{l=1}^{n}g_{0l}\eta_{l}-\eta_{n+1}},~{}j=1,\ldots,n.$
Since we know that
$\eta_{l}=\dfrac{2x_{l}}{1+|x|^{2}},~{}l=1\ldots,n,~{}\eta_{n+1}=\dfrac{|x|^{2}-1}{|x|^{2}+1},$
a simple computation gives
$y_{j}=\dfrac{\frac{(1+|x|^{2})}{2}g_{j0}+\sum_{l=1}^{n}g_{jl}x_{l}}{\frac{1-|x|^{2}}{2}+\frac{(1+|x|^{2})}{2}g_{00}+\sum_{l=1}^{n}g_{0l}x_{l}},~{}~{}j=1,\ldots,n.$
∎
As in the proof of the Euclidean case [EK], to characterize functions in
$Z(\mbox{Ann}(r,R),)$ it would be enough to characterize the spherical
harmonic coefficients of smooth functions in $Z(\mbox{Ann}(r,R))$. This can be
done using the following approximation argument.
Let $\varphi_{\epsilon}$ be nonnegative, $K$-biinvariant, smooth, compactly
supported approximate identity on $G/K$. Let $f\in Z(\mbox{Ann}(r,R))$. Then
$f$ can be thought as a right $K$-invariant function on $G$. Define
$S_{\epsilon}(f)(g)=\int_{G}f(gh^{-1})\varphi_{\epsilon}(h)dh,~{}g\in G.$
Then $S_{\epsilon}(f)$ is smooth and it is easy to see that
$S_{\epsilon}(f)\in Z(\mbox{Ann}(r+\epsilon,R-\epsilon))$ for each
$\epsilon>0.$ Since $f$ is continuous, $S_{\epsilon}(f)$ converges to $f$
uniformly on compact sets. Therefore, for each $k$,
$\lim\limits_{\epsilon\rightarrow 0}\Pi_{k}(S_{\epsilon}(f))=\Pi_{k}(f).$
Hence, we can assume, without loss of generality, that the functions in
$Z(\mbox{Ann}(r,R))$ are also smooth in the annulus $\mbox{Ann}(r,R).$
We next introduce right $K$-invariant differential operators on $G/K$ which
leave invariant the space $Z(\mbox{Ann}(r,R))$. These differential operators
arise naturally from the Lie algebra $\mathfrak{g}$ of $G$, in the following
way. They also appear prominently in the work of Volchkov on ball means in
real hyperbolic spaces, (see [V], p. 108).
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition of
the Lie algebra $\mathfrak{g}$ of $G$. Here $\mathfrak{k}$ is the Lie algebra
of $K$ and $\mathfrak{p}$ its orthogonal complement in $\mathfrak{g}$ with
respect to the killing form $B(-,-)$. Let
$X_{i}=E_{0i}+E_{i0},~{}i=1,\ldots,n$ and $X_{ij}=E_{ij}-E_{ji},~{}1\leq
i<j\leq n$, where $E_{ij}\in gl_{n+1}(\mathbb{R})$ is the matrix with entry
$1$ at the ${ij}^{th}$ place and zero elsewhere. Then
$\\{X_{i}:~{}i=1,\ldots,n\\}$ and $\\{X_{ij}:~{}1\leq i<j\leq n\\}$ form bases
of $\mathfrak{p}$ and $\mathfrak{k}$ respectively.
Let $f\in C^{\infty}(\mathbb{B}^{n})$. Then $f$ can be thought as the right
$K$-invariant function on $G$. For given $X\in\mathfrak{g}$, let $\tilde{X}$
be the differential operator given by
$\displaystyle(\tilde{X}f)(gK)=\left.\dfrac{d}{dt}\right\rvert_{t=0}f(\exp
tXgK).$ ( 3.6 )
For $X=X_{p}\in\mathfrak{p}$, let
$\tau_{t,p}=\exp tX_{p}=\left(\begin{array}[]{ccccccccc}\cosh t&0&\sinh t&0\\\
0&I_{p-1}&0&0\\\ \sinh t&0&\cosh t&0\\\ 0&0&0&I_{n-p}\end{array}\right),$
for $t\in\mathbb{R}$. Let $x\in\mathbb{B}^{n}$. Then by Lemma 3.2,
$\tau_{t,p}.x=y\in\mathbb{B}^{n},$ where
$y_{j}=x_{j}u(t,x),~{}\mbox{if}~{}j\neq p$ and $y_{p}=(x_{p}\cosh
t+(1+|x|^{2})\frac{\sinh t}{2})u(t,x)$,
$u(t,x)=(\cosh^{2}\frac{t}{2}+x_{p}\sinh t+|x|^{2}\sin^{2}\frac{t}{2})^{-1}$.
Rewrite $\tau_{t,p}.x$ as $\tau(t,x)$. Then $\tau$ is a differentiable
function on $\mathbb{R}\times\mathbb{R}^{n}$ into $\mathbb{R}^{n}$ and from
(3.6), we have
$\frac{\partial}{\partial
t}(fo\tau(t,x))=f^{\prime}(\tau(t,x))\frac{\partial\tau}{\partial
t}(t,x)=\sum_{j=1}^{n}\frac{\partial f}{\partial y_{j}}\frac{\partial
y_{j}}{\partial t}.$
Evaluating the above equation at $t=0$, we get
$\displaystyle\left.\frac{\partial}{\partial
t}(fo\tau(t,x))\right\rvert_{t=0}=\sum_{j=1}^{n}\left.\frac{\partial
f}{\partial y_{j}}\right\rvert_{t=0}\left.\frac{\partial y_{j}}{\partial
t}\right\rvert_{t=0}=\sum_{j=1}^{n}\frac{\partial f}{\partial
x_{j}}\left.\frac{\partial y_{j}}{\partial t}\right\rvert_{t=0}.$ ( 3.7 )
A straightforward calculation then gives,
$\left.\frac{\partial y_{j}}{\partial
t}\right\rvert_{t=0}=\left\\{\begin{array}[]{ll}-x_{p}x_{j}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}j\neq
p,\\\
\frac{1}{2}(1+|x|^{2})-x_{p}^{2}~{}~{}~{}~{}~{}~{}\textrm{if}~{}j=p.\end{array}\right.$
Substituting these values in (3.7), we get
$\tilde{X}_{p}=\frac{1}{2}(1+|x|^{2})\dfrac{\partial}{\partial
x_{p}}-\sum_{j=1}^{n}~{}x_{p}x_{j}\dfrac{\partial}{\partial
x_{j}},~{}p=1,\ldots,n.$
The following lemma is a crucial step towards the proof of our main result.
###### Lemma 3.3.
Suppose $f$ is a smooth function belonging to $Z(\mbox{Ann}(r,R))$. Then
$\tilde{X}_{p}f\in Z(\mbox{Ann}(r,R)),~{}\forall~{}p,~{}1\leq p\leq n.$
###### Proof.
For $t\in\mathbb{R}$, define
$\epsilon_{1}=\sup_{y\in B_{r}(0)}d(\tau_{t,p}.y,y)\text{ and
}\epsilon_{2}=\sup_{y\in B_{R}(0)}d(\tau_{t,p}.y,y).$
Then it is easy to see that the translated function $\tau_{t,p}f$ defined by
$\tau_{t,p}f(y)=f(\tau_{t,p}.y),~{}y\in\mathbb{B}^{n}$ belongs to
$Z(\mbox{Ann}(r+\epsilon_{1},R-\epsilon_{2})).$ Therefore,
$\int_{S_{s}(x)}~{}f(\tau_{t,p}.\xi)d\mu_{s}(\xi)=\int_{S_{s}(\tau_{t,p}.x)}~{}f(\xi)d\mu_{s}(\xi)=0,$
whenever $S_{s}(x)\subset\mbox{Ann}(r+\epsilon_{1},R-\epsilon_{2})$ and
$B_{r+\epsilon_{1}}(0)\subset B_{s}(x).$ As $t\rightarrow 0$, this implies
$\int_{S_{s}(x)}\left.\frac{\partial f}{\partial
t}\right\rvert_{t=0}(\tau_{t,p}.\xi)d\mu_{s}(\xi)=0,$
whenever $S_{s}(x)\subset\mbox{Ann}(r,R)$ and $B_{r}(0)\subseteq B_{s}(x).$
Hence $\tilde{X}_{p}f\in Z(\mbox{Ann}(r,R)).$ ∎
A repeated application of Lemma 3.3, leads naturally to a family of
differential operators which we now introduce. These operators also appear in
the work of Volchkov ([V], p.108) in the problems on averages over geodesic
balls in real hyperbolic spaces. Let $C^{1}(0,1)$ denote the space of all
differentiable functions on $(0,1)$. For $m\in\mathbb{Z}$, the set of
integers, define a differential operator $A_{m}$ on $C^{1}(0,1)$ by
$\mathbb{(}A_{m}f)(t):=\frac{t^{m}}{(1-t^{2})^{m-1}}\frac{d}{dt}\left[\left(\frac{1}{t}-t\right)^{m}f(t)\right].$
( 3.8 )
The Laplace-Beltrami operator $\mathcal{L}_{x}$ on $\mathbb{B}^{n}$ is given
by
$\mathcal{L}_{x}=\frac{(1-|x|^{2})^{n}}{4}\sum_{i}\frac{\partial}{\partial
x_{i}}\left(\sum_{i}(1-|x|^{2})^{2-n}\frac{\partial}{\partial x_{i}}\right).$
The radial part $\mathcal{L}_{s}$ of $\mathcal{L}_{x}$ is given by
$\mathcal{L}_{s}=\frac{\partial^{2}}{\partial s^{2}}+(n-1)\coth
s\;\frac{\partial}{\partial s}$
and satisfies the Darboux equation
$M_{s}\mathcal{L}_{x}=\mathcal{L}_{s}M_{s}$.
For any positive integer $k$, let
$\mathcal{L}_{k}=\mathcal{L}-4(k-1)(n+k-2)\mbox{Id}.$
Let $f(x)=a(\rho)Y_{k}(\omega),$ where $Y_{k}$ is a spherical harmonic of
degree $k$. Then, a simple calculation shows that
$\mathcal{L}_{k}f(x)=A_{k-1}A_{2-k-n}a(\rho)Y_{k}(\omega),~{}x=\rho\omega.$
###### Lemma 3.4.
Let $x=\rho\omega,~{}0<\rho<1$ and $\omega\in S^{n-1}$ and $k\geq 0$. Suppose
the function $f(x)=a(\rho)Y_{k}(\omega)\in Z(\mbox{Ann}(r,R))$. Then following
are true.
(i)
$A_{2-k-n}a(\rho)Y_{(k-1)j}(\omega)\in Z(\mbox{Ann}(r,R)),k\geq 1$ and $1\leq
j\leq d_{k-1}(n),$
(ii)
$A_{k}a(\rho)Y_{(k+1)i}(\omega)\in Z(\mbox{Ann}(r,R)),k\geq 0$ and $1\leq
i\leq d_{k+1}(n),$
(iii)
$A_{1-k-n}A_{k}a(\rho)Y_{k}(\omega)$ belongs to $Z(\mbox{Ann}(r,R)),k\geq 0$
and
(iv)
$\mathcal{L}_{k}f(x)=A_{k-1}A_{2-k-n}a(\rho)Y_{k}(\omega)\in
Z(\mbox{Ann}(r,R)),~{}k\geq 1.$
###### Proof.
Let $k\geq 1$. Let $P(x)=\rho^{k}Y_{k}(\omega)$ and
$\tilde{a}(\rho)=\rho^{-k}a(\rho)$. Then $f(x)=\tilde{a}(\rho)P(x)$, where
$P(x)\in H_{k}$. By Lemma 3.3, the function $2\tilde{X_{p}}f\in
Z(\mbox{Ann}(r,R))$ $\forall~{}1\leq p\leq n.$ A straightforward calculation
then gives
$\displaystyle
2\tilde{X_{p}}f=\left(\frac{(1-\rho^{2})}{\rho}\frac{\partial\tilde{a}}{\partial\rho}-2k\tilde{a}\right)x_{p}P+(1+\rho^{2})\tilde{a}\frac{\partial
P}{\partial x_{p}}.$ ( 3.9 )
Further,
$x_{p}P=P_{k+1}+\frac{|x|^{2}}{n+2(k-1)}\frac{\partial P}{\partial x_{p}},$
where $P_{k+1}\in H_{k+1}$ (for a proof, see [EK]). Let $l=2-k-n$, then (3.9)
gives
$2\tilde{X_{p}}f=\left(\frac{(1-\rho^{2})}{\rho}\frac{\partial\tilde{a}}{\partial\rho}-2k\tilde{a}\right)\left(P_{k+1}+\frac{\rho^{2}}{k-l}\frac{\partial
P}{\partial x_{p}}\right)+(1+\rho^{2})\tilde{a}\frac{\partial P}{\partial
x_{p}}.$
After a rearrangement of terms, we get
$\displaystyle 2(k-l)\tilde{X_{p}}f$ $\displaystyle=$
$\displaystyle(k-l)\left(\frac{(1-\rho^{2})}{\rho}\frac{\partial\tilde{a}}{\partial\rho}-2k\tilde{a}\right)P_{k+1}$
$\displaystyle+$
$\displaystyle\left(\rho(1-\rho^{2})\frac{\partial\tilde{a}}{\partial\rho}-2k\rho^{2}\tilde{a}+(k-l)(1+\rho^{2})\tilde{a}\right)\frac{\partial
P}{\partial x_{p}}.$
Since $\tilde{a}(\rho)=\rho^{-k}~{}a(\rho)$,
$\dfrac{\partial\tilde{a}}{\partial\rho}=-k\rho^{-k-1}a+\rho^{-k}\dfrac{\partial
a}{\partial\rho}.$ Using this in the above equation, we have
$\displaystyle 2(k-l)\tilde{X_{p}}f$ $\displaystyle=$
$\displaystyle(k-l)\left((1-\rho^{2})\frac{\partial
a}{\partial\rho}-k\frac{(1+\rho^{2})}{\rho}a\right)\rho^{-k-1}P_{k+1}$ ( 3.10
) $\displaystyle+$ $\displaystyle\left((1-\rho^{2})\frac{\partial
a}{\partial\rho}-l\frac{(1+\rho^{2})}{\rho}a\right)\rho^{-k}\frac{\partial
P}{\partial x_{p}}.$
Also the operator $A_{m}$, given by (3.8), can be rewritten as
$A_{m}=(1-t^{2})\frac{d}{dt}-m\frac{(1+t^{2})}{t}.$
Thus (3.10) can be rephrased as
$2(k-l)\tilde{X_{p}}f=A_{k}a(\rho)\rho^{-k-1}P_{k+1}+A_{2-k-n}a(\rho)\rho^{-k+1}\frac{\partial
P}{\partial x_{p}}\in Z(\mbox{Ann}(r,R)),$
whenever $1\leq p\leq n$. Consequently, by Lemma 3.1, we get
$A_{k}a(\rho)\rho^{-k-1}P_{k+1}\in Z(\mbox{Ann}(r,R))$ and
$A_{2-k-n}a(\rho)\rho^{-k+1}\dfrac{\partial P}{\partial x_{p}}$ are in
$Z(\mbox{Ann}(r,R))$ and in particular $A_{2-k-n}a(\rho)~{}Y_{(k-1)j}(\omega)$
and $A_{k}a(\rho)Y_{(k+1)i}(\omega)$ are in $Z(\mbox{Ann}(r,R)).$
The assertions (iii) and (iv) can be obtained by composing (i) and (ii). ∎
## 4 Proof of the main result
In this section we prove our main result Theorem 1.2. We first take up the
necessary part of the theorem.
###### Proposition 4.1.
Let $f$ be a radial function in $Z(\mbox{Ann}(r,R))$. Then $f\equiv 0$ on
$\mbox{Ann}(r,R)$.
###### Proof.
By hypothesis
$\int_{S_{s}(x)}~{}f(\rho)d\mu_{s}(y)=0,$
whenever $x\in\mathbb{B}^{n}$ is such that the sphere
$S_{s}(x)\subseteq\mbox{Ann}(r,R)$ and ball $B_{r}(0)\subseteq B_{s}(x).$
Evaluating at $x=0$, this implies
$\int_{S_{s}(0)}~{}f(|y|)d\mu_{s}(y)=0,\mbox{ whenever }~{}~{}R>s>r.$
Thus $f(\tanh\frac{s}{2})=0,~{}R>s>r$. ∎
###### Proposition 4.2.
Let $f(\rho\omega)=a(\rho)Y_{k}(\omega)\in Z(\mbox{Ann}(r,R)),k\geq 1.$ Then
$a(\rho)$ is given by is given by
$a(\rho)=\sum_{i=1}^{k}C_{i}\frac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-2}},C_{i}\in\mathbb{C},\mbox{
whenever }~{}~{}\tanh\frac{r}{2}<\rho<\tanh\frac{R}{2}.$ ( 4.11 )
###### Proof.
We use induction on $k$. For $k=1$, let $f(\rho\omega)=a(\rho)Y_{1}(\omega)\in
Z(\mbox{Ann}(r,R))$. Using Lemma 3.4(ii), it follows that
$A_{1-n}a(\rho)Y_{0}(\omega)$ belongs to $Z(\mbox{Ann}(r,R))$. Therefore, by
Proposition 4.1, $A_{1-n}a(\rho)=0,$ on $\mbox{Ann}(r,R).$ On solving this
differential equation, we get
$a(\rho)=C\left(\frac{1}{\rho}-\rho\right)^{n-1}.$
Next we assume the result is true for $k$. Suppose
$f(\rho\omega)=a(\rho)Y_{k+1}(\omega)\in Z(\mbox{Ann}(r,R))$. An application
of Lemma 3.4(ii) gives $A_{1-k-n}a(\rho)Y_{k}(\omega)\in Z(\mbox{Ann}(r,R))$.
Using the result for $k$ and the definition of $A_{1-k-n}$, it follows that
$\frac{\rho^{1-k-n}}{(1-\rho^{2})^{-k-n}}\frac{\partial}{\partial\rho}\left(\left(\frac{1}{\rho}-\rho\right)^{1-k-n}a(\rho)\right)=\sum_{i=1}^{k}C_{i}\frac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-2}}.$
Simplifying this equation and integrating both sides with respect to $\rho$,
we obtain
$\left(\frac{1}{\rho}-\rho\right)^{1-k-n}a(\rho)=\sum_{i=1}^{k}D_{i}\frac{1}{(1-\rho^{2})^{k-i+2}}+D_{k+1},D_{i}\in\mathbb{C}.$
Hence
$a(\rho)=\sum_{i=1}^{k+1}D_{i}\frac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-1}},$
whenever $\tanh\frac{r}{2}<\rho<\tanh\frac{R}{2}$. ∎
Now, we shall prove the sufficient part of Theorem 1.2. For this, without loss
of generality, we may assume that $R=\infty$. The idea of the proof is to use
the asymptotic behavior of the hypergeometric function and compare it with
that of the coefficients given in (4.11). In the proof, we need, the following
result from [EMOT], p. 75.
###### Lemma 4.1.
The general solution of the hypergeometric differential equation
$z(1-z)U^{\prime\prime}+\\{\gamma-(\alpha+\beta+1)z\\}U^{\prime}-\alpha\beta
U=0,$ ( 4.12 )
where $\alpha,\beta,\gamma$ are independent of $z$, in the neighborhood of
$\infty$ is given in the following way. If $\alpha-\beta$ is not an integer
then
$U(z)=\lambda_{1}z^{-\alpha}+\lambda_{2}z^{-\beta}+O\left(z^{-\alpha-1}\right)+O\left(z^{-\beta-1}\right),$
otherwise $z^{-\alpha}$ or $z^{-\beta}$ has to be multiplied by a factor of
$\log z$.
###### Theorem 4.2.
Let $y=\rho\omega,~{}\omega\in S^{n-1}$ and $\tanh\frac{r}{2}<\rho<\infty.$
Let $h(y)=a(\rho)Y_{k}(\omega)$ with
$a(\rho)=\sum_{i=1}^{k}C_{i}\frac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-2}},~{}C_{i}\in\mathbb{C}.\text{
Then }h\in Z(\mbox{Ann}(r,\infty)).$
###### Proof.
We use the induction hypothesis on $k$. Let $k=1$ and
$h(y)=a(\rho)Y_{1}(\omega)=\left(\frac{1}{\rho}-\rho\right)^{n-1}Y_{1}(\omega).$
Then the function $A_{1-n}\left(\frac{1}{\rho}-\rho\right)^{n-1}Y_{0}(\omega)$
is identically zero and therefore it belongs to $Z(\mbox{Ann}(r,\infty))$.
Using Lemma 3.4(ii), we have
$A_{0}A_{1-n}\left(\frac{1}{\rho}-\rho\right)^{n-1}Y_{1}(\omega)=A_{0}A_{1-n}h(y)=0.$
Thus $\mathcal{L}_{y}h(y)=0.$ Again by Darboux’s equation
$\mathcal{L}_{s}M_{s}h=M_{s}\mathcal{L}_{y}h,$ the above leads to
$\mathcal{L}_{s}(M_{s}h)=0.$ Define $F_{1}(s,x)=M_{s}h(x)$. For fixed
$x,F_{1}$ as a function of $s$ satisfies the differential equation
$\dfrac{\partial^{2}F_{1}}{\partial s^{2}}+(n-1)\coth s~{}\dfrac{\partial
F_{1}}{\partial s}=0.$ ( 4.13 )
Setting $z=-\sinh^{2}s$ then, we get
$\dfrac{\partial F_{1}}{\partial s}=\dfrac{\partial F_{1}}{\partial
z}\dfrac{\partial z}{\partial s}=-\sinh 2s\dfrac{\partial F_{1}}{\partial
z},~{}\dfrac{\partial^{2}F_{1}}{\partial s^{2}}={(\sinh
2s)}^{2}\dfrac{\partial^{2}F_{1}}{\partial z^{2}}-2\cosh 2s\dfrac{\partial
F_{1}}{\partial z}.$
After substituting these values in (4.13), we obtain
$-4z(1-z)\dfrac{\partial^{2}F_{1}}{\partial
z^{2}}-2\\{n-(n+1)\\}z\dfrac{\partial F_{1}}{\partial z}=0.$ ( 4.14 )
Comparing this equation with (4.12), we get
$\gamma=\frac{n}{2},\alpha+\beta+1=\frac{n+1}{2},\alpha\beta=0$. For
$\alpha=0$, $\beta=\frac{n-1}{2}$. The solution of (4.14) as
$|z|\rightarrow\infty$ is given by
$F_{1}(z,x)=\left\\{\begin{array}[]{ll}\lambda_{1}(x)z^{-\frac{(n-1)}{2}}+O\left(z^{-\frac{(n-1)}{2}-1}\right)~{}~{}~{}~{}~{}\mbox{if}~{}~{}\frac{(n-1)}{2}\not\in\mathbb{Z};\\\
\lambda_{2}(x)z^{-\frac{(n-1)}{2}}\log
z+O\left(z^{-\frac{(n-1)}{2}-1}\right)\mbox{ otherwise.}\\\
\end{array}\right.$ ( 4.15 )
On the other hand for $x=g.o,g\in G$
$\displaystyle M_{s}h(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{A(s)}\int_{S_{s}(x)}h(y)d\mu_{s}(y),$ $\displaystyle=$
$\displaystyle\dfrac{1}{A(s)}\int_{S_{s}(o)}h(g^{-1}.y)d\mu_{s}(y).$
From above the equation, it follows that
$M_{s}h(x)=O\left(\left|a(\tanh\frac{s}{2})\right|\right),\mbox{ as
}s\rightarrow\infty.$ ( 4.16 )
From (4.16) one can conclude that any function of type
$h(y)=a(\rho)Y_{k}(\omega),$ must satisfies the relation
$M_{s}h(x)=O(a(\tanh\frac{s}{2}))$. In fact, for $k=1$,
$\left|a(\tanh\frac{s}{2})\right|=\left|\cosh\frac{s}{2}\sinh\frac{s}{2}\right|^{-(n-1)}=2^{(n-1)}\left|z\right|^{-\frac{(n-1)}{2}}.$
( 4.17 )
From (4.16) and (4.17), we have $F_{1}(z,x)=O(z^{-\frac{(n-1)}{2}})$, as
$|z|\rightarrow\infty$. In view of (4.15), we infer that $F_{1}(z,x)=0$,
whenever $|z|>\sinh^{2}r$. Thus $M_{s}h(x)=0$, whenever $x\in\mathbb{B}^{n}$
is such that the ball $B_{r}(0)\subseteq B_{s}(x)$ and $r<s<\infty$, which
proves the result for $k=1$.
To complete the induction argument, we assume the result is true for $k-1$ and
then prove for $k$. For this, consider the function
$h(y)=a(\rho)Y_{k}(\omega)=\dfrac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-2}}Y_{k}(\omega),$
for each $i,~{}1\leq i\leq k.$ Using Lemma 3.4(i) and the case $(k-1)$, it
follows that
$(A_{2-k-n}a)(\rho)Y_{k-1}(\omega)=\dfrac{(1-\rho^{2})^{n+i-2}}{\rho^{n+k-3}}Y_{k-1}(\omega)\in
Z(\mbox{Ann}(r,\infty)).$
Applying Lemma 3.4(ii), it follows that
$\mathcal{L}_{k}h(y)=(A_{k-1}A_{2-k-n})(a)Y_{k}(\omega)$ belongs to
$Z(\mbox{Ann}(r,\infty)).$ Since we know that
$\mathcal{L}_{k}h(y)=L_{y}h(y)-4(k-1)(n+k-2)h(y),$
therefore, evaluating mean and using Darboux’s equation, we obtain
$\mathcal{L}_{s}(M_{s}h(x))-4(k-1)(n+k-2)M_{s}h(x)=0,$
whenever $x\in\mathbb{B}^{n}$ is such that the ball $B_{r}(0)\subseteq
B_{s}(x)$ and $r<s<\infty$. Let $F_{k}(s,x)=M_{s}h(x)$. For fixed $x,F_{k}$ as
a function of $s$ satisfies the differential equation
$\dfrac{\partial^{2}F_{k}}{\partial s^{2}}+(n-1)\coth s~{}\dfrac{\partial
F_{k}}{\partial s}-4(k-1)(n+k-2)F_{k}=0.$
Using the change of variable $z=-\sinh^{2}t$, the above equation becomes
$-4z(1-z)\dfrac{\partial^{2}F_{k}}{\partial
z^{2}}-2\\{n-(n+1)z\\}\dfrac{\partial F_{k}}{\partial
z}-4(k-1)(n+k-2)F_{k}=0.$ ( 4.18 )
Comparing this equation with (4.12), we have
$\gamma=\frac{n}{2},\alpha+\beta+1=\frac{n+1}{2},\alpha\beta=-(k-1)(n+k-2)$.
On solving, we find
$\alpha-\beta=\pm\nu,\nu=\frac{\sqrt{{(n-1)}^{2}+4(k-1)(n+k-2)}}{2}.$ Clearly,
$\nu\not\in\mathbb{Z}.$ Therefore, solution of (4.18) as
$|z|\rightarrow\infty$ is given by
$F_{k}(z,x)=\lambda_{1}(x)z^{-\alpha}+\lambda_{2}(x)z^{-\beta}+O\left(z^{-\alpha-1}\right)+O\left(z^{-\beta-1}\right),$
( 4.19 )
where $\alpha=\frac{n-1+2\nu}{4},~{}\beta=\frac{n-1-2\nu}{4}$. But from the
given expression of function $h$, one can find
$M_{s}h(x)=O\left(\left|a(\tanh\frac{s}{2})\right|\right),\mbox{ as
}s\rightarrow\infty.$
Using $z=-\sinh^{2}s$, it follows that
$\left|a(\tanh\frac{s}{2})\right|=2^{n+i-2}\frac{({1+\sqrt{1+|z|})}^{k-i}}{|z|^{\frac{n+k-2}{2}}}$
That is,
$F_{k}(z,x)=O(z^{\frac{-(n+i-2)}{2}}),i=1,\cdots,n\mbox{ as
}|z|\rightarrow\infty.$
In view of (4.19), we infer that $F_{k}(z,x)=0$, whenever $|z|>\sinh^{2}r$.
Thus $M_{s}h(x)=0$, whenever $x\in\mathbb{B}^{n}$ is such that the ball
$B_{r}(0)\subseteq B_{s}(x)$ and $r<s<\infty$, which proves the result for any
positive integer $k$. This completes the proof. ∎
As a corollary of Theorem 1.2, we have the following Helgason support theorem
( see [H], p. 156).
###### Theorem 4.3.
Let $f$ be a function on $\mathbb{B}^{n}$. Suppose for each
$m\in\mathbb{Z}^{+}$, the function $e^{md(x,~{}0)}f(x)$ is bounded. Then $f$
is supported in closed geodesic ball $B_{r}(0)$ if and only if $f\in
Z(\mbox{Ann}(r,\infty))$.
###### Proof.
The decay condition on function $f$ implies that for all $k$ and $j$,
$a_{kj}(|x|)=0,$ whenever $|x|>\tanh\frac{r}{2}$. This proves $f$ is supported
in the ball $B_{r}(0)$. ∎
Acknowledgements: The second author wishes to thank the MHRD, India, for the
senior research fellowship and IIT Kanpur for the support provided during the
preparation of this work.
## References
* [EK] C. L. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl.Math. 46 (1993), no.3, 441-451.
* [EMOT] Erd lyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Tricomi and Francesco G. Higher transcendental functions (Based on notes left by Harry Bateman), Vol. I. McGraw-Hill, New York, 1953.
* [H] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.
* [J] P. Jaming, Harmonic functions on the real hyperbolic ball. I. Boundary values and atomic decomposition of Hardy spaces, Colloq. Math. 80 (1999), no. 1, 63–82.
* [M] G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes tudes Sci. Publ. Math. No. 34 1968 53–104.
* [Re] H. M. Reimann, Invariant differential operators in hyperbolic space, Comment. Math. Helv. 57 (1982), no. 3, 412–444.
* [T] S. Thangavelu, An introduction to the uncertainty principle, Prog. Math. 217, Birkhauser, Boston (2004).
* [V] V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht, The Nertherlands, 2003.
Department of Mathematics and Statistics,
Indian Institute of Technology
Kanpur 208 016
India.
E-mail: rrawat@iitk.ac.in, rksri@iitk.ac.in
|
arxiv-papers
| 2009-08-17T14:15:46 |
2024-09-04T02:49:04.661619
|
{
"license": "Public Domain",
"authors": "Rama Rawat and R. K. Srivastava",
"submitter": "Rajesh Srivastava Dr.",
"url": "https://arxiv.org/abs/0908.2289"
}
|
0908.2339
|
11institutetext: Centro de Rádio Astronomia e Astrofísica Mackenzie, R. da
Consolação 896, 01302-907, São Paulo, SP, Brazil. 22institutetext: LESIA,
Observatoire de Paris, Section de Meudon, 92195 Meudon, France.
33institutetext: Space Sciences Laboratory, University of California,
Berkeley, USA. 44institutetext: Instituto Nacional de Pesquisas Espaciais, São
José dos Campos, Brazil. 55institutetext: Centro de Componentes
Semicondutores, Universidade Estadual de Campinas, Campinas, Brazil.
66institutetext: Complejo Astronómico El Leoncito, CONICET, San Juan,
Argentina.
# Submillimeter and X-ray observations of an X Class flare
C.G. Giménez de Castro 11 G. Trottet 22 A. Silva-Valio 11 S. Krucker 33 J.E.R.
Costa 44 P. Kaufmann 1155 E. Correia 1144 H. Levato 66
The GOES X1.5 class flare that occurred on August 30,2002 at 1327:30 UT is one
of the few events detected so far at submillimeter wavelengths. We present a
detailed analysis of this flare combining radio observations from 1.5 to 212
GHz (an upper limit of the flux is also provided at 405 GHz) and X-ray.
Although the observations of radio emission up to 212 GHz indicates that
relativistic electrons with energies of a few MeV where accelerated, no
significant hard X-ray emission was detected by RHESSI above $\sim$ 250 keV.
Images at 12–20 and 50–100 keV reveal a very compact, but resolved, source of
about $\sim 10\arcsec\ \times 10\arcsec$. EUV TRACE images show a multi-kernel
structure suggesting a complex (multipolar) magnetic topology. During the peak
time the radio spectrum shows an extended flatness from $\sim 7$ to 35 GHz.
Modeling the optically thin part of the radio spectrum as gyrosynchrotron
emission we obtained the electron spectrum (spectral index $\delta$,
instantaneous number of emitting electrons). It is shown that in order to keep
the expected X-ray emission from the same emitting electrons below the RHESSI
background at 250 keV, a magnetic field above 500 G is necessary. On the other
hand, the electron spectrum deduced from radio observations $\geq 50$ GHz is
harder than that deduced from $\sim 70-250$ keV X-ray data, meaning that there
must exist a breaking energy around a few hundred keV. During the decay of the
impulsive phase, a hardening of the X-ray spectrum is observed which is
interpreted as a hardening of the electron distribution spectrum produced by
the diffusion due to Coulomb collisions of the trapped electrons in a medium
with an electron density of $n_{e}\sim 3-5\ 10^{10}\ \mathrm{cm}^{-3}$.
###### Key Words.:
Sun:activity – flares – particle emission – radio radiation – X-ray gamma-ray
## 1 Introduction
During solar flares, a fraction of the released energy is used to accelerate
electrons with energies well above 1 MeV. The interaction of these particles
with the magnetic field of the flaring region produces gyrosynchrotron /
synchrotron radiation observed at cm or smaller wavelengths (see e.g. Bastian
et al., 1998; Pick & Vilmer, 2008, for reviews) and a bremsstrahlung continuum
caused by Coulomb collisions observed with X- and $\gamma$-ray detectors. It
was shown (e.g. Kundu et al., 1994) that the electron spectrum $N(E)$
determined by means of $\geq$ 30 GHz radio observations is harder than that
deduced from Hard X-ray (HXR) below a few hundred keV. However for a few
events, the electron spectra were found consistent with spectra inferred from
$\gamma$-ray continuum above $\sim$ 1 MeV (Trottet et al., 1998, 2000). Since
radio emission above 30 GHz is produced mainly by electrons of a few MeV (see
e.g. White & Kundu, 1992; Ramaty et al., 1994), these results have an impact
on acceleration mechanism models, which are still, an open question in solar
flare theory, and reinforces the need for good diagnostics of the $>$ 1 MeV
particles.
Continuum X- and $\gamma$-ray detectors may observe photons from a few keV up
to tens of MeV, but have as a limitation the low sensitivity and / or high
background in the high energy range. In the past three solar cycles only a few
tens of flares have been observed above 1 MeV. On the other hand,
radioastronomy at millimeter and submillimeter wavelengths is more efficient
than the $\gamma$-ray detectors. Routine solar flare observations at 212 and
405 GHz started in March 2001 with the Solar Submillimeter Telescope (SST,
Kaufmann et al., 2001), installed in the Argentinean Andes. A few flares were
also observed at 210, 230, and 345 GHz with a receiver array installed at the
focus of the Köln Observatory for Submillimeter and Millimeter Astronomy
(KOSMA) telescope (Lüthi et al., 2004b, a). The first observations using such
instruments showed that the spectrum above 100 GHz is a continuation of the
cm-wavelength optically thin spectrum (e.g. Trottet et al., 2002; Lüthi et
al., 2004b) and extended the diagnostic tools of radio observations to higher
energy (a few tens of MeV) electrons. However, an unexpected upturn of the
spectrum above 100 GHz was reported for other $\geq$ M class events (e.g.
Kaufmann et al., 2004; Lüthi et al., 2004a; Cristiani et al., 2008). The
physical processes responsible for the production of the spectrum upturn are
still unknown and are a subject of debate (Kaufmann & Raulin, 2006; Silva et
al., 2007; Trottet et al., 2008).
In this paper, we present a combined analysis of the impulsive phase of the
August 30, 2002, X class flare using RHESSI X-ray observations and spatially
unresolved radio data covering the range between 1.5 to 212 GHz (and an upper
limit for 405 GHz) obtained by different instruments. The event has been
analyzed by different authors. Karlický et al. (2004) related radio
observations between 0.8 and 2.0 GHz and X-ray spectra and images from RHESSI.
They found high-frequency drifting structures between 1327:38 and 1327:50 UT
with a global drift of -25 MHz s-1. The 10–20 keV X-ray sources show a north-
east displacement with a projected velocity of about 10 km s-1, while the
29–44 keV emission is delayed by about 0.5 to 0.7 s after the radio drifting
structure. Microwave observations of a short pulse during the onset of this
event was analyzed by Giménez de Castro et al. (2006) who found a strikingly
narrow spectrum that was explained as gyrosynchrotron emission of accelerated
electrons with a maximum energy (high energy cutoff) of about 250 keV. Another
event with similar properties was qualitatively discussed by Lüthi et al.
(2004b). In this work we extend the analysis of Giménez de Castro et al.
(2006) to the entire event. Moreover, we perform a quantitative analysis of
the data which allows us to estimate the characteristics of the emitting
electrons (energy spectrum, total number) and of the flaring region (density,
magnetic field strength) that are necessary to account for the apparent
discrepancy between X-ray and radio observations.
## 2 Instrumentation
Figure 1: Time evolution of the August 30, 2002 flare, at different radio
frequencies and in selected SXR and HXR energy channels. At 89.4 and 212 GHz
the dashed curve represents the computed contribution of an isothermal source.
A,B, and C indicate, respectively, characteristic time bins around the maximum
of the HXR emission, the maximum of the 212 GHz radiation, and the decay phase
of the burst. Horizontal bars denote the time intervals I through IV (see
text).
The hard X-ray (HXR) and radio data used in the present analysis of the August
30, 2002 event were obtained with the NASA Reuven Ramaty High Energy Solar
Spectroscopic Imager (RHESSI), the Solar Submillimeter Telescope (SST,
installed in the Argentinean Andes), the nulling interferometer and patrol
telescopes of the University of Bern (Switzerland), the Radio Solar Telescope
Network (RSTN), and the Solar Radio-polarimeter of the Radio Observatory of
Itapetinga (ROI, Brazil).
RHESSI provides imaging and spectral HXR/$\gamma$-ray observations, with high
spatial ($\sim$ 2 arc sec) and spectral ($\sim$ 1 keV) resolution in the
$\sim$ 3 keV – 17 MeV energy range (Lin & et al., 2002).
The SST (Kaufmann et al., 2000) operates simultaneously at 212 and 405 GHz and
with a time resolution of 1 ms for the present event. The focal system
consists of four receivers at 212 GHz and two at 405 GHz. At 212 GHz this
produces a cluster of beams that, in principle, allows us to determine the
centroid of the emitting region whenever an event is detected (see Giménez de
Castro et al., 1999, and references therein for details). During the August
30, 2002 flare, SST was tracking NOAA region 10095, with one of the two 405
GHz beams pointing at the active region. At 212 GHz, the event was observed
with only one beam so that it was not possible to estimate the centroid
position of the emitting region. The antenna temperatures have been corrected
for atmospheric attenuation (zenith optical depth $\tau_{212}=0.3$ nepers and
$\tau_{405}=1.35$ nepers) and converted to flux density assuming that (i) the
source is much smaller than the beam size and (ii) there is no important main-
lobe gain correction due to a possible offset pointing. Since the HPBW of the
beams are respectively $\sim$ 4$\arcmin$ and 2$\arcmin$ at 212 and 405 GHz,
hypothesis (i) is justified here, because this event is very compact in the
HXR domain (see Sect. 3.2). On the other hand, the projected position of the
beam that observed the burst in the sky is separated by less than 30$\arcsec$
from the HXR emitting region observed by RHESSI. As this is comparable to the
absolute position uncertainty of the SST antenna, hypothesis (ii) is also
justified. It should be emphasized that a misalignment of 30$\arcsec$ produces
a main-lobe gain correction of less than 5% at 212 GHz (HPBW $\sim$
4$\arcmin$).
The two-element nulling interferometer of the University of Bern provides
total flux measurements at 89.4 GHz with a sensitivity of $\sim$ 35 s.f.u. (1
s.f.u. = 10-22 W m-2 Hz-1) and a time resolution of 31 ms (Lüthi et al.,
2004b). Total flux densities at 11.8, 19.6, 35, and 50 GHz were recorded by
the patrol telescopes at Bumishus (Switzerland) with a time resolution of 100
ms. Total flux density measurements made at 1.415, 2.695, 4.995, 8.8, and 15.4
GHz by the RSTN with a time resolution of 1 s, and at 7 GHz by the Solar
Radio-polarimeter of the Radio Observatory of Itapetinga with 20 ms time
resolution have also been used.
Figure 2: Background subtracted count rate spectra measured by RHESSI (error
bars) during time bins A,B, and C marked in Fig. 1. The asterisks show the
background spectrum. The continuous lines show the best fit models (see text).
The vertical dashed lines indicate the fitted energy range.
## 3 Observations and data analysis
The August 30, 2002 HXR and radio event starting at $\sim$ 1327:30 UT is
associated with a GOES X1.5 SXR burst and, with a surprisingly small,
H$\alpha$ sub-flare which occurred in NOAA Active Region 10095 (N15 E74). This
event was observed by RHESSI up to $\sim$ 250 keV around its maximum and in a
large part of the radio spectrum ranging from submillimeter to decameter and
longer wavelengths. Figure 1 displays the time profile of the event in the 100
– 150 keV HXR band, in the SXR 1–8 Å channel; it also shows the total flux
densities at 11.8, 89.4, and 212 GHz. The flare comprises an impulsive phase
that starts at $\sim$ 1327:40 UT in the 100 – 150 keV band and which lasts for
about 60 s.
### 3.1 Flux and spectra
#### 3.1.1 Hard X-rays
Figure 3: From top to bottom: time evolution of the HXR spectral index
$\gamma$ and of the radio spectral index $\alpha$ (dots), of the radio flux
density at 212 GHz (the dot-dashed curve represents the computed contribution
of an isothermal source), and of the 100 – 150 keV emission. Horizontal bars
show the time intervals I through IV (see text).
Spectral analysis of RHESSI data was performed between 1327:28 and 1328:36 UT.
Count rate spectra were accumulated between 1327:38 and 1327:44 UT (before the
thick shutter came in) and in all 4 second intervals between 1327:52 and
1328:32 UT using front detectors 1, 3–6, 8 and 9. We applied pile up and
decimation corrections. Each spectra consists of 77 energy bands between 3 and
250 keV. For each interval, spectral fitting was carried out for energies
ranging from 40 keV to the highest energy where count rates in excess of 2
$\sigma$ above background are measured. It is found that the count rate
spectra could be reasonably represented by considering either a single power
law or a double power-law. In the latter case, the break energy lies around 70
keV. Since we are interested in the non-thermal X-ray emission, we have
restricted the analysis to the energies above 70 keV using a single power-law
of spectral index $\gamma$ for the trial photon spectra. Fig 2 displays
examples of the fitted spectra for the time bins labeled A, B, and C in Fig.
1. The time evolution of $\gamma$ is shown in Fig. 3.
#### 3.1.2 Radio
Figure 4: The radio spectrum at instants labeled A,B, and C. (see Figure 1)
Solid curves represent the homogeneous gyrosynchrotron solution fits discussed
in Sect. 4.3.
At frequencies above 20 GHz there is a time extended emission that lasts for
tens of minutes after the impulsive phase (see Fig 1 at 89.4 and 212 GHz).
This gradual component is most likely of thermal origin because it seems to
follow the SXR emission from GOES and has no HXR counterpart. The comparison
of the $>$ 20 GHz time evolution with that of the 1 – 8 Å SXR indicates that
this thermal component may have started at the beginning of the impulsive
phase and therefore should be subtracted from the total flux densities in
order to estimate the non thermal emission of the radio burst. For this we
consider that the thermal emission arises from the hot thermal source that
produces the SXR emission observed by GOES. We computed the free-free flux
density of an isothermal source with temperature and emission measure (EM)
derived from GOES 8 observations. A source size of 60″ provides a reasonable
agreement with the observations as illustrated in Fig. 1 (dot-dash lines at
89.4 and 212 GHz curves). Fig. 4 shows the non-thermal radio spectrum for the
three time bins marked A, B and C in Fig. 1 that correspond, respectively, to
the maximum of the 100–150 keV emission, to the maximum of the 212 GHz, and to
the decay of the impulsive phase. The main characteristics of the radio
spectra can be summarized as follows:
From 1 to 7 GHz:
the spectrum increases with frequency and can be represented by a power law
with spectral index $\sim 1$.
From 7 to 35 GHz:
there is a plateau observed during the impulsive phase (Fig. 4, A and B).
During the decay of the impulsive phase the spectrum gradually evolves
exhibiting a rather well defined turnover frequency around 5 GHz (Fig. 4, C).
Above 50 GHz:
the flux density is roughly proportional to $\nu^{-\alpha}$. Fig. 3 shows the
time evolution of $\alpha$.
### 3.2 HXR Images
Figure 5: RHESSI emitting regions of 50 – 100 keV (solid contours) and of 12 -
20 keV (dashed contours) for different time intervals superimposed on a
negative 195 Å image taken by TRACE at 1327:31 UT. Contour levels are: 50, 70,
90, and 99% of the image maximum.
Figure 5 displays RHESSI contours in the 12–20 keV and 50–100 keV bands
overlaid on a 195 Å TRACE image taken at 1327:31 UT for the four intervals
marked by horizontal bars in Fig. 1. Intervals I and II cover the first two
100 – 150 keV peaks, interval III spreads over the maximum of the 212 GHz,
while interval IV extends over the HXR decay phase. RHESSI images were
obtained by applying the PIXON algorithm (Metcalf et al., 1996) and by using
front detector segments 1 to 6. Detector 2 was not used for the 12 – 20 keV
images. The TRACE image was taken close to the onset of the radio and HXR
impulsive peaks. The figure shows that the impulsive phase of the flare was
triggered within a complex pattern of bright EUV features in a compact region
(10″$\times$ 10″). Unfortunately, subsequent 195 Å TRACE images are saturated
until 1350 UT. During intervals I and II, the 50–100 keV emission (solid
contours) arises from two compact regions ($\sim$ 3–7″) overlaying bright EUV
structures. During the rest of the event (intervals III to IV), only the
southeast source is observed at 50 – 100 keV. During the whole event the lower
energy HXR (12 – 20 keV, dashed contours) arise predominantly from a single
source close to the southeast of the 50 – 100 keV emitting region. However,
the interpretation of the HXR images at 50 keV for interval III and IV is
inconclusive as up to half of the counts are due to pile up of thermal
photons. To correct pile up in images is very difficult as the corrections
should be done in the modulated light curves. Currently, the RHESSI software
does not provide pile up correction in images.
## 4 Discussion
EUV observations close to the onset of the impulsive phase of the August 30,
2002 event reveal that this flare arose from a compact region with a multi
kernel structure that suggests a complex (multipolar) magnetic field topology.
The HXR emitting sources are observed in association with different EUV bright
structures, suggesting that they are located at the footpoints of different
magnetic loops (see Fig. 5). The compactness of this flare is further
supported by the fact that such an X class event produced only an H$\alpha$
sub-flare. In the following, we discuss some peculiarities of this HXR and
radio event:
* •
the unusually flat radio spectrum between 7 and 35 GHz during most of the
impulsive phase (intervals I to III on Fig. 1),
* •
the lack of significant $\gamma$ ray emission detected by RHESSI in the MeV
domain, although $\geq$ 90 GHz radio data indicate that relativistic electrons
were produced during the flare,
* •
the strong hardening of the $>$ 70 keV HXR spectrum during the decay of the
impulsive phase.
Fig. 3 (top) presents the time evolution of the HXR spectral index $\gamma$
obtained assuming that the photon distribution is a single power law from 70
keV to the highest energy where count rates are above the background (see
Sect. 3.1.1). During intervals I to III, the spectral index does not vary
significantly, remaining around $\gamma=4.3\pm 0.3$, however during interval
IV there is a spectral hardening. We consider that intervals I, II, and III
correspond to three successive different particle injections:
* •
during intervals I and II, the 100 – 150 keV emission is consistent with
thick-target emission because it arises predominantly from footpoint sources.
Furthermore there is no long-term coronal trapping because the HXR peaks occur
simultaneously at all energies within the RHESSI time resolution (4 s). Thus,
the time evolution of the HXR emission mimics that of the electron injection
into the thick-target region.
* •
Taking into account the above arguments, interval III corresponds to a third
injection which is reflected by the shoulder in the 100 – 150 keV time profile
range and corresponds to the maximum of the 212 GHz emission.
* •
The hardening of the HXR spectrum during interval IV may be indicative of some
trapping.
### 4.1 The flat radio spectrum
It is well documented that cm-mm emission comes from gyrosynchrotron radiation
of energetic relativistic electrons propagating in the magnetic structures
(e.g. Pick & Vilmer, 2008, and references therein). The radio spectra of the
present event are indeed reminiscent of this emission process. In a uniform
magnetic field, the emission would lead to a spectral index larger than 2.5 in
the optically thick region and a rather well defined turnover frequency.
However, the spectral index for the frequency range 1 – 7 GHz (optically thick
emission) remains around or below 1 during the whole event. And until the
decay of the impulsive phase (interval IV) the radio spectrum is flat between
7 and 35 GHz (see Fig. 4). Both these characteristics are indications that
radio emitting electrons propagate in a highly inhomogeneous magnetic field
(e.g. Dulk, 1985; Klein et al., 1986; Lee et al., 1994). This inhomogeneous
magnetic field interpretation is consistent with the complex magnetic
structure revealed by TRACE and RHESSI images. During the decay of the
impulsive phase, a well defined turnover frequency gradually becomes better
defined. At the same time, the X-ray emission arises from a single source, so
that the observed radio spectrum is closer to that expected from a homogeneous
source.
Ramaty & Petrosian (1972) explained flat microwave spectra as observed by
Hachenberg & Wallis (1961) by including the free-free absorption of a cold
medium uniformly mixed in the homogeneous gyrosynchrotron source region.
Indeed: (i) at low frequencies, where both gyrosynchrotron and free-free
opacities are $>$ 1, the radio flux increases with frequency; (ii) at
frequencies for which the gyrosynchrotron opacity is $<$ 1 while the free-free
opacity remains $>1$, a plateau is observed because while the gyrosynchrotron
emission starts to decrease the free-free emission is still increasing; (iii)
at higher frequencies both emissions are optically thin and the radio spectrum
decreases with frequency. The observation of simultaneous brightenings and
line broadening of hot ($\sim 10^{7}$ K) and cold ($\sim 10^{4}$ K) plasmas
during a solar limb flare (Kliem et al., 2002) provides some support to the
Ramaty & Petrosian hypothesis. In addition to magnetic field inhomogeneities,
free-free absorption may also contribute to provide the observed flat radio
spectrum. In that case, as the flare evolves, the free-free opacity should
decrease in order to allow lower frequency radiation to become optically thin.
Since free-free opacity is proportional to $n_{p}^{2}T_{p}^{-3/2}$ ($n_{p},\
T_{p}$ medium density and temperature, respectively, and assuming
$n_{e}=n_{p}$), this would imply either a decrease of the density, or an
increase of the temperature, or an increase of both. In the latter case the
temperature should increase faster than the density.
Above 50 GHz (optically thin emission), the radiation, which is mostly emitted
by highly relativistic electrons, is not affected by the medium, and the
spectral radio index $\alpha$ is only related to the index of the
instantaneous electron distribution $\delta$. For the present event $\alpha$
remains between 1.1 and 1.3 (Fig 3). Considering the ultra-relativistic case
as a gross approximation, this leads to $\delta=2\alpha+1=3.2-3.6$ (Dulk,
1985).
### 4.2 HXR spectral hardening during the decay of the impulsive phase
During interval IV the HXR spectral index $\gamma$ decreases from 4.1 to 2.8
(Fig. 3). This provides some indication that a significant amount of HXR was
produced by trapped electrons, the diffusion rate being governed by Coulomb
collisions. For the weak-diffusion limit case, the trapping time
$t^{trap}\simeq t^{def}$, with $t^{def}$ the characteristic deflection time,
which is given by Trubnikov (1965) and Melrose & Brown (1976)
$t^{def}\simeq\frac{E}{2}\left(\frac{dE}{dt}\right)^{-1}_{coll}\ ,$ (1)
where $(dE/dt)_{coll}$, the electron energy loss rate due to Coulomb
collisions, is approximated by (Bai & Ramaty, 1979)
$\left(\frac{dE}{dt}\right)_{coll}=\left\\{\begin{array}[]{l@{\quad}l}-4.9\
10^{-9}\ E^{-0.5}\ n_{e}&E\leq 160\ \mathrm{keV}\\\ -3.8\ 10^{-10}\
n_{e}&E>160\ \mathrm{keV}\\\ \end{array}\right.$ (2)
with $E$ the electron energy in keV and $n_{e}$ the medium electron density in
$cm^{-3}$.
We consider that the injection of electrons in the HXR emitting region stopped
at 1328:14 UT because it is the last time bin where $\gamma$ remains almost
constant (see Fig. 3). Under these conditions the time evolution of the
distribution of the trapped electrons is approximately given by (Aschwanden,
1998)
$N(E,t)\propto E^{-\delta}\exp{\left(-\frac{t}{t^{def}}\right)}\ ,$ (3)
with $\delta$ the electron index at the end of the injection. The photon flux
at energy $\epsilon$ as a function of time $t$ may be written as (see e.g.
Brown, 1971)
$F(\epsilon,t)=n_{e}\int_{\epsilon}^{\infty}\sigma(\epsilon,E)v(E)N(E,t)dE\ ,$
(4)
where $v(E)$ is the electron velocity corresponding to energy $E$,
$\sigma(\epsilon,E)$ is the bremsstrahlung differential cross-section per unit
photon energy $\epsilon$ for an electron of energy $E$. For $\sigma(E)$ we
adopted the Bethe-Heitler electron – proton bremsstrahlung cross-section. It
should be noted that the photon flux produced by precipitated electrons has a
time evolution as $F(\epsilon,t)$ (Melrose & Brown, 1976). Therefore, for a
given $\delta$, the time evolution of the photon flux depends only on $n_{e}$.
By using equation 4 we computed the expected photon flux $F(\epsilon,t)$ and
compared its time evolution at different energies $\epsilon$ with that
observed by RHESSI during the decaying of the impulsive phase. Comparing the
expected and observed time profiles at 70, 100, and 150 keV, we obtained
reasonable agreement for $n_{e}\sim 3-5\ 10^{10}\ \mathrm{cm}^{-3}$ when
$\delta$ ranges between 3.5 and 5, which correspond respectively to thin and
thick target limits for $\gamma\sim 4$. Krucker et al. (2008) find similar
electron densities for coronal $\gamma$-ray emission during flares.
Spectral hardening has been reported during the impulsive phase of long
duration GOES X class flares and associated with non thermal footpoint
bremsstrahlung (Qiu et al., 2004; Saldanha et al., 2008). Grigis & Benz (2008)
analyzed the spectral hardening during the gradual phase of great flares and
concluded that the cause is the continuing acceleration with longer trapping
in the accelerator before escape. Kiplinger (1995) has shown that the
hardening is associated with SEP events.
### 4.3 Relationship between HXR and radio emission
The absence of HXR emission $>$ 250 keV while we observe radio emission above
50 GHz can be used to constrain the high energy electron distribution, the
magnetic field, and the trapping time in the radio emitting region. For that,
we consider that non thermal electrons are injected in coronal loops. The
radio emission is produced in the coronal portion of these loops where they
become partially trapped while precipitating electrons produce the HXR
radiation by thick target interaction at the loop footpoints. Since no
spatially resolved radio data are available for this event and the optically
thin part of the radio spectrum does not depend on the structure details of
the medium, we used a homogeneous model to derive the mean parameters of the
radio source and emitting electrons. For that, we computed the radio optically
thin emission by using the numerical code for a gyrosynchrotron source with a
homogeneous ambient density and magnetic field and an isotropic electron
distribution developed by Ramaty (1969) and corrected by Ramaty et al.
(1994)111See http://lheawww.gsfc.nasa.gov/users/ramaty. The instantaneous
electron distribution in the radio source was taken as $N(E)=KE^{-\delta}$,
where $K$ is the number of electrons per MeV at 1 MeV and the energy $E$ is in
MeV. The angle between the observer and the magnetic field (view-angle) was
set to 84∘ (the maximum allowed value in Ramaty’s solution) in order to obtain
the lower limit of the total number of electrons necessary to produce a given
radio spectrum for a given magnetic field strength. For a view-angle of 45∘,
as is usually assumed, the computed number of electrons is roughly twice that
obtained for 84∘. Table 1 displays the values of the instantaneous total
number of electrons above 25 keV $N(>25\mathrm{keV})$ and $K$ obtained for
different values of the magnetic field at the maximum of the 100 – 150 keV HXR
(time bin A) . $N(>25\mathrm{keV})$ was computed for $\delta=3.4$ which
provides the best fit to the radio data. This is in agreement with the value
of $\delta$ inferred from the slope of the optically thin part of the radio
spectrum $\alpha$ (see sect. 4.1)
Table 1: Derived characteristic of radio radiating electrons for different magnetic field strengths and view angle equal to 84∘. B | N($>25$ keV) $\times 10^{33}$ | $K\times 10^{29}$
---|---|---
(Gauss) | | ($e^{-}$ MeV-1)
500 | 38.1 | 76.0
750 | 15.0 | 30.0
1000 | 7.5 | 15.0
1600 | 1.5 | 3.0
The mean electron flux $\dot{N}(E)$ entering the thick target HXR source is
$\dot{N}(E)\simeq N(E)/t^{trap}$, where $t^{trap}$ is the time spent by the
electrons in the radio source. The thick target photon emission from these
precipitating electrons was calculated by using a numerical code that takes
into account both electron-proton and electron-electron collisions. The photon
spectrum is then convolved with the RHESSI response matrix to get the
corresponding count rate spectrum. Figure 1 shows that within the RHESSI time
resolution (4 s), the 89.4 GHz, 212 GHz and 100-150 keV count rates show
simultaneous peaks during intervals I to III. The electron trapping time
$t^{trap}$ can thus be considered independent of energy as a first
approximation. This suggest that during intervals I to III the precipitation
rate is more likely governed by wave-particle interactions (turbulent
trapping) than by Coulomb collisions (e.g. Vilmer, 1987). The computations
were then carried out for $10^{-2}\leq t^{trap}\leq 4\ \mathrm{s}$, the lower
and upper limits correspond respectively to free streaming of the electrons in
a compact loop ($\sim 10$″), and to the RHESSI time resolution. Figure 6
displays the RHESSI expected count rates in the 250 – 265 keV band as a
function of $t^{trap}$ for different values of the magnetic field. The dashed
horizontal line corresponds to the RHESSI count rate in this channel, the
highest energy where this event was detected. We conclude that the mean
magnetic field strength should be greater than about 500 G to keep the thick
target photon flux expected from radio emitting electrons with trapping times
smaller than 1 s below the detection limit of RHESSI.
Figure 6: Expected RHESSI count rates at 250 keV as a function of the emitting
electron trapping time for different magnetic field strengths and view angle
equal to 84∘. Each solid line represents a different solution determined by a
different magnetic field. The dashed horizontal line represents the measured
background count rate at 250 keV.
It has been shown that the thick target spectral index $\gamma$ is bounded
between $\delta-1.5\leq\gamma\leq\delta-1$, the lower limit corresponding to
turbulent trapping of electrons with energies above a few 100 keV, whereas the
higher limit is set by free propagating electrons with energies below a few
100 keV (see e.g. Trottet et al., 1998, and references therein.). Therefore if
$\delta=3.4$, the HXR emission of these electrons should have an index between
$1.9\leq\gamma\leq 2.4$ which is significantly smaller than the observed
values, $\sim 4.3$, during intervals I to III (see Fig. 3). This is in
accordance with previous works that show that radio high frequency emission is
generated by electrons with energies above $\sim$ 500 keV with an electron
index harder than the $<$ 500 keV electrons (White & Kundu, 1992; Kundu et
al., 1994; Trottet et al., 1998; Silva et al., 2000).
## 5 Summary
In this paper we have analyzed X-ray observations from RHESSI and radio data
obtained at submillimeter wavelengths by the Solar Submillimeter Telescope
(SST) of the X1.5 event that occurred in Active Region 10095 on August 30,
2002, at 1327:30 UT, complemented with radio observations from 1.5 to 89.4 GHz
from other instruments. EUV images from TRACE provided information about the
source emitting region.
The radio spectrum above 100 GHz is the continuation of the optically thin
microwave spectrum, therefore does not belong to the so called THz bursts
(Kaufmann et al., 2004), although it is an X Class flare. We summarize below
our main findings:
* •
The magnetic structure of the flare is complex and highly inhomogeneous. This
is revealed by the 50 – 100 keV and EUV images. Such an inhomogeneous source
may produce the flatness in the radio spectrum observed between 7 and 35 GHz,
although we do not discard the free-free absorption.
* •
The electron spectrum $N(E)$ above 1 MeV is harder than that at energies below
a few hundred keV.
* •
Modeling simultaneously the expected RHESSI count rate and the expected
gyrosynchrotron emission, we obtain 500 G as a lower limit for the mean
magnetic field of the flaring region.
* •
The time evolution of the spectral index deduced from X-ray observations at
the end of interval III suggests that trapped electrons are diffused by
Coulomb collisions. This leads to a mean ambient electron density of $3-5\
10^{10}$ cm-3, typical of the low corona / upper chromosphere and is
compatible with previous results (Krucker et al., 2008) and with the small
size of the EUV pattern observed by TRACE, which also suggests that the
flaring region does not extend high in the corona.
Finally it should be emphasized that radio observation in the sub-THz domain
provide a unique tool to constrain acceleration model because they constitute
a more sensitive diagnostic of ultra-relativistic electrons than present
HXR/$\gamma$-ray measurements.
###### Acknowledgements.
This research was partially supported by Brazil Agencies FAPESP, CNPq and
Mackpesquisa, and Argentina Agency CONICET. CGGC also thanks the Observatory
of Paris in Meudon, that supported his stay to finish the present work. The
authors are in debt to A. Magun and T. Lüthi who provided the calibrated data
of the Bern patrol telescopes and of the nulling interferometer at 89.4 GHz.
They would also like to thank Dr. Lidia van Driel-Gesztelyi for helpful
discussions.
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|
arxiv-papers
| 2009-08-17T12:51:24 |
2024-09-04T02:49:04.669336
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.G. Gimenez de Castro, G. Trottet, A. Silva-Valio, S. Krucker, J.E.R.\n Costa, P. Kaufmann, E. Correia, H. Levato",
"submitter": "Carlos Guillermo Gim\\'enez de Castro",
"url": "https://arxiv.org/abs/0908.2339"
}
|
0908.2510
|
# Entropy of Partitions on Sequential Effect Algebras ††thanks: This project
is supported by Research Fund of Kumoh National Institute of Technology.
Wang Jiamei, Wu Junde, Cho Minhyung Department of Mathematics, Zhejiang
University, Hangzhou 310027, People’s Republic of China.Corresponding author:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s
Republic of China. E-mail: wjd@zju.edu.cnDepartment of Applied Mathematics,
Kumoh National Institute of Technology, Kyungbuk 730-701, Korea. E-mail:
mignon@kumoh.ac.kr
Abstract By using the sequential effect algebra theory, we establish the
partitions and refinements of quantum logics and study their entropies.
Key Words: Sequential effect algebra, Boolean algebra, entropy.
PACS numbers: 02.10-v, 02.30.Tb, 03.65.Ta.
1\. Introduction
Quantum entropy or Von Neumann entropy, which is a counterpart of the
classical Shannon entropy, is an important subject in quantum information
theory ([1]). In order to study the entropy of partition of quantum logics, in
[2], the author tried to define the partitions and refinements of quantum
logics, nevertheless, his methods are only suitable for classical logics, the
essential reasons are that the classical logics satisfy the distributive law
but quantum logics do not. In this paper, by using the sequential effect
algebra theory, we establish really effective refinement methods of quantum
logics and study their entropies.
2\. Classical logics and quantum logics
As we know, the classical logics can be described by the Boolean algebras and
the quantum logics can be described by the orthomodular lattices ([2-5]). The
classical probability or Shannon entropy was based on the classical logics and
quantum entropy was established on the quantum logics ([1]). Now, we recall
some elementary notions and conclusions of Boolean algebras and the
orthomodular lattices.
Let $(L,\leq)$ be a partially ordered set. If for any $a,b\in L$, its infimum
$a\wedge b$ and supremum $a\vee b$ exist, then $(L,\leq)$ is said to be a
lattice. If $(L,\leq)$ is a lattice and for any $a,b,c\in L$, we have
$\displaystyle a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c),$ (1)
$\displaystyle a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c),$ (2)
then we say that $(L,\leq)$ satisfies the distributive law. Let $(L,\leq)$ be
a lattice with the largest element $I$ and the smallest element $\theta$. If
there exists a mapping ${}^{\prime}:L\rightarrow L$ such that for each $a\in
L$, $a\vee a^{\prime}=I,a\wedge a^{\prime}=\theta$, $(a^{\prime})^{\prime}=a$
and $b^{\prime}\leq a^{\prime}$ whenever $a\leq b$, then $(L,\leq)$ is said to
be an orthogonal complement lattice. Let $(L,\leq)$ be an orthogonal
complement lattice. If $a,b\in L$ and $a\leq b$, we have
$b=a\vee(b\wedge a^{\prime}),$ $None$
then we say that $(L,\leq)$ satisfies the orthomodular law.
Definition 2.1 Let $(L,\leq)$ be an orthogonal complement lattice. If
$(L,\leq)$ satisfies the distributive law, then $(L,\leq)$ is said to be a
Boolean algebra; if $(L,\leq)$ satisfies the orthomodular law, then $(L,\leq)$
is said to be an orthomodular lattice.
Example 2.1 Let $X$ be a set and $2^{X}$ be its all subsets. Then
$(2^{X},\subseteq)$ is a Boolean algebra.
Example 2.2([4-5]) Let $H$ be a complex Hilbert space, $P(H)$ be the set of
all orthogonal projection operators on $H$, $P_{1},P_{2}\in P(H)$. If we
define $P_{1}\leq P_{2}$ if and only if $P_{1}P_{2}=P_{2}P_{1}=P_{1}$, then
$(P(H),\leq)$ is an orthomodular lattice.
Example 2.2 is the most important and famous quantum logic model which was
introduced in 1936 by Birkhoff and von Neumann ([4]).
Let $(L,\leq)$ be an orthomodular lattice and $a,b\in L$. If $a\leq
b^{\prime}$, then we say that $a$ and $b$ are orthogonal and denoted by
$a\perp b$. A subset $\\{a_{1},a_{2},\cdots,a_{n}\\}$ of $L$ is said to be an
orthogonal set if $a_{1}\perp a_{2},(a_{1}\vee a_{2})\perp
a_{3},\cdots,(a_{1}\vee a_{2}\vee\cdots a_{n-1})\bot a_{n}.$
Let $(L,\leq)$ be an orthomodular lattice and $s:L\rightarrow[0,1]$ be a
mapping from $L$ into the real number interval $[0,1]$. If $s(I)=1$ and
$s(a\vee b)=s(a)+s(b)$ whenever $a\bot b$, then $s$ is said to be a state of
$(L,\leq)$.
It is clear that if $s$ is a state of $(L,\leq)$ and
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ is a finite orthogonal subset of $L$, then
$s(\vee_{i=1}^{n}a_{i})=\sum_{i=1}^{n}s(a_{i})$.
In [2], the author defined the following three concepts:
Let $(L,\leq)$ be an orthomodular lattice and $s$ be a state of $(L,\leq)$,
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ be a finite orthogonal subset of $L$. If
$s(\vee_{i=1}^{n}a_{i})=1$, then $\\{a_{1},a_{2},\cdots,a_{n}\\}$ is said to
be a partition of $(L,\leq)$ with respect to the state $s$. If
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ is a partition of $(L,\leq)$ with respect to
the state $s$ and for each $b\in L$,
$s(b)=\sum_{i=1}^{n}s(a\wedge b),$
then $s$ is said to have the Bayes property. Moreover, let
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ and $\\{b_{1},b_{2},\cdots,b_{m}\\}$ be two
partitions of $(L,\leq)$ with respect to the state $s$. Then the set
$\\{a_{i}\wedge b_{j}:i=1,2,\cdots,n;j=1,2,\cdots,m\\}$ is said to be a
refinement of the partitions $\\{a_{1},a_{2},\cdots,a_{n}\\}$ and
$\\{b_{1},b_{2},\cdots,b_{m}\\}$ .
By the distributive law of Boolean algebra, it is clear that each state on the
Boolean algebra has the Bayes property. However, the following example shows
that there is no state $s$ on $(P(H),\leq)$ with the Bayes property, where $H$
is a complex Hilbert space with $\dim(H)=2$. Moreover, our example shows also
that the concept of refinement of partitions is also not effective for
$(P(H),\leq)$.
Example 2.3 Let $H$ be a complex Hilbert space with $\dim(H)=2$ and
$a_{1}=\\{(0,z):z\in\mathbb{C}\\}$, $a_{2}=\\{(z,0):z\in\mathbb{C}\\}$. If
$P_{1},P_{2}$ are the orthogonal projection operators from $H$ onto $a_{1}$
and $a_{2}$, respectively, then for any state $s$, $A=\\{P_{1},P_{2}\\}$ is a
partition of $(P(H),\leq)$ with respect to state $s$. Let
$b_{1}=\\{(\frac{\sqrt{2}z}{2},\frac{\sqrt{2}z}{2}):z\in\mathbb{C}\\},b_{2}=\\{(-\frac{\sqrt{2}z}{2},\frac{\sqrt{2}z}{2}):z\in\mathbb{C}\\}$
and $Q_{1},Q_{2}$ are the orthogonal projection operators on $b_{1}$ and
$b_{2}$, respectively. Then $P_{i}\wedge Q_{j}=0,i,j=1,2.$ So
$\vee_{i=1}^{n}(P_{i}\wedge Q_{j}))=0,j=1,2.$ If state $s$ has the Bayes
property, then we have $s(Q_{j})=s(0)=0,j=1,2$, so $s(Q_{1})+s(Q_{2})=0$. On
the other hand, note that $Q_{1}\bot Q_{2}$ and $Q_{1}\vee Q_{2}=I$, so
$1=s(I)=s(Q_{1})+s(Q_{2})=0$, this is a contradiction and so there is no state
$s$ on $(P(H),\leq)$ which has the Bayes property. Moreover, since
$P_{i}\wedge Q_{j}=0,i,j=1,2$, so $\\{P_{i}\wedge Q_{j}:i,j=1,2\\}$ cannot be
considered as a refinement of two partitions $\\{P_{1},P_{2}\\}$ and
$\\{Q_{1},Q_{2}\\}$.
Example 2.3 told us that we must redefine the refinement concept of partitions
of quantum logics.
In quantum theory, we have known that each orthogonal projection operator can
be looked as the sharp measurement. For two sharp measurements $P$ and $Q$, if
$P$ is performed first and $Q$ second, then $PQP$ have important physics
meaning ([6-8]). If $\\{P_{1},P_{2},\cdots,P_{n}\\}$ and
$\\{Q_{1},Q_{2},\cdots,Q_{m}\\}$ are two orthogonal sets of $(P(H),\leq)$ and
$\vee_{i=1}^{n}P_{i}=I$, $\vee_{i=1}^{m}Q_{i}=I$, then we may try to use
$\\{Q_{j}P_{i}Q_{j},i=1,2,\cdots,n,i=1,2,\cdots,n;j=1,2,\cdots,m\\}$
as the refinement of $\\{P_{1},P_{2},\cdots,P_{n}\\}$ and
$\\{Q_{1},Q_{2},\cdots,Q_{m}\\}$. However, note that, in general,
$Q_{j}P_{i}Q_{j}$ is not an orthogonal projection operator on $H$, that is,
$Q_{j}P_{i}Q_{j}\notin P(H)$, so we must to transfer the sharp measurements to
unsharp measurements. In 1994, Foulis and Bennett completed the famous
transformation, that is, they introduced the following algebra structure and
called it as the effect algebra ([9]):
Let $(E,\theta,I,\oplus)$ be an algebra system, where $\theta$ and $I$ be two
distinct elements of $E$, $\oplus$ be a partial binary operation on $E$
satisfying that:
(EA1) If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2) If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3) For every $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=I$.
(EA4) If $a\oplus I$ is defined, then $a=\theta$.
In an effect algebra $(E,\theta,I,\oplus)$, if $a\oplus b$ is defined, we
write $a\bot b$. For each $a\in E$, it follows from (EA3) that there exists a
unique element $b\in E$ such that $a\oplus b=1$, we denote $b$ by
$a^{\prime}$. Let $a,b\in E$, if there exists an element $c\in E$ such that
$a\bot c$ and $a\oplus c=b$, then we say that $a\leq b$. It follows from [9]
that $\leq$ is a partial order of $(E,0,1,\oplus)$ and satisfies that for each
$a\in E$, $0\leq a\leq 1$, $a\bot b$ if and only if $a\leq b^{\prime}$. If
$a\wedge a^{\prime}=0$, then $a$ is said to be a sharp element of $E$.
Let $H$ be a complex Hilbert space. A self-adjoint operator $A$ on $H$ such
that $0\leq A\leq I$ is called a quantum effect on $H$ ([6-9]). If a quantum
effect represents a measurement, then the measurement may be unsharp ([6, 9]).
The set of quantum effects on $H$ is denoted by $E(H)$. For $A,B\in E(H)$, if
we define $A\oplus B$ if and only if $A+B\leq I$ and let $A\oplus B=A+B$, then
$(E(H),\theta,I,\oplus)$ is an effect algebra, and its all sharp elements are
just $P(H)$ ([5-6, 9]).
Moreover, Professor Gudder introduced and studied the following sequential
effect algebra theory ([10-11]):
Let $(E,\theta,I,\oplus)$ be an effect algebra and another binary operation
$\circ$ defined on $(E,\theta,I,\oplus)$ satisfying that
(SEA1) The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2) $I\circ a=a$ for each $a\in E$.
(SEA3) If $a\circ b=\theta$, then $a\circ b=b\circ a$.
(SEA4) If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
for each $c\in E$, $a\circ(b\circ c)=(a\circ b)\circ c$.
(SEA5) If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,\theta,I,\oplus,\circ)$ be a sequential effect algebra. If $a,b\in E$
and $a\circ b=b\circ a$, then we say that $a$ and $b$ is sequentially
independent and denoted by $a|b$.
Now, we use the sequential effect algebra theory as tools to study the
partitions and refinements of quantum logics and their entropies.
3\. Partitions, refinements and their entropies
Let $(E,\theta,I,\oplus,\circ)$ be a sequential effect algebra. A set
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ is said to be a partition of
$(E,\theta,I,\oplus,\circ)$ if $\oplus_{i=1}^{n}a_{i}$ is defined and
$\oplus_{i=1}^{n}a_{i}=I$.
In following, we denote partitions $A=\\{a_{1},a_{2},\cdots,a_{n}\\}$,
$B=\\{b_{1},b_{2},\cdots,b_{m}\\}$, $C=\\{c_{1},c_{2},\cdots,c_{l}\\}$, and
$A\circ B=\\{a_{i}\circ b_{j}:a_{i}\in A,b_{j}\in
B,i=1,2,\cdots,n,j=1,2,\cdots,m\\}$. That $A\circ B\neq B\circ A$ are clear.
Let $(E,\theta,I,\oplus,\circ)$ be a sequential effect algebra, $A$ and $B$ be
two partitions of $(E,\theta,I,\oplus,\circ)$. Then it follows from (SEA1) and
([11, Lemma 3.1(i)]) that $A\circ B=\\{a_{i}\circ b_{j}:a_{i}\in A,b_{j}\in
B,i=1,2,\cdots,n,j=1,2,\cdots,m\\}$ is also a partition of
$(E,\theta,I,\oplus,\circ)$. We say that the partition $A\circ B$ is a
refinement of the partitions $A$ and $B$.
Example 3.1([11]) Let $(L,\leq)$ be a Boolean algebra, $a,b\in L$. Let
$a\oplus b$ be defined iff $a\wedge b=\theta$, in this case, $a\oplus b=a\vee
b$, and define $a\circ b=a\wedge b$. Then $(L,\theta,I,\oplus,\circ)$ is a
sequential effect algebra.
Example 3.2([11]) Let $X$ be a set and $\mathcal{F}$$(X)$ be the all fuzzy
sets of $X$, $\mu_{\tilde{A}},\mu_{\tilde{B}}\in\mathcal{F}$$(X)$. Let
$\mu_{\tilde{A}}\oplus\mu_{\tilde{B}}$ be defined iff
$\mu_{\tilde{A}}+\mu_{\tilde{B}}\leq 1$, in this case,
$\mu_{\tilde{A}}\oplus\mu_{\tilde{B}}=\mu_{\tilde{A}}+\mu_{\tilde{B}}$, and
define $\mu_{\tilde{A}}\circ\mu_{\tilde{B}}=\mu_{\tilde{A}}\mu_{\tilde{B}}$.
Then $(\mathcal{F}$$(X),0,1,\oplus,\circ)$ is a sequential effect algebra.
Example 3.3([11]) Let $H$ be a complex Hilbert space, if for any two quantum
effects $B$ and $C$, we define $B\circ C=B^{\frac{1}{2}}CB^{\frac{1}{2}}$,
then $({\cal E}(H),0,I,\oplus,\circ)$ is a sequential effect algebra. In
particular, for any two orthogonal projection operators $P$ and $Q$ on $H$,
$PQP=P^{\frac{1}{2}}QP^{\frac{1}{2}}$ is a sequential product of $P$ and $Q$.
The above three examples showed that our refinement methods of the partitions
are not only suitable for classical logics, but also effective for fuzzy
logics and quantum logics.
Now, we begin to study the entropies of partitions and refinements of
sequential effect algebras. First, we need the following:
Let $(E,\theta,I,\oplus,\circ)$ be a sequential effect algebra, $s$ be a state
of $(E,0,1,\oplus,\circ)$, that is, $s:E\rightarrow[0,1]$ be a mapping from
$E$ into the real number interval $[0,1]$ such that $s(I)=1$ and whenever
$a\oplus b$ be defined, $s(a\oplus b)=s(a)+s(b)$. Then for given $A$,
$s_{A}:b\rightarrow\sum_{i=1}^{n}s(a_{i}\circ b)$
defines a new state $s_{A}$, this is the resulting state after the system $A$
is executed but no observation is performed ([12]). Moreover, we denote
$s(b\mid a)$ by $s(a\circ b)/s(a)$ if $s(a)\neq 0$ and $0$ if $s(a)=0$.
The entropy of $A$ with respect to the state $s$ is defined by
$H_{s}(A)=-\sum_{i=1}^{n}s(a_{i})\log s(a_{i}).$
The refinement entropy of $A$ and $B$ with respect to the state $s$ is defined
by
$H_{s}(A\circ B)=-\sum_{i=1}^{n}\sum_{j=1}^{m}s(a_{i}\circ b_{j})\log
s(a_{i}\circ b_{j}).$
The conditional entropy of $B$ conditioned by $A$ with respect to the state
$s$ is defined by
$H_{s}(B|A)=-\sum_{i=1}^{n}\sum_{j=1}^{m}s(a_{i}\circ b_{j})\log
s(b_{j}|a_{i}).$
Lemma 3.1([13]) (log sum inequality) For non-negative numbers
$a_{1},a_{2},\cdots,a_{n}$ and $b_{1},b_{2},\cdots,b_{n}$,
$\displaystyle\sum_{i=1}^{n}a_{i}\log\frac{a_{i}}{b_{i}}\geq(\sum_{i=1}^{n}a_{i})\log(\frac{\sum_{i=1}^{n}a_{i}}{\sum_{i=1}^{n}b_{i}}).$
We use the convention that $0\log 0=0,a\log\frac{a}{0}=\infty$ if $a>0$ and
$0\log\frac{0}{0}=0.$
In this paper, our main result is the following theorem which generalizes the
classical entropy properties ([2, 13-14]) to the sequential effect algebras.
Theorem 3.1 (i). $H_{s}(A\circ B)=H_{s}(B|A)+H_{s}(A)$.
(ii). $H_{s}(A|C)\leq H_{s}(A\circ B|C)$.
(iii). $H_{s}(B|A)\leq H_{s_{A}}(B)$.
(iv). $H_{s}(A\circ B)\leq H_{s}(A)+H_{s_{A}}(B)$.
(v). $\max\\{H_{s_{A}}(B),H_{s}(A)\\}\leq H_{s}(A\circ B)$.
(vi). $H_{s}(B\circ A|C)\leq H_{s_{C}}(A|B)+H_{s}(B|C)$.
Proof. We only prove (vi). In fact, by Lemma 3.1, we have
$\displaystyle H_{s_{C}}(A|B)+H_{s}(B|C)$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\sum_{j=1}^{m}s_{C}(b_{j}\circ a_{i})\log
s_{C}(a_{i}|b_{j})-\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ b_{j})\log
s(b_{j}|c_{k})$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ(b_{j}\circ
a_{i}))\log\frac{\sum_{k=1}^{l}s(c_{k}\circ(b_{j}\circ
a_{i}))}{\sum_{k=1}^{l}s(c_{k}\circ b_{j})}$
$\displaystyle-\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ
b_{j})\log\frac{s(c_{k}\circ b_{j})}{s(c_{k})}$ $\displaystyle\geq$
$\displaystyle-\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ(b_{j}\circ
a_{i}))\log\frac{s(c_{k}\circ(b_{j}\circ a_{i}))}{s(c_{k}\circ b_{j})}$
$\displaystyle-\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ
b_{j})\log\frac{s(c_{k}\circ b_{j})}{s(c_{k})}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{l}s(c_{k}\circ(b_{j}\circ
a_{i}))\log\frac{s(c_{k}\circ(b_{j}\circ a_{i}))}{s(c_{k})}$ $\displaystyle=$
$\displaystyle H_{s}(B\circ A|C).$
That concludes the proof.
Finally, we would like to point out that for the advances of sequential effect
algebras, see [15-18].
References
[1] Ohya M. and Petz D. Quantum Entropy and its Use. Springer-Verlag, Berlin,
(1991)
[2] Yuan Hejun. J. Entropy of Partitions on Quantum Logic. Commun. Theor.
Phys. 43, 437-439, (2005)
[3] Kelly J. L. General Topology. Springer-Verlag, New York, (1955)
[4] Birkhoff G. and von Neumann J. The logic of quantum mechanics, Ann. Math.
37, 823-834, (1936)
[5] Dvure$\check{c}$enskij A. and Pulmannov$\check{a}$ S. New Trends in
Quantum Structures. Kluwer, (2002)
[6] Busch P., Grabowski M. and Lahti P. J. Operational Quantum Physics.
Springer-Verlag, Berlin, (1995)
[7] Busch P., Lahti P. J. and Mittlestaedt P. The Quantum Theory of
Measurements. Springer-Verlag, Berlin, (1991)
[8] Busch P. and Singh J. Lüders theorem for unsharp quantum measurements.
Phys. Lett. A. 249, 10-12, (1998)
[9] Foulis D. J. and Bennett M. K. Effect algebras and unsharp quantum logics.
Found. Phys. 24, 1331-1352, (1994)
[10] Gudder S. and Nagy G. Sequential quantum measurements. J. Math. Phys. 42,
5212-5222, (2001)
[11] Gudder S. and Greechie R. Sequential products on effect algebras. Rep.
Math. Phys. 49, 87-111, (2002)
[12] Arias A., Gheondea A. and Gudder S. Fixed points of quantum operations.
J. Math. Phys. 43, 5872-5881, (2002)
[13] Cover T. M. and Thomas J. A. Elements of Information Theory. Wiley, New
York, (1991)
[14] Zhao Yuexu and Ma Zhihao. Conditional Entropy of Partitions on Quantum
Logic. Commun. Theor. Phys. 48, 11-13 (2007)
[15] Shen Jun and Wu Junde. Not each sequential effect algebra is sharply
dominating. Phys. Letter A. 373, 1708-1712, (2009)
[16] Shen Jun and Wu Junde. Remarks on the sequential effect algebras. Report.
Math. Phys. 63, 441-446, (2009)
[17] Liu Weihua and Wu Junde. A uniqueness problem of the sequence product on
operator effect algebra ${\cal E}(H)$. J. Phys. A: Math. Theor. 42,
185206-185215, (2009)
[18] Shen Jun and Wu Junde. Sequential product on standard effect algebra
${\cal E}(H)$. J. Phys. A: Math. Theor. 44, (2009)
|
arxiv-papers
| 2009-08-18T07:39:44 |
2024-09-04T02:49:04.680089
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wang Jiamei, Wu Junde, Cho Minhyung",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0908.2510"
}
|
0908.2707
|
2010453-464Nancy, France 453 Edward A. Hirsch Dmitry Itsykson
# On optimal heuristic randomized semidecision procedures, with application to
proof complexity
E. A. Hirsch and D. Itsykson Steklov Institute of Mathematics at St.
Petersburg,
27 Fontanka, St.Petersburg, 191023, Russia
http://logic.pdmi.ras.ru/~hirschhttp://logic.pdmi.ras.ru/~dmitrits
###### Abstract.
The existence of a ($p$-)optimal propositional proof system is a major open
question in (proof) complexity; many people conjecture that such systems do
not exist. Krajíček and Pudlák [KP89] show that this question is equivalent to
the existence of an algorithm that is optimal111Recent papers [Mon09] call
such algorithms _$p$ -optimal_ while traditionally Levin’s algorithm was
called _optimal_. We follow the older tradition. Also there is some mess in
terminology here, thus please see formal definitions in Sect. 2 below. on all
propositional tautologies. Monroe [Mon09] recently gave a conjecture implying
that such algorithm does not exist.
We show that in the presence of errors such optimal algorithms _do_ exist. The
concept is motivated by the notion of heuristic algorithms. Namely, we allow
the algorithm to claim a small number of false “theorems” (according to any
polynomial-time samplable distribution on non-tautologies) and err with
bounded probability on other inputs.
Our result can also be viewed as the existence of an optimal proof system in a
class of proof systems obtained by generalizing automatizable proof systems.
###### Key words and phrases:
propositional proof complexity, optimal algorithm
###### 1991 Mathematics Subject Classification:
F.2
Partially supported by grants RFBR 08-01-00640 and 09-01-12066, and the
president of Russia grant “Leading Scientific Schools” NSh-4392.2008.1, by
Federal Target Programme “Scientific and scientific-pedagogical personnel of
the innovative Russia” 2009-2013 (contract N $\Pi$265 from 23.07.2009). The
second author is also supported by Russian Science Support Foundation.
## 1\. Introduction
Given a specific problem, does there exist the “fastest” algorithm for it?
Does there exist a proof system possessing the “shortest” proofs of the
positive solutions to the problem? Although the first result in this direction
was obtained by Levin [Lev73] in 1970s, these important questions are still
open for most interesting languages, for example, the language of
propositional tautologies.
#### Classical version of the problem.
According to Cook and Reckhow [CR79], a proof system is a polynomial-time
mapping of all strings (“proofs”) onto “theorems” (elements of certain
language $L$; if $L$ is the language of all propositional tautologies, the
system is called a _propositional_ proof system). The existence of a
_polynomially bounded_ propositional proof system (that is, a system that has
a polynomial-size proof for every tautology) is equivalent to
${\mathbf{NP}}={\mathbf{{\mathbf{co\,}\textbf{-}}NP}}$. In the context of
polynomial boundedness a proof system can be equivalently viewed as a function
that given a formula and a “proof”, verifies in polynomial time that a formula
is a tautology: it must accept at least one “proof” for each tautology
(_completeness_) and reject all proofs for non-tautologies (_soundness_).
One proof system $\Pi_{w}$ is _simulated_ by another one $\Pi_{s}$ if the
shortest proofs for every tautology in $\Pi_{s}$ are at most polynomially
longer than the shortest proofs in $\Pi_{w}$. The notion of _$p$ -simulation_
is similar, but requires also a polynomial-time computable function for
translating the proofs from $\Pi_{w}$ to $\Pi_{s}$. A _( $p$-)optimal_
propositional proof system is one that ($p$-)simulates all other propositional
proof systems.
The existence of an optimal (or $p$-optimal) propositional proof system is a
major open question. If one would exist, it would allow to reduce the
${\mathbf{NP}}$ vs ${\mathbf{{\mathbf{co\,}\textbf{-}}NP}}$ question to
proving proof size bounds for just one proof system. It would also imply the
existence of a complete disjoint $\mathbf{NP}$ pair [Raz94, Pud03]. Krajíček
and Pudlák [KP89] show that the existence of a $p$-optimal system is
equivalent to the existence of an algorithm that is optimal on all
propositional tautologies, namely, it always solves the problem correctly and
it takes for it at most polynomially longer to stop on every tautology than
for any other correct algorithm _on the same tautology_. Monroe [Mon09]
recently gave a conjecture implying that such algorithm does not exist. Note
that Levin [Lev73] showed that an optimal algorithm does exist for finding
witnesses to non-tautologies; however, (1) its behaviour on tautologies is not
restricted; (2) after translating to the decision problem by self-reducibility
the running time in the optimality condition is compared to the running time
for _all shorter formulas as well_.
An _automatizable_ proof system is one that has an automatization procedure
that given a tautology, outputs its proof of length polynomially bounded by
the length of the shortest proof in time bounded by a polynomial in the output
length. The automatizability of a proof system $\Pi$ implies polynomial
separability of its canonical $\mathbf{NP}$ pair [Pud03], and the latter
implies the automatizability of a system that $p$-simulates $\Pi$. This,
however, does not imply the existence of ($p$-)optimal propositional proof
systems in the class of automatizable proof systems. To the best of our
knowledge, no such system is known to the date.
#### Proving propositional tautologies heuristically.
An obvious obstacle to constructing an optimal proof system by enumeration is
that no efficient procedure is known for enumerating the set of all complete
and sound proof systems. Recently a number of papers overcome similar
obstacles in other settings by considering either computations with non-
uniform advice (see [FS06] for survey) or _heuristic_ algorithms [FS04, Per07,
Its09]. In particular, optimal propositional proof systems with advice do
exist [CK07]. We try to follow the approach of heuristic computations to
obtain a “heuristic” proof system. While our work is motivated by
propositional proof complexity, i.e., proof systems for the set of
propositional tautologies, our results apply to proof systems for any
recursively enumerable language.
We introduce a notion of a _randomized heuristic automatizer_ (a randomized
semidecision procedure that may have false positives) and a corresponding
notion of a _simulation_. Its particular case, a deterministic automatizer
(making no errors) for language $L$, along with deterministic simulations, can
be viewed in two ways:
* •
as an automatizable proof system for $L$ (note that such proof system can be
identified with its automatization procedure; however, it may not be the case
for randomized algorithms, whose running time may depend on the random coins),
where simulations are $p$-simulations of proof systems;
* •
as an algorithm for $L$, where simulations are simulations of algorithms for
$L$ in the sense of [KP89].
Given $x\in L$, an automatizer must return $1$ and stop. The question (handled
by simulations) is how fast it does the job. For $x\notin L$, the running time
does _not_ matter. Given $x\notin L$, a deterministic automatizer simply must
_not_ return $1$. A randomized heuristic automatizer may erroneously return
$1$; however, for “most” inputs it may do it only with bounded probability
(“good” inputs). The precise notion of “most” inputs is: given an integer
parameter $d$ and a sampler for $\overline{L}$, “bad” inputs must have
probability less than $1/d$ according to the sampler. The parameter $d$ is
handled by simulations in the way such that no automatizer can stop in time
polynomial in $d$ and the length of input unless an optimal automatizer can do
that.
In Sect. 2 we give precise definitions. In Sect. 3 we construct an optimal
randomized heuristic automatizer. In Sect. 4 we give a notion of heuristic
probabilistic proof system and discuss the relation of automatizers to such
proof systems.
## 2\. Preliminaries
### 2.1. Distributional proving problems
In this paper we consider algorithms and proof systems _that allow small
errors_ , i.e., claim a small amount of wrong theorems. Formally, we have a
probability distribution concentrated on non-theorems and require that the
probability of sampling a non-theorem accepted by an algorithm or validated by
the system is small.
###### Definition 2.1.
We call a pair $(D,L)$ a _distributional proving problem_ if $D$ is a
collection of probability distributions $D_{n}$ concentrated on
$\overline{L}\cap\\{0,1\\}^{n}$.
In what follows we write $\Pr_{x\leftarrow D_{n}}$ to denote the probability
taken over $x$ from such distribution, while $\Pr_{A}$ denotes the probability
taken over internal random coins used by algorithm $A$.
### 2.2. Automatizers
###### Definition 2.2.
A _$(\lambda,\epsilon)$ -correct automatizer_ for distributional proving
problem $(D,L)$ is a randomized algorithm $A$ with two parameters
$x\in\\{0,1\\}^{*}$ and $d\in\mathbb{N}$ that satisfies the following
conditions:
1. (1)
$A$ either outputs $1$ (denoted $A(\ldots)=1$) or does not halt at all
(denoted $A(\ldots)=\infty$);
2. (2)
For every $x\in L$ and $d\in\mathbb{N}$, $A(x,d)=1$.
3. (3)
For every $n,d\in\mathbb{N}$,
$\Pr_{r\leftarrow
D_{n}}\left\\{\Pr_{A}\\{A(r,d)=1\\}>\epsilon\right\\}<\frac{1}{\lambda d}.$
Here $\lambda>0$ is a constant and $\epsilon>0$ may depend on the first input
($x$) length. An _automatizer_ is a $(1,\frac{1}{4})$-correct automatizer.
###### Remark 2.3.
For recursively enumerable $L$, conditions 1 and 2 can be easily enforced at
the cost of a slight overhead in time by running $L$’s semidecision procedure
in parallel.
In what follows, all automatizers are for the same problem $(D,L)$.
###### Definition 2.4.
The _time_ spent by automatizer $A$ on input $(x,d)$ is defined as the median
time
$t_{A}(x,d)=\min\left\\{t\in\mathbb{N}\;\bigg{|}\;\Pr_{A}\\{\text{$A(x,d)$
stops in time at most $t$}\\}\geq\frac{1}{2}\right\\}.$
We will also use a similar notation for “probability $p$ time”:
$t^{(p)}_{A}(x,d)=\min\left\\{t\in\mathbb{N}\;\bigg{|}\;\Pr_{A}\\{\text{$A(x,d)$
stops in time at most $t$}\\}\geq p\right\\}.$
###### Definition 2.5.
Automatizer $S$ simulates automatizer $W$ if there are polynomials $p$ and $q$
such that for every $x\in L$ and $d\in\mathbb{N}$,
${t_{S}(x,d)}\leq\max_{d^{\prime}\leq
q(d\cdot|x|)}p(t_{W}(x,d^{\prime})\cdot|x|\cdot d).$
###### Definition 2.6.
An _optimal_ automatizer is one that simulates every other automatizer.
###### Definition 2.7.
Automatizer $A$ is _polynomially bounded_ if there is a polynomial $p$ such
that for every $x\in L$ and every $d\in\mathbb{N}$,
$t_{A}(x,d)\leq p(d\cdot|x|).$
The following proposition follows directly from the definitions.
###### Proposition 2.8.
1. (1)
If $W$ is polynomially bounded and is simulated by $S$, then $S$ is
polynomially bounded too.
2. (2)
An optimal automatizer is not polynomially bounded if and only if no
automatizer is polynomially bounded.
## 3\. Optimal automatizer
The optimal automatizer that we construct runs all automatizers in parallel
and stops when the first of them stops (recall Levin’s optimal algorithm for
SAT [Lev73]). A major obstacle to this simple plan is the fact that it is
unclear how to enumerate all automatizers efficiently (put another way, how to
check whether a given algorithm is a correct automatizer). The plan of
overcoming this obstacle (similar to constructing a complete public-key
cryptosystem [HKN+05] (see also [GHP09])) is as follows:
* •
Prove that w.l.o.g. a correct automatizer is very good: in particular, amplify
its probability of success.
* •
Devise a “certification” procedure that distinguishes very good automatizers
from incorrect automatizers with overwhelming probability.
* •
Run all automatizers in parallel, try to certify automatizers that stop, and
halt when the first automatizer passes the check.
The amplification is obtained by repeating and the use of Chernoff bounds.
###### Proposition 3.1 (Chernoff bounds (see, e.g., [MR95, Chapter 4])).
Let $X_{1},X_{2},\ldots,X_{n}\in\\{0,1\\}$ be independent random variables.
Then if $X$ is the sum of $X_{i}$ and if $\mu$ is $\mathbf{E}[X]$, for any
$\delta$, $0<\delta\leq 1$:
$\Pr\\{X<(1-\delta)\mu\\}<e^{-\mu\delta^{2}/2},\qquad\Pr\\{X>(1+\delta)\mu\\}<e^{-\mu\delta^{2}/3}.$
###### Corollary 3.2.
Let $X_{1},X_{2},\ldots,X_{n}\in\\{0,1\\}$ be independent random variables.
Then if $X$ is the sum of $X_{i}$ and if
$1\geq\mu_{1}\geq\mathbf{E}[X]\geq\mu_{2}\geq 0$, for any $\delta$,
$0<\delta\leq 1$:
$\Pr\\{X<(1-\delta)\mu_{2}\\}<e^{-\mu_{2}\delta^{2}/2},\qquad\Pr\\{X>(1+\delta)\mu_{1}\\}<e^{-\mu_{1}\delta^{2}/3}.$
###### Lemma 3.3 (amplification).
Every automatizer $W$ is simulated by a $(4,e^{-m/{48}})$-correct automatizer
$S$, where $m\in\mathbb{N}$ may depend at most polynomially on $d\cdot|x|$
(for input $(x,d)$). Moreover, there are polynomials $p$ and $q$ such that for
every $x\in L$ and $d\in\mathbb{N}$,
${t^{(1-e^{-m/{64}})}_{S}(x,d)}\leq\max_{d^{\prime}\leq
q(d\cdot|x|)}p(t_{W}(x,d^{\prime})).$ (1)
###### Proof 3.4.
$S(x,d)$ runs $m$ copies of $W(x,4d)$ in parallel and stops as soon as the
$\frac{3}{8}$ fraction of copies stop.
By Chernoff bounds, $S$ is $(4,e^{-m/{48}})$-correct. The “strong” simulation
condition (1) is satisfied because by Chernoff bounds the running time of the
fastest $\frac{3}{8}$ fraction of executions is less than median time with
probability at least $1-e^{-m/{64}}$.
###### Theorem 3.5 (optimal automatizer).
Let $(D,L)$ be a distributional proving problem, where $L$ is recursively
enumerable and $D$ is polynomial-time samplable, i.e., there is a polynomial-
time randomized Turing machine that given $1^{n}$ on input outputs $x$ with
probability $D_{n}(x)$ for every $x\in\\{0,1\\}^{n}$. Then there exists an
optimal automatizer for $(D,L)$.
###### Proof 3.6.
For algorithm $A$, we say that it is _$(\lambda,\epsilon)$ -correct for input
length $n$ and parameter $d$_ if it it satisfies condition 3 of Definition 2.2
for $n$ and $d$. If an algorithm is $(\lambda,\epsilon)$-correct for every $n$
(resp., every $d$), we omit $n$ (resp., $d$).
In order to check an algorithm for correctness, we define a _certification_
procedure that takes an algorithm $A$ and distinguishes between the cases
where $A$ is $(4,{\frac{1}{18d\log_{*}^{2}n}})$-correct for given $n,d$ (from
Lemma 3.3 we know that one can assume such correctness) or it is not
$(1,{\frac{1}{16d\log_{*}^{2}n}})$-correct
($(1,{\frac{1}{16d\log_{*}^{2}n}})$-correct automatizers suffice for the
correctness of further constructions). W.l.o.g. we may assume that
$A$ satisfies conditions 1 and 2 of Definition 2.2 (2)
(for the latter condition, notice that $L$ is recursively enumerable and one
may run its semidecision procedure in parallel).
The certification procedure has a subroutine Test that estimates the
probability of $A$’s error simply by repeating $A$ and couting its faults.
$\textsc{Test}(A,x,d^{\prime},T,l,f)$:
1. (1)
Repeat for each $i\in\\{1,\ldots,l\\}$
1. (a)
If $A(x,d^{\prime})$ stops in $T$ steps, let $c_{i}=1$; otherwise let
$c_{i}=0$.
2. (2)
If $\sum_{i}c_{i}\geq l/f$, then reject; otherwise accept.
###### Lemma 3.7.
For every $A,x,d^{\prime},T,l,f$,
1. (1)
If $A(x,d^{\prime})$ stops with probability less than $\frac{1}{1.01f}$, then
Test will reject it with probability less than $e^{-\frac{l}{3.03\cdot
10^{4}\cdot f}}$.
2. (2)
If $A(x,d^{\prime})$ stops in time at most $T$ with probability more than
$\frac{1}{0.99f}$, then Test will accept it with probability less than
$e^{-\frac{l}{2\cdot 10^{4}\cdot f}}$.
###### Proof 3.8.
Follows directly from Chernoff bounds.
$\textsc{Certify}(A,n,d^{\prime},T,k,l,f)$:
1. (1)
Repeat for each $i\in\\{1,\ldots,k\\}$
1. (a)
Generate $x_{i}$ according to $D_{n}$.
2. (b)
If $\textsc{Test}(A,x_{i},d^{\prime},T,l,f)$ rejects, let $b_{i}=1$; otherwise
let $b_{i}=0$.
2. (2)
If $\sum_{i}b_{i}\geq k/(2d^{\prime})$, then reject; otherwise accept.
###### Lemma 3.9.
Let $d,n,T\in\mathbb{N}$. Let $A$ be an algorithm pretending to be an
automatizer. Run
$\textstyle\textsc{Certify}(A,n,d^{\prime},T,k,l,f).$
Then
1. (1)
If $A$ is $(4,{\frac{1}{1.011f}})$-correct, then $A$ is accepted by Certify
almost for sure, failing with probability less than
${e^{-\frac{k}{12d^{\prime}}}+k\cdot e^{-\frac{l}{3.03\cdot 10^{4}\cdot f}}}$.
2. (2)
Let $A^{T}$ be a restricted version of $A$ that behaves similarly to $A$ for
$T$ steps and enters an infinite loop afterwards. If $A^{T}$ is not
$(1,{\frac{1}{0.99f}})$-correct for length $n$ and parameter $d$, then $A$ is
accepted by Certify with probability less than
${e^{-\frac{k}{8d^{\prime}}}+k\cdot e^{-\frac{l}{2\cdot 10^{4}\cdot f}}}$.
###### Proof 3.10.
1\. Let
$\Delta=\\{x\in\mathop{\mathrm{Im}}D_{n}\;|\;\Pr\\{A(x,d)=1\\}>{\frac{1}{1.011f}}\\}$.
By assumption, $D_{n}(\Delta)<\frac{1}{4d^{\prime}}$.
The certification procedure takes $k$ samples from $D_{n}$. For every sample
$x_{i}\in\overline{L}\setminus\Delta$, the probability that the corresponding
$b_{i}$ equals 1 is less than $e^{-\frac{l}{3.03\cdot 10^{4}\cdot f}}$. Thus,
the probability that there is a sample $x_{i}$ from
$\overline{L}\setminus\Delta$ that yields $b_{i}=1$ is less than $k\cdot
e^{-\frac{l}{3.03\cdot 10^{4}\cdot f}}$. Denote this unfortunate event by $E$.
If it does not hold, only samples from $\Delta$ may cause $b_{i}=1$ and by
Chernoff’s bound
$\Pr\\{\sum_{i}b_{i}\geq
k/(2d^{\prime})\;|\;\overline{E}\\}<e^{-\frac{k}{12d^{\prime}}}.$
Thus, the total probability of reject is as claimed.
2\. Let
$\Delta=\\{x\in\mathop{\mathrm{Im}}D_{n}\;|\;\Pr\\{A(x,d)=1\\}>{\frac{1}{0.99f}}\\}$.
By assumption, $D_{n}(\Delta)\geq\frac{1}{d^{\prime}}$.
The certification procedure takes $k$ samples from $D_{n}$. For every sample
$x_{i}\in\Delta$, the probability that the corresponding $b_{i}$ equals 0 is
less than $e^{-\frac{l}{2\cdot 10^{4}\cdot f}}$. Thus, the probability that
there is a sample $x_{i}$ from $\Delta$ that yields $b_{i}=0$ is less than
$k\cdot e^{-\frac{l}{2\cdot 10^{4}\cdot f}}$. Denote this unfortunate event by
$E$. Assuming it does not hold only samples outside $\Delta$ may cause
$b_{i}=0$ and by Chernoff’s bound
$\Pr\\{\sum_{i}b_{i}<k/(2d^{\prime})\;|\;\overline{E}\\}<e^{-\frac{k}{8d^{\prime}}}.$
We now define the optimal automatizer $U$. It works as follows:
$U(x,d)$:
1. (1)
Let
$\displaystyle n$ $\displaystyle=$ $\displaystyle|x|,$ $\displaystyle
d^{\prime}$ $\displaystyle=$ $\displaystyle 16d\log^{2}_{*}n,$ $\displaystyle
f$ $\displaystyle=$ $\displaystyle 17d\log_{*}^{2}n,$ $\displaystyle k$
$\displaystyle=$ $\displaystyle 12d^{\prime}\ln(16d\log_{*}^{2}n),$
$\displaystyle l$ $\displaystyle=$ $\displaystyle(3.03\cdot 10^{4})\cdot
f\cdot\ln(16kd\log_{*}^{2}n).$
2. (2)
Run the following processes for $i\in\\{1,\ldots,\log_{*}n\\}$ in parallel:
1. (a)
Run $A_{i}(x,d^{\prime})$, the algorithm with Turing number $i$ satisfying
assumption (2), and compute the number of steps $T_{i}$ made by it before it
stops.
2. (b)
If $\textsc{Certify}(A_{i},n,d^{\prime},T_{i},k,l,f)$ accepts,
then output $1$ and stop $U$ (all processes).
3. (3)
If none of the processes has stopped, go into an infinite loop.
#### Correctness.
We now show that $U$ errs with probability less than 1/4.
What are the inputs that cause $U$ to error? For every such input $x$ there
exists $i\leq\log_{*}n$ such that
$u^{i}_{x}=\sum_{T=1}^{\infty}p^{i}_{x,T}c_{T}^{i}\geq\frac{1}{4\log_{*}n},$
(3)
where
$\displaystyle p^{i}_{x,t}$ $\displaystyle=\Pr\\{\text{$A_{i}(x,d^{\prime})$
stops in exactly $t$ steps}\\},$ $\displaystyle c^{i}_{t}$
$\displaystyle=\Pr\\{\text{$\textsc{Certify}(A_{i},n,d^{\prime},t,k,l,f)$
accepts}\\}.$
Let $E_{i}$ be the set of inputs $x\notin L$ satisfying inequality (3).
We claim that $D(E_{i})<\frac{1}{d\log_{*}n}$, which suffices to show the
$(1,1/4)$-correctness.
Assume the contrary. Let
$T_{i}=\min\\{t\;|\;c^{i}_{t}<{e^{-\frac{k}{8d^{\prime}}}+k\cdot
e^{-\frac{l}{2\cdot 10^{4}\cdot f}}}\\}$. Note that by Lemma 3.9
$A_{i}^{T_{i}-1}$ is $(1,\frac{1}{0.99f})$-correct for $n$ and $d^{\prime}$,
i.e.,
$\Pr_{x\leftarrow
D_{n}}\\{\sum_{T<T^{i}_{*}}p^{i}_{x,T}>{\frac{1}{0.99f}}\\}<\frac{1}{d^{\prime}}.$
We omit $i$ and $n$ in the estimations that follow. Here is how we get a
contradiction:
$\frac{1}{4d\log_{*}^{2}n}\leq\frac{D(E_{i})}{4\log_{*}n}=\sum_{x\in
E_{i}}\frac{1}{4\log_{*}n}D(x)\leq\sum_{x\in E_{i}}u_{x}D(x)\leq\\\
\sum_{x\notin L}u_{x}D(x)=\sum_{x\notin
L}\sum_{T=1}^{\infty}p_{x,T}c_{T}D(x)=\\\ \sum_{x\notin
L}\left(\sum_{T<T_{*}}p_{x,T}c_{T}D(x)+\sum_{T\geq
T_{*}}p_{x,T}c_{T}D(x)\right)\leq\\\ \sum_{T<T_{*}}\left(\sum_{x\notin L,\
\sum\limits_{t<T_{*}}p_{x,t}\leq{\frac{1}{0.99f}}}p_{x,T}D(x)+\sum_{x\notin
L,\ \sum\limits_{t<T_{*}}p_{x,t}>{\frac{1}{0.99f}}}p_{x,T}D(x)\right)\\\
\mbox{\hfill}+e^{-\frac{k}{8d^{\prime}}}+k\cdot e^{-\frac{l}{2\cdot
10^{4}\cdot f}}\leq\\\
{\frac{1}{0.99f}}+\frac{1}{d^{\prime}}+e^{-\frac{k}{8d^{\prime}}}+k\cdot
e^{-\frac{l}{2\cdot 10^{4}\cdot
f}}<\frac{1}{16d\log_{*}^{2}n}+\frac{1}{16d\log^{2}_{*}n}+\frac{1}{8d\log_{*}^{2}n}=\frac{1}{4d\log^{2}_{*}n}.$
#### Simulation.
Assume we are give a correct automatizer $A^{s}$. Plug in
$m=48\cdot\ln(18d\log_{*}^{2}n)$ into Lemma 3.3. The lemma yields that $A^{s}$
is “strongly” simulated by a $(4,{\frac{1}{18d\log_{*}^{2}n}})$-correct
automatizer $A$. It remains to estimate, for given “theorem” $x\in L$, the
(median) running time of $U$ in terms of
$t^{(1-e^{-m/64})}_{A}(x,d)=t^{(1-\frac{1}{(18d\log_{*}^{2}n)^{3/4}})}_{A}(x,d)$
(as we know that the latter is bounded by $\max\limits_{d^{\prime}\leq
q(d\cdot|x|)}p(t_{A^{s}}(x,d^{\prime}))$ for a polynomials $p$ and $q$).
Since the definition of simulation is asymptotic, we consider only $x$ of
length greater than the Turing number of $A$. By Lemma 3.9, $A$ is not
certified with probability less than $e^{-\frac{k}{12d^{\prime}}}+k\cdot
e^{-\frac{l}{3.03\cdot 10^{4}\cdot f}}\leq\frac{1}{8d\log_{*}^{2}n}$. If $A$
is certified, $U$ stops in time upper bounded by a polynomial of the time
spent by $A$ with an overhead polynomial in $|x|$ and $d$ for running other
algorithms and the certification procedures. Thus the median time $t_{U}(x,d)$
is bounded by a polynomial in $|x|$, $d$, and
$t_{A}^{(\frac{1}{2}+\frac{1}{8d\log_{*}^{2}n})}(x,d)\leq
t^{(1-\frac{1}{(18d\log_{*}^{2}n)^{3/4}})}_{A}(x,d)$.
## 4\. Heuristic proof systems
In this section we define proof systems that make errors (claim a small
fraction of wrong theorems). We consider automatizable systems of this kind
and show that every such system defines an automatizer taking time at most
polynomially larger than the length of the shortest proof in the initial
system. This shows that automatizers form a more general notion than
automatizable heuristic proof systems. The opposite direction is left as an
open question.
###### Definition 4.1.
Randomized Turing machine $\Pi$ is a _heuristic proof system_ for
distributional proving problem $(D,L)$ if it satisfies the following
conditions.
1. (1)
The running time of $\Pi(x,w,d)$ is bounded by a polynomial in $d$, $|x|$, and
$|w|$.
2. (2)
(Completeness) For every $x\in L$ and every $d\in\mathbb{N}$, there exists a
string $w$ such that $\Pr\\{\Pi(x,w,d)=1\\}\geq\frac{1}{2}$. Every such string
$w$ is called a $\Pi^{(d)}$-proof of $x$.
3. (3)
(Soundness) $\Pr_{x\leftarrow D_{n}}\\{\exists
w:\Pr\\{\Pi(x,w,d)=1\\}>\frac{1}{4}\\}<\frac{1}{d}$.
###### Definition 4.2.
Heuristic proof system is _automatizable_ if there is a randomized Turing
machine $A$ satisfying the following conditions.
1. (1)
For every $x\in L$ and every $d\in\mathbb{N}$, with probability at least
$\frac{1}{2}$ algorithm $A(x,d)$ outputs a correct $\Pi^{(d)}$-proof of size
bounded by a polynomial in $d$, $|x|$, and $|w|$, where $w$ is the shortest
$\Pi^{(d)}$-proof of $x$.
2. (2)
The running time of $A(x,d)$ is bounded by a polynomial in $|x|$, $d$, and the
size of its own output.
###### Definition 4.3.
We say that heuristic proof system $\Pi_{1}$ _simulates_ heuristic proof
system $\Pi_{2}$ if there exist polynomials $p$ and $q$ such that for every
$x\in L$, the shortest $\Pi_{1}^{(d)}$-proof of $x$ has size at most
$p(d\cdot|x|\cdot\max_{d^{\prime}\leq q(|x|d)}\\{\mbox{the size of the
shortest $\Pi_{2}^{(d^{\prime})}$-proof of $x$}\\}).$
Note that this definition essentially ignores proof systems that have much
shorter proofs for some inputs than the inputs themselves. We state it this
way for its similarity to the automatizers case.
###### Definition 4.4.
Heuristic proof system $\Pi$ is _polynomially bounded_ if there exists a
polynomial $p$ such that for every $x\in L$ and every $d\in\mathbb{N}$, the
size of the shortest $\Pi^{(d)}$-proof of $x$ is bounded by $p(|x|d)$.
###### Proposition 4.5.
If heuristic proof system $\Pi_{1}$ simulates system $\Pi_{2}$ and $\Pi_{2}$
is polynomially bounded, then $\Pi_{1}$ is also polynomially bounded.
We now show how automatizers and automatizable heuristic proof systems are
related.
Consider automatizable proof system $(\Pi,A)$ for distributional proving
problem $(D,L)$ with recursively enumerable language $L$. Let us consider the
following algorithm $A_{\Pi}(x,d)$:
1. (1)
Execute 1000 copies of $A(x,d)$ in parallel.
For each copy,
1. (a)
if it stops with result $w$, then
* •
execute $\Pi(x,w,d)$ 10000 times;
* •
if there were at least 4000 accepts of $\Pi$ (out of 10000), stop all parallel
processes and output $1$.
2. (2)
Execute the enumeration algorithm for $L$; output 1 if this algorithm says
that $x\in L$; go into an infinite loop otherwise.
###### Proposition 4.6.
If $(\Pi,A)$ is a (correct) heuristic automatizable proof system for
recursively enumerable language $L$, then $A_{\Pi}$ is a correct automatizer
for $x\in L$ and $t_{A_{\Pi}}(x,d)$ is bounded by polynomial in size of the
shortest $\Pi_{d}$-proof of $x$.
###### Proof 4.7.
_Soundness (condition 3 in Def. 2.2)._ Let
$\Delta_{n}=\\{x\in\overline{L}\mid\exists
w:\Pr\\{\Pi(x,w,d)=1\\}>\frac{1}{4}\\}$. By definition,
$D_{n}(\Delta_{n})<\frac{1}{d}$. For $x\in\\{0,1\\}^{n}\setminus\Delta_{n}$
and specific $w$, Chernoff bounds imply that $\Pi(x,w,d)$ accepts in $0.4$ or
more fraction of executions with exponentially small probability, which
remains much smaller than $\frac{1}{4}$ even after multiplying by 1000.
_Completeness (conditions 2 and 1 in Def. 2.2)_ is guaranteed by the execution
of the semi-decision procedure for $L$.
_Simulation._ For $x\in L$, the probability that $A$ errs 1000 times is
negligible (at most $2^{-1000}$). Thus with high probability at least one of
the parallel executions of $A(x,d)$ outputs a correct $\Pi_{d}$-proof of size
bounded by a polynomial in the size of the shortest $\Pi_{d}$-proof of $x$.
For $x\in L$ and (correct) $\Pi^{(d)}$-proof $w$, Chernoff bounds imply that
$\Pi(x,w,d)$ accepts in at least $0.4$ fraction of executions with probability
close to $1$. Therefore, $t_{A_{\Pi}}(x,d)$ is bounded by a polynomial in
$|x|$, $d$, and the size of the shortest $\Pi_{d}$-proof of $x$.
## 5\. Further research
One possible direction is to show that automatizers are equivalent to
automatizable heuristic proof systems or, at least, that there is an optimal
automatizable heuristic proof system. That may require some tweak in the
definitions, because the first obstacle to proving the latter fact is the
inability to check a candidate proof system for the non-existence of a much
shorter (correct) proof than those output by a candidate automatizer.
Also Krajíček and Pudlák [KP89] and Messner [Mes99] list equivalent conditions
for the existence of (deterministic) optimal and $p$-optimal proof systems. It
seems promising (and, in some places, challenging) to prove similar statements
in the heuristic setting.
## Acknowledgements
During the work on the subject, we discussed it with many people. Our
particular thanks go to (in the alphabetical order) Dima Antipov, Dima
Grigoriev, and Sasha Smal.
## References
* [CK07] Stephen A. Cook and Jan Krajíček. Consequences of the provability of $NP\subseteq P/poly$. The Journal of Symbolic Logic, 72(4):1353–1371, 2007.
* [CR79] Stephen A. Cook and Robert A. Reckhow. The relative efficiency of propositional proof systems. The Journal of Symbolic Logic, 44(1):36–50, March 1979.
* [FS04] Lance Fortnow and Rahul Santhanam. Hierarchy theorems for probabilistic polynomial time. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pages 316–324, 2004.
* [FS06] Lance Fortnow and Rahul Santhanam. Recent work on hierarchies for semantic classes. SIGACT News, 37(3):36–54, 2006.
* [GHP09] Dima Grigoriev, Edward A. Hirsch, and Konstantin Pervyshev. A complete public-key cryptosystem. Groups, Complexity, Cryptology, 1(1):1–12, 2009.
* [HKN+05] Danny Harnik, Joe Kilian, Moni Naor, Omer Reingold, and Alon Rosen. On robust combiners for oblivious transfer and other primitives. In Proc. of EUROCRYPT-2005, 2005.
* [Its09] Dmitry M. Itsykson. Structural complexity of AvgBPP. In Proceedings of 4th International Computer Science Symposium in Russia, volume 5675 of Lecture Notes in Computer Science, pages 155–166, 2009.
* [KP89] Jan Krajíček and Pavel Pudlák. Propositional proof systems, the consistency of first order theories and the complexity of computations. The Journal of Symbolic Logic, 54(3):1063–1079, September 1989\.
* [Lev73] Leonid A. Levin. Universal sequential search problems. Problems of Information Transmission, 9:265–266, 1973.
* [Mes99] Jochen Messner. On optimal algorithms and optimal proof systems. In Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science, volume 1563 of Lecture Notes in Computer Science, pages 361–372, 1999.
* [Mon09] Hunter Monroe. Speedup for natural problems and coNP?=NP. Technical Report 09-056, Electronic Colloquium on Computational Complexity, 2009.
* [MR95] Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms. Cambridge University Press, 1995.
* [Per07] Konstantin Pervyshev. On heuristic time hierarchies. In Proceedings of the 22nd IEEE Conference on Computational Complexity, pages 347–358, 2007.
* [Pud03] Pavel Pudlák. On reducibility and symmetry of disjoint NP pairs. Theoretical Computer Science, 295(1–3):323–339, 2003.
* [Raz94] Alexander A. Razborov. On provably disjoint NP-pairs. Technical Report 94-006, Electronic Colloquium on Computational Complexity, 1994.
|
arxiv-papers
| 2009-08-19T09:18:25 |
2024-09-04T02:49:04.688274
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edward A. Hirsch and Dmitry Itsykson",
"submitter": "Edward Hirsch",
"url": "https://arxiv.org/abs/0908.2707"
}
|
0908.2895
|
# Semiclassical Methods for Hawking Radiation from a Vaidya Black Hole111Some
findings in this work had already been worked out on 26 .
Haryanto M. Siahaan anto˙102@students.itb.ac.id Theoretical Physics
Laboratory, THEPI Division
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung
Jalan Ganesha 10, Bandung 40132, Indonesia
and
St. Aloysius School
Jalan Batununggal Indah II/30
Bandung 40266, Indonesia Triyanta triyanta@fi.itb.ac.id Theoretical Physics
Laboratory, THEPI Division
and
INDONESIA Center for Theoretical and Mathematical Physics (ICTMP)
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung
Jalan Ganesha 10, Bandung 40132, Indonesia
###### Abstract
We derive the general form of Hawking temperature for Vaidya black hole in the
tunneling pictures. This kind of black hole is regarded as the description of
a more realistic one since it’s time dependent decreasing mass due to the
evaporation process. Clearly, the temperature would be time dependent as our
findings. We use the semiclassical methods, namely radial null geodesic and
complex paths methods. Both methods are found to give the same results. Then,
we discuss the possible form of corresponding entropy.
Semiclassical method; Vaidya black holes; Hawking radiation; tunneling
pictures
###### pacs:
04.70.Dy ; 03.65.Sq
## I Introduction
In 1974, Hawking startled the physics community by proving that black hole
evaporates particles 1 ; 2 . It contradicts with classical general
relativistic definition of a black hole, an object that nothing can escape
from it 3 . Hawking’s derivation was a quantum field theoretically. With
Hartle 22 , he also derived the black hole temperature by using Feynman path
integral which can be categorized as semiclassical method, but it still has
such mathematical complexity. Semiclassical methods for black hole radiation 4
; 7 were developed in the past decade and they attract many attentions, e.g.
Refs. 15, and 16, .
There are two ways to perform semi-classical analysis for a black hole
radiation. The first is by the use of radial null geodesic method developed by
Parikh and Wilczek 4 . In this method, one has to get the expression dr/dt
from the radial null geodesic condition, $ds^{2}=d\Omega=0$, for a metric that
has the form
$ds^{2}=-a\left(r\right)dt^{2}+b\left(r\right)dr^{2}+r^{2}d\Omega^{2}$. Then,
the obtained expression is used to calculate the imaginary part of the action
for the process of s-wave emission and relates it to the Boltzman factor for
emission to get Hawking temperature.
The second one is called complex paths method was developed by Padmanabhan
et.al. 6 ; 7 . In the method, the scalar wave function is determined by the
ansatz
$\phi\left({r,t}\right)=\exp\left[{{{-iS\left({r,t}\right)}\mathord{\left/{\vphantom{{-iS\left({r,t}\right)}\hbar}}\right.\kern-1.2pt}\hbar}}\right]$
where $S\left({r,t}\right)$ is the action for a single scalar particle.
Inserting this ansatz into the Klein-Gordon equation in a gravitational
background, one yields an equation for the action $S\left({r,t}\right)$ which
can be solved by the Hamilton-Jacobi method. After obtaining the action, one
can get the probability for outgoing and ingoing particles,
$P\left[{{\rm{out}}}\right]=\left|{\phi_{{\rm{out}}}}\right|^{2}$ and
$P\left[{{\rm{in}}}\right]=\left|{\phi_{{\rm{in}}}}\right|^{2}$, respectively.
The Hawking temperature can be obtained by using the ’principle of detailed
balance’ 6 ; 7 , $P\left[{{\rm{out}}}\right]=\exp\left[{-\beta
E}\right]P\left[{{\rm{in}}}\right]=\exp\left[{-\beta E}\right]$, since all
particles must be absorbed by the black hole.
In this paper, we consider a more general metric with the mass of black hole
is time $\left(t\right)$ and radius $\left(r\right)$ dependent. As generally
known, this subject is well described by the Vaidya metric. To get an exact
$\left(t-r\right)$ dependence of the Hawking temperature, we insist to work in
using Schwarzschild like metric for the Vaidya space time 17 rather than
standard Eddington-Finkelstein metric which is widely used for this case. We
derive the general form for Hawking temperature both in the radial null
geodesic and complex paths methods. We arrive only at the general form since
one has to get the exact $\left({t-r}\right)$ dependence of mass which of
course needs a specific model of the related black hole. Until now, even
though some models for dynamical black holes had been proposed, e.g. Refs. 23,
and 24, , but efforts to get better one are still performed.
The organization of our paper is as follows. In the second section, we will
derive the Hawking temperature for a metric with a varying mass by the use of
radial null geodesic method. In the third section, the Hawking temperature is
obtained by the complex paths method, by considering only the lowest order
expansion of action. At the fourth section, we show the possible expression of
entropy. In the last section, we give a conclusion for our work. For the rest
of this paper, we use the unit dimension: Newton constant, light velocity in
vacuum, and Boltzman constant, $G=c=k_{B}=1$.
## II Radial Null Geodesic Method
Previously, a tightly connected work has been performed by T. Clifton 8 , but
with different method of calculations. In this work, we keep that the metric
contains the mass of black hole as the function of standard radius-time
$\left(r-t\right)$ coordinate rather than Eddington-Finkelstein radiation
coordinate $\left(u-v\right)$. We start with the metric derived by Farley and
D’Eath 17 for Vaidya space-time
$\displaystyle
ds^{2}=-\left({\frac{{\dot{m}}}{{x\left(m\right)}}}\right)^{2}\left({1-\frac{{2m}}{r}}\right)dt^{2}+\left({1-\frac{{2m}}{r}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(1)
In the above, the black hole mass $m$ varies with time $t$ and radius $r$,
$m\equiv m\left({r,t}\right)$. The $x\left(m\right)$ is an arbitrary function
of mass and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the
metric of unit $2-$sphere. We will let the arbitrariness of $x\left(m\right)$
since the determination of this function depends on the model of corresponding
black hole. A short discussion of the model will be given at the last section
but will not the main subject of this paper. The metric (1) has the general
form
$\displaystyle
ds^{2}=-F\left({r,t}\right)dt^{2}+G\left({r,t}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(2)
where in our case
$F\left({r,t}\right)\equiv\dot{m}^{2}x\left(m\right)^{-2}\left({1-2mr^{-1}}\right)$
and $G\left({r,t}\right)\equiv 1-2mr^{-1}$. The general form (2) will simplify
our next calculation. Unless for specific purposes, we will write
$x\left(m\right)$, $F\left({r,t}\right)$, and $G\left({r,t}\right)$ as $x$,
$F$, and $G$ respectively for the sake of brevity.
It turns out that the metric (1) has a coordinate singularity at $r_{h}=2m$
which of course is time dependent. Painleve transformation that is used to
remove the coordinate singularity for a metric with time-like Killing vector,
is also applicable for this analysis. By transforming
$\displaystyle dt\to dt-\sqrt{\frac{{1-G}}{{FG}}}dr$ (3)
the metric (2) changes into
$\displaystyle
ds^{2}=-F\left({r,t}\right)dt^{2}+2F\sqrt{\frac{{1-G}}{{FG}}}dtdr+dr^{2}+r^{2}d\Omega^{2},$
(4)
and therefore no coordinate singularity is found. Thus, by such a Painleve
transformation, it is understandable that, in principle, a coordinate
singularity in general relativity can be removed only by changing coordinate,
without defining new physical condition or theorem. By this thought, it would
be alright if we still insist in using the metric (1) for carrying out a
tunneling process. For radial null geodesic, $ds^{2}=d\Omega^{2}=0$, the
differentiation of radius with respect to time can be obtained from (4) as
$\displaystyle\frac{{dr}}{{dt}}=\pm\sqrt{FG},$ (5)
where $+(-)$ signs denote outgoing(ingoing) radial null geodesics.
Near the horizon, we can expand the coefficient $F$ and $G$ by the use of
Taylor expansion. Since both $F$ and $G$ are $\left({t-r}\right)$ dependent,
and we only need their approximation values for short distances from a point
(horizon), we could apply the Taylor expansion at a fixed time. So, we can
write
$\displaystyle\left.{F\left({r,t}\right)}\right|_{t}\simeq\left.{F^{\prime}\left({r,t}\right)}\right|_{t}\left({r-r_{h}}\right)+\left.{O\left({\left({r-r_{h}}\right)^{2}}\right)}\right|_{t},$
(6)
and
$\displaystyle\left.{G\left({r,t}\right)}\right|_{t}\simeq\left.{G^{\prime}\left({r_{h},t}\right)}\right|_{t}\left({r-r_{h}}\right)+\left.{O\left({\left({r-r_{h}}\right)^{2}}\right)}\right|_{t},$
(7)
By the approximations (6) and (7) above, the dependence of radius to time in
(5) can be approached by
$\displaystyle\frac{{dr}}{{dt}}\simeq\frac{1}{2}\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}\left({r-r_{h}}\right).$
(8)
Now, we discuss the action of outgoing particle through the horizon. In the
original work by Parikh and Wilczek 4 , the imaginary action is written as
${\mathop{\rm Im}\nolimits}S={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{p_{r}dr}={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{\int\limits_{0}^{p_{r}}{dp_{r}^{\prime}dr}}$
$\displaystyle={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{\int\limits_{0}^{H}{\frac{{dH^{\prime}}}{{{\textstyle{{dr}\over{dt}}}}}}}dr.$
(9)
The above expression is due to the Hamilton equation $dr/dt=dH/dp_{r}|_{r}$
where $r$ and $p_{r}$ are canonical variables (in this case, the radial
component of the radius and the momentum). As a reminder, the action of a
tunneled particle in a potential barrier higher than the energy of the
particle itself will be imaginary, $p_{r}=\sqrt{2m\left({E-V}\right)}$.
Different from discussions of several authors for a static black hole mass,
e.g. Refs. 4, , 8, –15, , 21, and 22, , the outgoing particle’s energy must
be time dependent for black holes with varying mass. So, the $dH^{\prime}$
integration at (9) is for all values of outgoing particle’s energy, say from
zero to $+E\left(t\right)$.
By using approximation (8), we can perform the integration (9). For $dr$
integration, we can perform a contour integration for upper half complex plane
to avoid the coordinate singularity $r_{h}$. The result is
$\displaystyle{\mathop{\rm Im}\nolimits}S=\frac{{2\pi
E\left(t\right)}}{{\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}.$
(10)
Since the tunneling probability is given by
$\Gamma\sim\exp\left[{-{\textstyle{2\over\hbar}}{\mathop{\rm
Im}\nolimits}S}\right]$, equalizing it with the Boltzmann factor
$\exp\left[{-\beta E\left(t\right)}\right]$ for a system with time dependent
energy we obtain
$\displaystyle
T_{H}=\frac{{\hbar\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}{{4\pi}}.$
(11)
As seen in expression (11), the Hawking temperature $T_{H}$ is time dependent.
We will see it later that it is also radius dependent. Partial derivatives of
$F$ and $G$ with respect to $r$ are easily to be found in the following forms
$F^{\prime}=\frac{{2m}}{{x^{2}}}\left[{m^{\prime}\left({1-\frac{{2m}}{r}}\right)-\frac{{mx^{\prime}}}{x}\left({1-\frac{{2m}}{r}}\right)+m\left({-\frac{{m^{\prime}}}{r}+\frac{m}{{r^{2}}}}\right)}\right],$
and
$G^{\prime}=\frac{2}{r}\left[{-m^{\prime}+\frac{m}{r}}\right].$
Inserting the values of $r_{h}=2m$ to the last expressions yields Hawking
temperature (11) in a general form
$\displaystyle
T_{H}=\frac{{\hbar\left({{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}-m^{\prime}\left({r_{h},t}\right)}\right)}}{{4\pi
x\left({r_{h},t}\right)}}.$ (12)
In the above, the $m^{\prime}$ and $x$ are written has the complete expression
as $m^{\prime}\left({r_{h},t}\right)$ and $x\left({r_{h},t}\right)$. These
mean that both $m^{\prime}$ and $x$ are evaluated at $r_{h}$ (the radius of
event horizon). Our finding in (12) behaves as it is expected for radius
independent black hole, where it means that $m^{\prime}(r_{h},t)=0$ and we get
$T_{H}={\hbar\mathord{\left/{\vphantom{\hbar{4\pi
x\left(t\right)}}}\right.\kern-1.2pt}{4\pi x\left(t\right)}}$ which is close
to an expression that had been guessed by Ashtekar long time ago,
$T_{H}={\hbar\mathord{\left/{\vphantom{\hbar{8\pi
m\left(t\right)}}}\right.\kern-1.2pt}{8\pi m\left(t\right)}}$.
## III Complex Paths Method
Massless scalar particles under the gravitational background $g_{\mu\nu}$ obey
the Klein-Gordon equation
$\displaystyle\frac{{-\hbar^{2}}}{{\sqrt{-g}}}\partial_{\mu}\left[{g^{\mu\nu}\sqrt{-g}\partial_{\nu}}\right]\phi=0.$
(13)
For spherical symmetric black hole, we may reduce our attention only to
$\left({t-r}\right)$ sector in the space-time, or in other words, we reduce to
two dimensional black hole problems. Later, since we are dealing with massless
particle described in (13), we can employ the null geodesic condition
${{dr}\mathord{\left/{\vphantom{{dr}{dt}}}\right.\kern-1.2pt}{dt}}=\pm\sqrt{FG}$
for outgoing particle’s path. Equation (13) under the background metric (2)
simplifies to
$\displaystyle\partial_{t}^{2}\phi-\frac{1}{{2FG}}\left({\dot{F}G+\dot{G}F}\right)\partial_{t}\phi-\frac{1}{2}\left({F^{\prime}G+FG^{\prime}}\right)\partial_{r}\phi-
fg\partial_{r}^{2}\phi=0.$ (14)
By the standard ansatz for scalar wave function
$\phi\left({r,t}\right)=\exp\left[{-{\textstyle{i\over\hbar}}S\left({r,t}\right)}\right]$,
equation (14) leads to the equation for the action $S\left({r,t}\right)$
$\left({\frac{{-i}}{\hbar}\left({\frac{{\partial^{2}S}}{{\partial
t^{2}}}}\right)}\right)-\frac{1}{{\hbar^{2}}}\left({\frac{{\partial
S}}{{\partial
t}}}\right)^{2}-\frac{1}{{2FG}}\left({\dot{F}G+\dot{G}F}\right)\left({\frac{{-i}}{\hbar}}\right)\left({\frac{{\partial
S}}{{\partial t}}}\right)$
$\displaystyle-\frac{1}{2}\left({F^{\prime}g+fG^{\prime}}\right)\left({\frac{{-i}}{\hbar}}\right)\left({\frac{{\partial
S}}{{\partial
r}}}\right)-fg\left({\frac{{-i}}{\hbar}\left({\frac{{\partial^{2}S}}{{\partial
r^{2}}}}\right)-\frac{1}{{\hbar^{2}}}\left({\frac{{\partial S}}{{\partial
r}}}\right)^{2}}\right)=0.$ (15)
Now, our next step is to solve this equation. An approximation method can be
applied by expanding the action in the order of Planck constant power,
$\displaystyle
S\left({r,t}\right)=S_{0}\left({r,t}\right)+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}S_{n}\left({r,t}\right)},$
(16)
for $n=1,2,3,...$. The constant $\alpha_{n}$ is set to keep all the expansion
terms have the action’s dimension. Taking unit dimensions $G=c=k_{B}=1$,
$\alpha_{n}$ would have the dimension of $[m]^{-2n}$ which $m$ refers to the
mass. It is clear that this expansion would lead to a very long equation. Due
to the very small value of the Planck constant, many authors 8 ; 10 ; 21
neglect the terms for $n\geq 1$. This consideration is acceptable, and
including higher terms is just adding correction for semi-classical derivation
of Hawking temperature. By grouping all the terms into the same powers of
$\hbar$, and take the term with lowest order of Planck’s constant (zero
power), one can write
$\displaystyle
FG\left({\partial_{r}S_{0}}\right)^{2}-\left({\partial_{t}S_{0}}\right)^{2}=0,$
(17)
Our next task is to find the solution for the last equation which is not too
simple since our $FG$ is $(r-t)$ dependent. In the standard Hamilton-Jacobi
method, $S_{0}\left({r,t}\right)$ can be written into two parts, the time part
which has the form of $Et$ and the radius part $\tilde{S}_{0}\left(r\right)$
which is in general a radius dependent only. Since our metric coefficients are
both radius and time dependent, the standard method would not be applicable.
We could generalized the method by making an ansatz
$\displaystyle
S_{0}\left({r,t}\right)=\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+\tilde{S}_{0}\left({r,t}\right).$
(18)
At the first sight, it seems that the ansatz is rather strange, that is
$S\left({r,t}\right)$ and $\tilde{S}_{0}\left({r,t}\right)$ are both $t$ and
$r$ dependent. The term $\int{E\left({t^{\prime}}\right)dt^{\prime}}$ is more
understandable, since the emitted particle’s energy is continuum and time
dependent. Let see how it works.
From (18), one can write that
$\displaystyle\partial_{t}S_{0}\left({r,t}\right)=E\left(t\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)$
(19)
and
$\displaystyle\partial_{r}S_{0}\left({r,t}\right)=\partial_{r}\tilde{S}_{0}\left({r,t}\right).$
(20)
Since $\tilde{S}_{0}\left({r,t}\right)$ is $t$ and $r$ dependent, one can
write
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}=\partial_{r}\tilde{S}_{0}\left({r,t}\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)\frac{{dt}}{{dr}}.$
(21)
Eliminating $dt/dr$ by the use of
${{dr}\mathord{\left/{\vphantom{{dr}{dt}}}\right.\kern-1.2pt}{dt}}=\pm\sqrt{FG}$,
equation (21) can be written as
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}\mp\frac{1}{{\sqrt{FG}}}\partial_{t}\tilde{S}_{0}\left({r,t}\right)=\partial_{r}\tilde{S}_{0}\left({r,t}\right).$
(22)
Combining the first equation of (17), with equations (18) and (22) where we
should note that the action equation
$-\left({FG}\right)^{-1/2}\partial_{t}S_{0}\left({r,t}\right)=\partial_{r}S_{0}\left({r,t}\right)$
is belong to outgoing particle and with the radial evolution is
$dr/dt=\sqrt{FG}$, then we could write
$\displaystyle\mp\left({FG}\right)^{-1/2}\left({E\left(t\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)}\right)=\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}\mp\left({FG}\right)^{-1/2}\partial_{t}\tilde{S}_{0}\left({r,t}\right).$
(23)
From (23), we can get the exact differentiation of
$\tilde{S}_{0}\left({r,t}\right)$
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}=\mp\left({FG}\right)^{-1/2}E\left(t\right),$
(24)
and the solution of $\tilde{S}_{0}\left({r,t}\right)$ can be obtained by
integration
$\displaystyle\tilde{S}_{0}\left({r,t}\right)=\mp
E\left(t\right)\int{\frac{{dr}}{{\sqrt{FG}}}}.$ (25)
The integration (25) can be evaluated by adopting the value of
$\int{\left({FG}\right)^{-1/2}dr}$ as in obtaining expression (10) from (9)
along with it’s approximation method (near horizon Taylor expansion). The
result for the integration (25) is
$\displaystyle\tilde{S}_{0}\left({r,t}\right)=\mp
E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}.$ (26)
The equation gives the complete action
$\displaystyle
S\left({r,t}\right)=\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}\mp
E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}.$ (27)
The signs $+\left(-\right)$ in expression (27) refer to the action for ingoing
(outgoing) particle.
Back to our first ansatz for scalar wave function,
$\phi=\exp\left[{{\textstyle{{-i}\over\hbar}}S\left({r,t}\right)}\right]$, the
wave function for ingoing and outgoing massless scalar particle can be read of
as
$\displaystyle\phi_{in}\left({r,t}\right)=\exp\left[{-\frac{i}{\hbar}\left({\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right]$
(28)
and
$\displaystyle\phi_{out}\left({r,t}\right)=\exp\left[{-\frac{i}{\hbar}\left({\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}-E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right]$
(29)
respectively. Consequently, from (28) one can get the ingoing probability of
particle as below
$\displaystyle P_{in}=\exp\left[{\frac{2}{\hbar}\left({{\mathop{\rm
Im}\nolimits}\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+\frac{{\pi
E\left(t\right)}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right].$ (30)
This ingoing probability must be equal to unity since all particles including
the massless one are absorbed by the black hole. This consideration gives us
the relation
${\mathop{\rm
Im}\nolimits}\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}=-\frac{{\pi
E\left(t\right)}}{{\sqrt{F^{\prime}G^{\prime}}}},$
which leads the outgoing probability
$\displaystyle P_{out}=\exp\left[{-\frac{{4\pi
E\left(t\right)}}{{\hbar\sqrt{F^{\prime}G^{\prime}}}}}\right].$ (31)
Finally, to get the Hawking temperature from the outgoing probability (31), we
equate this probability expression with $\exp\left[{-\beta
E\left(t\right)}\right]$ which in Refs. 6, and 7, is called ’detailed
balance’ principle. It yields
$\displaystyle
T_{H}=\frac{{\hbar\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}{{4\pi}}.$
(32)
## IV General Form of Entropy
The law of black hole mechanics which expresses the conservation of energy by
relating the change in black hole mass m to the changes of its entropy
$S_{bh}$, angular momentum $J$, and electric charge $Q$, is given by 25
$\displaystyle dm=T_{H}dS_{bh}+\Phi dQ+\Omega dJ$ (33)
where $\Omega$ is the angular velocity and $\Phi$ is electrostatic potential.
By simplifying the black hole with no angular velocity and electrical charges,
one only has to pay attention on
$\displaystyle dm=T_{H}dS_{bh}.$ (34)
Inserting our previous result for $T_{H}$ from (12), we an integral form for
entropy as
$\displaystyle
S_{bh}=\frac{{8\pi}}{\hbar}\int{\frac{{x\left(m\right)}}{{1-2m^{\prime}}}dm}.$
(35)
To get an exact value of entropy which is intuitively should be equivalent to
dynamical area of black hole, we must write $x\left(m\right)$, $m^{\prime}$,
and $\dot{m}$ in their exact forms. This manner would be such an interesting
work to be pursued further.
## V Summary
To summarize our findings above, we can state that we have derived a general
expression for Hawking temperature (up to the model function $x\left(m\right)$
and partial derivative of black hole mass with respect to radius) at (11) and
(12) in two different semiclassical approaches, null geodesic and complex
paths methods for a Vaidya black hole. This kind of black hole is regarded as
the description of a more realistic one since it’s time dependent decreasing
mass due to the evaporation process. Clearly, the temperature would be time
dependent as our findings. Both methods are found to give the same results.
Then, we discuss he possible general form of discussed black hole entropy
(35). This last finding perhaps can be compared to some models that had been
proposed, e.g. Refs. 23, and 24, .
## Acknowledgments
H.M.S. is grateful to M. Siahaan and N. Sinurat for their supports and
encouragements.
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|
arxiv-papers
| 2009-08-20T10:10:21 |
2024-09-04T02:49:04.697746
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Haryanto M. Siahaan and Triyanta",
"submitter": "Haryanto Siahaan",
"url": "https://arxiv.org/abs/0908.2895"
}
|
0908.2948
|
††thanks: Present address: Department of Applied Physics, Yale University, New
Haven, CT 06520
# Reducing quantum-regime dielectric loss of silicon nitride for
superconducting quantum circuits
Hanhee Paik hanhee.paik@yale.edu Kevin D. Osborn Laboratory for Physical
Sciences
College Park, MD 20740
###### Abstract
The loss of amorphous hydrogenated silicon nitride (a-SiNx:H) is measured at
30 mK and 5 GHz using a superconducting LC resonator down to energies where a
single-photon is stored, and analyzed with an independent two-level system
(TLS) defect model. Each a-SiNx:H film was deposited with different
concentrations of hydrogen impurities. We find that quantum-regime dielectric
loss tangent $\tan\delta_{0}$ in a-SiNx:H is strongly correlated with N-H
impurities, including NH2. By slightly reducing $x$ we are able to reduce
$\tan\delta_{0}$ by approximately a factor of 50, where the best films show
$\tan\delta_{0}$ $\simeq$ 3 $\times$ 10-5.
###### pacs:
77.84.Bw, 84.40.Dc, 85.25.-j
††preprint: APS/123-QED
Superconducting quantum circuits use amorphous dielectric films for wiring
crossovers and capacitors Siddiqi05a ; Steffen06b ; Sillanpaa07a , but these
films often cause loss at low temperatures in the quantum-regime where the
resonator is occupied by a single photon. These defects can be described by a
tunneling two-level system (TLS) model Schickfus77a . There has been recent
interest in the low-temperature properties of dielectrics in superconducting
devices OConnell ; Barends ; Gao08a because they can increase the decoherence
rate $1/T_{1}$ in a superconducting phase qubits Martinis05a and the phase
noise in microwave kinetic inductance detectors Gao07 .
While both films are common in microelectronics, amorphous hydrogenated
silicon nitride (a-SiNx:H) is found to exhibit less dielectric loss than
silicon dioxide (SiO2) in the quantum regime Martinis05a . Close to the
stoichiometric point $x\simeq 4/3$, the electronic and optical properties of
this film are determined by the dominant hydrogen impurity which can be
changed by deposition conditions Martinu00a .
In this paper we first show the composition of five films grown under the
condition where there is a strong variation in relative hydrogen impurity
type. Then we present low-temperature loss measurements of the a-SiNx:H films
using superconducting resonators. The loss in the quantum-regime of the films
varied dramatically as the relative amount of N-H and Si-H impurities was
changed, and in particular we discovered that the NH2 concentration is
proportional to the TLS density indicating that NH2 may be responsible for the
microscopic loss. This research demonstrates the low temperature dielectric
loss in a-SiNx:H can be reduced by reducing nitrogen impurities.
Table 1: a-SiNx:H film and resonator sample properties: precursor gas ratio $f_{\textrm{N}_{2}/\textrm{SiH}_{4}}$, refractive index $n$ measured at 633 nm, compressive stress $\sigma_{c}$ (MPa), ratio of N-H bond concentration to total N-H and Si-H concentration $c{}_{N-H}/c_{H}$, and five resonator fit parameters. samples | $f_{\textrm{N}_{2}/\textrm{SiH}_{4}}$ | $n$ | $\sigma_{c}$ (MPa) | $c{}_{N-H}/c_{H}$ | tan$\delta_{0}\times 10^{6}$ | $V{}_{c}$ ($\mu$V) | $\Delta$ | $f{}_{R}$ (GHz)
---|---|---|---|---|---|---|---|---
A (Si rich) | 1 | 2.159 | 886 | 31 | 25 | 3.8 | 0.35 | 4.970
B (Si rich) | 1.1 | 2.094 | 1178 | 40 | 25 | 2.0 | 0.35 | 4.987
C (Si rich) | 1.12 | 2.065 | 1000 | 38 | 25 | 2.0 | 0.35 | 5.170
D (N rich) | 1.2 | 1.922 | 82 | 72 | 500 | 0.65 | 0.28 | 5.196
E (N rich) | 1.21 | 1.916 | 100 | 83 | 1200 | 0.12 | 0.28 | 5.350
The a-SiNx:H films were deposited using inductively-coupled plasma chemical
vapor deposition (ICP CVD) in an Oxford Plasmalab 100, using nitrogen (N2) and
100% silane (SiH4) precursor gases. The films were deposited at T = 300 ∘C at
a pressure of 5 mTorr, with an ICP power of 500 W, and a nominal rf power of 4
W. Table 1 summarizes the device parameters and loss tangent fit results in
terms of the precursor gas flow rate ratio of N2 to SiH4,
$f_{\textrm{N}_{2}/\textrm{SiH}_{4}}$. The SiH4 flow rate was set to 10 sccm
for all films except for E (9 sccm) while N2 flow rate was varied.
Figure 1: FT-IR absorbance versus wavenumber $\nu=1/\lambda$, shown with
offsets for better viewing. The nitrogen to silane ratio increases from film A
to E.
The refractive index and the compressive stress were used to determine whether
a film is N-rich or Si-rich. Parsons91a ; Martinu00a . An N$\&$K Analyzer
NKT1500 was used to measure the film refractive index at $\lambda=$633 nm and
the thickness, and a KLA Tencor P-10 profilometer was used to measure film
stress. For N-rich films, the refractive index is below 2 and the compressive
stress is significantly reduced. Only a 10 $\%$ change in
$f_{\textrm{N}_{2}/\textrm{SiH}_{4}}$ yielded a 1 GPa change in stress
implying a large change in the structure of the film when the stoichiometry
changes between N-rich and Si-rich. The refractive index and stress were also
consistent with a second set of 5 films that were measured with rf, and the
growth rate in the films also systematically depended on the stoichiometry
(not shown). Although the parameters may depend on the particular machine, a
similar procedure can be performed with another deposition system to find this
crossover stoichiometry.
To evaluate the impurity types of the a-SiNx:H films, we measured FT-IR
absorption with a Nicolet 670 attenuated total-reflection FT-IR spectrometer
with a ZnSe prism. The films for FT-IR analysis were deposited to 1 $\mu$m
thick to prevent absorption from the underlying substrate and were deposited
immediately after those of the resonators. Figure 1 shows the absorption
spectra with baseline correction of five SiNx:H films (A to E). The six
absorption bands identified include 1) the Si-N stretching mode appears at 890
cm-1, 2) the N-H bending mode at 1180 cm-1, 3) the H-N-H scissoring mode at
1545 cm-1, 4) the Si-H stretching mode at 2180 cm-1, 5) the N-H stretching
mode at 3340 cm-1 and 6) the H-N-H stretching mode at 3460 cm-1 Lucovsky86a ;
Yin90a ; Hanyaloglu98a . The relative concentration of N-H bond concentration
to the sum of the N-H and Si-H bond concentrations $c_{N-H}/c_{H}$ is
estimated from the corresponding stretching mode absorbances and the relative
absorptivity of the two modes (shown in Table I). The low value $(<40\%)$ of
$c_{N-H}/c_{H}$ for films A, B, and C and the high value $(>70\%)$ of D and E
indicates that the former films are Si-rich and the later films are N-rich. It
is interesting that NH2 absorption modes (both scissoring and stretching)
appear strongly in N-rich samples D and E, which shows a N-rich film has a
different bonding structure than a Si-rich film as reported previously Tsu86a
; Lucovsky86a .
Figure 2: a) Photograph of notch-type aluminum LC resonator with a SiNx
parallel plate capacitor. Size of the capacitor (shown in right) is 80$\mu$m
by 80$\mu$m. The substrate is sapphire which appears black in the photograph.
b) Loss tangent curves of samples A to E measured at 30 mK (shown with
markers) with the two-level system model fit in Eq. (2) (shown with fit
curves).
We measured dielectric loss of a-SiNx:H films within fabricated Al
superconducting LC resonators (see Figure 2). The LC resonator consists of a
meandering inductor $L$ and a parallel-plate capacitor $\hat{C}$ in parallel.
Here $\hat{C}=C(1-i\tan\delta)$ where $C$ is the real part of the capacitance
and tan$\delta$ is the loss tangent from dielectric. The $L$ and $C$ couple to
a coplanar waveguide [Figure 2(a)] to form a notch-filter resonator.
For loss tangent measurement, we used an Anritsu 68369A microwave synthesizer,
frequency locked to an Agilent E4440A spectrum analyzer. The driving line is
attenuated with 20 dB attenuators at both 1 K and the base temperature stages
and the input line is calibrated at room temperature. The return line is
isolated by a PAMTech circulator at the 1 K stage and the transmitted signal
is amplified with a 4-12 GHz Caltech HEMT amplifier at 4K. Thermal photons
from the circulator at 1 K, can create on the order of one thermal photon in
our resonators, allowing us to measure coherent response down to the quantum-
regime. We obtained the loaded quality factor $Q$ and the internal quality
factor $Q_{i}$ by fitting the measured power transmission to our model
function $|t|^{2}$ given by
$|t|^{2}=\left|1-e^{i\phi}\frac{1-Q/Q_{i}}{1+2iQ(f-f_{0})/f_{0}}\right|^{2}.$
(1)
A loaded $Q$ and an internal $Q_{i}$ are the fitting parameters together with
a resonance frequency $f_{0}$ and a loss tangent is given as
tan$\delta=1/Q_{i}$. The coupling quality factor $Q_{e}$ is
$\left(Q^{-1}-Q_{i}^{-1}\right)^{-1}\simeq 20000$ for our resonators, and the
parameter $\phi$ accounts for small impedance mismatches for waveguides near
our resonator. The capacitor dielectric has a thickness of 250 nm, and a
nominal capacitance of 1.47 pF assuming a relative permitivity of 6.5, which
at $<V^{2}>^{1/2}\approx 3\times 10^{-7}$ corresponds to an average photon
number of $\langle n\rangle\approx$ 0.1 (not including thermal photons).
However, from the the resonant frequency in Table I, it is apparent that the
dielectric constant changes with stoichiometry.
Figure 2(b) shows the loss tangent of a-SiNx:H films at 30 mK as a function of
RMS voltage $V$ on the resonator. The error bars are from our $\chi^{2}$ fit
of the resonance peaks to the model function $|t|^{2}$. The horizontal scale
error is limited by a calibration of our input line at room temperature. We
fit the low temperature loss tangent data to a two-level system defect model
with a parallel-plate geometry which is given as
$\tan\delta=\frac{\pi\rho(er)^{2}\textrm{tanh}(\hbar\omega/2k_{B}T)}{3\epsilon\sqrt{1+(\Omega_{R})^{2}T_{1}T_{2}}}\simeq\frac{tan\delta_{0}}{\sqrt{1+(V/V_{c})^{2-\Delta}}}$
(2)
where $\rho$ is the TLS density of states, $e$ is the electron charge, $r$ is
a distance between two level sites, $\varepsilon$ is a permittivity of the
film, $\Omega_{R}=eVr/\hbar d$ is a TLS Rabi frequency, $V$ is an RMS voltage,
d is the thickness of a-SiNx:H, $\omega$ is the angular resonance frequency
and $T$ is the temperature of the resonator Martinis05a . Here the intrinsic
loss tangent tan$\delta_{0}$ which is defined as the dielectric loss in a
single photon regime is a function of TLS density of states $\rho$ and $V_{c}$
is a threshold voltage where saturation starts to occur in TLS Phillips87a .
The fitting parameters are summarized in Table I.
Surprisingly, Si-rich SiNx:H films (A, B and C) showed a 20 and 50 lower
intrinsic loss than the N-rich films (D and E). This also implies that the TLS
density of states $\rho$ is correlated with the N-H bond concentration. We
believe the results are reproducible from the fact that the samples A, B (low
loss) and D (high loss) were fabricated 8 months after the samples C (low
loss) and E (high loss) were made. We confirmed our film’s reproducibility by
making another low-loss a-SiNx:H film 4 months after these films A, B, and D,
using the same recipe as the sample A. With that new film we obtained
tan$\delta_{0}\simeq$ 3$\times$ 10-5, which agrees with the original
measurement of film A.
Figure 3: Density of states as a function of NH2 bonding peak area in FT-IR
measurement.
In Figure 3, we plot $\rho$ normalized by the TLS density of states $\rho_{0}$
of the lowest loss film (from samples A, B and C) as a function of NH2
scissoring mode absorption band area A${}_{NH_{2}}$, which is proportional to
the concentration of the NH2 bond Lanford78a ; Yin90a ; Parsons91a . We find
that $\rho$ is proportional to A${}_{NH_{2}}$ and the amount of NH2 in the
three Si-rich films (samples A, B and C) is much smaller than the N-rich
samples. (Note that the NH2 stretching mode almost disappears for the Si-rich
films A, B and C.) Therefore, for the two N-rich a-SiNx:H films, NH2 is likely
responsible for the TLS dielectric loss.
The mechanism by which NH2 bonding causes low temperature loss can be
understood by comparing the film with one composed of silicon dioxide with
hydrogen impurities (SiOx:H). The N-rich a-SiNx:H is known to have a local
bonding arrangement that is usually seen in Si(NH)2 Tsu86a ; Lucovsky86a
which is isoelectric and isostructural with SiOx:H. In this case, $O$ is
substituted with $NH$ Tsu86a ; Lucovsky86a ; Martinu00a , thus the NH2
impurity in a-SiNx:H is analogous to the OH impurity in SiOx:H. The molecular
motion of the OH bond is believed to cause the low-temperature loss in a-SiO2
Schickfus77a ; Phillips87a ; and similarly, the molecular motion of NH2 may
cause the loss in a-SiNx:H.
The measured loss tangent includes the loss from the few nanometer thick
native aluminum oxide on top of the bottom Al electrode, which was not
removed. In our case with a parallel plate capacitor, the field energy that
resides in the native oxide is on the order of $10^{-2}$ of the total electric
field energy. If we assume that the observed loss is entirely from the native
oxide, then the loss tangent of the aluminum oxide is
$\tan\delta_{0,AlOx}\simeq$ 3$\times$10-3, which is similar to the previously
reported estimates Martinis05a .
In conclusion, we have measured the quantum-regime loss tangent of five
a-SiNx:H films at 30 mK. Each film was grown with a different precursor gas
flow rate ratio $f_{\textrm{N}_{2}/\textrm{SiH}_{4}}$, such that films were
rich in either Si-H or N-H impurities. We found that the NH2 bond in N-rich
a-SiNx:H is correlated with TLS-induced dielectric loss and were able to
reduce the quantum-regime loss tangent to 3 $\times$ 10-5 by making a-SiNx:H
films Si-rich.
The authors thank John Martinis, Ben Palmer, Dave Schuster, and Fred Wellstood
for useful discussions, and Dan Hinkle for his advice on amorphous silicon
nitride deposition. This work was funded by the National Security Agency.
## References
* (1) I. Siddiqi et al., Phys. Rev. Lett. 94, 027005 (2005).
* (2) M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006).
* (3) M. A. Sillanpaa et al., Nature 449, 438-442 (2007).
* (4) M. von Schickfus and S. Hucklinger, Phys. Lett. 64A, 14 (1977).
* (5) W. A. Phillips, Rep. Prog. Phys. 50, 1657 (1987).
* (6) Aaron D. O’Connell et al., APL 92, 112903 (2008).
* (7) J. Gao et al., Appl. Phys. Lett. 92, 152505 (2008).
* (8) J. Gao et al., Appl. Phys. Lett. 90, 102507 (2007).
* (9) R. Barends et al., Appl. Phys. Lett. 92, 223502 (2008).
* (10) J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005).
* (11) L. Martinu and D. Poitras, J. Vac. Sci. Technol. A 18, 2619 (2000).
* (12) G. Lucovsky and D. V. Tsu, J. Vac. Sci. Technol. A 4, 681 (1986).
* (13) Z. Yin and F. W. Smith, Phys. Rev. B 42, 3666 (1990).
* (14) B. F. Hanyaloglu and E. S. Aydil, J. Vac. Technol. A 16, 2794 (1998).
* (15) D. V. Tsu et al., Phys. Rev. B 33, 7069 (1986).
* (16) G. N. Parsons et al., J. Appl. Phys. 70, 1553 (1991).
* (17) W. A. Lanford and M. J. Rand, J. Appl. Phys. 49, 2473 (1978).
|
arxiv-papers
| 2009-08-20T16:16:44 |
2024-09-04T02:49:04.703788
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hanhee Paik, Kevin D. Osborn",
"submitter": "Hanhee Paik",
"url": "https://arxiv.org/abs/0908.2948"
}
|
0908.3266
|
# Extension and averaging operators for finite fields
Doowon Koh
###### Abstract.
In this paper we study $L^{p}-L^{r}$ estimates of both extension operators and
averaging operators associated with the algebraic variety
$S=\\{x\in{\mathbb{F}}_{q}^{d}:Q(x)=0\\}$ where $Q(x)$ is a nondegenerate
quadratic form over the finite field ${\mathbb{F}}_{q}.$ In the case when
$d\geq 3$ is odd and the surface $S$ contains a $(d-1)/2$-dimensional
subspace, we obtain the exponent $r$ where the $L^{2}-L^{r}$ extension
estimate is sharp. In particular, we give the complete solution to the
extension problems related to specific surfaces $S$ in three dimension. In
even dimensions $d\geq 2$, we also investigates the sharp $L^{2}-L^{r}$
extension estimate. Such results are of the generalized version and extension
to higher dimensions for the conical extension problems which Mochenhaupt and
Tao ([10]) studied in three dimensions. The boundedness of averaging operators
over the surface $S$ is also studied. In odd dimensions $d\geq 3$ we
completely solve the problems for $L^{p}-L^{r}$ estimates of averaging
operators related to the surface $S.$ On the other hand, in the case when
$d\geq 2$ is even and $S$ contains a $d/2$-dimensional subspace, using our
optimal $L^{2}-L^{r}$ results for extension theorems we, except for endpoints,
have the sharp $L^{p}-L^{r}$ estimates of the averaging operator over the
surface $S$ in even dimensions.
###### Contents
1. 1 Introduction
1. 1.1 Definition of extension and averaging problems in the finite field setting
2. 2 Statement of necessary conditions for $R^{*}(p\to r)\lesssim 1$ and $A(p\to r)\lesssim 1$
1. 2.1 Necessary conditions for the boundedness of extension operators
2. 2.2 Necessary conditions for the boundedness of averaging operators
3. 3 Statement of main results
1. 3.1 Results of extension problems
2. 3.2 Results of averaging problems
4. 4 Estimate of the Fourier transform of the surface measure $d\sigma$
5. 5 Proof of necessary conditions
1. 5.1 Proof of Theorem 2.1(Necessary conditions for the boundedness of extension operators)
2. 5.2 Proof of Theorem 2.2 (Necessary conditions for the boundedness of averaging operators)
6. 6 Proof of results for extension problems (Theorem 3.1 and Theorem 3.3)
7. 7 Proof of the results for averaging problems(Theorem 3.6)
1. 7.1 Proof of (3.8) in Theorem 3.6
2. 7.2 Proof of (3.9) in Theorem 3.6
## 1\. Introduction
In recent year the topics of harmonic analysis in the Euclidean setting have
been studied in the finite field settings. In this paper we study mapping
properties of extension operators and averaging operators related to surfaces
in $d$-dimensional vector space over a finite field ${\mathbb{F}}_{q}.$ These
topics are the analogues of the corresponding Euclidean problems. In the
Euclidean setting the extension theorem is the problem of determining the
optimal range of exponents $(p,r)$ such that the following estimate holds:
$\|(gd\sigma)^{\vee}\|_{L^{r}({\mathbb{R}}^{d})}\leq
C(p,r,d)\|g\|_{L^{p}(S,d\sigma)}~{}~{}\mbox{for all}~{}~{}g\in
L^{p}(S,d\sigma)$
where $d\sigma$ is a measure on the set $S$ in ${\mathbb{R}}^{d}.$ In 1967,
this problem was addressed by E.M.Stein and it has been extensively studied.
In the case when the set $S$ is a hypersurface such as a sphere, paraboloid,
or cone, the extension problems have received much attention. In higher
dimensions the problems are still open. For a comprehensive survey of these
problems, see [1], [13], [3], and [14] and the references therein.
Another interesting problem in classical harmonic analysis is to study the
boundedness of averaging operators associated with some surfaces in
${\mathbb{R}}^{d}.$ This problem is origin from investigating the regularity
of the fundamental solution of the wave equation for a fixed time. More
precisely, one asks that what is the optimal range of exponents $(p,r)$ such
that
(1.1) $\|f\ast d\sigma\|_{L^{r}({\mathbb{R}}^{d})}\leq
C(p,r,d)\|f\|_{L^{p}({\mathbb{R}}^{d})}~{}~{}\mbox{for all}~{}~{}f\in
L^{p}({\mathbb{R}}^{d}),$
where $d\sigma$ is a measure on a surface $S$ in ${\mathbb{R}}^{d}.$ For
example, if $S$ is a hypersurface with everywhere nonvanishing Gaussian
curvature, then the estimate (1.1) holds if and only if $({1}/{p},{1}/{r})$
lies in the triangle with vertices $(0,0),(1,1),$ and $(d/(d+1),1/(d+1))$ (see
[11], [12], and [8]). It is also known that the decay of the Fourier transform
of the surface measure $d\sigma$ determines the boundedness of the averaging
operator over the surface $S.$ For instance, if $S\subset{\mathbb{R}}^{d}$ is
a hypersurface with a surface measure $d\sigma$ and
$|\widehat{d\sigma}(\xi)|\leq C(1+|\xi|)^{-(d-1)/2},$ then the optimal result
for the boundedness of the averaging operator can be obtained by interpolating
the $L^{2}-L^{2}$ and $L^{1}-L^{\infty}$ estimates along with the trivial
$L^{1}-L^{1}$ and $L^{\infty}-L^{\infty}$ estimates. However, in the case when
the decay of the Fourier transform of the surface measure $d\sigma$ is
somewhat worse, the result obtained by the interpolations as before is not
sharp even if the decay is optimal. Thus, one may be interested in
investigating the optimal $L^{p}-L^{r}$ estimate for the averaging operator
related to the surface $S$ on which the Fourier decay is somewhat worse. For
example, A. Iosevich and E. Sawyer ([7]) obtained certain conditions which
give the sharp $L^{p}-L^{r}$ estimates for averaging operators related to a
graph of homogeneous function of degree $\geq 2.$ As one of the key ideas for
the optimal conditions, they considered a damped averaging operator and
applied the analytic interpolation theorem(see [7]). However, it seems that
such ideas do not work in finite field setting, in part because a complex
power of the element in the finite field is ambiguous to define.
In the finite field setting, both extension theorems and the boundedness of
averaging operators related to some algebraic varieties have recently been
studied extensively. It is very interesting to see that such problems for
finite fields sometimes exhibit unexpected phenomenas which never happen in
the Euclidean case. For example, the Fourier decay of certain algebraic
varieties in even dimensions can be distinguished with that in odd dimensions.
This yields some different results of the boundedness of operators between in
even dimensions and in odd dimensions. In this paper we shall study the
extension theorem and averaging operators associated with such algebraic
varieties in $d$-dimensional vector spaces over finite fields. Mochenhaupt and
Tao ([10]) first constructed and studied extension problems in the finite
field setting for various algebraic varieties. They mainly obtained quite good
results for paraboloids and cones in lower dimensions. Iosevich and Koh ([4],
[5]) developed aforementioned authors’ work for both paraboloids and spheres
in higher even dimensions. On the other hand, Carbery, Stones and Wright ([2])
recently introduced the averaging problems over algebraic varieties related to
vector-valued polynomials. In the case when the decay of the Fourier transform
of the surface measure is quite good they provided us of the best possible
$L^{p}-L^{r}$ mapping property of the averaging operators. In addition they
obtained partial results for averaging problems over surfaces with affine
subspaces of large dimension but the results seem to be far from the best
expected results.
The purpose of this paper is to introduce specific algebraic varieties of
which the Fourier decays are distinguished between odd dimensions and even
dimensions and to study the $L^{p}-L^{r}$ mapping properties of both extension
operators and averaging operators related to such surfaces in the finite field
setting. Doing these we shall achieve the following two goals. First, we
extend the conical extension theorems in three dimensions done by Mockenhaupt
and Tao ([10]) into higher dimensions. Second, we find the almost sharp result
of the boundedness of averaging operators associated with some sample
algebraic varieties containing large dimensional affine subspaces. In order to
state our results we begin by introducing some notation and definitions in the
finite field setting. We denote by ${\mathbb{F}}_{q}$ a finite field with $q$
elements and assume that the characteristic of ${\mathbb{F}}_{q}$ is greater
than $2$. Namely $q$ is a power of odd prime. As usual, ${\mathbb{F}}_{q}^{d}$
refers to the $d$-dimensional vector space over a finite field
${\mathbb{F}}_{q}$. Let $g:{\mathbb{F}}_{q}^{d}\to{\mathbb{C}}$ be a complex
valued function on ${\mathbb{F}}_{q}^{d}.$ We endow the space
${\mathbb{F}}_{q}^{d}$ with a counting measure $dm.$ Thus, the integral of the
function $g$ over $({\mathbb{F}}_{q}^{d},dm)$ is given by
$\int_{{\mathbb{F}}_{q}^{d}}g(m)~{}dm=\sum_{m\in{\mathbb{F}_{q}^{d}}}g(m).$
For a fixed non-trivial additive character
$\chi:{\mathbb{F}}_{q}\to{\mathbb{C}}$ and a complex valued function $g$ on
$({\mathbb{F}_{q}^{d}},dm)$, we define the Fourier transform of $g$ by the
following formula
(1.2) $\widehat{g}(x)=\int_{{\mathbb{F}_{q}^{d}}}\chi(-m\cdot
x)g(m)~{}dm=\sum_{m\in{\mathbb{F}_{q}^{d}}}\chi(-m\cdot x)g(m),$
where $x$ is an element of the dual space of $({\mathbb{F}_{q}^{d}},dm).$
Recall that the Fourier transform of the function $g$ on
$({\mathbb{F}_{q}^{d}},dm)$ is actually defined on the dual space
$({\mathbb{F}_{q}^{d}},dx).$ Here we endow the dual space
$({\mathbb{F}_{q}^{d}},dx)$ with a normalized counting measure $dx.$ We
therefore see that if $f:({\mathbb{F}_{q}^{d}},dx)\to{\mathbb{C}},$ then its
integral over $({\mathbb{F}_{q}^{d}},dx)$ is given by
$\int_{{\mathbb{F}_{q}^{d}}}f(x)~{}dx=\frac{1}{q^{d}}\sum_{x\in{\mathbb{F}_{q}^{d}}}f(x).$
Note that the Fourier transform of the function $f$ on
$({\mathbb{F}_{q}^{d}},dx)$ is given by the formula
$\widehat{f}(m)=\int_{{\mathbb{F}_{q}^{d}}}\chi(-x\cdot
m)f(x)~{}dx=\frac{1}{q^{d}}\sum_{x\in{\mathbb{F}_{q}^{d}}}\chi(-x\cdot
m)f(x),$
where $m$ is any element in $({\mathbb{F}}_{q}^{d},dm)$ with the counting
measure $dm$. We also recall that the Fourier inversion theorem holds: for
$x\in({\mathbb{F}_{q}^{d}},dx)$
$f(x)=\int_{m\in{\mathbb{F}_{q}^{d}}}\chi(m\cdot
x)\widehat{f}(m)dm=\sum_{m\in{\mathbb{F}_{q}^{d}}}\chi(m\cdot
x)\widehat{f}(m).$
Using the orthogonality relation of the non-trivial additive character, we see
that the Plancherel theorem holds:
$\|\widehat{f}\|_{L^{2}({\mathbb{F}}_{q}^{d},dm)}=\|f\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}.$
In other words, the Plancherel theorem yields the following equation:
(1.3)
$\sum_{m\in{\mathbb{F}_{q}^{d}}}|\widehat{f}(m)|^{2}=\frac{1}{q^{d}}\sum_{x\in{\mathbb{F}_{q}^{d}}}|f(x)|^{2}.$
Let $f$ and $h$ be the complex valued functions defined on
$({\mathbb{F}_{q}^{d}},dx).$ Then we can easily check that
$\widehat{(f\ast
h)}(m)=\widehat{f}(m)\cdot\widehat{h}(m)~{}~{}\mbox{and}~{}~{}\widehat{(f\cdot
h)}(m)=(\widehat{f}\ast\widehat{h})(m).$
###### Remark 1.1.
Throughout the paper we always consider the variable $``m"$ as an element of
$({\mathbb{F}_{q}^{d}},dm)$ with the counting measure $``dm".$ On the other
hand, we always use the variable $``x"$ as an element of
$({\mathbb{F}_{q}^{d}},dx)$ with the normalized counting measure $``dx".$
We now introduce algebraic varieties $S$ in $({\mathbb{F}_{q}^{d}},dx)$ on
which we shall work. Given $a_{j}\in{\mathbb{F}_{q}}\setminus\\{0\\}$ for
$j=1,2,\dots,d,$ we define an algebraic variety $S$ in
$({\mathbb{F}_{q}^{d}},dx)$ by the set
$S=\\{x\in{\mathbb{F}_{q}^{d}}:Q(x)=0\\},$
where $Q(x)$ denotes a nondegenerate quadratic polynomial. By a nonsingular
linear substitution, any nondegenerate quadratic polynomial $Q(x)$ can be
transformed into $a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}$ for some $a_{j}\neq
0,j=1,\dots,d.$ Therefore we may express the set $S$ as follows:
(1.4)
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}\subset({\mathbb{F}_{q}^{d}},dx).$
We endow the set $S$ with a normalized surface measure $d\sigma$ which is
given by the relation
$\int_{S}f(x)~{}d\sigma(x)=\frac{1}{|S|}\sum_{x\in S}f(x),$
where $|S|$ denotes the number of elements in $S.$ Note that the total mass of
$S$ is one and the measure $\sigma$ is just a function on
$({\mathbb{F}_{q}^{d}},dx)$ given by
$\sigma(x)=\frac{q^{d}}{|S|}S(x),$
here, and throughout the paper, we identify the set $S$ with the
characteristic function on the set $S.$ For example, we write $E(x)$ for
$\chi_{E}(x)$ where $E$ is a subset of ${\mathbb{F}_{q}^{d}}.$
### 1.1. Definition of extension and averaging problems in the finite field
setting
We introduce the definition of the extension problem related to the algebraic
variety $S$ in $({\mathbb{F}_{q}^{d}},dx).$ For $1\leq p,r\leq\infty,$ we
denote by $R^{*}(p\to r)$ the smallest constant such that the following
extension estimate holds:
$\|(fd\sigma)^{\vee}\|_{L^{r}({\mathbb{F}_{q}^{d}},dm)}\leq R^{*}(p\to
r)\|f\|_{L^{p}(S,d\sigma)}$
for every function $f$ defined on $S$ in $({\mathbb{F}_{q}^{d}},dx).$ By
duality, we see that the quantity $R^{*}(p\to r)$ is also the smallest
constant such that the following restriction estimate holds: for every
function $g$ on $({\mathbb{F}_{q}^{d}},dm),$
(1.5) $\|\widehat{g}\|_{L^{p^{\prime}}(S,d\sigma)}\leq R^{*}(p\to
r)\|g\|_{L^{r^{\prime}}({\mathbb{F}_{q}^{d}},dm)},$
here, throughout the paper, $p^{\prime}$ and $r^{\prime}$ denote the dual
exponents of $p$ and $r$ respectively. In other words, $1/p+1/p^{\prime}=1$
and $1/r+1/r^{\prime}=1.$ The constant $R^{*}(p\to r)$ may depend on $q,$ the
size of the underlying finite field ${\mathbb{F}_{q}}.$ However the extension
problem is to determine the exponents $(p,r)$ such that $R^{*}(p\to r)\lesssim
1$ where the constant in the notation $\lesssim$ is independent of $q$. Here
we recall that for positive numbers $A$ and $B$, the notation $A\lesssim B$
means that there exists a constant $C>0$ independent of the parameter $q$ such
that $A\leq CB.$ We also review that the notation $A\lessapprox B$ is used if
for every $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that $A\leq
C_{\varepsilon}q^{\varepsilon}B.$ We also use the notation $A\sim B$ which
means that there exist $C_{1}>0$ and $C_{2}>0$ such that $C_{1}A\leq B\leq
C_{2}A.$
###### Remark 1.2.
A direct calculation yields the trivial estimate, $R^{*}(1\to\infty)\lesssim
1.$ Using Hölder’s inequality and the nesting properties of $L^{p}$-norms, we
also see that
$R^{*}(p_{1}\to r)\leq R^{*}(p_{2}\to r)\quad\mbox{for}~{}~{}1\leq p_{2}\leq
p_{1}\leq\infty$
and
$R^{*}(p\to r_{1})\leq R^{*}(p\to r_{2})\quad\mbox{for}~{}~{}1\leq r_{2}\leq
r_{1}\leq\infty.$
Therefore, if we want to obtain the optimal result, then for any fixed
exponent $1\leq p(\mbox{or}~{}r)\leq\infty$, it is enough to find the smallest
exponent $1\leq r(\mbox{or}~{}p)\leq\infty$ such that $R^{*}(p\to r)\lesssim
1.$ By interpolating the result $R^{*}(p\to r)\lesssim 1$ with the trivial
bound $R^{*}(1\to\infty)\lesssim 1$, further results can be obtained.
We now introduce the averaging problems over the algebraic variety $S$ in
$({\mathbb{F}_{q}^{d}},dx).$ We denote by $A(p\to r)$ the best constant such
that the following estimate holds: for every $f$ defined on
$({\mathbb{F}_{q}^{d}},dx)$, we have
$\|f\ast d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d},dx})}\leq A(p\to
r)\|f\|_{L^{p}({\mathbb{F}_{q}^{d},dx})},$
where $d\sigma$ is the normalized surface measure on $S$ defined as before.
Then the averaging problem asks us to determine the optimal exponents $(p,r)$
such that $A(p\to r)\lesssim 1.$
## 2\. Statement of necessary conditions for $R^{*}(p\to r)\lesssim 1$ and
$A(p\to r)\lesssim 1$
### 2.1. Necessary conditions for the boundedness of extension operators
Mockenhaupt and Tao ([10]) introduced the necessary conditions for the
boundedness of extension operators related to general algebraic varieties in
$d$-dimensional vector spaces over finite fields. For example, if $|S|\sim
q^{d-1}$ and $d\sigma$ is the surface measure on $S$, then the necessary
conditions for $R^{*}(p\to r)\lesssim 1$ take the following:
(2.1) $r\geq\frac{2d}{d-1}$
and
(2.2) $r\geq\frac{dp}{(d-1)(p-1)}.$
In the case when $S$ contains a $k$-dimensional affine subspace
$H(|H|=q^{k})$, the necessary condition (2.2) can be improved by the condition
(2.3) $r\geq\frac{p(d-k)}{(p-1)(d-1-k)}$
For the proof of above necessary conditions, see pages $41-42$ in [10]. Let
$S$ be the algebraic variety defined as in (1.4) and $d\sigma$ the normalized
surface measure on $S$ in $({\mathbb{F}_{q}^{d}},dx).$ It is not so hard to
show $|S|\sim q^{d-1}.$ Here, we improve the necessary condition (2.1) when
the dimension $d\geq 2$ is even. In addition, we also improve the necessary
condition (2.1) for specific surfaces $S$ in odd dimensions $d\geq 3$. The
details are given by the following theorem which shall be proved in Section 5.
###### Theorem 2.1.
For $a_{j}\neq 0,j=1,\dots,d,$ let
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}.$ If
$d\geq 2$ is even, then it follows that if $R^{*}(p\to r)\lesssim 1$ then we
must have $r\geq\frac{2d-2}{d-2}.$ On the other hand, suppose that there
exists $l\in{\mathbb{F}_{q}}$ such that $-a_{i}a_{j}^{-1}=l^{2}$ for some
$i,j\in\\{1,\dots,d\\}$ with $i\neq j.$ If $d\geq 3$ is odd, then $R^{*}(p\to
r)\lesssim 1$ only if $r\geq\frac{2d-2}{d-2}.$
### 2.2. Necessary conditions for the boundedness of averaging operators
The authors in [2] introduced the necessary conditions for the boundedness of
the averaging operators related to polynomial surfaces in
$({\mathbb{F}_{q}^{d}},dx).$ Using the same arguments we have the following
necessary conditions for $A(p\to r)\lesssim 1$ associated with the surface $S$
defined as in (1.4). For readers’ convenience, we introduce the proof of the
following theorem in Section 5.
###### Theorem 2.2.
For $a_{j}\neq 0,j=1,\dots,d,$ let
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}.$
Then $A(p\to r)\lesssim 1$ only if $(1/p,1/r)$ lie in the convex hull of the
points
(2.4)
$(0,0),(0,1),(1,1),~{}\mbox{and}~{}\left(\frac{d}{d+1},\frac{1}{d+1}\right).$
Moreover, if $k>(d-1)/2$ and $S$ contains a $k-$dimensional affine subspace
$H(|H|=q^{k})$ of ${\mathbb{F}_{q}^{d}},$ then $A(p\to r)\lesssim 1$ only if
$(1/p,1/r)$ lie in the convex hull of the points $(0,0),(0,1),(1,1),$
(2.5)
$\left(\frac{d^{2}-(k+2)d+2k+1}{(d-1)(d-k)},\frac{k}{(d-1)(d-k)}\right),~{}\mbox{and}~{}\left(\frac{d(d-1-k)}{(d-1)(d-k)},\frac{d-1-k}{(d-1)(d-k)}\right).$
## 3\. Statement of main results
### 3.1. Results of extension problems
For $a_{j}\neq 0,j=1,\dots,d,$ we consider an algebraic variety
(3.1)
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}.$
Our first result below can be viewed as the generalization and extension to
higher odd dimensions for conical extension theorems which Mochenhaupt and Tao
([10]) studied in three dimensions.
###### Theorem 3.1.
If $d\geq 3$ is odd, then we have
(3.2) $R^{*}\left(2\to\frac{2d+2}{d-1}\right)\lesssim 1.$
Moreover, in the case when $S$ contains a $(d-1)/2$-dimensional subspace $H$,
then (3.2) gives a sharp $L^{2}-L^{r}$ estimate.
In odd dimensions, we can construct the surface $S$ which contains a
$(d-1)/2$-dimensional subspace $H$ of $({\mathbb{F}_{q}^{d}},dx).$ For
instance, if we take $a_{j}=1$ for $j$ odd and $a_{j}=-1$ for $j$ even, then
the surface $S$ contains a $(d-1)/2$-dimensional subspace $H(|H|=q^{(d-1)/2})$
which is given by
$H=\left\\{(t_{1},t_{1},t_{2},t_{2},\dots,t_{(d-1)/2},t_{(d-1)/2},0)\in{\mathbb{F}_{q}^{d}}:t_{k}\in{\mathbb{F}_{q}},~{}k=1,2,\dots,(d-1)/2\right\\}.$
In this specific case, if the dimension $d$ is three, then the result (3.2) in
Theorem 3.1 implies a complete solution to the extension problem related to
the surface
$S=\\{x\in{\mathbb{F}_{q}^{3}}:x_{1}^{2}-x_{2}^{2}+x_{3}^{2}=0\\}.$
This fact follows immediately from combining (2.3) and the second part of
Theorem 2.1 with the statements in Remark 1.2. However, this type of the
results was already studied by Mochenhaupt and Tao ([10]) who studied the
extension theorem for the cone ${C}$ in three dimensions where they defined
the cone ${C}$ as the set
${C}=\\{x\in{\mathbb{F}_{q}^{3}}:x_{1}^{2}=x_{2}x_{3}\\}.$
Note that by the nonsingular linear transform, the cone $C$ can be transformed
into the set $S$ above and so the spaces ${C}$ and $S$ yield the same
$L^{p}-L^{r}$ estimate. What we emphasize in our result is as follows.
Aforementioned authors calculated the number of solutions to the following
equation: for each $\eta\in{\mathbb{F}_{q}^{3}},$
$\xi_{1}+\xi_{2}=\eta,$
where $\xi_{1}$ and $\xi_{2}$ are elements in the cone $C$ of three
dimensional vector spaces over finite fields ${\mathbb{F}_{q}}.$ Consequently,
they obtained $L^{2}-L^{4}$ estimate which implies the complete answer to the
conical extension problem in three dimension. Such a method can apply to
conical extension problems in higher dimensions but in higher dimensions it
does not work for an improvement of the $L^{2}-L^{4}$ result (see [10]). On
the other hand, if we take $a_{j}=1$ for $j=1,\dots,(d-1)$ and $a_{d}=-1$, the
set $S$ in (3.1) takes the form
(3.3)
$S=\\{x\in{\mathbb{F}_{q}^{d}}:x_{1}^{2}+\cdots+x_{d-1}^{2}-x_{d}^{2}=0\\},$
which can be transformed by a nonsingular transform into the cone
$C\subset({\mathbb{F}_{q}^{d}},dx)$ given by
(3.4)
$C=\\{x\in{\mathbb{F}_{q}^{d}}:x_{1}^{2}+\cdots+x_{d-2}^{2}=x_{d-1}x_{d}\\}.$
Thus, Theorem 3.1 implies the partial results for conical extension theorems
in higher odd dimensions.
We now introduce the condition on $S$ defined as in (3.1) such that the result
(3.2) in Theorem 3.1 implies a complete solution to the extension problem
related to surface $S$ in three dimension. We have the following corollary.
###### Corollary 3.2.
Let $S$ be the surface defined as in (3.1). Suppose that $d=3$ and there
exists $l\in{\mathbb{F}_{q}}$ such that $-a_{i}a_{j}^{-1}=l^{2}$ for some
$i,j\in\\{1,2,3\\}$ with $i\neq j.$ Then $R^{*}(p\to r)\lesssim 1$ if and only
if the exponents $1\leq p,r\leq\infty$ satisfy the following two inequations:
$r\geq 4$ and $r\geq 2p/(p-1).$
###### Proof.
Without loss of generality, we may assume that $-a_{1}a_{2}^{-1}=l^{2}$ for
some $l\in{\mathbb{F}_{q}}.$ Assume that $R^{*}(p\to r)\lesssim 1$. Observe
that the surface $S$ always contains the line
$H=\\{(t,tl,0)\in{\mathbb{F}_{q}^{3}}:t\in{\mathbb{F}_{q}}\\}$ which is a
$1$-dimensional subspace of ${\mathbb{F}_{q}^{3}}.$ Therefore, using (2.3) and
the second part of Theorem 2.1 we see that $r\geq 4$ and $r\geq 2p/(p-1).$ On
the other hand, if we assume that $r\geq 4$ and $r\geq 2p/(p-1)$ for $1\leq
p,r\leq\infty,$ then $R^{*}(p\to r)\lesssim 1$ is obtained from the result
(3.2) in Theorem 3.1 and the statements in Remark 1.2. Thus, the proof of
Corollary 3.2 is complete. ∎
As the generalization of conical extension theorems in even dimensions, we
also have the following result.
###### Theorem 3.3.
If $d\geq 2$ is even, then
(3.5) $R^{*}\left(2\to\frac{2d}{d-2}\right)\lesssim 1.$
In addition, if we assume that the surface $S$ contains $d/2$-dimensional
subspace in $({\mathbb{F}_{q}^{d}},dx)$, then above estimate (3.5) is a sharp
$L^{2}-L^{r}$ extension estimate.
When we compare Theorem 3.1 with Theorem 3.3 we see that extension theorems in
even dimensional case seem to be worse than those in odd dimensional case.
This is due to lack of decay of the Fourier transform of the surface measure
on $S$ in even dimensions. In general, this is the best possible $L^{2}-L^{r}$
estimate in even dimensions. For example, if we take $a_{j}=1$ for $j$ odd and
$a_{j}=-1$ for $j$ even, then the surface $S$ contains $d/2$-dimensional
subspace $H$ of $({\mathbb{F}_{q}^{d}},dx)$ given by
$H=\left\\{(t_{1},t_{1},t_{2},t_{2},\dots,t_{d/2},t_{d/2})\in{\mathbb{F}_{q}^{d}}:t_{k}\in{\mathbb{F}_{q}},~{}k=1,2,\dots,d/2\right\\}.$
Thus the sharpness follows from (2.3) and the statement in Remark 1.2.
From Theorem 3.1 and Theorem 3.3, we have the following corollary which also
says that conical extension theorems in odd dimensions are different from the
conical extension theorems in even dimensions.
###### Corollary 3.4.
Let $C$ be the cone in $({\mathbb{F}_{q}^{d}},dx)$ defined as in (3.4). In
addition, assume that $-1\in{\mathbb{F}_{q}}$ is a square number. Namely,
there exists $i\in{\mathbb{F}_{q}}$ such that $i^{2}=-1.$ Then, the result
$R^{*}\left(2\to\frac{2d+2}{d-1}\right)\lesssim 1$ gives a sharp $L^{2}-L^{r}$
extension estimate related to cones in odd dimensions. On the other hand, the
result $R^{*}\left(2\to\frac{2d}{d-2}\right)\lesssim 1$ is the best possible
$L^{2}-L^{r}$ extension estimate in even dimensions.
###### Proof.
Recall that the set $S$ in (3.3) can be transfered into the cone in (3.4) by a
nonsingular linear transform. Therefore, using Theorem 3.1 and Theorem 3.3,
the statements in Corollary 3.4 follow immediately from the following
observation. If $d\geq 3$ is odd, then $S$ in (3.3) contains a
$(d-1)/2$-dimensional subspace $H$ of $({\mathbb{F}_{q}^{d}},dx)$ given by
$H=\left\\{(t_{1},it_{1},\dots,t_{(d-1)/2},it_{(d-1)/2},0)\in{\mathbb{F}_{q}^{d}}:t_{k}\in{\mathbb{F}_{q}}~{}~{}~{}k=1,2,\dots,(d-1)/2\right\\},$
and if $d\geq 2$ is even, then the $d/2$-dimensional subspace
$H=\left\\{(t_{1},it_{1},\dots,t_{(d-2)/2},it_{(d-2)/2},t_{d/2},t_{d/2})\in{\mathbb{F}_{q}^{d}}:t_{k}\in{\mathbb{F}_{q}}~{}~{}~{}k=1,2,\dots,d/2\right\\}$
is contained in the surface $S$ in (3.3).∎
As another application of Theorem 3.3, we have the following corollary which
makes an important role for proving the optimal result except for endpoints
for averaging problems over algebraic varieties with subspaces of large
dimension. In other words, we shall see that the boundedness of the averaging
operators can be obtained from the results of extension problems.
###### Corollary 3.5.
For any subset $E$ of $({\mathbb{F}_{q}^{d}},dx)$ and $b_{j}\neq 0$ for
$j=1,\dots,d,$ if $d\geq 2$ is even, then we have
$\sum_{m\in
S}|\widehat{E}(m)|^{2}\lesssim\min\left\\{q^{-(d+1)}|E|^{\frac{d+2}{d}},~{}q^{-d}|E|\right\\},$
where
$S=\\{m\in{\mathbb{F}_{q}^{d}}:b_{1}m_{1}^{2}+\cdots+b_{d}m_{d}^{2}=0\\}\subset({\mathbb{F}_{q}^{d}},dm)$
###### Proof.
It is clear that $\sum_{m\in S}|\widehat{E}(m)|^{2}\leq q^{-d}|E|$, because
the Plancherel theorem in (1.3) shows that $\sum_{m\in
S}|\widehat{E}(m)|^{2}\leq\sum_{m\in{\mathbb{F}_{q}^{d}}}|\widehat{E}(m)|^{2}=q^{-d}|E|.$
It therefore remains to show that
(3.6) $\sum_{m\in S}|\widehat{E}(m)|^{2}\lesssim
q^{-(d+1)}|E|^{\frac{d+2}{d}}.$
Since the space $({\mathbb{F}_{q}^{d}},dm)$ is isomorphic to the dual space
$({\mathbb{F}_{q}^{d}},dx)$ as an abstract group, we may identify the space
$({\mathbb{F}_{q}^{d}},dm)$ with the dual space $({\mathbb{F}_{q}^{d}},dx).$
Thus, they possess same algebraic structures. However, we endowed them with
different measures: the counting measure $dm$ for $({\mathbb{F}_{q}^{d}},dm)$
and the normalized counting measure for $({\mathbb{F}_{q}^{d}},dx).$ For these
reasons, the inequality (3.6) is essentially same as the following: for every
subset $E$ of $({\mathbb{F}_{q}^{d}},dm)$
(3.7) $\sum_{x\in S}q^{-2d}|\widehat{E}(x)|^{2}\lesssim
q^{-(d+1)}|E|^{\frac{d+2}{d}},$
where $S$ is considered as a subset of the dual space
$({\mathbb{F}_{q}^{d}},dx)$ given by
$S=\\{x\in{\mathbb{F}_{q}^{d}}:b_{1}x_{1}^{2}+\cdots+b_{d}x_{d}^{2}=0\\}$ and
$\widehat{E}(x)$ is defined as in (1.2). By duality (1.5), Theorem 3.3 implies
that the following restriction estimate holds: for every function $g$ on
$({\mathbb{F}_{q}^{d}},dm),$
$\|\widehat{g}\|^{2}_{L^{2}(S,d\sigma)}\lesssim\|g\|^{2}_{L^{\frac{2d}{d+2}}({\mathbb{F}_{q}^{d}},dm)}.$
If we take $g(m)=E(m)$, then we see
$\frac{1}{|S|}\sum_{x\in S}|\widehat{E}(x)|^{2}\lesssim|E|^{\frac{d+2}{d}}$
Since $|S|\sim q^{d-1}$, (3.7) follows and the proof of Corollary 3.5 is
complete.
∎
### 3.2. Results of averaging problems
###### Theorem 3.6.
Let $S$ be the algebraic variety in $({\mathbb{F}_{q}^{d}},dx)$ defined as in
(3.1). If $d\geq 3$ is odd, then we have
(3.8) $A(p\to r)\lesssim
1~{}~{}\iff~{}~{}\left(\frac{1}{p},\frac{1}{r}\right)\in{\mathbb{T}},$
where ${\mathbb{T}}$ denotes the convex hull of points $(0,0),(0,1),(1,1)$ and
$(d/(d+1),1/(d+1)).$ On the other hand, if $d\geq 2$ is even, then we have
(3.9) $A(p\to r)\lessapprox 1$
when $(1/p,1/r)$ lies in the convex hull of points $(0,0),(0,1),(1,1),$
$P_{1}=\left(\frac{d^{2}-2d+2}{d(d-1)},\frac{1}{(d-1)}\right)~{}~{}\mbox{and}~{}~{}P_{2}=\left(\frac{d-2}{d-1},\frac{d-2}{d(d-1)}\right).$
In particular, in the case when $S$ contains a $d/2$-dimensional subspace of
$({\mathbb{F}_{q}^{d}},dx)$, the result (3.9) gives the best possible result
up to endpoints for averaging problems in even dimensions.
###### Remark 3.7.
The proof of Theorem 3.6 shall be given in Section 7. Moreover, from the proof
for the statement (3.9) above, we shall see that $A(d/(d-1),d)\lesssim 1$
where the point $((d-1)/d,1/d)$ is the midpoint between $P_{1}$ and $P_{2},$
and that if $d=2$, then the condition $A(p\to r)\lessapprox 1$ in (3.9) can be
replaced by the condition $A(p\to r)\lesssim 1.$
As we shall see, the result (3.8) is mainly due to our observation that the
decay of Fourier transform of the surface measre $d\sigma$ in odd dimensions
is quite good. Note that the result (3.9) is worse than the result (3.8). In
general, the mapping property of averaging operators related to the surface
$S$ in odd dimensions is better than in even dimensions. This is partially due
to the lack of the Fourier decay of the surface measure in even dimensions.
One may ask why such difference happens between in odd dimensions and in even
dimensions in the finite field setting. Even if the algebraic variety $S$ in
(3.1) is defined by using a same equation, the surface $S$ has the
distinguished properties depending on whether the dimension is even or not.
For example, the surface $S$ in even dimensions may contain a
$d/2$-dimensional subspace but this never happens in odd dimensions because
the dimension of the subspace in $S$ must be an integer number. In fact, the
maximal possible dimension of a subspace contained in the surface $S$ in odd
dimensions is $(d-1)/2.$ This can be proved by using the decay estimate of the
surface measure $d\sigma$ (see Proposition $9$ in [2]). In the case when the
surface $S$ contains a subspace $H$ whose dimension is greater than $(d-1)/2$,
it is not easy to obtain a sharp $L^{p}-L^{r}$ estimate for averaging
operators, in part because the result can not be shown by simply using the
Fourier decay of the surface measure $d\sigma.$ We shall prove the result
$(\ref{sharpevena})$ using the explicit formula for the Fourier transform of
$d\sigma$ along with Corollary 3.5. Notice that our result (3.9) is a partial
evidence to show that the necessary conditions in (2.5) are in fact the
sufficient conditions.
## 4\. Estimate of the Fourier transform of the surface measure $d\sigma$
In this section we obtain the explicit formula for the Fourier transform of
the surface measure $d\sigma$ on the surface $S$ defined as in (3.1). We shall
see that the Fourier transform is closely related to the classical Gauss sums.
Moreover, it makes a key role to prove our main results for extension problems
and averaging problems. It is useful to review classical Gauss sums in the
finite field setting. In the remainder of this paper, we fix the additive
character $\chi$ as a canonical additive character of ${\mathbb{F}_{q}}$ and
$\eta$ always denotes the quadratic character of ${\mathbb{F}_{q}}.$ Recall
that $\eta(t)=1$ if $s$ is a square number in
${\mathbb{F}}_{q}\setminus\\{0\\}$ and $\eta(t)=-1$ if $t$ is not a square
number in ${\mathbb{F}_{q}}\setminus\\{0\\}.$ We also recall that
$\eta(0)=0,\eta^{2}\equiv 1,\eta(ab)=\eta(a)\eta(b)$ for
$a,b\in{\mathbb{F}_{q}},$ and $\eta(t)=\eta(t^{-1})$ for $t\neq 0.$ For each
$t\in{\mathbb{F}_{q}},$ the Gauss sum $G_{t}(\eta,\chi)$ is defined by
$G_{t}(\eta,\chi)=\sum_{s\in{\mathbb{F}_{q}}\setminus\\{0\\}}\eta(s)\chi(ts).$
The absolute value of the Gauss sum is given by the relation
$|G_{t}(\eta,\chi)|=\left\\{\begin{array}[]{ll}q^{\frac{1}{2}}&\mbox{if}~{}~{}t\neq
0\\\ 0&\mbox{if}~{}~{}t=0.\end{array}\right.$
In addition, we have the following formula
(4.1)
$\sum_{s\in{\mathbb{F}_{q}}}\chi(ts^{2})=\eta(t)G_{1}(\eta,\chi)~{}~{}\mbox{for
any}~{}~{}t\neq 0,$
because $\eta$ is the multiplicative character on ${\mathbb{F}_{q}}$ of order
two. For the nice proofs for the properties related to the Gauss sums, see
Chapter $5$ in [9] and Chapter $11$ in [6]. When we complete the square and
apply a change of variable, the formula (4.1) yields the following equation:
for each $a\in{\mathbb{F}_{q}}\setminus\\{0\\},b\in{\mathbb{F}_{q}}$
(4.2)
$\sum_{s\in{\mathbb{F}_{q}}}\chi(as^{2}+bs)=G_{1}(\eta,\chi)\eta(a)\chi\left(\frac{b^{2}}{-4a}\right).$
We shall name the skill used to obtain the formula (4.2) as the complete
square method. Relating the inverse Fourier transform of $d\sigma$ with the
Gauss sum, we shall obtain the explicit form of $(d\sigma)^{\vee}$, the
inverse Fourier transform of the surface measure on $S.$ We have the following
lemma.
###### Lemma 4.1.
Let $d\sigma$ be the surface measure on $S$ defined as in (3.1). If $d\geq 3$
is odd, then we have
$(d\sigma)^{\vee}(m)=\left\\{\begin{array}[]{ll}q^{d-1}|S|^{-1}&\mbox{if}~{}~{}m=(0,\dots,0)\\\
0&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
\frac{G_{1}^{d+1}}{q|S|}\eta(-a_{1}\cdots
a_{d})\eta\left(\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\right)&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0.\end{array}\right.$
If $d\geq 2$ is even, then we have
$(d\sigma)^{\vee}(m)=\left\\{\begin{array}[]{ll}q^{d-1}|S|^{-1}+\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m=(0,\dots,0)\\\
\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
-\frac{G_{1}^{d}}{q|S|}\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0,\end{array}\right.$
here, and throughout this paper, we write $G_{1}$ for the Gauss sum
$G_{1}(\eta,\xi)$ and $\eta$ denotes the quadratic character of
${\mathbb{F}_{q}}.$
###### Proof.
Using the definition of the inverse Fourier transform and the orthogonality
relations of the nontrivial additive character $\chi$ of ${\mathbb{F}_{q}}$,
we see
$\displaystyle(d\sigma)^{\vee}(m)$ $\displaystyle=|S|^{-1}\sum_{x\in
S}\chi(x\cdot m)$
$\displaystyle=|S|^{-1}q^{-1}\sum_{x\in{\mathbb{F}_{q}^{d}}}\sum_{s\in{\mathbb{F}_{q}}}\chi\left(s(a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2})\right)~{}\chi(x\cdot
m)$
$\displaystyle=q^{d-1}|S|^{-1}\delta_{0}(m)+|S|^{-1}q^{-1}\sum_{x\in{\mathbb{F}_{q}^{d}}}\sum_{s\neq
0}\chi\left(s(a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2})\right)~{}\chi(x\cdot m)$
$\displaystyle=q^{d-1}|S|^{-1}\delta_{0}(m)+|S|^{-1}q^{-1}\sum_{s\neq
0}\prod_{j=1}^{d}\sum_{x_{j}\in{\mathbb{F}_{q}}}\chi(sa_{j}x_{j}^{2}+m_{j}x_{j}).$
Using the complete square method (4.2), compute the sums over
$x_{j}\in{\mathbb{F}_{q}}$ and then we have
$(d\sigma)^{\vee}(m)=q^{d-1}|S|^{-1}\delta_{0}(m)+G_{1}^{d}|S|^{-1}q^{-1}\eta(a_{1}\cdots
a_{d})\sum_{s\neq
0}\eta^{d}(s)\chi\left(-\frac{1}{4s}\left(\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\right)\right).$
Case I. Suppose that $d\geq 3$ is odd. Then $\eta^{d}\equiv\eta$, because
$\eta$ is the multiplicative character of order two. Therefore, if
$\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0$, then the proof is
complete, because $\sum_{s\in{\mathbb{F}_{q}}\setminus\\{0\\}}\eta(s)=0.$ On
the other hand, if $\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0,$ then the statement follows from using a change of
variable,$-\frac{1}{4s}\left(\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\right)\to
s,$ and the facts that $\eta(4)=1,\eta(s)=\eta(s^{-1})$ for $s\neq 0$, and
$G_{1}=\sum_{s\neq 0}\eta(s)\chi(s).$
Case II. Suppose that $d\geq 2$ is even. Then $\eta^{d}\equiv 1.$ The proof is
complete, because $\sum_{s\neq 0}\chi(as)=-1$ for all $a\neq 0$, and
$\sum_{s\neq 0}\chi(as)=(q-1)$ if $a=0.$ ∎
Since the Fourier decay of the surface measure is distinguished between in odd
dimensions and in even dimensions, it is natural to see different results for
boundedness of operators between odd dimensions and even dimensions. Lemma 4.1
yields the following corollary which tells us the Fourier decay.
###### Corollary 4.2.
If $d\geq 3$ is odd, then it follows that
(4.3) $\begin{array}[]{ll}(d\sigma)^{\vee}(0,\dots,0)=1,&\\\
|(d\sigma)^{\vee}(m)|\lesssim
q^{-\frac{d-1}{2}}&\mbox{if}~{}~{}m\neq(0,\dots,0).\end{array}$
On the other hand, if $d\geq 2$ is even, then we have
(4.4) $\begin{array}[]{ll}(d\sigma)^{\vee}(0,\dots,0)=1,&\\\
|(d\sigma)^{\vee}(m)|\lesssim
q^{-\frac{d-2}{2}}&\mbox{if}~{}~{}m\neq(0,\dots,0).\end{array}$
###### Proof.
Recall that the Fourier inverse transform of the surface measure $d\sigma$ is
given by the relation
$(d\sigma)^{\vee}(m)=\int_{S}\chi(x\cdot m)d\sigma=\frac{1}{|S|}\sum_{x\in
S}\chi(x\cdot m)$
where $m\in({\mathbb{F}_{q}^{d}},dm).$ Therefore, it is clear that
$(d\sigma)^{\vee}(0,\ldots,0)=1$ for all $d\geq 2.$ If we compare this with
the values $(d\sigma)^{\vee}(0,\ldots,0)$ given by Lemma 4.1, then we see that
$|S|\sim q^{d-1}$ for $d\geq 2.$ Since the absolute of the Gauss sum $G_{1}$
is exactly $q^{1/2}$, the statements in Corollary 4.2 follows immediately from
Lemma 4.1. ∎
## 5\. Proof of necessary conditions
### 5.1. Proof of Theorem 2.1(Necessary conditions for the boundedness of
extension operators)
###### Proof.
First, we show that the first part of Theorem 2.1 holds. Suppose that the
dimension $d\geq 2$ is even. We consider a set
$M=\left\\{m\in{\mathbb{F}_{q}^{d}}:\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\right\\}.$
Note that $|M|\sim q^{d-1}.$ We test (1.5) by taking the function $g$ as a
characteristic function on the set $M\subset({\mathbb{F}_{q}^{d}},dm).$ Since
the measure $dm$ is the counting measure, we have
(5.1)
$\|M\|_{L^{r^{\prime}}({\mathbb{F}_{q}^{d}},dm)}=|M|^{\frac{1}{r^{\prime}}}\sim
q^{\frac{d-1}{r^{\prime}}}.$
We now aim to estimate the quantity
$\|\widehat{M}\|_{L^{p^{\prime}}(S,d\sigma)}.$ For each $x\in S,$ we have
$\displaystyle\widehat{M}(x)$
$\displaystyle=\int_{{\mathbb{F}_{q}^{d}}}M(m)\chi(-m\cdot x)dm=\sum_{m\in
M}\chi(-m\cdot x)$ $\displaystyle=\sum_{m\in{\mathbb{F}_{q}^{d}}}\chi(-m\cdot
x)~{}\delta_{0}\left(\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\right),$
where $\delta_{0}(\alpha)=1$ if $\alpha=0$ and $\delta_{0}(\alpha)=0$
otherwise. Using the orthogonality relation of the non-trivial additive
character $\chi$, we see that
$\widehat{M}(x)=q^{-1}\sum_{m\in{\mathbb{F}_{q}^{d}}}\sum_{s\in{\mathbb{F}_{q}}}\chi(-m\cdot
x)~{}\chi\left(s\left(\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\right)\right)$
$=q^{d-1}\delta_{0}(x)+q^{-1}\sum_{s\neq
0}\prod_{j=1}^{d}\sum_{m_{j}\in{\mathbb{F}_{q}}}\chi(sa_{j}^{-1}m_{j}^{2}-x_{j}m_{j}).$
Using the complete square method (4.2), we obtain
$\widehat{M}(x)=q^{d-1}\delta_{0}(x)+q^{-1}G_{1}^{d}\eta(a_{1}^{-1}\cdot\cdots
a_{d}^{-1})\sum_{s\neq
0}\chi\left(-(4s)^{-1}(a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2})\right),$
where we used the fact that $\eta^{d}\equiv 1$ for $d$ even. Thus, if $x\in
S\setminus\\{(0,\dots,0)\\},$ then we have
$|\widehat{M}(x)|=|q^{-1}G_{1}^{d}\eta(a_{1}^{-1}\cdot\cdots\cdot
a_{d}^{-1})(q-1)|\sim q^{\frac{d}{2}}.$
This implies the following estimate holds:
(5.2) $q^{\frac{d}{2}}\lesssim\|\widehat{M}\|_{L^{p^{\prime}}(S,d\sigma)}.$
Comparing (5.1) with (5.2), we must have
$q^{\frac{d}{2}}\lesssim q^{\frac{d-1}{r^{\prime}}},$
which completes the proof of the first part of Theorem 2.1.
We prove the second part of Theorem 2.1. Without loss of generality we may
assume that $-a_{d-1}a_{d}^{-1}=l^{2}$ for some $l\in{\mathbb{F}_{q}}.$ Then
the set $S$ is given by
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+\cdots+a_{d-2}x_{d-2}^{2}-a_{d}(lx_{d-1}+x_{d})(lx_{d-1}-x_{d})=0\\}.$
Since the mapping property of extension operators related to $S$ is invariant
under the non-singular linear transform of the surface $S$, we may assume
$S=\\{x\in{\mathbb{F}_{q}^{d}}:a_{1}x_{1}^{2}+\cdots+a_{d-2}x_{d-2}^{2}-x_{d-1}x_{d}=0\\}.$
Let ${\mathbb{D}}=\\{s\in{\mathbb{F}_{q}}\setminus\\{0\\}:s\mbox{ is a square
number}\\}.$ Then it is clear that $|{\mathbb{D}}|=(q-1)/2\sim q.$ Now, define
a set
$\Omega=\left\\{m\in{\mathbb{F}_{q}^{d-1}}\times{\mathbb{D}}:m_{d-1}=\frac{a_{1}^{-1}m_{1}^{2}+\cdots+a_{d-2}^{-1}m_{d-2}^{2}}{4m_{d}}\right\\}.$
Observe that $|\Omega|=q^{d-2}|{\mathbb{D}}|\sim q^{d-1}.$ We test (1.5) with
the characteristic function on the set
$\Omega\subset({\mathbb{F}_{q}^{d}},dm).$ We have
(5.3)
$\|\Omega\|_{L^{r^{\prime}}({\mathbb{F}_{q}^{d}},dm)}=|\Omega|^{\frac{1}{r^{\prime}}}\sim
q^{\frac{d-1}{r^{\prime}}}.$
Let us estimate the quantity
$\|\widehat{\Omega}\|_{L^{p^{\prime}}(S,d\sigma)}.$ For each $x\in S$ with
$x_{d-1}\neq 0,$ we have
$\displaystyle\widehat{\Omega}(x)$
$\displaystyle=\int_{{\mathbb{F}_{q}^{d}}}\Omega(m)\chi(-m\cdot
x)dm=\sum_{m\in\Omega}\chi(-m\cdot x)$
$\displaystyle=\sum_{m_{1},\dots,m_{d-2}\in{\mathbb{F}_{q}}}\sum_{m_{d}\in{\mathbb{D}}}\chi({\mathbb{P}}_{x_{1},\dots,x_{d}}(m_{1},\dots,m_{d-2},m_{d})),$
where
${\mathbb{P}}_{x_{1},\dots,x_{d}}(m_{1},\dots,m_{d-2},m_{d})=-m_{1}x_{1}-\cdots-
m_{d-2}x_{d-2}-\left(\frac{a_{1}^{-1}m_{1}^{2}+\cdots+a_{d-2}^{-1}m_{d-2}^{2}}{4m_{d}}\right)\cdot
x_{d-1}-m_{d}x_{d}.$ Therefore, the completing square method (4.2) yields that
for each $x\in S$ with $x_{d-1}\neq 0,$
$\widehat{\Omega}(x)=G_{1}^{d-2}\eta(a_{1}^{-1}\cdot\dots\cdot
a_{d-2}^{-1})\eta^{d-2}\left(\frac{-x_{d-1}}{4}\right)\sum_{m_{d}\in{\mathbb{D}}}\eta^{d-2}(m_{d}^{-1})\chi\left(m_{d}\left(\frac{a_{1}x_{1}^{2}+\cdots+a_{d-2}x_{d-2}^{2}}{x_{d-1}}-x_{d}\right)\right).$
Since $\eta^{d-2}(m_{d}^{-1})=1$ for $m_{d}\in{\mathbb{D}},$ and
$\frac{a_{1}x_{1}^{2}+\cdots+a_{d-2}x_{d-2}^{2}}{x_{d-1}}-x_{d}=0$ for $x\in
S$ with $x_{d-1}\neq 0$ we see that for $x\in S$ with $x_{d-1}\neq 0$,
$|\widehat{\Omega}(x)|=|G_{1}^{d-2}||{\mathbb{D}}|\sim q^{\frac{d}{2}}.$
Using this estimate we have
(5.4) $q^{\frac{d}{2}}\sim\left(\frac{1}{|S|}\sum_{x\in S:x_{d-1}\neq
0}q^{\frac{dp^{\prime}}{2}}\right)^{\frac{1}{p^{\prime}}}\lesssim\|\widehat{\Omega}\|_{L^{p^{\prime}}(S,d\sigma)}.$
From (5.3) and (5.4), the proof of the second part of Theorem 2.1 is complete.
∎
### 5.2. Proof of Theorem 2.2 (Necessary conditions for the boundedness of
averaging operators)
###### Proof.
Let $1\leq p,r\leq\infty.$ Suppose that the following estimate holds: for
every function $f$ on $({\mathbb{F}_{q}^{d}},dx),$ we have
(5.5) $\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d},dx})}\lesssim\|f\|_{L^{p}({\mathbb{F}_{q}^{d},dx})}.$
We test (5.5) with $f=\delta_{0}.$ It follows that
(5.6) $\|f\|_{L^{p}({\mathbb{F}_{q}^{d},dx})}=q^{-\frac{d}{p}}.$
Recall that $d\sigma(x)=q^{d}|S|^{-1}S(x).$ We have
$f\ast d\sigma(x)=\delta_{0}\ast d\sigma(x)=\frac{q^{d}}{|S|}\delta_{0}\ast
S(x)=\frac{1}{|S|}\delta_{S}(x),$
where $\delta_{S}(x)=1$ if $x\in S$, and $\delta_{S}(x)=0$ if $x\neq S.$ It
therefore follows that
(5.7) $\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d},dx})}=q^{-\frac{d}{r}}|S|^{\frac{1-r}{r}}\sim
q^{\frac{r(1-d)-1}{r}}.$
From (5.6) and (5.7), the inequality (5.5) holds only if
$\frac{d}{p}\leq\frac{1}{r}+d-1.$
By duality we also see that the inequality (5.5) holds only when
$\frac{d}{r^{\prime}}\leq\frac{1}{p^{\prime}}+d-1,$
and the statement (2.4) in Theorem 2.2 follows.
We now prove the statement (2.5) in Theorem 2.2. Assume that $k>(d-1)/2$ and
$H\subset S$ is a $k$-dimensional affine subspace of
$({\mathbb{F}_{q}^{d}},dx).$ From the statement (2.4), it suffices to show
that if $A(p\to r)\lesssim 1,$ then we must have
(5.8) $\frac{1}{r}\geq\frac{1}{p}+\frac{k-d+1}{d-k}.$
We also test (5.5) with $f(x)=H(x),$ the characteristic function on the set
$H$ with $|H|=q^{k}.$ We have
(5.9)
$\|f\|_{L^{p}({\mathbb{F}_{q}^{d},dx})}=\|H\|_{L^{p}({\mathbb{F}_{q}^{d},dx})}=q^{\frac{k-d}{p}}.$
Since $H$ is an affine subspace of $({\mathbb{F}_{q}^{d}},dx)$, we may assume
that for some $\alpha\in({\mathbb{F}_{q}^{d}},dx)$ and a subspace $\Lambda$ of
$({\mathbb{F}_{q}^{d}},dx),$ we have
$H=\Lambda+\alpha=\\{x+\alpha\in{\mathbb{F}_{q}^{d}}:x\in\Lambda\\}.$
Since $H\subset S$, we have
$\displaystyle f\ast d\sigma(x)$ $\displaystyle=H\ast
d\sigma(x)\geq\frac{1}{|S|}\sum_{y\in{\mathbb{F}_{q}^{d}}}H(y)H(x-y)$
$\displaystyle=\frac{1}{|S|}\sum_{y\in{\mathbb{F}_{q}^{d}}}(\Lambda+\alpha)(y)~{}(\Lambda+\alpha)(x-y)$
$\displaystyle=\frac{1}{|S|}\sum_{y\in\Lambda}(\Lambda+\alpha)(x-y-\alpha)$
Since $\Lambda$ is a subspace of $({\mathbb{F}_{q}^{d}},dx)$, we therefore see
that if $x\in\Lambda+2\alpha,$ then
$f\ast d\sigma(x)\geq\frac{|\Lambda|}{|S|}=\frac{|H|}{|S|}\sim q^{k-d+1}.$
From this estimation, we have
$\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}\gtrsim\left(q^{-d}q^{r(k-d+1)}|\Lambda+2\alpha|\right)^{\frac{1}{r}}$
Since $|\Lambda+2\alpha|=|H|=q^{k},$ we have
(5.10) $\|f\ast d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}\gtrsim
q^{-\frac{d}{r}+k-d+1+\frac{k}{r}}.$
From (5.9) and (5.10), our aim (5.8) holds and so the statement (2.5) in
Theorem 2.2 follows. ∎
## 6\. Proof of results for extension problems (Theorem 3.1 and Theorem 3.3)
Both Theorem 3.1 and Theorem 3.3 follow from Corollary 4.2 and the following
lemma. In addition, we note that the sharpness of $L^{2}-L^{r}$ estimates in
Theorem 3.1 and Theorem 3.3 can be justified by the necessary condition given
in (2.3).
###### Lemma 6.1 (Tomas-Stein Type argument in the finite field setting).
Let $d\sigma$ be the surface measure on the algebraic variety
$S\subset({\mathbb{F}_{q}^{d}},dx)$ defined as in (3.1). If
$|(d\sigma)^{\vee}(m)|\lesssim q^{-\frac{\alpha}{2}}$ for some $\alpha>0$ and
for all $m\in{\mathbb{F}_{q}^{d}}\setminus(0,\dots,0),$ then we have
$R^{*}\left(2\to\frac{2(\alpha+2)}{\alpha}\right)\lesssim 1.$
###### Proof.
By duality, it suffices to prove that the following restriction estimate
holds: for every function $g$ defined on $({\mathbb{F}_{q}^{d}},dm),$ we have
$\|\widehat{g}\|^{2}_{L^{2}(S,d\sigma)}\lesssim\|g\|^{2}_{L^{\frac{2(\alpha+2)}{\alpha+4}}({\mathbb{F}_{q}^{d}},dm)}.$
By the orthogonality principle and Hölder’s inequality, we see
$\|\widehat{g}\|^{2}_{L^{2}(S,d\sigma)}\leq\|g\ast(d\sigma)^{\vee}\|_{L^{\frac{2(\alpha+2)}{\alpha}}({\mathbb{F}_{q}^{d}},dm)}\|g\|_{L^{\frac{2(\alpha+2)}{\alpha+4}}({\mathbb{F}_{q}^{d}},dm)}.$
It therefore suffices to show that for every function $g$ on
$({\mathbb{F}_{q}^{d}},dm),$
$\|g\ast(d\sigma)^{\vee}\|_{L^{\frac{2(\alpha+2)}{\alpha}}({\mathbb{F}_{q}^{d}},dm)}\lesssim\|g\|_{L^{\frac{2(\alpha+2)}{\alpha+4}}({\mathbb{F}_{q}^{d}},dm)}.$
Define $K=(d\sigma)^{\vee}-\delta_{0}.$ Since $(d\sigma)^{\vee}(0,\dots,0)=1$,
we see that $K(m)=0$ if $m=(0,\dots,0)$, and $K(m)=(d\sigma)^{\vee}(m)$ if
$m\in{\mathbb{F}_{q}^{d}}\setminus\\{(0,\dots,0)\\}.$ In addition, we see that
$\begin{array}[]{ll}\|g\ast\delta_{0}\|_{L^{\frac{2(\alpha+2)}{\alpha}}({\mathbb{F}_{q}^{d}},dm)}&=\|g\|_{L^{\frac{2(\alpha+2)}{\alpha}}({\mathbb{F}_{q}^{d}},dm)}\\\
&\leq\|g\|_{L^{\frac{2(\alpha+2)}{\alpha+4}}({\mathbb{F}_{q}^{d}},dm)},\end{array}$
where the inequality follows from the fact that $dm$ is the counting measure
and $\frac{2(\alpha+2)}{\alpha}\geq\frac{2(\alpha+2)}{\alpha+4}.$ Thus, it is
enough to show that for every $g$ on $({\mathbb{F}_{q}^{d}},dm),$
(6.1) $\|g\ast
K\|_{L^{\frac{2(\alpha+2)}{\alpha}}({\mathbb{F}_{q}^{d}},dm)}\lesssim\|g\|_{L^{\frac{2(\alpha+2)}{\alpha+4}}({\mathbb{F}_{q}^{d}},dm)}.$
We now claim that the following two estimates hold: for every function $g$ on
$({\mathbb{F}_{q}^{d}},dm),$
(6.2) $\|g\ast K\|_{L^{2}({\mathbb{F}_{q}^{d}},dm)}\lesssim
q\|g\|_{L^{2}({\mathbb{F}_{q}^{d}},dm)}$
and
(6.3) $\|g\ast K\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dm)}\lesssim
q^{-\frac{\alpha}{2}}\|g\|_{L^{1}({\mathbb{F}_{q}^{d}},dm)}.$
Note that the estimate (6.1) follows by interpolating (6.2) with (6.3). It
therefore remains to show that both (6.2) and (6.3) hold. Using Plancherel,
the inequality (6.2) follows from the following observation:
$\begin{array}[]{ll}\|g\ast
K\|_{L^{2}({\mathbb{F}_{q}^{d}},dm)}&=\|\widehat{g}\widehat{K}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\\\
&\leq\|\widehat{K}\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\|\widehat{g}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\\\
&\lesssim q\|g\|_{L^{2}({\mathbb{F}_{q}^{d}},dm)},\end{array}$
where the last line is due to the observation that for each
$x\in({\mathbb{F}_{q}^{d}},dx)$
$\widehat{K}(x)=d\sigma(x)-\widehat{\delta_{0}}(x)=q^{d}|S|^{-1}S(x)-1\lesssim
q.$ On the other hand, the estimate (6.3) follows from Young’s inequality and
the assumption for the Fourier decay away from the origin. Thus, the proof is
complete. ∎
## 7\. Proof of the results for averaging problems(Theorem 3.6)
### 7.1. Proof of (3.8) in Theorem 3.6
From the necessary condition (2.4) in Theorem 2.2, it suffices to prove that
if $(1/p,1/r)\in{\mathbb{T}}$, then $A(p\to r)\lesssim 1,$ where
${\mathbb{T}}$ is the convex hull of points $(0,0),(0,1),(1,1)$, and
$(d/(d+1),1/(d+1)).$ Recall that both $d\sigma$ and
$({\mathbb{F}_{q}^{d}},dx)$ have total mass $1$. It therefore follows from
Young’s inequality and Hölder’s inequality that if $1\leq r\leq p\leq\infty$,
then
(7.1) $\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d},dx})}\leq\|f\|_{L^{r}({\mathbb{F}_{q}^{d},dx})}\leq\|f\|_{L^{p}({\mathbb{F}_{q}^{d},dx})}.$
Therefore, if $1\leq r\leq p\leq\infty,$ then $A(p\to r)\leq 1.$ Thus, we are
interested in the case when $1\leq p<r\leq\infty.$ In this case, combining
(4.3) in Corollary 4.2 with the following lemma below, we see that
$A(d/(d+1),1/(d+1))\lesssim 1.$ Consequently, the statement (3.8) in the first
part of Theorem 3.6 follows immediately from interpolating above results.
Thus, it remains to prove the following corollary.
###### Lemma 7.1.
Let $d\sigma$ be the surface measure on the algebraic variety
$S\subset({\mathbb{F}_{q}^{d}},dx)$ defined as in (3.1). If
$|(d\sigma)^{\vee}(m)|\lesssim q^{-\frac{\alpha}{2}}$ for all
$m\in{\mathbb{F}_{q}^{d}}\setminus(0,\dots,0)$ and for some $\alpha>0,$ then
we have
$A\left(\frac{\alpha+2}{\alpha+1}\to\alpha+2\right)\lesssim 1.$
###### Proof.
The proof of Lemma 7.1 is exactly same as that of Theorem 3 in [2]. For
readers’ convenience, we introduce the proof. Consider a function $K$ on
$({\mathbb{F}_{q}^{d}},dm)$ defined as $K=(d\sigma)^{\vee}-\delta_{0}.$ We
want to prove that for every function $f$ on $({\mathbb{F}_{q}^{d}},dx)$,
$\|f\ast
d\sigma\|_{L^{\alpha+2}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|f\|_{L^{\frac{\alpha+2}{\alpha+1}}({\mathbb{F}_{q}^{d}},dx)}.$
Since $d\sigma=\widehat{K}+\widehat{\delta_{0}}=\widehat{K}+1$ and $\|f\ast
1\|_{L^{\alpha+2}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|f\|_{L^{\frac{\alpha+2}{\alpha+1}}({\mathbb{F}_{q}^{d}},dx)}$,
it suffices to show that for every $f$ on $({\mathbb{F}_{q}^{d}},dx),$
(7.2)
$\|f\ast\widehat{K}\|_{L^{\alpha+2}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|f\|_{L^{\frac{\alpha+2}{\alpha+1}}({\mathbb{F}_{q}^{d}},dx)}.$
The inequality (7.2) can be proved by interpolating the following two
estimates:
(7.3) $\|f\ast\widehat{K}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\lesssim
q^{-\frac{\alpha}{2}}\|f\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}$
and
(7.4) $\|f\ast\widehat{K}\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\lesssim
q\|f\|_{L^{1}({\mathbb{F}_{q}^{d}},dx)}.$
The inequality (7.3) follows from Plancherel, the decay assumption of
$d\sigma$, and the definition of $K.$ On the other hand, the inequality (7.4)
follows from Young’s inequality and the observation that
$\|\widehat{K}\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\lesssim q.$ Thus, the
proof of Lemma 7.1 is complete. ∎
### 7.2. Proof of (3.9) in Theorem 3.6
Before we prove the statement (3.9) in Theorem 3.6, we notice that (4.4) in
Corollary 4.2 and Lemma 7.1 yield
$A\left(\frac{d}{d-1}\to d\right)\lesssim 1.$
Even if we interpolate this result with trivial results, in even dimensions
$d\geq 4$ we can not obtain the result (3.9) in Theorem 3.6. In the case when
the dimension $d\geq 4$ is even, it seems that the averaging problems are not
easy. In fact, we need more efforts to estimate the $L^{2}-$norm of $f\ast
d\sigma.$ To prove the statement (3.9) in Theorem 3.6 we begin by proving the
following lemma.
###### Lemma 7.2.
Let $d\sigma$ be the surface measure on $S\subset({\mathbb{F}_{q}^{d}},dx)$
defined as in (3.1). If the dimension $d\geq 2$ is even, then it follows that
for every set $E\subset({\mathbb{F}_{q}^{d}},dx),$
(7.5)
$\|E\ast\widehat{K}\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\lesssim\frac{|E|}{q^{d-1}}$
and
(7.6)
$\|E\ast\widehat{K}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-d+\frac{1}{2}}|E|^{\frac{d+2}{2d}}&\mbox{if}~{}~{}1\leq|E|\leq
q^{\frac{d}{2}}\\\
q^{-d+1}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}q^{\frac{d}{2}}\leq|E|\leq
q^{d},\end{array}\right.$
where $K=(d\sigma)^{\vee}-\delta_{0}.$
###### Proof.
The estimate (7.5) follows immediately from the inequality (7.4), because
$\|E\|_{L^{1}({\mathbb{F}_{q}^{d}},dx)}=q^{-d}|E|.$ Let us prove the estimate
(7.6). Using Plancherel, we have
$\displaystyle\|E\ast\widehat{K}\|^{2}_{L^{2}({\mathbb{F}_{q}^{d}},dx)}$
$\displaystyle=\|\widehat{E}K\|^{2}_{L^{2}({\mathbb{F}_{q}^{d}},dm)}$
$\displaystyle=\sum_{m\in{\mathbb{F}_{q}^{d}}}|\widehat{E}(m)|^{2}|K(m)|^{2}=\sum_{m\neq(0,\dots,0)}|\widehat{E}(m)|^{2}|(d\sigma)^{\vee}(m)|^{2},$
where the last line follows from the definition of $K$ and the fact that
$(d\sigma)^{\vee}(0,\dots,0)=1.$ Since $|S|\sim q^{d-1},|\eta|\equiv 1,$ and
the absolute value of the Gauss sum $G_{1}$ is $q^{1/2},$ using the explicit
formula for $(d\sigma)^{\vee}$ in the second part of Lemma 4.1 we see that
$\|E\ast\widehat{K}\|^{2}_{L^{2}({\mathbb{F}_{q}^{d}},dx)}$
$\sim\frac{1}{q^{d-2}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\end{subarray}}|\widehat{E}(m)|^{2}+\frac{1}{q^{d}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0\end{subarray}}|\widehat{E}(m)|^{2}=\mbox{I}+\mbox{II}.$
From the Plancherel theorem given in (1.3), it is not difficult to see that
$\mbox{II}\leq\frac{1}{q^{d}}\sum_{m\in{\mathbb{F}_{q}^{d}}}|\widehat{E}(m)|^{2}=q^{-2d}|E|.$
On the other hand, using Corollary 3.5, the upper bound of I is given by
$\mbox{I}\lesssim\min\left\\{q^{-2d+1}|E|^{\frac{d+2}{d}},q^{-2d+2}|E|\right\\}.$
Putting together above estimates yields
$\|E\ast\widehat{K}\|^{2}_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\lesssim\min\left\\{q^{-2d+1}|E|^{\frac{d+2}{d}},q^{-2d+2}|E|\right\\}+q^{-2d}|E|.$
By a direct calculation, this estimate implies that (7.6) in Lemma 7.2 holds.
Therefore, the proof of Lemma 7.2 is complete. ∎
We also need the following lemma.
###### Lemma 7.3.
Suppose that for every subset $E$ of $({\mathbb{F}_{q}^{d}},dx),$ it satisfies
that
(7.7) $\|E\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|E\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}.$
Then for every function $f$ defined on $({\mathbb{F}_{q}^{d}},dx),$ we have
$\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}\lesssim\log{q}~{}\|f\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}.$
###### Proof.
Without loss of generality, we may assume that $f\geq 0$ and
$\|f\|_{\infty}=1.$ For each nonnegative integer $k$, define
$E_{k}=\\{x\in{\mathbb{F}_{q}^{d}}:2^{-k-1}<f(x)\leq 2^{-k}\\}.$ Then the
function $f$ can be decomposed as $f=\sum_{k=0}^{\infty}f_{k}$ where
$f_{k}=f\cdot E_{k}.$ From the definition of $f_{k}$, we see that for every
nonnegative integer $k$,
(7.8)
$\|f\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}\geq\|f_{k}\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}\geq
2^{-k-1}\|E_{k}\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}$
Since $\|f\|_{\infty}=1$, we also have
(7.9) $\|f\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}\geq q^{-\frac{d}{p}}\geq
2^{-(N+1)},$
where $N$ is the nonnegative integer satisfying
$\frac{d\log{q}}{p\log{2}}-1\leq N<\frac{d\log{q}}{p\log{2}}.$
We now estimate the quantity $\|f\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}.$ It follows that
$\displaystyle\|f\ast d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}$
$\displaystyle\leq\|\sum_{k=0}^{N}f_{k}\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}+\|\sum_{k=N+1}^{\infty}f_{k}\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}$ $\displaystyle\leq N\max_{0\leq
k\leq N}\|f_{k}\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}+2^{-(N+1)}\|1\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}.$
From the definition of $f_{k}$ and the observation that $\|1\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}=1$, we see that above expression is
dominated by the quantity
$N\max_{0\leq k\leq N}2^{-k}\|E_{k}\ast
d\sigma\|_{L^{r}({\mathbb{F}_{q}^{d}},dx)}+2^{-(N+1)}.$
From the hypothesis (7.7) and the inequalities (7.8), (7.9), we therefore
obtain that
$\displaystyle\|f\ast d\sigma\|_{L^{r}({\mathbb{F}_{q}^{r}},dx)}$
$\displaystyle\lesssim N\max_{0\leq k\leq
N}2^{-k}\|E_{k}\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}+2^{-(N+1)}$
$\displaystyle\leq(2N+1)\|f\|_{L^{p}({\mathbb{F}_{q}^{d}},dx)}.$
Since $2N+1\sim\log{q}$, we complete the proof of Lemma 7.3. ∎
We are ready to prove the statement (3.9) in Theorem 3.6. From (7.1), recall
that if $1\leq r\leq p\leq\infty,$ then $A(p\to r)\leq 1.$ Thus, if $d=2$,
then there is nothing to prove. We therefore assume that the dimension $d\geq
4$ is even. By duality and interpolation, it suffices to show that for every
function $f$ on $({\mathbb{F}_{q}^{d}},dx),$
$\|f\ast
d\sigma\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\lessapprox\|f\|_{L^{\frac{d(d-1)}{d^{2}-2d+2}}({\mathbb{F}_{q}^{d}},dx)}.$
Since we allow the logarithmic growth of $q$ in this inequality, using Lemma
7.3 it is enough to prove that for every subset $E$ of
$({\mathbb{F}_{q}^{d}},dx),$
$\|E\ast
d\sigma\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|E\|_{L^{\frac{d(d-1)}{d^{2}-2d+2}}({\mathbb{F}_{q}^{d}},dx)}.$
Since $dx$ is the normalized counting measure, it is clear that
$\|1\|_{L^{a}({\mathbb{F}_{q}^{d}},dx)}=1$ for all $1\leq a\leq\infty.$ Using
Young’s inequality for convolutions together with this fact, we see that
$\|E\ast
1\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\leq\|E\|_{L^{\frac{d(d-1)}{d^{2}-2d+2}}({\mathbb{F}_{q}^{d}},dx)}.$
Since $d\sigma=\widehat{K}+\widehat{\delta_{0}}=\widehat{K}+1,$ it therefore
suffices to show that for every $E\subset{\mathbb{F}_{q}^{d}},$
(7.10)
$\|E\ast\widehat{K}\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\lesssim\|E\|_{L^{\frac{d(d-1)}{d^{2}-2d+2}}({\mathbb{F}_{q}^{d}},dx)}.$
Case 1. Assume that $1\leq|E|\leq q^{\frac{d}{2}}.$ In this case, the estimate
(7.10) follows by interpolating the following two estimates.
(7.11) $\|E\ast\widehat{K}\|_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\lesssim
q\|E\|_{L^{1}({\mathbb{F}_{q}^{d}},dx)}$
and
(7.12) $\|E\ast\widehat{K}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\lesssim
q^{\frac{-d+3}{2}}\|E\|_{L^{\frac{2d}{d+2}}({\mathbb{F}_{q}^{d}},dx)}.$
As before, the inequality (7.11) can be obtained by Young’s inequality and the
observation that $|\widehat{K}|\lesssim q.$ For the inequality (7.12), we
recall from (7.6) in Lemma 7.2 that if $1\leq|E|\leq q^{\frac{d}{2}},$ then
$\|E\ast\widehat{K}\|_{L^{2}({\mathbb{F}_{q}^{d}},dx)}\lesssim
q^{-d+\frac{1}{2}}|E|^{\frac{d+2}{2d}}.$
Since
$\|E\|_{L^{\frac{2d}{d+2}}({\mathbb{F}_{q}^{d}},dx)}=(q^{-d}|E|)^{\frac{d+2}{2d}},$
a direct calculation shows that if $1\leq|E|\leq q^{\frac{d}{2}},$ then the
right-hand side in (7.12) is same as $q^{-d+\frac{1}{2}}|E|^{\frac{d+2}{2d}}.$
Thus, the inequality (7.12) holds.
Case 2. Assume that $q^{\frac{d}{2}}\leq|E|\leq q^{d}.$ In this case, we want
to show that the estimate (7.10) holds. In other words, we must show that for
every $q^{\frac{d}{2}}\leq|E|\leq q^{d},$
(7.13)
$\|E\ast\widehat{K}\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\lesssim(q^{-d}|E|)^{\frac{d^{2}-2d+2}{d(d-1)}}.$
It follows that if $d\geq 4$ is even, then
$\displaystyle\|E\ast\widehat{K}\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}$
$\displaystyle=\|(E\ast\widehat{K})^{\frac{d-1}{2}}\|^{\frac{2}{d-1}}_{L^{2}({\mathbb{F}_{q}^{d}},dx)}$
$\displaystyle\leq\|E\ast\widehat{K}\|^{\frac{d-3}{d-1}}_{L^{\infty}({\mathbb{F}_{q}^{d}},dx)}\|E\ast\widehat{K}\|^{\frac{2}{d-1}}_{L^{2}({\mathbb{F}_{q}^{d}},dx)}.$
Since $q^{\frac{d}{2}}\leq|E|\leq q^{d}$, using Lemma 7.2 we obtain
$\|E\ast\widehat{K}\|_{L^{d-1}({\mathbb{F}_{q}^{d}},dx)}\lesssim(q^{-d+1}|E|)^{\frac{d-3}{d-1}}(q^{-d+1}|E|^{\frac{1}{2}})^{\frac{2}{d-1}}=q^{-d+1}|E|^{\frac{d-2}{d-1}}.$
Thus, the inequality (7.13) follows from an observation that if
$q^{\frac{d}{2}}\leq|E|\leq q^{d}$, then
$q^{-d+1}|E|^{\frac{d-2}{d-1}}\leq(q^{-d}|E|)^{\frac{d^{2}-2d+2}{d(d-1)}}.$
By Case 1 and Case 2, the inequality (7.10) holds and we complete the proof of
the statement (3.9) in Theorem 3.6. Finally, when $S$ contains a
$d/2$-dimensional subspace of $({\mathbb{F}_{q}^{d}},dx)$, the sharpness of
the result (3.9) in Theorem 3.6 follows from the necessary condition (2.5) in
Theorem 2.2.
## References
* [1] J. Bourgain, On the restriction and multiplier problem in ${\mathbb{R}}^{3}$, Lecture notes in Mathematics, no. 1469, Springer Verlag, 1991.
* [2] A. Carbery, B. Stones, and J. Wright, Averages in vector spaces over finite fields, Math. Proc. Camb. Phil. Soc. (2008), 144, 13, 13–27.
* [3] L. De Carli and A. Iosevich, Some sharp restriction theorems for homogeneous manifolds, J. Fourier Anal. Appl., 4 (1998), no.1, 105–128.
* [4] A. Iosevich and D. Koh, Extension theorems for paraboloids in the finite field setting, Math.Z., (to appear), (2009).
* [5] A. Iosevich and D. Koh, Extension theorems for spheres in the finite field setting, Forum Math., (to appear), (arXiv:0712.1627), (2009).
* [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloquium Publications, 53 (2004).
* [7] A. Iosevich and E. Sawyer, Sharp $L^{p}-L^{q}$ estimates for a class of averaging operators, Ann. Inst. Fourier, Grenoble, 46, 5 (1996), 1359–1384.
* [8] W. Littman, $L^{p}-L^{q}$ estimates for singular integral operators, Proc. Symp. Pure Math., 23 (1973), 479–481.
* [9] R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, (1997).
* [10] G. Mockenhaupt, and T. Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J. 121(2004), no. 1, 35–74.
* [11] R. Strichartz, Convolutions with kernels having singularities on the sphere, Trans. Amer. Math. Soc., 148 (1970), 461–471.
* [12] E. M. Stein, $L^{p}$ boundedness of certain convolution operators, Bull. Amer. Math. Soc., 77 (1971), 404–405.
* [13] E. M. Stein, Harmonic Analysis, Princeton University Press (1993).
* [14] T. Tao, Recent progress on the restriction conjecture, Fourier analysis and convexity, 217–243, Appl. Number. Harmon. Anal., Birkhuser Boston, Boston, MA 2004.
|
arxiv-papers
| 2009-08-24T11:01:43 |
2024-09-04T02:49:04.715227
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Doowon Koh and Chun-Yen Shen",
"submitter": "Doowon Koh",
"url": "https://arxiv.org/abs/0908.3266"
}
|
0908.3288
|
# Almost orthogonality and Hausdorff interval topologies of atomic lattice
effect algebras
Jan Paseka111This work was supported by by the Grant Agency of the Czech
Republic under the grant No. 201/06/0664 and by the Ministry of Education of
the Czech Republic under the project MSM0021622409 paseka@math.muni.cz Zdenka
Riečanová22footnotemark: 2 zdenka.riecanova@stuba.sk Department of Mathematics
and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37
Brno, Czech Republic Department of Mathematics, Faculty of Electrical
Engineering and Information Technology, Slovak University of Technology,
Ilkovičova 3, SK-812 19 Bratislava, Slovak Republic Department of
Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Wu Junde wjd@zju.edu.cn
###### Abstract
We prove that the interval topology of an Archimedean atomic lattice effect
algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost
orthogonal. In such a case $E$ is order continuous. If moreover $E$ is
complete then order convergence of nets of elements of $E$ is topological and
hence it coincides with convergence in the order topology and this topology is
compact Hausdorff compatible with a uniformity induced by a separating
function family on $E$ corresponding to compact and cocompact elements. For
block-finite Archimedean atomic lattice effect algebras the equivalence of
almost orthogonality and s-compact generation is shown. As the main
application we obtain the state smearing theorem for these effect algebras, as
well as the continuity of $\oplus$-operation in the order and interval
topologies on them.
###### keywords:
Non-classical logics , D-posets , effect algebras , MV-algebras , interval and
order topology, states
###### PACS:
02.10.De , 02.40.Pc
###### MSC:
: 03G12, 06F05, 03G25, 54H12, 08A55
fn1fn1footnotetext: This work was supported by the Slovak Resaerch and
Development Agency under the contract No. APVV–0071–06 and the grant
VEGA-1/3025/06 of MŠ SR.
## 1 Introduction, basic definitions and facts
In the study of effect algebras (or more general, quantum structures) as
carriers of states and probability measures, an important tool is the study of
topologies on them. We can say that topology is practically equivalent with
the concept of convergence. From the probability point of view the convergence
of nets is the main tool in spite of that convergence of filters is easier to
handle and preferred in the modern topology. It is because states or
probabilities are mappings (functions) defined on elements but not on subsets
of quantum structures. Note also, that connections between order convergence
of filters and nets are not trivial. For instance, if a filter order converges
to some point of a poset then the associated net need not order converge (see
e.g, [12]).
On the other hand certain topological properties of studied structures
characterize also their certain algebraic properties and conversely. For
instance a known fact is that a Boolean algebra $B$ is atomic iff the interval
topology $\tau_{i}$ on $B$ is Hausdorff (see [20, Corollary 3.4]). This is not
more valid for lattice effect algebras (even MV-algebras). By Frink’s Theorem
the interval topology $\tau_{i}$ on $B$ (more generally on any lattice $L$) is
compact iff it is a complete lattice [5]. In [16] it was proved that if a
lattice effect algebra $E$ (more generally any basic algebra) is compactly
generated then $E$ is atomic.
We are going to prove that on an Archimedean atomic lattice effect algebra $E$
the interval topology $\tau_{i}$ is Hausdorff and $E$ is (o)-continuous if and
only if $E$ is almost orthogonal. Moreover, if $E$ is complete then $\tau_{i}$
is compact and coincides with the order topology $\tau_{o}$ on $E$ and this
compact topology $\tau_{i}=\tau_{o}$ is compatible with a uniformity on $E$
induced by a separating function family on $E$ corresponding to compact and
cocompact elements of $E$.
As the main corollary of that we obtain that every Archimedean atomic block-
finite lattice effect algebra $E$ has Hausdorff interval topology and hence
both topologies $\tau_{i}$ and $\tau_{o}$ are Hausdorff and they coincide. In
this case almost orthogonality of $E$ and s-compact generation by finite
elements of $E$ are equivalent. As an application the state smearing theorem
for these effect algebras is formulated. Moreover, the continuity of
$\oplus$-operation in $\tau_{i}$ and $\tau_{o}$ on them is shown.
###### Definition 1.1.
A partial algebra $(E;\oplus,0,1)$ is called an effect algebra if $0$, $1$ are
two distinct elements and $\oplus$ is a partially defined binary operation on
$E$ which satisfy the following conditions for any $a,b,c\in E$:
(Ei)ii
$b\oplus a=a\oplus b$ if $a\oplus b$ is defined,
(Eii)i
$(a\oplus b)\oplus c=a\oplus(b\oplus c)$ if one side is defined,
(Eiii)
for every $a\in E$ there exists a unique $b\in E$ such that $a\oplus b=1$ (we
put $a^{\prime}=b$),
(Eiv)i
if $1\oplus a$ is defined then $a=0$.
We often denote the effect algebra $(E;\oplus,0,1)$ briefly by $E$. In every
effect algebra $E$ we can define the partial order $\leq$ by putting
$a\leq b$ and $b\ominus a=c$ iff $a\oplus c$ is defined and $a\oplus c=b$, we
set $c=b\ominus a$ .
If $E$ with the defined partial order is a lattice (a complete lattice) then
$(E;\oplus,0,1)$ is called a lattice effect algebra (a complete lattice effect
algebra).
Recall that a set $Q\subseteq E$ is called a sub-effect algebra of the effect
algebra $E$ if
1. (i)
$1\in Q$
2. (ii)
if out of elements $a,b,c\in E$ with $a\oplus b=c$ two are in $Q$, then
$a,b,c\in Q$.
If $Q$ is simultaneously a sublattice of $E$ then $Q$ is called a sub-lattice
effect algebra of $E$.
We say that a finite system $F=(a_{k})_{k=1}^{n}$ of not necessarily different
elements of an effect algebra $(E;\oplus,0,1)$ is $\oplus$-orthogonal if
$a_{1}\oplus a_{2}\oplus\dots\oplus a_{n}$ (written
$\bigoplus\limits_{k=1}^{n}a_{k}$ or $\bigoplus F$) exists in $E$. Here we
define $a_{1}\oplus a_{2}\oplus\dots\oplus a_{n}=(a_{1}\oplus
a_{2}\oplus\dots\oplus a_{n-1})\oplus a_{n}$ supposing that
$\bigoplus\limits_{k=1}^{n-1}a_{k}$ exists and
$\bigoplus\limits_{k=1}^{n-1}a_{k}\leq a^{\prime}_{n}$. An arbitrary system
$G=(a_{\kappa})_{\kappa\in H}$ of not necessarily different elements of $E$ is
$\oplus$-orthogonal if $\bigoplus K$ exists for every finite $K\subseteq G$.
We say that for a $\oplus$-orthogonal system $G=(a_{\kappa})_{\kappa\in H}$
the element $\bigoplus G$ exists iff $\bigvee\\{\bigoplus K\mid K\subseteq G$,
$K$ is finite$\\}$ exists in $E$ and then we put $\bigoplus
G=\bigvee\\{\bigoplus K\mid K\subseteq G\\}$ (we write $G_{1}\subseteq G$ iff
there is $H_{1}\subseteq H$ such that $G_{1}=(a_{\kappa})_{\kappa\in H_{1}}$).
Recall that elements $x$ and $y$ of a lattice effect algebra are called
compatible (written $x\leftrightarrow y$) if $x\vee y=x\oplus(y\ominus(x\wedge
y))$ [13]. For $x\in E$ and $Y\subseteq E$ we write $x\leftrightarrow Y$ iff
$x\leftrightarrow y$ for all $y\in Y$. If every two elements are compatible
then $E$ is called an MV-effect algebra. In fact, every MV-effect algebra can
be organized into an MV-algebra (see [2]) if we extend the partial $\oplus$ to
a total operation by setting $x\oplus y=x\oplus(x^{\prime}\wedge y)$ for all
$x,y\in E$ (also conversely, restricting a total $\oplus$ into partial
$\oplus$ for only $x,y\in E$ with $x\leq y^{\prime}$ we obtain a MV-effect
algebra).
Moreover, in [23] it was proved that every lattice effect algebra is a set-
theoretical union of MV-effect algebras called blocks. Blocks are maximal
subsets of pairwise compatible elements of $E$, under which every subset of
pairwise compatible elements is by Zorn’s Lemma contained in a maximal one.
Further, blocks are sub-lattices and sub-effect algebras of $E$ and hence
maximal sub-MV-effect algebras of $E$. A lattice effect algebra is called
block-finite if it has only finitely many blocks.
Finally note that lattice effect algebras generalize orthomodular lattices
[10] (including Boolean algebras) if we assume existence of unsharp elements
$x\in E$, meaning that $x\wedge x^{\prime}\neq 0$. On the other hand the set
$S(E)=\\{x\in E\mid x\wedge x^{\prime}=0\\}$ of all sharp elements of a
lattice effect algebra $E$ is an orthomodular lattice [8]. In this sense a
lattice effect algebra is a “smeared” orthomodular lattice, while an MV-effect
algebra is a “smeared” Boolean algebra. An orthomodular lattice $L$ can be
organized into a lattice effect algebra by setting $a\oplus b=a\vee b$ for
every pair $a,b\in L$ such that $a\leq b^{\perp}$.
For an element $x$ of an effect algebra $E$ we write ${\mathop{\rm
ord}}(x)=\infty$ if $nx=x\oplus x\oplus\dots\oplus x$ ($n$-times) exists for
every positive integer $n$ and we write ${\mathop{\rm ord}}(x)=n_{x}$ if
$n_{x}$ is the greatest positive integer such that $n_{x}x$ exists in $E$. An
effect algebra $E$ is Archimedean if ${\mathop{\rm ord}}(x)<\infty$ for all
$x\in E$. We can show that every complete effect algebra is Archimedean (see
[22]).
An element $a$ of an effect algebra $E$ is an atom if $0\leq b<a$ implies
$b=0$ and $E$ is called atomic if for every nonzero element $x\in E$ there is
an atom $a$ of $E$ with $a\leq x$. If $u\in E$ and either $u=0$ or
$u=p_{1}\oplus p_{2}\oplus\dots\oplus p_{n}$ for some not necessarily
different atoms $p_{1},p_{2},\dots,p_{n}\in E$ then $u\in E$ is called finite
and $u^{\prime}\in E$ is called cofinite. If $E$ is a lattice effect algebra
then for $x\in E$ and an atom $a$ of $E$ we have $a\leftrightarrow x$ iff
$a\leq x$ or $a\leq x^{\prime}$. It follows that if $a$ is an atom of a block
$M$ of $E$ then $a$ is also an atom of $E$. On the other hand if $E$ is atomic
then, in general, every block in $E$ need not be atomic (even for orthomodular
lattices [1]).
The following theorem is well known.
###### Theorem 1.2.
[25, Theorem 3.3] Let $(E;\oplus,0,1)$ be an Archimedean atomic lattice effect
algebra. Then to every nonzero element $x\in E$ there are mutually distinct
atoms $a_{\alpha}\in{E}$, $\alpha{\in\mathcal{E}}$ and positive integers
$k_{\alpha}$ such that
$x=\bigoplus\\{k_{\alpha}a_{\alpha}\mid\alpha\in{\mathcal{E}}\\}=\bigvee\\{k_{\alpha}a_{\alpha}\mid\alpha\in{\mathcal{E}}\\}$
under which $x\in S(E)$ iff $k_{\alpha}=n_{a_{\alpha}}={\mathop{\rm
ord}}(a_{\alpha})$ for all $\alpha\in{\mathcal{E}}$.
###### Definition 1.3.
(1) An element $a$ of a lattice $L$ is called compact iff, for any $D\subseteq
L$ with $\bigvee D\in L$, if $a\leq\bigvee D$ then $a\leq\bigvee F$ for some
finite $F\subseteq D$.
(2) A lattice $L$ is called compactly generated iff every element of $L$ is a
join of compact elements.
The notions of cocompact element and cocompactly generated lattice can be
defined dually. Note that compact elements are important in computer science
in the semantic approach called domain theory, where they are considered as a
kind of primitive elements.
## 2 Characterizations of interval topologies on bounded lattices
The order convergence of nets ((o)-convergence), interval topology $\tau_{i}$
and order-topology $\tau_{o}$ ((o)-topology) can be defined on any poset. In
our observations we will consider only bounded lattices and we will give a
characterization of interval topologies on them.
###### Definition 2.1.
Let $L$ be a bounded lattice. Let $\mathcal{H}=\\{[a,b]\subseteq L|a,b\in L$
with $a\leq b\\}$ and let
$\mathcal{G}=\\{\bigcup^{n}_{k=1}[a_{k},b_{k}]|[a_{k},b_{k}]\in\mathcal{H},k=1,2,...,n\\}$.
The _interval topology $\tau_{i}$ of $L$_ is the topology of $L$ with
$\mathcal{G}$ as a closed basis, hence with $\mathcal{H}$ as a closed
subbasis.
From definition of $\tau_{i}$ we obtain that $U\in\tau_{i}$ iff for each $x\in
U$ there is $F\in\mathcal{G}$ such that $x\in L\backslash F\subseteq U$.
###### Definition 2.2.
Let $L$ be a poset.
* (i)
A net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ _order
converges_ (_$(o)$ -converges_, for short) to a point $x\in L$ if there exist
nets $(u_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(v_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ such that
$x\uparrow u_{\alpha}\leq x_{\alpha}\leq v_{\alpha}\downarrow
x,\,{\alpha\in\mathcal{E}}$
where $x\uparrow u_{\alpha}$ means that $u_{\alpha_{1}}\leq u_{\alpha_{2}}$
for every ${\alpha_{1}}\leq{\alpha_{2}}$ and
$x=\bigvee\\{x_{\alpha}\mid\alpha\in\mathcal{E}\\}$. The meaning of
$v_{\alpha}\downarrow x$ is dual.
We write
$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}$
in $L$.
* (ii)
A topology $\tau_{0}$ on $L$ is called the _order topology_ on $L$ iff
1. (a)
for any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ and $x\in
L$: $x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}a$ in L
$\Rightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau_{0}}}{{\rightarrow}}x$,
${\alpha\in\mathcal{E}}$, where
$x_{\alpha}\stackrel{{\scriptstyle\tau_{0}}}{{\rightarrow}}x$ denotes that
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ _converges_ to $x$ in the topological
space $(L,\tau_{0})$,
2. (b)
if $\tau$ is a topology on $L$ with property (a) then $\tau\subseteq\tau_{o}$.
Hence $\tau_{o}$ is the strongest (finest, biggest) topology on $L$ with
property (a).
Recall that, for a directed set $(\mathcal{E},\leq)$, a subset
$\mathcal{E}^{\prime}\subseteq\mathcal{E}$ is called _cofinal_ in
$\mathcal{E}$ iff for every $\alpha\in\mathcal{E}$ there is
$\beta\in\mathcal{E}^{\prime}$ such that $\alpha\leq\beta$. A special kind of
a subnet of a net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ is net
$(x_{\beta})_{\beta\in\mathcal{E}^{\prime}}$ where $\mathcal{E}^{\prime}$ is a
cofinal subset of $\mathcal{E}$. This kind of subnets works in many cases of
our considerations.
In what follows we often use the following useful characterization of
topological convergence of nets:
###### Lemma 2.3.
For a net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of a topological
space $(X,\tau)$ and $x\in X$:
$x_{\alpha}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\alpha\in\mathcal{E}$ | iff | for all $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, where $\mathcal{E}^{\prime}$ is cofinal in $\mathcal{E}$ there exist
---|---|---
| | $\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$, $\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$ such that $x_{\gamma}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$.
###### Proof.
$\Rightarrow$: It is trivial.
$\Leftarrow$: Let for every $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, where
$\mathcal{E}^{\prime}$ is cofinal in $\mathcal{E}$ there exist
$\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$,
$\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$ and
$x_{\gamma}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$,
and let
$x_{\alpha}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\alpha\in\mathcal{E}$.
Then there exist $U(x)\in\tau$ such that for all $\alpha\in\mathcal{E}$ there
exist $\beta_{\alpha}\in\mathcal{E}$ with $\beta_{\alpha}\geq\alpha$ and
$x_{\beta_{\alpha}}\not\in U(x)$. Let
$\mathcal{E}^{\prime}=\\{\beta_{\alpha}\in\mathcal{E}|\alpha\in\mathcal{E},\beta_{\alpha}\geq\alpha,x_{\beta_{\alpha}}\not\in
U(x)\\}$ then
$x_{\beta_{\alpha}}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,{\beta_{\alpha}}\in\mathcal{E}^{\prime}$
and for all cofinal $\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$:
$x_{\gamma}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$.
Hence there exists $\mathcal{E}^{\prime}\subseteq\mathcal{E}$ cofinal in
$\mathcal{E}$ and for all
$\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$,
$\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$:
$x_{\gamma}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$
a contradiction. ∎
Further, let us recall the following well known facts:
###### Lemma 2.4.
Let $L$ be a bounded lattice. Then
1. (i)
$F\subseteq L$ is $\tau_{0}$-closed iff for every net
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ and $x\in L$:
$(x_{\alpha}\in
F,x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,\alpha\in\mathcal{E})\Rightarrow
x\in F$.
2. (ii)
For every $a,b\in L$ with $a\leq b$ the interval $[a,b]$ is $\tau_{0}$-closed.
3. (iii)
$\tau_{i}\subseteq\tau_{o}$.
4. (iv)
For any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of a $L$ and
$x\in L$:
$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}\
\Longrightarrow\
x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}.$
5. (v)
If $\tau_{i}$ is Hausdorff then $\tau_{0}=\tau_{i}$ (see [4]).
6. (vi)
The interval topology $\tau_{i}$ of a lattice $L$ is compact iff $L$ is a
complete lattice (see [5]).
Finally, let us note that compact Hausdorff topological space is always
normal. Thus separation axiom $T_{2},T_{3}$ and $T_{4}$ are trivially
equivalent for the interval topology of a complete lattice $L$.
###### Theorem 2.5.
Let $L$ be a complete lattice with interval topology $\tau_{i}$. If
$F\subseteq L$ is a complete sub-lattice of $L$ then
1. (a)
$\tau^{F}_{i}=\tau_{i}\cap F$ is the interval topology of $F$,
2. (b)
for any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $F$ and $x\in
F$:
$x_{\alpha}\stackrel{{\scriptstyle\tau^{F}_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}\
\Longleftrightarrow\
x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}.$
###### Proof.
(a): Let $\mathcal{H}$ and $\mathcal{H}_{F}$ be a closed subbasis of
$\tau_{i}$ and $\tau^{F}_{i}$ respectively. Then evidently $\mathcal{H}\cap
F=\\{[a,b]\cap F|[a,b]\in\mathcal{H}\\}$ is a closed subbasis of $\tau_{i}\cap
F$. Further for $[c,d]_{F}\in\mathcal{H}_{F}$ we have $[c,d]_{F}=\\{x\in
F|c\leq x\leq d\\}=[c,d]\cap F\in\mathcal{H}\cap F$. Conversely, since $F$ is
a complete sub-lattice of $L$, if $[a,b]\in\mathcal{H}$ then $[a,b]\cap
F=\\{x\in F|a\leq x\leq b\\}$ and either $[a,b]\cap F=\emptyset$ or there
exist $c=\wedge\\{x\in F|a\leq x\leq b\\}$ and $d=\vee\\{x\in F|a\leq x\leq
d\\}$ and $[a,b]\cap F=[c,d]_{F}\in\mathcal{H}_{F}$. This proves that
$\tau^{F}_{i}=\tau_{i}\cap F$.
(b): This is an easy consequence of (a). ∎
## 3 Hausdorff interval topology of almost orthogonal Archimedean atomic
lattice effect algebras and their order continuity
The atomicity of Boolean algebra $B$ is equivalent with Hausdorffness of
interval topology on $B$ (see [11], [29] and [20, Corollary 3.4]). This is not
more valid for lattice effect algebras, even also for MV-algebras.
###### Example 3.1.
Let $M=[0,1]\subseteq{\mathbb{R}}$ be a standard MV-effect algebra, i.e., we
define $a\oplus b=a+b$ iff $a+b\leq 1$, $a,b\in M$. Then $M$ is a complete
(o)-continuous lattice with $\tau_{i}=\tau_{o}$ being Hausdorff and with
(o)-convergence of nets coinciding with $\tau_{o}$-convergence. Nevertheless,
$M$ is not atomic.
We have proved in [16] that a complete lattice effect algebra is atomic and
(o)-continuous lattice iff $E$ is compactly generated. Nevertheless, in such a
case, the interval topology on $E$ need not be Hausdorff.
###### Example 3.2.
Let $E$ be a horizontal sum of infinitely many finite chains
$(P_{i},\bigoplus_{i},0_{i},1_{i})$ with at least 3 elements,
$i=1,2,...,n,\dots,$ (i. e., for $i=1,2,...,n,\dots,$, we identify all $0_{i}$
and all $1_{i}$ as well, $\bigoplus_{i}$ on $P_{i}$ are preserved and any
$a\in P_{i}\backslash\\{0_{i},1_{i}\\}$, $b\in
P_{j}\backslash\\{0_{j},1_{j}\\}$ for $i\not=j$ are noncomparable). Then $E$
is an atomic complete lattice effect algebra, $E$ is not block-finite and the
interval topology $\tau_{i}$ on $E$ is compact. Nevertheless, $\tau_{i}$ is
not Hausdorff because e.g., for $a\in P_{i},b\in P_{j},i\not=j$, $a,b$
noncomparable, we have $[a,1]\cap[0,b]=\emptyset$ and there is no finite
family $\mathcal{I}$ of closed intervals in $E$ separating $[a,1],[0,b]$
(i.e., the lattice $E$ can not be covered by a finite number of closed
intervals from $\mathcal{I}$ each of which is disjoint with at least one of
the intervals $[a,1]$ and $[0,b]$). This implies that $\tau_{i}$ is not
Hausdorff by [20, Lemma 2.2]. Further $E$ is compactly generated by finite
elements (hence (o)-continuous). It follows by [16] that the order topology
$\tau_{o}$ on $E$ is a uniform topology and (o)-convergence of nets on $E$
coincides with $\tau_{o}$-convergence.
In what follows we shall need an extension of [26, Lemma 2.1 (iii)].
###### Lemma 3.3.
Let $E$ be a lattice effect algebra, $x,y\in E$. Then $x\wedge y=0$ and $x\leq
y^{\prime}$ iff $kx\wedge ly=0$ and $kx\leq(ly)^{\prime}$, whenever $kx$ and
$ly$ exist in $E$.
###### Proof.
Let $x\leq y^{\prime}$, $x\wedge y=0$ and $2y$ exists in $E$. Then $x\oplus
y=(x\vee y)\oplus(x\wedge y)=x\vee y\leq y^{\prime}$ and hence there exists
$x\oplus 2y=(x\vee y)\oplus y=(x\oplus y)\vee 2y=x\vee y\vee 2y=x\vee 2y$,
which gives that $x\leq(2y)^{\prime}$ and $x\wedge 2y=0$. By induction, if
$ly$ exists then $x\oplus ly=x\vee ly$ and hence $x\leq(ly)^{\prime}$ and
$x\wedge ly=0$.
Now, $x\leq(ly)^{\prime}$ iff $ly\leq x^{\prime}$ and because $x\wedge ly=0$,
we obtain by the same argument as above that $ly\oplus kx=ly\vee kx$, hence
$kx\leq(ly)^{\prime}$ and $ly\wedge kx=0$ whenever $kx$ exists in $E$.
Conversely, $kx\wedge ly=0$ implies that $x\wedge y=0$ and
$kx\leq(ly)^{\prime}$ implies $x\leq kx\leq(ly)^{\prime}\leq y^{\prime}$. ∎
In next we will use the statement of Lemma 3.3 in the following form: For any
$x,y\in E$ with $x\wedge y=0$, $x\not\leq y^{\prime}$ iff
$kx\not\leq(ly)^{\prime}$, whenever $kx$ and $ly$ exist in $E$.
###### Definition 3.4.
Let $E$ be an atomic lattice effect algebra. $E$ is said to be almost
orthogonal if the set $\\{b\in E\mid b\not\leq a^{\prime},b\ \mbox{is an
atom}\\}$ is finite for every atom $a\in E$.
Note that our definition of almost orthogonality coincides with the usual
definition for orthomodular lattices (see e.g. [17, 18]).
###### Theorem 3.5.
Let $E$ be an Archimedean atomic lattice effect algebra. Then $E$ is almost
orthogonal if and only if for any atom $a\in E$ and any integer $l$, $1\leq
l\leq n_{a}$, there are finitely many atoms $c_{1},\dots,c_{m}$ and integers
$j_{1},\dots,j_{m}$, $1\leq j_{1}\leq n_{c_{1}},\dots,1\leq j_{m}\leq
n_{c_{m}}$ such that $j_{k}{}c_{k}\not\leq(la)^{\prime}$ for all
$k\in\\{1,\dots,m\\}$ and, for all $x\in E$, $x\not\leq(la)^{\prime}$ implies
$j_{k_{0}}{}c_{k_{0}}\leq x$ for some ${k_{0}}\in\\{1,\dots,m\\}$.
###### Proof.
$\Longrightarrow$: Assume that $E$ is almost orthogonal. Let $a\in E$ be an
atom, $1\leq l\leq n_{a}$. We shall denote $A_{a}=\\{b\in E\mid b\ \mbox{is}\
\mbox{an}\ \mbox{atom},b\not\leq a^{\prime}\\}$. Clearly, $A_{a}$ is finite
i.e. $A_{a}=\\{b_{1},\dots,b_{n}\\}$ for suitable atoms $b_{1},\dots,b_{n}$
from $E$.
Let $b\in E$ be an atom, $1\leq k\leq n_{b}$ and $kb\not\leq(la)^{\prime}$.
Either $b=a$ or $b\not={}a$ and in this case we have by Lemma 3.3 (iv) that
$b\not\leq a^{\prime}$. Hence either $b=a$ or $b\in A_{a}$. Let us put
$\\{c_{1},\dots,c_{m}\\}=\left\\{\begin{array}[]{l l}A_{a}&\mbox{if}\ a\in
S(E)\\\ A_{a}\cup\\{a\\}&\mbox{otherwise}\end{array}\right.$. In both cases we
have that $a\in\\{c_{1},\dots,c_{m}\\}$.
Now, let $x\in E$ and $x\not\leq(la)^{\prime}$. By Theorem 1.2 there is an
atom $c\in E$ and an integer $1\leq j\leq n_{c}$ such that $jc\leq x$ and
$jc\not\leq(la)^{\prime}$. Either $c=a$ or $c\not\leq a$. In the first case we
have that $j\geq(n_{a}-l+1)$ i.e. $x\geq(n_{a}-l+1)a$. In the second case we
get that $c\not\leq a^{\prime}$ i.e. $c\in A_{a}$ and $x\geq b_{i}$ for
suitable $i\in\\{1,\dots,n\\}$. Hence it is enough to put $j_{k}=1$ if
$c_{k}\in A_{a}$ and $j_{k}=(n_{a}-l+1)$ if $c_{k}=a$.
$\Longleftarrow$: Conversely, let $a\in E$ be an atom. Then there are finitely
many atoms $c_{1},\dots,c_{m}$ and integers $j_{1},\dots,j_{m}$, $1\leq
j_{1}\leq n_{c_{1}},\dots,1\leq j_{m}\leq n_{c_{m}}$ such that
$j_{k}{}c_{k}\not\leq a^{\prime}$ for all $k\in\\{1,\dots,m\\}$ and, for all
$x\in E$, $x\not\leq a^{\prime}$ implies $j_{k_{0}}{}c_{k_{0}}\leq x$ for some
${k_{0}}\in\\{1,\dots,m\\}$. Let us check that
$A_{a}\subseteq\\{c_{1},\dots,c_{m}\\}$. Let $b\in A_{a}$. Then $b\geq
j_{k_{0}}{}c_{k_{0}}\geq c_{k_{0}}$ for some ${k_{0}}\in\\{1,\dots,m\\}$.
Hence $b=c_{k_{0}}$. This yields $A_{a}$ is finite. ∎
###### Lemma 3.6.
Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then, for any atom $a\in E$ and any integer $l$, $1\leq l\leq n_{a}$ there are
finitely many atoms $b_{1},\dots,b_{n}$ and integers $j_{1},\dots,j_{n}$,
$1\leq j_{1}\leq n_{b_{1}},\dots,1\leq j_{n}\leq n_{b_{n}}$ such that
|
$E=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$
---|---
and |
|
$[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)=\emptyset$.
Hence $[0,(la)^{\prime}]$ is a clopen subset in the interval topology.
###### Proof.
Let $a\in E$ be an atom, $1\leq l\leq n_{a}$. By Definition 3.5, let
$\\{j_{1}{}b_{1},\dots,j_{n}{}b_{n}\\}$ be the finite set of non-orthogonal
finite elements to $la$ of the form $j_{k}b_{k}$, $1\leq j_{k}\leq n_{b_{k}}$
minimal such that $b_{1},\dots,b_{n}$ are atoms different from $a$. We put
$D=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$.
Let us check that $D=E$. Clearly, $D\subseteq E$. Now, let $z\in E$. Then by
Theorem 1.2 there are mutually distinct atoms $c_{\gamma}\in{E}$,
$\gamma{\in\mathcal{E}}$ and integers $t_{\gamma}$ such that
$z=\bigoplus\\{t_{\gamma}c_{\gamma}\mid\gamma\in{\mathcal{E}}\\}=\bigvee\\{t_{\gamma}c_{\gamma}\mid\gamma\in{\mathcal{E}}\\}.$
Either $t_{\gamma}c_{\gamma}\leq(la)^{\prime}$ for all
$\gamma\in{\mathcal{E}}$ and hence $z\in[0,(la)^{\prime}]$ or there exists
$\gamma_{0}\in{\mathcal{E}}$ such that
$t_{\gamma_{0}}c_{\gamma_{0}}\not\leq(la)^{\prime}$. Hence, by almost
orthogonality, either $j_{k_{0}}b_{k_{0}}\leq t_{\gamma_{0}}c_{\gamma_{0}}\leq
z$ for some $k_{0}\in\\{1,\dots,n\\}$ or $(n_{a}+1-l)a\leq
t_{\gamma_{0}}c_{\gamma_{0}}\leq z$. In both cases we get that $z\in D$.
Now, assume that
$y\in{}[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$.
Then $(n_{a}+1-l)a\leq y\leq(la)^{\prime}$ or $j_{k}{}b_{k}\leq
y\leq(la)^{\prime}$ for some $k\in\\{1,\dots,n\\}$. In any case we have a
contradiction. ∎
###### Proposition 3.7.
Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then, for any not necessarily different atoms $a,b\in E$ and any integers
$l,k$; $1\leq l\leq n_{a}$, $1\leq k\leq n_{b}$, the interval
$[kb,(la)^{\prime}]$ is clopen in the interval topology.
###### Proof.
From Lemma 3.6 we have that $[0,(la)^{\prime}]$ is a clopen subset. Since a
dual of an almost orthogonal Archimedean atomic lattice effect algebra is an
almost orthogonal Archimedean atomic lattice effect algebra as well, we have
that $[kb,1]$ is again clopen in the interval topology. Hence also
$[kb,(la)^{\prime}]$ is clopen in the interval topology. ∎
###### Theorem 3.8.
Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then the interval topology $\tau_{i}$ on $E$ is Hausdorff.
###### Proof.
Let $x,y\in E$ and $x\not=y$. Then (without loss of generality) we may assume
that $x\not\leq y$. Then by [25, Theorem 3.3] there exists an atom $b$ from
$E$ and an integer $k$, $1\leq k\leq n_{b}$ such that $kb\leq x$ and
$kb\not\leq y$. Applying the dual of [25, Theorem 3.3] there exists an atom
$a$ from $E$ and an integer $l$, $1\leq l\leq n_{a}$ such that
$y\leq(la)^{\prime}$ and $kb\not\leq(la)^{\prime}$. Clearly, $x\in[kb,1]$,
$y\in[0,(la)^{\prime}]$.
Assume that there is an element $z\in E$ such that
$z\in[kb,1]\cap[0,(la)^{\prime}]$. Then $kb\leq z\leq(la)^{\prime}$, a
contradiction. Hence by Proposition 3.7, $[kb,1]$ and $[0,(la)^{\prime}]$ are
disjoint open subsets separating $x$ and $y$. ∎
###### Theorem 3.9.
Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then $E$ is compactly generated and therefore (o)-continuous.
###### Proof.
It is enough to check that, for any atom $a\in E$ and any integer $l$, $1\leq
l\leq n_{a}$ the element $la$ is compact in $E$ since any element of $E$ is a
join of such elements (see Theorem 1.2 resp. [25, Theorem 3.3]).
Let $x=\bigvee_{\alpha\in\mathcal{E}}x_{\alpha}$ for some net
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $E$, $la\leq x$, i.e.,
$(la)^{\prime}\geq x^{\prime}\downarrow x^{{}^{\prime}}_{\alpha}$.
By Lemma 3.6 we have
$E=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$,
$[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[b_{k},1]\cup[(n_{a}+1-l)a,1]\right)=\emptyset$,
$b_{1},\dots,b_{n}$ are atoms of $E$, $1\leq j_{k}\leq n_{b_{k}}$, $1\leq
k\leq n$.
Since ${\mathcal{E}}$ is directed upwards, there exists a cofinal subset
$\mathcal{E}^{\prime}\subseteq\mathcal{E}$ such that
$x^{{}^{\prime}}_{\beta}\in[0,(la)^{\prime}]$ for all
$\beta\in\mathcal{E}^{\prime}$ or there exists $k_{0}\in\\{1,2,...,n\\}$ such
that $x_{\beta}\in[j_{k_{0}}b_{k_{0}},1]$ for all
$\beta\in\mathcal{E}^{\prime}$ or $x^{{}^{\prime}}_{\beta}\in[(n_{a}+1-l)a,1]$
for all $\beta\in\mathcal{E}^{\prime}$. If
$x^{{}^{\prime}}_{\beta}\in[0,(la)^{\prime}]$ for all
$\beta\in\mathcal{E}^{\prime}$ then clearly $la\leq x_{\beta}$ for all
$\beta\in\mathcal{E}^{\prime}$. If there exists $k_{0}\in\\{1,2,...,n\\}$ such
that $x^{{}^{\prime}}_{\beta}\in[j_{k_{0}}b_{k_{0}},1]$ for all
$\beta\in\mathcal{E}^{\prime}$ or $x^{{}^{\prime}}_{\beta}\in[(n_{a}+1-l)a,1]$
for all $\beta\in\mathcal{E}^{\prime}$ we obtain that
$x^{\prime}\in[j_{k_{0}}b_{k_{0}},1]$ or $x^{{}^{\prime}}\in[(n_{a}+1-l)a,1]$
which is a contradiction with $x^{{}^{\prime}}\in[0,(la)^{\prime}]$. ∎
Let $E$ be an Archimedean atomic lattice effect algebra. We put
${\mathcal{U}}=\\{x\in E\mid x=\bigvee_{i=1}^{n}l_{i}a_{i},a_{1},\dots,a_{n}\
\mbox{are atoms of }\ E,1\leq l_{i}\leq n_{a_{i}},1\leq i\leq n,n\
\mbox{natural}$ $\mbox{number}\\}$ and ${\mathcal{V}}=\\{x\in E\mid
x^{\prime}\in{\mathcal{U}}\\}$. Then by [25, Theorem 3.3], for every $x\in L$,
we have that
$x=\bigvee\\{u\in{\mathcal{U}}\mid u\leq
x\\}=\bigwedge\\{v\in{\mathcal{V}}\mid x\leq v\\}.$
Consider the function family $\Phi=\\{f_{u}\mid u\
\in{\mathcal{U}}\\}\cup\\{g_{v}\mid v\in{\mathcal{V}}\\}$, where
$f_{u},g_{v}:L\to\\{0,1\\}$, $u\in{\mathcal{U}},v\in{\mathcal{V}}$ are defined
by putting $f_{u}(x)=\left\\{\begin{array}[]{r c l}1&\hbox{iff}&u\leq x\\\
0&\hbox{iff}&u\not\leq x\end{array}\right.$
and $g_{v}(y)=\left\\{\begin{array}[]{r c l}1&\hbox{iff}&x\leq v\\\
0&\hbox{iff}&x\not\leq v\end{array}\right.$ for all $x,y\in L$.
Further, consider the family of pseudometrics on $L$:
$\Sigma_{\Phi}=\\{\rho_{u}\mid u\in{\mathcal{U}}\\}\cup\\{\pi_{v}\mid
v\in{\mathcal{V}}\\}$, where $\rho_{u}(a,b)=|f_{u}(a)-f_{u}(b)|$ and
$\pi_{v}(a,b)=|g_{v}(a)-g_{v}(b)|$ for all $a,b\in L$.
Let us denote by $\mathcal{U}_{\Phi}$ the uniformity on $L$ induced by the
family of pseudometrics $\Sigma_{\Phi}$ (see e.g. [3]). Further denote by
$\tau_{\Phi}$ the topology compatible with the uniformity
$\mathcal{U}_{\Phi}$.
Then for every net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$
$\begin{array}[]{l}x_{\alpha}\stackrel{{\scriptstyle\tau_{\Phi}}}{{\longrightarrow}}x\hbox{
iff }\varphi(x_{\alpha})\to\varphi(x)\hbox{ for any
}\varphi\in\Phi.\end{array}$
This implies, since $f_{u}$, $u\in{\mathcal{U}}$, and $g_{v}$,
$v\in{\mathcal{V}}$, is a separating function family on $L$, that the topology
$\tau_{\Phi}$ is Hausdorff. Moreover, the intervals
$[u,v]=[u,1]\cap[0,v]=f_{u}^{-1}(\\{1\\})\cap g_{v}^{-1}(\\{1\\})$ are clopen
sets in $\tau_{\Phi}$.
###### Definition 3.10.
Let $E$ be an Archimedean atomic lattice effect algebra. Let $\Phi$ be a
separating function family on $E$ defined above. We will denote by
$\tau_{\Phi}$ the uniform topology on $E$ defined by this function family,
that means for every net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of
$L$
$\begin{array}[]{l}x_{\alpha}\stackrel{{\scriptstyle\tau_{\Phi}}}{{\longrightarrow}}x\
\hbox{ iff }\ \varphi(x_{\alpha})\to\varphi(x)\hbox{ for any
}\varphi\in\Phi.\end{array}$
###### Theorem 3.11.
Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then $\tau_{i}=\tau_{o}=\tau_{\Phi}$.
###### Proof.
Since by Theorem 3.8, $\tau_{i}$ is Hausdorff we obtain by [4] that
$\tau_{i}=\tau_{o}$. Further if $O\in\tau_{o}$ and $x\in O$ then by Theorem
1.2 we have $x=\bigvee\\{u\in{\mathcal{U}}\mid u\leq
x\\}=\bigwedge\\{v\in{\mathcal{V}}\mid x\leq v\\}$, which by [12] implies that
there exist finite sets $F\subseteq{\mathcal{U}}$, $G\subseteq{\mathcal{V}}$
such that $x\in[\bigvee F,\bigwedge G]\subseteq O$. Hence
$\tau_{o}\subseteq\tau_{\Phi}$. By Theorem 3.9 and [16, Theorem 1] we obtain
$\tau_{o}=\tau_{\Phi}$. ∎
###### Theorem 3.12.
Let $E$ be an Archimedean atomic block-finite lattice effect algebra. Then
$\tau_{i}=\tau_{o}$ is a Hausdorff topology.
###### Proof.
As in [18], it suffices to show that for every $x,y\in E$, $x\not\leq y$ there
are finitely many intervals, none of which contains both $x$ and $y$ and the
union of which covers $E$.
By [15], $E$ is a union of finitely many atomic blocks $M_{i}$,
$i=1,2,\dots,n$. Choose $i\in\\{1,2,\dots,n\\}$. If $x,y\in M_{i}$ then there
is an atom $a_{i}\in M_{i}$ and an integer $l_{i}$, $1\leq l_{i}\leq
n_{a_{i}}$ such that $l_{i}a_{i}\leq x$, $l_{i}a_{i}\not\leq y$. Let us put
$k_{i}=n-l_{i}+1$. Since $M_{i}$ is almost orthogonal (the only possible non-
orthogonal $kb$ to $la$ for an atom $a$, $1\leq l\leq n_{a}$ is that $a=b$) we
have by Lemma 3.6 that $M_{i}=([0,(k_{i}a_{i})^{\prime}]\cap
M_{i})\cup([(n_{a_{i}}+1-k_{i})a_{i},1]\cap M_{i})$. Hence
$M_{i}\subseteq[0,(k_{i}a_{i})^{\prime}]\cup[(n_{a_{i}}+1-k_{i})a_{i},1]$. Let
us check that
$[0,(k_{i}a_{i})^{\prime}]\cap[(n_{a_{i}}+1-k_{i})a_{i},1]=\emptyset$. Assume
that $(n_{a_{i}}+1-k_{i})a_{i}\leq z\leq(k_{i}a_{i})^{\prime}$. Then
$(n_{a_{i}}+1-k_{i})a_{i}\leq(k_{i}a_{i})^{\prime}$, a contradiction. Put
$J_{i}=[0,(k_{i}a_{i})^{\prime}]$, $K_{i}=[(n_{a_{i}}+1-k_{i})a_{i},1]$. This
yields $x\in K_{i}$, $y\in J_{i}$, $M_{i}\subseteq J_{i}\cup K_{i}$ and
$J_{i}\cap K_{i}=\emptyset$. Let $x\not\in M_{i}$. Then there exists an atom
$a_{i}\in M_{i}$ that is not compatible with $x$. Let us check that
$x\not\in[0,(a_{i})^{\prime}]\cup[n_{a_{i}}a_{i},1]$. Assume that
$x\in[0,(a_{i})^{\prime}]$ or $x\in[n_{a_{i}}a_{i},1]$. Then
$x\leq(a_{i})^{\prime}$ or $a_{i}\leq n_{a_{i}}a_{i}\leq x$, i.e., in both
cases we get that $x\leftrightarrow a_{i}$, a contradiction. Let us put
$J_{i}=[0,(a_{i})^{\prime}]$, $K_{i}=[n_{a_{i}}a_{i},1]$. As above,
$M_{i}\subseteq J_{i}\cup K_{i}$, $J_{i}\cap K_{i}=\emptyset$ and moreover
$x\notin J_{i}\cup K_{i}$. The remaining case $y\not\in M_{i}$ can be checked
by similar considerations. We obtain
$E=\bigcup_{i=1}^{n}M_{i}\subseteq\bigcup_{i=1}^{n}(J_{i}\cup K_{i})\subseteq
E$ and none of the intervals $J_{i},K_{i}$, $i=1,2,\dots,n$ contains both $x$
and $y$. ∎
## 4 Order and interval topologies of complete atomic block-finite lattice
effect algebras
We are going to show that on every complete atomic block-finite lattice effect
algebra $E$ the interval topology is Hausdorff. Hence both topologies
$\tau_{i}$ and $\tau_{o}$ are in this case compact Hausdorff and they
coincide. Moreover, a necessary and sufficient condition for a complete atomic
lattice algebra $E$ to be almost orthogonal is given.
For the proof of Theorems 4.2 and 4.3 we will use the following statement,
firstly proved in the equivalent setting of D-posets in [19].
###### Theorem 4.1.
[19, Theorem 1.7] Suppose that $(E;\oplus,0,1)$ is a complete lattice effect
algebra. Let $\emptyset\not=D\subseteq E$ be a sub-lattice effect algebra. The
following conditions are equivalent:
1. (i)
For all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ such that $x_{\alpha}\in D$
for all ${\alpha\in\mathcal{E}}$
$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x\ \mbox{in}\ E\
\mbox{if and only if}\ x\in D\ \mbox{and}\
x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x\ \mbox{in}\ D.$
2. (ii)
For every $M\subseteq D$ with $\bigvee M=x$ in $E$ it holds $x\in D$.
3. (iii)
For every $Q\subseteq D$ with $\bigwedge Q=y$ in $E$ it holds $y\in D$.
4. (iv)
$D$ is a complete sub-lattice of $E$.
5. (v)
$D$ is a closed set in order topology $\tau_{o}$ on $E$.
Each of these conditions implies that $\tau_{o}^{D}=\tau_{o}^{E}\cap D$, where
$\tau_{o}^{D}$ is an order topology on $D$.
Important sub-lattice effect algebras are blocks, $S(E)$,
$B(E)=\bigcap\\{M\subseteq E\mid M\ \mbox{block\/ of}\ E\\}$ and
$C(E)=B(E)\cap S(E)$ (see [6, 7, 13, 21, 23]).
###### Theorem 4.2.
Let $E$ be a complete lattice effect algebra. Then for every
$D\in\\{S(E),C(E),B(E)\\}$ or $D=M$, where $M$ is a block of $E$, we have:
1. (1)
$x_{\alpha}\stackrel{{\scriptstyle\tau^{E}_{i}}}{{\rightarrow}}x$ $\Longleftrightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau^{D}_{i}}}{{\rightarrow}}x$, | for all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $D$ and all $x\in D$.
---|---
2. (2)
If $\tau_{i}$ is Hausdorff then
$x_{\alpha}\stackrel{{\scriptstyle\tau^{E}_{i}}}{{\rightarrow}}x$ $\Longleftrightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau^{D}_{i}}}{{\rightarrow}}x$, | for all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $D$ and all $x\in E$.
---|---
###### Proof.
The first part of the statement follows by Theorem 2.5 and the fact that if
$E$ is a complete lattice effect algebra then $M$, $S(E)$, $C(E)$ and $B(E)$
are complete sub-lattices of $E$ (see [9, 24]). The second part follows by [4]
since $\tau_{i}$ is Hausdorff implies $\tau_{i}=\tau_{o}$ and by Theorem 4.1.
∎
###### Theorem 4.3.
(i) The interval topology $\tau_{i}$ on every Archimedean atomic MV-effect
algebra $M$ is Hausdorff and $\tau_{i}=\tau_{o}=\tau_{\Phi}$.
(ii) For every complete atomic MV-effect algebra $M$ and for any net
$(x_{\alpha})$ of $M$ and any $x\in M$,
$x_{\alpha}\stackrel{{\scriptstyle\tau_{o}}}{{\longrightarrow}}x$ if and only
if $x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x$ (briefly
$\tau_{o}\equiv(o)$).
Moreover, $\tau_{o}$ is a uniform compact Hausdorff topology on $M$.
(iii) For every atomic block-finite lattice effect algebra $E$, $E$ is a
complete lattice iff $\tau_{i}=\tau_{o}$ is a compact Hausdorff topology.
###### Proof.
(i), (ii): This follows from the fact that every pair of elements of $M$ is
compatible, hence every pair of atoms is orthogonal. Thus for (i) we can apply
Theorem 3.11 and for (ii) we can use (i) and [16, Theorem 2] since $M$ is
compactly generated by finite elements and $\tau_{i}$ is compact.
(iii) From Theorem 3.12 we know that $\tau_{i}=\tau_{o}$ is a Hausdorff
topology. By Lemma 2.4 (vi) the interval topology $\tau_{i}$ on $E$ is compact
iff $E$ is a complete lattice. ∎
In what follows we will need Corollary 4.5 of Lemma 4.4.
###### Lemma 4.4.
Let $E$ be an Archimedean atomic lattice effect algebra. Then
1. (i)
If $c,d\in E$ are compact elements with $c\leq d^{\prime}$ then $c\oplus d$ is
compact.
2. (ii)
If $u=\bigoplus G$, where $G$ is a $\oplus$-orthogonal system of atoms of $E$,
and $u$ is compact then $G$ is finite.
###### Proof.
(i) Let $c\oplus d\leq\bigvee D$. Let ${\cal E}=\\{F\subseteq D:F\ \mbox{is
finite}\\}$ be directed by set inclusion and let for every $F\in{\cal E}$ be
$x_{F}=\bigvee F$. Then $x_{F}\uparrow x=\bigvee D$. Since $c\leq\bigvee D$
and $d\leq\bigvee D$ there is a finite subset $F_{1}\subseteq D$ such that
$c\vee d\leq\bigvee F_{1}$. Therefore, for $F\geq F_{1}$, $x_{F}\ominus
c\uparrow x\ominus c,d\leq x\ominus c$. Then there is a finite subset
$F_{2}\subseteq D$, $F_{1}\leq F_{2}$ such that $d\leq x_{F_{2}}\ominus c$.
Hence $c\oplus d\leq x_{F_{2}}$.
(ii) Let $u\in E$, $u=\bigoplus G=\bigvee\\{\bigoplus K\mid K\subseteq G$ is
finite$\\}$ where $G=(a_{\kappa})_{\kappa\in H}$ is a $\oplus$-orthogonal
system of atoms. Clearly if $K_{1},K_{2}\subseteq G$ are finite such that
$K_{1}\subseteq K_{2}$ then $\bigoplus K_{1}\leq\bigoplus K_{2}$.
Assume that $u$ is compact. Hence there are finite $K_{1},K_{2},$
$\dots,K_{n}\subseteq G$ such that $u\leq\bigvee\\{\bigoplus K_{i}\mid
i=1,2,\dots,n\\}$. Let $K_{0}=\bigcup\\{K_{i}\mid i=1,2,\dots,n\\}$. Then
$K_{0}\subseteq G$, $K_{0}$ is finite and $\bigoplus K_{i}\leq\bigoplus
K_{0}$, $i=1,2\dots,n$, which gives that $\bigvee\\{\bigoplus K_{i}\mid
i=1,2,\dots,n\\}\leq\bigoplus K_{0}$. It follows that $u\leq\bigoplus
K_{0}\leq u=\bigvee\\{\bigoplus K\mid K\subseteq G$ is finite$\\}$. Hence
$u=\bigoplus K_{0}$, $K_{0}\subseteq G$ is finite. Further, for every finite
$K\subseteq G\setminus K_{0}$ we have $\bigoplus
K_{0}\subseteq\bigoplus(K_{0}\cup K)=\bigoplus K_{0}\oplus\bigoplus K\leq
u=\bigoplus K_{0}$ , which gives that $\bigoplus K=0$. Hence $K=\emptyset$ and
thus $G\setminus K_{0}=\emptyset$ which gives that $K_{0}=G$.∎
###### Corollary 4.5.
Let $E$ be an o-continuous Archimedean atomic lattice effect algebra. Then
every finite element of $E$ is compact.
###### Proof.
Clearly, by [16, Theorem 7] we know that $E$ is compactly generated.
Therefore, any atom of $E$ is compact. The compactness of every finite element
follows by an easy induction. ∎
###### Theorem 4.6.
Let $E$ be an Archimedean atomic lattice effect algebra. Then the following
conditions are equivalent:
1. (i)
$\tau_{i}=\tau_{o}=\tau_{\Phi}$.
2. (ii)
$E$ is o-continuous and $\tau_{i}$ is Hausdorff.
3. (iii)
$E$ is almost orthogonal.
###### Proof.
(i)$\implies$(ii): Since $\tau_{o}=\tau_{\Phi}$ we have by [16, Theorem 1]
that $E$ is compactly generated and hence o-continuous. The condition
$\tau_{i}=\tau_{\Phi}$ implies that $\tau_{i}$ is Hausdorff because
$\tau_{\Phi}$ is Hausdorff.
(ii)$\implies$(i), (iii): Since $\tau_{i}$ is Hausdorff we obtain
$\tau_{i}=\tau_{o}$ by [4]. Moreover, from [16, Theorem 7] and Corollary 4.5
the (o)-continuity of $E$ implies that $E$ is compactly generated by the
elements from ${\mathcal{U}}$. This gives $\tau_{o}=\tau_{\Phi}$ from [16,
Theorem 1].
Let $a\in E$ be an atom, $1\leq l\leq n_{a}$. Then the interval
$[0,(la)^{\prime}]$ is a clopen set in the order topology
$\tau_{o}=\tau_{\Phi}=\tau_{i}$. Hence there is a finite set of intervals in
$E$ such that $0\in
E\setminus\bigcup_{i=1}^{n}[u_{i},v_{i}]\subseteq[0,(la)^{\prime}]$. Thus
$E\subseteq[0,(la)^{\prime}]\cup\bigcup_{i=1}^{n}[u_{i},v_{i}]\subseteq[0,(la)^{\prime}]\cup\bigcup_{i=1}^{n}[k_{i}{}b_{i},1]$,
where $b_{i}\in E$ are atoms such that $k_{i}{}b_{i}\leq u_{i}$, $1\leq
k_{i}\leq n_{b_{i}}$, $i=1,\dots,n$. This yields that $E$ is almost
orthogonal.
(iii)$\implies$(ii): From Theorems 3.8 and 3.9 we have that $\tau_{i}$ is
Hausdorff and $E$ is compactly generated, hence (o)-continuous. ∎
###### Corollary 4.7.
Let $E$ be a complete atomic lattice effect algebra. Then the following
conditions are equivalent:
1. (i)
$E$ is almost orthogonal.
2. (ii)
$\tau_{i}=\tau_{o}=\tau_{\Phi}\equiv(o)$.
3. (iii)
$E$ is (o)-continuous and $\tau_{i}$ is Hausdorff.
###### Proof.
It follows from Theorems 4.6 and the fact that by (o)-continuity of $E$ [27,
Theorem 8] we have $\tau_{o}\equiv(o)$. ∎
The next example shows that a complete block-finite atomic lattice effect
algebra need not be (o)-continuous and almost orthogonal in spite of that
$\tau_{i}=\tau_{o}$ is a compact Hausdorff topology.
###### Example 4.8.
Let $E$ be a horizontal sum of finitely many infinite complete atomic Boolean
algebras $(B_{i},\bigoplus_{i},0_{i},1_{i})$, $i=1,2,...,n$. Then $E$ is an
atomic complete lattice effect algebra, $E$ is not almost orthogonal, $E$ is
not compactly generated by finite elements (hence $\tau_{o}\not=\tau_{\Phi}$),
$E$ is block-finite, $\tau_{i}=\tau_{o}$ is Hausdorff by Theorem 3.12, and the
interval topology $\tau_{i}$ on $E$ is compact.
## 5 Applications
###### Theorem 5.1.
Let $E$ be a block-finite complete atomic lattice effect algebra. Then the
following conditions are equivalent:
1. (i)
$E$ is almost orthogonal.
2. (ii)
$E$ is compactly generated.
3. (iii)
$E$ is (o)-continuous.
4. (iv)
$\tau_{i}=\tau_{o}=\tau_{\Phi}\equiv(o)$.
###### Proof.
By Theorem 3.12, $\tau_{i}=\tau_{o}$ is a Hausdorff topology. This by [16,
Theorem 7] gives that (ii) $\Longleftrightarrow$ (iii) and by Corollary 4.7 we
obtain that (i) $\Longleftrightarrow$ (iii) $\Longleftrightarrow$ (iv). ∎
In Theorem 5.1, the assumption that $E$ is atomic can not be omitted. For
instance, every non-atomic complete Boolean algebra is (o)-continuous but it
is not compactly generated, because in such a case $E$ must be atomic by [16,
Theorem 6].
###### Remark 5.2.
If a $\oplus$-operation on a lattice effect algebra $E$ is continuous with
respect to its interval topology $\tau_{i}$ meaning that $x_{\alpha}\leq
y_{\alpha}^{{}^{\prime}}$,
$x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ implies $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$, then
$\tau_{i}$ is Hausdorff (see [14]). Hence $\oplus$-operation on complete
(o)-continuous atomic lattice effect algebras which are not almost orthogonal
cannot be $\tau_{i}$-continuous, by [14] and Corollary 4.7.
###### Theorem 5.3.
Let $E$ be a block-finite complete atomic lattice effect algebra. Let
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(y_{\alpha})_{\alpha\in\mathcal{E}}$ be nets of elements of $E$ such that
$x_{\alpha}\leq y_{\alpha}^{{}^{\prime}}$ for all ${\alpha\in\mathcal{E}}$.
If $x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ then $x\leq y^{{}^{\prime}}$ and $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$. Moreover, $\tau_{i}=\tau_{o}$.
###### Proof.
Since, by Theorem 3.12, $\tau_{i}$ is Hausdorff, we obtain that
$\tau_{i}=\tau_{o}$ by [4]. Let $\\{M_{1},\dots,M_{n}\\}$ be the set of all
blocks of $E$. Further, for every ${\alpha\in\mathcal{E}}$, elements of the
set $\\{x_{\alpha},y_{\alpha},x_{\alpha}\oplus y_{\alpha}\\}$ are pairwise
compatible. It follows that for every ${\alpha\in\mathcal{E}}$ there exists a
block $M_{k_{\alpha}}$ of $E$, ${k_{\alpha}}\in\\{1,\dots,n\\}$ such that
$\\{x_{\alpha},y_{\alpha},x_{\alpha}\oplus y_{\alpha}\\}\subseteq
M_{k_{\alpha}}$. Let ${\mathcal{E}}^{\prime}$ be any cofinal subset of
${\mathcal{E}}$. Since ${\mathcal{E}}^{\prime}$ is directed upwards, there
exists a block $M_{k_{0}}$ of $E$ and a cofinal subset
${\mathcal{E}}^{\prime\prime}$ of ${\mathcal{E}}^{\prime}$ such that
$\\{x_{\beta},y_{\alpha},x_{\beta}\oplus y_{\beta}\\}\subseteq M_{k_{0}}$ for
all $\beta\in{\mathcal{E}}^{\prime\prime}$. Otherwise we obtain a
contradiction with the finiteness of the set $\\{M_{1},\dots,M_{n}\\}$.
Further, by Theorem 2.5, we obtain that $\tau^{M_{k_{0}}}_{i}=\tau_{i}\cap
M_{k_{0}}$, as $M_{k_{0}}$ is a complete sublattice of $E$ (see Theorem 4.2).
It follows that the interval topology $\tau^{M_{k_{0}}}_{i}$ on the complete
MV-effect algebra $M_{k_{0}}$ is Hausdorff. The last by [14, Theorem 3.6]
gives that $x_{\beta}\oplus
y_{\beta}\stackrel{{\scriptstyle\tau^{M_{k_{0}}}_{i}}}{{\rightarrow}}x\oplus
y$, $\beta\in\mathcal{E}^{\prime\prime}$ and hence $x_{\beta}\oplus
y_{\beta}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\beta\in\mathcal{E}^{\prime\prime}$, as $\tau^{M_{k_{0}}}_{i}=\tau_{i}\cap
M_{k_{0}}$. It follows that $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$ by Lemma 2.3. ∎
In [22, Theorem 4.5] it was proved that a block-finite lattice effect algebra
$(E;\oplus,0,1)$ has a MacNeille completion which is a complete effect algebra
$(MC(E);\oplus,0,1)$ containing $E$ as a (join-dense and meet-dense) sub-
lattice effect algebra iff $E$ is Archimedean. In what follows we put
$\widehat{E}=MC(E)$.
###### Corollary 5.4.
Let $E$ be a block-finite Archimedean atomic lattice effect algebra. Then for
any nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(y_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $E$ with $x_{\alpha}\leq
y_{\alpha}^{{}^{\prime}}$, ${\alpha\in\mathcal{E}}$:
$x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ implies $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$.
###### Proof.
By [20, Lemma 1.1], for interval topologies $\widehat{\tau}_{i}$ on
$\widehat{E}$ and ${\tau_{i}}$ on $E$, we have $\widehat{\tau_{i}}\cap
E={\tau_{i}}$. Thus for $x_{\alpha},y_{\alpha},x,y\in E$ we obtain
$x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\widehat{\tau}_{i}}}{{\rightarrow}}x\oplus
y$, $\alpha\in\mathcal{E}$ which gives $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle{\tau}_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$ by the fact that $\widehat{\tau_{i}}\cap E={\tau_{i}}$.
∎
###### Definition 5.5.
Let $E$ be a lattice.Then
1. (i)
An element $u$ of $E$ is called strongly compact (briefly s-compact) iff, for
any $D\subseteq E$: $u\leq c\in E$ for all $c\geq D$ implies $u\leq\bigvee F$
for some finite $F\subseteq D$.
2. (ii)
$E$ is called s-compactly generated iff every element of $E$ is a join of
s-compact elements.
###### Theorem 5.6.
Let $E$ be a block-finite Archimedean atomic lattice effect algebra. Then the
following conditions are equivalent:
1. (i)
$E$ is almost orthogonal.
2. (ii)
$\widehat{E}=MC(E)$ is almost orthogonal.
3. (iii)
$\widehat{E}=MC(E)$ is compactly generated.
4. (iv)
${E}$ is s-compactly generated.
###### Proof.
By J. Schmidt [30] a MacNeille completion $\widehat{E}$ of $E$ is (up to
isomorphism) a complete lattice such that for every element $x\in\widehat{E}$
there exist $P,Q\subseteq E$ such that
$x=\bigvee_{\widehat{E}}P=\bigwedge_{\widehat{E}}Q$ (taken in
${}_{\widehat{E}}$). Here we identify $E$ with $\varphi(E)$, where
$\varphi:E\to{\widehat{E}}$ is the embedding (meaning that $E$ and
$\varphi(E)$ are isomorphic lattice effect algebras). It follows that $E$ and
${\widehat{E}}$ have the same set of all atoms and coatoms and hence also the
same set of all finite and cofinite elements, which implies that (i)
$\Longleftrightarrow$ (ii).
Moreover, for any $A\subseteq E$ and $u\in E$, we have ($d\in E$, $A\leq d$
implies $u\leq d$) iff $u\leq\bigvee_{\widehat{E}}A$. Then $u$ is s-compact in
$E$ iff $u$ is compact in $\widehat{E}$, which gives (iii)
$\Longleftrightarrow$ (iv).
Finally (ii) $\Longleftrightarrow$ (iii) by Theorem 5.1. ∎
###### Definition 5.7.
Let $E$ be an effect algebra. A map $\omega:E\to[0,1]$ is called a state on
$E$ if $\omega(0)=0$, $\omega(1)=1$ and $\omega(x\oplus
y)=\omega(x)+\omega(y)$ whenever $x\oplus y$ exists in $E$.
###### Theorem 5.8.
(State smearing theorem for almost orthogonal block-finite Archimedean atomic
lattice effect algebras) Let $(E;\oplus,0,1)$ be a block-finite Archimedean
atomic lattice effect algebra. If $E$ is almost orthogonal then:
1. (i)
$E_{1}=\\{x\in E\mid x\ \mbox{or}\ x^{\prime}\ \mbox{is finite}\\}$ is a sub-
lattice effect algebra of $E$.
2. (ii)
If there exists an $(o)$-continuous state $\omega$ on $E_{1}$ (or on
$S(E_{1})=S(E)\cap E_{1}$, or on $S(E)$) then there exists an $(o)$-continuous
state $\widetilde{\omega}$ on $E$ extending $\omega$ and an $(o)$-continuous
state $\widehat{\omega}$ on $\widehat{E}=MC(E)=MC(E_{1})$ extending
$\widetilde{\omega}$.
###### Proof.
(i) By Theorem 5.6, ${E}$ is s-compactly generated and thus by [28, Theorem
2.7] $E_{1}$ is a sub-lattice effect algebra of $E$.
(ii) Since ${E}$ is s-compactly generated, we obtain the existence of
$(o)$-continuous extensions $\widetilde{\omega}$ on $E$ and $\widehat{\omega}$
on $\widehat{E}$ by [28, Theorem 4.2]. ∎
## References
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|
arxiv-papers
| 2009-08-24T00:36:21 |
2024-09-04T02:49:04.725770
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Paseka, Zdenka Riecanova, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0908.3288"
}
|
0908.3294
|
# Sensitivity of rocky planet structures to the equation of state
Damian C. Swift PLS-CMMD, Lawrence Livermore National Laboratory, 7000 East
Avenue, Livermore, California 94550, USA
(June 10, 2009 – LLNL-TR-414058)
###### Abstract
Structures were calculated for Mercury, Venus, Earth, the Moon, and Mars,
using a core-mantle model and adjusting the core radius to reproduce the
observed mass and diameter of each body. Structures were calculated using Fe
and basalt equations of state of different degrees of sophistication for the
core and mantle. The choice of equation of state had a significant effect on
the inferred structure. For each structure, the moment of inertia ratio was
calculated and compared with observed values. Linear Grüneisen equations of
state fitted to limited portions of shock data reproduced the observed moments
of inertia significantly better than did more detailed equations of state
incorporating phase transitions, presumably reflecting the actual compositions
of the bodies. The linear Grüneisen equations of state and corresponding
structures seem however to be a reasonable starting point for comparative
simulations of large-scale astrophysical impacts.
## I Introduction
The pressure-temperature-compression equation of state (EOS) of condensed
matter is vital in understanding planetary structures, and the response of
astrophysical bodies to impacts. Even for Earth, it is necessary to infer the
internal structure of the planet from limited data. The situation is much more
difficult for other bodies, where seismic data is at best extremely limited
and more usually absent.
For planetary structures, we are usually interested in pressures from zero to
a few hundred gigapascals. In this range, EOS were originally deduced
empirically from shock wave measurements shock . Shock experiments typically
explore states along the principal Hugoniot of a material, which may be
significantly hotter than the corresponding compression or pressure occurring
in a self-gravitating body, and corrections are made to estimate the pressure
away from the Hugoniot shock . More recently, mechanical presses such as
diamond anvil cells have provided a source for isothermal compression in this
regime, although the pressure calibration is made with respect to shock
measurements on reference materials press . Samples may also be heated within
presses, allowing a wide range of states to be explored. In parallel with the
evolution of experimental methods, theoretical techniques have been developed
to predict the EOS from first principles, typically with an a priori accuracy
of a few percent in mass density, and better if adjusted to reproduce the STP
state Swift_SiEOS_2001 . Theoretical EOS are particularly useful as
thermodynamically complete states can be calculated for arbitrary loading
conditions that may be experimentally difficult either to induce or to
measure. Theoretical EOS can also be constructed fairly readily for different
material compositions Swift_NiAlEOS_2007 . The prediction of phase diagrams is
however typically less certain than the EOS for a single, known phase.
The strength, failure, and plastic flow behaviors of materials are also
important in planetary physics. The effects of strength must be taken into
account when deducing the EOS from dynamic loading measurements, as these are
at high strain rates and non-hydrostatic conditions in contrast to the
quasistatic conditions prevailing within self-gravitating bodies. Conversely,
these properties are important for understanding dynamical processes including
the formation of the body, the effect of impacts, cooling, volcanism, mantle
convection, and plate tectonics. These constitutive properties are very
challenging to predict theoretically, and models rely heavily on experimental
measurements. Recent studies have demonstrated the importance of strength in
the heating induced by shocks and other dynamic loading Swift_NRS_2008 .
Strength and flow stress can also change greatly with pressure and
temperature. Recent advances in the measurement and modeling of plastic flow
under extreme conditions has implications for astrophysical impacts, such as
the interpretation of previous simulation studies using simpler material
models. An example is the widely-quoted study that a liquid interior allow
impact energy to be transported more efficiently to the antipode of an impact
than would solid components Hughes1977 . This conclusion relies strongly on
the assumptions used for material strength. If the flow stress increases
sufficiently with compression, solid components can transport compression
waves more efficiently than can liquids.
We are constructing theoretical EOS for planetary materials of different
composition, for use in simulations of astrophysical impacts, focusing on
systematic differences in the shape and location of impact-produced structures
between the rocky bodies. As a baseline for trial simulations, we report here
the construction of structure models for the rocky planets using existing EOS
of different sophistication. There is significant uncertainty over the
composition and temperature profile of all planetary bodies; here we use EOS
for representative substances rather than for the current best estimates of
composition, because a wider variety of EOS of different degrees of
sophistication exist for our chosen materials. We optimize a single structure
parameter – the core radius – for each combination of EOS. This approach is
different from the commonly-used solution of the Emden equation Lyttleton1965
; Lyttleton1969 where the compressibility of the planetary material is
assumed to vary linearly with pressure, and the parameters in this relation –
i.e., the material properties – are deduced from the observable properties of
the planet.
## II Isostatic structures
Consider a spherical body comprising compressible material, of mass density
$\rho(r)$. Knowing or assuming in addition the distribution of composition and
temperature, $\rho(r)$ implies a pressure distribution $p(r)$. The condition
for isostatic equilibrium is that the stress induced by pressure variations is
balanced by the gravitational acceleration $g(r)$ and centripetal force:
$\mbox{grad}\,p(r)=\omega^{2}-\rho(r)g(r)$ (1)
where $\omega$ is the angular speed.
For Newtonian gravitation, by Gauss’ theorem, $g(r)$ can be expressed in terms
of the mass enclosed within a given radius $m(r)$:
$g(r)=\frac{Gm(r)}{r^{2}}$ (2)
so
$\frac{dg(r)}{dr}=G\left(\frac{1}{r^{2}}\frac{dm(r)}{dr}-2\frac{m(r)}{r^{3}}\right).$
(3)
$m(r)$ can be calculated simply from the distribution of mass density,
$m(r)=4\pi\int_{0}^{r^{\prime}}r^{\prime 2}\rho(r^{\prime})\,dr^{\prime}$ (4)
or
$\frac{dm(r)}{dr}=4\pi r^{2}\rho(r).$ (5)
In the isostatic equilibrium structure, Eq. 5 can be integrated from either
the center or the surface $r=R$ to obtain the complete structure. In practice,
the material state at the center depends on the material models used, so it is
more useful to integrate from the surface, where $p=0$ and $\rho$ can be
worked out simply for the outermost material.
Usually, the total mass $M$ and outer radius $R$ of the body are known fairly
well from observation of orbits and size respectively, and the problem is to
establish the internal structure. Starting with the assumption of a layered
structure, i.e. an ordering of composition, and the $p(\rho)$ relation for
each layer, any unknown parameter in the structure can be fitted to give the
desired total mass.
For example, consider a layered structure where the radius of one layer is
unknown. For a stable configuration, inner layers have a higher density.
Consider integration downward from the surface as above, assuming a trial
value for the unknown radius. If too large, the total mass $M$ will be reached
at some $r>0$; if too small, $m(0)>0$. These two conditions can be used for a
solution of this shooting problem by bracketing followed by bisection.
Another remotely observable structural parameter is the moment of inertia $I$,
which can be inferred from the rate of precession of the rotational pole about
the normal to the orbit Gubbins2007 . In the absence of seismic data, $I$ is a
useful constraint on the structure of the body. The moment of inertia for a
sphere is
$I=\frac{8\pi}{3}\int_{0}^{R}\rho(r)r^{4}\,dr,$ (6)
and for self-gravitating bodies is conveniently expressed as the dimensionless
moment of inertia ratio, $I/MR^{2}$.
Similar approaches assuming the EOS and deducing isostatic planetary
structures have been reported previously. Kerley has calculated structures for
Jupiter and Saturn, including rotational flattening, using the theory of
figures and very detailed EOS Kerley2004 . Seager et al have predicted mass-
radius relationships for solid exoplanets, using EOS which are less suited to
subsequent impact studies and a different solution approach involving
integration outward from the center of the planet Seager2007 . Grasset et al
have similarly predicted mass-radius relationships for rocky and icy
exoplanets, again using EOS which are less suited to impact studies
Grasset2009 .
## III Compressibility curves for different equations of state
Two types of EOS were considered: linear Grüneisen EOS using a representative
part of the principal shock Hugoniot as reference, and more elaborate EOS
constructions incorporating phase transitions and reproducing detailed
features of the shock Hugoniot. The shock Hugoniot data used to calibrate the
Grüneisen EOS are reliable in the sense of being absolute, mechanical
measurements, but require the additional assumption of a thermal correction
model to calculate states off the principal Hugoniot, as are required here.
There is considerable uncertainty in the composition of planetary interiors,
and detailed EOS are not available for all compositions of interest. In order
to calculate structures for all the rocky planets without introducing
additional degrees of freedom in the individual compositions, the compositions
were taken to be pure Fe in the core, and basalt in the mantle. At zero
pressure, basalt typically comprises $\sim$70% plagioclase (often andesite)
and $\sim$30% olivine, and is thus reasonably representative of the
composition thought to occur through the mantle. Further subdivision of layers
was not considered; the structure chosen allowed the core radius to be used as
a parameter to ensure the correct total mass for each body. Similar structure
calculations using detailed EOS for other compositions will be reported
separately.
For both Fe and basalt, the shock speed-particle speed relation does not
follow a straight line over the full pressure range of interest. For basalt, a
straight line was chosen that reproduced the principal Hugoniot at pressures
of a few tens of gigapascals vanThiel1966 , as a representative range for
planetary mantles. Similarly, the fit to Fe shock data was for pressures
$\sim$50-500 GPa vanThiel1966 . The more detailed EOS for basalt was a
tabulated Grüneisen form using three piecewise linear fits to the principal
Hugoniot Barnes-Lyon1988 . The detailed Fe EOS was an equilibrium construction
comprising four solid phases and the fluid, with mixed-phase regions
Kerley1993 , and should be much more accurate away from the principal
Hugoniot.
In the absence of accurately-known composition profiles, the use of simple EOS
is desirable as it allows planetary structures to be optimized via the EOS
parameters, though this optimization was not done here. It is much more
cumbersome to optimize detailed EOS in this was, particularly when presented
in tabular form.
Table 1: Linear Grüneisen equation of state parameters. | Fe | basalt
---|---|---
$\rho_{0}$ (g/cm3) | 7.86 | 2.67
$c_{0}$ (km/s) | 3.635 | 1.45
$s_{1}$ | 1.802 | 1.97
$\Gamma$ | 2.604 | 1.5
The compression curve used was the principal isentrope. This is a reasonable
approach for large, self-gravitating objects. One would expect $p(\rho)$ to be
close to an isentrope on initial formation, as accretion proceeds by multiple
small impacts followed by some stratification as dense components sink toward
the center of the body. After formation, the decay of radionuclides (and
possibly other processes such as tidal deformation and ohmic heating) may
supply additional heat, but the surface of the body cools by radiation. In the
adiabatic compression of non-porous condensed matter, the thermal contribution
to pressure is usually much smaller than the repulsion between atoms, so the
difference between the isentrope and the isotherm as extreme limits is small
compared with the other uncertainties in planetary structure such as the
composition.
Each EOS was used in the form $p(\rho,e)$, and integrated numerically starting
at STP conditions to obtain $p(\rho)$ along the principal isentrope. The use
of the principal isentrope for each constituent gives a discontinuity in
temperature where different layers meet, but the effect on pressure is small
compared with the uncertainties in composition and temperature. It would be
straightforward, though less efficient, to calculate the isentrope starting at
the temperature outside each layer. The isentropes for Fe agreed quite closely
up to $\sim$120 GPa; at higher pressures, the linear Grüneisen EOS was
significantly stiffer. The linear Grüneisen EOS for basalt was much softer
than the tabular EOS at low pressures, and much stiffer at high pressures,
with a crossover at 100 GPa. (Fig. 1.)
Figure 1: Comparison between isentropes.
## IV Planetary structures
The planetary structure equation, Eq. 5, was integrated numerically using the
4th order Runge-Kutta algorithm with a constant step size in radius. The
effect of rotation was ignored as it depends on latitude; rotation found to be
a negligible effect in a trial equatorial integration. The core radius was
adjusted to give a density profile consistent with the specified total mass of
the body. The step size was chosen such that core radius was converged
numerically to 1 km. The resulting moment of inertia ratio was converged to
0.1%, which is better than measured values. The density near the center of the
body was very sensitive to small variations in the core radius or any other
parameters, often giving unphysical variations for the innermost couple of
tens of kilometers. The volume and mass affected were negligible in comparison
with the overall body.
Using the linear Grüneisen EOS for core and mantle, the numerical scheme was
very robust, and the core radius converged reliably for very wide ranges of
the initial bracket. Using the tabular EOS, the initial bracket had to be
chosen much more carefully for a physical core radius to be found, because of
the increased compressibility of the core. compressibility of core. Converged
solutions were however found for all the bodies considered and compared with
observations of the moment of inertia ratio Yoder1995 and the usually-quoted
values of the core radius (Table 2). We also summarize pressures calculated at
the key locations of the mantle-core boundary and the center (Table 3) as an
indication of the regimes desirable to explore experimentally, for each body.
Table 2: Structure parameters for rocky bodies. body | reference | linear | tabular
---|---|---|---
| | Grüneisen |
| $r_{c}$ | $I/MR^{2}$ | $r_{c}$ | $I/MR^{2}$ | $r_{c}$ | $I/MR^{2}$
| (km) | | (km) | | (km) |
Mercury | 1600 | 0.33 | 1740 | 0.3196 | 1802 | 0.3140
Venus | 3000 | 0.33 | 3014 | 0.3348 | 3167 | 0.3146
Earth | 3481 | 0.33 | 3361 | 0.3290 | 3405 | 0.3118
Moon | 350 | 0.393 | 620 | 0.3647 | 752 | 0.3572
Mars | 1700$\pm$500 | 0.366 | 1188 | 0.3621 | 1706 | 0.3316
Table 3: Key pressures in rocky bodies. body | linear Grüneisen | tabular
---|---|---
| center | boundary | center | boundary
| (GPa) | (GPa) | (GPa) | (GPa)
Mercury | 46.5 | 9.6 | 47.5 | 9.6
Venus | 318 | 121 | 365 | 120
Earth | 390 | 137 | 448 | 143
Moon | 8.8 | 5.0 | 10.0 | 4.4
Mars | 45.2 | 27.2 | 60.0 | 20.7
Given that the EOS used were not for the most likely individual core
compositions that have been postulated (Fe-Ni, or Fe-S for Mars), and the
planetary structures were simplified, it is interesting to note that the core
radius inferred for Earth – the only body for which it can be estimated using
multiple techniques including seismology – is too small by only 2% for the
tabular EOS and 3% for the linear Grüneisen. The core radius for the other
bodies did not agree very well with the nominal values in the literature, with
the exception of Venus, but the difference was commensurate with the
uncertainty in radius. The exceptions were Venus, where the agreement was good
at 0.5-2% (the linear Grüneisen EOS being the better), and the Moon, where the
present calculations were larger by a factor $\sim$2.
For the linear Grüneisen EOS, the moment of inertia ratios were within 3%
except for the Moon, for which it was too small by 7%. For the tabular EOS,
the ratio was systematically smaller: 4-5% except for the Moon and Mars (9 and
10% respectively). Too small a ratio implies that the body is stiffer than the
EOS used in the calculation, in the sense that $p$ would be greater for a
given $\rho$. Such an increase in stiffness could reflect a difference in
composition rather than too low a stiffness for the material used. Thus the
discrepancy for Mars could be attributed to a significant proportion of S,
reducing the mass density for a given compression.
The linear Grüneisen EOS gives pressures at the center of the Earth that are
within the range of values in the literature (360-390 GPa).
We next compare in more detail the structures predicted for each body by the
different EOS.
### IV.1 Mercury
For Mercury, the EOS gave very similar pressure distributions. The core radius
from each EOS was similar. The density distribution differed somewhat,
particularly in the mantle. The pressure to induce the $\alpha$-$\epsilon$
phase change occurred just within the core, and showed up clearly in the
density distribution. The gravitational acceleration from each EOS was
similar. At the level of detail considered here, the EOS were equivalent.
(Figs 2 to 4.)
Figure 2: Radial pressure distribution in Mercury calculated using different
equations of state. Figure 3: Radial density distribution in Mercury
calculated using different equations of state. Figure 4: Radial gravity
distribution in Mercury calculated using different equations of state.
### IV.2 Venus
For Venus, the EOS gave very similar pressure distributions in the mantle, and
deviated monotonically and significantly in the core. The core radius from
each EOS was similar. The density distribution differed somewhat. The core
pressure for both EOS was well above the $\alpha$-$\epsilon$ phase transition.
The gravitational acceleration from each EOS was similar within the core, but
differed somewhat in the mantle. At the level of detail considered here, the
EOS were not equivalent but conversely not hugely different. (Figs 5 to 7.)
Figure 5: Radial pressure distribution in Venus calculated using different
equations of state. Figure 6: Radial density distribution in Venus calculated
using different equations of state. Figure 7: Radial gravity distribution in
Venus calculated using different equations of state.
### IV.3 Earth
The results for Earth were very similar to those for Venus, as one might
expect for bodies of similar mass and volume. The EOS gave similar pressure
distributions in the mantle, and deviated monotonically and significantly in
the core. The core radius from each EOS was very similar. The density
distribution differed somewhat. The core pressure for both EOS was well above
the $\alpha$-$\epsilon$ phase transition. The gravitational acceleration from
each EOS differed somewhat over the whole profile (except for the ends, which
are constrained to be equal), although the shapes were very similar. At the
level of detail considered here, the EOS were not equivalent but not hugely
different. (Figs 8 to 10.)
Figure 8: Radial pressure distribution in Earth calculated using different
equations of state. Figure 9: Radial density distribution in Earth calculated
using different equations of state. Figure 10: Radial gravity distribution in
Earth calculated using different equations of state.
### IV.4 Moon
For the Moon, the EOS gave very similar pressure distributions in the mantle,
and deviated monotonically and significantly in the core. The core radii
differed significantly. The density distributions were very similar,
particularly in the core, except for the difference in core radius. differed
somewhat, particularly in the mantle. The core pressures were all below the
$\alpha$-$\epsilon$ phase transition. The gravitational acceleration from each
EOS was very similar in the core, but deviated substantially where the core
radii differed and in the inner mantle. At the level of detail considered
here, the EOS were fairly equivalent. (Figs 11 to 13.)
Figure 11: Radial pressure distribution in the Moon calculated using different
equations of state. Figure 12: Radial density distribution in the Moon
calculated using different equations of state. Figure 13: Radial gravity
distribution in the Moon calculated using different equations of state.
### IV.5 Mars
For Mars, the EOS gave similar pressure distributions in the mantle, and
deviated greatly in the core. The core radii differed significantly. The
density distributions were significantly different, but mostly because of the
difference in core radius. The core pressures were all above the
$\alpha$-$\epsilon$ phase transition. The gravitational acceleration from each
EOS was very similar in the core, but deviated substantially where the core
radii differed and in the mantle. For simulation purposes, the EOS not
equivalent. (Figs 14 to 16.)
Figure 14: Radial pressure distribution in Mars calculated using different
equations of state. Figure 15: Radial density distribution in Mars calculated
using different equations of state. Figure 16: Radial gravity distribution in
Mars calculated using different equations of state.
## V Conclusions
The isostatic equations were used to calculate density distributions for
Mercury, Venus, Earth, the Moon, and Mars, assuming different EOS for the
structural components, and optimizing the core radius to constrain the overall
mass to be correct. Two pairs of EOS were considered for Fe and basalt, either
linear Grüneisen fits to subsets of published shock data, or more detailed EOS
constructions including phase changes and nonlinearities in the shock data.
The core radius of Earth, which is by far the best known, was reproduced
reasonably well using both types of EOS. The other radii, and structures for
all the bodies, varied significantly between the EOS, though to a varying
degree. The structures were used to calculate the moment of inertia ratio for
each body, as an independent test of the accuracy of each structure.
Better overall agreement with the observed properties of the rocky bodies
considered was obtained with the linear Grüneisen EOS than the more carefully
constructed tabular EOS. This does not imply that the linear Grüneisen EOS are
more accurate models of Fe and basalt, but that the properties of the bodies
are represented more closely by the simpler EOS, presumably because the
compositions deviate significantly from Fe and basalt in the core and mantle
respectively. The simpler EOS do not reproduce the moment of inertia ratios
perfectly, and deviate particularly for the Moon and Mars.
The linear Grüneisen EOS seem to be a reasonable starting point for
simulations of large-scale impacts on these bodies, more so than the tabular
EOS. These results also suggest that more detailed EOS for the postulated
compositions of the bodies can be evaluated to some degree by their
improvement over Fe and basalt in calculating the moment of inertia ratios.
## References
* (1) For instance, R.G. McQueen, S.P. Marsh, T.W. Taylor, J.N. Fritz, and W.J. Carter, in R. Kinslow (Ed.), “High Velocity Impact Phenomena” (Academic Press, New York, 1970).
* (2) For instance, A. Dewaele, P. Loubeyre, F. Occelli, M. Mezouar, P.I. Dorogokupets, and M. Torrent, Phys. Rev. Lett. 97, 215504 (2006).
* (3) For instance, D.C. Swift, G.J. Ackland, A. Hauer, and G.A. Kyrala, Phys. Rev. B 64, 214107 (2001).
* (4) For instance, D.C. Swift, D.L. Paisley, K.J. McClellan, and G.J. Ackland, Phys. Rev. B 76, 134111 (2007).
* (5) D.C. Swift, A. Seifter, D.B. Holtkamp, V.W. Yuan, D. Bowman, and D.A. Clark, Phys. Rev. B 77, 092102 (2008).
* (6) H.G. Hughes, F.N. App, and T.R. McGetchin, Proc. Conf. on Comparisons of Mercury and the Moon, Houston, TX, 15-17 Nov. 1976 (Lunar Science Institute, 1977).
* (7) R.A. Lyttleton, Mon. Not. Roy. Astr. Soc. 129, 21 (1965).
* (8) R.A. Lyttleton, Astrophys. Space Sci. 5, 18 (1969).
* (9) D. Gubbins and E. Herrero-Bervera, “Encyclopedia of geomagnetism and paleomagnetism” (Springer, New York, 2007).
* (10) G.I. Kerley, Structures of the Planets Jupiter and Saturn, report KTS04-1 (Kerley Technical Services, Appomattox, 2004).
* (11) S. Seager, M. Kuchner, C.A. Hier-Majumder, and B. Militzer, ApJ 669, pp 1279-1297 (2007).
* (12) O. Grasset, J. Schneider, and C. Sotin, ApJ 693, pp 722-733 (2009).
* (13) M. van Thiel, Compendium of Shock Wave Data, Lawrence Livermore National Laboratory report UCRL-50108 (1966).
* (14) J.F. Barnes and S.P. Lyon, SESAME Equation of State Number 7530, Basalt, Los Alamos National Laboratory report LA-11253-MS (1988).
* (15) G.I. Kerley, Multiphase Equation of State for Iron, Sandia National Laboratories report SAND93-0227 (1993).
* (16) C.F. Yoder in T.J. Ahrens (Ed.), “Global Earth Physics: A Handbook of Physical Constants” (American Geophysical Union, Washington DC, 1995).
|
arxiv-papers
| 2009-08-23T10:50:27 |
2024-09-04T02:49:04.733314
|
{
"license": "Public Domain",
"authors": "Damian C. Swift",
"submitter": "Damian Swift",
"url": "https://arxiv.org/abs/0908.3294"
}
|
0908.3350
|
# Order Topology and Frink Ideal Topology of Effect Algebras††thanks: This
project is supported by Natural Science Foundation of China (10771191 and
10471124) and Natural Science Foundation of Zhejiang Province of China
(Y6090105).
Qiang Lei Department of Mathematics, Harbin Institute of Technology, Harbin,
China; e-mail: leiqiang@hit.edu.cn Junde Wu Department of Mathematics,
Zhejiang University, Hangzhou, China; e-mail: wjd@zju.edu.cn Ronglu Li
Department of Mathematics, Harbin Institute of Technology, Harbin, China.
Abstract. In this paper, the following results are proved: (1) If $E$ is a
complete atomic lattice effect algebra, then $E$ is (o)-continuous iff $E$ is
order-topological iff $E$ is totally order-disconnected iff $E$ is algebraic.
(2) If $E$ is a complete atomic distributive lattice effect algebra, then its
Frink ideal topology $\tau_{id}$ is Hausdorff topology and $\tau_{id}$ is
finer than its order topology $\tau_{o}$, and $\tau_{id}=\tau_{o}$ iff $1$ is
finite iff every element of $E$ is finite iff $\tau_{id}$ and $\tau_{o}$ are
both discrete topologies. (3) If $E$ is a complete (o)-continuous lattice
effect algebra and the operation $\oplus$ is order topology $\tau_{o}$
continuous, then its order topology $\tau_{o}$ is Hausdorff topology. (4) If
$E$ is a (o)-continuous complete atomic lattice effect algebra, then $\oplus$
is order topology continuous.
Key words: Effect algebras, order topology, Frink ideal topology.
1\. Introduction
Effect algebra is an important model in studying the unsharp quantum logic
theory, it is also an important carrier of quantum states and quantum measures
([1]). As an important tool of studying the quantum states and quantum
measures, the topological structures of effect algebras not only can help us
to describe the convergence properties of quantum states and quantum measures,
but also can help us to characterize some algebra properties of effect
algebras. This paper contributes to the understanding of the topological
properties and algebraic properties of effect algebras, it both promotes some
classical results, for example, Theorem 2.1, and obtain several new
interesting conclusions, for example, Theorem 3.1, Theorem 4.1 and Theorem
4.3, etc. Now, we show them in details in the following three sections.
The structure $(E,\oplus,0,1)$ is said to be an effect algebra if 0, 1 are two
distinguished elements of $E$ and $\oplus$ is a partially defined binary
operation on $E$ which satisfies the following conditions for any $a,b,c\in E$
([1]):
(1) If $a\oplus b$ is defined, then $b\oplus a$ is defined and $a\oplus
b=b\oplus a$.
(2) If $a\oplus b$ and $(a\oplus b)\oplus c$ are defined, then $b\oplus c$ and
$a\oplus(b\oplus c)$ are defined and $(a\oplus b)\oplus c=a\oplus(b\oplus c)$.
(3) For each $a\in E$ there exists a unique $b\in E$ such that $a\oplus b$ is
defined and $a\oplus b=1$.
(4) If $a\oplus 1$ is defined, then $a=0$.
The effect algebra $(E,\oplus,0,1)$ is often denoted by $E$. For every $a\in
E$, we denote the unique $b$ in condition (3) by $a^{\prime}$ and call it the
orthosupplement of $a$. The sense is that if $a$ presents a proposition, then
$a^{\prime}$ corresponds to its negation. The operation $\oplus$ of an effect
algebra $E$ can induce a new partial operation $\ominus$ and a partial order
$\leq$ as follows: $a\ominus b$ is defined iff there exists $c\in E$ such that
$b\oplus c$ is defined and $b\oplus c=a$, in which case we denote $c$ by
$a\ominus b$; $d\leq e$ iff there exists $f\in E$ such that $d\oplus f$ is
defined and $d\oplus f=e$. This showed that every effect algebra is a partial
order set. If $(E,\leq)$ is a lattice, then $E$ is called a lattice effect
algebra, similarly, we can define the complete lattice effect algebras. If a
lattice effect algebra is a distributive lattice, then it is called a
distributive lattice effect algebra. For more details on effect algebras, for
example, $a\oplus b$ is defined iff $a\leq b^{\prime}$, we refer to [1].
Let $E$ be an effect algebra and $a,b\in E$ with $a\leq b$, denote
$[a,b]=\\{p\in E|a\leq p\leq b\\}$. A nonzero element $a\in E$ is said to be
an atom of $E$ if $[0,a]=\\{0,a\\}$, $E$ is said to be atomic iff for every
nonzero element $p\in E$, there is an atom $a\in E$ such that $a\leq p$ ([1]).
An element $x\in E$ is said to be a sharp element of $E$ if $x\wedge
x^{\prime}=0$, that is, proposition $x$ and its negation $x^{\prime}$ have no
overlaps.
2\. The order-continuity and order-topological effect algebras
Assume that $P$ is a partial order set and
$(a_{\alpha})_{\alpha\in\varepsilon}$ is a net of $P$. If for any
$\alpha,\beta\in\varepsilon$, when $\alpha\preceq\beta$, $a_{\alpha}\leq
a_{\beta}$, then we denote $a_{\alpha}\uparrow$, moreover, if
$a=\bigvee\\{a_{\alpha}|\alpha\in\varepsilon\\}$, then we denote
$a_{\alpha}\uparrow a$. Dual, we have $a_{\alpha}\downarrow$ and
$a_{\alpha}\downarrow a$. A net $(a_{\alpha})_{\alpha\in\varepsilon}$ is said
to be order convergent ((o)-convergent, for short) to a point $a\in P$ if
there are nets $(u_{\alpha})_{\alpha\in\varepsilon}$ and
$(v_{\alpha})_{\alpha\in\varepsilon}$ of $P$ such that $a\uparrow
u_{\alpha}\leq a_{\alpha}\leq v_{\alpha}\downarrow a$, and we denote
$a_{\alpha}\xrightarrow{(o)}a$. If $\tau$ is a topology equipped on $P$ such
that every (o)-convergent net of $P$ is $\tau$\- convergent, then $\tau$ is
said to have $C$ property. The strongest topology of $P$ which has $C$
property is called the order topology of $P$ and denoted by $\tau_{o}$. It is
obvious that the (o)-convergence of nets implies $\tau_{o}$-convergence, but
the converse does not hold ([2]). The following Theorem 2.1 answer when they
are equal.
A lattice $L$ is said to be (o)-continuous if $x_{\alpha},x,y\in L$ and
$x_{\alpha}\uparrow x$ implies that $x_{\alpha}\wedge y\uparrow x\wedge y$.
It can be proved that if $E$ is an (o)-continuous lattice effect algebra and
$x_{\alpha}\xrightarrow{(o)}x$ and $y_{\alpha}\xrightarrow{(o)}y$, then
$x_{\alpha}\vee y_{\alpha}\xrightarrow{(o)}x\vee y$ and $x_{\alpha}\wedge
y_{\alpha}\xrightarrow{(o)}x\wedge y$ ([2]).
A complete lattice effect algebra $E$ is said to be order-topological
((o)-topological) if (o)-convergence of nets of elements coincides with
$\tau_{o}$-convergence and $E$ is (o)-continuous ([2]).
Lemma 2.1 ([2]). A complete atomic (o)-continuous lattice effect algebra $E$
is (o)-topological iff $\tau_{o}$ of $E$ is Hausdorff.
A partial order set $P$ is said to be down-directed if every finite subset of
$P$ has a lower bound in $P$.
Lemma 2.2 ([3]). A subset $U$ of a lattice $L$ is open in $\tau_{o}$ iff for
every directed subset $Y$ of $L$ and every down-directed subset $Z$ of $L$
with $\bigvee Y=\bigwedge Z\in U$, there exist elements $y\in Y$ and $z\in Z$
such that $[y,z]$ is contained in $U$.
An element $u$ of an effect algebra $E$ is called finite if there is a finite
sequence $\\{p_{1},\cdots,p_{n}\\}$ of atoms of $E$ such that
$u=p_{1}\oplus\cdots\oplus p_{n}$. If $E$ is complete and atomic, then for
every $x\in E$ we have $x=\vee\\{u\in E|u\leq x$, $u$ is finite$\\}$.
Moreover, if $E$ is (o)-continuous, then the join of two finite elements is
also finite ([4]).
An element $u$ of an effect algebra $E$ is called compact if $u\leq\vee D$ for
$D\subseteq E$ implies that $u\leq\vee F$ for some finite subset $F\subseteq
D$, and $E$ is called algebraic (or compactly generated) if every $x\in E$ is
a join of compact elements of $E$ ([4]).
Lemma 2.3 ([4]). Let $E$ be a complete atomic (o)-continuous lattice effect
algebra. Then for every finite element $u$ of $E$, if $u\leq\vee D$ for
$D\subseteq E$ implies that $u\leq\vee F$ for some finite subset $F$ of $D$.
Lemma 2.4. Let $E$ be a complete atomic (o)-continuous effect algebra. Then
for each finite element $u$, $[u,1]$ and $[0,u^{\prime}]$ are
$\tau_{o}$-clopen sets.
Proof. Evidently, $[u,1]$ and $[0,u^{\prime}]$ are $\tau_{o}$-closed sets. Let
$Y$ be a directed subset of $E$ and $Z$ be a down-directed subset of $E$ and
satisfy $\bigvee Y=\bigwedge Z\in[u,1]$. As $u$ is finite, by Lemma 2.3, there
exists a subset $\\{y_{1},\cdots,y_{n}\\}\subseteq Y$ such that
$u\leq\vee_{i=1}^{n}y_{i}.$ Since $Y$ is directed, there is a $y_{0}\in Y$
such that $u\leq\vee_{i=1}^{n}y_{i}\leq y_{0}.$ For every fixed $z_{0}\in Z$,
$y_{0}\leq\bigvee Y=\bigwedge Z\leq z_{0}$, so $[y_{0},z_{0}]\subseteq[u,1]$.
By Lemma 2.2, $[u,1]$ is $\tau_{o}$-open.
Similarly, we can prove $[0,u^{\prime}]$ is $\tau_{o}$-open set.
A lattice $L$ is said to be totally order-disconnected if its lattice
operations are order topology $\tau_{o}$ continuous and for any two elements
$x,y$ with $x\nleq y$, there exists a clopen upper set $U$ containing $x$ but
not $y$, where $U$ is an upper set iff $u\in U$ implies $\\{x\in L:x\geq
u\\}\subseteq U$.
Lemma 2.5. Let $E$ be a complete atomic (o)-continuous lattice effect algebra.
Then $E$ is totally order-disconnected.
Proof. Let $a,b$ be two elements of $E$ with $a\nleq b$. As $a=\vee\\{u\in
E|u\leq a$, $u$ is finite}, there exists a finite element $u_{0}\in E$ such
that $u_{0}\leq a$ and $u_{0}\nleq b$. Let $U_{1}=[u_{0},1]$. Then $a\in
U_{1}$ and $b\notin U_{1}$. By Lemma 2.4, $U_{1}$ is an upper set and
$\tau_{o}$-clopen. Thus, $E$ is totally order-disconnected.
It is clear that for every totally order-disconnected lattice, its order
topology is disconnected and Hausdorff.
The following theorem establishes the equivalent relation among
(o)-continuity, (o)-topological, totally order-disconnected and algebra
property of a complete atomic lattice effect algebra $E$.
Theorem 2.1 Let $E$ be a complete atomic lattice effect algebra. The following
statements are equivalent:
(1) $E$ is order-continuous.
(2) $E$ is order-topological.
(3) $E$ is totally order-disconnected.
(4) $E$ is algebraic.
Proof. (1) $\Rightarrow$ (2). It follows from Lemma 2.5 that $\tau_{o}$ is
Hausdorff, so by Lemma 2.1, (2) holds. (1) $\Rightarrow$ (3) can be proved by
Lemma 2.5 and (1) implies (2). (1) $\Rightarrow$ (4) is obtained by Lemma 2.3.
(3) $\Rightarrow$ (2): Since (3) implies that $\tau_{o}$ is Hausdorff and the
binary operations of $\wedge$ and $\vee$ are $\tau_{o}$ continuous, by the
similar method with the proof of Theorem 8 in [2], (2) holds. For (4)
$\Rightarrow$ (1), we refer to [3]. (2) $\Rightarrow$ (1) is clear.
3\. The relation of order topology and Frink ideal topology of effect algebras
The Frink ideal topology is an important intrinsic topology of partial order
set. Frink pointed out that the topology is the correct topology for chains
and direct products of a finite numbers of chains. Atherton in [5] asked:
Whether the Frink ideal topology is Hausdorff topology in every distributive
lattice. Ward pointed out that in a Boolean algebra, the Frink ideal topology
is Hausdorff topology and it is quite usual for the Frink ideal topology to be
strictly finer than the order topology ([6]). Now, we show that for every
complete atomic distributive lattice effect algebra $E$, the Frink ideal
topology $\tau_{id}$ is Hausdorff and is finer than its order topology
$\tau_{o}$, and $\tau_{id}=\tau_{o}$ iff $1$ is finite iff every element of
$E$ is finite iff $\tau_{id}$ and $\tau_{o}$ are discrete topologies.
Let $L$ be a lattice and $I\subseteq L$. Then $I$ is said to be an ideal of
$L$ if the following conditions are satisfied:
(i) When $a\in I$, $x\in L$ and $x\leq a$, $x\in I$.
(ii) When $a\in I,\ b\in I$, $a\vee b\in I$.
The ideal $I$ of $L$ is said to be a completely irreducible ideal if it is not
the intersection of a collection of ideals all distinct from it, i.e., if
$(I_{\alpha})_{\alpha\in\Lambda}$ is a collection of ideals such that
$I=\cap_{\alpha\in\Lambda}I_{\alpha}$, then $I=I_{\alpha_{0}}$ for some
$\alpha_{0}\in\Lambda$. It is clear that every maximal ideal is a completely
irreducible ideal.
Similarly, the dual ideal and completely irreducible dual ideal of $L$ can be
defined, too.
Let $L$ be a lattice. The Frink ideal topology $\tau_{id}$ of $L$ can be
described as following: Take all completely irreducible ideals and completely
irreducible dual ideals of $L$ as a subbasis of the open sets of the topology
$\tau_{id}$ ([7]).
Elements $a,b$ of a lattice effect algebra $E$ are said to be compatible iff
$a\vee b=a\oplus(b\ominus(a\wedge b))$ and denoted by $a\leftrightarrow b$. If
for any $a,b\in E$, $a\leftrightarrow b$, then $E$ is said to be a $MV$-effect
algebra ([8]).
In order to prove our main results in this section, we first need the
following:
Lemma 3.1 ([9]). Let $E$ be a lattice effect algebra.
(i) If $x\oplus y$ exists, then $x\oplus y=(x\vee y)\oplus(x\wedge y)$.
(ii) If $x\wedge y=0$ and for $m,n\in\bf N$, the elements $mx$, $ny$ and
$(mx)\oplus(ny)$ exist in $E$, then $(kx)\wedge(ly)=0$ and
$(kx)\vee(ly)=(kx)\oplus(ly)$ for all $k\in\\{1,\cdots,m\\}$,
$l\in\\{1,\cdots,n\\}$.
(iii) Let $Y\subseteq E$. If $\bigvee Y$ exists in $E$ and $x\in E$ such that
$x\leftrightarrow y$ for every $y\in Y$, then $x\wedge(\bigvee
Y)=\bigvee\\{x\wedge y:y\in Y\\}$ and $x\leftrightarrow\bigvee Y$.
A finite subset $F=(a_{k})_{k=1}^{n}$ of effect algebra $E$ is said to be
$\oplus$-orthogonal if $a_{1}\oplus a_{2}\oplus\cdots\oplus a_{n}$ exists in
$E$ and denote $a_{1}\oplus a_{2}\oplus\cdots\oplus a_{n}$ with
$\bigoplus_{k=1}^{n}a_{k}$ or $\bigoplus F$. An arbitrary subset
$G=(a_{k})_{k\in H}$ of $E$ is said to be $\oplus$-orthogonal if $\bigoplus K$
exists for every finite subset $K\subseteq G$. Let $G=(a_{k})_{k\in H}$ be a
$\oplus$-orthogonal subset of $E$. If $\bigvee\\{\bigoplus K|K\subseteq G$
finite$\\}$ exists in $E$, we denote $\bigvee\\{\bigoplus K|K\subseteq G$
finite$\\}$ with $\bigoplus G$ ([9]).
Lemma 3.2 ([9]). Let $E$ be a complete effect algebra and $(a_{k})_{k\in H}$ a
$\oplus$-orthogonal subset of $E$. If $H_{1}\subseteq H$, $H_{2}=H\backslash
H_{1}$, then
$\bigoplus_{k\in H}a_{k}=(\bigoplus_{k\in H_{1}}a_{k})\oplus(\bigoplus_{k\in
H_{2}}a_{k}).$
For an element $x$ of an effect algebra $E$, we define ord$(x)=\infty$ if
$nx=x\oplus\cdots\oplus x$ ($n$ times) exists for every $n\in N$ and
ord$(x)=n_{x}\in N$ (called isotropic index) if $n_{x}$ is the greatest
integer such that $n_{x}x$ exists in $E$. It is clear that in a complete
lattice effect algebra $E$, $n_{x}<\infty$ for every $x\in E$.
The set of sharp elements of $E$ is denoted by $S(E)$. It has been shown that
in every lattice effect algebra $E$, $S(E)$ is an orthomodular lattice, is a
sub-effect algebra and a sublattice of $E$ ([4]). Moreover, $S(E)$ is a full
sub-lattice of $E$, that is, $S(E)$ inherits all suprema and infima of subsets
of $S(E)$ if they exist in $E$. In a complete atomic distributive lattice
effect algebra $E$, $S(E)$ is a complete atomic Boolean algebra ([9]).
An element $x\in E$ is said to be principle if $a\leq x,b\leq x$ and $a\oplus
b$ is defined, then $a\oplus b\leq x$.
In a lattice effect algebra $E$, $x$ is principle iff $x$ is sharp.
Lemma 3.3 ([9]). If $E$ is an atomic lattice effect algebra and $a\in E$ is an
atom with ord$(a)=n_{a}$, then
(i) $(ka)\wedge(ka)^{\prime}\neq 0$ for all $k\in\\{1,2,\cdots,n_{a}-1\\}$.
(ii) $n_{a}a\in S(E)$.
(iii) If $u=(k_{1}a_{1})\oplus(k_{2}a_{2})\oplus\cdots\oplus(k_{n}a_{n})$,
where $\\{a_{1},a_{2},\cdots,a_{n}\\}$ is a set of mutually different atoms of
$E$, then $u=\vee_{i=1}^{n}(k_{i}a_{i})$.
(iv) If $E$ is complete and $x\neq 0$, then there are mutually different atoms
$a_{\alpha}\in E,\alpha\in\Gamma$, and positive integers $k_{\alpha}$ such
that
$x=\bigoplus\\{k_{\alpha}a_{\alpha}|\alpha\in\Gamma\\}=\bigvee\\{k_{\alpha}a_{\alpha}|\alpha\in\Gamma\\}.$
Moreover, $x\in S(E)$ iff $k_{\alpha}=n_{a_{\alpha}}=$ ord$(a_{\alpha})$ for
all $\alpha\in\Gamma$.
Lemma 3.4 ([9]). Let $E$ be a complete atomic effect algebra. Then for every
$x\in E$ with $x\neq 0$, there exists a unique $w_{x}\in S(E)$, a unique set
$\\{a_{\alpha}|\alpha\in\mathcal{A}\\}$ of atoms of $E$ and unique positive
integers $k_{\alpha}$, $\alpha\in\mathcal{A}$, such that
$x=w_{x}\oplus(\bigoplus\\{k_{\alpha}a_{\alpha}|\alpha\in\mathcal{A}\\}).$
Moreover, if $w\in S(E)$ with $w\leq x\ominus w_{x}$, then $w=0$.
Definition 3.1 ([10-12]). An effect algebra $E$ is called sharply dominating
if for every $a\in E$ there exists a smallest sharp element $\hat{a}\in E$
such that $a\leq\hat{a}$. A sharply dominating effect algebra $E$ is called
$S$-dominating if $a\wedge p$ exists for every $a\in E$ and $p\in S(E)$.
It is clear that a lattice effect algebra $E$ is $S$-dominating iff $E$ is
sharply dominating. Every complete lattice effect algebra is sharply
dominating.
Lemma 3.5. Let $E$ be a complete atomic distributive lattice effect algebra.
If $u$ is a finite element of $E$, then the smallest sharp element $\hat{u}$
dominating $u$ is also finite. If $u_{1},u_{2}\in E$ with $u_{1}\leq u_{2}$,
then $\hat{u_{1}}\leq\hat{u_{2}}$.
Proof. By Lemma 3.3, there are mutually distinct atoms
$\\{a_{1},\cdots,a_{m}\\}$ and positive integers $\\{k_{1},\cdots,k_{m}\\}$
such that $u=\oplus_{i=1}^{m}k_{i}a_{i}=\vee_{i=1}^{m}k_{i}a_{i}$. We claim
$\hat{u}=\oplus_{i=1}^{m}n_{i}a_{i}=\vee_{i=1}^{m}n_{i}a_{i}$, where $n_{i}$
is the isotropic index of $a_{i}$. Let $b\in S(E)$ with $u\leq b$. For every
$i$ with $k_{i}\neq n_{i}$, we have $k_{i}a_{i}\oplus b^{\prime}$ is defined
and since $(k_{i}a_{i})\wedge b^{\prime}\leq b\wedge b^{\prime}=0$, we obtain,
by Lemma 3.1, that $(k_{i}a_{i})\oplus b^{\prime}=(k_{i}a_{i})\vee
b^{\prime}$. As $k_{i}a_{i}\leq a_{i}^{\prime}$ and
$b^{\prime}\leq(k_{i}a_{i})^{\prime}\leq a_{i}^{\prime}$, $(k_{i}a_{i})\oplus
b^{\prime}=(k_{i}a_{i})\vee b^{\prime}\leq a_{i}^{\prime}$. It follows that
there exists $(k_{i}+1)a_{i}\oplus b^{\prime}=(k_{i}+1)a_{i}\vee
b^{\prime}\leq a_{i}^{\prime}$. Hence, $(k_{i}+2)a_{i}\oplus b^{\prime}$
exists, by induction, $n_{i}a_{i}\oplus b^{\prime}$ exists and so
$n_{i}a_{i}\leq b$. Thus, $\hat{u}=\vee_{i=1}^{m}n_{i}a_{i}\leq b$.
Let $u_{1}\leq u_{2}$. Then $u_{1}\leq u_{2}\leq\hat{u_{2}}$. By the
definition of sharply dominating effect algebra, we get
$\hat{u_{1}}\leq\hat{u_{2}}$.
Lemma 3.6. Let $E$ be a complete atomic distributive lattice effect algebra
and $F$ the set of all finite elements and $0$ of $E$. Then $F$ is an ideal of
$E$.
Proof. Note that in every complete atomic distributive lattice effect algebra,
the join of two finite elements is finite as well, so we only need to prove
the fact that if $u\in E$ is finite and $x\in E$ with $0\neq x\leq u$, then
$x$ is finite.
Let $u$ be finite and $0\neq x\leq u$. Then, by Lemma 3.5, $\hat{u}\in S(E)$
and $\hat{u}$ is finite with $x\leq\hat{u}$. It follows from Lemma 2.3 that
$\hat{u}$ is a compact element of $E$, so it is a compact element of $S(E)$.
(i) If $x\in S(E)$, by Lemma 3.3, we can assume
$x=\oplus_{\alpha\in\Lambda}n_{\alpha}a_{\alpha}=\vee_{\alpha\in\Lambda}n_{\alpha}a_{\alpha}$,
where $\\{a_{\alpha}:\alpha\in\Lambda\\}$ is a set of atoms and $n_{\alpha}$
is the isotropic index of $a_{\alpha}$. Note that $S(E)$ is a complete atomic
Boolean algebra and $x\leq\hat{u}$, so $x$ is a compact element of $S(E)$.
Thus, there exist $\\{\alpha_{1},\cdots,\alpha_{m}\\}\subseteq\Lambda$ such
that
$x=\vee_{\alpha\in\Lambda}n_{\alpha}a_{\alpha}\leq\vee_{i=1}^{m}n_{\alpha_{i}}a_{\alpha_{i}}$,
so
$x=\vee_{i=1}^{m}n_{\alpha_{i}}a_{\alpha_{i}}=\oplus_{i=1}^{m}n_{\alpha_{i}}a_{\alpha_{i}}$.
That is, $x$ is a finite element of $E$.
(ii) If $x\notin S(E)$. There exists $x_{1}\in E$ such that $\hat{u}=x\oplus
x_{1}$. By Lemma 3.4, we can assume that
$x=w_{x}\oplus(\bigvee_{\alpha\in\Lambda}k_{\alpha}b_{\alpha})=w_{x}\oplus(\bigoplus_{\alpha\in\Lambda}k_{\alpha}b_{\alpha}),$
$x_{1}=w_{x_{1}}\oplus(\bigvee_{\beta\in\Gamma}l_{\beta}c_{\beta})=w_{x_{1}}\oplus(\bigoplus_{\beta\in\Gamma}l_{\beta}c_{\beta}),$
where $w_{x},w_{x_{1}}\in S(E)$, $\\{b_{\alpha}:\alpha\in\Lambda\\}$ and
$\\{c_{\beta}:\beta\in\Gamma\\}$ are sets of atoms and $k_{\alpha}\neq
n_{\alpha}$, $l_{\beta}\neq n_{\beta}$ for every $\alpha\in\Lambda$ and
$\beta\in\Gamma$. Note that $S(E)$ is a sub-effect algebra, denote
$x_{0}=(\bigoplus_{\alpha\in\Lambda}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in\Gamma}l_{\beta}c_{\beta})$,
we obtain $x_{0}=\hat{u}\ominus w_{x}\ominus w_{x_{1}}\in S(E)$. Denote
$\Lambda_{1}=\\{\alpha\in\Lambda:$ there exists $c_{\beta}$ such that
$b_{\alpha}=c_{\beta}\\}$ and $\Gamma_{1}=\\{\beta\in\Gamma:$ there exists
$b_{\alpha}$ such that $c_{\beta}=b_{\alpha}\\}$. For every
$\beta\in\Gamma_{1}$, if $c_{\beta}=b_{\alpha}$, then we denote
$l_{\alpha}=l_{\beta}$. Thus, by Lemma 3.2,
$(\bigoplus_{\alpha\in\Lambda}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in\Gamma}l_{\beta}c_{\beta})=\bigoplus_{\alpha\in\Lambda_{1}}(k_{\alpha}+l_{\alpha})b_{\alpha}\oplus(\bigoplus_{\alpha\in{\Lambda\setminus\Lambda_{1}}}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta}).$
So $b_{\alpha}\neq c_{\beta}$ for every
$\alpha\in(\Lambda\setminus\Lambda_{1})$ and
$\beta\in(\Gamma\setminus\Gamma_{1})$. It follows from Lemma 3.1 that
$(k_{\alpha}b_{\alpha})\wedge(l_{\beta}c_{\beta})=0$ for every
$\alpha\in(\Lambda\setminus\Lambda_{1})$ and
$\beta\in(\Gamma\setminus\Gamma_{1})$. As
$(k_{\alpha}b_{\alpha})\oplus(l_{\beta}c_{\beta})$ is defined,
$(k_{\alpha}b_{\alpha})\leftrightarrow(l_{\beta}c_{\beta})$, where
$\alpha\in(\Lambda\setminus\Lambda_{1})$ and
$\beta\in(\Gamma\setminus\Gamma_{1})$. By Lemma 3.1, we have
$k_{\alpha}b_{\alpha}\leftrightarrow\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta}),\
\
\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1}}(k_{\alpha}b_{\alpha})\leftrightarrow\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta}).$
So
$(\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1}}(k_{\alpha}b_{\alpha}))\wedge(\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta}))=\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1},\beta\in\Gamma\setminus\Gamma_{1}}((k_{\alpha}b_{\alpha})\wedge(l_{\beta}c_{\beta}))=0.$
Hence
$(\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1}}(k_{\alpha}b_{\alpha}))\oplus(\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta}))=(\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1}}(k_{\alpha}b_{\alpha}))\vee(\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta})).$
That is
$(\bigoplus_{\alpha\in{\Lambda\setminus\Lambda_{1}}}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta})=(\bigvee_{\alpha\in\Lambda\setminus\Lambda_{1}}(k_{\alpha}b_{\alpha}))\vee(\bigvee_{\beta\in\Gamma\setminus\Gamma_{1}}(l_{\beta}c_{\beta})).$
Similarly, we can prove
$x_{0}=(\bigvee_{\alpha\in\Lambda_{1}}(k_{\alpha}+l_{\alpha})b_{\alpha})\vee(\bigvee_{\alpha\in{\Lambda\setminus\Lambda_{1}}}k_{\alpha}b_{\alpha})\vee(\bigvee_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta})$.
For every fixed $\alpha_{0}\in\Lambda\setminus\Lambda_{1}$,
$(k_{\alpha_{0}}a_{\alpha_{0}})\wedge x_{0}^{\prime}\leq x_{0}\wedge
x_{0}^{\prime}=0$. As
$\bigoplus_{\alpha\in\Lambda_{1}}(k_{\alpha}+l_{\alpha})b_{\alpha}\oplus(\bigoplus_{\alpha\in{\Lambda\setminus\Lambda_{1}}}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta})$
is defined, we have
$k_{\alpha_{0}}a_{\alpha_{0}}\leq(\bigvee_{\alpha\in\Lambda_{1}}(k_{\alpha}+l_{\alpha})b_{\alpha})^{\prime},\
k_{\alpha_{0}}a_{\alpha_{0}}\leq(\bigvee_{\alpha\in{\Lambda\setminus\Lambda_{1}},\alpha\neq\alpha_{0}}k_{\alpha}b_{\alpha})^{\prime},\
k_{\alpha_{0}}a_{\alpha_{0}}\leq(\bigvee_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta})^{\prime}.$
So $(k_{\alpha_{0}}a_{\alpha_{0}})\wedge
x_{0}^{\prime}=(k_{\alpha_{0}}a_{\alpha_{0}})\wedge((\bigvee_{\alpha\in\Lambda_{1}}(k_{\alpha}+l_{\alpha})b_{\alpha})\vee(\bigvee_{\alpha\in{\Lambda\setminus\Lambda_{1}}}k_{\alpha}b_{\alpha})\vee(\bigvee_{\beta\in{\Gamma\setminus\Gamma_{1}}}l_{\beta}c_{\beta}))^{\prime}=(k_{\alpha_{0}}a_{\alpha_{0}})\wedge(k_{\alpha_{0}}a_{\alpha_{0}})^{\prime}=0$.
By Lemma 3.3, we have $k_{\alpha_{0}}=n_{\alpha_{0}}$. Note that we have
assumed that $k_{\alpha}\neq n_{\alpha}$ for every $\alpha\in{\Lambda}$, so
$\Lambda=\Lambda_{1}$. Similarly, we have $\Gamma=\Gamma_{1}$ and
$k_{\alpha}+l_{\alpha}=n_{\alpha}$ for every $\alpha\in\Lambda$. That is
$(\bigoplus_{\alpha\in\Lambda}k_{\alpha}b_{\alpha})\oplus(\bigoplus_{\beta\in\Gamma}l_{\beta}c_{\beta})=\bigvee_{\alpha\in\Lambda}(k_{\alpha}+l_{\alpha})b_{\alpha}=\bigvee_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}=\hat{u}\ominus
w_{x}\ominus w_{x_{1}}\in S(E).$
Since $\bigvee_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}\leq\hat{u}$, $\hat{u}$
is a compact element of $S(E)$ and $S(E)$ is a complete atomic Boolean
algebra, we get $\bigvee_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}$ is compact.
Thus, there exists $\\{\alpha_{1},\cdots,\alpha_{m}\\}\subseteq\Lambda$ such
that
$\bigvee_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}=\bigvee_{i=1}^{m}n_{\alpha_{i}}b_{\alpha_{i}}=\bigoplus_{i=1}^{m}n_{\alpha_{i}}b_{\alpha_{i}}.$
As
$\bigvee_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}=\bigoplus_{\alpha\in\Lambda}n_{\alpha}b_{\alpha}=\bigoplus_{i=1}^{m}n_{\alpha_{i}}b_{\alpha_{i}}$,
$\Lambda$ is finite. Therefore
$x=w_{x}\oplus(\bigvee_{\alpha\in\Lambda}k_{\alpha}b_{\alpha})$ is finite. The
lemma is proved.
Recall that the interval topology $\tau_{i}$ of an effect algebra $E$ is the
topology which is defined by taking all closed interval $[a,b]$ as a sub-basis
of closed sets of $E$. It is well known that $\tau_{id}\geq\tau_{i}$ and
$\tau_{o}\geq\tau_{i}$ in every lattice.
Lemma 3.7 ([2]). Let $E$ be a complete atomic distributive lattice effect
algebra. Then its interval topology $\tau_{i}$ is compact Hausdorff topology
and $\tau_{o}=\tau_{i}$.
Theorem 3.1. Let $E$ be a complete atomic distributive lattice effect algebra.
Then $\tau_{id}$ is Hausdorff and $\tau_{id}\geq\tau_{o}$. Moreover, the
following conditions are equivalent:
(i) $\tau_{id}=\tau_{o}$.
(ii) $1$ is finite.
(iii) Every element of $E$ is finite.
(iv) $\tau_{id}$ and $\tau_{o}$ are both discrete topologies.
Proof. It follows from Lemma 3.7 and $\tau_{id}\geq\tau_{i}$ that $\tau_{id}$
is Hausdorff topology and $\tau_{id}\geq\tau_{o}$.
(ii) $\Leftrightarrow$ (iii) can be proved by Lemma 3.6 easily.
(iii) $\Rightarrow$ (iv). For every $x\in E$, $x$ and $x^{\prime}$ are both
finite. By Lemma 2.4, $[0,x]$ and $[x,1]$ are $\tau_{o}$-open, so
$\\{x\\}=[0,x]\cap[x,1]$ is $\tau_{o}$ open, this showed that $\tau_{o}$ is
discrete. Note that $\tau_{id}\geq\tau_{o}$, we have $\tau_{id}$ and
$\tau_{o}$ are both discrete.
(iv) $\Rightarrow$ (i) is obvious.
(i) $\Rightarrow$ (ii). Assume $1$ is not finite. Let $F_{0}$ be the set of
all finite elements and $0$. Then it follows from Lemma 3.6 that $F_{0}$ is an
ideal and $1\notin F_{0}$. By the Zorn’s Lemma, $F_{0}$ is contained in an
ideal $F$ maximal subject to not containing $1$. It is easy to prove that
$F^{\prime}=\\{f^{\prime}\in E:f\in F\\}$ is a maximal dual ideal and so
$\tau_{id}$-open with $1\in F^{\prime}$. As $1=\vee\\{u\in E:u$ is finite}, we
can choose a net of finite elements $(u_{\alpha})_{\alpha\in\Lambda}$ of $E$
such that $u_{\alpha}\uparrow 1$, thus, by Lemma 3.5,
$(\hat{u}_{\alpha})_{\alpha\in\Lambda}$ is also a net of finite elements of
$E$ and $\hat{u}_{\alpha}\uparrow 1$. Note that
$(u_{\alpha})_{\alpha\in\Lambda}\subseteq F$ and
$\hat{u}_{\alpha}\vee\hat{u}_{\alpha}^{\prime}=1$, we have
$\hat{u}_{\alpha}^{\prime}\notin F$, otherwise $1\in F$. Hence
$\hat{u}_{\alpha}\notin F^{\prime}$. That is, $\hat{u}_{\alpha}$ is not
$\tau_{id}$-convergent to $1$. However,
$\hat{u}_{\alpha}\xrightarrow{(\tau_{o})}1$. This contradicts (i). So $1$ is
finite.
4\. The order topology continuity of operation $\oplus$ of effect algebras
For the order topology continuity of operation $\oplus$, Wu had only presented
a sufficient condition under a very strictly assumption ([13]). Now, we study
this question continuously.
Lemma 4.1 ([14]). Let $(X,T)$ be a topology space. Then $(X,T)$ is Hausdorff
space iff every convergent net in $X$ has exactly one limit point.
Lemma 4.2 ([2]). Let $E$ be a complete (o)-continuous lattice effect algebra,
$x_{\alpha},x,y\in E$. Then
(i) $x_{\alpha}\xrightarrow{(\tau_{o})}x\Rightarrow x_{\alpha}\vee
y\xrightarrow{(\tau_{o})}x\vee y$.
(ii) $x_{\alpha}\xrightarrow{(\tau_{o})}x\Rightarrow x_{\alpha}\wedge
y\xrightarrow{(\tau_{o})}x\wedge y$.
(iii) $x_{\alpha}\xrightarrow{(\tau_{o})}x\Rightarrow
x_{\alpha}^{\prime}\xrightarrow{(\tau_{o})}x^{\prime}$.
Our main results in this section are the following:
Theorem 4.1. Let $(E,\oplus,0,1)$ be a complete (o)-continuous lattice effect
algebra. If $\oplus$ is order topology $\tau_{o}$ continuous, then $\tau_{o}$
is Hausdorff topology.
Proof. Let $(x_{\alpha})_{\alpha\in\Lambda}$ be a net in $E$ and
$x_{\alpha}\xrightarrow{(\tau_{o})}x$, $x_{\alpha}\xrightarrow{(\tau_{o})}y$.
By Lemma 4.2,
$x_{\alpha}\vee x\xrightarrow{(\tau_{o})}x,\ x_{\alpha}\vee
x\xrightarrow{(\tau_{o})}x\vee y,\ (x_{\alpha}\vee
x)^{\prime}\xrightarrow{(\tau_{o})}(x\vee y)^{\prime}.$
It follows from $\oplus$ is $\tau_{o}$ continuous that
$(x_{\alpha}\vee x)\oplus(x_{\alpha}\vee
x)^{\prime}\xrightarrow{(\tau_{o})}x\oplus(x\vee y)^{\prime}.$
That is, $1\xrightarrow{(\tau_{o})}x\oplus(x\vee y)^{\prime}$. Note that
$\tau_{o}\geq\tau_{i}$, $1\xrightarrow{(\tau_{i})}x\oplus(x\vee y)^{\prime}$
and $\\{1\\}$ is $\tau_{i}$-closed, we have $x\oplus(x\vee y)^{\prime}=1$, so
$x=x\vee y$, $y\leq x$. Similarly, we can prove that $x\leq y$. Thus $x=y$ and
$\tau_{o}$ is Hausdorff topology.
Example 4.1. Let $L$ be the complete Boolean algebra of all regular open
subsets of the unit interval $I$. It follows from $L$ is (o)-continuous and
the order topology $\tau_{o}$ of $L$ is not Hausdorff topology ([15]) and
Theorem 4.1 that $\oplus$ is not order topology $\tau_{o}$ continuous.
Theorem 4.2. Let $(E,\oplus,0,1)$ be a complete atomic (o)-continuous lattice
effect algebra. If $a_{\alpha}\xrightarrow{(\tau_{o})}0$ and
$b_{\alpha}\xrightarrow{(\tau_{o})}0$ with $a_{\alpha}\leq
b_{\alpha}^{\prime}$ for every $\alpha$, then $a_{\alpha}\oplus
b_{\alpha}\xrightarrow{(\tau_{o})}0$.
Proof. Suppose $\wedge_{\beta}\vee_{\alpha\geq\beta}(a_{\alpha}\oplus
b_{\alpha})=c$. For every finite element $u\in E$, there exists sharply
dominating element $\hat{u}$ such that $u\leq\hat{u}$. It follows from Lemma
3.5 that $\hat{u}$ is also finite. So
$[0,\hat{u}^{\prime}]\subseteq[0,u^{\prime}]$ and by Lemma 2.4 that they are
both $\tau_{o}$-open. Note that $0\in[0,\hat{u}^{\prime}]$, there exists
$\alpha_{0}$ such that for every $\alpha\geq\alpha_{0}$,
$a_{\alpha}\in[0,\hat{u}^{\prime}]$ and $b_{\alpha}\in[0,\hat{u}^{\prime}]$.
Thus, for every $\alpha\geq\alpha_{0}$, $a_{\alpha}\oplus
b_{\alpha}\leq\hat{u}^{\prime}$ since $\hat{u}^{\prime}$ is principle, so
$\vee_{\alpha\geq\alpha_{0}}(a_{\alpha}\oplus b_{\alpha})\leq\hat{u}^{\prime}$
and $c\leq\hat{u}^{\prime}$, this showed that $c^{\prime}\geq\hat{u}\geq u$.
As $1=\vee\\{u\in E:u$ is finite $\\}$, $c^{\prime}\geq 1$. That is,
$c^{\prime}=1$ and $c=0$. Hence, we have $a_{\alpha}\oplus
b_{\alpha}\xrightarrow{(o)}0$, thus, $a_{\alpha}\oplus
b_{\alpha}\xrightarrow{(\tau_{o})}0$.
Lemma 4.3 ([16]). Let $E$ be a lattice effect algebra. Then
$x_{\alpha}\xrightarrow{(o)}x$ iff $(x\vee x_{\alpha})\ominus
x\xrightarrow{(o)}0$ and $x\ominus(x\wedge x_{\alpha})\xrightarrow{(o)}0$.
Theorem 4.3. Let $(E,\oplus,0,1)$ be a complete atomic (o)-continuous lattice
effect algebra. If $a_{\alpha}\xrightarrow{(\tau_{o})}a$ and
$b_{\alpha}\xrightarrow{(\tau_{o})}b$ with $a_{\alpha}\leq
b_{\alpha}^{\prime}$ and $a_{\alpha}\leq b^{\prime}$ and $b_{\alpha}\leq
a^{\prime}$ for every $\alpha$, then $a_{\alpha}\oplus
b_{\alpha}\xrightarrow{(\tau_{o})}a\oplus b$.
Proof. By Theorem 2.1 and Lemma 4.3, we only need to prove that
$((a_{\alpha}\oplus b_{\alpha})\vee(a\oplus b))\ominus(a\oplus
b)\xrightarrow{(\tau_{o})}0,\ (a\oplus b)\ominus((a\oplus
b)\wedge(a_{\alpha}\oplus b_{\alpha}))\xrightarrow{(\tau_{o})}0.$
Note that $a_{\alpha}\xrightarrow{(\tau_{o})}a$ implies that
$a_{\alpha}\xrightarrow{(\tau_{i})}a$, and since $a_{\alpha}\leq b^{\prime}$
for every $\alpha$, we have $a\leq b^{\prime}$. Moreover, since
$a_{\alpha}\leq(b_{\alpha}^{\prime}\wedge b^{\prime})$ and
$a\leq(b_{\alpha}^{\prime}\wedge b^{\prime})$, $(a_{\alpha}\vee
a)\oplus(b_{\alpha}\vee b)$ is defined. As
$a_{\alpha}\xrightarrow{(\tau_{o})}a$ and
$b_{\alpha}\xrightarrow{(\tau_{o})}b$, it follows from Lemma 4.3 that
$(a_{\alpha}\vee a)\ominus a\xrightarrow{(\tau_{o})}0,\ (b_{\alpha}\vee
b)\ominus b\xrightarrow{(\tau_{o})}0.$
Note that $((a_{\alpha}\vee a)\ominus a)\oplus((b_{\alpha}\vee b)\ominus
b)=((a_{\alpha}\vee a)\oplus(b_{\alpha}\vee b))\ominus(a\oplus
b)\geq((a_{\alpha}\oplus b_{\alpha})\vee(a\oplus b))\ominus(a\oplus b)$ and
Theorem 4.2, we have
$((a_{\alpha}\vee a)\ominus a)\oplus((b_{\alpha}\vee b)\ominus
b)\xrightarrow{(\tau_{o})}0,$
so
$((a_{\alpha}\oplus b_{\alpha})\vee(a\oplus b))\ominus(a\oplus
b)\xrightarrow{(\tau_{o})}0.$
Similarly, we can prove that $(a\oplus b)\ominus((a\oplus
b)\wedge(a_{\alpha}\oplus b_{\alpha}))\xrightarrow{(\tau_{o})}0$ and the
theorem is proved.
Corollary 4.1. Let $(E,\oplus,0,1)$ be a complete atomic $MV$-effect algebra,
$a_{\alpha}\xrightarrow{(\tau_{o})}a$ and
$b_{\alpha}\xrightarrow{(\tau_{o})}b$ with $a_{\alpha}\leq
b_{\alpha}^{\prime}$ for every $\alpha$. Then $a_{\alpha}\oplus
b_{\alpha}\xrightarrow{(\tau_{o})}a\oplus b$.
Acknowledgement. The authors wish to express their thanks to the referees for
their valuable comments and suggestions.
## References
* [1] Foulis, D. and Bennett, M. K. (1994), Effect Algebras and Unsharp Quantum Logics. Found. Phys.24, 1331–1352.
* [2] Rie$\breve{c}$anov$\acute{a}$, Z. (2003), Order-topological Lattice Effect Algebras. Contributions to General Algebra 15, Proceeding of the Klagenfurt Workshop 2003 on General Algebra, Klagenfurt, Austria, 151–159.
* [3] Ern$\acute{e}$, M and Rie$\breve{c}$anov$\acute{a}$, Z. (1995), Order-topological Complete Orthomodular Lattices. Topology and its Applications 37, 215–227.
* [4] Rie$\breve{c}$anov$\acute{a}$, Z. (2002), Smearings of States Defined on Sharp Elements onto Effect Algebras. Internat. J. Theor. Phys. 41, 1511–1524.
* [5] Atherton, C. R. (1970), Concerning Intrinsic Topologies on Boolean Algebras and Certain Bicompactly Generated Lattices. Glasgow Mathematical Journal 11, 156–161.
* [6] Ward, A. J. (1955), On Relations between Certain Intrinsic Topologies in Partially Ordered Sets. Proc. Cambriage Philos. Soc. 51, 254–261.
* [7] Frink, O. (1954), Ideals in Partially Ordered Sets. Amer. Math. Mon. 61, 2319–2323.
* [8] K$\hat{o}$pka, F. (1995), Compatibility in D-posets. Internat. J. Theor. Phys. 34, 1525–1531.
* [9] Rie$\breve{c}$anov$\acute{a}$, Z. (2005), Basic Decomposition of Elements and Jauch-Piron Effect Algebras. Fuzzy Sets and Systems 155, 138–149.
* [10] Gudder, S. P. (1994), Sharply Dominating Effect Algebras. Found. Phys. 24, 1331–1352.
* [11] Gudder, S. P. (1998), S-dominating Effect Algebra. Internat. J. Theor. Phys. 37, 1525–1531.
* [12] Rie$\breve{c}$anov$\acute{a}$, Z. and Wu, J. D. (2008), States on Sharply Dominating Effect Algebras. Science in China (A) 51, 907–914.
* [13] Wu, J. D. and Tang, Z. F. and Cho, M. H. (2005), Two Variables Operation Continuity of Effect Algebras. Internat. Theory. Phys. 44, 581–586.
* [14] Wilansky, A. (1970), Topology for Analysis. Waltham. Mass., Ginn.
* [15] Floyd, E. E. (1955), Boolean Algebras with Pathological Order Topologies. Pacific. J. Math. 5, 687–699.
* [16] Rie$\breve{c}$anov$\acute{a}$, Z. (2002), States, Uniformities and Metrics on Lattice Effect Algebras. Int. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems 10, 125–133.
|
arxiv-papers
| 2009-08-24T01:11:08 |
2024-09-04T02:49:04.740432
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Qiang, Wu Junde, Li Ronglu",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0908.3350"
}
|
0908.3366
|
# Interplay between antiferromagnetic order and spin polarization in
ferromagnetic metal/electron-doped cuprate superconductor junctions
Pok-Man Chiu,1 C. S. Liu,2,1 and W. C. Wu1 1Department of Physics, National
Taiwan Normal Univesity, Taipei 11650, Taiwan
2Department of Physics, Yanshan University, Qinhuangdao 066004, China
###### Abstract
Recently we proposed a theory of point-contact spectroscopy and argued that
the splitting of zero-bias conductance peak (ZBCP) in electron-doped cuprate
superconductor point-contact spectroscopy is due to the coexistence of
antiferromagnetic (AF) and $d$-wave superconducting orders [Phys. Rev. B 76,
220504(R) (2007)]. Here we extend the theory to study the tunneling in the
ferromagnetic metal/electron-doped cuprate superconductor (FM/EDSC) junctions.
In addition to the AF order, the effects of spin polarization, Fermi-wave
vector mismatch (FWM) between the FM and EDSC regions, and effective barrier
are investigated. It is shown that there exits midgap surface state (MSS)
contribution to the conductance to which Andreev reflections are largely
modified due to the interplay between the exchange field of ferromagnetic
metal and the AF order in EDSC. Low-energy anomalous conductance enhancement
can occur which could further test the existence of AF order in EDSC. Finally,
we propose a more accurate formula in determining the spin polarization value
in combination with the point-contact conductance data.
###### pacs:
74.20.-z, 74.25.Ha, 74.45.+c, 74.50.+r
## I Introduction
Using point contact technique to measure the spin polarization in
ferromagnetic metal/conventional superconductor (FM/CS) junctions was
pioneeringly done by Soulen et al. Soulen98 and Upadhyay et al. Upadhyay98
in 1998. Their works showed that determining the spin polarization at Fermi
surface is essentially not an easy task. That leads to some definitions of
spin polarization such as “tunneling polarization” proposed by Tedrow and
Meservey Tedrow94 and “point-contact polarization” proposed by Soulen et al.
Soulen98 . One year later, Zhu et al. Zhu99 ; Zhu00 and Kashiwaya et al.
Kashiwaya99 have utilized the ideas to study the spin-polarized quasiparticle
transport in ferromagnet/$d$-wave superconductor junctions. Zhu et al. Zhu99 ;
Zhu00 predicted that conductance resonances occur in a normal-metal-
ferromagnet/$d$-wave superconductor junction and in a following paper, they
further studied the junctions by solving the Bogoliubov-de Gennes (BdG)
equations within an extended Hubbard model which included the proximity
effect, the spin-flip interfacial scattering at the interface, and the local
magnetic moment. They have reported that the proximity can induce order
parameter oscillation in the ferromagnetic region. In contrast, Kashiwaya et
al. Kashiwaya99 focused on the spin current and spin filtering effects at the
magnetic interface. In the works of Zutic and Valls Zutic99 ; Zutic00 , they
first considered the effect of Fermi-wave vector mismatch (FWM) and have
pointed out that if one neglects FWM, the effect of spin polarization
invariably leads to the suppression of Andreev reflection (AR). Among many
other junction studies, Dong et al. Dong01 studied a little different
junction which forms a four layer sandwich, i.e., FM/I/$d+is$/$d$-wave
junctions, by taking into account the roughness of the interfacial barrier and
broken time-reversal symmetry states.
The pioneering works of Soulen et al. and Upadhyay et al. have inspired
several experimental studies Ji01 ; Strijkers01 ; Kant02 ; Raychaudhuri03 ;
Perez-Willard04 ; Woods04 ; Mukhopadhyay07 ; Chalsani07 as well. Especially
normal and ferromagnetic metal/conventional superconductor or $s$-wave
superconductor (FM/$s$-wave SC) junctions have been intensely studied
experimentally and theoretical modelings (Blonder-Tinkham-Klapwijk (BTK)
formula Blonder82 or its extension) had a good fitting with the conductance
data. Recently Linder and Sudbø Linder07 presented a theoretical study of
FM/$s$-wave SC junction that investigated the possibility of induced triplet
pairing state in the ferromagnetic metal side. They have also used the BTK
approach but allowed for arbitrary magnetization strength and direction in the
ferromagnet, arbitrary spin-active barrier, arbitrary FWM, and different
effective masses in the two sides of the junction. As is expected, there is no
retroreflection process when an exchange field is present. However, they
pointed out that retroreflection can occur under some conditions Linder07 .
If one replaces the conventional superconductor by the high-temperature or
$d$-wave superconductor into the junction, it will occur several novel
phenomena due to its $d$-wave pairing symmetry, complex band structure, and
rich magnetic properties. Of particular interest, in the electron-doped side
of cuprate superconductors (EDSC), it is strongly suggested that
antiferromagnetic (AF) order may coexist with the $d$-wave superconducting
order, especially in the underdoped and optimally-doped regimes Liu07 . In
this paper, we shall explore the possible novel phenomena in the FM/EDSC
junction case, taking into account the interplay between antiferromagnetic
order and spin polarization. The ideas and models developed in FM/CS junctions
in the literature will be applied to the current FM/EDSC junction cases.
This paper is organized as follows. In Sec. II, the basic formulation is
given. We set up the condition of the junction and generalize the BdG
equations to include AF order parameter. As the formal process, we utilize
WKBJ approximation to obtain the more simple Andreev-like equations, which are
then solved to determine the four spin-dependent reflection coefficients
(detailed derivations are given in Appendix A). Formulas of charge and spin
conductances are derived . Sec. III are our main results and discussions. In
Sec. III.A, the condition of midgap surface states was derived (details are
given in Appendix B). In Sec. III.B, the effect of FWM was studied. In Secs.
III.C and III.D, we discuss the effects of spin-polarization and generalized
effective barrier, respectively. It is shown that anomalous conductance
enhancement can occur at low energies which could provide a further test for
the existence of AF order in EDSC. In Sec. III.E, a more general formula for
determining the spin polarization is proposed in terms of the experimental
zero-bias conductance data. Finally in Sec. IV, a brief conclusion is given.
## II Formalism
Our formulation is given based on the following assumptions. We consider a
point contact or planar FM/I/EDSC junction where the superconductor overlayer
is coated with a clean, size-quantized, ferromagnetic-metal overlayer of
thickness $d$, that is much shorter than the mean free path $l$ of normal
electrons. The interface is assumed to be perfectly flat and infinitely large.
Considering $l\rightarrow\infty$ limit, the discontinuity of all parameters at
the interface can be neglected, except for the SC order parameter to which the
proximity effect is ignored Hu94 . When SC and AF orders coexist,
quasiparticle (QP) excitations of an inhomogeneous superconductor can have a
coupled electron-hole character associated with the coupled $\mathbf{k}$ and
${\bf k}+{\bf Q}$ [$\mathbf{Q}=(\pi,\pi)$] subspaces. QP states are thus
governed by the generalized BdG equations Gennes66 ; Liu07
$\displaystyle Eu_{1\sigma}({\bf{x}})$ $\displaystyle=$
$\displaystyle\hat{H}_{\sigma}u_{1\sigma}({\bf{x}})+\int{d{\bf{x}^{\prime}}\Delta({\bf{s}},{\bf{r}})v_{1\bar{\sigma}}({\bf{x}^{\prime}})+\Phi
u_{2\sigma}({\bf{x}})}$ $\displaystyle Ev_{1\bar{\sigma}}({\bf{x}})$
$\displaystyle=$
$\displaystyle\int{d{\bf{x}^{\prime}}\Delta^{*}({\bf{s}},{\bf{r}})u_{1\sigma}({\bf{x}^{\prime}})-\hat{H}_{\sigma}v_{1\bar{\sigma}}({\bf{x}})+\Phi
v_{2\bar{\sigma}}({\bf{x}})}$ $\displaystyle Eu_{2\sigma}({\bf{x}})$
$\displaystyle=$ $\displaystyle\Phi
u_{1\sigma}({\bf{x}})+\hat{H}_{\sigma}u_{2\sigma}({\bf{x}})-\int{d{\bf{x}^{\prime}}\Delta({\bf{s}},{\bf{r}})v_{2\bar{\sigma}}({\bf{x}^{\prime}})}$
$\displaystyle Ev_{2\bar{\sigma}}({\bf{x}})$ $\displaystyle=$
$\displaystyle\Phi
v_{1\bar{\sigma}}({\bf{x}})-\int{d{\bf{x}^{\prime}}\Delta^{*}({\bf{s}},{\bf{r}})u_{2\sigma}({\bf{x}^{\prime}})-\hat{H}_{\sigma}v_{2\bar{\sigma}}({\bf{x}})},$
where ${\bf{s}}\equiv{\bf{x}}-{\bf{x}^{\prime}}$,
${\bf{r}}\equiv({\bf{x}}+{\bf{x}^{\prime}})/2$,
$\hat{H}_{{}_{\sigma}}\equiv-\hbar^{2}\nabla^{2}/2m-E_{F}^{F,S}-\sigma J$ with
$J$ the exchange energy and $\sigma=\uparrow$ ($\downarrow$) for up (down)
spin ($\bar{\sigma}=-\sigma$), and $\Phi$ is the AF order parameter.
$\Delta({\bf{s}},{\bf{r}})$ is the Cooper pair order parameter in terms of
relative and center-of-mass coordinates. In the FM region, we define
$E_{F}^{F}\equiv\hbar^{2}q_{F}^{2}/2m=(\hbar^{2}q_{F\uparrow}^{2}/2m+\hbar^{2}q_{F\downarrow}^{2}/2m)/2$
as the spin averaged value. It differs from the value in the superconductor,
$E_{F}^{S}\equiv\hbar^{2}k_{F}^{2}/2m$, to which a FWM can occur between the
FM and EDSC regions Zutic00 . In (LABEL:eq:BdG), the wave functions $u_{1}$
and $v_{1}$ are considered related to the $\mathbf{k}$ subspace, while $u_{2}$
and $v_{2}$ are related to the ${\bf k}+{\bf Q}$ subspace. Comparing with the
first and second lines of Eq. (LABEL:eq:BdG), minus signs associated with the
$\Delta({\bf{s}},{\bf{r}})$ term in the third and fourth lines occur due to
the symmetry requirement, $\Delta({\bf k}+{\bf Q})=-\Delta({\bf{k}})$, for a
$d_{x^{2}-y^{2}}$-wave superconductor in ${\bf k}$ space. At Fermi level, the
$d_{x^{2}-y^{2}}$-wave SC gap is
$\Delta({\bf{\hat{k}}}_{F})\equiv\Delta_{0}\sin 2\theta$ with $\Delta_{0}$ the
gap magnitude and $\theta$ the azimuthal angle relative to the $x$-axis.
Figure 1: Schematic plot showing all possible reflection and transmission
processes for an up-spin electron incident into the FM/I/EDSC junction. An AF
order is assumed to exist in the EDSC. For convenience for a $d$-wave
superconductor, $\mathbf{k_{x}}$ axis is chosen to be along the [110]
direction. The right-bottom inset shows a given Fermi wave vector
$\mathbf{k}_{F}=\left(k_{F},k_{y},k_{z}\right)$ and its coupled AF wave vector
$\mathbf{k}_{F}+\mathbf{Q}\equiv\mathbf{k}_{F\mathbf{Q}}=\left(-k_{F},k_{y},k_{z}\right)$.
Both vectors are tied to the Fermi surface, which is approximated by a square
(thick line). NR, AR, AF-NR, and AF-AR stand for normal reflection, Andreev
reflection, antiferromagnetic-normal reflection, and antiferromagnetic-Andreev
reflection respectively. Their corresponding reflection angles are also shown.
For the case of an incident down-spin electron, all spin indices just reverse.
In a $d$-wave superconductor, it’s useful to consider a junction to which the
superconductor surface is allied along the [110] direction. A thin insulating
layer exists between the ferromagnetic metal and the superconductor (see Fig.
1) to which the barrier potential is assumed to take a delta function,
$V(x)=H\delta\left(x\right)$. Considering that an up-spin electron is injected
into the FM/I/EDSC junction from the ferromagnetic metal side, there are four
possible reflections as follows: (a) Normal reflection (NR): reflected as
electrons. (b) Andreev reflection (AR): reflected as holes, due to electron
and hole coupling in the $\mathbf{k}$ subspace. (c) Antiferromagnetic-Normal
reflection (AF-NR): reflected as electrons, due to the coupling of
$\mathbf{k}$ and ${\bf k}+{\bf Q}$ subspaces. (d) Antiferromagnetic-Andreev
reflection (AF-AR): reflected as holes, due to electron and hole coupling in
the ${\bf k}+{\bf Q}$ subspace (see Fig. 1).
In addition to the effect of AF order, AR is largely modified due to the
exchange field of ferromagnetic metal when electron is not normally incident
into the EDSC region. Owing to the momentum conserved parallel to the
interface, Snell’s law Jackson75 ; Zutic00 ; Kashiwaya99 requires that
$\displaystyle
q_{F\sigma}\sin\theta_{N\sigma}=q_{F\bar{\sigma}}\sin\theta_{A\bar{\sigma}}=$
$\displaystyle k_{F}\sin\theta_{S\sigma},$ (2)
where $\theta_{N\sigma}$, $\theta_{A\bar{\sigma}}$, and $\theta_{S\sigma}$ are
the angles of NR, AR, and transmission into the SC respectively (see Fig. 1).
Incident angle $\theta_{N\sigma}$ is typically not equal to the AR angle
$\theta_{A\bar{\sigma}}$ except when $J=0$ or for normal incidence. Assuming
that there is no FWM and $q_{F\downarrow}<k_{F}<q_{F\uparrow}$, ranges of six
normal and Andreev reflection angles are
$0<\theta_{N\uparrow}<\sin^{-1}(k_{F}/q_{F\uparrow})\equiv\theta_{c2}$,
$0<\theta_{A\uparrow}<\sin^{-1}(q_{F\downarrow}/q_{F\uparrow})\equiv\theta_{c1}$,
and
$0<\theta_{S\uparrow},\theta_{S\downarrow}<\sin^{-1}(q_{F\downarrow}/k_{F})$,
while $\theta_{A\downarrow}$ and $\theta_{N\downarrow}$ can be any angles. For
AF reflections, the angles $\theta_{A\sigma}^{\rm AF}=\pi-\theta_{A\sigma}$
and $\theta_{N\sigma}^{\rm AF}=\theta_{N\sigma}$ respectively. It is noted
that when $\theta_{N\uparrow}$ is within the range
$\theta_{c1}<\theta_{N\uparrow}<\theta_{c2}$, $x$ component of the wave
vector, $\sqrt{q_{F\downarrow}^{2}-k_{F}^{2}\sin^{2}\theta_{S\uparrow}}$,
becomes purely imaginary for the AR process Zutic00 ; Kashiwaya99 . Although
spin down electron as a propagating wave is impossible for AR, it can still
transmit into the superconductor side.
As emphasized by Kashiwaya et al. Kashiwaya99 , one can define two types of
conductance in a FM, namely the charge and spin conductances. As a matter of
fact, the normalized angle and spin dependent tunneling charge conductance is
given by
$\displaystyle
C_{q\sigma}=1-\left|{R_{N\sigma}}\right|^{2}+a_{\sigma}\left|{R_{A\bar{\sigma}}}\right|^{2}+\left|{R^{AF}_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R^{AF}_{A\bar{\sigma}}}\right|^{2},$
(3)
where $a_{\downarrow}\equiv 1$ and $a_{\uparrow}\equiv
L_{\downarrow}\lambda_{2\downarrow}/L_{\uparrow}\lambda_{1\uparrow}$ with
$\lambda_{1\uparrow}=\cos\theta_{N\uparrow}/\cos\theta_{S\uparrow}$,
$\lambda_{2\downarrow}=\cos\theta_{A\downarrow}/\cos\theta_{S\uparrow}$, and
$L_{\sigma}=\sqrt{(q_{F}/k_{F})(1-\sigma J/E_{F}^{F})}$. Detailed derivations
of all four reflection coefficients ($R_{N\sigma}$, $R_{A\bar{\sigma}}$,
$R^{AF}_{N\sigma}$, and $R^{AF}_{A\bar{\sigma}}$) are given in Appendix A.
Similarly, the normalized angle and spin dependent spin conductance is given
by
$\displaystyle
C_{s\sigma}=1-\left|{R_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R_{A\bar{\sigma}}}\right|^{2}+\left|{R^{AF}_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R^{AF}_{A\bar{\sigma}}}\right|^{2}.$
(4)
Comparing with the results of charge conductance in (3), due to the spin
imbalance induced by the exchange field, different sign of $R_{A\bar{\sigma}}$
terms occurs in the spin conductances. Consequently normalized total charge
(spin) conductance is given by
$\displaystyle G_{q(s)}(E)=G_{q(s)\uparrow}(E)\pm G_{q(s)\downarrow}(E),$ (5)
where $+$ ($-$) sign is for charge (spin) channel and
$\displaystyle
G_{q(s)\sigma}(E)=\frac{1}{{G^{N}_{q(s)}}}\int_{\alpha}^{\beta}{d\theta_{N\sigma}\cos\theta_{N\sigma}C_{q(s)\sigma}(E,\theta_{N\sigma})P_{\sigma}}.$
(6)
The lower and upper integration limits of $\alpha$ and $\beta$ are restricted
by Snell’s law (as discussed before) or experimental setup. In practice,
integration over two separate ranges of incident angle, i.e.,
$0<\left|{\theta_{N\sigma}}\right|<\theta_{c1}$ and
$\theta_{c1}<\left|{\theta_{N\sigma}}\right|<\theta_{c2}$ should be carried
and results are added up to the total conductance. In (6), the normal-state
charge (spin) conductance
$\displaystyle
G^{N}_{q(s)}=\int_{-\pi/2}^{\pi/2}{d\theta_{N\sigma}\cos\theta_{N\sigma}[C_{N\uparrow}P_{\uparrow}\pm
C_{N\downarrow}P_{\downarrow}]},$ (7)
where
$\displaystyle
C_{N\sigma}(\theta_{N\sigma})=\frac{{4\lambda_{1}L_{\sigma}}}{{\left|{1+\lambda_{1}L_{\sigma}+2iZ}\right|^{2}}}$
(8)
with $Z=mH/\hbar^{2}k_{F}$ the barrier. In both (6) and (7), we have
introduced a factor $P_{\sigma}=(E_{F}^{F}+\sigma J)/2E_{F}^{F}$ which can be
interpreted as the probability of spin-$\sigma$ incident electron as a
function of the exchange energy Kashiwaya99 ; Zutic00 ; Linder07 . When $J=0$,
$P_{\uparrow}=P_{\downarrow}=1/2$.
In addition to the conductances, the normalized total charge (spin) current
can be given by
$\displaystyle\begin{array}[]{l}I_{q(s)}=I_{q(s)\uparrow}\pm
I_{q(s)\downarrow},\\\ \end{array}$ (10)
where
$\displaystyle I_{q(s)\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{I^{N}_{q(s)}}\int_{-\infty}^{\infty}{dE}\int_{\alpha}^{\beta}d\theta_{N\sigma}\cos\theta_{N\sigma}C_{q(s)\sigma}(E,\theta_{N\sigma})P_{\sigma}q_{F\sigma}$
with
$\displaystyle{I^{N}_{q(s)}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{dE}\int_{-\pi/2}^{\pi/2}{d\theta_{N\sigma}\cos\theta_{N\sigma}[C_{N\uparrow}P_{\uparrow}q_{F\uparrow}\pm
C_{N\downarrow}P_{\downarrow}q_{F\downarrow}]}.$
Charge and spin currents and their conversion are important probes for spin-
related phenomena such as those in spin Hall effect.
## III Results and Discussions
Both charge and spin conductances are important probes for tunneling in spin-
polarized junctions. In this paper, we will focus on the charge conductance
however. Moreover, for simplicity, all the results presented are for normal
incidence ($\theta_{N\sigma}=0$).
### III.1 Midgap Surface States
Detailed derivations of the midgap surface states (MSS) in the current FM/EDSC
junction are give in Appendix B. Basically it is an extension of Hu’s Hu94
and Liu and Wu’s Liu07 works. The boundary condition that leads to the MSS is
the wave function $\psi_{N\sigma}(x=-d)=0$ for a free boundary at $x=-d$.
Consequently one obtains the following condition for the MSS (see Appendix B):
$\displaystyle e^{-2ik_{1\sigma}d}E_{+}+e^{2ik_{1\sigma}d}E_{-}=2\Phi,$ (13)
where $E_{\pm}\equiv E\pm\varepsilon^{\prime}_{\sigma}$ with
$\varepsilon^{\prime}_{\sigma}=\sqrt{(E+\sigma J)^{2}-\Delta^{2}-\Phi^{2}}$
incident spin-$\sigma$ electron is assumed to have wave vector $k_{1\sigma}$
along the $x$ direction.
In case of $J=0$, the result is reduced to our previous case without spin
polarization Liu07 . In case of $J=\Phi=0$, the result is reduced to Hu’s case
Hu94 , i.e.,
$\displaystyle
e^{4ik_{1}d}=-\frac{{E+{\varepsilon^{\prime}}}}{{E-{\varepsilon^{\prime}}}},$
(14)
where $\varepsilon^{\prime}\equiv\sqrt{E^{2}-\Delta^{2}}$. The most crucial
result of the above is that there exists a zero-energy state which is
responsible for the ZBCP widely observed in hole-doped $d_{x^{2}-y^{2}}$-wave
cuprate superconductors Hu94 . When $J=0$ but $\Phi\neq 0$, zero-energy state
no longer exists such that the energy of the existing state is always finite
($E=\Phi$ in the limit of $d=0$). This leads to the splitting of the ZBCP.
When $J$ is also finite, there will be further effect caused by spin
polarization although the splitting peak remains at $E=\Phi$ in the limit of
$d=0$. It is interesting to note that beyond the quasiclassical approximation,
a more accurate calculation for the surface bound-state energies in
$d_{x^{2}-y^{2}}$-wave and other unconventional cuprate superconductors was
reported by Walker et al. Walker97 .
### III.2 Effect of Fermi-Wave-Vector Mismatch
Tunneling conductances are in general strongly modified by the effect of
Fermi-wave-vector mismatch (FWM) Zutic00 ; Linder07 . In our case, due to the
presence of the AF order, the conductance spectra are somewhat different from
those obtained by Žutić and Valls Zutic00 and Linder et al. Linder07 . Here
we introduce a parameter
$\displaystyle L_{0}\equiv{q_{F}\over k_{F}}$ (15)
to account for the effect of FWM. Both $L_{0}$ greater and smaller than one
cases are considered. As shown in Figs. 2-4 for the normalized charge
conductance $G_{q}$, the effect of FWM is typically strong when $L_{0}<1$,
while it has relatively minor effect when $L_{0}>1$ (That is, $G_{q}$ changes
little from the no FWM $L_{0}=1$ case.). As first pointed out by Blonder and
Tinkham Blonder83 , FWM can be interpreted as a type of barrier which could
enhance the conductance near zero bias.
Figure 2: Effect of FWM on normalized charge conductance spectra $G_{q}$ for
various wave-vector mismatch value $L_{0}$ with fixed barrier $Z=0$, AF order
$\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$. Figure 3: Effect of FWM
on normalized charge conductance spectra $G_{q}$ for various wave-vector
mismatch value $L_{0}$ with fixed barrier $Z=1$, AF order $\Phi=0$, and spin
polarization $X=0$. This can be considered as the case of hole-doped cuprate
superconductors without AF order and in the absence of spin polarization.
In our previous paper Liu07 , it was shown that ZBCP of a
$d_{x^{2}-y^{2}}$-wave superconductor can be split by the AF order $\Phi$. No
spin-active barrier Kashiwaya99 ; Linder07 , external magnetic field, and spin
polarization effects were considered in our previous case though. Previously
Žutić and Valls Zutic00 had given a detailed analysis of the FWM effect on
the conductance in ferromagnet/s-wave and d-wave superconductor junctions.
Here we show how FWM influences the conductance in the current case and point
out the key physics. Fig. 2 plots $G_{q}$ for various $L_{0}$ with barrier
$Z=0$, AF order $\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$ [see Eq.
(18) for the definition of $X$]. One sees that the effect of FWM is most
noticeable at large FWM ($L_{0}=0.2$ case) to which a ZBCP is developed, while
the spectra are humdrum when $L_{0}\geq 1$. Since no barrier ($Z=0$) is
considered, no effect of AF order and spin polarization is seen in terms of
peak splitting. Note that normalized zero-bias conductance is not equal to 2
due to the presence of AF order and spin polarization. In order to compare
with the case of hole-doped high-$T_{c}$ superconductors (without AF order),
Fig. 3 plots $G_{q}$ at different values of $L_{0}$ with $\Phi=X=0$ and $Z=1$.
One sees that ZBCP is largely enhanced by the FWM effect (see $L_{0}=0.2$
case). Thus FWM can significantly enhance the number of midgap surface states
near zero-bias voltage.
Figure 4: Effect of FWM on normalized charge conductance spectra $G_{q}$ for
various wave-vector mismatch value $L_{0}$ with fixed barrier $Z=1$, AF order
$\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$. FWM causes the reduction
of conductance at zero bias, while enhances the splitting peak associated with
the AF order.
Aiming to electron-doped cuprate superconductors, Fig. 4 shows the effect of
FWM on the splitting peak when the AF order is present ($\Phi=0.5\Delta_{0}$).
Here barrier $Z=1$ and spin polarization $X=0.5$. In contrast to the case of
$\Phi=0$ in Fig. 3, FWM actually reduces the number of midgap surface states
near zero bias. At the same time, it enhances the strength of the splitting
peak associated with the AF order. Following the idea of Blonder and Tinkham
Blonder83 such that FWM can be interpreted as a type of barrier, the
enhancement of ZBCP in Fig. 3 and the reduction of zero-bias conductance in
Fig. 4 is a natural outcome at large FWM.
In principle, the effect of FWM should be included when a serious calculation
is performed for spin-polarized conductances.
### III.3 Effect of Spin Polarization
In the literature, there exists different definitions of spin polarization.
One example is the “tunneling polarization” proposed by Tedrow and Meservey
Tedrow94 . In point contact experiment, the more suitable definition is the
so-called “contact polarization” Soulen98
$\displaystyle
P_{c}=\frac{{N_{\uparrow}(E_{F})v_{F\uparrow}-N_{\downarrow}(E_{F})v_{F\downarrow}}}{{N_{\uparrow}(E_{F})v_{F\uparrow}+N_{\downarrow}(E_{F})v_{F\downarrow}}},$
(16)
where $v_{F\sigma}$ and $N_{\sigma}(E_{F})$ are respectively the Fermi
velocity and DOS at Fermi level for spin-$\sigma$ electron. Since
$I_{\sigma}\propto N_{\sigma}(E_{F})v_{F\sigma}$, Eq. (16) is identical to
$\displaystyle
P_{c}=\frac{{I_{\uparrow}-I_{\downarrow}}}{{I_{\uparrow}+I_{\downarrow}}}.$
(17)
However, the most natural definition of spin polarization is
$\displaystyle
X\equiv\frac{{N_{\uparrow}(E_{F})-N_{\downarrow}(E_{F})}}{{N_{\uparrow}(E_{F})+N_{\downarrow}(E_{F})}}.$
(18)
In ballistic point contact situation, the electron density of states in the
presence of an exchange field can be written as
$N_{\sigma}(E_{F})=q_{F\sigma}^{2}A/4\pi$, where $A$ is the area of the
interface. Thus $X=J/E_{F}^{F}$ with
$E_{F}^{F}\equiv\hbar^{2}q_{F}^{2}/2m=(\hbar^{2}q_{F\uparrow}^{2}/2m+\hbar^{2}q_{F\downarrow}^{2}/2m)/2$
Chalsani07 . In Sec. III.E, we will show that spin polarization $X$ can be
determined by a general formula in combination with the experimental
conductance data.
Figure 5: Effect of spin polarization on normalized charge conductance spectra
$G_{q}$ for various spin polarization value $X$ with fixed barrier $Z=0$, AF
order $\Phi=0.5\Delta_{0}$, and without FWM ($L_{0}=1$). Figure 6: Effect of
spin polarization on normalized charge conductance spectra $G_{q}$ for various
spin polarization value $X$ with fixed barrier $Z=1$, AF order $\Phi=0$, and
without FWM ($L_{0}=1$). This is considered an example of the hole-doped
cuprate superconductor without AF order and FWM. Figure 7: Effect of spin
polarization on normalized charge conductance spectra $G_{q}$ for various spin
polarization value $X$ with fixed barrier $Z=1$, AF order
$\Phi=0.5\Delta_{0}$, and without FWM ($L_{0}=1$). Low-energy anomalous
conductance enhancement arises due to AF contributions (see text for details).
Note that current quasiparticle wave function of BdG equations has four
components which involve two components associated with the AF order. In the
limit of $Z=0$ and without spin polarization ($X=0$), normalized charge
conductance has value 2 as expected (see Fig. 5). With a finite AF order
($\Phi=0.5\Delta_{0}$), the resulting effective gap magnitude is about
$\tilde{\Delta}\simeq 1.12\Delta_{0}$ (see Fig. 5). In general, at
$E<\tilde{\Delta}$, effect of spin polarization is to suppress the
conductance. When FWM is absent ($L_{0}=1$) together with $Z=0$, normal
reflection has no contribution and Andreev reflection actually dominates the
tunneling process for $E<\tilde{\Delta}$ Blonder82 . In our current case,
Andreev reflection involves contributions from both $R_{A}$ and $R_{A}^{AF}$
channels.
The most interesting results occur when the barrier $Z$ is finite. When the AF
order $\Phi=0$ (as for the case of hole-doped cuprate superconductors) to
which $R_{N}^{AF}=R_{A}^{AF}=0$, ZBCP appears whose (normalized) strength is
largely suppressed due to the strong spin polarization effect (see Fig. 6).
However, as seen in Fig. 7, when AF order is finite ($\Phi=0.5\Delta_{0}$), in
contrast, the strengths of both the zero-bias conductance and the splitting
peak turn out to get enhanced by the strong spin polarization effect. This
“anomalous conductance enhancement” phenomenon is in drastic contrast as
compared to the ZBCP associated with $\Phi=0$ case (Fig. 6). These somewhat
surprising results arise due to a significant increase of $|R_{N}^{AF}|$ and
at the same time, a significant decrease of $|R_{A}^{AF}|$ for large $X$ cases
– a consequence of the interplay between AF order and spin polarization. Since
$|R_{N}^{AF}|$ contributes positively to the conductance, while $|R_{A}^{AF}|$
contributes negatively to the conductance [see Eq. (3)], resultantly they
cause the anomalous conductance enhancement at low energies ($E\leq\Phi$). It
should be emphasized that this low-energy conductance enhancement is not due
to the spin-flip effect which is not considered in this paper. At higher
energies, $E>\Phi$, the conductances behave more normally such that they get
suppressed due to the spin polarization effect. Anomalous conductance
enhancement at low energies can serve as a test to see whether there is an
significant AF order in electron-doped cuprate superconductors.
Interface barrier and band structure are in general having strong effect on
spin polarization. Kant et al. have built an“extended interface” model to
illustrate the decay of spin polarization Kant02 . Besides, Mazin had a
detailed discussion on the definition of spin polarization and band structure
effects in spin polarization Mazin99 .
### III.4 Effect of Effective Barrier
In the study of the tunneling transition in Cu-Nb point contacts, Blonder and
Tinkham Blonder83 pointed out that barrier is not the only source for normal
reflection and in a more realistic system, one should consider “impedance” or
FWM as well which results in normal reflection even with no barrier present.
They proposed an effective barrier $Z_{\rm eff}=[Z^{2}+(1-r)^{2}/4r]^{1/2}$
where $r$ is the Fermi velocity ratio. They showed that effective barrier has
an obvious effect on the conductance when $E<\Delta_{0}$, as shown in Fig. 2
of Ref. Blonder83 . Here we generalize their idea to consider a spin, FWM, and
angle dependent effective barrier $Z_{\rm eff}$ Blonder83 ; Zutic00 :
$\displaystyle Z_{\rm
eff}\equiv[Z^{2}+(1-L_{\sigma})^{2}/4L_{\sigma}]^{1/2}/\cos\theta_{S\sigma},$
(19)
where $L_{\sigma}=q_{F\sigma}/k_{F}$ corresponds to the spin-dependent FWM. It
is noted that we are not considering the spin-active barrier which has spin
filtering effects and can lead to the ZBCP splitting Kashiwaya99 ; Linder07 .
Instead we propose a possible alternative mechanism to account for the decay
of spin polarization. Based on the generalized effective barrier, spin-up and
-down particles experience different strength of effective barrier that causes
spin-up and -down currents to decrease at different speed as compared to the
current in the absence of barrier. Consequently, $Z_{\rm eff}$ can modify the
values of $I_{\uparrow}-I_{\downarrow}$ (and thus $P_{c}$) dramatically. With
this strong effect at work, the decay of spin polarization should not be
dominant by the spin-flitting process in the point contact spin polarization
case.
Figure 8: Effect of effective barrier $Z_{\rm eff}$ on normalized charge
conductance spectra $G_{q}$ for various values of FWM $L_{0}$. AF order
$\Phi=0.5\Delta_{0}$ and spin polarization $X=0.5$. The bare barrier $Z$ is
set to zero, while $Z_{\rm eff}$ is given by Eq. (19).
As seen in Eq. (19), $Z_{\rm eff}$ can differ significantly from $Z$,
especially when $Z$ is small. Essentially their difference can be measured by
spin-polarized tunneling experiments. In Fig. 8, we compare the effects of $Z$
and $Z_{\rm eff}$ on the conductance with bare barrier $Z$ set to zero and
vary the FWM $L_{0}$ value. For $Z=0$ and $\theta_{S\sigma}=0$, $Z_{\rm
eff}=[(1-L_{\sigma})^{2}/4L_{\sigma}]^{1/2}$ [see (19)]. In our case, we have
also included AF order and spin polarization. The difference is most
noticeable when FWM is large ($L_{0}=0.2$ case). Since $Z=0$, AF order and
spin polarization have little effect at small FWM. However, when FWM is large,
AF order and spin polarization can have a strong effect such that a splitting
peak can develop at $E\approx\Phi=0.5\Delta_{0}$ with the effective barrier
$Z_{\rm eff}$ (see Fig. 8). This supports Blonder and Tinkham’s idea of
“impedance” mismatch which enhances the normal reflection.
### III.5 A General Formula for Determining the Spin Polarization
Based on the phenomenon of Andreev reflection, Soulen et al. Soulen98
proposed a formula for determining the point contact spin polarization $P_{c}$
[see Eqs. (16) and (17)] when the normalized zero-bias conductance data is
compared. Their original form was
$\displaystyle G(0)/G_{N}=2(1-P_{c}),$ (20)
which is valid only when FWM is absent Strijkers01 . Since Andreev reflection
could be strongly modified due to the FWM effect, it’s useful to replace Eq.
(20) by
$\displaystyle
G(0)/G_{N}=\left[1+\left|{R_{A}}\right|^{2}-\left|{R_{A}^{AF}}\right|^{2}\right](1-P_{c}),$
(21)
where $R_{A}$ and $R^{AF}_{A}$ are the AR and AF-AR coefficients respectively.
Eq. (21) can be reduced to Eq. (20) when the exchange energy $J$ is set to
zero in $R_{A}$ and the AF order $\Phi$ is set to zero in $R^{AF}_{A}$. Note
also that the parameter $X$ should be set to zero when the “contact
polarization” $P_{c}$ is determined under the idea of Soulen et al.
Here we propose a more general formula for determining the spin polarization:
$\displaystyle G(0)/G_{N}=A_{\uparrow}+A_{\downarrow},$ (22)
where
$\displaystyle
A_{\uparrow}=\int_{\alpha}^{\beta}d\theta_{N\sigma}\cos\theta_{N\sigma}(1+a_{\uparrow}|{R_{A\downarrow}}|^{2}-a_{\uparrow}|{R_{A\downarrow}^{AF}}|^{2})P_{\uparrow}$
(23)
and
$\displaystyle
A_{\downarrow}=\int_{\alpha}^{\beta}{d\theta_{N\sigma}\cos\theta_{N\sigma}(1+|{R_{A\uparrow}}|^{2}-|{R_{A\uparrow}^{AF}}|^{2})}P_{\downarrow}.$
(24)
Here $R_{A\sigma}=R_{A\sigma}(L_{0},X,\Phi,\theta_{N\sigma})$ and
$R_{A\sigma}^{AF}=R_{A\sigma}^{AF}(L_{0},X,\Phi,\theta_{N\sigma})$ with $E=0$.
Eq. (22) is a natural result of our earlier formalism. It is regarded as the
generalization of Eq. (20) of Soulen et al., which includes the effects of
FWM, spin polarization, AF order, as well as the incident angle.
## IV Conclusions
Tunneling experiment provides a useful tool for probing the properties of a
superconductor such as the magnitude and symmetry of the superconducting order
parameter, quasiparticle density of states, and any existing competing orders.
In fact, tunneling experiment is also a powerful probe for investigating the
spin-charge separation in connection with the spin-injection techniques. This
involves both charge imbalance and spin imbalance studies.
In this paper, we have presented a detailed study of the tunneling conductance
spectra of a ferromagnetic metal/electron-doped superconductor junctions,
taking into account an AF order existing in the the electron-doped
superconductor. Interesting result, such as low-energy anomalous conductance
enhancement, occurs as a result of the interplay between AF order and spin
polarization (see Fig. 7). These results in turn provide a further opportunity
to test whether there is an significant AF order in electron-doped cuprate
superconductors.
###### Acknowledgements.
This work is supported by National Science Council of Taiwan (Grant No.
96-2112-M-003-008) and National Natural Science Foundation of China (Grant No.
10347149). We also acknowledge the support from the National Center for
Theoretical Sciences, Taiwan.
## Appendix A Reflection Coefficients
Under the WKBJ approximation Bardeen69 ; Bar-Sagi72 ; Hu75 ; Bruder90 ;
Blonder82 ; Hu94 ; Tanaka95 ; Kashiwaya96 , the wave functions in the
generalized BdG equations (LABEL:eq:BdG) can be approximated by
$\displaystyle\left({\begin{array}[]{*{20}c}{u_{1\sigma}}\\\
{v_{1\bar{\sigma}}}\\\ {u_{2\sigma}}\\\ {v_{2\bar{\sigma}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{e^{i{\bf{k}}_{F}\cdot{\bf{r}}}\tilde{u}_{1\sigma}}\\\
{e^{i{\bf{k}}_{F}\cdot{\bf{r}}}\tilde{v}_{1\bar{\sigma}}}\\\
{e^{i{\bf{k}}_{F{\bf Q}}\cdot{\bf{r}}}\tilde{u}_{2\sigma}}\\\
{e^{i{\bf{k}}_{F{\bf Q}}\cdot{\bf{r}}}\tilde{v}_{2\bar{\sigma}}}\\\
\end{array}}\right).$ (33)
Thus one obtains a set of Andreev equations in the $x$ direction,
$\displaystyle E\tilde{u}_{1\sigma}(x)$ $\displaystyle=$ $\displaystyle
H_{\sigma}\tilde{u}_{1\sigma}(x)+\Delta({\bf{\hat{k}}}_{F})\tilde{v}_{1\bar{\sigma}}(x)+\Phi\tilde{u}_{2\sigma}(x)$
$\displaystyle E\tilde{v}_{1\bar{\sigma}}(x)$ $\displaystyle=$
$\displaystyle\Delta^{*}({\bf{\hat{k}}}_{F})\tilde{u}_{1\sigma}(x)-H_{\sigma}\tilde{v}_{1\bar{\sigma}}(x)+\Phi\tilde{v}_{2\bar{\sigma}}(x)$
$\displaystyle E\tilde{u}_{2\sigma}(x)$ $\displaystyle=$
$\displaystyle\Phi\tilde{u}_{1\sigma}(x)-H_{\sigma}\tilde{u}_{2\sigma}(x)+\Delta({\bf{\hat{k}}}_{F{\bf
Q}})\tilde{v}_{2\bar{\sigma}}(x)$ $\displaystyle
E\tilde{v}_{2\bar{\sigma}}(x)$ $\displaystyle=$
$\displaystyle\Phi\tilde{v}_{1\bar{\sigma}}(x)+\Delta^{*}({\bf{\hat{k}}}_{F{\bf
Q}})\tilde{u}_{2\sigma}(x)+H_{\sigma}\tilde{v}_{2\bar{\sigma}}(x),$
where $H_{\sigma}=-\frac{{i\hbar^{2}k_{F}}}{m}\frac{d}{{dx}}-\sigma J$ and $x$
is the coordinate normal to the interface. The $d_{x^{2}-y^{2}}$-wave SC gap
$\Delta({\bf{\hat{k}}}_{F})=-\Delta({\bf{\hat{k}}}_{F{\bf
Q}})\equiv\Delta_{0}\sin 2\theta$ with $\theta$ the azimuthal angle relative
to the $x$-axis. In obtaining Eq. (LABEL:eq:BdG2), the Fourier transform of
the Cooper pair order parameter $\Delta({\bf s},{\bf r})$ from relative
coordinate ${\bf s}$ to ${\bf k}$ space is assumed to take the form,
$\Delta({\bf{k}},{\bf{r}})=\Delta({\bf\hat{k}}_{F})\Theta(x)$, with
$\Theta\left(x\right)$ the Heaviside step function Hu94 ; Kashiwaya96 .
Solving Eq. (LABEL:eq:BdG2), one obtains four eigenvectors which build up the
spin-$\sigma$ wave function in the superconductor region ($x>0$) Zutic00 ,
$\displaystyle\psi_{S\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}\Delta\\\
{E_{-}}\\\ 0\\\ \Phi\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{E_{+}}\\\
\Delta\\\ \Phi\\\ 0\\\ \end{array}}\right)}\right]e^{ik^{+}x}$ (43)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{E_{-}}\\\
{-\Delta}\\\ \Phi\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-\Delta}\\\
{E_{+}}\\\ 0\\\ \Phi\\\ \end{array}}\right)}\right]e^{-ik^{-}x}.$ (52)
Here $E_{\pm}\equiv E\pm\varepsilon_{\sigma}$ with
$\varepsilon_{\sigma}=\sqrt{E^{2}-\Delta^{2}-\Phi^{2}}$,
$\Delta\equiv\Delta({\bf{\hat{k}}}_{F})$,
$k^{+}=k^{-}=k_{F}\cos\theta_{S\sigma}$, and $c_{i\sigma}$ are coefficients of
the corresponding waves. As pointed out by Blonder et al. Blonder82 , there is
no need to normalize the coefficients as it just complicates the calculation.
If we set $\Phi=J=0$ and normalize the coefficients, it will reduce to the
case for a typical N/I/S junction Blonder82 ; Tanaka95 ; Kashiwaya96 .
Since we consider that there is an AF order in the EDSC side, an incident
electron from the FM side will have four possible reflections Liu07 . The
spin-$\sigma$ wave function in the FM side $(x<0)$ with incident angle
$\theta_{N\sigma}$ can thus be written as Blonder82 ; Kashiwaya96 ;
Kashiwaya99
$\displaystyle\Psi_{N\sigma}(x)=\left({\begin{array}[]{*{20}c}{e^{iq_{F\sigma}\cos\theta_{N\sigma}x}+R_{N\sigma}e^{-iq_{F\sigma}\cos\theta_{N\sigma}x}}\\\
{R_{A\bar{\sigma}}e^{iq_{F\bar{\sigma}}\cos\theta_{A\bar{\sigma}}x}}\\\
{R^{AF}_{N\sigma}e^{iq_{F\sigma}\cos\theta_{N\sigma}x}}\\\
{R^{AF}_{A\bar{\sigma}}e^{-iq_{F\bar{\sigma}}\cos\theta_{A\bar{\sigma}}x}}\\\
\end{array}}\right),$ (57)
where $R_{N\sigma}$, $R_{A\bar{\sigma}}$, $R^{AF}_{N\sigma}$, and
$R^{AF}_{A\bar{\sigma}}$ are amplitudes of NR, AR, AF-NR, and AF-AR
respectively. Applying the following boundary conditions:
$\displaystyle\psi_{N\sigma}\left(x\right)|_{x=0^{-}}$ $\displaystyle=$
$\displaystyle\psi_{S\sigma}\left(x\right)|_{x=0^{+}}$ (58)
$\displaystyle\frac{2mH}{\hbar^{2}}\psi_{S\sigma}\left(x\right)|_{x=0^{+}}$
$\displaystyle=$
$\displaystyle\frac{d\psi_{S\sigma}\left(x\right)}{dx}|_{x=0^{+}}-\frac{d\psi_{N\sigma}\left(x\right)}{dx}|_{x=0^{-}},$
the four reflection amplitudes (coefficients) are solved to be
$\displaystyle R_{N\sigma}$ $\displaystyle=$
$\displaystyle\frac{{E_{-}(1-L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})B}}{{(1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})D}}$
$\displaystyle-$
$\displaystyle\frac{{\Delta(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}+2iZ_{\theta})A}}{{(1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})D}}$
$\displaystyle-$
$\displaystyle\frac{{1-L_{\sigma}\lambda_{1\sigma}-2iZ_{\theta}}}{{1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta}}}$
$\displaystyle R_{A\bar{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{{\Delta(1+L_{\sigma}\lambda_{1\sigma}-2iZ_{\theta})B}}{{(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})D}}$
$\displaystyle+$
$\displaystyle\frac{{E_{-}(1-L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})A}}{{(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})D}}$
$\displaystyle R^{AF}_{N\sigma}$ $\displaystyle=$ $\displaystyle\frac{{\Phi
B}}{D}$ $\displaystyle R^{AF}_{A\bar{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{{\Phi A}}{D},$ (59)
where
$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\Delta
L_{\sigma}\lambda_{1\sigma}[1-L_{\sigma}L_{\bar{\sigma}}\lambda_{1\sigma}\lambda_{2\bar{\sigma}}+4Z_{\theta}^{2}$
$\displaystyle+$ $\displaystyle
2iZ_{\theta}(L_{\sigma}\lambda_{1\sigma}+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}})]$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
2L_{\sigma}\lambda_{1\sigma}[2L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}E+\varepsilon(1+L_{\bar{\sigma}}^{2}\lambda_{2\bar{\sigma}}^{2})]$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\Delta^{2}[(1-L_{\sigma}L_{\bar{\sigma}}\lambda_{1\sigma}\lambda_{2\bar{\sigma}}+4Z_{\theta}^{2})^{2}$
(60) $\displaystyle+$ $\displaystyle
4Z_{\theta}^{2}(L_{\sigma}\lambda_{1\sigma}+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}})^{2}]$
$\displaystyle+$ $\displaystyle[2L_{\sigma}\lambda_{1\sigma}E+4\varepsilon
Z_{\theta}^{2}+\varepsilon(1+L_{\sigma}^{2}\lambda_{1\sigma}^{2})]$
$\displaystyle\times$
$\displaystyle[2L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}E+4\varepsilon
Z_{\theta}^{2}+\varepsilon(1+L_{\bar{\sigma}}^{2}\lambda_{2\bar{\sigma}}^{2})].$
Moreover $Z_{\theta}=Z/\cos\theta_{S\sigma}$ with the barrier
$Z=mH/\hbar^{2}k_{F}$,
$\lambda_{1\sigma}=\cos\theta_{N\sigma}/\cos\theta_{S\sigma}$,
$\lambda_{2\bar{\sigma}}=\cos\theta_{A\bar{\sigma}}/\cos\theta_{S\sigma}$, and
${\rm{}}L_{\sigma}=\sqrt{q_{F}/k_{F}-\sigma(q_{F}/k_{F})(J/E_{FN})}$. It is
interesting to note in (59) that $R^{AF}_{N\sigma}$ and
$R^{AF}_{A\bar{\sigma}}$ are proportional to the AF order $\Phi$, as is
expected.
## Appendix B Midgap Surface States
Following Ref. Hu94 , we first assume that
$\displaystyle\left(\begin{array}[]{c}\tilde{u}_{l\sigma}\\\
\tilde{v}_{l\sigma}\end{array}\right)=e^{-\gamma_{\sigma}x}\left(\begin{array}[]{c}\hat{u}_{l\sigma}\\\
\hat{v}_{l\sigma}\end{array}\right),$ (65)
where $\gamma_{\sigma}$ is the attenuation constant for
$|{E({\bf{q}}_{F\sigma})}|<\sqrt{|{\Delta({\bf{\hat{k}}}_{F})}|^{2}+\Phi^{2}}$.
With (65), Eq. (LABEL:eq:BdG2) becomes
$\displaystyle E\left({\begin{array}[]{*{20}c}{\hat{u}_{1\sigma}}\\\
{\hat{v}_{1\bar{\sigma}}}\\\ {\hat{u}_{2\sigma}}\\\
{\hat{v}_{2\bar{\sigma}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{h}&{\Delta}&\Phi&0\\\
{\Delta}&{-h}&0&\Phi\\\ \Phi&0&{-h}&{-\Delta}\\\ 0&\Phi&{-\Delta}&{h}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\hat{u}_{1\sigma}}\\\
{\hat{v}_{1\bar{\sigma}}}\\\ {\hat{u}_{2\sigma}}\\\
{\hat{v}_{2\bar{\sigma}}}\\\ \end{array}}\right)$ (78)
for the superconducting overlayer ($x>0$). Here
$h={\varepsilon^{\prime}_{\sigma}}-\sigma J$ with
${\varepsilon^{\prime}_{\sigma}}=i\hbar^{2}m^{-1}\gamma_{\sigma}q_{F}\cos\theta_{N\sigma}$.
The wave-vector components parallel to the interface are conserved for all
possible processes.
Solving Eq. (78), one obtains double degenerate eigenvalues
$E=\pm\sqrt{\Delta^{2}+\Phi^{2}+{\varepsilon^{\prime}_{\sigma}}^{2}}-\sigma
J$, where $+$ ($-$) corresponds to the electron- (hole-) like QP excitation.
Similar to the wave function (52), superposition of the four eigenstates makes
up the formal wave function for the superconductor overlayer ($x>0$)
$\displaystyle\psi_{S\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}\Delta\\\
{E_{-}}\\\ 0\\\ \Phi\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{E_{+}}\\\
\Delta\\\ \Phi\\\ 0\\\
\end{array}}\right)}\right]e^{-\gamma_{\sigma}x}e^{ik^{+}x}$ (87)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{E_{-}}\\\
{-\Delta}\\\ \Phi\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-\Delta}\\\
{E_{+}}\\\ 0\\\ \Phi\\\
\end{array}}\right)}\right]e^{-\gamma_{\sigma}x}e^{-ik^{-}x}.$ (96) (97)
Here $E_{\pm}\equiv E\pm\varepsilon^{\prime}_{\sigma}$ with
$\varepsilon^{\prime}_{\sigma}=\sqrt{(E+\sigma J)^{2}-\Delta^{2}-\Phi^{2}}$,
$c_{i}$ are coefficients of the corresponding waves, and
$k^{+}=k^{-}=k_{F}\cos\theta_{S\sigma}$. At the interface, the wave functions
of FM and superconductor meet ideal continuity
$\psi_{N\sigma}(x=0)=\psi_{S\sigma}(x=0)$. After some algebra, the formal wave
function for the FM overlayer is obtained to be ($x<0$):
$\displaystyle\psi_{N\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}{e^{ik_{1\sigma}x}\Delta}\\\
{e^{-ik_{1\sigma}x}E_{-}}\\\ 0\\\ {e^{ik_{1\sigma}x}\Phi}\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{e^{ik_{1\sigma}x}E_{+}}\\\
{e^{-ik_{1\sigma}x}\Delta}\\\ {e^{-ik_{1\sigma}x}\Phi}\\\ 0\\\
\end{array}}\right)}\right]e^{ik^{+}x}$ (104)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{e^{-ik_{1\sigma}x}E_{-}}\\\
{-e^{ik_{1\sigma}x}\Delta}\\\ {e^{ik_{1\sigma}x}\Phi}\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-e^{-ik_{1\sigma}x}\Delta}\\\
{e^{ik_{1\sigma}x}E_{+}}\\\ 0\\\ {e^{-ik_{1\sigma}x}\Phi}\\\
\end{array}}\right)}\right]e^{-ik^{-}x},$ (112) (113)
where it is assumed that incident spin-$\sigma$ electron has the wave vector
$k_{1\sigma}$ along the $x$ direction. Considering the free boundary at
$x=-d$, $\psi_{N\sigma}(x=-d)=0$, one thus obtains the condition for the
surface bound states:
$\displaystyle e^{-2ik_{1\sigma}d}E_{+}+e^{2ik_{1\sigma}d}E_{-}=2\Phi.$ (114)
## References
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* (18) G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982).
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* (20) C. S. Liu and W. C. Wu, Phys. Rev. B 76, 220504(R) (2007), and references therein.
* (21) C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994).
* (22) P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
* (23) J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
* (24) M. B. Walker, P. Pairor, and M. E. Zhitomirsky, Phys. Rev. B 56, 9015 (1997).
* (25) G. E. Blonder and M. Tinkham, Phys. Rev. B 27, 112 (1983).
* (26) I. I. Mazin, Phys. Rev. Lett. 83, 1427 (1999).
* (27) S. Kashiwaya et al., Phys. Rev. B 53, 2667 (1996).
* (28) J. Bardeen et al., Phys. Rev. B 12, 3635 (1969).
* (29) J. Bar-Sagi and C. G. Kuper, Phys. Rev. Lett. 28, 1556 (1972).
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|
arxiv-papers
| 2009-08-24T06:27:29 |
2024-09-04T02:49:04.746616
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pok-Man Chiu, C. S. Liu, and W. C. Wu",
"submitter": "Cheng Shi Liu",
"url": "https://arxiv.org/abs/0908.3366"
}
|
0908.3401
|
Kolmogorov-Sinai entropy from recurrence times M. S. Baptista
# Kolmogorov-Sinai entropy from recurrence times
M. S. Baptista, E. J. Ngamga , Paulo R. F. Pinto[CMUP], Margarida Brito[CMUP],
J. Kurths[aberdeen,potsdam] partially supported by the “Fundação para a
Ciência e Tecnologia” (FCT).partially supported by SFB555.partially supported
by the “Fundação para a Ciência e Tecnologia” (FCT).partially supported by
SFB555. Institute for Complex Systems and Mathematical Biology, King’s
College, University of Aberdeen, AB24 3UE Aberdeen, United Kingdom Centro de
Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto,
Portugal Potsdam Institute for Climate Impact Research, Telegraphenberg,
14412 Potsdam, Germany
###### Abstract
ABSTRACT: Observing how long a dynamical system takes to return to some state
is one of the most simple ways to model and quantify its dynamics from data
series. This work proposes two formulas to estimate the KS entropy and a lower
bound of it, a sort of Shannon’s entropy per unit of time, from the recurrence
times of chaotic systems. One formula provides the KS entropy and is more
theoretically oriented since one has to measure also the low probable very
long returns. The other provides a lower bound for the KS entropy and is more
experimentally oriented since one has to measure only the high probable short
returns. These formulas are a consequence of the fact that the series of
returns do contain the same information of the trajectory that generated it.
That suggests that recurrence times might be valuable when making models of
complex systems.
## 1 Introduction
Recurrence times measure the time interval a system takes to return to a
neighborhood of some state, being that it was previously in some other state.
Among the many ways time recurrences can be defined, two approaches that have
recently attracted much attention are the first Poincaré recurrence times
(FPRs) [1] and the recurrence plots (RPs) [2].
While Poincaré recurrences refer to the sequence of time intervals between two
successive visits of a trajectory (or a signal) to one particular interval (or
a volume if the trajectory is high dimensional), a recurrence plot refers to a
visualization of the values of a square array which indicates how much time it
takes for two points in a trajectory with $M$ points to become neighbors
again. Both techniques provide similar results but are more appropriately
applicable in different contexts. While the FPRs are more appropriated to
obtain exact dynamical quantities (Lyapunov exponents, dimensions, and the
correlation function) of dynamical systems [3], the RPs are more oriented to
estimate relevant quantities and statistical characteristics of data coming
from complex systems [4].
The main argument in order to use recurrence times to model complex systems
[5] is that one can easily have experimental access to them. In order to know
if a model can be constructed from the recurrence times, it is essential that
at least the series of return times contains the same amount of information
generated by the complex system, information being quantified by the entropy.
Entropy is an old thermodynamic concept and refers to the disorganized energy
that cannot be converted into work. It was first mathematically quantified by
Boltzmann in 1877 as the logarithm of the number of microstates that a gas
occupies. More recently, Shannon [6] proposed a more general way to measure
entropy $H_{S}$ in terms of the probabilities $\rho_{i}$ of all possible $i$
states of a system:
$H_{S}=-\sum_{i}\rho_{i}\log{(\rho_{i})}.$ (1)
Applied to non-periodic continuous trajectories, e.g. chaotic trajectories,
$H_{S}$ is an infinite quantity due to the infinitely many states obtained by
partitioning the phase space in arbitrarily small sites. Therefore, for such
cases it is only meaningful to measure entropy relative to another trajectory.
In addition, once a dynamical system evolves with time, it is always useful
for comparison reasons to measure its entropy production per unit of time.
Such an ideal entropy definition for a dynamical system was introduced by
Kolmogorov in 1958 [7] and reformulated by Sinai in 1959\. It is known as the
Kolmogorov-Sinai (KS) entropy, denoted by $H_{KS}$, basically the Shannon’s
entropy of the set per unit of time [8], and it is the most successful
invariant quantity that characterize a dynamical system [9]. However, the
calculation of the KS entropy to systems that might possess an infinite number
of states is a difficult task, if not impossible. For a smooth chaotic system
[10] (typically happens for dissipative systems that present an attractor),
Pesin [11] proved an equality between $H_{KS}$ and the sum of all the positive
Lyapunov exponents. However, Lyapunov exponents are difficult or even
impossible to be calculated in systems whose equations of motion are unknown.
Therefore, when treating data coming from complex systems, one should use
alternative ways to calculate the KS entropy, instead of applying Pesin’s
equality.
Methods to estimate the correlation entropy, $K_{2}$, a lower bound of
$H_{KS}$, and to calculate $H_{KS}$ from time series were proposed in Refs.
[12, 13]. In Ref. [12] $K_{2}$ is estimated from the correlation decay and in
Ref. [13] by the determination of a generating partition of phase space that
preserves the value of the entropy. But while the method in Ref. [12]
unavoidably suffers from the same difficulties found in the proper calculation
of the fractal dimensions from data sets, the method in Ref. [13] requires the
knowledge of the generating partitions, information that is not trivial to be
extracted from complex data [14]. In addition, these two methods and similar
others as the one in Ref. [15] require the knowledge of a trajectory. Our work
is devoted to systems whose trajectory cannot be measured.
A convenient way of determining all the relevant states of a system and their
probabilities (independently whether such a system is chaotic) is provided by
the FPRs and the RPs. In particular to the Shannon’s entropy, in Refs. [16,
17, 18, 4] ways were suggested to estimate it from the RPs. In Refs. [16, 17,
4] a subset of all the possible probabilities of states, the probabilities
related to the level of coherence/correlation of the system, were considered
in Eq. (1). Therefore, as pointed out in Ref. [18], the obtained entropic
quantity does not quantify the level of disorganization of the system. Remind
that unavoidably Shannon’s entropy calculated from RPs or FPRs depends on the
resolution with which the returns are measured.
The main result of this contribution is to show how to easily estimate the KS-
entropy from return times, without the knowledge of a trajectory. We depart
from similar ideas as in Refs. [16, 17, 18, 4] and show that the KS entropy is
the Shannon entropy [in Eq. (1)] calculated considering the probabilities of
all the return times observed divided by the length of the shortest return
measured. This result is corroborated with simulations on the logistic map,
the Hénon map, and coupled maps. We also show how to estimate a lower bound
for the KS entropy using for that the returns with the shortest lengths (the
most probable returns), an approach oriented to the use of our ideas in
experimental data. Finally, we discuss in more details the intuitive idea of
Lettelier [18] to calculate the Shannon’s entropy from a RP and show the
relation between Letellier’s result and the KS entropy.
## 2 Estimating the KS entropy from time returns
Let us start with some definitions. By measuring two subsequent returns to a
region, one obtains a series of time intervals (FPRs) denoted by $\tau_{i}$
(with $i=1,\ldots,N$). The characterization of the FPRs is done by the
probability distribution $\rho(\tau,{\mathcal{B}})$ of $\tau_{i}$, where
${\mathcal{B}}$ represents the volume within which the FPRs are observed. In
this work, ${\mathcal{B}}$ is a $D$-dimensional box, with sides
$\epsilon_{1}$, and $D$ is the phase space dimension of the system being
considered. We denote the shortest return to the region ${\mathcal{B}}$ as
$\tau_{min}({\mathcal{B}})$.
Given a trajectory $\\{\bf x_{i}\\}_{i=1}^{M}$, the recurrence plot is a two-
dimensional graph that helps the visualization of a square array $R_{ij}$:
$R_{ij}=\theta(\epsilon_{2}-\|{\bf x_{i}}-{\bf x_{j}}\|)$ (2)
where $\epsilon_{2}$ is a predefined threshold and $\theta$ is the Heaviside
function [2]. In the coordinate $(i,j)$ of the RP one plots a black point if
$R_{ij}=1$, and a white point otherwise.
There are many interesting ways to characterize a RP, all of them related to
the lengths (and their probabilities of occurrence) of the diagonal,
horizontal, and vertical segments of recurrent points (black points) and of
nonrecurrent points (white points). Differently from Ref. [18] where it was
used the nonrecurrent diagonal segments, we consider here the vertical
nonrecurrent and recurrent segments because they provide a direct link to the
FPRs [19].
Given a column $i$, a vertical segment of $Q$ white points starting at $j=p$
and ending at $j=p+Q-1$, indicates that a trajectory previously in the
neighborhood of the point ${\bf x}_{i}$ returns to it firstly after $Q+1$
iterations in the neighborhood of the point ${\bf x}_{i}$, basically the same
definition as the FPR to a volume centered at ${\bf x}_{i}$. However, the
white points represent returns to the neighborhood of ${\bf x}_{i}$ which are
larger than 1. In order to obtain the returns of length 1, one needs to use
the recurrent segments, the segments formed by black points. A recurrent
vertical segment at the column $i$, starting at $j=p$ and ending at $j=p+Q$,
means that it occurred $Q$ first returns of length 1 to the neighborhood of
the point ${\bf x}_{i}$. The probability density of the return times observed
in the RP is represented also by $\rho(\tau,{\mathcal{B}})$. It is constructed
considering the first returns observed in all columns of the RP and it
satisfies $\int\rho(\tau,{\mathcal{B}})d\tau=1$.
Notice that the Shannon’s entropy of first returns of non-periodic continuous
systems becomes infinite [20] as the size $\epsilon$ of the volume
${\mathcal{B}}$ approaches zero. For chaotic systems (as well as for
stochastic systems) the reason lies on the fact that the probability density
$\rho(\tau,{\mathcal{B}})$ approaches the exponential form $\mu e^{-\mu\tau}$
[21], where $\mu$ is the probability of finding the trajectory within the
volume ${\mathcal{B}}$.
Placing in Eq. (1) the probabilities of returns $\rho(\tau,{\mathcal{B}})$, we
can write that $H_{KS}=H_{S}/T$, where $T$ is some characteristic time of the
returns [8] that depends on how the returns are measured. For the FPRs there
exists three characteristic times: the shortest, the longest and the average
return. The quantity $T$ cannot be the longest return since it is infinite. It
cannot be the average return, since one would arrive to
$H_{KS}\cong\mu\log{(\mu)}$ which equals zero as $\epsilon\rightarrow 0$.
Therefore, $T=\tau_{min}$ is the only remaining reasonable characteristic time
to be used which lead us to
$H_{KS}({\mathcal{B}}[\epsilon])=\frac{1}{\tau_{min}({\mathcal{B}}[\epsilon])}\sum_{\tau}\rho(\tau,{\mathcal{B}}[\epsilon])\log{\left(\frac{1}{\rho(\tau,{\mathcal{B}}[\epsilon])}\right)}.$
(3)
For uniformly hyperbolic chaotic systems (tent map, for example), we can prove
the validity of Eq. (3). From Ref. [26] we have that
$H_{KS}=-\lim_{\epsilon\rightarrow
0}\frac{1}{\tau_{min}}\log(\rho(\tau_{min},{\mathcal{B}}[\epsilon]))$ (4)
a result derived from the fact that the KS entropy exponentially increases
with the number of unstable periodic orbits embedded in the chaotic attractor.
Since $\rho(\tau,\epsilon)\rightarrow\mu e^{-\mu\tau}$ as $\epsilon\rightarrow
0$, assuming $\tau_{min}$ to be very large, and noticing that $\int-\mu
e^{-\mu\tau}log{[\mu e^{-\mu\tau}]}d\tau=-log{[\mu]}+1$, assuming that
$\tau_{min}\rightarrow\infty$ and noticing that for such systems
$\mu[{\mathcal{B}}]=\rho(\tau_{min},\epsilon)$, we finally arrive that
$-\frac{1}{\tau_{min}}\log{[\rho(\tau_{min})]}=-\frac{1}{\tau_{min}}\sum_{\tau}\rho(\tau)\log{[\rho(\tau)]}$
(5)
and therefore, the right-hand side of Eq. (3) indeed reflects the KS entropy.
But notice that Eq. (3) is being applied not only to non-uniformly hyperbolic
systems (Logistic and Hénon maps) but also to higher dimensional systems (two
coupled maps).
This result can also be derived from Ref. [27] where it was shown that the
positive Lyapunov exponent $\lambda$ in hyperbolic 1D maps is
$\lambda=\lim_{\epsilon\rightarrow
0}\frac{-log{[\mu(\epsilon)]}}{\tau_{min}(\mathcal{B}[\epsilon])}.$ (6)
Since $\rho(\tau,\epsilon)\rightarrow\mu e^{-\mu\tau}$ as $\epsilon\rightarrow
0$, and using that $\lambda=H_{KS}$ (Pesin’s equality), and finally noticing
that $\int-\mu e^{-\mu\tau}log{[\mu e^{-\mu\tau}]}d\tau=-log{[\mu]}+1$, one
can arrive to the conclusion that $T=\tau_{min}$ in Eq. (3).
The quantity in Eq. (3) is a local estimation of the KS entropy. To make a
global estimation we can define the average
$\langle
H_{KS}\rangle=\frac{1}{L}\sum_{\mathcal{B}(\epsilon)}H_{KS}[\mathcal{B}(\epsilon)]$
(7)
representing an average of $H_{KS}[\mathcal{B}(\epsilon)]$ calculated
considering $L$ different regions in phase space.
In order to estimate the KS entropy in terms of the probabilities obtained
from the RPs, one should use $T=\langle\tau_{min}\rangle$, i.e., replace
$\tau_{min}$ in Eq. (3) by $\langle\tau_{min}\rangle$, where
$\langle\tau_{min}\rangle=\frac{1}{M}\sum_{i}\tau_{min}(i)$, the average value
of the shortest return observed in every column of the RP. The reason to work
with an average value instead of using the shortest return considering all
columns of the RP is that every vertical column in the RP defines a shortest
return $\tau_{min}(i)$ ($i=1,\ldots,M$), and it is to expect that there is a
nontypical point $i$ for which $\tau_{min}(i)=1$.
Imagining that the RP is constructed considering arbitrarily small regions
($\epsilon_{2}\rightarrow 0$) and that we could treat an arbitrarily long data
set, the column of the RP which would produce $\tau_{min}=1$ would be just one
out of infinite others which produce $\tau_{min}>>1$. There would be also a
finite number of columns which would produce $\tau_{min}$ of the order of one
(but larger than one), but also those could be neglect when estimating the KS-
entropy from the RPs. The point we want to make in here is that the possible
existence of many columns for which one has $\tau_{min}=1$ are a consequence
of the finite resolution with which one constructs a RP. In order to minimize
such effect in our calculation we just ignore the fact that we have indeed
found in the RP $\tau_{min}=1$, and we consider as $\tau_{min}$ any return
time longer than 1 as the minimal return time. In fact, neglecting the
existence of returns of length one is a major point in the work of Ref. [18],
since there only the nonrecurrent diagonal segments are considered [19], and
thus, the probability of having a point returning to its neighborhood after
one iteration is zero.
From the conditional probabilities of returns, a lower bound for the KS
entropy can be estimated in terms of the FPRs and RPs by
$H_{KS}({\mathcal{B}}[\epsilon])\geq-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{P_{i}}\frac{\rho(\tau_{i}+P_{i})}{\rho(\tau_{i})}\log{\left[\frac{\rho(\tau_{i}+P_{i})}{\rho(\tau_{i})}\right]}$
(8)
where we consider only the returns $\tau_{i}$ for which
$\rho(\tau_{i}+P_{i})/\rho(\tau_{i})>0$ and $\tau_{i}+P_{i}<2\tau_{min}$, with
$P_{i}\in\mathcal{N}$.
The derivation of Eq. (8) is not trivial because it requires the use of a
series of concepts and quantities from the Ergodic Theory. In the following,
we describe the main steps to arrive at this inequality.
First we need to understand the way the KS-entropy is calculated via a spatial
integration. In short, the KS-entropy is calculated using the Shannon’s
entropy of the conditional probabilities of trajectories within the partitions
of the phase space as one iterates the chaotic system backward [2]. More
rigorously, denote a phase space partition $\delta_{N}$. By a partition we
refer to a space volume but that is defined in terms of Markov partitions.
Denote $S$ as $S=S_{0}\cap S_{1}\cap S_{k-1}$ where $S_{j}\in
F^{-j}\delta_{N}$ ($j=0,\ldots,k-1$), where $F$ is a chaotic transformation.
Define $h_{N}(k)=\frac{\mu(S\cap S_{k})}{\mu(S)}log{\frac{\mu(S\cap
S_{k})}{\mu(S)}}$ and $\mu(S)$ represents the probability measure of the set
$S$. The KS-entropy is defined as
$H_{KS}=\lim_{l\rightarrow\infty}\frac{1}{l}\sum_{k=0}^{l-1}\int\rho(dx)h_{N}(k)$,
where the summation is taken over $l$ iterations.
Assume now that the region ${\mathcal{B}}$ represents the good partition
$\delta_{N}$. The region $S_{j}$ is the result of $F^{-j}\delta_{N}$, i.e., a
$j$-th backward iteration of ${\mathcal{B}}$. So, clearly, if one applies $j$
forward iterations to $S_{j}$, then $F^{j}S_{j}\rightarrow{\mathcal{B}}$. The
quantities $\mu(S\cap S_{k})$ and $\mu(S)$ refer to the measure of the chaotic
attractor inside $S\cap S_{k}$ and $S$, respectivelly. By measure we mean the
natural measure, i.e. the frequency with which a typical trajectory visits a
region. ${\mu(S\cap S_{k})}$ refers to the measure that remained in
${\mathcal{B}}$ after $k$ iterations and ${\mu(S)}$ the measure that remained
in ${\mathcal{B}}$ after $k-1$ iterations.
Figure 1: [color online] Results from Eq. (3) and (6). The probability
function $\rho(\tau,{\mathcal{B}})$ of the FPRs (RPs) were obtained from a
series of 500.000 FPRs (from a trajectory of length 15.000 points). The brown
line represents the values of the positive Lyapunov exponent. In (A) we show
results for the Logistic map as we vary the parameter $c$,
$\epsilon_{2}=0.002$ for the brown stars and $\epsilon_{1}$=0.001 for the
green diamonds. In (B) we show results for the Hénon map as we vary the
parameter $a$ for $b$=0.3, $\epsilon_{2}=[0.002-0.03]$ for the brown stars and
$\epsilon_{1}$=0.002 for all the other results, and in (C) results for the
coupled maps as we vary the coupling strength $\sigma$, $\epsilon_{2}$=0.05
for the brown stars and $\epsilon_{1}$=0.02 for green diamonds.
For $k\rightarrow\infty$, we have that $\frac{\mu(S\cap
S_{k})}{\mu({\mathcal{B}})}\rightarrow\mu({\mathcal{B}})$. Also for finite
values of $k$, one has that $\frac{\mu(S\cap
S_{k})}{\mu({\mathcal{B}})}\approx\mu({\mathcal{B}})$. For any finite $k$, we
can split this fraction into two components: $\frac{\mu(S\cap
S_{k})}{\mu({\mathcal{B}})}=\mu_{REC}(k,{\mathcal{B}})+\mu_{NR}(k,{\mathcal{B}})$.
$\mu_{REC}$ refers to the measure in ${\mathcal{B}}$ associated with unstable
periodic orbits (UPOs) that return to ${\mathcal{B}}$, after $k$ iteration of
$F$, at least twice or more times. $\mu_{NR}$ refers to the measure in
${\mathcal{B}}$ associated with UPOs that return to ${\mathcal{B}}$ only once.
As it is shown in Ref. [26],
$\rho(\tau,{\mathcal{B}})=\mu_{NR}(\tau,{\mathcal{B}})$, which in other words
means that the probability density of the FPRs in ${\mathcal{B}}$ is given by
$\mu_{NR}(k,{\mathcal{B}})$. But, notice that for $\tau<2\tau_{min}$,
$\mu_{REC}(k,{\mathcal{B}})=0$ since only returns associated with UPOs that
return once can be observed inside ${\mathcal{B}}$, and therefore
$\rho(\tau,{\mathcal{B}})$ = $\frac{\mu(S\cap S_{\tau})}{\mu({\mathcal{B}})}$,
if $\tau<2\tau_{min}$. Consequently, we have that $\frac{\mu(S\cap
S_{\tau})}{\mu(S)}=\frac{\rho(\tau,{\mathcal{B}})}{\rho(\tau-1,{\mathcal{B}})}$,
since $\frac{\mu(S\cap
S_{\tau})}{\mu({\mathcal{B}})}=\rho(\tau,{\mathcal{B}})$ and
$\frac{\mu(S)}{\mu({\mathcal{B}})}=\rho(\tau-1,{\mathcal{B}})$.
The remaining calculations to arrive in Eq. (8) consider the measure of the
region $S_{\tau}\cap S_{\tau+P}$ (instead of $S\cap S_{\tau}$) in order to
have a positive condition probability, i.e. $\frac{\mu(S_{\tau}\cap
S_{\tau+P})}{\mu(S_{\tau})}>0$, with $\mu(S_{\tau})$ representing the measure
of the trajectories that return to ${\mathcal{B}}$ after $\tau$ iterations and
$\mu(S_{\tau}\cap S_{\tau+P})$ the measure of the trajectories that return to
${\mathcal{B}}$ after $\tau+P$ iterations. The inequality in Eq. (8) comes
from the fact that one neglects the infinitely many terms coming from the
measure $\mu_{REC}(\tau,{\mathcal{B}})$ that would contribute positively to
this summation.
## 3 Estimation of errors in $H_{KS}$ and $\langle H_{KS}\rangle$
In order to derive Eq. (5), we have assumed that $\int-\mu
e^{-\mu\tau}log{[\mu e^{-\mu\tau}]}d\tau=-\log{[\mu]}+1$, which is only true
when $\tau_{min}$=0. In reality, for $\tau_{min}>0$, we have
$\int_{\tau_{min}}^{\infty}-\mu e^{-\mu\tau}log{[\mu e^{-\mu\tau}]}d\tau$ =
$e^{-\mu\tau_{min}}[\mu\tau_{min}-\log{\mu}]+1$, but as $\epsilon$ tends to
zero $\mu\tau_{min}\rightarrow 0$ and therefore, as assumed $\int-\mu
e^{-\mu\tau}log{[\mu e^{-\mu\tau}]}d\tau\approxeq-\log{[\mu]}+1$.
Making the same assumptions as before that $\rho(\tau,\epsilon)\rightarrow\mu
e^{-\mu\tau}$ as $\epsilon\rightarrow 0$, and using Eq. (6), then Eq. (3) can
be written as
$H_{KS}({\mathcal{B}}[\epsilon])\approxeq\lambda+\frac{1}{\tau_{min}({\mathcal{B}}[\epsilon])}.$
(9)
Theoretically, one can always imagine a region $\epsilon$ with an arbitrarily
small size, which would then make the term $\frac{1}{\tau_{min}}$ to approach
zero. But, in practice, for the considered values of $\epsilon$, we might have
(for atypical intervals) shortest returns as low as $\tau_{min}=4$. As a
result, we expect that numerical calculations of the quantity in Eq. (3) would
lead us to a value larger than the positive Lyapunov exponent, as estimated
from the returns of the trajectory to a particular region.
Naturally, $\frac{1}{\tau_{min}}$ would provide a local deviation of the
quantity in Eq. (3) with respect to the KS entropy. To have a global
estimation of the error we are making by estimating the KS entropy, we should
consider the error in the average quantity $\langle H_{KS}\rangle$ which is
given by
$E=\sum_{\mathcal{B}(\epsilon)}\frac{1}{\tau_{min}({\mathcal{B}}[\epsilon])}$
(10)
where the average is taken over $L$ different regions in phase space, and thus
for chaotic systems with no more than one positive Lyapunov exponent
$\langle H_{KS}\rangle\approxeq\lambda+E$ (11)
To generalize this result to higher dimensional systems, we make the same
assumptions as the ones to arrive to Eq. (9), but now we use Eq. (5). We
arrive that
$\langle H_{KS}({\mathcal{B}}[\epsilon])\rangle\approxeq H+E,$ (12)
where $H$ denotes the exact value of the KS entropy.
Finally, it is clear from Eq. (12) that $\langle
H_{KS}({\mathcal{B}}[\epsilon])\rangle$ is an upper bound for the KS entropy.
Thus,
$H\leq\langle H_{KS}({\mathcal{B}}[\epsilon])\rangle.$ (13)
## 4 Estimating the KS entropy and a lower bound of it in maps
Figure 2: [color online] Results from Eq. (8). The probability function
$\rho(\tau,{\mathcal{B}})$ of the FPRs (RPs) were obtained from a series of
500.000 FPRs (from a trajectory of length 15.000 points). The brown line
represents the values of the positive Lyapunov exponent. In (A) we show
results for the Logistic map as we vary the parameter $c$,
$\epsilon_{2}=0.002$ for the black circles and $\epsilon_{1}$=0.001 for the
red squares. In (B) we show results for the Hénon map as we vary the parameter
$a$ for $b$=0.3, $\epsilon_{2}=[0.002-0.03]$ for the black circles and
$\epsilon_{1}$=0.002 for the red squares, and in (C) results for the coupled
maps as we vary the coupling strength $\sigma$, $\epsilon_{2}$=0.05 for the
black circles and $\epsilon_{1}$=0.02 the red squares.
Figure 3: [color online] Results from Eq. (3) applied to the FPRs coming from
the Logistic map (A-B), as we vary the parameter $c$ and
$\epsilon_{1}$=0.00005, and from the Hénon map (C), as we vary the parameter
$a$ and $\epsilon_{1}=0.001$. These quantities were estimated considering 10
randonmly selected regions. The brown line represents the values of the
positive Lyapunov exponent. The probability density function
$\rho(\tau,{\mathcal{B}})$ was obtained from a series of 500.000 FPRs. Green
diamonds represent in (A) the values of $H_{KS}$ calculated for each one of
the 10 randonmly selected regions, in (B) the average value $\langle
H_{KS}\rangle$ and in (C) the minimal value of $H_{KS}$.
In order to illustrate the performance of our formulas we use the Logistic map
[$x_{n+1}=cx_{n}(1-x_{n})$], the Hénon map [$x_{n+1}=a-x_{n}^{2}+by_{n}$, and
$y_{n+1}=x_{n}$], and a system of two mutually coupled linear maps
[$x_{n+1}=2x_{n}-2\sigma(y_{n}-x_{n})$ and
$y_{n+1}=2y_{n}-2\sigma(x_{n}-y_{n})$, $mod(1)$], systems for which Pesin’s
equality holds. The parameter $\sigma$ in the coupled maps represents the
coupling strength between them, chosen to produce a trajectory with two
positive Lyapunov exponents.
Using Eqs. (3) and (6) to estimate $H_{KS}$ and $\lambda$ furnishes good
values if the region ${\mathcal{B}}$ where the returns are being measured is
not only sufficiently small but also well located such that $\tau_{min}$ is
sufficiently large. In such a case the trajectories that produce such a short
return visit the whole chaotic set [28]. For that reason we measure the FPRs
for 50 different regions with a sufficiently small volume dimension, denoted
by $\epsilon_{1}$, and use the FPRs that produce the largest $\tau_{min}$,
minimizing $H_{KS}$. Since the lower bound of $H_{KS}$ in Eq. (8) is a minimal
bound for the KS entropy, the region chosen to calculate it is the one for
which the lower bound is maximal. This procedure makes $H_{KS}$ and its lower
bound (calculated using the FPRs) not to depend on ${\mathcal{B}}$.
As pointed out in Ref. [18], one should consider volume dimensions (also known
as thresholds) which depend linearly on the size of the attractor [28], in
order to calculate the Shannon’s entropy. In this work, except for the Hénon
map, we could calculate well $H_{KS}$, $\lambda$ and a lower bound for
$H_{KS}$ from the FPRs and RPs, considering for every system fixed values
$\epsilon_{1}$ and $\epsilon_{2}$. For the Hénon map, as we increase the
parameter $b$ producing more chaotic attractors, we increase linearly the size
of the volume dimension $\epsilon_{2}$ within the interval $[0.002-0.03]$.
We first compare $H_{KS}$ (see Fig. 1), calculated from Eq. (3) in terms of
the probabilities coming from the FPRs and RPs, in green diamonds and brown
stars, respectively, with the value of the KS entropy calculated from the sum
of the positive Lyapunov exponents, represented by the brown straight line. As
expected $H_{KS}$ is close to the sum of all the positive Lyapunov exponents.
When the attractor is a stable periodic orbit we obtain that $H_{KS}$ is small
if calculated from the RPs. In such a case, we assume that $H_{KS}=0$ if
calculated from the FPRs. This assumption has theoretical grounds, since if
the region is centered in a stable periodic attractor and
$\epsilon_{1}\rightarrow 0$ (what can be conceptually make), one will clearly
obtain that the attractor is periodic.
Figure 4: [color online] The same quantities shown in Fig. 3, but now
considering only the Logistc map, with $\epsilon_{1}$=0.0002 and 500 randonmly
selected regions.
The value of the Lyapunov exponent calculated from the formula (6) is
represented in Fig. 1 by the blue up triangles. As it can be checked in this
figure, Eq. (6) holds only for 1D hyperbolic maps. So, it works quite well for
the logistic map (a 1D “almost” uniformly hyperbolic map) and somehow good for
the Hénon map. However, it is not appropriate to estimate the sum of the
positive Lyapunov exponents coming from 2D coupled systems. This formula
assumes sufficient hyperbolicity and one-dimensionality such that
$e^{\tau_{min}\lambda}=1/\epsilon$.
To compare our approach with the method in Ref. [12], we consider the Hénon
map with $a$=1.4 and $b=0.3$ for which the positive Lyapunov exponent equals
0.420. Therefore, by using Ruelle equality, $H_{KS}=0.420$. In Ref. [12] it is
obtained that the correlation entropy $K_{2}$ equals 0.325, with $H_{KS}\geq
K_{2}$ and in Ref. [13] $H_{KS}=0.423$. From Eq. (3), we obtain $H_{KS}=0.402$
and from Eq. (8), we obtain $H_{KS}\geq 0.342$, for $\epsilon_{1}$=0.01.
In Fig. 2(A-C), we show the lower bound estimation of $H_{KS}$ [in Eq. (8)] in
terms of the RPs (black circles) and in terms of FPRs (red squares). As
expected, both estimations follow the tendency of $H_{KS}$ as we increase $a$.
Another possible way Eq. (3) can be used to estimate the value of the KS-
entropy is by averaging all the values obtained for different intervals, the
quantity $\langle H_{KS}\rangle$ in Eq. (7). In Fig. 3(A), we show the values
of $H_{KS}$ as calculated from Eq. (3) considering a series of FPRs with
500.000 returns of trajectories from the Logistic map. For each value of the
control parameter $c$, we randomnly pick 10 different intervals with
$\epsilon_{1}$=0.00005. The average $\langle H_{KS}\rangle$ is shown in Fig.
3(B). As one can see, $\langle H_{KS}\rangle$ is close to the Lyapunov
exponent $\lambda$. Notice that from Fig. 3(A) one can see that the minimal
value of $H_{KS}$ (obtained for the largest $\tau_{min}$) approaches well the
value of $\lambda$.
In order to have a more accurate estimation of the KS-entropy for the Hénon
map, we have used in Figs. 1(B) and 2(B) a varying $\epsilon_{2}$ depending on
the value of the parameter $a$, exactly as suggested in [18], but similar
results would be obtained considering a constant value. As an example, in Fig.
3(C) we show the minimal value of $H_{KS}$ considering regions with
$\epsilon_{1}=0.001$, for a large range of the control parameter $a$.
In order to illustrate how the number of regions as well as the size of the
regions alter the estimation of the KS-entropy, we show, in Fig. 4(A-C), the
same quantities shown in Fig. 3(A-B), but now from FPRs exclusively coming
from the Logistic map, considering 500 randonmly selected regions all having
sizes $\epsilon_{1}$=0.0002. Recall that in Figs. 1 and 3, the minimal value
of $H_{KS}$ was chosen out of no more than 50 randonmly selected regions.
Comparing Figs. 3(B) and 4(B) one notices that an increase in the number of
selected regions is responsible to smooth the curve of $\langle H_{KS}\rangle$
with respect to $c$. Concerning the minimal value of $H_{KS}$, the use of
intervals with size $\epsilon_{1}=0.0002$ provides values close to the
Lyapunov exponent if this exponent is sufficiently low (what happens for
$b<3.7$). Otherwise, these values deviate when this exponent is larger (what
happens for $b>3.7$). This deviation happens because for these chaotic
attractors the size of the chosen interval was not sufficiently small [28].
Notice that the estimated KS entropy deviates from $\lambda$. See, for
example, Figs. 3(B) and 4(B). One sees two main features in these figures. The
first is that for most of the simulations, $\langle H_{KS}\rangle>\lambda$.
The second is that the larger $\lambda$ is, the larger the deviation is. The
reason for the first feature can be explained by Eqs. (11) and (13). The
reason for the second is a consequence of the fact that the larger the
Lyapunov exponent is, the smaller $\tau_{min}$ is, and therefore the larger
the error in the estimation of the KS entropy.
To see that our error estimate provides reasonable results, we calculate the
quantities $\langle H_{KS}\rangle$ (green diamonds in Fig. 5), for the
Logistic map considering a series of 250.000 FPRs to $L$=100 randomly selected
regions of size $\epsilon_{1}=0.0002$, and the average error $E$, in Eq. (11)
[shown in Fig. 5 by the error bars]. The value of the positive Lyapunov
exponent is shown in the full brown line.
The error in our estimation is inversely proportional to the shortest return.
Had we considered smaller $\epsilon$ regions, $\tau_{min}$ would be typically
larger and as a consequence we would obtain a smaller error $E$ in our
estimation for the KS entropy. Had we consider a larger number of FPRs, the
numerically obtained value of $\tau_{min}$ would be typically slightly
smaller, making the error $E$ to become slightly larger. So, the reason of why
the positive Lyapunov exponent in Fig. 5 is located bellow the error bars for
the quantity $\langle H_{KS}\rangle$ is a consequence of the fact that we have
only observed 250.000 returns, producing an overestimation for the value of
$\tau_{min}$. Had we considered a larger number of FPRs would make the error
$E$ to become slightly larger.
Figure 5: [color online] Results obtained considering FPRs coming from the
Logistic map, as we vary the parameter $c$ and $\epsilon_{1}$=0.0002. The
probability density function $\rho(\tau,{\mathcal{B}})$ was obtained from a
series of 250.000 FPRs. Green diamonds represent the values of $\langle
H_{KS}\rangle$ calculated for each one of the 100 randomly selected regions.
The error bar indicates the value of the average error $E$ in Eq. (11). These
quantities were estimated considering 100 randomly selected regions. The brown
line represents the values of the positive Lyapunov exponent.
The considered maps are Ergodic. And therefore, the more (less) intervals
used, the shorter (the longer) the time series needed in order to calculate
the averages from the FPR as well as from the RP, as the average $\langle
H_{KS}\rangle$.
## 5 Conclusions
Concluding, we have shown how to estimate the Kolmogorov-Sinai entropy and a
lower bound of it using the Poincaré First Return Times (FPRs) and the
Recurrence Plots. This work considers return times in discrete systems. The
extension of our ideas to systems with a continuous description can be
straightforwardly made using the ideas in Ref. [29].
We have calculated the expected error in our estimation for the KS entropy and
shown that this error appears due to the fact that FPRs can only be physically
measured considering finite sized regions and only a finite number of FPRs can
be measured. This error is not caused by any fundamental problems in the
proposed Eq. (3). Nevertheless, even for when such physical limitations are
present, the global estimator of the KS entropy [Eq. (7)] can be considered as
an upper bound for the KS entropy [see Eq. (13)].
## References
* [1] H. Poincaré, Acta Matematica, 13, 1 (1890).
* [2] J.-P. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 4, 973 (1987).
* [3] V. Afraimovich, Chaos, 7, 12 (1997); N. Hadyn, J. Luevano, G. Mantica, S. Vaienti, Phys. Rev. Lett., 88 (2002); B. Saussol, Discrete and Continuous Dynamical Systems A, 15, 259 (2006); N. Hadyn, et al., Phys. Rev. Lett., 88, 224502 (2002).
* [4] N. Marwan, M. C. Romano, M. Thiel, et al., Phys. Reports, 438, 237 (2007); M. Thiel, M. C. Romano, J. Kurths, et al., Europhys. Lett. 75, 535 (2006); M. C. Romano, M. Thiel, J. Kurths, and C.Grebogi, Phys. Rev. E 76, 036211 (2007).
* [5] M. S. Baptista, I. L. Caldas, M. S. Baptista, et al., Physica A, 287, 91 (2000).
* [6] C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (The University of Illinois Press, 1949).
* [7] A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 119, 861 (1958); 124, 754 (1959).
* [8] The time version of the KS entropy is calculated from the average value of the difference between the Shannon entropies (per unit of time) of a trajectory with a length $n\delta t$ and of a trajectory with a length $(n+1)\delta t$, for all possible values of $n$. More rigorously, $\lim_{n\rightarrow\infty}\lim_{(\epsilon,\delta t)\rightarrow 0}\frac{1}{n\delta\tau}\sum_{i=0}^{n-1}[K_{i+1}-K_{i}]$, where $K_{i}$ is the shannon’s entropy of a trajectory that visits $(n+1)$ volumes of sides $\epsilon$ in phase space. In each volume the trajectory remains during a time interval $\delta t$. As $i\rightarrow\infty$, the terms $K_{i+1}$ are infinities, but not the difference $[K_{i+1}-K_{i}]$. Since $\sum_{i=0}^{n-1}[K_{i+1}-K_{i}]=K_{n}-K_{0}$ and that $K_{0}/{n\delta\tau}\rightarrow 0$, then $H_{KS}=\lim_{n\rightarrow\infty}\lim_{(\epsilon,\delta t)\rightarrow 0}\frac{1}{n\delta\tau}K_{n}$. So, basically the time version of the KS-entropy can be thought as the Shannon’s entropy divided by a characteristic time, $T$, yet to be determined.
* [9] P. Walters, An introduction to the Ergodic Theory, (Springer-Verlag Berlin, 1982).
* [10] L.-S. Young, J. Stat. Phys. 108, 733 (2002).
* [11] Y. B. Pesin Russian Math. Surveys, 32, 55 (1977).
* [12] P. Grassberger and I. Procaccia, Phys. Rev. A 28, 2591 (1983).
* [13] A. Cohen and I. Procaccia, Phys. Rev. A 31, 1872 (1985).
* [14] M. S. Baptista, C. Grebogi, R. Köberle, Phys. Rev. Lett. 97, 178102 (2006).
* [15] A. Wolf, J. B. Swift, H. L. Swinney et al., Physica D 16, 285 (1985).
* [16] L. L. Trulla, A. Giuliani, J, P. Zbilut, and C. L. Webber, Jr., Phys. Lett. A 223, 255 (1996).
* [17] P. Faure and H. Korn, Physica D, 122, 265 (1998).
* [18] C. Letellier, Phys. Rev. Lett. 96, 254102 (2006).
* [19] While the white vertical segments correspond to the first Poincaré returns to an interval centered at a point, the white diagonal segments provide the $n$-th Poincaré returns. As an example, imagine that there are two black points placed at the coordinates (10,20) and (20,30) in the RP. In the white diagonal segment connecting these two black points there are no black points, which means that we have a nonrecurrent diagonal segment of length $10$. That can only be possible if two first Poincaré returns of length 10 happened, or if one second Poincaré return of length 10 happened.
* [20] Assuming that $\rho(\tau,{\mathcal{B}})=\mu e^{-\mu\tau}$, this leads $H_{S}=-\log{(\mu)}+1$. Since $\mu\sim\epsilon$, if $\epsilon\rightarrow 0$, then $H_{S}\rightarrow\infty$.
* [21] The exponential form of $\rho$ relies on the fact that the first returns can be imagined to be uncorrelated random variables due to the fast decay of correlation that chaotic systems have. The exponential form of $\rho$ to arbitrarely small volumes is proved for a large class of uniformly hyperbolic maps (see Ref. [22]), for one-dimensional non-uniform hyperbolic maps (unimodal maps, see Ref. [23] and multimodal maps, see [24]). For finite sized volumes $\rho$ still preserves the exponential form [25].
* [22] B. Saussol, Nonlinearity, 14, 179 (2001); M. Hirata, B. Saussol, and S. Vaienti, Comm. Math. Phys. 206, 33 (1999).
* [23] H. Bruin, B. Saussol, S. Troubetzkoy, S. Vaienti, Ergodic Theory and Dynamical Systems, 23, 991 (2003).
* [24] H. Bruin and M. Todd, arXiv:0708.0379.
* [25] M. S. Baptista, S. Kraut, C. Grebogi, Phys. Rev. Lett., 95, 094101 (2005).
* [26] P. R. F. Pinto, M. S. Baptista, I. Laboriaou, “Density of First Poincaré returns and Periodic Orbits”, preprints can be downloaded from http://arxiv.org/abs/0908.4575.
* [27] B.Saussol, S.Troubetzkoy, S.Vaienti, Moscow Mathematical Journal 3, 189 (2003).
* [28] Usually, the larger an attractor, the larger the Lyapunov exponent. As discussed in [23], $1=\epsilon e^{\tau_{min}\lambda}$. Assuming that a sufficiently small threshold $\epsilon$ provides a sufficiently large $\tau_{min}$, this leads to $\epsilon=e^{-\tau_{min}\lambda}$ and therefore the more chaotic an attractor, the larger $\epsilon$ must be in order to have a sufficiently large $\tau_{min}$. Since the series of FPRs can be (in principle theoretically) calculated from arbitrarily large trajectories, we can consider (from a theoretical perspective) regions sufficiently small so that one can obtain FPRs with sufficiently large $\tau_{min}$, even for different values of the control parameters. On the other hand, RPs are constructed with trajectories (time series) shorter than the ones used for the FPRs. In order to have a RP for which $\langle\tau_{min}\rangle$ is sufficiently large and at the same time producing a reasonable continuous distribution $\rho(\tau)$, the volume dimensions $\epsilon_{2}$ considered to construct the RPs should be reasonably larger than $\epsilon_{1}$. In addition, as one changes the control parameters producing more complex chaotic attractors, we might have to increase the value of $\epsilon_{2}$.
* [29] J. B. Gao, Phys. Rev. Lett. 83, 3178 (1999).
|
arxiv-papers
| 2009-08-24T10:12:22 |
2024-09-04T02:49:04.753751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Baptista, E. J. Ngamga, Paulo R. F. Pinto, Margarida Brito, J.\n Kurths",
"submitter": "Murilo Baptista S.",
"url": "https://arxiv.org/abs/0908.3401"
}
|
0908.3416
|
# Taylor Expansion and Discretization Errors in Gaussian Beam Superposition
Mohammad Motamed Department of Mathematics, Simon Fraser University, Burnaby,
BC V5A 1S6, Canada, email: mmotamed@math.sfu.ca Olof Runborg Department of
Numerical Analysis, Royal Institute of Technology, 10044 Stockholm, Sweden,
email: olofr@nada.kth.se
(Feb 4 2010)
###### Abstract
The Gaussian beam superposition method is an asymptotic method for computing
high frequency wave fields in smoothly varying inhomogeneous media. In this
paper we study the accuracy of the Gaussian beam superposition method and
derive error estimates related to the discretization of the superposition
integral and the Taylor expansion of the phase and amplitude off the center of
the beam. We show that in the case of odd order beams, the error is smaller
than a simple analysis would indicate because of error cancellation effects
between the beams. Since the cancellation happens only when odd order beams
are used, there is no remarkable gain in using even order beams. Moreover,
applying the error estimate to the problem with constant speed of propagation,
we show that in this case the local beam width is not a good indicator of
accuracy, and there is no direct relation between the error and the beam
width. We present numerical examples to verify the error estimates.
Keywords: wave propagation, high frequency, asymptotic approximation, Gaussian
beam superposition, accuracy, error estimates
## 1 Introduction
Simulation of wave propagation is expensive when the frequency of the waves is
high. In this case, a large number of grid points are needed to resolve the
wave oscillations, and the computational cost to maintain constant accuracy
grows algebraically with the frequency. At sufficiently high frequencies,
therefore, direct simulations are no longer feasible.
Instead one can use high frequency asymptotic models for wave propagation. The
most popular one is geometrical optics, which is obtained when the frequency
tends to infinity. The unknowns in geometrical optics are the phase and
amplitude which are independent of the frequency and vary on a much coarser
scale than the full wave solution. They can therefore be computed at a
computational cost independent of the frequency. However, a main drawback of
geometrical optics is that the model breaks down at caustics, where
geometrical optics rays intersect and the predicted amplitude is unbounded.
Gaussian beams approximation is another high frequency asymptotic model which
is valid also at caustics. It was introduced by Popov [1], based on an earlier
work of Babic and Pankratova [2]. A Gaussian beam is an approximate high
frequency solution to the linear wave equation which is concentrated close to
a standard ray of geometrical optics, called the central ray of the beam.
Although the phase function is real-valued along the central ray, Gaussian
beams accept complex-valued phase functions off their central ray. The
imaginary part of the phase is chosen such that the solution decays
exponentially away from the central ray, maintaining a Gaussian-shaped
profile. The main advantage of this construction is that it gives the correct
solution also at caustics. It has recently been proved to be beneficial in
seismic imaging, [6, 7].
Numerical methods based on Gaussian beams use the superposition principle.
Individual beams are computed via ray tracing like equations, where quantities
such as the curvature and width of beams are calculated from ordinary
differential equations (ODEs) along the central rays, and the contribution of
the beams concentrated close to their central rays are determined by Taylor
expansion. The full wave field is then obtained by a superposition integral
over all beams. This integral is replaced by a discrete summation of beams in
practical computations. See for example [3, 4, 5, 6, 7]. Numerical techniques
based on both Lagrangian and Eulerian formulations of the problem have been
devised [8, 9, 10, 11, 12]. For a rigorous mathematical analysis of Gaussian
beams we refer to [13].
In this paper we derive error estimates for the beam superposition method. We
study the discretization error, caused by replacing the superposition integral
by the summation of beams, and the error related to Taylor expansion of the
phase and amplitude off the center of the beam. Some error estimates for this
method have been derived earlier, [14, 15]. We aim to give a more complete
picture of the error by also including the error due to the spreading of the
beams, which is related to the Taylor expansions. This error is recognized as
important in e.g. [14]. It turns out that, in the case of using odd order
beams, the error is smaller than a simple analysis would indicate because of
error cancellation effects between the beams. Since the cancellation happens
only when odd order beams are used, there is no remarkable gain in using even
order beams. Moreover, we show that in the case of constant coefficient
equations, i.e. when the speed of propagation is constant, the local beam
width is not a good indicator of accuracy, and there is no direct relation
between the error and the beams’ width. However, this may not be true in the
case of varying speed of propagation, where the beam width can be an important
factor in the Taylor expansion error. For other recent results on error
estimates see [16, 17].
In Section 2, we review the construction of Gaussian beams and the Gaussian
beam superposition method. The accuracy of Gaussian beam superposition is
studied in Section 3, where the main result is formulated together with
numerical examples verifying the obtained error estimates. In Section 4, the
proof of the main theorem is given in detail. Finally, in Section 5, we
compute the errors analytically in the case of constant coefficient equations
and give some remarks on how to select the Gaussian beam parameters.
## 2 Gaussian beam superposition method
Gaussian beams are obtained when the linear wave equation is solved with
oscillatory initial or boundary data with an amplitude in the shape of a
Gaussian bell. A Gaussian beam is an asymptotic solution concentrated on its
central ray in the domain. By the superposition principle for linear
equations, such solutions can be added to find the full wave field. The
initial/boundary data for beams are obtained such that the wave data at the
source is well approximated. In this section, we consider the Helmholtz
equation and review the construction of Gaussian beams and their
superposition.
### 2.1 Construction of Gaussian beams
Consider the Helmholtz equation
$\Delta
u({{\mbox{\boldmath$x$}}})+\frac{\omega^{2}}{c({{\mbox{\boldmath$x$}}})^{2}}\,u({{\mbox{\boldmath$x$}}})=0,\qquad{{\mbox{\boldmath$x$}}}\in\Omega\subset\mathbb{R}^{2},$
where $\omega\gg 1$ and $c({{\mbox{\boldmath$x$}}})$ are the frequency and
speed of propagation, respectively. Boundary conditions are given on
$\partial\Omega$, which we assume is divided in two parts: one where ingoing
waves are specified, and one with outgoing radiation condition, typically at
infinity. We call the first, ingoing, part of $\partial\Omega$ the source
curve. We substitute the WKBJ ansatz
${}u({{\mbox{\boldmath$x$}}})=e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}})}\sum_{k=0}^{\infty}A_{k}({{\mbox{\boldmath$x$}}})(i\omega)^{-k},$ (1)
into the Helmholtz equation. Here, the phase function $\phi$ and the amplitude
functions $A_{k}$ are assumed to be smooth and independent of $\omega$.
Equating coefficients of powers of $\omega$ to zero gives us the _eikonal
equation_ and the _transport equation_ for the phase and the first amplitude
term in the frequency domain,
$|\nabla\phi|=1/c({{\mbox{\boldmath$x$}}}),\qquad 2\,\nabla
A_{0}\cdot\nabla\phi+A_{0}\,\Delta\phi=0.$
For the remaining amplitude terms, we get additional transport equations
$2\,\nabla A_{k+1}\cdot\nabla\phi+A_{k+1}\,\Delta\phi+\Delta A_{k}=0.$
When $\omega$ is large, only the first terms in the WKBJ expansion are
significant. We henceforth denote the high frequency approximation taking $K$
terms in (1) by $u_{\rm GO}({{\mbox{\boldmath$x$}}})$,
${}u_{\rm
GO}({{\mbox{\boldmath$x$}}})=A({{\mbox{\boldmath$x$}}})e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}})},\qquad
A({{\mbox{\boldmath$x$}}}):=\sum_{k=0}^{K-1}A_{k}({{\mbox{\boldmath$x$}}})(i\omega)^{-k}.$
(2)
This approximation is usually called the geometrical optics approximation, in
particular when $K=1$. It introduces an error of the order $O(\omega^{-K})$.
The Gaussian beam approximation has the same form as the geometrical optics
approximation,
${}u_{\rm
GB}({{\mbox{\boldmath$x$}}})=A({{\mbox{\boldmath$x$}}})e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}})},$ (3)
where the phase $\phi$ and amplitude terms $A_{k}$ satisfy the same PDEs.
There are, however, two important differences. First, while in geometrical
optics $\phi$ is globally defined for all rays, for Gaussian beams it is
constructed based on one specific ray (the beam’s central ray). Second, in
geometrical optics, $\phi$ is real-valued, but in the Gaussian beam
construction it is real-valued only on the central ray of the beam. Away from
the central ray, it is complex-valued with _positive imaginary part_. The
solution will then be exponentially decreasing away from the central ray,
maintaining its Gaussian shape. Note that since $\phi$ is complex valued, it
actually satisfy the _complex eikonal equation_ , [18, 19]. Unfortunately, the
question of existence and uniqueness of the complex eikonal equation is to a
certain extent still open. In particular what precise boundary conditions are
well-posed for the above setting is not known.
As in geometrical optics, the Gaussian beam approximation breaks down when
$\phi({{\mbox{\boldmath$x$}}})$ becomes non-smooth. This is typical for
solutions to both the standard and the complex eikonal equation. It happens in
general some distance away from the central beam. On the other hand, away from
the beam the solution rapidly goes to zero and the precise value of the phase
is not important. One usually deals with this problem by multiplying the
amplitude with a smooth cut-off function that is one close to the central ray,
and zero for some fixed distance away from it. In practice, (3) is thus
replaced by
$u_{\rm
GB}({{\mbox{\boldmath$x$}}})=\varphi({{\mbox{\boldmath$x$}}})A({{\mbox{\boldmath$x$}}})e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}})},$
where $\varphi({{\mbox{\boldmath$x$}}})$ is smooth and compactly supported
around the central ray.
For a beam starting at point ${{\mbox{\boldmath$x$}}}_{0}$ with direction
${{\mbox{\boldmath$p$}}}_{0}$, the corresponding central ray satisfies the ray
tracing ODEs
${}\frac{d{{{\mbox{\boldmath$x$}}}}}{dt}=c^{2}({{\mbox{\boldmath$x$}}}){{\mbox{\boldmath$p$}}},\quad\frac{d{{{\mbox{\boldmath$p$}}}}}{dt}=-\frac{\nabla
c({{\mbox{\boldmath$x$}}})}{c({{\mbox{\boldmath$x$}}})},\quad{{\mbox{\boldmath$x$}}}(0)={{\mbox{\boldmath$x$}}}_{0},\quad{{\mbox{\boldmath$p$}}}(0)=\frac{{{\mbox{\boldmath$p$}}}_{0}}{|{{\mbox{\boldmath$p$}}}_{0}|c({{\mbox{\boldmath$x$}}}_{0})},$
(4)
with $t$ being the real-valued travel time along the ray. If we set
${{\mbox{\boldmath$p$}}}=(\cos\theta,\;\sin\theta)^{\top}/c({{\mbox{\boldmath$x$}}})$
and ${{\mbox{\boldmath$x$}}}=(x,\;y)^{\top}$ we can reduce (4) to
${}\frac{dx}{dt}=c({{\mbox{\boldmath$x$}}})\cos\theta,\qquad\frac{dy}{dt}=c({{\mbox{\boldmath$x$}}})\sin\theta,\qquad\frac{d\theta}{dt}=c_{x}({{\mbox{\boldmath$x$}}})\sin\theta-
c_{y}({{\mbox{\boldmath$x$}}})\cos\theta.$ (5)
The complex-valued $A_{k}$ and $\phi$ close to the central ray are then
approximated by Taylor expansions around the ray,
$\displaystyle{}A_{k}({{\mbox{\boldmath$x$}}})$ $\displaystyle\approx
A_{k}({{\mbox{\boldmath$x$}}}^{*})+({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})\cdot\nabla
A_{k}({{\mbox{\boldmath$x$}}}^{*})+\frac{1}{2}({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})^{\top}D^{2}A_{k}({{\mbox{\boldmath$x$}}}^{*})\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})+\cdots,$
(6) $\displaystyle{}\phi({{\mbox{\boldmath$x$}}})$
$\displaystyle\approx\phi({{\mbox{\boldmath$x$}}}^{*})+({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})\cdot\nabla\phi({{\mbox{\boldmath$x$}}}^{*})+\frac{1}{2}({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})^{\top}D^{2}\phi({{\mbox{\boldmath$x$}}}^{*})\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*})+\cdots,$
(7)
where ${{\mbox{\boldmath$x$}}}^{*}={{\mbox{\boldmath$x$}}}(t)$ for some $t$.
The Taylor coefficients $\phi({{\mbox{\boldmath$x$}}}(t))$,
$\nabla\phi({{\mbox{\boldmath$x$}}}(t))$, $A_{k}({{\mbox{\boldmath$x$}}}(t))$,
etc. on the central ray can be computed. The lowest ones are real on the beam
and given directly,
$\phi({{\mbox{\boldmath$x$}}}(t))=\phi({{\mbox{\boldmath$x$}}}_{0})+t,\qquad\nabla\phi({{\mbox{\boldmath$x$}}}(t))={{\mbox{\boldmath$p$}}}(t).$
The higher order ones can be obtained by solving ODEs similar to (4), and may
have a complex part on the beam. The most common approximation by far is to
take $K=1$ in (2), so that
$A({{\mbox{\boldmath$x$}}})=A_{0}({{\mbox{\boldmath$x$}}})$, and to
approximate $A({{\mbox{\boldmath$x$}}})$ to zeroth order and
$\phi({{\mbox{\boldmath$x$}}})$ to second order. In this case we have, [5],
${}A({{\mbox{\boldmath$x$}}}(t))=A({{\mbox{\boldmath$x$}}}_{0})\,\left(\frac{c({{\mbox{\boldmath$x$}}}(t))}{c({{\mbox{\boldmath$x$}}}_{0})}\,\frac{Q(0)}{Q(t)}\right)^{1/2},\qquad
D^{2}\phi({{\mbox{\boldmath$x$}}}(t))=HNH^{\top},$ (8)
with
${}H=\left(\begin{matrix}\sin\theta&\cos\theta\\\
-\cos\theta&\sin\theta\end{matrix}\right),\quad
N=\left(\begin{matrix}P/Q&-c_{1}/c^{2}\\\
-c_{1}/c^{2}&-c_{2}/c^{2}\end{matrix}\right),\quad\left(\begin{matrix}c_{1}\\\
c_{2}\end{matrix}\right)=H^{\top}\nabla c,$ (9)
and the complex-valued scalar functions $P$ and $Q$ satisfy the dynamic ray
tracing ODEs
$\displaystyle\frac{dQ}{dt}$
$\displaystyle=c^{2}({{\mbox{\boldmath$x$}}})\,P,\qquad Q(0)=Q_{0},$ (10)
$\displaystyle\frac{dP}{dt}$
$\displaystyle=-\frac{c_{xx}\sin^{2}\theta-2c_{xy}\sin\theta\cos\theta+c_{yy}\cos^{2}\theta}{c({{\mbox{\boldmath$x$}}})}\,Q,\quad
P(0)=P_{0}.$ (11)
The quantities $P$ and $Q$ determine the leading order wavefront curvature and
the beam width. For example, if
${{\mbox{\boldmath$y$}}}={{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$x$}}}^{*}$
is orthogonal to the beam at ${{\mbox{\boldmath$x$}}}^{*}$, then by (3) and
(7)
$|u_{\rm
GB}({{\mbox{\boldmath$x$}}}^{*}+{{\mbox{\boldmath$y$}}})|\sim\left|e^{i\omega{{\mbox{\boldmath$\scriptstyle
y$}}}^{\top}D^{2}({{\mbox{\boldmath$\scriptstyle
x$}}}^{*})\phi{{\mbox{\boldmath$\scriptstyle
y$}}}/2}\right|=e^{-\omega(H^{\top}{{\mbox{\boldmath$\scriptstyle
y$}}})^{\top}\Im(N)(H^{\top}{{\mbox{\boldmath$\scriptstyle
y$}}})/2}=e^{-\omega|{{\mbox{\boldmath$\scriptstyle y$}}}|^{2}\Im(P/Q)/2},$
showing that the effective beam width is proportional to
$[\omega\Im(P/Q)/2]^{-1/2}$. It can be proved that if $Q_{0}\neq 0$ and
$\Im(P_{0}/Q_{0})>0$, then $Q(t)\neq 0$ and $\Im(P(t)/Q(t))>0$ along the
central ray for all $t>0$, [1]. Therefore, by a proper choice of initial data
$Q_{0}$ and $P_{0}$, each beam will be regular (with finite amplitude at
caustics) and concentrate along the central ray. A common choice is $Q_{0}>0$
and $P_{0}=i$.
### 2.2 Beam superposition
Let the source curve be given by ${{\mbox{\boldmath$x$}}}_{0}(s)$ in
${\mathbb{R}}^{2}$ parameterized by $s$. We introduce the notation
$A({{\mbox{\boldmath$x$}}},s)$, $\phi({{\mbox{\boldmath$x$}}},s)$ and
$\varphi({{\mbox{\boldmath$x$}}},s)$ for the amplitude, phase and cut-off of a
beam with initial position ${{\mbox{\boldmath$x$}}}_{0}(s)$. In the Gaussian
beam superposition method, the boundary condition on
${{\mbox{\boldmath$x$}}}_{0}(s)$ for the wave field is asymptotically expanded
into Gaussian beams, [5]. Individual Gaussian beams are computed by solving
the ODEs (4) and (10,11). The contributions of the beams concentrated close to
their central rays are determined by the approximations (6, 7) entered in (3).
The wave field is then obtained by the superposition integral over the beams,
${}u_{s}({{\mbox{\boldmath$x$}}})=\omega^{1/2}\int\varphi({{\mbox{\boldmath$x$}}},s)\,A({{\mbox{\boldmath$x$}}},s)\,e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}},s)}ds.$ (12)
In practical computations, this integral is replaced by a discrete sum of
individual beams, the trapezoidal rule approximation,
${}u_{s}^{D}({{\mbox{\boldmath$x$}}})=\omega^{1/2}h\sum_{j\in\mathbb{Z}}\varphi({{\mbox{\boldmath$x$}}},s_{j})A({{\mbox{\boldmath$x$}}},s_{j})e^{i\omega\phi({{\mbox{\boldmath$\scriptstyle
x$}}},s_{j})},$ (13)
where $h$ is the initial spacing of the beams.
The initial conditions for the Taylor coefficient ODEs are chosen such that
$u_{s}^{D}$ well approximates the exact ingoing boundary data. This can be
done in different ways. In particular the initial width of the beams can be
varied to give different approximations. As an example, we consider a plane
wave in the $y$-direction as boundary condition on the $x$-axis,
${{\mbox{\boldmath$x$}}}_{0}(s)=(s,0)$. This will be approximated by a sum of
beams starting in the same direction, [6]. The approximation is based on the
relationship
${}1=\frac{1}{\sqrt{\pi}{{\eta}}_{0}}\int
e^{-(x-s)^{2}/{{\eta}}_{0}^{2}}ds=\sum_{j}\frac{1}{\sqrt{\pi}}\,\frac{h}{{{\eta}}_{0}}e^{-(x-s_{j})^{2}/{{\eta}}_{0}^{2}}+{\mathcal{O}}(e^{-({{\eta}}_{0}/h)^{2}}),\quad
s_{j}=jh,$ (14)
with ${{\eta}}_{0}$ representing the initial beam widths, see Figure 1.
Figure 1: The sum of several Gaussian functions is almost constant. A plane
wave can therefore be decomposed approximately to a sum of parallel Gaussian
beams.
Identifying (14) with (12, 13), assuming $\varphi\equiv 1$ we see that
$A(x,0,s)=\frac{1}{\sqrt{\pi\omega}{{\eta}}_{0}},\qquad\phi(x,0,s)=i\frac{(x-s)^{2}}{\omega{{\eta}}_{0}^{2}}.$
To properly choose the initial data, one must take the parameters
${{\eta}}_{0}$ and $h$ such that ${{{\eta}}_{0}}>h$ by (14). Then the wave
field (13) will produce an accurate plane wave on
${{\mbox{\boldmath$x$}}}_{0}(s)=(s,0)$. The condition ${{{\eta}}_{0}}>h$ can
be related to the initial data $(P_{0},Q_{0})$ of the dynamic ray tracing ODEs
(10,11). Since beams go in the $y$-direction $\theta=\pi/2$ and we have $H=I$
and $\phi_{xx}=P/Q$ by (8) and (9). Thus,
$\frac{P_{0}}{Q_{0}}=\frac{2i}{\omega{{\eta}}_{0}^{2}}.$
Chosing $P_{0}=i$ we therefore get
$h<{{\eta}}_{0}=\left(\frac{2Q_{0}}{\omega}\right)^{1/2}.$
With this relation between $h$ and $Q_{0}$ we get an accurate approximation.
In particular we need a spacing $h$ of order $O(1/\sqrt{\omega})$ for a fixed
$Q_{0}$. Note also that for computational efficiency, $h$ should not be taken
much smaller. These restrictions were derived for a plane wave but similar
scalings will be necessary also for more general boundary data.
In what follows, in order to simplify the calculations, we assume that all
beams, originating from ${{\mbox{\boldmath$x$}}}_{0}(s)$, shoot out
orthogonally. We denote by ${{\mbox{\boldmath$X$}}}(t,s)$ the location of the
center ray originating in ${{\mbox{\boldmath$x$}}}_{0}(s)$ after time $t$. We
further assume that $\phi({{\mbox{\boldmath$x$}}}_{0}(s),s)=0$.
We make one observation that will be used in the analysis below. It is well-
known that ${{\mbox{\boldmath$X$}}}_{t}\parallel\nabla_{x}\phi$,
${{\mbox{\boldmath$X$}}}_{t}\cdot\nabla_{x}\phi=1$ and
${{\mbox{\boldmath$X$}}}_{s}\perp{{\mbox{\boldmath$X$}}}_{t}$ under the
assumptions made above. Therefore, since
$\phi({{\mbox{\boldmath$X$}}}(t,s),s)=t$,
${}0=\frac{d}{ds}\phi({{\mbox{\boldmath$X$}}}(t,s),s)={{\mbox{\boldmath$X$}}}_{s}\cdot\nabla_{x}\phi+\phi_{s}=\phi_{s}({{\mbox{\boldmath$X$}}}(t,s),s)$
(15)
Hence $\phi_{s}=0$ everywhere on the central rays.
## 3 Accuracy of Gaussian beams summation
In this section we study the accuracy of summation of Gaussian beams. One can
distinguish six different types of errors in the approximation:
1. 1.
High frequency approximation.
2. 2.
Error in initial data.
3. 3.
Discretization error.
4. 4.
Taylor expansion error.
5. 5.
Cut-off error.
6. 6.
Error in numerical integrators for solving Taylor coefficient ODEs.
The first error depends on the number of terms used in the WKBJ approximation,
i.e. the difference
$u({{\mbox{\boldmath$x$}}})-u_{s}({{\mbox{\boldmath$x$}}})$. For example, for
standard beams it is of the order $O(1/\omega)$ since one amplitude term is
used, meaning that each beam is a solution to the Helmholtz equation up to
order $O(1/\omega)$. The second error represents how well the exact boundary
data is approximated by a superposition of Gaussian beams. The third error is
caused by replacing the superposition integral by a discrete summation of
beams, i.e.
$u_{s}({{\mbox{\boldmath$x$}}})-u_{s}^{D}({{\mbox{\boldmath$x$}}})$. The
fourth error is due to the fact that $A$ and $\phi$ are not computed globally,
and only their derivatives on the central beams are computed. One therefore
needs to approximate their values around the central beams by Taylor
expansions. The fifth error is caused by multiplying the solution by a smooth
cut-off function in order to account for possible irregularities away from the
central rays. Finally, the last error is the numerical error in solving the
ODEs for computing Taylor coefficients. For example the global error in a
fourth order Runge-Kutta method is $O(\Delta t^{4})$, with $\Delta t$ being
the time-step.
Here, we will only concentrate on the Discretization and Taylor expansion
errors.
### 3.1 Motivation and preliminaries
Let the source be a curve ${{\mbox{\boldmath$x$}}}_{0}(s)$ and assume that we
look for the solution along a line ${{\mbox{\boldmath$x$}}}=(x,y^{*})$. We
simplify the notation by setting
$A_{k}({{\mbox{\boldmath$x$}}},s)=A_{k}(x,s)$,
$\phi({{\mbox{\boldmath$x$}}},s)=\phi(x,s)$,
$\varphi({{\mbox{\boldmath$x$}}},s)=\varphi(x,s)$ and write
$u_{s}(x)=\omega^{1/2}\int\varphi(x,s)\,A(x,s)\,e^{i\omega\phi(x,s)}ds,$
$u_{s}^{D}(x)=\omega^{1/2}h\sum_{j\in\mathbb{Z}}\varphi(x,s_{j})A(x,s_{j})e^{i\omega\phi(x,s_{j})},\qquad
s_{j}=jh.$
We now let $X(s)$ denote the location of the center beam on the line
$(x,y^{*})$ when the initial data is given at
${{\mbox{\boldmath$x$}}}_{0}(s)$. Hence, $X(s)$ is implicitly defined by
${{\mbox{\boldmath$X$}}}(t(s),s)=(X(s),y^{*}),$
for some function $t(s)$. Figure 2 shows the setting for
${{\mbox{\boldmath$x$}}}_{0}(s)=(s,0)$, as an example.
Figure 2: A schematic representation of the initial source and a beam central
ray.
We will now explain the approximation of $A(x,s)$ and $\phi(x,s)$ for a
$(q+1)$-th order Gaussian beam, complying with the standard notation that the
basic choice $q=0$ is a first order beam. We then take $K=\lfloor
q/2\rfloor+1$ terms in the WKBJ expansion (2) and observe that the high
frequency approximation error will be of the order
${}O(\omega^{-q^{*}/2})\quad{\rm where}\quad q^{*}=2(\lfloor
q/2\rfloor+1)=\begin{cases}q+2,&\text{$q$ even},\\\ q+1,&\text{$q$
odd}.\end{cases}$ (16)
For the term $A_{k}(x,s)$ we make a Taylor expansion up to order $q-2k$ around
$X(s)$,
${}A_{k}(x,s)\approx\tilde{A}_{k,q-2k}(x,s):=A_{k}(X(s),s)+\cdots+\frac{(x-X(s))^{q-2k}}{(q-2k)!}\partial_{x}^{q-2k}A_{k}(X(s),s),$
(17)
with $0\leq k\leq\lfloor q/2\rfloor$. We also set
${}\tilde{A}_{q}(x,s):=\sum_{k=0}^{\lfloor
q/2\rfloor}\tilde{A}_{k,q-2k}(x,s)(i\omega)^{-k}\approx A(x,s).$ (18)
Furthermore, we approximate $\phi(x,s)$ up to level $q+2$,
${}\phi(x,s)\approx\tilde{\phi}_{q}(x,s):=\phi(X(s),s)+\cdots+\frac{(x-X(s))^{q+2}}{(q+2)!}\partial_{x}^{q+2}\phi(X(s),s).$
(19)
The approximate Gaussian beam solution is then given by
$\tilde{u}_{s}(x)=\omega^{1/2}\int\varphi(x,s)\,\tilde{A}_{q}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}ds,$
$\tilde{u}_{s}^{D}(x)=\omega^{1/2}h\sum_{j\in\mathbb{Z}}\varphi(x,s_{j})\tilde{A}_{q}(x,s_{j})e^{i\omega\tilde{\phi}_{q}(x,s_{j})}.$
The reason why the phase is approximated to two orders higher than the
amplitude is to balance the Taylor expansion errors; the phase error is
multiplied by the frequency $\omega$, which is proportional to one over the
beam width squared (cf. (20) below). Note that for $q\geq 2$, one needs to
take $K>1$ in (2), i.e. to include more terms in the WKBJ expansion in order
to also balance the high frequency approximation error and the Taylor
expansion error, cf. Remark 2 below and the discussion in [15].
Our motivation for considering the Taylor expansion error comes from the
following observation. We define the width of the Gaussian beam passing
through $(x,y^{*})$ as
${\eta}(x):=\frac{1}{\sqrt{\omega\,\Im\phi_{xx}(x,X^{-1}(x))}}.$
Because of the term $e^{i\omega(x-X(s))^{2}\phi_{xx}/2}$ the solution will be
close to zero for $|x-X(s)|>{\eta}(x)$. A simple error analysis would
therefore give the following result. Using (2) we have
$\displaystyle u_{s}-\tilde{u}_{s}$
$\displaystyle=(A-\tilde{A}_{q})e^{i\omega\tilde{\phi}_{q}}+Ae^{i\omega\tilde{\phi}_{q}}(e^{i\omega(\phi-\tilde{\phi}_{q})}-1)$
$\displaystyle=\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega)^{-k}(A_{k}-\tilde{A}_{k,q-2k})e^{i\omega\tilde{\phi}_{q}}+Ae^{i\omega\tilde{\phi}_{q}}(e^{i\omega(\phi-\tilde{\phi}_{q})}-1)$
$\displaystyle=\sum_{k=0}^{\lfloor
q/2\rfloor}O(\omega^{-k}{{\eta}}^{q-2k+1})e^{i\omega\tilde{\phi}_{q}}+Ae^{i\omega\tilde{\phi}_{q}}(e^{iO(\omega{{\eta}}^{q+3})}-1).$
Hence, since ${{\eta}}=O(\omega^{-1/2})$ we would have
${}u_{s}-\tilde{u}_{s}\sim\sum_{k=0}^{\lfloor
q/2\rfloor}O(\omega^{-(q+1)/2})+Ae^{i\omega\tilde{\phi}_{q}}O(\omega^{-(q+3)/2+1})\sim
O(\omega^{-(q+1)/2}).$ (20)
In particular, for first order beams with $q=0$, the convergence rate in
$\omega$ would be half order, i.e. proportional to $1/\sqrt{\omega}$. This is
also what is observed numerically for a single Gaussian beam. However, we now
consider two numerical examples using superposition of first order beams to
verify the convergence rate for this case. Since it is difficult to obtain the
exact Gaussian beam superposition solution $u_{s}$ we here instead compare
$\tilde{u}_{s}$ with first order ($K=1$) geometrical optics $u_{\rm GO}$,
which is close enough to $u_{s}$ to verify or refute the sharpness of (20);
the high-frequency error in both $u_{s}$ and first order geometrical optics is
of the order $O(1/\omega)$. Hence, the predicted error of first order beams,
$O(1/\sqrt{\omega})$, would dominate if it was sharp since
$|\tilde{u}_{s}-u_{\rm GO}|\leq|\tilde{u}_{s}-u_{s}|+|u_{s}-u|+|u-u_{\rm
GO}|\leq C(1/\sqrt{\omega}+1/\omega)\leq C/\sqrt{\omega}.$
In the first example, a plane wave generated on the line $y=0$ propagates
orthogonally into the computational domain with a variable speed of
propagation. Figure 3a shows the central rays of Gaussian beams, and Figure 3c
shows the absolute value of the Gaussian beams and geometrical optics
solutions along the line $y=0.6$, shown in bold in Figure 3a. Figure 3e shows
the logarithmic scale of the maximum error between the Gaussian beams solution
and the geometrical optics solution. As can be seen, the convergence rate of
the error is surprisingly proportional to $\omega^{-1}$, which is half order
better than what we expected.
In the second example, a plane wave generated on the line $x=0$ propagates
with an angle of $45^{o}$ into the computational domain with a variable speed
of propagation. The convergence rate of the error, shown in Figure 3f, is
again proportional to $\omega^{-1}$.
Figure 3: Left and right top figures show the central rays of Gaussian beams
by an initial plane wave on $x$\- and $y$-axis, respectively. Middle figures
show the absolute value of the Gaussian beams and geometrical optics solutions
along the lines $y=0.6$ and $y=2$. Bottom figures show the logarithmic scale
of the maximum error between the Gaussian beams solution and the geometrical
optics solution. The convergence rate of the maximum error is $\omega^{-1}$.
We will therefore study the Taylor expansion and discretization errors more
carefully to describe why it is smaller than what we expected.
### 3.2 Main result
For our results we make the following precise assumptions
* (A1)
Smoothness of all coefficients. We assume $A_{k}(x,s)\in
C_{b}^{p+q+2-2k}(\mathbb{R}^{2})$, the space of functions with $p+q+2-2k$
continuous and bounded derivatives. Similarly $\phi(x,s)\in C^{p+q+4}$ and
${{\mbox{\boldmath$X$}}}(t,s)\in C^{p+1}$, with $p\geq 2$.
* (A2)
Algebraic growth of phase off center beam. For $p_{1},p_{2}\leq p+q+4$, we
have for some $\bar{p}$,
$\partial_{x}^{p_{1}}\partial_{s}^{p_{2}}\phi(x,s)\leq
C(1+|x-X(s)|^{\bar{p}}).$
In particular, all derivatives are bounded on the center beam, $x=X(s)$.
* (A3)
No caustics. The derivative $X^{\prime}(s)$ is bounded away from zero,
$0<c_{0}\leq X^{\prime}(s)\leq c_{1}<\infty$.
* (A4)
Non-degeneracy of each beam. The imaginary part of $\phi_{xx}$ on the beam is
strictly positive and bounded,
${}0<c_{0}\leq\Im\phi_{xx}(X(s),s)\leq c_{1}<\infty.$ (21)
Moreover, the frequency is non-vanishing, $\omega>c_{2}$. This means that the
approximate beams will have a fast decay off the central beam for high
frequencies, and also that the beam width never vanishes or becomes infinite.
The last point is an important feature of Gaussian beams, related to the fact
that Gaussian beams can approximate the exact field at caustics.
* (A5)
Cut-off of fixed size. We use $\varphi(x,s)=\varphi(X(s)-x)$ with $\varphi\in
C^{\infty}$ such that $\varphi(x)=1$ for $|x|\leq\alpha/2$ and $\varphi(x)=0$
for $|x|>\alpha$. The size of $\alpha$ will be chosen ”small enough” depending
on $\phi$ but independent of $\omega$.
The error that we want to estimate is given by
$E(x)=u_{s}(x)-\tilde{u}_{s}^{D}(x)=u_{s}(x)-\tilde{u}_{s}(x)+\tilde{u}_{s}(x)-\tilde{u}_{s}^{D}(x)=:E^{T}+E^{D},$
where $E^{T}=u_{s}(x)-\tilde{u}_{s}(x)$ and
$E^{D}=\tilde{u}_{s}(x)-\tilde{u}_{s}^{D}(x)$ represent the Taylor expansion
error and the discretization error, respectively.
Then we can show
###### Theorem 1
(Main Theorem) For the $(q+1)$-th order Gaussian beams, we have
$|u_{s}(x)-\tilde{u}^{D}_{s}(x)|\leq|E^{T}|+|E^{D}|,$
where
${}|E^{T}|\leq C\,\omega^{-\frac{q^{*}}{2}},\qquad
q^{*}=\begin{cases}q+2,&\text{\rm$q$ even},\\\ q+1,&\text{\rm$q$
odd},\end{cases},$ (22)
and
${}|E^{D}|\leq C\,\left(\frac{h}{{\eta}(x)}\right)^{p}.$ (23)
The constants depend on $p$, the initial data, $P_{0}$ and $Q_{0}$, for the
beams but does not depend on $x$, $\omega$ or $h$. For the Taylor expansion
error we have
$\left|E^{T}-C^{*}(x)\,\omega^{1/2}\,{{\eta}}^{q^{*}+1}\right|\leq
C^{\prime}\,\omega^{1/2}\,{{\eta}}^{q^{*}+2},$
i.e., the leading order term of the error $E^{T}$ in $\omega$ is
$C^{*}(x)\,\omega^{1/2}\,{{\eta}}^{q^{*}+1}\sim\omega^{-q^{*}/2}$, with
$C^{*}(x)$ given by (39), (37), (34) and (36).
###### Remark 1
If we take $h<{\eta}(x)$, the discretization error $E^{D}$ is typically
smaller than the Taylor expansion error $E^{T}$ because of the ”spectral”
accuracy in (23). For first order beams (with $q=0$), the observed convergence
rate is therefore first order in $\omega$, which is the same as geometrical
optics. However, $h$ should not be chosen too small for computational
complexity reasons. It is also important to note that to balance the error
with the error in initial data, $h$ should also relate to the initial beam
width ${{\eta}}_{0}$.
###### Remark 2
The estimate (22) shows that the Taylor expansion error indeed balances the
high frequency approximation error (16). Moreover, it suggests that there is
no remarkable gain in using even order beams (with an odd $q$); neither the
high frequency nor the Taylor expansion error get a better convergence rate in
$1/\omega$ with these beams. However, one should note that this is only true
in the case of the superposition of beams, where error in adjacent beams
cancel. If we only have one beam, this does not hold and the simple error
estimate in (20) is sharp. In this case the Taylor expansion error dominates
the high frequency approximation error for even order beams.
## 4 Proof of main result
Before going on to the proof of Theorem 1, we prove the following utility
lemma concerning estimates for the composition of two functions.
###### Lemma 1
Suppose $g_{\delta}(z)$ belongs to $C^{p}(\mathbb{R})$ for each value of the
parameter $\delta$. If
${}|g^{(k)}_{\delta}(z)|\leq C_{k}(1+|z|^{q}),\qquad 1\leq k\leq p,$ (24)
where $C_{k}$ and $q\geq 0$ are constants independent of $z$ and $\delta$,
then there are functions $h_{m,k}\in C^{p-k}(\mathbb{R})$ and constants
$C_{m,k}$ independent of $z$ and $\delta$, such that
${}\frac{d^{k}}{dz^{k}}e^{g_{\delta}(z)}=e^{g_{\delta}(z)}\,\sum_{m=1}^{k}h_{m,k}(z),\qquad\max_{0\leq
n\leq p-k}|h^{(n)}_{m,k}(z)|\leq C_{m,k}(1+|z|^{qk}).$ (25)
Proof: We show (25) by induction. For $k=1$ we have
$h_{1,1}=g^{\prime}_{\delta}(z)\in C^{p-1}$ and the statement clearly holds.
Suppose (25) is true for $1\leq k<p$. Then
$\frac{d^{k+1}}{dz^{k+1}}e^{g_{\delta}(z)}=e^{g_{\delta}(z)}\,\sum_{m=1}^{k}h^{\prime}_{m,k}(z)+g_{\delta}^{\prime}(z)\,h_{m,k}(z)=e^{g_{\delta}(z)}\,\sum_{m=1}^{k+1}h_{m,k+1}(z).$
Thus
$h_{m,k+1}(z)=\begin{cases}h^{\prime}_{m,k},&m=1,\\\
h^{\prime}_{m,k}+g_{\delta}^{\prime}h_{m-1,k},&1<m\leq k,\\\
g_{\delta}^{\prime}h_{m-1,k},&m=k+1.\end{cases}$
Using the induction hypothesis, we immediately get that $h_{m,k+1}(z)\in
C^{p-k-1}(\mathbb{R})$. Moreover,
$\max_{0\leq n\leq p-k-1}|h^{(n)}_{m,k+1}(z)|\leq\max_{0\leq n\leq
p-k-1}|h^{(n+1)}_{m,k}(z)|+\max_{0\leq n\leq
p-k-1}\sum_{j=0}^{n}c_{jn}|h^{(j)}_{m-1,k}(z)||g^{(n+1-j)}_{\delta}(z)|$
The first term is bounded by $C_{1,1}(1+|z|^{qk})$ by assumption, and for the
second term we can estimate
$|h^{(j)}_{m-1,k}(z)||g^{(n+1-j)}_{\delta}(z)|\leq
C_{m-1,k}(1+|z|^{qk})\,C_{k}(1+|z|^{q})\leq C^{\prime}(1+|z|^{q(k+1)}),$
which proves (25). $~{}\Box$
We can now start with the main proof. We consider each error separately.
### 4.1 Taylor expansion error
The Taylor expansion error is given by
$\displaystyle{}E^{T}$
$\displaystyle=u_{s}(x)-\tilde{u}_{s}(x)=\omega^{1/2}\,\int\varphi(X(s)-x)\,\left(A(x,s)\,e^{i\omega\phi(x,s)}-\tilde{A}_{q}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}\right)ds$
(26) $\displaystyle=\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega)^{-k}\omega^{1/2}\int\varphi(X(s)-x)\left(A_{k}(x,s)\,e^{i\omega\phi(x,s)}-\tilde{A}_{k,q-2k}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}\right)ds.$
In this subsection we will start by studying the general integral
approximation
${}\bar{E}^{T}=\omega^{1/2}\int\varphi(X(s)-x)\left(A(x,s)\,e^{i\omega\phi(x,s)}-\tilde{A}_{q_{a}}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}\right)ds,$
(27)
where, with a slight abuse of notation, $A$ and $\tilde{A}_{q_{a}}$ will
represent one of $A_{k}$ and $\tilde{A}_{k,q-2k}$ respectively in the sum
above. We can therefore also assume that $q_{a}\leq q$ and that $q-q_{a}$ is
even.
Let us denote $X^{-1}(x)$ by $m(x)$ and then, since $X^{\prime}(s)$ is bounded
away from zero we can use the change of variables
${}z=\frac{X(s)-x}{{\eta}(x)}\quad\Rightarrow\quad s=m(x+{\eta}(x)z).$ (28)
We obtain
$\displaystyle\bar{E}^{T}$
$\displaystyle=\omega^{1/2}\,{\eta}\,\int\varphi({{\eta}}z)\,\Big{(}A(x,m(x+{{\eta}}z))\,e^{i\omega\phi(x,m(x+{{\eta}}z))}-$
$\displaystyle\tilde{A}_{q_{a}}(x,m(x+{{\eta}}z))\,e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}\Big{)}\,m^{\prime}(x+{{\eta}}z)\,dz.$
Now, letting
$D_{A}(x,s):=A(x,s)-\tilde{A}_{q_{a}}(x,s),\qquad
D_{\phi}(x,s):=\phi(x,s)-\tilde{\phi}_{q}(x,s).$
and recalling that supp $\varphi\subset[-\alpha,\alpha]$, we can write the
integral as
${}\bar{E}^{T}=\omega^{1/2}\,{\eta}\,\int_{|z|\leq\frac{\alpha}{w}}\varphi\,\left(D_{A}+A(e^{i\omega
D_{\phi}}-1)\right)e^{i\omega\tilde{\phi}_{q}}m^{\prime}dz.$ (29)
We will now approximate the terms in the integral (29) by their Taylor
expansion. Let us use the shorthand
$\tilde{a}_{p}(x)=\frac{(-1)^{p}}{p!}\partial_{x}^{p}A(x,m(x)),\qquad\tilde{b}_{p}(x)=\left.\frac{1}{p!}\frac{d^{p}}{dz^{p}}A(x,m(x+z))\right|_{z=0}.$
and
${}\tilde{p}_{p}(x)=\frac{(-1)^{p}}{p!}\partial_{x}^{p}\phi(x,m(x)),\qquad\tilde{r}_{p}(x)=\left.\frac{1}{p!}\frac{d^{p}}{dz^{p}}\phi(x,m(x+z))\right|_{z=0}.$
(30)
We note that, in this notation
$\tilde{A}_{q_{a}}(x,m(x+z))=\sum_{j=0}^{q_{a}}\tilde{a}_{j}(x+z)\,z^{j},\qquad\tilde{\phi}_{q}(x,m(x+z))=\sum_{j=0}^{q+2}\tilde{p}_{j}(x+z)\,z^{j}.$
Let
$a_{1}(x)=\tilde{a}_{q_{a}+1}(x),\qquad
a_{2}(x)=\tilde{a}_{q_{a}+2}(x)+\tilde{a}^{\prime}_{q_{a}+1}(x),$
$b_{1}(x)=i\frac{\tilde{p}_{q+3}(x)}{\Im\phi_{xx}(x,m(x))},\qquad
b_{2}(x)=i\frac{\tilde{p}_{q+4}(x)+\tilde{p}^{\prime}_{q+3}(x)}{\Im\phi_{xx}(x,m(x))},$
$c_{1}(x)=\Re\frac{\tilde{r}_{2}(x)}{\Im\phi_{xx}(x,m(x))},\qquad
c_{2}(x)=i\frac{\tilde{r}_{3}(x)-\sigma\tilde{p}_{3}(x)}{\Im\phi_{xx}(x,m(x))}.$
where $\sigma=1$ for $q=0$ and $\sigma=0$ for $q>0$. We then approximate
$\displaystyle D_{A}(x,m(x+{{\eta}}z))$
$\displaystyle\approx{{\eta}}^{q_{a}+1}\tilde{D}_{A}(x,z):=({{\eta}}z)^{q_{a}+1}a_{1}(x)+({{\eta}}z)^{q_{a}+2}a_{2}(x),$
$\displaystyle e^{i\omega D_{\phi}(x,m(x+{{\eta}}z))}-1$
$\displaystyle\approx{{\eta}}^{q+1}\tilde{B}(x,z):={{\eta}}^{q+1}b_{1}(x)z^{q+3}+{{\eta}}^{q+2}(b_{2}(x)z^{q+4}+\sigma
b_{1}^{2}(x)z^{2q+6}/2),$ $\displaystyle
e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}$
$\displaystyle\approx\tilde{C}(x,z)=:e^{i\omega\phi(x,m(x))+iz^{2}c_{1}(x)-z^{2}/2}(1+c_{2}(x){\eta}z^{3}).$
The residual terms are denoted
$D_{A}(x,m(x+{{\eta}}z))-{{\eta}}^{q_{a}+1}\tilde{D}_{A}(x,z)=:{{\eta}}^{q_{a}+3}R_{A}(x,z),$
$e^{i\omega
D_{\phi}(x,m(x+{{\eta}}z))}-1-{{\eta}}^{q+1}\tilde{B}(x,z)=:{{\eta}}^{q+3}R_{B}(x,z),$
$e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}-\tilde{C}(x,z)=:{{\eta}}^{2}R_{C}(x,z).$
Then we have
###### Lemma 2
Let the residual terms $R_{A}$, $R_{B}$ and $R_{C}$ be defined as above. Under
assumptions (A1) and (A2), for small enough $\alpha$,
$\left|R_{A}\right|\leq C|z|^{q_{a}+3},\qquad\left|R_{B}\right|\leq
Ce^{z^{2}/7},\qquad\left|R_{C}\right|\leq
Ce^{-z^{2}/4},\quad\forall|z|\leq\alpha/{\eta},$
where the constant $C$ is independent of $x$, $\omega$ and $z$.
Proof: We note that $\tilde{a}_{q_{a}}(x+z)$ are the first $q_{a}$
coefficients in the Taylor expansion of $A(x+z-x^{\prime},m(x+z))$ around
$x^{\prime}=0$. Therefore, by Taylor’s theorem and assumption (A1)
$\left|D_{A}(x,m(x+z))-z^{q_{a}+1}\tilde{a}_{q_{a}+1}(x+z)-z^{q_{a}+2}\tilde{a}_{q_{a}+2}(x+z)\right|\leq
C|z|^{q_{a}+3}.$
Expanding the second and third terms around $z=0$ gives the bound for $R_{A}$.
We now estimate $\omega D_{\phi}$ in two different ways. By Taylor’s theorem
as above, for some $\xi$ with $|\xi-x|\leq{{\eta}}z$,
$|\omega
D_{\phi}(x,m(x+{{\eta}}z))|\leq\left|\partial^{(q+3)}_{x}\phi(\xi,m(x+{{\eta}}z))\right|\frac{\omega|{{\eta}}z|^{q+3}}{(q+3)!}\leq
C\omega|{{\eta}}z|^{q+3}|(1+|{{\eta}}z|^{\bar{p}}),$
where we used the growth condition (A2) for $\phi$ to bound the error term.
Then, for $|z|\leq\alpha/{\eta}$, and small enough $\alpha$,
${}|\omega D_{\phi}|\leq C{{\eta}}^{q+1}|z|^{q+3},\qquad|\omega D_{\phi}|\leq
Cz^{2}\alpha^{q+1}(1+\alpha^{\bar{p}})\leq\frac{z^{2}}{8},$ (31)
implying
$\left|e^{i\omega D_{\phi}}-1-i\omega D_{\phi}-\frac{(i\omega
D_{\phi})^{2}}{2}\right|\leq\frac{1}{6}|\omega D_{\phi}|^{3}e^{|\omega
D_{\phi}|}\leq C{{\eta}}^{3q+3}|z|^{3q+9}e^{z^{2}/8}.$
Moreover, the same steps as for $D_{A}$ together with (A2) gives
$\left|D_{\phi}(x,m(x+z))-z^{q+3}\tilde{p}_{q+3}(x)-z^{q+4}(\tilde{p}_{q+4}(x)+\tilde{p}^{\prime}_{q+3}(x))\right|\leq
C|z|^{q+5}(1+|z|^{\bar{p}}),$
and since $\omega=1/{{\eta}}^{2}\Im\phi_{xx}$, when $|z|\leq\alpha/{\eta}$,
$\left|i\omega
D_{\phi}(x,m(x+{{\eta}}z))-{{\eta}}^{q+1}z^{q+3}b_{1}(x)-{{\eta}}^{q+2}z^{q+4}b_{2}(x)\right|\leq
C{{\eta}}^{q+3}|z|^{q+5}.$
Finally, for $q>0$, clearly $|\omega D_{\phi}|^{2}\leq
C{{\eta}}^{2q+2}|z|^{2q+6}\leq C{{\eta}}^{q+3}|z|^{2q+6}$ and for $q=0$ we get
$|(i\omega
D_{\phi})^{2}-{{\eta}}^{2}b_{1}^{2}z^{6}|=\frac{|D_{\phi}^{2}-({{\eta}}^{3}z^{3}\tilde{p}_{3})^{2}|}{{{\eta}}^{4}\Im\phi_{xx}^{2}}=\frac{|D_{\phi}-{{\eta}}^{3}z^{3}\tilde{p}_{3}||D_{\phi}+{{\eta}}^{3}z^{3}\tilde{p}_{3}|}{{{\eta}}^{4}\Im\phi_{xx}^{2}}\leq
C{{\eta}}^{3}|z|^{7}.$
Thus,
$|R_{B}|\leq
C{{\eta}}^{2q}|z|^{3q+9}e^{z^{2}/8}+C|z|^{q+5}+(1-\sigma)C|z|^{2q+6}+\sigma
C|z|^{2q+7}\leq C^{\prime}e^{z^{2}/7}.$
To show the third inequality, we note that since $\phi_{s}(x,m(x))\equiv 0$ by
(15), we have $\tilde{r}_{1}(x)=0$. Therefore by Taylor’s theorem and
assumption (A2), for $q^{\prime}\geq 2$,
${}\left|\phi(x,m(x+z))-\phi(x,m(x))-\sum_{p=2}^{q^{\prime}}z^{p}\tilde{r}_{p}(x)\right|\leq
C|z|^{q^{\prime}+1}(1+|z|^{\bar{p}}).$ (32)
Let $v(x,z)=\tilde{\phi}_{q}(x,m(x+z))-\phi(x,m(x))-z^{2}\tilde{r}_{2}(x)$.
Then, by (31) and (32),
$|v(x,z)|=|\phi(x,m(x+z))-\phi(x,m(x))-D_{\phi}(x,m(x+z))-z^{2}\tilde{r}_{2}(x)|\leq
C|z|^{3}(1+|z|^{\bar{p}}).$
Moreover,
$|v(x,z)-z^{3}(\tilde{r}_{3}(x)+\sigma\tilde{p}_{3}(x))|\leq
C|z|^{4}(1+|z|^{\bar{p}}).$
As above, if $|z|\leq\alpha/{\eta}$,
$|e^{i\omega v(x,{{\eta}}z)}-1-{{\eta}}z^{3}c_{2}(x)|\leq|i\omega
v(x,{{\eta}}z)-{{\eta}}z^{3}c_{2}(x)|+\frac{1}{2}|\omega v|^{2}e^{|\omega v|}$
$\leq
C\omega|{{\eta}}z|^{4}(1+|{{\eta}}z|^{\bar{p}})+\frac{1}{2}\left|C\omega|{{\eta}}z|^{3}(1+|{{\eta}}z|^{\bar{p}})\right|^{2}e^{C\omega|{{\eta}}z|^{3}(1+|{{\eta}}z|^{\bar{p}})}$
$\leq
C{{\eta}}^{2}|z|^{4}(1+\alpha^{\bar{p}})+C{{\eta}}^{2}|z|^{6}(1+\alpha^{\bar{p}})^{2}e^{Cz^{2}\alpha(1+\alpha^{\bar{p}})}\leq
C{{\eta}}^{2}e^{z^{2}/4},$
for small enough $\alpha$. It remains to note that, since
$\phi_{s}(x,m(x))=\Im\phi_{x}(x,m(x))\equiv 0$,
$\Im\tilde{r}_{2}=\frac{1}{2}m^{\prime}(x)^{2}\Im\phi_{ss}=\frac{1}{2}m^{\prime}(x)\Im\left(\frac{d}{dx}\phi_{s}-\phi_{sx}\right)=-\frac{1}{2}\Im\left(\frac{d}{dx}\phi_{x}-\phi_{xx}\right)=\frac{1}{2}\Im\phi_{xx},$
which shows that $i\omega{{\eta}}^{2}z^{2}\tilde{r}_{2}=iz^{2}c_{1}-z^{2}/2$.
Therefore,
$\displaystyle{{\eta}}^{2}|R_{C}|=\left|e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}-\tilde{C}(x,z)\right|$
$\displaystyle=\left|\left(e^{i\omega
v(x,{{\eta}}z)}-1-{{\eta}}z^{3}c_{2}(x)\right)\,e^{i\omega\phi(x,m(x))+iz^{2}c_{1}(x)-z^{2}/2}\right|$
$\displaystyle\leq C{{\eta}}^{2}e^{z^{2}/4}e^{-z^{2}/2},$
and the estimate for $|R_{C}|$ follows. $~{}\Box$
We Taylor expand the remaining quantities in (29) and use the assumption (A5)
to get
$\displaystyle\varphi({{\eta}}z)$ $\displaystyle\approx 1,$ $\displaystyle
A(x,m(x+{{\eta}}z))$
$\displaystyle\approx\tilde{A}(x,z):=A(x,m(x))+{{\eta}}z\tilde{b}_{1}(x),$
$\displaystyle m^{\prime}(x+{{\eta}}z)$
$\displaystyle\approx\tilde{m}(x,z):=m^{\prime}(x)+{{\eta}}zm^{\prime\prime}(x).$
It is easy to see that the residual terms for these approximations can all be
bounded by $C{{\eta}}^{2}z^{2}$. Since these residual terms as well as $R_{A}$
and $R_{B}$ above all grow slower than $\exp(z^{2}/4)$, we can replace the
terms in the integral in (29) by their approximations and control the error by
$O({{\eta}}^{q_{a}+3})$,
${}\left|\bar{E}^{T}-\omega^{1/2}\,{\eta}\,{{\eta}}^{q_{a}+1}\int_{|z|\leq\frac{\alpha}{{\eta}}}\tilde{D}_{A}\tilde{C}\tilde{m}dz-\omega^{1/2}\,{\eta}\,{{\eta}}^{q+1}\int_{|z|\leq\frac{\alpha}{{\eta}}}\tilde{A}\tilde{B}\tilde{C}\tilde{m}dz\right|\leq
C\omega^{1/2}\,{\eta}\,{{\eta}}^{q_{a}+3}.$ (33)
Moreover, since the $\tilde{C}(x,z)$ is exponentially small in ${\eta}$ for
$|z|\geq\alpha/{\eta}$ this estimate holds also when taking the integral over
all of $\mathbb{R}$. We can now compute the leading error terms in ${\eta}$.
The first one, $e_{1}=e_{11}+{\eta}e_{12}$, is
$\displaystyle\int\tilde{D}_{A}\tilde{C}\tilde{m}dz$
$\displaystyle=\int(z^{q_{a}+1}a_{1}(x)+{{\eta}}z^{q_{a}+2}a_{2}(x))e^{i\omega\phi(x,m(x))+iz^{2}c_{1}(x)-z^{2}/2}(1+c_{2}(x){\eta}z^{3})(m^{\prime}(x)+{{\eta}}zm^{\prime\prime}(x))dz$
$\displaystyle=e^{i\omega\phi(x,m(x))}\int
z^{q_{a}+1}a_{1}m^{\prime}+{{\eta}}z^{q_{a}+2}(a_{2}m^{\prime}+a_{1}m^{\prime\prime}+a_{1}c_{2}m^{\prime}z^{2})e^{iz^{2}c_{1}(x)-z^{2}/2}dz+O({{\eta}}^{2})$
$\displaystyle=:e^{i\omega\phi(x,m(x))}(e_{11}+{\eta}e_{12})+O({{\eta}}^{2}),$
where
${}e_{11}=d_{q_{a}+1}a_{1}m^{\prime},\qquad
e_{12}=a_{2}m^{\prime}d_{q_{a}+2}+a_{1}m^{\prime\prime}d_{q_{a}+2}+a_{1}c_{2}m^{\prime}d_{q_{a}+4},$
(34)
and
$d_{p}(x)=\int
z^{p}e^{iz^{2}c_{1}(x)-z^{2}/2}dz=\begin{cases}N_{p}(1-2ic_{1}(x))^{-(p+1)/2},&\text{\rm$p$
even},\\\ 0,&\text{\rm$p$ odd},\end{cases}$ (35)
with $N_{p}$ being a constant. We note that $d_{p}(x)\equiv 0$ when $p$ is odd
and it is bounded in $x$ when $p$ is even. The second term is
$e_{2}=e_{21}+{\eta}e_{22}$,
$\displaystyle\tilde{A}\tilde{B}\tilde{C}\tilde{m}dz=$
$\displaystyle\int[A(x,m(x))+{{\eta}}z\tilde{b}_{1}(x)]\left[b_{1}(x)z^{q+3}+{\eta}\left(b_{2}(x)z^{q+4}+\frac{\sigma}{2}b_{1}^{2}(x)z^{2q+6}\right)\right]$
$\displaystyle\times
e^{i\omega\phi(x,m(x))+iz^{2}c_{1}(x)-z^{2}/2}(1+{\eta}c_{2}(x)z^{3})(m^{\prime}(x)+{{\eta}}zm^{\prime\prime}(x))dz$
$\displaystyle=$ $\displaystyle\
e^{i\omega\phi(x,m(x))}\int\Bigr{[}Ab_{1}z^{q+3}m^{\prime}+{\eta}\Bigl{(}\tilde{b}_{1}b_{1}z^{q+4}m^{\prime}+A\left(b_{2}z^{q+4}+\frac{\sigma}{2}b_{1}^{2}z^{2q+6}\right)m^{\prime}$
$\displaystyle+Ab_{1}z^{q+6}c_{2}m^{\prime}+Ab_{1}z^{q+4}m^{\prime\prime}\Bigl{)}\Bigr{]}e^{iz^{2}c_{1}(x)-z^{2}/2}dz+O({{\eta}}^{2})$
$\displaystyle=:$ $\displaystyle\
e^{i\omega\phi(x,m(x))}(e_{21}+{\eta}e_{22})+O({{\eta}}^{2}),$
with
${}e_{21}=Ab_{1}d_{q+3}m^{\prime},\qquad
e_{22}=\tilde{b}_{1}b_{1}d_{q+4}m^{\prime}+A\left(b_{2}d_{q+4}+\frac{\sigma}{2}b_{1}^{2}d_{2q+6}\right)m^{\prime}+Ab_{1}d_{q+6}c_{2}m^{\prime}+Ab_{1}d_{q+4}m^{\prime\prime}.$
(36)
To find an expression for the leading order error term we now have to consider
four cases depending on $q_{a}$. First, if $q_{a}<q$ then the second term in
(33) is of the same order or smaller than the right hand side and we can
disregard $e_{2}$. Second, if $q_{a}$ is even then $e_{11}=e_{21}=0$ since
$d_{p}=0$ for $p$ odd, and we gain an additional order in ${\eta}$. Upon also
noting that $q-q_{a}$ is even, we can therefore write
$\left|\bar{E}^{T}-\bar{C}(x)\omega^{1/2}\,{{\eta}}^{\bar{q}+1}\right|\leq
C\omega^{1/2}\,{{\eta}}^{\bar{q}+2},$
where
${}\bar{q}=\begin{cases}q_{a}+2,&\text{\rm$q_{a}$ even},\\\
q_{a}+1,&\text{\rm$q_{a}$
odd},\end{cases}\qquad\bar{C}(x)=e^{i\omega\phi(x,m(x))}\begin{cases}e_{11}+e_{21},&\text{\rm$q_{a}$
odd and $q=q_{a}$},\\\ e_{12}+e_{22},&\text{\rm$q_{a}$ even and $q=q_{a}$},\\\
e_{11},&\text{\rm$q_{a}$ odd and $q_{a}<q$},\\\ e_{12},&\text{\rm$q_{a}$ even
and $q_{a}<q$}.\end{cases}$ (37)
Moreover, $\bar{C}(x)$ is independent of $\omega$ and $h$ and can be bounded
by a constant independent of $x$.
We now go back to the full Taylor expansion error in (26) and use the results
that were obtained above for (27). Clearly, all parameters and functions will
depend on the term number $k$, and we indicate this with a subscripted $k$.
Since $q-2k$ is even if and only if $q$ is even, we have
$\bar{q}_{k}=\bar{q}_{0}-2k=q^{*}-2k$ with $q^{*}$ defined in (22). Then,
$\displaystyle{}\left|E^{T}-\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega)^{-k}\bar{C}_{k}(x)\omega^{1/2}\,{{\eta}}^{\bar{q}_{k}+1}\right|$
$\displaystyle=\left|\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega)^{-k}(\bar{E}^{T}_{k}-\bar{C}_{k}(x)\omega^{1/2}\,{{\eta}}^{\bar{q}_{k}+1})\right|$
(38) $\displaystyle\leq C\omega^{1/2}\sum_{k=0}^{\lfloor
q/2\rfloor}\omega^{-k}{{\eta}}^{\bar{q}_{k}+2}=C\omega^{1/2}\sum_{k=0}^{\lfloor
q/2\rfloor}({{\eta}}^{2}\omega)^{-k}{{\eta}}^{q^{*}+2}.$
The result therefore follows with
${}{C}^{*}(x)=\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega)^{-k}\bar{C}_{k}(x)\,{{\eta}}^{\bar{q}_{k}-q^{*}}=\sum_{k=0}^{\lfloor
q/2\rfloor}(i\omega{{\eta}}^{2})^{-k}\bar{C}_{k}(x).$ (39)
Since $\omega{{\eta}}^{2}=O(1)$, the leading order term of the error $E^{T}$
in $\omega$ is $\omega^{-q^{*}/2}$.
### 4.2 Discretization error
The discretization error is given by
$E^{D}=\tilde{u}_{s}(x)-\tilde{u}_{s}^{D}(x)=\omega^{1/2}\,\int\varphi(X(s)-x)\,\tilde{A}_{q}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}ds-\omega^{1/2}\,h\sum_{j\in\mathbb{Z}}f(j),$
with
$f(j)=\varphi(X(s_{j})-x)\,\tilde{A}_{q}(x,s_{j})\,e^{i\omega\tilde{\phi}_{q}(x,s_{j})},$
for a fixed $x$. The Poisson summation formula gives
$\sum_{j\in\mathbb{Z}}f(j)=\sum_{k\in\mathbb{Z}}\hat{f}(k),$
where
$\displaystyle\hat{f}(k)$ $\displaystyle=\int f(s)e^{-2\pi
isk}ds=\int\varphi(X(sh)-x)\,\tilde{A}_{q}(x,sh)\,e^{i\omega\tilde{\phi}_{q}(x,sh)}e^{-2\pi
isk}ds$
$\displaystyle=\frac{1}{h}\int\varphi(X(s)-x)\,\tilde{A}_{q}(x,s)\,e^{i\omega\tilde{\phi}_{q}(x,s)}e^{-2\pi
isk/h}ds.$
Therefore
$E^{D}=-\omega^{1/2}\,h\sum_{k\neq 0}\hat{f}(k).$
Using the change of variables (28) we obtain
${}\hat{f}(k)=\frac{{\eta}}{h}\int\varphi({{\eta}}z)\,\tilde{A}_{q}(x,m(x+{{\eta}}z))\,e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}\,e^{-2\pi
im(x+{{\eta}}z)k/h}\,m^{\prime}(x+{{\eta}}z)\,dz.$ (40)
We will now show that the integrand functions in (40) are smooth, with bounded
derivatives. Then the non-stationary phase lemma can be used to bound
$\hat{f}(k)$ since the phase derivative $m^{\prime}(x)$ never vanishes.
We need
###### Lemma 3
Under assumptions (A1), (A2) and (A4), for $0\leq\ell\leq p$ and
$|z|\leq\alpha/{\eta}$ with small enough $\alpha$,
$\displaystyle{}\left|\frac{d^{\ell}}{dz^{\ell}}\tilde{A}_{q}(x,m(x+{{\eta}}z))\right|$
$\displaystyle\leq C,$ (41)
$\displaystyle{}\left|\frac{d^{\ell}}{dz^{\ell}}e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}\right|$
$\displaystyle\leq C^{\prime}e^{-z^{2}/5}.$ (42)
The constants $C$ and $C^{\prime}$ are independent of of $\ell$, $x$, $\omega$
and $z$.
Proof: For the first inequality we can consider the individual terms in the
sum (18) separately. They will each be of the form
$\tilde{A}_{k,q-2k}(x,m(x+{{\eta}}z))=\sum_{j=0}^{q-2k}\tilde{a}_{j}(x+{{\eta}}z)\,({\eta}z)^{j}.$
where
$\tilde{a}_{j}(x)=\frac{(-1)^{j}}{j!}\partial_{x}^{j}A_{k}(x,m(x)).$
By assumption (A1), $\tilde{a}_{j}\in C_{b}^{p+2}$ are bounded, for
$0\leq\ell\leq p$, uniformly in $x$ and, after noting that
$|{{\eta}}z|\leq\alpha$ and that $|{\eta}|$ is bounded by a constant because
of assumption (A4), the result (41) follows.
For $\ell=0$ the second inequality is obtained by writing
$e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}=\tilde{C}(x,z)+{{\eta}}^{2}R_{C},\qquad|\tilde{C}(x,z)|=|1+{{\eta}}z^{3}c_{2}(x)|\,e^{-z^{2}/2}.$
Now since $|{{\eta}}z|\leq\alpha$ and $c_{2}(x)$ grows algebraically by
assumptions (A2), (A3), (A4), and since ${\eta}$ is bounded by a constant, we
have by Lemma 2,
${}|e^{i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))}|\leq|\tilde{C}(x,z)|+{{\eta}}^{2}|R_{C}|\leq
Ce^{-z^{2}/4}.$ (43)
Now consider $1\leq\ell\leq p$. We write
$\tilde{\phi}_{q}(x,m(x+{{\eta}}z))=\sum_{j=0}^{q+2}\tilde{p}_{j}(x+{{\eta}}z)\,({\eta}z)^{j},$
with $\tilde{p}_{j}(x)$ defined in (30). Then, since
$\tilde{p}_{0}^{\prime}+\tilde{p}_{1}\equiv 0$ by (15), we have
$\frac{d}{dz}\tilde{\phi}_{q}(x,m(x+{{\eta}}z))={{\eta}}^{2}\tilde{p}_{1}^{\prime}z+\sum_{j=2}^{q+2}\left({{\eta}}^{j+1}z^{j}\tilde{p}_{j}^{\prime}(x+{{\eta}}z)+j{{\eta}}^{j}z^{j-1}\tilde{p}_{j}(x+{{\eta}}z)\right),$
and therefore
$\frac{d}{dz}\left(i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))\right)=\frac{i}{\Im\phi_{xx}(x,m(x))}\,\sum_{j=1}^{q+2}\gamma_{j}(x+{{\eta}}z)\,{{\eta}}^{j-1}z^{j},$
where
$\gamma_{j}:=\tilde{p}_{j}^{\prime}+(j+1)\tilde{p}_{j+1},\quad 1\leq j\leq
q+1,\qquad\gamma_{q+2}:=\tilde{p}_{q+2}^{\prime}.$
Since the phase derivatives are evaluated on a center beam, $\gamma_{j}\in
C_{b}^{p}$ are bounded, for $0\leq\ell\leq p$, uniformly in $x$ by assumption
(A2) and we therefore have
$\left|\frac{d^{\ell}}{dz^{\ell}}\left(i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))\right)\right|\leq
C_{\ell}(1+|z|^{q+2}),\qquad 1\leq\ell\leq p.$
Thus, by Lemma 1 with $g_{w}=i\omega\tilde{\phi}_{q}(x,m(x+{{\eta}}z))$ and
$\delta={\eta}$, using (25) and (43), the inequality (42) follows for
$1\leq\ell\leq p$. This completes the proof. $~{}\Box$
The remaining terms in (40), i.e. $\varphi({{\eta}}z)$ and
$m^{\prime}(x+{\eta}z)$, are all assumed to be smooth with derivatives of
order up to $p$ bounded uniformly in $x$ by the assumptions (A1) and (A5).
Since the growth in (41) is offset by the rapid decay in (42), the above Lemma
shows that all $z$-derivatives of the integrand,
$g(x,z):=\varphi\,\tilde{A}_{q}\,e^{i\omega\tilde{\phi}_{q}}\,m^{\prime},$
up to order $p$ belongs to $L_{1}$ and $||\partial^{k}_{z}g(x,\cdot)||\leq
C_{k}$ for $0\leq k\leq p$. The constants $C_{k}$ are independent of $x$ and
$\omega$. We can then use the following version of the non-stationary phase
lemma.
###### Lemma 4
Suppose $\psi(z)\in C^{p+1}(\mathbb{R})$ with $\psi^{\prime}(z)\in
C_{b}^{p}(\mathbb{R})$ and $\psi^{\prime}(z)\geq c_{0}>0$. Moreover, let
$\epsilon<\delta<1$ and suppose $g(z)\in W^{p,1}$. Then
${}\left|\int g(z)e^{-i\psi(\delta z)/\varepsilon}dz\right|\leq
C||g||_{W^{p,1}}\left(\frac{\varepsilon}{\delta}\right)^{p},$ (44)
where $C$ depends on $\psi(x)$ and $p$, but not on $g(z)$, $\delta$ and
$\varepsilon$.
For the proof we refer to [20]. It is an easy adaptation of the proof of
theorem 7.7.1.
Taking $\psi$ as $2\pi m(x+\cdot)$, $\delta$ as ${\eta}$ and $\varepsilon$ as
$h/k$ we can apply this to (40),
$|\hat{f}(k)|=\frac{{\eta}}{h}\left|\int g(x,z)e^{-2\pi
im(x+{{\eta}}z)k/h}dz\right|\leq
C\frac{{\eta}}{h}||g(x,\cdot)||_{W^{p,1}}\left(\frac{h}{k{\eta}}\right)^{p}.$
Consequently,
$\left|\sum_{k\neq 0}\hat{f}(k)\right|\leq
C\frac{{\eta}}{h}||g(x,\cdot)||_{W^{p,1}}\sum_{k\neq
0}\left(\frac{h}{k{\eta}}\right)^{p}\leq
C\frac{{\eta}}{h}\left(\frac{h}{{\eta}}\right)^{p}\sum_{k=1}^{\infty}k^{-p}\leq
C^{\prime}\frac{{\eta}}{h}\left(\frac{h}{{\eta}}\right)^{p}.$
Thus since by the assumptions (A1) $p\geq 2$,
$|E^{D}|=\omega^{1/2}\,h\,|\sum_{k\neq 0}\hat{f}(k)|\leq
C^{\prime}\omega^{1/2}\,{\eta}\left(\frac{h}{{\eta}}\right)^{p}.$
Together with (38) this shows the theorem.
## 5 Constant coefficient equations
It is often claimed that the beam width is important in the accuracy of
Gaussian beams, because for wide beams the Taylor expansion error should be
large. See for example [4, 6]. We therefore in this section consider the
constant coefficient Helmholtz equation, with the speed of propagation
$c({{\mbox{\boldmath$x$}}})\equiv 1$, for which exact Gaussian beam solutions
and the Taylor expansion error $|E^{T}|$ can be computed. We investigate the
importance of the beam width on Taylor error in this particular case. Our
conclusion is that the local beam width is not a good indicator of accuracy,
and there is no direct relation between the error and the beams’ width. We
show the main steps of the derivation and the final expression for $C^{*}(x)$
and the leading relative error terms below. For more details we refer to [12].
We consider first order beams where $q=0$. These only contain one amplitude
term $A_{0}(x)$ which we for simplicity call $A(x)$ here. The source curve
will be denoted ${{\mbox{\boldmath$x$}}}_{0}(s)=(s,y_{0}(s))$ and we assume
all beams originating from ${{\mbox{\boldmath$x$}}}_{0}$ shoot out
orthogonally. Therefore
$\theta_{0}(s)=\frac{\pi}{2}+\tan^{-1}(y_{0}^{\prime}(s))$. In the constant
coefficient case the central ray $\Omega$ is a straight line. With
$x(0)=x_{0}(s)=s$, $y(0)=y_{0}(s)$ and $\theta(0)=\theta_{0}(s)$, we get from
(5) at $y=y^{*}$,
$\displaystyle\theta(t(s))=\frac{\pi}{2}+\tan^{-1}(y_{0}^{\prime}(s)),$ (45)
$\displaystyle x(t(s))=X(s)=s-y_{0}^{\prime}(s)\,(y^{*}-y_{0}(s)),$ (46)
$\displaystyle t(s)=\left((X(s)-s)^{2}+(y^{*}-y_{0}(s))^{2}\right)^{1/2}.$
(47)
Here we will only compute the error at ${{\mbox{\boldmath$x$}}}=(0,y^{*})$.
For this point, let $s^{*}:=m(0)=X^{-1}(0)$. To simplify the calculations, and
without loss of generality, we assume $y_{0}(s^{*})=y_{0}^{\prime}(s^{*})=0$.
Therefore, by (45)-(47), the central ray starting at
${{\mbox{\boldmath$x$}}}_{0}(s^{*})$ will lie on the $y$-axis, and we have
$s^{*}=X(s^{*})=0$ and $t(s^{*})=y^{*}$. See Figure 4.
Figure 4: A schematic representation of the initial source and central beam
rays which are straight lines.
Assuming the initial phase on ${{\mbox{\boldmath$x$}}}_{0}(s)$ to be zero,
$\phi({{\mbox{\boldmath$x$}}}_{0})=0$, we also get
${}\phi(X(s),s)=t(s),\qquad\phi(0,m(0))=y^{*}.$ (48)
To obtain ODEs for higher order Taylor coefficients, we introduce the
orthogonal ray-centered coordinates $t,n$, where $n$ is the axis perpendicular
to the ray at point $t$ with the origin on the ray. In this coordinate system,
$\phi(t,n=0)$ and $A(t,n=0)$ correspond to $\phi(X(s),s)$ and $A(X(s),s)$ in
the Cartesian coordinate, respectively. The eikonal equation and transport
equation in the ray-centered coordinates read
${}\phi_{t}^{2}+\phi_{n}^{2}=1,$ (49) ${}2\nabla
A\cdot\nabla\phi+A\Delta\phi=0,\qquad\nabla\phi=(\phi_{t}\,\,\phi_{n})^{\top}.$
(50)
Set $\phi^{(j)}(t):=\partial_{n}^{j}\phi(t,n=0)$ and
$A^{(j)}(t):=\partial_{n}^{j}A(t,n=0)$, with $j=0,1,2,\dotsc$. We first note
that by (48),
$\phi^{(0)}(t)=t,\qquad\partial_{t}\phi(t,n=0)=1,\qquad\partial_{t}^{j}\phi(t,n=0)=0,\quad
j=2,3,\dotsc.$
Moreover, by (49) and (50) and taking several of their derivatives with
respect to $t$ and $n$,
$\displaystyle\phi^{(1)}(t)=0,\quad\partial_{t}\partial_{n}\phi(t,n=0)=0,\quad\partial_{t}\partial_{n}^{2}\phi(t,n=0)=-{\phi^{(2)}}^{2}(t),$
$\displaystyle\partial_{t}\partial_{n}^{3}\phi(t,n=0)=0,\quad\partial_{t}^{2}\partial_{n}\phi(t,n=0)=0,\quad\partial_{t}^{3}\partial_{n}\phi(t,n=0)=0,$
$\displaystyle\partial_{t}^{2}\partial_{n}^{2}\phi(t,n=0)=2{\phi^{(2)}}^{3}(t),\quad\partial_{t}A(t,n=0)=-\frac{1}{2}A^{(0)}(t)\,\phi^{(2)}(t),$
$\displaystyle\partial_{t}^{2}A(t,n=0)=\frac{3}{4}A^{(0)}(t)\,{\phi^{(2)}}^{2}(t),\quad\partial_{t}\partial_{n}A(t,n=0)=0.$
Now, let
${}\phi(t,n)\approx
t+\frac{n^{2}}{2}\phi^{(2)}(t)+\frac{n^{3}}{6}\phi^{(3)}(t)+\frac{n^{4}}{24}\phi^{(4)}(t),$
(51)
and
${}A(t,n)\approx A^{(0)}(t)+nA^{(1)}(t)+\frac{n^{2}}{2}A^{(2)}(t).$ (52)
Putting (51) and (52) into (49) and (50), we obtain the following ODEs for the
Taylor coefficients,
$\displaystyle\frac{d}{dt}\phi^{(2)}+{\phi^{(2)}}^{2}=0,$
$\displaystyle\frac{d}{dt}\phi^{(3)}+3\phi^{(2)}\phi^{(3)}=0,$
$\displaystyle\frac{d}{dt}\phi^{(4)}+4\phi^{(2)}\phi^{(4)}+3{\phi^{(2)}}^{4}+3{\phi^{(3)}}^{2}=0,$
$\displaystyle\frac{d}{dt}A^{(0)}+\frac{1}{2}\phi^{(2)}A^{(0)}=0,$
$\displaystyle\frac{d}{dt}A^{(1)}+\frac{3}{2}\phi^{(2)}A^{(1)}+\frac{1}{2}\phi^{(3)}A^{(0)}=0,$
$\displaystyle\frac{d}{dt}A^{(2)}+\frac{5}{2}\phi^{(2)}A^{(2)}+2\phi^{(3)}A^{(1)}+\frac{1}{2}\phi^{(4)}A^{(0)}+\frac{3}{2}{\phi^{(2)}}^{3}A^{(0)}=0$
We then solve these ODEs with $A^{(0)}(0)=1$ and zero initial conditions for
the rest of Taylor coefficients. At our observation point
${{\mbox{\boldmath$x$}}}=(0,y^{*})$, we have that
$\partial_{x}^{j}=\partial_{n}^{j}$, since the $n$-axis is parallel to the
$x$-axis. We can therefore easily transform the solutions in the ray-centered
coordinates $t,n$ back to the coordinate system $x,s$. In the end we note that
all terms with odd $x$-derivatives are zero. Hence, we obtain that
$a_{1}(0)=b_{1}(0)=0$ and $e_{12}+e_{22}$ in (34) and (36) simplifies to
$e_{12}(0)+e_{22}(0)=m^{\prime}(0)\,a_{2}(0)\,d_{2}(0)+m^{\prime}(0)\,A(0,0)\,b_{2}(0)\,d_{4}(0).$
After some additional algebraic manipulations and assuming that $P_{0}=i$,
$\Im Q_{0}=0$, $\Re Q_{0}>0$, we get
$\displaystyle a_{2}(0)$
$\displaystyle=i\frac{3Q_{0}^{1/2}\,y^{*}-2Q_{0}^{1/2}\,(Q_{0}+iy^{*})^{2}m^{\prime}(0)\frac{d}{ds}\theta(y^{*})}{4(Q_{0}+iy^{*})^{7/2}},$
(53) $\displaystyle b_{2}(0)$
$\displaystyle=i\frac{(Q_{0}^{2}+{y^{*}}^{2})\left(-y^{*}+4(Q_{0}+iy^{*})^{2}m^{\prime}(0)\frac{d}{ds}\theta(y^{*})\right)}{8Q_{0}(Q_{0}+iy^{*})^{4}},$
(54) $\displaystyle c_{1}(0)$
$\displaystyle=\frac{y^{*}+(Q_{0}^{2}+{y^{*}}^{2})m^{\prime}(0)\frac{d}{ds}\theta(y^{*})}{2Q_{0}},$
(55)
and
$A(0,0)=\frac{Q_{0}^{1/2}}{(Q_{0}+iy^{*})^{1/2}},\qquad{\eta}(0)=\left(\frac{Q_{0}^{2}+{y^{*}}^{2}}{\omega\,Q_{0}}\right)^{1/2}.$
(56)
Moreover, by (45-47),
$\frac{d}{ds}\theta(y^{*})=y_{0}^{\prime\prime}(0),\quad
m^{\prime}(0)={(X^{-1})}^{\prime}(0)=\left(1-y^{*}y_{0}^{\prime\prime}(0)\right)^{-1}.$
(57)
Therefore, knowing $y_{0}(s)$ and by (53-57) and (35), we can calculate
$e_{12}(0)+e_{22}(0)=e^{-i\omega\phi(0,m(0))}C^{*}(0)=e^{-i\omega
y^{*}}C^{*}(0).$
Note that $C^{*}(0)$ only depends on $Q_{0}$, $y^{*}$ and
$y_{0}^{\prime\prime}(0)$.
We now consider the following two canonical cases:
* (1)
$y_{0}^{\prime\prime}(0)=0$,
* (2)
$y_{0}^{\prime\prime}(0)=-1$.
The first case corresponds to a line $y_{0}=0$. The second case corresponds to
a circle $y_{0}(s)=-1+\sqrt{1-s^{2}}$ or a parabola $y_{0}(s)=-s^{2}/2$. Note
that with an initial curve with positive second derivative, the rays will
intersect and form a caustic, and then our theory does not hold.
For the first case, we obtain the simple expression
${}C^{*}_{\rm l}(0)=e^{-i\omega
y^{*}}\frac{n_{0}\,y^{*}\,Q_{0}^{2}}{(Q_{0}+iy^{*})^{2}\,(Q_{0}^{2}+{y^{*}}^{2})^{3/2}},$
(58)
where $n_{0}$ is a constant complex number. For the second case, the
expression is much more complicated. In the small and large $Q_{0}$-limit we
have
${}C^{*}_{\rm c}(0)=e^{-i\omega
y^{*}}\frac{n_{1}+n_{2}y^{*}+n_{3}{y^{*}}^{2}}{{y^{*}}^{4}\sqrt{1+y^{*}}}Q_{0}^{2}+\mathcal{O}(Q_{0}^{3}),\qquad
C^{*}_{\rm c}(0)=e^{-i\omega
y^{*}}\frac{n_{4}}{\sqrt{1+y^{*}}}Q_{0}^{-5/2}+\mathcal{O}(Q_{0}^{-7/2}),$
(59)
where $n_{j}$, with $j=1,\dotsc,4$, are constant complex numbers. The
amplitude of the geometrical optics solution is proportional to
$|1-y^{*}\,y_{0}^{\prime\prime}(0)|^{-1/2}$, and by (38), the relative error
will be
$|E_{\rm
rel}|=|E^{T}|\,|1-y^{*}\,y_{0}^{\prime\prime}(0)|^{1/2}+\mathcal{O}(\omega^{-3/2})=\omega^{1/2}{{\eta}}^{3}(0)\,|C^{*}(0)|\,|1-y^{*}\,y_{0}^{\prime\prime}(0)|^{1/2}+\mathcal{O}(\omega^{-3/2}).$
We therefore obtain the leading order term
$|E_{\rm rel}^{\rm
l}|=\omega^{-1}\;\left|\frac{n_{0}\,y^{*}Q_{0}^{1/2}}{(Q_{0}+iy^{*})^{2}}\right|,$
and for small and large $Q_{0}$,
$|E_{\rm rel}^{\rm
c}|=\omega^{-1}\,\frac{n_{1}+n_{2}y^{*}+n_{3}{y^{*}}^{2}}{{y^{*}}}\sqrt{Q_{0}}+\mathcal{O}(\omega^{-1}Q_{0}^{3/2}),\qquad|E_{\rm
rel}^{\rm
c}|=\omega^{-1}\,\frac{n_{4}}{Q_{0}}+\mathcal{O}(\omega^{-1}Q_{0}^{-2}),$
corresponding to (58) and (59), respectively.
Figure 5: Absolute value of relative error as a function of $Q_{0}$ (top) and
of the width ${\eta}$ (bottom) in the case when the initial source is a line
(left) and a circle (right).
Figure 5 shows the absolute values of the relative errors at $y^{*}=3$. As it
can be seen from the formulas and figures, the relative error has a direct
relation with $Q_{0}$, tending to zero both for small and large $Q_{0}$. A
reduced error with large $Q_{0}$ has also been noticed in [14] for the
oscillatory part of the error (or the discretization error). However, there is
no clear connection between the error and the beam width; the same width can
correspond to different errors.
In many approximations, the optimal $Q_{0}$, corresponding to the minimum beam
width at a receiver point is chosen for computations, see [4] for instance.
Figure 6 shows the beam width as a function of $Q_{0}$. In our case the
minimum width is attained at $Q_{0}=y^{*}$. With $Q_{0}=y^{*}$ and $y*\gg 1$
we get
$|E_{\rm rel}^{\rm l}|=\frac{N}{\omega\,{y^{*}}^{1/2}},\qquad|E_{\rm rel}^{\rm
c}|\approx\frac{N^{\prime}}{\omega y^{*}},$
with $N$ and $N^{\prime}$ being constant numbers. When using this $Q_{0}$, we
do not obtain the minimum error as was seen above. However, importantly, the
relative error decreases as the distance from the source increases.
We conclude that in the case analysed here large and very small $Q_{0}$ will
improve the Taylor expansion error. From Figure 6 we see that this corresponds
to having wide beams, not narrow beams. One should keep in mind, however, that
this is not the whole story. The approximation of the initial data where the
source curve is not flat and/or the amplitude is not constant will in general
deteriorate when wider beams are used. Hence, this restricts the beam widths
that can be used. Wider beams also mean that the wave field will be more
expensive to evaluate since beams contribute more globally to the solution.
Moreover, our result is strictly for constant coefficients. In the presence of
a varying speed of propagation where the properties may change dramatically as
we get farther from the central rays, the Taylor expansion error could be
large for wide beams. In addition, when the rays can bend, it may not be
possible to have very wide beams, since as was noted before, the Gaussian beam
approximation may break down when the phase becomes non-smooth, and this
happens at some distance away from the central ray (outside the regularity
region). In the general case, finding the optimal $Q_{0}$ for a given
observation point is an open problem.
Figure 6: The beam width as a function of $Q_{0}$ at $y^{*}=3$.
## References
* [1] M. M. Popov, A New Method of Computation of Wave Fields Using Gaussian Beams, Wave Motion, vol. 4, 1982, 85-97.
* [2] V. M. Babic and T. F. Pankratova, On Discontinuities of Green’s Function of the Wave Equation with Variable Coefficient, Problemy Matem. Fiziki, Leningrad University, Saint-Petersburg, vol. 6, 1973.
* [3] A. P. Katchalov and M. M. Popov, Application of the Method of Summation of Gaussian Beams for Calculation of High-frequency Wave Fields, Sov. Phys. Dokl., vol. 26, 1981, 604-606.
* [4] V. Cerveny, M. M. Popov and I. Psencik, Computation of Wave Fields in Inhomogeneous Media - Gaussian Beam Approach, Geophys. J. R. Astr. Soc., vol. 70, 1982, 109-128.
* [5] L. Klimes, Expansion of a High-frequency Time-harmonic Wavefield Given on an Initial Surface into Gaussian Beams, Geophys. J. R. astr. Soc., vol. 79, 1984, 105-118.
* [6] N. R. Hill, Gaussian Beam Migration, Geophysics, vol. 55, no. 11, 1990, 1416-1428.
* [7] N. R. Hill, Prestack Gaussian-Beam Depth Migration, Geophysics, vol. 66, no. 4, 2001, 1240-1250.
* [8] S. Jin, H. Wu and X. Yang, Gaussian Beam Methods for the Schrödinger Equation in the Semi-Classical Regime: Larangian and Eulerian Formulations, Comm. Math. Sci., vol. 6, 2008, 995-1020.
* [9] S. Jin, H. Wu and X. Yang, A Numerical Study of the Gaussian Beam Methods for One-Dimensional Schrödinger-Poisson Equations, J. Comp. Math., to appear.
* [10] S. Leung, J. Qian and R. Burridge, Eulerian Gaussian Beams for High-Frequency Wave Propagation, Geophysics, vol. 72, no. 5, 2007, SM61-SM76.
* [11] S. Leung and J. Qian, Eulerian Gaussian Beams for Schrödinger Equations in the Semi-Classical Regime, J. Comp. Phys., vol. 228, 2009, 2951-2977.
* [12] M. Motamed, Topics in Analysis and Computation of Linear Wave Propagation, PhD Thesis, CSC, KTH, Stockholm, 2008.
* [13] J. Ralston, Gaussian Beams and the Propagation of Singularities, In Studies in partial differential equations, Math. Assoc. America, Washington, DC, vol. 23, 1982, 206-248.
* [14] L. Klimes, Discretization Error for the Superposition of Gaussian Beams, Geophys. J. R. astr. Soc., vol. 86, 1986, 531-551.
* [15] N. M. Tanushev, Superpositions and Higher Order Gaussian Beams, Comm. Math. Sci, 6(2):449–475, 2008.
* [16] H. Liu and J. Ralston, Recovery of High Frequency Wave Fields from Phase Space Based Measurements, Preprint, 2009.
* [17] . S. Bougacha, J.-L. Akian and R. Alexandre, Gaussian Beams Summation for the Wave Equation in a Convex Domain, Preprint, 2009.
* [18] R. Magnanini and G. Talenti, On Complex-Valued Solutions to a 2-D Eikonal Equation. I. Qualitative Properties, Contemporary Mathematics, vol. 238, 1999, 203-229
* [19] R. Magnanini and G. Talenti, On Complex-Valued Solutions to a Two-Dimensional Eikonal Equation. II. Existence Theorems, SIAM Journal on Mathematical Analysis, vol. 34, no. 4, 2003, 805-835.
* [20] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin Heidelberg New York, 1983.
|
arxiv-papers
| 2009-08-24T11:58:23 |
2024-09-04T02:49:04.761489
|
{
"license": "Public Domain",
"authors": "Mohammad Motamed, Olof Runborg",
"submitter": "Olof Runborg",
"url": "https://arxiv.org/abs/0908.3416"
}
|
0908.3531
|
# Mechanism of unidirectional emission of ultrahigh Q Whispering Gallery mode
in microcavities
C.-L. Zou F.-W. Sun fwsun@ustc.edu.cn C.-H. Dong X.-W. Wu J.-M. Cui Y.
Yang G.-C. Guo Z.-F. Han zfhan@ustc.edu.cn Key Lab of Quantum Information,
University of Science and Technology of China, Hefei 230026
###### Abstract
The mechanism of unidirectional emission of high Q Whispering Gallery mode in
deformed circular micro-cavities is studied and firstly presented in this
paper. In phase space, light in the chaotic sea leaks out the cavity through
the refraction regions, whose positions are controlled by stable islands.
Moreover, the positions of fixed points according to the critical line in
unstable manifolds mainly determines the light leak out from which refraction
region and the direction of the emission. By controlling the cavity shape, we
can tune the leaky regions, as well as the positions of fixed points, to
achieve unidirectional emission high Q cavities with narrow angular divergence
both in high and low refractive index materials. Especially for high index
material, almost all Gibbous-shaped cavitiess have unidirectional emission.
###### pacs:
42.55.Sa, 05.45.Mt, 42.25.-p,42.60.Da
In recent years, being the high potential elements for integrated devices,
dielectric microcavities have attracted more and more attentions. Especial
attentions are focused on the Whispering Gallery (WG) micro-resonators, such
as micro-sphere, micro-disk, micro-toroid, etc, where energy can be well
confined in these rotational symmetrical structures by total internal
reflections. The unique high quality (Q) factor and low mode volume features
of WG Modes lead to wild applications, ranging from nonlinear optics, low
threshold lasers, sensitive sensors, to cavity quantum electrodynamics vahala
. However, the isotropic emission property of WGMs makes very low efficient
coupling from free space. Also, the near field coupling component is a big
obstacle for practical applications. To solve the coupling problem,
researchers found that the microcavity with slightly deformed circular
boundary, also called asymmetric resonant cavity (ARC), can lead directional
emission. The ARCs, such as the extensively studied quadruple qua1 ; qua2 ;
qua3 ; qua4 ; qua5 and stadium microcavities sta1 ; sta2 ; sta3 , can be
efficiently pumping through free space focused beam, which makes them have
been well used in the realization of strong coupling pump1 and laser pump2 .
Besides the easy free space coupling in the optical experiments, the deformed
microcavities can also serve as open billiards for experimental research on
quantum chaos chaos ; chaos1 .
In order to get more efficient free space excitation and collection in
practical application, people are pursuing single directional emission with
narrow angular divergence, such as the spiral shaped chaos1 ; sprial1 ;
spiral4 and the rounded isosceles triangle shaped triangle microcavities.
However, their low Q factors highly limit the applications in low threshold
laser and cavity quantum electronic dynamics. Although there are several
theoretical approaches annular ; coupledisk , the trade-off between the high Q
and unidirectional emission still blocks its further development. It is till
recently that fabrication-friendly limaçon-shaped microcavity with both
unidirectional emission and high Q was theoretically proposed limacon and
experimentally realized by several groups limaconexp1 ; limaconexp2 ;
limaconexp3 . However, the mechanism to design such cavity is not clear yet.
Moreover, this limaçon-shaped microcavity is based on the high refractive
index material. There is no report for high Q and unidirectional emission
cavity with low index material.
In this paper, we will show the mechanism to get unidirectional emission high
Q mode both in the high and low refractive index ARC. At the beginning, we
summarize the necessary conditions to achieve the high Q unidirectional
emission ARC. First, the cavity boundary should be continuous, smooth, and
slightly deformed from a circle to support the WG-like modes. In such deformed
microcavity, the wave is mainly localized high above the critical line and
well confined in the cavity to support high Q. The light may leak out when the
ray is lower than critical line in phase space. In this case, the rays follow
the unstable manifolds, cross the critical line, and perform the directional
emission tangential to the cavity boundary.
Second, the boundary shape should have less symmetry. For planar deformed
cavity with two or more axial symmetries, there are at least two pairs of
symmetric tangential emission spots on the boundary because they support both
clockwise and counterclockwise traveling modes qua1 ; qua2 ; qua3 ; qua4 ;
qua5 ; sta1 ; sta2 ; sta3 . The only way to achieve unidirectional emission is
to design the cavity with at most one symmetric axis and make the emission
direction along the axis direction.
In the cavity with continuous, smooth, and slightly deformed circular
boundary, there are stable and unstable orbits, corresponding to the stable
islands and fixed points of unstable manifolds in phase space. The mechanism
for the unidirectional emission in the single-axis cavity is: The light only
along the manifolds may leak out the cavity. Isolated by the stable islands,
the manifolds have several leaky regions when they cross the critical
refractive line. By controlling the cavity shape, we can tune the leaky
regions, as well as the position of fixed points according to the critical
line. The light through the lower fixed point has higher probability to leak
out than through the higher point. The parallel leaky lights along the axis
direction shows unidirectional emission from the high Q ARC. Based on this
mechanism, we succeed in designing the unidirectional high Q ARC both in high
and low refractive index materials. We find that almost all Gibbous-shape ARCs
have unidirectional emission for high refractive index material. By precisely
controlling the boundary shape, we can achieve far field emission with very
narrow divergence angle.
Figure 1: (color online) (a) The real space illustration of the rays
reflection inside HQHC deformed microcavity with $\epsilon=0.11$. The dashed
line outside is the circle boundary. The blue lines and red dash lines are the
stable and unstable period-3 orbits. (b) The calculated field spatial
distribution of TM polarized fundamental WG-like mode in HQHC, with the
wavenumber in vacuum $kr\approx 26.15$ and quality factor $Q=3.3\times
10^{5}$. (c) The phase space structure of HQHC. The blue circles and red
diamonds are the stable islands and fixed points, corresponding to the stable
and unstable period-3 orbits in (a). The red line is the critical refraction
line with $\sin\chi=1/n$, and the green points are the unstable manifolds from
the unstable period-3 orbits. Magenta points is the collection records of
leaky rays location from the original rays upon $\sin x>0.7$. It shows that
most rays leak out from this refraction region. (d) the far field patterns for
TM polarized ray (the red line, lifted by $0.4$) and wave simulation (blue
line and green line for even and odd parity mode respectively).
At first, we consider a boundary shape of Half-Quadruple-Half-Circle (HQHC) to
illustrate the unidirectional emission in high refractive index cavity, which
have been demonstrated in silica microsphere hqhc . The slightly deformed
circular boundary shape, as shown in Fig. 1(a), can be read as
$R(\phi)=\left\\{\begin{array}[]{c}R_{0},\text{\ \ }\\\
R_{0}\left(1-\epsilon\text{cos}^{2}\phi\right),\end{array}\text{\
}\begin{array}[]{c}\cos\phi<0\text{,}\\\ \cos\phi\geq
0\text{,}\end{array}\text{\ }\right.$ (1)
where $\phi$ is the polar angle according to the symmetric X-axis and
$\epsilon$ is the deformation factor. In the limit of $R\gg\lambda$, the light
in cavity can be semiclassically treated as ray. Although wave nature of light
is important in microcavity, the ray dynamics is still an efficient tool to
understand and analyze the emission properties of ARCs ray1 ; ray2 ; ray3 ;
ray4 ; um1 ; um2 ; limaconexp2 ; limaconexp3 .
The sequence of ray reflection inside the cavity in real space can be
represented in the phase space as Poincaré Surface of Section (SOS). Each ray
reflection on the boundary is recorded in Birkhoff coordinates reichl as
$(\phi,\sin\chi)$, where $\phi$ and $\chi$ denote the polar angle of
reflection position on boundary and the incidence angle, respectively. By
setting rays initially random on boundary (here we only consider the
counterclockwise propagation rays, the clockwise SOS can be obtained by
symmetric transformation), we can get the SOS by treating the cavity as
billiard chaos , which is shown in Fig. 1(c).
In SOS, the phase space structure can be divided into two parts, the chaotic
sea and the stable islands. The two regions are isolated to each other. The
ray in chaotic sea [black points in Fig. 1(c)] has chaotic motions in real
space, while the ray in islands (in blue) is conserved and will never move
into chaotic sea. In the real space, the stable islands are corresponding to
the stable (blue solid lines) period-3 orbits reichl , as shown in Fig. 1(a).
The light in the chaotic sea will run along the unstable manifolds ray2
[green curve in Fig. 1(c)] and leak out when it crosses critical line, which
is denoted by the red line in Fig. 1(c). The critical line shows the critical
incidence angle for total internal reflection $\sin\chi=1/n$. We set the
refractive index $n=3.3$ in our case.
In this slightly deformed circular cavity, the light is mainly localized upon
the critical line in phase space. In the real space, the light runs in the
unstable periodic orbits or high Q WG-like mode, as shown in Fig. 1(b) ray1 ;
limacon ; limaconexp1 . Initially, we set rays uniformly distributed in the
region $\sin\chi>0.7$, which can be considered as energy of high Q modes
dynamics localized above the critical line. Rays will be reflected on the
boundary in sequence until they hit the refraction regions. In the SOS, the
refraction regions are the parts of chaotic sea below the critical line
($\sin\chi<1/n$). As the manifolds enclose the stable islands, the locations
of the stable islands determine the positions of these refraction regions. In
our case, the center two stable islands are lower than the third island. So,
there are two refraction regions near $\phi\approx\pi/2$ and $3\pi/2$ at
boundary, corresponding to two directional emissions with far field angles of
180 and 360 degrees.
In order to get the unidirectional emission, we need to analyze the relative
positions of the three fixed points in the manifolds, as shown with red
diamonds in Fig.1(c). It is corresponding to the unstable (red dashed lines)
period-3 orbit reichl in Fig. 1(a). They are sandwiched by the stable
islands. All lights following the manifolds will transit through/near these
three points. However, the fixed point (marked as $1$) before refraction
region near $\phi\approx\pi/2$ is lower than the point (marked as $2$) before
the other refraction region near $\phi\approx 3\pi/2$. Also, it is closer to
the refraction region. So the manifolds here are shorter and steeper to cross
the critical line. As a result, the light through/near fixed point $1$ has the
higher probability to enter the fraction region and leak out than the other
one. It will give rise to the most energy leaky here and the directional
emission with far field angle of 180 degrees. From the symmetry, the clockwise
propagation rays leak out around $\phi\approx 3\pi/2$ and show the same
directional far field emission. Therefore, the leaky beams from the
counterclockwise and clockwise propagations are parallel and finally show
unidirectional emission from this HQHC cavity.
With the numerical solution of Maxwell equations through boundary elements
method bem1 ; bem2 , we can observe the high Q WG-like mode in this HQHC
microcavity. Fig. 1(b) shows the electric field intensity distribution of WG-
like transverse magnetic (TM) polarized mode with radiation quantum number
$q=1$ (the transverse electric polarized modes have the similar properties).
The near field pattern shown tangent light emission from $\phi\approx\pi/2$
and $3\pi/2$, corresponding to the two refraction regions. Obviously
directional emission beams for (counter)clockwise propagation are along the
X-axis.
The far field distribution of the Odd and Even polarity WG-mode is shown in
Fig. 2(d), as well as the ray-wave correspondence with ray dynamics. In the
ray simulation, with the weighted refractive coefficient from Fresnel law, we
can get the far field amplitude by summing up the all refractions, as shown in
Fig. 2(d) with the red curve. In order to get clear illustration, the curve is
lifted by 0.4. The far field pattern shows the unidirectional emission at 180
degree with divergence about 30 degrees.
As discussed above, the refraction regions and fixed points in unstable
manifolds play the most important roles to determine the cavity directional
emission. So, we can tune the cavity shape to adjust the locations of stable
islands and the positions of fixed points to control the refraction regions
and the emission direction, simultaneously. To achieve the unidirectional
emission of high Q modes, we can design the X-axis symmetric cavity and lead
the unidirectional emission along the axis. In general, the boundary of X-axis
symmetric shape can be expressed as
$R(\phi)=\left\\{\begin{array}[]{c}R_{0}\sum a_{i}\cos^{i}\phi,\\\ R_{0}\sum
b_{i}\cos^{i}\phi,\end{array}\text{\ }\begin{array}[]{c}\text{ }\cos\phi\geq
0\text{\ \ }\\\ \cos\phi<0\end{array}\text{\ }\right.$ (2)
By setting $a_{0}=b_{0}=1$, and $a_{1}=b_{1}=0$, the norm direction is
continuous and the cavity boundary is smooth. In addition, we should keep the
boundary slow varying, so simply we cut off the high order terms, only keep
$a_{2}(b_{2})$ and $a_{3}(b_{3})$ nonzero note . Also, to break Y-axis
symmetry, it needs $a_{2}\neq b_{2}$ or/and $a_{3}\neq-b_{3}$. Moreover, we
can set $b_{2}=b_{3}=0$ and $a_{2}+a_{3}<0$ to form Gibbous shape for further
simplification.
For the this kind of Gibbous shape cavity with high refractive index material,
there will always exist stable and unstable period-3 orbits with vertexes near
($0,2\pi/3,4\pi/3$) and ($\pi/3,\pi,5\pi/3$), respectively. By randomly set
the parameters $a_{i}$, we can always get similar phase space structures to
Fig. 1(c) in any Gibbous-shaped cavity. That is because, without Y-axis
symmetric, the two sets of period-3 orbits in Gibbous shape cavities always
have smaller convex angles near $\pi/2$ in real space. Correspondingly, the
fixed points of manifolds and stable islands are lower in phase space, which
make the refraction regions fixed at about $\phi\approx\pi/2$ and $3\pi/2$ and
unidirectional emission from the first refraction region. We carried out ray
and wave simulations on lots of Gibbous shape cavities. Results indicate the
single emission with far field divergence angle ranges from $20$ to $50$
degrees, confirming to our conjecture from the orbits. We find the cavity with
$a_{2}=0.0486,a_{3}=-0.1258$ gives good unidirectional far field pattern. The
wave simulation gives a high Q ($Q=5.78\times 10^{6}$) TM polarized WG mode
with $kr\approx 25.1842$, and the far field divergence is only about $24$
degrees, which is much smaller, comparing to $40$ degrees reported in limaçon
cavity limacon .
Thus, we have shown how to get unidirectional emission in the Gibbous-shaped
cavity. The limaçon cavity limacon , as well as the cavity in Song’s
experiments limaconexp1 , is a generalized Gibbous-shaped cavity. Their
mechanisms for unidirectional emission can also be explained with our
approach.
Figure 2: (color online)(a) (b) the SOS of two cavities (a) HQHC
$\epsilon=0.05$ and (b) the cavity shape support stable rectangle orbit and
unidirectional emission designed in the text. (c) the rectangle and diamond
shape period-4 orbits in the X-symmetric cavity. (d) The near field
distribution of the unidirectional emission with $kr\approx 52.5284$. (e) The
far field pattern of the cavity, red line is the ray simulation result (lifted
by 1), and the blue line is corresponding to the result in (d).
Now, we will generalize the above mechanism for unidirectional emission to low
refractive index microcavities. Here we take the silica ($n=1.45$) for
example. Similar to the period-3 orbits in high index cavities, the silica
cavity’s directional emission is basically influenced by period-4 orbits. The
former experimental study on HQHC micro-sphere showed that the emission
direction could be reduced to 2 nearly perpendicular direction hqhc . The SOS
of such HQHC is present in Fig. 2(a), where the stable period-4 islands [the
red diamond orbit in Fig.2(c)] around the critical line prevent the rays
unidirectional emission around $\phi=\pi/2$.
A strategy to solve the problem in this low index cavity is to find a shape
with stable islands away from $\pi/2$. That is to make the green rectangle
period-4 orbit be stable and the red diamond orbit be unstable. Corresponding
SOS is presented in Fig. 2(b). Rays can go cross the critical line near
$\pi/2$. Similar to the analysis in high index cavity, the conditions for
unidirectional emission in silica microcavity is: (i) The cavity supports the
stable rectangle orbit. (ii) The stable island near $\phi=3\pi/4$ and $5\pi/4$
is lower, and the fixed point (the red triangle in SOS) is lower near
$\phi=\pi/2$.
In the numerical simulation of ray dynamics, we randomly choose parameters to
set the boundary shape, judge that the rectangle orbit stable or not, and
compare the islands position in SOS to fulfill the condition (ii). We can
easily find a good cavity shape with
$a_{2}=-0.1329,a_{3}=0.0948,b_{2}=-0.0642,b_{3}=-0.0224$ satisfies the above
conditions, and gives good unidirectional far field pattern. Fig. 2(b) and
Fig. 2(d) shows the SOS and field distribution of TM mode. Fig.2(e) is the
corresponding far field pattern compares to the ray simulation. As we expect,
the Q factor is up to $2.75\times 10^{6}$ and divergence is only about 30
degree.
Thanks to the ray-wave correspondence, an efficient route to design a
unidirectional emission cavity shape could be: using the ray dynamics to give
the SOS of the selective boundary shape, predicting the emission properties,
and carrying out the wave simulation to confirm it. Here, we only gives
examples on the typically materials, such as semiconductor and silicon
($n\approx 3.3$) and silica glass ($n\approx 1.45$). Other popular solid
materials for photonics have similar refractive index. There are also many
materials not at this range, such as some doped glass with the refractive
index around $2.0$. We have also successfully designed the cavity shapes to
achieve high Q and unidirectional emission (not presented here). By adjusting
the periodic-3 orbits or periodic-4 orbits, we can get appropriate emission
direction and expect the unidirectional emission in the materials with
refractive index range from $1.4$ to $4$.
There are also some deviations between the simulations with ray and wave,
especially when $kr$ is small. The actions of light could not totally be
described by rays. Better ray-wave correspondence could be expected in larger
cavity limaconexp3 . In smaller cavity, wave properties should be included,
such as, the diffractions and interference. In microcavities, researches have
illustrated that some modifications should be included in the ray dynamics
wave , such as the corrected Fresnel’s law at curve interface fresnel , the
Goos-Hänchen shift gh and the Fresnel Filtering effect filt .
In conclusion, we examined and presented the mechanism for high Q
unidirectional emission WG modes in micro-cavities. Based on the necessary
condition for the continuous and single axis symmetric boundary shape, we can
well control the directional emission from ARC by setting appropriate boundary
shape to adjust the stable islands and fixed points of unstable orbits in
phase space. With the assistance of ray dynamics, it is easy to design
fabrication-friendly simple cavity boundary shape to achieve high Q and
unidirectional emission in different materials with the refractive index range
from $1.4$ to $4$. We expect our approaches presented here to discuss the
light in cavities can be applied to the study of the chaotic transport in two
dimensional phase space Shim08 , as well as other opening non-integrable
systems, such as quantum dots and nano-structures.
###### Acknowledgements.
C.-L. Zou thanks Juhee Yang and J. Wiersig for discussions. Financial support
by the National fundamental Research Program of China under Grant No
2006CB921900; National Science Foundation of China under Grant No. 60537020
and 60621064; The Knowledge Innovation Project of Chinese Academy of Sciences
& Chinese Academy of Sciences and International Partnership Project; Y. Yang
is also funded by the China Postdoctoral Science Foundation. F.-W. Sun is also
supported by the starting funds from USTC for new faculty.
## References
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* (25) J. U. Nöckel and A. Douglas Stone, Nature 385, 45(1997).
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* (34) It is not required for general cases. Here we just set the values of high orders zero for simplicity in simulation, as well as simplification in fabrication.
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|
arxiv-papers
| 2009-08-25T03:50:27 |
2024-09-04T02:49:04.771889
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.-L. Zou, F.-W. Sun, C.-H. Dong, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C.\n Guo, and Z.-F. Han",
"submitter": "Fangwen Sun",
"url": "https://arxiv.org/abs/0908.3531"
}
|
0908.3543
|
# Exact spectral dimension of the random surface
I S Goncharenko School of Natural Sciences, University of California, Merced,
CA 95343 igoncharenko@ucmerced.edu
###### Abstract
We propose a new method of the analytical computation of the spectral
dimension which is based on the equivalence of the random walk and the q-state
Potts model with non-zero magnetic field in the limit $q\to 0$. Calculating
the critical exponent of the magnetization $\delta$ of this model on the
dynamically triangulated random surface by means of a matrix model technique
we obtain that the spectral dimension of this surface is equal to two.
###### pacs:
04.60.Nc, 05.40.Fb, 05.50.+q, 05.70.Jk
## 1 Introduction
A diffusion in the dynamic medium is an important problem due to its wealth of
applications in many branches of physics. Examples include the diffusion
through fluid membranes [1, 2], the diffusion in the presence of two-
dimensional quantum gravity [3, 4] and others. The main characteristic of the
diffusion (random walk) is the probability that the particle returns to the
starting site at time $t$. At large times one expects this probability to
decay as $t^{-d_{s}/2},$ where $d_{s}$ is the spectral dimension of the
underlying geometry. In the case of ${\bf Z}^{d}$ we have $d_{s}=d$, which
simply gives the dimension of the regular lattice. It is interesting how the
fractal structure of random lattices affects the spectral dimension. Despite
the fact that the spectral dimension of many random graphs has been calculated
analytically, in such cases, as branched polymers or generic trees $d_{s}=4/3$
[5, 6], non-generic trees or multi-critical branched polymers with $k$ phases
[7]:
$d_{s}={2k+2\over 2k+1},\quad\mbox{$k=2,3,4\dots$, }$
random combs $d_{s}=(4-b)/2,b<2$ [8], where $b$ is a power law exponent for
the length of the tooth of the random comb, there is no theoretical derivation
of the spectral dimension of random lattices of a given, for instance planar
or toroidal, topology. Dynamical triangulations, dual to random lattices,
arise as a discretization of the integral over the metrics of some smooth two-
dimensional manifold [9]. Each triangulation (see Fig. 1) is in one-to-one
correspondence with a vacuum diagram of some $N\times N$ Hermitian matrix
model. In the large $N$ limit only lattices with planar topology survive [10,
11].
We show that the random walk is exactly equivalent to the $q$-state Potts
model with non-zero magnetic field taking in the limit $q\to 0$ [12]. On a
random lattice this model is defined by the partition function:
$Z^{(q)}_{n}(\beta,H)=\sum_{G_{n}}\sum_{{\sigma}}\exp\left({\beta\over
2}\sum^{n}_{i,j=1}G^{(n)}_{ij}\delta(\sigma_{i},\sigma_{j})+Hq^{a}\sum^{n}_{i=1}\delta(1,\sigma_{i})\right),$
(1.1)
where $G^{(n)}_{ij}$ is the adjacency matrix of the graph, the upper index $n$
is the number of vertices in the graph, indices $i,j=1\dots n$ enumerate
vertices, spin variables $\sigma_{i}$ associated with the vertex $i$ take $q$
different values (colours) enumerating independent components of the spin,
$\sum_{G_{n}}$ represents the sum over all configurations of graphs in the
ensemble, $\sum_{\sigma}$ represents the sum over all configurations of spins,
$H$ represents the magnetic field normalized by the temperature, $0<a<1$ is an
auxiliary parameter which is essential in the limit of small $q$, $\beta$ is
the product of the inverse temperature and the coupling constant of spins. The
limit $q\to 0$ can be better understood through the cluster representation
[13] of the model (1.1). It also can be defined in terms of the tree-like
percolations (spanning forests) on a random graph [14]. In this representation
the partition function (1.1) for zero magnetic field $H=0$ is given by
$Z^{(q)}_{n}(\beta)=\sum_{G_{n}}\sum_{trees}B^{b(tree)},$ (1.2)
where $\sum_{trees}$ is the sum over all trees spanning the lattice $G_{n}$,
$b(tree)$ is the number of bonds in a given tree on the lattice $G_{n}$ and
the constant $B$ is connected to $\beta$ in (1.1) through the equality
$e^{\beta}=1+q^{a}B$.
The behaviour of the return probability, the conditional probability and the
square displacement of the random walk at large times can be found by
computing critical exponents of the magnetization $\delta$, the two-point
correlation function $\eta$ and the correlation length $\nu$ of the spin model
(1.1) correspondingly [15].
The Potts model on a random lattice [16, 17] belongs to the long list of
exactly solvable models which could be reformulated as matrix models. The list
includes the Ising model [18, 19], bond-percolations [20], tree-percolations
[21], the $O(n)$ vector model [22, 23], dilute Potts model [24] and many
others [25, 26]. The limit $q\to 0$ of the Potts model which is relevant for
our consideration had been solved by the saddle point technique [21], by the
loop equation technique [17] and, recently, by the combinatorial method [28].
We generalize this results to the case with non-zero magnetic field which
breaks the symmetry of the model. In this scenario there will be two different
saddle points. However we shall show how it can be simplified in the limit
$q\to 0$.
Fig.1: Duality between fat graphs arising from the perturbation expansion of
the $\phi^{3}$ of one-matrix model and triangulated surface. Each $\phi^{3}$
vertex of the fat graph corresponds to the triangle. Gluing together triangles
edge to edge is equivalent to the Wick’s contraction of $\phi$ matrices.
We derive the exact result for the spectral dimension of a random surface and
show that in the case of non-zero field there are two phases. One phase
corresponds to Hamiltonian walks [29] or branched polymer phase $d_{s}=4/3$
($\gamma_{str}=-1$) another is dilute or pure gravity phase $d_{s}=2$
($\gamma_{str}=-1/2$).
This paper is organized as follows. In section 2 we establish the equivalence
of the random walk and the Potts model. We probe this equivalence computing
the critical exponent of the Potts model magnetization on the Bethe lattice
and comparing it to the spectral dimension of the corresponding lattice. In
section 3 we reformulate the Potts model on a random surface as a random
multi-matrix model. In section 4 we find the solution of this model and
construct the phase diagram. In section 5 we compare our results with
numerical simulations and present our conclusions.
## 2 Random walks and the Potts model
A lattice is a set of $n$ vertices connected by links. It is uniquely defined
by the $n\times n$ adjacency matrix $G^{(n)}_{ij}$, whose entries are
$G^{(n)}_{ij}=0$ if there is no link between $i$ and $j$ and $G^{(n)}_{ij}>0$
otherwise. The coordination number or the degree of a vertex is defined as the
total number of links connected to it. Consider the ensemble of lattices
$G_{n}$ with $n$ vertices and the random walk jumping on the sites of the such
lattices. At each time step the walker with equal probability must jump to the
nearest-neighbour site. This process is independent of what can happen to the
lattice bonds. We consider that lattice changes its configuration by choosing
new one from $G_{n}$ at random every time step. A jump can occur only if the
sites are connected by the bonds at the time the walker attempts to jump. Let
$G^{(n)}_{ij}(k)$ denote the adjacency matrix at time step $k$. Suppose that
the walker starts at time $k=0$ at the site $0$. Given a particular bond
history
$G=G(k)=\\{G^{(n)}_{ij}(0)\dots G^{(n)}_{ij}(k)\\}$ (2.1)
let $p_{i}(k;G)$ be the probability to find the random walk at the vertex $i$
after time $t$ and $p_{i}(0)=\delta_{0,i}$. The master equation for
conditional probabilities of the random walk on the dynamical lattice of $n$
sites can be cast into the following system of equations
$p_{i}(k+1;G)=\sum^{k}_{j=1}\left(G^{(n)}_{ij}(k)p_{j}(k;G)+[1-G^{(n)}_{ij}(k)]p_{i}(k;G)\right),$
(2.2)
where the lower case indices $i,j$ enumerate vertices and $k$ counts the
number of jumps made by the walker. The random walk is non-Markovian because
jumps depend on bond histories $G$.
It was shown numerically in [30] that conditional probabilities $p_{i}(k;G)$
in the long time limit $k\to+\infty$ did not depend on the particular bond
history and approached some average value $\bar{p}_{i}(k)$. This universal
behaviour is very similar to the behaviour of the spin system on the
fluctuating lattice.
We show that there is an explicit correspondence between the random walk and
the Potts model. Firstly, we consider the simplest case when the bond
configuration is static. Then conditional probabilities $p_{i}(k)$ become
Markovian because all jumps are independent of earlier events. Then the master
equation (2.2) simplifies and after taking the continuum time limit it can be
written as the system of differential equations
$\dot{p}_{i}=\sum_{j}G_{ij}(p_{j}(t)-p_{i}(t)),$ (2.3)
where $G_{ij}$ is the adjacency matrix of some fixed lattice. After the
Laplace transform
$P_{i}(H)=\int_{0}^{+\infty}p_{i}(t)e^{-Ht}dt$ (2.4)
the master equation (2.3) becomes the system of linear equations for the
quantities $P_{i}=P_{i}(H)$:
$L_{ij}P_{j}=\delta_{0,i}\qquad L_{ij}=H\delta_{ij}-G_{ij},$ (2.5)
where $L_{ij}$ is the Laplacian of a graph with the the adjacency matrix
$G_{ij}$. This system can be easily solved by inverting the Laplacian. The
return probability is defined by the determinant of the Laplacian
$P_{0}={1\over n}{\partial\ln\det L\over\partial H}.$ (2.6)
On the other hand, it was rigorously proven in [31] that the determinant of
the Laplacian is the sum over all spanning forests on a lattice. The forest,
including $l$ trees which span $m_{i}$, $i=1\dots l$ vertices correspondingly,
gains the weight $H^{l}\prod_{i}m_{i}$, where $\sum_{i}m_{i}=n$ and $n$ is the
total number of vertices in a lattice. Thus it can be interpreted as the
cluster representation of the partition function of $q\to 0$ Potts model with
non-zero magnetic field. The return probability is the magnetization $M$ and
it scales as $M\sim H^{1/\delta}$ at the critical point. The spectral
dimension is defined by
$d_{s}=2\left(1+{1\over\delta}\right).$ (2.7)
We consider a generalization of the above result to the case of a random
lattice. The key conjecture is
$P_{0}(H)={\partial\over\partial H}\ln\left[\lim_{q\to
0}{Z^{(q)}_{n}(\beta,H)\over q^{aN}}\right],$ (2.8)
where $Z^{(q)}_{n}(\beta,H)$ is the partition function (1.1). To see if it is
true we consider the graph associated with the path of the walk on a random
lattice. The edges of the graph are those bonds on a lattice that the random
walk crossed for the first time on its path. It is easy to see that this graph
is a tree which is called the forward tree (see Fig. 2). Thus the partition
function of the random walk which is the sum over all possible paths of the
random walk is dual to the partition function of all trees on a random lattice
[4].
Our conjecture (2.8) is actually a powerful tool for computing exponents of
the random walk on different graphs. It allows to employ a critical phenomena
technique to the random walk problem which is much broader than methods of a
direct solution of the equation (2.2).
Fig.2: Forward tree is pictured for the path of the random walk on the square
regular lattice. The walker starts at the lower left corner and ends at the
upper right corner. Jumps through the diagonal are also allowed. Arrows denote
the direction of jumps. Edges, belonging to the path of the walker, are shown
by broken lines. The edge belongs to the forward tree if and only if the
walker, jumping through this edge, gets to the vertex it had never visited
before. All such edges are shown by solid lines.
As an example we consider the Bethe lattice with coordination number $z$ (see
Fig. 3). We put Potts spins at each site of the lattice. The solution of this
model is well-known [32]. In the thermodynamic limit the magnetization of the
central site of the lattice is
$M=\coth[(H-zs)/2],$ (2.9)
where $s$ is a parameter defined by $x=e^{s}$ and $x$ is the fixed point of
the recurrence relation which for the case of $q\to 0$ can be written as:
$x={e^{H}+(e^{\beta}-2)x^{z-1}\over e^{H+\beta}-x^{z-1}}.$ (2.10)
Recasting the exponent of the magnetic field from (2.10) we find that
$e^{H}=x^{z-1}{e^{\beta}-2+x\over e^{\beta}x-1}.$ (2.11)
Now we can write $H$ as Taylor series of the small parameter $s$. Up to two
leading terms the expansion proceeds
$H=(z-2)s+(e^{\beta}-1)^{-1}s^{2}+\dots.$ (2.12)
On the other hand from the expansion of the magnetization (2.9) one has $M\sim
s^{-1}$. Treating $s$ as a function of $M$ we obtain
$H=(z-2)M^{-1}+(e^{\beta}-1)^{-1}M^{-2}+\dots$ (2.13)
Using the scaling hypothesis $H=M^{\delta}f_{s}(M^{-1/\beta})$ one has
$H=M^{-2}f_{s}(M).$ (2.14)
It gives the values for critical exponents $\delta$ and $\beta$ of the
$q$-state Potts model $q\to 0$ on the Bethe lattice $\delta=-2$, $\beta=-1$.
Knowing $\delta$ and using formula (2.7) we derive that $d_{s}=1$ in agreement
with the result [33, 34].
Fig.3: Bethe lattice with coordination number $z=3$.
## 3 Correspondence with the Matrix Model
From now on we restrict ourselves to the ensemble of random lattices with
coordination number 3 and the topology of the sphere. Consider the q-matrix
model defined by the partition function
$Z=\int dM_{1}\dots dM_{q}\exp(N\tr[c\sum^{q}_{i\neq
j}M_{i}M_{j}-\sum^{q}_{i=1}M^{2}_{i}+{ge^{Hq^{a}}\over
3}M_{1}^{3}+\sum^{q}_{i=2}{g\over 3}M_{i}^{3})]),$ (3.1)
where $M_{i},i=1\dots q$ are $N\times N$ Hermitian matrices, $H$ is the
magnetic field, $0<a<1$ and
$c=1/(e^{\beta}+q-2),$ (3.2)
where $e^{\beta}-1=q^{a}B$. This model generalizes the model of [17, 21] to
the case of non-zero magnetic field.
We note that all matrices in (3.1) are coupled to each other. Physically each
matrix $M_{i}$ represents one component of spin. It can be shown that the
propagator is
$\langle\tr
M_{i}M_{j}\rangle_{0}=N{c\over(c+1)(1-c(q-1))}\left\\{{(1-c(q-2))/c,i=j\atop
1,i\neq j}\right.$ (3.3)
where $\langle\dots\rangle_{0}$ denotes the Gaussian average ($g=0$). Using
Feynman diagrammatic expansion one would get that the free energy
corresponding to (3.1) is equal to the generating function:
$Z_{T}=\lim_{q\to
0}\sum_{n=1}^{+\infty}\left({cg\over(c+1)(1-c(q-1))}\right)^{n}e^{Hq^{a}}Z^{(q)}_{n}(\beta,H)$
(3.4)
where $Z^{(q)}_{n}(\beta,H)$ is the partition function (1.1).
After the change of variables $M_{i}\to M_{i}(2(1+c))^{-1/2}$ we have
$Z=\int dM_{1}\dots dM_{q}\exp(N\tr[{h^{2}\over 2}Y^{2}-{1\over
2}\sum^{q}_{i=1}M^{2}_{i}+{\bar{g}e^{Hq^{a}}\over
3}M_{1}^{3}+\sum^{q}_{i=2}{\bar{g}\over 3}M_{i}^{3})]),$ (3.5)
where $\bar{g}=g(2(1+c))^{-3/2}$, $h^{2}=c/(1+c)$ and $Y=M_{1}+\dots+M_{q}$.
By introducing in (3.5) new Gaussian-distributed random matrix variable $X$ we
replace the first term in the exponent by the matrix polynomial linear in
$M_{i}$:
$\int
dX\prod_{i}dM_{i}\exp(N\tr[-X^{2}/2+hX\sum^{q}_{i=1}M_{i}-\sum^{q}_{i=1}M^{2}_{i}+{\bar{g}e^{Hq^{a}}\over
3}M_{1}^{3}+\sum^{q}_{i=2}{\bar{g}\over 3}M_{i}^{3})]).$ (3.6)
We want to express the integral over matrices (3.6) by the integral over
eigenvalues. As it was demonstrated in [35] the integral over the matrix in
the external field can be reduced to the integral over eigenvalues by the
following formula:
$\int
dM\exp(N\tr[-M^{2}/2+MX])=\int\prod_{i=1}^{N}dm_{i}{\Delta(m)\over\Delta(x)}\exp(N[-m_{i}^{2}/2+m_{i}x_{i}])$
(3.7)
Using (3.7) and noticing that all integrals over $M_{i}$ in (3.6) are similar,
the partition function can be rewritten as:
$Z=\int\prod^{N}_{i=1}dx_{i}\Delta(x)^{2-q}\exp\left(N\sum^{N}_{i=1}x_{i}^{2}/2\right)\Theta_{+}(x)\Theta_{-}(x)^{q-1},$
(3.8)
where
$\Theta_{+}(x)=\int\prod^{N}_{i=1}dm_{i}\Delta(m)\exp\left(N\sum^{N}_{i=1}[hx_{i}m_{i}-{1\over
2}m_{i}^{2}+{\bar{g}e^{Hq^{a}}\over 3}m_{i}^{3}]\right),$ (3.9)
$\Theta_{-}(x)=\int\prod^{N}_{i=1}dm_{i}\Delta(m)\exp\left(N\sum^{N}_{i=1}[hx_{i}m_{i}-{1\over
2}m_{i}^{2}+{\bar{g}\over 3}m_{i}^{3}]\right).$ (3.10)
In the present paper we will not give the general solution of (3.8). For our
purposes it is enough to find the partition function when $q\to 0$. In the
absence of the magnetic field $H=0$ functionals $\Theta_{+}(x)$ and
$\Theta_{-}(x)$ are equal to each other. Hence the value of this functionals
will be governed by the same saddle point equation as shown in [20, 21]. By
noticing that $h^{2}=1/q^{a}B$ and after the change of variables $m_{i}\to
m_{i}/\sqrt{q^{a}B}$ one has
$\Theta_{+}(x)=\int dm_{i}\Delta(m)\exp({N\over
q^{a}B}\sum^{N}_{i=1}[x_{i}m_{i}-{1\over 2}m_{i}^{2}-{Ge^{Hq^{a}}\over
3}m_{i}^{3}]),$ (3.11)
where $G=g(\sqrt{2}q^{a}B)^{-3}$. When $q$ is small the prefactor in the
exponent becomes large and the steepest descent method can be used to compute
(3.11). Unlike the usual large-N limit the contribution from the Van-der-Monde
determinant will be small and can be neglected. To the leading order we have
$\Theta_{+}(x)=\exp\left({N\over q^{a}B}\sum^{N}_{i=1}[x_{i}u_{i}-{1\over
2}u_{i}^{2}-{Ge^{Hq^{a}}\over 3}u_{i}^{3}]\right)$ (3.12)
Similarly we have
$\Theta_{-}(x)=\exp\left({N\over q^{a}B}\sum^{N}_{i=1}[x_{i}v_{i}-{1\over
2}v_{i}^{2}-{G\over 3}v_{i}^{3}]\right)$ (3.13)
where $u_{i}$ and $v_{i}$ are given by the saddle point condition:
$x_{i}=u_{i}+Ge^{Hq^{a}}u_{i}^{2}\quad x_{i}=v_{i}+Gv_{i}^{2}.$ (3.14)
In the limit of small $q$ one can express the solution $u_{i}$ in terms of
$v_{i}$ as perturbation series. Choosing the ansatz
$u_{i}=v_{i}+\epsilon v^{(1)}_{i}\quad\epsilon=Hq^{a}$ (3.15)
and expanding the exponent $e^{Hq^{a}}$ we have
$u_{i}=v_{i}-\epsilon{v^{2}_{i}\over 1+2v_{i}}$ (3.16)
By doing this the ratio of (3.12) and (3.13) is significantly simplified:
${\Theta_{+}(x)\over\Theta_{-}(x)}=\exp\left(-{NGH\over
3B}\sum^{N}_{i=1}v_{i}^{3}\right)$ (3.17)
It follows that the partition function is
$Z=\int\prod^{N}_{i=1}dx_{i}\Delta(x)^{2}\exp\left(N\sum^{N}_{i=1}x_{i}^{2}/2\right)\exp\left(-{NGH\over
3B}\sum^{N}_{i=1}v_{i}^{3}\right)$ (3.18)
After changing variables from $x$ to $v$ and using (3.14) the integral (3.18)
becomes
$Z=\int
dv_{i}\prod_{i,j}(1+G(v_{i}+v_{j}))\Delta(v)^{2}\exp[-N\sum_{i}V(v_{i})],$
(3.19) $V(x)=\left({1\over 2}(x+Gx^{2})^{2}+{HGx^{3}\over 3B}\right).$ (3.20)
Again, we apply the steepest descent method but now with respect to large $N$.
The saddle point equation is
$2\sum_{j\neq i}{1\over v_{i}-v_{j}}+\sum_{j}{2G\over
1+G(v_{i}+v_{j})}=NV^{\prime}(v_{i}).$ (3.21)
The distribution of the eigenvalues becomes continuous with density
$\rho(x)=(1/N)\sum\delta(x-v_{i})$. We restrict ourselves to the one-cut case
where all eigenvalues $v_{i}$ belong to the support consisting of one interval
$[a,b]$, $ab>0$. The equation for the density is the integral equation:
$\mbox{P}\int{\rho(y)dy\over x-y}+\int{\rho(y)dy\over 1/G+x+y}={1\over
2}V^{\prime}(x),$ (3.22)
where P denotes the principal value of the integral. Introducing the trace of
the resolvent
$\omega(x)={1\over N}\tr{1\over M-x}=\int{\rho(y)dy\over y-x}$ (3.23)
(3.22) can be equivalently written as
$\omega(x+i0)+\omega(x-i0)+2\omega(-x-1/G)={1\over 2}V^{\prime}(x)$ (3.24)
The integral equation that governs the eigenvalue density is
$\int{dy\rho(y)\over(x-y)(1+G(x+y))}={1\over 2}(1+Gx)x+{HG\over 2B}{x^{2}\over
1+2Gx}$ (3.25)
## 4 Solution
The equation (3.25) represents the Riemann-Hilbert problem. The solution can
be found [36]:
$\rho(x-1/2G)={1\over\pi}[(x^{2}-a^{2})(b^{2}-x^{2})]^{1/2}\int_{a}^{b}{f(y)dy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}(x^{2}-y^{2})},$
(4.1)
where
$f(x)=Gx^{3}+HGx^{2}/(2B)-(1/4+H/(2B))x+H/(8BG),$ (4.2)
supplied with additional condition:
$\int_{a}^{b}{f(y)dy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}}=0$ (4.3)
and with the normalization condition:
$\int_{a}^{b}dx\rho(x)=1.$ (4.4)
The ends of the interval $[a,b]$ are functions of $G,B,H$, to be determined
from transcendental equations (4.3) and (4.4) on $a=a(G,B,H)$ and
$b=b(G,B,H)$.
We will need the first equation which can be resolved as
$G{\pi\over 2}\left(1+{1\over 2}(b^{2}-a^{2})\right)+{HGa\over
2B}E\left(1-{b^{2}\over a^{2}}\right)-\left({1\over 4}+{H\over
2B}\right){\pi\over 2}+{H\over 8BGa}K\left(1-{b^{2}\over a^{2}}\right)=0,$
(4.5)
where $K(x),E(x)$ are elliptic integrals of the first and second kind
respectively. We used the following integrals:
$\int_{a}^{b}{dy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}}=a^{-1}K\left(1-{b^{2}\over
a^{2}}\right),$ (4.6)
$\int_{a}^{b}{ydy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}}={\pi\over 2},$ (4.7)
$\int_{a}^{b}{y^{2}dy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}}=aE\left(1-{b^{2}\over
a^{2}}\right),$ (4.8)
$\int_{a}^{b}{y^{3}dy\over[(y^{2}-a^{2})(b^{2}-y^{2})]^{1/2}}={\pi\over
2}\left(1+{1\over 2}(b^{2}-a^{2})\right)$ (4.9)
Then the critical behaviour of the partition function (3.1) is obtained by
taking the double scaling limit $B\to B_{c}$ (infinite random surface) and
$G\to G_{c}$ (infinite trees). The latter occurs near the upper edge $b$ of
the support. We have $b_{c}=-1/2G_{c}$. It can be recast in the following form
$w=1\pm 1/h,h>1\qquad 1=\cos(\pi/h)\quad h=2,4\dots$ (4.10)
Then the critical density is
$\rho(x)\sim(g-g_{*})(b-x)^{1-1/h}+(b-x)^{1+1/h}$ (4.11)
and string susceptibility is $\gamma_{str}=-1/h$. The solution is singular
over $[a-1/2G,b-1/2G]$ and $[-b-1/2G,-a-1/2G]$. Branching points coincide when
$a=0$ [21]. It can be shown that the Potts model is in the critical point if
this condition is satisfied. Thus there are two phases of the model: dilute
and dense. In the dilute phase the problem is equivalent to the
(-2)-dimensional dynamical triangulated surface. The critical exponent of the
string susceptibility is $\gamma_{str}=-1$. In the dense phase defined by the
condition $b=-1/2G$ one has $\gamma_{str}=-1/2$. From the equation (4.5) we
can see that
${\partial b\over\partial H}\sim{1\over H}.$ (4.12)
Finally, let us compute the magnetization $M=-\partial F/\partial H$. The free
energy is
$F=\int_{a}^{b}dx\rho(x)V(x)+\int_{a}^{b}\int_{a}^{b}dxdy\rho(x)\rho(y)[\ln|x-y|+\ln(1+G(x+y))].$
(4.13)
It is linear in $H$ for the small magnetic field. Taking the derivative one
can obtain that
$M\sim{\partial b\over\partial H}(C_{1}H+C_{2}H^{2})=(C_{1}+C_{2}H)=f_{s}(H).$
(4.14)
It means that $1/\delta=0$. Substituting the exponent $\delta$ in (2.7) one
has
$d_{s}=2.$ (4.15)
## 5 Conclusion
In conclusion, we have demonstrated an alternative derivation of the spectral
dimension of the random surface. We note that our results match the numerical
simulations [37]. It should also coincide with the KPZ [38] result for the
conformal field theory with central charge $c=-2$ coupled to the two-
dimensional gravity. The result can be generalized to topologies with higher
genus by the DDK [39, 40] formula. We notice that it should not change the
value of the spectral dimension $d_{s}$. The only sensitive exponent to DDK is
the string susceptibility $\gamma_{str}$.
## Acknowledgment
I would like to thank Dmitry Krotov and Sergei Alexandrov for valuable
discussions. I would like to acknowledge support from Ajay Gopinathan via his
start-up funds and his James S. McDonnell Foundation Award.
## References
## References
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* [33] Cassi D 1989 Europhys. Lett. 9 627-631
* [34] Samukhin A N, Dorogovtsev S N and Mendes J F F 2008 Phys. Rev.E 77, 036115
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* [36] Gakhov F D 1990 Boundary value problems (Mineola: Dover Publications)
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|
arxiv-papers
| 2009-08-25T06:04:23 |
2024-09-04T02:49:04.777917
|
{
"license": "Public Domain",
"authors": "Igor Goncharenko",
"submitter": "Igor Goncharenko",
"url": "https://arxiv.org/abs/0908.3543"
}
|
0908.3614
|
# Detrended Fluctuation Analysis of Systolic Blood Pressure Control Loop
C.E.C. Galhardo1, T.J.P.Penna1, M. Argollo de Menezes1 and P.P.S. Soares2 1
Instituto de Física, Universidade Federal Fluminense, Av. Litoranea, s/n,
24210-340, Niteroi, RJ ,Brazil 2 Instituto Biomédico, Universidade Federal
Fluminense, R. Prof. Hernani Melo n. 101, 24210-130, Niteroi, RJ, Brazil
###### Abstract
We use detrended fluctuation analysis (DFA) to study the dynamics of blood
pressure oscillations and its feedback control in rats by analyzing systolic
pressure time series before and after a surgical procedure that interrupts its
control loop. We found, for each situation, a crossover between two scaling
regions characterized by exponents that reflect the nature of the feedback
control and its range of operation. In addition, we found evidences of
adaptation in the dynamics of blood pressure regulation a few days after
surgical disruption of its main feedback circuit. Based on the paradigm of
antagonistic, bipartite (vagal and sympathetic) action of the central nerve
system, we propose a simple model for pressure homeostasis as the balance
between two nonlinear opposing forces, successfully reproducing the crossover
observed in the DFA of actual pressure signals.
###### pacs:
05.40.-a,87.19.Hh,87.80.Vt,89.75.Da
††: New J. Phys.
## 1 Introduction
Negative feedback loops are ubiquitous in living systems, with important
examples like the lac-operon in gene regulation [1], which inhibits lactose
consumption in the presence of glucose, and serve as efficient ways of
maintaining stability and suppressing fluctuations in noisy environments [2,
3, 4, 5, 6] 111 Negative feedback loops also appear in electronic circuits as
a tool for the stabilization of laser beams (see [7]). On a much larger
physical scale, the autonomous nerve system is able to sustain (without
external supervision) basic life signals like temperature, water and
metabolite concentrations at safe levels by the action of a pair of nerve
branches, called sympathetic and parasympathetic (or vagal). These nerve
branches have cooperative and “antagonistic” roles in our body: while the
sympathetic prepares our body for “flight-or-fight” situations (increasing
heart rate, dilating pupils and cancelling digestive functions, for instance),
the vagal, or parasympathetic, decreases heart rate, constricts pupils and
stimulate salivary glands. The balance between these “forces”, which keeps
living systems operating close to optimal levels, is called homeostasis [8,
9]. Alterations of a given control mechanism can perturb such balance and lead
to pathological conditions such as Diabetes Mellitus, which results from a
malfunctional insulin metabolism [6].
A major feature of the autonomous nerve system is that stimulation of the
vagal branch results in a inhibition of the sympathetic branch, which acts
continuously on organs and veins at an approximately steady level when not
inhibited. These nerve branches are controlled at the Nucleus Tractus
Solitarius (NTS) of the medulla by integration of neural information coming
from afferent neural fibers, which carry information from sensory neurons
spread around the body. Among other sensory information carried by those
fibers, one of utmost importance regards arterial blood pressure: through
these afferent nerve fibers, stretch-sensitive mechanoreceptors spread around
veins and arteries of the heart return to the NTS (in a timescale of few
seconds) information about the current status of pressure (and its variation).
111There are also baroreceptors at the kidneys, which change body fluid volume
at the timescale of hours or days [10]. Those are responsible for very low
frequency fluctuations and will not be analyzed here. The NTS, in turn, excite
(when pressure is high) or inhibit (when pressure is low) the vagal branch,
closing the circuit for what can be regarded as a self-inhibitory feedback
loop called baroreflex [8, 9, 11] (See figure 1).
Figure 1: Schematics of the negative feedback loop for pressure control, or
baroreflex. Stimulus from afferent neurons excite the vagal branch of the
autonomous nerve system, which in turn slows down heart rate. At the same
time, the sympathetic branch, which acts to increase heart rate, is inhibited
by the vagal branch. As a result, a surge in blood pressure tends to stimulate
the vagal branch and inhibit the sympathetic branch, decreasing heart rate
and, consequently, decreasing blood pressure.
As a result of this balance the body, although continuously perturbed by
external factors, is able to keep homeostasis, a stationary state where, among
other things, arterial pressure, temperature, water and metabolite
concentrations are kept at optimal levels [8, 9]. One can think of homeostasis
as a locally optimal state sustained by feedback loops in a noisy environment.
The reasonably controlled flow of nutrients throughout veins and arteries is
achieved with the aid of the blood system and the heart, whose pumping action
is monitored and controlled by the autonomous nerve system. Arterial blood
pressure (ABP) is one of the vital signals that can be continuously monitored,
which carries a large amount of information about the mechanisms responsible
for homeostasis and the different timescales for their responses [12, 13].
Given a continuous set of recordings of ABP, $\\{p(t)\\}$, over a given period
of time, one defines the $n$-th diastolic blood pressure as the $n$-th local
minimum $p_{n}$, the systolic blood pressure as the $n$-th local maximum
$p_{n}$ and the time interval between two neighboring ABP minima,
$b_{n}=t(p_{n+1})-t(p_{n})$, as the instantaneous inter-beat heart rate (IR),
as depicted in figure 2.
Figure 2: Time evolution of arterial blood pressure (ABP). The local maxima
are called systolic blood pressure, the minima are the diastolic pressure and
the time interval between two neighboring ABP minima is the instantaneous
inter-beat heart rate. In this work we focus on the arterial systolic blood
pressure and its variation in time.
These quantities have long been characterized by spectral methods [14, 15],
where peaks in the power spectrum $S_{\omega}=\left|\frac{1}{2\pi}\sum
b_{n}e^{i\omega n}\right|^{2}$ reveal the timescales for the response of
different control mechanisms [16, 17, 18, 19, 20]. Nevertheless, in order to
assess the long-range correlations [21] emerging from these feedback control
systems, or to characterize disruptive and abnormal states, one must recur to
methods which account for the strong non-stationarity of those signals [22],
such as detrended fluctuation analysis (DFA) [23, 24, 25, 26, 27, 28, 29]. In
this work we analyze the dynamics of baroreflex, the negative feedback loop
providing a rapid and powerful reflex control of blood pressure, which is by
far the most studied cardiovascular reflex in physiological and clinical
settings. For such purpose we apply DFA to experimental time series consisting
of continuous arterial systolic blood pressure measurements.
We report results of experiments on rats with surgical disruption of the nerve
fibers connecting the baroreceptors to the medulla, a procedure called
sinoaortic denervation (SAD) [30], and find that other mechanisms might be
responsible for arterial blood pressure control, although at different time
scales, possibly due to synaptic plasticity at the NTS [31, 32, 33]. Following
this recovery, average blood pressure is kept at almost the same levels as
before denervation, a determinant condition for the kidneys to work properly
[8, 9]. We apply detrended fluctuation analysis to our experimental time
series and find that fluctuations in systolic blood pressure cross over from
non-stationary to stationary, long-range correlated at a characteristic time
scale $\tau$. Surgical denervation of baroreceptors significantly changes the
correlation patterns of pressure signals but, after $20$ days, correlation
patterns typical of non-operated rats are recovered, only with larger
crossover times $\tau^{\prime}>\tau$. This suggests that the control loop is
reestablished, possibly due to adaptation to sensory information coming from
other less effective receptors.
To model such feedback control loop we develop a model of a random walker
forced by two opposing nonlinear (sigmoidal) forces, representing the
sympathetic action and its inhibition by the vagal (parasympathetic) branch.
We find the same crossover from non-stationary to stationary, long-range
correlated noise observed in actual pressure measurements. Moreover, by
changing the difference between the sensitivity of each branch, we find the
same shift in the crossover time scale, as observed in rats $20$ days after
surgery, when adaptation occurs and homeostasis is recovered.
## 2 Experiments and Measurements
Adult male Wistar rats were maintained on a $12$-hour light/dark lighting
schedule at $23^{o}$C, food and water ad libitum. All procedures were
performed according to [34]. The animals were divided in three groups: control
rats (ctr, $N=11$ rats), acute sinoaortic denervated rats (1d, $N=5$ rats),
i.e, animals surgically denervated one day before measurements, and chronic
sinoaortic denervated rats (20d, $N=8$ rats), animals surgically denervated
$20$ days before measurements. SAD was performed using the methods described
by Krieger et al [35], and basically consists of full disruption of the nerve
fibers connecting the baroreceptors spread in veins and arteries of the heart
to the medulla. Blood pressure was recorded from the left femoral artery for
$90$ minutes in conscious rats. Before the analog to digital conversion, blood
pressure was low-pass filtered (fc= $50$ Hz) for high-frequency noise removal,
and recorded with a $2$kHz sampling frequency. Systolic (maximum) and
diastolic (minimum) values were detected after parabolic interpolation and
signal artifacts were visually identified and removed. Pulse intervals were
measured in milliseconds (ms), considering intervals between consecutive
diastole and the heart rate was calculated as the inverse of pulse interval
and measured in beats per minute (bpm) (A more detailed account of this
experiment can be found in [36]). Since the measurements were made in awake,
conscious unrestrained rats, some distortions in the blood pressure signal
might arise due to their movements. To reduce this problem we discard series
that show any kind of discontinuities or jumps. After this selection we keep
six time series for the control group, five time series for the chronic
denervated group and four time series for the acute denervated group. Each
time series consists of $10^{4}$ data points, equivalent to $30$ minutes of
continuous measurements.
In figure 3 we depict the series of systolic blood pressure values for the
three groups: while pressure in non-operated rats fluctuates in a stationary
fashion about $116.55\pm 10.15$ mm Hg (Figure 3a), it is non-stationary in
rats with disrupted baroreflex (Figure 3b), fluctuating about a much higher
average value of $178.31\pm 31.15$ mm Hg. After a period of $20$ days, average
blood pressure falls back to safe levels, $129.95\pm 9.32$ mm Hg, and
fluctuations are again stationary (Figure 3c), indicating that baroreflex is
recovered. In order to understand the underlying principles behind blood
pressure regulation and the sources of fluctuations in blood pressure levels
we give, in the next section, a precise, quantitative meaning to such
fluctuations with detrended fluctuations analysis (DFA).
Figure 3: 3 Fluctuations of arterial systolic blood pressure from a rat in the
control group. Blood pressure oscillates about safe, steady levels. 3 One day
after disrupting the pressure control loop with a surgical procedure, pressure
fluctuates in a non-stationary fashion, reaching dangerously high values. 3 As
a result of physiological adaptation, 20 days after surgical denervation of
baroreceptors average blood pressure returns to safe levels and fluctuations
are again stationary.
## 3 Fluctuation Analysis and Computer Modelling
We used detrended fluctuation analysis (DFA) [25, 23] to characterize long
term correlations in arterial systolic blood pressure. This method has been
successfully applied to analyze diverse non-stationary physiological signals
[37, 38, 25, 26, 27, 28] and we briefly describe it in the following: Let
$\\{P(t)\\}$ be the systolic blood pressure time series and $P_{ave}$ its time
average. Define the integrated time series $\\{y(t)\\}$ with
$y(t)=\sum_{k=1}^{t}(P(k)-P_{ave})$ (1)
Divide the integrated series in boxes of equal sizes $n$ and, for each box,
calculate the detrended profile subtracting from the original signal a
$l$-degree polynomial least-squares fit, $y^{l}_{n}(t)$ (In the following
DFA$-l$ will stand for detrended fluctuation analysis with $l$-degree
polynomials [39]). At each box of size $n$, calculate the fluctuation
$F(n)=\sqrt{\frac{1}{N}\sum_{t=1}^{N}\left(y(t)-y^{l}_{n}(t)\right)^{2}}.$ (2)
A power-law relation $F(n)\sim n^{\alpha}$ implies different correlation
patterns for different values of $\alpha$: When $0<\alpha<1/2$ the signal is
stationary and long-range anti-correlated, with $\alpha=1/2$ for a white noise
(and $\alpha=3/2$ for its integral, the Brownian motion), $\alpha>1/2$ for
long-range correlated signals, while the paradigmatic $1/f$ noise corresponds
to $\alpha=1$. This value of $\alpha$ also marks the borderline between
stationary and non-stationary behavior: For $\alpha\geq 1$ one has non-
stationary signals, with sub-diffusive ($\alpha<3/2$), diffusive
($\alpha=3/2$) or super diffusive ($\alpha>3/2$) behavior.
Figure 4: 4 Detrended fluctuation analysis of systolic blood pressure time
series for a typical rat in the control group. There is a crossover from non-
stationary to stationary, long-range correlated behavior at $n\approx 35$: For
short time scales we have $\alpha\approx 1.18$ and for large time scales
$\alpha\approx 0.93$. We apply DFA-1 (red crosses), DFA-2 (green times), DFA-3
(blue stars) and DFA-4 (pink empty squares) to the series and find that the
crossover always exists, although at different scales. We also applied DFA-1
to shuffled data (bottom curve), for which $\alpha\approx 0.5$ as for a white
noise. 4 DFA-1 for all rats in the control group. In both figures curves are
shifted vertically for better visibility. The curves $y=Ax^{\alpha}$ with
$\alpha=0.5$ (full black line), $0.9$ (dashed blue line) and $1.3$ (dashed
green line) are plotted as guides to the eye.
Results for a typical time series from the control group are depicted in
figure 4a. With DFA-1 we obtain a crossover from $\alpha=1.18$ to
$\alpha=0.93$ at $n\approx 35$. To check that the crossover is not an artifact
of a specific polynomial fit or non-stationarities [40, 24, 41], we also
employed DFA-2, DFA-3 and DFA-4 on the time series. For all orders $l$ there
is a crossover, although at slightly shifted time scales. We also show
surrogate data, where data points are randomly shuffled, and applied DFA-1 to
it (Figure 4a, bottom curve) to find that fluctuations scale with
$\alpha\approx 0.5$, as in a typical white noise. We depict in figure 4b
results for all rats in the control group, evidencing the same behavior in all
curves.
With sinoartic denervation stationarity is lost, as DFA indicates (Figure 5).
On pressure series from rats analyzed $24$ hours after denervation (acute
group) the crossover disappears, and the series is non-stationary at all time
scales ($\alpha\approx 1.25$), severely affecting homeostatic regulation of
blood pressure. Again we use higher order DFA check that no trends or non-
stationarities are shaping the results. The surrogate test is also shown at
the bottom curve of figure 5a.
It is interesting to note that the same change of behavior has been observed
in the DFA analysis of fluctuations in blood glucose levels of healthy humans
and in patients with Diabetes Mellitus [6]: The damaged insulin metabolism
controlling blood sugar levels is reflected in the disappearance of the
crossover observed in the DFA curves of healthy subjects. In other study [42],
this has been connected to the loss of short-term adaptability of the cerebral
blood flow control system of migraineurs patients.
Figure 5: 5 Detrended fluctuation analysis of systolic blood pressure time
series for a typical rat in the acute group. One day past surgical denervation
of baroreceptors, fluctuations in blood pressure are non-stationary at all
time scales and there is no crossover in the $F(n)$ curve. We apply DFA-1 (red
crosses), DFA-2 (green times), DFA-3 (blue stars) and DFA-4 (pink empty
squares) and find $\alpha\approx 1.25$, indicating a disruption of short-term
homeostatic control of blood pressure, or baroreflex. We also applied DFA-1 to
shuffled data (bottom curve,full squares), for which $\alpha\approx 0.5$ as
for white noise. 5 DFA-1 for all rats in the acute group. In both figures
curves are shifted vertically for better visibility. The curves
$y=Ax^{\alpha}$ with $\alpha=0.5$ (full black line) and $1.25$ (dashed green
line) are plotted as guides to the eye.
Twenty day past the denervation procedure, average blood pressure returns to
safe levels and stationarity is recovered (Figure 6): there is again a
crossover from non-stationary ($\alpha\approx 1.42$) to stationary
($\alpha\approx 0.99$) fluctuations, although at a larger timescale $n\approx
100$. Again we use DFA-$1$ up to DFA-$4$ to insure that the crossover is not
an artifact of nonstationarities (Figure 6a) and depict in figure 6b results
for each rat in the chronic group. The average blood pressure and the
stationary, long-range correlated fluctuations (as measured by $\alpha$ in the
region after the crossover) are statistically equivalent, as summarized in
figure 7. When comparing the exponent $\alpha$ in the control and chronic
groups with a paired t-test [43] we find statistical equivalence with p-value
$p=0.04$, the same test for average blood pressure giving $p=0.07$.
Figure 6: 6 Detrended fluctuation analysis of systolic blood pressure time
series for a typical rat in the chronic group: $20$ days after surgical
denervation, stationarity is recovered at large timescales and fluctuations
cross over from non-stationary ($\alpha\approx 1.42$) to stationary, long-
range correlated ($\alpha\approx 0.99$) at $n\approx 100$. This result
suggests that, although the fast response from the baroreceptors in the heart
is lost, physiological adaptation reestablishes homeostatic regulation. We
apply DFA-1 (red crosses), DFA-2 (green times), DFA-3 (blue stars) and DFA-4
(pink empty squares) to the series and find that the crossover always exists,
although at different scales. We also applied DFA-1 to shuffled data (bottom
curve), for which $\alpha\approx 0.5$ as in white noise. 6 DFA-1 for all rats
in the chronic group. In both figures curves are shifted vertically for better
visibility. The curves $y=Ax^{\alpha}$ with $\alpha=0.5$ (full black line),
$1.0$ (dashed blue line) and $1.4$ (dashed green line) are plotted as guides
to the eye. Figure 7: Results for the mean arterial systolic pressure (MASP)
(light gray bars) and the exponent $\alpha$ of long-term fluctuations averaged
over all rats in each group, showing homeostasis adaptation of mean pressure
and its fluctuations. In the control group (ctr), MASP have basal levels of
$116.55\pm 10.15$ mm Hg. One day after denervation (1d), MASP rises to
$178.31\pm 31.15$ mm Hg and, after 20 days (20d), get back to a basal level of
$129.95\pm 9.32$, closer to basal levels of the control group. The long-range
correlations observed both in control and chronic groups are statistically
equivalent ($\alpha=0.96\pm 0.05$ for the first and $\alpha=1.03\pm 0.05$ for
the latter group) with p-value $p=0.04$. Acute (1d) denervated rats have
nonstationary fluctuations, with $\alpha=1.23\pm 0.09$ on all timescales.
Baroreflex recovery can be associated to the adaptation of sensory neurons,
most possibly at the Nucleus Tractus Solitarius (NTS) [31, 32, 33] (the
mechanisms underlying this learning or synaptic plasticity are not completely
understood, but are already present in the adaptation of stretch sensitivity
in baroreceptors during the execution of simple tasks such as sitting or head
tilting for a reasonable amount of time [44]). In rats with intact
baroreceptors, baroreflex sensitivity can be evaluated, both with vasoactive
drugs or by spontaneous fluctuations of heart rate and blood pressure, by
means of the _Oxford method_ [45]: Beat-to-beat variation of systolic blood
pressure is plotted against variation of the heart rate at the subsequent beat
interval. The slope of a linear regression of this relation provides an index
of arterial baroreflex sensitivity (the same measure can be achieved by
correlating blood flow and heart rate variation and is known as the _Trieste
method_ [46]). These methods assume that the two signals are coupled, mostly
at oscillatory frequencies of $0.4$ Hz [47] and give a sigmoidal-like relation
between afferent nerve activity and blood pressure [48, 44]. In order to model
the action of both vagal and sympathetic branches on blood pressure we devise
a model of a Brownian particle forced by opposing nonlinear forces, an idea
briefly touched upon in [49]. Pressure information merges through afferents
and is integrated at the NTS, stimulating the vagal branch, which further
inhibits the sympathetic branch of the autonomous nerve system. This coupled
action can be modelled by sigmoidal-like pressure-activity curves, as depicted
in figure 8: at each time step, pressure changes due to the action of the
forces $f_{v}(p)$ and $f_{s}(p)$ as
$p(t+1)=p(t)+\left(f_{s}(p+\xi(t))-f_{v}(p+\xi(t))\right)$ (3)
where $\xi(t)$ represents the background noise integrated together with
afferent signal at the NTS, and the response curve $f_{k}(p)$ is modeled by
sigmoid-like curves [48, 44]:
$f_{k}(p)=A_{k}\pm\frac{1}{B_{k}+e^{-(p-thr_{k})}}$ (4)
where $k=s,v$ stands for sympathetic and vagal, respectively. In the first
case one subtracts and in the latter one adds the sigmoidal curve to the base
level of operation of each branch, called _tone_ , represented by $A_{k}$. The
parameters $thr_{s}$ and $thr_{v}$ give the pressure values for the optimal
response of each branch: the more different they are the larger is the region
where pressure fluctuates randomly. In order to understand the role of the
antagonistic regulation of average blood pressure in our model, we arbitrarily
set $A_{v}=0.1$, $A_{s}=1.0$, $B_{v}=1.1$ and $B_{s}=1.0$. 222One could
simplify the problem substituting the sigmoidal forces by step functions. We
chose, however, to keep the biologically motivated sigmoidal responses.
Figure 8: To model homeostatic blood pressure control we propose a simple
model of a Brownian particle driven by noisy sigmoidal antagonistic forces
$f_{s}$ (in red) and $f_{v}$ (in green). Pressure information is sent through
afferents to the NTS of the medulla, stimulating the vagal branch in a sigmoid
fashion (green curve). The otherwise constant action of the sympathetic branch
is modified by its vagal inhibition, resulting in the red curve depicted
above. The equilibrium condition $f_{s}=f_{v}$ sets the average pressure.
We analyze artificial systolic blood pressure series generated by such forced
random walk with DFA. After some transient behavior we store a time series
$\\{p(t)\\}$ with the same number of points as the experimental datasets,
$T=10^{4}$. We find, with this simple model, the same crossover observed in
the actual pressure time series of intact rats from the control group.
Moreover, keeping the same mechanism for pressure control, but changing the
sensitivity difference $thr_{s}-thr_{v}$, we are able to reproduce the
increase in the crossover scale observed in chronic SAD rats (figure 9).
Figure 9: Detrended fluctuation analysis (DFA) of the artificial systolic
blood pressure time series generated by the forced random walk model (equation
3). We plot 5 values of the sensitivity difference: $thr_{s}-thr_{v}=3$ (red
cross),$thr_{s}-thr_{v}=5$ (green times), $thr_{s}-thr_{v}=8$ (blue
stars),$thr_{s}-thr_{v}=11$ (pink empty squares) and $thr_{s}-thr_{v}=15$
(cyan full squares). The sensitivity difference increase as the same the
crossover scale. To guide the eye we show the curve with $\alpha=1.5$ (black
full line). A large threshold for the action of autonomous system forces also
means that more information (afferent signals) needs to be integrated at NTS
to respond to a change in blood pressure.
This result can be understood by the following simple argument: substituting
the sigmoidal curves by step functions, the problem reduces to one of a
particle in a confining square-well potential of width $L\approx
thr_{s}-thr_{v}$. The first-passage-time of the random walker to the walls of
the potential sets a timescale for a crossover between random, non-stationary
fluctuations and confined motion [50]. Thus, with an increase of the width of
the potential well one should expect an increase of the range of the scaling
region related to non-stationary fluctuations.
## 4 Discussion
We analyzed the dynamics of baroreflex, the negative feedback loop providing
reflex control of blood pressure by the autonomous nerve system, with
detrended fluctuation analysis of continuous measurements of arterial systolic
blood pressure. We report results of our experiments with three groups of
rats: a control group, another group where baroreflex is surgically disrupted
one day before measurements and a third one, again with baroreflex surgically
impaired but whose measurements were made $20$ days after clinical
intervention. With DFA, we find on intact rats from the control group a
crossover from non-stationary to stationary, long-range correlated
fluctuations in arterial systolic blood pressure time series. This crossover
indicates that baroreflex sets in for pressure control at a characteristic
timescale. One day after surgery one finds that the feedback control,
previously provided by baroreceptors, is impaired: no crossover is found, and
pressure fluctuations are non-stationary. Nevertheless, after $20$ days of
surgical intervention we find evidence for physiological adaptation, and
fluctuations scale in a fashion which is statistically similar to those from
the time series of rats in the control group, only with the crossover from
non-stationary to stationary fluctuations occurring at a larger timescale. We
also design a model for baroreflex which has the same dynamical behavior of
both normal and chronic SAD rats, qualitatively reproducing the crossover in
the scaling of fluctuations. The main feature of the model is its self-
inhibitory behavior, which illustrates the main principles underlying
homeostatic control in living systems, and has been observed at very different
organizational levels as an efficient mechanism for the maintenance of
regularity in a fluctuating environment.
We thank A.Tavares Costa Jr. for a critical reading of the manuscript and an
anonymous referee for suggestions significantly improving manuscript
presentation. This work is partially supported by Brazilian Agencies CNPq,
CAPES and FAPERJ.
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|
arxiv-papers
| 2009-08-25T13:12:10 |
2024-09-04T02:49:04.784741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.E.C. Galhardo, T.J.P.Penna, M. Argollo de Menezes and P.P.S. Soares",
"submitter": "Carlos Eduardo Galhardo",
"url": "https://arxiv.org/abs/0908.3614"
}
|
0908.3653
|
# Chaotic Transitions in Wall Following Robots ††thanks: The research
reported in this document/presentation was performed in connection with
contract/instrument W911QX-09-C-0055 with the U.S. Army Research Laboratory.
The views and conclusions contained in this document/presentation are those of
the authors and should not be interpreted as presenting the official policies
or position, either expressed or implied, of the U.S. Army Research Laboratory
or the U.S. Government unless so designated by other authorized documents.
Citation of manufacturer’s or trade names does not constitute an official
endorsement or approval of the use thereof. The U.S. Government is authorized
to reproduce and distribute reprints for Government purposes notwithstanding
any copyright notation hereon.
Harry W. Bullen IV and Priya Ranjan
###### Abstract
In this paper we examine how simple agents similar to Braitenberg vehicles can
exhibit chaotic movement patterns. The agents are wall following robots as
described by Steve Mesburger and Alfred Hubler in their paper “Chaos in Wall
Following Robots”. These agents uses a simple forward facing distance sensor,
with a limited field of view ($\Phi$) for navigation. An agent drives forward
at a constant velocity and uses the sensor to turn right when it is too close
to an object and left when it is too far away.
For a flat wall the agent stays a fixed distance from the wall and travels
along it, regardless of the sensor’s capabilities. But, if the wall represents
a periodic function, the agent drives on a periodic path when the sensor has a
narrow field of view. The agent’s trajectory transitions to chaos when the
sensor’s field of view is increased. Numerical experiments were performed with
square, triangle, and sawtooth waves for the wall, to find this pattern. The
bifurcations of the agents were analyzed, finding both border collision and
period doubling bifurcations.
Detailed experimental results will be reported in the final version.
## 1 Introduction
In this paper we look at how a simple wall following system leads to chaos,
when modeled as a mathematical system. This shows the importance of
understanding chaos in the creation of artificial intelligences for robots.
The model was originally described in the paper “Chaos in Wall Following
Robots” by Steve Mesburger and Alfred Hubler. A model for a simple wall
following robot was described and it was shown that when following a
sinusoidal wall the robot’s behavior would undergo a transition to chaos via
period doubling as the robot’s field of view increased [2]. In this paper we
extend this research by walls with discontinuous corners and show that the
robot still experiences a transition into chaos.
## 2 Model
The model for this simulation is small robot agent with an independent drive
train and a simple forward pointing boundary sensor [2]. An independent drive
train 111commonly found on tanks and the Mars Exploration Rovers allows the
vehicle to turn in place by driving the wheels on each side in opposite
directions. The boundary sensor is only able to minimum distance to the
boundary and not the shape or direction of the boundary.
Furthermore the sensor has field of view limited to $2\Phi$. Where $\Phi$ is
one of parameters varied in the experiment.
The most unrealistic part of this robot is that it can change its angular
velocity (the rate at which it is turning) instantaneously, allowing the robot
to switch from a hard left turn to a hard right turn and back. This can happen
when the robot’s sensor detects a wall at the edge of it’s vision and then
turns or moves away from the wall. This is not physically possible but as long
as the speed is low this approximates how real vehicles behave. For the
purpose of modeling the robot also maintains a constant velocity which we
convivially set to one.
As such the robots position can be represented by an $(x,y,\theta)$ vector
where $\theta$ is the counter-clockwise angle of the robot’s direction of
travel from the positive x-axis. The velocity of the robot is represented by
$v$ and the angular velocity is $\omega$. (this makes the last equations
somewhat trivial) The robot can the be represented as a series of deferential
equations [2].
$\displaystyle\frac{dx}{dt}$ $\displaystyle=v\cos\theta$ (1)
$\displaystyle\frac{dy}{dt}$ $\displaystyle=v\sin\theta$ (2)
$\displaystyle\frac{\delta\theta}{\delta t}$ $\displaystyle=\omega$ (3)
Also the model dose not enforce the boundary by preventing the robot from
crossing it. So it may be better to think of the boundary as the edge of a no
entry zone for the robot instead of a physical barrier.
Figure 1: Four common types of waves. The triangle, square, and sawtooth waves
were the focus of this study.
The boundary is modeled as one of the waves. The wave travels along the X axis
with an amplitude of one and a wavelength ($\lambda$) of 10. There waves are
studied in this paper, the square wave, the triangle wave and the sawtooth
wave. A sinusoidal wave has been studied previously [2].
These waves can be represented using the following equations.
$\displaystyle y_{sawtooth}(x)$ $\displaystyle=x-\lfloor x\rfloor$ (4)
$\displaystyle y_{triangle}(x)$ $\displaystyle=\arcsin(\sin(x))$ (5)
$\displaystyle y_{square}(x)$ $\displaystyle=-1^{\lfloor x\rfloor}$ (6)
However, for efficiency of the calculations, and because the boundaries the
vertical disjoint parts of the wave are part of the boundaries each wave was
represented as a series of points representing one period of the wave.
(Actually more than one period is represented to avoid issues with the robot
around the boundaries.) For example the square wave with $\lambda$ being the
wavelength was represented by the following series of points.
$(-.5\lambda,-1),(0,-1),(0,1),(.5\lambda,1),(.5\lambda,-1),(\lambda,-1)$
## 3 Procedures
A model for the robot was created using Mathematica™. Each step of the robot
was calculated using Runge-Kutta 4 with a $\Delta t=0.1$. Smaller values for
$\Delta t$ did not improve the results significantly.
The largest calculation of the problem it to determine the sensor value, or
the minimum distance to the boundary. For the each segment of the wall with
the robot’s range the closest point to the robot was calculated as well as any
intersections with the edges of the robot’s field of vision. The closest of
all these points was then used to calculate the sensor value.
$R$ represented the maximum range of the sensor. $r$ represented the range at
which the robot would move straight or how far it tried to say away from the
wall. A scaling value $\alpha$ of the angular velocity might also be
appropriate. Finally $\Phi$ was half of the robot’s angular vision, it could
range $(0,\pi]$. $R$ was taken to be 4, $r$ was taken to be 3, and $\alpha$
was taken to be $1.0$ .
The results were then graphed on X-Y, X-$\Theta$, Y-$\Theta$. With the X
values were taken modulo $\lambda$ (10) so that they coincided with the period
of the boundary.
### 3.1 Calculation of Sensor Value
In order calculate the closest point on the boundary to the robot agent with
in it’s sensor range several different distances were calculated and the
minimum of them was used. The first, trivially, distance is the maximum range
of the sensor $R$. Secondly the distance to each selected points on the
boundary that have been tested to be in the robot’s sensor range. These points
are divided into two further groups, points on the intersection of the
boundary and the edge of the robot’s vision, and the closest point on each
segment to the robot’s location. It can be shown that these points will always
contain the closest point to the robot within its vision.
The point on a segment closest to another point (the robot’s location in this
case) was found using parametric equations to reduce machine rounding errors
and minimize the calculation time.
$\frac{|(s_{2}-s_{1})\times(p-s_{1})|}{|s_{2}-s_{1}|^{2}}=u$ Then:
$\text{Closest Point}=\begin{cases}s_{1}&\text{if }u\leq 0\\\ s_{2}&\text{if
}u\geq 1\\\ (s_{2}-s_{1})u+s_{1}&\text{i}f0<u<1\end{cases}$
These points are then tested to make sure that they have the right angle from
the robot to be seen by its sensors and are close enough to the robot to be
seen. Point that are not in the robots sensor range are removed. Each segment
is then tested to see if any point on it intersects with either edge of the
robot’s sensor range. The intersection of segments $(\hat{p_{1}},\hat{p_{2}})$
and $(\hat{p_{3}},\hat{p_{4}})$ was then found by:[1].
$\displaystyle u$
$\displaystyle=\frac{(x_{4}-x_{3})(y_{1}-y_{3})-(y_{4}-y_{3})(x_{1}-x_{3})}{(y_{4}-y_{3})(x_{2}-x_{1})-(x_{4}-x_{3})(y_{2}-y_{1})}$
(7) intersection $\displaystyle=\hat{p_{1}}+u(\hat{p_{2}}-\hat{p_{1}})$ (8)
## 4 Numerical Results
The tree different boundary waves all followed the same pattern as $\Phi$
increased from $0$ to $\pi$ When $\Phi$ was close to zero the robot tends to
hug the wall very closely. In places where the is a long straight wall with no
obstacles you can see the robot follow the wall all along it’s path. The
figures here were made by taking the point form each time step and plotting
$(x\mod\lambda,y)$ where $\lambda=10$ in this case.
Figure 2: Plot of the robot following a sawtooth wave, when $\Phi=1/64\pi$.
Notice how the robot is willing to follow the straight segment as long as it
goes.
You also see the robot tends to head directly for corners, just narrowly
passing them and then quickly stabilizing on the segment. Furthermore as $R$
remains constant decreasing $r$ causes the robot to follow the wall more
closely. However when $r=R$ the robot no longer turns counter clockwise.
Similarly when $r=0$ the robot will only turn counter clockwise. Both of these
cases are less interesting as the robot tends to either go in circles or drive
off in a straight line.
Figure 3: Plots of the xy movement when the sensor range $\Phi=1/64\pi$,
$R=4$ and $r$ in the range $[1.5,3.5]$.
As the value of $\Phi$ increases robot’s path becomes slightly chaotic. While
the robot’s path remains in a very narrow, and stable channel within that
channel the path is chaotic. With the sawtooth wave this behavior becomes most
pronounced when $\Phi\approx 19\pi/64$ .
Figure 4: Path of robot against the sawtooth wave show in experience chaos
even at low values of $\Phi$.
Figure 5: Cobweb diagram of the robot’s $y$ position when as it passes a
point where $x\mod\lambda=0$. The plot show $\log(y+2.65)$ as this makes the
lower left hand section of the graph easier to read.
Verhulst diagrams or more commonly cobweb plots were created of a single path
to better view robot’s behavior. To do this a point from each period of the
robot’s path was selected. For simplicity the points were when the
x-coordinate was a multiple of the wavelength of the boundary. Then either the
$y$ or $\theta$ values could be used to make the Verhulst diagram. The $y$
value was chosen arbitrary.
In fact it was found that the robot tends to exhibit this chaotic behavior
even at low $\Phi$ where it appears stable. This may be a property of having
sharp corners in the boundary wave that are not present in a sinusoidal wave.
Figure 6: A Bifurcation diagram of sawtooth wall showing a transition to
chaos through period doubling
For all three waves the chaotic patterns appeared to die down around
$\Phi\approx.5\pi$. Then after that all three waveforms undergo a transition
to chaos. To view the transition bifurcation diagrams were created for each
type of wall. Like the Verhulst diagrams the value of $y$ at the when $x$ is a
multiple of $\lambda(10)$ were taken from each simulation of the robot. The
first 30% of the values were discarded to ensure that the system and converged
to it’s attractor.
The sawtooth boundary produced an period doubling bifurcation diagram. Where
the full transition to chaos occurred when $\Phi\approx 121/128\pi$ However an
exception was also found when $\Phi=15/16\pi$. The robot moved in the negative
direction by turning counter clockwise in ellipses. It may be that this is
another attractor for the sawtooth boundary regardless of the $\Phi$ value.
Figure 7: A plot of the robot’s movement backwards along a sawtooth boundary.
When the robot was following a triangle wave it was found that the $\theta$
values provided a better bifurcation diagram than the $y$ values. The triangle
wave boundary first cause chaotic behavior around $\Phi\approx 3/16\pi$ The
sharp disconnect when $\Phi\approx 3/4\pi$ in the bifurcation diagram appears
to be an indicator of a second attractor.
Figure 8: A Bifurcation diagram of triangle wall
The square boundary gave results looking more similar to that of the triangle
wave then the sawtooth wave. It’s first expression of chaos occurs around
$\Phi\approx\pi/4$ Also around $\Phi\approx 49/64\pi$ it too has a disconnect,
with initial starting values of $(0,-2,0)$. In order to make the bifurcation
diagram simulations were run for each $\Phi$ value with a starting points at
$(0,y,0)$ where for every half unit between $-1$ and $-5$. These two strange
attractors actually curl around each other when visualized in 3d.
Figure 9: A Bifurcation diagram of Square wall showing a very sharp
transition around $\Phi=\frac{3\pi}{4}\approx 2.3$
Figure 10: A Poincare Plot of the y and $\theta$ values when x is multiple of
10.
## 5 Conclusion
In this work it was show that the behavior of a simple control system can
under go a chaotic transition from a change in the environment. A consistent
pattern was still observable through out these changes.
Further improvements to the algorithm could be made by creating a spatial hash
table of segments that maps the location of the robots to a list of segments
that are with in the range of the robot. This would reduce the number of tests
the simulation performs by as much as 50%.
## 6 Future Work
Further work beyond optimizations of the algorithm involve adding additional
robot agents for make a swarm. The swarm of wall following robots would
interact with each other and the wall, which leads to a rich dynamical system.
Research on the conditions when the swarm behaved chaotically and when it did
not would be done.
* •
Build a swarm of wall following robots
* •
Experiment with different cognitive behaviors in the agents
* •
Use agents with differing abilities in both sensors and cognition
* •
Combine real and virtual robot agents in a simulation
## References
* [1] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithm; 2nd Ed. McGraw Hill Book Company, 2001.
* [2] Steve Mesburger and Alfred Hubler. Chaos in wall following robots. September 2006.
|
arxiv-papers
| 2009-08-25T18:03:22 |
2024-09-04T02:49:04.792020
|
{
"license": "Public Domain",
"authors": "Harry W. Bullen IV and Priya Ranjan",
"submitter": "Priya Ranjan",
"url": "https://arxiv.org/abs/0908.3653"
}
|
0908.3733
|
# Induced interactions and the superfluid transition temperature in a three-
component Fermi gas
J.-P. Martikainen NORDITA, Roslagstullsbacken 21, 106 91 Stockholm, Sweden
J. J. Kinnunen Department of Applied Physics, Helsinki University of
Technology, P.O. Box 5100, 02015 HUT, Finland P. Törmä Department of Applied
Physics, Helsinki University of Technology, P.O. Box 5100, 02015 HUT, Finland
C. J. Pethick NORDITA, Roslagstullsbacken 21, 106 91 Stockholm, Sweden The
Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17,
DK-2100 Copenhagen Ø, Denmark
###### Abstract
We study many-body contributions to the effective interaction between fermions
in a three-component Fermi mixture. We find that effective interactions
induced by the third component can lead to a phase diagram different from that
predicted if interactions with the third component are neglected. As a result,
in a confining potential a superfluid shell structure can arise even for equal
populations of the components. We also find a critical temperature for the BCS
transition in a ${}^{6}{\rm Li}$ mixture which can deviate strongly from the
one in a weakly interacting two-component system.
###### pacs:
03.75.Ss, 71.10.-w, 03.65.-w
By using Feshbach resonances to change the effective interaction between
ultracold atoms several groups have probed the crossover from the Bardeen-
Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate of molecules
Jochim et al. (2003); Regal et al. (2003a); Cubizolles et al. (2003); Regal et
al. (2003b); Zwierlein et al. (2004); Chin et al. (2004); Kinast et al.
(2004); Zwierlein et al. (2005, 2006); Partridge et al. (2006). Studies of the
crossover have provided important insights into fermionic superfluids around
the unitarity limit of strong interactions. Importantly, it has become
experimentally feasible to study also more complicated mixtures than Fermi
gases with two different atomic internal states. Bose-Fermi mixtures Roati et
al. (2002); Ospelkaus et al. (2006) and Bose-Einstein condensates with many
components have been created using many different setups Myatt et al. (1997);
Stenger et al. (1998). Also, heteronuclear Fermi-Fermi mixtures Wille et al.
(2008); Spiegelhalder et al. (2009) and even heteronuclear Fermi-Fermi-Bose
mixtures Taglieber et al. (2008) have been recently demonstrated. In yet
another breakthrough a three-component Fermi mixture of atoms in the three
lowest hyperfine states of ${}^{6}{\rm Li}$ Ottenstein et al. (2008); Huckans
et al. (2009) has also been demonstrated. Such multi-component systems have
some intriguing similarities with quark matter counterparts where color
superconductivity may appear Alford et al. (2008).
The purpose of this Letter is to explore how the induced interactions due to
the third component modify the expected behavior of three-component mixtures.
This is important because, depending on parameters, the many-body effects can
change the effective interaction between atoms substantially and on occasion
even change the relative magnitudes of couplings between different components.
Such changes imply that phase diagrams predicted using only two-body
scattering properties can be incorrect. Also, even when the corrections to the
effective interactions are weak, they can cause large changes to the critical
temperature for the BCS transition. Indeed, for a two component system Gorkov
and Melik-Barkhudarov (GM) showed Gorkov and Melik-Barkhudarov (1961) that the
perturbative correction to the effective interaction can reduce the critical
temperature by a constant factor of $(4e)^{1/3}\approx 2.22$ in the weak-
coupling limit. Also in spin-density imbalanced systems such corrections have
been shown to have a considerable effect Gubbels et al. (2008). Here we will
analyze the effects of analogous corrections on the three component system and
find important changes to the GM result. As an interesting consequence of
these many-body corrections we predict in a spatially varying confining
potential (typically harmonic trap) the appearance of superfluid shell
structures even in the absence of population imbalance (polarization) of the
components. These shell structures are due to many-body effects only and
therefore fundamentally different from earlier predictions of shell structures
due to population, mass, or trapping potential imbalance Paananen et al.
(2007); Lin et al. (2006) We also point out that many body effects due to the
third component provide a new way to tune the effective interaction between
the two other fermions and that this contribution can dominate over the usual
GM contribution.
Earlier, intriquing results have been found experimentally for the critical
temperature of iron-based multiband superconductors Terashima et al. (2009a)
and degenerate three-component Fermi gases have been studied theoretically in
a lattice Honerkamp and Hofstetter (2004); Rapp et al. (2008). Furthermore,
pairing Paananen et al. (2006, 2007); Bedaque et al. (2009), stability Blume
et al. (2008), and breached pairing Errea et al. (2009) have recently been
studied in a three-component fermionic mixtures. However, these theoretical
approaches did not consider situations directly relevant to ongoing
experiments and also did not study how the many-body effects due to the
presence of the third component influence the properties of the other two
components. Some aspects of the many flavor problem were discussed by
Heiselberg et al. Heiselberg et al. (2000).
|
---|---
Figure 1: Diagrams of second order in the interactions for the induced
interaction between components $1$ and $2$. Solid lines represent atoms and
the dashed lines interactions between them, and numbers indicate the fermionic
component. The interactions are taken to be antisymmetrized with respect to
interchange of spins and consequently interactions between atoms of the same
species are absent.
The relevant second order diagrams which give rise to induced interactions
between fermions of type $1$ and $2$ are shown in Fig. 1. In these diagrams
the arrows are the component propagators and dashed line is a contact
interaction with strength $U_{\alpha\beta}$ between components labeled by
$\alpha,\beta\in\\{1,2,3\\}$ ($\alpha\neq\beta$). These couplings can be
expressed in terms of the scattering lengths $a_{\alpha\beta}$ through
$U_{\alpha\beta}=2\pi\hbar^{2}a_{\alpha\beta}/m_{\alpha\beta}$, where
$m_{\alpha\beta}=(1/m_{\alpha}+1/m_{\beta})^{-1}$ is the reduced mass. Of the
diagrams shown, the diagram (a) is relevant in the case of a two component
system with a contact interaction between unlike fermions Baranov et al.
(2008) and for equal mass fermions gives rise to the GM correction mentioned
earlier. In a three-component system the diagram (b) describes the induced
effect of the third component. Similar loop diagrams with the mediating
fermion in component $1$ or $2$ are forbidden in the s-wave scattering channel
for symmetry reasons. More formally the diagram (a) indicates the induced
interaction
$V^{G}({\bf p},{\bf p^{\prime}})=-U_{12}^{2}\sum_{\bf
k}\frac{f\left[\xi_{1}({\bf k}+{\bf q}/2)\right]-f\left[\xi_{2}({\bf k}-{\bf
q}/2)\right]}{\xi_{1}({\bf k}+{\bf q}/2)-\xi_{2}({\bf k}-{\bf q}/2)},$
where ${\bf q}={\bf p}+{\bf p^{\prime}}$ and (b) describes the induced
interaction
$V^{3c}({\bf p},{\bf p^{\prime}})=U_{13}U_{23}\sum_{\bf
k}\frac{f\left[\xi_{3}({\bf k}+{\bf q^{\prime}}/2)\right]-f\left[\xi_{3}({\bf
k}-{\bf q^{\prime}}/2)\right]}{\xi_{3}({\bf k}+{\bf
q^{\prime}}/2)-\xi_{3}({\bf k}-{\bf q^{\prime}}/2)},$
with ${\bf q^{\prime}}={\bf p}-{\bf p^{\prime}}$. In these formulas
$\xi_{\alpha}({\bf k})=\hbar^{2}k^{2}/2m_{\alpha}-\mu_{\alpha}$ are the free
atom dispersion relations and $f(\epsilon)$ is the Fermi distribution.
In the weak coupling limit the scattering processes around the Fermi surfaces
dominate and to find the effective coupling the induced interactions are
averaged over the Fermi surfaces. In this way we find that the effective
coupling between fermions of types $1$ and $2$ becomes
$\begin{split}&U_{12}^{\rm
eff}=\frac{4\pi\hbar^{2}a_{12}}{m_{1}}\left\\{1+\frac{2}{\pi}\left[a_{12}k_{F,1}F\left(1\right)\right.\right.\\\
&\left.\left.-\frac{a_{13}a_{23}}{a_{12}}\frac{(m_{3}+m_{1})^{2}}{4m_{1}m_{3}}\left(\frac{k_{F,3}^{3}}{k_{F,1}^{2}}\right)F\left(\frac{k_{F,1}}{k_{F,3}}\right)\right]\right\\},\end{split}$
(1)
where $k_{F,\alpha}$ is the Fermi wavevector for the component $\alpha$ and we
have assumed that fermions $1$ and $2$ both have a mass $m_{1}$ while the
third component has a mass $m_{3}$. The function $F(y)$ is given by the
integral
$F(y)=\int_{0}^{y}dw2w\left[\frac{1}{2}+\frac{(1-w^{2})}{4w}\ln\left(\frac{|1+w|}{|1-w|}\right)\right],$
(2)
whose analytical solution is given by
$F(y)=\frac{1}{6}\left[-y\left(y^{2}-3\right)\log\left|\frac{y+1}{y-1}\right|+2\left(y^{2}+\log\left|y^{2}-1\right|\right)\right]$
The effective interactions in other channels can be found in the same way.
In Eq. (1) the first term describes the two-body scattering in the absence of
Fermi seas, the second term gives rise to the GM correction, and the third
term describes the effect of the interactions with the third component and its
Fermi sea. The correction due to the second term always suppresses the
critical temperature for the BCS transition. However, the last term is
proportional to the product $a_{23}a_{13}$ and can have either sign.
Therefore, the presence of the third component can either suppress or enhance
the critical temperature.
Since three component systems have been demonstrated using ${}^{6}{\rm Li}$
atoms, let us now investigate these many-body effects using the coupled
channel scattering data for ${}^{6}{\rm Li}$ Jul . In Fig. 2 we show the
scattering lengths between different components of the ${}^{6}{\rm Li}$
mixture ($|1\rangle$, $|2\rangle$, $|3\rangle$ refer to the states
$|F,m_{F}\rangle=|1/2,1/2\rangle$, $|1/2,-1/2\rangle$, $|3/2,-3/2\rangle$,
respectively).
---
Figure 2: Scattering lengths in units of the Bohr radius between ${}^{6}{\rm
Li}$ atoms as a function of magnetic field. (The figure is taken from Ref. Jul
.)
It can be seen that, in the absence of many-body corrections, the $1-3$
channel has the most negative scattering length for weaker magnetic fields,
while at magnetic fields above the Feshbach resonances the $1-2$ channel
eventually becomes dominant. In the simple mean-field picture one would infer
that these channels are also the ones with highest critical temperatures.
However, when we include induced interactions, density dependencies appear in
the effective coupling strengths and change the simple picture in which the
third component is neglected.
Let us first explore the case where all ${}^{6}{\rm Li}$ components have the
same density. In Fig. 3 we show the dominant coupling channel in the magnetic
field–density plane.
|
---|---
Figure 3: The dominant interaction channel including induced interactions for
an equal-density ${}^{6}{\rm Li}$ mixture. The solid black lines indicate
phase boundaries. Below the (red) dashed line both $|k_{F}a_{13}|$ and
$|k_{F}^{2}a_{23}a_{12}|$ are less than one, while below the (red) dot-dashed
line both $|k_{F}a_{12}|$ and $|k_{F}^{2}a_{23}a_{13}|$ are less than one. In
(a) we show the interesting regions at low magnetic fields and in (b) the
behavior at higher magnetic fields.
It can be seen that below the Feshbach resonance the $1-3$ channel dominates
for smaller densities, but for densities higher than about $2\cdot
10^{14}/{\rm cm^{3}}$ there is a possibility that the $1-2$ channel becomes
dominant. At higher magnetic fields we find a possibility of dominant $1-3$
coupling in the region where the scattering lengths would predict the $1-2$
channel. In experiments the atoms are trapped, and applying a local density
approximation (which has been sufficient to describe many experiments) with
our results suggests the interesting possibility of different superfluid
phases appearing in different parts of the cloud even in the absence of
polarization or unequal trapping potentials/masses Paananen et al. (2007).
This possibility is a many-body effect caused by the induced interactions
only; for a balanced system at zero temperature, simple mean field theory
would not predict the shell structures that arise from the density dependence
of the GM correction as shown here.
It is important to investigate what these results imply for the critical
temperature. In attractive dilute Fermi gases, the critical temperature for
the BCS transition in the weak coupling limit is
$k_{B}T_{c}/\epsilon_{F}\propto\exp[-\pi/(2k_{F}|a^{\rm eff}|)]$ where $a^{\rm
eff}=U^{\rm eff}N(\epsilon_{F})\pi/(2k_{F})$ is the effective scattering
length and $N(\epsilon_{F})$ is the density of states at the Fermi level. We
now use this result to estimate the many-body correction to $T_{c}$ in a
three-component ${}^{6}{\rm Li}$ system. For simplicity we use the above
functional dependence in all regions where the coupling is attractive, but
indicate the regions where $|k_{F}a|>1$ in the figures. In those regions the
weak-coupling formula is only suggestive. In Fig. 4 we show the fraction
$T_{c}/T_{c,0}$ for the equal density ${}^{6}{\rm Li}$ mixture ($T_{c,0}$ is
the critical temperature in the absence of induced interactions). As can be
seen, the correction to $T_{c}$ is often very different from the
$1/2.22\approx 0.45$ GM result and shows a non-trivial behavior as a function
of the magnetic field and density due to complicated variation of the
scattering lengths. This also makes it possible that the pairing channel is
changed due to many-body effects. At high fields above the Feshbach resonances
it is possible that the critical temperature is enhanced since the effective
scattering length there becomes more negative due to induced interactions.
However, this happens in the region of stronger interactions where our results
are not necessarily quantitatively accurate.
---
Figure 4: The highest fraction $T_{c}/T_{c,0}$ of the critical temperatures
with and without induced effects for the equal-density ${}^{6}{\rm Li}$
mixture. The fraction is only computed in the regions where the dominant
effective interaction is attractive and set to zero elsewhere. Below the
yellow line $|k_{F}a^{\rm eff}|<1$. Regions where the optimal pairing channel
is changed are visible as kinks in the fraction $T_{c}/T_{c,0}$. The inset
shows a close-up into the region of high magnetic fields and low densities.
In Fig. 5 we demonstrate another possibility for changing the critical
temperature: the use of density imbalance. In Fig. 5 (a) we show an example of
how $T_{c}$ in the $1-3$ channel is changed as the density of the component
$2$ is varied. It is again clear that the result deviates substantially from
the GM prediction, but the $T_{c}$ is nevertheless suppressed by the component
not involved in pairing. In Fig. 5 (b) we show the similar result in the $1-2$
channel which dominates at higher magnetic fields. Due to the different
behavior of the scattering lengths, here the induced interactions can act to
enhance $T_{c}$ above the value predicted by the usual mean-field theory.
|
---|---
Figure 5: The fraction $T_{c}/T_{c,0}$ of the critical temperatures with and
without induced effects for the ${}^{6}{\rm Li}$ mixture. In (a) we show an
example of how the critical temperature is changed in the $1-3$ channel when
the density of the component $2$ is varied while in (b) we show the same for
the $1-2$ channel when the density of the component $3$ is varied. We only
focus on those magnetic fields where the pairing channel in question is the
dominant one.
Finally, since heteronuclear fermionic mixtures are experimentally feasible
Wille et al. (2008); Taglieber et al. (2008) let us briefly discuss what our
results imply in that case. A mass imbalance can be realized if the third
component is a different isotope, but also if the third component experiences
an optical lattice which changes its effective mass. In the latter case, for
the formulas derived here to be valid, the filling fraction of all the
components should be much less than one. For higher filling fractions the
Fermi surface is no longer spherical and the result would change considerably
Kim et al. (2009). We focus on a scenario with equal masses for atoms of type
$1$ and $2$, since it is known that unequal mass of the interacting fermions
suppresses the critical temperature Baranov et al. (2008); Paananen et al.
(2007) and for this reason the equal mass superfluidity appears more generic.
The effective interaction between $1$ and $2$ is given by Eq. (1). Note that
since $(m_{3}+m_{1})^{2}/4m_{1}m_{3}>1$, induced interactions become
relatively stronger in an unequal mass mixture and mass imbalance can be used
to enhance the role of many-body corrections. If
$b=\left(a_{12}^{2}/|a_{13}a_{23}|\right)\left(k_{F,1}/k_{F,3}\right)^{3}F(1)/F(k_{F,1}/k_{F,3})$
and $b>1$, the contribution of the third component to the induced interaction
becomes larger than the GM contribution when
$m_{3}/m_{1}>(2b-1)+2\sqrt{b(b-1)}$ or when
$m_{3}/m_{1}<(2b-1)-2\sqrt{b(b-1)}$. If $b<1$ the contribution due to the
third component is dominant for all mass ratios. However, for mass ratios
larger than about $13.6$ other physics can come into play since then weakly
bound diatomic molecules might become collisionally unstable Petrov et al.
(2009).
In this Letter we have explored the induced interactions and their role in the
BCS pairing in a three-component Fermi mixture. We found striking differences
from physics ignoring these many-body corrections. In particular, we found
that when the induced interactions are taken into account, the phase-diagram
can change drastically, that shell structures in traps can appear even without
number,mass, or trap imbalance, and that the critical temperature for the BCS
transition is strongly dependent on the induced interactions in the three-
component systems.
We thank Academy of Finland (Projects No. 213362, 217041, 217043, and 210953).
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|
arxiv-papers
| 2009-08-26T06:36:36 |
2024-09-04T02:49:04.798201
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.-P. Martikainen, J. J. Kinnunen, P. Torma, C. J. Pethick",
"submitter": "Jani-Petri Martikainen",
"url": "https://arxiv.org/abs/0908.3733"
}
|
0908.3749
|
# Multiple-relaxation-time lattice Boltzmann model for compressible fluids
Feng Chen1, Aiguo Xu2111 Corresponding author. E-mail: Xu_Aiguo@iapcm.ac.cn,
Guangcai Zhang2, Yingjun Li1 1, China University of Mining and Technology
(Beijing), Beijing 100083
2, National Key Laboratory of Computational Physics,
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009-26,
Beijing 100088, P.R.China
###### Abstract
We present an energy-conserving multiple-relaxation-time finite difference
lattice Boltzmann model for compressible flows. This model is based on a
16-discrete-velocity model. The collision step is first calculated in the
moment space and then mapped back to the velocity space. The moment space and
corresponding transformation matrix are constructed according to the group
representation theory. Equilibria of the nonconserved moments are chosen
according to the need of recovering compressible Navier-Stokes equations
through the Chapman-Enskog expansion. Numerical experiments showed that
compressible flows with strong shocks can be well simulated by the present
model. The used benchmark tests include (i) shock tubes, such as the Sod, Lax,
Colella explosion wave, collision of two strong shocks and a new shock tube
with high mach number, (ii) regular and Mach shock reflections, (iii) shock
wave reaction on cylindrical bubble problems, and (iv)Couette flow. The new
model works for both low and high speeds compressible flows. It contains more
physical information and has better numerical stability and accuracy than its
single-relaxation-time version.
###### pacs:
47.11.-j, 51.10.+y, 05.20.Dd
Keywords:lattice Boltzmann method; compressible flows; multiple-relaxation-
time; von Neumann stability analysis
## I Introduction
The Lattice Boltzmann (LB) method is an innovative numerical scheme originated
from Lattice Gas Automata (LGA)1 and aim to simulate various hydrodynamics2 .
The LB method was introduced to overcome some serious deficiencies of LGA,
such as intrinsic noise, limited values of transport coefficients, non-
Galilean invariance, and implementation difficulty in three dimensions. In the
past two decades, most of the LB models are based on the famous Bhatnagar-
Gross-Krook (BGK) approximation5 where a Single Relaxation Time (SRT) is
used. Due to its validity and simplicity, the SRT LB method has been widely
used to simulate various fluid flow problems, such as the multiphase
flowShanChen ; Swift ; XGL1 ; XGL2 ; XGL6 ; XSB ; SS2007 ; Xu2009 ,
magnetohydrodynamicsPRL1991 ; 7 ; 8 ; PRA1991 ; PRE2002 ; PRE2004 ; CiCP2008 ,
flows through porous media9 ; 10 ; 10a and thermal fluid dynamics11 ; 11a ;
12 , etc.
However, the extreme simplicity of the SRT leads also to some constraints for
the SRT LB model. For example, the simulation will be unstable when the
relaxation time $\tau$ is close to 0.5, the model works only for low Mach
number flows. One possible remedy is to use the Multiple Relaxation Time (MRT)
methodHSB ; 13 . In real fluid the equilibrating rates of mass, momentum,
energy, etc. are generally different. This difference can be manifested by the
non-unique adjustable parameters in the MRT LB model. In contrast to the SRT
model, the MRT version has much more adjustable parameters and degrees of
freedom. The relaxation rates of various processes owing to particle
collisions may be adjusted independently. The main strategy of the MRT LB
scheme is that the collision step is first calculated in the moment space and
then mapped back to the velocity space. The advection step is still computed
in the velocity space. In many cases, it has been shown by Luo, et al17 ; 19
that the MRT LB model has better numerical stability.
Recently, the MRT LB method has attracted considerable interest and much
progress has been achieved. For example, MRT models for viscoelastic fluids15
; 16 ; 16a , multiphase flows20 ; 200 , flow with free surfaces18 , etc. were
developed; optimal boundary condition for MRT LB was composed14 . To simulate
system with temperature field, Luo, et al.19 suggested a hybrid thermal MRT
LB model. These models work only for nearly incompressible fluids with very
low Mach number.
LB community has long been attempting to construct models for compressible
fluidsAlexander1 ; li1 ; sun1 ; yan1 . Alexander and Chen et al.Alexander1
constructed a model where the sound speed is adjustable so that the Mach
number can be enhanced. Li, et al.li1 gave a model by reforming the velocity
space. Sun, et al.sun1 formulated adaptive LB models where the particle
velocities are determined by the mean velocity and internal energy. Yan, et
al. yan1 proposed three-speed-three-energy-level models. Besides the standard
LB mentioned above, some researchers have also tried to develop Finite
Difference (FD) LB for compressible fluids20a ; 21a ; 2a , but in the real
simulations the accessible Mach number is still not large. The model
introduced by Kataoka and Tsutahara21a uses only sixteen discrete velocities
and hence has a high computational efficiency.
The low-Mach number constraint is generally related to a numerical stability
problem. The latter has been partly addressed by a number of techniques, such
as the entropic method16+ ; 17+ , the fix-up scheme16+ ; 18+ , Flux-
limitersSofonea1 and dissipation43 ; Brownlee1 techniques. In existing SRT
models, it seems that the most effective solution to overcome the low Mach
number constraint is to introduce artificial viscosity. But with the
artificial viscosity, some fundamental kinetics are not very clear. In many
cases, the MRT formulation has been shown to offer improved numerical
stability, and provide additional physics. In this paper we present an energy-
conserving multiple relaxation time finite difference lattice Boltzmann model
for compressible flows with high Mach number. This model is based on the one
proposed by Kataoka and Tsutahara21a . The moment space and transformation
matrix are constructed according to the group representation theory.
Equilibria of the nonconserved moments in the moment space are chosen when
recovering compressible Navier-Stokes (NS) equations through the Chapman-
Enskog (CE) expansion.
This paper is organized as follows. In Sect. II a brief review to the MRT LB
model is presented. In Sect. III the new model is constructed. The von Neumann
stability analysis is given in Sect. IV. Section V shows the numerical tests
and some simulation results. Section VI provides a summary and concludes the
paper.
## II Brief review of the MRT LB model
The evolution of the distribution function $f_{i}$ for the particle velocity
$v_{i}$ is governed by the following equation:
$\frac{\partial f_{i}}{\partial t}+v_{i\alpha}\frac{\partial f_{i}}{\partial
x_{\alpha}}=-\mathbf{S}_{ik}\left[f_{k}-f_{k}^{eq}\right]\text{,}$ (1)
where $f_{i}$ ($f_{i}^{eq}$) is the particle (equilibrium) distribution
function, $v_{i}$ represents a group of particle velocities, subscript $i$
indicates the particle’s direction, $i=1,\ldots,N$, $N$ is the number of
discrete velocities, the subscript $\alpha$ indicates $x$ or $y$ component,
$\mathbf{S}$ is the collision matrix. The equation reduces to the usual
lattice BGK equation if all the relaxation parameters are set to be a single
relaxation time $\tau$, namely $\mathbf{S}=\frac{1}{\tau}\mathbf{I}$, where
$\mathbf{I}$ is the identity matrix.
The discrete (equilibrium) distribution function $f_{i}$ ($f_{i}^{eq}$) in Eq.
(1) can be listed with the following matrixes:
$\mathbf{f}=\left(f_{1},f_{2},\cdots,f_{N}\right)^{T}\text{,}$ (2a)
$\mathbf{f}^{eq}=\left(f_{1}^{eq},f_{2}^{eq},\cdots,f_{N}^{eq}\right)^{T}\text{,}$
(2b) where $T$ is the transpose operator.
Given a set of discrete velocities $v_{i}$, and corresponding distribution
functions $f_{i}$, we can get a velocity space $S^{V}$, spanned by discrete
velocities $v_{i}$, and a moment space $S^{M}$, spanned by moments of particle
distribution function $f_{i}$, where $i=1,\cdots,N$. Similarly, we also
express the moments of distribution function with the column matrix:
$\hat{\mathbf{f}}=\left(\hat{f}_{1},\hat{f}_{2},\cdots,\hat{f}_{N}\right)^{T}$,
where $\hat{f}_{i}=m_{ij}f_{j}$, $m_{ij}$ is an element of the matrix
$\mathbf{M}$ and is a polynomial of discrete velocities. Obviously, the
moments are simply linear combination of distribution functions$\ f_{i}$, and
the mapping between moment space and velocity space is defined by the linear
transformation $\mathbf{M}$, i.e., $\hat{\mathbf{f}}=\mathbf{Mf}$,
$\mathbf{f=M}^{-1}\hat{\mathbf{f}}$, where
$\mathbf{M}=\left(m_{1},m_{2},\cdots,m_{N}\right)^{T},m_{i}=(m_{i1},m_{i2},\cdots,m_{iN})$.
The LB simulation consists of two steps: the collision step and the advection
one. In the MRT LB method, the advection step is computed in the velocity
space. The collision step is first calculated in the moment space and then
mapped to the velocity space. So, the MRT LB equation can be described as:
$\frac{\partial f_{i}}{\partial t}+v_{i\alpha}\frac{\partial f_{i}}{\partial
x_{\alpha}}=-\mathbf{M}_{il}^{-1}\hat{\mathbf{S}}_{lk}(\hat{f}_{k}-\hat{f}_{k}^{eq})\text{,}$
(3)
where $\hat{\mathbf{S}}=\mathbf{MSM}^{-1}=diag(s_{1},s_{2},\cdots,s_{N})$ is
the diagonal relaxation matrix. $\hat{f}_{i}^{eq}$ is the equilibrium value of
the moment $\hat{f}_{i}$. The moments can be divided into two groups. The
first group consists of the moments locally conserved in the collision
process, i.e. $\hat{f}_{i}=\hat{f}_{i}^{eq}$. The second group consists of the
moments not conserved, i.e. $\hat{f}_{i}\neq\hat{f}_{i}^{eq}$. The equilibrium
$\hat{f}_{i}^{eq}$ is a function of conserved moments.
## III Energy-conserving MRT LB model
We use the two-dimensional discrete velocity model by Kataoka and Tsutahara21a
(see Fig. 1). It can be expressed as:
$\left(v_{i1},v_{i2}\right)=\left\\{\begin{array}[]{cc}\mathbf{cyc}:\left(\pm
1,0\right)\text{,}&\text{for }1\leq i\leq 4\text{,}\\\ \mathbf{cyc}:\left(\pm
6,0\right)\text{,}&\text{for }5\leq i\leq 8\text{,}\\\ \sqrt{2}\left(\pm 1,\pm
1\right)\text{,}&\text{for }9\leq i\leq 12\text{,}\\\
\frac{3}{\sqrt{2}}\left(\pm 1,\pm 1\right)\text{,}&\text{for }13\leq i\leq
16\text{,}\end{array}\right.$ (4)
where cyc indicates the cyclic permutation.
Figure 1: Distribution of $\mathbf{v}_{i\alpha}$ for the discrete velocity
model.
### III.1 Construction of transformation matrix $\mathbf{M}$
The transformation matrix $\mathbf{M}$ is constructed according to the
irreducible representations of SO(2) group:
$\displaystyle 1\text{,}$ $\displaystyle\cos\theta\text{,}\sin\theta\text{,}$
$\displaystyle\sin^{2}\theta+\cos^{2}\theta\text{,}\cos 2\theta\text{,}\sin
2\theta\text{,}$
$\displaystyle\cos\theta(\sin^{2}\theta+\cos^{2}\theta)\text{,}\sin\theta(\sin^{2}\theta+\cos^{2}\theta)\text{,}\cos
3\theta\text{,}\sin 3\theta\text{,}$
$\displaystyle(\sin^{2}\theta+\cos^{2}\theta)^{2}\text{,}\cos
2\theta(\sin^{2}\theta+\cos^{2}\theta)\text{,}\sin
2\theta(\sin^{2}\theta+\cos^{2}\theta)\text{,}\cos 4\theta\text{,}\sin
4\theta\text{,}$ $\displaystyle\cdots$
Let $v_{ix}$ and $v_{iy}$ play the roles of $\cos\theta$ and $\sin\theta$,
respectively. Then we define
$m_{1i}=1\text{,}$ (5a) $m_{2i}=v_{ix}\text{,}$ (5b) $m_{3i}=v_{iy}\text{,}$
(5c) $m_{4i}=(v_{ix}^{2}+v_{iy}^{2})/2\text{,}$ (5d)
$m_{5i}=v_{ix}^{2}-v_{iy}^{2}\text{,}$ (5e) $m_{6i}=v_{ix}v_{iy}\text{,}$ (5f)
$m_{7i}=v_{ix}(v_{ix}^{2}+v_{iy}^{2})/2\text{,}$ (5g)
$m_{8i}=v_{iy}(v_{ix}^{2}+v_{iy}^{2})/2\text{,}$ (5h)
$m_{9i}=v_{ix}(v_{ix}^{2}-3v_{iy}^{2})\text{,}$ (5i)
$m_{10i}=v_{iy}(3v_{ix}^{2}-v_{iy}^{2})\text{,}$ (5j)
$m_{11i}=(v_{ix}^{2}+v_{iy}^{2})^{2}/4\text{,}$ (5k)
$m_{12i}=v_{ix}^{4}-6v_{ix}^{2}v_{iy}^{2}+v_{iy}^{4}\text{,}$ (5l)
$m_{13i}=(v_{ix}^{2}+v_{iy}^{2})(v_{ix}^{2}-v_{iy}^{2})\text{,}$ (5m)
$m_{14i}=(v_{ix}^{2}+v_{iy}^{2})v_{ix}v_{iy}\text{,}$ (5n)
$m_{15i}=v_{ix}(v_{ix}^{2}-3v_{iy}^{2})(v_{ix}^{2}+v_{iy}^{2})\text{,}$ (5o)
$m_{16i}=v_{iy}(3v_{ix}^{2}-v_{iy}^{2})(v_{ix}^{2}+v_{iy}^{2})\text{.}$ (5p)
where $i=1,\cdots,16$.
For two-dimensional compressible models, we have four conserved moments,
density $\rho$, momentums $j_{x}$, $j_{y}$, and energy $e$. They are denoted
by $\hat{f}_{1}$, $\hat{f}_{2}$, $\hat{f}_{3}$ and $\hat{f}_{4}$,
respectively. Specifically, $\hat{f}_{1}=\rho=\sum f_{i}m_{1i}$,
$\hat{f}_{2}=j_{x}=\sum f_{i}m_{2i}$, $\hat{f}_{3}=j_{y}=\sum f_{i}m_{3i}$,
$\hat{f}_{4}=e=\sum f_{i}m_{4i}$. To be consistent with the idiomatic
expression of energy, in the definitions of $m_{4i}$, $m_{7i}$ and $m_{8i}$, a
coefficient $1/2$ is used. Similarly, a coefficient $1/4$ is used in the
definition of $m_{11i}$. Thus, the transformation matrix $\mathbf{M}$ can be
expressed as follows:
$\mathbf{M}=(m_{1},m_{2},\cdots,m_{16})^{T}\text{,}$
where
$m_{1}=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)\text{,}$
$m_{2}=(1,0,-1,0,6,0,-6,0,\sqrt{2},-\sqrt{2},-\sqrt{2},\sqrt{2},\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}},\frac{3}{\sqrt{2}})\text{,}$
$m_{3}=(0,1,0,-1,0,6,0,-6,\sqrt{2},\sqrt{2},-\sqrt{2},-\sqrt{2},\frac{3}{\sqrt{2}},\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}})\text{,}$
$m_{4}=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},18,18,18,18,2,2,2,2,\frac{9}{2},\frac{9}{2},\frac{9}{2},\frac{9}{2})\text{,}$
$m_{5}=(1,-1,1,-1,36,-36,36,-36,0,0,0,0,0,0,0,0)\text{,}$
$m_{6}=(0,0,0,0,0,0,0,0,2,-2,2,-2,\frac{9}{2},-\frac{9}{2},\frac{9}{2},-\frac{9}{2})\text{,}$
$m_{7}=(\frac{1}{2},0,-\frac{1}{2},0,108,0,-108,0,2\sqrt{2},-2\sqrt{2},-2\sqrt{2},2\sqrt{2},\frac{27}{2\sqrt{2}},-\frac{27}{2\sqrt{2}},-\frac{27}{2\sqrt{2}},\frac{27}{2\sqrt{2}})\text{,}$
$m_{8}=(0,\frac{1}{2},0,-\frac{1}{2},0,108,0,-108,2\sqrt{2},2\sqrt{2},-2\sqrt{2},-2\sqrt{2},\frac{27}{2\sqrt{2}},\frac{27}{2\sqrt{2}},-\frac{27}{2\sqrt{2}},-\frac{27}{2\sqrt{2}})\text{,}$
$m_{9}=(1,0,-1,0,216,0,-216,0,-4\sqrt{2},4\sqrt{2},4\sqrt{2},-4\sqrt{2},-\frac{27}{\sqrt{2}},\frac{27}{\sqrt{2}},\frac{27}{\sqrt{2}},-\frac{27}{\sqrt{2}})\text{,}$
$m_{10}=(0,-1,0,1,0,-216,0,216,4\sqrt{2},4\sqrt{2},-4\sqrt{2},-4\sqrt{2},\frac{27}{\sqrt{2}},\frac{27}{\sqrt{2}},-\frac{27}{\sqrt{2}},-\frac{27}{\sqrt{2}})\text{,}$
$m_{11}=(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},324,324,324,324,4,4,4,4,\frac{81}{4},\frac{81}{4},\frac{81}{4},\frac{81}{4})\text{,}$
$m_{12}=(1,1,1,1,1296,1296,1296,1296,-16,-16,-16,-16,-81,-81,-81,-81)\text{,}$
$m_{13}=(1,-1,1,-1,1296,-1296,1296,-1296,0,0,0,0,0,0,0,0)\text{,}$
$m_{14}=(0,0,0,0,0,0,0,0,8,-8,8,-8,\frac{81}{2},-\frac{81}{2},\frac{81}{2},-\frac{81}{2})\text{,}$
$m_{15}=(1,0,-1,0,7776,0,-7776,0,-16\sqrt{2},16\sqrt{2},16\sqrt{2},-16\sqrt{2},-\frac{243}{\sqrt{2}},\frac{243}{\sqrt{2}},\frac{243}{\sqrt{2}},-\frac{243}{\sqrt{2}})\text{,}$
$m_{16}=(0,-1,0,1,0,-7776,0,7776,16\sqrt{2},16\sqrt{2},-16\sqrt{2},-16\sqrt{2},\frac{243}{\sqrt{2}},\frac{243}{\sqrt{2}},-\frac{243}{\sqrt{2}},-\frac{243}{\sqrt{2}})\text{.}$
### III.2 Determination of $\hat{f}_{i}^{eq}$
We perform the Chapman-Enskog expansion20 ; 22 ; 23 on the two sides of
Eq.(1). We define
$f_{i}=f_{i}^{(0)}+f_{i}^{(1)}+f_{i}^{(2)}\text{,}$ (6a)
$\frac{\partial}{\partial t}=\frac{\partial}{\partial
t_{1}}+\frac{\partial}{\partial t_{2}}\text{,}$ (6b) $\frac{\partial}{\partial
x}=\frac{\partial}{\partial x_{1}}\text{,}$ (6c) where $f_{i}^{(0)}$ is the
zeroth order, $f_{i}^{(1)}$, $\frac{\partial}{\partial t_{1}}$and
$\frac{\partial}{\partial x_{1}}$ are the first order, $f_{i}^{(2)}$ and
$\frac{\partial}{\partial t_{2}}$ are the second order terms of the Knudsen
number $\epsilon$. Equating the coefficients of the zeroth, the first, and the
second order terms in $\epsilon$ gives $f_{i}^{(0)}=f_{i}^{eq}\text{,}$ (7a)
$(\frac{\partial}{\partial t_{1}}+v_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}})f_{i}^{(0)}=-\mathbf{S}_{il}f_{l}^{(1)}\text{,}$ (7b)
$\frac{\partial}{\partial t_{2}}f_{i}^{(0)}+(\frac{\partial}{\partial
t_{1}}+v_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}})f_{i}^{(1)}=-\mathbf{S}_{il}f_{l}^{(2)}\text{.}$ (7c) They can be
converted into moment space to obtain:
$\hat{f}_{i}^{(0)}=\hat{f}_{i}^{eq}\text{,}$ (8a) $(\frac{\partial}{\partial
t_{1}}+\hat{\mathbf{E}}_{\alpha}\frac{\partial}{\partial
x_{1\alpha}})\hat{f}_{i}^{(0)}=-\hat{\mathbf{S}}_{il}\hat{f}_{l}^{(1)}\text{,}$
(8b) $\frac{\partial}{\partial
t_{2}}\hat{f}_{i}^{(0)}+(\frac{\partial}{\partial
t_{1}}+\hat{\mathbf{E}}_{\alpha}\frac{\partial}{\partial
x_{1\alpha}})\hat{f}_{i}^{(1)}=-\hat{\mathbf{S}}_{il}\hat{f}_{l}^{(2)}\text{,}$
(8c) where
$\hat{\mathbf{E}}_{\alpha}=\mathbf{M}(v_{i\alpha}\mathbf{I})\mathbf{M}^{-1}$.
The equilibria of the moments in the moment space can be defined as :
$\hat{\mathbf{f}}^{eq}=(\rho,j_{x},j_{y},e,\hat{f}_{5}^{eq},\hat{f}_{6}^{eq},\cdots,\hat{f}_{16}^{eq})^{T}$.
The first order deviations from equilibria are defined as :
$\hat{\mathbf{f}}^{(1)}=(0,0,0,0,\hat{f}_{5}^{(1)},\hat{f}_{6}^{(1)},\cdots,\hat{f}_{16}^{(1)})^{T}$.
In the same way, the second order deviations are
$\hat{\mathbf{f}}^{(2)}=(0,0,0,0,\hat{f}_{5}^{(2)},\hat{f}_{6}^{(2)},\cdots,\hat{f}_{16}^{(2)})^{T}$.
From Eq.(8b) we obtain
$\frac{\partial\rho}{\partial t_{1}}+\frac{\partial j_{x}}{\partial
x_{1}}+\frac{\partial j_{y}}{\partial y_{1}}=0\text{,}$ (9a) $\frac{\partial
j_{x}}{\partial t_{1}}+\frac{\partial}{\partial
x_{1}}(e+\frac{1}{2}\hat{f}_{5}^{eq})+\frac{\partial}{\partial
y_{1}}\hat{f}_{6}^{eq}=0\text{,}$ (9b) $\frac{\partial j_{y}}{\partial
t_{1}}+\frac{\partial}{\partial
x_{1}}\hat{f}_{6}^{eq}+\frac{\partial}{\partial
y_{1}}(e-\frac{1}{2}\hat{f}_{5}^{eq})=0\text{,}$ (9c) $\frac{\partial
e}{\partial t_{1}}+\frac{\partial}{\partial
x_{1}}\hat{f}_{7}^{eq}+\frac{\partial}{\partial
y_{1}}\hat{f}_{8}^{eq}=0\text{,}$ (9d) $\frac{\partial}{\partial
t_{1}}\hat{f}_{5}^{eq}+\frac{\partial}{\partial
x_{1}}(\hat{f}_{7}^{eq}+\frac{1}{2}\hat{f}_{9}^{eq})+\frac{\partial}{\partial
y_{1}}(-\hat{f}_{8}^{eq}+\frac{1}{2}\hat{f}_{10}^{eq})=-s_{5}\hat{f}_{5}^{(1)}\text{,}$
(9e) $\frac{\partial}{\partial
t_{1}}\hat{f}_{6}^{eq}+\frac{1}{4}\frac{\partial}{\partial
x_{1}}(2\hat{f}_{8}^{eq}+\hat{f}_{10}^{eq})+\frac{1}{4}\frac{\partial}{\partial
y_{1}}(2\hat{f}_{7}^{eq}-\hat{f}_{9}^{eq})=-s_{6}\hat{f}_{6}^{(1)}\text{,}$
(9f) $\frac{\partial}{\partial t_{1}}\hat{f}_{7}^{eq}+\frac{\partial}{\partial
x_{1}}(\hat{f}_{11}^{eq}+\frac{1}{4}\hat{f}_{13}^{eq})+\frac{1}{2}\frac{\partial}{\partial
y_{1}}\hat{f}_{14}^{eq}=-s_{7}\hat{f}_{7}^{(1)}\text{,}$ (9g)
$\frac{\partial}{\partial
t_{1}}\hat{f}_{8}^{eq}+\frac{1}{2}\frac{\partial}{\partial
x_{1}}\hat{f}_{14}^{eq}+\frac{\partial}{\partial
y_{1}}(\hat{f}_{11}^{eq}-\frac{1}{4}\hat{f}_{13}^{eq})=-s_{8}\hat{f}_{8}^{(1)}\text{,}$
(9h) $\frac{\partial}{\partial
t_{1}}\hat{f}_{9}^{eq}+\frac{1}{2}\frac{\partial}{\partial
x_{1}}(\hat{f}_{12}^{eq}+\hat{f}_{13}^{eq})-\frac{\partial}{\partial
y_{1}}\hat{f}_{14}^{eq}=-s_{9}\hat{f}_{9}^{(1)}\text{,}$ (9i)
$\frac{\partial}{\partial t_{1}}\hat{f}_{10}^{eq}+\frac{\partial}{\partial
x_{1}}\hat{f}_{14}^{eq}+\frac{1}{2}\frac{\partial}{\partial
y_{1}}(-\hat{f}_{12}^{eq}+\hat{f}_{13}^{eq})=-s_{10}\hat{f}_{10}^{(1)}\text{,}$
(9j) $\frac{\partial}{\partial
t_{1}}\hat{f}_{11}^{eq}+\frac{\partial}{\partial
x_{1}}(-9j_{x}+\frac{25}{2}\hat{f}_{7}^{eq}+3\hat{f}_{9}^{eq})+\frac{\partial}{\partial
y_{1}}(-9j_{y}+\frac{25}{2}\hat{f}_{8}^{eq}-3\hat{f}_{10}^{eq})=-s_{11}\hat{f}_{11}^{(1)}\text{,}$
(9k) $\frac{\partial}{\partial
t_{1}}\hat{f}_{12}^{eq}+\frac{\partial}{\partial
x_{1}}\hat{f}_{15}^{eq}-\frac{\partial}{\partial
y_{1}}\hat{f}_{16}^{eq}=-s_{12}\hat{f}_{12}^{(1)}\text{,}$ (9l)
$\frac{\partial}{\partial t_{1}}\hat{f}_{13}^{eq}+\frac{\partial}{\partial
x_{1}}(-18j_{x}+\frac{1}{2}\hat{f}_{15}^{eq}+25\hat{f}_{7}^{eq}+6\hat{f}_{9}^{eq})+\frac{\partial}{\partial
y_{1}}(18j_{y}+\frac{1}{2}\hat{f}_{16}^{eq}-25\hat{f}_{8}^{eq}+6\hat{f}_{10}^{eq})=-s_{13}\hat{f}_{13}^{(1)}\text{,}$
(9m) $\frac{\partial}{\partial
t_{1}}\hat{f}_{14}^{eq}+\frac{\partial}{\partial
x_{1}}(-9j_{y}+\frac{1}{4}\hat{f}_{16}^{eq}+\frac{25}{2}\hat{f}_{8}^{eq}-3\hat{f}_{10}^{eq})-\frac{\partial}{\partial
y_{1}}(9j_{x}+\frac{1}{4}\hat{f}_{15}^{eq}-\frac{25}{2}\hat{f}_{7}^{eq}-3\hat{f}_{9}^{eq})=-s_{14}\hat{f}_{14}^{(1)}\text{,}$
(9n) $\frac{\partial}{\partial
t_{1}}\hat{f}_{15}^{eq}+\frac{\partial}{\partial
x_{1}}(75e-54\rho-18\hat{f}_{5}^{eq}+18\hat{f}_{11}^{eq}+\frac{25}{2}\hat{f}_{12}^{eq}+\frac{37}{2}\hat{f}_{13}^{eq})+\frac{\partial}{\partial
y_{1}}(36\hat{f}_{6}^{eq}-13\hat{f}_{14}^{eq})=-s_{15}\hat{f}_{15}^{(1)}\text{,}$
(9o) $\frac{\partial}{\partial
t_{1}}\hat{f}_{16}^{eq}-\frac{\partial}{\partial
x_{1}}(36\hat{f}_{6}^{eq}-13\hat{f}_{14}^{eq})-\frac{\partial}{\partial
y_{1}}(75e-54\rho+18\hat{f}_{5}^{eq}+18\hat{f}_{11}^{eq}+\frac{25}{2}\hat{f}_{12}^{eq}-\frac{37}{2}\hat{f}_{13}^{eq})=-s_{16}\hat{f}_{16}^{(1)}\text{.}$
(9p) From Eq.(8c) we obtain $\frac{\partial\rho}{\partial t_{2}}=0\text{,}$
(10a) $\frac{\partial j_{x}}{\partial
t_{2}}+\frac{1}{2}\frac{\partial}{\partial
x_{1}}\hat{f}_{5}^{(1)}+\frac{\partial}{\partial
y_{1}}\hat{f}_{6}^{(1)}=0\text{,}$ (10b) $\frac{\partial j_{y}}{\partial
t_{2}}+\frac{\partial}{\partial
x_{1}}\hat{f}_{6}^{(1)}-\frac{1}{2}\frac{\partial}{\partial
y_{1}}\hat{f}_{5}^{(1)}=0\text{,}$ (10c) $\frac{\partial e}{\partial
t_{2}}+\frac{\partial}{\partial
x_{1}}\hat{f}_{7}^{(1)}+\frac{\partial}{\partial
y_{1}}\hat{f}_{8}^{(1)}=0\text{.}$ (10d) Adding Eq.(10) and the first four
formulas of Eq.(9) leads to the following equations,
$\frac{\partial\rho}{\partial t}+\frac{\partial j_{x}}{\partial
x}+\frac{\partial j_{y}}{\partial y}=0\text{,}$ (11a) $\frac{\partial
j_{x}}{\partial t}+\frac{\partial}{\partial
x}(e+\frac{1}{2}\hat{f}_{5}^{eq})+\frac{\partial}{\partial
y}\hat{f}_{6}^{eq}=-\frac{1}{2}\frac{\partial}{\partial
x}\hat{f}_{5}^{(1)}-\frac{\partial}{\partial y}\hat{f}_{6}^{(1)}\text{,}$
(11b) $\frac{\partial j_{y}}{\partial t}+\frac{\partial}{\partial
x}\hat{f}_{6}^{eq}+\frac{\partial}{\partial
y}(e-\frac{1}{2}\hat{f}_{5}^{eq})=-\frac{\partial}{\partial
x}\hat{f}_{6}^{(1)}+\frac{1}{2}\frac{\partial}{\partial
y}\hat{f}_{5}^{(1)}\text{,}$ (11c) $\frac{\partial e}{\partial
t}+\frac{\partial}{\partial x}\hat{f}_{7}^{eq}+\frac{\partial}{\partial
y}\hat{f}_{8}^{eq}=-\frac{\partial}{\partial
x}\hat{f}_{7}^{(1)}-\frac{\partial}{\partial y}\hat{f}_{8}^{(1)}\text{.}$
(11d)
To obatin the NS equations, we choose
$\hat{f}_{5}^{eq}=(j_{x}^{2}-j_{y}^{2})/\rho\text{,}$ (12a)
$\hat{f}_{6}^{eq}=j_{x}j_{y}/\rho\text{,}$ (12b)
$\hat{f}_{7}^{eq}=(e+P)j_{x}/\rho\text{,}$ (12c)
$\hat{f}_{8}^{eq}=(e+P)j_{y}/\rho\text{,}$ (12d)
$\hat{f}_{9}^{eq}=(j_{x}^{2}-3j_{y}^{2})j_{x}/\rho^{2}\text{,}$ (12e)
$\hat{f}_{10}^{eq}=(3j_{x}^{2}-j_{y}^{2})j_{y}/\rho^{2}\text{,}$ (12f)
$\hat{f}_{11}^{eq}=2e^{2}/\rho-(j_{x}^{2}+j_{y}^{2})^{2}/4\rho^{3}\text{,}$
(12g) $\hat{f}_{13}^{eq}=(6\rho
e-2j_{x}^{2}-2j_{y}^{2})(j_{x}^{2}-j_{y}^{2})/\rho^{3}\text{,}$ (12h)
$\hat{f}_{14}^{eq}=(6\rho e-2j_{x}^{2}-2j_{y}^{2})j_{x}j_{y}/\rho^{3}\text{.}$
(12i) The definitions of $\hat{f}_{12}^{eq}$, $\hat{f}_{15}^{eq}$,
$\hat{f}_{16}^{eq}$ have no effect on macroscopic equations, so we can choose
$\hat{f}_{12}^{eq}=\hat{f}_{15}^{eq}=\hat{f}_{16}^{eq}=0$. In this way the
recovered NS equations are as follows: $\frac{\partial\rho}{\partial
t}+\frac{\partial j_{x}}{\partial x}+\frac{\partial j_{y}}{\partial
y}=0\text{,}$ (13a) $\frac{\partial j_{x}}{\partial
t}+\frac{\partial}{\partial
x}\left(j_{x}^{2}/\rho\right)+\frac{\partial}{\partial
y}\left(j_{x}j_{y}/\rho\right)=-\frac{\partial P}{\partial
x}+\frac{\partial}{\partial x}[\mu_{s}(\frac{\partial u_{x}}{\partial
x}-\frac{\partial u_{y}}{\partial y})]+\frac{\partial}{\partial
y}[\mu_{v}(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial
y})]\text{,}$ (13b) $\frac{\partial j_{y}}{\partial
t}+\frac{\partial}{\partial
x}\left(j_{x}j_{y}/\rho\right)+\frac{\partial}{\partial
y}\left(j_{y}^{2}/\rho\right)=-\frac{\partial P}{\partial
y}+\frac{\partial}{\partial x}[\mu_{v}(\frac{\partial u_{y}}{\partial
x}+\frac{\partial u_{x}}{\partial y})]-\frac{\partial}{\partial
y}[\mu_{s}(\frac{\partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial
y})]\text{,}$ (13c) $\displaystyle\frac{\partial e}{\partial
t}+\frac{\partial}{\partial x}[(e+P)j_{x}/\rho]+\frac{\partial}{\partial
y}[(e+P)j_{y}/\rho]$ (13d) $\displaystyle=$
$\displaystyle\frac{\partial}{\partial x}[\lambda_{1}(2\frac{\partial
T}{\partial x}+u_{y}\frac{\partial u_{y}}{\partial x}+u_{x}\frac{\partial
u_{x}}{\partial x}-u_{x}\frac{\partial u_{y}}{\partial y}+u_{y}\frac{\partial
u_{x}}{\partial y})]$ $\displaystyle+\frac{\partial}{\partial
y}[\lambda_{2}(2\frac{\partial T}{\partial y}+u_{x}\frac{\partial
u_{x}}{\partial y}-u_{y}\frac{\partial u_{x}}{\partial x}+u_{x}\frac{\partial
u_{y}}{\partial x}+u_{y}\frac{\partial u_{y}}{\partial y})]\text{,}$ where
$\mu_{s}=$ $\rho RT/s_{5}$, $\mu_{v}=$ $\rho RT/s_{6}$, $\lambda_{1}=\rho
RT/s_{7}$, $\lambda_{2}=\rho RT/s_{8}$. When $\mu_{s}=$ $\mu_{v}=\mu$,
$\lambda_{1}=\lambda_{2}=\lambda$, the above NS equations reduce to
$\frac{\partial\rho}{\partial t}+\frac{\partial j_{\alpha}}{\partial
x_{\alpha}}=0\text{,}$ (14a) $\frac{\partial j_{\alpha}}{\partial
t}+\frac{\partial\left(j_{\alpha}j_{\beta}/\rho\right)}{\partial
x_{\beta}}=-\frac{\partial P}{\partial x_{\alpha}}+\frac{\partial}{\partial
x_{\beta}}[\mu(\frac{\partial u_{\alpha}}{\partial x_{\beta}}+\frac{\partial
u_{\beta}}{\partial x_{\alpha}}-\frac{\partial u_{\chi}}{\partial
x_{\chi}}\delta_{\alpha\beta})]\text{,}$ (14b) $\frac{\partial e}{\partial
t}+\frac{\partial}{\partial
x_{\alpha}}[(e+P)j_{\alpha}/\rho]=\frac{\partial}{\partial
x_{\alpha}}[\lambda(2\frac{\partial T}{\partial
x_{\alpha}}+u_{\beta}(\frac{\partial u_{\alpha}}{\partial
x_{\beta}}+\frac{\partial u_{\beta}}{\partial x_{\alpha}}-\frac{\partial
u_{\chi}}{\partial x_{\chi}}\delta_{\alpha\beta})]\text{.}$ (14c)
## IV Von Neumann Stability Analysis
In this section we perform the von Neumann stability analysis on the new MRT
LB model. In the stability analysis, we write the solution of FD LB equation
in the form of Fourier series. If all the eigenvalues of the coefficient
matrix are less than 1, the algorithm is stable.
The distribution function $f_{i}(x_{\alpha},t)$ is split into two parts:
$f_{i}(x_{\alpha},t)=\bar{f_{i}^{0}}+\Delta f_{i}(x_{\alpha},t)\text{,}$ (15)
where $\bar{f_{i}^{0}}$ is the global equilibrium distribution function. It is
a constant and does not vary with time or space, depends only on the average
density, velocity and temperature. Similarly, the distribution function
$\hat{f}_{i}(x_{\alpha},t)$ is split into two parts:
$\hat{f}_{i}(x_{\alpha},t)=\hat{\bar{f_{i}^{0}}}+\Delta\hat{f}_{i}(x_{\alpha},t)\text{,}$
(16)
where $\hat{\bar{f_{i}^{0}}}$ is a constant. Putting the Eq.(15) and Eq.(16)
into Eq.(3) gives
$\Delta f_{i}(x_{\alpha},t+\Delta t)=\Delta f_{i}(x_{\alpha},t)-\Delta
tv_{i\alpha}\frac{\partial\Delta f_{i}}{\partial x_{\alpha}}-\Delta
t\mathbf{M}_{il}^{-1}\hat{\mathbf{S}}_{lk}(\Delta\hat{f}_{k}-\Delta\hat{f}_{k}^{eq})\text{.}$
(17)
The solution can be written as
$\Delta
f_{i}(x_{\alpha},t)=F_{i}^{t}\mathrm{exp}(\mathbf{i}k_{\alpha}x_{\alpha})\text{,}$
(18)
where $F_{i}^{t}$ is an amplitude of sine wave at lattice point $x_{\alpha}$
and time $t$, $k_{\alpha}$ is the wave number. From the above two equations we
can obtain $F_{i}^{t+\Delta t}=G_{ij}F_{j}^{t}.$ Coefficient matrix $G_{ij}$
describes the growth rate of amplitude $F_{i}^{t}$ in each time step $\Delta
t$. If $\omega$ denotes the eigenvalue of coefficient matrix $G_{ij}$, the von
Neumann stability condition is $\mathrm{max}|\omega|\leq 1$. Coefficient
matrix $G_{ij}$ can be expressed as follows,
$\displaystyle G_{ij}$ $\displaystyle=\delta_{ij}-\frac{v_{i\alpha}\Delta
t}{2\Delta x_{\alpha}}(e^{\mathbf{i}k_{\alpha}\Delta
x_{\alpha}}-e^{-\mathbf{i}k_{\alpha}\Delta
x_{\alpha}})\delta_{ij}+\frac{1}{2}(\frac{v_{i\alpha}\Delta t}{\Delta
x_{\alpha}})^{2}(e^{\mathbf{i}k_{\alpha}\Delta x_{\alpha}}-2$
$\displaystyle+e^{-\mathbf{i}k_{\alpha}\Delta x_{\alpha}})\delta_{ij}-\Delta
t\mathbf{M}_{il}^{-1}\hat{\mathbf{S}}_{lk}(\frac{\partial\hat{f}_{k}}{\partial
f_{j}}-\frac{\partial\hat{f}_{k}^{eq}}{\partial f_{j}})\text{,}$ (19)
where
$\hat{f}_{k}=\mathbf{M}_{kp}f_{p}\text{,}$
$\frac{\partial\hat{f}_{k}^{eq}}{\partial
f_{j}}=\frac{\partial\hat{f}_{k}^{eq}}{\partial\rho}\frac{\partial\rho}{\partial
f_{j}}+\frac{\partial\hat{f}_{k}^{eq}}{\partial T}\frac{\partial T}{\partial
f_{j}}+\frac{\partial\hat{f}_{k}^{eq}}{\partial u_{\alpha}}\frac{\partial
u_{\alpha}}{\partial f_{j}}\text{.}$ (20)
Coefficient matrix $G_{ij}$ contains a large number of matrix elements (as
many as $16\times 16$), and every matrix element correlates with the
macroscopic quantities and model parameters, so analytic analysis is very
difficult. We conduct a quantitative analysis using Mathematica software.
In Fig.2 we show an example of stability comparison for the new MRT model and
its SRT version. The abscissa is for $kdx$, and the vertical axis is for
$|\omega|_{max}$ which is the largest eigenvalue of coefficient matrix
$G_{ij}$. The macroscopic values in stability analysis are chosen as follows:
$(\rho,u_{1},u_{2},T)$ = $(2.0,10.0,0.0,2.0)$, other common parameters are:
$dx=dy=2\times 10^{-3}$, $dt=10^{-5}$. The relaxation time in SRT is
$\tau=10^{-5}$, while the collision parameters in MRT are $s_{5}=6500$,
$s_{7}=s_{8}=9\times 10^{4}$, $s_{9}=8\times 10^{4}$, $s_{13}=7\times 10^{4}$,
$s_{14}=8\times 10^{3}$, $s_{15}=2.5\times 10^{4}$, the others are $10^{5}$.
In this case, the MRT scheme is stable, while the SRT version is not. It is
clear that, by choosing appropriate collision parameters, the stability of MRT
can be much better than the SRT.
Figure 2: Stability comparison for the new MRT model and its SRT version.
## V Numerical Simulations
In this section we study the following problems using the new MRT LB model:
One-dimensional Riemann problems, shock reflections, shock wave reaction on
cylindrical bubble, and Couette flow.
### V.1 One-dimensional Riemann problems
Here, we study several typical one-dimensional Riemann problems, including the
Sod, Lax shock tube, Colella explosion wave, collision of two strong shocks
and a new shock tube with high Mach number. In the $x$ direction, $f_{i}=$
$f_{i}^{eq}$ is set on the boundary nodes before the disturbance reaches the
two ends. In the $y$ direction, the periodic boundary condition is adopted. In
the following part, subscripts “L” and “R” indicate the macroscopic variables
at the left and right sides of discontinuity.
(a) Sod shock tube problem
The initial condition is
$\left\\{\begin{array}[]{cc}(\rho,u_{1},u_{2},T)|_{L}=(1.0,0.0,0.0,1.0)\text{,}&x\leq
0\text{.}\\\
(\rho,u_{1},u_{2},T)|_{R}=(0.125,0.0,0.0,0.8)\text{,}&x>0\text{.}\end{array}\right.$
(21)
Figure 3 shows a comparison of the MRT LB simulation results and exact
solutions for the density, pressure, velocity and temperature of the Sod shock
tube problem at time $t=0.18$. Here, in the collision matrix
$s_{5}=s_{6}=5\times 10^{2}$, $s_{7}=s_{8}=10^{3}$, $s_{11}=2500$, and other
collision parameters are$\ 10^{5}$. The red circles correspond to simulation
results with the grid size $dx=dy=0.002$ and time step $dt=2\times 10^{-6}$
(case 1), the green triangles correspond to simulation results with
$dx=dy=0.001$, $dt=10^{-6}$ (case 2), and solid lines represent the exact
solutions. The relative errors of the density, pressure, velocity and
temperature for case 1 are $0.234\%$, $0.182\%$, $3.32\%$ and $0.327\%$,
respectively. The relative errors of the density, pressure, velocity and
temperature for case 2 are $0.225\%$, $0.171\%$, $3.16\%$ and $0.322\%$,
respectively. Here, the relative error is defined as
$E=\sum_{I}\left|\varsigma_{{}_{I,J,num}}-\varsigma_{{}_{I,J,exa}}\right|/\sum_{I}\left|\varsigma_{{}_{I,J,exa}}\right|$,
where $\varsigma_{{}_{I,J,num}}$ denotes the variables at the node of
$(x_{I},y_{J})$ obtained from the numerical simulation, and
$\varsigma_{{}_{I,J,exa}}$ is the exact solution at the same node. The
simulation results successfully capture the main structure of the waves.
Figure 3: Comparison of the MRT LB simulation results and exact solutions for
Sod shock tube at time $t=0.18$.
b) Lax shock tube problem
The initial condition of the problem is:
$\left\\{\begin{array}[]{cc}(\rho,u_{1},u_{2},T)|_{L}=(0.445,0.698,0.0,7.928)\text{,}&x\leq
0\text{.}\\\
(\rho,u_{1},u_{2},T)|_{R}=(0.5,0.0,0.0,1.142)\text{,}&x>0\text{.}\end{array}\right.$
(22)
Figure 4 shows the MRT LB numerical results and exact solutions for the Lax
shock tube problem at time $t=0.2$. The red squares, green circles and blue
triangles correspond to simulation results with different grid sizes and time
steps: $dx=dy=0.004$, $dt=4\times 10^{-6}$ (case 1), $dx=dy=0.002$,
$dt=2\times 10^{-6}$ (case 2), $dx=dy=0.001$, $dt=10^{-6}$ (case 3),
respectively, and solid lines represent the exact solutions. The used
parameters are $s_{7}=s_{8}=3\times 10^{3}$, $s_{13}=10^{2}$, other collision
parameters are$\ 10^{5}$. The relative errors of the density, pressure,
velocity and temperature for case 1 are $0.398\%$, $0.205\%$, $0.592\%$ and
$0.310\%$, respectively. The relative errors for case 2 are $0.344\%$,
$0.130\%$, $0.408\%$ and $0.287\%$, respectively. And the relative errors for
case 3 are $0.334\%$, $0.117\%$, $0.372\%$ and $0.283\%$, respectively.
Figure 4: The MRT LB numerical and exact solutions for Lax shock tube at time
$t=0.2$.
(c) Colella explosion wave
For the problem, the initial condition is
$\left\\{\begin{array}[]{cc}(\rho,u_{1},u_{2},T)|_{L}=(1.0,0.0,0.0,1000.0)\text{,}&x\leq
0\text{.}\\\
(\rho,u_{1},u_{2},T)|_{R}=(1.0,0.0,0.0,0.01)\text{,}&x>0\text{.}\end{array}\right.$
(23)
This is a strong temperature discontinuity problem that can be used to study
the robustness and precision of numerical methods. Figure 5 gives density,
pressure, velocity and temperature results at $t=0.1$. The red squares and
green circles correspond to simulation results with different grid sizes and
time steps: $dx=dy=0.002$, $dt=2\times 10^{-6}$ (case 1), and $dx=dy=0.001$,
$dt=10^{-6}$ (case 2), respectively, and solid lines represent the exact
solutions. Here, the parameters are $s_{7}=s_{8}=5\times 10^{4}$,
$s_{11}=s_{13}=5\times 10^{5}$, other values of $s$ adopt $10^{5}$. The
relative errors of the density, pressure, velocity and temperature for case 1
are $1.69\%$, $1.11\%$, $1.60\%$ and $0.779\%$, respectively. The relative
errors for case 2 are $1.68\%$, $1.11\%$, $1.59\%$ and $0.777\%$,
respectively. The oscillations at the interface are difficult to eliminate
completely in our simulations.
Figure 5: The MRT simulation results and exact solutions for the Colella
explosion wave at time $t=0.1$.
(d) Collision of two strong shocks
The initial condition can be write as:
$\left\\{\begin{array}[]{cc}(\rho,u_{1},u_{2},T)|_{L}=(5.99924,19.5975,0.0,76.8254)\text{,}&x\leq
0\text{.}\\\
(\rho,u_{1},u_{2},T)|_{R}=(5.99242,-6.19633,0.0,7.69222)\text{,}&x>0\text{.}\end{array}\right.$
(24)
The MRT and SRT numerical results and exact solutions at time $t=0.1$ are
shown in Fig.6, where the common parameters are $dx=dy=0.003$, $dt=10^{-5}$,
the collision matrix in MRT is $s_{5}=s_{6}=5\times 10^{3}$,
$s_{7}=s_{8}=3\times 10^{4}$, other values of $s$ are $10^{5}$, and the
relaxation time in SRT is $\tau=10^{-5}$. The red squares and green circles
correspond to the MRT and SRT simulation results, respectively, and solid
lines represent the exact solutions. Compared with the simulation results of
SRT, we can find that the oscillations at the discontinuity are weaker in MRT
simulation.
Figure 6: The MRT and SRT numerical results and exact solution for collision
of two strong shocks at time $t=0.1$.
(e) High Mach number shock
In order to test the stability of the new model, we construct a new shock tube
problem with the initial condition,
$\left\\{\begin{array}[]{cc}(\rho,u_{1},u_{2},T)|_{L}=(5.0,45.0,0.0,10.0)\text{,}&x\leq
0\text{.}\\\
(\rho,u_{1},u_{2},T)|_{R}=(6.0,-20.0,0.0,5.0)\text{,}&x>0\text{.}\end{array}\right.$
(25)
The Mach number of the left side is $10.1$, and the right is $6.3$. And this
test is successfully passed by the MRT LB, but failed by the SRT. Figure 7
shows comparison of the MRT LB results and exact solutions at $t=0.018$, where
the parameters are $dx=dy=0.003$, $dt=10^{-5}$, $s_{5}=s_{6}=1.5\times
10^{4}$, $s_{10}=5\times 10^{4}$, other values of $s$ are $10^{5}$. Circle
symbols correspond to MRT simulation results, and solid lines represent the
exact solutions.
Figure 7: MRT LB results and exact solutions for the high Mach number shock
tube problem at $t=0.018$.
### V.2 Shock reflections
We adopt the macroscopic variable boundary conditions in Figs. 8-11. The
values of the distribution functions on boundaries are assigned with the
corresponding values of the equilibrium distribution functions. The
determination methods of macroscopic quantities are explained combining with
specific problems. Here we study two gas dynamic problems: regular and Mach
shock reflections.
(a) Regular shock reflection
In the problem, we have performed a $30^{\circ}$ shock reflection, and the
corresponding Mach number is $5$. The computational domain is a rectangle with
length $0.6$ and height $0.2$, which is divided into $300\times 100.$ The
reflecting wall lies at the bottom of the domain (reflecting boundary
condition denotes the $y$ component of the fluid velocity on the boundary is
reverse to that of interior point), and the linear extrapolation technique is
applied to define the values of the macroscopic quantities on the right-hand
boundary. The other two sides adopt the Dirichlet boundary conditions:
$\left\\{\begin{array}[]{l}(\rho\text{, }u_{1}\text{, }u_{2}\text{,
}T)|_{0\text{, }y\text{, }t}=(1.0\text{, }5.0\text{, }0.0\text{,
}0.5)\text{,}\\\ (\rho\text{, }u_{1}\text{, }u_{2}\text{, }T)|_{x\text{,
}0.2\text{, }t}=(2.27273\text{, }4.3\text{, -1.21244,
1.76})\text{.}\end{array}\right.$ (26)
Initially, the entire interior zone is set the values of the left boundary.
Parameters in the simulation are as follows: $dx=dy=0.002$, $dt=10^{-5}$,
$s_{5}=10^{4}$, $s_{6}=2\times 10^{3}$, $s_{7}=s_{8}=10^{4}$, other collision
parameters are $10^{5}$. Figure 8 shows the pressure contour at time $t=0.3$,
and the density, temperature contours have similar results. From black to
white, the grey level corresponds to the increase of the pressure. The result
shows that the new MRT model has the ability to accurately capture the shock
front.
Figure 8: Pressure contour of regular shock reflection at time $t=0.3$.
(b) Mach reflection problem
This problem is on an unsteady shock reflection. A planar shock impacts an
oblique surface which is at a $30^{\circ}$ angle to the propagation direction
of the shock. The fluid in front of the shock has zero velocity, and the shock
Mach number is $1.5$. The initial condition is as follows:
$\left(\rho,u_{1},u_{2},T\right)\mid_{x\text{,}y\text{,}0}=\left\\{\begin{array}[]{ll}(3.17647,0.555556\cos
30^{\circ},-0.555556\sin 30^{\circ},0.839506)\text{,}&\text{ if }y\geq
h(x,0)\text{.}\\\ (2.0,0.0,0.0,0.5)\text{,}&\text{ if
}y<h(x,0)\text{.}\end{array}\right.$ (27)
where $h(x,t)=\tan 60^{\circ}(x-150dx)-1.5t/\sin 30^{\circ}.$ The
computational domain is divided into $600\times 300$. At the bottom boundary,
reflecting boundary condition is used from the 150th grid, and the left side
adopts the values of the initial post-shock flow; the left boundary is also
assigned values of the initial post-shock flow, and at the right boundary the
extrapolation technique is applied; at the top boundary, the macroscopic
variables are assigned using the same method of the right boundary when
$x>g(t)$, and are set the same values as the left boundary’s when $x\leq
g(t)$, where $g(t)=150dx+\tan 30^{\circ}(0.9+1.5t/\sin 30^{\circ})$, $dx$ is
the grid size in simulation. In Fig.9 we show the results of density,
pressure, velocity in $x$ direction and temperature contours at $t=0.25$ in
the part of $\left[50,450\right]\times\left[0,200\right]$. Parameters in the
simulation are $dx=dy=0.003$, $dt=10^{-5}$, $s_{5}=10^{3}$, $s_{6}=5\times
10^{2}$, $s_{7}=s_{8}=10^{3}$, and other collision parameters are$\ 10^{5}$.
The simulation results are accordant with those of other numerical methods41 ;
42 ; 43 ; CShuPRE .
Figure 9: Mach reflection of shock wave. Figs. (a)-(d) shows the density,
pressure, velocity in $x$ direction, and temperature at time $t=0.25$,
respectively. From black to white, the grey level corresponds to the increase
of values.
### V.3 Shock wave reaction on cylindrical bubble problem
The problems are as follows: A planar shock wave with the Mach number $1.22$
impinges on a cylindrical bubble with different densities. In the first case
the bubble has a lower density. In the second case the bubble’s density is
higher.
Initial conditions for the first case are
$\left(\rho,u_{1},u_{2},p\right)\mid_{x\text{,}y\text{,}0}=\left\\{\begin{array}[]{cc}\left(1,0,0,1\right)\text{,}&\mathbf{pre-
shock}\text{{,}}\\\ \left(1.28,-0.3774,0,1.6512\right)\text{,}&\mathbf{post-
shock}\text{{,}}\\\
\left(0.1358,0,0,1\right)\text{,}&\mathbf{bubble}\text{{,}}\end{array}\right.$
(28)
and for the second case are
$\left(\rho,u_{1},u_{2},p\right)\mid_{x\text{,}y\text{,}0}=\left\\{\begin{array}[]{cc}\left(1,0,0,1\right)\text{,}&\mathbf{pre-
shock}\text{{,}}\\\ \left(1.28,-0.3774,0,1.6512\right)\text{,}&\mathbf{post-
shock}\text{{,}}\\\
\left(3.1538,0,0,1\right)\text{,}&\mathbf{bubble}\text{{.}}\end{array}\right.$
(29)
In the simulations, the right side adopts the values of the initial post-shock
flow; the extrapolation technique is applied at the left boundary, and
reflecting boundary conditions are imposed on the top and bottom. The common
parameters are as follows: $dx=dy=0.003$, $dt=10^{-5}$. When simulate the low
density cylindrical bubble, the collision parameters are
$s_{5}=s_{6}=s_{7}=s_{8}=10^{4}$, and others are $10^{5}$; when simulate the
high density bubble, the collision parameters are $s_{5}=10^{3}$, and
$s=10^{5}$ for the others. In Fig. 10(a), from top to bottom, the three plots
show the density contours at the times $t=0$, $0.5$, $0.65$, respectively. In
Fig. 10(b), from top to bottom, the three plots show the density contours at
the times $t=0$, $0.6$, $0.9$, respectively. These results are accordant with
those from other methods24 and experiment24aa . The surface of bubbles is
comparatively smooth, which indicates that the MRT modle has high accuracy and
resolution.
Figure 10: Snapshots of shock wave reaction on single bubble. The left column
is for the process with initial condition (28), and the right column is for
the process with initial condition (29). From black to white, the density
value increases.
### V.4 Couette flow
In order to demonstrate the accuracy of the model, numerical simulations of
the incompressible Couette flow are carried out. Consider a viscous fluid flow
between two infinite parallel flat plates, separated by a distance of $D$. The
initial state of the fluid is $\rho=1$, $T=1$, $U=0$. At time $t=0$ the two
plates start to move at velocities $U$, $-U$, respectively, ($U=0.1$). The
periodic boundary condition is adopted in the x direction. The top and bottom
boundaries are constant speed and constant temperature boundaries
($U=0.1,T=1$). The analytical solution of horizontal velocity along a vertical
line is as follows:
$u=2yU/D-\sum_{j}(-1)^{j+1}\frac{2U}{j\pi}\exp[-\frac{4j^{2}\pi^{2}\mu}{\rho
D^{2}}t]\sin(\frac{2j\pi}{D}y)\text{,}$
where $j$ is an integer, the two walls locate at $y=\pm D/2$.
We carried out a set of simulations: $dx=dy=0.004$, $dt=10^{-5}$, $NX\times
NY=16\times 32$ (case 1), $dx=dy=0.002$, $dt=5\times 10^{-6}$, $NX\times
NY=32\times 64$(case 2), $dx=dy=0.001$, $dt=2.5\times 10^{-6}$, $NX\times
NY=64\times 128$ (case 3). All of the collision parameters are$\ 10^{5}$.
Figure 11 shows a comparison of the MRT LB simulation results and exact
solution for the horizontal velocity distribution at time $t=57.5$. The black
squares, red triangles and green circles correspond to simulation results of
case 1, case 2, and case 3, respectively, and solid line represents the exact
solution. The relative errors of the horizontal velocity for the three cases
are $7.02\%$, $3.86\%$ and $2.06\%$, respectively. So this model is of first
order accuracy.
Figure 11: Comparison of the MRT LB simulation results and exact solutions for
the horizontal velocity distribution of Couette fiow at time $t=57.5$.
## VI Conclusion
An energy-conserving multiple-relaxation-time finite difference lattice
Boltzmann model for compressible flows is proposed. This model is based on a
16-discrete-velocity model designed by Kataoka and Tsutahara21a . The
collision step is first calculated in the moment space and then mapped back to
the velocity space. The moment space and corresponding transformation matrix
are constructed according to the group representation theory. Equilibria of
the nonconserved moments are chosen according to the need of recovering
compressible Navier-Stokes equations through the Chapman-Enskog expansion. In
the new model different transport coefficients, such as viscosity and heat
conductivity, are related to different collision parameters. Consequently,
they can be controlled independently. This flexibility makes the MRT model
contain more physical information and easier to satisfy the von Neumann
stability condition than its SRT counterpart. The new model passed well-known
benchmark tests, including (i) shock tubes, such as the Sod, Lax, Colella
explosion wave, collision of two strong shocks and a new shock tube with high
Mach number, (ii) regular and Mach shock reflections, (iii) shock wave
reaction on cylindrical bubble problems, and (iv) Couette flow. This model
works for both low and high speeds compressible flows. Both the LB and
traditional CFD approach work when the Knudsen number is very small, but in
the vicinity of shock wave, the system is in a nonequilibrium state, and the
traditional Euler and Navier-Stokes descriptions are problematic. For such
problems, LB has more sound physical ground.
## Acknowledgments
The authors would like to thank Drs. Michael McCracken and Zhaoli Guo for
helpful discussions. This work is supported by the Science Foundations of LCP
and CAEP [under Grant Nos. 2009A0102005, 2009B0101012], National Basic
Research Program (973 Program) [under Grant No. 2007CB815105], National
Natural Science Foundation [under Grant Nos. 10775018, 10702010] of China.
## Appendix. Spatial and temporal discretization effects
In our simulations, the time evolution is based on the usual first-order
upwind scheme, while space discretization is performed through a Lax-Wendroff
scheme.
$\frac{\partial}{\partial t}f_{i}=[f_{i}(t+\delta t)-f_{i}(t)]/\delta
t\text{,}$ (30a) $\displaystyle\overrightarrow{v}_{i}\cdot\nabla f_{i}$
$\displaystyle=$ $\displaystyle v_{ix}[f_{i}(x+\delta x,y)-f_{i}(x-\delta
x,y)]/2\delta x$ (30b) $\displaystyle+v_{iy}[f_{i}(x,y+\delta
y)-f_{i}(x,y-\delta y)]/2\delta y$ $\displaystyle-v_{ix}^{2}\delta
t[f_{i}(x+\delta x,y)-2f_{i}(x,y)+f_{i}(x-\delta x,y)]/2\delta x^{2}$
$\displaystyle-v_{iy}^{2}\delta t[f_{i}(x,y+\delta
y)-2f_{i}(x,y)+f_{i}(x,y-\delta y)]/2\delta y^{2}\text{.}$ If we perform the
following series expansion up to second order in Eq.(30a) and Eq.(30b)
$f_{i}(t+\delta t)=f_{i}(t)+\delta t\frac{\partial}{\partial
t}f_{i}+\frac{1}{2}\delta t^{2}\frac{\partial^{2}}{\partial
t^{2}}f_{i}\text{,}$ (31a) $f_{i}(x+\delta x,y,t)=f_{i}(x,y,t)+\delta
x\frac{\partial}{\partial x}f_{i}+\frac{1}{2}\delta
x^{2}\frac{\partial^{2}}{\partial x^{2}}f_{i}\text{,}$ (31b) $f_{i}(x-\delta
x,y,t)=f_{i}(x,y,t)-\delta x\frac{\partial}{\partial x}f_{i}+\frac{1}{2}\delta
x^{2}\frac{\partial^{2}}{\partial x^{2}}f_{i}\text{,}$ (31c) we can get
$\frac{\partial}{\partial t}f_{i}\Rightarrow\frac{\partial}{\partial
t}f_{i}+\frac{1}{2}\delta t\frac{\partial^{2}}{\partial t^{2}}f_{i}\text{,}$
(32a) $\overrightarrow{v}_{i}\cdot\nabla f_{i}\Rightarrow
v_{ix}\frac{\partial}{\partial x}f_{i}+v_{iy}\frac{\partial}{\partial
y}f_{i}-\frac{v_{ix}^{2}\delta t}{2}\frac{\partial^{2}}{\partial
x^{2}}f_{i}-\frac{v_{iy}^{2}\delta t}{2}\frac{\partial^{2}}{\partial
y^{2}}f_{i}\text{,}$ (32b) So a more accurate LB equation solved using the
updating schemes mentioned above is, $\frac{\partial}{\partial
t}f_{i}+\frac{1}{2}\delta t\frac{\partial^{2}}{\partial
t^{2}}f_{i}+v_{i\beta}\frac{\partial}{\partial
r_{\beta}}f_{i}-\frac{1}{2}\delta tv_{i\beta}^{2}\frac{\partial^{2}}{\partial
r_{\beta}^{2}}f_{i}=-\mathbf{S}_{ik}\left[f_{k}-f_{k}^{eq}\right]\text{,}$
(33)
To investigate the effect of the supplementary terms in Eq.(33), a similar
Chapman-Enskog expansion procedure may be considered. The zeroth and first
order LB equations Eq.(7a), Eq.(7b) remain unchanged when performing the
Chapman–Enskog expansion, while the second-order equation (7c) becomes,
$\displaystyle\frac{\partial}{\partial
t_{2}}f_{i}^{(0)}+\frac{\partial}{\partial t_{1}}f_{i}^{(1)}-\frac{\delta
t}{2}\frac{\partial}{\partial
t_{1}}(\mathbf{S}_{il}f_{l}^{(1)})+v_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}}f_{i}^{(1)}+\frac{\delta t}{2}v_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}}(\mathbf{S}_{il}f_{l}^{(1)})+{}$ $\displaystyle\frac{\delta
t}{2}v_{i\alpha}v_{i\beta}\frac{\partial}{\partial
x_{1\alpha}}\frac{\partial}{\partial x_{1\beta}}f_{i}^{(0)}-\frac{\delta
t}{2}v_{i\alpha}^{2}\frac{\partial^{2}}{\partial
x_{1\alpha}^{2}}f_{i}^{(0)}=-\mathbf{S}_{il}f_{l}^{(2)}\text{.}$ (34)
It can be converted into moment space to obtain:
$\displaystyle\frac{\partial}{\partial
t_{2}}\hat{f}_{i}^{(0)}+\frac{\partial}{\partial
t_{1}}\hat{f}_{i}^{(1)}-\frac{\delta t}{2}\frac{\partial}{\partial
t_{1}}(\hat{\mathbf{S}}_{il}\hat{f}_{l}^{(1)})+\hat{\mathbf{E}}_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}}\hat{f}_{i}^{(1)}+\frac{\delta
t}{2}\hat{\mathbf{E}}_{i\alpha}\frac{\partial}{\partial
x_{1\alpha}}(\hat{\mathbf{S}}_{il}\hat{f}_{l}^{(1)})+{}$
$\displaystyle\frac{\delta
t}{2}\hat{\mathbf{E}}_{i\alpha}\hat{\mathbf{E}}_{i\beta}\frac{\partial}{\partial
x_{1\alpha}}\frac{\partial}{\partial x_{1\beta}}\hat{f}_{i}^{(0)}-\frac{\delta
t}{2}\hat{\mathbf{E}}_{i\alpha}^{2}\frac{\partial^{2}}{\partial
x_{1\alpha}^{2}}\hat{f}_{i}^{(0)}=-\hat{\mathbf{S}}_{il}\hat{f}_{l}^{(2)}\text{,}$
(35)
where
$\hat{\mathbf{E}}_{\alpha}=\mathbf{M}(v_{\alpha}\mathbf{I})\mathbf{M}^{-1}$.
From the equation, we obtain
$\frac{\partial\rho}{\partial t_{2}}+\delta t\frac{\partial}{\partial
x_{1}}\frac{\partial}{\partial y_{1}}\hat{f}_{6}^{eq}=0\text{,}$ (36a)
$\frac{\partial j_{x}}{\partial t_{2}}+\frac{1}{2}\frac{\partial}{\partial
x_{1}}(1+\frac{\delta t}{2}s_{5})\hat{f}_{5}^{(1)}+\frac{\partial}{\partial
y_{1}}(1+\frac{\delta t}{2}s_{6})\hat{f}_{6}^{(1)}+\frac{\delta
t}{4}\frac{\partial}{\partial x_{1}}\frac{\partial}{\partial
y_{1}}(\hat{f}_{10}^{eq}+2\hat{f}_{8}^{eq})=0\text{,}$ (36b) $\frac{\partial
j_{y}}{\partial t_{2}}+\frac{\partial}{\partial x_{1}}(1+\frac{\delta
t}{2}s_{6})\hat{f}_{6}^{(1)}-\frac{1}{2}\frac{\partial}{\partial
y_{1}}(1+\frac{\delta t}{2}s_{5})\hat{f}_{5}^{(1)}+\frac{\delta
t}{4}\frac{\partial}{\partial x_{1}}\frac{\partial}{\partial
y_{1}}(2\hat{f}_{7}^{eq}-\hat{f}_{9}^{eq})=0\text{,}$ (36c) $\frac{\partial
e}{\partial t_{2}}+\frac{\partial}{\partial x_{1}}(1+\frac{\delta
t}{2}s_{7})\hat{f}_{7}^{(1)}+\frac{\partial}{\partial y_{1}}(1+\frac{\delta
t}{2}s_{8})\hat{f}_{8}^{(1)}+\frac{\delta t}{2}\frac{\partial}{\partial
x_{1}}\frac{\partial}{\partial y_{1}}\hat{f}_{14}^{eq}=0\text{.}$ (36d) In
this way the recovered NS equations are as follows:
$\frac{\partial\rho}{\partial t}+\frac{\partial j_{x}}{\partial
x}+\frac{\partial j_{y}}{\partial y}=-\delta t\frac{\partial}{\partial
x}\frac{\partial}{\partial y}(j_{x}j_{y}/\rho)\text{,}$ (37a) $\frac{\partial
j_{x}}{\partial t}+\frac{\partial}{\partial
x}\left(j_{x}^{2}/\rho\right)+\frac{\partial}{\partial
y}\left(j_{x}j_{y}/\rho\right)=-\frac{\partial P}{\partial
x}+\frac{\partial}{\partial x}[\mu_{s}^{{}^{\prime}}(\frac{\partial
u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial
y})]+\frac{\partial}{\partial y}[\mu_{v}^{{}^{\prime}}(\frac{\partial
u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y})]-\delta
t\frac{\partial}{\partial x}\frac{\partial}{\partial y}[(\rho
u_{x}^{2}+P)u_{y}]\text{,}$ (37b) $\frac{\partial j_{y}}{\partial
t}+\frac{\partial}{\partial
x}\left(j_{x}j_{y}/\rho\right)+\frac{\partial}{\partial
y}\left(j_{y}^{2}/\rho\right)=-\frac{\partial P}{\partial
y}+\frac{\partial}{\partial x}[\mu_{v}^{{}^{\prime}}(\frac{\partial
u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial
y})]-\frac{\partial}{\partial y}[\mu_{s}^{{}^{\prime}}(\frac{\partial
u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial y})]-\delta
t\frac{\partial}{\partial x}\frac{\partial}{\partial y}[(\rho
u_{y}^{2}+P)u_{x}]\text{,}$ (37c) $\displaystyle\frac{\partial e}{\partial
t}+\frac{\partial}{\partial x}[(e+P)j_{x}/\rho]+\frac{\partial}{\partial
y}[(e+P)j_{y}/\rho]$ (37d) $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x}[\lambda_{1}^{{}^{\prime}}(2\frac{\partial T}{\partial
x}+u_{y}\frac{\partial u_{y}}{\partial x}+u_{x}\frac{\partial u_{x}}{\partial
x}-u_{x}\frac{\partial u_{y}}{\partial y}+u_{y}\frac{\partial u_{x}}{\partial
y})]$ $\displaystyle+\frac{\partial}{\partial
y}[\lambda_{2}^{{}^{\prime}}(2\frac{\partial T}{\partial
y}+u_{x}\frac{\partial u_{x}}{\partial y}-u_{y}\frac{\partial u_{x}}{\partial
x}+u_{x}\frac{\partial u_{y}}{\partial x}+u_{y}\frac{\partial u_{y}}{\partial
y})]$ $\displaystyle-\delta t\frac{\partial}{\partial
x}\frac{\partial}{\partial y}[(3P+\frac{1}{2}\rho u^{2})u_{x}u_{y}]\text{,}$
where $\mu_{s}^{{}^{\prime}}=\rho RT(\frac{1}{s_{5}}+\frac{\delta
t}{2})\text{, }\mu_{v}^{{}^{\prime}}=\rho RT(\frac{1}{s_{6}}+\frac{\delta
t}{2})\text{, }\lambda_{1}^{{}^{\prime}}=\rho
R^{2}T(\frac{1}{s_{7}}+\frac{\delta t}{2})\text{,
}\lambda_{2}^{{}^{\prime}}=\rho R^{2}T(\frac{1}{s_{8}}+\frac{\delta
t}{2})\text{.}$
When $\delta t$ approaches $0$, equations (37a)-(37d) reduce to the
Eqs.(13a)-(13d).
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arxiv-papers
| 2009-08-26T07:54:18 |
2024-09-04T02:49:04.804306
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng Chen, Aiguo Xu, Guangcai Zhang, Yingjun Li",
"submitter": "Aiguo Xu Dr.",
"url": "https://arxiv.org/abs/0908.3749"
}
|
0908.3776
|
# The CSM extension for description of the positive and negative parity bands
in even-odd nuclei
A. A. Radutaa),b),c), C. M. Radutab) and Amand Faesslerd) a) Department of
Theoretical Physics and Mathematics, Bucharest University, Bucharest, POBox
MG11, Romania b) Department of Theoretical Physics, Institute of Physics and
Nuclear Engineering, Bucharest, POBox MG6, Romania c) Academy of Romanian
Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania d)Institut
für Theoretische Physik der Universität Tübingen, Auf der Morgenstelle 14,
Germany
###### Abstract
A particle-core Hamiltonian is used to describe the lowest parity partner
bands $K^{\pi}=1/2^{\pm}$ in 219Ra, 237U and 239Pu, and three parity partner
bands, $K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$, in 227Ra. The core is
described by a quadrupole and octupole boson Hamiltonian which was previously
used for the description of four positive and four negative parity bands in
the neighboring even-even isotopes. The particle-core Hamiltonian consists of
four terms: a quadrupole-quadrupole, an octupole-octupole, a spin-spin and a
rotational $\hat{I}^{2}$ interaction, with $\hat{I}$ denoting the total
angular momentum. The single particle space for the odd nucleon consists of
three spherical shell model states, two of positive and one of negative
parity. The product of these states with a collective deformed ground state
and the intrinsic gamma band state generate, through angular momentum
projection, the bands with $K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$,
respectively. In the space of projected states one calculates the energies of
the considered bands. The resulting excitation energies are compared with the
corresponding experimental data as well as with those obtained with other
approaches. Also, we searched for some signatures for a static octupole
deformation in the considered odd isotopes. The calculated branching ratios in
219Ra agree quite well with the corresponding experimental data.
###### pacs:
21.10.Re,21.60.Ev,27.80.+w,27.90.+b
## I Introduction
The coherent state model (CSM)Rad81 describes in a realistic fashion three
interacting bands, ground, beta and gamma, in terms of quadrupole bosons. The
formalism was later extended Rad97 ; Rad02 ; Rad03 ; Rad003 ; Rad06 ; Rad006
by considering the octupole degrees of freedom. The most recent extension
describes eight rotational bands, four of positive and four of negative
parity. Observable like excitation energies, intraband E2 and interband E1, E2
and E3 reduced transition probabilities have been calculated and the results
were compared with the corresponding experimental data. The formalism works
well for both near spherical and deformed nuclei excited in low and high
angular momentum states. Indeed, we considered all states with $J\leq 30$ in
both, the positive and the negative parity bands. Signatures for a static
octupole deformation in ground as well as in excited bands have been pointed
out in several even-even nuclei.
The aim of this paper is to extend CSM for the even-odd nuclei which exhibit
both quadrupole and octupole deformation.
The formalism concerning the excitation energies in the positive and negative
parity bands is presented in Sections II and III. The E1 and E2 transitions
are considered in Section IV, while the numerical application to four even-odd
nuclei, is described in Section V. The final conclusions are drawn in Section
VI.
## II The model Hamiltonian
We suppose that the rotational bands in even-odd nuclei may be described by a
particle-core Hamiltonian:
$H=H_{sp}+H_{core}+H_{pc},$ (2.1)
where $H_{sp}$ is a spherical shell model Hamiltonian associated to the odd
nucleon, while $H_{core}$ is a phenomenological Hamiltonian which describes
the collective motion of the core in terms of quadrupole and octupole bosons.
This term is identical to that used in Ref.Rad006 to describe eight
rotational bands in even-even nuclei. The two subsystems interact with each
other by $H_{pc}$, which has the following expression:
$\displaystyle H_{pc}=$ $\displaystyle-$ $\displaystyle
X_{2}\sum_{\mu}r^{2}Y_{2,-\mu}(-)^{\mu}\left(b^{\dagger}_{2\mu}+(-)^{\mu}b_{2,-\mu}\right)$
(2.2) $\displaystyle-$ $\displaystyle
X_{3}\sum_{\mu}r^{3}Y_{3,-\mu}(-)^{\mu}\left(b^{\dagger}_{3\mu}+(-)^{\mu}b_{3,-\mu}\right)$
$\displaystyle+$ $\displaystyle
X_{jJ}\vec{j}\cdot\vec{J}+X_{I^{2}}\vec{I}^{2}.$
$b^{\dagger}_{\lambda\mu}$ denotes the components of the $\lambda$-pole (with
$\lambda$=2,3) boson operator. The term $\vec{j}\cdot\vec{J}$ is similar to
the spin-orbit interaction from the shell model and expresses the interaction
between the angular momenta of the odd-particle and the core. The last term is
due to the rotational motion of the whole system, $\vec{I}$ denoting the total
angular momentum of the particle-core system.
The core states are described by eight sets of mutually orthogonal functions,
obtained by projecting out the angular momentum and the parity from four
quadrupole and octupole deformed functions: one is a product of two coherent
states:
$\Psi_{g}=e^{f(b_{30}^{+}-b_{30})}e^{d(b^{+}_{20}-b_{20})}|0\rangle_{2}|0\rangle_{3}\equiv\Psi_{o}\Psi_{q}|0\rangle_{2}|0\rangle_{3},$
(2.3)
while the remaining three are polynomial boson excitations of $\Psi_{g}$. The
parameters $d$ and $f$ are real numbers and simulate the quadrupole and
octupole deformations, respectively. The vacuum state for the $\lambda$-pole
boson, $\lambda=2,3$, is denoted by $|0\rangle_{\lambda}$.
The particle-core interaction generates a deformation for the single particle
trajectories. Indeed, averaging the model Hamiltonian with $\Psi_{g}$, one
obtains a deformed single particle Hamiltonian, $H_{mf}$ which plays the role
of the mean field for the particle motion:
$H_{mf}={\cal C}+H_{sp}-2dX_{2}r^{2}Y_{20}-2fX_{3}r^{3}Y_{30},$ (2.4)
where ${\cal C}$ is a constant determined by the average of $H_{core}$. The
Hamiltonian $H_{mf}$ represents an extension of the Nilsson Hamiltonian by
adding the octupole deformation term. In Ref.Rad99 we have shown that in
order to get the right deformation dependence of the single particle energies,
$H_{mf}$ must be amended with a monopole-monopole interaction,
$M\omega^{2}r^{2}\alpha_{00}Y_{00}$, where the monopole coordinate
$\alpha_{00}$ is determined from the volume conservation restriction. This
term has a constant contribution within a band. The constant value is,
however, band dependent.
In order to find the eigenvalues of the model Hamiltonian we follow several
steps:
1) In principle the single particle basis could be determined by diagonalizing
$H_{mf}$ amended with the monopole interaction. The product basis for particle
and core may be further used to find the eigenvalues of $H$. Due to some
technical difficulties in restoring the rotation and space reversal symmetries
for the composite system wave function, this procedure is however tedious and
therefore we prefer a simpler method. Thus, the single particle space consists
of three spherical shell model states with angular momenta
$j_{1},j_{2},j_{3}$. We suppose that $j_{1}$ and $j_{2}$ have the parity
$\pi=+$, while $j_{3}$ has a negative parity $\pi=-$. Due to the quadrupole-
quadrupole interaction the odd particle from the state $j_{1}$ can be promoted
to $j_{2}$ and vice-versa. The octupole-octupole interaction connects the
states $j_{1}$ and $j_{2}$ with $j_{3}$. Due to the above mentioned effects,
the spherical and space reversal symmetries of the single particle motion are
broken. To be more specific, by diagonalizing $H$ (2.1) in a projected
spherical particle-core basis with the spherical single particle state factors
mentioned above, the eigenstates could be written as a projected spherical
particle-core state having as single particle state factor a function without
good rotation and parity symmetries. Therefore, one could start with a coupled
basis where the single particle state is a linear combination of the spherical
states, where the mixing coefficients are to be determined by a least square
fitting procedure as to obtain an optimal description of the experimental
excitation energies. Thus, instead of dealing with a spherical shell model
state coupled to a deformed core without reflection symmetry, as the
traditional particle-core approaches proceed, here the single particle orbits
are lacking the spherical and space reversal symmetries and by this, their
symmetry properties are consistent with those of the phenomenological core.
2) We remark that $\Psi_{g}$ is a sum of two states of different parities.
This happens due to the specific structure of the octupole coherent state:
$\Psi_{o}=\Psi_{o}^{(+)}+\Psi_{o}^{(-)}.$ (2.5)
The states of a given angular momentum and positive parity can be obtained
through projection from the intrinsic states:
$|n_{1}l_{1}j_{1}K\rangle\Psi^{(+)}_{o}\Psi_{q},\;\;|n_{2}l_{2}j_{2}K\rangle\Psi^{(+)}_{o}\Psi_{q},\;\;|n_{3}l_{3}j_{3}K\rangle\Psi^{(-)}_{o}\Psi_{q}.$
(2.6)
The projected states of negative parity are obtained from the states:
$|n_{1}l_{1}j_{1}K\rangle\Psi^{(-)}_{o}\Psi_{q},\;\;|n_{2}l_{2}j_{2}K\rangle\Psi^{(-)}_{o}\Psi_{q},\;\;|n_{3}l_{3}j_{3}K\rangle\Psi^{(+)}_{o}\Psi_{q}.$
(2.7)
The angular momentum and parity projected states are denoted by:
$\displaystyle\varphi^{(+)}_{IM}(j_{i}K;d,f)$ $\displaystyle=$ $\displaystyle
N^{(+)}_{i;IK}P^{I}_{MK}|n_{i}l_{i}j_{i}K\rangle\Psi^{(+)}_{o}\Psi_{q}\equiv
N^{(+)}_{i;IK}\psi^{(+)}_{IM}(j_{i}K;d,f),i=1,2,$
$\displaystyle\varphi^{(+)}_{IM}(j_{3}K;d,f)$ $\displaystyle=$ $\displaystyle
N^{(+)}_{3;IK}P^{I}_{MK}|n_{3}l_{3}j_{3}K\rangle\Psi^{(-)}_{o}\Psi_{q}\equiv
N^{(+)}_{3;IK}\psi^{(+)}_{IM}(j_{3}K;d,f),$
$\displaystyle\varphi^{(-)}_{IM}(j_{i}K;d,f)$ $\displaystyle=$ $\displaystyle
N^{(-)}_{i;IK}P^{I}_{MK}|n_{i}l_{i}j_{i}K\rangle\Psi^{(-)}_{o}\Psi_{q}\equiv
N^{(-)}_{i;IK}\psi^{(-)}_{IM}(j_{i}K;d,f),i=1,2,$
$\displaystyle\varphi^{(-)}_{IM}(j_{3}K;d,f)$ $\displaystyle=$ $\displaystyle
N^{(-)}_{3;IK}P^{I}_{MK}|n_{3}l_{3}j_{3}K\rangle\Psi^{(+)}_{o}\Psi_{q}\equiv
N^{(-)}_{3;IK}\psi^{(-)}_{IM}(j_{3}K;d,f).$ (2.8)
The factors $N^{(\pm)}_{i,IK}$ assure that the projected states
$\varphi^{(\pm)}$ are normalized to unity. Obviously the unnormalized
projected states are denoted by $\psi^{(\pm)}$. For the quantum number $K$ we
consider the lowest three values, i.e. $K=1/2,3/2,5/2$. Note that the earlier
particle-core approaches RaCea ; Lea restrict the single particle space to a
single $j$, which results in eliminating the contribution of the octupole-
octupole interaction.
3) Note that for a given $j_{i}$, the projected states with different $K$ are
not orthogonal. Indeed, the overlap matrices :
$\displaystyle A^{(+)}_{K,K^{\prime}}(Ij_{l};d,f)$ $\displaystyle=$
$\displaystyle\langle\psi^{(+)}_{IM}(j_{l}K;d,f)|\psi^{(+)}_{IM}(j_{l}K^{\prime};d,f)\rangle,$
$\displaystyle l$ $\displaystyle=$ $\displaystyle
1,2,3;\;K,K^{\prime}=1/2,3/2,5/2,$ $\displaystyle
A^{(-)}_{K,K^{\prime}}(Ij_{l};d,f)$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{IM}(j_{l}K;d,f)|\psi^{(-)}_{IM}(j_{l}K^{\prime};d,f)\rangle,$
$\displaystyle l$ $\displaystyle=$ $\displaystyle
1,2,3;\;K,K^{\prime}=1/2,3/2,5/2,$ (2.9)
are not diagonal. By diagonalization, one obtains the eigenvalues
$a^{(\pm)}_{Ip}(j_{l})$ and the corresponding eigenvectors
$V^{(\pm)}_{IK}(j_{l},p)$, with $K=1/2,3/2,5/2$ and $p=1,2,3$. Then, the
functions:
$\displaystyle\Psi^{(+)}_{IM}(j_{l},p;d,f)=N^{(+)}_{l;Ip}\sum_{K}V^{(+)}_{IK}(j_{l},p)\psi^{(+)}_{IM}(j_{l}K;d,f),$
$\displaystyle\Psi^{(-)}_{IM}(j_{l},p;d,f)=N^{(-)}_{l;Ip}\sum_{K}V^{(-)}_{IK}(j_{l},p)\psi^{(-)}_{IM}(j_{l}K;d,f),$
(2.10)
are mutually orthogonal. The norms are given by:
$\left(N^{(\pm)}_{l;Ip}\right)^{-1}=\sqrt{a^{(\pm)}_{Ip}(j_{l})}.$ (2.11)
For each of the new states, there is a term in the defining sum (2.10), which
has a maximal weight. The corresponding quantum number $K$ is conventionally
assigned to the mixed state.
4) In order to simulate the core deformation effect on the single particle
motion, in some cases the projected states corresponding to different $j$ must
be mixed up.
$\displaystyle\Phi^{(+)}_{IM}(p;d,f)=\sum_{l=1,2,3}{\cal
A}^{(+)}_{pl}\Psi^{(+)}_{IM}(j_{l}p;d,f),$
$\displaystyle\Phi^{(-)}_{IM}(p;d,f)=\sum_{l=1,2,3}{\cal
A}^{(-)}_{pl}\Psi^{(-)}_{IM}(j_{l}p;d,f).$ (2.12)
The amplitudes ${\cal A}^{(\pm)}_{pl}$ can be obtained either by diagonalizing
$H_{mf}$ or, as we mentioned before, by a least square fitting procedure
applied to the excitation energies.
The energies of the odd system are approximated by the average values of the
model Hamiltonian corresponding to the projected states:
$\displaystyle
E^{(+)}_{I}(p;d,f)=\langle\Phi^{(+)}_{IM}(p;d,f)|H|\Phi^{(+)}_{IM}(p;d,f)\rangle,$
$\displaystyle
E^{(-)}_{I}(p;d,f)=\langle\Phi^{(-)}_{IM}(p;d,f)|H|\Phi^{(-)}_{IM}(p;d,f)\rangle.$
(2.13)
The matrix elements involved in the above equations can be analytically
calculated. Note that due to the structure of the particle-core projected
states, the energies for the odd system are determined by the coupling of the
odd particle to the excited states of the core ground band.
The approach presented in this section was applied for the description of the
$K^{\pi}=1/2^{\pm}$ bands. However, this procedure can be extended by
including the $K\neq 0$ states in the space describing the deformed core.
## III Description of the $K^{\pi}=\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}$
bands.
In principle the method presented in the previous section may work for the
description of bands with the quantum number $K$ larger than $1/2$. However
the intrinsic reference frame for the odd system is determined by the deformed
core and therefore one expects that this brings an important contribution to
the quantum number $K$. To be more specific, we cannot expect that projecting
out the good angular momentum from $|j,5/2\rangle\otimes\Psi_{g}$, a realistic
description of the $K=5/2$ bands is obtained. Therefore we assume that the
$K^{\pi}=\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}$ bands are described by
projecting out the angular momentum from a product state of a low $K$ single
particle state and the intrinsic gamma band state.
We recall that within CSM, the states of the gamma band are obtained by
projection from the intrinsic state:
$\Psi^{(\gamma;\pm)}_{2}=\Omega^{\dagger}_{\gamma,2}\Psi^{(\pm)}_{o}\Psi_{q},$
(3.1)
where the excitation operator for the gamma intrinsic state is defined as:
$\Omega^{\dagger}_{\gamma,2}=\left(b^{\dagger}_{2}b^{\dagger}_{2}\right)_{22}+d\sqrt{\frac{2}{7}}b^{\dagger}_{22}.$
(3.2)
The low index of $\Psi$ in Eq. (3.1) is the the $K$ quantum number for the
$\gamma$ intrinsic state. Thus, a simultaneous description of the bands with
$K=1/2,3/2,5/2$ can be achieved with the projected states:
$\displaystyle\varphi^{(\pm)}_{IM;1/2}$ $\displaystyle=$ $\displaystyle
N^{(\pm)}_{I,1/2}\sum_{J}\left(N^{(g,\pm)}_{J}\right)^{-1}C^{j_{1}~{}J~{}I}_{1/2\;0\;1/2}\left[|n_{1}l_{1}j_{1}\rangle\otimes\varphi^{(g;\pm)}_{J}\right]_{IM},$
$\displaystyle\varphi^{(\pm)}_{IM;3/2}$ $\displaystyle=$ $\displaystyle
N^{(\pm)}_{I,3/2}\sum_{J}\left(N^{(\gamma,\pm)}_{J}\right)^{-1}C^{\;\;j_{2}~{}\;\;J~{}\;I}_{-1/2\;\;2\;3/2}\left[|n_{2}l_{2}j_{2}\rangle\otimes\varphi^{(\gamma;\pm)}_{J}\right]_{IM},$
$\displaystyle\varphi^{(\pm)}_{IM;5/2}$ $\displaystyle=$ $\displaystyle
N^{(\pm)}_{I,5/2}\sum_{J}\left(N^{(\gamma,\pm)}_{J}\right)^{-1}C^{j_{3}~{}J~{}I}_{1/2\;2\;5/2}\left[|n_{3}l_{3}j_{3}\rangle\otimes\varphi^{(\gamma;\mp)}_{J}\right]_{IM}.$
(3.3)
In the above expressions the notation $N^{(i,\pm)}_{J}$ with $i=g,\gamma$ is
used for the normalization factors of the projected states describing the
ground and the gamma bands, respectively, of the even-even core. Note that for
each angular momentum $I$ the above set of three projected states is
orthogonal.
The energies for the six bands with $K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$
are obtained by averaging the model Hamiltonian (2.1) with the projected
states defined above.
$E_{I,K}=\langle\varphi^{(\pm)}_{IM;K}|H|\varphi^{(\pm)}_{IM;K}\rangle,K=1/2,3/2,5/2.$
(3.4)
The matrix elements of the particle-core interaction are given in Appendix A
## IV Transition probabilities
For some $K=1/2$ bands, results for the reduced $E1$ and $E2$ transition
probabilities are available. They are given in terms of the branching ratios:
$R_{I^{\pi}}=\frac{B(E1;I^{\pi}\to(I-1)^{\pi^{{}^{\prime}}})}{B(E2;I^{\pi}\to(I-2)^{\pi})},\pi^{{}^{\prime}}\neq\pi.$
(4.1)
To describe these data we use the wave functions defined in Section II. We
recall that the positive parity states are obtained by coupling the spherical
shell model state $j_{1}$ or $j_{2}$ to a positive parity core with a small
admixture of a state coupling $j_{3}$ and a negative parity core. On the other
hand the negative parity states are given by coupling one of the states
$j_{1}$ or $j_{2}$ to a negative parity core and a small component consisting
in a product state of $j_{3}$ and a positive parity core-state. Thus, the
single particle E1 transition operator may connect the leading term of the
initial state with the small component of the final state. One expects that
the contribution of this term to the E1 transition is negligible comparing it
with the contribution of collective dipole operator. Therefore the dipole
transition operator considered in the present paper is the boson operator:
$Q_{1\mu}=eq_{1}\left((b^{\dagger}_{2}b^{\dagger}_{3})_{1\mu}+(b_{3}b_{2})_{\widetilde{1\mu}}\right).$
(4.2)
Concerning the quadrupole transition operator, this has the structure:
$Q_{2\mu}=eq_{2}\left(b^{\dagger}_{2\mu}+(-)^{\mu}b_{2,-\mu}+ar^{2}Y_{2\mu}\right).$
(4.3)
The branching ratio (4.1) for the initial state $I^{\pi}$ is:
$R_{I^{\pi}}=\left[\frac{\langle
I^{\pi}||Q_{1}||(I-1)^{\pi^{{}^{\prime}}}\rangle}{\langle
I^{\pi}||Q_{2}||(I-2)^{\pi}\rangle}\right]^{2}.$ (4.4)
Here the initial and final states are mixture of different $K$ states as well
as mixture of the $j$ states defined by Eq.(2.12). The matrix elements of the
transition operators between the basis states are given in Appendix B1
11footnotetext: Throughout this paper the reduced matrix elements are defined
according to Rose’s convention Rose ..
## V Numerical results
The results obtained in Section II have been used to calculate the excitation
energies for one positive and one negative parity bands in three even-odd
isotopes: 219Ra, 237U and 239Pu. The parameters defining $H_{core}$, as well
as the deformation parameters $d$ and $f$ are those used to describe the
properties of eight rotational bands in the even-even neighboring isotopes.
The single particle states are spherical shell model states with the
appropriate parameters for the $(N,Z)$ region of the considered isotopes Ring
. Our calculations for the mentioned odd isotopes correspond to the single
particle states: $(j_{1},j_{2},j_{3})=(2g_{7/2},2g_{9/2},1h_{9/2})$. In order
to obtain the best agreement between the calculated excitation energies and
the corresponding experimental data, in the expansion (2.12) a small admixture
of the states $(j_{1};j_{3})$ and $(j_{2};j_{3})$ was considered: $|{\cal
A}^{(+)}_{i,3}|^{2}$ and $|{\cal A}^{(-)}_{i,3}|^{2}$, are both equal to 0.001
for for 219Ra, while for 237U and 239Pu the amplitudes take the common value:
$|A^{(+)}_{i,3}|^{2}=|A^{(-)}_{1,3}|^{2}=0.04$. The mixing amplitude of the
states $(j_{1},j_{2})$ is negligible small. Energies (2.13) depend on the
interaction strengths $X_{2},X_{3},X_{jJ}$ and $X_{I^{2}}$. These were
determined by fitting four particular energies in the two bands of different
parities, i.e. $K^{\pi}=\frac{1}{2}^{\pm}$. The results of the fitting
procedure are given in Table I. Inserting these in Eqs. (2.13), the energies
in the two bands with $K=1/2$ are readily obtained.
$E(I^{\pm})=E^{(\pm)}_{I}(1;d,f)-E^{(+)}_{\frac{1}{2}}(1;d,f).$ (5.1)
The theoretical results for excitation energies, listed in Tables II and III,
agree quite well with the corresponding experimental data. The levels for the
three isotopes have been populated by different experiments. Indeed, the
$K^{\pi}=1/2^{\pm}$ bands have been identified in 219Ra with the reaction
208Pb(14C,3n)219Ra Cottle , in 237U via a pickup reaction on a 238U target,
while in 239Pu with the so-called ”unsafe” Coulomb excitation technique Zhu .
Our results suggest that the dominant $K$ component is $K=1/2$ while the
dominant $j$ component is $g_{9/2}$. To have a measure for the agreement
quality, we calculated the r.m.s. values for the deviations of the calculated
values from the experimental ones. The results for 219Ra, 237U and 239Pu are
66.24 keV, 48.97 keV and 31.8 keV, respectively. In calculating the $r.m.s.$
value for 219Ra we ignored the data for the states $53/2^{\pm}$ since the spin
assignment is unsure. It is interesting to mention that the spectrum of 219Ra
has been measured by two groups Cottle ; Wiel by the same reaction,
208Pb(14C,3n)219Ra. However they assign for the ground state different angular
momenta, $9/2^{+}$ Cottle and $7/2^{+}$Wiel . In our approach both
assignments yield good description of the data. However we made the option for
$9/2^{+}$ since the corresponding results agree better with the experimental
data than those obtained with the other option. The results and experimental
data for 219Ra are plotted in Fig.1.
The case of 227Ra was treated with the formalism presented in Section III. The
single particle basis is: $2g_{7/2},2g_{9/2},2f_{5/2}$. The first state
coupled to the coherent state describing the unprojected ground state, i.e.
$2g_{7/1,1/2}\Psi_{g}$, generates the parity partner bands
$K^{\pi}=1/2^{\pm}$. The bands $K^{\pi}=3/2^{\pm}$ are obtained through
projection from the product state $2g_{9/2,-1/2}\Psi^{(\gamma;\pm)}_{2}$ while
the bands $K^{\pi}=5/2^{\pm}$ originate from the intrinsic state
$2f_{5/2,1/2}\Psi^{(\gamma;\mp)}_{2}$. Concerning the bands characterized by
$K^{\pi}=1/2^{\pm}$ one could consider also the mixing of components with
different K in the manner discussed in Section II. However, our numerical
application suggests that such a mixing is not really necessary in order to
obtain a realistic description of the available data. The calculated energies
in the three bands are compared with the corresponding experimental data in
Fig.2. The plotted values are collected in Table IV. The states for 227Ra have
been obtained in Ref.Egidy by using the $(n,\gamma),(d,p)$ and $(\vec{t},d)$
reactions and the $\beta^{-}$ decay of 227Fr. The spectrum yielded by the
mentioned experiments was interpreted in Ref. LeaChen in terms of a particle-
core interaction.
Parameters | 219Ra | 227Ra | 237U | 239Pu
---|---|---|---|---
$X_{2}b^{2}$[keV] | 22.714 | -1.992 | 1.080 | -2.515
$X_{3}b^{3}$[keV] | -8.823 | 169.511 | 2.227 | 4.937
$X_{jJ}$[keV] | -0.230 | 8.553 | -5.817 | -3.985
$X_{I^{2}}$[keV] | 3.778 | 4.390 | 4.634 | 5.050
Table 1: Parameters involved in the particle-core Hamiltonian obtained by fitting four excitation energies. Here $b$ denotes the oscillator length: $b=(\frac{\hbar}{M\omega})^{1/2};\,\hbar\omega=41A^{-1/3}$. The usual notations for nucleon mass (M) and atomic number (A) were used. | 219Ra
---|---
| $\pi=+$ | $\pi=-$
J | Exp. | Th. | Exp. | Th.
9/2 | 0.0 | 0.0 | |
13/2 | 234.3 | 235.4 | |
15/2 | | | 495.4 | 496.0
17/2 | 529.1 | 526.7 | |
19/2 | | | 733.7 | 729.1
21/2 | 876.6 | 863.4 | |
23/2 | | | 1035.6 | 1029.0
25/2 | 1271.6 | 1238.1 | |
27/2 | | | 1393.6 | 1388.4
29/2 | 1684.7 | 1646.8 | |
31/2 | | | 1815.6 | 1800.2
33/2 | 2113.4 | 2088.4 | |
35/2 | | | 2272.1 | 2258.2
37/2 | 2563.6 | 2564.2 | |
39/2 | | | 2750.8 | 2756.7
41/2 | 3029.0 | 3076.5 | |
43/2 | | | 3255.8 | 3291.6
45/2 | 3505.0 | 3627.8 | |
47/2 | | | 3776.5 | 3859.8
49/2 | 4009.6 | 4219.9 | |
51/2 | | | 4328.9 | 4459.6
53/2 | 4540.4 | 4784.7 | |
55/2 | | | 4913.6 | 5078.5
Table 2: Excitation energies in 219Ra for the bands characterized by $K^{\pi}=\frac{1}{2}^{+}$ and $K^{\pi}=\frac{1}{2}^{-}$ respectively, are given in keV. The results of our calculations (Th.) are compared with the corresponding experimental data (Exp.) taken from Ref.Cottle . | 237U | 239Pu
---|---|---
| $\pi=+$ | $\pi=-$ | $\pi=+$ | $\pi=-$
J | Exp. | Th | Exp. | Th. | Exp. | Th. | Exp. | Th.
1/2 | 0.0 | 0.0 | | 398.5 | 0.0 | 0.0 | 469.8 | 469.8
3/2 | 11.4 | 11.4 | | 454.4 | 7.9 | 7.9 | 492.1 | 477.7
5/2 | 56.3 | 74.6 | | 475.5 | 57.3 | 62.8 | 505.6 | 498.3
7/2 | 82.9 | 106.9 | | 550.3 | 75.7 | 108.4 | 556.0 | 549.8
9/2 | 162.3 | 191.2 | | 581.3 | 163.8 | 183.5 | 583.0 | 572.0
11/2 | 204.1 | 231.8 | | 680.9 | 193.5 | 222.0 | 661.2 | 655.2
13/2 | 317.3 | 347.7 | | 721.9 | 318.5 | 338.1 | 698.7 | 685.7
15/2 | 375.1 | 393.1 | 846.4 | 846.4 | 359.2 | 386.5 | 806.4 | 799.9
17/2 | 518.2 | 544.2 | 930.0 | 899.1 | 519.5 | 534.9 | 857.5 | 839.5
19/2 | 592.0 | 592.0 | 1027.5 | 1046.6 | 570.9 | 592.2 | 992.5 | 984.2
21/2 | 762.8 | 780.3 | 1131.0 | 1113.3 | 764.7 | 773.7 | 1058.1 | 1033.3
23/2 | 853.0 | 829.0 | 1250.7 | 1281.3 | 828.0 | 839.2 | 1219.4 | 1208.3
25/2 | 1048.7 | 1065.8 | 1376.1 | 1364.8 | 1053.1 | 1054.4 | 1300.9 | 1267.2
27/2 | 1155.1 | 1108.8 | 1515.7 | 1550.2 | 1127.8 | 1127.8 | 1487.4 | 1472.2
29/2 | 1372.2 | 1378.3 | 1662.3 | 1654.0 | 1381.5 | 1377.0 | 1584.9 | 1541.2
31/2 | 1494.1 | 1421.6 | 1821.8 | 1852.8 | 1467.8 | 1458.0 | 1795.4 | 1776.0
33/2 | 1729.2 | 1728.7 | 1987.7 | 1981.0 | 1748.5 | 1744.2 | 1908.9 | 1855.4
35/2 | 1868.2 | 1772.5 | 2166.5 | 2188.9 | 1847.0 | 1831.3 | 2143.4 | 2119.8
37/2 | 2117.2 | 2117.2 | 2349.7 | 2346.1 | 2152.2 | 2150.2 | 2272.0 | 2209.8
39/2 | 2272.2 | 2161.7 | 2547.5 | 2558.3 | 2263.0 | 2245.0 | 2529.4 | 2503.6
41/2 | 2530.1 | 2544.1 | 2746.7 | 2749.4 | 2590.1 | 2597.9 | 2672.0 | 2604.4
43/2 | 2702.5 | 2589.4 | 2960.5 | 2960.5 | 2714.0 | 2700.5 | 2951.4 | 2927.5
45/2 | 2963.8 | 3009.5 | 3174.7 | 3191.3 | 3060.1 | 3087.5 | 3108.0 | 3039.3
47/2 | 3154.5 | 3055.6 | 3401.5 | 3395.3 | 3198.0 | 3198.0 | 3407.0 | 3395.3
49/2 | 3415.8 | 3513.7 | 3630.0 | 3671.7 | 3559.1 | 3619.1 | 3578.0 | 3514.4
51/2 | 3625.5 | 3560.5 | 3865.0 | 3862.4 | 3713.0 | 3737.0 | 3895.0 | 3895.8
53/2 | 3886.8 | 4057.8 | 4105.0 | 4190.9 | 4087.1 | 4194.0 | 4080.0 | 4029.9
55/2 | 4115.0 | 4104.8 | 4344.0 | 4350.0 | 4256.0 | 4319.8 | 4413.0 | 4436.7
Table 3: Excitation energies in 237U and 239Pu, for the bands characterized by $K^{\pi}=\frac{1}{2}^{+}$ and $K^{\pi}=\frac{1}{2}^{-}$ respectively, are given in keV. The results of our calculations (Th.) are compared with the corresponding experimental data (Exp.) taken from Ref.Zhu . Figure 1: Calculated (Th.) and experimental (Exp.) excitation energies for the $K^{\pi}=\frac{1}{2}^{\pm}$ bands in 219Ra. The data were taken from Ref.Cottle . Figure 2: Calculated and experimental excitation energies for the bands with $K^{\pi}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}$ in 227Ra. The experimental data were taken from Ref.Egidy . | 227Ra
---|---
| K=1/2 | K=3/2 | K=5/2
J | $\pi=+$ | $\pi=-$ | $\pi=+$ | $\pi=-$ | $\pi=+$ | $\pi=-$
| Exp. | Th. | Exp. | Th. | Exp. | Th. | Exp. | Th. | Exp. | Th. | Exp. | Th.
1/2 | 121 | 96.6 | 297 | 251.8 | | | | | | | |
3/2 | 161 | 145.5 | 284 | 232.4 | 0.0 | 0.0 | 90 | 90 | | | |
5/2 | 177 | 177.0 | | 359.1 | 26 | 26.0 | 102 | 102 | 2 | 2. | | 107.6
7/2 | 268 | 283.6 | | 310.6 | 64 | 40.33 | | 104.6 | 26. | 26.5 | | 86.6
9/2 | 300 | 304.6 | | | | 66.2 | | 115.1 | 84 | 61.0 | | 82.8
11/2 | | 574.5 | | | | 97.9 | 139 | 139.1 | 187 | 107.5 | | 99.9
13/2 | | | | | | 140.5 | | 176.9 | | 160.1 | | 131.1
15/2 | | | | | | | 228 | 226.6 | | 221.0 | | 177.5
17/2 | | | | | | | | 288.4 | | 291.4 | | 239.6
19/2 | | | | | | | | | | 372.3 | | 317.6
Table 4: Excitation energies in 227Ra for the bands characterized by
$K^{\pi}=\frac{1}{2}^{\pm},\frac{3}{2}^{\pm},\frac{5}{2}^{\pm}$ respectively,
are given in keV. The results of our calculations (Th.) are compared with the
corresponding experimental data (Exp.) taken from Ref.Egidy . Figure 3: The
theoretical and experimental energy displacement functions $\delta E(I)$ and
$\Delta E_{1,\gamma}(I)$ given by Eqs.(LABEL:delta) and (5.3) respectively,
characterizing the isotope 239Pu, are plotted as a function of the angular
momentum $I$. Experimental data are taken from Ref.Zhu . In the lower panel,
the theoretical and experimental $\Delta E_{1,\gamma}(I)$ corresponding to the
states $I^{\pi}=\left(\frac{1}{2}+2k\right)^{+}$ with k=1,2,3,…, are
represented by the symbols labeled by $Th.I$ and $Exp.I$ respectively, while
those associated with the negative parity states
$I^{\pi}=\left(\frac{1}{2}+2k\right)^{-}$ with k=1,2,3,… bear the labels
$Th.II$ and $Exp.II$ , respectively.
| $\frac{B(E1;J\to(J-1))}{B(E2;J\to(J-2))}[10^{-6}fm^{-2}]$
---|---
$J^{\pi}-J_{g.s.}$ | Exp. | present | Ref.Zub1
5- | 2.52(18) | 2.52 | 1.195
6+ | 1.12(08) | 1.09 | 0.314
7- | 1.49(10) | 3.97 | 1.318
8+ | 1.23(16) | 1.23 | 0.313
9- | 1.16(08) | 4.56 | 1.442
10+ | 2.77(64) | 1.44 | 0.312
11- | 1.41(9) | 4.59 | 1.567
12+ | 3.68(26) | 1.69 | 0.313
13- | 2.14(30) | 4.39 | 1.691
14+ | 1.96(14) | 1.96 | 0.314
15- | 1.76(18) | 4.11 | 1.814
16+ | 1.06(17) | 2.22 | 0.315
17- | 2.08(28) | 3.84 | 1.936
18+ | 3.34(48) | 2.45 | 0.317
19- | 1.34(42) | 3.62 | 2.057
20+ | 2.38(44) | 2.63 | 0.318
21- | 4.01(94) | 3.44 | 2.177
Average | 2.09(9) | 2.97 | 1.072
Table 5: The experimental (Exp.) and calculated (present) ratios $B(E1)/B(E2)$
for initial state $J^{\pi}$ running from $19/2^{-}$ up to $51/2^{-}$. As
mentioned in the text, $J_{g.s.}=9/2$. Experimental data are from Ref.Cottle .
The results are given in units of $10^{-6}fm^{-2}$. For comparison the results
of Ref.Zub1 are also given in the column 3.
From Fig. 2 we note that our approach reproduces the experimental energies
ordering in the band $K^{\pi}=1/2^{-}$. The energy split of the states
$3/2^{-},1/2^{-}$ is nicely described although the doublet is shifted down by
an amount of about 50 keV. In the band $5/2^{+}$ there exists an energy level
which is tentatively assigned with $11/2^{+}$. Our calculations suggests that
this level could be assigned as $13/2^{+}$. No experimental data are available
for the band $5/2^{-}$. In Fig. 2 we gave however the results of our
calculations for this band. Note that the ordering for the lowest levels is
not the natural one. However starting with $13/2^{-}$ the normal ordering is
restored. It is interesting to note that the heading states for the bands
$1/2^{+}$ and $5/2^{+}$ are almost degenerate. The same it is true for the
lowest angular momenta states in their negative parity partner bands. The
deviations r.m.s. for this nucleus is 23 keV.
Now we would like to comment on the parameters yielded by the fitting
procedure, for the considered isotopes. Except for 237U, where both
quadrupole-quadrupole and octupole-octupole interactions are attractive, the
two interactions have different characters for the rest of nuclei. In the
first situation the $\lambda$ (=2,3)-pole moments of the odd nucleon and that
of the collective core have different signs. In the remaining cases the two
moments are of similar sign. We also remark the large strength for the
$q_{3}Q_{3}$ interaction in 227Ra which is consistent with the fact that the
neighboring even-even isotope exhibits a relatively large octupole
deformation. Indeed, according to Ref.Rad006 for this nucleus we have
$f=0.8$. The large value of the strength $X_{3}$ determines large mixing
amplitudes of the states $[g_{9/2}\Psi^{(+)}_{g};f_{5/2}\Psi^{(-)}_{g}]$ as
well as of the states $[g_{9/2}\Psi^{(-)}_{g};f_{5/2}\Psi^{(+)}_{g}]$. Indeed,
the value obtained for this amplitude is:
$|A^{(+)}_{i,3}|^{2}=|A^{(-)}_{1,3}|^{2}=0.07425$. Another distinctive feature
for 227Ra consists in the fact that the $jJ$ interaction strength has a sign
which is different from that associated to other nuclei. In fact the repulsive
character of this interaction in 227Ra is necessary in order to compensate the
large attractive contribution of the $q_{3}Q_{3}$ interaction.
Further, we addressed the question whether one could identify signatures for
static octupole deformation in the two bands. To this goal, in Fig. 3, we
plotted the energy displacement functions Rad02 ; Rad03 ; Bona0 :
$\displaystyle\delta E(I)$ $\displaystyle=$ $\displaystyle
E(I^{-})-\frac{(I+1)E((I-1)^{+})+IE((I+1)^{+})}{2I+1},$ $\displaystyle\Delta
E_{1,\gamma}(I)$ $\displaystyle=$
$\displaystyle\frac{1}{16}[6E_{1,\gamma}(I)-4E_{1,\gamma}(I-1)-4E_{1,\gamma}(I+1)$
(5.3) $\displaystyle+E_{1,\gamma}(I-2)+E_{1,\gamma}(I+2)],$ $\displaystyle
E_{1,\gamma}(I)$ $\displaystyle=$ $\displaystyle E(I+1)-E(I).$
The first function, $\delta E$, vanishes when the excitation energies of the
parity partner bands depend linearly on $I(I+1)$ and, moreover, the moments of
inertia of the two bands are equal. Thus, the vanishing value of $\delta E$ is
considered to be a signature for octupole deformation. If the excitation
energies depend quadratically on $I(I+1)$ and the coefficients of the
$[I(I+1)]^{2}$ terms for the positive and negative parity bands are equal, the
second energy displacement function $\Delta E_{1,\gamma}$ vanishes, which
again suggests that a static octupole deformation shows up. The parities
associated to the angular momenta, involved in $\Delta E_{1,\gamma}$ are as
follows: the levels $I$ and $I\pm 2$ have the same parity, while levels $I$
and $I\pm 1$ are of opposite parities. The results plotted in Fig. 3
correspond to 239Pu. We choose this nucleus, since more data are available.
The plot suggests that a static octupole deformation is possible for the
states with angular momenta $I\geq\frac{51}{2}$ belonging to the two parity
partner bands.
Finally we calculated the branching ratio $R_{J}$ defined by Eq.(4.1), for
219Ra. There are two parameters involved which were fixed so that two
particular experimental data are reproduced. The values obtained for these
parameters are:
$\frac{q_{1}}{q_{2}}=18.377\times 10^{-3}fm^{-1},\;\;ab^{2}=-0.63616fm^{2}$
(5.4)
where $b$ denotes the oscillator length characterizing the spherical shell
model states for the odd nucleon. As shown in Table V, the theoretical results
agree reasonably well to the corresponding experimental data. Our results show
an oscillating behavior with maxima for the negative parity states. Note that
some off the data are well described while others deviate from the data by a
factor ranging from 2 to 3. In the third column of Table V we listed the
results obtained in Ref.Zub1 by a different model. In the quoted reference
the ratios corresponding to positive parity states are almost constant and
small.
The spectra of the odd isotopes, considered here, have been previously studied
in Refs.Zub ; Zub1 ; Deni ; Bona using a quadrupole-octupole Hamiltonian in
the intrinsic deformation variables $\beta_{2}$ and $\beta_{3}$ separated in a
kinetic energy, a potential energy term and a Coriolis interaction. Due to the
specific structure of the model Hamiltonian, an analytical solution for the
excitation energies in the two bands of opposite parities was possible. It was
shown that the split of the parity partner bands is determined by a combined
effect coming from the Coriolis interaction, which affects the $K=\frac{1}{2}$
bands, and a quantum number $k$ associated to the motion of a phase angle
$\phi$, characterizing both the quadrupole and the octupole deformation
variables. Based on analytical calculations, some conclusions concerning the
$B(E2)$ values associated to the intraband transitions between states of
similar parities, have been drawn. Thus, if the odd particle state is of
positive parity, the transitions between positive parity states are enhanced
with respect to those connecting negative parity states. If the parity of the
odd particle state is negative the ordering of the mentioned transitions is
reversed.
Comparison between the present formalism and that of Ref. Bona reveals the
following features:
a) Having in mind the asymptotic behavior of the coherent states written in
the intrinsic frame of reference Rad81 , one may anticipate that the wave
function describing the odd system from Ref. Bona , might be recovered in the
asymptotic limit of the present approach. Due to the fact that our formalism
is associated to the laboratory reference frame, the Coriolis interaction does
not show up explicitly. The split of the states of different parities is
determined by the matrix elements of $H_{pc}$. Indeed, the quadrupole-
quadrupole interaction has different matrix elements in the space of positive
parity states $\Phi^{(+)}_{IM}(p;d,f)$ than in the space spanned by
$\Phi^{(-)}_{IM}(p;d,f)$ (see (2.12)). Note that the octupole-octupole
interaction does not connect the states $\Phi^{(+)}_{IM}(p;d,f)$ and
$\Phi^{(-)}_{IM}(p;d,f)$ but the diagonal elements corresponding to the
mentioned states are different since so are the mixing amplitudes ${\cal
A}^{(+)}_{pl}$ and ${\cal A}^{(-)}_{pl}$. The spin-orbit like interaction is
also very important in determining the band parity split. The set of angular
momenta $J$ characterizing the core system, which participate in building up
the angular momentum $I^{+}$ is very different from that involved in the
structure of the state $I^{-}$. Therefore, summing the quantity
$\frac{1}{2}[I(I+1)-j(j+1)-J(J+1)]$ with different weights for the parity
partner states $I^{+}$ and $I^{-}$ and then comparing the results, one
certainly obtains the energy split caused by the term $\vec{j}\cdot\vec{J}$.
Concluding, one may say that we identified three distinct sources generating a
split for the parity partner states in the even-odd nuclei. While in Ref.Bona
$K$ is a good quantum number here $K$ labels the leading component in an
expansion characterizing a wave function with a definite angular momentum and
a definite angular momentum z-projection, in the laboratory frame. Thus,
although one says that $K=\frac{1}{2}$ since the corresponding component in
the above mentioned expansion prevails, the mixing of different $K$ components
due to the single particle mixed states as well as due to the core projected
states is considered in a natural manner. Therefore, one expects that the
complex structure of the model states might be suitable for the description of
the transition probabilities between states from the two bands.
b) Since the coherent states are axially symmetric functions (only the boson
components $b^{\dagger}_{\lambda 0}$, $\lambda=2,3,$ are used in Eq. (2.3)
defining the coherent states) we don’t account for the motion of the
$\gamma$-like deformation. Again the two formalisms are on a par with each
other.
c) In general, the quadrupole-octupole boson descriptions overestimate the
contribution to the system energy coming from the rotational degrees of
freedom, since the Euler angles defining the intrinsic reference frame are
involved in both the quadrupole and octupole bosons. This ambiguity is however
removed in our approach due to the angular momentum projection operation. The
description used in Ref.Bona is also not confronted with such a difficulty.
d) The approach of Ref. Bona is of a strong coupling type and therefore $K$
is a good quantum number, which is not the case in the present paper. Indeed,
we use the laboratory frame and the meaning of the quantum number $K$ is given
by the fact that the $K$-component of the spherical function prevails over the
components with $K^{\prime}\neq K$.
e) The Hamiltonian describing the odd system (2.1) involves a term $H_{core}$
which describes in a realistic fashion the neighboring even-even system.
Indeed, this has been used in Ref.Rad06 to describe simultaneously eight
rotational bands, four of positive and four of negative parity. By contrast,
in Ref.Bona the terms associated to the core are not appropriate for
describing the complex structure of the even-even sub-system.
f) The agreement obtained in our approach for 239Pu is better than that shown
in Ref.Bona . However, the results from Ref.Bona for 237U agree better, with
the corresponding data, than ours. Indeed, the $r.m.s$. values for the
deviations of theoretical results from experimental data, reported in Ref.Bona
, are 30 keV and 60 keV for 237U and 239Pu, which are to be compared with 48.9
keV and 31.8 keV respectively, obtained with our approach.
g) For some isotopes, in Ref.Bona , the bands with $K=5/2$ are solely
considered. By contradistinction we treated simultaneously the bands with
$K=1/2,3/2,5/2$, respectively. Moreover, a distinctive feature is the fact
that here the bands with $K=3/2$ and $K=5/2$ are generated by coupling a
single particle state to the states belonging to the $\gamma$ band of the core
system.
## VI Conclusions
In the previous sections we proposed a new formalism for the description of
parity partner bands in even-odd nuclei. Our approach uses a particle-core
Hamiltonian, with a phenomenological core described in terms of quadrupole and
octupole bosons. The single particle space consists of three spherical shell
model states, two of them having positive parity while the third one a
negative parity. The particle-core coupling terms cause the excitation of the
odd particle from one state to any of the remaining two. Thus, the particle-
core interaction might break two symmetries for the single particle motion,
the rotation and space reflection, which, as a matter of fact, is consistent
with the structure of the mean field obtained by averaging the model
Hamiltonian with a quadrupole and octupole boson coherent state. For $K=1/2$
bands the single particle states are coupled to the ground state of a deformed
core while for $K=3/2,5/2$ the single particle states are coupled to the gamma
intrinsic state. The bands are generated through angular momentum projection
from the particle-core intrinsic states mentioned above. In this way the
influence of the excited states from the ground band on the structure of the
$K^{\pi}=1/2^{\pm}$ and that of the excited states from the $\gamma$ band on
the $K^{\pi}=3,2^{\pm},5/2^{\pm}$ bands are taken into account. The
contribution of various terms of the model Hamiltonian are analyzed in terms
of the magnitude and the sign of the interaction strengths yielded by the
fitting procedure. Approaches which treat the particle-core interaction in the
intrinsic frame of reference stress on the role played by the Coriolis
interaction, through the decoupling parameter, in determining the energy
splitting of the parity partner states with $K=1/2$. For example, in 227Ra the
decoupling factor is quite high (0.7) Egidy . In the laboratory frame we
identified the interaction which determine the energy parity split.
Application to 219Ra, 237U and 239Pu shows a good agreement between the
calculated excitation energies in the bands with $K^{\pi}=\frac{1}{2}^{\pm}$
and the corresponding experimental data. The branching ratios of 219Ra have
been also calculated. The agreement with the available data is quite good.
Finally the results for a simultaneous treatment of six bands,
$K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$, were presented for 227Ra. The plot
for the energy displacement functions, or energy staggering factors, made for
239Pu, indicates that a static octupole deformation might be set on for states
with angular momentum larger than $\frac{51}{2}\hbar$.
Before closing, we would like to add few remarks about the possible
development of the present formalism. Choosing for the core unprojected
states, the generating states for the parity partner bands with
$K^{\pi}=0_{\beta}^{\pm},1^{\pm}$ states, otherwise keeping the same single
particle basis for the odd nucleon, the present formalism can be extended to
another four bands, two of positive and two of negative parity. Another
noteworthy remark refers to the chiral symmetry Frau for the composite
particle and core system. Indeed, in Ref.Rad006 we showed that starting from
a certain total angular momentum of the core, the angular momenta carried by
the quadrupole ($\vec{J}_{2}$) and octupole ($\vec{J}_{3}$) bosons
respectively, are perpendicular on each other. Naturally, we may ask ourselves
whether there exists a strength for the particle-core interaction such that
the angular momentum of the odd particle becomes perpendicular to the plane
($\vec{J}_{2},\vec{J}_{3}$). This would be a signature that the three
component system exhibits a chiral symmetry.
As a final conclusion, one may say that the present CSM extension to odd
nuclei can describe quite well the excitation energies in the parity partner
bands with $K^{\pi}=\frac{1}{2}^{\pm}$.
Acknowledgment. This paper was supported by the Romanian Ministry of Education
and Research under the contracts PNII, No. ID-33/2007 and ID-1038/2009.
## VII Appendix A
The diagonal matrix elements of the quadrupole-quadrupole ($q_{2}Q_{2}$) and
octupole-octupole ($q_{3}Q_{3}$) particle-core interactions in the basis
defined in Section III are:
$\displaystyle\langle\varphi^{(\pm)}_{IM;j_{i}K}|q_{2}Q_{2}|\varphi^{(\pm)}_{IM;j_{i}K}=-X_{2}C^{j_{i}\;J\;I}_{k-2\;2\;K}C^{j_{i}\;J^{{}^{\prime}}\;I}_{k-2\;2\;K}\hat{I}^{2}\hat{j_{i}}\hat{J}W(j_{i}I2J;J^{{}^{\prime}}j_{i})$
(A.1) $\displaystyle\times$
$\displaystyle\left(N^{(\pm)}_{I,K}\right)^{2}\left(N^{(\gamma,\pm)}_{J}N^{(\gamma,\pm)}_{J^{{}^{\prime}}}\right)^{-1}\langle
j_{i}||r^{2}Y_{2}||j_{i}\rangle\langle\varphi^{(\gamma;\pm)}_{J}||b^{\dagger}_{2}+b_{2}||\varphi^{(\gamma;\pm)}_{J^{{}^{\prime}}}\rangle,i=2,3;K=i-1/2,$
$\displaystyle\langle\varphi^{(\pm)}_{IM;j_{3}5/2}|q_{3}Q_{3}|\varphi^{(\pm)}_{IM;j_{2}3/2}=X_{3}C^{j_{3}\;J\;I}_{1/2\;2\;5/2}C^{j_{2}\;J^{{}^{\prime}}\;I}_{-1/2\;2\;3/2}\hat{I}^{2}\hat{j_{3}}\hat{J}W(j_{2}I3J;J^{{}^{\prime}}j_{3})$
$\displaystyle\times$ $\displaystyle
N^{(\pm)}_{I,5/2}N^{(\pm)}_{I,3/2}\left(N^{(\gamma,\pm)}_{J}N^{(\gamma,\pm)}_{J^{{}^{\prime}}}\right)^{-1}\langle
j_{3}||r^{3}Y_{3}||j_{i}\rangle\langle\varphi^{(\gamma;\pm)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(\gamma;\mp)}_{J^{{}^{\prime}}}\rangle.$
The expectation value for the $q_{2}Q_{2}$ term, in the state
$\varphi^{\pm}_{IM;j_{1}1/2}$ can be obtained from the expression given above
by replacing $j_{i}$ by $j_{1}$ and $\varphi^{(\gamma;\pm)}_{J}$ by
$\varphi^{(g;\pm)}_{J}$. Also, the projections associated to $J$ and
$J^{{}^{\prime}}$ in the two Clebsch Gordan coefficients should be equal to
zero and not 2. It is easy to check that this state is not connected by the
$q_{3}Q_{3}$ interaction to the state $\varphi^{\pm}_{IM;j_{3}5/2}$. The
reduced matrix elements of the boson operators involved in the above equations
have the expressions:
$\displaystyle\langle\varphi^{(\gamma;\pm)}_{J}||b^{\dagger}_{2}+b_{2}||\varphi^{(\gamma;\pm)}\rangle$
$\displaystyle=$ $\displaystyle
dC^{J^{{}^{\prime}}\;2\;J}_{2\;0\;2}\left[\frac{N^{(\gamma;\pm)}_{J}}{N^{(\gamma;\pm)}_{J^{{}^{\prime}}}}+\frac{2J^{{}^{\prime}}+1}{2J+1}\frac{N^{(\gamma;\pm)}_{J^{{}^{\prime}}}}{N^{(\gamma;\pm)}_{J}}+\frac{6}{7}\sum_{J^{{}^{\prime}}}\frac{N^{(\gamma;\pm)}_{J}N^{(\gamma;\pm)}_{J^{{}^{\prime}}}}{N^{(g;\pm)}_{J_{1}}}\right.$
$\displaystyle\times\left.\left((C^{J_{1}\;2\;J^{{}^{\prime}}}_{0\;2\;2})^{2}+\frac{2J^{{}^{\prime}}+1}{2J+1}(C^{J_{1}\;2\;J}_{0\;2\;2})^{2}\right)\right],$
$\displaystyle\langle\varphi^{(g;\pm)}_{J}||b^{\dagger}_{2}+b_{2}||\varphi^{(g;\pm)_{J^{{}^{\prime}}}}\rangle$
$\displaystyle=$ $\displaystyle
dC^{J^{{}^{\prime}}\;2\;J}_{0\;0\;0}\left[\frac{N^{(g;\pm)}_{J}}{N^{(g;\pm)}_{J^{{}^{\prime}}}}+\frac{2J^{{}^{\prime}}+1}{2J+1}\frac{N^{(g;\pm)}_{J^{{}^{\prime}}}}{N^{(g;\pm)}_{J}}\right],$
$\displaystyle\langle\varphi^{(\gamma;+)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(\gamma;-)}_{J^{{}^{\prime}}}\rangle$
$\displaystyle=$ $\displaystyle
fC^{J^{{}^{\prime}}\;2\;J}_{2\;0\;2}\left[\frac{N^{(\gamma;+)}_{J}}{N^{(\gamma;-)}_{J^{{}^{\prime}}}}+\frac{2J^{{}^{\prime}}+1}{2J+1}\frac{N^{(\gamma;-)}_{J^{{}^{\prime}}}}{N^{(\gamma;+)}_{J}}\right],$
$\displaystyle\langle\varphi^{(\gamma;-)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(\gamma;+)}_{J^{{}^{\prime}}}\rangle$
$\displaystyle=$
$\displaystyle(-)^{J^{{}^{\prime}}-J}\frac{\hat{J^{{}^{\prime}}}}{\hat{J}}\langle\varphi^{(\gamma;+)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(\gamma;-)}\rangle,$
$\displaystyle\langle\varphi^{(g;+)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(g;-)}_{J^{{}^{\prime}}}\rangle$
$\displaystyle=$ $\displaystyle
fC^{J^{{}^{\prime}}\;2\;J}_{0\;0\;0}\left[\frac{N^{(g;+)}_{J}}{N^{(g;-)}_{J^{{}^{\prime}}}}+\frac{2J^{{}^{\prime}}+1}{2J+1}\frac{N^{(g;-)}_{J^{{}^{\prime}}}}{N^{(g;+)}_{J}}\right],$
$\displaystyle\langle\varphi^{(g;-)}_{J}||b^{\dagger}_{3}+b_{3}||\varphi^{(g;+)}_{J^{{}^{\prime}}}\rangle$
$\displaystyle=$
$\displaystyle(-)^{J^{{}^{\prime}}-J}\frac{\hat{J^{{}^{\prime}}}}{\hat{J}}\langle\varphi^{(g;+)}_{J^{{}^{\prime}}}||b^{\dagger}_{3}+b_{3}||\varphi^{(g;-)}_{J}\rangle.$
(A.2)
The matrix elements of $H_{core}$ have the expressions:
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{2}3/2}|H_{core}|\varphi^{(\pm)}_{IM,j_{2}3/2}\rangle$
$\displaystyle=$ $\displaystyle
N^{(\pm)}_{I;j_{2}3/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{-1/2\;2\;3/2}\right)^{2}\left(N^{(\gamma;\pm)}_{J}\right)^{-2}E^{(\gamma,\pm)}_{J},$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{3}5/2}|H_{core}|\varphi^{(\pm)}_{IM,j_{3}5/2}\rangle$
$\displaystyle=$ $\displaystyle
N^{(\pm)}_{I;j_{3}5/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{1/2\;2\;5/2}\right)^{2}\left(N^{(\gamma;\pm)}_{J}\right)^{-2}E^{(\gamma,\pm)}_{J},$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{1}1/2}|H_{core}|\varphi^{(\pm)}_{IM,j_{1}1/2}\rangle$
$\displaystyle=$ $\displaystyle
N^{(\pm)}_{I;j_{1}1/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{1/2\;0\;1/2}\right)^{2}\left(N^{(g;\pm)}_{J}\right)^{-2}E^{(g,\pm)}_{J},$
(A.3)
where $E^{(g,\pm)}_{J}$ and $E^{(\gamma,\pm)}_{J}$ denote the energies of the
state $J^{\pm}$ belonging to the bands $g^{\pm}$ and $\gamma^{\pm}$,
respectively. Obviously, the term $H_{sp}$ is diagonal in the chosen basis:
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{1}1/2}|H_{sp}|\varphi^{(\pm)}_{IM,j_{1}1/2}\rangle$
$\displaystyle=$ $\displaystyle\epsilon_{j_{1}},$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{2}3/2}|H_{sp}|\varphi^{(\pm)}_{IM,j_{2}3/2}\rangle$
$\displaystyle=$ $\displaystyle\epsilon_{j_{2}},$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{3}5/2}|H_{sp}|\varphi^{(\pm)}_{IM,j_{3}5/2}\rangle$
$\displaystyle=$ $\displaystyle\epsilon_{j_{3}}.$ (A.4)
Here $\epsilon_{j_{k}}$ denotes the energies of the spherical shell model
states $|n_{k},l_{k},j_{k},m_{k}\rangle$ with $k=1,2,3.$ The matrix elements
of the term $\vec{j}\cdot\vec{J}$ are:
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{1}1/2}|\vec{j}\cdot\vec{J}|\varphi^{(\pm)}_{IM,j_{1}1/2}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[I(I+1)-j_{1}(j_{1}+1)-N^{(\pm)}_{I;j_{1}1/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{1/2\;0\;1/2}\right)^{2}\left(N^{(g;\pm)}_{J}\right)^{-2}J(J+1)\right],$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{2}3/2}|\vec{j}\cdot\vec{J}|\varphi^{(\pm)}_{IM,j_{2}3/2}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[I(I+1)-j_{2}(j_{2}+1)-N^{(\pm)}_{I;j_{2}3/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{-1/2\;2\;3/2}\right)^{2}\left(N^{(\gamma;\pm)}_{J}\right)^{-2}J(J+1)\right],$
$\displaystyle\langle\varphi^{(\pm)}_{IM,j_{3}5/2}|\vec{j}\cdot\vec{J}|\varphi^{(\pm)}_{IM,j_{3}5/2}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[I(I+1)-j_{3}(j_{3}+1)-N^{(\pm)}_{I;j_{3}5/2}\sum_{J}\left(C^{j_{2}\;J\;I}_{1/2\;2\;5/2}\right)^{2}\left(N^{(\gamma;\pm)}_{J}\right)^{-2}J(J+1)\right].$
## VIII Appendix B
The matrix elements involved in the expression of the branching ratios are:
$\displaystyle\langle\varphi^{(\pi)}_{I}(j_{i}K;d,f)||r^{2}Y_{2}||\varphi^{(\pi)}_{I^{{}^{\prime}}}(j_{i}K^{{}^{\prime}};d,f)\rangle$
$\displaystyle=$ $\displaystyle-\sqrt{\frac{5}{4\pi}}\langle
r^{2}\rangle\hat{I^{{}^{\prime}}}\hat{j_{i}}N^{(\pi)}_{i,IK}N^{(\pi)}_{i,I^{{}^{\prime}}K^{{}^{\prime}}}$
$\displaystyle\times\sum_{J}C^{j_{i}\;J\;I}_{K\;0\;K}C^{j_{i}\;J\;I^{{}^{\prime}}}_{K^{{}^{\prime}}\;0\;K^{{}^{\prime}}}\left(N^{(g,\sigma)}_{J}\right)^{-2},$
$\displaystyle\langle\varphi^{(\pi)}_{I}(j_{i}K;d,f)||b^{\dagger}_{2}+b_{2}||\varphi^{(\pi)}_{I^{{}^{\prime}}}(j_{i}K^{{}^{\prime}};d,f)\rangle$
$\displaystyle=$ $\displaystyle
dC^{I^{{}^{\prime}}\;2\;I}_{K\;0\;K}\left(\frac{N^{(\pi)}_{i;IK}}{N^{(\pi)}_{i;I^{{}^{\prime}}K^{{}^{\prime}}}}+\frac{2I^{{}^{\prime}}+1}{2I+1}\frac{N^{(\pi)}_{i;I^{{}^{\prime}}K^{{}^{\prime}}}}{N^{(\pi)}_{i;IK}}\right),$
$\displaystyle\langle\varphi^{(\pi)}_{I}(j_{i}K;d,f)||\left(b^{\dagger}_{2}b^{\dagger}_{3}\right)_{1}+\left(b_{3}b_{2}\right)_{1}||\varphi^{(\pi^{{}^{\prime}})}_{I^{{}^{\prime}}}(j_{i}K^{{}^{\prime}};d,f)\rangle$
$\displaystyle=$ $\displaystyle
dfC^{I^{{}^{\prime}}\;1\;I}_{K\;0\;K}C^{2\;3\;1}_{0\;0\;0}\left[\frac{N^{(\pi)}_{i;IK}}{N^{(\pi^{{}^{\prime}})}_{i;I^{{}^{\prime}}K^{{}^{\prime}}}}+\frac{2I^{{}^{\prime}}+1}{2I+1}\frac{N^{(\pi^{{}^{\prime}})}_{i;I^{{}^{\prime}}K^{{}^{\prime}}}}{N^{(\pi)}_{i;IK}}\right].$
## References
* (1) A. A. Raduta, V. Ceausescu, A. Gheorghe and R.M. Dreizler, Phys. Lett. B99 , 444 (1981); Nucl. Phys. A381, 253 (1982).
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* (8) A. A. Raduta, Ad. R. Raduta and Al. H. Raduta, Phys. Rev. B 59, 8209 (1999).
* (9) A. A. Raduta, V. Ceausescu and R. M. Dreizler, Nucl. Phys. A 272, 11 (1976).
* (10) G. A. Leander, et al., Nucl. Phys. A 388, 452 (1982).
* (11) M. E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957.
* (12) Peter Ring and Peter Schuck, The Nuclear Many-Body Problem, Springer-Verlag, Berlin, Heidelberg, New York, 2000, pp. 76.
* (13) D. Bonatsos et al., Phys. Rev. C62, 024301 (2000).
* (14) S. Zhu et al., Phys. Lett. B 618, 51 (2005).
* (15) P. D. Cottle et al., Phys. Rev. C36, 2286 (1987).
* (16) M. Wieland et al., Phys. Rev. C 45, 1035 (1992).
* (17) T. von Egidy et al., Nucl. Phys. A365, 26 (1981).
* (18) G. A. Leander, Y. S. Chen, Phys. Rev. C37, 2744 (1988).
* (19) A.Ya Dzyublik and V. Yu Denisov, Yad. Fiz. 56, 30 (1993) [Phys. At. Nucl. 56, 303 (1993) ].
* (20) V. Yu. Denisov and A. Ya. Dzyublik, Yad. Fiz. 56, 96 (1993) [Phys. At. Nucl. 56, 477 (1993)].
* (21) V. Yu Denisov and A.Ya Dzyublik, Nucl. Phys. A 589, 17 (1995).
* (22) N. Minkov, S. Drenska, P. Yotov, S. Lalkovski, D. Bonatsos and W. Scheid, Phys. Rev. C76, 034324 (2007).
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|
arxiv-papers
| 2009-08-26T09:12:04 |
2024-09-04T02:49:04.812262
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. A. Raduta, C. M. Raduta and Amand Faessler",
"submitter": "Apolodor Aristotel Raduta",
"url": "https://arxiv.org/abs/0908.3776"
}
|
0908.3805
|
# A noncommutative version of the Fejér-Riesz theorem
Yuriĭ Savchuk Mathematisches Institut, Universität Leipzig, Johannisgasse 26,
04103 Leipzig, Germany savchuk@math.uni-leipzig.de and Konrad Schmüdgen
Mathematisches Institut, Universität Leipzig, Johannisgasse 26, 04103 Leipzig,
Germany schmuedgen@math.uni-leipzig.de
###### Abstract.
Let ${\mathcal{X}}$ be the unital $*$-algebra generated by the unilateral
shift operator. It is shown that for any nonnegative operator
$X\in{\mathcal{X}}$ there is an element $Y\in{\mathcal{X}}$ such that
$X=Y^{*}Y$.
###### Key words and phrases:
Fejér-Riesz theorem, noncommutative Positivstellensatz, Toeplitz algebra.
###### 2010 Mathematics Subject Classification:
Primary 14A22, 47A68. Secondary 42A05
## 1\. Introduction and Main Result
Let ${\mathcal{P}}$ denote the $*$-algebra of complex Laurent polynomials
$p(z,z^{-1})\\\ =\sum_{k=-n}^{n}a_{k}z^{k}$ with involution
$p\to\overline{p}(z):=\sum_{k=-n}^{n}\overline{a}_{k}z^{-k},$ where
$n\in{\mathbb{N}}_{0}.$ Setting $z=e^{it}$ with $t\in[0,2\pi]$ we see that
${\mathcal{P}}$ is isomorphic to the $*$-algebra of all trigonometric
polynomials. There is a faithful $*$-representation $\pi$ of the $*$-algebra
${\mathcal{P}}$ on the Hilbert space $l^{2}({\mathbb{Z}})$ such that
$\pi(z)=U$, where $U$ is the bilateral shift operator.
The classical Fejér-Riesz theorem states that if a polynomial
$p\in{\mathcal{P}}$ takes only nonnegative values on the unit circle
${\mathbb{T}}=\\{z\in{\mathbb{C}}:|z|{=}1\\}$ or equivalently if the operator
$\pi(p)$ on $l^{2}({\mathbb{Z}})$ is nonnegative, then $p$ is of the form
$p=\overline{q}\cdot{q}$ for some $q\in{\mathcal{P}}$. Further, if $p$ has
degree $d\in{\mathbb{N}}_{0}$, then $q$ can be chosen to be an analytic
polynomial $q(z)=\sum_{k=0}^{d}b_{k}z^{k}$ of degree $d$ such that
$q(z)\not=0$ for $|z|<1$ and $q(0)>0.$ The latter conditions determine the
polynomial $q$ uniquely. A simple proof of this theorem can be found in [Sz],
see Theorems 1.2.1 and 1.2.2 therein.
The aim of this paper is to prove an analog of the Fejér-Riesz theorem if
${\mathcal{P}}$ is replaced by the unital $*$-algebra
$\mathcal{A}={\mathbb{C}}\langle s,s^{*}\ |\ s^{*}s=1\rangle$
and the bilateral shift $U$ is replaced by the unilateral shift
(1) $\displaystyle
S(\varphi_{0},\varphi_{1},\varphi_{2},\dots)=(0,\varphi_{0},\varphi_{1},\varphi_{2},\dots)$
on the Hilbert space $l^{2}({\mathbb{N}}_{0})$. Let $\pi_{0}$ denote the
$*$-representation of $\mathcal{A}$ on $l^{2}({\mathbb{N}}_{0})$ determined by
$\pi_{0}(s)=S$. Our main result is the following
###### Theorem 1.
For any element $x=x^{*}\in\mathcal{A}$ the following statements are
equivalent:
(i) $x=y^{*}y$ for some $y\in\mathcal{A}.$
(ii) $\pi(x)\geq 0$ for any $*$-representation of the $*$-algebra
$\mathcal{A}$ on a Hilbert space.
(iii) $\pi_{0}(x)\geq 0$ on the Hilbert space $l^{2}({\mathbb{N}}_{0})$.
If this holds, then $y$ can be chosen such that the matrix of $\pi_{0}(y)$
with respect to the standard base of $l^{2}({\mathbb{N}}_{0})$ is lower-
triangular.
The implications (i)$\to$ (ii)$\to$ (iii) are trivial, so it remains to prove
that (iii) implies (i). This will be done in the next section.
If ${\mathcal{H}}$ and ${\mathcal{K}}$ are Hilbert spaces,
${\mathcal{B}}({\mathcal{H}},{\mathcal{K}})$ are the bounded operators from
${\mathcal{H}}$ into ${\mathcal{K}}$ and
${\mathcal{B}}({\mathcal{H}}){:=}{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}).$
For $x,y\in{\mathcal{H}},\ x\otimes y$ denotes the rank one operator
$\langle\cdot,x\rangle y$. By a $*$-representation of a unital $*$-algebra on
${\mathcal{H}}$ we mean a unit preserving $*$-homomorphism into the
$*$-algebra ${\mathcal{B}}({\mathcal{H}})$.
## 2\. Proof of the Main Implication
For the main proof we need three simple lemmas. The second lemma is a well-
known fact on outer analytic polynomials, while the third is the crucial
factorization lemma. To make the exposition in this section as elementary as
possible we include complete proofs.
We identify the Hilbert spaces $l^{2}({\mathbb{N}}_{0})$ and
$H^{2}({\mathbb{T}})$ in the obvious way by identifying their standard
orthonormal bases $\left\\{e_{k};k\in{\mathbb{N}}_{0}\right\\}$ and
$\left\\{z^{k};\ k\in{\mathbb{N}}_{0}\right\\}.$ Let ${\mathcal{T}}_{p}$
denote the set of all Toeplitz operators $T_{p}$ on $H^{2}({\mathbb{T}})$ with
symbol $p\in{\mathcal{P}}$ and ${\mathcal{F}}$ the set of all bounded
operators on $l^{2}({\mathbb{N}}_{0})$ which have finite matrices with respect
to the base $\left\\{e_{k}\right\\}.$ That is,
$F\in{\mathcal{B}}(l^{2}({\mathbb{N}}_{0}))$ is in ${\mathcal{F}}$ if and only
if there exists a natural number $k$ such that $\langle Fe_{i},e_{j}\rangle=0$
if $i>k$ or $j>k.$
Set ${\mathcal{X}}:=\pi_{0}(\mathcal{A}).$
###### Lemma 1.
${\mathcal{X}}={\mathcal{T}}_{p}+{\mathcal{F}}.$
###### Proof.
Put ${\mathcal{Y}}:={\mathcal{T}}_{p}+{\mathcal{F}}$. Since
$\pi_{0}(z^{n}){=}\pi_{0}(z)^{n}{=}S^{n}=T_{z^{n}}$ and
$\pi_{0}(z^{-n}){=}\pi_{0}((z^{n})^{*})){=}(T_{z^{n}})^{*}{=}T_{z^{-n}}$ for
$n\in{\mathbb{N}}_{0}$, we have ${\mathcal{T}}_{p}\subseteq{\mathcal{X}}$.
From the relations
$\pi_{0}(s^{n}(1-ss^{*})s^{*k})=S^{n}(I-SS^{*})S^{*k}=e_{k}\otimes e_{n}$,
$k,n\in{\mathbb{N}}_{0}$, we conclude that
${\mathcal{F}}\subseteq{\mathcal{X}}$. Thus,
${\mathcal{Y}}={\mathcal{T}}_{p}+{\mathcal{F}}\subseteq{\mathcal{X}}$.
Now we prove the converse inclusion ${\mathcal{X}}\subseteq{\mathcal{Y}}$. For
$n\in{\mathbb{N}}$ we have $ST_{z^{n}}=T_{z^{n+1}}$,
$S^{*}T_{z^{n}}=T_{z^{n-1}}$, $ST_{z^{-n}}=T_{z^{1-n}}-e_{n-1}\otimes e_{0}$,
and $S^{*}T_{z^{-n}}=T_{z^{-n-1}}$. These relations imply that
$S\cdot{\mathcal{T}}_{p}\subseteq{\mathcal{Y}}$ and
$S^{*}\cdot{\mathcal{T}}_{p}\subseteq{\mathcal{Y}}$. Since obviously
$S\cdot{\mathcal{F}}\subseteq{\mathcal{F}}$ and
$S^{*}\cdot{\mathcal{F}}\subseteq{\mathcal{F}}$, we obtain
$S\cdot{\mathcal{Y}}\subseteq{\mathcal{Y}}$ and
$S^{*}\cdot{\mathcal{Y}}\subseteq{\mathcal{Y}}$. Because ${\mathcal{X}}$ is
generated as an algebra by $S=\pi_{0}(z)$ and $S^{*}=\pi_{0}(z^{-1})$, the
latter yields ${\mathcal{X}}\cdot{\mathcal{Y}}\subseteq{\mathcal{Y}}$, so
${\mathcal{X}}\subseteq{\mathcal{Y}}$. Consequently,
${\mathcal{X}}={\mathcal{Y}}.$ ∎
###### Lemma 2.
Let $q(z)=\sum_{k=0}^{d}b_{k}z^{k}$ be an analytic polynomial such that
$q(z)\not=0$ for $|z|<1$. Then $\mathrm{Ran}T_{q}=q(z)H^{2}({\mathbb{T}})$ is
dense in $H^{2}({\mathbb{T}})$.
###### Proof.
We proceed by induction on the degree of $q$. Suppose that the assertion holds
when $\deg q\leq d$. Let $q_{0}(z)=\sum_{k=0}^{d{+}1}b_{k}z^{k}$ be as above
and $\deg q_{0}={d{+}1}.$ We write $q_{0}(z)=(z{-}\lambda)q(z).$ Then $q(z)$
satisfies also the assumptions, so $qH^{2}({\mathbb{T}})$ is dense by the
induction hypothesis. Therefore it suffices to show that
$\mathrm{Ran}(S{-}\lambda I)=(z{-}\lambda)H^{2}({\mathbb{T}})$ is dense or
equivalently that $\mathrm{Ker}(S^{*}{-}\overline{\lambda}I)=\\{0\\}$. For let
$\varphi{=}(\varphi_{n})\in\mathrm{Ker}(S^{*}{-}\overline{\lambda}I)$. Then we
have $\varphi_{n}-\overline{\lambda}\varphi_{n{-}1}=0$ and hence
$\varphi_{n}=\overline{\lambda}^{n}\varphi_{0}$ for $n\in{\mathbb{N}}$. Since
$q_{0}(z)\not=0$ for $|z|<1$, $|\lambda|\geq 1$ and hence $\varphi=0$, because
$\varphi\in l^{2}({\mathbb{N}}_{0})$. ∎
###### Lemma 3.
Let ${\mathcal{H}}$ and ${\mathcal{K}}$ be Hilbert spaces,
$A\in{\mathcal{B}}({\mathcal{H}})$, $W\in{\mathcal{B}}({\mathcal{K}})$, and
$V\in{\mathcal{B}}({\mathcal{H}},{\mathcal{K}})$. Let $P_{W}$ denote the
orthogonal projection of ${\mathcal{K}}$ onto the closure of $\mathrm{Ran}W$.
Suppose that the block matrix
$\displaystyle\left(\begin{array}[]{ll}\ \ A&V^{*}W\\\
W^{*}V&W^{*}W\end{array}\right)$
defines a nonnegative operator on ${\mathcal{H}}\oplus{\mathcal{K}}$. Then we
have $A\geq V^{*}P_{W}V.$
For any $U\in{\mathcal{B}}({\mathcal{H}})$ such that $A-V^{*}P_{W}V=U^{*}U$,
we have
(8) $\displaystyle\left(\begin{array}[]{ll}\ \ A&V^{*}W\\\
W^{*}V&W^{*}W\end{array}\right)=\left(\begin{array}[]{ll}\ \ U&0\\\
P_{W}V&W\end{array}\right)^{*}\left(\begin{array}[]{ll}\ \ U&0\\\
P_{W}V&W\end{array}\right).$
###### Proof.
Fix $\varphi\in{\mathcal{H}}$ and let $\psi\in{\mathcal{K}}$. Since the block
matrix is nonnegative, for all $\lambda\in{\mathbb{C}}$ we have the inequality
$\displaystyle\left\langle\left(\begin{array}[]{ll}\ \ A&W^{*}V\\\
V^{*}W&W^{*}W\end{array}\right)\left(\begin{array}[]{l}\ \varphi\\\
\lambda\psi\end{array}\right),\left(\begin{array}[]{l}\ \varphi\\\
\lambda\psi\end{array}\right)\right\rangle\geq 0$
which can be written as
(9) $\displaystyle\langle A\varphi,\varphi\rangle+\lambda\langle
V^{*}W\psi,\varphi\rangle+\overline{\lambda}\langle
W^{*}V\varphi,\psi\rangle+\lambda\overline{\lambda}\langle
W^{*}W\psi,\psi\rangle\geq 0.$
Since the latter inequality holds for arbitrary $\lambda\in{\mathbb{C}}$, we
conclude that
$\displaystyle\langle A\varphi,\varphi\rangle\langle
W^{*}W\psi,\psi\rangle=\langle
A\varphi,\varphi\rangle\left\|W\psi\right\|^{2}\geq|\langle
V^{*}W\psi,\varphi\rangle|^{2}=|\langle W\psi,V\varphi\rangle|^{2},$
By the definition of $P_{W}$, there is a sequence $\psi_{n}\in{\mathcal{K}}$,
$n\in{\mathbb{N}}$, such that $W\psi_{n}\to P_{W}V\varphi$. Setting
$\psi=\psi_{n}$ in the preceding inequality and passing to the limit we obtain
$\displaystyle\langle
A\varphi,\varphi\rangle\left\|P_{W}V\varphi\right\|^{2}\geq|\langle
P_{W}V\varphi,V\varphi\rangle|^{2}=\left\|P_{W}V\varphi\right\|^{4}.$
Hence $\langle A\varphi,\varphi\rangle\geq\langle
V^{*}P_{W}V\varphi,\varphi\rangle$ when $P_{W}V\varphi\not=0$. Since $\langle
A\varphi,\varphi\rangle\geq 0$ by setting $\lambda=0$ in (9), we have $\langle
A\varphi,\varphi\rangle\geq 0=\langle V^{*}P_{W}V\varphi,\varphi\rangle$ when
$P_{W}P\varphi=0$. Therefore, $A\geq V^{*}P_{W}V.$ Equation (8) is obvious. ∎
Now we are ready to prove the implication (iii)$\to$(i).
First we note that by Lemma 1 a bounded operator $X$ on
$l^{2}({\mathbb{N}}_{0})$ belongs to $\pi_{0}(\mathcal{A})$ if and only if it
has a matrix representation
(17) $\displaystyle
X=\left(\begin{array}[]{lllllll}x_{00}&x_{01}&\dots&x_{0,n}&0&\dots&\dots\\\
x_{10}&x_{11}&\dots&x_{1,n}&x_{-n}&0&\dots\\\
\texttimes&\texttimes&\ddots&\texttimes&\vdots&x_{-n}&\ddots\\\
x_{n,0}&x_{n,1}&\dots&x_{n,n}&x_{-1}&\vdots&\ddots\\\
0&x_{n}&\dots&x_{1}&x_{0}&x_{-1}&\texttimes\\\
\vdots&0&x_{n}&\dots&x_{1}&x_{0}&\ddots\\\
\vdots&\vdots&\ddots&\ddots&\texttimes&\ddots&\ddots\end{array}\right)$
with respect to the standard basis $\\{e_{k}\\}$ of $l^{2}({\mathbb{N}}_{0}).$
(If $X=F+T_{p}$ with $F\in{\mathcal{F}}$ and $p\in{\mathcal{P}}$, by adding
zeros we can find a common $n$ such that $p=\sum_{k{=}-n}^{n}a_{k}z^{k}$ and
$F$ has the size $(n{+}1)\times(n{+}1)$.) For simplicity we use the same
notation for operators and the corresponding matrices.
Suppose that $x{=}x^{*}\in\mathcal{A}$ and $X:=\pi_{0}(x)\geq 0$. Let (17) be
the matrix of $X$. Since $X$ is symmetric, we have $x_{ij}=\overline{x_{ji}}$
and $x_{k}=\overline{x_{-k}}$ for all $i,j,k.$ We will prove that there is a
lower-triangular matrix
(25) $\displaystyle Y=\left(\begin{array}[]{lllllll}y_{00}&0&&&&&\texttimes\\\
y_{10}&y_{11}&0&&&&\texttimes\\\ \vdots&\vdots&\ddots&\ddots&&&\\\
y_{n,0}&y_{n,1}&\dots&y_{n,n}&0&&\\\ 0&y_{n}&\dots&y_{1}&y_{0}&0&\texttimes\\\
\vdots&0&y_{n}&\dots&y_{1}&y_{0}&0\\\
\texttimes&\vdots&\ddots&\ddots&\texttimes&\ddots&\ddots\end{array}\right)$
such that $X=Y^{*}Y$. Since the matrix (25) of $Y$ is also of the form (17),
we conlude that $Y\in{\mathcal{X}}$.
Let $P_{n}$ be the projection of $l^{2}({\mathbb{N}}_{0})$ onto the linear
span of $e_{0},\dots,e_{n}.$ Then we write $X$ and $Y$ as block matrices
(30) $\displaystyle X=\left(\begin{array}[]{ll}A&B\\\
B^{*}&C\end{array}\right)\ \mbox{and}\ Y=\left(\begin{array}[]{ll}U&0\\\
V&W\end{array}\right)$
where the blocks $A,B,C$ and $U,V,W$ correspond to the matrices of
$P_{n}XP_{n}$, $(I-P_{n})XP_{n}$, $(I-P_{n})X(I-P_{n})$ and $P_{n}YP_{n}$,
$(I-P_{n})YP_{n}$, $(I-P_{n})Y(I-P_{n})$, respectively.
Define $p(z){:=}\sum_{k=-n}^{n}x_{k}z^{k}\in{\mathcal{P}}$. If $p\equiv 0$,
then $X$ has the positive semi-definite matrix $A$ in the left-upper corner
and zeros elsewhere. By the Cholesky UL-decomposition (see e.g. [H], p.13) we
have $A=U^{*}U$ for some lower-triangular matrix $U.$ Putting $V=0,W=0$ in
(30) the assertion is proven in this case.
Assume now that $p\not\equiv 0$. The assumption $X\geq 0$ implies that
$C=(I-P_{n})X(I-P_{n})\geq 0$. By (17), $C$ has the same matrix as the
Toeplitz operator $T_{p}$, so that $T_{p}\geq 0$. Since nonnegative Toeplitz
operators have nonnegative symbols (see e.g. [Do], 7.19), it follows that
$p(z)\geq 0$ for all $z\in{\mathbb{T}}.$ Therefore, by the Fejér-Riesz theorem
there is a polynomial $q(z)=\sum_{k=0}^{n}y_{k}z^{k}\in{\mathcal{P}}$ such
that $p=\overline{q}q$, $q(z)\not=0$ for $|z|<1$ and $q(0)>0.$ (Note the the
degrees of $p$ and $q$ may be smaller than $n$.)
Having the polynomial $q(z)=\sum_{k=0}^{n}y_{k}z^{k},$ we define $V$ and $W$
as above (see (25)). Then $W^{*}W=T_{\overline{q}}T_{q}=T_{p}$. A direct
computation shows that $V^{*}W=B.$ Since $q(z)\not=0$ for $|z|<1$, it follows
from Lemma 2 that $\mathrm{Ran}T_{q}\equiv\mathrm{Ran}W$ is dense in the
corresponding Hilbert space $(I-P_{n})H^{2}({\mathbb{T}})$, so that $P_{W}=I$.
Therefore, $A\geq V^{*}V=V^{*}P_{W}V$ by Lemma 3. Applying the Cholesky UL-
decomposition to the finite positive semi-definite matrix $A-V^{*}V$, it
follows that there is a lower-triangular matrix $U$ such that
$A-V^{*}V=U^{*}U$. Then we have $X=Y^{*}Y.$
Since $Y\in{\mathcal{X}},$ there is a $y\in\mathcal{A}$ such that
$Y=\pi_{0}(y).$ Then we have
$\pi_{0}(x)=X=Y^{*}Y=\pi_{0}(y)^{*}\pi_{0}(y)=\pi_{0}(y^{*}y).$ Since the
representation $\pi_{0}$ of $\mathcal{A}$ is faithful, it follows that
$x=y^{*}y$ and statement (i) is proven.
## 3\. Concluding Remarks
1\. For any finite positive semi-definite complex matrix there are an LU-
decomposition and an UL-decomposition, see [G],[H]. The UL-decomposition was
used in the above proof. It might be worth to emphasize that a direct
generalization of the LU-decomposition algorithm would not work to prove our
result. For let
$\displaystyle Y_{1}{=}\left(\begin{array}[]{cccccc}3&\ 2&&&&\\\ 1&\ 1&\
0&&&\\\ 0&\ 1&\ 1&0&&\\\ &\ 0&\ 1&1&0&\\\ &&\ 0&1&1&\ddots\\\
&&&\ddots&\ddots&\ddots\end{array}\right),\
X{=}\left(\begin{array}[]{cccccc}10&\ 7&\ 0&&&\\\ 7&\ 6&\ 1&0&&\\\ 0&\ 1&\
2&1&0&\\\ &\ 0&\ 1&2&1&\ddots\\\ &&\ 0&1&2&\ddots\\\
&&&\ddots&\ddots&\ddots\end{array}\right).$
Then we have $X=Y_{1}^{*}Y_{1}$ and hence $X\geq 0$. By Lemma 1, $Y_{1}$ and
$X$ are in the $*$-algebra ${\mathcal{X}}.$ Applying the row-by-row algorithm
for the LU-decomposition (see e.g. [G]) to the infinite matrix $X$ we obtain
$Y_{2}=\left(\begin{array}[]{cccccc}\sqrt{10}&\frac{7}{\sqrt{10}}&0&&&\\\
0&\sqrt{\frac{11}{10}}&\sqrt{\frac{10}{11}}&0&&\\\
&0&\sqrt{\frac{12}{11}}&\sqrt{\frac{11}{12}}&0&\\\
&&0&\sqrt{\frac{13}{12}}&\sqrt{\frac{12}{13}}&\ddots\\\
&&&0&\sqrt{\frac{14}{13}}&\ddots\\\ &&&&\ddots&\ddots\end{array}\right)$
and $X=Y_{2}^{*}Y_{2}.$ But Lemma 1 implies that $Y_{2}$ is not in
${\mathcal{X}}$! It was crucial for our result to get a factorization inside
the $*$-algebra ${\mathcal{X}}$. However the UL-construction yields the
decomposition $X=Y^{*}Y,$ where
$Y=\left(\begin{array}[]{cccccc}\frac{1}{\sqrt{5}}&0&&&&\\\
\frac{7}{\sqrt{5}}&\sqrt{5}&0&&&\\\ 0&1&1&\ 0&&\\\ &0&1&\ 1&0&\\\ &&0&\
1&1&\ddots\\\ &&&\ddots&\ddots&\ddots\end{array}\right)\in{\mathcal{X}}.$
2\. The main result of [HMP] implies that each nonnegative $X\in{\mathcal{X}}$
is a finite sum of hermitian squares $Y^{*}Y$ in the $*$-algebra
${\mathcal{X}}$. Our theorem says that such an $X$ is a single hermitian
square. Results of this kind can be interpreted in the context of
noncommutative real algebraic geometry [S].
3\. There is the following operator version of our main result:
Let ${\mathcal{H}}\not=\\{0\\}$ be a Hilbert space and let ${\mathcal{X}}$
denote the set of all bounded operators on the Hilbert space
${\mathcal{K}}=\oplus_{k=0}^{\infty}{\mathcal{H}}_{k}$, where
${\mathcal{H}}_{k}{=}{\mathcal{H}}$, which are given by operator block
matrices of the form (17) with entries $x_{ij}$ and $x_{k}$ from
${\mathcal{B}}({\mathcal{H}})$. Then, for any element
$X=X^{*}\in{\mathcal{X}}$ such that $X\geq 0$ on ${\mathcal{K}}$ there is an
element $Y\in{\mathcal{X}}$ such that $X=Y^{*}Y.$ Moreover,
$Y\in{\mathcal{X}}$ can be chosen as a lower-triangular block matrix.
A proof of this result can be given along the lines of the proof in Section 2
with the following modifications. All bars of numbers are replaced by the
adjoints of operators. The Hilbert space ${\mathcal{K}}$ is identified with
the Hardy space $H^{2}_{\mathcal{H}}({\mathbb{T}})$ of ${\mathcal{H}}$-valued
functions. Then the operator Laurent polynomial
$p(z)=\sum_{k=-n}^{n}x_{k}z^{k}$ is nonnegative on ${\mathcal{H}}$ for all
$z\in{\mathbb{T}}$. Therefore, by Rosenblum’s operator Fejér-Riesz theorem
([R], see e.g. [D] for a nice approach) there exists an operator-valued outer
function $q(z)=\sum_{k=0}^{n}y_{k}z^{k}$, where
$y_{0},\dots,y_{n}\in{\mathcal{B}}({\mathcal{H}})$, such that
$p(z)=q(z)^{*}q(z)$ for all $z\in{\mathbb{T}}$. That $q$ is outer means that
there is a closed subspace ${\mathcal{G}}$ of ${\mathcal{H}}$ such that
$H^{2}_{\mathcal{G}}({\mathbb{T}})$ is the closure of the range of the
multiplication operator by $q$ on $H^{2}_{\mathcal{H}}({\mathbb{T}})$. If
$P_{\mathcal{G}}$ denotes the projection of ${\mathcal{H}}$ onto
${\mathcal{G}}$, then the projection $P_{W}$ from Lemma 3 acts as
$P_{W}(\varphi_{n})=(P_{\mathcal{G}}\varphi_{n})$. Therefore, the operator
matrix $Y$, defined by (30) with $V$ replaced by $P_{W}V$, is in
${\mathcal{X}}$. By Lemma 3 the finite block matrix $A-V^{*}P_{W}V$ on the
n-fold sum ${\mathcal{H}}\oplus\cdots\oplus{\mathcal{H}}$ is nonnegative.
Hence there exists a lower triangular block matrix $U$ such that
$A-V^{*}P_{W}V=U^{*}U$ (see [FF], Remark 7.5). Then we have $X=Y^{*}Y$ by (8)
which completes the proof.
## References
* [Do] R.Douglas, Banach algebra techniques in operator theory, Academic Press, New Yorck, 1972.
* [D] M.Dritschel, On factorization of trigonometric polynomials, Integral Equ. Operator Theory 49 (2004), 11-42.
* [FF] C.Foias, Frazho, A., The commutant lifting approach to interpolation problems, Birkhäuser Verlag, Basel, 1990.
* [G] G.H. Golub, Van Loan, C.F., Matrix computations, Johns Hopkins University Press, Baltimore, MD, 1996.
* [H] A.S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co., New York-Toronto-London, 1964.
* [HMP] J. Helton, McCullough, S., Putinar, M., A noncommutative Positivstellensatz on isometries, J. Reine Angew. Math. 568 (2004), 71–80.
* [R] M. Rosenblum, Vectorial Toeplitz operators and the Fejer-Riesz theorem, J. Math. Anal. Appl. 23(1968), 139–147.
* [S] K. Schmüdgen, Noncommutative real algebraic geometry – some basic concepts and first ideas, In: Emerging applications of algebraic geometry. M. Putinar and S. Sullivant (eds.), Springer-Verlag, 2009.
* [Sz] G. Szegő, Orthogonal polynomials, Colloquium Publications, Vol. XXIII, Amer. Math. Soc., Providence, R.I., 1975.
|
arxiv-papers
| 2009-08-26T12:59:00 |
2024-09-04T02:49:04.819580
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yurii Savchuk, Konrad Schm\\\"udgen",
"submitter": "Savchuk Yurii",
"url": "https://arxiv.org/abs/0908.3805"
}
|
0908.3866
|
# Pythagorean Triangles with Repeated Digits – Different Bases
Habib Muzaffar
Department of Mathematics
International Islamic University Islamabad
P.O. Box 1244, Islamabad, Pakistan
Konstantine Zelator
Department of Mathematics
College of Arts and Sciences
Mail Stop 942
University of Toledo
Toledo, OH 43606-3390
USA
## 1 Introduction
In 1998, in the winter issue of Mathematics and Computer Education ([1]) Monte
Zerger posed the following problem. He had noticed or discovered the
Pythagorean triple $(216,630,666);\ (216)^{2}+(630)^{2}=(666)^{2}$. Note that
$216=6^{3}$ and $666$ is the hypotenuse length of this Pythagorean triangle.
The question was, then whether there existed a digit $d$ (in the decimal
system) and a positive integer $k$ (other than the above) such that $d^{k}$ is
a leg length and $\underset{k\,{\rm times}}{\left(\underbrace{d\ldots
d}\right)}$ is the hypotenuse length of a Pythagorean triangle. The symbol or
notation $(\underset{k\,{\rm times}}{\underbrace{d\ldots d}})$ stands for a
natural number which in the base $10$ or decimal system has $k$ digits all of
which are equal to $d$. In other words $(\underset{k\,{\rm
times}}{\underbrace{d\ldots d}})=d\cdot 10^{k-1}+d\cdot
10^{k-2}+\ldots+10d+d$.
In 1999, F. Luca and Paul Bruckman ([2]), answered the above question in the
negative. They proved that the above Pythagorean triple is the only one with
this base $b=10$ property. In 2001, K. Zelator took this question further and
showed that there exists no Pythagorean triangle one of whose legs having leg
length $d^{k}$ while the other leg length being equal to $\underset{k\,{\rm
times}}{(}\underbrace{d\ldots d})$ (again with base $b=10$) ([3]). Note by the
way, that any such triangle (of either type) must be non-primitive.
The purpose of this work is to explore such questions in general, when the
base $b$ is no longer $10$. In Section 2, we give definitions and introduce
some notation, while in Section 5 we prove the five theorems of this paper. We
present a summary of results in Section 3. In Section 4, we state the very
well-known parametric formulas that generate the entire family of Pythagorean
triangles. In Section 6, we give five families of Pythagorean triangles with
certain properties (similar to the triple $(216,630,666)$ above; but with
respect to bases $b$ other than $10$); and in Section 7, we offer some
corollaries to these families.
Also, let us point out that in the proofs found in this work, we only use
elementary number theory.
## 2 Notation and Definitions
Let $b$ and $d$ be positive integers such that $b\geq 3$ and $2\leq d\leq
b-1$.
Notation: By $d_{k,b}$, where $k$ is a positive integer, $k\geq 2$, we will
mean the positive integer which in the base $b$ system has $k$ digits all
equal to $d$. In other words,
$\begin{array}[]{rcl}{d_{k,b}}&=&d\cdot\left(b^{k-1}+\ldots+b+1\right)=\dfrac{d\cdot(b^{k}-1)}{b-1}\\\
&=&d\cdot b^{k-1}+d\cdot b^{k-2}+\ldots+d\cdot b+d\end{array}$
$d_{k,b}=(\underset{k\,{\rm times}}{\underbrace{d\ldots d}})_{b}.$ Also, we
denote a Type 1 triangle (see definition below) by $\mathbf{T_{1}(k,b,d)}$;
and a Type 2 triangle by $\mathbf{T_{2}(k,b,d)}$. Note that for $k\geq 2,\
d_{k,b}>db^{k-1}>d\cdot d^{k-1}=d^{k}$.
Definition 2: (i) A Pythagorean triangle is called a Type 1 triangle with base
$b$ repeated digits, if there exist positive integers $k,b,d$ such that $b\geq
3,\ 2\leq d\leq b-1,\ k\geq 2$; and with one of its two legs having length
$d^{k}$, while the hypotenuse having length $d_{k,b}$. We denote such a
triangle by $T_{1}(k,b,d)$. (ii) A Pythagorean triangle is called a Type 2
triangle with base $b$ repeated digits, if there exist positive integers
$k,b,d$ such that $b\geq 3,\ 2\leq d\leq b-1,\ k\geq 2$; and with one of its
legs having length $d^{k}$, while the other length being equal to $d_{k,b}$.
We denote such a triangle by $T_{2}(k,b,d)$.
Remarks
1. 1.
Note that the inequalities in the above definition are justified by
inspection, by the fact that no Pythagorean triangle can have a side whose
length is equal to $1$. Consequently, $d\geq 2$ and thus $b$ must be at least
$3$ in value. Also, as trivially, one can see that $k$ must be at least $2$ in
value. (No Pythagorean triangle can be isosceles.)
2. 2.
Since two side lengths completely determine a Pythagorean triangle, it follows
that both notations $T_{1}(k,b,d)$ and $T_{2}(k,b,d)$ are unambiguous. In
other words, $T_{1}(k,b,d)$ or $T_{2}(k,b,d)$ can only represent one
Pythagorean triangle.
## 3 Summary of Results
In Theorem 1, we prove that if $b=4$, there exist no Type 2 Pythagorean
triangles. In other words, there exists no Pythagorean triangle which is a
$T_{2}(k,4,d)$, for some $k$ and $d$. In Theorem 2, it is shown that the only
Pythagorean triangle which is a $T_{1}(k,4,d)$ is the triangle $T_{1}(2,4,3)$,
which has side lengths $9,12,$ and $15$. In Theorem 3, we prove that there
exists no Pythagorean triangle which is a $T_{2}(k,3,d)$. Likewise, for
triangles $T_{1}(k,3,d)$. No such triangle exists, according to Theorem 4.
Finally, Theorem 5 says that there exist no Pythagorean triangles of the form
$T_{2}(2,b,d)$ with $2\leq d\leq 4$, and for any value of $b$ (remember that
always, $2\leq d\leq b-1$). Of the five families presented in Section 5, three
are families of triangles which are $T_{1}(k,b,d)$. The other two families
consist of Type 2 triangles or $T_{2}(k,b,d)$.
## 4 Pythagorean triples-parametric formulas
If $(a,b,c)$ is a Pythagorean triple with $c$ being the hypotenuse length then
(without loss of generality; $a$ and $b$ can be switched)
$a=\delta(m^{2}-n^{2}),\ b=\delta(2mn),\ c=\delta(m^{2}+n^{2})$, where
$\delta,m,n$ are positive integers such that $m>n\geq 1$, $(m,n)=1$ (i.e., $m$
and $n$ are relatively prime) and $m+n\equiv 1({\rm mod\,}2)$ (i.e., $m$ and
$n$ have different parities; one is even, the other odd). (1)
The above formulas are very well known, and they generate the entire family of
Pythagorean triangles. They can be found in almost every number theory book;
certainly in any undergraduate number theory textbook.
## 5 The Five Theorems and Their Proofs
Theorem 1: There exists no Pythagorean triangle which is a $T_{2}(k,4,d)$.
Proof: If such a triangle existed, we would have $b=4,k\geq 2$, and $2\leq
d\leq 3$. According to (1),
$\left.\left.\begin{array}[]{lll}{\rm
either}&d_{k,4}=\delta(m^{2}-n^{2}),&d^{k}=\delta(2mn)\\\ {\rm
or}&d^{k}=\delta(m^{2}-n^{2}),&d_{k,4}=\delta(2mn)\end{array}\right\\}\begin{array}[]{l}(2a)\\\
(2b)\end{array}\right\\}$ (2)
We distinguish between cases, according to whether $d=2$ or $d=3$.
Case 1: d=2
Since $m$ and $n$ have different parities, it follows that $mn\equiv 0({\rm
mod\,}2)$ and thus $\delta\cdot(2mn)\equiv 0({\rm mod\,}4)$. Also,
$d_{k,4}=2_{k,4}=\dfrac{2\cdot(4^{k}-1)}{4-1}=\dfrac{2(4^{k}-1)}{3}\equiv
2({\rm mod\,}4),$
which shows that possibility (2b) is ruled out. Thus, we consider (2a) with
$d=2$:
$\dfrac{2(4^{k}-1)}{3}=\delta(m^{2}-n^{2}),\ \ 2^{k}=\delta(2mn)$ (2c)
The second equation (2c) implies, by virtue of $m>n$, $(m,n)=1,$ and
$m+n\equiv 1({\rm mod\,}2)$, that
$\left.\begin{array}[]{l}m=2^{u},\ n=1,\ \delta=2^{v};\ {\rm for\ some}\\\
{\rm integers}\ \ u\ {\rm and}\ v\ {\rm such\ that\ }v\geq 0,u\geq 1\\\ {\rm
and\ with}\ u+v+1=k\end{array}\right\\}$ (2d)
By the first equation in (2c) and (2d) we obtain,
$2\cdot(4^{k}-1)=3\cdot 2^{v}(2^{2u}-1)$ (2e)
which easily implies $v=1$ (consider the power of $2$ on both sides). Thus,
from $k=u+v+1\Rightarrow k=u+2$; and by (2e),
$2^{2u+4}+2=3\cdot 2^{2u}\Leftrightarrow 2^{2u+3}+1=3\cdot 2^{2u-1},$
which is impossible modulo 2 since $2u-1\geq 1$ (by (2d)).
Case 2: d=3
We have $d_{k,4}=3_{k,4}=\dfrac{3(4^{k}-1)}{3}=4^{k}-1$. Obviously, since
$4^{k}-1$ is odd, it cannot equal $\delta(2mn)$. Thus again, as in the
previous case, this leads us to (2a) in (2);
$4^{k}-1=\delta(m^{2}-n^{2}),\ \ 3^{k}=\delta(2mn),$
an impossibility again since $3^{k}$ is odd, while $\delta(2mn)$ is even. The
proof is complete. $\Box$
Theorem 2: The only Type 1 triangle which is a $T_{1}(k,4,d)$, is the triangle
$T_{1}(2,4,3)$, which has side lengths $9,12,$ and $15$.
Proof: Let $T_{1}(k,4,d)$ be such a triangle; it’s hypotenuse length being
$d_{k,4}$. We must have
$\begin{array}[]{l}{\rm either}\\\ {\rm
or}\end{array}\left.\left\\{\begin{array}[]{ll}d^{k}=\delta(m^{2}-n^{2}),&d_{k,4}=\delta(m^{2}+n^{2}))\\\
d^{k}=\delta(2mn),&d_{k,4}=\delta(m^{2}+n^{2})\end{array}\right\\}\begin{array}[]{l}(3a)\\\
(3b)\end{array}\right\\}$ (3)
Since $b=4$, we must have $d=2$ or $3$. We distinguish between two cases.
Case 1: d=3
Obviously, possibility (3b) cannot hold true since $3^{k}\not\equiv 0({\rm
mod\,}2)$. Thus, we need only consider (3a). Since
$d_{k,4}=3_{k,4}=\dfrac{3(4^{k}-1)}{3}=4^{k}-1$; equations (3a) imply
$\left.\begin{array}[]{ll}&\delta(m^{2}-n^{2})=\delta(m-n)(m+n)=3^{k}\\\ {\rm
and}&\delta(m^{2}+n^{2})=4^{k}-1\end{array}\right\\}$ (3c)
Since $(m,n)=1$ and $m,n$ have different parities, we conclude that
$(m-n,m+n)=1$ and also $1\leq m-n<m+n$. This, then combined with the first
equation in (3c), implies
$\left.\begin{array}[]{l}\delta=3^{v},m-n=1,m+n=3^{w},w+v=k\\\ {\rm for\
integers}\ v,w\ {\rm with}\ v\geq 0\ {\rm and}\ w\geq 1\end{array}\right\\}$
(4)
From (4) we obtain $m=\dfrac{3^{w}+1}{2}$ and $n=\dfrac{3^{w}-1}{2}$ and thus,
the second equation in (3c) gives
$2(4^{k}-1)=3^{v}\cdot\left[3^{2w}+1\right]$ (5)
By virtue of the fact that $4^{3}=64$; $4^{3}-1\equiv 0({\rm mod\,}9)$, the
following three statements can be easily verified.
If $k\equiv 0({\rm mod\,}3)\Rightarrow 4^{k}-1\equiv 0({\rm mod\,}9)$
If $k\equiv 1({\rm mod\,}3)\Rightarrow 4^{k}-1\equiv 3({\rm mod\,}9)$
If $k\equiv 2({\rm mod\,}3)\Rightarrow 4^{k}-1\equiv 6({\rm mod\,}9)$
This then shows, by (5), that if $v\geq 2$, $k$ must be a multiple of $3$.
Accordingly, we distinguish between two subcases: $v\geq 2$ being one subcase,
while $v<2$ (i.e., $v=0$ or $1$) the other.
Subcase 1a: $v\geq 2$
In this subcase, we must have $k\equiv 0({\rm mod\,}3)$ (see above). Since
$k\geq 2$, if we consider (5) modulo (8), we see that on account of
$3^{2w}\equiv 9^{w}\equiv 1({\rm mod\,}8)$; we have
$\begin{array}[]{rcl}2(0-1)&\equiv&3^{v}(1+1)({\rm mod\,}8);\\\
-2&\equiv&3^{v}\cdot 2({\rm mod\,}8);\\\ -1&\equiv&3^{v}({\rm
mod\,}4)\Rightarrow v\equiv 1({\rm mod\,}2)\end{array}$
(Clearly, if $v$ were even, $3^{v}$ would be congruent to $1({\rm mod\,}4)$.
Thus, $v$ must be an odd integer. The next observation shows that $w$ must
also be odd. To see why, observe that if $w$ were even; then $2w\equiv 0({\rm
mod\,}4)\Rightarrow 3^{2w}\equiv 1({\rm mod\,}16)$, since $3^{4}\equiv 1({\rm
mod\,}16)$. But then, if we consider (5) modulo 16 we see that
$2(0-1)=3^{v}(1+1)({\rm mod\,}16)\Rightarrow 14\equiv 2\cdot 3^{v}({\rm
mod\,}16),$
which is impossible because $2\cdot 3^{v}\equiv 6({\rm mod\,}16)$, in view of
$v\equiv 1({\rm mod\,}2)$. Indeed, to make this a bit more clear, put $v=4t+1$
or alternatively $v=4t+3$ for some integer $t$. If $v=4t+1\Rightarrow 2\cdot
3^{v}=2\cdot 3^{4t+1}\equiv 2\cdot 3^{4t}\cdot 3({\rm mod\,}16)$ so that
$2\cdot 3^{v}\equiv 2\cdot 1\cdot 3\equiv 6({\rm mod\,}16)$. If, on the other
hand, $v=4t+3$, we have $2\cdot 3^{v}=2\cdot 3^{4t+3}\equiv 2\cdot 3^{4t}\cdot
3^{3}\equiv 2\cdot 1\cdot 11\equiv 22\equiv 6({\rm mod\,}16)$. Therefore, both
$v$ and $w$ must be odd; and so, by $k=v+w$ in (4), it follows that $k$ must
be even. Thus, since $k\equiv 0({\rm mod\,}2)$ and $k\equiv 0({\rm mod\,}3)$
(see beginning of this subcase), it follows that $k\equiv 0({\rm mod\,}6)$.
Next, we apply Fermat’s Little Theorem for the prime $p=7$:
$4^{k}-1\equiv({\rm mod\,}7)$ and thus, by (5) we see that
$3^{2w}+1\equiv 0({\rm mod\,}7)$
which is impossible by virtue of the fact that $w$ is odd. Indeed, $w\equiv
1,3,$ or $5({\rm mod\,}6)\Rightarrow 2w\equiv 2,0,$ or $4({\rm mod\,}6)4$ and
so
$\begin{array}[]{rcl}3^{2w}+1&\equiv&3^{2}+1,3^{0}+1\ {\rm or}\ 3^{4}+1\\\
&\equiv&9+1,\ 1+1\ {\rm or}\ 4+1;\\\ &\equiv&3,2,\ {\rm or}\ 5({\rm
mod\,}7).\end{array}$
This concludes the proof of Subcase 1a. $\Box$
Subcase 1b: $v<2;v=0$ or $1$
If $v=0$, then from (4) we have $w=k$ and hence from (5),
$2\cdot(4^{k}-1)=3^{2k}+1\Rightarrow 2\cdot 4^{k}=3\cdot(3^{2k-1}+1),$
which is impossible modulo 8; since $2\cdot 4^{k}\equiv 0({\rm mod\,}8)$ (in
view of $k\geq 2$), and $3^{2k-1}+1\equiv 4({\rm mod\,}8)$; and so
$3(3^{2k-1}+1)\equiv 3\cdot 4\equiv 4({\rm mod\,}8)$ as well. Note that when
the exponent is odd, say $2\rho+1$ (like in the case of $2k-1$);
$3^{2\rho+1}+1=3^{2\rho}\cdot 3+1\equiv 1\cdot 3+1\equiv 4({\rm mod\,}8)$. If
$v=1$, equation (4) gives $k=w+1$ and by (5)
$2(4^{k}-1)=3\cdot\left[3^{2(k-1)}+1\right]\Leftrightarrow
2^{2k+1}=3^{2k-1}+5.$
We claim that $k$ must equal $2$; for if to the contrary $k\geq 3$, the last
equation implies $2\cdot 2^{2k}=\dfrac{3^{2k}}{3}+5;\
6=\left(\dfrac{9}{4}\right)^{k}+\dfrac{15}{4^{k}}\Rightarrow\left(\dfrac{9}{4}\right)^{k}<6$
which is impossible since for $k\geq 3;\
\left(\dfrac{9}{4}\right)^{k}\geq\left(\dfrac{9}{4}\right)^{3}>6$. Thus,
$k=2$, and so from (4) we obtain $w=1$.
Altogether, $v=1=w$ and $k=2$. By (4) $\Rightarrow\delta=3,\ m=2,\ n=1$, we
obtain the Pythagorean triangle whose side lengths are $9,12,$ and $15$; and
since $\delta=3$, this is the triangle $T_{1}(2,4,3)$. This concludes the
proof of Subcase 1b. $\Box$
Case 2: d=2
Going back to (3), we easily see that possibility (3a) cannot hold true, since
the first equation in (3a) would imply that both $\delta$ and $m^{2}-n^{2}$
are powers of $2$. But $m^{2}-n^{2}$ is an odd integer (since $m$ and $n$ have
different parities) and $m^{2}-n^{2}>1$.
Now, consider (3b). We have,
$2^{k}=\delta(2mn),\ \dfrac{2(4^{k}-1)}{3}=\delta(m^{2}+n^{2})$ (6)
and since $m>n\geq 1$, $(m,n)=1$ and $m+n\equiv 1({\rm mod\,}2)$, the first
equation in (6) implies
$\left.\begin{array}[]{l}\delta=2^{u},\ m=2^{t},\ n=1,\ k=u+t+1\\\ {\rm for\
integers}\ u,t,\ {\rm with}\ u\geq 0\ {\rm and}\ t\geq 1\end{array}\right\\}$
(7)
Combining (7) with the second equation in (6) yields
$4^{k}-1=3\cdot 2^{u-1}\cdot(2^{2t}+1).$
The possibility $u=0$ is impossible since $3\cdot(2^{2t}+1)$ is odd.
It follows that we must have $u=1$; and by (7), $t=k-2$. Thus, the last
equation above gives, after some algebra, $4^{k-1}-1=3\cdot 2^{2k-6}$. Recall
that $k\geq 2$. Clearly, the last equation requires $2k-6=0$ since its left-
hand side is an odd integer. Thus, $k=3$, which in turn implies $4^{2}-1=3;\
15=3$, a contradiction. This concludes the proof of Theorem 2. $\Box$
Theorem 3: Let $b=3$. There exists no Pythagorean Type 2 triangle with base 3
repeated digits. In other words, there exists no triangle which is a
$T_{2}(k,3,d)$.
Proof: First observe that since $2\leq d\leq b-1,\ 2\leq d\leq 3-1\Rightarrow
d=2$; and so, $d^{k}=2^{k}$ and
$d_{k,3}=2_{k,3}=\dfrac{2(3^{k}-1)}{2}=3^{k}-1$. If such a triangle exists,
one leg will have length $2^{k}$, the other $3^{k}-1$. Thus, there are two
possibilities.
$\left.\left.\begin{array}[]{lll}{\rm
Either}&3^{k}-1=\delta(2mn),&2^{k}=\delta(m^{2}-n^{2})\\\ {\rm
or}&3^{k}-1=\delta(m^{2}-n^{2}),&2^{k}=\delta(2mn)\end{array}\right\\}\begin{array}[]{l}(8a)\\\
(8b)\end{array}\right\\}$ (8)
Case 1: Assume possibility (8a) to hold.
From the second equation in (8a), it follows that both positive integers
$\delta$ and $m^{2}-n^{2}$ are powers of $2$. However,
$m^{2}-n^{2}=(m-n)(m+n)\geq 3$, on account of $m>n\geq 1$. In fact,
$m^{2}-n^{2}\geq(n+1)^{2}-n^{2}=2n+1\geq 3$. Since $m,n$ have different
parities, $m^{2}-n^{2}$ is an odd integer greater than or equal to $3$. Thus,
it cannot equal to a power of $2$, which renders the second equation in (8a)
contradictory.
Case 2: Assume possibility (8b)
The second equation implies that each of the positive integers $\delta,m,n$,
must be a power of $2$; and since $m>n\geq 1$ and $m+n\equiv 1({\rm mod\,}2)$,
it follows that
$\left.\begin{array}[]{ll}&n=1,\ m=2^{v},\ \delta=2^{p}\\\ {\rm for\
integers}&v\ {\rm and}\ p\ {\rm such\ that}\ v\geq 1,p\geq
0\end{array}\right\\}$ (9)
Combining (9) with the first equation in (8b) yields,
$3^{k}-1=2^{p}\cdot(2^{2v}-1)$ (10)
Consider equation (10) modulo 3. We have,
$2^{2v}-1=4^{v}-1\equiv 1-1\equiv 0({\rm mod\,}3)\Rightarrow
2^{p}\cdot(2^{2v}-1)\equiv 0({\rm mod\,}3)$
while
$3^{k}-1\equiv 0-1\equiv-1\equiv 2({\rm mod\,}3).$
We have a contradiction. The proof is complete. $\Box$
Theorem 4: Let $b=3$. There exists no Type 1 triangle with base 3 repeated
digits. In other words, there exists no triangle which is a $T_{1}(k,3,d)$
Proof:
Again, as in the previous proof of Theorem 3, observe that in view of $2\leq
d\leq b-1$, we have $d=2$. If a triangle $T_{2}(k,3,2)$ exists, it would be a
Pythagorean triangle with the hypotenuse having length
$d_{k,3}=\left(\dfrac{3^{k}-1}{2}\right)\cdot 2=3^{k}-1$, and with one of the
two legs having length $2^{k}$. Which means that,
$\left.\left.\begin{array}[]{ll}{\rm
Either}&2^{k}=\delta(m^{2}-n^{2}),3^{k}-1=\delta(m^{2}+n^{2})\\\ {\rm
or}&2^{k}=\delta(2mn),3^{k}-1=\delta(m^{2}+n^{2})\end{array}\right\\}\begin{array}[]{l}(11a)\\\
(11b)\end{array}\right\\}$ (11)
The first possibility (11a) is ruled out at once by the first equation in
(11a), since $m^{2}-n^{2}$ is an odd integer and $m^{2}-n^{2}\geq 3$; we
already saw this in the proof of Theorem 3.
Next, let us consider (11b), the other possibility. As in the previous proof,
in view of $m>n\geq 1$, we easily infer from the first equation that
$\left.\begin{array}[]{ll}&\delta=2^{p},m=2^{v},n=1\\\ {\rm for\ some\
integers}&p,v\ {\rm with}\ p\geq 0,v\geq 1\\\ {\rm
and}&p+v+1=k.\end{array}\right\\}$ (12)
Recall that always $k\geq 2$. From (12) and the second equation in (11b), we
obtain
$3^{p+v+1}-1=2^{p}\cdot(2^{2v}+1)$ (13)
Consider (13) modulo $3$. Since $2^{2v}+1=4^{v}+1\equiv 1+1\equiv 2({\rm
mod\,}3)$, (13) implies $-1\equiv 2^{p+1}({\rm mod\,}3)\Leftrightarrow 2\equiv
2^{p+1}({\rm mod\,}3)\Leftrightarrow$ (since $(2,3)=1$) $1\equiv 2^{p}({\rm
mod\,}3)\Leftrightarrow p\equiv 0({\rm mod\,}2)$. $p$ must be an even integer.
Moreover, since the left-hand side of (13) is an even integer, while
$2^{2v}+1$ is odd; $2^{p}$ must be even; which means $p\geq 1$. But $p$ is
even, so we must have $p\geq 2$.
Since $p\geq 2$, the right-hand side of (13) must be a multiple of $4$;
$3^{p+v+1}-1\equiv 0({\rm mod\,}4)$ (13a)
However, if $\ell$ is a positive integer, then $3^{\ell}-1\equiv 0({\rm
mod\,}4)$ if $\ell$ is even; while $3^{\ell}-1\equiv 2({\rm mod\,}4)$ if
$\ell$ is odd, as it can be easily verified. Thus, (13a) implies that the
exponent $p+v+1$ must be an even integer; and in view of $p\equiv 0({\rm
mod\,}2)$, we see that $v$ must be odd:
$p\equiv 0({\rm mod\,}2),\ \ v\equiv 1({\rm mod\,}2)$ (13b)
Accordingly by (13b), $p\equiv 0,2,$ or $4({\rm mod\,}6)$, while $v\equiv
1,3,$ or $5({\rm mod\,}6)$
We will use this, by considering equation (13) modulo $7$. To facilitate this
end, observe that, if $r$ is a positive integer and $r\equiv i({\rm mod\,}6)$,
with $0\leq i\leq 5$, then $2^{r}\equiv 2^{i}({\rm mod\,}7)$ and $3^{r}\equiv
3^{i}({\rm mod\,}7)$. This observation leads to the following table:
$\begin{array}[]{|c|c|c|c|}\hline\cr&&&\\\ {\rm Value\ of}\ p&{\rm Value\ of}\
v&{\rm Value\ of}\ 2^{p}\cdot(2^{2v}+1)&{\rm Value\ of}\ 3^{p+v+1}-1\\\ {\rm
modulo}\ 6&{\rm mod\,}\ 6&{\rm mod\,}\ 7&{\rm mod\,}\ 7\\\ \hline\cr&&&\\\
0&1&5&1\\\ \hline\cr&&&\\\ 0&3&2&3\\\ \hline\cr&&&\\\ 0&5&3&0\\\
\hline\cr&&&\\\ 2&1&6&3\\\ \hline\cr&&&\\\ 2&3&1&0\\\ \hline\cr&&&\\\
2&5&5&1\\\ \hline\cr&&&\\\ 4&1&3&0\\\ \hline\cr&&&\\\ 4&3&4&1\\\
\hline\cr&&&\\\ 4&5&6&3\\\ \hline\cr\end{array}$
The results on the last two columns (columns 3 and 4) clearly render (13)
impossible modulo $7$; a contradiction. The proof is complete. $\Box$
Theorem 5: Let $k=2,\ 2\leq d\leq 4$ (i.e., $d=2,3,$ or $4$). There exists no
Type 2 triangle with base $b$ repeated digits. In other words, there exists no
triangle which is a $T_{2}(2,b,d)$. That is no triangle which is a
$T_{2}(2,b,2)$, a $T_{2}(2,b,3)$, or a $T_{2}(2,b,4)$.
Note: Due to $2\leq d\leq b-1$, when $d=2$, we must have $b\geq 3$; when
$d=3,\ b\geq 4$; while for $d=4,\ b\geq 5$.
Proof: We give a simple proof, without making use of parametric formulas (1).
If a triangle $T_{2}(2,b,d)$ exists, with $2\leq d\leq 4$, then it must have
one leg length equal to $d_{2,b}=d(b+1)$, while the other leg length is equal
to $d^{2}$. Thus, $\left[d(b+1)\right]^{2}+(d^{2})^{2}=m^{2}$, for some
positive integer $m$; and so,
$d^{2}\cdot\left[(b+1)^{2}+d^{2}\right]=m^{2}.$
In the last equation, $d^{2}$ is a divisor of $m^{2}$, so $d$ must be a
divisor of $m$. Put $m=d\cdot c$, for some positive integer $c$, in order to
obtain
$d^{2}+(b+1)^{2}=c^{2}\Leftrightarrow\left[c+(b+1)\right]\cdot\left[c-(b+1)\right]=d^{2}$
(14)
Next, we use the conditions $2\leq d\leq 4$ and $2\leq d\leq b-1$. If $d=2$,
(14) $\Rightarrow$ (since $c+b+1>c-(b-1)$) $c+b+1=4$ and $c-(b+1)=1$ which, in
turn, gives $c=\dfrac{5}{2}$, a contradiction since $c$ is an integer. If
$d=3$, (14) $\Rightarrow c+b+1=9$ and $c-(b+1)=1$, which gives $b=3$,
contradicting $d\leq b-1$.
If $d=4$, we have either $c+b+1=8$ and $c-(b+1)=2$ or $c+b+1=16$ and
$c-(b+1)=1$. In the first case we obtain $b=2$, a contradiction since $b\geq
5$. In the second case, $c=\dfrac{17}{2}$ a contradiction once more. The proof
is complete. $\Box$
## 6 Five Families with $k=2$
A basic principle used for the construction of all five families below, is the
identity $(r^{2}-q^{2})^{2}+(2rq)^{2}=(r^{2}+q^{2})^{2}$.
Basic Principle: $(r^{2}-q^{2})^{2}+(2rq)^{2}=(r^{2}+q^{2})^{2}$, for any
positive integer $r$ and $q$.
1. A.
Two families of Type 2 triangles
In both families below, we take $k=2$; and so $d_{k,b}=d_{2,b}=d\cdot
b+d=d(b+1)$. To see how the first family comes about, let $\ell,q$ be positive
integers such that $\ell^{2}\leq 2q^{2}-2$, and let $r=q+\ell$. Take
$b=2rq-1,\ d=r^{2}-q^{2}$. Note that
$d=r^{2}-q^{2}\geq(q+1)^{2}-q^{2}=2q+1\geq 3$ and also that,
$b-1-d=2rq-1-1-(r^{2}-q^{2})=2(q+\ell)q-2-(q+\ell)^{2}+q^{2}=2q^{2}-\ell^{2}-2\geq
0$. Thus $b-1-d\geq 0;\ d\leq b-1$. Altogether, $3\leq d\leq b-1$. Also, by
our basic principle above one can easily verify that indeed
$(d_{2,b})^{2}+(d^{2})^{2}=$ integer square. This then establishes the first
family.
Family F1 The following family of Type 2 triangles is described in terms of
two independent positive integer parameters $\ell$ and $q$ which satisfy the
condition $\ell^{2}\leq 2q^{2}-2$. This family consists of all Type 2
triangles of the form $T_{2}(2,b,d)$, where the integers $b$ and $d$ are
defined as follows: $b=2rq-1,\ d=r^{2}-q^{2},\ r=q+\ell$.
To construct the second family, let $\ell$ and $q$ be positive integers such
that, this time, $\ell^{2}\geq 2q^{2}+2$. Also, let $r=q+\ell,\
b=r^{2}-q^{2}-1,\ d=2rq$. By inspection, $d\geq 2$. Clearly, $\ell^{2}\geq
2q^{2}+2>2q^{2}$, which shows that $\ell>q$. So,
$b=r^{2}-q^{2}-1=(\ell+q)^{2}-q^{2}-1\geq 2q\ell\geq 4$, since $\ell>q$. Thus
$b\geq 4$. Moreover,
$b-1-d=r^{2}-q^{2}-1-1-2rq=(q+\ell)^{2}-q^{2}-2-2(q+\ell)q=\ell^{2}-2-2q^{2}\geq
0$. Thus, altogether $2\leq d\leq b-1$; and by our basic principle above, we
also have $(d_{2,b})^{2}+(d^{2})^{2}=$ integer square.
Family F2 The following family of Type 2 triangles is described in terms of
two independent positive integer parameters which satisfy the condition
$\ell^{2}\geq 2q^{2}+2$. This family consists of all Type 2 triangles of the
form $T_{2}(2,b,d)$, where the integers $b$ and $d$ are defined as follows:
$b=r^{2}-q^{2}-1,\ d=2rq,\ r=q+\ell$.
2. B.
Three families of Type 1 Triangles
Let $r,q$ be positive integers with $r>q$. Let $b=r^{2}+q^{2}-1,\
d=r^{2}-q^{2}$. Then $d\geq(q+1)^{2}-q^{2}=2q+1\geq 3$; and let
$b+1-d=2q^{2}\geq 2$, so that $3\leq d\leq b-1$. Also, by our basic principle,
$(b+1)^{2}-d^{2}=(2rq)^{2}$ and thus $(d_{2,b})^{2}-(d^{2})^{2}=$ square.
Family S1 The following family of Type 1 triangles is described in terms of
two independent positive integer parameters $r$ and $q$ satisfying $r>q$. This
family consists of all Type 1 triangles of the form $T_{1}(2,b,d)$ where $b$
and $d$ are defined as follows: $b=r^{2}+q^{2}-1,\ d=r^{2}-q^{2},\ r>q\geq 1$.
Next, let $r,q$ be positive integers with $r\geq q+2$. Take $b=r^{2}+q^{2}-1$,
$d=2rq$. Then, clearly $d\geq 2$ and $b+1-d=(r-q)^{2}\geq 4$, so that $2\leq
d\leq b-3<b-1$. As before, by our basic principle, we have
$(d_{2,b})^{2}-(d^{2})^{2}=$ square.
Family S2 The following family of Type 1 triangles is described in terms of
two independent positive integer parameters. This family consists of all Type
1 triangles of the form $T_{1}(2,b,d)$; where $b$ and $d$ are defined as
follows: $b=r^{2}+q^{2}-1,\ d=2rq,\ r\geq q+2$.
Finally, let $b\geq 4$, with $b$ being an integer square, $b=k^{2},\
k\in{\mathbb{Z}}^{+}$. Let $d=b-1$. Then $(d_{2,b})^{2}-(d^{2})^{2}=$ square,
since $(b+1)^{2}-d^{2}=(b+1)^{2}-(b-1)^{2}=4b=4k^{2}$.
Family U The following family of Type 1 triangles is described in terms of one
positive integer parameter $t$. This family consists of all Type 1 triangles
of the form $T_{1}(2,b,d)$ where $b=t^{2},\ d=b-1=t^{2}-1,\ t\geq 2,\
t\in{\mathbb{Z}}^{+}$.
## 7 Corollaries
Corollaries of Family $S_{1}$
Let $d$ be an odd integer with $d\geq 3$. Then, there exists a triangle which
is a $T_{1}(2,b,d)$ for some positive integer $b$ with $d\leq b-1$.
Proof:
We have $d=2v+1$, for some integer $v\geq 1$. Set $u=v+1$, so that
$d=u^{2}-v^{2}$. By Family S1, if we take $b=u^{2}+v^{2}-1=\dfrac{d^{2}-1}{2}$
then $d\leq b-1$ and $(d_{2,b})^{2}-(d^{2})^{2}=$ square $\Box$
Corollaries of Family S2
Let $d$ be an even integer with $d\geq 6$. Then, there exists a triangle which
is $T_{1}(2,b,d)$ for some positive integer $b$ with $d\leq b-1$.
Proof:
We have $d=2v$, for some integer $v\geq 3$. Let $u=1$, so that $v-u\geq 2$.
Then, if we take $b=v^{2}=\dfrac{d^{2}}{4}$, by Family S2, we have $d\leq b-1$
and $(d_{2,b})^{2}-(d^{2})^{2}=$ square. $\Box$
Remark: It is easy to show that for $d=2,4$ there exists no integer $b$ with
$d\leq b-1$ and $(d_{2,b})^{2}-(d^{2})^{2}=$ square.
Corollary of Family F1
Let $d$ be an odd integer with $d\geq 5$. Then, there exists a triangle which
is $T_{2}(2,b,d)$ for some positive integer $b$ with $d\leq b-1$.
Proof:
We have $d=2q+1$, for some positive integer $q\geq 2$.
Let $\ell=1$, so that $\ell^{2}\leq 2q^{2}-2$. Also, note that if we take
$r=q+\ell,\ r=q+1$, then $d=r^{2}-q^{2}=(q+1)^{2}-q^{2}$. Therefore, by Family
F1, if we set $b=2(q+1)q-1=2rq-1=\dfrac{d^{2}-3}{2}$. It follows that $d\leq
b-1$ and $(d_{2,b})^{2}+(d^{2})^{2}=$ square. $\Box$
Corollary to Family F2
Let $d$ be an even integer with $d\geq 6$. Then, there exists a triangle which
is $T_{2}(2,b,d)$ for some positive integer $b$ with $d\leq b-1$.
Proof: We have $d=2r$, for some integer $r\geq 3$. Let $q=1$, so that $d=2rq$;
and $\ell=r-q=r-1\geq 2$. Then $\ell^{2}\geq 2q^{2}+2=4$. From Family F2, it
is now clear that if we set $b=r^{2}-2=\dfrac{d^{2}-8}{4}$, then $d\leq b-1$
and $(d_{2,b})^{2}+(d^{2})^{2}=$ square. $\Box$
Note that the corollaries of F1 and F2 complement the result of Theorem 5.
## References
* [1] Repeated Digits in Pythagorean Triples, Problem 337, Proposed by Monte J. Zerger Mathematics and Computer Engineering, Vol. 32, No.1 (winter 1998), p. 86.
* [2] Repeated Digits in Pythagorean Triples, Problem 337, Solution by Paul Bruckman and Florian Luca, Mathematics and Computer Education, Vol. 33, (Spring 1998), pp. 291-292.
* [3] Konstantine D. Zelator, Pythagorean Triangles with Repeated Digits: A Solution to a Problem, Mathematics and Computer Education, Vol. 36, No. 1 (Winter 2001), pp. 38-42.
|
arxiv-papers
| 2009-08-26T17:05:25 |
2024-09-04T02:49:04.827624
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Habib Muzaffar, Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/0908.3866"
}
|
0908.3933
|
11institutetext: Max-Planck-Institute f$\rm\ddot{u}$r Radioastronomie, Auf
dem H$\rm\ddot{u}$gel 69, 53121 Bonn, Germany
11email: xuye@pmo.ac.cn 22institutetext: Purple Mountain Observatory, Chinese
Academy of Sciences, Nanjing 210008, China 33institutetext: Australia
Telescope National Facility CSIRO, PO Box 76, Epping, NSW 1710, Australia
44institutetext: Astro Space Centre, Profsouznaya st. 84/32, 117997 Moscow,
Russia 55institutetext: Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai,
20030, China 66institutetext: Ural State University, Ekaterinburg, 620083,
Russia
# Absolute positions of 6.7-GHz methanol masers
Y. Xu 1122 M. A. Voronkov 3344 J. D. Pandian 11 J. J. Li 55 A. M. Sobolev 66
A. Brunthaler 11 B. Ritter 11 K. M. Menten 11
(Received date; accepted date)
The ATCA, MERLIN and VLA interferometers were used to measure the absolute
positions of 35 6.7-GHz methanol masers to subarcsecond or higher accuracy.
Our measurements represent essential preparatory data for Very Long Baseline
Interferometry, which can provide accurate parallax and proper motion
determinations of the star-forming regions harboring the masers. Our data also
allow associations to be established with infrared sources at different
wavelengths. Our findings support the view that the 6.7 GHz masers are
associated with the earliest phases of high-mass star formation.
###### Key Words.:
masers — ISM: techniques: interferometric — astrometry — spiral arm: distances
††offprints: Y. Xu
## 1 Introduction
The $5_{1}-6_{0}$ A+ transition of methanol at 6.7-GHz produces the brightest
methanol masers (Menten 1991). The masers are widespread in the Galaxy and
more than 550 sources have been detected to date, including the compilations
of Xu et al. (2003), Malyshev & Sobolev (2003), and Pestalozzi et al. (2005),
and the searches of Caswell et al. (1995a), Caswell (1996a, 1996b), MacLeod et
al. (1998), Szymczak et al. (2000), Pandian et al. (2007), Ellingsen (2007),
and Xu et al. (2008).
It has been shown that 12.2-GHz methanol masers are excellent tools for
determining the distances to massive star-forming regions by measuring their
trigonometric parallax using Very Long Baseline Interferometry (VLBI) (e.g.,
Xu et al. 2006a). The $2_{0}-3_{-1}$ E transition at 12.2-GHz is the second
brightest methanol maser transition and the locations of the 6.7-GHz and
12.2-GHz methanol maser spots largely overlap, with several features showing a
one-to-one correspondence within milliarcseconds and the spectra of the two
transitions typically covering similar velocity ranges (Menten et al. 1992,
Norris et al. 1993, Minier et al. 2000). Since 6.7 GHz masers are almost
always stronger, i.e., much stronger than their 12.2-GHz counterparts, they
are also expected to be a useful for probing distances to massive star-forming
regions in the Galaxy. Measuring accurate distances are critical for studying
the massive star-forming regions individually and understanding the
distribution of these regions in the context of our perception of the Galaxy’s
spiral structure.
For phase-referenced VLBI observations (mandatory for high precision
astrometry), one usually employs strong masers as the phase-reference, and
synthesizes images of nearby extragalactic continuum sources. The astrometric
precision scales with the source separation and, statistically, weaker sources
are found closer to masers. These sources can be detected, e.g., by VLA
observations; see Xu et al. 2006b). On the other hand, for a successful VLBI
astrometric measurement, one requires a position estimate of the maser
position that is accurate to at least 1′′, as input in the correlator. This
means that a large number of masers in the literature have positions
determined with single dish observations that are not accurate enough for VLBI
observations. Here we report absolute position measurements of 35 6.7-GHz
methanol masers with subarcsecond accuracy using the Australia Telescope
Compact Array (ATCA), MERLIN and the NRAO Very Large Array (VLA). Most of the
sources in our study are associated with 12.2-GHz counterparts.
## 2 Observations and data reduction
The ATCA observations were completed in 2006 April in the 6C configuration,
which produces baselines from 153 to 6000 m. The observations were done in
snap-shot mode. Each source was observed in six 5-minute scans spread over a
range of hour angles to ensure a good uv-coverage. The correlator was
configured to have a 4 MHz bandwidth with 1024 spectral channels. Two
orthogonal linear polarizations were observed and averaged together during the
data processing. The full width at half-maximum (FWHM) of the primary beam was
$7.2^{\prime}$. The default pointing model was used, giving an rms pointing
accuracy of around of 5–10′′. The accuracy of the pointing model affects the
accuracy of the flux density measurements, particularly for sources that are
offset from the pointing center. The absolute flux density scale was
determined from observations of PKS B1934–638. The accuracy of the flux
density calibration is expected to be approximately 3% 111For details of the
calibration using 1934–638, query this calibrator at the ATCA calibrators
webpage (http://www.narrabri.atnf.csiro.au/calibrators). The bandpass
calibration was carried out using observations of the continuum source
1921–293. The data were reduced with the MIRIAD package using standard
procedures.
The MERLIN observations were carried out in 2007 February using six
telescopes, and in 2007 March using five telescopes. The correlator was used
in two modes. The phase reference sources and the primary calibrator (3C84)
were observed in wide-band mode with a bandwidth of 16 MHz and 32 spectral
channels. The bandpass calibrator (also 3C84) and the targets were observed in
the narrow-band mode with 2 MHz bandwidth and 256 spectral channels. The total
on-source integration time per source was over 1 hour, divided into a number
of 5 min scans to achieve good uv-coverage. The flux density of 3C84 was
assumed to be 14.5 Jy in February 2007 and 15 Jy in March 2007. The initial
calibration and conversion of the data to FITS format was done using the local
MERLIN software, and subsequent analysis was done using the Astronomical Image
Processing System (AIPS). The instrumental phase offset between the wide band
and narrow band data was derived using 3C84 (see the MERLIN User Guide for
details; Diamond et al. 2003).
The VLA observations were conducted on 2009 January 27 in BnA configuration
using 18 EVLA antennas. The observations were done in single IF mode (A1, RCP)
with each source being observed in three scans with an integration time of
2.75 minutes per scan. The full 3.125 MHz bandwidth was divided into 256
channels. For all target sources the bandwidth was always centered on the
velocity of 14.75 km s-1 with respect to the local standard of rest. The
primary beam was 6.75′. The total flux density of the flux calibrator 3C 286
was calculated to be 6.072 Jy. The source 2007+404 served as a bandpass
calibrator and 2084+431 was the phase reference calibrator for all target
sources.
The spectral resolution was 0.18, 0.35 and 0.55 km s-1 for the ATCA, MERLIN
and VLA observations, respectively. The rest frequency of 6668.5192 MHz was
assumed for all observations.
After imaging the targets, the AIPS package task “JMFIT” was used to determine
the fluxes and positions of each maser feature in all observations.
## 3 Results
The accuracy of the absolute maser positions is limited by several factors,
such as source elevation, weather conditions, length and type of observations,
position accuracy of the phase calibrator, and signal-to-noise ratio. For the
ATCA data, the typical signal-to-noise ratio is over 500, and the phase
calibrators have positions accurate to better than 0.15′′. Hence, we assume
that most of the target sources have an absolute position accuracy of 0.5′′ or
higher, except for sources that are close to the celestial equator. This is a
typical position accuracy of ATCA data (Caswell et al. 1995c; Walsh et al.
1998; Phillips et al. 1998; Minier et al. 2001, Caswell 2009). For the MERLIN
data, the typical signal-to-noise ratio is at least 100, and all but one phase
calibrator are from the Jodrell Bank–VLA Astrometric Survey, which has a
position accuracy of higher than 5 milliarcseconds. Hence, we estimate that
the absolute positions of the target sources are accurate to within 0.1′′. The
source $IRAS$ 20290+4052 was observed by both the VLA and MERLIN, and the VLA
positions deviate from the MERLIN positions by 0.05′′. On the other hand, our
previous observations with the VLA B configuration have a position uncertainty
of better than 0.1′′ (Xu et al. 2006b). Therefore, a position uncertainty of
better than 0.1′′ is expected for sources observed with the VLA. A position
uncertainty of better than $1^{\prime\prime}$ is sufficient for successful
observations using the European VLBI Network (EVN) and the Very Long Baseline
Array (VLBA), for example to determine parallax measurements.
Tables 1 - 3 list the properties of the 6.7-GHz methanol sources observed with
the ATCA, MERLIN, and VLA, respectively. For sources that exhibit multiple
masing spots, the properties of individual spots are given in separate rows.
In the three tables, the first three columns show the source name and J2000
equatorial coordinates. Columns 4 and 5 give their Galactic coordinates.
Columns 6 and 7 show the radial velocity of the maser peak and that of the
molecular lines, respectively. Column 8 presents the peak flux density.
Columns 9 to 11 present the distance of the source from the Galactic center
and its heliocentric kinematical distance. The kinematical distances were
calculated using the velocities of molecular lines, such as CO, CS, and NH3
(where these data are available; for other sources, the maser peak velocity
was used) and the Galactic rotation model of Wouterloot & Brand (1989),
assuming $R_{0}$ = 8.5 kpc and $\Theta_{0}$ = 220 km s-1. The uncertainties in
kinematic distances were estimated by applying $\pm$10 km s-1 velocity
offsets.
### 3.1 Notes on selected sources
The spectra of the 6.7-GHz methanol masers are shown in Figs. 3–5, which is
available on line, for sources observed with the ATCA, MERLIN, and VLA,
respectively . The ATCA data have a velocity resolution of 0.18 km s-1 which
is sufficient to resolve the multiple velocity components of each source. The
velocity resolution of the VLA data is 0.55 km s-1. The velocity resolution of
the MERLIN data is only 0.7 km s-1 after Hanning smoothing (because of the
limitations of the correlator), which results in the blending of individual
components in the spectra. Here we present notes on some sources.
G8.8316–0.0281. There are at least six features within the velocity range of
10 km s-1 that have peak flux densities exceeding 10 Jy/beam. The strongest
feature is at the LSR velocity of $-$3.9 km s-1.
G8.8722–0.4928. There are only two features stronger than 10 Jy/beam, at
$+$23.4 and $+$24.1 km s-1. The two features are spectrally blended together.
G14.1014+0.0869. There are a number of features located within a
0.2${}^{\prime\prime}\times 0.2^{\prime\prime}$ region. The strongest 6.7 GHz
peak is at the velocity of the second strongest 12.2-GHz feature (Blaszkiewicz
& Kus 2004).
G23.0099–0.4107. The features span the velocity range from 70.1 to 83.2 km
s-1, as do the 12.2-GHz maser features in this source, although they do not
coincide precisely. There are at least 5 features for which the peak flux
density exceeds 10 Jy/beam. The strongest feature is at the same velocity as
the strongest 12.2-GHz maser (Caswell et al. 1995b). However, the 6.7-GHz
feature corresponding to the second strongest 12.2-GHz feature at +76.6 km s-1
is not clearly distinguishable from the other lines.
G23.2068–0.3777. This source has two prominent peaks. The strongest feature is
at the same velocity, 81.7 km s-1, as the strongest feature at 12.2-GHz
(Blaszkiewicz & Kus 2004).
G24.1480–0.0092. The spectra at both 6.7-GHz and 12.2-GHz (Blaszkiewicz & Kus
2004) are similar and dominated by a single feature.
G27.3652–0.1659. There are clearly five features with peak flux densities
exceeding 10 Jy/beam. The emission has a spatial extent of 0.1′′ and is
confined to a narrow velocity range of approximately 5 km s-1. The 12.2-GHz
spectrum is dominated by a single feature (Caswell et al. 1995b). The peak
velocity is the same for both 6.7 and 12.2-GHz masers. Our observations
detected an unresolved 8.6 GHz continuum source that is offset by about 2.4′′
from the maser.
G29.8630–0.0442. There are three strong features within an area of
0.1${}^{\prime\prime}\times 0.1^{\prime\prime}$. The strongest feature at
6.7-GHz matches the velocity of the second strongest 12.2-GHz feature and vice
versa (Caswell et al. 1995b).
G30.1987–0.1687 and G30.2251-0.1796. The two sources are separated by 103′′.
The peak velocities are $+$108.6 km s-1 and $+$113.5 km s-1 for
G30.1987–0.1687 and G30.2251–0.1796, respectively. There are two corresponding
12.2-GHz features at $+$108.5 and +110.2 km s-1, respectively (Caswell et al.
1995b), which are likely to originate in the same two emission centres.
However, a high angular resolution study at 12.2-GHz is required to confirm
this. Although there are a number of features in both sources, only these two
peaks have peak flux densities that exceed 10 Jy/beam.
G30.8987+0.1616. There are two features with peak flux densities exceeding 10
Jy/beam. The emission peaks at 12.2 and 6.7 GHz are close in velocity.
S255. The spectrum of Szymczak et al. (2000) shows multiple spectral features,
which are not visible in the VLA spectrum because of its poor velocity
resolution. However, imaging shows two maser sites separated by 0.2′′.
18556+0136. The spectrum of Szymczak et al. (2000) shows multiple spectral
features. However, in this study we found only a single feature. The 12.2-GHz
spectrum also has multiple features, but is dominated by just two peaks
(Caswell et al. 1995b).
G43.15+0.02. The spectrum of Caswell et al. (1995a) indicated that multiple
features were present. We detected only one spectral feature. Its velocity
corresponds to that of the strongest 12.2 GHz feature (Caswell et al. 1995b).
19120+0917. This sources exhibits multiple features that coincide in velocity
with the 12.2-GHz features (Blaszkiewicz & Kus 2004).
19186+1440. This source displays multiple features at 6.7-GHz in the range
from -16 to -9 km s-1. Szymczak et al. (2000) also reported 6.7-GHz emission
in the velocity range from $-$31 to $-$25 km s-1, which was not detected in
our observations.
19303+1651 and 20290+4052. The spectra of these sources are both dominated by
a single feature. Each velocity component has its 12.2-GHz counterpart
(Blaszkiewicz & Kus 2004).
ON1. This source consists of two separate masing sites with a separation of
around 1′′. The peak velocities for these two sites are 15.7 and $-$0.1 km
s-1.
21381+5000. There is only one feature detected in the MERLIN observations,
while Szymczak et al. (2000) detected multiple features.
## 4 Methanol masers and spiral arms
Since 6.7-GHz methanol masers appear to be exclusively associated with massive
star-forming regions, they are reliable tracers of the spiral arms of the
Galaxy. This is especially so since the lifetime of methanol masers is
understood to be about 104 yr (van der Walt 2005). To investigate whether any
information about the spiral structure of the Galaxy can be inferred from the
data of methanol masers detected to date, we compiled a table of all known
6.7-GHz methanol masers (Table 5, on line). The LSR velocities in Table 5
originate in molecular lines such as CS, CO, and NH3, where such data are
available. For other sources, the velocity of the maser peak was used. The
kinematical distances were calculated using the Galactic rotation model of
Wouterloot & Brand (1989), assuming $R_{0}$ = 8.5 kpc and $\Theta_{0}$ = 220
km s-1. We made no attempt to calculate the distances for those sources, which
are located in the two Galactic longitude ranges $0^{\circ}\pm 10^{\circ}$ and
$180^{\circ}\pm 10^{\circ}$, where the uncertainty in the kinematic method is
large.
A significant fraction of the sources in the first and the fourth Galactic
quadrants are affected by an ambiguity between two distances, the near and the
far distance. This kinematical distance ambiguity has been resolved for only a
small number of methanol masers (Pandian et al. 2008). Sobolev et al. (2005)
proposed that statistically, it is preferable to assume a more nearby
kinematic distance than a far distance. The left panel of Fig. 1 shows a face-
on diagram of the Galaxy, where the near kinematic distance is assumed for all
sources affected by a distance ambiguity. Spiral arm loci from the NE2001
model of Cordes & Lazio (2002) are superimposed. It can be seen that there is
little if any correlation between the location of methanol masers and the
spiral arm model. The right panel of Fig. 1 shows the same diagram, but with
the far distance being assumed for all masers affected by a distance
ambiguity. Qualitatively, there appears to be a stronger correlation with the
spiral arm loci in the right-hand panel than in the left-hand one. Keeping in
mind that in reality there are only a fraction of sources located at the near
distance, and that the kinematic distances have relatively large
uncertainties, it seems possible to reconcile the spiral arm model with the
distribution of methanol masers in the Galaxy. However, this exercise does
suggest that the assumption of the majority of sources being at the near
kinematic distance may be flawed. This suggestion can be corroborated by a
general observation that the majority of young massive star-forming regions
associated with HII regions appear to be at the far distance in the studies
able to resolve the ambiguity (e.g., Kolpak et al. 2003).
Figure 1 also shows that there is a poor correspondence between the spiral arm
model and the massive star-forming regions in the outer Galaxy where there is
no distance ambiguity. This is mostly caused by a significant deviation from
the circular rotation in the Perseus arm region (Xu et al. 2006a). Based on
the VLBI parallax measurements for a number of massive star forming-regions,
Reid et al. (2009) found that these regions orbit the Galactic center $\sim$
15 km s-1 slower than the Galaxy itself, if one assumes circular rotation. In
addition, the motion of the Sun towards the local standard of rest (LSR) was
found to be consistent with that derived by Dehnen & Binney (1998) from
Hipparcos data. We hence recalculated kinematic distances using the
methodology explained in sect. 4 of Pandian et al. (2008) – the radial
velocities were recalculated to the new frame of solar motion, and kinematic
distances were calculated using the Galactic rotation curve of Wouterloot &
Brand (1989) with $R_{0}$ = 8.4 kpc and $\Theta_{0}$ = 254 km s-1 (Reid et al.
2009), assuming that the massive star forming-regions were rotating 15 km s-1
slower than predicted by the rotation curve. The left-hand and the right-hand
panels of Fig. 2 show the equivalent of Fig. 1 for the new kinematic
distances. It can be seen that there is little difference between Figs. 1 and
2 for the inner Galaxy, but there is now much closer agreement between the
model and the data in the Perseus arm region of the outer Galaxy.
## 5 Association with star formation tracers
It is well established that the 6.7-GHz methanol masers are associated with
high-mass stars (e.g., W3(OH); see Menten et al. 1992), which are able to pump
the masers by heating a sufficient amount of surrounding dust to temperatures
higher than 100 K, or producing hypercompact HII regions with extremely high
emission measures (Sobolev et al. 2007). No 6.7-GHz methanol masers have been
found to be associated with low-mass young stellar objects (Minier et al.
2003, Bourke et al. 2005).
Maser surveys suggest that the 6.7-GHz methanol masers are associated with
different phases of development in the HII regions (Ellingsen 2007). Almost
all 6.7-GHz masers are found to be associated with 1.2 mm emission (Hill et
al. 2005), while many have no associated 8.6-GHz continuum emission (Walsh et
al. 1998). Relevant cases can be found even within one star-forming region,
e.g., NGC6334 I, which possesses a maser cluster associated with a prominent
ultracompact HII region and another one associated with the sub-mm core and a
candidate hypercompact HII region with very weak radio continuum emission
(Hunter et al. 2006).
Maser positions measured to subarcsecond accuracy allow us to study the
connection between the methanol masers and the other signposts of massive star
formation. However, there are no published high resolution continuum surveys
in radio or submillimeter wavelengths that cover all or a significant fraction
of the sources in our sample. Hence, we focus on infrared counterparts from
all sky or Galactic plane surveys.
Table 4 presents the association with infrared sources. Among the 35 sources
in our sample, 25 are in the inner Galaxy, while 10 are located in the outer
Galaxy. In the inner Galaxy, 19 of 25 sources are covered by the GLIMPSE
survey, which is limited to Galactic longitudes, $|l|\leq 65^{\circ}$.
Seventeen sources have a GLIMPSE point source within 5 arcsec, dropping to 11
within 2 arcsec. Most sources with no nearby point source in the GLIMPSE
catalog or archive data are associated with extended emission, and one source
(G23.2068-0.3777) is associated with an infrared dark cloud. Only four sources
have flux measurements in all four bands (often due to extended emission in
the other bands), and hence we do not attempt to compare the properties of the
sources with those published previously (e.g., Ellingsen 2006).
Twenty-one masers have a 2MASS point-source counterpart within 5 arcsec,
dropping to 9 within 2 arcsec. Most of the sources show an infrared excess
based on their JHK colors, which is indicative of an association with
protostars. Four sources show no infrared excess, suggesting that they are
either foreground stars or more evolved objects. By Cross-correlating with the
GLIMPSE catalog, only five GLIMPSE sources are found to have 2MASS
counterparts. This strongly suggests that most of the 6.7-GHz methanol masers
do not have 2MASS counterparts, and that most of the nearby 2MASS sources are
more evolved young stellar objects in the star-forming region. This supports
the results of Ellingsen (2005, 2006).
Since the black-body emission of the warm dust, hypothesized to be the pump
source for the masers, peaks at around 25 $\mu$m (Ostrovskii & Sobolev, 2002),
one expects all methanol masers to have mid/far infrared counterparts. Two
surveys with data at this wavelength range are the all sky survey of $IRAS$,
and the MIPSGAL survey using the Spitzer space telescope. Keeping in mind that
the $IRAS$ point source catalog is limited by both source confusion and poor
resolution, 20 methanol masers have an $IRAS$ point source within 30 arcsec.
We note that 9 of 10 sources located in the outer Galaxy, where source
confusion is not as severe as in the inner Galaxy, have an $IRAS$ point source
with an infrared luminosity greater than 103 $L_{\odot}$. However, due to the
poor resolution of the $IRAS$ satellite, it is possible that a single point
source in the $IRAS$ catalog may correspond to multiple star-forming sites in
the molecular cloud. Hence, higher spatial resolution data is required to
infer properties such as the luminosity and mass of the source associated with
the maser.
MIPSGAL is a Galactic plane survey at 24 and 70 $\mu$m using the MIPS camera
of the Spitzer Space Telescope (Rieke et al. 2004, Carey et al. 2005). The
survey is limited to Galactic longitudes between 5 and 63 degrees in the first
Galactic quadrant and 298 and 355 degrees in the fourth quadrant for Galactic
latitudes $|b|\leq 1^{\circ}$. Eighteen sources in our sample are covered by
the survey, and all sources are associated with 24 $\mu$m emission, as shown
in Fig. 6 (available on line). An association with point sources (which are
occasionally saturated) is evident for 16 sources, while for 2 sources
(G43.15+0.02 and $W51e2$) the images are completely saturated. It is thus
reasonable to expect that all the 6.7-GHz methanol masers have MIPSGAL
counterparts. However, image artifacts such as saturation make it difficult to
determine 24 $\mu$m fluxes. Further observations will be required before we
will be able to determine their spectral energy distributions and dust
properties. Fifteen sources in our sample have an MSX point source within 5′′.
When restricted to sources that are covered by the MIPSGAL survey, only six
sources have a nearby MSX source. The far poorer statistics of the
associations with MSX sources is probably caused by the coarser spatial
resolution of the MSX satellite and its poorer sensitivity.
## 6 Conclusions
Absolute positions with an accuracy of 1 arcsecond or higher have been
determined for 35 6.7-GHz methanol masers. Our measurements are essential to a
future VLBI astrometric follow-up observations. Kinematic distances to the
masers imply that they do not trace the spiral arms well irrespective of
whether they are at the near or far kinematic distances, although there is a
small improvement if the rotation curve of Reid et al. (2009) is used.
Although our sample is not statistically complete, the number of associations
with infrared sources is consistent with the expectation that the 6.7 GHz
masers are associated with the early phases of massive star formation.
###### Acknowledgements.
We would like to thank the referee, Simon Ellingsen, for many useful
suggestions and comments which help us to improve this paper. The Australia
Telescope is funded by the Commonwealth of Australia for operation as a
National Facility managed by CSIRO. We thank Drs. A. M. S. Richards and R.
Beswick for help in reducing MERLIN data. This work was supported by the
Chinese NSF through grants NSF 10673024, NSF 10733030, NSF 10703010 and NSF
10621303, and NBRPC (973 Program) under grant 2007CB815403. AMS was supported
by RFBR grants 07-02-00628-a and 08-02-00933-a. This work used the NASA/IPAC
Infrared Science Archive.
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Table 1: 6.7-GHz methanol masers observed with ATCA. The first column lists
the source name. The next two columns give their J2000 equatorial coordinates.
Columns. (4) and (5) give their galactic coordinates. Columns. (6) and (7)
show the radial velocity of peak emission from the maser and molecular lines.
Col. (8) presents the peak flux density. Columns. (9) - (11) present the
distance from the Galactic center, and the far and near kinematic distances.
$\begin{array}[]{lccrrrrccrr}\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr{\parbox[t]{54.06023pt}{\centering Source\\\
Name\@add@centering}}&{\parbox[t]{54.06023pt}{\centering R.A.(2000)\\\
\mbox{$\mathrm{(^{h}\;\;\;{}^{m}\;\;\;{}^{s})}$}\@add@centering}}&{\parbox[t]{54.06023pt}{\centering
DEC(2000) \\\
\mbox{$(\degr\;\;\;\arcmin\;\;\;\arcsec)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering
l \\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering b
\\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{LSR}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{mol}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{31.29802pt}{\centering$\rm
S_{peak}$\\\
\mbox{\scriptsize(Jy/beam)}\@add@centering}}&{\parbox[t]{25.60747pt}{\centering$\rm
R$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{far}$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{near}$\\\ \mbox{\scriptsize(kpc)}\@add@centering}}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\small\par$G$8.832-0.028&18\ 05\ 25.66&-21\ 19\
25.5&8.832&-0.028&-3.9&0.5(1)&132.0&8.4&16.7(\pm 3.6)&0.1(\pm 2.0)\\\
$G$8.872-0.493&18\ 07\ 15.32&-21\ 30\
54.4&8.872&-0.493&23.4&&30.9&4.9&13.2(\pm 1.1)&3.6(\pm 1.1)\\\
$G$14.101+0.087&18\ 15\ 45.80&-16\ 39\
09.7&14.101&0.087&15.2&9.3(2)&90.2&7.2&15.1(\pm 1.5)&1.3(\pm 1.1)\\\ &18\ 15\
45.80&-16\ 39\ 09.5&14.101&0.087&5.9&&15.4&&&\\\ &18\ 15\ 45.81&-16\ 39\
09.5&14.102&0.087&10.1&&25.1&&&\\\ &18\ 15\ 45.81&-16\ 39\
09.6&14.101&0.087&11.0&&22.8&&&\\\ &18\ 15\ 45.80&-16\ 39\
09.7&14.101&0.087&13.1&&16.3&&&\\\ &18\ 15\ 45.80&-16\ 39\
09.7&14.101&0.087&13.6&&22.6&&&\\\ $G$23.010-0.411&18\ 34\ 40.29&-09\ 00\
38.1&23.010&-0.411&74.8&74.8(2)&415.4&4.4&10.8(\pm 0.5)&4.9(\pm 0.5)\\\ &18\
34\ 40.29&-09\ 00\ 38.2&23.010&-0.411&72.7&&43.4&&&\\\ &18\ 34\ 40.28&-09\ 00\
38.4&23.010&-0.411&80.6&&49.1&&&\\\ &18\ 34\ 40.28&-09\ 00\
38.4&23.010&-0.411&81.6&&52.0&&&\\\ &18\ 34\ 40.27&-09\ 00\
38.5&23.010&-0.411&82.3&&43.4&&&\\\ $G$23.207-0.378&18\ 34\ 55.20&-08\ 49\
14.2&23.207&-0.378&81.7&&38.2&4.3&10.5(\pm 0.5)&5.2(\pm 0.5)\\\ &18\ 34\
55.21&-08\ 49\ 14.4&23.207&-0.378&76.5&&19.4&&&\\\ &18\ 34\ 55.21&-08\ 49\
14.6&23.207&-0.378&77.0&&35.4&&&\\\ $G$24.148-0.009&18\ 35\ 20.94&-07\ 48\
55.6&24.148&-0.009&17.7&23.1(3)&26.8&6.7&13.5(\pm 0.8)&2.0(\pm 0.8)\\\
$G$24.329+0.144&18\ 35\ 08.14&-07\ 35\
04.0&24.329&0.144&110.2&112.0(2)&5.0&3.7&8.9(\pm 0.9)&6.6(\pm 0.9)\\\
$G$27.220+0.260&18\ 40\ 03.72&-04\ 57\ 45.6&27.220&0.260&9.3&&6.2&7.8&14.3(\pm
0.9)&0.8(\pm 0.8)\\\ $G$27.365-0.166&18\ 41\ 51.06&-05\ 01\
42.8&27.365&-0.166&100.0&92.2(4)&28.0&4.2&9.2(\pm 0.7)&5.9(\pm 0.7)\\\ &18\
41\ 51.06&-05\ 01\ 42.8&27.365&-0.166&98.0&&10.2&&&\\\ &18\ 41\ 51.06&-05\ 01\
42.8&27.365&-0.166&98.9&&13.7&&&\\\ &18\ 41\ 51.06&-05\ 01\
42.8&27.365&-0.166&100.7&&14.2&&&\\\ &18\ 41\ 51.06&-05\ 01\
42.7&27.365&-0.166&101.5&&10.8&&&\\\ $G$29.863-0.044&18\ 45\ 59.57&-02\ 45\
04.4&29.863&-0.044&101.4&100.4(4)&76.5&4.3&8.2(\pm 0.8)&6.6(\pm 0.8)\\\ &18\
45\ 59.57&-02\ 45\ 04.5&29.863&-0.044&100.3&&52.0&&&\\\ &18\ 45\ 59.57&-02\
45\ 04.5&29.863&-0.044&101.7&&54.2&&&\\\ $G$30.199-0.169&18\ 47\ 03.07&-02\
30\ 33.6&30.199&-0.169&108.6&103.3(4)&16.0&4.3&7.8(\pm 1.0)&6.9(\pm 1.0)\\\
$G$30.225-0.180&18\ 47\ 08.30&-02\ 29\
27.1&30.225&-0.180&113.5&104.5(4)&10.8&4.3&7.7(\pm 1.1)&7.0(\pm 1.1)\\\
$G$30.899+0.162&18\ 47\ 09.13&-01\ 44\
08.8&30.899&0.162&101.8&&25.7&4.4&7.7(\pm 1.1)&6.9(\pm 1.1)\\\ &18\ 47\
09.13&-01\ 44\ 08.7&30.899&0.162&103.0&&14.2&&&\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$
References for the velocities:
1 Zhang et al. 2005. 2 Larionov et al. 1999. 3 Szymczak et al. 2007. 4 van der
Walt et al. 2007.
Table 2: Same as Table 1 for the 6.7-GHz methanol masers observed with MERLIN.
$\begin{array}[]{lccrrrrccrr}\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr{\parbox[t]{54.06023pt}{\centering Source\\\
Name\@add@centering}}&{\parbox[t]{54.06023pt}{\centering R.A.(2000)\\\
\mbox{$\mathrm{(^{h}\;\;\;{}^{m}\;\;\;{}^{s})}$}\@add@centering}}&{\parbox[t]{54.06023pt}{\centering
DEC(2000) \\\
\mbox{$(\degr\;\;\;\arcmin\;\;\;\arcsec)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering
l \\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering b
\\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{LSR}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{mol}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{31.29802pt}{\centering$\rm
S_{peak}$\\\
\mbox{\scriptsize(Jy/beam)}\@add@centering}}&{\parbox[t]{25.60747pt}{\centering$\rm
R$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{far}$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{near}$\\\ \mbox{\scriptsize(kpc)}\@add@centering}}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\small\par$L1287$&00\ 36\ 47.358&63\ 29\
02.18&121.298&0.659&-23.2&-17.6(3)&7.5&9.4&1.6(\pm 0.9)&\\\ $NGC281-N$&00\ 52\
24.196&56\ 33\ 43.17&123.066&-6.309&-29.3&-31.8(3)&32.4&10.4&2.9(\pm 1.0)&\\\
$S231$&05\ 39\ 13.066&35\ 45\
51.29&173.482&2.446&-12.8&-15.8(3)&32.1&24.1&15.6(\pm 12.9)&\\\ $AFGL5180$&06\
08\ 53.342&21\ 38\ 29.09&188.946&0.886&10.7&3.1(1)&194&9.4&0.9(\pm 4.7)&\\\
$AFGL6366$&06\ 08\ 40.671&21\ 31\
06.89&189.030&0.784&8.8&2.5(1)&1.6&9.2&0.7(\pm 4.4)&\\\ $S255$&06\ 12\
54.006&17\ 59\ 23.21&192.600&-0.048&5.5&8.2(3)&10.0&10.3&1.9(\pm 3.6)&\\\ &06\
12\ 54.020&17\ 59\ 23.27&192.600&-0.048&4.4&&8.5&&&\\\ $S269$&06\ 14\
37.051&13\ 49\ 36.16&196.454&-1.677&15.3&17.9(1)&3.6&12.1&3.7(\pm 3.7)&\\\
18556+0136&18\ 58\ 13.053&01\ 40\
35.68&35.197&-0.743&28.5&35.0(3)&72.8&6.6&11.4(\pm 0.6)&2.5(\pm 0.6)\\\
$G43.15+0.02$&19\ 10\ 11.049&09\ 05\
20.49&43.149&0.013&13.4&2.9(1)&6.0&8.3&12.2(\pm 0.8)&0.2(\pm 0.8)\\\
19120+0917&19\ 14\ 26.393&09\ 22\
36.53&43.890&-0.784&52.0&54.2(1)&3.0&6.2&8.1(\pm 1.6)&4.2(\pm 1.6)\\\ &19\ 14\
26.392&09\ 22\ 36.62&43.890&-0.784&47.7&&4.5&&&\\\ 19186+1440&19\ 20\
59.212&14\ 46\ 49.65&49.416&0.326&-12.1&&7.0&9.2&12.1(\pm 0.9)&\\\ W51e2&19\
23\ 43.949&14\ 30\ 34.44&49.490&-0.388&59.2&56.5(3)&217&&&\\\ 19303+1651&19\
32\ 36.071&16\ 57\ 38.46&52.663&-1.092&65.7&59.7(1)&1.3&&&\\\ $ON1A$&20\ 10\
09.047&31\ 31\ 35.06&69.540&-0.976&14.7&11.7(3)&31.0&8.0&4.0(\pm 1.7)&2.0(\pm
1.8)\\\ $ON1B$&20\ 10\ 09.073&31\ 31\
35.94&69.540&-0.976&-0.1&11.7(3)&8.8&8.0&4.0(\pm 1.7)&2.0(\pm 1.8)\\\
20290+4052&20\ 30\ 50.673&41\ 02\
27.55&79.736&0.991&-5.2&-1.4(1)&18.2&8.6&3.3(\pm 1.5)&\\\ 21381+5000&21\ 39\
58.263&50\ 14\ 20.96&94.602&-1.796&-40.7&-45.6(3)&4.2&10.8&6.1(\pm 1.1)&\\\
&21\ 39\ 58.262&50\ 14\ 21.05&94.603&-1.796&-43.6&&4.0&&&\\\ $L1206$&22\ 28\
51.408&64\ 13\ 41.30&108.184&5.519&-11.0&-9.9(2)&29.4&8.9&1.2(\pm 1.1)&\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$
References for the velocities:
1 Bronfman et al. 1996. 2 Molinari et al. 1996. 3 Plume et al. 1992.
Table 3: Same as Table 1 for the 6.7-GHz methanol masers observed with VLA.
$\begin{array}[]{lccrrrrccrr}\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr{\parbox[t]{54.06023pt}{\centering Source\\\
Name\@add@centering}}&{\parbox[t]{54.06023pt}{\centering R.A.(2000)\\\
\mbox{$\mathrm{(^{h}\;\;\;{}^{m}\;\;\;{}^{s})}$}\@add@centering}}&{\parbox[t]{54.06023pt}{\centering
DEC(2000) \\\
\mbox{$(\degr\;\;\;\arcmin\;\;\;\arcsec)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering
l \\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{45.5244pt}{\centering b
\\\ \mbox{$(\degr)$}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{LSR}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{28.45274pt}{\centering$\rm
V_{mol}$\\\ \mbox{\scriptsize(km
s${}^{-1}$)}\@add@centering}}&{\parbox[t]{31.29802pt}{\centering$\rm
S_{peak}$\\\
\mbox{\scriptsize(Jy/beam)}\@add@centering}}&{\parbox[t]{25.60747pt}{\centering$\rm
R$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{far}$\\\
\mbox{\scriptsize(kpc)}\@add@centering}}&{\parbox[t]{39.83385pt}{\centering$\rm
d_{near}$\\\ \mbox{\scriptsize(kpc)}\@add@centering}}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\small\par$DR20$&20\ 37\ 00.960&41\ 34\
55.74&80.861&0.383&-4.44&3.98(1)&1.24&8.4&1.7(\pm 2.1)&1.0(\pm 2.1)\\\
$W75$&20\ 38\ 36.428&42\ 37\ 34.82&81.871&0.781&6.76&9.8(2)&152.28&&&\\\
$DR21B\\_1$&20\ 39\ 01.989&42\ 24\
59.30&81.752&0.591&-9.07&-2.5(2)&4.00&8.6&3.0(\pm 1.5)&\\\ $DR21B\\_2$&20\ 39\
01.057&42\ 22\ 49.18&81.722&0.571&-3.03&-2.5(2)&2.65&8.6&3.0(\pm 1.5)&\\\
$DR21B\\_3$&20\ 39\ 00.374&42\ 24\
37.13&81.744&0.591&3.56&-2.5(2)&3.16&8.6&3.0(\pm 1.5)&\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\end{array}$
References for the velocities:
1 Schneider et al. 2006. 2 Plume et al. 1992.
Table 4: Association of the 6.7-GHz methanol masers with the GLIMPSE, 2MASS, MSX, and $IRAS$ point sources. Source | 2MASS | | GLIMPSE | | MSX | | $IRAS$
---|---|---|---|---|---|---|---
Name | Name | Separation | | Name | Separation | | Name | Separation | | Name | Separation
| | (arcsec) | | | (arcsec) | | | (arcsec) | | | (arcsec)
G8.832$-$0.028 | | | | GLMC G008.8315$-$00.0278 | 0.5 | | | | | 18024$-$2119 | 15.9
G8.872$-$0.493 | 18071532$-$2130540 | 0.4 | | GLMA G008.8721$-$00.4926 | 0.3 | | G008.8725$-$00.4929 | 2.4 | | 18042$-$2131 | 6.9
G14.101$+$0.087 | | | | GLMC G014.1017$+$00.0872 | 1.7 | | | | | 18128$-$1640 | 11.8
G23.010$-$0.411 | | | | GLMC G023.0097$-$00.4105 | 0.3 | | | | | |
G23.207$-$0.378 | | | | GLMC G023.2075$-$00.3772 | 3.1 | | | | | |
G24.148$-$0.009 | | | | GLMC G024.1479$-$00.0091 | 0.3 | | | | | 18326$-$0751 | 11.3
G24.329$+$0.144 | | | | GLMC G024.3285$+$00.1440 | 0.4 | | | | | 18324$-$0737 | 5.2
G27.220$+$0.260 | | | | GLMC G027.2197$+$00.2607 | 0.9 | | | | | |
G27.365$-$0.166 | | | | GLMA G027.3650$-$00.1656 | 0.8 | | | | | 18391$-$0504 | 2.3
G29.863$-$0.044 | 18455955$-$0245061 | 1.8 | | GLMA G029.8623$-$00.0442 | 2.1 | | G029.8620$-$00.0444 | 3.5 | | |
G30.199$-$0.169 | 18470306$-$0230361 | 2.6 | | GLMA G030.1981$-$00.1691 | 2.5 | | G030.1981$-$00.1691 | 2.4 | | |
G30.225$-$0.180 | 18470824$-$0229302 | 3.3 | | extended emission | | | | | | |
G30.899$+$0.162 | | | | GLMC G030.8988$+$00.1615 | 1.1 | | | | | |
L1287 | 00364719$+$6329058 | 3.8 | | | | | | | | 00338$+$6312 | 1.0
NGC281-N | 00522425$+$5633471 | 4.0 | | | | | | | | 00494$+$5617 | 4.3
S231 | 05391330$+$3545538 | 3.8 | | | | | G173.4815$+$02.4459 | 1.4 | | |
AFGL5180 | 06085340$+$2138281 | 1.3 | | | | | | | | 06058$+$2138 | 12.0
AFGL6366 | 06084091$+$2131056 | 3.6 | | | | | | | | 06056$+$2131 | 7.6
S255 | 06125385$+$1759242 | 2.5 | | | | | G192.6005$-$00.0479 | 0.8 | | 06099$+$1800 | 10.0
S269 | 06143706$+$1349364 | 0.4 | | | | | G196.4542$-$01.6777 | 1.8 | | 06117$+$1350 | 7.1
18556$+$0136 | | | | GLMA G035.1973$-$00.7430 | 1.1 | | G035.1979$-$00.7427 | 3.7 | | 18556$+$0136 | 1.6
G43.15$+$0.02 | 19101091$+$0905176 | 3.5 | | GLMC G043.1480$+$00.0131 | 3.5 | | G043.1492$+$00.0130 | 1.1 | | |
19120$+$0917 | 19142616$+$0922346 | 3.8 | | extended emission | | | G043.8896$-$00.7835 | 3.9 | | 19120$+$0917 | 3.8
19186$+$1440 | | | | GLMC G049.4152$+$00.3253 | 2.1 | | | | | |
W51e2 | | | | GLMC G049.4892$-$00.3879 | 2.3 | | | | | |
19303$+$1651 | 19323607$+$1657384 | 0.1 | | GLMC G052.6625$-$01.0919 | 0.4 | | | | | 19303$+$1651 | 12.1
ON1A | 20100886$+$3131392 | 4.8 | | | | | G069.5395$-$00.9754 | 1.2 | | 20081$+$3122 | 1.4
ON1B | 20100886$+$3131392 | 4.2 | | | | | G069.5395$-$00.9754 | 2.0 | | 20081$+$3122 | 1.8
20290$+$4052 | 20305058$+$4102298 | 2.5 | | | | | G079.7358$+$00.9905 | 0.9 | | 20290$+$4052 | 2.9
21381$+$5000 | 21395825$+$5014209 | 0.1 | | | | | G094.6028$-$01.7966 | 2.7 | | 21381$+$5000 | 7.0
L1206 | | | | | | | | | | 22272$+$6358A | 5.8
DR20 | 20370116$+$4134545 | 2.6 | | | | | G080.8624$+$00.3827 | 4.6 | | 20352$+$4124 | 2.2
W75 | | | | | | | | | | |
DR21B_1 | 20390200$+$4225008 | 1.6 | | | | | G081.7522$+$00.5906 | 0.7 | | |
DR21B_2 | 20390101$+$4222502 | 1.2 | | | | | G081.7220$+$00.5699 | 4.1 | | |
DR21B_3 | 20390047$+$4224369 | 1.2 | | | | | | | | |
Figure 1: Positions of 495 6.7-GHz masers at the near (open triangles), far
(open squares) kinematic distances, and of the outer Galaxy (filled circles)
in the Galactic plane. The distances calculated using the Galactic rotation
model of Wouterloot & Brand (1989), assuming $R_{0}$ = 8.5 kpc and
$\Theta_{0}$ = 220 km s-1. There is a poor correspondence between the spiral
arm models and massive star-forming regions for both the near and far
distances and also in the outer Galaxy. The positions of the spiral arms are
taken from Cordes & Lazio (2002). We do not show error bars for clarity.
Figure 2: Same as in Fig. 1, but the revised distances from Reid et al. (2009)
have been used. There is little difference between Figs. 1 and 2 in the inner
Galaxy, but there is a far closer agreement between the model and the data in
the Perseus arm region of the outer Galaxy.
| | |
---|---|---|---
| | |
| | |
| | |
| | |
Figure 3: Spectra of the 6.7-GHz methanol masers from the ATCA observations. The spectral resolution is approximately 0.18 km s-1. | | |
---|---|---|---
| | |
| | |
| | |
| | |
---|---|---|---
| | |
Figure 4: Same as fig. 3, but for the spectra obtained with the MERLIN. The spectral resolution is approximately 0.7 km s-1. | | |
---|---|---|---
| | |
Figure 5: Same as fig. 3, but for the VLA spectra. The spectral resolution is
approximately 0.55 km s-1.
---
---
Figure 6: Gray scale is the MIPSGAL 24 $\mu$m emission, the pluses are the positions of 6.7-GHz methanol masers, the triangles are the positions of GLIMPSE point sources, the squares are the positions of 2MASS point sources, the stars are the positions of MSX point sources, and the crosses are the positions of $IRAS$ point sources. Moving from the top left to the bottom right, the sources are G8.8316-0.0281, G8.8722-0.4928, G14.1014+0.0869, G23.0099-0.4107, G23.2068-0.3777, G24.1480-0.0092, G24.3287+0.1440, G27.2198+0.2604, G27.3652-0.1659, G29.8630-0.0442, G30.1987-0.1687, G30.2251-0.1796, G30.8987+0.1616, 18556+0136, G43.15+0.02, 19120+0917, 19186+1440, and W51e2. Table 5: Parameters of 592 6.7-GHz methanol masers. The first column lists the source name. The next two columns give their J2000 equatorial coordinates. Column. (4) shows the radial velocity of peak emission from molecular lines, such as CS, CO and NH3, etc., if available, otherwise from the maser peak. Columns. (5) - (7) present the Galactic center distance, far and near kinematic distances calculated using the Galactic rotation model of Wouterloot & Brand (1989), assuming $R_{0}$ = 8.5 kpc and $\Theta_{0}$ = 220 km s-1. Columns. (8) - (10) are the same as Cols. (5) - (7), but revised using new $R_{0}$ and $\Theta_{0}$ from Reid et al. (2009). | | | | $\Theta_{0}$ = 220 km s-1 | | $\Theta_{0}$ = 254 km s-1 |
---|---|---|---|---|---|---|---
Source | R.A.(2000) | DEC(2000) | $\rm V_{LSR}$ | R | $\rm d_{far}$ | $\rm d_{near}$ | | R | $\rm d_{far}$ | $\rm d_{near}$ | ref
Name | $\mathrm{(^{h}\;\;\;{}^{m}\;\;\;{}^{s})}$ | ($\degr\;\;\;\arcmin\;\;\;\arcsec$) | (km s-1) | (kpc) | (kpc) | (kpc) | | (kpc) | (kpc) | (kpc) |
0.21$-$0.00 | 17 46 07.66 | $-$28 45 20.0 | 37.519 | 0.2 | 8.7 | 8.3 | | 0.2 | 8.6 | 8.2 | 6,43,44
0.31$-$0.20 | 17 47 09.12 | $-$28 46 16.0 | 18.73 | 0.5 | 9.0 | 8.0 | | 0.5 | 8.9 | 7.9 | 6,43,44
0.37$+$0.04 | 17 46 21.42 | $-$28 35 39.5 | 84.43 | 0.1 | 8.6 | 8.4 | | 0.1 | 8.5 | 8.3 | 4,6,43
0.39$-$0.03 | 17 46 41.12 | $-$28 37 05.5 | 28.727 | 0.4 | 8.9 | 8.1 | | 0.4 | 8.8 | 8.0 | 6
0.49$+$0.18 | 17 46 04.00 | $-$28 24 51.5 | $-$3.83 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 6,43,44
0.53$+$0.18 | 17 46 09.8 | $-$28 23 29 | $-$3.83 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 32,43,33
0.54$-$0.85 | 17 50 14.53 | $-$28 54 31.0 | 16.630 | 0.9 | 9.4 | 7.6 | | 0.9 | 9.3 | 7.5 | 20,4,43,44
0.64$-$0.04 | 17 47 18.65 | $-$28 24 25.0 | 60.530 | 0.3 | 8.8 | 8.2 | | 0.3 | 8.7 | 8.1 | 20,4,6,43,44 33
0.66$-$0.02 | 17 47 18.61 | $-$28 22 56.0 | 60.230 | 0.3 | 8.8 | 8.2 | | 0.3 | 8.7 | 8.1 | 20,4,6,33
0.67$-$0.02 | 17 47 19.23 | $-$28 22 14.5 | 66.130 | 0.3 | 8.8 | 8.2 | | 0.3 | 8.7 | 8.1 | 20,4,6,33
0.69$-$0.03 | 17 47 24.81 | $-$28 21 43.5 | 66.130 | 0.3 | 8.8 | 8.2 | | 0.3 | 8.7 | 8.1 | 20,4,6,43,33
0.83$+$0.18 | 17 46 52.82 | $-$28 07 35.0 | 5.43 | 3.1 | 11.5 | 5.4 | | 3.2 | 11.6 | 5.2 | 6
1.14$-$0.12 | 17 48 48.47 | $-$28 01 12.0 | $-$15.63 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 43,44
2.14$+$0.01 | 17 50 35.5 | $-$27 05 55 | 63.027 | 0.9 | 9.3 | 7.6 | | 1.0 | 9.3 | 7.5 | 20,4
2.53$+$0.19 | 17 50 46.48 | $-$26 39 45.0 | 4.527 | 5.7 | 14.2 | 2.8 | | 5.8 | 14.2 | 2.6 | 32,43,44
3.91$+$0.00 | 17 54 39.0 | $-$25 34 48 | 17.027 | 3.9 | 12.3 | 4.7 | | 4.0 | 12.3 | 4.4 | 20,4
5.90$-$0.42 | 18 00 40.87 | $-$24 04 20.5 | 10.530 | 5.7 | 14.1 | 2.8 | | 5.8 | 14.1 | 2.7 | 4,43,44
6.60$-$0.08 | 18 00 54.05 | $-$23 17 02.0 | 1.027 | 8.2 | 16.5 | 0.3 | | 8.0 | 16.3 | 0.4 | 32,43,44
6.78$-$0.27 | 18 01 57.2 | $-$23 12 37 | 21.13 | 4.6 | 12.9 | 4.0 | | 4.7 | 12.9 | 3.8 | 39,33
8.13$+$0.22 | 18 03 01.0 | $-$21 48 47 | 18.630 | 5.2 | 13.5 | 3.3 | | 5.3 | 13.5 | 3.2 | 32,43
8.68$-$0.36 | 18 06 23.5 | $-$21 37 23 | 33.741 | 4.1 | 12.3 | 4.5 | | 4.2 | 12.3 | 4.3 | 20,4,43
8.83$-$0.03 | 18 05 25.66 | $-$21 19 25.5 | 0.549 | 8.4 | 16.7 | 0.1 | | 8.2 | 16.4 | 0.2 | 52
8.87$-$0.49 | 18 07 15.32 | $-$21 30 54.4 | 23.452 | 4.9 | 13.2 | 3.6 | | 5.0 | 13.1 | 3.5 | 52
9.62$+$0.19 | 18 06 14.659 | $-$20 31 31.57 | 4.330 | 7.6 | 15.8 | 0.9 | | 7.5 | 15.6 | 1.0 | 20,4,43,23
9.98$-$0.02 | 18 07 50.11 | $-$20 18 57.0 | 48.941 | 3.6 | 11.7 | 5.1 | | 3.7 | 11.7 | 4.9 | 32,43,44
10.09$+$0.71 | 18 05 18.18 | $-$19 51 14.5 | 2.027 | 8.1 | 16.3 | 0.4 | | 7.9 | 16.0 | 0.5 | 43
10.32$-$0.26 | 18 09 24.2 | $-$20 08 07 | 31.33 | 4.6 | 12.7 | 4.0 | | 4.7 | 12.7 | 3.8 | 33
10.32$-$0.15 | 18 09 01.46 | $-$20 05 08.0 | 14.741 | 6.1 | 14.3 | 2.4 | | 6.1 | 14.2 | 2.3 | 32,43,44
10.44$-$0.01 | 18 08 44.9 | $-$19 54 38 | 71.412 | 2.9 | 10.9 | 5.9 | | 3.0 | 10.9 | 5.6 | 20,4,43,44
10.47$+$0.02 | 18 08 38.21 | $-$19 51 49.5 | 67.130 | 3.1 | 11.0 | 5.7 | | 3.2 | 11.0 | 5.5 | 20,4,43,44
10.62$-$0.29 | 18 10 09.9 | $-$19 53 22 | $-$8.027 | 10.7 | 18.9 | | | 10.2 | 18.3 | | 20,4,43
10.62$-$0.38 | 18 10 29.24 | $-$19 55 41.5 | $-$3.441 | 9.3 | 17.5 | | | 9.0 | 17.1 | | 4,43,44
10.89$+$0.14 | 18 09 03.3 | $-$19 26 26 | 21.747 | 5.5 | 13.6 | 3.1 | | 5.5 | 13.5 | 3.0 | 35
10.95$+$0.02 | 18 09 39.3 | $-$19 26 28 | 20.63 | 5.6 | 13.7 | 3.0 | | 5.6 | 13.7 | 2.8 | 32,43,44
11.03$+$0.06 | 18 09 39.7 | $-$19 21 36 | 15.941 | 6.1 | 14.2 | 2.5 | | 6.1 | 14.1 | 2.4 | 4,43
11.15$-$0.14 | 18 10 28.1 | $-$19 22 40 | 29.23 | 4.9 | 13.0 | 3.7 | | 5.0 | 13.0 | 3.5 | 9
11.49$-$1.48 | 18 16 22.13 | $-$19 41 27.5 | 10.641 | 6.8 | 14.9 | 1.8 | | 6.7 | 14.8 | 1.7 | 32,43,44
11.90$-$0.14 | 18 12 11.44 | $-$18 41 29.0 | 37.841 | 4.5 | 12.5 | 4.1 | | 4.6 | 12.5 | 3.9 | 20,4,43,44
11.93$-$0.61 | 18 14 00.89 | $-$18 53 26.5 | 38.43 | 4.5 | 12.5 | 4.2 | | 4.6 | 12.5 | 4.0 | 20,4,43,44
11.99$-$0.27 | 18 12 51.20 | $-$18 40 39.5 | 59.541 | 3.6 | 11.4 | 5.2 | | 3.7 | 11.5 | 5.0 | 43,44
12.02$-$0.03 | 18 12 01.851 | $-$18 31 55.50 | 109.83 | 2.4 | 9.9 | 6.7 | | 2.5 | 10.0 | 6.4 | 20,4,43,44
12.20$-$0.10 | 18 12 39.92 | $-$18 24 17.5 | 24.241 | 5.5 | 13.5 | 3.1 | | 5.5 | 13.4 | 3.0 | 20,4,43,44
12.25$-$0.04 | 18 12 31.9 | $-$18 20 11 | 48.727 | 4.1 | 11.9 | 4.7 | | 4.1 | 11.9 | 4.5 | 35
12.62$+$0.00 | 18 13 07.4 | $-$17 59 09 | 21.13 | 5.8 | 13.8 | 2.8 | | 5.8 | 13.7 | 2.7 | 33
12.68$-$0.18 | 18 13 54.2 | $-$18 01 44 | 55.430 | 3.8 | 11.7 | 4.9 | | 3.9 | 11.7 | 4.7 | 20,4
12.71$-$0.11 | 18 13 43.4 | $-$17 58 06 | 57.727 | 3.8 | 11.6 | 5.0 | | 3.9 | 11.6 | 4.8 | 35
12.79$-$0.19 | 18 14 11.1 | $-$17 55 57 | 35.716 | 4.8 | 12.7 | 3.9 | | 4.9 | 12.7 | 3.7 | 20
12.89$+$0.49 | 18 11 51.4 | $-$17 31 30 | 33.730 | 4.9 | 12.8 | 3.7 | | 5.0 | 12.8 | 3.6 | 20,4,43,44
12.90$-$0.26 | 18 14 39.52 | $-$17 52 00.0 | 37.630 | 4.7 | 12.6 | 4.0 | | 4.8 | 12.6 | 3.8 | 20,4,43,44
12.93$-$0.07 | 18 14 01.1 | $-$17 45 05 | 58.727 | 3.8 | 11.5 | 5.0 | | 3.9 | 11.6 | 4.8 | 35
13.18$+$0.06 | 18 14 00.8 | $-$17 28 05 | 50.81 | 4.1 | 11.9 | 4.6 | | 4.2 | 11.9 | 4.4 | 35
13.66$-$0.60 | 18 17 24.4 | $-$17 22 13 | 49.930 | 4.2 | 12.0 | 4.5 | | 4.3 | 12.0 | 4.3 | 35,51
13.70$-$0.05 | 18 15 30.2 | $-$17 03 58 | 51.927 | 4.2 | 11.9 | 4.6 | | 4.2 | 11.9 | 4.4 | 35
14.09$+$0.10 | 18 15 45.80 | $-$16 39 09.7 | 9.341 | 7.2 | 15.1 | 1.3 | | 7.1 | 15.0 | 1.3 | 33,52
14.33$-$0.63 | 18 18 53.37 | $-$16 47 39.5 | 22.230 | 6.0 | 13.8 | 2.7 | | 6.0 | 13.7 | 2.6 | 43
14.45$-$0.05 | 18 16 59.4 | $-$16 24 52 | 27.027 | 5.6 | 13.4 | 3.0 | | 5.6 | 13.4 | 2.9 | 35
14.60$+$0.01 | 18 17 01.14 | $-$16 14 39.0 | 24.43 | 5.8 | 13.6 | 2.8 | | 5.8 | 13.6 | 2.7 | 43,44
15.03$-$0.67 | 18 20 24.79 | $-$16 11 35.5 | 19.841 | 6.2 | 14.0 | 2.4 | | 6.2 | 13.9 | 2.3 | 20,4,43,44
15.08$-$0.09 | 18 18 22.7 | $-$15 52 46 | 45.827 | 4.6 | 12.3 | 4.2 | | 4.7 | 12.3 | 4.0 | 35
15.67$-$0.48 | 18 20 58.3 | $-$15 32 34 | $-$3.427 | 9.0 | 16.9 | | | 8.7 | 16.5 | | 35
16.36$-$0.19 | 18 21 15.6 | $-$14 47 33 | 45.031 | 4.8 | 12.3 | 4.0 | | 4.9 | 12.3 | 3.8 | 35
16.58$-$0.05 | 18 21 09.13 | $-$14 31 48.5 | 59.341 | 4.3 | 11.7 | 4.6 | | 4.3 | 11.7 | 4.4 | 20,4,43,44
16.86$-$2.15 | 18 29 24.41 | $-$15 16 04.0 | 17.541 | 6.6 | 14.3 | 2.0 | | 6.6 | 14.1 | 2.0 | 4,43,44
17.01$-$2.39 | 18 30 33.9 | $-$15 14 53 | 19.841 | 6.4 | 14.1 | 2.2 | | 6.4 | 13.9 | 2.1 | 35
17.02$-$0.08 | 18 22 08.6 | $-$14 09 29 | 90.827 | 3.4 | 10.5 | 5.8 | | 3.5 | 10.5 | 5.5 | 35
17.65$+$0.16 | 18 22 25.7 | $-$13 29 28 | 22.541 | 6.3 | 13.8 | 2.4 | | 6.3 | 13.7 | 2.3 | 17,4
18.06$+$0.08 | 18 23 31.3 | $-$13 09 23 | 55.227 | 4.6 | 11.8 | 4.3 | | 4.7 | 11.9 | 4.1 | 35
18.15$+$0.10 | 18 23 37.2 | $-$13 04 23 | 57.927 | 4.5 | 11.7 | 4.4 | | 4.6 | 11.7 | 4.2 | 35
18.26$-$0.27 | 18 25 13.3 | $-$13 09 16 | 75.227 | 4.0 | 11.0 | 5.1 | | 4.0 | 11.0 | 4.9 | 35
18.34$+$1.78 | 18 17 54.1 | $-$12 06 48 | 32.925 | 5.7 | 13.1 | 3.1 | | 5.7 | 13.0 | 2.9 | 39,33
18.46$-$0.00 | 18 24 36.35 | $-$12 51 08.0 | 52.23 | 4.8 | 12.0 | 4.1 | | 4.8 | 12.0 | 3.9 | 20,4,43,44
18.65$+$0.04 | 18 24 48.7 | $-$12 39 17 | 80.227 | 3.9 | 10.8 | 5.3 | | 3.9 | 10.9 | 5.1 | 35
18.84$-$0.30 | 18 26 24.7 | $-$12 39 10 | 42.541 | 5.2 | 12.5 | 3.6 | | 5.3 | 12.5 | 3.4 | 33
18.89$+$0.05 | 18 25 12.4 | $-$12 26 16 | 38.227 | 5.4 | 12.7 | 3.3 | | 5.5 | 12.7 | 3.2 | 35
18.99$-$0.04 | 18 25 44.1 | $-$12 24 15 | 55.49 | 4.7 | 11.8 | 4.3 | | 4.8 | 11.8 | 4.1 | 9
19.36$-$0.03 | 18 26 25.79 | $-$12 03 53.0 | 26.225 | 6.2 | 13.5 | 2.5 | | 6.2 | 13.4 | 2.4 | 43,44,35
19.48$+$0.15 | 18 26 00.39 | $-$11 52 22.5 | 24.441 | 6.3 | 13.7 | 2.4 | | 6.3 | 13.5 | 2.3 | 20,4,43,44,35
19.61$-$0.13 | 18 27 16.52 | $-$11 53 38.5 | 57.641 | 4.7 | 11.7 | 4.3 | | 4.7 | 11.7 | 4.1 | 4,43
19.61$-$0.23 | 18 27 37.2 | $-$11 56 27 | 42.730 | 5.3 | 12.5 | 3.5 | | 5.3 | 12.4 | 3.4 | 4,43,44,35
19.70$-$0.26 | 18 27 55.52 | $-$11 52 39.0 | 43.627 | 5.3 | 12.4 | 3.6 | | 5.3 | 12.4 | 3.4 | 35,44
19.88$-$0.52 | 18 29 11.7 | $-$11 49 58 | 43.63 | 5.3 | 12.4 | 3.6 | | 5.3 | 12.4 | 3.4 | 35
20.08$-$0.13 | 18 28 10.32 | $-$11 28 47.0 | 41.930 | 5.4 | 12.5 | 3.5 | | 5.4 | 12.5 | 3.3 | 20,4,43,44,36
20.24$+$0.08 | 18 27 41.0 | $-$11 14 05 | 71.041 | 4.3 | 11.1 | 4.9 | | 4.4 | 11.1 | 4.6 | 20,4,35,36
21.41$-$0.25 | 18 31 06.3 | $-$10 21 37 | 90.737 | 3.9 | 10.2 | 5.6 | | 4.0 | 10.3 | 5.3 | 35,36
21.57$-$0.03 | 18 30 36.5 | $-$10 06 44 | 97.725 | 3.7 | 10.0 | 5.9 | | 3.8 | 10.1 | 5.6 | 35,36
21.87$+$0.01 | 18 31 01.75 | $-$09 49 01.0 | 22.43 | 6.6 | 13.7 | 2.1 | | 6.6 | 13.6 | 2.0 | 20,4,43,44,35
22.05$+$0.22 | 18 30 35.7 | $-$09 34 26 | 50.034 | 5.2 | 12.0 | 3.8 | | 5.2 | 12.0 | 3.6 | 35,36
22.34$-$0.16 | 18 32 32.1 | $-$09 29 10 | 30.937 | 6.1 | 13.1 | 2.6 | | 6.1 | 13.0 | 2.6 | 36
22.36$+$0.06 | 18 31 44.13 | $-$09 22 12.5 | 84.641 | 4.1 | 10.4 | 5.3 | | 4.2 | 10.5 | 5.0 | 32,43,44,35,36
22.43$-$0.16 | 18 32 41.2 | $-$09 24 34 | 28.741 | 6.3 | 13.2 | 2.5 | | 6.2 | 13.1 | 2.4 | 20,4,35,36
22.45$-$0.17 | 18 32 45.8 | $-$09 24 06 | 29.527 | 6.2 | 13.2 | 2.5 | | 6.2 | 13.1 | 2.5 | 36
23.01$-$0.41 | 18 34 40.29 | $-$09 00 38.1 | 74.816 | 4.4 | 10.8 | 4.9 | | 4.5 | 10.8 | 4.6 | 20,4,52
23.04$-$0.32 | 18 34 26.2 | $-$08 56 35 | 76.031 | 4.4 | 10.7 | 4.9 | | 4.5 | 10.8 | 4.7 | 35,36
23.18$-$0.45 | 18 35 08.1 | $-$08 52 32 | 75.227 | 4.4 | 10.7 | 4.9 | | 4.5 | 10.8 | 4.7 | 35
23.19$-$0.38 | 18 34 55.20 | $-$08 49 14.2 | 81.827 | 4.3 | 10.5 | 5.2 | | 4.3 | 10.5 | 4.9 | 35,36,52
23.24$-$0.24 | 18 34 31.26 | $-$08 42 47.0 | 63.241 | 4.8 | 11.3 | 4.3 | | 4.9 | 11.3 | 4.1 | 32,43,44,33,35,36
23.32$-$0.29 | 18 34 49.9 | $-$08 40 34 | 63.827 | 4.8 | 11.2 | 4.4 | | 4.9 | 11.3 | 4.2 | 35
23.39$+$0.19 | 18 33 13.1 | $-$08 23 56 | 74.127 | 4.5 | 10.8 | 4.8 | | 4.6 | 10.8 | 4.6 | 35,36
23.43$-$0.18 | 18 34 39.27 | $-$08 31 39.0 | 104.23 | 3.8 | 9.5 | 6.1 | | 3.9 | 9.6 | 5.8 | 20,4,43,44,35,36
23.48$+$0.08 | 18 33 44.05 | $-$08 21 20.5 | 84.43 | 4.2 | 10.3 | 5.3 | | 4.3 | 10.4 | 5.0 | 32,43,44,33,35
23.66$-$0.13 | 18 34 51.7 | $-$08 18 16 | 82.327 | 4.3 | 10.4 | 5.2 | | 4.4 | 10.5 | 4.9 | 36
23.70$-$0.19 | 18 35 12.37 | $-$08 17 39.5 | 68.937 | 4.7 | 11.0 | 4.6 | | 4.7 | 11.0 | 4.4 | 32,43,44,33,35
23.80$+$0.40 | 18 33 13.0 | $-$07 55 41 | 76.427 | 4.5 | 10.6 | 4.9 | | 4.5 | 10.7 | 4.7 | 35
23.91$+$0.07 | 18 34 38.2 | $-$07 59 35 | 35.748 | 6.0 | 12.7 | 2.9 | | 6.0 | 12.6 | 2.8 | 48
23.98$-$0.08 | 18 35 22.9 | $-$08 01 11 | 72.737 | 4.6 | 10.8 | 4.8 | | 4.6 | 10.8 | 4.5 | 35,36
24.14$+$0.00 | 18 35 20.94 | $-$07 48 55.6 | 23.137 | 6.7 | 13.5 | 2.0 | | 6.6 | 13.3 | 2.0 | 35,36,52
24.33$+$0.14 | 18 35 08.14 | $-$07 35 04.0 | 112.016 | 3.7 | 8.9 | 6.6 | | 3.8 | 9.2 | 6.1 | 20,4,35,36,52
24.50$-$0.03 | 18 36 08.7 | $-$07 30 55 | 109.830 | 3.7 | 9.0 | 6.5 | | 3.8 | 9.2 | 6.0 | 39,33,35,36
24.54$+$0.32 | 18 34 53.0 | $-$07 19 11 | 107.837 | 3.8 | 9.1 | 6.4 | | 3.9 | 9.3 | 6.0 | 35,36
24.64$-$0.33 | 18 37 25.2 | $-$07 31 53 | 42.737 | 5.7 | 12.2 | 3.2 | | 5.7 | 12.2 | 3.1 | 36
24.67$-$0.14 | 18 36 48.4 | $-$07 24 25 | 113.03 | 3.7 | 8.8 | 6.7 | | 3.8 | 9.1 | 6.2 | 33,35
24.78$+$0.08 | 18 36 12.57 | $-$07 12 11.5 | 110.741 | 3.8 | 8.9 | 6.5 | | 3.8 | 9.2 | 6.1 | 20,4,43,44,35
24.92$+$0.10 | 18 36 24.0 | $-$07 04 27 | 109.327 | 3.8 | 8.9 | 6.5 | | 3.9 | 9.2 | 6.0 | 35
24.93$+$0.08 | 18 36 29.2 | $-$07 05 05 | 53.427 | 5.3 | 11.6 | 3.8 | | 5.3 | 11.6 | 3.6 | 36
25.38$+$0.00 | 18 37 35.52 | $-$06 42 34.5 | 95.727 | 4.1 | 9.6 | 5.8 | | 4.2 | 9.7 | 5.4 | 8
25.38$-$0.18 | 18 38 14.3 | $-$06 47 47 | 68.27 | 4.8 | 10.9 | 4.5 | | 4.9 | 10.9 | 4.3 | 48
25.41$+$0.10 | 18 37 16.92 | $-$06 38 28.0 | 96.037 | 4.1 | 9.6 | 5.8 | | 4.2 | 9.7 | 5.5 | 39,8,35,36
25.53$+$0.38 | 18 36 32.6 | $-$06 24 26 | 92.027 | 4.2 | 9.7 | 5.6 | | 4.3 | 9.9 | 5.3 | 8
25.65$+$1.04 | 18 34 20.91 | $-$05 59 40.5 | 42.225 | 5.8 | 12.2 | 3.2 | | 5.8 | 12.1 | 3.0 | 39,44,33,35
25.70$+$0.04 | 18 38 03.15 | $-$06 24 15.0 | 101.315 | 4.0 | 9.2 | 6.1 | | 4.1 | 9.4 | 5.7 | 8,43,44,35,36
25.82$-$0.17 | 18 39 03.63 | $-$06 24 09.5 | 91.227 | 4.2 | 9.7 | 5.6 | | 4.3 | 9.9 | 5.3 | 32,8,43,44,35,36
26.52$-$0.26 | 18 40 40.23 | $-$05 49 07.5 | 104.231 | 4.0 | 8.9 | 6.3 | | 4.1 | 9.2 | 5.9 | 8
26.59$-$0.00 | 18 39 52.8 | $-$05 38 48 | 23.337 | 6.8 | 13.3 | 1.9 | | 6.7 | 13.1 | 1.9 | 35,36
26.60$-$0.22 | 18 40 38.55 | $-$05 43 56.0 | 107.841 | 3.9 | 8.6 | 6.6 | | 4.0 | 9.0 | 6.1 | 39,33,8,35,36
26.65$+$0.02 | 18 39 51.8 | $-$05 34 52 | 111.53 | 3.9 | 8.3 | 6.9 | | 4.0 | 8.7 | 6.3 | 35,36
27.21$+$0.26 | 18 40 03.72 | $-$04 57 45.6 | 9.127 | 7.8 | 14.3 | 0.8 | | 7.6 | 14.0 | 0.9 | 36,52
27.22$+$0.13 | 18 40 30.43 | $-$05 00 59.0 | 112.637 | 3.9 | 7.9 | 7.3 | | 4.0 | 8.5 | 6.4 | 8,36
27.28$+$0.15 | 18 40 34.48 | $-$04 57 13.5 | 31.841 | 6.4 | 12.6 | 2.5 | | 6.4 | 12.5 | 2.4 | 8,33,35,36
27.36$-$0.16 | 18 41 50.98 | $-$05 01 28.0 | 92.241 | 4.3 | 9.4 | 5.7 | | 4.4 | 9.6 | 5.3 | 20,4,8,35,36,52
27.78$+$0.05 | 18 41 47.5 | $-$04 33 11 | 111.827 | 4.0 | 7.8 | 7.3 | | 4.0 | 8.4 | 6.4 | 35
27.78$-$0.25 | 18 42 57.2 | $-$04 41 59 | 45.73 | 5.8 | 11.8 | 3.3 | | 5.8 | 11.7 | 3.2 | 35,36
27.86$-$0.24 | 18 43 01.1 | $-$04 36 41 | 20.17 | 7.1 | 13.4 | 1.7 | | 7.0 | 13.2 | 1.7 | 48
28.02$-$0.44 | 18 44 02.1 | $-$04 34 14 | 16.027 | 7.3 | 13.6 | 1.4 | | 7.2 | 13.4 | 1.4 | 36
28.14$+$0.00 | 18 42 42.59 | $-$04 15 32.0 | 100.927 | 4.2 | 8.7 | 6.2 | | 4.3 | 9.0 | 5.8 | 4,44,8,36
28.20$-$0.05 | 18 42 58.20 | $-$04 13 59.5 | 96.816 | 4.3 | 9.0 | 6.0 | | 4.4 | 9.2 | 5.6 | 20,4,43,35,36
28.30$-$0.38 | 18 44 21.99 | $-$04 17 38.5 | 48.13 | 5.7 | 11.6 | 3.4 | | 5.7 | 11.5 | 3.3 | 32,8,43,44,33,36
28.39$+$0.08 | 18 42 54.5 | $-$04 00 04 | 79.228 | 4.7 | 9.9 | 5.0 | | 4.8 | 10.0 | 4.7 | 35,36
28.53$+$0.12 | 18 42 57.7 | $-$03 51 59 | 27.127 | 6.7 | 12.8 | 2.1 | | 6.6 | 12.7 | 2.1 | 36
28.69$+$0.41 | 18 42 13.6 | $-$03 35 07 | 94.227 | 4.4 | 9.1 | 5.8 | | 4.5 | 9.3 | 5.5 | 36
28.81$+$0.36 | 18 42 37.49 | $-$03 30 12.5 | 87.037 | 4.6 | 9.5 | 5.4 | | 4.6 | 9.6 | 5.1 | 8,36
28.82$+$0.48 | 18 42 12.43 | $-$03 25 39.5 | 83.327 | 4.7 | 9.7 | 5.2 | | 4.7 | 9.8 | 4.9 | 8
28.83$-$0.25 | 18 44 51.09 | $-$03 45 48.0 | 86.530 | 4.6 | 9.5 | 5.4 | | 4.6 | 9.6 | 5.1 | 20,4,8,44,35,36
28.84$-$0.23 | 18 44 47.1 | $-$03 44 39 | 86.530 | 4.6 | 9.5 | 5.4 | | 4.6 | 9.6 | 5.1 | 36
28.85$+$0.50 | 18 42 12.8 | $-$03 24 26 | 83.527 | 4.7 | 9.6 | 5.2 | | 4.7 | 9.8 | 4.9 | 35,36
28.86$+$0.07 | 18 43 45.1 | $-$03 35 29 | 103.830 | 4.2 | 8.3 | 6.6 | | 4.3 | 8.7 | 6.0 | 4,43
29.31$-$0.16 | 18 45 24.97 | $-$03 17 44.5 | 45.527 | 5.9 | 11.6 | 3.2 | | 5.9 | 11.5 | 3.1 | 8,36
29.86$-$0.04 | 18 45 59.57 | $-$02 45 04.4 | 100.441 | 4.3 | 8.3 | 6.5 | | 4.4 | 8.7 | 5.9 | 4,8,36,52
29.91$-$0.05 | 18 46 05.9 | $-$02 42 27 | 104.348 | 4.2 | 7.7 | 7.1 | | 4.3 | 8.4 | 6.2 | 8
29.91$-$0.03 | 18 46 03.69 | $-$02 41 52.5 | 97.43 | 4.4 | 8.5 | 6.2 | | 4.5 | 8.8 | 5.7 | 48
29.92$+$0.05 | 18 45 44.18 | $-$02 39 03.5 | 99.027 | 4.4 | 8.4 | 6.3 | | 4.4 | 8.7 | 5.8 | 8
29.95$-$0.02 | 18 46 03.741 | $-$02 39 21.43 | 97.341 | 4.4 | 8.5 | 6.2 | | 4.5 | 8.8 | 5.7 | 20,4,8,43,44,36,23
29.98$-$0.04 | 18 46 11.6 | $-$02 38 42 | 97.341 | 4.4 | 8.5 | 6.2 | | 4.5 | 8.8 | 5.7 | 4,36
30.00$-$0.01 | 18 46 09.85 | $-$02 36 30.5 | 97.715 | 4.4 | 8.5 | 6.2 | | 4.5 | 8.8 | 5.8 | 8
30.20$-$0.17 | 18 47 03.07 | $-$02 30 33.6 | 103.341 | 4.3 | 7.7 | 7.0 | | 4.4 | 8.3 | 6.2 | 4,36,52
30.22$-$0.18 | 18 47 08.30 | $-$02 29 27.1 | 104.541 | 4.3 | 7.7 | 7.0 | | 4.3 | 8.2 | 6.3 | 4,36,52
30.30$+$0.06 | 18 46 23.7 | $-$02 17 56 | 45.337 | 6.0 | 11.5 | 3.2 | | 5.9 | 11.4 | 3.1 | 35,36
30.30$-$0.18 | 18 47 18.6 | $-$02 25 40 | 102.63 | 4.3 | 7.8 | 6.9 | | 4.4 | 8.4 | 6.1 | 35
30.40$-$0.29 | 18 47 51.2 | $-$02 23 15 | 102.437 | 4.3 | 7.7 | 6.9 | | 4.4 | 8.4 | 6.1 | 36
30.53$+$0.01 | 18 47 00.1 | $-$02 07 26 | 48.041 | 5.9 | 11.3 | 3.3 | | 5.9 | 11.3 | 3.2 | 43
30.59$-$0.13 | 18 47 37.5 | $-$02 08 46 | 115.527 | | | | | | | | 36
30.59$-$0.04 | 18 47 18.89 | $-$02 06 07.0 | 42.341 | 6.1 | 11.6 | 3.0 | | 6.1 | 11.5 | 2.9 | 20,4,35,44,36
30.70$-$0.07 | 18 47 36.9 | $-$02 01 05 | 88.816 | 4.6 | 9.0 | 5.7 | | 4.7 | 9.2 | 5.3 | 20,4,35,36
30.76$-$0.05 | 18 47 39.73 | $-$01 57 22.0 | 90.530 | 4.6 | 8.8 | 5.8 | | 4.7 | 9.0 | 5.4 | 20,4,43,44,35,36
30.78$+$0.23 | 18 46 41.52 | $-$01 48 32.0 | 41.641 | 6.1 | 11.6 | 3.0 | | 6.1 | 11.6 | 2.9 | 20,35
30.78$+$0.00 | 18 47 29.9 | $-$01 54 39 | 95.13 | 4.5 | 8.5 | 6.1 | | 4.6 | 8.8 | 5.7 | 4,44,35,36
30.79$+$0.20 | 18 46 48.09 | $-$01 48 46.0 | 86.027 | 4.7 | 9.1 | 5.5 | | 4.8 | 9.3 | 5.1 | 4,44,36
30.82$+$0.27 | 18 46 37.4 | $-$01 45 14 | 97.616 | 4.5 | 8.2 | 6.4 | | 4.5 | 8.6 | 5.8 | 20,4,35,36
30.82$-$0.05 | 18 47 46.2 | $-$01 54 14 | 98.730 | 4.4 | 8.1 | 6.5 | | 4.5 | 8.5 | 5.9 | 20,4,43,35,36
30.86$+$0.16 | 18 47 09.13 | $-$01 44 10.5 | 102.027 | 4.4 | 7.7 | 6.9 | | 4.4 | 8.2 | 6.2 | 43,44,52
30.91$+$0.14 | 18 47 15.0 | $-$01 44 07 | 104.027 | 4.4 | 7.7 | 6.9 | | 4.4 | 8.0 | 6.4 | 32
30.94$+$0.11 | 18 47 23.8 | $-$01 42 39 | 101.527 | 4.4 | 7.4 | 7.1 | | 4.4 | 8.3 | 6.2 | 35
30.96$-$0.14 | 18 48 19.7 | $-$01 48 59 | 74.827 | 5.0 | 9.8 | 4.8 | | 5.1 | 9.8 | 4.6 | 36
30.98$-$0.05 | 18 48 05.8 | $-$01 45 36 | 74.727 | 5.0 | 9.8 | 4.8 | | 5.1 | 9.8 | 4.6 | 35
31.04$+$0.36 | 18 46 41.5 | $-$01 30 42 | 77.637 | 4.9 | 9.6 | 5.0 | | 5.0 | 9.7 | 4.7 | 35,36
31.06$+$0.09 | 18 47 41.34 | $-$01 37 21.5 | 38.43 | 6.3 | 11.8 | 2.8 | | 6.2 | 11.7 | 2.7 | 32,43,44,35,36
31.16$+$0.06 | 18 47 59.6 | $-$01 32 37 | 38.937 | 6.3 | 11.7 | 2.8 | | 6.2 | 11.7 | 2.7 | 35
31.28$+$0.06 | 18 48 12.39 | $-$01 26 22.6 | 109.93 | | | | | 4.4 | 7.5 | 6.8 | 20,4,43,44,35,36,23
31.41$+$0.31 | 18 47 34.31 | $-$01 12 47.0 | 97.030 | 4.5 | 8.0 | 6.5 | | 4.6 | 8.5 | 5.9 | 32,4,43,44,35,36
31.58$+$0.08 | 18 48 44.3 | $-$01 11 18 | 96.037 | 4.5 | 8.1 | 6.4 | | 4.6 | 8.5 | 5.8 | 35,36
32.03$+$0.06 | 18 49 37.3 | $-$00 45 47 | 96.030 | 4.6 | 7.9 | 6.5 | | 4.6 | 8.3 | 5.9 | 39,35,36
32.11$+$0.09 | 18 49 36.7 | $-$00 41 05 | 96.53 | 4.6 | 7.8 | 6.6 | | 4.6 | 8.3 | 5.9 | 33
32.74$-$0.08 | 18 51 22.8 | $-$00 12 15 | 36.930 | 6.4 | 11.6 | 2.7 | | 6.4 | 11.5 | 2.6 | 20,4,35,36
32.80$+$0.19 | 18 50 31.1 | $-$00 01 54 | 15.03 | 7.5 | 13.1 | 1.2 | | 7.4 | 12.9 | 1.3 | 48
32.97$+$0.04 | 18 51 23.0 | $+$00 03 46 | 83.437 | 4.9 | 8.8 | 5.5 | | 5.0 | 9.0 | 5.1 | 39,33,35,36
33.09$-$0.07 | 18 51 58.9 | $+$00 07 27 | 78.116 | 5.1 | 9.1 | 5.1 | | 5.1 | 9.3 | 4.8 | 4,35,36
33.13$-$0.09 | 18 52 07.1 | $+$00 07 56 | 78.116 | 5.1 | 9.1 | 5.1 | | 5.1 | 9.3 | 4.8 | 20,4,35,36
33.40$+$0.01 | 18 52 10.6 | $+$00 25 09 | 74.616 | 5.2 | 9.3 | 4.9 | | 5.2 | 9.4 | 4.6 | 33,35,36
33.64$-$0.21 | 18 53 28.7 | $+$00 31 58 | 61.537 | 5.6 | 10.0 | 4.1 | | 5.6 | 10.1 | 3.9 | 36
33.68$-$0.26 | 18 53 45.2 | $+$00 32 47 | 62.627 | 5.5 | 10.0 | 4.2 | | 5.5 | 10.0 | 4.0 | 35
33.74$-$0.15 | 18 53 26.9 | $+$00 39 01 | 53.627 | 5.8 | 10.5 | 3.7 | | 5.8 | 10.5 | 3.5 | 36
33.86$+$0.01 | 18 53 05.2 | $+$00 49 36 | 64.027 | 5.5 | 9.8 | 4.3 | | 5.5 | 9.9 | 4.1 | 36
33.97$-$0.00 | 18 53 24.1 | $+$00 55 13 | 61.137 | 5.6 | 10.0 | 4.1 | | 5.6 | 10.0 | 3.9 | 35,36
34.10$+$0.01 | 18 53 31.9 | $+$01 02 26 | 55.927 | 5.8 | 10.3 | 3.8 | | 5.8 | 10.3 | 3.6 | 36
34.24$+$0.13 | 18 53 21.5 | $+$01 13 43 | 58.930 | 5.7 | 10.1 | 4.0 | | 5.7 | 10.1 | 3.8 | 20,4,35,36
34.39$+$0.24 | 18 53 21.5 | $+$01 24 09 | 57.225 | 5.7 | 10.2 | 3.9 | | 5.7 | 10.2 | 3.7 | 32,35,36
34.74$-$0.09 | 18 55 03.4 | $+$01 34 17 | 51.137 | 6.0 | 10.5 | 3.5 | | 5.9 | 10.4 | 3.4 | 36
34.78$-$1.38 | 18 59 44.7 | $+$01 01 12 | 45.83 | 6.2 | 10.8 | 3.2 | | 6.1 | 10.7 | 3.1 | 39,33,35
34.82$+$0.35 | 18 53 37.4 | $+$01 50 32 | 56.911 | 5.8 | 10.1 | 3.9 | | 5.8 | 10.1 | 3.7 | 26
35.02$+$0.35 | 18 54 00.6 | $+$02 00 50 | 52.516 | 5.9 | 10.3 | 3.6 | | 5.9 | 10.3 | 3.5 | 20,4,35,36
35.18$-$0.74 | 18 58 09.8 | $+$01 39 36 | 35.010 | 6.6 | 11.4 | 2.5 | | 6.5 | 11.3 | 2.5 | 9
35.20$-$0.74 | 18 58 13.053 | $+$01 40 35.68 | 35.030 | 6.6 | 11.4 | 2.5 | | 6.5 | 11.3 | 2.5 | 20,4 35,22,52
35.20$-$1.74 | 19 01 45.6 | $+$01 13 28 | 43.63 | 6.3 | 10.8 | 3.1 | | 6.2 | 10.8 | 3.0 | 20,4
35.25$-$0.24 | 18 56 30.9 | $+$01 57 11 | 72.426 | 5.3 | 9.0 | 4.9 | | 5.3 | 9.1 | 4.6 | 26
35.39$+$0.02 | 18 55 51.2 | $+$02 11 37 | 96.926 | | | | | 4.9 | 7.3 | 6.4 | 26
35.40$+$0.03 | 18 55 51.1 | $+$02 12 25 | 89.126 | 4.9 | 7.1 | 6.7 | | 5.0 | 7.8 | 5.9 | 26
35.59$+$0.06 | 18 56 04.3 | $+$02 23 28 | 49.63 | 6.1 | 10.4 | 3.4 | | 6.0 | 10.4 | 3.3 | 26
35.79$-$0.17 | 18 57 16.1 | $+$02 27 44 | 61.937 | 5.7 | 9.6 | 4.2 | | 5.7 | 9.6 | 4.0 | 36
36.02$-$0.20 | 18 57 45.8 | $+$02 39 15 | 93.026 | | | | | 4.9 | 6.8 | 6.7 | 26
36.10$+$0.56 | 18 55 15.6 | $+$03 04 42 | 76.037 | 5.3 | 8.5 | 5.2 | | 5.3 | 8.7 | 4.9 | 35,36
36.64$-$0.21 | 18 58 55.9 | $+$03 12 05 | 77.326 | 5.3 | 8.2 | 5.4 | | 5.3 | 8.4 | 5.1 | 26
36.70$+$0.09 | 18 58 00.9 | $+$03 23 30 | 59.837 | 5.8 | 9.5 | 4.1 | | 5.8 | 9.6 | 3.9 | 36
36.84$-$0.02 | 18 58 39.3 | $+$03 27 55 | 61.726 | 5.7 | 9.3 | 4.2 | | 5.7 | 9.4 | 4.0 | 26
36.90$-$0.41 | 19 00 08.6 | $+$03 20 35 | 84.726 | 5.1 | 7.0 | 6.6 | | 5.1 | 7.6 | 5.8 | 26
36.92$+$0.48 | 18 57 00.7 | $+$03 46 01 | $-$35.926 | 11.8 | 17.4 | | | 11.0 | 16.5 | | 26
37.02$-$0.03 | 18 58 59.9 | $+$03 37 40 | 80.137 | 5.2 | 7.7 | 5.8 | | 5.2 | 8.1 | 5.4 | 36
37.38$-$0.09 | 18 59 52.3 | $+$03 55 12 | 70.626 | 5.5 | 8.6 | 4.9 | | 5.5 | 8.7 | 4.6 | 26
37.40$+$1.52 | 18 54 10.5 | $+$04 40 49 | 43.725 | 6.3 | 10.4 | 3.1 | | 6.3 | 10.4 | 3.0 | 32,35
37.47$-$0.11 | 19 00 06.7 | $+$03 59 27 | 59.137 | 5.8 | 9.4 | 4.1 | | 5.8 | 9.4 | 3.9 | 36
37.52$-$0.10 | 19 00 14.4 | $+$04 02 35 | 52.93 | 6.0 | 9.8 | 3.7 | | 6.0 | 9.8 | 3.5 | 35,36
37.54$+$0.21 | 18 59 11.6 | $+$04 12 08 | 85.125 | 5.2 | 7.2 | 6.3 | | 5.1 | 7.1 | 6.2 | 33,35,36
37.60$+$0.42 | 18 58 28.5 | $+$04 20 34 | 90.037 | | | | | 5.1 | 7.1 | 6.2 | 36
37.74$-$0.12 | 19 00 38.0 | $+$04 13 18 | 50.326 | 6.1 | 9.9 | 3.5 | | 6.1 | 9.9 | 3.4 | 26
37.76$-$0.19 | 19 00 56.5 | $+$04 12 08 | 55.126 | 6.0 | 9.6 | 3.8 | | 5.9 | 9.6 | 3.7 | 26
37.77$-$0.22 | 19 01 03.0 | $+$04 12 13 | 61.53 | 5.8 | 9.2 | 4.3 | | 5.8 | 9.2 | 4.1 | 26
38.03$-$0.30 | 19 01 50.0 | $+$04 23 54 | 55.727 | 5.9 | 9.5 | 3.9 | | 5.9 | 9.5 | 3.7 | 36
38.08$-$0.27 | 19 01 48.4 | $+$04 27 25 | 67.526 | 5.6 | 8.6 | 4.7 | | 5.6 | 8.7 | 4.5 | 26
38.10$-$0.21 | 19 01 47.6 | $+$04 30 32 | 83.53 | 5.3 | 7.2 | 6.2 | | 5.2 | 7.0 | 6.2 | 35,36
38.20$-$0.08 | 19 01 22.8 | $+$04 39 10 | 79.827 | 5.3 | 7.0 | 6.4 | | 5.3 | 7.6 | 5.6 | 36
38.26$-$0.08 | 19 01 28.7 | $+$04 42 02 | 15.427 | 7.6 | 12.2 | 1.2 | | 7.4 | 11.9 | 1.3 | 36
38.26$-$0.20 | 19 01 54.0 | $+$04 38 38 | 70.226 | 5.5 | 8.3 | 5.0 | | 5.5 | 8.5 | 4.7 | 26
38.56$+$0.15 | 19 01 10.3 | $+$05 04 26 | 31.526 | 6.9 | 11.0 | 2.3 | | 6.8 | 10.9 | 2.3 | 26
38.60$-$0.21 | 19 02 34.0 | $+$04 56 40 | 62.626 | 5.8 | 8.9 | 4.4 | | 5.8 | 8.9 | 4.2 | 26
38.66$+$0.08 | 19 01 36.7 | $+$05 07 42 | $-$31.526 | 11.1 | 16.4 | | | 10.5 | 15.6 | | 26
38.92$-$0.36 | 19 03 39.7 | $+$05 09 36 | 40.225 | 6.5 | 10.4 | 2.9 | | 6.5 | 10.3 | 2.8 | 35,26
39.10$+$0.48 | 19 00 59.5 | $+$05 42 28 | 23.137 | 7.2 | 11.5 | 1.7 | | 7.1 | 11.3 | 1.8 | 36
39.54$-$0.38 | 19 04 53.5 | $+$05 41 59 | 47.826 | 6.3 | 9.7 | 3.4 | | 6.2 | 9.7 | 3.3 | 26
39.39$-$0.14 | 19 03 45.3 | $+$05 40 39 | 65.83 | 5.7 | 8.4 | 4.7 | | 5.7 | 8.5 | 4.5 | 26
40.28$-$0.22 | 19 05 42.1 | $+$06 26 08 | 74.027 | 5.5 | 6.8 | 6.2 | | 5.5 | 7.4 | 5.4 | 27,35,26
40.41$+$0.70 | 19 02 37.9 | $+$06 58 25 | 12.03 | 7.8 | 12.0 | 0.9 | | 7.6 | 11.7 | 1.0 | 32,35
40.62$-$0.14 | 19 06 01.7 | $+$06 46 25 | 32.916 | 6.9 | 10.5 | 2.4 | | 6.8 | 10.4 | 2.4 | 20,4,35
40.94$-$0.04 | 19 06 16.1 | $+$07 06 00 | 36.626 | 6.7 | 10.2 | 2.7 | | 6.6 | 10.1 | 2.6 | 26
41.08$-$0.13 | 19 06 49.3 | $+$07 11 01 | 57.526 | 6.0 | 8.6 | 4.2 | | 6.0 | 8.7 | 4.0 | 26
41.12$-$0.22 | 19 07 15.4 | $+$07 10 54 | 63.427 | 5.8 | 8.1 | 4.7 | | 5.8 | 8.2 | 4.5 | 35,26
41.12$-$0.11 | 19 06 50.7 | $+$07 13 57 | 36.626 | 6.7 | 10.1 | 2.7 | | 6.7 | 10.0 | 2.6 | 26
41.16$-$0.20 | 19 07 15.1 | $+$07 13 20 | 63.626 | 5.8 | 8.0 | 4.8 | | 5.8 | 8.1 | 4.5 | 26
41.23$-$0.20 | 19 07 21.9 | $+$07 17 06 | 57.127 | 6.0 | 8.6 | 4.2 | | 6.0 | 8.6 | 4.0 | 35,26
41.27$+$0.37 | 19 05 24.6 | $+$07 35 02 | 20.326 | 7.4 | 11.2 | 1.5 | | 7.3 | 11.0 | 1.6 | 26
41.34$-$0.14 | 19 07 21.870 | $+$07 25 17.34 | 14.027 | 7.7 | 11.7 | 1.1 | | 7.6 | 11.4 | 1.2 | 27
41.58$+$0.04 | 19 07 09.4 | $+$07 42 19 | 11.926 | 7.8 | 11.8 | 0.9 | | 7.6 | 11.5 | 1.0 | 26
41.87$-$0.10 | 19 08 10.8 | $+$07 54 04 | 15.826 | 7.6 | 11.5 | 1.2 | | 7.5 | 11.2 | 1.3 | 26
42.07$+$0.24 | 19 07 20.8 | $+$08 14 12 | 12.527 | 7.8 | 11.7 | 1.0 | | 7.6 | 11.4 | 1.1 | 27
42.30$-$0.30 | 19 09 44.2 | $+$08 11 33 | 28.126 | 7.1 | 10.5 | 2.1 | | 7.0 | 10.3 | 2.1 | 26
42.43$-$0.26 | 19 09 50.2 | $+$08 19 32 | 65.63 | 5.8 | 7.2 | 5.3 | | 5.8 | 7.5 | 4.9 | 26
42.70$-$0.15 | 19 09 55.8 | $+$08 36 56 | $-$42.926 | 12.1 | 16.9 | | | 11.3 | 15.9 | | 26
43.04$-$0.46 | 19 11 39.7 | $+$08 46 32 | 58.025 | 6.1 | 8.0 | 4.5 | | 6.0 | 8.0 | 4.3 | 35,26,51
43.08$-$0.08 | 19 10 22.4 | $+$08 59 01 | 10.226 | 7.9 | 11.6 | 0.8 | | 7.7 | 11.3 | 0.9 | 26
43.15$+$0.02 | 19 10 11.049 | $+$09 05 20.49 | 2.93 | 8.3 | 12.2 | 0.2 | | 8.1 | 11.8 | 0.4 | 20,4,27,52
43.18$-$0.01 | 19 10 20.2 | $+$09 06 06 | 12.116 | 7.8 | 11.5 | 0.9 | | 7.7 | 11.2 | 1.1 | 26
43.80$-$0.13 | 19 11 55.1 | $+$09 36 00 | 44.330 | 6.5 | 8.9 | 3.3 | | 6.5 | 8.9 | 3.2 | 20,4,35,27
43.87$-$0.77 | 19 14 26.393 | $+$09 22 36.53 | 54.23 | 6.2 | 8.1 | 4.2 | | 6.2 | 8.1 | 4.0 | 35,52
44.31$+$0.04 | 19 12 16.4 | $+$10 07 44 | 55.726 | 6.2 | 7.7 | 4.4 | | 6.1 | 7.8 | 4.2 | 26
44.64$-$0.52 | 19 14 54.6 | $+$10 10 02 | 49.326 | 6.4 | 8.3 | 3.8 | | 6.3 | 8.3 | 3.7 | 26
45.07$+$0.13 | 19 13 22.4 | $+$10 50 45 | 58.930 | 6.1 | 7.0 | 5.0 | | 6.1 | 7.1 | 4.7 | 20,4,35,27
45.44$+$0.07 | 19 14 17.4 | $+$11 08 45 | 58.03 | 6.1 | 6.9 | 5.0 | | 6.1 | 7.1 | 4.7 | 4,35
45.47$+$0.13 | 19 14 07.8 | $+$11 12 01 | 61.816 | 6.1 | 6.4 | 5.5 | | 6.0 | 6.3 | 5.5 | 20,4,35,27
45.49$+$0.13 | 19 14 10.1 | $+$11 13 05 | 61.83 | 6.1 | 6.4 | 5.6 | | 6.0 | 6.2 | 5.5 | 4
45.57$-$0.12 | 19 15 13.2 | $+$11 10 25 | 1.626 | 8.4 | 11.8 | 0.1 | | 8.2 | 11.4 | 0.4 | 26
45.81$-$0.36 | 19 16 31.9 | $+$11 16 22 | 59.926 | 6.1 | 6.4 | 5.4 | | 6.1 | 6.5 | 5.2 | 26
46.07$+$0.22 | 19 14 55.6 | $+$11 46 12 | 23.326 | 7.4 | 10.0 | 1.8 | | 7.2 | 9.8 | 1.9 | 26
46.12$+$0.38 | 19 14 26.4 | $+$11 53 24 | 59.026 | 6.2 | 6.4 | 5.3 | | 6.1 | 6.5 | 5.1 | 26
48.89$-$0.17 | 19 21 47.5 | $+$14 04 58 | 57.326 | | | | | 6.3 | 6.0 | 5.1 | 26
48.90$-$0.27 | 19 22 10.1 | $+$14 02 38 | 72.026 | | | | | | | | 26
48.99$-$0.30 | 19 22 26.3 | $+$14 06 37 | 71.626 | | | | | | | | 26
49.03$-$1.06 | 19 25 18.5 | $+$13 46 59 | 38.73 | 6.8 | 7.9 | 3.2 | | 6.7 | 7.8 | 3.2 | 39,35
49.27$+$0.31 | 19 20 44.8 | $+$14 38 29 | $-$3.226 | 8.7 | 11.4 | | | 8.4 | 10.9 | 0.0 | 26
49.35$+$0.41 | 19 20 33.2 | $+$14 45 48 | 68.026 | | | | | | | | 26
49.41$+$0.33 | 19 20 59.212 | $+$14 46 49.65 | $-$12.427 | 9.2 | 12.1 | | | 8.9 | 11.6 | | 35,26,52
49.47$-$0.37 | 19 23 39.821 | $+$14 31 04.47 | 60.930 | | | | | | | | 20,4,23
49.48$-$0.40 | 19 23 46.1 | $+$14 29 38 | 56.316 | | | | | 6.4 | 5.5 | 5.4 | 26
49.49$-$0.39 | 19 23 43.949 | $+$14 30 34.44 | 56.530 | | | | | | | | 4,52
49.60$-$0.25 | 19 23 26.7 | $+$14 40 19 | 62.827 | | | | | | | | 35,26
49.57$-$0.38 | 19 23 53.6 | $+$14 34 54 | 59.327 | | | | | | | | 35
49.62$-$0.36 | 19 23 52.8 | $+$14 38 10 | 49.326 | 6.5 | 6.1 | 4.9 | | 6.4 | 6.2 | 4.7 | 26
49.66$-$0.45 | 19 24 19.7 | $+$14 38 02 | 68.83 | | | | | | | | 33
50.00$+$0.59 | 19 21 10.5 | $+$15 25 42 | $-$5.027 | 8.8 | 11.3 | | | 8.5 | 10.9 | | 35
50.29$+$0.69 | 19 21 22.2 | $+$15 43 43 | 25.83 | 7.3 | 8.8 | 2.1 | | 7.2 | 8.5 | 2.2 | 35
50.78$+$0.15 | 19 24 17.2 | $+$15 53 54 | 49.126 | 6.6 | 5.8 | 4.9 | | 6.5 | 5.9 | 4.8 | 26,51
52.66$-$1.09 | 19 32 36.071 | $+$16 57 38.46 | 59.73 | | | | | | | | 40,52
52.92$+$0.41 | 19 27 35.2 | $+$17 54 26 | 39.126 | 6.9 | 6.4 | 3.9 | | 6.8 | 6.2 | 3.9 | 26
53.04$+$0.11 | 19 28 55.7 | $+$17 52 01 | 5.03 | 8.3 | 9.8 | 0.4 | | 8.0 | 9.4 | 0.7 | 35,26
53.14$+$0.07 | 19 29 17.5 | $+$17 56 24 | 23.827 | 7.5 | 8.1 | 2.1 | | 7.3 | 7.9 | 2.2 | 35,26
53.62$+$0.04 | 19 30 22.6 | $+$18 20 28 | 24.125 | 7.4 | 8.0 | 2.1 | | 7.3 | 7.7 | 2.2 | 35,26
58.75$+$0.65 | 19 38 43.9 | $+$23 07 54 | 32.93 | 7.3 | 4.8 | 4.0 | | 7.2 | 4.7 | 4.0 | 35
59.78$+$0.06 | 19 43 11.2 | $+$23 44 03 | 22.330 | 7.6 | 6.1 | 2.4 | | 7.4 | 5.7 | 2.7 | 20,4,35,23
59.84$+$0.66 | 19 40 59.4 | $+$24 04 39 | 34.925 | | | | | | | | 32,33
60.56$-$0.17 | 19 45 48.8 | $+$24 17 22 | 4.93 | 8.3 | 7.9 | 0.5 | | 8.0 | 7.5 | 0.8 | 35
69.52$-$0.97 | 20 10 09.047 | $+$31 31 35.06 | 11.730 | 8.0 | 4.0 | 2.0 | | | | | 20,52
70.12$+$1.72 | 20 00 49.6 | $+$33 28 20 | $-$21.43 | 9.5 | 8.1 | | | 9.1 | 7.4 | | 33,35
71.51$-$0.38 | 20 12 56.8 | $+$33 30 05 | 16.447 | | | | | | | | 35
73.04$+$1.80 | 20 08 04.6 | $+$35 58 47 | 0.725 | 8.5 | 4.9 | 0.1 | | 8.2 | 4.1 | 0.8 | 33,35
75.76$+$0.34 | 20 21 40.1 | $+$37 25 37 | $-$1.230 | 8.5 | 4.4 | | | 8.3 | 3.5 | 0.6 | 20,35
78.10$+$3.64 | 20 14 26.044 | $+$41 13 33.39 | $-$3.216 | 8.6 | 4.1 | | | 8.3 | 3.2 | 0.3 | 17,33,35,22,23
78.62$+$0.98 | 20 27 26.8 | $+$40 07 50 | $-$39.027 | 10.5 | 8.0 | | | 9.9 | 7.2 | | 27
78.89$+$0.71 | 20 29 24.9 | $+$40 11 21 | $-$6.216 | 8.8 | 4.3 | | | 8.5 | 3.5 | | 48
79.75$+$0.99 | 20 30 50.673 | $+$41 02 27.55 | $-$1.43 | 8.6 | 3.3 | | | 8.3 | 1.9 | 1.1 | 40,52
80.85$+$0.43 | 20 36 47.8 | $+$41 36 30 | $-$3.13 | 8.6 | 3.4 | | | 8.3 | 2.2 | 0.5 | 33
81.72$+$0.57 | 20 39 01.057 | $+$42 22 49.18 | $-$2.530 | 8.6 | 3.0 | | | 8.3 | 1.4 | 1.0 | 20,52
81.76$+$0.59 | 20 39 01.989 | $+$42 24 59.30 | $-$3.230 | 8.6 | 3.2 | | | 8.3 | 1.9 | 0.5 | 20,52
81.87$+$0.78 | 20 38 36.452 | $+$42 37 36.06 | 9.830 | | | | | | | | 20,23,52
85.40$-$0.00 | 20 54 13.710 | $+$44 54 07.85 | $-$29.527 | 9.9 | 5.8 | | | 9.4 | 5.0 | | 27
90.90$+$1.50 | 21 09 03.7 | $+$50 01 13 | $-$70.527 | 12.7 | 9.3 | | | 11.7 | 8.1 | | 35
94.58$-$1.79 | 21 39 58.263 | $+$50 14 20.96 | $-$45.630 | 10.8 | 6.1 | | | 10.2 | 5.2 | | 33,52
97.52$+$3.17 | 21 32 13.0 | $+$55 52 56 | $-$71.13 | 12.8 | 8.5 | | | 11.8 | 7.3 | | 48
98.02$+$1.44 | 21 42 58.1 | $+$54 55 46 | $-$63.93 | 12.2 | 7.6 | | | 11.3 | 6.5 | | 35
106.80$+$5.31 | 22 19 18.3 | $+$63 18 48 | $-$15.548 | 9.2 | 1.9 | | | 8.8 | 1.2 | | 48
108.18$+$5.51 | 22 28 51.408 | $+$64 13 41.30 | $-$9.925 | 8.9 | 1.2 | | | 8.6 | 0.6 | | 33,52
108.75$-$0.96 | 22 58 40.3 | $+$58 46 05 | $-$50.33 | 11.3 | 5.2 | | | 10.6 | 4.3 | | 35
109.86$+$2.10 | 22 56 18.095 | $+$62 01 49.45 | $-$10.230 | 9.0 | 1.2 | | | 8.6 | 0.6 | | 20,35,23
109.92$+$1.98 | 22 57 11.2 | $+$61 56 03 | $-$8.746 | 8.9 | 1.0 | | | 8.6 | 0.4 | | 33
110.21$+$2.62 | 22 57 05.2 | $+$62 37 44 | $-$2.751 | 8.6 | 0.3 | | | 8.3 | | | 51
111.24$-$0.76 | 23 16 05.4 | $+$59 55 22 | $-$44.73 | 11.0 | 4.5 | | | 10.3 | 3.7 | | 35
111.53$+$0.76 | 23 13 45.364 | $+$61 28 10.55 | $-$55.730 | 11.8 | 5.7 | | | 11.0 | 4.7 | | 20,21,23
121.28$+$0.65 | 00 36 47.358 | $+$63 29 02.18 | $-$17.630 | 9.4 | 1.6 | | | 9.0 | 1.1 | | 51,33,52
123.05$-$6.31 | 00 52 24.196 | $+$56 33 43.17 | $-$31.830 | 10.4 | 2.9 | | | 9.8 | 2.3 | | 33,52
133.94$+$1.04 | 02 27 03.849 | $+$61 52 25.42 | $-$47.430 | 12.3 | 4.8 | | | 11.4 | 3.8 | | 20,33,35
136.84$+$1.12 | 02 49 23.1 | $+$60 46 26 | $-$39.83 | 11.7 | 3.9 | | | 10.9 | 3.1 | | 33
168.06$+$0.82 | 05 17 13.3 | $+$39 22 14 | $-$25.425 | 19.8 | 11.4 | | | 16.8 | 8.5 | | 48
173.49$+$2.42 | 05 39 13.059 | $+$35 45 51.29 | $-$15.830 | 24.1 | 15.6 | | | 19.4 | 11.1 | | 20,22,52
173.69$+$2.87 | 05 41 33.8 | $+$35 48 27 | $-$19.230 | 25.0 | 16.6 | | | 25.0 | 16.7 | | 40,35
173.71$+$2.35 | 05 39 27.6 | $+$35 30 58 | $-$13.516 | 20.0 | 11.5 | | | 16.7 | 8.4 | | 20
174.19$-$0.09 | 05 30 42.0 | $+$33 47 14 | $-$3.53 | 10.2 | 1.7 | | | 9.5 | 1.1 | | 13,33,35
183.34$-$0.59 | 05 51 06.0 | $+$25 45 45 | $-$9.646 | | | | | 4.9 | | | 33,35
188.79$+$1.02 | 06 09 06.5 | $+$21 50 26 | $-$0.73 | 9.1 | 0.7 | | | 8.1 | | | 35
188.95$+$0.89 | 06 08 53.342 | $+$21 38 29.09 | 3.13 | 9.4 | 0.9 | | | 9.1 | 0.7 | | 20,4,35,23,52
189.03$+$0.76 | 06 08 40.671 | $+$21 31 06.89 | 2.53 | 9.2 | 0.7 | | | 8.9 | 0.5 | | 4,33,35
189.78$+$0.34 | 06 08 34.5 | $+$20 38 50 | 9.430 | 11.5 | 3.0 | | | 10.8 | 2.5 | | 20,4,35
192.60$-$0.05 | 06 12 54.006 | $+$17 59 23.21 | 8.230 | 10.3 | 1.9 | | | 9.9 | 1.5 | | 20,4,22,23,52
196.45$-$1.68 | 06 14 37.051 | $+$13 49 36.16 | 17.93 | 12.1 | 3.7 | | | 11.3 | 3.0 | | 20,4,23,52
203.32$+$2.05 | 06 41 09.7 | $+$09 29 35 | 7.616 | 9.4 | 0.9 | | | 9.0 | 0.6 | | 20
206.54$-$16.4 | 05 41 44.15 | $-$01 54 44.9 | 5.046 | 9.0 | 0.6 | | | 8.7 | 0.3 | | 24
212.06$-$0.74 | 06 47 12.9 | $+$00 26 07 | 45.03 | 14.1 | 6.1 | | | 12.9 | 4.9 | | 48
213.70$-$12.6 | 06 07 47.87 | $-$06 22 57.0 | 10.430 | 9.3 | 1.0 | | | 9.0 | 0.7 | | 20,4,43,44,35,23
232.62$+$0.99 | 07 32 09.79 | $-$16 58 12.5 | 16.63 | 9.4 | 1.4 | | | 9.0 | 1.0 | | 17,4,43,44,35
259.94$-$0.04 | 08 35 31.09 | $-$40 38 24.0 | 7.73 | 8.8 | 1.3 | | | 8.5 | 0.5 | | 43,44
263.25$+$0.52 | 08 48 47.85 | $-$42 54 28.0 | 12.03 | 9.0 | 2.2 | | | 8.7 | 1.4 | | 32,43,44
264.29$+$1.46 | 08 56 24.8 | $-$43 06 05 | 5.53 | 8.7 | 1.3 | | | 8.4 | 0.2 | | 4,43
269.15$-$1.13 | 09 03 32.3 | $-$48 28 00 | 10.33 | 8.9 | 2.6 | | | 8.6 | 1.7 | | 4,43
269.45$-$1.47 | 09 03 14.85 | $-$48 55 11.5 | 59.745 | 11.8 | 8.1 | | | 11.0 | 7.0 | | 43,44
270.25$+$0.84 | 09 16 41.4 | $-$47 55 46 | 9.33 | 8.9 | 2.6 | | | 8.6 | 1.7 | | 4,43
284.35$-$0.42 | 10 24 10.0 | $-$57 52 38 | 5.750 | 8.7 | 5.0 | | | 8.4 | 4.3 | | 4
285.35$-$0.00 | 10 32 09.62 | $-$58 02 04.5 | $-$2.03 | 8.4 | 4.2 | 0.3 | | 8.2 | 3.1 | 1.3 | 39,43,44
287.37$+$0.64 | 10 48 04.44 | $-$58 27 01.0 | 0.93 | 8.5 | 5.2 | | | 8.3 | 4.5 | 0.5 | 4,39,43,44
290.37$+$1.66 | 11 12 16.1 | $-$58 46 18 | $-$18.53 | | | | | | | | 4,43
290.41$-$2.91 | 10 57 33.98 | $-$62 59 03.0 | $-$16.53 | 8.0 | 3.7 | 2.3 | | | | | 43,44
291.27$-$0.70 | 11 11 53.37 | $-$61 18 23.5 | $-$23.43 | | | | | | | | 43,44
291.58$-$0.43 | 11 15 06.4 | $-$61 09 37 | 13.550 | 9.1 | 7.7 | | | 8.8 | 7.1 | | 13,4
293.82$-$0.74 | 11 32 05.61 | $-$62 12 25.5 | 32.93 | 10.2 | 10.1 | | | 9.7 | 9.3 | | 43,44
293.95$-$0.89 | 11 32 43.14 | $-$62 23 02.5 | 32.031 | 10.2 | 10.0 | | | 9.7 | 9.3 | | 43
294.51$-$1.62 | 11 35 32.22 | $-$63 14 42.5 | $-$15.43 | 7.9 | 5.0 | 2.1 | | 7.7 | 4.1 | 2.9 | 17,4,43,44
294.98$-$1.71 | 11 39 22.85 | $-$63 28 26.0 | $-$8.129 | 8.2 | 6.3 | 0.9 | | 7.9 | 5.7 | 1.4 | 43,44
296.90$-$1.31 | 11 56 50.1 | $-$63 32 11 | 20.027 | 9.5 | 9.6 | | | 9.1 | 8.9 | | 40
298.22$-$0.33 | 12 09 59.73 | $-$62 49 19.0 | 35.050 | 10.5 | 11.3 | | | 9.9 | 10.5 | | 4
298.26$+$0.74 | 12 11 47.66 | $-$61 46 21.0 | $-$30.027 | 7.5 | 4.4 | 3.7 | | | | | 43,44
299.01$+$0.12 | 12 17 24.6 | $-$62 29 04 | 18.027 | 9.4 | 9.9 | | | 9.0 | 9.3 | | 13,4,43,44
300.51$-$0.17 | 12 30 03.58 | $-$62 56 49.0 | 28.43 | 10.1 | 11.2 | | | 9.6 | 10.5 | | 13,4,43,44
300.96$+$1.14 | 12 34 53.25 | $-$61 39 40.5 | $-$43.03 | | | | | | | | 13,4,43,44
301.14$-$0.23 | 12 35 36.9 | $-$63 02 48 | $-$39.43 | | | | | | | | 4,43
302.03$-$0.06 | 12 43 31.96 | $-$62 55 09.5 | $-$35.43 | 7.2 | 5.1 | 3.9 | | 7.1 | 4.7 | 4.2 | 32,43,44
305.20$+$0.01 | 13 11 16.91 | $-$62 45 55.5 | $-$35.63 | 7.0 | 6.1 | 3.7 | | 6.9 | 5.8 | 3.9 | 4,43,44
305.21$+$0.21 | 13 11 14.4 | $-$62 34 26 | $-$41.03 | 7.0 | 5.4 | 4.4 | | 6.9 | 5.4 | 4.3 | 4
305.25$+$0.25 | 13 11 33.6 | $-$62 31 52 | $-$32.027 | 7.2 | 6.7 | 3.1 | | 7.0 | 6.4 | 3.3 | 4
305.36$+$0.15 | 13 12 35.89 | $-$62 37 18.5 | $-$38.529 | 6.9 | 5.4 | 4.5 | | 6.9 | 5.4 | 4.4 | 13,4,44
305.55$+$0.01 | 13 14 21.5 | $-$62 44 24 | $-$39.03 | 6.9 | 5.4 | 4.5 | | 6.9 | 5.7 | 4.1 | 43
305.80$-$0.24 | 13 16 43.4 | $-$62 58 15 | $-$33.13 | 7.1 | 6.8 | 3.2 | | 7.0 | 6.5 | 3.3 | 4,43
305.89$+$0.00 | 13 17 18.4 | $-$62 43 14 | $-$34.33 | 7.1 | 6.6 | 3.3 | | 7.0 | 6.4 | 3.5 | 39
306.33$-$0.34 | 13 21 27.6 | $-$63 00 48 | $-$20.031 | 7.6 | 8.3 | 1.7 | | 7.4 | 8.0 | 1.9 | 4,43
308.74$+$0.55 | 13 40 51.23 | $-$61 45 28.0 | $-$48.13 | 6.6 | 5.7 | 4.9 | | 6.6 | 5.5 | 5.0 | 39
308.92$+$0.12 | 13 43 03.14 | $-$62 09 00.0 | $-$50.914 | 6.6 | 6.0 | 4.7 | | | | | 4
309.39$-$0.14 | 13 47 27.72 | $-$62 18 25.0 | $-$50.027 | 6.6 | 6.0 | 4.8 | | 6.5 | 5.9 | 4.7 | 4
309.92$+$0.47 | 13 50 41.85 | $-$61 35 11.0 | $-$58.714 | | | | | | | | 4,43,44
310.13$+$0.75 | 13 51 54.2 | $-$61 16 18 | $-$55.33 | | | | | | | | 39
310.17$-$0.11 | 13 54 00.59 | $-$62 06 22.0 | 9.33 | 9.0 | 11.7 | | | 8.7 | 11.2 | | 43
311.62$+$0.29 | 14 04 54.11 | $-$61 20 08.5 | $-$56.23 | 6.4 | 5.8 | 5.5 | | 6.3 | 6.0 | 5.1 | 43
311.64$-$0.38 | 14 06 38.82 | $-$61 58 24.0 | 35.13 | 10.9 | 14.5 | | | 10.3 | 13.7 | | 17,4,43,44
311.96$+$0.14 | 14 07 56.3 | $-$61 23 01 | $-$42.214 | 6.7 | 7.9 | 3.5 | | 6.6 | 7.8 | 3.5 | 4
312.10$+$0.26 | 14 08 49.3 | $-$61 13 26 | $-$50.027 | 6.4 | 7.0 | 4.4 | | 6.4 | 7.0 | 4.3 | 40,43,44
312.59$+$0.04 | 14 13 15.00 | $-$61 16 54.5 | $-$62.73 | | | | | | | | 17,4,43,44
313.46$+$0.19 | 14 19 41.01 | $-$60 51 48.0 | $-$4.73 | 8.2 | 11.3 | 0.4 | | 8.0 | 10.9 | 0.6 | 17,4,43,44
313.57$+$0.32 | 14 20 05.4 | $-$60 42 40 | $-$46.53 | 6.5 | 8.0 | 3.7 | | 6.4 | 7.9 | 3.7 | 39
313.76$-$0.86 | 14 25 01.62 | $-$61 44 58.0 | $-$51.33 | 6.4 | 7.5 | 4.2 | | 6.3 | 7.5 | 4.1 | 43,44
313.78$-$0.89 | 14 25 12.6 | $-$61 46 14 | $-$48.027 | 6.5 | 7.9 | 3.9 | | 6.4 | 7.8 | 3.8 | 32
314.26$+$0.11 | 14 25 59.1 | $-$60 39 55 | $-$51.93 | 6.3 | 7.7 | 4.2 | | 6.3 | 7.6 | 4.1 | 3243
316.35$-$0.36 | 14 43 11.21 | $-$60 17 14.0 | $-$1.23 | 8.4 | 12.2 | 0.1 | | 8.2 | 11.8 | 0.3 | 43,44
316.38$-$0.38 | 14 43 23.8 | $-$60 17 43 | $-$2.614 | 8.4 | 12.1 | 0.2 | | 8.1 | 11.7 | 0.4 | 32,4
316.41$-$0.31 | 14 43 22.9 | $-$60 13 09 | $-$1.23 | 8.4 | 12.2 | 0.1 | | 8.2 | 11.8 | 0.3 | 17,4
316.64$-$0.08 | 14 44 18.43 | $-$59 55 12.0 | $-$20.314 | 7.5 | 10.8 | 1.5 | | 7.3 | 10.6 | 1.6 | 4,43,44
316.81$-$0.06 | 14 45 26.44 | $-$59 49 16.5 | $-$38.114 | 6.7 | 9.6 | 2.8 | | 6.6 | 9.4 | 2.8 | 17,4,43,44
317.47$-$0.41 | 14 51 22.5 | $-$59 51 07 | $-$37.79 | 6.7 | 9.8 | 2.8 | | 6.6 | 9.6 | 2.8 | 9
317.70$+$0.10 | 14 51 08.1 | $-$59 17 32 | $-$42.63 | 6.5 | 9.5 | 3.1 | | 6.5 | 9.3 | 3.1 | 39
318.05$+$0.08 | 14 53 44.3 | $-$59 09 14 | $-$50.214 | 6.3 | 9.0 | 3.7 | | 6.2 | 8.9 | 3.6 | 13,4,43
318.05$-$1.40 | 14 59 10.5 | $-$60 28 00 | 46.027 | 12.5 | 17.5 | | | 11.6 | 16.4 | | 4,43
318.77$-$0.15 | 14 59 35.32 | $-$59 01 21.5 | $-$36.63 | 6.7 | 10.1 | 2.7 | | 6.6 | 10.0 | 2.7 | 43
318.94$-$0.19 | 15 00 55.40 | $-$58 58 53.5 | $-$35.027 | 6.8 | 10.3 | 2.5 | | 6.7 | 10.1 | 2.6 | 4,43,44
319.84$-$0.20 | 15 06 57.1 | $-$58 33 02 | $-$9.027 | 8.0 | 12.3 | 0.7 | | 7.8 | 12.0 | 0.9 | 4,43
320.12$-$0.50 | 15 10 00.13 | $-$58 40 16.0 | $-$8.73 | 8.0 | 12.4 | 0.7 | | 7.8 | 12.0 | 0.9 | 43,44
320.23$-$0.29 | 15 09 51.96 | $-$58 25 38.0 | $-$66.23 | 5.7 | 8.2 | 4.8 | | 5.7 | 8.3 | 4.6 | 17,4,43,44
321.03$-$0.48 | 15 15 52.52 | $-$58 11 07.5 | $-$61.027 | 5.8 | 8.9 | 4.3 | | 5.8 | 8.9 | 4.1 | 32,43,44
321.15$-$0.53 | 15 16 49.2 | $-$58 09 49 | $-$66.027 | 5.7 | 8.5 | 4.7 | | 5.7 | 8.6 | 4.5 | 13,4
321.71$+$1.17 | 15 13 46.5 | $-$56 25 18 | $-$41.33 | 6.5 | 10.4 | 2.9 | | 6.4 | 10.3 | 2.9 | 4,43
322.16$+$0.64 | 15 18 34.3 | $-$56 38 10 | $-$63.027 | 5.7 | 9.0 | 4.4 | | 5.7 | 9.1 | 4.2 | 17,4
323.45$-$0.07 | 15 29 19.32 | $-$56 31 22.5 | $-$67.53 | 5.5 | 9.0 | 4.6 | | 5.5 | 9.1 | 4.4 | 4,43,44
323.74$-$0.26 | 15 31 45.41 | $-$56 30 50.0 | $-$49.514 | 6.1 | 10.3 | 3.4 | | 6.0 | 10.2 | 3.3 | 4,43,44
324.72$+$0.34 | 15 34 59.2 | $-$55 27 21 | $-$47.027 | 6.1 | 10.6 | 3.3 | | 6.1 | 10.5 | 3.2 | 13,4
324.92$-$0.56 | 15 39 57.62 | $-$56 04 07.5 | $-$73.83 | 5.3 | 8.9 | 5.0 | | 5.3 | 9.0 | 4.7 | 43,44
326.47$+$0.70 | 15 43 18.0 | $-$54 07 57 | $-$40.93 | 6.3 | 11.3 | 2.9 | | 6.2 | 11.2 | 2.8 | 39,8
326.64$+$0.61 | 15 44 33.24 | $-$54 05 30.5 | $-$38.83 | 6.4 | 11.4 | 2.8 | | 6.3 | 11.3 | 2.7 | 32
326.64$-$0.60 | 15 49 42.7 | $-$55 02 44 | $-$40.027 | 6.3 | 11.4 | 2.8 | | 6.3 | 11.2 | 2.8 | 8,43,44
326.66$+$0.52 | 15 45 02.9 | $-$54 09 03 | $-$41.03 | 6.3 | 11.3 | 2.9 | | 6.2 | 11.2 | 2.8 | 8
326.85$-$0.67 | 15 51 14.2 | $-$54 58 04 | $-$57.63 | 5.7 | 10.4 | 3.9 | | 5.7 | 10.3 | 3.7 | 8
327.11$+$0.51 | 15 47 32.72 | $-$53 52 38.0 | $-$83.63 | 4.9 | 8.8 | 5.5 | | 4.9 | 8.9 | 5.2 | 17,4,8,43,44
327.29$-$0.58 | 15 53 07.9 | $-$54 37 15 | $-$53.042 | 5.8 | 10.7 | 3.6 | | 5.8 | 10.6 | 3.5 | 4
327.39$+$0.19 | 15 50 18.5 | $-$53 57 06 | $-$84.627 | 4.9 | 8.8 | 5.5 | | 4.9 | 9.0 | 5.2 | 8
327.40$+$0.44 | 15 49 19.5 | $-$53 45 14 | $-$79.73 | 5.0 | 9.1 | 5.2 | | 5.0 | 9.2 | 4.9 | 4,8,43
327.59$-$0.09 | 15 52 36.8 | $-$54 03 18 | $-$86.327 | 4.8 | 8.7 | 5.6 | | 4.8 | 8.9 | 5.3 | 8
327.61$-$0.11 | 15 52 50.2 | $-$54 03 00 | $-$97.527 | 4.6 | 7.6 | 6.8 | | 4.6 | 8.1 | 6.1 | 8
327.94$-$0.11 | 15 54 33.9 | $-$53 50 44 | $-$51.727 | 5.8 | 10.9 | 3.5 | | 5.8 | 10.8 | 3.4 | 8
328.23$-$0.54 | 15 57 58.28 | $-$53 59 22.5 | $-$44.927 | 6.1 | 11.3 | 3.2 | | 6.0 | 11.2 | 3.1 | 4,8,43,44
328.80$+$0.63 | 15 55 48.37 | $-$52 43 06.0 | $-$42.33 | 6.1 | 11.5 | 3.0 | | 6.1 | 11.4 | 2.9 | 4,8,43,44
328.95$+$0.56 | 15 56 49.50 | $-$52 40 26.5 | $-$93.03 | 4.6 | 8.6 | 6.0 | | 4.6 | 8.8 | 5.6 | 43
329.02$-$0.20 | 16 00 33.3 | $-$53 13 02 | $-$43.414 | 6.1 | 11.5 | 3.1 | | 6.0 | 11.4 | 3.0 | 4,8
329.03$-$0.19 | 16 00 30.7 | $-$53 12 35 | $-$43.414 | 6.1 | 11.5 | 3.1 | | 6.0 | 11.4 | 3.0 | 4,8
329.06$-$0.30 | 16 01 09.9 | $-$53 16 02 | $-$42.63 | 6.1 | 11.5 | 3.0 | | 6.1 | 11.5 | 3.0 | 8
329.18$-$0.31 | 16 01 45.0 | $-$53 11 40 | $-$55.727 | 5.6 | 10.8 | 3.8 | | 5.6 | 10.8 | 3.6 | 17,4,8
329.33$+$0.14 | 16 00 33.11 | $-$52 44 39.5 | $-$107.13 | 4.3 | 7.6 | 7.0 | | 4.3 | 7.7 | 6.8 | 8,43,44
329.40$-$0.46 | 16 03 30.65 | $-$53 10 00.5 | $-$75.03 | 5.0 | 9.8 | 4.8 | | 5.0 | 9.9 | 4.6 | 17,4,8,44
329.46$+$0.50 | 15 59 40.71 | $-$52 23 27.5 | $-$67.63 | 5.2 | 10.2 | 4.4 | | 5.2 | 10.2 | 4.2 | 32,8,43,44
329.61$+$0.11 | 16 02 06.7 | $-$52 36 32 | $-$60.127 | 5.4 | 10.6 | 4.0 | | 5.4 | 10.6 | 3.9 | 39,8
330.07$+$1.05 | 16 00 18.0 | $-$51 35 08 | $-$50.03 | 5.8 | 11.3 | 3.5 | | 5.7 | 11.2 | 3.4 | 39
330.88$-$0.37 | 16 10 21.21 | $-$52 06 12.0 | $-$62.814 | 5.3 | 10.7 | 4.2 | | 5.3 | 10.7 | 4.0 | 4,43
330.95$-$0.18 | 16 09 52.36 | $-$51 54 57.5 | $-$92.514 | 4.4 | 9.1 | 5.8 | | 4.5 | 9.3 | 5.4 | 13,4,8,43,44
331.12$-$0.11 | 16 10 23.17 | $-$51 45 19.2 | $-$93.227 | 4.4 | 9.1 | 5.8 | | 4.5 | 9.3 | 5.5 | 5,8
331.13$-$0.24 | 16 10 59.77 | $-$51 50 22.5 | $-$87.03 | 4.6 | 9.5 | 5.4 | | 4.6 | 9.6 | 5.1 | 17,4,5,8,43,44
331.27$-$0.18 | 16 11 26.56 | $-$51 41 56.5 | $-$89.314 | 4.5 | 9.3 | 5.6 | | 4.6 | 9.5 | 5.3 | 4,5,8,43,44
331.33$-$0.29 | 16 12 12.15 | $-$51 44 12.5 | $-$67.427 | 5.1 | 10.5 | 4.4 | | 5.1 | 10.5 | 4.2 | 17,4,8,43,44
331.42$+$0.26 | 16 10 09.4 | $-$51 16 04 | $-$88.627 | 4.5 | 9.4 | 5.5 | | 4.6 | 9.5 | 5.2 | 8
331.44$-$0.18 | 16 12 12.56 | $-$51 35 11.1 | $-$88.527 | 4.5 | 9.4 | 5.5 | | 4.6 | 9.6 | 5.2 | 39,5,8
331.54$-$0.06 | 16 12 08.81 | $-$51 25 47.0 | $-$91.042 | 4.4 | 9.3 | 5.6 | | 4.5 | 9.4 | 5.3 | 4,5,8
331.55$-$0.12 | 16 12 27.22 | $-$51 27 37.0 | $-$100.73 | 4.2 | 8.7 | 6.3 | | 4.3 | 8.9 | 5.8 | 17,4,5,8,43
332.09$-$0.42 | 16 16 16.49 | $-$51 18 25.0 | $-$61.427 | 5.2 | 10.9 | 4.1 | | 5.2 | 10.9 | 3.9 | 8,44
332.29$+$2.26 | 16 05 44.5 | $-$49 12 20 | $-$24.027 | 6.8 | 13.1 | 2.0 | | 6.7 | 12.9 | 2.0 | 39
332.29$-$0.09 | 16 15 45.4 | $-$50 55 53 | $-$48.43 | 5.7 | 11.6 | 3.4 | | 5.7 | 11.6 | 3.3 | 8
332.35$-$0.43 | 16 17 31.6 | $-$51 08 21 | $-$53.127 | 5.5 | 11.4 | 3.7 | | 5.5 | 11.3 | 3.6 | 8
332.55$-$0.14 | 16 17 12.09 | $-$50 47 12.0 | $-$46.73 | 5.7 | 11.7 | 3.4 | | 5.7 | 11.7 | 3.2 | 5,8,43,44
332.60$-$0.16 | 16 17 29.19 | $-$50 46 11.6 | $-$50.927 | 5.6 | 11.5 | 3.6 | | 5.6 | 11.5 | 3.5 | 5,8
332.65$-$0.62 | 16 19 43.48 | $-$51 03 36.5 | $-$49.514 | 5.6 | 11.6 | 3.5 | | 5.6 | 11.5 | 3.4 | 4,43,44
332.72$-$0.62 | 16 20 01.3 | $-$51 00 44 | $-$46.027 | 5.8 | 11.8 | 3.3 | | 5.7 | 11.7 | 3.2 | 4
332.94$-$0.68 | 16 21 19.0 | $-$50 54 10 | $-$52.927 | 5.5 | 11.4 | 3.7 | | 5.5 | 11.4 | 3.6 | 8
332.96$+$0.77 | 16 15 01.06 | $-$49 50 40.0 | $-$44.53 | 5.8 | 11.9 | 3.2 | | 5.8 | 11.8 | 3.1 | 43
332.96$-$0.67 | 16 21 22.9 | $-$50 52 58 | $-$45.927 | 5.7 | 11.8 | 3.3 | | 5.7 | 11.7 | 3.2 | 8
333.02$-$0.01 | 16 18 44.06 | $-$50 21 51.7 | $-$55.227 | 5.4 | 11.3 | 3.8 | | 5.4 | 11.3 | 3.7 | 5,8
333.02$-$0.06 | 16 18 56.71 | $-$50 23 55.1 | $-$53.627 | 5.4 | 11.4 | 3.7 | | 5.4 | 11.4 | 3.6 | 8,5
333.06$-$0.44 | 16 20 50.3 | $-$50 38 43 | $-$59.042 | 5.2 | 11.1 | 4.0 | | 5.3 | 11.1 | 3.9 | 4,8
333.12$-$0.43 | 16 20 59.71 | $-$50 35 52.0 | $-$52.014 | 5.5 | 11.5 | 3.7 | | 5.5 | 11.5 | 3.5 | 17,4,8,43,44
333.12$-$0.56 | 16 21 35.4 | $-$50 40 57 | $-$56.827 | 5.3 | 11.3 | 3.9 | | 5.3 | 11.2 | 3.8 | 8
333.16$-$0.10 | 16 19 42.68 | $-$50 19 54.5 | $-$91.83 | 4.3 | 9.5 | 5.6 | | 4.4 | 9.7 | 5.3 | 4,5,8,43
333.18$-$0.09 | 16 19 45.67 | $-$50 18 35.5 | $-$91.83 | 4.3 | 9.6 | 5.6 | | 4.4 | 9.7 | 5.3 | 4,5,8
333.23$-$0.06 | 16 19 51.39 | $-$50 15 15.0 | $-$92.042 | 4.3 | 9.5 | 5.6 | | 4.4 | 9.7 | 5.3 | 4,5,8
333.31$+$0.10 | 16 19 29.09 | $-$50 04 42.0 | $-$45.027 | 5.8 | 11.9 | 3.3 | | 5.7 | 11.8 | 3.2 | 5,8
333.46$-$0.16 | 16 21 20.17 | $-$50 09 49.0 | $-$45.33 | 5.7 | 11.9 | 3.3 | | 5.7 | 11.8 | 3.2 | 17,4,5,8,43,44
333.56$-$0.02 | 16 21 08.86 | $-$49 59 49.0 | $-$35.827 | 6.2 | 12.5 | 2.8 | | 6.1 | 12.3 | 2.7 | 5,8
333.64$+$0.05 | 16 21 09.23 | $-$49 52 44.2 | $-$87.327 | 4.4 | 9.8 | 5.4 | | 4.4 | 9.9 | 5.1 | 5,8
333.68$-$0.43 | 16 23 29.8 | $-$50 12 08 | $-$5.227 | 8.1 | 14.7 | 0.5 | | 7.8 | 14.4 | 0.7 | 8
333.93$-$0.13 | 16 23 15.00 | $-$49 48 48.7 | $-$36.727 | 6.1 | 12.4 | 2.8 | | 6.0 | 12.3 | 2.8 | 5,8
334.63$-$0.01 | 16 25 45.78 | $-$49 13 38.2 | $-$30.027 | 6.4 | 12.9 | 2.4 | | 6.3 | 12.8 | 2.4 | 5,8
334.93$-$0.09 | 16 27 24.26 | $-$49 04 10.6 | $-$19.527 | 7.0 | 13.7 | 1.7 | | 6.9 | 13.5 | 1.8 | 5,8
335.06$-$0.42 | 16 29 23.1 | $-$49 12 27 | $-$47.027 | 5.6 | 12.0 | 3.5 | | 5.6 | 11.9 | 3.3 | 8
335.55$-$0.31 | 16 30 55.3 | $-$48 46 14 | $-$116.027 | 3.6 | 8.6 | 6.9 | | 3.7 | 8.9 | 6.4 | 4
335.58$-$0.29 | 16 30 58.79 | $-$48 43 53.0 | $-$46.63 | 5.5 | 12.0 | 3.5 | | 5.5 | 12.0 | 3.3 | 17,4,5,44
335.60$-$0.07 | 16 30 07.20 | $-$48 34 22.1 | $-$50.427 | 5.4 | 11.8 | 3.7 | | 5.4 | 11.8 | 3.5 | 5
335.72$+$0.19 | 16 29 27.25 | $-$48 17 54.3 | $-$44.327 | 5.6 | 12.2 | 3.3 | | 5.6 | 12.1 | 3.2 | 5
335.78$+$0.17 | 16 29 47.34 | $-$48 15 52.0 | $-$47.527 | 5.5 | 12.0 | 3.5 | | 5.5 | 11.9 | 3.4 | 4,5
336.01$-$0.82 | 16 35 09.3 | $-$48 46 47 | $-$48.33 | 5.4 | 12.0 | 3.6 | | 5.4 | 11.9 | 3.5 | 39,43,44
336.35$-$0.13 | 16 33 29.15 | $-$48 03 44.0 | $-$79.63 | 4.4 | 10.5 | 5.1 | | 4.4 | 10.6 | 4.8 | 17,4,5,43,44
336.41$-$0.26 | 16 34 14.1 | $-$48 06 26 | $-$92.042 | 4.0 | 10.0 | 5.6 | | 4.1 | 10.1 | 5.3 | 4
336.43$-$0.26 | 16 34 19.0 | $-$48 05 33 | $-$92.042 | 4.0 | 10.0 | 5.6 | | 4.1 | 10.1 | 5.3 | 4,43
336.82$+$0.02 | 16 34 38.18 | $-$47 36 33.3 | $-$76.727 | 4.4 | 10.7 | 5.0 | | 4.5 | 10.7 | 4.7 | 17,4,5
336.83$-$0.38 | 16 36 26.4 | $-$47 52 28 | $-$22.33 | 6.7 | 13.6 | 2.0 | | 6.6 | 13.4 | 2.0 | 39
336.86$+$0.00 | 16 34 54.43 | $-$47 35 37.5 | $-$73.33 | 4.5 | 10.8 | 4.8 | | 4.5 | 10.9 | 4.6 | 4,5,43,44
336.94$-$0.15 | 16 35 55.13 | $-$47 38 44.7 | $-$67.327 | 4.7 | 11.1 | 4.5 | | 4.7 | 11.1 | 4.4 | 5
336.98$-$0.18 | 16 36 12.31 | $-$47 37 58.4 | $-$80.827 | 4.3 | 10.5 | 5.1 | | 4.3 | 10.6 | 4.9 | 5
336.99$-$0.02 | 16 35 33.98 | $-$47 31 11.5 | $-$120.93 | 3.4 | 8.6 | 7.0 | | 3.5 | 9.0 | 6.5 | 4,5,43,44
337.09$-$0.21 | 16 36 46.36 | $-$47 34 24.1 | 0.027 | 8.5 | 15.7 | 0.1 | | 8.2 | 15.3 | 0.2 | 44
337.15$-$0.40 | 16 37 50.9 | $-$47 39 09 | $-$49.49 | 5.3 | 12.0 | 3.7 | | 5.3 | 11.9 | 3.6 | 9
337.17$-$0.03 | 16 36 18.78 | $-$47 23 18.6 | $-$65.63 | 4.7 | 11.2 | 4.5 | | 4.7 | 11.2 | 4.3 | 5
337.25$-$0.10 | 16 36 56.43 | $-$47 22 27.0 | $-$69.327 | 4.6 | 11.0 | 4.6 | | 4.6 | 11.0 | 4.4 | 5
337.39$-$0.20 | 16 37 55.9 | $-$47 20 58 | $-$56.49 | 5.0 | 11.6 | 4.1 | | 5.0 | 11.6 | 3.9 | 9
337.40$-$0.40 | 16 38 50.51 | $-$47 28 00.5 | $-$41.314 | 5.6 | 12.4 | 3.3 | | 5.6 | 12.3 | 3.2 | 17,4,43,44
337.61$-$0.05 | 16 38 09.53 | $-$47 05 00.0 | $-$49.73 | 5.2 | 12.0 | 3.7 | | 5.3 | 11.9 | 3.6 | 4,5,43,44
337.68$+$0.13 | 16 37 35.41 | $-$46 53 49.5 | $-$74.927 | 4.4 | 10.8 | 4.9 | | 4.4 | 10.9 | 4.7 | 5
337.70$-$0.05 | 16 38 29.62 | $-$47 00 35.0 | $-$48.13 | 5.3 | 12.1 | 3.7 | | 5.3 | 12.0 | 3.5 | 5,43,44
337.71$+$0.08 | 16 37 53.58 | $-$46 54 41.0 | $-$72.627 | 4.4 | 10.9 | 4.8 | | 4.5 | 10.9 | 4.6 | 5
337.92$-$0.46 | 16 41 06.05 | $-$47 07 02.0 | $-$38.714 | 5.7 | 12.6 | 3.1 | | 5.7 | 12.5 | 3.1 | 17,4,43,44
337.96$-$0.16 | 16 40 01.07 | $-$46 53 33.2 | $-$60.427 | 4.8 | 11.5 | 4.3 | | 4.8 | 11.5 | 4.1 | 5,51
338.00$+$0.13 | 16 38 50.8 | $-$46 40 05 | $-$32.027 | 6.0 | 13.0 | 2.7 | | 6.0 | 12.9 | 2.7 | 4
338.07$+$0.01 | 16 39 39.04 | $-$46 41 28.0 | $-$40.53 | 5.6 | 12.5 | 3.3 | | 5.6 | 12.4 | 3.2 | 17,5,44
338.28$+$0.54 | 16 38 09.6 | $-$46 11 09 | $-$57.027 | 4.9 | 11.7 | 4.1 | | 4.9 | 11.6 | 4.0 | 4
338.28$+$0.12 | 16 40 00.07 | $-$46 27 38.3 | $-$39.327 | 5.6 | 12.6 | 3.2 | | 5.6 | 12.5 | 3.1 | 5
338.43$+$0.05 | 16 40 49.81 | $-$46 23 36.5 | $-$32.33 | 6.0 | 13.0 | 2.8 | | 6.0 | 12.9 | 2.7 | 39,5,43,44
338.46$-$0.24 | 16 42 15.48 | $-$46 34 18.5 | $-$50.427 | 5.1 | 12.0 | 3.8 | | 5.2 | 11.9 | 3.7 | 17,4,5,43,44
338.47$+$0.29 | 16 39 58.1 | $-$46 12 39 | $-$30.027 | 6.1 | 13.2 | 2.6 | | 6.1 | 13.0 | 2.6 | 13,4
338.56$+$0.11 | 16 41 06.98 | $-$46 15 30.1 | $-$75.027 | 4.3 | 10.9 | 5.0 | | 4.4 | 10.9 | 4.7 | 5
338.87$-$0.08 | 16 43 08.21 | $-$46 09 13.9 | $-$41.427 | 5.5 | 12.5 | 3.4 | | 5.5 | 12.4 | 3.2 | 17,4,5
338.92$+$0.55 | 16 40 34.1 | $-$45 42 06 | $-$66.042 | 4.5 | 11.3 | 4.6 | | 4.6 | 11.3 | 4.4 | 17,4
338.92$+$0.38 | 16 41 17.71 | $-$45 48 21.0 | $-$26.027 | 6.3 | 13.5 | 2.4 | | 6.3 | 13.3 | 2.4 | 43
338.93$-$0.06 | 16 43 16.1 | $-$46 05 39 | $-$41.927 | 5.5 | 12.5 | 3.4 | | 5.5 | 12.4 | 3.3 | 4,5
339.05$-$0.31 | 16 44 49.09 | $-$46 10 14.0 | $-$111.627 | 3.4 | 9.5 | 6.4 | | 3.5 | 9.6 | 6.1 | 5
339.06$+$0.15 | 16 42 49.65 | $-$45 51 22.7 | $-$85.627 | 4.0 | 10.5 | 5.4 | | 4.0 | 10.5 | 5.2 | 5
339.28$+$0.13 | 16 43 43.05 | $-$45 42 08.5 | $-$69.127 | 4.4 | 11.2 | 4.7 | | 4.5 | 11.2 | 4.5 | 5
339.29$+$0.13 | 16 43 44.97 | $-$45 41 28.8 | $-$74.627 | 4.2 | 10.9 | 5.0 | | 4.3 | 11.0 | 4.8 | 5
339.47$+$0.04 | 16 44 51.00 | $-$45 36 56.4 | $-$9.327 | 7.6 | 14.9 | 1.0 | | 7.4 | 14.6 | 1.1 | 5
339.58$-$0.12 | 16 45 58.84 | $-$45 38 47.0 | $-$33.23 | 5.9 | 13.0 | 2.9 | | 5.8 | 12.9 | 2.8 | 5,44
339.62$-$0.12 | 16 46 05.98 | $-$45 36 43.5 | $-$33.23 | 5.9 | 13.0 | 2.9 | | 5.8 | 12.9 | 2.8 | 17,4,5,43,44
339.68$-$1.21 | 16 51 06.21 | $-$46 16 03.0 | $-$21.027 | 6.6 | 13.9 | 2.1 | | 6.5 | 13.7 | 2.1 | 17,4,43,44
339.76$+$0.05 | 16 45 51.72 | $-$45 23 32.2 | $-$51.027 | 5.0 | 12.0 | 3.9 | | 5.0 | 12.0 | 3.8 | 5
339.88$-$1.25 | 16 52 04.67 | $-$46 08 34.0 | $-$31.63 | 5.9 | 13.1 | 2.8 | | 5.9 | 13.0 | 2.8 | 4,43,44
339.94$-$0.53 | 16 49 08.00 | $-$45 37 58.5 | $-$89.031 | 3.8 | 10.4 | 5.6 | | 3.9 | 10.5 | 5.3 | 43,44
340.03$-$0.29 | 16 48 21.44 | $-$45 24 33.5 | $-$60.027 | 4.6 | 11.6 | 4.4 | | 4.7 | 11.6 | 4.2 | 17,4,43,44
340.25$-$0.05 | 16 48 07.37 | $-$45 05 10.0 | $-$121.93 | 3.1 | 9.2 | 6.8 | | 3.2 | 9.4 | 6.4 | 43
340.53$-$0.15 | 16 49 33.7 | $-$44 56 03 | $-$48.351 | 5.0 | 12.2 | 3.9 | | 5.1 | 12.1 | 3.7 | 51
340.79$-$0.10 | 16 50 17.0 | $-$44 42 22 | $-$107.027 | 3.3 | 9.8 | 6.2 | | 3.4 | 9.9 | 6.0 | 4
340.97$-$1.03 | 16 54 58.7 | $-$45 09 23 | $-$31.49 | 5.8 | 13.2 | 2.9 | | 5.8 | 13.1 | 2.8 | 9
341.22$-$0.21 | 16 52 17.9 | $-$44 26 41 | $-$43.43 | 5.2 | 12.4 | 3.7 | | 5.2 | 12.4 | 3.5 | 4
341.28$+$0.06 | 16 51 20.8 | $-$44 13 36 | $-$74.027 | 4.0 | 11.0 | 5.1 | | 4.1 | 11.1 | 4.9 | 13,4
342.36$+$0.14 | 16 54 49.14 | $-$43 20 03.0 | $-$6.027 | 7.8 | 15.4 | 0.8 | | 7.6 | 15.1 | 0.9 | 43,44
342.50$+$0.17 | 16 55 07.2 | $-$43 12 54 | $-$41.851 | 5.1 | 12.5 | 3.7 | | 5.1 | 12.5 | 3.5 | 51
343.52$-$0.50 | 17 01 27.7 | $-$42 49 44 | $-$33.93 | 5.4 | 13.0 | 3.3 | | 5.4 | 12.9 | 3.2 | 9
343.92$+$0.12 | 17 00 10.90 | $-$42 07 19.5 | 14.027 | 11.1 | 19.1 | | | 10.4 | 18.2 | | 17,4,43,44
344.22$-$0.56 | 17 04 07.7 | $-$42 18 39 | $-$23.23 | 6.0 | 13.8 | 2.6 | | 6.0 | 13.6 | 2.5 | 18,4,43,44
344.42$+$0.04 | 17 02 08.76 | $-$41 46 58.0 | $-$65.93 | 3.9 | 11.4 | 5.0 | | 4.0 | 11.4 | 4.8 | 18,4,43,44
344.58$-$0.02 | 17 02 57.73 | $-$41 41 54.0 | $-$1.03 | 8.4 | 16.2 | 0.2 | | 8.1 | 15.8 | 0.4 | 20,4,43,44
345.00$-$0.22 | 17 05 10.90 | $-$41 29 06.5 | $-$26.214 | 5.7 | 13.5 | 2.9 | | 5.7 | 13.4 | 2.8 | 20,4,43,44
345.01$+$1.79 | 16 56 47.56 | $-$40 14 25.5 | $-$13.414 | 6.8 | 14.7 | 1.8 | | 6.7 | 14.5 | 1.8 | 20,4,43,44
345.20$-$0.04 | 17 05 01.0 | $-$41 13 09 | $-$0.79 | 8.4 | 16.3 | 0.1 | | 8.1 | 15.9 | 0.3 | 9
345.42$-$0.95 | 17 09 38.57 | $-$41 35 04.0 | $-$21.214 | 6.1 | 13.9 | 2.5 | | 6.0 | 13.8 | 2.5 | 4,43,44
345.50$+$0.34 | 17 04 22.89 | $-$40 44 23.0 | $-$16.314 | 6.5 | 14.4 | 2.1 | | 6.4 | 14.2 | 2.1 | 20,4,43,44
345.69$-$0.09 | 17 06 51.3 | $-$40 50 59 | 0.027 | 8.5 | 16.5 | 0.1 | | 8.2 | 16.1 | 0.2 | 20
345.83$+$0.04 | 17 06 42.28 | $-$40 39 26.0 | $-$10.031 | 7.1 | 15.1 | 1.4 | | 7.0 | 14.8 | 1.5 | 43,44
346.42$+$0.28 | 17 07 33.7 | $-$40 03 04 | $-$18.751 | 6.2 | 14.1 | 2.4 | | 6.1 | 13.9 | 2.4 | 51
346.48$+$0.13 | 17 08 23.1 | $-$40 05 33 | 3.031 | 9.0 | 17.1 | | | 8.7 | 16.6 | | 13,4
346.52$+$0.08 | 17 08 43.1 | $-$40 05 25 | 3.031 | 9.0 | 17.1 | | | 8.7 | 16.6 | | 4,43
347.58$+$0.21 | 17 11 26.8 | $-$39 09 37 | $-$103.027 | 2.6 | 10.1 | 6.5 | | 2.7 | 10.1 | 6.3 | 4,43
347.63$+$0.21 | 17 11 36.14 | $-$39 07 06.5 | $-$97.027 | 2.7 | 10.2 | 6.4 | | 2.8 | 10.3 | 6.1 | 17,4,44
347.82$+$0.02 | 17 12 58.0 | $-$39 04 43 | $-$25.027 | 5.4 | 13.4 | 3.2 | | 5.4 | 13.3 | 3.1 | 4
347.86$+$0.01 | 17 13 06.22 | $-$39 02 40.5 | $-$30.031 | 5.1 | 13.0 | 3.6 | | 5.1 | 13.0 | 3.5 | 17,4,44
347.90$+$0.05 | 17 13 05.1 | $-$38 59 46 | $-$30.031 | 5.1 | 13.0 | 3.6 | | 5.1 | 13.0 | 3.5 | 4,43
348.23$-$0.97 | 17 18 23.88 | $-$39 19 10.0 | $-$13.73 | 6.4 | 14.5 | 2.1 | | 6.4 | 14.3 | 2.1 | 43
348.55$-$0.98 | 17 19 20.8 | $-$39 03 53 | $-$15.714 | 6.2 | 14.3 | 2.4 | | 6.1 | 14.1 | 2.4 | 13,4,43
348.70$-$1.04 | 17 20 04.02 | $-$38 58 30.0 | $-$12.314 | 6.6 | 14.7 | 2.0 | | 6.4 | 14.5 | 2.0 | 20,4,43,44
348.88$+$0.10 | 17 15 48.4 | $-$38 10 18 | $-$75.027 | 3.0 | 10.8 | 5.9 | | 3.0 | 10.8 | 5.7 | 13,4
348.89$-$0.18 | 17 17 00.24 | $-$38 19 29.5 | 7.93 | 10.5 | 18.7 | | | 9.9 | 18.0 | | 4,43,44
349.07$-$0.02 | 17 16 51.9 | $-$38 05 12 | 7.027 | 10.3 | 18.5 | | | 9.7 | 17.8 | | 4
349.09$+$0.10 | 17 16 24.72 | $-$37 59 47.0 | $-$77.027 | 2.9 | 10.7 | 6.0 | | 3.0 | 10.7 | 5.8 | 17,4,43,44
350.01$+$0.43 | 17 17 45.47 | $-$37 03 12.0 | $-$31.53 | 4.6 | 12.7 | 4.1 | | 4.6 | 12.6 | 3.9 | 13,4,43,44
350.10$+$0.08 | 17 19 27.03 | $-$37 10 53.5 | $-$69.53 | 2.9 | 10.9 | 5.9 | | 3.0 | 10.9 | 5.7 | 20,4,43,44
350.29$+$0.12 | 17 19 50.86 | $-$37 00 00.5 | $-$64.027 | 3.0 | 11.0 | 5.7 | | 3.1 | 11.0 | 5.5 | 32,43,44
350.69$-$0.50 | 17 23 31.6 | $-$37 01 40 | $-$18.63 | 5.5 | 13.7 | 3.1 | | 5.5 | 13.6 | 3.0 | 39
351.16$+$0.69 | 17 19 58.84 | $-$35 57 40.5 | $-$6.114 | 7.2 | 15.4 | 1.4 | | 7.0 | 15.2 | 1.4 | 20,4,43,44
351.23$+$0.66 | 17 20 17.8 | $-$35 54 58 | $-$3.530 | 7.7 | 16.0 | 0.8 | | 7.4 | 15.6 | 1.0 | 20
351.41$+$0.64 | 17 20 53.37 | $-$35 47 02.0 | $-$6.93 | 7.0 | 15.3 | 1.5 | | 6.8 | 15.0 | 1.6 | 20,4,43,44
351.58$-$0.35 | 17 25 25.19 | $-$36 12 45.5 | $-$98.53 | 2.0 | 10.0 | 6.9 | | 2.1 | 10.0 | 6.6 | 17,4,43,44
351.77$-$0.53 | 17 26 42.56 | $-$36 09 17.5 | $-$3.43 | 7.6 | 16.0 | 0.9 | | 7.4 | 15.6 | 1.0 | 20,4,43,44
352.08$+$0.16 | 17 24 42.4 | $-$35 30 42 | $-$67.027 | 2.5 | 10.7 | 6.2 | | 2.6 | 10.7 | 6.0 | 4
352.11$-$0.17 | 17 26 08.0 | $-$35 40 19 | $-$55.027 | 2.9 | 11.1 | 5.8 | | 3.0 | 11.1 | 5.6 | 4
352.12$-$0.93 | 17 29 17.6 | $-$36 04 41 | $-$12.027 | 6.0 | 14.3 | 2.5 | | 5.9 | 14.1 | 2.5 | 39
352.52$-$0.16 | 17 27 13.0 | $-$35 19 34 | $-$49.23 | 3.0 | 11.2 | 5.6 | | 3.1 | 11.2 | 5.4 | 13,4,43
352.60$-$0.18 | 17 27 32.04 | $-$35 16 15.0 | $-$86.53 | 2.0 | 10.1 | 6.8 | | 2.1 | 10.1 | 6.6 | 43
352.63$-$1.06 | 17 31 13.88 | $-$35 44 08.0 | $-$0.43 | 8.4 | 16.7 | 0.1 | | 8.0 | 16.3 | 0.4 | 32,43,44
353.21$-$0.23 | 17 29 25.45 | $-$34 47 27.0 | $-$17.33 | 5.0 | 13.3 | 3.5 | | 5.0 | 13.3 | 3.4 | 43
353.40$-$0.36 | 17 30 26.18 | $-$34 41 45.0 | $-$16.73 | 5.0 | 13.4 | 3.5 | | 5.0 | 13.3 | 3.4 | 20,4,43,44
353.46$+$0.56 | 17 26 51.5 | $-$34 08 40 | $-$47.13 | 2.8 | 11.1 | 5.8 | | 2.9 | 11.1 | 5.6 | 17,4,43
354.61$+$0.47 | 17 30 16.8 | $-$33 14 13 | $-$21.03 | 4.1 | 12.5 | 4.4 | | 4.1 | 12.4 | 4.3 | 20,4,43
354.72$+$0.30 | 17 31 15.55 | $-$33 14 06.0 | 93.027 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 32,43,44
355.34$+$0.14 | 17 33 28.92 | $-$32 47 49.0 | 14.63 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 20,4,43,44
356.66$-$0.27 | 17 38 30.5 | $-$31 54 57 | $-$52.33 | 1.6 | 10.0 | 7.0 | | 1.6 | 10.0 | 6.8 | 13,4,43
357.96$-$0.17 | 17 41 20.2 | $-$30 45 47 | $-$5.13 | 5.1 | 13.5 | 3.4 | | 5.0 | 13.4 | 3.4 | 39
358.27$-$2.08 | 17 49 41.6 | $-$31 29 30 | 5.227 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 39,33
358.37$-$0.46 | 17 43 31.95 | $-$30 34 11.0 | $-$3.03 | 5.7 | 14.2 | 2.8 | | 5.5 | 13.9 | 2.9 | 32,43,44
359.13$+$0.03 | 17 43 25.67 | $-$29 39 17.5 | $-$4.027 | 3.7 | 12.2 | 4.8 | | 3.7 | 12.1 | 4.7 | 20,4,6,43,44
359.43$-$0.10 | 17 44 40.66 | $-$29 28 17.6 | $-$52.027 | 0.3 | 8.8 | 8.2 | | 0.3 | 8.7 | 8.1 | 20,4,6
359.61$-$0.24 | 17 45 39.08 | $-$29 23 29.0 | 22.527 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 20,4,6
359.89$-$0.06 | 17 45 38.2 | $-$29 03 28 | 10.027 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 20
359.97$-$0.45 | 17 47 20.15 | $-$29 11 57.2 | 19.114 | 25.0 | 33.5 | | | 25.0 | 33.4 | | 20,4,6,43
References:
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Caswell et al. 1995. 5 Caswell et al. 1996a. 6 Caswell et al. 1996b. 7
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Felli et al. 1992. 11 Fontani et al. 2006. 12 Garay et al. 1993. 13 Gaylard &
Macleod 1993. 14 Juvela et al. 1996. 15 Kim & Koo 2003. 16 Larionov et al.
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Slysh et al. 1999. 34 Solomon et al. 1987. 35 Szymczak et al. 2000. 36
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der Walt et al. 1995\. 40 van der Walt et al. 1996. 41 van der Walt et al.
2007\. 42 Vilas-Boas et al. 2000. 43 Walsh et al. 1997. 44 Walsh et al. 1998.
45 Wouterloot et al. 1989. 46 Wouterloot et al. 1993. 47 Wu et al. 2006. 48 Xu
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|
arxiv-papers
| 2009-08-27T04:29:21 |
2024-09-04T02:49:04.835304
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y.Xu, M.A.Voronkov, J.D.Pandian, J.J.Li, A.M.Sobolev, A.Brunthaler,\n B.Ritter, K.M.Menten",
"submitter": "Ye Xu",
"url": "https://arxiv.org/abs/0908.3933"
}
|
0908.3986
|
DO-TH-09/12
# Explaining LSND using extra-dimensional shortcuts
Sebastian Hollenberg Octavian Micu111Speaker. Heinrich Päs Thomas J. Weiler
###### Abstract
We explore the possibility to explain the LSND Aguilar:2001ty result in the
context of extra-dimensional theories. If sterile neutrinos take shortcuts
through extra dimensions, this results in altered neutrino dispersion
relations. Active-sterile neutrino oscillations thus are modified and new
types of resonances occur.
###### Keywords:
Neutrino oscillations, extra dimensions, altered dispersion relations.
13.15.+g, 14.60.Pq, 14.60.St
In theories with large extra dimensions, the Standard Model particles are
typically confined to the $3+1$-dimensional brane, which is embedded in an
extra-dimensional bulk ArkaniHamed:1998rs ; Randall:1999ee . Singlets under
the gauge group such as gravitons or sterile neutrinos however are allowed to
travel freely on the brane as well as in the bulk. While the active neutrinos
are confined to the brane, sterile states can take shortcuts through the extra
dimension and the active-sterile neutrino oscillations Pas:2005rb generate
new resonances.
There are different ways in which these bulk shortcuts can be realized, one
being the case of asymmetrically-warped extra dimensions. Warp factors shrink
the space dimensions $x$ parallel to the brane but leave the time and bulk
dimension $t$ and $u$ unaffected Chung:1999xg ; Csaki:2000dm
$\displaystyle d\tau^{2}=dt^{2}-e^{-2ku}dx^{2}-du^{2}.$ (1)
When the sterile neutrinos take shortcuts through an extra dimension a
resonance arises due to an additional phase difference $\delta(Ht)=t\delta
H+H\delta t$. Thus, in these models there are two sources of phase difference,
the standard one $t\delta H=L\Delta m^{2}/2E$, and a new one $Ht\,(\delta
t/t)$ arising from temporal shortcuts through the bulk available to sterile
neutrinos. The two phase differences may beat against one another to produce
resonant oscillations phenomena.
By introducing the shortcut parameter $\epsilon\equiv(t^{\rm brane}-t^{\rm
bulk})/t^{\rm brane}=\delta t/t$ the effective Hamiltonian for the two state
system (one active and one sterile neutrino) is given by
$\displaystyle H_{\rm eff}=\frac{\Delta
m^{2}}{4E}\left(\begin{array}[]{cc}-\cos 2\theta&\sin 2\theta\\\ \ \ \ \sin
2\theta&\cos
2\theta\end{array}\right)-E~{}\frac{\epsilon}{2}\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)\,.$ (6)
One notices that the diagonal terms in the effective Hamiltonian cancel out
for a resonance energy $E_{\rm Res}=\sqrt{\frac{\Delta m^{2}\cos
2\theta}{2\epsilon}}$. The effective mixing angle becomes
$\displaystyle\sin^{2}2\tilde{\theta}=\frac{\sin^{2}2\theta}{\sin^{2}2\theta+\cos^{2}2\theta\left[1-\frac{E^{2}}{E_{\rm
Res}^{2}}\right]^{2}}.$ (7)
It can be seen that there are three distinct energy domains. Below the
resonance energy one recovers vacuum mixing, at the resonance energy the
effective mixing becomes maximal ($\tilde{\theta}=\pi/4$), while above the
resonance energy oscillations are suppressed.
Starting from the asymmetrically-warped metric in Eq. (1), one needs to first
solve the geodesic equations to calculate the time it takes the sterile
neutrinos to travel through the bulk. For the sterile neutrinos to leave the
brane, they must have a nonzero initial velocity $\dot{u}_{0}$ along the extra
dimension. When this happens, the geodesics are parabolic and they oscillate
about the brane. The distance between two consecutive points where a geodesic
crosses the brane is proportional to $\dot{u}_{0}$. Translational invariance
is maintained on the brane and the Minkowski metric on the brane assures that
Lorentz invariance is maintained on the brane. Therefore, momentum components
are conserved on the brane, and we cannot generate a nonzero $\dot{u}_{0}$ on
the brane except as an initial condition. The uncertainty principle applied to
the $u$ dimension allows for such a nonzero velocity. Momentum conservation in
the $u$-direction is a non-issue, as translational invariance in the
$u$-direction is broken by the brane itself. For a baseline $L$, as measured
on the brane, there are more geodesic paths which cross the brane at the
position of the detector, and the shortcut parameter for each of those paths
is given by
$\displaystyle\epsilon_{n}(v)$ $\displaystyle=$ $\displaystyle
1-\left(\frac{n}{v}\right)\,{\rm arcsinh}\left(\frac{v}{n}\right),$ (8)
where the scaling variable $v\equiv kL/2$ was introduced, and the subscript
$n$ refers to neutrinos which enter the detector upon intersecting the brane
for the $n^{\rm th}$ time.
These different modes have to be accounted for when calculating the
probability of oscillation. Each mode is weighted by the quantum mechanical
weight $e^{iS_{n}}$, with $S_{n}$ being the classical action for the free
particle.
While the initial momenta are mostly on the brane, the uncertainty principle
requires a nonzero $p_{u}$ component as well. Thus we assume a normalized
Gaußian distribution for the momentum component along the extra dimension with
a width $\sigma$ which is related to the thickness of the brane. The
distribution can be written in terms of the mode number through its dependence
on velocity component $\dot{u}_{0}$. In order to be able to select only the
geodesics which cross the brane at baseline length $L$, the integral over the
momenta needs to be approximated with the corresponding sum with a measure
$\Delta n$.
When counting for all neutrinos $\Delta n$ equals one, which is an upper
bound. When looking only at the neutrinos which cross through the detector,
the value of $\Delta n$ is found by varying the action about the classical
extremum. We account only for deviations $\Delta S=|S-S_{\rm cl}|$ smaller
than or of the order of $\hbar$ from the classical action, because larger
variations lead to rapid oscillations and the integration averages to zero.
Putting all the pieces together, the probability of oscillation including the
weights mentioned above is given by
$\displaystyle P_{\rm as}=\left|\sum_{n=1}^{\infty}\Delta n\;e^{iS_{\rm
cl}(n)}\,\frac{vn}{(n^{2}+v^{2})^{3/2}}~{}\left[\sqrt{\frac{2}{\pi}}\,\frac{\beta
E}{\sigma}~{}e^{-\frac{(\beta Ev)^{2}}{2\sigma^{2}(n^{2}+v^{2})}}\right]\sin
2\tilde{\theta}_{n}\ \sin\frac{L\delta\tilde{H}_{n}}{2}\,\right|^{2},$ (9)
with $\sin 2\tilde{\theta}_{n}$ and $\delta\tilde{H}_{n}$ obtained by
replacing $E_{\rm Res}$ with the resonant energy of the mode $n$. This now
depends both on the energy $E$ and the baseline $L$. The factor $\beta$ is the
velocity of the sterile neutrinos.
Particularly interesting is the ”Near Zone”, defined for values $v/n\ll 1$. A
detailed motivation can be found in Hollenberg:2009ws . After making the small
$v/n$ expansion, a new feature which emerges is that the resonance peaks are
functions of the energy and of the baseline through the combination $LE$
rather than the energy alone as in the MSW matter-resonance Wolfenstein:1977ue
; Mikheev:1986wj . The $LE$ dependence of the resonances is a novel feature of
our model.
The plot in Fig. 1 shows the probability of oscillation as a function of the
baseline for the same value of energy. One can see the consecutive peaks
corresponding to consecutive modes $n$. In our case the term
$\sin(L\delta\tilde{H}_{n}/2)$ oscillates fast, and phase-averaging then sets
$\langle\sin\frac{L\delta\tilde{H}_{n}}{2}\rangle$ to zero and
$\langle\sin^{2}\frac{L\delta\tilde{H}_{n}}{2}\rangle$ to $\frac{1}{2}$.
On the plot in Fig. 2, the dependence on the product $LE$ of the oscillation
probability becomes obvious. The resonance peaks are distributed in hyperbolic
patterns on the $L$ vs. $E$ plane, with the peak closest to the origin of the
axes being the one for $n=1$. The relative height of consecutive peaks depends
on the thickness of the brane $\sigma$. Higher $LE$ resonances are suppressed,
and active-sterile neutrino mixing is suppressed for $LE$ above the resonant
values.
The resonance encountered in our model might explain the observed excess in
the LSND data. Sterile neutrinos decouple from the active neutrinos for long-
baseline experiments as well as for high energies. Thus, no active-sterile
mixing is expected in atmospheric data, in MINOS Michael:2006rx or CDHS
Abramowicz:1984yk . All explanations proposed so-far for the LSND and
MiniBooNE anomalies assumed baseline-independent oscillations and mixing. Our
model relies on metric shortcuts, and it does not discriminate between
particles and antiparticles. Thus it will be difficult to accommodate the
recent MiniBooNE claims that an excess of flavor changing events exists in the
neutrino channel AguilarArevalo:2007it but not in the antineutrino channel
AguilarArevalo:2009xn . It might still be possible though to explain the
MiniBooNE data by non-standard matter effects Hollenberg:2009bq . The failure
of previous models to reconcile short baseline data such as LSND with longer
baseline data might be construed as favoring the extra-dimensional shortcut
scenario. Finally, the bulk shortcut scenario might even relieve some of the
remaining tension between the LSND and KARMEN Armbruster:2002mp experiments
since LSND has almost twice the baseline of KARMEN.
${P_{as}}$
Figure 1: Oscillation probability as a function of the experimental baseline,
for a Gaußian distribution for $\dot{u}_{0}$ (red and green curves). The green
curve presents the phase-averaged oscillation probability, and the sinusoidal
blue curve presents the probability as given by the standard 4D vacuum formula
for oscillations between sterile and active neutrinos. Parameter choices are
$\sin^{2}2\theta=0.003$, $k=5/(10^{8}~{}m)$, $E=15$ MeV, $\Delta
m^{2}=64~{}eV^{2}$, and $\sigma=100~{}eV$. The resulting value of $(LE)_{\rm
Res}$ is 550 m MeV. For our choice of $E$, the resonance peaks are found at
the multiples $L=n(LE)_{\rm Res}/E=37n$ m, $n=1,2,3\cdots$, with the principal
resonance corresponding to $n=1$.
$P_{as}$
Figure 2: Oscillation probability (vertical) in the $L$-$E$ plane, with the
same parameters as in Fig. 1. Units of $L$ and $E$ are m and eV, respectively.
## References
* (1) A. Aguilar et al. [LSND Collaboration], Phys. Rev. D 64, 112007 (2001) [arXiv:hep-ex/0104049].
* (2) N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998) [arXiv:hep-ph/9803315]; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257 (1998) [arXiv:hep-ph/9804398]; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D 59, 086004 (1999) [arXiv:hep-ph/9807344].
* (3) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221]; L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690 [arXiv:hep-th/9906064].
* (4) H. Päs, S. Pakvasa and T. J. Weiler, Phys. Rev. D 72, 095017 (2005) [arXiv:hep-ph/0504096].
* (5) D. J. H. Chung and K. Freese, Phys. Rev. D 62, 063513 (2000) [arXiv:hep-ph/9910235]; D. J. H. Chung and K. Freese, Phys. Rev. D 61, 023511 (2000) [arXiv:hep-ph/9906542].
* (6) C. Csaki, J. Erlich and C. Grojean, Nucl. Phys. B 604, 312 (2001) [arXiv:hep-th/0012143]. R. R. Volkas, Prog. Part. Nucl. Phys. 48, 161 (2002) [arXiv:hep-ph/0111326].
* (7) S. Hollenberg, O. Micu, H. Päs and T. J. Weiler, arXiv:0906.0150 [hep-ph].
* (8) L. Wolfenstein, Phys. Rev. D 17, 2369 (1978).
* (9) S. P. Mikheev and A. Y. Smirnov, Nuovo Cim. C 9, 17 (1986);
* (10) D. G. Michael et al. [MINOS Collaboration], Phys. Rev. Lett. 97, 191801 (2006) [arXiv:hep-ex/0607088].
* (11) H. Abramowicz et al., Z. Phys. C 25, 29 (1984).
* (12) A. A. Aguilar-Arevalo et al. [The MiniBooNE Collaboration], Phys. Rev. Lett. 98, 231801 (2007) [arXiv:0704.1500 [hep-ex]]; A. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Rev. Lett. 102 (2009) 101802 [arXiv:0812.2243 [hep-ex]].
* (13) A. A. Aguilar-Arevalo et al., arXiv:0904.1958 [hep-ex].
* (14) S. Hollenberg and H. Päs, arXiv:0904.2167 [hep-ph].
* (15) B. Armbruster et al. [KARMEN Collaboration], Phys. Rev. D 65, 112001 (2002) [arXiv:hep-ex/0203021].
|
arxiv-papers
| 2009-08-27T13:36:59 |
2024-09-04T02:49:04.865184
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian Hollenberg, Octavian Micu, Heinrich P\\\"as, Thomas J. Weiler",
"submitter": "Octavian Micu",
"url": "https://arxiv.org/abs/0908.3986"
}
|
0908.3997
|
# Thermodynamic witness of quantum probing
H.Dong Institute of Theoretical Physics,Chinese Academy of Science, Beijing
100080, China X.F. Liu Department of Mathematics, Peking University, Beijing
100871, China C.P. Sun suncp@itp.ac.cn http://power.itp.ac.cn/~suncp/
Institute of Theoretical Physics,Chinese Academy of Science, Beijing 100080,
China
###### Abstract
The thermodynamic influence of quantum probing on an object is studied. Here,
quantum probing is understood as a pre-measurement based on a non-demolition
interaction, which records some information of the probed object, but does not
change its energy state when both the probing apparatus and the probed object
are isolated from the environment. It is argued that when the probing
apparatus and the probed object are immersed in a same equilibrium
environment, the probing can affect the effective temperature of the object or
induce a quantum isothermal process for the object to transfer its energy.
This thermodynamic feature can be regarded as a witness of the existence of
quantum probing even if the quantum probing would not disturb the object if
the environment were not present.
###### pacs:
03.65.Ta, 03.65.Yz , 05.30.Ch
Introduction\- The Landauer’s principle that the erasure of one bit of
information requires a minimum heat generation of $k_{B}T\ln 2$, which is
based on the second law of thermodynamics landaure ; book_maxwelldeamon2 ,
underlies the thermodynamics of information processing. This principle
eventually resolves the Maxwell’s demon paradox: why a demon can assist a
binary thermal medium to do extra work book_maxwelldeamon2 . For the science
and technology of quantum information, the Landauer’s principle is undoubtedly
crucial since it gives a physical limitation on the spatial-time scales of
logical devices on chips.
However, in the arguments for and against the Landauer’s principle, the
conventional question in the thermodynamics of information processing that
whether the measurement process requires a cost of heat generation has never
been convincingly answered. Surely, earlier authors had touched on this
problem, but it has not been clarified yet because the measurement process is
not defined properly. Particularly, in the quantum approach of measurement,
people can not be unanimous for some fundamental problems, such as whether or
not there exist wave function collapse zurek . In fact, to clarify the
situation, we need to answer the following subtle questions in an unambiguous
way: when can we say a system is performing a measurement on another system
and what kind of measurement can dissipate information?
In this paper, we refer quantum probing to a pre-measurement book_measurement
; sun based on the non-demolition coupling of the probed system $S$ to the
apparatus $A$, which is only a unitary process to produce entanglement between
$S$ and $A$, and does not concerns the subtle, seemingly philosophical
arguments, such as wave function collapses. Generally speaking, when the
energy state of the measured system $S$ is not influenced by the coupling to
the measuring apparatus $A$, but $A$ can record some information of the system
$S$, the measurement performed by $A$ on $S$ is called a pre-measurement. The
so called non-demolition (pre-) measurement book_measurement is an ideal
measurement under some circumstances. In the case of non-demolition
measurement, the system-apparatus coupling $V_{SA}$ commutes with the
Hamiltonian $H_{S}$ of the measured system, but does not commute with the
Hamiltonian $H_{A}$ of the measuring apparatus. When studying non-demolition
measurement it is usually assumed that both $A$ and $S$ are isolated from an
environment. In this paper, we will investigate the effect of environment on
the measured system in a non-demolition measurement. In the following
discussion, we will use the name quantum probing or just probing when
regarding such non-demolition measurement. In the presence of environment, the
practical thermalization efftem of the total system $A+S$ will have
thermodynamic effects on $S$. These effects can be regarded as the
thermodynamic witness of quantum probing.
When the existence of an environment $E$ is considered, quantum probing is
understood in two steps(illustrated in Fig. 1) :
* •
Due to the extreme-weakness of the coupling between $S+A$ and $E$, it can be
neglected within the dephasing time $\tau_{2}$ of the system, the dephasing
being the result of the interaction between $S$ and $A$. In the end of this
step, the initial factorized state of $S+A$ becomes a state assuming the form
of an ideal Schmidt decomposition $\sum
c_{n}\left|n\right\rangle\otimes\left|D_{n}\right\rangle$sun , with
$\left\langle D_{m}\right|\left.D_{n}\right\rangle=\delta_{mn}$.
* •
If the probing apparatus continues to probe the state of the system $S$, the
non-demolition coupling should hold for a longer time, and the communication
with the environment will result in the thermalization of the total system
$S+A$. In this step, the information of the initial state should be erased
totally, but the correlation between $A$ and $S$ needs to exist to leave the
witness of quantum probing.
As shown as follows, it is the comment environment of the probed system and
the probing apparatus that selects a special set of ideal entanglement states
of $S+A$ to be thermalized, so that these witnesses of quantum probing are of
thermodynamics, and thus observable in the classical or macroscopic level.
Figure 1: Time scale of the non-demolition pre-measurement and thermalization.
(1) In the time interval $\left(0,\tau_{2}\right)$, the apparatus makes the
pre-measurement: $\sum
c_{n}\left|n\right\rangle\otimes\left|D\right\rangle\rightarrow\sum
c_{n}\left|n\right\rangle\otimes\left|D_{n}\right\rangle$ and the environment
does not play the role, since its coupling to $S+A$ is weaker than that
between $S$ and $A$. (2) In the time interval
$\left(\tau_{2},\tau_{1}\right)$, the effect of thermalization due to the
environment becomes prominent. The total system $S+A$ is finally thermalized
in the canonical state $\rho_{\mathrm{can}}\left(T\right)$ with temperature
$T$.
Universal Setup for quantum probing apparatus\- Let $A$ be a general apparatus
weakly coupling to the system $S$ to be probed. We require that the energy
spectrum of $A$ be denser than that of $S$. The following heuristic argument
may help to justify this requirement: to measure the spatial scale of an
object, the ruler should have a much finer graduation than the size of this
object.
To be precise, let $H_{A}=\sum_{k}\epsilon_{k}\left|k\right\rangle\left\langle
k\right|$ be the spectrum decomposition of the Hamiltonian of $A$ and
$H_{S}=\sum_{n}E_{n}\left|n\right\rangle\left\langle n\right|$ the spectrum
decomposition of the Hamiltonian of $S$, the requirement then can be expressed
as $\min\\{|E_{n}-E_{n+M}|\\}>>\max\\{|\epsilon_{k}-\epsilon_{k+1}|\\}$. Here,
$\left|k\right\rangle$ is the eigenvector of $H_{A}$ corresponding to the
eigenvalue $\epsilon_{k}$, and $\left|n\right\rangle$ is the eigenvector of
$H_{S}$ corresponding to the eigenvalue $E_{n}$ Let $V_{AS}$ be a weak
coupling between $S$ and $A$. The weakness of $V_{AS}$ means that its effect
on the dynamics of the total system $S+A$ can be well studied by the
perturbation method frohlich .
To investigate the behavior of the total system $S+A$ immersed in an
environment, we consider the partition function
$Z=\mathrm{Tr}\left(e^{-\beta H}\right)=\mathrm{Tr}\left(W^{\dagger}e^{-\beta
H}W\right)$
Here, as a trick, we have introduced a unitary transformation
$W=\exp\left(-S\right)$, defined by an anti-Hermitian operator $S$, which is a
perturbation quantity of the same order as $V_{AS}$. If
$V_{AS}+\left[H_{A}+H_{S},S\right]=0,$
then the partition function can be approximated as
$Z=\mathrm{Tr}\exp\left(-\beta H_{\mathrm{eff}}\right)$ where the effective
Hamiltonian
$H_{\mathrm{eff}}=H_{A}+H_{S}+\frac{1}{2}\left[V_{AS},S\right],$ (1)
is just the Frohlich-Nakajima Hamiltonian in solid state physics frohlich .
Here, we re-derive it to justify its applicability in thermodynamics.
Since the minimal energy level spacing of the system is much larger than the
energy level spacing of the apparatus, the effective interaction
$V_{\mathrm{eff}}$ can be obtained as
$V_{\mathrm{eff}}=\sum_{n}\mathcal{H}\left(n\right)\left|n\right\rangle\left\langle
n\right|$ where
$\mathcal{H}\left(n\right)=\sum_{k}G(n)\left|k\right\rangle\left\langle
k^{\prime}\right|$ (2)
is a branched effective Hamiltonian of the apparatus corresponding to the
situation that the system is prepared in the state $\left|n\right\rangle$ with
the coupling
$G(n)=\frac{\left\langle
nk\right|V_{AS}\left|nk^{\prime}\right\rangle}{\left(\epsilon_{k}-\epsilon_{k^{\prime}}\right)}.$
(3)
The above obtained effective interaction $V_{\mathrm{eff}}$ satisfies
$\left[H_{S},V_{\mathrm{eff}}\right]=0$. Thus the total effective Hamiltonian
$H_{\mathrm{eff}}=H_{S}+H_{A}+V_{\mathrm{eff}}$ describes a non-demolition
measurement without the presence of an environment. In this case, the
factorized initial state
$\left|\varphi\left(0\right)\right\rangle=\sum_{n}c_{n}\left|n\right\rangle\otimes\left|D\right\rangle$
will evolve into
$\left|\varphi\left(t\right)\right\rangle=\sum_{n}c_{n}\left|n\right\rangle\otimes\left|D_{n}\left(t\right)\right\rangle$
where $\left|D_{n}\left(t\right)\right\rangle$ takes the form
$\left|D_{n}\left(t\right)\right\rangle=\exp\left[-iG\left(n\right)t\right]\left|D\right\rangle.$
If $\left\\{\left|D_{n}\left(t\right)\right\rangle\right\\}$ becomes an
orthogonal set when $t$ approaches infinity, then the time evolution
$\left|\varphi\left(t\right)\right\rangle$ represents a process of ideal pre-
measurement.
Thermodynamic effects of measurements\- Now we study the change in the
thermodynamic features of the system caused by the apparatus. We assume that
both the probed system and the measuring apparatus are immersed in the same
thermal bath with temperature $T$ or inverse temperature $\beta=1/k_{B}T$.
After or during the measuring process, the total system $S+A$ will reach the
state with the same temperature $T$, if the total system is non-degenerate.
Then we can calculate the reduced density matrix
$\rho_{S}=\mathrm{Tr_{A}}\left[\exp\left(-\beta\left(H_{S}+H_{A}+V_{\mathrm{eff}}\right)\right)\right]$,
obtaining
$\rho_{S}=\frac{1}{Z_{S}^{\prime}}\sum_{n}e^{-\beta
E_{n}}\xi\left(n\right)\left|n\right\rangle\left\langle n\right|,$ (4)
where $Z_{S}^{\prime}=\sum_{n}\exp\left(-\beta E_{n}\right)\xi\left(n\right)$
and
$\xi\left(n\right)=\mathrm{Tr_{A}}\left\\{\left\langle
n\right|\exp\left[-\beta\left(H_{A}+V_{\mathrm{eff}}\right)\right]\left|n\right\rangle\right\\}$
(5)
is a formal factor depending on the system state $\left|n\right\rangle$ and
vanishing trivially when no coupling exists.
Here, we consider a manipulation process. (i) Initially, no probing apparatus
is coupled to the system, which is in equilibrium with the heat bath with
temperature $T$. (ii) Then, the apparatus begin to probe the system at time
$t=0$. As the evolution described in Fig. 2, the total system $S+A$ reaches
the state with the same temperature $T$. We observe that the thermodynamic
effect of measurement implied in Eq. (6) allows two interpretations. These two
interpretations are illustrated in Fig. 2 as those in Ref. q1 ; quanPRE2007 .
Figure 2: Two interpretations. (i) Isometric process ($A\rightarrow C$, the
solid line): the inverse temperature $\beta$ is fixed in the process. (ii)
Isothermal process ($A\rightarrow B$, the dotted line): The level spacing
$\Delta$ is fixed in the process.
The first interpretation goes as follows. The change from the close thermal
state of the system $\rho_{Sc}=\exp\left(-\beta H_{S}\right)/Z_{S}$ to the
modified thermal state $\rho_{S}$ is understood as a quantum isometric
process, represented by the solid line between the points $A$ and $C$ in Fig.
2. In this process, the temperature keeps unchanged, but the energy level
spacings are alternated by the interaction with the apparatus. Accordingly, we
define $\xi\left(n\right)=\exp\left(-\beta\Delta E_{n}\right)$, then
$\rho_{S}$ can be written in the form with clear physical meaning:
$\rho_{S}=\frac{1}{Z^{\prime}_{S}}\sum_{n}e^{-\beta\left(E_{n}+\Delta
E_{n}\right)}\left|n\right\rangle\left\langle n\right|.$ (6)
In the present case, it is rather natural to regard the inner energy change
$\Delta
U=\mathrm{Tr_{S}}\left[H_{\mathrm{S}}\left(\rho_{S}-\rho_{Sc}\right)\right]$
as a witness of the thermodynamic role of measurement.
In the second interpretation the change from $\rho_{Sc}$ to $\rho_{S}$ is
understood as a quantum isothermal process, represented by the dotted line
between the points $A$ and $B$ in Fig. 2. In this process, the energy level
spacings are fixed. To justify our using the term “isothermal process” here,
at least to some extent, we define the effective temperature efftem via
$\beta\left(n\right)=\ln(P_{n}/P_{n+1})/\Delta_{n}$ or
$\beta\left(n\right)=\beta+\frac{1}{\Delta_{n}}\ln\frac{\xi\left(n\right)}{\xi\left(n+1\right)},$
(7)
where $\Delta_{n}=E_{n+1}-E_{n}$ is the $n-th$ energy level spacing. We notice
that generally this generalized temperature cannot be regarded as an effective
temperature since it depends on the energy levels of the system. But if
$\beta\left(n+1\right)-\beta\left(n\right)=0,$ i.e.,
$\left[\frac{\xi\left(n+1\right)}{\xi\left(n+2\right)}\right]^{\Delta_{n}}=\left[\frac{\xi\left(n\right)}{\xi\left(n+1\right)}\right]^{\Delta_{n+1}},$
(8)
then $\beta\left(n\right)$ becomes a well defined thermodynamic parameter
independent of $n$, which we denote by $\beta_{\mathrm{eff}}$. In this case,
the above defined generalized temperature allows the physical interpretation
of effective temperature, and we have
$\rho_{S}=\sum_{n}\exp\left(-\beta_{\mathrm{eff}}E_{n}\right)\left|n\right\rangle\left\langle
n\right|/Z_{S}^{\prime}$, and the inner energy change
$\Delta U=\sum_{n}E_{n}\left[e^{-\beta_{\mathrm{eff}}E_{n}}-e^{-\beta
E_{n}}\right]$ (9)
reflects the thermalization effect.
The quantum role of measurement and renormalization\- As an explicit
illustration, we model the apparatus with weak couplings to the probed system
as a collection of harmonic oscillators. According to the results of Ref.
leggett and Ref. suncp98pra (where he arguments are carried out for the bath
modeling, but can work well for our setup for the apparatus), the coupling of
the system to the apparatus is linear with respect to the coordinates of the
bath harmonic oscillators. Let $b_{j}^{\dagger}\left(b_{j}\right)$ be the
creation(annihilation) operator of the $j-th$ mode of the bath with eigen-
frequency $\omega_{j}$, and $\lambda_{n}g_{j}$ the coupling coefficient of the
system state $\left|n\right\rangle$ to the $j-th$ mode. Then, the total
Hamiltonian is obtained as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}\left|n\right\rangle\left\langle
n\right|+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$ (10)
$\displaystyle+\sum_{n}\lambda_{n}\left|n\right\rangle\left\langle
n\right|\sum_{k}\left(g_{k}a_{k}^{\dagger}+\mathrm{h.c}\right).$
In this case, the above defined generalized inverse temperature reads
$\beta\left(n\right)=\beta\left(1-\frac{|\lambda_{n+1}|^{2}-|\lambda_{n}|^{2}}{E_{n+1}-E_{n}}\varepsilon\right),$
(11)
where
$\varepsilon=\sum_{k}\frac{|g_{k}|^{2}}{\omega_{k}}=\int\rho\left(\omega\right)\frac{|g_{k}|^{2}}{\omega_{k}}dk$
(12)
represents the self-energy of the apparatus, which causes the Lamb shift of
its coupled system. As pointed above, to define reasonably an effective
temperature for the system, the generalized inverse temperature
$\beta\left(n\right)$ should be independent of the energy level. Here is a
simple example satisfying this condition: the system is a harmonic oscillator
with the energy level $E_{n}=\left(n+1/2\right)\omega$ and the coupling
strength $\lambda_{n}=\sqrt{n}$. In this example, the well defined effective
inverse temperature is
$\beta_{\mathrm{eff}}=\beta(1-\sum_{k}\frac{|g_{k}|^{2}}{\omega\omega_{k}}),$
(13)
and the corresponding effective temperature
$T_{\mathrm{eff}}=\left(k_{B}\beta_{\mathrm{eff}}\right)^{-1}$ of the system
is higher than that of the environment $T=\left(k_{B}\beta\right)^{-1}$.
For a two-level system with the excited state $\left|e\right\rangle$ and the
ground state $\left|g\right\rangle$ and the energy level spacing $\Delta$, the
effective inverse temperature is also well defined. It reads
$\beta_{\mathrm{eff}}=\beta+\frac{1}{\Delta}\ln\frac{\xi_{g}}{\xi_{e}}.$ (14)
If the apparatus is a single mode cavity with frequency $\omega_{b}$ and the
two-level system is coupled to it by the dipole interaction with coupling
strength $g$, then in the large detuning case $\omega_{b}\gg\Delta$, the
formal factor $\xi_{e}$ and $\xi_{g}$ can be explicitly calculated as follows:
$\displaystyle\xi_{e}$ $\displaystyle=$
$\displaystyle\sum_{n}\exp\left\\{-\beta\left[\omega_{b}n+\frac{\left|g\right|^{2}}{\omega_{b}-\Delta}\left(n+1\right)\right]\right\\},$
$\displaystyle\xi_{g}$ $\displaystyle=$
$\displaystyle\sum_{n}\exp\left\\{-\beta\left[\omega_{b}n+\frac{\left|g\right|^{2}}{\omega_{b}-\Delta}n\right]\right\\}.$
(15)
Since $\xi_{g}>\xi_{e}$, we are led to the conclusion that the measurement
will decrease the temperature of the system by
$\Delta
T=\frac{\frac{1}{\Delta}\ln\frac{\xi_{g}}{\xi_{e}}}{\beta+\frac{1}{\Delta}\ln\frac{\xi_{g}}{\xi_{e}}.}T.$
(16)
Figure 3: Fidelity. (a) Fidelity $\mathcal{F}$ vs the level shift $\lambda$.
(b)Fidelity $\mathcal{F}$ vs the temperature shift $\Delta T$. Here, we take
$\omega=1$ and $\beta=1$.
Finally, to quantitatively evaluate the thermalized state resulting from the
interaction with the environment, let us check the fidelity fed $\mathcal{F}$
of such a state to the ordinary canonical state
$\rho_{\mathrm{can}}=\exp\left(-\beta
H_{S}\right)/\mathrm{Tr}\left[\exp\left(-\beta H_{S}\right)\right]$. Generally
for the reduced density matrix in Eq. 4, the fidelity between the initial
state and the final state reads
$\mathcal{F}=\frac{\sum_{n}e^{-\beta
E_{n}}\sqrt{\xi\left(n\right)}}{\left(\sum_{n}e^{-\beta
E_{n}}\right)^{1/2}\left(\sum_{n}e^{-\beta
E_{n}}\xi\left(n\right)\right)^{1/2}}.$ (17)
When the probed system is also a harmonic oscillator discussed above, the
fidelity can be analytically obtained as
$\mathcal{F}=\frac{\sqrt{\sinh\frac{\beta\omega}{2}\sinh\frac{\beta\left(\omega-\lambda\right)}{2}}}{\sinh\frac{\omega-\lambda/2}{2}},$
(18)
where $\lambda=\sum_{k}|g_{k}|^{2}/\omega_{k}$ characterizes the shift of the
energy level of the harmonic oscillator, which reflects the effect due to the
coupling to the apparatus. In Fig. 3(a), the fidelity is plotted against the
level shift. As the coupling between the apparatus and the system is turn on,
the energy level for the system is effective
$E^{\prime}_{n}=n(\omega-\lambda)$ based on the first interpreting of the
reduced density matrix. The fidelity between the canonical thermal and the
reduced density matrix decreases as the coupling becomes strong. Therefore,
the system gradually deviates from canonical thermal state, leaving an
evidence of witness of the apparatus. For the second interpreting, we plot the
fidelity $\mathcal{F}$ as the function of the temperature shift $\Delta
T=1/\beta_{\mathrm{eff}}-1/\beta=\left[\beta(\omega/\lambda-1)\right]^{-1}$ in
Fig. 3(b).
Conclusion\- In summary, for the weak coupling case, we show from the
generalized approach of Frohlich-Nakajima transformation the universality of a
non-demolition Hamiltonian in connection with the probing process. Based on
this general non-demolition measurement, we investigate the probing effect on
the system when the total measurement happens in a reservoir. It is concluded
that the probing of the system can be witted through the change of the
effective temperature, even though there is no direct energy exchange between
the system and the detector. To characterize the change of state for the
system, we evaluate the fidelity of the modified canonical thermal state to
the original canonical state without being probed.
The work is supported by National Natural Science Foundation of China and the
National Fundamental Research Programs of China under Grant No. 10874091 and
No. 2006CB921205.
## References
* (1) R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
* (2) H.S. Leff, & A.F. Rex(eds), Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing. Institute of Physics. (2003).
* (3) W. H. Zurek, Rev. Mod. Phys. 75,715 (2003).
* (4) V. B. Braginsky and F. Y. Khalili, Quantum Measurement, Camebridge University Press, (1992).
* (5) C.P. Sun, Phys. Rev. A 48, 898 (1993).
* (6) H. Dong, S. Yang, X. F. Liu, and C. P. Sun, Phys. Rev. A 76, 044104 (2007); H. T. Quan, P. Zhang, and C. P. Sun, Phys. Rev. E 73, 036122 (2006).
* (7) H. Fröhlich, Phys. Rev. 79, 845(1950); S. Nakajima, Adv. Phys. 4, 363(1953).
* (8) H. T. Quan, P. Zhang, and C. P. Sun, Phys. Rev. E 72, 056110 (2005)
* (9) H.T. Quan, Yu-xi Liu, C. P. Sun and F. Nori, Phys. Rev. E 76, 031105 (2007); H.T. Quan, Phys. Rev. E 79, 041129 (2009)
* (10) A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); A. J. Leggett, S. Chakravarty, A. T. Dosey, M. P. A. Fisher, and W. Zwerger, Rev. Mod. Phys. 59, 187 (1987).
* (11) C.P. Sun, H. Zhan and X.F. Liu, Phys.Rev. A 58, 1810 (1998).
* (12) P. Zanardi, H. T. Quan, X. Wang, C. P. Sun Phys. Rev. A 75, 032109 (2007) and references therein.
|
arxiv-papers
| 2009-08-27T12:19:03 |
2024-09-04T02:49:04.869176
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H.Dong, X.F. Liu and C.P. Sun",
"submitter": "H. Dong",
"url": "https://arxiv.org/abs/0908.3997"
}
|
0908.3999
|
arxiv-papers
| 2009-08-27T12:30:49 |
2024-09-04T02:49:04.874343
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ping Zhu",
"submitter": "Ping Zhu",
"url": "https://arxiv.org/abs/0908.3999"
}
|
|
0908.4059
|
# Compactification projective de $\mathrm{Spec\ }\mathbb{Z}$
(d’après Durov)
Javier Fresán111Il s’agit d’une version très préliminaire d’un survey sur la
thèse de Durov. Tout commentaire est bienvenu au courriel de l’auteur:
javierfresan@gmail.com.
###### Contents
1. 1 Introduction
2. 2 L’analogie entre les corps de nombres et les corps de fonctions
3. 3 La catégorie des anneaux généralisés
1. 3.1 La monade associée à un anneau
2. 3.2 Les monades algébriques
3. 3.3 Présentation d’une monade algébrique
4. 3.4 L’additivité
5. 3.5 Les monades commutatives
4. 4 Exemples
1. 4.1 $\mathbb{N}$
2. 4.2 $\mathbb{Z}_{\infty}$
3. 4.3 Anneaux de valuation
4. 4.4 $\mathbb{F}_{1}:$ le corps à un élément
5. 4.5 Extensions cyclotomiques
6. 4.6 $\mathbb{Z}$ admet une présentation finie sur $\mathbb{F}_{1}$
7. 4.7 Les anneaux généralisés $A_{N}$ et $B_{N}.$
5. 5 Vers la compactification de $\mathrm{Spec\ }\mathbb{Z}$
1. 5.1 Schémas généralisés
2. 5.2 Compactification à la main
3. 5.3 Description en termes de la construction $\mathrm{Proj}$
4. 5.4 Compactification d’Arakelov des variétés et fibrés en droite
6. 6 Questions ouvertes
7. A Construction fonctorielle des schémas sur $\mathbb{F}_{1}$
8. B Annexe: quelques outils catégoriques
## 1 Introduction
Parmi les nombreux outils mathematiques developpés par Gauss, certains ont eu
un destin tout particulier. C’est le cas des $q$-entiers, introduits par lui
afin d’évaluer ce que l’on appelle aujourd’hui les sommes quadratiques
gaussiennes. Ils servent, par exemple, à formuler des analogues non
commutatifs du théorème du binôme et ils jouent un rôle proéminent dans la
mécanique quantique. Mais c’est leur apparition dans le cadre des géométries
finies qui a ouvert une voie de recherche tout à fait inattendue. En effet,
lorsque $q$ est une puissance du nombre premier $p,$ le cardinal des espaces
projectifs sur le corps à $q$ éléments n’est autre que
$\lvert\mathbb{P}^{n-1}(\mathbb{F}_{q})\rvert=\frac{\lvert\mathbb{A}^{n}(\mathbb{F}_{q})-\\{0\\}\rvert}{\lvert\mathbb{G}_{m}(\mathbb{F}_{q})\rvert}=\frac{q^{n}-1}{q-1}=[n]_{q},$
et l’on obtient aussi une formule pour le cardinal de la Grassmanienne:
$\lvert\mathrm{Gr}(n,j)(\mathbb{F}_{q})\rvert=\lvert\\{\mathbb{P}^{j}(\mathbb{F}_{q})\subset\mathbb{P}^{n}(\mathbb{F}_{q})\\}\rvert={n\choose
j}_{q}=\frac{[n]_{q}!}{[j]_{q}![n-j]q!}.$
La remarque que les expressions ci-dessus ont encore un sens lorsque $q=1$ a
conduit Tits en 1957 à postuler l’existence d’un objet algébrique, le corps de
caractéristique un, sur lequel on pourrait écrire formellement
$\lvert\mathbb{P}^{n-1}(\mathbb{F}_{1})\rvert=[n]_{1}=n,\quad\lvert\mathrm{Gr}(n,j)(\mathbb{F}_{1})\rvert={n\choose
j}_{1}={n\choose j}.$
Ainsi, $\mathbb{P}^{n-1}(\mathbb{F}_{1})$ serait juste un ensemble fini $P$ à
$n$ éléments, et $\mathrm{Gr}(n,j)(\mathbb{F}_{1}),$ l’ensemble des parties de
$P$ ayant cardinal $j.$ En niant l’axiome d’après lequel une droite projective
possède au moins trois points, on obtient des espaces où la géométrie devient
combinatoire. Notamment, on peut imaginer $\mathrm{GL}_{n}(\mathbb{F}_{1})$
comme le group symétrique $\mathcal{S}_{n}$ et
$\mathrm{SL}_{n}(\mathbb{F}_{1})$ comme étant formé par les permutations
paires. Le programme de Tits consiste alors à interpréter les groupes de Weyl
comme des groupes de Chevalley sur le corps à un élément [Ti].
Il a fallu attendre jusqu’en 1994 pour la suite des événements. Dans ses
conférences sur la fonction zêta et les motifs [Ma1], largement inspirées des
travaux de Deninger et Kurokawa, Manin se réfère à $\mathrm{Spec\
}\mathbb{F}_{1}$ comme le point absolu dont on aurait besoin pour reproduire
la preuve de Weil de l’hypothèse de Riemann pour les courbes sur un corps
fini. Les fonctions zêta des variétés sur $\mathbb{F}_{1}$ auraient la forme
la plus simple, comme celle-ci:
$\zeta_{\mathbb{P}^{N}(\mathbb{F}_{1})}(s)=s(s-1)\ldots(s-N).$ En essayant de
donner un sens au produit infini régularisé
$\frac{(2\pi)^{s}}{\Gamma(s)}=\prod_{n\geq 0}\frac{s+n}{2\pi},$
Manin a conjecturé l’existence d’une catégorie de motifs sur $\mathbb{F}_{1}$
et encouragé les mathématiciens à développer une géométrie algébrique sur le
corps à un élément.
Même si cette suggestion a pris forme dans de très diverses notions de schémas
sur $\mathbb{F}_{1},$ il y a quelques traits communs. D’abord, on est tous
d’accord sur le fait que la catégorie des anneaux commutatifs est très
restrictive pour accueillir les nouveaux êtres, une limitation qui a été aussi
mise en évidence, pour des propos différents, dans le contexte de la géométrie
analytique globale [Pa]. Autrement dit, $\mathrm{Spec\ }\mathbb{Z}$ est trop
compliqué comme objet final de la catégorie des schémas affines. Ainsi, Durov
[Du] et Shai-Haran [Ha] se proposent de l’élargir jusqu’à avoir $\mathrm{Spec\
}\mathbb{F}_{1}$, ou même un spectre encore plus simple, comme objet final. La
tendance générale, c’est d’oublir la somme afin d’obtenir une géométrie non-
additive. C’est le point de vue de Deitmar, qui imite la théorie des schémas
juste en remplacent les anneaux par les monoïdes commutatifs. Dans son
approche, un sous-monoïde $P$ de $M$ es premier si $xy\in P$ implique $x\in P$
ou $y\in P.$ Alors, un schéma affine est l’ensemble $\mathrm{Spec\ }M$ des
sous-monoïdes premiers de $M$ muni d’un faisceau de monoïdes localisés (voir
[De1] pour les détails de cette construction).
Dans leur très beau article Au dessous de $\mathrm{Spec\ }\mathbb{Z}$ [TV],
Töen et Vaquié ont développé la géométrie algébrique sur $\mathbb{F}_{1}$
comme un cas particulier de géométrie algébrique relative à une catégorie
monoïdale symétrique vérifiant un certain nombre de propriétés techniques sur
l’existence de limites. Si $(C,\times,\textbf{1})$ est une telle catégorie, on
définit $\mathrm{Comm}(C)$ comme la catégorie de monoïdes commutatifs à
l’intérieur de $C,$ ce qui va remplacer les anneaux commutatifs classiques.
Alors, en posant $\mathrm{Aff}_{C}:=\mathrm{Comm}(C)^{\text{op}},$ avec
$\mathrm{Spec}$ le foncteur d’une catégorie vers la catégorie duale, on
obtient des schémas affines ayant une topologie de Grothendieck que l’on peut
recoller pour avoir des schémas généraux. Les schémas sur $\mathbb{F}_{1}$
sont les schémas relatifs à la catégorie monoïdale symétrique
$(\mathfrak{Ens},\times,\ast),$ où $\times$ est le produit cartésien
d’ensembles. Lorsque l’on fait un certain relèvement, on obtient des schémas
classiques, toujours une façon de tester que l’on manipule de bons candidats.
Une approche plus terre à terre, mais qui permet d’étudier les fonctions zêta,
est celle de Soulé [So2] raffinée ensuite dans les travaux de Connes et
Consani [CC, CC2]. De leur point de vue, un schéma sur $\mathbb{F}_{1}$ est
défini par quelques données de descente d’un $\mathbb{Z}-$schéma de type fini
$X_{\mathbb{Z}}.$ Une variété affine sur $\mathbb{F}_{1}$ est un triplet
$(\underline{X},X_{\mathbb{C}},e_{X})$ formé d’un foncteur covariant
$\underline{X}=\amalg_{k\geq
0}\underline{X}^{k}:\mathcal{F}_{ab}\longrightarrow\mathfrak{Ens}$ de la
catégorie des groupes abéliens finis vers la catégorie des ensembles gradués,
d’une variété affine $X_{\mathbb{C}}$ et d’une transformation naturelle entre
$\underline{X}$ et le foncteur $D\mapsto\mathrm{Hom}(\mathrm{Spec\
}\mathbb{C}[D],X_{\mathbb{C}}).$ À travers d’un caractère $\chi$ qui permet
d’interpréter les éléments de $\underline{X}(D)$ comme des points de
$X_{\mathbb{C}},$ ils requièrent quelques hypothèses fortes garantissant que
ces donnés définissent une seule variété affine sur $\mathbb{Z}.$ Très
récemment, James Borger a proposée les $\lambda-$anneaux comme la structure
codifiant ces données de descente de $\mathbb{Z}$ vers $\mathbb{F}_{1}$ [Bo].
Dans ce mémoire, on aborde les constructions de Durov dans sa thèse [Du], dans
laquelle il a développé tout un cadre algébrique permettant de surmonter les
difficultés qui se présentent dès que l’on essaie d’appliquer la théorie
d’Arakelov aux variétés non lisses, non propres ou ayant des métriques
singulières. Outre qu’obtenir une nouvelle description de la géométrie
d’Arakelov, qui pourrait servir à démontrer des analogues arithmétiques du
théorème de Riemann-Roch, on propose un cadre suffisamment général pour
traiter de la même manière la géométrie algébrique à la Grothendieck, la
géométrie tropicale ou la géométrie sur le corps à un élément, juste en
faisant varier l’anneau généralisé de base sur lequel on travaille. Cela
permet notamment de définir d’une façon rigoureuse $\mathbb{F}_{1}$ et ses
extensions, non seulement les structures sur eux, ainsi que de construire la
compactification de $\mathrm{Spec\ }\mathbb{Z}$ comme un pro-schéma généralisé
projectif sur $\mathbb{F}_{1}.$
### Plan du mémoire
On commence par exposer quelques limitations de l’analogie entre les corps de
nombres et le corps de fonctions provenant du fait que $\mathrm{Spec\
}\mathbb{Z}$ ne soit pas un schéma propre et qu’il n’y ait pas un corps de
base chez les entiers. Dans l’exposé, on rencontre trois objets à la recherche
d’une définition satisfaisante: le corps à un élément $\mathbb{F}_{1},$
l’anneau local à l’infini $\mathbb{Z}_{\infty}$ et la compactification de
${\mathrm{Spec\ }\mathbb{Z}}.$ C’est la tâche à laquelle on se consacre dans
la section suivante, où l’on voit un anneau $R$ comme la monade sur les
ensembles qui fait correspondre à chaque $X$ toutes les $R-$combinaisons
linéaires formelles à support fini d’éléments dans $X.$ On extrait deux
propriétés importantes de ce type de monades: l’algébricité, qui garantit
essentiellement que l’on peut trouver une présentation par des opérations et
des relations, et la commutativité, une des nouveautés introduites par Durov.
On dit qu’une monade algébrique commutative est un anneau généralisé. Ensuite,
on peut définir le spectre d’un anneau généralisé en calquant le cas
classique. Dans la section suivante, on construit de nombreux exemples
d’anneaux généralisés, parmi lesquels il faut souligner $\mathbb{N},$
$\mathbb{Z}_{\infty},$ $\mathbb{Z}_{(\infty)},$ ainsi que le corps à un
élément $\mathbb{F}_{1}$ et ses extensions $\mathbb{F}_{1^{n}.}$ On obtient
dans chaque cas des descriptions de la catégorie des modules sur ces anneaux.
La seconde partie du mémoire présente la construction de
$\widehat{\mathrm{Spec\ }\mathbb{Z}}.$ On définit d’abord les schémas de
Zariski sur $\mathbb{F}_{1}$ en suivant le point de vue fonctoriel
grothendieckien, d’après lequel un schéma sur $\mathbb{F}_{1}$ est grosso modo
un foncteur de la catégorie des anneaux généralisés vers la catégorie des
ensembles qui admet un recouvrement par des ouverts affines. Ensuite, on
compactifie $\mathrm{Spec\ }\mathbb{Z}$ à la main à partir de deux anneaux
généralisés $A_{N}$ et $B_{N},$ et l’on obtient une description plus
intrinsèque en termes de la construction $\mathrm{Proj}$ des monades graduées.
Finalement, on fait le lien avec la géométrie d’Arakelov classique en
discutant la compactification des variétés algébriques définies sur
$\mathbb{Q}$ et le concept de fibré vectoriel sur $\widehat{\mathrm{Spec\
}\mathbb{Z}}$. On finit avec une discussion brève de quelques questions
ouvertes. Puisque tout au long du travail il y a un usage intensif de la
théorie des catégories, on a considéré opportun d’ajouter un annexe où l’on
explique d’une façon concise les notions employés, sauf celles de catégorie,
foncteur et transformation naturelle, que l’on suppose connues. Les termes
définis dans l’annexe sont indiqués avec une petite étoile en haut. On
présente de même une bibliographie exhaustive des articles concernant
$\mathbb{F}_{1}.$
###### Remerciements.
Je tiens à remercier Frédéric Paugam, princeps functorum, pour m’avoir proposé
ce travail et pour les nombreuses heures qu’il a passé avec moi au tableau.
J’ai discuté quelques idées du texte avec Javier López Peña (Queen Mary
University of London), auquel je remercie l’envoie du préprint [LL], et avec
Peter Arndt (Göttingen), lors de la conférence i-Math School on Derived
Algebraic Geometry (Salamanca, juin 2009) où l’on s’est rencontrés par hasard.
J’ai aussi bénéficié du cours d’Alain Connes au Collège de France sur la
Thermodynamique des espaces non commutatifs, où le contenu de [CC2] a été
présenté pour la première fois, et d’une conférence de Yuri Manin à l’Institut
de Mathématiques de Jussieu [Ma3], dont il m’a envoyé les transparences très
gentiment. Une partie de ce travail a fait objet de la conférence ¿Es
$\mathbb{Z}$ un anillo de polinomios? au Séminaire de Géométrie Algébrique de
l’Universidad Complutense de Madrid (séance du 26 mai) et de l’exposé court On
Durov’s compactification of $\mathrm{Spec\ }\mathbb{Z}$, the field with one
element, the local ring at infinity and other rara avis dans l’école doctorale
en Géométrie Diophantienne tenue à l’Université de Rennes du 15 au 26 juin
2009.
## 2 L’analogie entre les corps de nombres et les corps de fonctions
Depuis les travaux constitutifs de plusieurs visionnaires qui ont rêvé d’une
correspondance parfaite entre la géométrie et l’arithmétique (Kummer,
Kronecker, Dedekind, Hensel, Hasse, Minkowski, Artin, Weil, etc.), l’analogie
entre les corps de nombres et les corps de fonctions est l’une des idées les
plus fructueuses de la mathématique moderne. Typiquement, des énoncés simples
mais inaccessibles sur les corps de nombres (par exemple, la distribution des
zéros des fonctions L ou les conjectures de Langlands) sont transformés en des
théorèmes plus techniques dont la démonstration est possible grâce au nouveau
cadre géométrique-analytique de la traduction dans les corps de fonctions
(dans ce cas, les conjectures de Weil ou la correspondance de Langlands pour
$GL_{n}$ établie par Lafforgue). Un exemple significatif, c’est le théorème de
Mason, un exercice facile mettant en relation le degré de trois polynômes avec
celui du radical de leur produit, dont l’analogue arithmétique, la conjecture
abc, a des conséquences qui vont du dernier théorème de Fermat en forme
asymptotique et de quelques résultats d’infinitude des nombres premiers
jusqu’au théorème de Mordell-Faltings ou la conjecture de Lang [Ni].
On rappel qu’un corps de nombres est une extension finie de $\mathbb{Q}$ et
qu’un corps de fonctions est une extension finie $K$ de degré de transcendance
1 sur un corps $k,$ que l’on supposera en général fini ou algébriquement clos.
Soit $K/k$ un corps de fonctions. À chaque valuation discrète de $K/k$ (i.e.
une application $v:K\longrightarrow\mathbb{Z}\cup\\{\infty\\}$ telle que
$v(xy)=v(x)+v(y),$ $v(x+y)\geq\min\\{v(x),v(y)\\}$ et que $v(x)=0$ quel que
soit $x\in k),$ on associe l’anneau local
$\mathcal{O}_{v}=\\{x\in K:v(x)\geq 0\\}.$
Alors, l’ensemble $C_{k}$ d’anneaux de valuation discrète de $K/k$ est une
courbe algébrique abstraite isomorphe à une courbe algébrique projective lisse
$C$ ayant corps de fonctions $k(C)=K.$ Soit $\rho$ un nombre réel plus grand
que 1. Étant donné une valuation discrète $v_{p},$ si l’on pose
$\lvert\cdot\rvert_{p}=\rho^{-v_{p}(\cdot)},$ on obtient une norme sur $K.$ On
a alors une bijection entre les points fermés de $C$ et les normes sur $K$
triviales sur $k$ à équivalence près (cf. [Har, I.6] pour les détails).
L’analogie entre les corps de nombres et les corps de fonctions devient plus
forte lorsque l’on considère les entiers $\mathbb{Z}$ et les polynômes $k[T]$
sur un corps fini $k=\mathbb{F}_{q},$ ainsi que leurs corps de fractions
$\mathbb{Q}$ et $k(T).$ Ce sont des domaines euclidiens de dimension de Krull
1, dont les spectres sont formés par
$\displaystyle\mathrm{Spec\ }\mathbb{Z}=\\{p\mathbb{Z}:\text{p
premier}\\}\cup\\{0\\},$ $\displaystyle\mathbb{A}^{1}_{k}:=\mathrm{Spec\ }$
$\displaystyle k[T]=\\{f\in k[T]\text{\ unitaire irr\'{e}ductible}\\}.$
De plus, si $k=\mathbb{F}_{q},$ les corps résiduels
$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ et $\mathbb{F}_{q}[T]/(f)$ sont finis.
D’autre part, si $f\in\mathbb{A}^{1}_{k},$ l’anneau des séries de puissances
$Z_{f}:=\varprojlim k[T]/(f^{n})$ et le corps de séries de Laurent
$Q_{f}:=Z_{f}[\frac{1}{f}]$ ont des équivalents $p-$adiques
$\mathbb{Z}_{p}=\varprojlim\mathbb{Z}/p^{n}\mathbb{Z}$ et
$\mathbb{Q}_{p}=\mathbb{Z}_{p}[\frac{1}{p}]$ ayant des plongements denses
$\mathbb{Z}\subset\mathbb{Z}_{p}$ et $\mathbb{Q}\subset\mathbb{Q}_{p}.$ On
peut même regarder les nombres comme fonctions sur $\mathrm{Spec\
}\mathbb{Z},$ où l’image du premier $p$ par l’entier $n$ est la classe de
congruence de $n$ modulo $p.$ Notons que l’espace d’arrivée de cette fonction
dépend essentiellement du point où elle est évaluée.
Quelles sont les limites de cette analogie? Le premier problème qui se pose,
c’est l’existence d’un équivalent de la droite projective
$\mathbb{P}^{1}_{k}:=\mathbb{A}^{1}_{k}\amalg_{\mathbb{G}_{m}}\mathbb{A}^{1}_{k}$
sur un corps de nombres. En effet, pour avoir la correspondance citée ci-
dessus dans le cas plus simple $k(T)=k(\mathbb{P}^{1}_{k}),$ c’est nécessaire
de passer au projective car l’on la norme
$\lvert\frac{f(T)}{g(T)}\rvert=\rho^{\deg f-\deg g}$
correspondant au point de l’infini de $\mathbb{P}^{1}_{k}.$ Cela nous permet
notamment d’avoir des formules produit comme
$\prod_{p\in\mathbb{P}^{1}_{k},\ p\neq\xi}\lvert f\rvert_{p}=1\quad\forall
f\in K^{\ast},$
où $\xi$ est le point générique de $\mathbb{P}^{1}_{k}.$ Ces formules sont
extrêmement importantes en géométrie diophantienne, où elles permettent, par
exemple, de définir la hauteur d’un point d’un espace projectif en choisissant
un représentant quelconque [BG].
En ce qui concerne les rationnels, les normes sur $\mathbb{Q}$ sont
classifiées par le théorème d’Ostrowski, d’après lequel toute norme non-
triviale est équivalente soit à une norme $p$-adique $\lvert\cdot\rvert_{p}$
soit à la valeur absolue archimédienne usuelle. On rappel que $\lvert
x\rvert_{p}=p^{-v_{p}(x)},$ $v_{p}(x)$ étant le seul entier tel que
$x=p^{v_{p}(x)}\frac{a}{b}$ avec $\mathrm{pgcd}(p,ab)=1,$ vérifie la propriété
ultramétrique
$\lvert x+y\rvert_{p}\leq\max\\{\lvert x\rvert_{p},\lvert y\rvert_{p}\\}.$
On a de nouveau
$\prod_{(p)\in\mathrm{Spec\ }\mathbb{Z},\ p\neq 0}\lvert
x\rvert_{p}=\frac{1}{\lvert x\rvert_{\infty}}\quad\forall
x\in\mathbb{Q}^{\ast},$
ce qui suggère qu’il faut penser à une compactification
$\widehat{\mathrm{Spec\ }\mathbb{Z}}$ pour avoir des résultats analogues aux
théorèmes sur les corps de fonctions. C’est le point de vue de la théorie
d’Arakelov comme développée par exemple dans [La] ou [SABK].
Cette brisure de la symétrie entre les places archimédiennes et non
archimédiennes est mise en évidence par la nécessité d’introduire un facteur à
l’infini afin d’obtenir l’équation fonctionnelle de la zêta de Riemann. En
effet, selon l’interprétation de Tate [Ta], le produit d’Euler des facteurs
locaux
$L_{p}(s)=\int_{\mathbb{Q}_{p}^{\ast}}\mathbbm{1}_{\mathbb{Z}_{p}}(x)\lvert
x\rvert_{p}^{s}\ d^{\ast}x=\frac{1}{1-\frac{1}{p^{s}}},$
où $d^{\ast}x=\frac{1}{1-\frac{1}{p^{s}}}\frac{dx}{\lvert x\rvert_{p}}$ est la
mesure de Haar sur $\mathbb{Q}_{p},$ ne donne une équation fonctionnelle que
lorsque l’on le complète avec un facteur à l’infini
$L_{\infty}(s)=\int_{\mathbb{R}^{\ast}}e^{-\pi x^{2}}\lvert
x\rvert_{\infty}^{s}\ \frac{dx}{\lvert x\rvert},$
la gaussienne étant l’analogue mystérieux de la fonction indicatrice de
$\mathbb{Z}_{p}$ dans $\mathbb{Q}_{p}.$
Une autre difficulté qui se présente, c’est la construction des anneaux de
valuation d’un corps de nombres. Pour $p$ premier,
$\mathbb{Z}_{p}=\\{x\in\mathbb{Q}_{p}:\lvert x\rvert_{p}\leq 1\\}$ est bien
défini en vertu de la propriété ultramétrique, mais ce que jouerai le rôle
d’anneau local à l’infini
$\mathcal{O}_{v_{\infty}}:=\mathbb{Z}_{\infty}=\\{x\in\mathbb{R}=\mathbb{Q}_{\infty}:\lvert
x\rvert\leq 1\\}=[-1,1]$
n’est pas fermé sous l’addition. L’idée sera de la remplacer par des
combinaisons linéaires convexes dans un cadre moins restrictif que l’algèbre
commutative usuelle. Dans cette catégorie d’anneaux généralisés que l’on va
construire dans la section suivante, $\mathbb{Z}_{\infty}$ aura le même status
que les anneaux classiques.
On peut attendre que cette façon d’envelopper les problèmes par le haut jette
une nouvelle lumière sur la manque de produits limitant l’analogie entre les
corps de nombres et les corps de fonctions. Tandis qu’en géométrie on peut
définir l’espace affine comme le produit de $n$ copies de la droite affine, ce
qui revient à considérer le spectre du coproduit⋆ des $k-$algèbres
$k[X_{1},\ldots,X_{n}]=k[X]\otimes_{k}\ldots\otimes_{k}k[X],$ chaque tentative
de définir une surface arithmétique ne donne pas de résultats. En effet,
$\mathbb{Z}$ étant l’objet initial de la catégorie des anneaux, $\mathrm{Spec\
}\mathbb{Z}\times\mathrm{Spec\ }\mathbb{Z}$ est réduit à la diagonale. Cet
obstacle est lié traditionnellement à l’absence d’un corps de base chez les
entiers, sur lequel on pourrait prendre des produits
$C\otimes_{\mathbb{F}_{1}}C$ comme celui dont Weil s’est servi dans sa
démonstration de l’hypothèse de Riemann pour les courbes sur un corps fini.
Dans la suite, on montrera que dans la catégorie des anneaux généralisés,
l’élusive corps à un élément admet une définition naturelle. Malgré cela, le
produit de deux copies de $\mathrm{Spec\ }\mathbb{Z}$ est encore trivial sur
cette nouvelle catégorie.
## 3 La catégorie des anneaux généralisés
On rappel que la classe de toutes les catégories $\mathfrak{Cat}$ est elle
même une $2-$catégorie, ce qui entraîne notamment que l’ensemble des foncteurs
entre deux catégories quelconques est de nouveau une catégorie, dont les
morphismes sont donnés par les transformations naturelles. Lorsque les deux
catégories coïncident, la composition usuelle munit les endofoncteurs
$\mathrm{End}(\mathcal{C}):=\mathrm{Hom}_{\mathfrak{Cat}}(\mathcal{C},\mathcal{C})$
d’une structure de catégorie monoïdale${}^{\star}.$ Si $F$ et $G$ sont des
endofoncteurs, on pose $FG:=F\otimes G$ et $F^{n}:=F\otimes\cdots\otimes F,$
ce qui est bien défini grâce à l’associativité. On appelle monade sur
$\mathcal{C}$ toute algèbre dans $\mathrm{End}(\mathcal{C})$ qui soit
compatible avec la structure de catégorie monoïdale. Plus explicitement:
###### Définition 1.
Une monade sur une catégorie $\mathcal{C}$ est un triplet
$\Sigma=(\Sigma,\mu,\epsilon)$ formé d’un endofoncteur
$\Sigma:\mathcal{C}\longrightarrow\mathcal{C},$ d’une multiplication
$\mu:\Sigma^{2}\longrightarrow\Sigma$ et d’un morphisme identité
$\epsilon:\mathrm{id}_{\mathcal{C}}\longrightarrow\Sigma$ tels que
1. 1.
$\mu$ et $\epsilon$ sont des transformations naturelles, i.e. pour tout couple
d’objets $X,Y$ dans $\mathcal{C}$ et tout morphisme $f:X\longrightarrow Y,$
les diagrammes suivants commutent
$\textstyle{\Sigma^{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{2}(f)}$$\scriptstyle{\mu_{X}}$$\textstyle{\Sigma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma(f)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\epsilon_{X}}$$\textstyle{\Sigma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma(f)}$$\textstyle{\Sigma^{2}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{Y}}$$\textstyle{\Sigma(Y)}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{Y}}$$\textstyle{\Sigma(Y)}$
2. 2.
$\mu$ et $\epsilon$ respectent les axiomes d’associativité et d’unité,
autrement dit, les diagrammes suivants sont commutatifs pour tout objet $X$
dans $\mathcal{C}$
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Un morphisme de monades $\varphi:\Sigma_{1}\longrightarrow\Sigma_{2}$ est une
transformation naturelle des endofoncteurs sous-jacents $\Sigma_{1}$ et
$\Sigma_{2}$ telle que
$\displaystyle\varphi_{Y}\circ\Sigma_{1}(Y)=\Sigma_{2}(f)\circ\varphi_{X},\quad\varphi_{X}\circ\epsilon_{\Sigma_{1,X}}=\epsilon_{\Sigma_{2,X}}\
\ \text{et que}$ $\displaystyle\mu_{\Sigma_{2},X}\circ$
$\displaystyle\Sigma_{2}(\varphi_{X})\circ\varphi_{\Sigma_{1}(X)}=\varphi_{X}\circ\mu_{\Sigma_{1},X}=\mu_{\Sigma_{2},X}\circ\varphi_{\Sigma_{2}(X)}\circ\Sigma(\varphi_{X})$
(3.1)
pour tout couple d’objets $X,Y\in\text{Ob}(\mathcal{C})$ et tout morphisme
$f:X\longrightarrow Y.$ On obtient ainsi la catégorie
$\mathfrak{Monades}(\mathcal{C})$ des monades sur $\mathcal{C}.$
###### Exemple 1.
La catégorie $\mathfrak{Monades}(\mathfrak{Ens})$ admet un objet initial qui
sera essentiel dans la suite. C’est le foncteur identité
$\mathrm{id}_{\mathfrak{Ens}}$ muni des applications $\mu$ et $\epsilon$
évidentes. On le note $\mathbb{F}_{\emptyset}$ et on l’appelle le corps sans
éléments. Les monades sur les ensembles possèdent aussi un objet final 1 qui
vaut $\\{0\\}$ sur tout ensemble $X$ et qui va jouer le rôle de l’anneau
trivial dans la construction des anneaux généralisés.
Étant donnée une monade $\Sigma=(\Sigma,\mu,\epsilon)$ sur $\mathcal{C},$ on
peut lui associer de façon naturelle une notion de sous-structure et de module
sur elle:
###### Définition 2.
Une sous-monade $\Sigma^{\prime}$ de $\Sigma$ est un sous-foncteur⋆
$\Sigma^{\prime}\subset\Sigma$ stable par $\mu$ et $\epsilon,$ i.e. une monade
$\Sigma^{\prime}$ telle que
1. 1.
$\Sigma^{\prime}(X)\subset\Sigma(X)$ pour tout objet $X$ dans $\mathcal{C}.$
2. 2.
la transformation naturelle $\Sigma^{\prime}\longrightarrow\Sigma$ dont les
composantes sont données par l’inclusion est un morphisme de monades.
Si l’on se donne deux sous-monades $\Sigma_{1}$ et $\Sigma_{2}$ de la même
monade $\Sigma,$ leur intersection est définie comme le produit fibré⋆
$\Sigma_{1}\times_{\Sigma}\Sigma_{2}$ dans la catégorie
$\mathrm{End}(\mathcal{C}).$ Plus terre à terre, $\Sigma_{1}\cap\Sigma_{2}$
est la monade dont l’image de chaque ensemble $X$ est l’intersection
$\Sigma_{1}(X)\cap\Sigma_{2}(X)$ et dont les composantes des morphismes
d’identité et de multiplication sont les restrictions de $\mu_{X}$ et
$\epsilon_{X}$ à $\Sigma_{1}(X)\cap\Sigma_{2}(X).$
###### Définition 3.
Un module sur $\Sigma$ est la donné d’un couple $M=(M,\alpha)$ formé d’un
objet $M\in\text{Ob}(\mathcal{C})$ et d’un morphisme
$\alpha:\Sigma(M)\longrightarrow M$ tels que
$\alpha\circ\mu_{M}=\alpha\circ\Sigma(\alpha)$ et que
$\alpha\circ\epsilon_{M}=\mathrm{id}_{M}.$ Un morphisme de $\Sigma-$modules
$f:(M,\alpha_{M})\longrightarrow(N,\alpha_{N})$ est un morphisme
$f\in\text{Hom}_{\mathcal{C}}(M,N)$ tel que
$f\circ\alpha_{M}=\alpha_{N}\circ\Sigma(f).$ On construit ainsi la catégorie
$\mathcal{C}^{\Sigma}$ des $\Sigma-$modules, qui peut être pensée plus
intrinsèquement comme l’objet qui représente le foncteur
$\mathrm{Hom}_{\mathfrak{Cat}}(\cdotp,\mathcal{C})^{\Sigma}$ dans
$\mathfrak{Cat}.$
###### Définition 4.
Soit $\varphi:\Sigma_{1}\longrightarrow\Sigma_{2}$ un morphisme de monades sur
une catégorie $\mathcal{C}.$ On appelle restriction des scalaires par rapport
à $\varphi$ le foncteur
$\varphi^{\ast}:\mathcal{C}^{\Sigma_{2}}\longrightarrow\mathcal{C}^{\Sigma_{1}}$
qui fait correspondre à un $\Sigma_{2}-$module $(N,\alpha)$ le
$\Sigma_{1}-$module $(N,\alpha^{\prime})$ ayant le même objet sous-jacent et
dont l’action de $\Sigma_{1}$ est définie par
$\alpha^{\prime}=\alpha\circ\varphi_{N}.$ Lorsque ce foncteur admet un adjoint
à gauche${}^{\star},$ on obtient aussi une notion d’extension des scalaires.
Le foncteur d’oubli
$\underline{O}:\mathcal{C}^{\Sigma}\longrightarrow\mathcal{C},$ qui associe à
chaque $\Sigma-$module $(M,\alpha)$ l’objet sous-jacent $M,$ admet un adjoint
à gauche, que l’on appellera comme d’habitude foncteur libre. C’est simplement
$\underline{L}:\mathcal{C}\longrightarrow\mathcal{C}^{\Sigma}$ défini par la
formule $\underline{L}(X)=(\Sigma(X),\mu_{X})$ pour tout
$X\in\text{Ob}(\mathcal{C}).$ La proposition suivante montre que ces deux
foncteurs encodent toute la structure de la monade.
###### Proposition 1.
Une monade $\Sigma$ sur $\mathcal{C}$ est entièrement déterminée par le couple
de foncteurs adjoints
$\underline{L}:\mathcal{C}\longrightarrow\mathcal{C}^{\Sigma}$ et
$\underline{O}:\mathcal{C}^{\Sigma}\longrightarrow\mathcal{C}.$
###### Proof.
On se réfère à l’annexe pour la définition du produit de transformations
naturelles. Si l’on considère l’unité et la co-unité de l’adjonction de
$\underline{L}$ et $\underline{O},$ qui sont des transformations naturelles
$\xi:\text{id}_{\mathcal{C}}\longrightarrow\underline{O}\underline{L}$ et
$\eta:\underline{L}\underline{O}\longrightarrow\text{id}_{\mathcal{C}^{\Sigma}}$,
on a
$\Sigma=\underline{O}\underline{L},\quad\epsilon=\xi,\quad\mu=\underline{O}\star\eta\star\underline{L}.$
En effet, on a bien que
$\mu:\underline{OLOL}=\Sigma^{2}\longrightarrow\underline{OL}=\Sigma$ et pour
tout objet $X$ dans $\mathcal{C}:$
$(\underline{O}\star\eta\star\underline{L})_{X}=\underline{O}(\eta_{\underline{L}(X)})=\underline{O}(\eta_{(\Sigma(X),\mu_{X})})=\underline{O}(\mu_{X})=\mu_{X}.\qed$
Cette construction admet une généralisation immédiate fournissant de nombreux
exem-ples de monades. Supposons que $F:\mathcal{C}\longrightarrow\mathcal{D}$
et $G:\mathcal{D}\longrightarrow\mathcal{C}$ sont des foncteurs adjoints, avec
des transformations $\xi:\text{id}_{\mathcal{C}}\longrightarrow GF$ et
$\eta:FG\longrightarrow\text{id}_{\mathcal{D}}.$ Alors, si l’on pose
$\Sigma=GF,\epsilon=\xi$ et $\mu=F\star\eta\star G,$ il est facile de vérifier
que le triplet $(\Sigma,\epsilon,\mu)$ définit une monade sur $\mathcal{C}.$
Lorsque $\mathcal{C}=\mathfrak{Ens},$ $\mathcal{D}$ est une certaine catégorie
d’ensembles munis de structure algébrique et
$G:\mathcal{D}\longrightarrow\mathfrak{Ens}$ est le foncteur d’oubli, on
obtiendra des monades dans la catégorie des ensembles, comme celle qui suit:
###### Exemple 2 (La monade des mots).
Soit $\mathcal{D}=\mathfrak{Mon}$ la catégorie dont les objets sont les
monoïdes (i.e. les semi-groupes avec identité) et dont les morphismes sont les
morphismes de semi-groupes qui préservent l’identité. Dans ce cas, le foncteur
d’oubli vers les ensembles
$\underline{O}:\mathfrak{Mon}\longrightarrow\mathfrak{Ens}$ admet comme
adjoint à gauche le foncteur libre
$\underline{L}:\mathfrak{Ens}\longrightarrow\mathfrak{Mon},$ qui associe à
chaque ensemble $X$ le monoïde libre $L(X),$ ceci étant formé par les mots de
longueur fini $x_{1}\ldots x_{n},$ dans l’alphabet $X$ et ayant comme
multiplication la concaténation des mots et comme identité le mot vide. On
obtient ainsi une monade $M=(M,\mu,\epsilon)$ sur la catégorie des ensembles
que l’on appellera la monade des mots. On donne ensuite une description plus
explicite.
Si $X\in\text{Ob}(\mathfrak{Ens}),$ on identifie $X^{0}$ au mot vide et
$X^{n}$ à l’ensemble des mots de longueur $n:$
$\\{x_{1}\\}\\{x_{2}\\}\ldots\\{x_{n}\\}.$ Alors $M(X)=\coprod_{n\geq
0}X^{n}.$ La correspondance entre $X$ et les mots de longueur un définit le
morphisme d’unité $\epsilon_{X}(x)=\\{x\\},$ et le morphisme de multiplication
$\mu_{X}:M^{2}(X)\longrightarrow M(X)$ agit de la façon suivante:
$\mu_{X}(\\{\\{x_{1}\\}\ldots\\{x_{n}\\}\\}\ldots\\{\\{z_{1}\\}\ldots\\{z_{m}\\}\\})=\\{x_{1}\\}\ldots\\{x_{n}\\}\ldots\\{z_{1}\\}\ldots\\{z_{m}\\}$
La preuve que $M=(M,\mu,\epsilon)$ est en fait une monade se réduit donc à la
vérification de l’associativité de l’opération "enlever les accolades".
### 3.1 La monade associée à un anneau
Étant donné un anneau $R,$ que l’on suppose unitaire mais pas forcément
commutatif, l’adjonction du foncteur d’oubli
$\underline{O}_{R}:R-\mathfrak{Mod}\longrightarrow\mathfrak{Ens}$ de la
catégorie des $R-$modules vers les ensembles et du foncteur libre
$\underline{L}_{R}:\mathfrak{Ens}\longrightarrow R-\mathfrak{Mod}$ définit une
monade $\Sigma_{R}=(\Sigma_{R},\mu,\epsilon)$ sur $\mathfrak{Ens}.$ Pour
chaque $X,$
$\Sigma_{R}(X)=\text{Hom}_{\mathfrak{Ens}}^{\text{fini}}(X,\underline{O}_{R}(R))$
est l’ensemble des applications de $X$ dans $R$ à support fini, qui
s’identifie à l’ensemble des $R-$combinaisons linéaires formelles
$\Sigma_{R}(X)=\\{\lambda_{1}\\{x_{1}\\}+\ldots+\lambda_{n}\\{x_{n}\\}:\lambda_{i}\in
R,x_{i}\in X,n\geq 0\\}.$
Ensuite, on définit le morphisme d’unité comme dans le cas de la monade des
mots, i.e. $\epsilon_{X}(x)=\\{x\\}$ pour tout $x\in X,$ et la multiplication
comme la transformation naturelle de $\Sigma_{R}^{2}$ dans $\Sigma_{R}$ dont
les composantes calculent les combinaisons linéaires des combinaisons
linéaires formelles. Autrement dit:
$\mu_{X}(\sum_{i}\lambda_{i}\\{\sum_{j}\mu_{ij}\\{x_{j}\\}\\})=\sum_{i,j}\lambda_{i}\mu_{ij}\\{x_{j}\\}$
Si $(X,\alpha)$ est un $\Sigma_{R}-$module, l’anneau $R$ agit sur l’ensemble
$X$ au moyen des opérations $x+y:=\alpha(\\{x\\}+\\{y\\})$ et $\lambda
x:=\alpha(\lambda\\{x\\}),$ qui vérifient bien les axiomes de $R-$module.
Réciproquement, si l’on se donne un $R-$module $X,$ l’application qui évalue
chaque combinaison linéaire formelle
$\lambda_{1}\\{x_{1}\\}+\ldots+\lambda_{n}\\{x_{n}\\}\in\Sigma_{R}(X)$ en
$\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n}\in X$ définit une structure de
$\Sigma_{R}-$module sur $X.$ Il est facile de montrer que cette correspondance
est en fait une équivalence des catégories $R-\mathfrak{Mod}$ et
$\mathfrak{Ens}^{\Sigma_{R}},$ ce qui revient à dire que le foncteur
$\underline{O}_{R}$ est monadique.
###### Proposition 2.
Soient $R$ un anneau unitaire et $\Sigma_{R}$ la monade sur les ensembles
construite comme ci-dessus. Alors:
1. 1.
Si $S$ est un sous-anneau de $R,$ alors $\Sigma_{S}$ est une sous-monade de
$\Sigma_{R}.$
2. 2.
$\Sigma_{R}$ commute avec les limites inductives filtrées⋆.
3. 3.
La structure de monoïde multiplicatif de $R$ peut être récupérée à partir de
$\Sigma_{R}.$
4. 4.
L’application $R\mapsto\Sigma_{R}$ est un foncteur pleinement fidèle⋆ de la
catégorie des anneaux unitaires dans la catégorie des monades. De plus, il
préserve les suites exactes.
###### Proof.
La première assertion découle simplement du fait qu’un sous-anneau $S$ de $R$
contient l’unité et il est fermé pour le produit. Par conséquent, $\Sigma_{S}$
est compatible avec $\epsilon$ et $\mu,$ donc c’est une sous-monade de
$\Sigma_{R}.$ Pour prouver la seconde affirmation, soit
$X=\varinjlim_{I}X_{i}$ la limite du système inductive filtré $(X_{i})_{i\in
I}.$ On a par définition
$\mathrm{Hom}_{\mathfrak{Ens}}(X,Z)=\varinjlim_{I}\mathrm{Hom}_{\mathfrak{Ens}}(X_{i},Z)$
pour tout ensemble $Z.$ Notamment, lorsque l’on prend $Z=\underline{O}_{R}(R)$
et on se restreint aux applications à support fini, on obtient
$\Sigma_{R}(X)=\varinjlim_{I}\Sigma_{R}(X_{i}).$
Montrons dans ce qui reste la nature des relations entre un anneau $R$ et sa
monade associée $\Sigma_{R}.$ Soit $\textbf{1}=\\{1\\}$ l’objet final de
$\mathfrak{Ens}.$ Alors $\Sigma_{R}(\textbf{1})=\\{\lambda\\{1\\}:\lambda\in
R\\}$ est l’ensemble sous-jacent à l’anneau, que l’on note
$\lvert\Sigma_{R}\rvert.$ On peut le voir comme un module à gauche sur lui
même par l’égalité $R_{s}:=\underline{L}_{R}(\textbf{1}),$ ce qui nous permet
de récupérer la multiplication de $R.$ Notons que l’unité est l’image
$\epsilon_{\textbf{1}}(1)\in\lvert\Sigma_{R}\rvert.$
Maintenant, si l’on considère un morphisme d’anneaux $f:R\longrightarrow S,$
la famille d’applications ensemblistes
$\Sigma_{f,X}:\Sigma_{R}(X)\longrightarrow\Sigma_{R}(S)$ définies par
$\sum_{i=1}^{n}\lambda_{i}\\{x_{i}\\}\longmapsto\sum_{i=1}^{n}f(\lambda_{i})\\{x_{i}\\}$
induit un morphisme $\Sigma_{f}$ entre les monades $\Sigma_{R}$ et
$\Sigma_{S}.$ En effet, la première condition de compatibilité dans (1) se
vérifie automatiquement, la seconde équivaut à dire que $f(e_{R})=e_{S}$ et la
dernière découle du fait que $f(x+y)=f(x)+f(y)$ et que $f(xy)=f(x)f(y).$ Donc,
$\Sigma_{f}$ est un morphisme de monades.
Pour prouver que ce foncteur est pleinement fidèle, il suffit de montrer que,
étant donnés deux anneaux $R$ et $S,$ tous les morphismes de monades
$\zeta:\Sigma_{R}\longrightarrow\Sigma_{S}$ sont de la forme
$\zeta=\Sigma_{f}$ pour un unique morphisme d’anneaux $f:R\longrightarrow S.$
En effet, on sait déjà que si $\zeta:\Sigma_{R}\longrightarrow\Sigma_{S}$ est
un morphisme de monades, $f:=\lvert\zeta\rvert$ est un morphisme entre les
monoïdes sous-jacents $R$ et $S.$ Il suffit de montrer que
$\zeta:R\longrightarrow S$ respecte aussi l’addition et que
$\Sigma_{f,X}=\zeta_{X}$ pour chaque ensemble $X$. ∎
###### Corollaire 1.
La catégorie des anneaux unitaires $\mathfrak{Ann}$ s’identifie à une sous-
catégorie pleine⋆ de $\mathfrak{Monades}.$
###### Remarque 1.
Même si l’on vient de montrer que tout sous-anneau de $R$ fournit une sous-
monade de $\Sigma_{R},$ il n’y a pas une correspondance bijective entre les
sous-anneaux de $R$ et les sous-monades de $\Sigma_{R}.$ Si l’on prend, par
exemple, $R=\mathbb{Z},$ il n’existe pas de sous-anneaux non-isomorphes à
$\mathbb{Z}$, mais par contre on peut construire la sous-monade $\mathbb{N},$
qui n’est pas isomorphe à $\mathbb{Z}$ (voir 4.1).
###### Remarque 2.
Si l’on calcule le foncteur de restriction des scalaires par rapport au
morphisme $\Sigma_{f}$ introduit ci-dessus, on obtient le foncteur usuel de
$S-\mathfrak{Mod}$ vers $R-\mathfrak{Mod}$ qui à chaque $S-$module $M$ fait
correspondre le $R-$module $M$ dont l’action est donné par $r\cdot m=f(r)\cdot
m.$ Par unicité de l’adjonction, le foncteur d’extension est dans ce cas
l’application $M\longmapsto S\otimes_{R}M$ de $R-\mathfrak{Mod}$ vers
$S-\mathfrak{Mod}.$
En vue de la proposition 2, on définit l’ensemble sous-jacent à une monade
$\Sigma$ dans la catégorie des ensembles comme
$\lvert\Sigma\rvert:=\Sigma(\textbf{1}).$ L’isomorphisme canonique
$\varphi:\lvert\Sigma\rvert=\mathrm{Hom}_{\mathfrak{Ens}}(\textbf{1},\Sigma(\textbf{1}))\cong\mathrm{End}_{\Sigma-\mathfrak{Mod}}(\underline{L}_{R}(\textbf{1}))$
munit $\lvert\Sigma\rvert$ d’une structure de monoïde. En effet, étant donnés
deux éléments $x$ et $y$ de $\lvert\Sigma\rvert,$ en posant $x\cdot
y:=\varphi^{-1}(\varphi(x)\circ\varphi(y))$ on obtient une loi associative
dont l’unité est $\epsilon_{\textbf{1}}(1).$ Il n’y a aucune raison pour
attendre que cela soit un monoïde commutatif, car la composition
d’endomorphismes ne l’est pas.
### 3.2 Les monades algébriques
Pour définir une catégorie d’anneaux généralisés adéquate, il faudra imposer
quelques restrictions aux monades. La première d’entre elles, l’algébricité,
nous permettra d’obtenir une notion d’anneau généralisé non commutatif.
###### Définition 5.
Un endofoncteur $\Sigma:\mathfrak{Ens}\longrightarrow\mathfrak{Ens}$ est dit
algébrique s’il commute avec des limites inductives filtrées, c’est-à-dire, si
$\Sigma(\varinjlim_{I}X_{i})\cong\varinjlim_{I}\Sigma(X_{i})$ pour chaque
famille d’ensembles indexée par un ensemble partiellement ordonné filtré. Une
monade est algébrique si son foncteur sous-jacent l’est.
###### Exemple 3 (Une monade non algébrique).
Comme on a montré dans la proposition 2, la monade associée à un anneau est
algébrique. Si l’on pense à $\Sigma(X)$ comme l’ensemble des
$\Sigma-$combinaisons linéaires formelles d’éléments de $X,$ il faut trouver
des combinaisons à support infini pour avoir un exemple de monade non
algébrique. Soit $\hat{\mathbb{Z}}_{\infty}$ (voir la section 4.2 pour le
choix de cette notation) la monade définie par
$\hat{\mathbb{Z}}_{\infty}(X):=\\{\sum_{x\in X}\lambda_{x}:\
\lambda_{x}\in\mathbb{R},\ \sum_{x\in X}\lvert\lambda_{x}\rvert\leq 1\\}.$
On montre que ce n’est pas une monade algébrique en la testant sur le système
$X_{\bullet}=((X_{n})_{n\in\mathbb{N}},(i_{n,m})_{n\leq m})$ formé des
ensembles $X_{n}=\\{0,\ldots,n\\}$ et des inclusions de $X_{n}$ dans $X_{m}$,
la limite inductive étant $\mathbb{N}.$ D’un côté,
$\hat{\mathbb{Z}}(\mathbb{N})$ est l’ensemble des séries convergentes dont la
somme des modules est plus petite ou égale à 1. De l’autre côté,
$\hat{\mathbb{Z}}(X_{n})$ est formé des $(n+1)$-uplets
$(\lambda_{0},\ldots,\lambda_{n})$ telles que
$\lvert\lambda_{0}\rvert+\ldots+\lvert\lambda_{n}\rvert\leq 1$ et la limite
inductive est la réunion de tous ces ensembles. C’est clair que dans ce cas on
ne peut pas échanger la limite car
$\hat{\mathbb{Z}}(\mathbb{N})-\varinjlim\hat{\mathbb{Z}}(X_{n})$ contient
toutes les séries convergentes avec un nombre infini de termes non nuls dont
la somme des modules est plus petite ou égale que 1, par exemple
$\sum_{n=0}^{\infty}\frac{1}{(n+1)^{2}\pi^{2}}$.
Puisque la composition de deux foncteurs algébriques est encore algébrique,
ils forment une sous-catégorie monoïdale de $\text{End}(\mathfrak{Ens}),$ qui
est en plus pleine. On remarque aussi qu’un foncteur algébrique est
complètement déterminé par sa valeur dans les ensembles finis, car tout
ensemble $X$ est la limite inductive filtrée de ses sous-ensembles finis. Si
$\Sigma$ est un endofoncteur algébrique et l’on note par
$\underline{\mathbb{N}}$ la catégorie des ensembles finis standard
$\textbf{n}=\\{1,\ldots,n\\}$ dont les morphismes sont les applications entre
ensembles finis, la restriction
$\Sigma\longmapsto\Sigma_{|\underline{\mathbb{N}}}$ induit une équivalence
catégorique entre
$\mathfrak{Ens}^{\underline{\mathbb{N}}}:=\text{Hom}_{\mathfrak{Cat}}(\underline{\mathbb{N}},\mathfrak{Ens})$
et les endofoncteurs algébriques. On peut les imaginer donc comme une
collection d’ensembles $\\{\Sigma(\textbf{n})\\}_{n\geq 0}$ munie des
applications fonctorielles
$\Sigma(\varphi):\Sigma(\textbf{m})\longrightarrow\Sigma(\textbf{n})$ qui sont
définies pour chaque $\varphi:\textbf{m}\longrightarrow\textbf{n}.$
Étant donné un foncteur
$G:\underline{\mathbb{N}}\longrightarrow\mathfrak{Ens}$ et un ensemble
quelconque $X,$ on pose:
$H_{0}(X):=\bigsqcup_{n\geq 0}G(\textbf{n})\times X^{n},\quad
H_{1}(X):=\bigsqcup_{\varphi:\textbf{m}\longrightarrow\textbf{n}}G(\textbf{m})\times
X^{n}.$
On a alors des applications $p,q:H_{1}(X)\longrightarrow H_{0}(X)$ définies de
la façon suivante: si $X^{\varphi}$ représente l’application
$(x_{1},\ldots,x_{n})\mapsto(x_{\varphi(1)},\ldots,x_{\varphi(n)}),$ alors les
restrictions de $p$ et $q$ à la composante indexée par
$\varphi:\textbf{m}\longrightarrow\textbf{n}$ sont donnés par
$\displaystyle id_{G(\textbf{m})}\times X^{\varphi}$
$\displaystyle:G(\textbf{m})\times X^{n}\longrightarrow G(\textbf{m})\times
X^{m},$ $\displaystyle G(\varphi)\times\mathrm{id}_{X^{n}}$
$\displaystyle:G(\textbf{m})\times X^{n}\longrightarrow G(\textbf{m})\times
X^{n}.$
Le lemme suivant, dont la démonstration se trouve dans [Du, 4.1.4], permet de
calculer $\Sigma(X)$ comme le coégaliseur⋆ du système ci-dessus:
###### Lemme 1.
Soit $\Sigma$ une monade algébrique et soit
$G=\Sigma|_{\underline{\mathbb{N}}}\in\mathfrak{Ens}^{\underline{\mathbb{N}}}$
le foncteur image de $\Sigma$ par l’équivalence de catégories. Alors
$\Sigma(X)=\mathrm{Coeg}(p,q:H_{1}(X)\rightrightarrows H_{0}(X))$
Par conséquent, si $X$ et $Y$ sont des ensembles, une application
$\alpha:\Sigma(X)\longrightarrow Y$ est la donnée d’une famille
$\\{\alpha^{(n)}:\Sigma(\textbf{n})\times X^{n}\longrightarrow Y\\}_{n\geq 0}$
qui fait commuter le diagramme
$\textstyle{\Sigma(\textbf{m})\times
X^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma(\varphi)\times
id_{X^{n}}}$$\scriptstyle{id_{\Sigma(\textbf{m})}\times
X^{\varphi}}$$\textstyle{\Sigma(\textbf{m})\times
X^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{(m)}}$$\textstyle{\Sigma(\textbf{n})\times
X^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{(n)}}$$\textstyle{Y}$
(3.2)
pour tout $\varphi:\textbf{m}\longrightarrow\textbf{n}.$ Cela montre que
$\Sigma(\textbf{n})$ paramètre en quelque mode les opérations n-aires de $X$
vers $Y$ et on peut écrire $[t]_{\alpha}(x_{1},\ldots,x_{n})$ au lieu de
$\alpha(t,x_{1},\ldots,x_{n}).$ On dit que $t\in\Sigma(\textbf{n})$ est une
opération n-aire formelle $[t]_{\alpha}:X^{n}\longrightarrow Y.$ Dans le
nouveau symbolisme, la commutativité du diagramme ci-dessus se traduit en
$[\Sigma(\varphi)(t)]_{\alpha}(x_{1},\ldots,x_{n})=[t]_{\alpha}(x_{\varphi(1)},\ldots,x_{\varphi(m)})$
pour toute opération formelle $t\in\Sigma(\textbf{m})$ et pour toute
application $\varphi:\textbf{m}\longrightarrow\textbf{n}.$
D’après le lemme, il existe une surjection
$H_{0}(X)\twoheadrightarrow\Sigma(X)$ dont la restriction à
$\Sigma(\textbf{n})\times X^{n}$ applique $(t,x_{1},\ldots,x_{n})$ dans
$\Sigma(x)(t)\in\Sigma(X),$ $x:\textbf{n}\longrightarrow X$ étant
l’application $k\mapsto x_{k}.$ Notons $t(\\{x_{1}\\},\ldots,\\{x_{n}\\})$
cette image. Alors pour tout $s$ dans $\Sigma(X)$ il existe $n\leq
0,t\in\Sigma(\textbf{n})$ et $x_{1},\ldots,x_{n}\in X$ tels que
$s=t(\\{x_{1}\\},\ldots,\\{x_{n}\\}).$ En fait, $\Sigma(X)$ est l’ensemble de
toutes ces expressions modulo les relations
$(\Sigma(\varphi))(t)(\\{x_{1}\\},\ldots,\\{x_{n}\\})\sim
t(\\{x_{\varphi(1)}\\},\ldots,\\{x_{\varphi(n)}\\}).$
Ce point de vue nous permet de déterminer univoquement une monade algébrique à
partir des données suivantes:
1. 1.
Une collection d’ensembles $\\{\Sigma(\textbf{n})\\}_{n\geq 0}$ et
d’applications
$\Sigma(\varphi):\Sigma(\textbf{m})\longrightarrow\Sigma(\textbf{n})$ définies
pour chaque $\varphi:\textbf{m}\longrightarrow\textbf{n}$ et telles que
$\Sigma(id_{\textbf{n}})=id_{\Sigma(\textbf{n})},\quad\Sigma(\psi\circ\varphi)=\Sigma(\psi)\circ\Sigma(\varphi).$
Puisque $\Sigma(X)=\varinjlim\Sigma(\textbf{n}),$ après avoir choisi
d’identifications convenables entre les sous-ensembles finis de $X$ et les
ensembles finis standard n, on peut reconstruire à partir de cette famille la
valeur de la monade algébrique dans tous les ensembles.
2. 2.
Une famille de morphismes de multiplication
$\mu_{n}^{(k)}:\Sigma(\textbf{k})\times\Sigma(\textbf{n})^{k}\longrightarrow\Sigma(\textbf{n})$
qui vérifient les relations
$\displaystyle\mu_{n}^{(k)}\circ(id$
${}_{\Sigma(\textbf{k})}\times\Sigma(\textbf{n})^{\varphi})=\mu_{n}^{(k^{\prime})}\circ(\Sigma(\varphi)\times
id_{\Sigma(\textbf{n})^{k^{\prime}}})$
$\displaystyle\mu_{n}^{(k)}\circ(id_{\Sigma(\textbf{k})}\times\Sigma(\psi)^{\varphi})=\mu_{m}^{(k)}$
pour chaque $\varphi:\textbf{k}\longrightarrow\textbf{k}^{\prime}$ et
$\psi:\textbf{m}\longrightarrow\textbf{n}.$ En effet, d’après ce que l’on a vu
précédemment, se donner le morphisme
$\mu_{X}:\Sigma^{2}(X)\longrightarrow\Sigma(X),$ c’est équivalent lorsque l’on
remplace $X$ et $Y$ par $\Sigma(X)$ dans (3.2) à se donner une famille
$\mu^{(k)}:\Sigma(\textbf{k})\times\Sigma(X)^{k}\longrightarrow\Sigma(X).$
Dans une seconde étape, on considère les restrictions à chaque
$\Sigma(\textbf{n}),$ ayant ainsi
$\mu_{n}^{(k)}:\Sigma(\textbf{k})\times\Sigma(\textbf{n})^{k}\longrightarrow\Sigma(\textbf{n}).$
3. 3.
Un élément distingué $\textbf{e}\in\Sigma(1).$ Comme
$\mathrm{id}_{\mathfrak{Ens}}=\mathrm{Hom}_{\mathfrak{Ens}}(\textbf{1},\cdot),$
par le lemme de Yoneda⋆ le morphisme d’unité
$\epsilon:\mathrm{id}_{\mathfrak{Ens}}\longrightarrow\Sigma$ est déterminé
univoquement par un élément
$\textbf{e}:=\epsilon_{\textbf{1}}(\textbf{1})\in\Sigma(\textbf{1}).$ Si
$\tilde{x}:\textbf{1}\longrightarrow X$ est la seule application ayant image
$x,$ alors $\epsilon_{X}(x)=(\Sigma(\tilde{x}))(\textbf{e}).$
Ces données doivent vérifier de plus les axiomes de monade. Ce n’est pas
difficile à voir que les conditions de la définition 1 se traduisent en:
1. 1.
$\mu_{n}^{(1)}(\textbf{e},t)=t$ pour tout $t\in\Sigma(\textbf{n})$ et pour
tout $n\geq 0.$
2. 2.
$\mu_{n}^{(n)}(t,\\{1\\}_{\textbf{n}},\ldots,\\{n\\}_{\textbf{n}})=t$ pour
tout $t\in\Sigma(\textbf{n}).$
3. 3.
Pour tout $n,k,m\geq 0,$ le diagramme suivant commute:
$\textstyle{\Sigma(\textbf{k})\times\Sigma(\textbf{n})^{k}\times\Sigma(\textbf{m})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{\Sigma(\textbf{k})}\times(\mu_{m}^{(n)})^{(k)}}$$\scriptstyle{\mu_{n}^{(k)}\times\mathrm{id}_{\Sigma(\textbf{m})^{n}}}$$\textstyle{\Sigma(\textbf{k})\times\Sigma(\textbf{m})^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{m}^{(k)}}$$\textstyle{\Sigma(\textbf{n})\times\Sigma(\textbf{m})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{m}^{(n)}}$$\textstyle{\Sigma(\textbf{m})}$
(3.3)
###### Notation.
Soit $\Sigma$ une monade algébrique définie par les données que l’on vient de
décrire. Posons $||\Sigma||:=\bigsqcup_{n\geq 0}\Sigma(\textbf{n}).$ On
appelle constantes de $\Sigma$ les éléments de $\Sigma(\textbf{0}),$
opérations unaires les éléments de $\Sigma(\textbf{1})$ et en général
opérations n-aires les éléments de $\Sigma(\textbf{n})$ pour $n\geq 2.$
Lorsque l’on veut souligner que $u$ est une opération $n-$aire, on écrit
$u^{[n]}.$ Si $\Sigma(\textbf{0})$ n’a qu’un élément, on dit que $\Sigma$ est
une monade avec zéro. Si $\lambda$ est une opération unaire, on écrit
normalement $\lambda x$ au lieu de $\lambda_{X}(x);$ par exemple, $-x$ au lieu
de $[-]_{X}(x).$ De même, si $+$ est une opération binaire, on n’écrit pas
$[+]_{X}(x_{1},x_{2})$ mais $x_{1}+x_{2}.$
### 3.3 Présentation d’une monade algébrique
Étant donnés une monade algébrique $\Sigma$ et une partie
$U\subset||\Sigma||,$ on considère la plus petite sous-monade contenant $U,$
ce qui revient à dire l’intersection de toutes les sous-monades
$\Sigma_{\alpha}$ de $\Sigma$ dont leurs ensembles d’opérations contiennent
$U.$ Notons-la
$\mathbb{F}_{\emptyset}\langle
U\rangle:=\bigcap_{U\subset||\Sigma_{\alpha}||}\Sigma_{\alpha}$
###### Proposition 3.
Soit $U$ une partie d’une monade algébrique $\Sigma.$ Alors
$\Sigma^{\prime}:=\mathbb{F}_{\emptyset}\langle U\rangle$ est la sous-monade
de $\Sigma$ dont l’ensemble $\Sigma^{\prime}(\textbf{n})$ est formé des
élément obtenus en appliquant un nombre fini de fois les règles suivantes:
1. 1.
$\\{k\\}_{\textbf{n}}\in\Sigma^{\prime}(\textbf{n})$ pour tout $1\leq k\leq
n.$
2. 2.
(Propriété de remplacement) Soit $u\in U$ une opération k-aire. Alors, pour
tout $t_{1},\ldots,t_{k}\in\Sigma^{\prime}(\textbf{n}),$ on a
$[u]_{\Sigma(\textbf{n})}(t_{1},\ldots,t_{k})\in\Sigma^{\prime}(\textbf{n}).$
###### Proof.
La preuve se réduit à la remarque que la collection de sous-ensembles
$\Sigma^{\prime\prime}(\textbf{n})\subset\Sigma^{\prime}(\textbf{n})\subset\Sigma(\textbf{n})$
obtenus en appliquant ces deux règles un nombre fini de fois définit une sous-
monade de $\Sigma$ contenant $U.$ Par conséquent
$\Sigma^{\prime\prime}=\Sigma^{\prime}$. ∎
De plus, Durov montre que l’on peut toujours réaliser
$\mathbb{F}_{\emptyset}\langle U\rangle$ comme une certaine sous-monade de la
monade $M_{U}$ des mots à constants dans $U$ [Du, 4.5.2]. On peut de même
généraliser la construction ci-dessus de la façon suivante: soit $\Sigma_{0}$
une autre monade algébrique et soit $\rho:\Sigma_{0}\longrightarrow\Sigma$ un
morphisme de monades. Si $U$ est une partie de $||\Sigma||,$ on appelle sous-
monade engendrée par $U$ sur $\Sigma_{0}$
$\Sigma_{0}\langle U\rangle=\mathbb{F}_{\emptyset}\langle
U\cup||\rho(\Sigma_{0})||\rangle.$
###### Définition 6.
Soit $\Sigma$ une monade algébrique. On dit que $\Sigma$ est de type fini sur
$\Sigma_{0}$ s’il existe une partie finie $U$ de $||\Sigma||$ telle que
$\Sigma=\Sigma_{0}\langle U\rangle.$ Une monade est dite absolument de type
fini si elle de type fini sur $\mathbb{F}_{\emptyset}.$
On peut construire de cette façon des monades libres, mais pour encoder
l’information de la plupart des objets algébriques, on aura besoin d’un
certain type de conditions de torsion, comme celles qui imposent, par exemple,
les axiomes de la définition d’anneau. On le fait en définissant des relations
d’équivalence algébrique compatibles avec la structure de monade:
###### Définition 7.
Une relation $R\subset\Sigma\times\Sigma$ compatible avec la structure de
monade de $\Sigma$ est la donnée d’un système de relations d’équivalence
$\\{R(\textbf{n})\subset\Sigma(\textbf{n})\times\Sigma(\textbf{n})\\}_{n\geq
0}$ tel que si $t\equiv_{R(\textbf{k})}s,t_{i}\equiv_{R(\textbf{n})}s_{i}$
pour tout $1\leq i\leq k,$ alors:
$[t]_{\Sigma(\textbf{n})}(t_{1},\ldots,t_{k})\equiv_{R(\textbf{n})}[s]_{\Sigma(\textbf{n})}(s_{1},\ldots,s_{k})$
En particulier, $R$ est compatible avec les applications induites
$\Sigma(\varphi).$
Soit $\Sigma$ une monade algébrique et $R$ une relation compatible avec
$\Sigma.$ On définit alors le quotient $\Sigma^{\prime}=\Sigma/R$ comme la
monade dont les images des ensembles finis sont
$\Sigma^{\prime}(\textbf{n})=\Sigma(\textbf{n})/R(\textbf{n})$ et dont les
morphismes de multiplication et d’unité sont les induits par passage au
quotient. C’est la seule structure de monade dans $\Sigma^{\prime}$ pour
laquelle l’application évidente $\Sigma\longrightarrow\Sigma^{\prime}$ est un
morphisme de monades. Dans la suite, on appliquera cette construction en
prenant comme relations les définies par un système d’équations, c’est-à-dire,
par l’intersection $R=\cap_{\alpha}R_{\alpha}$ d’un certain nombre de
relations algébriques compatibles. On note
$\Sigma_{0}\langle U|R\rangle:=\Sigma_{0}\langle U\rangle/R$
la monade engendré sur $\Sigma_{0}$ par les opérations $U$ et par l’ensemble
$R$ de relations.
###### Définition 8.
Une monade algébrique $\Sigma$ admet une présentation finie sur $\Sigma_{0}$
si elle de type fini et s’il existe un ensemble fini de relations $R$ tel que
$\Sigma\cong\Sigma_{0}\langle U|R\rangle$
### 3.4 L’additivité
Comme on a montré à la fin de l’épigraphe 3.1, l’ensemble sous-jacent à une
monade algébrique a toujours une structure de monoïde multiplicatif. On
voudrait explorer maintenant la possibilité de définir une addition. Soit
$\Sigma$ une monade algébrique ayant une constante $0\in\Sigma(\textbf{0}).$
On a des applications, dites de comparaison,
$\pi_{n}:\Sigma(\textbf{n})\longrightarrow\Sigma(\textbf{1})^{n}$ dont la
composante $k-$ième applique
$t\mapsto
t(0_{\Sigma(\textbf{1})},\ldots,\\{1\\}_{\textbf{1}}\ldots,0_{\Sigma(\textbf{1})}).$
###### Définition 9.
Soit $\Sigma$ une monade algébrique avec constante. Une opération binaire
$[+]\in\Sigma(\textbf{2})$ est une pseudoaddition si
$\pi_{2}([+])=(\textbf{e},\textbf{e})\in\Sigma(1)^{2}.$ S’il s’agit de plus de
la seule pseudoaddition que l’on peut définir sur $\Sigma,$ on dit que $[+]$
est une addition. On appelle $\Sigma$ monade hypoadditive (resp.
hyperadditive, additive) si toutes les applications
$\pi_{n}:\Sigma(\textbf{n})\longrightarrow\Sigma(\textbf{1})^{n}$ sont
injectives (resp. surjectives, bijectives).
###### Exemple 4.
Soit $\Sigma$ la monade définie par $\Sigma(\textbf{0})=\mathbb{Z}$ et par
$\Sigma(\textbf{n})=\mathbb{Z}[T_{1},\ldots,T_{n}]$ pour tout $n\geq 1$. Après
avoir fixée $c=0$ comme constante, $\pi_{2}$ applique un polynôme $F(X,Y)$
dans $(F(T,0),F(0,T))\in\Sigma(\textbf{1})^{2}$. Cette application étant
surjective, on conclut que $\Sigma$ est une monade hyperadditive. Elle n’est
pas additive car il y a plusieurs façons de définir un polynôme à deux
variables dont ses évaluations en $X=0$ et en $Y=0$ sont données. Par exemple,
tous les polynômes de la forme $f=X.g+Y.h$, où $g(X,0)=h(0,Y)=1$, ont la même
image par $\pi_{2}.$
La définition de pseudoaddition, qui peut sembler bizarre à première vue, ne
dit que $x+0=x=0+x$ lorsque l’on la lit dans un $\Sigma-$module $X.$ Si
$\Sigma$ est une monade additive, on obtient une addition en posant
$[+]=\pi_{2}^{-1}(\textbf{e},\textbf{e}).$ Dans une monade hypoadditive,
l’application $\pi_{0}:\Sigma(\textbf{0})\longrightarrow\textbf{1}$ est
injective, donc, il n’y a qu’un constante $0\in\Sigma(\textbf{0}).$ De plus,
le fait que $\pi_{2}$ soit injective implique que s’il existe une
pseudoaddition $[+],$ elle est nécessairement unique, c’est-à-dire, une
addition. Alors, l’associativité et la commutativité de $[+]$ découlent
automatiquement des identités pour $\pi_{2}(\\{1\\}+\\{2\\})$ et
$\pi_{3}((\\{1\\}+\\{2\\})+\\{3\\})$.
###### Proposition 4.
Les monades algébriques additives sont en correspondance bijective avec les
semi-anneaux. De plus, une monade additive est la monade associée à un anneau
s’il existe une symétrie $[-]$ (i.e. une opération unaire d’ordre deux).
###### Proof.
On rappel d’abord qu’un semi-anneau est un ensemble $R$ muni de deux
opérations binaires $+$ et $\cdot$ telles que $(R,+)$ est un monoïde
commutatif ayant élément neutre $0$ et que $(R,\cdot)$ est un monoïde avec
unité $1.$ On demande de plus que la multiplication soit distributive par
rapport à l’addition et que $0\cdot r=0=r\cdot 0$ pour tout $r\in R.$ Cette
liste d’axiomes nous permet d’écrire toute opération n-aire
$t\in\Sigma(\textbf{n})$ d’une façon unique sous la forme
$\lambda_{1}\\{1\\}+\ldots+\lambda_{n}\\{n\\},$ ce qui revient à dire que
l’application $\pi_{n}$ est bijective. Par conséquent, un semi-anneau définit
une monade algébrique additive.
Réciproquement, une monade algébrique additive est engendrée par la seule
constante $0\in\Sigma(\textbf{0})$, l’ensemble des opérations unaires
$\lvert\Sigma\rvert=\Sigma(\textbf{1})$ et l’addition
$[+]\in\Sigma(\textbf{2})$. En utilisant la structure de monoïde multiplicatif
de $\Sigma(\textbf{1})$, l’injectivité des applications $\pi_{2}$ et
$\pi_{3},$ ainsi que le fait que la monade n’admet qu’une seule constante, on
obtient la présentation suivante:
$\displaystyle\Sigma=\mathbb{F}_{\emptyset}\langle$ $\displaystyle
0^{[0]},\lvert\Sigma\rvert,[+]^{[2]}|x+y=y+x,\ (x+y)+z=x+(y+z),\ x+0=x,$
$\displaystyle\lambda(x+y)=\lambda x+\lambda y,\
(x+y)\lambda=x\lambda+y\lambda,\ \forall\lambda\in\lvert\Sigma\rvert\rangle$
Alors, si l’on muni $\lvert\Sigma\rvert$ du produit provenant de la structure
de monoïde multiplicatif et de l’addition
$+:\lvert\Sigma\rvert^{2}\longrightarrow\lvert\Sigma\rvert$, les relations ci-
dessus impliquent que $(\lvert\Sigma\rvert,+,\cdot)$ est un semi-anneau. C’est
clair que les inverses pour l’addition sont la seule donnée supplémentaire
dont on a besoin pour avoir en fait un anneau. ∎
###### Remarque 3.
Le nouveau cadre des monades algébriques permet de considérer des géométries
relatives à des semi-anneaux tels que $\mathbb{N}$ ou $\mathbb{R}_{\geq 0}$
qui ne puissent pas être traitées avec les techniques usuelles de la géométrie
algébrique. Cela a un spécial intérêt étant donnée que la géométrie tropicale,
très en vogue depuis la fin des années 90, repose sur le semi-anneau
$\mathbb{T}=(\mathbb{R},\oplus,\otimes)$, où $x\oplus y:=\mathrm{min\
}\\{x,y\\}$ et $x\otimes y:=x+y$ (voir [RST] pour une introduction). Puisque
on n’as pas encore trouvé une définition fonctorielle des variétés tropicales,
il serait intéressant d’étudier les propriétés de la géométrie relative à
$\mathbb{T}$ au sens de Durov et voir si cela correspond en quelque sens à ce
que l’on attend.
### 3.5 Les monades commutatives
Malgré le succès du langage des monades algébriques pour encoder la géométrie
sur de nouvelles structures, ce cadre est encore trop vaste pour avoir des
équivalents des théorèmes classiques. De la même manière que l’on a introduit
les monades algébriques en généralisant une propriété agréable des monades
$\Sigma_{R}$ associées à un anneau, on va s’inspirer d’un autre trait des
opérations définissant $\Sigma_{R}$ pour un anneau $R$ commutatif afin de
trouver une sous-catégorie pleine des monades algébriques sur laquelle on
travaillera dans la suite. En effet, dans un anneau classique
$R=(R,+,\times)$, on a toujours le groupe abélien $(R,+)$ et on peut voir la
distributivité comme une sorte de relation de commutation entre la somme et le
produit. De plus, lorsque $R$ est commutatif, le monoïde sous-jacent
$(R,\times)$ est aussi commutatif. Alors, dans un certain sens, un anneau
usuel est commutatif si toutes les opérations commutent entre elles. Cela
justifie la définition suivante:
###### Définition 10.
Soit $\Sigma$ une monade algébrique. On dit que les opérations
$t\in\Sigma(\textbf{n})$ et $s\in\Sigma(\textbf{m})$ commutent si pour tout
$\Sigma-$module $X$ et pour toute famille d’éléments $\\{x_{ij}\\}$ indexés
par $1\leq i\leq n$ et par $1\leq j\leq m$ on a
$t(s(x_{11},\ldots,x_{1m}),\ldots,s(x_{n1},\ldots,x_{nm}))=s(t(x_{11},\ldots,x_{1n}),\ldots,t(x_{m1},\ldots,x_{mn}))$
Une monade algébrique est commutative si tout couple d’opérations dans
$||\Sigma||$ commute.
Cette définition, tout à fait effective, a un goût clairement matriciel. En
effet, si l’on note par $M$ la matrice $n\times m$ formée des éléments
$\\{x_{ij}\\}$, on peut faire agir les opérations $t$ et $s$ sur $M$ de deux
façons: premièrement, on évalue $s$ dans chaque ligne et on obtient $n$
valeurs auxquels on applique $t$; deuxièmement, on évalue $t$ dans chaque
colonne et puis on applique $s$ aux $m$ éléments obtenus. Alors $t$ et $s$
sont commutatives si les deux procédés donnent le même résultat. Dans les
exemples suivants on montre ce que cette égalité entraîne pour les opérations
d’arité basse.
###### Exemple 5 (Commutativité des constantes).
Soient $c,d\in\Sigma(\textbf{0})$ deux constantes dans $\Sigma.$ La condition
ci-dessus se lit $c=d$, donc une monade commutative admet au plus une
constante. Cela permet de se référer sans ambiguïté à la constante $0$ d’une
monade commutative. Soit maintenant $t$ une opération n-aire. Si $t$ et $c$
commutent, alors $t(c,\ldots,c)=c.$ Par conséquent, dans une monade algébrique
$0$ reste invariant par toutes les opérations.
###### Exemple 6 (Commutativité des opérations unaires).
Soient $u,v\in\lvert\Sigma\rvert$ deux opérations unaires. Si elles commutent,
on a $uv=vu.$ On conclut que le monoïde sous-jacent à une monade commutative
est aussi commutatif. Soit maintenant $t$ une opération n-aire. La
commutativité de $u$ et $t$ se traduit en
$t(ux_{1},\ldots,ux_{n})=ut(x_{1},\ldots,x_{n})$, qui devient la
distributivité de la somme lorsque $t=[+]$.
###### Exemple 7 (Commutativité des opérations binaires).
Soient $t,s\in\Sigma(\textbf{2})$ deux opérations binaires. On dit que $t$ et
$s$ commutent si pour tout $x,y,z,t$ d’un $\Sigma-$module $X$ quelconque on a
$t(s(x,y),s(z,t))=s(t(x,z),t(y,t))$. Lorsque $t=s=[+]$ cela signifie
simplement que $(x+y)+(z+t)=(x+z)+(y+t)$.
Il existe un rapport étroit entre l’additivité et la commutativité d’une
monade algébrique. D’abord, c’est très facile à montrer qu’une monade
commutative admet au plus une pseudoaddition (voir la fin de la paragraphe
4.6). Par conséquent, toute monade commutative hyperadditive est
automatiquement additive. De plus, d’après la proposition 4, une monade
additive est isomorphe à $\Sigma_{\lvert\Sigma\rvert}$. On en déduit qu’une
monade algébrique additive $\Sigma$ est commutative si et seulement si le
semi-anneau $\lvert\Sigma\rvert$ est commutatif. En vue de toutes ces
considérations, les monades algébriques commutatives ressemblent suffisamment
aux anneaux commutatifs pour introduire:
###### Définition 11.
On appelle anneau généralisé une monade algébrique commutative sur les
ensembles. On note par $\mathfrak{Gen}$ la sous-catégorie pleine des monades
algébriques formé par les anneaux généralisés.
Une fois que l’on est arrivés à la définition d’anneau généralisé, on peut
essayer de trouver la généralisation d’un corps. Évidemment, il ne suffit pas
de demander que tous les éléments non nuls du monoïde commutatif sous-jacent
soient inversibles. Entre les propriétés usuels des corps classiques, on
choisit la non-existence d’idéaux propres comme définition. On dit qu’un
anneau généralisé est sous-trivial s’il est isomorphe à une sous-monade de la
monade finale 1 décrite dans l’exemple 1. Ainsi,
###### Définition 12.
Un anneau généralisé non sous-trivial $K$ est un corps généralisé si tout
quotient strict de $K$ différent de lui même est sous-trivial.
Cette définition généralise la notion de corps au sens que tout corps
classique est un corps généralisé. Comme on va montrer dans la section
suivante, le corps à un élément et le corps résiduel de $\mathbb{Z}_{\infty}$
sont des corps généralisés.
## 4 Exemples
Dans cette section, on montre à travers d’exemples d’anneaux généralisés la
puissance du nouveau cadre algébrique, justifiant ainsi toute la machinerie
qui a été développée afin de construire la catégorie des monades algébriques
commutatives. On identifie toujours un anneau classique $R$ avec sa monade
associée $\Sigma_{R}.$ Comme on avait promis dans la remarque 1, on commence
par décrire la sous-monade de $\mathbb{Z}$ formée des combinaisons linéaires
formelles à coefficients non négatifs. Ensuite, on construit l’anneau local à
l’infini $\mathbb{Z}_{\infty}$ et on décrit les $\mathbb{Z}_{\infty}-$modules.
Ces deux objets étant définis, on est prêts pour identifier le corps à un
élément $\mathbb{F}_{1}$ avec leur intersection. On étudie de même les
extensions cyclotomiques de $\mathbb{F}_{1}$ et l’on montre que $\mathbb{Z}$
admet une présentation finie sur $\mathbb{F}_{1}.$ Dans la partie finale, on
définit, pour tout entier $N\geq 1,$ deux anneaux généralisés $A_{N}$ et
$B_{N}$ qui sont à la base de la construction de la compactification de
$\mathrm{Spec\ }\mathbb{Z}.$
### 4.1 $\mathbb{N}$
Soit $\mathbb{N}$ l’endofoncteur des ensembles dont l’image de chaque $X$ est
donnée par les combinaisons linéaires à support fini d’éléments de $X$ à
coefficients non-négatifs:
$\mathbb{N}(X)=\\{\lambda_{1}\\{x_{1}\\}+\ldots+\lambda_{n}\\{x_{n}\\}:\
\lambda_{i}\in\mathbb{Z},\lambda_{i}\geq 0,x_{i}\in X,n\geq 0\\}.$
Puisque $\epsilon(x)=\\{x\\}\in\mathbb{N}$ et le produit de deux entiers non-
négatifs est encore non-négatif, $\mathbb{N}$ est compatible avec la
multiplication et l’unité de $\mathbb{Z},$ donc il s’agit d’une sous-monade
dont l’ensemble sous-jacent est le monoïde des entiers non-négatifs.
Considérons un $\mathbb{N}-$module: c’est la donnée d’un ensemble $X$ et d’un
morphisme $\alpha:\mathbb{N}(X)\longrightarrow X$ tel que
$\alpha\circ\mu_{X}=\alpha\circ\mathbb{N}(\alpha)$ et que
$\alpha\circ\epsilon_{X}=\mathrm{id}_{X}.$ Alors, l’opération $x\cdot
y=\alpha(\\{x\\}+\\{y\\})$ munit $X$ d’une structure de monoïde commutatif et
l’on montre comme dans (3.1) que la catégorie des $\mathbb{N}-$modules est en
fait équivalente à celle des monoïdes commutatifs. La restriction de scalaires
par rapport à l’inclusion $\mathbb{N}\longrightarrow\mathbb{Z}$ est donc le
foncteur d’oubli des groupes abéliens vers les monoïdes commutatifs, qui admet
comme adjoint le foncteur libre $D\longmapsto G[D]$.
### 4.2 $\mathbb{Z}_{\infty}$
Soit maintenant $\Sigma_{\infty}$ la sous-monade de $\mathbb{R}$ définie par
les combinaisons octahedrales, c’est-à-dire, par les combinaisons linéaires
formelles à support fini $\Sigma_{i}\lambda_{i}\\{x_{i}\\},$ où les
$\lambda_{i}\in\mathbb{R}$ satisfont $\sum_{i}\lvert\lambda_{i}\rvert\leq 1.$
En effet, c’est une sous-monade de $\mathbb{R}$ car
$\\{x\\}\in\Sigma_{\infty}$ pour tout élément de la base et l’on a l’inégalité
$\sum_{i,j}\lvert\lambda_{i}\mu_{ij}\rvert=\sum_{i}\lvert\lambda_{i}\rvert\sum_{j}\lvert\mu_{ij}\rvert\leq\sum_{i}\lvert\lambda_{i}\rvert\leq
1,$
ce qui revient à dire que les combinaisons formelles des éléments dans
$\mathbb{Z}_{\infty}$ appartiennent de nouveau à $\mathbb{Z}_{\infty}.$ Le
monoïde sous-jacent est $\lvert\mathbb{Z}_{\infty}\rvert=[-1,1]$ avec la
multiplication induite par celle de $\mathbb{R}.$ En remplacent les nombres
réels par les complexes, on obtient la sous-monade
$\overline{\mathbb{Z}}_{\infty}\subset\mathbb{C}.$
Quels sont les $\mathbb{Z}_{\infty}-$modules? Par définition, un
$Z_{\infty}-$module est un ensemble $X$ muni d’une application
$\alpha:\Sigma_{\infty}(X)\longrightarrow X$ telle que $\alpha(\\{x\\})=x$
pour tout $x\in X$ et que
$\alpha(\sum_{i}\lambda_{i}\\{\alpha(\sum_{i}\mu_{ij}\\{x_{j}\\})\\})=\alpha(\sum_{i,j}\lambda_{i}\mu_{ij}\\{x_{j}\\})$
Remarquons que lorsque $X$ est un ensemble à $n$ points dans le plan affine,
$\Sigma_{\infty}(X)$ s’identifie à l’enveloppe convexe des points. Cela nous
amène à étudier les réseaux et les corps convexes dans un espace vectoriel.
Afin de donner une description plus parlant de $\mathbb{Z}_{\infty}$-module,
on construit d’abord la catégorie de $\mathbb{Z}_{\infty}-$réseaux, que l’on
plonge ensuite dans la catégorie de $\mathbb{Z}_{\infty}$-modules sans
torsion.
Commençons par rappeler le cas $p$-adique. Fixons donc un nombre premier $p$
et soit $E$ un $\mathbb{Q}_{p}$-espace vectoriel de dimension finie. Alors, il
existe une bijection entre l’ensemble de $\mathbb{Z}_{p}$-réseaux modulo
l’action multiplicative de $\mathbb{Q}_{p}^{\ast}$ et les sous-monoïdes
maximaux compactes (pour la topologie $p$-adique) de $\mathrm{End}(E)$, la
correspondance étant donné par
$A\longmapsto M_{A}=\\{g\in\mathrm{End}(E):g(A)\subset A\\}$
Si $E$ est un espace vectoriel réel de dimension finie, on peut associer à
toute forme quadratique définie positive $Q$ le sous-monoïde compact maximal
$M_{Q}=\\{g\in\mathrm{End}(E):Q(g(x))\leq Q(x)\ \forall x\in E\\}.$
Cependant, il existe des sous-monoïdes compacts maximaux n’ayant pas cette
structure. Puisque $Q$ définit la norme $||x||=\sqrt{Q(x)}$, on peut remplacer
les espaces vectoriels quadratiques par le cadre plus général des espaces
vectoriels normés. En fait, on obtient ainsi une équivalence entre les normes
sur $E$ et les corps compacts symétriques convexes inclus dans $E$ modulo
l’action multiplicative de $R^{\ast}$. En effet, si l’on se donne une
norme$||\cdot||$ sur $E$, l’ensemble $A_{||\cdot||}=\\{x\in E:||x||\leq 1\\}$
est un corps compact symétrique convexe et, réciproquement, tout corps $A$
vérifiant ces propriétés définit la norme
$||x||=\inf\\{\lambda>0:\lambda^{-1}x\in A\\}$. D’après [Du, 2.3.3], les corps
compacts symétriques convexes sont en bijection avec le sous-monoïdes maximaux
compacts (pour la topologie euclidienne usuelle) de $\mathrm{End}(E)$. Par
analogie avec le cas $p$-adique, on dit que:
###### Définition 13.
Un $\mathbb{Z}_{\infty}$-réseau est la donnée d’un couple
$A=(A_{\mathbb{Z}_{\infty}},A_{\mathbb{R}})$ formé d’un espace vectoriel
$A_{\mathbb{R}}$ de dimension finie et d’un corps compact symétrique convexe
$A_{\mathbb{Z}_{\infty}}\subset A_{\mathbb{R}}$. Un morphisme de
$\mathbb{Z}_{\infty}$-réseaux est un couple
$f=(f_{\mathbb{Z}_{\infty}},f_{\mathbb{R}})$ formé d’une application linéaire
$f_{\mathbb{R}}:A_{\mathbb{R}}\longrightarrow B_{\mathbb{R}}$ et d’une
application $f_{\mathbb{Z}_{\infty}}:A_{\mathbb{Z}_{\infty}}\longrightarrow
B_{\mathbb{Z}_{\infty}}$ telle que
$f_{\mathbb{Z}_{\infty}}=f_{{\mathbb{R}|A}_{\mathbb{Z}_{\infty}}}$.
Ensuite, on peut élargir la catégorie en oubliant le fait que l’espace
vectoriel $A_{\mathbb{R}}$ soit de dimension finie et que le corps symétrique
convexe soit compact, même fermé. On appelle $\mathbb{Z}_{\infty}$-modules
plats (ou sans torsion) les nouveaux couples. Maintenant on utilise la
construction généralisant la proposition 1 pour obtenir une notion de
$\mathbb{Z}_{\infty}$-module. Puisque le foncteur d’oubli de la catégorie des
$\mathbb{Z}_{\infty}$-modules sans torsion vers les ensembles
$\Gamma_{\mathbb{Z}_{\infty}}:\mathbb{Z}_{\infty}-\mathfrak{Modpl}\longrightarrow\mathfrak{Ens},\quad
A\longmapsto
A_{\mathbb{Z}_{\infty}}=\mathrm{Hom}_{\mathbb{Z}_{\infty}}(\mathbb{Z}_{\infty},A)$
admet comme adjoint à gauche le foncteur libre
$L_{\mathbb{Z}_{\infty}}:\mathfrak{Ens}\longrightarrow\mathbb{Z}_{\infty}-\mathfrak{Modpl},\quad
X\longmapsto\mathbb{Z}_{\infty}(X),$
on obtient une monade $\Sigma_{\infty}$ sur les ensembles. À différence de ce
qui se passait avec les monades associées à des anneaux, dans ce cas la
catégorie $\mathbb{Z}_{\infty}-\mathfrak{Modpl}$ n’est pas équivalente à
$\mathfrak{Ens}^{\Sigma_{\infty}}$, mais une sous-catégorie pleine. On définit
alors:
###### Définition 14.
Un $\mathbb{Z}_{\infty}$-module est un objet de la catégorie
$\mathfrak{Ens}^{\Sigma_{\infty}}$.
On rappel que dans la théorie standard des anneaux commutatifs, la
localisation de $\mathbb{Z}$ dans le nombre premier $p,$ que l’on note
$\mathbb{Z}_{(p)},$ est l’ensemble des fractions $\frac{m}{n}$ tels que $p$ ne
divise pas $n.$ Cela correspond à considérer les éléments de $\mathbb{Z}_{p}$
qui appartient à $\mathbb{Q},$ c’est-à-dire, l’intersection
$\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap\mathbb{Q}.$ Par analogie, on définit
$\mathbb{Z}_{(\infty)}=\mathbb{Z}_{\infty}\cap\mathbb{Q}.$
Alors, $\mathbb{Z}_{(\infty)}(X)$ est l’ensemble des combinaisons linéaires
octahedrales à coefficients rationnels. Dans la construction de
$\widehat{\mathrm{Spec\ }\mathbb{Z}},$ on aura besoin d’exiger que les
dénominateurs des coefficients soient seulement des puissances d’un entier
$N.$ Cela revient à considérer l’intersection
$A_{N}:=\mathbb{Z}[\frac{1}{N}]\cap\mathbb{Z}_{(\infty)}.$
### 4.3 Anneaux de valuation
L’exemple précédent suggère la nécessité d’étudier les anneaux de valuation
dans toute leur généralité. Soit $K$ un corps et soit $\lvert\cdot\rvert_{v}$
une valuation sur $K$ archimédienne ou non. On pose $N_{v}:=\\{x\in K:\lvert
x\rvert_{v}\leq 1\\}$ et on définit l’anneau de valuation de
$\lvert\cdot\rvert_{v}$ comme étant la plus grande sous-monade algébrique
$\mathcal{O}_{v}$ de $K$ telle que $N_{v}$ soit un $\mathcal{O}_{v}-$module.
Alors, $\mathcal{O}_{v}$ est un anneau classique si et seulement si
$\lvert\cdot\rvert_{v}$ est une valuation archimédienne. En effet,
$\mathcal{O}_{v}(\textbf{n})=\\{(\lambda_{1},\ldots,\lambda_{n})\in K^{n}:\
\lvert\sum_{i=1}^{n}\lambda_{i}x_{i}\rvert_{v}\leq 1\ \text{si}\ \lvert
x_{i}\rvert_{v}\leq 1\ \forall i=1,\ldots,n\\}$
Lorsque $\lvert\cdot\rvert_{v}$ est archimédienne, la condition ci-dessus
équivaut à $\lvert\lambda_{i}\rvert_{v}\leq 1$ pour tout $i=1,\ldots,n,$ de
sorte que $\mathcal{O}_{v}(\textbf{n})=N_{v}^{n}$. Si par contre
$\lvert\cdot\rvert_{v}$ n’est pas archimédienne, $\mathcal{O}_{v}(\textbf{n})$
dévient l’ensemble des n-uplets telles que
$\lvert\lambda_{1}\rvert_{v}+\ldots+\lvert\lambda_{n}\rvert_{v}\leq 1$. En
prenant la valeur absolue archimédienne dans $\mathbb{Q},\mathbb{R}$ et
$\mathbb{C}$ on obtient $\mathbb{Z}_{(\infty)},\mathbb{Z}_{\infty}$ et
$\overline{\mathbb{Z}}_{\infty}$ respectivement comme anneaux de valuation. On
montre comme dans l’exemple 4.7 que l’on récupère les anneaux initiaux en
localisant les anneaux de valuations dans le système multiplicatif engendré
par un élément $0<\lvert f\rvert<1$ quelconque.
Si $\lvert\cdot\rvert_{v}$ est une valuation archimédienne, on sait que
$\mathcal{O}_{v}$ est un anneau local d’idéal maximal
$\mathfrak{m}_{v}=\\{x\in K:\ \lvert x\rvert_{v}<1\\}$ et l’on appelle corps
résiduel le quotient de $\mathcal{O}_{v}$ par $\mathfrak{m}_{v}$. Étudions le
cas non-archimédien sur $\mathbb{R}$. Cela revient à quotienter
$\mathbb{Z}_{\infty}$ par les points intérieurs de $[-1,1],$ qui sont ceux à
valeur absolue strictement plus petit que $1.$ Il s’agit d’un idéal de
l’anneau généralisé $\mathbb{Z}_{\infty}$ au sens que l’on va introduire dans
la section suivante. Comme ensemble sous-jacent
$\mathbb{F}_{\infty}:=\mathbb{Z}_{\infty}/\mathfrak{m}_{\infty}$ est réduit à
$\\{-1,0,1\\}$, qui admet la structure de $\mathbb{Z}_{\infty}$-module
suivante:
$\alpha(\lambda_{-1}{-1}+\lambda_{0}\\{0\\}+\lambda_{1}\\{1\\})=\left\\{\begin{array}[]{lr}1&\lambda_{1}-\lambda_{-1}=1\\\
-1&\lambda_{1}-\lambda_{-1}=-1\\\
0&\lvert\lambda_{1}-\lambda_{-1}\rvert<1\end{array}\right.$
Soit maintenant $x=\lambda_{1}\\{1\\}+\ldots+\lambda_{n}\\{n\\}$ un élément de
$\mathbb{Z}_{\infty}(\textbf{n})$. Si
$\sum_{i=1}^{n}\lvert\lambda_{i}\rvert<1$, alors $[x]=[0]$ dans
$\mathbb{F}_{\infty}(\textbf{n})$. Considérons $x$ et $y$ tels que
$\sum_{i=1}^{n}\lvert\lambda_{i}\rvert=1$. Dans ce cas, $[x]=[y]$ si et
seulement si les suites des signes des coefficients $\lambda_{i}$ coïncident.
Cette monade n’est pas additive, car tous les éléments différents de
$\pm\\{i\\}$, $i=1,\ldots,n$ sont dans le noyau de l’application de
comparaison
$\pi_{n}:\mathbb{F}_{\infty}(\textbf{n})\longrightarrow\lvert\mathbb{F}_{\infty}\rvert^{n}$.
Notamment, $\mathbb{F}_{\infty}$ est engendré par une constante $0^{[0]}$, une
opération unaire $[-1]^{[1]}$ et une opération binaire $\ast^{[2]}$ défini
dans chaque $\mathbb{F}_{\infty}$-module par $[x]\ast[y]=[(1-\lambda)x+\lambda
y]$, $0<\lambda<1$ étant un nombre réel quelconque. L’opération $\ast$ vérifie
les propriétés suivantes:
$\displaystyle 0\ast 0=0,\quad x\ast 0=0,\quad x\ast x=x,\quad x\ast(-x)=0,$
$\displaystyle(-x)\ast(-y)=-(x\ast y),\quad(x\ast y)\ast z=x\ast(y\ast z)$
Montrons, par exemple, la commutativité:
$x\ast y-y\ast x=(1-\lambda)x+\lambda y-(1-\lambda)y-\lambda
x=(1-2\lambda)x+(2\lambda-1)y.$
Puisque $\lvert 1-2\lambda\rvert+\lvert 2\lambda-1\rvert<1$, l’élément $x\ast
y-y\ast x$ est nul dans le quotient. Cela nous permet d’écrire la présentation
suivante, où l’on utilise une extension de $\mathbb{F}_{1}$ que l’on va
définir dans le paragraphe suivant:
$\mathbb{F}_{\infty}=\mathbb{F}_{1^{2}}[\ast^{[2]}|\textbf{e}\ast\textbf{e}=0,\
\textbf{e}\ast\textbf{e}=\textbf{e},\ x\ast y=y\ast x,\ (x\ast y)\ast
z=x\ast(y\ast z)]$ (4.1)
Remarquons que les relations découlant de la commutativité ont été omises.
### 4.4 $\mathbb{F}_{1}:$ le corps à un élément
On peut finalement définir le corps à un élément comme la monade
$\mathbb{F}_{1}:=\mathbb{N}\cap\mathbb{Z}_{\infty}.$
Quand on regarde l’image d’un ensemble $X$ par $\mathbb{F}_{1},$ la condition
de positivité implique que les seules combinaison linéaires permises sont
$\\{x\\}$ pour quelque $x\in X,$ ou celle où tous les coefficients sont nuls.
Autrement dit: $\mathbb{F}_{1}(X)=X\sqcup 0.$
Un $\mathbb{F}_{1}-$module est donc un couple $(X,\alpha)$ où $\alpha:X\sqcup
0\longrightarrow X$ coïncide avec l’identité sur $X$ et $\alpha(0)=0_{X}\in
X.$ Par conséquent, on identifie les $\mathbb{F}_{1}-$modules aux ensembles
ayant un point distingué. Pour obtenir une présentation de $\mathbb{F}_{1},$
il suffit d’ajouter à la monade initiale la constante
$0\in\mathbb{F}_{1}(\textbf{0}),$ de sorte que
$\mathbb{F}_{1}=\mathbb{F}_{\emptyset}\langle 0^{[0]}\rangle$ est la monade
libre engendrée par une constante. Cela entraîne que se donner une monade
algébrique ayant une constante $0\in\Sigma(\textbf{0})$ soit équivalent à se
donner un morphisme de monades $\rho:\mathbb{F}_{1}\longrightarrow\Sigma.$ De
la présentation ci-dessus, on déduit que $\mathbb{F}_{1}$ est un corps
généralisé, car le seul quotient strict, $\mathbb{F}_{\emptyset}$, est sous-
trivial.
Puisque le monoïde sous-jacent à $\mathbb{F}_{1}$ est $\\{0,1\\},$ le seule
idéal propre est $\\{0\\},$ autrement dit, $\mathrm{Spec\ }\mathbb{F}_{1}$ est
réduit à un point. C’est pour cela que l’on se réfère au spectre du corps à un
élément comme le point absolu. On obtient un résultat analogue pour
$\mathbb{F}_{\emptyset}$, ainsi que pour les extensions cyclotomiques de
l’exemple suivant. Cependant, il y a encore des géométries au-dessus de
$\mathbb{F}_{1}$. En effet, dans [TV] les auteurs montrent qu’il y a un
foncteur d’extension de base des schémas relatifs à la catégorie monoïdale
$(\mathfrak{SEns},\times,\ast)$ des ensembles simpliciaux⋆ munis du produit
direct vers les schémas sur $\mathbb{F}_{1}$. Le spectre de
$\mathbb{F}_{\emptyset}$ étant l’objet initial de la catégorie des schémas
affines généralisés, il serait intéressant d’élucider le rôle de cette
géométrie dans l’approche de Durov.
### 4.5 Extensions cyclotomiques
De même, l’extension $\mathbb{F}_{1^{2}}$ de $\mathbb{F}_{1}$ est définie
comme la monade
$\mathbb{F}_{1^{2}}:=\mathbb{Z}\cap\mathbb{Z}_{(\infty)}=\mathbb{Z}\cap\mathbb{Z}_{\infty}.$
Ainsi, $F_{1^{2}}(X)$ est formé des combinaisons formelles à au plus un
coefficient entier de valeur absolue $1,$ ce qui donne $0,$ les éléments de la
base $\\{x\\}$ et leurs opposés $-\\{x\\},$ de sorte que
$\mathbb{F}_{1^{2}}(X)=X\sqcup-X\sqcup 0.$ On a
$\mathbb{F}_{1}=\mathbb{N}\cap\mathbb{F}_{1^{2}}.$ Évidemment,
$\mathbb{F}_{1^{2}}$ est de type fini sur $\mathbb{F}_{1},$ engendré par une
opération unaire:
$\mathbb{F}_{1^{2}}=\mathbb{F}_{\emptyset}\langle 0^{[0]},[-]^{[1]}\ |\
-(-x)=x\rangle=\mathbb{F}_{1}\langle[-]^{[1]}\ |\ -(-x)=x\rangle$
Aux données qui définissent un $\mathbb{F}_{1}-$module, il faut ajouter
l’action de $\alpha$ sur $-X$ afin d’avoir un module sur $\mathbb{F}_{1^{2}}.$
Puisque on peut la voir comme une involution laissant invariant $0_{X},$
$\mathbb{F}_{1^{2}}-$Mod est la catégorie des ensembles avec un point
distingué munis d’une involution qui préserve ce point. Cette extension est
d’une spéciale importance depuis les travaux d’Alain Connes et Caterina
Consani, qui ont montré dans [CC] que les groupes de Chevalley n’ont pas une
structure de variété sur $\mathbb{F}_{1}$ mais sur $\mathbb{F}_{1^{2}},$ en
répondant de cette façon inattendue à la question posée par Soulé dans [So2].
Est-ce qu’il y a d’extensions de $\mathbb{F}_{1}$ de degré plus grand que
deux? Même si cela pourrait paraître paradoxale, la réponse à cette question a
été trouvée plus d’une dizaine d’ans avant d’avoir une définition précise de
$\mathbb{F}_{1}.$ En effet, dans le manuscrit non publié [KS] Kapranov et
Smirnov ont proposé de penser à $\mathbb{F}_{1^{n}}$ comme le monoïde formé de
zéro et les racines n-ièmes de l’unité $\mu_{n}$. En suivant des idées de Weil
et d’Iwasawa selon lesquelles ajouter des racines de l’unité est équivalent à
faire une extension du corps de base, ils suggèrent qu’un schéma $X$ est
défini sur $\mathbb{F}_{1^{n}}$ lorsque l’anneau des fonctions régulières sur
$X$ contient les racines n-ièmes de l’unité. On peut de même penser à la
clôture algébrique de $\mathbb{F}_{1}$ ou à la droite affine
$\mathbb{A}^{1}_{\mathbb{F}_{1}}$ comme le monoïde contenant zéro et toutes
les racines de l’unité. Ces idées, ainsi que le travail [Hab] de Habiro, ont
inspiré les définitions de coordonnées cyclotomiques et de fonctions
analytiques sur $\mathbb{F}_{1}$ [Ma3].
Dans le langage des monades,
$\mathbb{F}_{1^{n}}=\mathbb{F}_{1}\langle\zeta_{n}^{[1]}|\zeta_{n}^{n}x=x\rangle$.
Remarquons que $\mathbb{F}_{1}$ étant un quotient strict de
$\mathbb{F}_{1^{n}},$ il ne s’agit pas d’un corps généralisé si $n\geq 2$. Un
$\mathbb{F}_{1^{n}}-$module est un ensemble muni d’une action libre de
$\mathbb{Z}/n\mathbb{Z}$. La correspondance
$\varphi_{m,n}:\zeta_{n}\longmapsto\zeta_{nm}^{m}$ induit un plongement de
$\mathbb{F}_{1^{n}}$ dans $\mathbb{F}_{1^{nm}}$ pour des entiers $n,m\geq 1$
quelconques. Si l’on prend la limite inductive par rapport à $\varphi_{m,n}$,
on obtient l’anneau généralisé
$\mathbb{F}_{1^{\infty}}:=\varinjlim\mathbb{F}_{1^{n}}=\mathbb{F}_{1}\langle\zeta_{1},\zeta_{2},\ldots|\zeta_{1}=\textbf{e},\
\zeta_{n}=\zeta_{nm}^{m}\rangle$
Ce nouveau objet amusant acquiert dans [CCM] une interprétation en termes du
système de Bost-Connes (cf. [BC]).
### 4.6 $\mathbb{Z}$ admet une présentation finie sur $\mathbb{F}_{1}$
On montre d’abord que $\mathbb{Z}$ est de type fini sur
$\mathbb{F}_{\emptyset}.$ En effet, il suffit de prendre comme opérations
$0\in\Sigma_{\mathbb{Z}}(\textbf{0}),[-]\in\Sigma_{\mathbb{Z}}(\textbf{1})$ et
$[+]\in\Sigma_{\mathbb{Z}}(\textbf{2}),$ soumis à l’ensemble $R$ de relations
exprimant que $(\mathbb{Z},+)$ est un groupe abélien:
$x+(-x)=0,\quad x+0=x=0+x,\quad(x+y)+z=x+(y+z),\quad x+y=y+x.$
On a ainsi
$\mathbb{Z}=\mathbb{F}_{\emptyset}\langle[0]^{[0]},[-]^{[1]},[+]^{[2]}\ |\
R\rangle=\mathbb{F}_{1}\langle[-]^{[1]},[+]^{[2]}\ |\
R\rangle=\mathbb{F}_{1^{2}}\langle[+]^{[2]}\ |\ R\rangle.$ (4.2)
Si l’on sous-entend que $\mathbb{Z}$ est une monade algébrique commutative, on
n’a besoin que de deux premières relations, de sorte que
$\mathbb{Z}=\mathbb{F}_{1}\langle[-]^{[1]},[+]^{[2]}\ |\
0+\textbf{e}=\textbf{e}=\textbf{e}+0,\ (-\textbf{e})+\textbf{e}=0\rangle$
Cette présentation peut être interprété comme donnant une réponse positive à
la question si $\mathbb{Z}$ est un anneau de polynômes qui se pose en vue de
l’analogie entre les corps de nombres et les corps de fonctions discuté au
début. En effet, on peut voir $\mathbb{Z}$ comme un quotient d’un anneau de
polynômes sur $\mathbb{F}_{1},$ dont les deux variables sont des opérations,
par l’idéal engendré par les relations ci-dessus. En particulier, $\mathbb{Z}$
est de type fini sur $\mathbb{F}_{1}.$ C’est alors immédiat de montrer que:
###### Proposition 5.
Un anneau $R$ est absolument de type fini si et seulement s’il est de type
fini sur $\mathbb{Z}$ au sens classique, c’est-à-dire, s’il existe des
éléments $x_{1},\cdots,x_{n}\in R$ tels que
$R=\mathbb{Z}[x_{1},\ldots,x_{n}].$
De même, on obtient la présentation
$\mathbb{N}=\mathbb{F}_{1}\langle[+]^{[2]}\ |\
0+\textbf{e}=\textbf{e}=\textbf{e}+0\rangle.$ (4.3)
Le fait que $\mathbb{Z}$ soit de type fini sur $\mathbb{F}_{1}$ entraînera
évidement que $\mathrm{Spec\ }\mathbb{Z}$ est de type fini sur $\mathrm{Spec\
}\mathbb{F}_{1}$, après avoir donné un sens rigoureux au point absolu
$\mathrm{Spec\ }\mathbb{F}_{1}$. Cela contredit les prévision de Manin dans
[Ma4], où la question sur quelle est la dimension de $\mathrm{Spec\
}\mathbb{Z}$ est abordée. Du point de vue plus orthodoxe, la dimension de
Krull de $\mathrm{Spec\ }\mathbb{Z}$ est un et les nombres premiers sont des
points zéro-dimensionnels dans $\mathrm{Spec\ }\mathbb{Z}$, que l’on peut
considérer géométriquement comme les images du morphismes $\mathrm{Spec\
}\mathbb{F}_{p}\longrightarrow\mathrm{Spec\ }\mathbb{Z}$. Cependant, quelques
analogies surprenantes entre les nombres premiers et les noeuds [Mo], ainsi
que l’existence d’une dualité de Poincaré en dimension $3$ pour la topologie
étale de $\mathrm{Spec\ }\mathbb{Z}$, suggèrent qu’une autre possible réponse
est $\dim\mathrm{Spec\ }\mathbb{Z}=3$. Finalement, Manin a conjecturé que la
dimension de $\mathrm{Spec\ }\mathbb{Z}$ est infini lorsque l’on regarde le
spectre des entiers sur le point absolu. Dans l’approche de Durov,
$\mathrm{Spec\ }\mathbb{F}_{1}$ admet une présentation de type fini, mais ce
n’est pas clair comment on peut définir une notion de dimension. Remarquons,
en tout cas, que $\mathbb{Z}$ est engendré par trois opérations.
On montre l’utilité de décrire une monade par une présentation par des
opérations et par des relations en calculant le produit tensoriel
$\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ dans la catégorie des anneaux
généralisés. Prenons d’abord la première présentation dans 4.2. La monade
commutative $\mathbb{Z}\otimes_{\mathbb{F}_{\emptyset}}\mathbb{Z}$ est donc
engendré par deux constantes $c,d$ deux opérations unaires $u,v$ et deux
opérations binaires $s$ et $t$ soumis aux mêmes relations. D’après l’exemple
5, la commutativité implique $c=d.$ On peut donc considérer
$\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z}$. En remplacent $y$ et $z$ par
la constante $0$ dans l’équation de l’exemple 7 et en utilisant les relations,
on en déduit $s(x,t)=t(x,t)$ pour tout $x,t$ dans un module $X,$ donc $s=t$.
Un raisonnement pareil montre que $u$ et $v$ doivent aussi coïncider, de sorte
que $\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}.$ Il s’agit d’un
résultat en quelque mode décevant car l’on s’attendait de trouver une
catégorie où le produit tensoriel de $\mathbb{Z}$ par soi même n’était pas
trivial.
### 4.7 Les anneaux généralisés $A_{N}$ et $B_{N}.$
Dans cette épigraphe on étudie deux anneaux qui vont être essentiels dans la
construction de la compactification de $\mathrm{Spec\ }\mathbb{Z}.$ Soit
$N\geq 1$ entier, définissons $f=\frac{1}{N}.$ D’abord, on pose
$B_{N}=\mathbb{Z}[f],$ qui est donc un anneau au sens classique ayant
$\mathrm{Spec\ }B_{N}=(\bigcup_{p\nmid N}pB_{N})\cup(0)$
On définit ensuite $A_{N}$ comme l’intersection
$A_{N}:=B_{N}\cap\mathbb{Z}_{(\infty)}=B_{N}\cap\mathbb{Z}_{\infty}$. Tout
élément de $B_{N}(\textbf{n})$ peut être écrit comme un $n$-uplet
$(\frac{u_{1}}{N^{k}},\ldots,\frac{u_{n}}{N^{k}}),$ où $u_{i}\in\mathbb{Z}$ et
$k\geq 0$ est le plus grand exposant qui apparaît dans les dénominateurs.
Alors, l’intersection avec $\mathbb{Z}_{\infty}$ impose la contrainte
$\sum_{i=1}^{n}\lvert u_{i}\rvert\leq N^{k}$ et l’on en déduit:
$\displaystyle A_{N}(\textbf{n})$
$\displaystyle=\\{(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Q}^{n}:\lambda_{i}\in
B_{N},\sum_{i=1}^{n}\lvert\lambda_{i}\rvert\leq 1\\}=$
$\displaystyle=\\{(\frac{u_{1}}{N^{k}},\ldots,\frac{u_{n}}{N^{k}}):k\geq
0,u_{i}\in\mathbb{Z},\sum_{i=1}^{n}\lvert u_{i}\rvert\leq N^{k}\\}$ (4.4)
Notamment le monoïde sous-jacent à $A_{N}$ est l’ensemble $\lvert A_{N}\rvert$
des éléments de $B_{N}$ à valeur absolue plus petite ou égale à 1. On a un
plongement canonique $A_{N}\longrightarrow B_{N},$ induisant donc un morphisme
injectif des localisations (voir la section suivante pour une définition
précise) $A_{N}[f^{-1}]\longrightarrow B_{N}[f^{-1}]=B_{N}.$ Soit
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in B_{N}(\textbf{n}).$ Il existe un
entier $k\geq 0$ tel que
$\lvert\lambda_{1}\rvert+\ldots+\lvert\lambda_{n}\rvert\leq N^{k}.$ Alors
$f^{k}\lambda=N^{-k}\lambda$ appartient à $A_{N}(\textbf{n}),$ ce qui revient
à dire que $\lambda\in A_{N}[f^{-1}](\textbf{n}).$ Par conséquent, $B_{N}$ est
une sous-monade de $A_{N}[f^{-1}],$ d’où l’égalité $B_{N}=A_{N}[f^{-1}].$
###### Proposition 6.
L’anneau généralisé $A_{N}$ satisfait les propriétés suivantes:
1. 1.
C’est un anneau local généralisé d’idéal maximal
$\mathfrak{p}_{\infty}=\\{\lambda\in B_{N}:\lvert\lambda\rvert\leq 1\\}$.
2. 2.
À l’exception de $\mathfrak{p}_{\infty}$, les idéaux premiers non nuls de
$A_{N}$ sont en bijection avec les nombres premiers $p$ ne divisant pas $N$.
###### Proof.
Pour la première partie de la proposition, il suffit de considérer le
plongement $j:A_{N}\longrightarrow\mathbb{Z}_{(\infty)}.$ Comme on a vu dans
l’exemple 4.2, $\mathbb{Z}_{(\infty)}$ est un anneau local généralisé d’idéal
maximal
$\mathfrak{m}_{(\infty)}=\\{\lambda\in\mathbb{Q}:\lvert\lambda\rvert<1\\}.$
L’image réciproque de $\mathfrak{m}_{(\infty)}$ par $j$ est l’idéal maximal
$\mathfrak{p}_{\infty}:=j^{-1}(\mathfrak{m}_{(\infty)})=\\{\lambda\in\lvert
B_{N}\rvert:\lvert\lambda\rvert<1\\}$. Comme $\lvert
A_{N}\rvert-\mathfrak{p}_{\infty}=\\{-1,1\\}$ est l’ensemble des éléments
inversibles de $\lvert A_{N}\rvert,$ on conclut que $A_{N}$ est local. Pour la
deuxième partie, soit $p$ premier ne divisant pas $N$. Alors $pB_{N}$ est un
idéal premier de $B_{N}.$ Si l’on pose $\mathfrak{p}_{p}:=i^{-1}(pB_{N})$, $i$
étant le plongement canonique de $A_{N}$ dans $B_{N}$, on obtient un idéal
premier de $A_{N}$. Puisque $B_{N}$ est la localisation de $A_{N}$ dans le
système multiplicatif engendré par $\frac{1}{N}$, les idéaux premiers de
$A_{N}$ ne contenant pas $\frac{1}{N}$ sont en bijection avec $\mathrm{Spec\
}B_{N}$. La preuve se réduit donc à montrer que le seul idéal premier de
$A_{N}$ contenant $\frac{1}{N}$ est l’idéal maximal, ce qui revient à dire que
si $\mathfrak{p}$ est un tel idéal, alors
$\mathfrak{p}_{\infty}\subset\mathfrak{p}$. Soit
$\lambda\in\mathfrak{p}_{\infty}$. Comme $\lvert\lambda\rvert<1$ il existe $k$
suffisamment grand tel que $\lvert\lambda\rvert^{k}<\frac{1}{N}$. Alors
$\mu=N\lambda^{k}\in\lvert A_{N}\rvert,$ donc $\lambda^{k}\in\mathfrak{p}$.
Mais $\mathfrak{p}$ étant premier, cela implique $\lambda\in\mathfrak{p}$,
donc $\mathfrak{p}_{\infty}\subset\mathfrak{p}.$ ∎
La conséquence la plus importante de la proposition ci-dessus est la
description du spectre de l’anneau généralisé $A_{N}$, où l’on voit apparaître
pour la première fois une réalisation concrète du premier infini:
$\mathrm{Spec\ }A_{N}=\\{(0)\\}\cup\\{\mathfrak{p}_{p}:p\nmid
N\\}\cup\\{\mathfrak{p}_{\infty}\\}$
Dorénavant, on note $\xi:=(0)$, $p:=\mathfrak{p}_{p}$ et
$\infty:=\mathfrak{p}_{\infty}$ par analogie avec une courbe projective. Alors
$\mathrm{Spec\ }A_{N}$ contient un point générique $\xi$ et un seul point
fermé $\infty$, dont le complément est homéomorphe à $\mathrm{Spec\
}B_{N}=\mathrm{Spec\ }\mathbb{Z}[\frac{1}{N}]$. Par conséquent,
$\overline{p}=\\{p,\infty\\}$.
###### Corollaire 2.
Soit $U\subsetneq\mathrm{Spec\ }A_{N}$ une partie non-vide. Alors $U$ est
ouverte si et seulement si $\mathrm{Spec\ }A_{N}-U$ contient $\infty$ et un
nombre fini d’idéaux $p$.
On finit l’étude de l’anneau généralisé $A_{N}$ en énonçant un théorème, dont
la démonstration se trouve dans [Du, 7.1.26], qui sera important dans la
discussion sur la dimension de la compactification de $\mathrm{Spec\
}\mathbb{Z}$.
###### Theorème 1.
L’anneau généralisé $A_{N}$ admet une présentation finie sur $\mathbb{F}_{1},$
engendrée par les opérations
$s_{p}(\\{1\\},\ldots,\\{p\\})=\frac{1}{p}\\{1\\}+\ldots+\frac{1}{p}\\{p\\},$
où $p$ est un diviseur premier de $N,$ et les relations
$\displaystyle s_{p}(\\{1\\},\ldots,\\{1\\})$ $\displaystyle=\\{1\\}$
$\displaystyle s_{p}(\\{1\\},\ldots,\\{p\\})$
$\displaystyle=s_{n}(\\{\sigma(1)\\},\ldots,\\{\sigma(p)\\}),\
\quad\sigma\in\mathcal{S}_{p}$ $\displaystyle
s_{p}(\\{1\\},\ldots,\\{p-1\\},-\\{p-1\\})$
$\displaystyle=s_{n}(\\{1\\},\ldots,\\{p-2\\},0,0)$
Table 1: Exemples d’anneaux généralisés Monade | Hypoadditive | Hyperadditive | Corps | Présentation
---|---|---|---|---
$\mathbb{Z}$ | oui | oui | non | (4.2)
$\mathbb{N}$ | oui | oui | non | (4.3)
$\mathbb{Z}_{\infty}$ | oui | non | non | ?
$\mathbb{Z}_{(\infty)}$ | oui | non | non | $\mathbb{F}_{1^{2}}\langle\\{s_{n}\\}_{n>1}\rangle$
$\overline{\mathbb{Z}}_{\infty}$ | oui | non | non | ?
$\mathbb{F}_{\emptyset}$ | non | non | non | monade initiale
$\mathbb{F}_{1}$ | oui | non | oui | $\mathbb{F}_{\emptyset}[0^{[0]}]$
$\mathbb{F}_{1^{n}}$ | oui | non | non | $\mathbb{F}_{1}\langle\zeta_{n}^{[1]}|\zeta_{n}^{n}x=x\rangle$
$\mathbb{F}_{1^{\infty}}$ | oui | non | non | non finie
$\mathbb{F}_{\infty}$ | non | non | oui | (4.1)
$A_{N}$ | oui | non | non | theor. 1
$B_{N}$ | oui | oui | non | $B_{N}=\mathbb{Z}[N^{-1}]$
## 5 Vers la compactification de $\mathrm{Spec\ }\mathbb{Z}$
### 5.1 Schémas généralisés
On a déjà remarqué dans l’introduction que l’étape délicate de la nouvelle
théorie consistait en définir une catégorie juste pour remplacer les anneaux
commutatifs usuels. Après cela, la construction d’un schéma affine généralisé
est totalement analogue à celle provenant de la géométrie algébrique
classique. Le travail le plus dur a été, donc, déjà fait. Dans cette section,
on définit d’abord ce que l’on entend par localisation et par idéaux premiers,
puis on construit le spectre d’un anneau généralisé comme espace topologique
et on le munit d’un faisceau d’anneaux généralisés, ce qui nous permet
d’arriver à la définition de espace annelé généralisé.
Soit $A$ un anneau généralisé et soit $S\subset\lvert A\rvert$ un sous-
monoïde. On appelle localisation de $A$ l’objet initial $A[S^{-1}]$ dans la
catégorie des couples $(B,\rho),$ où $B$ est un anneau généralisé et
$\rho:A\longrightarrow B$ est un morphisme tel que tous les éléments de
$\rho_{1}(S)\subset\lvert B\rvert$ sont inversibles. Lorsque $S$ est le
système multiplicatif engendré par un élément $f\in\lvert A\rvert,$ on écrit
$A[f^{-1}]$ ou $A_{f}$. Une description plus explicite est la donnée par la
présentation suivante, dans laquelle on considère les opérations unaires
$s^{-1}$ pour tout $s\in S:$
$A[S^{-1}]=A\langle(s^{-1})_{s\in S}|ss^{-1}=\textbf{e}=s^{-1}s\rangle.$
Comme dans le cas classique, $A[S^{-1}](\textbf{n})$ contient les classes
d’équivalence des couples $(a,s)\in A(\textbf{n})\times S$ modulo la relation
$(x,s)\sim(y,t)\Leftrightarrow\exists u\in S$ tel que $utx=usy$.
Soit $A$ un anneau généralisé. On appelle idéal de $A$ tout $A$-sous-module du
monoïde sous-jacent $\lvert A\rvert$. En imitant la théorie usuelle, les
idéaux premiers sont les idéaux $\mathfrak{p}$ tels que
$S_{\mathfrak{p}}:=\lvert A\rvert-\mathfrak{p}$ est un système multiplicatif,
et les idéaux maximaux sont les éléments maximaux pour l’ordre partiel défini
par l’inclusion dans l’ensemble d’idéaux propres de $A$. Comme dans l’algèbre
commutative, tout idéal maximal est premier et tout anneau généralisé admet au
moins un idéal maximal. Quand il n’y en a plus, l’anneau est dit local. Cela
équivaut à dire que tous les éléments du complément de l’idéal dans $\lvert
A\rvert$ sont inversibles. On est alors en état de construire le spectre
premier d’un anneau généralisé:
###### Définition 15.
Soit $A$ un anneau généralisé. On appelle spectre premier de $A$, et on le
note $\mathrm{Spec\ }A,$ l’ensemble d’idéaux premiers de $A$ muni de la
topologie, dite de Zariski, dont les fermés sont les ensembles
$V(M):=\\{\mathfrak{p}\in\mathrm{Spec\ }A:M\subset\mathfrak{p}\\},$
$M$ étant un sous-ensemble quelconque de $\lvert A\rvert$. Une base de cette
topologie est formée des ouverts principaux
$D(f):=\\{\mathfrak{p}:f\notin\mathfrak{p}\\},$ où $f\in\lvert A\rvert$.
Chaque $A-$module $M$ définit un faisceau d’anneaux généralisés $\tilde{M}$
sur $\mathrm{Spec\ }A$ dont les sections sont $\Gamma(D(f),M)=M_{f}$ pour
chaque ouvert principal $D(f)$ correspondant à un élément $f$ du monoïde
$\lvert A\rvert.$ Lorsque $M=A$ on obtient le faisceau structural
$\mathcal{O}_{\mathrm{Spec}A}.$ Tous les $\tilde{M}$ sont alors des
$\mathcal{O}_{\mathrm{Spec}A}$-modules.
###### Définition 16.
Un espace annelé généralisé est la donnée d’un couple $(X,\mathcal{O}_{X})$
formé d’un espace topologique et d’un faisceau d’anneaux généralisés sur $X$.
Un morphisme d’espaces annelés généralisés est un couple
$(f,f^{\sharp}):(X,\mathcal{O}_{X})\longrightarrow(Y,\mathcal{O}_{Y})$ formé
d’une application continue $f:X\longrightarrow Y$ et d’un morphisme de
faisceaux $f^{\sharp}:\mathcal{O}_{Y}\longrightarrow f_{\ast}\mathcal{O}_{X}$,
où $f_{\ast}\mathcal{O}_{X}$ est l’image directe du faisceau $\mathcal{O}_{X}$
par $f$. On dit que $(X,\mathcal{O}_{X})$ est localement annélé si pour tout
$P\in X$ l’anneau des germes $\mathcal{O}_{X,P}$ est local.
On obtient de même une notion d’isomorphisme d’espaces annelés généralisés.
Les schémas affines généralisés sont alors les espaces annelés les plus
simples et l’on va construire les schémas généralisés en les recollant:
###### Définition 17.
Un schéma affine généralisé $X$ est un espace localement annelé généralisé qui
est isomorphe à $(\mathrm{Spec\ }A,\mathcal{O}_{\mathrm{Spec}A})$ pour un
certain anneau généralisé $A$. Un schéma généralisé est alors un espace annélé
généralisé $(X,\mathcal{O}_{X})$ qui admet un recouvrement ouvert par des
schémas affines généralisés. Lorsque $\Gamma(X,\mathcal{O}_{X})$ admet une
constante, ce qui revient à dire que $\Gamma(X,\mathcal{O}_{X}(\textbf{0}))$
est non-vide, on dit que $X$ est un schéma sur $\mathbb{F}_{1}$.
Un morphisme entre deux schémas généralisés $X$ et $Y$ est un morphisme $f$
des espaces annélés généralisés $(X,\mathcal{O}_{X})$ et $(Y,\mathcal{O}_{Y})$
tel que pour tout ouvert affine $\mathrm{Spec\ }B=V\subset Y$ et
$\mathrm{Spec\ }A=U\subset X$ vérifiant $f(V)\subset U$, la restriction
$f|V:V\longrightarrow U$ soit induite par un morphisme d’anneaux généralisés
$\varphi:A\longrightarrow B$. On obtient ainsi la catégorie des schémas
généralisés. Comme dans la géométrie algébrique usuelle du point de vue
fonctoriel, un schéma généralisé peut être défini d’une manière plus
intrinsèque comme un foncteur contravariant
$\mathcal{S}:\mathfrak{Gen}\longrightarrow\mathfrak{Ens}$ de la catégorie des
anneaux généralisés vers la catégorie des ensembles qui est localement
isomorphe au spectre d’un anneau pour la topologie de Zariski. On se refère à
l’annexe A pour la construction fonctorielle complète, inspiré de [TV], d’un
schéma généralisé sur $\mathbb{F}_{1}$.
### 5.2 Compactification à la main
Dans les exemples précédents, on a obtenu des descriptions complètes des
spectres
$\mathrm{Spec\ }\mathbb{Z}=\\{\xi,p,\ldots\\},\quad\mathrm{Spec\
}A_{N}=\\{\xi,p,\ldots,\infty\\}_{p\nmid N},\quad\mathrm{Spec\
}B_{N}=\\{\xi,p,\ldots\\}_{p\nmid N}.$
On définit $\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$ comme le schéma affine
généralisé obtenu en recollant $\mathrm{Spec\ }\mathbb{Z}$ et $\mathrm{Spec\
}A_{N}$ le long de leurs sous-ensembles ouverts principaux homéomorphes à
$\mathrm{Spec\ }B_{N}$:
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}:=\mathrm{Spec\
}\mathbb{Z}\coprod_{\mathrm{Spec\ }B_{N}}\mathrm{Spec\ }A_{N}$
Ainsi, $\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}=\mathrm{Spec\
}\mathbb{Z}\cup\\{\infty\\}$ contient déjà un point additionnel $\infty$
correspondant à la valuation archimédienne de $\mathbb{Q}$. Le point $\xi$ est
de nouveau générique et les points fermés sont $\infty$ et les premiers $p$
divisant $N$. Par conséquent:
###### Lemme 2.
Une partie non-vide $U\subsetneq\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$ est
ouverte si et seulement si elle contient $\xi$, son complément est fini et
soit $\infty\notin U$ soit $\mathrm{Spec\ }B_{N}\subset U.$
###### Proof.
C’est clair que le point générique est inclus dans toutes les parties ouvertes
non vides et que le complément de $U$ doit être fini, puisque $\mathrm{Spec\
}\mathbb{Z}$ est ouvert dans $\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$. Soit
alors $U$ ouvert contenant $\xi$ à complément fini $\\{p_{1},\ldots,p_{n}\\}$.
Supposons qu’un des $p_{i}$ ne divise pas $N$. Alors,
$\overline{\\{p_{i}\\}}=\\{p_{i},\infty\\}$, donc $\infty\notin U$.
Réciproquement, si $U$ est une partie de $\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}$ vérifiant les trois conditions de l’énoncé, son complément
est un ensemble fini $S=\\{p_{1},\ldots,p_{n}\\}$ tel que $p_{i}\neq\xi$ pour
tout $1\leq i\leq n$ et que, soit $\infty$ est un des $p_{i}$ et alors $S$ est
fermé, soit tous les premiers $p_{i}$ sont parmi les diviseurs de $N$, donc
$S$ est une réunion de points fermés. ∎
D’après le lemme, la topologie de $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ dépend
fortement du choix de $N$. Puisque $\xi$ et $p$ appartiennent au sous-schéma
ouvert $\mathrm{Spec\ }\mathbb{Z}$, lorsque l’on calcule les anneaux des
germes $\mathcal{O}_{p}$, on obtient le résultat attendu
$\mathcal{O}_{\xi}=\mathbb{Q}$ et $\mathcal{O}_{p}=\mathbb{Z}_{(p)}$.
Cependant, $\mathrm{Spec\ }A_{N}$ étant le voisinage ouvert à l’infini,
l’anneau des germes n’est pas $\mathbb{Z}_{(\infty)}$ comme l’on voudrait,
mais $\mathcal{O}_{\infty}=A_{N,\mathfrak{p}}=A_{N}$. Ces deux observations
motivent l’idée de faire disparaître $N$ en prenant une certaine limite
projective⋆. Ensuite, on décrit le système filtré que l’on va considérer.
Soient $N,M>1$ des entiers. On construit un morphisme
$f:=f_{N}^{NM}:\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(NM)}\longrightarrow\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$
en choisissant d’abord des sous-schémas ouverts $U_{1}\approx\mathrm{Spec\
}\mathbb{Z},\ U_{2}\approx\mathrm{Spec\ }A_{N}$ de $\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}$ et $U_{1}^{\prime}\approx\mathrm{Spec\ }\mathbb{Z},\
U_{2}^{\prime}\approx\mathrm{Spec\ }A_{NM}$ de $\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(NM)}$ respectivement. Ainsi,
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}=U_{1}\cup
U_{2},\quad\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(NM)}=U_{1}^{\prime}\cup
U_{2}^{\prime},$
l’intersection étant $W=U_{1}\cap U_{2}=U_{1}^{\prime}\cap
U_{2}^{\prime}=\mathrm{Spec\ }B_{N}$. On pose alors $V_{1}=U_{1}^{\prime}$ et
$V_{2}=U_{2}^{\prime}\cup W$. On a de nouveau $V_{1}\cap V_{2}=W$. Se donner
le morphisme f est donc équivalent à se donner des morphismes
$f_{i}:V_{i}\longrightarrow U_{i},\ i=1,2$, qui coïncident sur $W=V_{1}\cap
V_{2}$ et tels que $f_{i}^{-1}(U_{1}\cap U_{2})=V_{1}\cap V_{2}$.
Puisque $U_{1}=V_{1}=\mathrm{Spec\ }\mathbb{Z}$, on peut choisir $f_{1}$ comme
étant l’identité sur $\mathrm{Spec\ }\mathbb{Z}.$ Alors,
$f_{1|W}=\mathrm{id}_{W}$. Pour $f_{2}$ on peut faire à peu près la même
chose, quitte à utiliser la correspondance bijective entre les morphismes
$f:V_{2}\longrightarrow U_{2}$ dans la catégorie des schémas généralisés et
les morphismes
$\varphi:\Gamma(U_{2},\mathcal{O})\longrightarrow\Gamma(V_{2},\mathcal{O})$
dans la catégorie des anneaux généralisés. En effet, $U_{2}$ étant
$\mathrm{Spec\ }A_{N}$, le premier terme est juste
$\Gamma(U_{2},\mathcal{O})=A_{N}$. En ce qui concerne le deuxième:
$\Gamma(V_{2},\mathcal{O})=\Gamma(U_{2}^{\prime},\mathcal{O})\times_{\Gamma(U_{2}^{\prime}\cap
W,\ \mathcal{O}_{\mathrm{Spec}\mathbb{Z}})}\Gamma(W,\mathcal{O}).$
Puisque $U_{2}^{\prime}=\mathrm{Spec\ }A_{NM}$ et $W=\mathrm{Spec\ }B_{N},$ il
suffit de calculer les sections de l’intersection. On déduit de l’égalité
$\mathrm{Spec\ }A_{NM}\cap\mathrm{Spec\ }\mathbb{Z}\cap\mathrm{Spec\
}B_{N}=\mathrm{Spec\ }B_{NM}\cap\mathrm{Spec\ }B_{N}=\mathrm{Spec\ }B_{N}$
que $\Gamma(U_{2}^{\prime}\cap W,\mathcal{O}_{\mathrm{Spec}Z})=B_{NM}$. Par
conséquent:
$\Gamma(V_{2},\mathcal{O})=A_{NM}\times_{B_{NM}}B_{N}=A_{NM}\times_{\mathbb{Q}}B_{N}=A_{NM}\cap
B_{N}=A_{N},$
où l’on a appliqué encore une fois le fait que $\Gamma(\mathrm{Spec\
}\mathrm{R},\mathcal{O})=\mathrm{R}$ (cf. [Har, Prop. 2.2]), qui reste encore
valable dans le cas des anneaux généralisés. Cela montre que les morphismes
$f_{2}:V_{2}\longrightarrow U_{2}$ sont en bijection avec les morphismes
$\varphi:A_{N}\longrightarrow B_{N}$. On choisit donc l’image $f_{2}$ de
$\varphi=\mathrm{id}_{A_{N}}$ par cette correspondance. La même construction
prouve que $f_{2|W}=\mathrm{id}_{W}$. En effet, $f_{2|W}$ est induite par le
plongement canonique de $A_{N}$ dans $B_{N}$, donc il s’agit d’une immersion
ouverte de $\mathrm{Spec\ }B_{N}$ dans $\mathrm{Spec\ }A_{N}$.
Il reste à vérifier la condition sur les images réciproques. Dans le premier
cas, on a
$f_{1}^{-1}(U_{1}\cap U_{2})=f_{1}^{-1}(D_{U_{1}}(N))=D_{V_{1}}(N)=W=V_{1}\cap
V_{2}.$
Dans le deuxième cas,
$\displaystyle f_{2}^{-1}(U_{1}\cap U_{2})$
$\displaystyle=f_{2}^{-1}(D_{U_{2}}(1/N))=D_{U_{2}^{\prime}}(1/N)\cup
D_{W}(1/N)=\mathrm{Spec\ }A_{NM}[(1/N)^{-1}]\cup W$
$\displaystyle=\mathrm{Spec\ }B_{NM}\cup\mathrm{Spec\ }B_{N}=\mathrm{Spec\
}B_{N}=W,$
car d’après 4.7, $B_{NM}$ est la localisation de $A_{NM}$ dans $\frac{1}{N}$.
Cela complète la construction du morphisme $f$. Lorsque $M|N^{k}$ pour quelque
$k\geq 1$, les anneaux généralisés $A_{NM}$ et $B_{NM}$ coïncident avec
$A_{N}$ et $B_{N}$, donc $f_{N}^{NM}$ est l’identité. Autrement, il existe un
premier $p$ divisant $M$ mais pas $N$. Ce point est fermé dans
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(NM)}$ mais pas dans
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$, ce qui entraîne que l’application
$f_{N}^{NM}$ n’est pas un homéomorphisme, donc elle n’est pas un isomorphisme
non plus.
Ordonnons l’ensemble des entiers plus grands que 1 par la relation de
divisibilité. Si $M$ divise $N$ et l’on note par $K$ le quotient, soit
$f_{M}^{N}:=f_{M}^{MK}$, avec la convention $f_{N}^{N}=\mathrm{id}$ lorsque
$N=M$. Ces fonctions étant transitives, on obtient le système projectif filtré
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{\bullet}=(\\{\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}\\}_{N>1},\\{f_{M}^{N}\\}_{M|N})$
###### Définition 18.
On appelle compactification du spectre de $\mathbb{Z}$ la limite projective du
système ci-dessus dans la catégorie des pro-schémas généralisés, c’est-à-dire:
$\widehat{\mathrm{Spec\ }\mathbb{Z}}:=\varprojlim_{N>1}\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}$
On étudie d’abord la topologie de $\widehat{\mathrm{Spec\ }\mathbb{Z}}$.
Puisque tous les objets du système ont le même ensemble sous-jacent, au niveau
des ensembles, la compactification de $\mathrm{Spec\ }\mathbb{Z}$ n’est autre
chose que $\mathrm{Spec\ }\mathbb{Z}\cup\\{\infty\\}$. Dans
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$ un point $p$ était fermé si et
seulement si $p|N$. En traitant tous les entiers plus grand que 1 en même
temps dans la construction de la limite, on peut toujours choisir un multiple
$M$ de $p$ de sorte que le point soit fermé dans tous les
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{N}$ tels que $M|N$. Cela démontre le
lemme suivant:
###### Lemme 3.
Tous les points de $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ sauf le point
générique $\xi$ sont fermés. Par conséquent, une partie non-vide $U$ de
$\widehat{\mathrm{Spec\ }\mathbb{Z}}$ est ouverte si et seulement si elle
contient $\xi$ et son complément est fini.
En ce sens, $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ ressemble beaucoup à une
courbe projective sur $\mathbb{Z}$, ce que l’on voulait avoir pour renforcer
l’analogie entre les corps des nombres et les corps de fonctions. De plus,
lorsque l’on calcule l’anneau des germes à l’infini, on trouve la limite
$\mathcal{O}_{\widehat{\mathrm{Spec\
}\mathbb{Z}},\infty}=\varinjlim_{N>1}\mathcal{O}_{\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)},\infty}=\varinjlim A_{N}=\mathbb{Z}_{(\infty)},$
car l’on a défini $A_{N}$ comme étant l’intersection
$\mathbb{Z}_{(\infty)}\cap B_{N}$ et la limite de $B_{N}=\mathbb{Z}[1/N]$
lorsque $N\rightarrow\infty$ est bien $\mathbb{Z}$.
### 5.3 Description en termes de la construction $\mathrm{Proj}$
Le fait que la topologie de $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ soit la même
que celle d’une courbe projective suggère la possibilité d’imiter la
construction $\mathrm{Proj}$ des schémas projectives afin d’avoir une
description plus intrinsèque de la compactification de $\mathrm{Spec\
}\mathbb{Z}$. Soient $\mathcal{C}$ une catégorie monoïdale et $\Delta$ un
monoïde commutatif tels que les coproduits indexés par des sous-ensembles de
$\Delta$ existent dans $\mathcal{C}$ et qu’ils commutent avec $\otimes$. Une
algèbre $\Delta$-graduée dans $\mathcal{C}$ est alors une famille
$\\{S^{\alpha}\\}_{\alpha\in\Delta}$ d’objets de $\mathcal{C}$ munis d’un
morphisme d’identité $\epsilon:\mathrm{id}_{\mathcal{C}}\longrightarrow S^{0}$
et de morphismes de multiplication $\mu_{\alpha,\beta}:S^{\alpha}\otimes
S^{\beta}\longrightarrow S^{\alpha+\beta}$ vérifiant les relations
d’associativité et d’unité évidentes. Lorsque l’on considère la catégorie
monoïdale des endofoncteurs sur les ensembles, on obtient des monades
graduées:
###### Définition 19.
Soit $\Delta$ un monoïde commutatif, par exemple $\mathbb{N}$ ou $\mathbb{Z}$.
Une collection d’ensembles
$\\{R^{\alpha}(\textbf{n})\\}_{\alpha\in\Delta,n\geq 0}$ et d’applications
$S^{\alpha}(\varphi):S^{\alpha}(\textbf{n})\longrightarrow
S^{\alpha}(\textbf{m})$ définies pour chaque
$\varphi:\textbf{n}\longrightarrow\textbf{m}$ tels que le foncteur
$S^{\alpha}:\underline{\mathbb{N}}\longrightarrow\mathfrak{Ens},\quad\textbf{n}\longmapsto
S^{\alpha}(\textbf{n})$
admet une extension unique à un endofoncteur algébrique $S^{\alpha}$ est une
monade algébrique $R$ s’il existe un morphisme d’identité
$\epsilon:\mathrm{id}_{\mathfrak{Ens}}\longrightarrow S^{0}$ et des morphismes
de multiplication $\mu_{n,\beta}^{(k,\alpha)}:S^{\alpha}(\textbf{k})\times
S^{\beta}(\textbf{n})^{k}\longrightarrow S^{\alpha+\beta}(\textbf{n})$
vérifiant les versions graduées des axiomes de monade algébrique. Le monoïde
sous-jacent à $R$ est le monoïde gradué $\lvert
R\rvert=\sqcup_{\alpha\in\Delta}\lvert R^{\alpha}\rvert$ et l’ensemble
d’opérations est l’ensemble gradué
$||R||=\sqcup_{\alpha\in\Delta}||R^{\alpha}||$. On dit que $R$ est commutative
si toutes les opérations de $||R||$ commutent au sens de la définition 10. On
obtient alors des anneaux généralisés gradués.
Soit $R$ un anneau généralisé $\mathbb{N}$-gradué. Notons par $\lvert
R\rvert^{+}$ le sous-monoïde de $\lvert R\rvert$ formé des éléments de degré
non nul. Pour chaque $f\in\lvert R\rvert^{+}$, soit $R_{(f)}$ la composante de
degré zéro de la localisation $R_{f}=R[f^{-1}]$. Remarquons que si $g\in\lvert
R_{(f)}\rvert$, alors $(R_{(f)})_{(g)}$ est isomorphe à $R_{(fg)}$. Le monoïde
sous-jacent à $R$ étant commutatif, on obtient le même résultat lorsque l’on
localise d’abord en $g$ et puis en $f$. Soit $D_{+}(f)$ le schéma affine
généralisé $\mathrm{Spec\ }R_{(f)}$. Si $g\in\lvert R\rvert^{+}$ est un autre
élément, $D_{+}(f)$ et $D_{+}(g)$ contiennent des sous-schémas ouverts
isomorphes à $D_{+}(fg)$, donc on peut les recoller le long $D_{+}(fg)$.
###### Définition 20.
Soit $R$ un anneau généralisé gradué. On appelle spectre projectif de $R$
$\mathrm{Proj\ }R=\coprod_{D_{+}(fg),\ f,g\in\lvert R\rvert^{+}}D_{+}(f),$
qui est un schéma généralisé ayant un recouvrement ouvert affine
$\\{D_{+}(f)\\}_{f\in\lvert R\rvert^{+}}$ tel que $D_{+}(f)\cap D_{+}(g)\cong
D_{+}(fg)$.
###### Exemple 8.
La construction ci-dessus permet de définir l’espace projectif sur un anneau
généralisé $R$ quelconque comme le spectre projectif de l’algèbre de polynômes
à $n+1$ indéterminées de degré un, autrement dit:
$\mathbb{P}_{R}^{n}:=\mathrm{Proj\ }R[T_{0}^{[1]},\ldots,T_{n}^{[1]}].$
D’après la définition, la famille $\\{D_{+}(T_{i})\\}_{0\leq i\leq n}$
constitue un recouvrement ouvert de $\mathbb{P}_{R}^{n}$. Ce n’est pas
difficile à voir que chaque $D_{+}(T_{i})$ est isomorphe à
$\mathbb{A}^{n}_{R}:=\mathrm{Spec\ }[T_{1}^{[1]},\ldots,T_{n}^{[1]}]$.
Notamment, la droite projective $\mathbb{P}_{\mathbb{F}_{1}}^{1}=\mathrm{Proj\
}\mathbb{F}_{1}[T_{0}^{[1]},T_{1}^{[1]}]$ est obtenu en recollant deux copies
de $\mathbb{A}^{1}_{\mathbb{F}_{1}}=\mathrm{Spec\
}\mathbb{F}_{1}[T]=\\{(0),(T)\\}$ le long du point générique. Elle contient
donc trois points $\\{\xi,1,\infty\\}$ correspondant aux idéaux $(T_{0})$,
$(T_{1})$ et $(0)$, dont les deux premiers sont fermés. En général, le
cardinal de l’espace projectif $\mathbb{P}^{n}(\mathbb{F}_{1})$ construit
ainsi est $2^{n+1}-1$, ce qui n’accorde pas avec le point de vue de Tits.
Après cet interlude théorique, on est en état de décrire
$\widehat{\mathrm{Spec\ }\mathbb{Z}}$ comme un (pro)schéma projectif. En vue
de la seconde égalité dans 4.7, définissons $R$ comme l’anneau généralisé
gradué dont la composante en degré $d$ est donné par
$R_{d}(\textbf{n})=\\{(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}:\sum_{i=1}^{n}\lvert\lambda_{i}\rvert\leq
N^{d}\\}$
On peut plonger $R$ dans $\mathbb{Z}[T]$ en identifiant
$(\lambda_{1},\ldots,\lambda_{n})$ avec
$T^{d}(\lambda_{1},\ldots,\lambda_{n})$. Les éléments de degré un de $R$ sont
alors ceux de la forme $uT^{d}$, où $d\geq 0,u\in\mathbb{Z},\lvert u\rvert\leq
N^{d}$. En particulier, $f_{1}:=T$ et $f_{2}:=NT$ appartient à $\lvert
R\rvert$. Alors, on peut montrer [Du, 7.1.44]:
###### Lemme 4.
$\lvert R\rvert^{+}$ coïncide avec le radical de l’idéal engendré par $f_{1}$
et $f_{2}$.
Par conséquent, il suffit de calculer les localisations
$R_{(f_{1})},R_{(f_{2})}$ et $R_{(f_{1}f_{2})}$. Commençons par
$R_{(f_{1})}(\textbf{n})$, qui est la réunion
$\bigcup_{d\geq
0}\\{T^{d}(\lambda_{1},\ldots,\lambda_{n})/T^{d}:\sum_{i=1}^{n}\lvert\lambda_{i}\rvert\leq
N^{d}\\}$
L’inclusion
$R_{(f_{1})}(\textbf{n})\subset\mathbb{Z}^{n}=\mathbb{Z}(\textbf{n})$ étant
évidente, il suffit de montrer la réciproque. Or, si l’on se donne un
$n$-uplet $(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}$, il existe
$d\geq 0$ tel que $\sum_{i=1}^{n}\lvert\lambda_{i}\rvert\leq N^{d}$, donc
$\mathbb{Z}(\textbf{n})\subset R_{(f_{1})}$. On en déduit que
$R_{(f_{1})}=\mathbb{Z}$. Des raisonnements tout à fait analogues montrent que
$R_{(f_{2})}=A_{N}$ et que $R_{(f_{1}f_{2})}=B_{N}$. Alors, $\mathrm{Proj\ }R$
est le recollement des ouverts $D_{+}(f_{1})=\mathrm{Spec\ }\mathbb{Z}$ et
$D_{+}(f_{2})=\mathrm{Spec\ }A_{N}$ le long de
$D_{+}(f_{1}f_{2})=\mathrm{Spec\ }B_{N}$, de sorte que
$\mathrm{Proj\
}R=D_{+}(f_{1})\coprod_{D_{+}(f_{1}f_{2})}D_{+}(f_{2})=\mathrm{Spec\
}\mathbb{Z}\coprod_{\mathrm{Spec\ }B_{N}}\mathrm{Spec\ }A_{N}$
est isomorphe au schéma généralisé $\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}$. Cela démontre le théorème suivant:
###### Theorème 2.
La compactification de $\mathrm{Spec\ }\mathbb{Z}$ est un pro-schéma projectif
sur $\mathbb{F}_{1}$.
### 5.4 Compactification d’Arakelov des variétés et fibrés en droite
Dans cette section, on commence par revisiter l’analogie entre les corps de
nombres et les corps de fonctions. Si $X$ est une variété algébrique lisse et
projective sur un corps de fonctions $K=k(C)$, on appelle modèle de $X$ un
schéma projectif plat $\mathscr{X}\longrightarrow C$ dont la fibre générique
$\mathscr{X}_{\xi}$ est isomorphe à $X$. On a déjà remarque que pour un
certain nombre d’arguments (par exemple, si l’on veut utiliser la théorie de
l’intersection afin d’estimer le nombres de points rationnels de $X$), il est
indispensable que la courbe $C$ soit propre. Puisque tout corps de fonctions
est une extension finie de $k(T)$, le problème consiste essentiellement en
construire un modèle propre sur $K=k(T)$, puis on peut considérer la
normalisation dans des corps plus grands. En procédant par analogie, le
premier pas consiste en définir un modèle propre de $\mathbb{Q}$. Le premier
candidat était $\mathrm{Spec\ }\mathbb{Z}$, dont le corps de fonctions
rationnelles est $\mathbb{Q}$, mais on a montré qu’il n’est pas propre. Cela
est à l’origine de la construction de $\widehat{\mathrm{Spec\ }\mathbb{Z}}$,
que l’on peut voir comme un modèle propre et lisse de $\mathbb{Q}$. La
première étape étant accomplie, il s’agit maintenant de trouver des modèles
d’une variété algébrique définie sur $\mathbb{Q}$, que l’on peut appeler sa
compactification d’Arakelov. Remarquons que dans le cadre des corps de
nombres, on ne peut pas appliquer des procédés géométriques comme la
normalisation.
Soit $X$ une variété algébrique définie sur $\mathbb{Q}$. Puisque la catégorie
des schémas $\mathscr{X}$ sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ admettant
une présentation finie est équivalente à la catégorie des triplets
$(\mathscr{X},\mathscr{X}_{\infty},\theta)$, où $\mathscr{X}$ et
$\mathscr{X}_{\infty}$ sont des schémas ayant une présentation finie sur
$\mathrm{Spec\ }\mathbb{Z}$ et $\mathrm{Spec\ }\mathbb{Z}_{(\infty)}$
respectivement et
$\theta:\mathscr{X}(\mathbb{Q})\longrightarrow\mathscr{X}_{\infty}(\mathbb{Q})$
est un isomorphisme de $\mathbb{Q}$-schémas [Du, 7.1.23], trouver un modèle de
$X$ sur la compactification de $\mathrm{Spec\ }\mathbb{Z}$ est en fait la même
chose que trouver des modèles de $X$ sur $\mathrm{Spec\ }\mathbb{Z}$ et sur
$\mathrm{Spec\ }\mathbb{Z}_{(\infty)}$. Étant donné que la géométrie
algébrique classique s’est occupé avec succès de la première partie, on se
limite ici à décrire les modèles sur $\mathrm{Spec\ }\mathbb{Z}_{(\infty)}$.
###### Theorème 3.
Toute variété algébrique affine ou projective sur $\mathbb{Q}$ admet au moins
un modèle ayant présentation finie sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}$.
###### Proof.
D’après la remarque précédente, il suffit de construire un modèle sur
$\mathrm{Spec\ }\mathbb{Z}_{(\infty)}$, qui est le spectre de l’anneau de
valuation archimédien sur $\mathbb{Q}$. On rappelle que l’anneau de polynômes
$\mathbb{Q}[T_{1},\ldots,T_{k}]$ admet une norme
$||P||=\sum_{\alpha=(\alpha_{1},\ldots,\alpha_{k})}\lvert
c_{\alpha}\rvert,\quad\textit{o\\`{u}}\quad
P=\sum_{\alpha=(\alpha_{1},\ldots,\alpha_{k})}c_{\alpha}T^{\alpha}.$
Donc, on peut identifier l’anneau généralisé de polynômes
$\mathbb{Z}_{(\infty)}[T_{1},\ldots,T_{k}]$ avec les polynômes dans
$\mathbb{Q}[T_{1},\ldots,T_{n}]$ ayant norme plus petit ou égale à 1.
1. 1.
(Cas affine). Soit $X=\mathrm{Spec\ }A$, où
$A=\mathbb{Q}[T_{1},\ldots,T_{k}]/(f_{0},\ldots,f_{m})$, une variété
algébrique affine sur $\mathbb{Q}$. Quitte à multiplier par des scalaires non
nuls dans $\mathbb{Q}$, on peut supposer que $||f_{j}||\leq 1$, ce qui revient
à dire que $f_{j}\in\mathbb{Z}_{(\infty)}[T_{0},\ldots,T_{k}]$. Soit
$B=\mathbb{Z}_{(\infty)}[T_{0},\ldots,T_{k}|f_{1}=0,\ldots,f_{m}=0]$ (5.1)
Si l’on pose $\mathscr{X}=\mathrm{Spec\ }B$, on obtient un schéma affine
généralisé ayant une présentation finie sur $\mathbb{Z}_{(\infty)}$ dont la
fibre générique est
$\mathscr{X}_{(K)}=\mathrm{Spec\
}(B\otimes_{\mathbb{Z}_{(\infty)}\mathbb{Q}})=\mathrm{Spec\ }A=X.$
Par conséquent, $\mathscr{X}/\mathbb{Z}_{(\infty)}$ est un modèle de
$X/\mathbb{Q}$.
2. 2.
(Cas projectif). Soit maintenant $X/\mathbb{Q}$ une variété projective. Alors
$X=\mathrm{Proj\ }A$, où $A$ est le quotient de
$\mathbb{Q}[T_{0},\ldots,T_{k}]$ par un idéal homogène $(f_{1},\ldots,f_{m})$,
et l’on peut supposer de nouveau que $||f_{j}||\leq 1$. Dans ce cas,
$f_{j}\in\mathbb{Z}_{(\infty)}[T_{0},\ldots,T_{k}]$. Définissons $B$ comme
dans (5.1). Alors, le spectre projectif de l’anneau généralisé gradué $B$ est
un modèle $\mathscr{X}=\mathrm{Proj\ }B$ de $X$ sur $\mathbb{Z}_{(\infty)}$. ∎
###### Remarque 4.
Comme le lecteur aura déjà remarqué, la preuve du théorème ci-dessus n’utilise
aucune propriété spéciale de $\mathbb{Q}$. En fait, on peut étendre le
résultat à un corps $K$ quelconque et à son anneau de valuation archimédien
$\mathcal{O}_{\infty}$ au sens de l’exemple 4.3.
Puisque dans le cadre géométrique des corps de fonctions les fibrés en droite
sur une courbe projective lisse se sont révélés un outil indispensable, on
voudrait avoir des analogues arithmétiques. D’après la théorie d’Arakelov
classique, un fibré en droite sur l’hypothétique compactification de
$\mathrm{Spec\ }\mathbb{Z}$ est un fibré en droite sur $\mathrm{Spec\
}\mathbb{Z}$ muni de quelques données archimédiennes supplémentaires. Plus
précisément, un fibré en droite sur $\mathrm{Spec\ }\mathbb{Z}$ est un couple
$L=(L,<\cdot>_{E})$ formé d’un $\mathbb{Z}$-module libre de rang fini $L$ et
d’un produit scalaire hermitien sur $E\otimes_{\mathbb{Z}}\mathbb{C}$ qui est
invariant sous la conjugaison complexe, donc qui définit un produit scalaire
euclidien sur $E\otimes_{\mathbb{Z}}\mathbb{R}$. En posant comme d’habitude
$||x||=<x,x>_{E}$, on peut voir un fibré en droite sur la compactification de
$\mathrm{Spec\ }\mathbb{Z}$ comme un réseau euclidien
$(\mathbb{Z}^{d},||\cdot||)$. Cela suggère du coup une interprétation de
l’information additionnelle en termes des $\mathbb{Z}_{\infty}$-structures
introduites dans 4.2.
###### Définition 21.
Un fibré en droite sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ est un couple
$L=(L,A)$ formé d’un fibré en droite sur $\mathrm{Spec\ }\mathbb{Z}$, i.e.
d’un $\mathbb{Z}$-module libre de rang fini, et d’un
$\mathbb{Z}_{\infty}$-réseau $A$ (i.e. d’un corps compact symétrique convexe)
dans le $\mathbb{R}$-espace vectoriel $L\otimes_{\mathbb{Z}}\mathbb{R}$.
Alors, les sections globales correspondent à l’intersection $L\cap A$. Le
comptage du nombre de points d’un réseau appartenant à un corps compact
symétrique est un sujet d’étude classique. On a, par exemple, le théorème de
Minkowski, d’après lequel un corps convexe symétrique dans $\mathbb{R}^{n}$ de
volume plus grand que $2^{n}$ contient au moins un point du réseau
$\mathbb{Z}^{n}$. Ce résultat, qui implique que chaque classe du groupe de
classes d’idéaux d’un corps de nombres $K$ admet un représentant intégral de
norme borné par une constante qui ne dépend que du discriminant et de la
signature de $K$, peut être imaginé comme un théorème de Riemann-Roch pour les
fibrés en droite sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}$. On rencontre de
nouveau le leitmotiv la géométrie devient combinatoire sur le corps à un
élément.
La définition de fibré en droite sur la compactification de $\mathrm{Spec\
}\mathbb{Z}$ admet une extension à tout schéma généralisé. On se réfère à [Du,
7.1.22] pour la preuve que lorsque l’on pose $X=\widehat{\mathrm{Spec\
}\mathbb{Z}}$, ces deux définitions en apparence si différentes sont en fait
équivalentes:
###### Définition 22.
Soit $(X,\mathcal{O}_{X})$ un schéma généralisé. On définit les fibrés en
droite sur $X$ comme étant les $\mathcal{O}_{X}$-modules $L$ localement
isomorphes à $\lvert\mathcal{O}_{X}\rvert=\mathcal{O}_{X}(1)$.
L’ensemble des classes d’isomorphie de fibrés en droite est un groupe pour le
produit tensoriel $\otimes_{\mathcal{O}_{X}}$, l’élément neutre étant
$\lvert\mathcal{O}_{X}\rvert$. Si l’on pose
$L^{-1}=\mathrm{Hom}(L,\lvert\mathcal{O}_{X}\rvert)$ pour tout fibré en droite
$L$, il existe un isomorphisme
$L\otimes_{\mathcal{O}_{X}}L^{-1}\longrightarrow\lvert\mathcal{O}_{X}\rvert$,
donc $L^{-1}$ est l’inverse de $L$. On appelle groupe de Picard d’un schéma
généralisé $X$ le groupe de classes d’isomorphie de fibrés en droite sur $X$
par rapport au produit $\otimes_{\mathcal{O}_{X}}$. L’objectif de ce dernière
paragraphe est calculer le groupe de Picard de $\widehat{\mathrm{Spec\
}\mathbb{Z}}$. Pour faire cela, on remarque d’abord que le $\mathrm{Pic}(X)$
provenant de la définition 23 coïncide avec le groupe de Picard d’un schéma
classique. Notamment, pour un anneau de Dedekind usuel
$\mathrm{Pic}(\mathrm{Spec\ }R)=0$ si et seulement si $R$ est factoriel [Har,
6.2]. On en déduit que $\mathrm{Pic}(\mathrm{Spec\ }\mathbb{Z})=0$. De même,
on peut montrer que tout fibré en droite sur le spectre de l’anneau généralisé
$A_{N}$ est trivial à l’aide du lemme suivant, dont la preuve se trouve dans
[Du, 7.1.33]:
###### Lemme 5.
Soit $P$ un $A_{N}$-module projectif de type fini. Si
$\dim_{\mathbb{Q}}P_{(\mathbb{Q})}=1$, alors $P$ est isomorphe à $\lvert
A_{N}\rvert$.
En effet, soit $L\in\mathrm{Pic}(\mathrm{Spec\ }A_{N})$. Il existe un
$A_{N}$-module projectif de type fini $P$ tel que $L=\tilde{P}$. Puisque $L$
est un fibré en droite, la fibre du point générique $L_{\xi}\cong
P_{(\mathbb{Q})}$ est unidimensionnelle. D’après le lemme $P\cong\lvert
A_{N}\rvert$, donc $L=\widetilde{\lvert
A_{N}\rvert}=\mathcal{O}_{\mathrm{Spec}A_{N}}$ est le fibré trivial sur
$\mathrm{Spec\ }A_{N}$. Après avoir montré que tous les fibrés en droite sur
les spectres de $\mathbb{Z}$ et de $A_{N}$ sont triviaux, ce n’est pas
difficile à prouver que:
###### Theorème 4.
Le groupe de Picard de la compactification de $\mathrm{Spec\ }\mathbb{Z}$ est
isomorphe au groupe additive des rationnels positifs. En particulier, c’est un
groupe abélien libre de rang infini engendré par $\mathcal{O}(\log p)$ lorsque
$p$ parcourt l’ensemble des nombres premiers.
###### Proof.
La compactification de $\mathrm{Spec\ }\mathbb{Z}$ étant la limite projective
des schémas généralisés $\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$, la
catégorie des fibrés en droite sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}$ est
la limite inductive des catégories des fibrés en droite sur
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$. Par conséquent:
$\mathrm{Pic}(\widehat{\mathrm{Spec\
}\mathbb{Z}})=\varinjlim_{N>1}\mathrm{Pic}(\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}).$
Pour calculer ces groupes, on remarque que le schéma généralisé
$\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$ est la réunion des ouverts
$U_{1}=\mathrm{Spec\ }\mathbb{Z}$ et $U_{2}=\mathrm{Spec\ }A_{N}$ le long de
$U_{1}\cap U_{2}=\mathrm{Spec\ }B_{N}$. Puisque les groupes de Picard de
$\mathrm{Spec\ }\mathbb{Z}$ et de $\mathrm{Spec\ }A_{N}$ sont triviaux, étant
donné un fibré en droite $L$ sur $\widehat{\mathrm{Spec\ }\mathbb{Z}}^{(N)}$,
on peut choisir des trivialisations:
$\varphi_{1}:\mathcal{O}_{\mathrm{Spec\ }\mathbb{Z}}\longrightarrow
L_{|U_{1}},\quad\varphi_{1}:\mathcal{O}_{\mathrm{Spec\ }A_{N}}\longrightarrow
L_{|U_{2}}$
Ce choix est canonique quitte à fixer le signe de $\varphi_{1}$ et
$\varphi_{2}$, car l’on peut toujours multiplier les trivialisations par des
éléments inversibles
$s_{1}\in\Gamma(U_{1},\mathcal{O}_{U_{1}}^{\times})=\mathbb{Z}^{\times}=\\{\pm
1\\}$ et $s_{2}\in\Gamma(U_{2},\mathcal{O}_{U_{2}}^{\times})=\lvert
A_{N}\rvert^{\times}=\\{\pm 1\\}$. Sur l’intersection $U_{1}\cap U_{2}$ ces
trivialisations vérifient la condition de compatibilité
${\varphi_{2}}_{|U_{1}\cap U_{2}}=\lambda{\varphi_{1}}_{|U_{1}\cap U_{2}}$
pour un élément $\lambda$ inversible dans $B_{N}$. Réciproquement, si l’on se
donne $\lambda\in B_{N}^{\times}$ positif, on peut construire un fibré en
droite $L=\mathcal{O}(\log\lambda)$ en recollant des fibrés en droite triviaux
sur $U_{1}$ et sur $U_{2}$ selon la formule ci-dessus. Lorsque le signe de
$\lambda$ est fixé, cette correspondance est bijective. Donc,
$\mathrm{Pic}(\widehat{\mathrm{Spec\ }\mathbb{Z}}^{N})$ est isomorphe à $\log
B_{N}^{\times}/\\{\pm 1\\}$, c’est-à-dire, au group abélien $\log
B_{N,+}^{\times}$ des éléments positifs inversibles de l’anneau
$B_{N}=\mathbb{Z}[N^{-1}]$. Si l’on note par $p_{1},\ldots,p_{r}$ l’ensemble
des diviseurs premiers de $N$, le groupe de Picard est un groupe abélien libre
de rang r ayant base $\\{\mathcal{O}(\log p_{i}\\}_{1\leq i\leq r}$. On en
déduit:
$\mathrm{Pic}(\widehat{\mathrm{Spec\ }\mathbb{Z}})=\varinjlim_{N>1}\log
B_{N,+}^{\times}=\log\mathbb{Q}_{+}^{\ast}.\qed$
###### Remarque 5.
Le fait que le rang du groupe de Picard de $\widehat{\mathrm{Spec\
}\mathbb{Z}}^{(N)}$ soit le nombre de diviseurs premiers de $N$ fournit une
preuve alternative de l’énoncé affirmant que $f_{M}^{N}$ n’est pas un
isomorphisme lorsqu’il existe $p$ premier tel que $p|M$, mais $p\nmid N$.
## 6 Questions ouvertes
1. 1.
Comme on a remarqué dans la section concernant l’analogie entre les corps de
nombres et les corps de fonctions, c’est nécessaire d’introduire un facteur
local à l’infini afin d’avoir une équation fonctionnelle pour la fonction
zêta. L’expression intégrale de ce nouveau facteur est tout à fait analogue à
celle des facteurs correspondant aux premiers finis sauf pour l’apparition de
la gaussienne $e^{-\pi x^{2}}$ au lieu de la fonction indicatrice de
$\mathbb{Z}_{\infty}$, c’est-à-dire, de l’intervalle $[-1,1]$, que l’on
s’attendrait naivement. L’intervention de la gaussienne ne peut pas être sans
relation avec une interprétation probabiliste du passage à l’infini. À l’avis
de l’auteur, bien comprendre ce facteur local à l’infini mérite toute notre
attention, étant donné que la preuve de l’équation fonctionnelle des fonctions
zêta des variétés de dimension supérieure sur $\mathbb{Z}$ reste toujours une
question ouverte.
2. 2.
Dans ce mémoire, on ne s’est pas occupé de la question si les groupes
linéaires algébriques sont définis (au sens de Durov) sur $\mathbb{F}_{1}$ ou
sur une extension convenable. En termes de foncteurs représentables [Du,
5.1.21] et puis en introduisant la notion de déterminant, Durov montre [Du,
5.5.17] que les seules schémas affines en groupes définis sur
$\mathbb{F}_{\emptyset}$ sont les tores $\mathbb{G}^{n}_{m}$, que l’on peut
définir le groupe général linéaire sur $\mathbb{F}_{1}$ et que
$\mathrm{SL}_{n}$ existe sur l’extension quadratique $\mathbb{F}_{1^{2}}$.
Cependant, on ne sait pas si l’on peut définir, par exemple, le groupe
orthogonal dans ce cadre, car le concept de transposée d’une matrice pose des
problèmes quand on travaille dans l’algèbre non-additive. Notamment, l’auteur
s’intéresse à une possible application du concept récent de variété torifiée
introduit dans [LL] à la recherche de modèles d’autres groupes sur
$\mathbb{F}_{1}$ ou $\mathbb{F}_{1^{2}}$. Dans ce même papier, J. López Peña
et O. Lorscheid abordent la comparaison des notions de schéma sur le corps à
un élément de Deitmar, Soulé et Connes-Consani. Il serait nécessaire d’étendre
cette comparaison aux géométries de Töen-Vaquié, Shai-Haran et Durov.
3. 3.
L’espace projectif $\mathbb{P}^{n-1}(\mathbb{F}_{1})$ construit par Durov en
termes du spectre projectif d’un anneau généralisé gradué ne contient pas $n$
points comme l’on s’attendait, mais $2^{n}-1$. Cependant, étant donné que
$2^{n}-1$ est le nombre de parties non-vides d’un ensemble de $n$ points,
l’auteur pense que c’est possible de trouver une certaine relation
d’équivalence ou opération $n$-aire à $n$ orbites fermées. Notamment, il peut
être intéressant de comparer le $\mathbb{P}^{n-1}(\mathbb{F}_{1})$ de Durov
avec l’espaces projectifs définis par Connes-Consani dans [CC] comme le
foncteur gradué
$\mathbb{P}^{d}:\mathcal{F}_{ab}\longrightarrow\mathfrak{Ens},\quad\mathbb{P}^{d}(D)^{(k)}=\coprod_{Y\subset\\{1,\ldots,d+1\\},\
\lvert Y\rvert=k+1}D^{Y}/D,$
où $D$ agit à droite sur $D^{Y}$ par l’action diagonale. Ainsi, la partie de
degré nul sur $\mathbb{F}_{1^{n}}$ est juste $\\{1,\ldots,d+1\\}$, donc
$\lvert\mathbb{P}^{n-1}(\mathbb{F}_{1^{n}})^{(0)}\rvert=n$. D’après la
définition de Connes-Consani, la structure du projectif sur le corps à un
élément est plus compliquée que celle d’un ensemble fini, mais la partie de
degré nul coïncide avec la prévision de Tits. On peut attendre que cela soit
pareil après une légère modification de la construction de Durov.
4. 4.
Comme on a déjà signalé, le fait que le produit tensoriel
$\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ soit $\mathbb{Z}$ de nouveau,
et donc $\mathrm{Spec\ }\mathbb{Z}\times_{\mathrm{Spec\
}\mathbb{F}_{1}}\mathrm{Spec\ }\mathbb{Z}=\mathrm{Spec\ }\mathbb{Z}$, est en
quelque mode décevant, car une des motivations initiales pour développer toute
la théorie des anneaux et des schémas généralisés était la recherche d’une
catégorie ayant des produits arithmétiques. D’après Durov, cela rend inutile
sur son approche en ce qui concerne l’étude de la fonction zêta. De l’autre
côte, Connes-Consani, inspirés des idées de Soulé, ont réussi à construire les
fonctions zêta de schémas sur $\mathbb{F}_{1}$. La compactification de
$\mathrm{Spec\ }\mathbb{Z}$ obtenu dans ce mémoire a des très bonnes
propriétés topologiques (tous les points sur $\xi$ sont fermés comme dans une
courbe projective), mais par exemple son groupe de Picard n’est pas
$\mathbb{R}_{>0}$ comme l’analogie avec le cas des corps de fonctions suggère,
mais le groupe additif des rationnels positifs. Une façon definitive de tester
que l’on a trouvé la compactification correcte serait construire un cadre
suffisamment vaste pour définir les fonctions zêta des schémas au sens de
Durov et vérifier si cela donne
$Z(\widehat{\mathrm{Spec\ }\mathbb{Z}},\
s)=\xi(s)=\pi^{-\frac{s}{2}}\Gamma(s/2)\zeta(s).$
En quelque sens, on peut dire que Durov a la courbe et Connes et Consani ont
la fonction. À l’avis de l’auteur, la topologie de $\widehat{\mathrm{Spec\
}\mathbb{Z}}$ mérite encore plus d’étude. En effet, puisque la restriction à
$\mathrm{Spec\ }\mathbb{Z}$ de chacun des morphismes $f_{N}^{M}$ du système
projectif définissant la compactification est un isomorphisme, $f_{N}^{M}$
n’agit que sur le point à l’infini. Étant donné que le résultat final est un
modèle lisse de $\mathbb{Q}$, cette limite inductive est une sorte de
résolution infinie de singularités. Une autre question qui se pose est le
calcul du produit tensoriel
$\mathbb{Z}_{\infty}\otimes_{\mathbb{F}_{1}}\mathbb{Z}_{\infty}$.
## Appendix A Construction fonctorielle des schémas sur $\mathbb{F}_{1}$
Dans cet annexe, on présente une définition fonctorielle des schémas
généralisés par analogie avec la géométrie algébrique classique. Rappelons
d’abord qu’un faisceau $\mathcal{F}$ sur un espace topologique $X$ est la
donnée
1. 1.
d’un ensemble (peut-être muni d’une certaine structure algébrique
additionnelle) $\mathcal{F}(U)$, que l’on appelle les sections du faisceau,
défini pour chaque ouvert $U\subset X$
2. 2.
d’une application de restriction des sections
$p_{UV}:\mathcal{F}(U)\longrightarrow\mathcal{F}(V)$ définie pour chaque
inclusion $V\subset U$, que l’on note $f\longmapsto f_{|V}$.
Ensuite, on requiert la condition de recollement suivante: pour chaque
recouvrement ouvert $\\{U_{i}\\}_{i\in I}$ de $U$, si
$f_{i}\in\mathcal{F}(U_{i})$ sont des sections dont les restrictions
$(f_{j})_{|(U_{i}\cap U_{j})}=(f_{i})_{|(U_{i}\cap U_{j})}$
aux intersections des $U_{i}$ coïncident, il existe une unique section
$f\in\mathcal{F}(U)$ dont les restrictions $f_{|U_{i}}=f_{i}$ aux $U_{i}$ sont
égales aux sections données au départ.
Essayons de récrire cette définition en termes catégoriques. Étant donné
l’espace topologique $X$, on défini une catégorie $\mathrm{Ouv}(X)$ dont les
objets sont les ouverts de $X$ et dont les morphismes sont les inclusions.
Alors un faisceau $\mathcal{F}$ à valeurs dans une catégorie $\mathcal{C}$
basée sur les ensembles est un foncteur contravariant
$\mathcal{F}:\mathrm{Ouv}(X)^{\text{op}}\longrightarrow\mathcal{C}$, où
$\mathrm{Ouv}(X)^{\text{op}}$ représente, comme d’habitude, la catégorie
duale⋆ de $\mathrm{Ouv}(X)$. Si $f:X\longrightarrow Y$ est une application
continue entre les espaces topologiques $X$ et $Y$ et $\mathcal{F}$ est un
faisceau sur $X$, on définit le faisceau $f_{\ast}\mathcal{F}$ sur $Y$ par la
formule $f_{\ast}\mathcal{F}(V)=\mathcal{F}(f^{-1}(V))$ pour tout ouvert $V$
de $Y$. Maintenant, la condition de recollement signifie qu’une section
globale est déterminée par ses valeurs locales et que si l’on a des sections
locales qui se recollent, on peut construire une section globale. Cela se
traduit dans l’exactitude de la suite
$\mathcal{F}(U)\longrightarrow\prod_{i\in
I}\mathcal{F}(U_{i})\rightrightarrows\prod_{i\in I}\mathcal{F}(U_{i}\cap
U_{j}),$
où les flèches sont donnés par des restrictions des sections, pour tout
recouvrement ouvert $\\{U_{i}\\}_{i\in I}$ de $U$. En effet, cela signifie que
$\mathcal{F}(U)$ s’injecte dans le produit $\prod_{i\in I}\mathcal{F}(U_{i})$
et que $\mathcal{F}(U)$ est l’égaliseur⋆ du couple des flèches $\prod_{i\in
I}\mathcal{F}(U_{i})\rightrightarrows\prod_{i\in I}\mathcal{F}(U_{i}\cap
U_{j})$, autrement dit, que si l’on a des $f_{i}\in\mathcal{F}(U_{i})$ dont
les images par les deux restrictions possibles à $U_{i}\cap U_{j}$ coïncident,
il existe une seule section $f\in\mathcal{F}(U)$ telle que $f_{|U_{i}}=f_{i}$.
Notons que dans la catégorie des espaces topologiques, $U_{i}\cap U_{j}$ est
isomorphe à $U_{i}\times_{U}U_{j}$.
Pour généraliser la notion de faisceau à une catégorie quelconque, on a
d’abord besoin de définir ce que l’on appelle une topologie de Grothendieck,
une notion qui étend de façon naturelle les propriétés des recouvrements
ouverts dans un espace topologique. Pour notre but, il suffit de définir une
structure un peu plus générale:
###### Définition 23.
Une prétopologie de Grothendieck $\mathcal{T}$ dans une catégorie
$\mathcal{C}$ est la donnée d’un ensemble $\mathrm{Rec}(\mathcal{T})$ de
familles de morphismes (dites recouvrements)
$\\{\varphi_{i}:U_{i}\longrightarrow U\\}_{i\in I}$ dans $\mathcal{C}$
définies pour chaque objet $U$ et telles que:
1. 1.
Si $\varphi$ est un isomorphisme, la famille $\\{\varphi\\}$ appartient à
$\mathrm{Rec}(\mathcal{T})$.
2. 2.
Si $\\{\varphi_{i}:U_{i}\longrightarrow U\\}$ et
$\\{\psi_{ij}:V_{ij}\longrightarrow U_{i}\\}$ sont deux recouvrements dans
$\mathcal{T}$, alors le recouvrement composé
$\\{\varphi_{i}\circ\psi_{ij}:V_{ij}\longrightarrow U\\}$ est de nouveau dans
$\mathrm{Rec}(\mathcal{T})$.
3. 3.
Si $\\{U_{i}\longrightarrow U\\}$ est un recouvrement et $V\longrightarrow U$
est un morphisme quelconque, le produit fibré $U_{i}\times_{U}V$ existe et la
famille $\\{U_{i}\times_{U}V\longrightarrow V\\}$ appartient à
$\mathrm{Rec}(\mathcal{T})$.
On dit qu’une catégorie $\mathcal{C}$ muni d’une prétopologie de Grothendieck
est un prétopos.
Notamment, la catégorie $\mathrm{Ouv}(X)$ définie au début est un prétopos
dont les familles recouvrantes sont les données par des inclusions. Les
prétopos constituent, en quelque sens, le cadre minimal où l’on peut définir
la notion de faisceau.
###### Définition 24.
Soit $(\mathcal{C},\mathcal{T})$ un prétopos et soit $\mathcal{D}$ une
catégorie ayant des produits. On appelle faisceau sur $\mathcal{C}$ à valeurs
dans $\mathcal{D}$ un foncteur
$F:\mathcal{C}^{\text{op}}\longrightarrow\mathcal{D}$ tel que pour tout
famille recouvrante $\\{\varphi_{i}:U_{i}\longrightarrow U\\}$, la suite
$F(U)\longrightarrow\prod F(U_{i})\rightrightarrows\prod
F(U_{i}\times_{U}U_{j}),$
où toutes les flèches sont données par des restrictions des sections, soit
exacte.
Cette définition va nous permettre de construire des faisceaux sur les schémas
affines généralisés. Avant de les définir, on montre que c’est naturel de
définir la catégorie des schémas affines généralisés comme la catégorie duale⋆
de celle des anneaux généralisés en rappelant le cas classique. Soit $A$ un
anneau commutatif. La topologie de Zariski muni l’ensemble d’idéaux premiers
de $A$ d’une structure d’espace topologique, que l’on note $\mathrm{Spec\ }A$.
On veut définir maintenant un faisceau d’anneaux sur $\mathrm{Spec\ }A$ en
imitant les propriétés des fonctions régulières sur une variété. Soit
$X\subset\mathbb{A}^{n}_{k}$ une variété algébrique affine définie sur un
corps $k$. Une fonction $f:X\longrightarrow k$ est dite régulière en un point
$P\in X$ s’il existe un voisinage ouvert de $P$ dans $X$ tel que $f$ est le
quotient de deux polynômes $g,h\in k[X_{1},\ldots,X_{n}]$ dont $h$ ne s’annule
pas sur $U$. Si l’on note par $\mathcal{O}(U)$ l’anneau des fonctions
régulières sur $U$, on obtient ce que l’on appelle le faisceau structurel de
$X$. Dans le cas du spectre, il n’y a pas un corps de base distingué comme but
des fonctions, ce qui nous amène à prendre en même temps toutes les
localisations $A_{\mathfrak{p}}$, où $\mathfrak{p}\in\mathrm{Spec\ }A$. Alors,
étant donné un ouvert $U\subset\mathrm{Spec\ }A$, on considère les fonctions
$s:U\longrightarrow\coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$ satisfaisant
la condition de régularité suivante: soit $\mathfrak{p}\in U$, on requiert
qu’il existe un voisinage ouvert $\mathfrak{p}\in V\subset U$ et deux éléments
$a,f\in A$ tels que $f\notin\mathfrak{q}$ pour tout $\mathfrak{q}\in V$
(l’analogue de la non-annulation du polynôme $h$) et que
$s(\mathfrak{q})=a/f$. L’ensemble $\mathcal{O}(U)$ formé des fonctions
vérifiant la condition ci-dessus pour tout $\mathfrak{p}\in U$ admet une
structure d’anneau, et l’on obtient ainsi un faisceau d’anneaux $\mathcal{O}$
sur $\mathrm{Spec\ }A$.
Cette construction associe à chaque anneau commutatif $A$ un couple
$(\mathrm{Spec\ }A,\mathcal{O})$ formé d’un espace topologique et d’un
faisceau sur lui. Une question qui se pose immédiatement est s’il existe une
catégorie $\mathcal{C}$ tel que $(\mathrm{Spec\ }A,\mathcal{O})$ soit un objet
dans $\mathcal{C}$ pour tout anneau commutatif $A$ et que la correspondance
$A\longmapsto(\mathrm{Spec\ }A,\mathcal{O})$ soit fonctorielle. Ce sont les
espaces localement annélés. Les objets de la catégorie sont les couples
$(X,\mathcal{O}_{X})$ formés d’un espace topologique $X$ et d’un faisceau
d’anneaux sur $X$ tel que l’anneau des germes
$\mathcal{O}_{X,P}=\varinjlim_{P\in
U}\mathcal{O}_{P}^{\bullet},\quad\text{o\\`{u}}\quad\mathcal{O}_{P}^{\bullet}=((\mathcal{O}(U))_{P\in
U},(p_{UV})_{V\subset U})$
soit local pour tout $P\in X$. Les morphismes entre deux espaces localement
annélés $(X,\mathcal{O}_{X})$ et $(Y,\mathcal{O}_{Y})$ sont les couples
$(f,f^{\sharp})$, où $f:X\longrightarrow Y$ est une application continue et
$f^{\sharp}:\mathcal{O}_{Y}\longrightarrow f_{\ast}\mathcal{O}_{X}$ est un
morphisme de faisceaux telle que l’application induite
$f^{\sharp}:\mathcal{O}_{Y,f(P)}\longrightarrow\mathcal{O}_{X,P}$ soit un
homomorphisme local d’anneaux locaux pour tout $P\in X$.
Un schéma affine est un espace localement annélé qui est isomorphe à
$(\mathrm{Spec\ }A,\mathcal{O})$ pour un certain anneau commutatif $A$. On
obtient ainsi une application des anneaux commutatifs vers les schémas
affines, qui est fonctorielle au sens suivant: pour chaque homomorphisme
d’anneaux $\varphi:A\longrightarrow B$, la formule
$f(\mathfrak{p})=\varphi^{-1}(\mathfrak{p})$ définit une application continue
$f:=\mathrm{Spec}(\varphi):\mathrm{Spec\ }B\longrightarrow\mathrm{Spec\ }A$.
De même, en localisant $\varphi$, on obtient un homomorphisme local
$\varphi_{\mathfrak{p}}:A_{\varphi^{-1}(\mathfrak{p})}\longrightarrow
B_{\mathfrak{p}}$ qui peut s’étendre, par la construction du faisceau
structurel d’un spectre, à un morphisme de faisceaux
$f^{\sharp}:\mathcal{O}_{\mathrm{Spec}A}\longrightarrow\mathcal{O}_{\mathrm{Spec}B}$
dont la restriction à chaque anneau de germes est $\varphi_{\mathfrak{p}}$.
Par conséquent, $(f,f^{\sharp})$ est un morphisme d’espaces localement
annelés. On en déduit que le foncteur $\mathrm{Spec}$ est une équivalence
contravariante entre la catégorie des anneaux commutatifs et la catégorie des
schémas affines, donc les schémas affines constituent la catégorie duale des
anneaux commutatifs. Finalement, un schéma est un espace localement annélé
$(X,\mathcal{O}_{X})$ où chaque point $P$ admet un voisinage ouvert $U$ dans
$X$ tel que $(U,\mathcal{O}_{X|U})$ soit un schéma affine.
Soit maintenant $\mathfrak{Gen}$ la catégorie des anneaux généralisés. D’après
la discussion précédente, on définit les schémas affines généralisés comme la
catégorie duale de $\mathfrak{Gen}$, i.e.
$\mathfrak{Aff}:=\mathfrak{Gen}^{\textit{op}}.$ On note $\mathrm{Spec\ }R$, où
$R$ est un anneau généralisé, les objets de cette catégorie. En particulier,
un schéma affine généralisé est un foncteur contravariant de $\mathfrak{Gen}$
vers les ensembles. Étant donné un schéma affine généralisé $\mathrm{Spec\
}R$, pour définir une prétopologie de Grothendieck on considère les
localisations de l’anneau généralisé $R$ dans des éléments $f_{i}\in\lvert
R\rvert$. Si l’on pose $U=\mathrm{Spec\ }R$ et $U_{i}=\mathrm{Spec\
}R_{f_{i}}$, on définit les familles recouvrantes comme les
$\\{U_{i}\longrightarrow U\\}_{i\in I}$ telles que $\coprod_{i\in
I}U_{i}\longrightarrow U$ soit une surjection au niveau d’espaces
topologiques. Ce n’est pas difficile à vérifier que ces données satisfont les
axiomes d’une prétopologie de Grothendieck, que l’on appellera la topologie de
Zariski. Par conséquent, un faisceau sur le prétopos $\mathfrak{Aff}$ est un
foncteur
$F:\mathfrak{Aff}^{\text{op}}=\mathfrak{Gen}\longrightarrow\mathfrak{Ens}$
telle que la suite
$F(U)\longrightarrow\prod F(U_{i})\rightrightarrows\prod
F(U_{i}\times_{U}U_{j})$
soit exacte pour toute famille recouvrante $\\{U_{i}\longrightarrow U\\}$ dans
$\mathfrak{Aff}$.
Il nous reste seulement à définir une condition analogue au fait qu’un schéma
soit localement affine. On remarque que si l’on voit le spectre d’un anneau
commutatif classique $R$ comme le foncteur de $\mathfrak{Ann}$ vers
$\mathfrak{Ens}$ qu’au niveau des objets vaut $A\longrightarrow\mathrm{Spec\
}R(A)=\mathrm{Hom}_{\mathfrak{Ann}}(R,A)$, les ouverts de Zariski sont alors
de sous-foncteurs de $\mathrm{Spec\ }R$ vérifiant quelques propriétés liées à
la localisation. Cette notion se formalise de la façon suivante:
###### Définition 25.
Soit $S:\mathfrak{Gen}\longrightarrow\mathfrak{Ens}$ un foncteur
contravariant. Un sous-foncteur $U$ de $S$ est dit ouvert de Zariski si pour
tout anneau généralisé $B$ et pour tout transformation naturelle
$u:\mathrm{Spec\ }B\longrightarrow S$, il existe un système multiplicatif
$T\subset B$ tel que le produit fibré $U\times_{S}\mathrm{Spec\ }B$ soit
naturellement isomorphe à $\mathrm{Spec\ }B[T^{-1}]$.
Alors, un schéma généralisé est un faisceau qui admet un recouvrement par des
ouverts isomorphes à des schémas affines généralisés. Plus précisément:
###### Définition 26.
Un schéma généralisé est un foncteur contravariant
$S:\mathfrak{Gen}\longrightarrow\mathfrak{Ens}$ qui est un faisceau pour la
topologie de Zariski et qui est localement représentable, au sens où il existe
un recouvrement $\\{S_{i}\\}_{i\in I}$ de $S$ par des sous-foncteurs ouverts
de Zariski qui sont naturellement isomorphes à des schémas affines
généralisés, c’est-à-dire, tels que $S_{i}\cong\mathrm{Spec\ }R_{i}$ pour des
anneaux généralisés $R_{i}$.
## Appendix B Annexe: quelques outils catégoriques
Catégorie duale Soit $\mathcal{C}$ une catégorie. On appelle catégorie duale
de $\mathcal{C}$ la catégorie $\mathcal{C}^{\text{op}}$ dont les objets
coïncident avec ceux de $\mathcal{C}$ et dont les morphismes sont obtenus en
inversant les buts et sources des morphismes de $\mathcal{C}$, autrement dit
$\mathrm{Hom}_{\mathcal{C}^{\text{op}}}(X,Y)\cong\mathrm{Hom}_{\mathcal{C}}(Y,X)$
pour tout couple d’objets $X,Y$ dans $\mathcal{C}$. La notion de catégorie
duale permet par exemple d’éliminer la distinction entre foncteur covariant et
contravariant, car un foncteur contravariant
$F:\mathcal{C}\longrightarrow\mathcal{D}$ est un foncteur covariant
$F:\mathcal{C}^{\text{op}}$ sur la catégorie duale.
Catégorie monoïdale. Une catégorie $\mathcal{C}$ est munie d’une structure de
catégorie monoïdale (ou de $\otimes$-catégorie associative unitaire) s’il
existe
1. 1.
un bifoncteur $\otimes:\mathcal{C}\times\mathcal{C}\longrightarrow\mathcal{C}$
associative à isomorphisme naturel près (i.e. $X\otimes(Y\otimes
Z)\cong(X\otimes Y)\otimes Z$ pour tout triplet d’objets $(X,Y,Z)$)
2. 2.
une unité $\bf{1}\in\text{Ob(}\mathcal{C})$ à isomorphisme naturel près (i.e.
${\bf{1}}\otimes X\cong X\cong X\otimes{\bf{1}}$ pour tout objet $X$)
tels que $\otimes$ vérifie le diagramme du pentagone et que $\bf{e}$ soit
compatible avec l’associativité (voir [McL, XI.1]). Des exemples de catégories
monoïdales sont les ensembles avec le produit cartésien
$(\mathfrak{Ens},\times,\\{1\\})$ ou les $R-$modules avec le produit tensoriel
$(R-\mathfrak{Mod},\otimes,R).$ On dit qu’une catégorie monoïdale est
symétrique s’il y a un isomorphisme naturel entre $X\otimes Y$ et $Y\otimes X$
pour tout $X,Y$ dans $\mathcal{C}.$
(Co)égaliseur. Soient $X$ et $Y$ deux objets d’une catégorie $\mathcal{C}$ et
soient $f,g:X\rightrightarrows Y$ deux morphismes de $X$ vers $Y.$ On appelle
égaliseur du système le seul couple (à isomorphisme près) $(E,e)$ formé d’un
objet $E$ de $\mathcal{C}$ et d’un morphisme $u:E\longrightarrow X$ tels que
$f\circ e=g\circ e$ et que
$\mathrm{Hom}_{\mathcal{C}}(P,E)\cong\\{i\in\mathrm{Hom}_{\mathcal{C}}(P,X):f\circ
i=g\circ i\\}$
soient naturellement isomorphes pour tout $P$ dans $\mathcal{C}.$ On appelle
coégaliseur du système le seul couple (à isomorphisme près) $(Q,q)$ formé d’un
objet de $\mathcal{C}$ et d’un morphisme $q:Y\longrightarrow Q$ tels que
$q\circ f=q\circ g$ et que
$\mathrm{Hom}_{\mathcal{C}}(Q,P)\cong\\{i\in\mathrm{Hom}_{\mathcal{C}}(Y,P):i\circ
f=i\circ g$
soient naturellement isomorphes pour tout $P$ dans $\mathcal{C}.$ Autrement
dit, l’égaliseur et le coégaliseur ont la propriété universelle de rendre les
flèches $f$ et $g$ égales.
Composition de transformations naturelles. Soient
$\mathcal{C},\mathcal{D},\mathcal{E}$ des catégories,
$F,F^{\prime}:\mathcal{C}\longrightarrow\mathcal{D}$ et
$G,G^{\prime}:\mathcal{D}\longrightarrow\mathcal{E}$ des foncteurs et
$\xi:F\longrightarrow F^{\prime},\eta:G\longrightarrow G^{\prime}$ des
transformations naturelles. On définit $\eta\star\xi:GF\longrightarrow
G^{\prime}F^{\prime}$ comme la transformation naturelle dont les composantes
sont
$(\eta\star\xi)_{X}=\eta_{F^{\prime}(X)}\circ
G(\xi_{X})=G^{\prime}(\xi_{X})\circ\eta_{F(X)}$
pour chaque objet $X$ dans $\mathcal{C}.$ Si $F$ est un foncteur, on pose
$\eta\star F:=\eta\star\mathrm{id}_{F}.$
Coproduit. Soit $\mathcal{C}$ une catégorie et soient $X$ et $Y$ deux objets
de $\mathcal{C}.$ On appelle coproduit (ou somme catégorique) de $X$ et $Y$ le
seul objet $X\amalg Y$ tel que pour tout $Z\in\mathrm{Ob}(\mathcal{C})$ il
existe un isomorphisme naturel
$\mathrm{Hom}(X\amalg Y,Z)\cong\mathrm{Hom}(X,Z)\times\mathrm{Hom}(Y,Z).$
Par exemple, dans la catégorie des ensembles $\mathfrak{Ens}$, la somme est la
réunion disjointe, dans la catégorie des espaces topologiques pointés
$\mathfrak{Top}_{\ast}$, c’est le produit wedge, dans la catégorie des groupes
$\mathfrak{Grp}$ (resp. abéliens), c’est le produit libre (resp. la somme
directe), et dans la catégorie des anneaux commutatifs $\mathfrak{Ann}$, c’est
le produit tensoriel.
Foncteurs adjoints. Soient $\mathcal{C}$ et $\mathcal{D}$ deux catégories. On
dit que les foncteurs $F:\mathcal{C}\longrightarrow\mathcal{D}$ et
$G:\mathcal{D}\longrightarrow\mathcal{C}$ sont adjoints si et seulement si
$\mathrm{Hom}_{\mathcal{C}}(X,G(Y))\cong\mathrm{Hom}_{\mathcal{D}}(F(X),Y),$
pour tout couple d’objets
$X\in\text{Ob(}\mathcal{C}),Y\in\text{Ob(}\mathcal{D}),$ l’isomorphisme ci-
dessus étant naturel en $X$ et $Y.$ Dans ce cas, le foncteur $F$ (resp. $G$)
est dit l’adjoint à gauche (resp. à droite) du foncteur $G$ (resp. $F$).
Lorsque l’on pose $Y=F(X)$ dans la définition, $\mathrm{id}_{F(X)}$ appartient
au deuxième ensemble. Son image par l’isomorphisme est une application
$\eta_{X}:X\longrightarrow GF(X)$ naturelle en $X.$ On appelle unité de
l’adjonction la transformation naturelle
$\eta:\mathrm{id}_{\mathcal{C}}\longrightarrow GF$ dont les composantes sont
données par $\eta_{X}.$ De même, une adjonction admet une co-unité
$\xi:FG\longrightarrow\mathrm{id}_{\mathcal{D}}$ définie de manière analogue
en prenant $X=G(Y)$ dans la formule.
Foncteur (pleinement) fidèle. Considérons deux catégories $\mathcal{C}$ et
$\mathcal{D}$ et un foncteur covariant
$F:\mathcal{C}\longrightarrow\mathcal{D}$. Comme d’habitude, étant donné un
couple d’objets $(X,Y)$ dans $\mathcal{C}$, F induit une application
$F_{X,Y}:\mathrm{Hom}_{\mathcal{C}}(X,Y)\longrightarrow\mathrm{Hom}_{\mathcal{D}}(F(X),F(Y))$
On dit alors que F est fidèle (resp. plein, pleinement fidèle) si $F_{X,Y}$
est injective (resp. surjective, bijective) pour tout $X,Y$ dans
$\mathcal{C}.$
Lemme de Yoneda. Soient $X$ et $Y$ deux objets dans une catégorie
$\mathcal{C}.$ Notons $h_{A}:=\mathrm{Hom}_{\mathcal{C}}(A,\cdot)$ le foncteur
représenté par un objet $A$. Avec ces notations, le lemme de Yoneda affirme
que pour toute transformation naturelle $\eta:h_{Y}\longrightarrow h_{X},$ il
existe un morphisme $f:X\longrightarrow Y$ tel que $\eta=h_{f},$ où
$h_{f}:\mathrm{Hom}_{\mathcal{C}}(Y,\cdot)\longrightarrow\mathrm{Hom}_{\mathcal{C}}(X,\cdot)$
est la transformation naturelle ayant composantes $h_{f,A}$ qui appliquent
$\varphi:Y\longrightarrow A$ dans $\varphi\circ f:X\longrightarrow Y$.
Notamment, cela implique que les objets représentant des foncteurs isomorphes
sont ils mêmes isomorphes.
Limite inductive. Soit $\mathcal{C}$ une catégorie et soit $(I,\leq)$ un
ensemble partiellement ordonné. Un système inductive d’objets dans
$\mathcal{C}$ indexé par $I$ est une famille
$A_{\bullet}=((A_{i})_{i\in I},(f_{i,j})_{i\leq j})$
d’objets et pour chaque $i\leq j$ de morphismes $f_{i,j}:A_{i}\longrightarrow
A_{j}$ tels que $f_{i,i}=\mathrm{id}_{A_{i}}$ et que $f_{i,k}=f_{i,j}\circ
f_{j,k}.$ Une limite inductive pour $A_{\bullet}$ est alors un objet
$\varinjlim_{I}A_{\bullet}$ tel que pour tout objet $Z$ dans $\mathcal{C},$ on
ait une bijection naturelle
$\mathrm{Hom}(\varinjlim_{I}A_{\bullet},Z)\cong\varprojlim_{I}\mathrm{Hom}(A_{i},Z),$
cette dernière limite étant la famille incluse dans
$\prod_{i}\mathrm{Hom}(Z,A_{i})$ de morphismes $h_{i}$ tels que $f_{i,j}\circ
h_{j}=h_{i}.$ On dit que la limite est filtrée lorsque pour chaque couple
$(p,q)$ d’éléments de $I,$ il existe $r\in I$ tel que $p\leq r,q\leq r.$
Limite projective Soit $\mathcal{C}$ une catégorie et soit $(I,\leq)$ un
ensemble partiellement ordonné. On définit les limites projectives en
inversant les buts et sources des morphismes dans la définition de limite
inductive. Ainsi un système projectif d’objets dans $\mathcal{C}$ indexé par
$I$ est une famille $A_{\bullet}=((A_{i})_{i\in I},(f_{i,j})_{i\leq j})$
d’objets et pour chaque $i\leq j$ de morphismes $f_{i,j}:A_{j}\longrightarrow
A_{i}$ tels que $f_{i,i}=\mathrm{id}_{A_{i}}$ et que $f_{i,k}=f_{i,j}\circ
f_{j,k}.$ Une limite projective pour $A_{\bullet}$ est un objet
$\varprojlim_{I}A_{\bullet}$ tel que pour tout $Z$ dans $\mathcal{C},$ on ait
une bijection naturelle
$\mathrm{Hom}(Z,\varprojlim_{I}A_{\bullet})\cong\varprojlim_{I}\mathrm{Hom}(Z,A_{i}),$
cette dernière limite étant l’ensemble des
$h_{i}\in\prod_{i}\mathrm{Hom}(A_{i},Z)$ tels que $f_{i,j}\circ h_{j}=h_{i}.$
Objet (co)simplicial. Soit $\Delta$ la catégorie des ensembles finis
totalement ordonnés munis des applications croissantes. Elle est équivalente à
la catégorie dont les objets sont les ensembles n et dont les morphismes sont
les applications croissantes entre eux. On appelle objet (co)simplicial dans
une catégorie $\mathcal{C}$ un foncteur (co)contravariant de $\Delta$ dans
$\mathcal{C}.$ D’après l’équivalence des catégories mentionnées, un objet
(co)simplical peut être pensé comme une famille d’objets $X_{\textbf{n}}$
munis des morphismes entre eux correspondant aux applications croissantes
$\varphi:\textbf{n}\longrightarrow\textbf{m}$. En ce sens, une monade
algébrique est un objet cosimplicial dans la catégorie des ensembles: la
famille d’objets est $\\{\Sigma(\textbf{n})\\}_{n\geq 0}$ et les morphismes
sont les applications
$\Sigma(\varphi):\Sigma(\textbf{n})\longrightarrow\Sigma(\textbf{m})$ induites
par $\varphi:\textbf{n}\longrightarrow\textbf{m}$.
Produit fibré. Soient $X,Y,S$ trois objets d’une catégorie $\mathcal{C}$ ayant
des morphismes $f:X\longrightarrow S$ et $g:Y\longrightarrow S$ avec le même
but. On appelle produit fibré de $X$ et $Y$ sur $S$ le seul triplet (à
isomorphisme près) $(X\times_{S}Y,p,q)$ formé d’un objet $X\times_{S}Y$ et de
deux morphismes $p:X\times_{S}Y\longrightarrow X,q:X\times_{S}Y\longrightarrow
Y$ tels que pour tout objet $Z$ dans $\mathcal{C},$ l’on ait une bijection
naturel d’ensembles
$\mathrm{Hom}(Z,X\times_{S}Y)\cong\\{(h,k)\in\mathrm{Hom}(Z,X)\times\mathrm{Hom}(Z,Y):f\circ
h=g\circ k\\}.$
Par exemple, dans la catégorie des ensembles, le produit fibré n’est autre
chose que $X\times_{Z}Y=\\{(x,y)\in X\times Y:f(x)=g(y)\\}$ muni des
restrictions des projections usuelles $\text{pr}_{1}$ et $\text{pr}_{2}$ à cet
ensemble.
Sous-catégorie pleine. Soit $\mathcal{C}$ une catégorie. Une sous-catégorie
$\mathcal{D}$ de $\mathcal{C}$ est la donnée de quelques objets et de quelques
morphismes dans $\mathcal{C}$ tels que
1. 1.
Si $f:X\longrightarrow Y$ est un morphisme dans $\mathcal{D},$ alors
$X,Y\in\mathrm{Ob}(\mathcal{D}).$
2. 2.
Si $X$ est un objet dans $\mathcal{D},$ alors $\mathrm{id}_{X}$ appartient aux
morphismes de $\mathcal{D}.$
3. 3.
Si $f,g$ sont des morphismes composables dans $\mathcal{D},$ alors leur
composition appartient à $\mathcal{D}.$
On dit que $\mathcal{D}$ est pleine si le foncteur d’inclusion
$i:\mathcal{D}\longrightarrow\mathcal{C}$ est plein d’après la définition ci-
dessus, ce qui revient à dire que
$\mathrm{Hom}_{\mathcal{D}}(X,Y)=\mathrm{Hom}_{\mathcal{C}}(X,Y)$ pour tout
couple $(X,Y)$ d’objets dans $\mathcal{D}.$
Sous-foncteur. Soit $F$ un foncteur d’une catégorie $\mathcal{C}$ vers les
ensembles. On dit qu’un foncteur $G:\mathcal{C}\longrightarrow\mathfrak{Ens}$
est un sous-foncteur de $F$, et on le note $G\subset F$ par analogie avec les
ensembles, si et seulement si
1. 1.
pour tout objet $X$ dans $\mathcal{C}$, $F(X)=G(X)$ et
2. 2.
pour tout morphisme $f:X\longrightarrow Y$, $G(f)=F(f)_{|G(X)}$.
## References
* [1]
* [2]
### Bibiliographie sur $\mathbb{F}_{1}.$
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|
arxiv-papers
| 2009-08-27T18:35:31 |
2024-09-04T02:49:04.881878
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Javier Fresan",
"submitter": "Javier Fresan",
"url": "https://arxiv.org/abs/0908.4059"
}
|
0908.4121
|
Mapping class group
and a global Torelli theorem
for hyperkähler manifolds
Misha Verbitsky111The work is partially supported by RFBR grant 09-01-00242-a,
Science Foundation of the SU-HSE award No. 09-09-0009 and by RFBR grant
10-01-93113-NCNIL-a
Abstract
A mapping class group of an oriented manifold is a quotient of its
diffeomorphism group by the isotopies. We compute a mapping class group of a
hypekähler manifold $M$, showing that it is commensurable to an arithmetic
lattice in $SO(3,b_{2}-3)$. A Teichmüller space of $M$ is a space of complex
structures on $M$ up to isotopies. We define a birational Teichmüller space by
identifying certain points corresponding to bimeromorphically equivalent
manifolds. We show that the period map gives the isomorphism between connected
components of the birational Teichmüller space and the corresponding period
space $SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$. We use this result to obtain
a Torelli theorem identifying each connected component of the birational
moduli space with a quotient of a period space by an arithmetic group. When
$M$ is a Hilbert scheme of $n$ points on a K3 surface, with $n-1$ a prime
power, our Torelli theorem implies the usual Hodge-theoretic birational
Torelli theorem (for other examples of hyperkähler manifolds, the Hodge-
theoretic Torelli theorem is known to be false).
###### Contents
1. 1 Introduction
1. 1.1 Hyperkähler manifolds and their moduli
2. 1.2 Teichmüller space of a hyperkähler manifold
3. 1.3 The birational Teichmüller space
4. 1.4 The mapping class group of a hyperkähler manifold
5. 1.5 Teichmüller space and Torelli-type theorems
6. 1.6 A Hodge-theoretic Torelli theorem for $K3^{[n]}$
7. 1.7 Moduli of polarized hyperkähler varieties
2. 2 Hyperkähler manifolds
1. 2.1 Hyperkähler structures
2. 2.2 The Bogomolov’s decomposition theorem
3. 2.3 Kähler cone for hyperkähler manifolds
4. 2.4 The structure of the period space
3. 3 Mapping class group of a hyperkähler manifold
4. 4 Weakly Hausdorff manifolds and Hausdorff reduction
1. 4.1 Weakly Hausdorff manifolds
2. 4.2 Inseparable points in weakly Hausdorff manifolds
3. 4.3 Hausdorff reduction for weakly Hausdorff manifolds
4. 4.4 The birational Teichmüller space for a hyperkähler manifold
5. 5 Subtwistor metric on the period space
1. 5.1 Hyperkähler lines and hyperkähler structures
2. 5.2 Generic hyperkähler lines and the Teichmüller space
3. 5.3 GHK lines and subtwistor metrics
4. 5.4 Connected sequences of GHK lines
5. 5.5 Lipschitz homogeneous metric spaces
6. 5.6 Gleason-Palais theorem and its applications
6. 6 GHK lines and exceptional sets
1. 6.1 Lifting the GHK lines to the Teichmüller space
2. 6.2 Exceptional sets of etale maps
3. 6.3 Subsets covered by GHK lines
7. 7 Monodromy group for $K3^{[n]}$.
1. 7.1 Monodromy group for hyperkähler manifolds
2. 7.2 The Hodge-theoretic Torelli theorem for $K3^{[n]}$
8. 8 Appendix: A criterion for a covering map (by Eyal Markman)
## 1 Introduction
### 1.1 Hyperkähler manifolds and their moduli
Throughout this paper, a hyperkähler manifold is a compact, holomorphically
symplectic manifold of Kähler type, simply connected and with
$H^{2,0}(M)={\mathbb{C}}$. In the literature, such manifolds are often called
simple, or irreducible. For an explanation of this term and an introduction to
hyperkähler structures, please see Subsection 2.1.
We shall say that a complex structure $I$ on $M$ is of hyperkähler type if
$(M,I)$ is a hyperkähler manifold.
There are many different ways to define the moduli of complex structures. In
this paper we use the earliest one, which is due to Kodaira-Spencer. Let $M$
be an oriented manifold, ${\mathfrak{I}}$ the space of all complex structures
of hyperkähler type, compatible with orientation, and ${\cal
M}:={\mathfrak{I}}/\operatorname{\sf Diff}$ its quotient by the group of
oriented diffeomorphisms.111Throughout this paper, we speak of oriented
diffeomorphisms, but the reasons for this assumption are purely historical. We
could omit the mention of orientation, and most of the results will remain
valid. We call ${\cal M}$ the moduli space of complex structures of
hyperkähler type (or just “the moduli space”) of $M$. This space is a complex
variety (as shown by Kodaira-Spencer), usually non-Hausdorff.
For a hyperkähler manifold, the non-Hausdorff points of $\cal M$ are easy to
control, due to a theorem of D. Huybrechts (4.4). If $I_{1},I_{2}\in{\cal M}$
are inseparable points in ${\cal M}$, then the corresponding hyperkähler
manifolds are bimeromorphic.
In many cases, the moduli of complex structures on $M$ can be described in
terms of Hodge structures on cohomology of $M$. Such results are called
Torelli theorems. In this paper, we state a Torelli theorem for hyperkähler
manifolds, using the language of mapping class group and Teichmüller spaces.
This approach to the Torelli-type problems was pioneered by A. Todorov in
several important preprints and papers ([T1], [T2]; see also [LTYZ]).
### 1.2 Teichmüller space of a hyperkähler manifold
To define the period space for hyperkähler manifolds, one uses the so-called
Bogomolov-Beauville-Fujiki (BBF) form on the second cohomology. Historically,
it was the BBF form which was defined in terms of the period space, and not
vice versa, but the other way around is more convenient.
Let $\Omega$ be a holomorphic symplectic form on $M$. Bogomolov and Beauville
([Bo2], [Bea]) defined the following bilinear symmetric 2-form on $H^{2}(M)$:
$\displaystyle\widetilde{q}(\eta,\eta^{\prime}):=$
$\displaystyle\int_{M}\eta\wedge\eta^{\prime}\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$
(1.1)
$\displaystyle-\frac{n-2}{n}\frac{\left(\int_{M}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{M}\eta^{\prime}\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)}{\int_{M}\Omega^{n}\wedge\overline{\Omega}^{n}}$
where $n=\dim_{\mathbb{H}}M$.
Remark 1.1: The form $\widetilde{q}$ is compatible with the Hodge
decomposition, which is seen immediately from its definition. Also,
$\widetilde{q}(\Omega,\overline{\Omega})>0$.
The form $\widetilde{q}$ is topological by its nature.
Theorem 1.2: ([F]) Let be a simple hyperkähler manifold of real dimension
$4n$. Then there exists a bilinear, symmetric, primitive non-degenerate
integer 2-form $q:\;H^{2}(M,{\mathbb{Z}})\otimes
H^{2}(M,{\mathbb{Z}}){\>\longrightarrow\>}{\mathbb{Z}}$ and a constant
$c\in{\mathbb{Z}}$ such that
$\int_{M}\eta^{2n}=cq(\eta,\eta)^{n},$ (1.2)
for all $\eta\in H^{2}(M)$. Moreover, $q$ is proportional to the form
$\widetilde{q}$ of (1.1), and has signature $(+,+,+,-,-,-,...)$.
Remark 1.3: If $n$ is odd, the equation (1.2) determines $q$ uniquely,
otherwise – up to a sign. To choose a sign, we use (1.1).
Definition 1.4: Let $M$ be a hyperkähler manifold, and $\Omega$ a holomorphic
symplectic form on $M$. A Beauville-Bogomolov-Fujiki form on $M$ is a form
$q:\;H^{2}(M,{\mathbb{Q}})\otimes
H^{2}(M,{\mathbb{Q}}){\>\longrightarrow\>}{\mathbb{Q}}$ which satisfies (1.2),
and has $q(\Omega,\overline{\Omega})>0$.
Remark 1.5: The Beauville-Bogomolov-Fujiki form is integer, but not unimodular
on $H^{2}(M,{\mathbb{Z}})$.
Definition 1.6: Let $(M,I)$ be a compact hyperkähler manifold,
${\mathfrak{I}}$ the set of oriented complex structures of hyperkähler type on
$M$, and $\operatorname{\sf Diff}_{0}(M)$ the group of isotopies. The quotient
space $\operatorname{\sf Teich}:={\mathfrak{I}}/\operatorname{\sf
Diff}_{0}(M)$ is called the Teichmüller space of $(M,I)$, and the quotient of
$\operatorname{\sf Teich}$ over a whole oriented diffeomorphism group the
coarse moduli space of $(M,I)$.
In a similar way one defines the moduli of Kähler structures or of complex
structures on a given Kähler or complex manifold.
Definition 1.7: Let $(M,I)$ be a simple hyperkähler manifold, and
$\operatorname{\sf Teich}$ its Teichmüller space. For any
$J\in\operatorname{\sf Teich}$, $(M,J)$ is also a simple hyperlähler manifold,
as seen from 2.2 below, hence $H^{2,0}(M,J)$ is one-dimensional. Consider a
map $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$, sending $J$ to
a line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$. Clearly,
$\operatorname{\sf Per}$ maps $\operatorname{\sf Teich}$ into an open subset
of a quadric, defined by
$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\
\ |\ \ q(l,l)=0,q(l,\overline{l})>0\\}.$ (1.3)
The map $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ is called the
period map, and the set $\operatorname{{\mathbb{P}}\sf er}$ the period space.
The following fundamental theorem is due to F. Bogomolov [Bo2].
Theorem 1.8: Let $M$ be a simple hyperkähler manifold, and $\operatorname{\sf
Teich}$ its Teichmüller space. Then the period map $\operatorname{\sf
Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ is locally a
unramified covering (that is, an etale map).
Remark 1.9: Bogomolov’s theorem implies that $\operatorname{\sf Teich}$ is
smooth. However, it is not necessarily Hausdorff (and it is non-Hausdorff even
in the simplest examples).
Remark 1.10: D. Huybrechts has shown that $\operatorname{\sf Per}$ is
surjective ([H1], Theorem 8.1).
Remark 1.11: Using the boundedness results of Kollar and Matsusaka ([KM]), D.
Huybrechts has shown that the space $\operatorname{\sf Teich}$ has only a
finite number of connected components ([H5], Theorem 2.1).
The moduli ${\cal M}$ of complex structures on $M$ is a quotient of
$\operatorname{\sf Teich}$ by the action of the mapping class group
$\Gamma:=\operatorname{\sf Diff}/\operatorname{\sf Diff}_{0}$ of
diffeomorphisms up to isotopies. There is an interesting intermediate group
$\operatorname{\sf Diff}_{H}$ of all diffeomorphisms acting trivially on
$H^{2}(M)$. One has $\operatorname{\sf Diff}_{0}\subset\operatorname{\sf
Diff}_{H}\subset\operatorname{\sf Diff}$. The corresponding quotient
$\operatorname{\sf Teich}/\operatorname{\sf Diff}_{H}$ is called the coarse,
marked moduli of complex structures, and its points – marked hyperkähler
manifolds. To choose a marking it means to choose a basis in the cohomology of
$M$. The period map is well defined on $\operatorname{\sf
Teich}/\operatorname{\sf Diff}_{H}$.
We don’t use the marked moduli space in this paper, because the Teichmüller
space serves the same purpose. In the literature on moduli spaces, the marked
moduli space is used throughout, but these results are easy to translate to
the Teichmüller spaces’ language using the known facts about the mapping class
group.
For a K3 surface, the Teichmüller space is not Hausdorff. However, a quotient
of polarized moduli space by the mapping class group is Hausdorff. Moreover, a
version of Torelli theorem is valid, providing an isomorphism between
$\operatorname{\sf Teich}/\Gamma$ and $\operatorname{{\mathbb{P}}\sf
er}/O^{+}(H^{2}(M,{\mathbb{Z}}))$.222For an explanation of $O^{+}$, please see
7.2. This result has a long history, with many people contributing to
different sides of the picture, but its conclusion could be found in [BR] and
[Si].
One could state this Torelli theorem as a result about the Hodge structures,
as follows. The Torelli theorem claims that there is a bijective
correspondence between isomorphism classes of K3 surfaces and the set of
isomorphism classes of appropriate Hodge structures on a 22-dimensional space
equipped with an integer lattice, a spin orientation (7.2) and an integer
quadratic form.
It is natural to expect that this last result would be generalized to other
hyperkähler manifolds, but this straightforward generalization is invalid. In
[De], O. Debarre has shown that there exist birational hyperkähler manifolds
which are non-isomorphic, but have the same periods. A hope to have a Hodge
theoretic Torelli theorem for birational moduli was extinguished in early
2000-ies. As shown by Yo. Namikawa in a beautiful (and very short) paper [Na],
there exist hyperkähler manifolds $M,M^{\prime}$ which are not
bimeromorphically equivalent, but their second cohomology have equivalent
Hodge structures.
For the benefit of the reader, we give here a brief reprise of the Namikawa’s
construction. Let $T$ be a compact, complex, 2-dimensional torus, and
$T^{[n]}$ its Hilbert scheme. The torus $T$ acts on $T^{[n]}$, and its
quotient $T^{[n]}/T$ is called a generalized Kummer manifold. When $n=2$, this
quotient is a K3 surface obtained from the torus using the Kummer
construction. For $n>2$, the Hodge structure on $H^{2}(T^{[n]}/T)$ is easy to
describe. One has
$H^{2}(T^{[n]}/T)\cong\operatorname{Sym}^{2}(H^{1}(T))\oplus{\mathbb{R}}\eta,$
where $\eta$ is the fundamental class of the exceptional divisor of
$M:=T^{[n]}/T$. Therefore, $H^{2}(M)$ has the same Hodge structure as
$M^{\prime}=(T^{*})^{[n]}/T^{*}$, where $T^{*}$ is the dual torus. However,
the manifolds $M$ and $M^{\prime}$ are not bimeromorphically equivalent, when
$T$ is generic. This is easy to see, for instance, for $n=3$, because the
exceptional divisor of $M=T^{[3]}/T$ is a trivial
${\mathbb{C}}P^{1}$-fibration over $T$, and the exceptional divisor of
$M^{\prime}=(T^{*})^{[3]}/T^{*}$ is fibered over $T^{*}$ likewise. Since
bimeromorphic maps of holomorphic symplectic varieties are non-singular in
codimension 2, any bimeromorphic isomorphism between $M$ and $M^{\prime}$
would bring a bimeromorphic isomorphism between these divisors, and therefore
between $T$ and $T^{\prime}$, which is impossible, for general $T$.
A less elementary construction, due to E. Markman, gives a counterexample to
Hodge-theoretic global Torelli theorem when $M=K3^{[n]}$ is the Hilbert scheme
of points on a K3 surface, and $n-1$ is not a prime power ([M2]). When $n-1$
is a prime power, a Hodge-theoretic birational Torelli theorem holds true
(Subsection 7.2).
We are going to prove a different version of Torelli theorem, using the
language of Teichmüller spaces and the mapping class groups.
### 1.3 The birational Teichmüller space
The Teichmüller space approach allows one to state the Torelli theorem for
hyperkähler manifolds as it is done for curves. However, before any theorems
can be stated, we need to resolve the issue of non-Hausdorff points.
Definition 1.12: Let $M$ be a topological space. We say that points $x,y\in
M$ are inseparable (denoted $x\sim y$) if for any open subsets $U\ni x,V\ni
y$, one has $U\cap V\neq\emptyset$.
Remark 1.13: As shown by Huybrechts (4.4), inseparable points on a Teichmüller
space correspond to bimeromorphically equivalent hyperkähler manifolds.
Theorem 1.14: Let $\operatorname{\sf Teich}$ be a Teichmüller space of a
hyperkähler manifold, and $\sim$ the inseparability relation defined above.
Then $\sim$ is an equivalence relation. Moreover, the quotient
$\operatorname{\sf Teich}_{b}:=\operatorname{\sf Teich}\\!/{}_{\sim}$ is a
smooth, Hausdorff complex analytic manifold.
Proof: 4.2, 4.3.
We call the quotient $\operatorname{\sf Teich}\\!/{}_{\sim}$ the birational
Teichmüller space, denoting it as $\operatorname{\sf Teich}_{b}$. The
operation of taking the quotient $.../{}_{\sim}$ as above has good properties
in many situations, and brings similar results quite often. We call
$W\\!/{}_{\sim}$ the Hausdorff reduction of $W$ whenever it is Hausdorff (see
Subsection 4.3 for a detailed exposé).
### 1.4 The mapping class group of a hyperkähler manifold
Define the mapping class group $\Gamma:=\operatorname{\sf
Diff}/\operatorname{\sf Diff}_{0}$ of a manifold $M$ as a quotient of a group
of oriented diffeomorphisms of $M$ by isotopies. Clearly, $\Gamma$ acts on
$H^{2}(M,{\mathbb{R}})$ perserving the integer structure. We are able to
determine the group $\Gamma$ up to commensurability, proving that it is
commensurable to an arithmetic group $O(H^{2}(M,{\mathbb{Z}}),q)$ of finite
covolume in $O(3,b_{2}(M)-3)$.
Theorem 1.15: Let $M$ be a compact, simple hyperkähler manifold, and
$\Gamma=\operatorname{\sf Diff}/\operatorname{\sf Diff}_{0}$ its mapping class
group. Then $\Gamma$ acts on $H^{2}(M,{\mathbb{R}})$ preserving the Bogomolov-
Beauville-Fujiki form. Moreover, the corresponding homomorphism
$\Gamma{\>\longrightarrow\>}O(H^{2}(M,{\mathbb{Z}}),q)$ has finite kernel, and
its image has finite index in $O(H^{2}(M,{\mathbb{Z}}),q)$ .
Proof: This is 3.
Using results of E. Markman ([M2]), it is possible to compute the mapping
class group for a Hilbert scheme of points on a K3 surface $M=K3^{[n]}$, when
$n-1$ is a prime power (7.2).
### 1.5 Teichmüller space and Torelli-type theorems
The following version of Torelli theorem is proven in Section 6.
Theorem 1.16: Let $M$ be a compact, simple hyperkähler manifold, and
$\operatorname{\sf Teich}_{b}$ its birational Teichmüller space. Consider the
period map $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}_{b}:\;{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$, where
$\operatorname{{\mathbb{P}}\sf er}$ is the period space defined as in (1.3).
Then $\operatorname{\sf Per}$ is a diffeomorphism, for each connected
component of $\operatorname{\sf Teich}_{b}$.
Proof: This is 4.4.
The proof of 1.5 is obtained by using the quaternionic structures, associated
with holomorphic symplectic structures by the Calabi-Yau theorem, and the
corresponding rational lines in $\operatorname{\sf Teich}$ and
$\operatorname{{\mathbb{P}}\sf er}$.
If one wants to obtain a more traditional Torelli-type theorem, one should
consider the set of equivalence classes of complex structures up to birational
equivalence. This set can be interpreted in terms of the Teichmüller space as
follows.
Consider the action of the mapping class group $\Gamma$ on the Teichmüller
space $\operatorname{\sf Teich}$, and let $\operatorname{\sf Teich}^{I}$ be a
connected component of $\operatorname{\sf Teich}$ containing a given complex
structure $I$. Denote by $\Gamma_{I}\subset\Gamma$ a subgroup of $\Gamma$
preserving $\operatorname{\sf Teich}^{I}$. Since $\operatorname{\sf Teich}$
has only a finite number of connected components ([H5], Theorem 2.1),
$\Gamma_{I}$ has a finite index in $\Gamma$. The coarse moduli space of
complex structures on $M$ is $\operatorname{\sf Teich}^{I}/\Gamma_{I}$, and
the birational moduli is $\operatorname{\sf Teich}^{I}_{b}/\Gamma_{I}$, where
$\operatorname{\sf Teich}_{b}^{I}$ is the appropriate connected component of
$\operatorname{\sf Teich}_{b}$. 1.5 immediately implies the following Torelli-
type result.
Theorem 1.17: Let $M$ be a compact, simple hyperkähler manifold, ${\cal
M}_{b}:=\operatorname{\sf Teich}^{I}_{b}/\Gamma_{I}$ a connected component of
the birational moduli space defined above, and
${\cal M}_{b}\stackrel{{\scriptstyle\operatorname{\sf
Per}}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf er}/{\Gamma_{I}}$
(1.4)
the corresponding period map. Then (1.4) is an isomorphism of complex analytic
spaces.
Remark 1.18: The image $i(\Gamma_{I})$ of $\Gamma_{I}$ in
$O(H^{2}(M,{\mathbb{Z}}),q)$ has finite index (1.4). Therefore, it is an
arithmetic sugbroup of finite covolume.
Comparing this with 1.5, we immediately obtain the following corollary.
Corollary 1.19: Let $M$ be a compact, simple hyperkähler manifold, and ${\cal
M}_{b}$ a connected component of its birational moduli space, obtained as
above. Then ${\cal M}_{b}$ is isomorphic to a quotient of a homogeneous space
$\operatorname{{\mathbb{P}}\sf er}=\frac{O(b_{2}-3,3)}{SO(2)\times
O(b_{2}-3,1)}$
by an action of an arithmetic subgroup $i(\Gamma_{I})\subset
O(H^{2}(M,{\mathbb{Z}}),q)$.333For this interpretation of
$\operatorname{{\mathbb{P}}\sf er}$, please see Subsection 2.4.
In a traditional version of Torelli theorem, one takes a quotient of
$\operatorname{{\mathbb{P}}\sf er}$ by $O^{+}(H^{2}(M,{\mathbb{Z}}),q)$
instead of $i(\Gamma_{I})\subset
O^{+}(H^{2}(M,{\mathbb{Z}}),q)$.444$O^{+}(H^{2}(M,{\mathbb{Z}}),q)$ is a group
of orthogonal maps with positive spin norms (7.2). However, such a result
cannot be valid, as shown by Namikawa. 1.5 explains why this occurs: the group
$i(\Gamma_{I})$ is a proper subgroup in $O^{+}(H^{2}(M,{\mathbb{Z}}),q)$, and
the composition
${\cal M}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}/\Gamma_{I}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}/O^{+}(H^{2}(M,{\mathbb{Z}}),q)$ (1.5)
is a finite quotient map. We obtained the following corollary.
Corollary 1.20: Let $M$ be a compact, simple hyperkähler manifold, ${\cal
M}_{b}$ a connected component of its birational moduli space, and
${\cal M}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}/O^{+}(H^{2}(M,{\mathbb{Z}}),q)$ (1.6)
the corresponding period map. Then (1.6) is a finite quotient.
Remark 1.21: Please notice that the space $\operatorname{{\mathbb{P}}\sf
er}/O^{+}(H^{2}(M,{\mathbb{Z}}),q)$ is usually non-Hausdorff. However, it can
be made Hausdorff if one introduces additional structures (such as a
polarization), and then 1.5 becomes more useful.
For the Hilbert scheme of $n$ points on a K3 surface, the image of
$\Gamma_{I}$ in $O^{+}(H^{2}(M,{\mathbb{Z}}))$ was computed by E. Markman in
[M2] (see 7.2). When $n-1$ is a prime power,
$i(\Gamma_{I})=O^{+}(H^{2}(M,{\mathbb{Z}}))$, and the composition (1.6) is an
isomorphism, which is used to obtain the usual (Hodge-theoretic) version of
Torelli theorem.
### 1.6 A Hodge-theoretic Torelli theorem for $K3^{[n]}$
In [M1], [M2], E. Markman has proved many vital results on the way to
computing the mapping class group of a Hilbert scheme of points on K3 (denoted
by $K3^{[n]}$). Markman’s starting point was the notion of a monodromy group
of a hyperkähler manifold. A monodromy group of $M$ is the group generated by
monodromy of the Gauss-Manin local systems for all deformations of $M$ (see
Subsection 7.1 for a more detailed description). In Subsection 7.1, we relate
the monodromy group $\operatorname{\sf Mon}$ to the mapping class group
$\Gamma^{I}$, showing that $\operatorname{\sf Mon}$ is isomorphic to an image
of $\Gamma^{I}$ in $PGL(H^{2}(M,{\mathbb{C}}))$. For $M=K3^{[n]}$, Markman has
computed the monodromy group, using the action of Fourier-Mukai transform in
the derived category of coherent sheaves. He used this computation to show
that the standard (Hodge-theoretic) global Torelli theorem fails on
$K3^{[n]}$, unless $n-1$ is a prime power. We complete Markman’s analysis of
global Torelli problem for $K3^{[n]}$, proving the following.
Theorem 1.22: Let $M=K3^{[n]}$ be a Hilbert scheme of points on a K3 surface,
where $n-1$ is a prime power, and $I_{1},I_{2}$ deformations of complex
structures on $M$. Assume that the Hodge structures on $H^{2}(M,I_{1})$ and
$H^{2}(M,I_{2})$ are isomorphic, and this isomorphism is compatible with the
Bogomolov-Beauville-Fujiki form and the natural spin orientation on
$H^{2}(M,I_{1})$ and $H^{2}(M,I_{2})$. (7.2). Then $(M,I_{1})$ is
bimeromorphic to $(M,I_{2})$.
Proof: This is 7.2.
Remark 1.23: E. Markman in [M2] constructed counterexamples to the Hodge-
theoretic global Torelli problem for $K3^{[n]}$, where $n-1$ is not a prime
power.
### 1.7 Moduli of polarized hyperkähler varieties
For another application of 1.5, fix an integer class $\eta\in
H^{2}(M,{\mathbb{Z}})$, $q(\eta,\eta)>0$, and let $\operatorname{\sf
Teich}_{\eta}$ be a divisor in the connected component of the Teichmüller
space consisting of all $I$ with $\eta\in H^{1,1}(M,I)$. For a general
$I\in\operatorname{\sf Teich}_{\eta}$, $\eta$ or $-\eta$ is a Kähler class on
$(M,I)$ ([H3]; see also 2.3). However, there could be special points where
$\pm\eta$ is not Kähler.
Let $\operatorname{\sf Teich}_{\eta}^{I}$ be a connected component of
$\operatorname{\sf Teich}_{\eta}$. Denote by $\overline{\cal M}_{\eta}$ the
quotient of $\operatorname{\sf Teich}_{\eta}^{I}$ by the subgroup
$\Gamma_{\eta}^{I}$ of the mapping class group fixing $\eta$ and preserving
the component $\operatorname{\sf Teich}_{\eta}^{I}$. The same argument as
above can be used to show that $\Gamma_{\eta}^{I}$ is commensurable to an
arithmetic subgroup in $SO(\eta^{\bot})$, where $\eta^{\bot}\subset
H^{2}(M,{\mathbb{R}})$ is an orthogonal complement to $\eta$.
We call $\overline{\cal M}_{\eta}$ a connected component of the moduli space
of weakly polarized hyperkähler manifolds. A corresponding component ${\cal
M}_{\eta}$ of the moduli polarized hyperkähler manifolds is an open subset of
$\overline{\cal M}_{\eta}$ consisting of all $I$ for which $\eta$ is Kähler.
It is known (due to general theory which goes back to Viehweg, Grothendieck
and Kodaira-Spencer) that ${\cal M}_{\eta}$ is Hausdorff and quasiprojective
(see e.g. [Vi], [GHS2]).
The period space for weakly polarized hyperkähler manifolds is
$\operatorname{{\mathbb{P}}\sf
er}_{\eta}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\ \ |\ \ q(l,l)=0,\
q(\eta,l)=0,\ q(l,\overline{l})>0\\}.$ (1.7)
and the corresponding period map $\operatorname{\sf
Teich}_{\eta}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}_{\eta}$
induces an isomorphism from the Hausdorff reduction $\operatorname{\sf
Teich}_{\eta,b}^{I}$ of the component $\operatorname{\sf Teich}_{\eta}^{I}$ to
$\operatorname{{\mathbb{P}}\sf er}_{\eta}$, as follows from 1.5.
We define a connected component of the birational moduli space of weakly
polarized hyperkähler manifolds $\overline{\cal M}_{b,\eta}$ as a quotient of
the component $\operatorname{\sf Teich}_{b,\eta}^{I}$ by the corresponding
mapping class group $\Gamma_{\eta}^{I}$. It is obtained from $\overline{\cal
M}_{\eta}$ by identifying inseparable points.
Just as in Subsection 2.4, we may identify the period space
$\operatorname{{\mathbb{P}}\sf er}_{\eta}$ with the Grassmannian of positive
2-planes in $\eta^{\bot}$. This gives
$\operatorname{{\mathbb{P}}\sf er}_{\eta}\cong SO(b_{2}-3,2)/SO(2)\times
SO(b_{2}-3).$
This is significant, because $\operatorname{{\mathbb{P}}\sf er}_{\eta}$
(unlike $\operatorname{{\mathbb{P}}\sf er}$) is a symmetric space. The
corresponding result for the moduli spaces can be stated as follows.
Corollary 1.24: Let $(M,\eta)$ be a compact, simple, polarized hyperkähler
manifold, $\overline{\cal M}_{b,\eta}$ a connected component of the weakly
polarized birational moduli space, defined above, $G$ the group of integer
orthogonal automorphisms of the lattice $\eta^{\bot}$ of primitive elements in
$H^{2}(M)$, and
$\overline{\cal M}_{b,\eta}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}_{\eta}/G$ (1.8)
the corresponding period map. Then (1.8) is a finite quotient. Moreover,
${\cal M}_{b,\eta}$ is isomorphic to a quotient of a symmetric domain
$\operatorname{{\mathbb{P}}\sf er}_{\eta}$ by an arithmetic group
$\Gamma_{\eta}^{I}$ acting as above.
The quotients of such symmetric spaces by arithmetic lattices were much
studied by Gritsenko, Hulek, Nikulin, Sankaran and many others (see e.g.
[GHS1], [GHS2] and references therein). The geometry of
$\operatorname{{\mathbb{P}}\sf er}_{\eta}/G$ is in many cases well understood.
Using the theory of automorphic forms, many sections of pluricanonial (or, in
some cases, plurianticanonical) class can be found, depending on
$q(\eta,\eta)$ and other properties of the lattice $\eta^{\bot}$. In such
cases, 1.7 can be used to show that the weakly polarized birational moduli
space has ample (or antiample) canonical class.555See [DV] for an alternative
approach to the same problem.
The automorphic forms on polarized moduli were also used to show non-existence
of complete families of polarized K3 surfaces ([BKPS]). This program was
proposed by J. Jorgensen and A. Todorov in 1990-ies, in a string of influental
(but, sometimes, flawed) preprints, culminating with [JT].
## 2 Hyperkähler manifolds
In this Section, we recall a number of results about hyperkähler manifolds,
used further on in this paper. For more details and reference, please see
[Bes].
### 2.1 Hyperkähler structures
Definition 2.1: Let $(M,g)$ be a Riemannian manifold, and $I,J,K$
endomorphisms of the tangent bundle $TM$ satisfying the quaternionic relations
$I^{2}=J^{2}=K^{2}=IJK=-\operatorname{Id}_{TM}.$
The triple $(I,J,K)$ together with the metric $g$ is called a hyperkähler
structure if $I,J$ and $K$ are integrable and Kähler with respect to $g$.
Consider the Kähler forms $\omega_{I},\omega_{J},\omega_{K}$ on $M$:
$\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \
\omega_{J}(\cdot,\cdot):=g(\cdot,J\cdot),\ \
\omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$
An elementary linear-algebraic calculation implies that the 2-form
$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ is of Hodge type $(2,0)$ on
$(M,I)$. This form is clearly closed and non-degenerate, hence it is a
holomorphic symplectic form.
In algebraic geometry, the word “hyperkähler” is essentially synonymous with
“holomorphically symplectic”, due to the following theorem, which is implied
by Yau’s solution of Calabi conjecture ([Bes], [Bea]).
Theorem 2.2: Let $M$ be a compact, Kähler, holomorphically symplectic
manifold, $\omega$ its Kähler form, $\dim_{\mathbb{C}}M=2n$. Denote by
$\Omega$ the holomorphic symplectic form on $M$. Suppose that
$\int_{M}\omega^{2n}=\int_{M}(\operatorname{Re}\Omega)^{2n}$. Then there
exists a unique hyperkähler metric $g$ with the same Kähler class as $\omega$,
and a unique hyperkähler structure $(I,J,K,g)$, with
$\omega_{J}=\operatorname{Re}\Omega$, $\omega_{K}=\operatorname{Im}\Omega$.
Further on, we shall speak of “hyperkähler manifolds” meaning “holomorphic
symplectic manifolds of Kähler type”, and “hyperkähler structures” meaning the
quaternionic triples.
Every hyperkähler structure induces a whole 2-dimensional sphere of complex
structures on $M$, as follows. Consider a triple $a,b,c\in R$,
$a^{2}+b^{2}+c^{2}=1$, and let $L:=aI+bJ+cK$ be the corresponging quaternion.
Quaternionic relations imply immediately that $L^{2}=-1$, hence $L$ is an
almost complex structure. Since $I,J,K$ are Kähler, they are parallel with
respect to the Levi-Civita connection. Therefore, $L$ is also parallel. Any
parallel complex structure is integrable, and Kähler. We call such a complex
structure $L=aI+bJ+cK$ a complex structure induced by a hyperkähler structure.
There is a 2-dimensional holomorphic family of induced complex structures, and
the total space of this family is called the twistor space of a hyperkähler
manifold.
### 2.2 The Bogomolov’s decomposition theorem
The modern approach to Bogomolov’s decomposition is based on Calabi-Yau
theorem (2.1), Berger’s classification of irreducible holonomy and de Rham’s
splitting theorem for holonomy reduction ([Bea], [Bes]). It is worth mention
that the original proof of decomposition theorem (due to [Bo1]) was much more
elementary.
Theorem 2.3: Let $(M,I,J,K)$ be a compact hyperkähler manifold. Then there
exists a finite covering $\widetilde{M}{\>\longrightarrow\>}M$, such that
$\widetilde{M}$ is decomposed, as a hyperkähler manifold, into a product
$\widetilde{M}=M_{1}\times M_{2}\times\dots M_{n}\times T,$
where $(M_{i},I,J,K)$ satisfy $\pi_{i}(M_{i})=0$,
$H^{2,0}(M_{i},I)={\mathbb{C}}$, and $T$ is a hyperkähler torus. Moreover,
$M_{i}$ are uniquely determined by $M$ and simply connected, and $T$ is unique
up to isogeny.
Proof: See [Bea], [Bes].
Definition 2.4: Let $(M,I,J,K)$ be a compact hyperkähler manifold which
satisfies $H^{1}(M)=0$, $H^{2,0}(M,I)={\mathbb{C}}$. Then $M$ is called a
simple hyperkähler manifold, or an irreducible hyperkähler manifold
Remark 2.5: Notice that 2.2 implies that irreducible hyperkähler manifolds are
simly connected. In particular, they do not admit a further decomposition.
This explains the term “irreducible”.
As we mentioned in the Inroduction, all hyperkähler manifolds considered
further on are assumed to be simple. Since the Hodge numbers are invariant
under deformations, the deformations of simple manifolds are always simple.
However, the irreducibility is a topological property, as implied by the
following lemma.
Lemma 2.6: Let $M$ be a compact hyperkähler manifold, which is homotopy
equivalent to a simple hyperkähler manifold. Then $M$ is also simple.
Proof: Let $A^{*}$ be the part of the rational cohomology of $M$ generated by
$H^{2}(M)$. It is well known (see [V2] and [V3]) that $A^{*}$ is up to the
middle dimension a symmetric algebra. Since $M$ is simply connected, it is
diffeomorphic to a product of simple hyperkähler manifolds. Denote by
$A^{*}_{i}$ the corresponding subalgebras in cohomology generated by
$H^{2}(M_{i})$. These subalgebras are described in a similar way, and are
symmetric up to the middle. Then $A^{*}\cong\bigotimes A_{i}^{*}$ by Künneth
formula. Since the algebras $A^{*}$, $A^{*}_{i}$ are symmetric up to the
middle, this is impossible, as follows from an easy algebraic computation.
### 2.3 Kähler cone for hyperkähler manifolds
The following theorem is implied by results of S. Boucksom, using the
characterization of a Kähler cone due to J.-P. Demailly and M. Paun (see also
[H3]).
Notice that the Beauville-Bogomolov-Fujiki form $q$ on
$H^{1,1}(M,{\mathbb{R}}):=H^{1,1}(M)\cap H^{2}(M,{\mathbb{R}})$
has signature $(+,-,-,-,...)$, hence the set of vectors $\nu\in
H^{1,1}(M,{\mathbb{R}})$ with $q(\nu,\nu)>0$ has two connected components.
Theorem 2.7: Let $M$ be a simple hyperkähler manifold such that all integer
$(1,1)$-classes satisfy $q(\nu,\nu)\geqslant 0$. Then its Kähler cone is one
of two components $K_{+}$ of a set $K:=\\{\nu\in H^{1,1}(M,{\mathbb{R}})\ \ |\
\ q(\nu,\nu)>0\\}$.
Proof: This is [V6], Corollary 2.6.
For us, the case of trivial Neron-Severi lattice is of most interest.
Corollary 2.8: Let $M$ be a compact, simple hyperkähler manifold such that
$H^{1,1}(M)\cap H^{2}(M,{\mathbb{Q}})=0$. Then its Kähler cone is one of two
components of a set $K:=\\{\nu\in H^{1,1}(M,{\mathbb{R}})\ \ |\ \
q(\nu,\nu)>0\\}.$
### 2.4 The structure of the period space
Let $M$ be a hyperkähler manifold, and $b_{2}=\dim H^{2}(M)$. It is well known
that its period space $\operatorname{{\mathbb{P}}\sf er}$ (see (1.3)) is
diffeomorphic to the Grassmann space $Gr(2)=O(b_{2}-3,3)/SO(2)\times
O(b_{2}-3,1)$ of 2-dimensional oriented planes $V\subset
H^{2}(M,{\mathbb{R}})$ with $q{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$
positive definite. Indeed, for any line
$l\in\operatorname{{\mathbb{P}}\sf
er}\subset{\mathbb{P}}H^{2}(M,{\mathbb{C}}),$
let $V_{l}$ be the span of
$\langle\operatorname{Re}l,\operatorname{Im}l\rangle$. From (1.3) it follows
that $l\cap H^{2}(M,{\mathbb{R}})=0$, hence $V_{l}$ is an oriented
2-dimensional plane. Since $q(l,\overline{l})>0$, the restriction
$q{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ is positive definite. This
gives a map from $\operatorname{{\mathbb{P}}\sf er}$ to $Gr(2)$. To construct
the inverse map, we start from a 2-dimensional plane $V\subset
H^{2}(M,{\mathbb{R}})$ and consider the quadric
$\\{v\in{\mathbb{P}}(V\otimes{\mathbb{C}})\ \ |\ \ q(v,v)=0\\}$. This quadric
is actually a union of 2 points in
${\mathbb{P}}(V\otimes{\mathbb{C}})\cong{\mathbb{C}}P^{1}$, with each of these
points corresponding to a different choice of orientation on $V$. This gives
an inverse map from $Gr(2)$ to $\operatorname{{\mathbb{P}}\sf er}$.
The following claim will be used later on.
Claim 2.9: The period space $\operatorname{{\mathbb{P}}\sf er}$ is connected
and simply connected.
Proof: We represent $\operatorname{{\mathbb{P}}\sf er}$ as
$Gr(2)=O(b_{2}-3,3)/SO(2)\times O(b_{2}-3,1)$. The group $O(b_{2}-3,3)$ is
disconnected, but $O(b_{2}-3,1)$ is also disconnected, hence the connected
components cancel each other, and $Gr(2)$ is naturally isomorphic to
$SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$.
To see that it is simply connected, we take a long exact sequence of homotopy
groups
$...{\>\longrightarrow\>}\pi_{2}(Gr(2)){\>\longrightarrow\>}\pi_{1}(SO(2)\times
SO(b_{2}-3,1))\stackrel{{\scriptstyle(*)}}{{{\>\longrightarrow\>}}}\\\
\stackrel{{\scriptstyle(*)}}{{{\>\longrightarrow\>}}}\pi_{1}(SO(b_{2}-3,3)){\>\longrightarrow\>}\pi_{1}(Gr(2)){\>\longrightarrow\>}0,$
and notice that the map (*) above is surjective (it is easy to see from the
corresponding maps of spinor groups and Clifford algebras).
## 3 Mapping class group of a hyperkähler manifold
Definition 3.1: A connected CW-complex $M$ is called nilpotent if its
fundamental group $\pi_{1}(M)$ is nilpotent, acting nilpotently on homotopy
groups of $M$.
Definition 3.2: Let $M$ be an oriented manifold, $\operatorname{\sf Diff}$ the
group of oriented diffeomorphisms, and $\operatorname{\sf Diff}_{0}$ the group
of isotopies, that is, the connected component of the group $\operatorname{\sf
Diff}$. Then the quotient $\operatorname{\sf Diff}/\operatorname{\sf
Diff}_{0}$ is called the mapping class group of $M$ (see e.g. [LTYZ]).
Definition 3.3: Let $A,A^{\prime}$ be subgroups in a group $B$. Recall that
$A$ is commensurable with $A^{\prime}$ if $A\cap A^{\prime}$ has finite index
in $A$ and $A^{\prime}$. Let $G_{\mathbb{Z}}$ a group scheme over
${\mathbb{Z}}$, and
$G_{\mathbb{R}}=G_{\mathbb{Q}}\otimes\operatorname{Spec}{\mathbb{R}}$ be the
corresponding real algebraic group. A subgroup $\Gamma\subset G_{\mathbb{R}}$
is called arithmetic if $\Gamma$ is commensurable with the group of integer
points in $G_{\mathbb{R}}$.
Using rational homotopy theory, formality of Deligne-Griffiths-Morgan-Sullivan
and Smale’s h-cobordism, D. Sullivan proved the following general result.
Theorem 3.4: Let $M$ be a compact simply connected (or nilpotent) Kähler
manifold, $\dim_{\mathbb{C}}M\geqslant 3$. Denote by $\Gamma$ the group of
automorphisms of an algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin
classes $p_{i}(M)$. Then the natural map $\operatorname{\sf
Diff}/\operatorname{\sf Diff}_{0}{\>\longrightarrow\>}\Gamma$ has finite
kernel, and its image has finite index in $\Gamma$. Finally, $\Gamma$ is an
arithmetic subgroup in the group $\Gamma_{\mathbb{Q}}$ preserving $p_{i}(M)$.
Proof: Theorem 13.3 of [Su] is stated for general smooth manifolds of
$\dim_{\mathbb{R}}\geqslant 5$; to apply it to Kähler manifolds, one needs to
use [Su, Theorem 12.1]. The final statement is [Su, Theorem 10.3].
For hyperkähler manifolds, the group
$\operatorname{Aut}(H^{*}(M,{\mathbb{Q}}))$ is determined (up to
commensurability), which leads to the following application of Sullivan’s
theorem.
Theorem 3.5: Let $M$ be a compact, simple hyperkähler manifold, its dimension
$\dim_{\mathbb{C}}M=2n$, and $\Gamma_{A}$ the group of automorphisms of an
algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin classes $p_{i}(M)$.
Consider the action of $\Gamma_{A}$ on $H^{2}(M,{\mathbb{Q}})$ and let
$\Gamma_{2}$ be an image of $\Gamma_{A}$ in $GL(H^{2}(M,{\mathbb{Q}}))$. Then
(i)
$\Gamma_{2}$ preserves the Bogomolov-Beauville-Fujiki form $q$ on
$H^{2}(M,{\mathbb{Q}})$.
(ii)
$\Gamma_{2}$ is an arithmetic subgroup of $O(H^{2}(M,{\mathbb{Q}}),q)$.
(iii)
The natural projection $\Gamma_{A}{\>\longrightarrow\>}\Gamma_{2}$ has finite
kernel.
(iv)
The mapping class group $\operatorname{\sf Diff}/\operatorname{\sf Diff}_{0}$
acts on $H^{*}(M,{\mathbb{Z}})$ with finite kernel, and the image of
$\operatorname{\sf Diff}/\operatorname{\sf Diff}_{0}$ in $\Gamma_{2}$ has
finite index.
Proof: From the Fujiki formula $v^{2n}=q(v,v)^{n}$, it is clear that
$\Gamma_{A}$ preserves the Bogomolov-Beauville-Fujiki, up to a sign. The sign
is also fixed, because $\Gamma_{A}$ preserves $p_{1}(M)$, and (as Fujiki has
shown) $v^{2n-2}\wedge p_{1}(M)=q(v,v)^{n-1}c$, for some $c\in{\mathbb{R}}$.
The constant $c$ is positive, because the degree of $c_{2}(B)$ is positive for
any Yang-Mills bundle with $c_{1}(B)=0$ (this argument is based on [H5],
section 4; see also [Ni]).
In [V2] (see also [V3]) it was shown that the group
$Spin(H^{2}(M,{\mathbb{Q}}),q)$ acts on the cohomology algebra
$H^{*}(M,{\mathbb{Q}})$ by automorphisms, preserving the Pontryagin
classes.111In these two papers, the action of the corresponding Lie algebra
was obtained, giving a $\operatorname{Spin}(H^{2}(M,{\mathbb{Q}}),q)$-action
by the general Lie group theory. In [V4] the action of the centre of
$\operatorname{Spin}(H^{2}(M,{\mathbb{Q}}),q)$ was computed. It was shown that
it acts as $-1$ on odd cohomology and trivially on even cohomology. Therefore,
$\Gamma_{2}\subset O(H^{2}(M,{\mathbb{Q}}),q)$ is an arithmetic subgroup. This
gives 3, (ii).
To see that the map $\Gamma_{A}{\>\longrightarrow\>}\Gamma_{2}$ has finite
kernel, we notice that the subgroup
$K\subset\operatorname{Aut}(H^{*}(M,{\mathbb{Q}}))$ acts trivially on
$H^{2}(M)$, hence preserves all Lefschetz ${\mathfrak{s}l}(2)$-triples
$(L_{\omega},\Lambda_{\omega},H)$ associated with different $\omega\in
H^{1,1}(M)$. The commutators of $[L_{\omega},\Lambda_{\omega}]$ generate the
Lie algebra $\mathfrak{so}(H^{2}(M,{\mathbb{Q}}),q)$ acting by derivations on
$H^{*}(M,{\mathbb{Q}})$, as shown in [V2] (see also [V3]), hence $K$
centralizes $Spin(H^{2}(M,{\mathbb{Q}}),q)$. The complexification of this
group contains the complex structure operators associated with any complex,
hyperkähler structure on $M$ (see [V2], [V3]). Since $K$ centralizes
$Spin(H^{2}(M,{\mathbb{Q}}),q)$, $K$ preserves the Hodge decomposition, for
any complex structure $I$ on $M$ of hyperkähler type. Using the Hodge
decomposition and the Lefschetz ${\mathfrak{s}l}(2)$-action, one defines the
Riemann-Hodge pairing, writing down the Riemann-Hodge formulas as usual; it is
positive definite. Since $K$ commutes with the ${\mathfrak{s}l}(2)$-triples
and the Hodge decomposition, it preserves the Riemann-Hodge pairing $h$.
Therefore, $K$ is an intersection of a lattice and a compact group
$\operatorname{Spin}(H^{*}(M),h)$, hence finite. We proved 3, (iii). 3, (iv)
follows directly from (iii) and 3.
Remark 3.6: Let $V_{\mathbb{Q}}$ be a rational vector space equipped with a
quadratic form $q$, and $V_{\mathbb{R}}:=V_{\mathbb{Q}}\otimes{\mathbb{R}}$.
By [VGO], Example 7.5, the following conditions are equivalent:
(i)
For any arithmetic subgroup $\Gamma\subset SO(V_{\mathbb{R}},q)$, $\Gamma$ has
finite covolume (that is, the quotient $SO(V_{\mathbb{R}},q)/\Gamma$ has
finite Haar measure).
(ii)
The algebraic group $SO(V_{\mathbb{Q}},q)$ has no non-trivial homomorphisms to
the multiplicative group ${\mathbb{Q}}^{>0}$ of rational numbers (in this case
we say that $SO(V_{\mathbb{Q}},q)$ has no non-trivial rational characters).
For $V_{\mathbb{Q}}=H^{2}(M,{\mathbb{Q}})$ with the Beauville-Bogomolov-Fujiki
form, the latter condition always holds, hence the mapping class group is
mapped to a discrete subgroup of finite covolume $\Gamma_{2}\subset
SO(H^{2}(M,{\mathbb{R}}),q)$.
## 4 Weakly Hausdorff manifolds and Hausdorff reduction
### 4.1 Weakly Hausdorff manifolds
Definition 4.1: Let $M$ be a topological space, and $x\in M$ a point. Suppose
that for each $y\neq x$, there exist non-intersecting open neighbourhoods
$U\ni x,V\ni y$. Then $x$ is called a Hausdorff point of $M$.
Remark 4.2: The topology induced on the set of all Hausdorff points in $M$ is
clearly Hausdorff.
Definition 4.3: Let $M$ be an $n$-dimensional real analytic manifold, not
necessarily Hausdorff. Suppose that the set $Z\subset M$ of non-Hausdorff
points is contained in a countable union of real analytic subvarieties of
$\operatorname{codim}\geqslant 2$. Suppose, moreover, that the following
assumption (called “assumption S” in the sequel) is satisfied.
(S)
For every $x\in M$, there is a closed neighbourhood $B\subset M$ of $x$ and a
continuous surjective map $\Psi:\;B{\>\longrightarrow\>}{\mathbb{R}}^{n}$ to a
closed ball in ${\mathbb{R}}^{n}$, inducing a homeomorphism from an open
neighbourhood of $x$ in $B$ onto an open neighbourhood of $\Psi(x)$ in
${\mathbb{R}}^{n}$.
Then $M$ is called a weakly Hausdorff manifold.
Definition 4.4: Two points $x,y\in M$ are inseparable (denoted $x\sim y$) if
for any open subsets $U\ni x,V\ni y$, one has $U\cap V\neq\emptyset$.
Remark 4.5: A closure of an open set $U$ contains all points which are
inseparable from some $x\in U$. To extend a homeomorphism from
$\Psi_{0}:\;B_{0}{\>\longrightarrow\>}{\mathbb{R}}^{n}$ from an open
neighbourhood $B_{0}$ to its closure $B$ in order to fulfill the assertion of
S above, we need to extend $\Psi_{0}$ to all points which are inseparable from
some $x\in B$.
Remark 4.6: Throughout this paper, we could work in much weaker assumptions.
Instead of real analytic, we could demand that $M$ is a Lipschitz manifold,
and $Z$ has Hausdorff codimension $>1$. All the proofs in the sequel would
remain valid in this general situation. Also, the assumption S seems to be
unnecessary, though convenient. In fact, counterexamples to S are hard to
find, and it might possibly follow from the rest of assumptions.
Example 4.7: Let $\operatorname{\sf Teich}$ be a Teichmüller space of a
hyperkähler manifold $M$, and $Z\subset M$ the set of all
$I\in\operatorname{\sf Teich}$ such that the corresponding Neron-Severi
lattice $H^{1,1}(M,I)\cap H^{2}(M,{\mathbb{Z}})$ has rank $\geqslant 1$.
Clearly, $Z=\bigcup_{\eta}Z_{\eta}$, with the union taken over all elements
$\eta\in H^{2}(M,{\mathbb{Z}})$,111The group $H^{2}(M,{\mathbb{Z}})$ is
torsion-free, by the Universal Coefficients Theorem, because $M$ is simply
connected. and
$Z_{\eta}=\\{I\in\operatorname{\sf Teich}\ \ |\ \ \eta\in H^{1,1}(M,I)\\}.$
As follows from [H1] (see 4.4 below), the complement $\operatorname{\sf
Teich}\\!\backslash Z$ is Hausdorff. The period map $\operatorname{\sf
Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ is locally a
diffeomorphism, hence the assumption S is also satisfied. Therefore,
$\operatorname{\sf Teich}$ is weakly Hausdorff.
The following definition is straightforward; it is a non-Hausdorff version of
a notion of a manifold with smooth boundary. We have to give it in precise
detail, because the notion of a “boundary” is ambiguous in non-Hausdorff
situation.
Definition 4.8: We say that an open subset $U\subset M$ of a smooth manifold
$M$ has smooth boundary, if locally in a neighbourhood $V$ of each point in
$M$, there is a diffeomorphism mapping the $V$ to ${\mathbb{R}}^{n}$, with the
closure $\overline{U\cap V}_{V}$ of $U\cap V$ in $V$ mapping to
$[0,\infty]\times{\mathbb{R}}^{n-1}$, and the complement $\overline{U\cap
V}_{V}\backslash U\cap V$ mapping to the hyperplane
$\\{0\\}\times{\mathbb{R}}^{n-1}$. Denote by $\overline{U}$ the closure of $U$
in $M$, and by $\overline{U}^{\circ}$ the set of interior points of
$\overline{U}$. The closure $\overline{U}$ is called a smooth submanifold with
boundary. The boundary
$\partial_{M}U:=\overline{U}\backslash\overline{U}^{\circ}$ is by this
definition a smooth codimension 1 submanifold of $M$.
Further on, we shall need the following claim. It can be (roughly) stated as
follows. Take a subset $B$ in a weakly Hausdorff $n$-manifold, diffeomorphic
to a closed ball in $U\cong{\mathbb{R}}^{n}$ with smooth boundary
$\partial_{U}B$. Then its closure $\overline{B}$ in $M$ is obtained by adding
two kinds of extra points: those in the closure $\overline{\partial_{U}B}$ of
$\partial_{U}B$ in $M$ and those which are interior to $\overline{B}$.
Claim 4.9: Let $M$ be a weakly Hausdorff manifold, $U\subset M$ a subset
diffeomorphic to ${\mathbb{R}}^{n}$, and $B\subset U$ a connected, open subset
of $U$ which has compact closure in $U$ with smooth boundary
$\partial_{U}B\subset U$. Consider the set
$\overline{B}\backslash\overline{B}^{\circ}$ of all points in the closure
$\overline{B}$ of $B$ in $M$ which are not interior in $\overline{B}$. Then
$\overline{B}\backslash\overline{B}^{\circ}$ coincides with the closure
$\overline{\partial_{U}B}$ of $\partial_{U}B$ in $M$.
Proof: Clearly, $\partial_{U}B$ contains no interior points of $\overline{B}$.
Therefore,
$\overline{\partial_{U}B}\subset\overline{B}\backslash\overline{B}^{\circ}.$
We need only to prove the opposite inclusion.
Denote by $W$ the set of Hausdorff points of $M$. Since $M\backslash W$ has
codimension $\geqslant 2$, $W\cap\partial_{U}B$ is dense in $\partial_{U}B$.
The boundary $W\cap\partial_{U}B$ separates $W$ onto two disjoint open
subsets, $W_{1}:=W\cap B$ and $W_{2}:=W\backslash\overline{B}$. Since $W$ is
dense, $\overline{B}=\overline{W}_{1}$, and
$\overline{B}\backslash\overline{B}^{\circ}\subset\overline{W}_{2}$.
Therefore, 4.1 would follow if we prove an inclusion
$\overline{W}_{1}\cap\overline{W}_{2}\subset\overline{\partial_{U}B}.$ (4.1)
Let $z\in\overline{W}_{1}\cap\overline{W}_{2}$. Then in any neighbourhood of
$z$ there are points of $W_{1}$ and $W_{2}$. Since $W$ is a smooth manifold
with countably many codimension $\geqslant 2$ subvarieties removed, and
$W_{1}$, $W_{2}$ are disjoint open subsets of $W$ separated by a smooth
boundary $W\cap\partial_{U}B$, this implies that any neighbourhood of $z$
contains a point in $W\cap\partial_{U}B$. Indeed, by 4.1 below, for any path
connected open subset $B\subset M$, the intersection $W\cap B$ is also
connected. Unless $W\cap\partial U$ is non-empty, the open set $W\cap B$ is
represented as a union of two non-empty disjoint open subsets $W_{1}\cap B$
and $W_{2}\cap B$, which is impossible, because it is connected. This implies
(4.1), and finishes the proof of 4.1.
The following trivial lemma, used in the proof of 4.1, is well-known; we
include it here for completeness.
Lemma 4.10: Let $M$ be a path connected real analytic manifold, and
$W=M\backslash\bigcup Z_{i}$, where $\bigcup Z_{i}$ is a union of countably
many real analytic manifolds of codimension at least 2. Then $W$ is path
connected.
Proof: This result is clearly local. Therefore, we may assume that $M$ is
isomorphic to ${\mathbb{R}}^{n}$. Given two points $x,y\in W$, we shall prove
that there is $z\in W$ such that a straight segment of a line connecting $z$
to $x$ and the one connecting $z$ to $y$ belong to $W$. Let
$P^{x}\cong{{\mathbb{R}}P^{n}}$ be the set of all lines passing through $x$,
and $P_{W}^{x}$ the set of these lines which belong to $W$. Clearly, the set
$P_{Z_{i}}^{x}$ of lines $l\in P$ intersecting $Z_{i}$, being a projection of
$Z_{i}$ to $P$, has real codimension 1 in $P$. Therefore, the complement to a
set $P_{W}^{x}$ is of measure 0 in $P^{x}$. Similarly one defines $P_{W}^{y}$
and proves that it is dense. Let now $Q$ be the set of all pairs of lines
$l^{x}\in P^{x},l^{y}\in P^{y}$ which intersect. Clearly, $Q$ is equipped with
smooth projections $\pi_{x}$, $\pi_{y}$ to $P^{x}$ and $P^{y}$, with
1-dimensional fibers. Since the complements to $P_{W}^{x}$ and $P_{W}^{y}$ in
$P^{x}$ and $P^{y}$ have measure 0, the intersection
$\pi_{x}^{-1}(P_{W}^{x})\cap\pi_{y}^{-1}(P_{W}^{y})$ is non-empty. For each
pair of lines
$(l^{x},l^{y})\in\pi_{x}^{-1}(P_{W}^{x})\cap\pi_{y}^{-1}(P_{W}^{y})\subset Q,$
$l^{x}$ and $l^{y}$ are lines which belong to $W$, intersect and connect $x$
to $y$.
### 4.2 Inseparable points in weakly Hausdorff manifolds
Lemma 4.11: Let $M$ be a weakly Hausdorff manifold, $x,y\in M$ inseparable
points, and $U\ni x,V\ni y$ open sets. Then $x$ and $y$ are interior points of
$\overline{U}\cap\overline{V}$, where $\overline{U},\overline{V}$ denotes the
closure of $U$, $V$.
Remark 4.12: This statement is false without the weak Hausdorff assumption.
Indeed, take as $M$ the union of two real lines, with $t<0$ identified, $x$
the 0 of the first line, $y$ the 0 of the second line. Choose a neighbourhood
$U$ of $x$ and $V$ of $y$. The points $x$ and $y$ are clearly inseparable, but
the intersection of $\overline{U}\cap\overline{V}$ is a closure of an interval
$[-a,0[$, with $a>0$ a positive number, hence $x$ and $y$ and not interior
points of $\overline{U}\cap\overline{V}$.
Proof of 4.2: Consider an open ball $B\subset U$ with smooth boundary
$\partial_{U}B$ containing $x$. Since $x$ and $y$ are inseparable, $y$ belongs
to a closure $\overline{B}$ of $B$. Then either $y$ is interior in
$\overline{B}$, or $y$ lies in the closure of its boundary $\partial_{U}B$, as
follows from 4.1. To prove 4.2 it remains to show that the second option is
impossible. Using the assumption “S” of the definition of weakly Hausdorff
manifolds, we obtain that $\Psi(y)=\Psi(x)$, where
$\Psi:\;B{\>\longrightarrow\>}{\mathbb{R}}^{n}$ is the map defined in “S”.
Choosing $B$ sufficiently small, we can always assume that
$\Psi{\left|{}_{{\phantom{|}\\!\\!}_{B}}\right.}$ is a homeomorphism. Then
$\Psi(x)=\Psi(y)$ is in the interior of $\Psi(B)$, hence
$\Psi(y)\notin\Psi(\partial_{U}B)$. Since $\Psi$ is continuous,
$\Psi^{-1}(\Psi(\partial_{U}B))$ contains the closure of $\partial_{U}B$.
Therefore, $y\notin\overline{\partial_{U}B}$ by 4.1. We proved 4.2.
We shall also need the following trivial lemma.
Lemma 4.13: In assumptions of 4.1, let $W\subset M$ the set of Hausdorff
points of $M$. Then the intersection $W\cap\overline{B}^{\circ}$ lies in $B$.
Proof: Denote by $B_{cl}$ the union of $B$ and $\partial_{U}B$, homeomorphic
to a closed ball. Let $x\in W\cap\overline{B}^{\circ}$. Then $x$ is a limit of
a sequence $\\{x_{i}\\}\in B$. Since $B_{cl}$ is compact, $\\{x_{i}\\}$ has a
limit point $x^{\prime}$ in $B_{cl}$. Since $x$ is Hausdorff, and $x\sim
x^{\prime}$, one has $x=x^{\prime}$. Therefore, $x\in B_{cl}$. By 4.1, one and
only one of two things happens: either $x$ is interior in $\overline{B}$, or
it belongs to the closure $\overline{\partial_{U}B}$ of the smooth sphere
$\partial_{U}B=B_{1}\backslash B$. The later case is impossible, because $x$
is interior in $\overline{B}$. Therefore, $x$ is interior in $B_{cl}$.
Proposition 4.14: Let $M$ be a weakly Hausdorff manifold, and $\sim$ be
inseparability relation defined above. Then $\sim$ is an equivalence relation.
Remark 4.15: Without the weak Hausdorff assumption, $\sim$ is not an
equivalence relation. Indeed, consider for example a union
${\mathbb{R}}\coprod{\mathbb{R}}\coprod{\mathbb{R}}$ of three real lines and
glue $t<0$ for the first two lines, and $t>0$ for the second two. Then $0_{1}$
(the zero on the first line) is inseparable from $0_{2}$, and $0_{2}$ from
$0_{3}$, but $0_{1}\not\sim 0_{3}$.
Proof of 4.2: Only transitivity needs to be proven. Let $x_{1}\sim x_{2}$,
$x_{2}\sim x_{3}$ be points in $M$, $U_{1}\ni x_{1}$, $U_{3}\ni x_{3}$ their
neighbourhoods. By 4.2, $x_{2}$ is an interior point of $\overline{U}_{1}$ and
$\overline{U}_{3}$. Therefore, $\overline{U}_{1}\cap\overline{U}_{3}$ is non-
empty, and contains an open subset $A$. The intersection $A\cap W$ of $A$ with
the set of Hausdorff points in non-empty, because $W$ is dense. Let
$\overline{U}_{i}^{\circ}$ be the set of interior points of
$\overline{U}_{i}$. The intersection $\overline{U}_{i}^{\circ}\cap W$ lies in
$U_{i}$, as follows from 4.2, hence $A\cap W$ lies in $U_{1}$ and $U_{3}$, and
these two open sets have non-trivial intersection.
Further on, we shall be interested in the quotient $M/\\!{}_{\sim}$, equipped
with a quotient topology. By definition, a subset $U\subset M/\\!{}_{\sim}$ is
open if its preimage in $M$ is open, and closed if its preimage in $M$ is
closed.
Claim 4.16: Let $M$ be a weakly Hausdorff manifold, and $B\subset M$ an open
subset with smooth boundary. Consider its closure $\overline{B}$, and let
$\overline{B}^{\circ}$ be the set of its interior points. Then
$\overline{B}^{\circ}$ is a set of all points $y\in M$ which are inseparable
from some $x\in B$.
Proof: Let $x\in B$ be any point, and $y\in M$ a point inseparable from $x$.
By 4.2, for any neighbourhood $U\ni y$, $y$ is an interior point of
$\overline{U}\cap\overline{B}$. Therefore, $y$ is an interior point of
$\overline{B}$.
To finish the proof of 4.2, it remains to show that any interior point
$z\in\overline{B}$ is inseparable from some $z^{\prime}\in B$.
Choose a diffeomorphism
$B\stackrel{{\scriptstyle\Psi}}{{{\>\longrightarrow\>}}}B^{n}$ to an open ball
$B^{n}\subset{\mathbb{R}}^{n}$. Using the property S of 4.1, we may assume
that $\Psi$ can be extended to a continuous map from the closure
$\overline{B}$ to the closed ball $\overline{B}^{n}$.
Any point $z\in\overline{B}$ can be obtained as a limit of a sequence of
points $\\{z_{i}\\}\subset B$. Let $\zeta\in\overline{B}^{n}$ be a limit of
$\\{\Psi(z_{i})\\}$ in $\overline{B}^{n}$, which exists because
$\overline{B}^{n}$ is compact. Choosing a subsequence, we may also assume that
$\lim\\{\Psi(z_{i})\\}$ is unique. Then $\zeta=\Psi(z)$, and it is an interior
point of $\overline{B}^{n}$, as follows from 4.1. Since
$B\stackrel{{\scriptstyle\Psi}}{{{\>\longrightarrow\>}}}B^{n}$ is a
diffeomorphism, the sequence $\\{z_{i}\\}$ has a limit $z^{\prime}\in B$.
Since $\Psi(z)=\Psi(z^{\prime})=\lim\\{\Psi(z_{i})\\}$, the point $z$ is
inseparable from $z^{\prime}$.
Theorem 4.17: Let $M$ be a weakly Hausdorff manifold, and $\sim$ the
inseparability relation. Consider the quotient space $M/\\!{}_{\sim}$ equipped
with a natural quotient topology. Then $M/\\!{}_{\sim}$ is Hausdorff, and the
projection map
$M\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}M/\\!{}_{\sim}$ is
open.
Proof: Since $M$ is a manifold, we can choose a base of open subsets $U\subset
M$ with smooth boundary. By 4.2,
$\varphi^{-1}(\varphi(U))=\overline{U}^{\circ}$, where $\overline{U}^{\circ}$
is the set of all interior points of the closure $\overline{U}$. Therefore,
the image of $U$ is open in $M/\\!{}_{\sim}$, and $\varphi$ is an open map.
Denote by $\Gamma_{\sim}\subset M\times M$ the graph of $\sim$. It is well
known that a topological space $X$ is Hausdorff if and only if the diagonal
$\Delta$ is closed in $X\times X$. Since the projection $M\times
M\stackrel{{\scriptstyle\varphi\times\varphi}}{{{\>\longrightarrow\>}}}M/\\!{}_{\sim}\times
M/\\!{}_{\sim}$ is open, and
$\varphi(M\times M\backslash\Gamma_{\sim})=\left(M/\\!{}_{\sim}\times
M/\\!{}_{\sim}\right)\backslash\Delta,$
to prove that $M/\\!{}_{\sim}$ is Hausdorff it remains to show that
$\Gamma_{\sim}$ is closed in $M\times M$.
Let $(x,y)\notin\Gamma_{\sim}$, equivalently, $x\not\sim y$. Choose open
neighbourhoods $U\ni x,V\ni y$, $U\cap V=\emptyset$. Then $U\times
V\cap\Gamma_{\sim}=\emptyset$. This implies that $\Gamma_{\sim}$ is closed. We
proved that $M/\\!{}_{\sim}$ is Hausdorff.
### 4.3 Hausdorff reduction for weakly Hausdorff manifolds
Definition 4.18: Let
$X\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}Y$ be a surjective
morphism of topological spaces, with $Y$ Hausdorff. Suppose that for any map
$X\stackrel{{\scriptstyle\varphi^{\prime}}}{{{\>\longrightarrow\>}}}Y^{\prime}$,
with $Y^{\prime}$ Hausdorff, the map $\varphi^{\prime}$ is factorized through
$\varphi$. Then $\varphi$ is called the Hausdorff reduction map, and $Y$ the
Hausdorff reduction of $X$. Being an initial object in the category of
diagrams
$X\stackrel{{\scriptstyle\varphi^{\prime}}}{{{\>\longrightarrow\>}}}Y^{\prime}$
(with $Y^{\prime}$ Hausdorff), the Hausdorff reduction if obviously unique, if
it exists.
Remark 4.19: If $x\sim y$ are inseparable points of $M$, any morphism
$M\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}M^{\prime}$ to a
Hausdorff space $M^{\prime}$ satisfies $\varphi(x)=\varphi(y)$. Therefore,
whenever the quotient $M/\\!{}_{\sim}$ is Hausdorff, it is a Hausdorff
reduction of $M$.
Example 4.20: By 4.2, for any weakly Hausdorff manifold $M$, the quotient
$M/\\!{}_{\sim}$ is its Hausdorff reduction.
Definition 4.21: A local homeomorphism is a continuous map
$X\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}Y$ such that for all
$x\in X$ there is a neighbourhood $U\ni x$ such that
$\psi{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is a homeomorphism onto its
image, which is open in Y. If $\psi$ is also a smooth, it is called a local
diffeomorphism, or etale map.
Theorem 4.22: Let $M$ be a weakly Hausdorff manifold, and
$\varphi:\;M{\>\longrightarrow\>}M/\\!{}_{\sim}$
its Hausdorff reduction. Then $\varphi$ is etale, and $M/\\!{}_{\sim}$ is a
Hausdorff manifold.
Proof: Let $U\subset M$ be an open neighbourhood of a given point $x$,
diffeomorphic to ${\mathbb{R}}^{n}$, and $B\subset U$ a closed neighbourhood
diffeomorphic to a closed ball. Since $U$ is Hausdorff, the restriction
$\varphi{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is injective. An
injective map from a compact $B$ to a Hausdorff space is a homeomorphism to
its image. Then the restriction of $\varphi$ to interior of $B$ is a
homeomorphism.
### 4.4 The birational Teichmüller space for a hyperkähler manifold
The following result is due to D. Huybrechts.
Theorem 4.23: ([H3]) Let $M$ be a hyperkähler manifold, $\operatorname{\sf
Teich}$ its Teichmüller space, and $x,y\in\operatorname{\sf Teich}$ points
corresponding to hyperkähler manifolds $M_{x}$ and $M_{y}$. Suppose that $x$
and $y$ are inseparable, in the sense of 1.3. Then the manifolds $M_{x}$ and
$M_{y}$ are bimeromorphically equivalent. Conversely, if $M_{1}$ and $M_{2}$
are bimeromorphically equivalent, they can be realised as inseparable points
on the Teichmüller space.
Remark 4.24: Let $M_{1},M_{2}$ be bimeromorphically equivalent hyperkähler
manifolds. By [H1, Proposition 9.2], the Neron-Severi lattice
$\operatorname{\sf NS}(M_{i})=H^{1,1}(M,{\mathbb{Z}})$ has rank $\geqslant 1$,
unless the bimeromorphism $M_{1}\rightsquigarrow M_{2}$ is biregular.
Therefore, a point $I\in\operatorname{\sf Teich}$ with
$\operatorname{rk}\operatorname{\sf NS}(M,I)=0$ must be separable. This
argument was used earlier in this section to prove that $\operatorname{\sf
Teich}$ is weakly Hausdorff.
Remark 4.25: The Hausdorff reduction $\operatorname{\sf Teich}/\\!{}_{\sim}$
classifies complex structures on $M$ up to “bimeromorphic equivalence” and the
action of the isotopy group. We call $\operatorname{\sf Teich}/\\!{}_{\sim}$
the birational Teichmüller space, denoting it as $\operatorname{\sf
Teich}_{b}$. However, the term “bimeromorphic equivalence” is vague. Clearly,
there are distinct points in $\operatorname{\sf Teich}/\\!{}_{\sim}$ which
represent bimeromorphic (and biholomorphic) hyperkähler manifolds. A better
description of this equivalence might be gleaned from [H3] and [Bou] (I am
grateful to Eyal Markman for this observation). Consider the Hodge isometry
$f:\;H^{2}(M_{1},{\mathbb{Z}}){\>\longrightarrow\>}H^{2}(M_{2},{\mathbb{Z}})$
between the second cohomology corresponding to two inseparable points in
$\operatorname{\sf Teich}$. In the language of Boucksom, $f$ maps the Kähler
cone to one of the “rational chambers” of the positive cone. As shown in [Bou,
Theorem 4.3], there are three possibilities:
(i)
$f$ could map the Kähler cone to the Kähler cone, which means that $f$ is
induced by an isomorphism. In this case $M_{1}$ and $M_{2}$ correspond to the
same points of the marked moduli space.
(ii)
$f$ could map the Kähler cone onto a different rational chamber, which belongs
to the fundamental uniruled chamber. In this case $f$ is induced by a graph of
a bimermorphic morphism.
(iii)
$f$ could map the Kähler cone onto a different rational chamber, which does
not belong to the fundamental uniruled chamber. In this case $f$ is induced by
a reducible correspondence. One of its irreducible components is a graph of a
birational morphism. Other components, which necessarily exist, will appear as
certain fiber products of uniruled divisors.
Clearly, the map $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ is well
defined (it follows directly from the definition of the Hausdorff reduction).
The main result of this paper is the following theorem
Theorem 4.26: (global Torelli theorem) Let $M$ be a simple hyperkähler
manifold, $\operatorname{\sf Teich}_{b}$ its birational Teichmüller space, and
$\operatorname{\sf Per}:\;\operatorname{\sf
Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ (4.2)
the period map defined as above. Then (4.2) is a diffeomorphism, for each
connected component of $\operatorname{\sf Teich}_{b}$.
4.4 follows from 4.4, because $\operatorname{{\mathbb{P}}\sf er}$ is simply
connected (2.4).
Proposition 4.27: Consider the map $\operatorname{\sf
Per}:\;\operatorname{\sf
Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ defined as
in 4.4. Then $\operatorname{\sf Per}$ is a covering.
4.4 is implied by 6.1 and 6.3 below. Indeed, by 6.1, $\operatorname{\sf Per}$
is compatible with the generic hyperkähler lines (5.1), and by 6.3, any such
map is necessarily a covering.
## 5 Subtwistor metric on the period space
### 5.1 Hyperkähler lines and hyperkähler structures
Definition 5.1: Let $M$ be a simple hyperkähler manifold,
$\operatorname{{\mathbb{P}}\sf er}$ its period space, and $W\subset
H^{2}(M,{\mathbb{R}})$ an oriented 3-dimensional subspace, such that
$q{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}$ is positive definite. Consider
a 2-dimensional sphere $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$
consisting of all oriented 2-dimensional planes $V\subset W$. Using an
isomorphism
$\operatorname{{\mathbb{P}}\sf er}\cong Gr_{+,+}(H^{2}(M,{\mathbb{R}}))$
constructed in Subsection 2.4, we can consider $S_{W}$ as a subvariety in
$\operatorname{{\mathbb{P}}\sf er}$. This subvariety is called a hyperkähler
line associated with a 3-dimensional plane $W\subset H^{2}(M,{\mathbb{R}})$.
Remark 5.2: Let $(M,g,I,J,K)$ be a hyperkähler structure,
$S\subset\operatorname{\sf Teich}$ the sphere of induced complex structures
defined as in Subsection 2.1, and
$W:=\langle\omega_{I},\omega_{J},\omega_{K}\rangle\subset
H^{2}(M,{\mathbb{R}})$ the corresponding 3-dimensional plane. It is easy to
see that the sphere $\operatorname{\sf
Per}(S)\subset\operatorname{{\mathbb{P}}\sf er}$ coincides with the
hyperkähler line $S_{W}$ defined as above. This explains the term.
Definition 5.3: Let $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ be a
hyperkähler line associated with a 3-dimensional subspace $W\subset
H^{2}(M,{\mathbb{R}})$. We say that $S_{W}$ is a generic hyperkähler line if
the orthogonal complement to $W$ has no rational points:
$W^{\bot}\cap H^{2}(M,{\mathbb{Q}})=0.$
Often, we shall abbreviate “generic hyperkähler line” to “GHK line”
### 5.2 Generic hyperkähler lines and the Teichmüller space
Let $(M,I)$ be a hyperkähler manifold. The Hodge structure on $H^{2}(M,I)$ is
determined from the Bogomolov-Beauville-Fujiki form $q$ and the corresponding
1-dimensional space $l=\operatorname{\sf Per}(I)\subset
H^{2}(M,{\mathbb{C}})$: one has $H^{2,0}(M,I)=l$, $H^{0,2}(M,I)=\overline{l}$,
and $H^{1,1}(M,I)=\langle l,\overline{l}\rangle^{\bot}$, where $\bot$ denotes
the orthogonal complement. We define the Neron-Severi lattice of $(M,I)$ as
$\operatorname{\sf NS}(M,I):=H^{1,1}(M,I)\cap H^{2}(M,{\mathbb{Z}})$. Since
$H^{1,1}(M,I)=\langle l,\overline{l}\rangle^{\bot}$, the lattice
$\operatorname{\sf NS}(M,I)$ depends only on the point $\operatorname{\sf
Per}(I)\in\operatorname{{\mathbb{P}}\sf er}$. We shall often consider the
Neron-Severi lattice of a point $l\in\operatorname{{\mathbb{P}}\sf er}$,
defined as above. Since a simple hyperkähler manifold is simply connected,
$\operatorname{\sf NS}(M,I)=\operatorname{Pic}(M,I)$. This allows us to define
the Picard group $\operatorname{Pic}(l)$ for
$l\in\operatorname{{\mathbb{P}}\sf er}$:
$\operatorname{Pic}(l)=NS(l)=\langle l,\overline{l}\rangle^{\bot}\cap
H^{2}(M,{\mathbb{Z}}).$
Claim 5.4: Let $S\subset\operatorname{{\mathbb{P}}\sf er}$ be a hyperkähler
line, associated with a 3-dimensional subspace $W\subset
H^{2}(M,{\mathbb{R}})$. Then the following assumptions are equivalent.
(i)
$S$ is a GHK line
(ii)
For some $l\in S$, the corresponding Neron-Severi lattice $\operatorname{\sf
NS}(M,l)$ is trivial.
(iii)
For some $w\in W$, its orthogonal complement $w^{\bot}\subset
H^{2}(M,{\mathbb{R}})$ has no non-zero rational points.
Proof: The points of $S$ are parametrized by oriented 2-dimensional planes
$V\subset W$, and the corresponding Neron-Severi lattice $\operatorname{\sf
NS}(M,V)$ is $V^{\bot}\cap H^{2}(M,{\mathbb{Z}})$. Now, the chain of
inclusions
$W^{\bot}\cap H^{2}(M,{\mathbb{Q}})\subset V^{\bot}\cap
H^{2}(M,{\mathbb{Q}})\subset w^{\bot}\cap H^{2}(M,{\mathbb{Q}})$
immediately brings the implications (iii) $\Rightarrow$ (ii) $\Rightarrow$
(i). To finish the proof, it remains to deduce (iii) from (i). Let
$R:=\bigcup\limits_{\stackrel{{\scriptstyle\eta\in
H^{2}(M,{\mathbb{Q}})}}{{\eta\neq 0}}}\eta^{\bot}$
be the union of all hyperplanes orthogonal to non-zero rational vectors. Since
$W^{\bot}\cap H^{2}(M,{\mathbb{Q}})=0$, $W$ does not lie in $R$. Therefore,
$W\cap R$ is a countable union of planes of positive codimension. Take $w\in
W\backslash R$. Clearly, $w^{\bot}\cap H^{2}(M,{\mathbb{Q}})=0$.
Remark 5.5: The same proof also implies that for any generic hyperkähler
line, the set of all $I\in S$ with $\operatorname{\sf NS}(M,I)\neq 0$ is
countable. Indeed, it is a countable union of closed complex subvarieties of
positive codimension in ${\mathbb{C}}P^{1}$.
### 5.3 GHK lines and subtwistor metrics
Let $\operatorname{{\mathbb{P}}\sf er}$ be a period space of a hyperkähler
manifold $M$, identified with a Grassmanian $SO(b_{2}-3,3)/SO(2)\times
SO(b_{2}-1,3)$ of oriented, positive 2-planes in $H^{2}(M,{\mathbb{R}})$. We
shall consider $\operatorname{{\mathbb{P}}\sf er}$ as a complex manifold, with
the complex structure obtained as in (1.3). Fix an auxiliary Euclidean metric
$g$ on $H^{2}(M)$. Given a positive 3-dimensional plane $W\subset H^{2}(M)$,
denote by $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ the set of all
2-dimensional oriented planes contained in $W$. Clearly, $S_{W}$ is a complex
curve in $\operatorname{{\mathbb{P}}\sf er}$. The metric $g$ induces a Fubini-
Study metric on $S_{W}$.
Consider a sequence $S_{1},...,S_{n}$ of intersecting hyperkähler lines
connecting $x\in\operatorname{{\mathbb{P}}\sf er}$ to
$y\in\operatorname{{\mathbb{P}}\sf er}$, with $s_{i}\in S_{i}\cap S_{i+1}$,
$i=1,...,n-1$ the intersection points, and $s_{0}:=x$, $s_{n+1}:=y$. Denote by
$l_{S_{1},...,S_{n}}(x,y)$ the sum $\sum_{i=0}^{n}d_{g}(s_{i},s_{i+1})$, where
the distance $d_{g}(s_{i},s_{i+1})$ is computed on the hyperkähler line
$S_{i+1}$ using the metric induced by $g$ as above. Let
$d_{tw}(x,y):=\inf_{S_{1},...,S_{n}}l_{S_{1},...,S_{n}}(x,y)$
where the infimum is taken over all appropriate sequences of GHK lines,
connecting $x$ to $y$.
The following theorem is stated for periods of hyperkähler manifolds, but in
fact it could be stated abstractly for $\operatorname{{\mathbb{P}}\sf
er}=SO(m-3,3)/SO(2)\times SO(m-1,3)$, for any $m>0$. No results of geometry or
topology of hyperkähler manifolds are used in its proof.
Theorem 5.6: Let $\operatorname{{\mathbb{P}}\sf er}$ be a period space of a
hyperkähler manifold, and $d_{tw}:\;\operatorname{{\mathbb{P}}\sf
er}\times\operatorname{{\mathbb{P}}\sf
er}{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}\cup\infty$ the function
defined above. Then
(i)
$d_{tw}$ is a metric on $\operatorname{{\mathbb{P}}\sf er}$
(ii)
The metric $d_{tw}$ induces the usual topology on
$\operatorname{{\mathbb{P}}\sf er}$.
The rest of this section is taken by the proof of 5.3.
Definition 5.7: The metric $d_{tw}$ is called the subtwistor metric on the
period space, and a piecewise geodesic connecting $x$ to $y$ and going over
$S_{i}$ is called a subtwistor path.
Remark 5.8: It is in many ways similar to the sub-Riemannian metrics known in
metric geometry (see e.g. [BBI]).
The triangle inequality for $d_{tw}$ is clear from its definition. To prove
that $d_{tw}$ is a metric, we need only to show that $d_{tw}<\infty$ and
$d_{tw}(x,y)>0$ for $x\neq y$. The latter condition is clear, because
$d_{tw}\geqslant d_{g}$, where $d_{g}$ is a geodesic distance function on
$\operatorname{{\mathbb{P}}\sf er}$ associated with the Riemannian metric $g$.
Remark 5.9: To prove 5.3 (i) it remains to show that $d_{tw}<\infty$, that
is, to prove that any two points in $\operatorname{{\mathbb{P}}\sf er}$ are
connected by a sequence of GHK lines.
### 5.4 Connected sequences of GHK lines
Further on in this subsection, we shall use the following trivial linear-
algebraic lemma.
Lemma 5.10: Let $A$ be a real vector space, equipped with a non-degenerate
scalar product $q$, $W\subset A$ a $d$-dimensional subspace on which $q$ is
positive definite,111Further on, such spaces will be called positive. and
$W^{\prime}\subset A$ a positive subspace of dimension $d^{\prime}<d$. Then
there exists a non-zero vector $b\in W$, such that the subspace generated by
$b$ and $W^{\prime}$ is also positive.
Proof: Assume that $W\cap W^{\prime}=0$ (otherwise, we could choose $b\in
W\cap W^{\prime}$). Then $\dim{W^{\prime}}^{\bot}\cap W>0$. Choose
$b\in{W^{\prime}}^{\bot}\cap W$.
Remark 5.11: Of course, the set of such $b$ is open in $W$.
As follows from 5.3, 5.4 below implies 5.3 (i).
Proposition 5.12: Let $x,y\in\operatorname{{\mathbb{P}}\sf er}$. Then $x$ can
be connected to $y$ by a sequence of 4 sequentially intersecting GHK lines.
Proof. Step 0: Using the identification between $\operatorname{{\mathbb{P}}\sf
er}$ and the Grassmann space $Gr_{+,+}(H^{2}(M,{\mathbb{R}}))$ (Subsection
2.4), we shall consider points of $\operatorname{{\mathbb{P}}\sf er}$ as
2-dimensional subspaces $V\subset H^{2}(M,{\mathbb{R}})$ with
$q{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ positive definite. The
hyperkähler lines are understood as 3-dimensional spaces $W\subset
H^{2}(M,{\mathbb{R}})$ with $q{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}$
positive definite. Under this identification, the incidence relation is
translated into $V\subset W$.
Step 1: Let $x,y\in\operatorname{{\mathbb{P}}\sf er}$ be distinct points, and
$V_{x},V_{y}\subset H^{2}(M,{\mathbb{R}})$ the associated 2-planes. Then
$V_{x}$ and $V_{y}$ belong to the same hyperkähler line $S$ if and only if
$V_{x}\cap V_{y}$ is non-zero, and the space $\langle V_{x},V_{y}\rangle$
generated by $V_{x},V_{y}$ is positive. This is an immediate consequence of
Step 0.
Step 2: Let $x\in\operatorname{{\mathbb{P}}\sf er}$ be a point, and
$V_{x}\subset H^{2}(M,{\mathbb{R}})$ the corresponding 2-plane. A vector
$\omega\in V_{x}^{\bot}$ in the positive cone of $V_{x}$ defines a
3-dimensional plane $\langle\omega,V_{x}\rangle$. This gives a hyperkähler
line $C_{\omega}\subset\operatorname{\sf Teich}$ passing through $x$, whenever
$q(\omega,\omega)>0$. Clearly, for generic $\omega\in V_{x}^{\bot}$, all
rational points of $\omega^{\bot}$ lie in $(H^{2,0}\oplus H^{0,2})\cap
H^{2}(M,{\mathbb{Q}})$. Therefore, the orthogonal complement to $H^{2,0}\oplus
H^{0,2}\oplus{\mathbb{R}}\omega$ has no rational points (see also 5.2).
Step 3: Let $W_{1}$ and $W_{2}$ be 3-dimensional positive subspaces in the
space $H^{2}(M,{\mathbb{R}})$, containing $a\in H^{2}(M,{\mathbb{R}})$. Assume
that $a^{\bot}\cap H^{2}(M,{\mathbb{Q}})=0$. By 5.2 this implies, in
particular, that the subspaces $W_{i}$ correspond to GHK lines $S_{W_{1}}$,
$S_{W_{2}}$. Then there exists a GHK line intersecting $S_{W_{1}}$ and
$S_{W_{2}}$. Indeed, from 5.4 it follows that there exists a positive
2-dimensional plane $V:=\langle a,z\rangle\subset H^{2}(M,{\mathbb{R}})$
generated by $a,z$, with $z\in W_{1}$, Applying 5.4, again, we find a positive
3-dimensional plane $W:=\langle a,z,z^{\prime}\rangle\subset
H^{2}(M,{\mathbb{R}})$, with $z^{\prime}\in W_{2}$. By 5.2, $W$ corresponds to
a GHK line $S_{W}$. Now, Step 1 immediately implies that $S_{W}$ intersects
$S_{W_{1}}$ and $S_{W_{2}}$.
Step 4: Let $x,t\in\operatorname{{\mathbb{P}}\sf er}$. Using Step 2, we find
GHK lines passing through $x$ and $t$. Denote by $W_{x}$, $W_{t}\subset
H^{2}(M,{\mathbb{R}})$ the corresponding 3-planes, and let $a\in W_{x}$ be a
vector which satisfies $a^{\bot}\cap H^{2}(M,{\mathbb{Q}})=0$ (such $a$ exists
by 5.2.) Using 5.4, we choose a non-zero $b\in W_{t}$, in such a way that
$q|_{\langle a,b\rangle}$ is positive definite. Now, let $W$ be a positive
3-plane in $H^{2}(M,{\mathbb{R}})$ containing $a$ and $b$. By Step 3, there
exist a GHK line intersecting $S_{W_{x}}$ and $S_{W}$, and another GHK line
intersecting $S_{W_{t}}$ and $S_{W}$. We proved 5.4.
### 5.5 Lipschitz homogeneous metric spaces
Let $(\operatorname{{\mathbb{P}}\sf er},d_{g})$ be the space
$\operatorname{{\mathbb{P}}\sf er}$ equipped with a Riemannian metric
associated with a scalar product $g$ on $H^{2}(M,{\mathbb{R}})$, and $d_{tw}$
its subtwistor metric. To finish the proof of 5.3, we have to show that the
identity map $(\operatorname{{\mathbb{P}}\sf
er},d_{tw})\stackrel{{\scriptstyle\tau}}{{{\>\longrightarrow\>}}}(\operatorname{{\mathbb{P}}\sf
er},d_{g})$ is a homeomorphism.
Definition 5.13: Let $\gamma:\;[a,b]{\>\longrightarrow\>}M$ be a continuous
path in a metric space $(M,d)$. We define an arc length of $\gamma$ as a
supremum
$|\gamma|:=\sup_{a=x_{0}<x_{1}<...<x_{n+1}=b}\sum_{i=0}^{n}d(x_{i},x_{i+1}).$
We define the induced intrinsic metric on a metric space $(M,d)$ as
$d_{i}(x,y):=\inf|\gamma|$, where infimum is taken over all continuous paths
from $x$ to $y$. A metric $d$ on $M$ is called intrinsic, or a path metric if
it is equal to the corresponding induced intrinsic metric.
Example 5.14: From its definition it is clear that
$(\operatorname{{\mathbb{P}}\sf er},d_{tw})$ is a path metric space. In
particular, every two points of $\operatorname{{\mathbb{P}}\sf er}$ are
connected by a continuous path of finite arc length. Also, any Riemannian
manifold with a usual (geodesic) metric is a path metric space.
For an introduction to path metric spaces, see [G] and [BBI].
Recall that a map $\varphi$ of metric spaces is called a Lipschitz map, if
$d(\varphi(x),\varphi(y))\leqslant Cd(x,y),$
for some constant $C\geqslant 1$.
Definition 5.15: Let $M$ be a metric space, equipped with a transitive action
of a group $G$. We say that $M$ is Lipschitz homogeneous if for each
$\gamma\in G$, $\gamma$ acts on $M$ by Lipschitz maps.
The following lemma is the main application of the Lipschitz homogeneity.
Lemma 5.16: Let $M$ be a metric space, equipped with a transitive, free
action of a group $G$, and a Lipschitz homogeneous metric $d$. For each $x\in
M$, the map $G{\>\longrightarrow\>}M$, mapping $g\in G$ to $gx$, identifies
$M$ with $G$. Therefore, $d$ defines a topology on $G$, also denoted by $d$
(this topology is idependent from the choice of $x$, as follows from the
Lipschitz homogeneity). Then, $(G,d)$ is a topological group.
Proof: Given a sequence of points $\\{z_{i}=(x_{i},y_{i})\\}\in G\times G$
converging to $z=(x,y)$, its image in $G$ under the product map $\mu:\;G\times
G{\>\longrightarrow\>}G$ converges to $\mu(z)$. Indeed, it would suffice to
show that $\\{(x_{i},y)\\}$ converges to $(x,y)$, but the multiplication by
$y$ is Lipschitz, hence continuous in $d$.
An obvious example of a Lipschitz homogeneous manifold is given by the
positive Grassmannian $\operatorname{{\mathbb{P}}\sf
er}=Gr_{+,+}(H^{2}(M,{\mathbb{R}}))$ equipped with a Riemannian metric induced
from an auxuliary scalar product $g$ on $H^{2}(M)$. The group
$G:=SO(H^{2}(M,{\mathbb{R}}),q)$ acts on $\operatorname{{\mathbb{P}}\sf er}$
transitively, and each $\gamma\in G$ distorts the metric in a way which is
bounded by the biggest eigenvalue of $\gamma(g)g^{-1}$. The same argument
shows that $(\operatorname{{\mathbb{P}}\sf er},d_{tw})$ is also Lipschitz
homogeneous, where $d_{tw}$ is a subtwistor metric. Therefore, 5.3 (ii) is
implied by the following theorem.
Theorem 5.17: Let $M$ be a smooth manifold, equipped with a smooth,
transitive action of a Lie group $G$ and an intrinsic metric $d$, satisfying
an inequality $d\geqslant d_{g}$ for some Riemannian metric $d_{g}$. Suppose
that $(M,d)$ is Lipschitz homogeneous. Then the identity map
$(M,d){\>\longrightarrow\>}(M,d_{g})$ is a homeomorphism.
We prove 5.5 in Subsection 5.6.
Remark 5.18: To prove 5.5, it would suffice to show that $(M,d)$ is
homeomorphic to a manifold. Indeed, a continuous bijective map of manifolds is
always a homeomorphism.
### 5.6 Gleason-Palais theorem and its applications
We start the proof of 5.5 from its special case when the action of $G$ on $M$
is free. In this case, it follows directly from a theorem of Gleason and
Palais about transformation groups ([GP]; for a more recent treatment and
reference, see [BZ]).
Definition 5.19: Let $M$ be a topological space. We say that $M$ has Lebesgue
covering dimension $\leqslant n$ if every open covering of $M$ has a
refinement $\\{U_{i}\\}$ such that each point of $M$ belongs to at most $n+1$
element of $\\{U_{i}\\}$. A Lebesgue covering dimension of $M$ (denoted by
$\dim M$) is an infimum of all such $n$.
The following two well-known claims are easy to prove.
Claim 5.20: If $M$ is an $n$-manifold, $\dim M=n$.
Claim 5.21: If $X\subset M$ is a subset of a topological space, with induced
topology, one has $\dim X\leqslant\dim M$.
Theorem 5.22: (Gleason-Palais) Let $G$ be a topological group, which locally
path connected, and has $\dim K<\infty$ for each compact, metrizable subset
$K\subset G$. Then $G$ is homeomorphic to a Lie group.
Proof: [GP, Theorem 7.2].
Remark 5.23: Let $(M,d)$ be a metric space equipped with an intrinsic metric
$d$, and another metric $d_{g}\leqslant d$, such that $(M,d_{g})$ is a
manifold. Then $(M,d)$ satisfies all topological assumptions of 5.6. Indeed,
$(M,d)$ is locally path connected, because $d$ is intrinsic. For any compact
$K\subset(M,d)$, the identity map $(K,d){\>\longrightarrow\>}(K,d_{g})$ is a
continuous bijection, hence a homeomorphism. Therefore, $(K,d)$ is
homeomorphic to a subset of a manifold, and has finite dimension by 5.6 and
5.6.
Remark 5.24: When the action of $G$ on $M$ is free, the Gleason-Palais theorem
implies 5.5 directly. Indeed, $(M,d)$ is a topological group (5.5), and by 5.6
it satisfies assumptions of 5.6. From 5.6 we obtain that $(M,d)$ is
homeomorphic to a Lie group, and the bijection
$(M,d){\>\longrightarrow\>}(M,d_{g})$ is a homeomorphism, because it is a
bijective, continuous map of manifolds.
Now we can finish the proof of 5.5. Consider the natural projection
$G\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}G/H=M$. Let
$G_{M}:=(M,d)\times_{(M,d_{g})}G$ be a fibered product of $(M,d)$ and $G$ over
$(M,d)$, that is, the preimage of the diagonal under the natural map
$(M,d)\times G\ext@arrow
0099\arrowfill@\relbar\relbar\longrightarrow{}{\operatorname{Id}\times\pi}(M,d_{g})\times(M,d_{g}).$
Since $G{\>\longrightarrow\>}(M,d_{g})$ is a locally trivial fibration,
$G_{M}{\>\longrightarrow\>}(M,d)$ is also a locally trivial fibration, with
the same fibers, which are manifolds, hence locally path connected. This
implies that $G_{M}$ is locally path connected. Now, the natural bijection
$G_{M}{\>\longrightarrow\>}G$ is continuous, hence all compacts $K\subset
G_{M}$ are finite-dimensional (5.6). By 5.6, $G_{M}$ is a Lie group, and the
continuous bijection of manifolds $G_{M}{\>\longrightarrow\>}G$ is a
homeomorphism. Now, $G_{M}=(M,d)\times_{(M,d_{g})}G$, with $(M,d_{g})=G/H$,
hence the quotient $G_{M}/H$ is homeomorphic to $(M,d)$. However, $G\cong
G_{M}$, hence this quotient is homeomorphic to $G/H\cong(M,d_{g})$. We proved
5.5. This finishes the proof of 5.3 (ii)
## 6 GHK lines and exceptional sets
### 6.1 Lifting the GHK lines to the Teichmüller space
The following proposition insures that GHK lines are in some sense “liftable”
to the Teichmüller space. This is a key idea used to prove that the period map
is a covering.
Proposition 6.1: Let $I\in\operatorname{\sf Teich}$ be a point in the
Teichmüller space of a hyperkähler manifold, $\operatorname{\sf NS}(M,I)=0$,
and $S\subset\operatorname{{\mathbb{P}}\sf er}$ a hyperkähler line passing
through $\operatorname{\sf Per}(I)$.111Such a hyperkähler line is necessarily
generic, by 5.2. Then there exists a holomorphic curve
$S_{I}\subset\operatorname{\sf Teich}$ passing through $I$ and satisfying
$\operatorname{\sf Per}(S_{I})=S$.
Proof: Denote by $W\subset H^{2}(M,{\mathbb{R}})$ the 3-dimensional space used
to define $S$. Let $\Omega$ be the holomorphic symplectic form of $(M,I)$, and
$V:=\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega\rangle\subset
H^{2}(M,{\mathbb{R}})$ the corresponding 2-dimensional space. Then $V\subset
W$, and the 1-dimensional orthogonal complement $V^{\bot}\cap W$ intersects
both components of the cone $\\{x\in H^{1,1}_{I}(M,{\mathbb{R}})\ \ |\ \
q(x,x)>0\\}$. One of these components coincides with the Kähler cone (2.3).
Choose a Kähler form $\omega\in V^{\bot}_{W}$, normalize it in such a way that
$q(\operatorname{Re}\Omega,\operatorname{Re}\Omega)=q(\operatorname{Im}\Omega,\operatorname{Im}\Omega)=q(\omega,\omega),$
and let $(M,I,J,K)$ be the hyperkähler structure associated with $\omega$ as
in 2.1. Denote by $S_{I}$ the line of complex structures associated with this
hyperkähler structure. As shown above (5.1), the period map $\operatorname{\sf
Per}$ maps $S_{I}$ isomorphically to $S$.
Abusing the language, we call a ${\mathbb{C}}P^{1}$ of induced complex
structures associated with a hyperkähler structure “a hyperkähler line” as
well. These “hyperkähler lines” lie in the Teichmüller space, and the
hyperkähler lines defined previously lie in the period space. Then 6.1 can be
restated saying that a GHK line passing through a point
$l\in\operatorname{{\mathbb{P}}\sf er}$, satisfying $\operatorname{\sf
NS}(M,l)=0$, can be always lifted to a hyperkähler line
$S\subset\operatorname{\sf Teich}$ for each $I\in\operatorname{\sf Teich}$
such that $\operatorname{\sf Per}(I)=l$.
Definition 6.2: Let $\operatorname{{\mathbb{P}}\sf er}$ be a period space for
a hyperkähler manifold $M$, and
$\psi:\;D{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ an etale map
from a Hausdorff manifold $D$. Given a hyperkähler line
$S\subset\operatorname{{\mathbb{P}}\sf er}$, denote by
$S_{\operatorname{Pic}>0}$ the set of all $I\in S$ satisfying
$\operatorname{rk}\operatorname{Pic}(M,I)>0$. We say that $\psi$ is compatible
with generic hyperkähler lines if for any GHK line
$S\subset\operatorname{{\mathbb{P}}\sf er}$, the space $X:=\psi^{-1}(S)$ is a
union of several disjoint copies of $S$, which are closed and open in $X$, and
another subset $Y\subset X$, which satisfies $\psi(Y)\subset
S_{\operatorname{Pic}>0}$.
Proposition 6.3: Let $M$ be a hyperkähler manifold, and
$\operatorname{\sf Per}:\;\operatorname{\sf
Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$
its period map. Then $\operatorname{\sf Per}$ is compatible with generic
hyperkähler lines.
Proof: Let $S\subset\operatorname{{\mathbb{P}}\sf er}$ be a GHK line, $l\in S$
a point with $\operatorname{\sf NS}(M,l)=0$, and $I\in\operatorname{\sf
Teich}$ its preimage. By 6.1, $S$ can be lifted to a hyperkähler line
$S_{I}\subset\operatorname{\sf Teich}$ passing through $I$. Since
$\operatorname{\sf Per}$ is etale, the restriction $\operatorname{\sf
Per}:\;S_{I}{\>\longrightarrow\>}S$ is a diffeomorphism. By 6.1 below, $S_{I}$
is a connected component of $\operatorname{\sf Per}^{-1}(S)$.
The following claim is completely trivial.
Claim 6.4: Let $X\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}Y$ be
a local homeomorphism of Hausdorff spaces, $S\subset Y$ a compact subset, and
$S_{1}\subset X$ a subset of $\psi^{-1}(S)$, with
$\psi{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}:\;S_{1}{\>\longrightarrow\>}S$
a homeomorphism. Then $S_{1}$ is closed and open in $\psi^{-1}(S)$
Proof: The set $S_{1}$ is closed because it is homeomorphic to $S$ which is
compact, and $X$ is Hausdorff. Suppose that $S_{1}$ is not open in
$\psi^{-1}(S)$; then, there exists a sequence of points
$\\{x_{i}\\}\subset\psi^{-1}(S)\backslash S_{1}$ converging to $x\in S_{1}$.
Choose a neighbourhood $U\ni x$ such that
$\psi{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is a homeomorphism.
Replacing $\\{x_{i}\\}$ by a subsequence, we may assume that
$\\{x_{i}\\}\subset U$. Then $\psi{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}\cap
U}}\right.}$ is a homeomorphism onto its image $S_{U}$, which is a
neighbourhood of $\psi(x)$ in $S$. Replacing $\\{x_{i}\\}$ by a subsequence
again, we may assume that all $\psi(x_{i})$ lie in $S_{U}$. Since
$\psi{\left|{}_{{\phantom{|}\\!\\!}_{U}}\right.}$ is bijective onto its image,
this map induces a bijection from $S_{1}\cap U$ to $S_{U}$. Therefore,
$\\{x_{i}\\}\subset S_{1}\cap U$. We obtained a contradiction, proving that
$S_{1}$ is open in $\psi^{-1}(S)$.
### 6.2 Exceptional sets of etale maps
In [Br], F. Browder has discovered several criteria which can be used to prove
that a given etale map is a covering. Unfortunately, in our case neither of
his theorems can be applied, and we are forced to devise a new criterion,
which is then applied to the period map.
Definition 6.5: Let $X\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}Y$
be a local homeomorphism of Hausdorff topological spaces, e.g. an etale map.
Consider a connected, simply connected subset $R\subset Y$, and let
$\\{R_{\alpha}\\}$ be the set of connected components of $\psi^{-1}(R)$. An
exceptional set of $(\psi,R)$ is $R\backslash\psi(R_{\alpha})$.
Remark 6.6: The following topological criterion is the main technical engine
of this section. Its proof is complicated, but completely abstract, and we
hope that this result might have independent uses outside of hyperkähler
geometry. We include an alternative proof of this proposition in the Appendix
by Eyal Markman (Section 8).
Proposition 6.7: Let
$X\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}Y$ be a local
diffeomorphism of Hausdorff manifolds. Assume that for any open subset
$U\subset Y$, the closure $\overline{U}\subset Y$ has empty exceptional sets,
provided that $U$ has smooth boundary. Then $\psi$ is a covering.
Proof: 6.2 is local in $Y$, hence it will suffice to prove it when $Y$ is
diffeomorphic to ${\mathbb{R}}^{n}$. Choose a flat Riemannian metric on
$Y\cong{\mathbb{R}}^{n}$. Lifting the corresponding Riemannian tensor to $X$,
we can consider $X$ as a Riemannian manifold, also flat. The Riemannian tensor
defines a metric structure on $Y$ and $X$ as usual. For a point $x$ in a
metric space $M$, a closed $\varepsilon$-ball with center in $x$ is the set
$\overline{B}_{\varepsilon}(x):=\\{m\in M\ \ |\ \
d(x,m)\leqslant\varepsilon\\}.$
Taking strict inequality, we obtain an open ball,
$B_{\varepsilon}(x):=\\{m\in M\ \ |\ \ d(x,m)<\varepsilon\\}.$
Clearly, $\overline{B}_{\varepsilon}(x)$ is closed, $B_{\varepsilon}(x)$ is
open, and $\overline{B}_{\varepsilon}(x)$ is the closure of
$B_{\varepsilon}(x)$, and its completion, in the sense of metric geometry.
For any $x\in X$, $y=\psi(x)$, let $D_{x}\subset{\mathbb{R}}^{>0}$ be the set
of all $\varepsilon\in{\mathbb{R}}^{>0}$ such that the corresponding
$\varepsilon$-ball $\overline{B}_{\varepsilon}(x)$ is mapped to
$\overline{B}_{\varepsilon}(y)$ bijectively. Clearly, $D_{x}$ is an initial
interval of ${\mathbb{R}}^{>0}$. We are going to show that $D_{x}$ is open and
closed in ${\mathbb{R}}^{>0}$.
Step 1: The interval $D_{x}$ is open, for any etale map
$X\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}{\mathbb{R}}^{n}$.
Indeed, for any $\varepsilon\in D_{x}$, the corresponding $\varepsilon$-ball
$\overline{B}_{\varepsilon}(x)$ is compact, because it is isometric to
$\overline{B}_{\varepsilon}(y)$. Every point of
$\overline{B}_{\varepsilon}(x)$ has a neighbourhood which is isometrically
mapped to its image in $Y$. Take a covering
$\\{B_{\varepsilon}(x),U_{1},U_{2},...\\}$ of $\overline{B}_{\varepsilon}(x)$
where $U_{i}$ are open balls with this property, centered in a point on the
boundary of $\overline{B}_{\varepsilon}(x)$. Since
$\overline{B}_{\varepsilon}(x)$ is compact, $\\{B_{\varepsilon}(x),U_{i}\\}$
has a finite subcovering $U_{1},...,U_{n}$. By construction, for each point
$z\in W:=B_{\varepsilon}(x)\cup\bigcup_{i}U_{i}$, the set $W$ contains a
straight line (geodesic) from $x$ to $z$. Indeed, $W$ is a union of an open
ball $B_{\varepsilon}(x)$ and several open balls centered on its boundary, and
all these balls are isometric to open balls in ${\mathbb{R}}^{n}$. Since
$\psi$ maps straight lines to straight lines, it maps
$B_{\varepsilon^{\prime}}(x)$ surjectively to $B_{\varepsilon^{\prime}}(y)$.
To show that this map is also injective, consider two points $a_{1},a_{2}\in
B_{\varepsilon^{\prime}}(x)$, mapped to $b\in B_{\varepsilon^{\prime}}(y)$,
and let $[x,a_{1}]$ and $[x,a_{2}]$ be the corresponding intervals of a
straight line. Since $\psi(a_{1})=\psi(a_{2})=b$, one has
$\psi([x,a_{1}])=\psi([x,a_{2}])$, and these intervals have the same length.
Also, $[x,a_{1}]\cap B_{\varepsilon}(x)=[x,a_{2}]\cap B_{\varepsilon}(x)$,
because $\psi{\left|{}_{{\phantom{|}\\!\\!}_{B_{\varepsilon}(x)}}\right.}$ is
injective. Therefore, the intervals $[x,a_{1}]$ and $[x,a_{2}]$ coincide, and
$a_{1}=a_{2}$.
Step 2: Let $\psi:\;X{\>\longrightarrow\>}{\mathbb{R}}^{n}$ be an etale map,
$y=\psi(x)$, and suppose that $\psi:\;B_{s}(x){\>\longrightarrow\>}B_{s}(y)$
is bijective, for some $s>0$. Then
$\varphi:\;B_{s}(x){\>\longrightarrow\>}B_{s}(y)$ is an isometry, with respect
to the metric on $B_{s}$ induced from the ambient manifold. Indeed, $\psi$ is
etale, hence any piecewise geodesic path in $X$ is projected to such one in
${\mathbb{R}}^{n}$. Therefore, $\psi$ does not increase distance:
$d(a,b)\geqslant d(\psi(a),\psi(b))$. The open ball $B_{s}(y)$ is geodesically
convex, hence for any $y_{1},y_{2}\in B_{s}(y)$, the geodesic interval
$[y_{1},y_{2}]$ can be lifted to a geodesic in $B_{s}(x)$. This implies an
inverse inequality: $d(a,b)\leqslant d(\psi(a),\psi(b))$. We proved that
$\varphi:\;B_{s}(x){\>\longrightarrow\>}B_{s}(y)$ is an isometry. This implies
that the map
$\psi:\;\overline{B}_{s}(x){\>\longrightarrow\>}\overline{B}_{s}(y)$ of their
metric completions is also an isometry. In particular, this map is injective.
Step 3. In the assumptions of Step 2, we prove that $\overline{B}_{s}(x)$ is a
connected component of $\psi^{-1}(\overline{B}_{s}(y))$. Notice that
$\overline{B}_{s}(x)$ is a closure of $B_{s}(x)$, which is homeomorphic to a
ball in ${\mathbb{R}}^{n}$, hence $\overline{B}_{s}(x)$ is connected. To prove
that it is a connected component, we need only to show that it is open in
$\psi^{-1}(\overline{B}_{s}(y))$.
The corresponding map of open balls
$\psi:\;B_{s}(x){\>\longrightarrow\>}B_{s}(y)$ is by definition bijective. The
closed ball $\overline{B}_{s}(x)$ is closed in
$\psi^{-1}(\overline{B}_{s}(y))$. For any $z\in\partial\overline{B}_{s}(x)$ on
the boundary of $\overline{B}_{s}(x)$, an open ball $S$ centered in $z$ is
split by the boundary
$\partial\overline{B}_{s}(x)=\\{x^{\prime}\in X\ \ |\ \ d(x,x^{\prime})=s\\}$
onto two open components, $S_{o}:=\\{x^{\prime}\in X\ \ |\ \
d(x,x^{\prime})>s\\}$ and $S_{1}:=\\{x^{\prime}\in X\ \ |\ \
d(x,x^{\prime})<s\\}$, with $S_{1}$ mapping to $B_{s}(y)$,
$\partial\overline{B}_{s}(x)$ mapping to its boundary, and $S_{o}$ to
$Y\backslash\overline{B}_{s}(y)$.222Here we use the fact that
$\psi{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}$ is a bijection, for $S$
sufficiently small, hence the image of $S$ cannot wrap on itself. This implies
that
$\psi^{-1}(\overline{B}_{s}(y)\cap\psi(S))=S\cap\overline{B}_{s}(x).$
Therefore, $\overline{B}_{s}(x)$ is open in $\psi^{-1}(\overline{B}_{s}(y))$.
Step 4. Now we can show that $D_{x}$ is closed. This argument uses the
triviality of exceptional sets (the first time in this proof, the rest follows
just from the etaleness of $\psi$). Let $s:=\sup D_{x}$, and
$\overline{B}_{s}(x)$ the corresponding closed ball. We prove that
$\psi:\;\overline{B}_{s}(x){\>\longrightarrow\>}\overline{B}_{s}(y)$ is a
homeomorphism.
From Step 3, it follows that $\overline{B}_{s}(x)$ is a connected component of
the preimage $\psi^{-1}(\overline{B}_{s}(y))$. Since the exceptional sets of
$\overline{B}_{s}(y)$ are all empty, the map
$\psi:\;\overline{B}_{s}(x){\>\longrightarrow\>}\overline{B}_{s}(y)$ is
surjective. It is injective as follows from Step 2.
We proved that $D_{x}$ is open and closed, hence $D_{x}={\mathbb{R}}^{>0}$,
and $\psi$ maps any connected component of $X$ bijectively to $Y$.
Remark 6.8: An exceptional set of $(\psi,U)$ is always closed in $U$.
Lemma 6.9: Let $M$ be a Hausdorff manifold,
$M\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf
er}$ a local diffeomorphism, compatible with GHK lines,
$U\subset\operatorname{{\mathbb{P}}\sf er}$ an open, simply connected subset,
$U_{\alpha}$ a component of $\psi^{-1}(U)$, and $K_{\alpha}$ the corresponding
exceptional set. Consider a GHK line $C\subset\operatorname{{\mathbb{P}}\sf
er}$, and let $C_{1}$ be a connected component of $C\cap U$. Then
$C_{1}\subset K_{\alpha}$, or $C_{1}\cap K_{\alpha}=\emptyset$.
Proof: Suppose that $D:=C_{1}\cap(U\backslash K_{\alpha})$ is non-empty. Since
$K_{\alpha}$ is closed in $U$, $D$ is open in $C_{1}$. Then $D$ contains
points $l\in D$ with $\operatorname{\sf NS}(M,l)=\emptyset$ (5.2). The set
$\psi^{-1}(l)$ is non-empty, because $l\notin K_{\alpha}$. Since $\psi$ is
compatible with GHK lines, for any $I\in\psi^{-1}(l)$, there is a curve
$C_{I}\subset M$ passing through $I$ and projecting bijectively to $C$.
Clearly, the connected component of $C_{I}\cap\psi^{-1}(U)\ni I$ is
bijectively mapped to $C_{1}$, hence $C_{1}\cap K_{\alpha}=\emptyset$.
Remark 6.10: A version of 6.2 is also true if $\overline{U}$ is a closed set,
obtained as a closure of an open subset $U\subset\operatorname{{\mathbb{P}}\sf
er}$, and $C_{1}$ a connected component of $\overline{U}\cap C$, for a GHK
curve $C$. If $C_{1}$ contains interior points, the same argument as above can
be used to show that $C_{1}\subset K_{\alpha}$, or $C_{1}\cap
K_{\alpha}=\emptyset$.
### 6.3 Subsets covered by GHK lines
Let $U\subset\operatorname{{\mathbb{P}}\sf er}$ be an open subset, or a
closure of an open subset with smooth boundary, and $K\subset U$ a subset of
$U$. Given a GHK line $C\subset\operatorname{{\mathbb{P}}\sf er}$, denote by
$C_{U}$ a connected component of $C\cap U$. This component is non-unique for
some $C$ and $U$. Denote by $\Omega_{U}(K)$ the union of all segments
$C_{U}\subset U$ intersecting $K$, for all GHK lines
$C\subset\operatorname{{\mathbb{P}}\sf er}$. In other words, $\Omega_{U}(K)$
is the set of all points connected to $K$ by a connected segment of $C\cap U$,
with $C\subset\operatorname{{\mathbb{P}}\sf er}$ a GHK line. Let
$\Omega^{*}_{U}(X)$ be a union of
$\Omega_{U}(X),\Omega_{U}(\Omega_{U}(X)),\Omega_{U}(\Omega_{U}(\Omega_{U}(X))),...$.
Proposition 6.11: Let $U\subset\operatorname{{\mathbb{P}}\sf er}$ be an open
subset, and $x\in U$ a point. Assume that the closure of $U$ is compact. Then
$\Omega_{U}^{*}(x)$ is open in $U$.
Proof: By 5.3, the metric $d_{tw}$ induces the usual topology on
$\operatorname{{\mathbb{P}}\sf er}$. For any $x\in U$, the distance
$d_{tw}(x,\partial U)$ to the boundary of $U$ is positive, because $\partial
U$ is compact. Then the open ball $B_{r}(x,d_{tw})$ is contained in
$\Omega_{U}^{*}(x)$. Indeed, let $y\in B_{r}(x,d_{tw})$. Then $y=s_{n+1}$ is
connected to $x=s_{0}$ by a sequence of GHK lines $S_{1},...,S_{n}$, such that
$\sum_{i=0}^{n}d_{g}(s_{i},s_{i+1})<r$. Consider the corresponding piecewise
geodesic path $\gamma\subset\bigcup S_{i}$ of length $<r$. Since
$d_{tw}(x,\partial U)>r$, the whole of $\gamma$ belongs to $U$. Therefore, $y$
is connected to $x$ by a union of connected segments of GHK lines which lie in
$U$.
To apply 6.2 to the period map using the exceptional sets, we also need closed
subsets with smooth boundary. In this situation the following lemma can be
used.
Lemma 6.12: Let $K\subset\operatorname{{\mathbb{P}}\sf er}$ be a compact
closure of an open subset with smooth boundary, and $x\in K$ a point. Then
$\Omega_{K}(x)$ contains an interior point of $K$.
Proof: Let $V_{x}$ be the 2-plane in $H^{2}(M,{\mathbb{R}})$ corresponding to
$x$ via the identification $Gr(2)=\operatorname{{\mathbb{P}}\sf er}$. Then the
tangent space $T_{x}\operatorname{\sf Per}$ is identified with
$\operatorname{Hom}(V_{x},V_{x}^{\bot})$, where $V_{x}^{\bot}$ is an
orthogonal complement. For a hyperkähler line $C$ associated with a
3-dimensional space $W$, the corresponding 2-dimensional space $T_{x}C\subset
T_{x}\operatorname{{\mathbb{P}}\sf er}$ is the space
$\operatorname{Hom}(V_{x},(V_{x}^{\bot}\cap W))$. Since
$V_{x}^{\bot}=H^{1,1}_{x}(M)$ and $W$ can be chosen by adding to $V_{x}$ any
Kähler class in $H^{1,1}(M)$, the set of all tangent vectors $T_{x}C\subset
T_{x}\operatorname{{\mathbb{P}}\sf er}$ is open in the space
$P:=\\{l\in\operatorname{Hom}(V_{x},H^{1,1}_{x}(M))\ \ |\ \
\operatorname{rk}l=1\\}$
The condition $\operatorname{rk}l=1$ is quadratic, and it is easy to check
that an open subset $U_{P}\subset P$ cannot be contained inside a linear
subspace of positive codimension. In particular, $U_{P}$ cannot lie in the
tangent space to the boundary of $K$,
$U_{P}\not\subset T_{x}\partial K\subset\operatorname{Hom}(V_{x},H^{1,1}(M)).$
(6.1)
Take for $U_{P}$ the set of all vectors tangent to GHK lines passing through
$x$. Then (6.1) implies that for a generic GHK line $C$ passing through $x$,
$C$ intersects with the interior points of $K$.
Corollary 6.13: Let $K\subset\operatorname{{\mathbb{P}}\sf er}$ be a closure
of an open, connected subset $U\subset\operatorname{{\mathbb{P}}\sf er}$ with
smooth boundary, and $\Omega_{K}$ the operation on subsets of $K$ defined
above. Then $\Omega_{K}^{*}(x)=K$, for any point $x\in K$.
Proof: Clearly, $\Omega_{U}^{*}(x)$ is the set of all points in $U$ which can
be connected to $x$ within $U$ by a finite sequence of connected segments of
GHK lines. By 6.3, $\Omega_{U}^{*}(x)$ is open in $U$. If
$y\notin\Omega_{U}^{*}(x)$, then $\Omega_{U}^{*}(y)$ does not intersect
$\Omega_{U}^{*}(x)$. Then $U$ is represented as a disconnected union of open
sets $\Omega_{U}^{*}(x_{i})$, for some $\\{x_{i}\\}\subset U$. This is
impossible, because $U$ is connected. We proved that $\Omega^{*}_{U}(x)=U$.
Then $\Omega_{K}^{*}(x)=K$, because every point on a boundary of $K$ is
connected to some point of $U$ by a connected segment of a GHK line (6.3).
The main result of this section is the following theorem
Theorem 6.14: Let $M$ be a Hausdorff manifold and
$M\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf
er}$ a local diffeomorphism compatible with GHK lines. Then $\psi$ is a
covering.
Remark 6.15: It is well known that $\operatorname{{\mathbb{P}}\sf er}$ is
simply connected (2.4). Then 6.3 implies that $\psi$ is a diffeomorphism.
Proof of 6.3: To prove that $\psi$ is a covering, it suffices to show that all
its exceptional sets of $(\psi,K)$, are empty provided that $K$ is a closure
of a simply connected open subset $U\subset\operatorname{{\mathbb{P}}\sf er}$
which has a smooth boundary (6.2). Let $K_{\alpha}$ be an exceptional set,
associated with a closure $K\subset\operatorname{{\mathbb{P}}\sf er}$ of an
open subset $U\subset\operatorname{{\mathbb{P}}\sf er}$ with smooth boundary.
From 6.2 and 6.2 it follows that $\Omega_{K}(K_{\alpha})=K_{\alpha}$, where
$\Omega_{K}(Z)$ is a union of all connected segments of $C\cap K$ intersecting
$Z$, for all GHK lines $C\subset\operatorname{{\mathbb{P}}\sf er}$. Then
$\Omega_{K}^{*}(K_{\alpha})=K_{\alpha}$, where $\Omega_{K}^{*}(Z)$ is a union
of all iterations $\Omega_{K}^{i}(Z)$. However, for any non-empty $Z\subset
K$, one has $\Omega_{K}^{*}(Z)=K$ by 6.3. Therefore, any exceptional set
$K_{\alpha}$ of $(\psi,K)$ for $K$ as above is empty, and 6.3 follows.
## 7 Monodromy group for $K3^{[n]}$.
When $M=K3^{[n]}$ is a Hilbert scheme of points on a K3 surface, fundamental
results about its moduli were obtained by E. Markman ([M1], [M2]), using the
Fourier-Mukai action on the derived category of coherent sheaves. In this
section we relate these results with our computation of $\operatorname{\sf
Teich}_{b}$ to obtain a global Torelli theorem for $M=K3^{[p^{\alpha}+1]}$,
$p$ prime.
### 7.1 Monodromy group for hyperkähler manifolds
Let $M$ be a complex manifold, and ${\cal M}$ a coarse moduli space of its
deformations.
Definition 7.1: A monodromy group $\operatorname{\sf Mon}(M)$ of a hyperkähler
manifold $M$ is a subgroup of $GL(H^{*}(M,{\mathbb{Z}}))$ generated by the
monodromy of the Gauss-Manin local systems, for all deformations of $M$.
Consider the universal fibration ${\cal F}$ on ${\cal M}$, with the fiber in
$I\in{\cal M}$ corresponding to the associated complex manifold $(M,I)$. The
universal fibration does not always exist (it exists for fine moduli spaces,
in the category of stacks). One could consider the monodromy of the
corresponding Gauss-Manin connection, and relate it to the monodromy group of
$M$, as follows (please see [Z] and [No] for the definition and properties of
the fundamental groups of stacks).
Claim 7.2: Let $M$ be a hyperkähler manifold, and ${\cal M}_{st}$ its fine
moduli space, equipped with the universal fibration. Then the monodromy group
of $M$ is the image of the fundamental group $\pi_{1}({\cal M}_{st})$ in
$GL(H^{*}(M))$, under the monodromy action.
Remark 7.3: The stack ${\cal M}_{st}$ admits a natural projection to the
coarse moduli space ${\cal M}$. This map is an isomorphism for any open set
$U\subset{\cal M}_{st}$ such that all fibers of the universal fibration at $U$
have no automorphisms. In particular, ${\cal M}_{st}={\cal M}$ whenever all
deformations of $M$ have no automorphisms; the stack structure is a way of
taking automorphisms of $M$ into account.
Using the global Torelli theorem (4.4), the monodromy group can be related to
the mapping class group, as follows.
Theorem 7.4: Let $(M,I)$ be a hyperkähler manifold, and $\operatorname{\sf
Teich}^{I}$ the corresponding connected component of a Teichmüller space.
Denote by $\Gamma^{I}$ the subgroup of the mapping class group preserving the
component $\operatorname{\sf Teich}^{I}$, and let $\operatorname{\sf Mon}$ be
the monodromy group of $(M,I)$ defined above. Suppose that a general
deformation of $M$ has no automorphisms. Then $\operatorname{\sf Mon}$ is
naturally isomorphic to the image $i(\Gamma^{I})$ of $\Gamma^{I}$ in
$PGL(H^{2}(M,{\mathbb{C}}))$, under the natural action of $\Gamma^{I}$ on
$H^{2}(M)$.
Proof: Clearly, every loop in the birational moduli space ${\cal M}_{b}$ can
be lifted to a loop in ${\cal M}$. Therefore, $\operatorname{\sf Mon}$ is
isomorphic to an image of $\pi_{1}({\cal M}_{b})$ in
$GL(H^{*}(M,{\mathbb{Z}}))$, under the natural monodromy action. From 4.4 we
obtain that $\pi_{1}({\cal M}_{b})=i(\Gamma^{I})$, and the monodromy action on
$H^{*}(M)$ is factorized through $\pi_{1}({\cal M}_{b})=i(\Gamma^{I})$.
Therefore $\operatorname{\sf Mon}=i(\Gamma^{I})$.
This result allows one to answer the question asked in [M2] (Conjecture 1.3).
Corollary 7.5: Let $\gamma\in\operatorname{\sf Mon}$ be an element of the
monodromy group acting trivially on the projectivization
${\mathbb{P}}H^{2}(M,{\mathbb{C}})$. Suppose that a general deformation of $M$
has no automorphisms. Then $\gamma$ is trivial.
Proof: Let $\gamma\in{\cal M}$ be a loop, and $\sigma_{\gamma}$ the
corresponding element of the mapping class group of $M$, defined in the same
way as $\tau_{\gamma}$ above. If $\gamma$ acts trivially on
${\mathbb{P}}H^{2}(M,{\mathbb{C}})$, the corresponding loop in ${\cal
M}_{b}=\operatorname{{\mathbb{P}}\sf
er}\subset{\mathbb{P}}H^{2}(M,{\mathbb{C}})$ is contractible. Since the
Hausdorff reduction map $\operatorname{\sf
Teich}{\>\longrightarrow\>}\operatorname{\sf Teich}_{b}$ is etale, $\gamma$
can be lifted to a contractible loop in ${\cal M}$. Therefore, $\gamma$ is
contractible, and $\sigma_{\gamma}$ is trivial.
Remark 7.6: In the above corollary, a stronger result is actually proven.
Instead of defining the monodromy group as above, we could define
$\widetilde{\operatorname{\sf Mon}}$ as the image of $\pi_{1}({\cal M})$ in
the mapping class group of $M$. Then 7.1 implies that the natural map of
$\widetilde{\operatorname{\sf Mon}}$ to $PGL(H^{2}(M,{\mathbb{C}}))$ is
injective.
Remark 7.7: The kernel of the natural projection
$\Gamma_{I}{\>\longrightarrow\>}PGL(H^{2}(M,{\mathbb{C}})$ is identified with
the group of holomorphic automorphisms of a generic deformation of a
hyperkähler manifold $M$. When $M=K3^{[n]}$, this group is trivial, which can
be easily seen e.g. from the results of [V5]. When $M$ is a generalized Kummer
variety, it is known to be non-trivial ([KV]).
Remark 7.8: 7.1 is false when a generic deformation of $M$ has automorphisms
(e.g. for a generalized Kummer variety). Indeed, in this case we could take an
isotrivial deformation of $M$ with monodromy inside this automorphism group.
The corresponding elements in the monodromy group may have trivial action on
$H^{2}(M,{\mathbb{C}})$, which is, indeed, the case for a generalized Kummer
variety ([M1], last paragraph of Section 4.2).
### 7.2 The Hodge-theoretic Torelli theorem for $K3^{[n]}$
Definition 7.9: Let $V$ be a vector space, $g$ a non-degenerate quadratic
form, and $v\in V$ a vector which satisfies $g(v,v)=\pm 2$. Consider the
pseudo-reflection map $\rho_{v}:\;V{\>\longrightarrow\>}V$,
$\rho_{v}(x):=\frac{-2}{g(v,v)}x+g(x,v)v.$
Clearly, $\rho_{v}$ is a reflection when $g(v,v)=2$, and $-\rho_{v}$ is a
reflection when $g(v,v)=-2$. Given an integer lattice in $V$, consider the
group $\operatorname{Ref}(V)$ generated by $\rho_{v}$ for all integer vectors
$v$ with $g(v,v)=\pm 2$. We call $\operatorname{Ref}$ a reflection group.
The following fundamental theorem was proven by E. Markman in [M2].
Theorem 7.10: ([M2, Theorem 1.2]) Let $M=K3^{[n]}$ be a Hilbert scheme of
points on a K3 surface, and $\operatorname{\sf Mon}^{2}$ be the image of the
monodromy group in $GL(H^{2}(M,{\mathbb{Z}}))$. Then $\operatorname{\sf
Mon}^{2}=\operatorname{Ref}(H^{2}(M,{\mathbb{Z}}),q)$.
Comparing this with 7.1 and using the global Torelli theorem (4.4), we
immediately obtain the following result.
Theorem 7.11: Let $M=K3^{[n]}$ be a Hilbert scheme of points on K3, ${\cal
M}_{b}$ its birational Teichmüller space, and ${\cal M}_{b}(I)$ a connected
component of ${\cal M}_{b}$. Then ${\cal
M}_{b}(I)\cong\operatorname{{\mathbb{P}}\sf er}/\operatorname{Ref}$, where
$\operatorname{{\mathbb{P}}\sf er}$ is the period domain defined as in (1.3),
and $\operatorname{Ref}=\operatorname{Ref}(H^{2}(M,{\mathbb{Z}}),q)$ the
corresponding reflection group, acting on $\operatorname{{\mathbb{P}}\sf er}$
in a natural way.
The reflection group was computed in [M2] (Lemma 4.2). When $n-1$ is a prime
power, this computation is particularly effective.
Definition 7.12: Let $(V,g)$ be a real vector space equipped with a non-
degenerate quadratic form of signature $(m,n)$, and
$S:=\\{v\in V\ \ |\ \ g(v,v)>0\\}.$
It is easy to see that $S$ is homotopy equivalent to a sphere $S^{m-1}$.
Define the spinorial norm of $\eta\in O(V)$ as $\pm 1$, where the sign is
positive if $\eta$ acts as 1 on $H^{m-1}(S)$, and negative if $\eta$ acts as
-1. Let $O^{+}(V)$ denote the set of all isometries with spinorial norm 1.
Remark 7.13: It is easy to see that $\operatorname{Ref}\subset O^{+}(V)$,
where $\operatorname{Ref}$ is a reflection group.
Proposition 7.14: ([M2, Lemma 4.2]). Let $M=K3^{[n]}$ be a Hilbert scheme of
K3, and $\operatorname{Ref}=\operatorname{Ref}(H^{2}(M,{\mathbb{Z}}),q)$ the
corresponding reflection group. Then
$\operatorname{Ref}=O^{+}(H^{2}(M,{\mathbb{Z}}),q)$ if and only if $n-1$ is a
prime power.
Definition 7.15: Let $V$ be a real vector space equipped with a non-
degenerate quadratic form of signature $(m,n)$. A choice of spin orientation
on $V$ is a choice of a generator of the cohomology group $H^{m-1}(S)$ (7.2).
Clearly, $O^{+}(V)$ is a group of orthogonal maps preserving the spin
orientation.
Remark 7.16: For a space $V$ with signature $(m,n)$, the group $O(V)$ has 4
connected components, which are given by a choice of orientation and spin
orientation. Alternatively, these 4 components are distinguished by a choice
of orientation on positive $m$-dimensional planes and negative $n$-dimensional
planes.
Remark 7.17: Donaldson ([Do]) has shown that any diffeomorphism of a K3
surface $M$ preserves the spin orientation, and the global Torelli theorem
implies that every integer isometry of $H^{2}(M)$ preserving the spin
orientation is induced by a diffeomorphism ([Bor]). This implies that the
mapping class group $\Gamma_{M}$ is mapped to $O^{+}(H^{2}(M,{\mathbb{Z}}))$
surjectively.
Remark 7.18: Let $V=H^{2}(M,{\mathbb{R}})$ be the second cohomology of a
hyperkähler manifold, equipped with the Hodge structure and the BBF form, and
$V^{1,1}\subset V$ the space of real (1,1)-classes. The set of vectors
$R:=\\{v\in V^{1,1}\ \ |\ \ q(v,v)>0\\}$
is disconnected, and has two connected components. Since the orthogonal
complement $(V^{1,1})^{\bot}$ is oriented, a spin orientation on $V$ is
uniquely determined by a choice of one of two components of $R$. The Kähler
cone of $M$ is contained in one of two components of $R$. This gives a
canonical spin orientation on $H^{2}(M,{\mathbb{R}})$.
Definition 7.19: Let $M$ be a hyperkähler manifold. We say that the Hodge-
theoretic Torelli theorem holds for $M$, if for any $I_{1},I_{2}$ inducing
isomorphic Hodge structures on $H^{2}(M)$, the manifold $(M,I_{1})$ is
bimeromorphically equivalent to $(M,I_{2})$, provided that this isomorphism of
Hodge structures is also compatible with the spin orientation and the
Bogomolov-Beauville-Fujiki form, and $I_{1},I_{2}$ lie in the same connected
component of the moduli space.
Remark 7.20: This is the most standard version of global Torelli theorem.
The following claim immediately follows from 7.1.
Claim 7.21: Let $M$ be a hyperkähler manifold. Then the following statements
are equivalent.
(i)
The Hodge-theoretic Torelli theorem holds for $M$.
(ii)
The monodromy group $\operatorname{\sf Mon}$ of $M$ is surjectively mapped to
the group $O^{+}(H^{2}(M,{\mathbb{Z}}),q)$, under the natural action of
$\operatorname{\sf Mon}$ on $H^{2}(M)$.
Comparing this with the Markman’s computation of the monodromy group (7.2), we
immediately obtain the following theorem.
Theorem 7.22: Let $M=K3^{[p^{\alpha}+1]}$. Then the Hodge-theoretic Torelli
theorem holds.
Remark 7.23: For other examples of hyperkähler manifolds, the Hodge-theoretic
global Torelli theorem is known to be false. For generalized Kummer varieties
this was proven by Namikawa ([Na]), and for $M=K3^{[n]}$ this observation is
due to Markman ([M2]). For O’Grady’s examples of hyperkähler manifolds ([O]),
it is unknown.
## 8 Appendix: A criterion for a covering map (by Eyal Markman)
Another version of the proof of 6.2 was proposed by E. Markman; with his kind
permission, I include it here.
Proposition 8.1: (6.2) Let $\psi:X\rightarrow Y$ be a local homeomorphism of
Hausdorff topological manifolds. Assume that every open subset $U\subset Y$,
whose closure $\overline{U}$ is homeomorphic to a closed ball in
${\mathbb{R}}^{n}$, and such that $U$ is the interior of its closure,
satisfies the following property. For every connected component $C$ of
$\psi^{-1}(\overline{U})$, the equality $\psi(C)=\overline{U}$ holds. Then
$\psi$ is a covering map.
Verbitsky stated the above proposition in the category of differentiable
manifolds and provided a proof of the proposition, involving Riemannian-
geometric constructions on the domain $X$. We translate his proof to an
elementary point set topology language. The natural translation of the
statement and its proof to the category of differentiable manifolds is valid
as well. In that case $\psi$ is a local diffeomorphism and it suffices for the
assumption to hold for open subsets $U$, such that the boundary $\partial U$
is smooth, and there exists a homeomorphism from $\overline{U}$ onto a closed
ball in ${\mathbb{R}}^{n}$, which restricts to a diffeomorphism between the
two interiors and between the two boundaries. We will need the following well
known fact (see [Br], Lemma 1).
Lemma 8.2: Let $f:X\rightarrow Y$ be a local homeomorphism of topological
spaces, $W$ a connected Hausdorff topological space, $h:W\rightarrow Y$ a
continuous map, $x_{0}$ a point of $X$, and $w_{0}$ a point of $W$ satisfying
$h(w_{0})=f(x_{0})$. Then there exists at most one continuous map
$\widetilde{h}:W\rightarrow X$, satisfying $\widetilde{h}(w_{0})=x_{0}$, and
$f\circ\widetilde{h}=h$.
Proof of 8: The statement is local, so we may assume that
$Y={\mathbb{R}}^{n}$. Let $x$ be a point of $X$ and set $y:=\psi(x)$.
Definition 8.3: An open subset $U\subset{\mathbb{R}}^{n}$ is said to be
$x$-star-shaped, if it satisfies the following conditions.
1. 1.
$y$ belongs to $U$.
2. 2.
For every point $u\in U$, the line segment from $y$ to $u$ is contained in
$U$.
3. 3.
There exists a continuous map $\gamma:U\rightarrow X$, satisfying
$\gamma(y)=x$, and $\psi\circ\gamma:U\rightarrow{\mathbb{R}}^{n}$ is the
inclusion.
Claim 8.4:
1. 1.
Let $\\{U_{i}\\}_{i\in I}$ be a finite collection of $x$-star-shaped open
subsets of ${\mathbb{R}}^{n}$. Then their intersection $\cap_{i\in I}U_{i}$ is
$x$-star-shaped.
2. 2.
Let $\\{U_{i}\\}_{i\in I}$ be an arbitrary collection of $x$-star-shaped open
subsets of ${\mathbb{R}}^{n}$. Then their union $U:=\cup_{i\in I}U_{i}$ is
$x$-star-shaped.
3. 3.
Let $U\subset{\mathbb{R}}^{n}$ be an $x$-star-shaped open subset,
$W\subset{\mathbb{R}}^{n}$ a connected open subset satisfying the following
conditions. a) $W\cap U$ is connected. b) For every point $t\in W\cup U$, the
line segment from $t$ to $y$ is contained in $W\cup U$. c) There exists a
continuous map $\eta:W\rightarrow X$, such that
$\psi\circ\eta:W\rightarrow{\mathbb{R}}^{n}$ is the inclusion. d) There exists
a point $t\in W\cap U$, such that $\eta(t)=\gamma(t)$, where
$\gamma:U\rightarrow X$ is the lift of the inclusion satisfying $\gamma(y)=x$.
Then $W\cup U$ is $x$-star-shaped.
Proof: Part 1 is clear. Proof of part 2: Let $\gamma_{i}:U_{i}\rightarrow X$
be the unique lift of the inclusion, satisfying $\gamma_{i}(y)=x$. Define
$\gamma:U\rightarrow X$ by $\gamma(t)=\gamma_{i}(t)$, if $t$ belongs to
$U_{i}$. It sufficed to prove that $\gamma$ is well defined. If $t$ belongs to
$U_{i}\cap U_{j}$, then $U_{i}\cap U_{j}$ is connected, being $x$-star-shaped,
and $\gamma_{i}(t)=\gamma_{j}(t)$, by 8.
The proof of part 3 is similar to that of part 2.
Given a positive real number $\varepsilon$, set
$B_{\varepsilon}(y):=\\{y^{\prime}\in{\mathbb{R}}^{n}\ :\
d(y,y^{\prime})<\varepsilon\\}$, where $d(y^{\prime},y)$ is the Eucleadian
distance from $y^{\prime}$ to $y$. Let $\overline{B}_{\varepsilon}(y)$ be the
closure of $B_{\varepsilon}(y)$.
Claim 8.5: Assume that $B_{\varepsilon}(y)$ is $x$-star-shaped and let
$\gamma:B_{\varepsilon}(y)\rightarrow X$ be the lift of the inclusion
satisfying $\gamma(y)=x$, as in 8. Then there exists an open connected subset
$V\subset X$, such that $V$ contains the closure
$\overline{\gamma[B_{\varepsilon}(y)]}$, $\psi:V\rightarrow\psi(V)$ is
injective, and $\psi(V)$ is $x$-star-shaped.
Proof: Let $z$ be a point on the boundary
$\partial\gamma[B_{\varepsilon}(y)]$. Then $\psi(z)$ belongs to the boundary
of $B_{\varepsilon}(y)$. Now $\psi(z)$ has a basis of open neighborhoods $W$
with the property that $U_{z}:=W\cup B_{\varepsilon}(y)$ is $x$-star-shaped
(use 8 part 3). Let ${\cal U}_{z}$ denote the collection of all such $U_{z}$.
The collection
$\\{B_{\varepsilon}(y)\\}\cup\left[\cup_{z\in\partial\gamma[B_{\varepsilon}(y)]}{\cal
U}_{z}\right]$ is thus a collection of $x$-star-shaped open subsets. Their
union $U$ is $x$-star-shaped, by 8, so the inclusion
$U\subset{\mathbb{R}}^{n}$ admits a lift $\gamma:U\rightarrow X$ satisfying
$\gamma(y)=x$. Set $V:=\gamma[U]$. Then $V$ is open, since $\gamma$ is a
local-homeomorphism, and $V$ contains the closure of
$\gamma[B_{\varepsilon}(y)]$, by construction.
Let $D_{x}\subset{\mathbb{R}}^{>0}$ be the set of all
$\varepsilon\in{\mathbb{R}}^{>0}$, such that there exists a continuous map
$\gamma:\overline{B}_{\varepsilon}(y)\rightarrow X$, satisfying $\gamma(y)=x$,
and such that
$\psi\circ\gamma:\overline{B}_{\varepsilon}(y)\rightarrow{\mathbb{R}}^{n}$ is
the inclusion. Clearly, $D_{x}$ is a non-empty connected interval having $0$
as its left boundary point. We need to show that $D_{x}={\mathbb{R}}^{>0}$. It
suffices to show that $D_{x}$ is both open and closed.
Claim 8.6: $D_{x}$ is open.
Proof: Let $\varepsilon$ be a point of $D_{x}$. The image
$\gamma[\overline{B}_{\varepsilon}(y)]$ is compact and $X$ is Hausdorff.
Hence, $\gamma[\overline{B}_{\varepsilon}(y)]$ is closed and is thus equal to
the closure of $\gamma[B_{\varepsilon}(y)]$. Then
$\psi\left(\overline{\gamma[B_{\varepsilon}(y)]}\right)=\overline{B}_{\varepsilon}(y)$.
Hence, there exists an open $x$-star-shaped subset $U\subset{\mathbb{R}}^{n}$,
containing $\overline{B}_{\varepsilon}(y)$, by 8. Compactness of
$\overline{B}_{\varepsilon}(y)$ implies that $U$ contains
$\overline{B}_{\varepsilon_{1}}(y)$, for some $\varepsilon_{1}>\varepsilon$.
Now $\varepsilon_{1}$ belongs to $D_{x}$, since $U$ is $x$-star-shaped. Hence,
$D_{x}$ is open.
Set $s:=\sup(D_{x})$. If $s$ is infinite, we are done. Assume that $s$ is
finite. $B_{s}(y)$ is $x$-star-shaped, by 8. Let $\gamma:B_{s}(y)\rightarrow
X$ be the lift of the inclusion satisfying $\gamma(y)=x$.
Claim 8.7: The closure $C:=\overline{\gamma[B_{s}(y)]}$ is a connected
component of the preimage $\psi^{-1}[\overline{B}_{s}(y)]$. Furthermore,
$\psi:C\rightarrow\overline{B}_{s}(y)$ is injective.
Proof: There exists an open subset $V$ of $X$, containing $C$, such that
$\psi:V\rightarrow\psi(V)$ is a homeomorphism, by 8. Hence,
$V\cap\psi^{-1}[\overline{B}_{s}(y)]=C$, and $C$ is both open and closed in
$\psi^{-1}[\overline{B}_{s}(y)]$.
Up to now we used only the assumption that $\psi$ is a local homeomorphism. We
now use the assumption that $\psi:C\rightarrow\overline{B}_{s}(y)$ is
surjective, for every connected component of $\psi^{-1}[\overline{B}_{s}(y)]$,
and in particular for $C:=\overline{\gamma[B_{s}(y)]}$. We conclude that $s$
belongs to $D_{x}$. A contradiction, since $D_{x}$ is open. This completes the
proof of 8.
Acknowledgements: This paper owes much to Eyal Markman, whose efforts to clean
up my arguments were invaluable. Many thanks to E. Loojienga and F. Bogomolov
for their remarks and interest. An early version of this paper was used as a
source of a mini-series of lectures at a conference “Holomorphically
symplectic varieties and moduli spaces”, in Lille, June 2-6, 2009. I am
grateful to the organizers for this opportunity and to the participants for
their insight and many useful comments. This paper was written at Institute of
Physics and Mathematics of the Universe (University of Tokyo), and I would
like to thank my colleagues and the staff at IPMU for their warmth and
hospitality. Many thanks to Eyal Markman for many excellent suggestions,
remarks and corrections to several earlier versions of the manuscript, and to
Brendan Hassett and Yuri Tschinkel for finding gaps in the argument.
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* [V1] Verbitsky M., Hyperkähler embeddings and holomorphic symplectic geometry II, GAFA 5 no. 1 (1995), 92-104, alg-geom/9403006.
* [V2] Verbitsky, M., Cohomology of compact hyperkähler manifolds, alg-geom electronic preprint 9501001, 89 pages, LaTeX.
* [V3] Verbitsky, M., Cohomology of compact hyperkähler manifolds and its applications, alg-geom electronic preprint 9511009, 12 pages, LaTeX, also published in: GAFA vol. 6 (4) pp. 601-612 (1996).
* [V4] Verbitsky, M., Mirror Symmetry for hyperkähler manifolds, Mirror symmetry, III (Montreal, PQ, 1995), 115–156, AMS/IP Stud. Adv. Math., 10, Amer. Math. Soc., Providence, RI, 1999, alg-geom/9512195.
* [V5] Verbitsky, M., Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface, Geom. Funct. Anal. 8 (1998), no. 4, 732–782, arXiv:alg-geom/9705004.
* [V6] Verbitsky, M., Parabolic nef currents on hyperkähler manifolds, 19 pages, arXiv:0907.4217.
* [Vi] Viehweg, E., Quasi-projective Moduli for Polarized Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 30, also available at http://www.uni-due.de/ mat903/books.html.
* [VGO] E. B. Vinberg, V. V. Gorbatsevich, and O. V. Shvartsman, Discrete Subgroups of Lie Groups, in “Lie Groups and Lie Algebras II”, Springer-Verlag, 2000.
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Misha Verbitsky
Higher School of Economics
Faculty of Mathematics, 7 Vavilova Str. Moscow, Russia,
verbit@mccme.ru
|
arxiv-papers
| 2009-08-28T11:09:36 |
2024-09-04T02:49:04.904275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/0908.4121"
}
|
0908.4234
|
# Classes of Exact Solutions to the Teukolsky Master Equation
P. P. Fiziev Department of Theoretical Physics, University of Sofia, Boulevard
5 James Bourchier, Sofia 1164, Bulgaria, E-mail: fiziev@phys.uni-sofia.bg, and
BLTF, JINR, Dubna, 141980 Moscow Region, Rusia, E-mail: fiziev@theor.jinr.ru
###### Abstract
The Teukolsky Master Equation is the basic tool for study of perturbations of
the Kerr metric in linear approximation. It admits separation of variables,
thus yielding the Teukolsky Radial Equation and the Teukolsky Angular
Equation. We present here a unified description of all classes of exact
solutions to these equations in terms of the confluent Heun functions. Large
classes of new exact solutions are found and classified with respect to their
characteristic properties. Special attention is paid to the polynomial
solutions which are singular ones and introduce collimated one-way-running
waves. It is shown that a proper linear combination of such solutions can
present bounded one-way-running waves. This type of waves may be suitable as
models of the observed astrophysical jets.
## 1 Introduction
At present the study of different types of perturbations of the gravitational
field of black holes, neutron stars and other compact astrophysical objects is
a very active field for analytical, numerical, experimental and astrophysical
research. Ongoing and nearest future experiments based on perturbative and/or
numerical analysis of relativistic gravitational dynamics are expected to
provide critical tests of the existing theories of gravity [1].
The study of perturbations of rotating relativistic objects in Einstein GR was
pioneered by Teukolsky [2] by making use of the famous Teukolsky Master
Equation (TME). It describes the perturbations
${}_{s}\Psi(t,r,\theta,\varphi)$ of all physically interesting spin-weights
$s=0,\pm 1/2,\pm 1,\pm 3/2,\pm 2$ to the Kerr background metric in terms of
the corresponding Newman-Penrose scalars. The pairs of spin-weights $s$ with
opposite signs $\sigma=\text{sign}(s)=\pm 1$ correspond to two different
perturbations with opposite helicity and spin $|s|=0,1/2,1,3/2$, or $2$. Under
proper boundary conditions for the TME one obtains quasi-normal modes (QNM) of
the Kerr black holes. Various significant results and additional references
can be found in [3]-[6].
The key feature of the TME is that in the Boyer-Lindquist coordinates one can
separate the variables using the ansatz $\Psi(t,r,\theta,\varphi)=e^{-i\omega
t}e^{im\varphi}S(\theta)R(r)$, i.e. looking for solutions in a specific
factorized form. Thus, one obtains a pair of two connected ordinary
differential equations for the nontrivial factors
${}_{s}S_{\omega,E,m}(\theta)$ and ${}_{s}R_{\omega,E,m}(r)$ – the Teukolsky
angular equation (TAE) [2, 3, 7]
$\displaystyle{\frac{1}{\sin\theta}}{\frac{d}{d\theta}}\left(\sin\theta{\frac{d}{d\theta}}\,{}_{s}S_{\omega,E,m}(\theta)\right)+{}_{s}W_{\omega,E,m}(\theta){}_{s}S_{\omega,E,m}(\theta)\\!=0,\hskip
62.59596pt$ (1.1a)
$\displaystyle{}_{s}W_{\omega,E,m}(\theta)=E+a^{2}\omega^{2}\cos^{2}\theta-2sa\omega\cos\theta-(m^{2}+s^{2}+2ms\cos\theta)/\sin^{2}\theta;$
(1.1b)
and the Teukolsky radial equation (TRE) [2, 3]
$\displaystyle{\Delta}^{-s}{\frac{d}{dr}}\left({\Delta}^{s+1}{\frac{d}{dr}}\,{}_{s}R_{\omega,E,m}(r)\right)+{}_{s}V_{\omega,E,m}(r)\,{}_{s}R_{\omega,E,m}(r)=0,\hskip
0.0pt$ (1.2a)
$\displaystyle{}_{s}V_{\omega,E,m}(r)={\frac{1}{\Delta}}{{K}^{2}}-is{\frac{1}{\Delta}}{\frac{d\Delta}{dr}}K-L.\hskip
56.9055pt$ (1.2b)
The azimuthal number may have values $m=0,\pm 1,\pm 2,\dots$ for integer spin,
or $m=\pm 1/2,\pm 3/2,\dots$ for half-integer spin [4]. In Eq. (1.2) we use
the expressions $\Delta\\!=\\!r^{2}\\!-\\!2Mr\\!+\\!a^{2}$,
$K\\!=\\!\omega(r^{2}\\!+\\!a^{2})\\!-\\!ma$,
$L\\!=\\!E\\!-\\!s(s\\!+\\!1)\\!+\\!a^{2}\omega^{2}\\!-\\!2ma\omega\\!-\\!4\,is\omega\,r$.
The real parameter $a\\!=\\!J/M$ is related with the angular momentum $J$ of
the Kerr metric, $M$ being its Keplerian mass. The two complex parameters
$\omega$ and $E$ – the constants of separation, are to be determined using the
boundary conditions of the problem.
The negativity of the imaginary part $\omega_{I}=\Im(\omega)<0$ of the complex
frequency $\omega=\omega_{R}+i\omega_{I}$ ensures linear stability of the
solutions in the exterior domain of the Kerr metric with respect to the future
time direction $t\to+\infty$ [2, 8]. In the interior domain the solutions to
the TME are not stable [9].
From a mathematical point of view the function
${}_{s}{\mathcal{K}}_{\omega,E,m}(t,r,\theta,\varphi)\sim e^{-i\omega
t}e^{im\varphi}\,{}_{s}S_{\omega,E,m}(\theta)\,{}_{s}R_{\omega,E,m}(r)$
actually defines a factorized kernel of the general integral representation
for the solutions to the TME:
${}_{s}\Psi(t,r,\theta,\varphi)={1\over
2\pi}\int\\!\\!d\omega\int\\!\\!dE\,\sum_{m}{}_{s}A_{\omega,E,m}\,e^{-i\omega
t}\,e^{im\varphi}\,{}_{s}S_{\omega,E,m}(\theta)\,{}_{s}R_{\omega,E,m}(r).$
(1.3)
The formal mathematical representation (1.3) is written ad hoc as the most
general superposition of all particular solutions. In it a summation on all
admissible values of the two separation constants $\omega$ and $E$ is assumed.
Its usefulness will be illustrated by different examples in what follows.
It is well known [10] that the Carter separation constant (which is equivalent
to the constant $E$, used here) may be related with the total angular momentum
of the solution ${}_{s}\Psi(t,r,\theta,\varphi)$. Under proper boundary
conditions for the TAE this momentum has discrete values defined by an
(half)integer $l$ [2]. If we are interested in superpositions
${}_{s}\Psi(t,r,\theta,\varphi)$ of solutions with a definite total angular
momentum, the integration with respect to the constant $E$ must be replaced
with summation over the (half)integer $l$. Thus, instead of the most general
linear mixture (1.3) we have to use the representation of the solutions
${}_{s}\Psi(t,r,\theta,\varphi)={1\over
2\pi}\int\\!\\!d\omega\sum_{l}\,\sum_{m}{}_{s}A_{\omega,l,m}\,e^{-i\omega
t}\,e^{im\varphi}\,{}_{s}S_{\omega,l,m}(\theta)\,{}_{s}R_{\omega,l,m}(r),$
(1.4)
introduced in the problem at hand for the first time in [2]. The transition
from the representation (1.3) to representation (1.5) is formally equivalent
to the use of a singular kernel proportional to the sum of Dirac
$\delta$-functions: $\sum_{l}\delta(E-{}_{s}E(\omega,l,m))$ in (1.3). Here
${}_{s}E(\omega,l,m)$ belongs to some spectrum which is specific for the given
problem and is completely defined by the corresponding boundary conditions,
see section 7.
Further on, the boundary conditions may fix some discrete spectrum for the
frequencies $\omega$ in (1.5). Then the integral on $\omega$ will be replaced
by discrete summation over some $\omega_{n}$. This is equivalent to the use
once more of a singular kernel, now proportional to
$\sum_{n}\delta(\omega-\omega_{n})$. As a result one obtains
${}_{s}\Psi(t,r,\theta,\varphi)=\sum_{n}\,\sum_{l}\,\sum_{m}{}_{s}A_{n,l,m}\,e^{-i\omega_{n}t}\,e^{im\varphi}\,{}_{s}S_{n,l,m}(\theta)\,{}_{s}R_{n,l,m}(r).$
(1.5)
The lack of a rigorous mathematical theory explaining the Teukolsky separation
is an old and well-known problem111The author is grateful to unknown referee
for this important remark.. The study of the QNM [6] not only illustrates the
above situation but also shows that the kernel
${}_{s}{\mathcal{K}}_{\omega,E,m}(t,r,\theta,\varphi)$ can be singular with
respect to the variable $r$ at infinity and at the horizons. In the existing
literature only regular with respect to the variable $\theta$ kernels
${}_{s}{\mathcal{K}}_{\omega,E,m}(t,r,\theta,\varphi)$ are in use. In the
present paper, we start the consideration of both regular and singular with
respect to the angle $\theta$ kernels in the integral representation (1.3) of
the solutions to the TME. Different types of kernels are to be used for
solution of different boundary problems. Note that from a physical point of
view the regularity of the very solution ${}_{s}\Psi(t,r,\theta,\varphi)$ in
equations (1.3), (1.4) and (1.5) is important. The kernels like
${}_{s}{\mathcal{K}}_{\omega,E,m}(t,r,\theta,\varphi)$ are auxiliary
mathematical objects. One is often forced to use singular kernels in the
natural integral representations of the solution to physical problems. The
regularity of the very physical solution ${}_{s}\Psi(t,r,\theta,\varphi)$ with
respect to the variable $\theta$ depends on the choice of the amplitudes
${}_{s}A_{\omega,E,m}$. It can be guaranteed by a suitable choice of these
amplitudes, as shown in section 10.
Despite the essential progress both in the numerical study [11] of the
solutions to equations (1.1a) and (1.2a) and in the investigation of their
analytical properties [12], at present there exists a number of basic
questions remaining unanswered. For example, it has been well known for a long
time [13] that the TAE (1.1a) and TRE (1.2a) can be reduced to the confluent
Heun ordinary differential equation [14] written here in the following
simplest uniform shape [17, 18]:
$\displaystyle
H^{\prime\prime}+\left(\alpha+{{\beta+1}\over{z}}+{{\gamma+1}\over{z-1}}\right)H^{\prime}+\left({\mu\over
z}+{\nu\over{z-1}}\right)H=0.$ (1.6)
The constants $\mu$ and $\nu$ in Eq. (1.6) are related with the constants
$\alpha,\beta,\gamma,\delta,\eta$, accepted in the notation
$\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ as follows:
$\displaystyle\delta=\mu+\nu-\alpha\,{\frac{\beta+\gamma+2}{2}},$ (1.7a)
$\displaystyle{\eta={\frac{\alpha(\beta+1)}{2}}-\mu-\frac{\beta+\gamma+\beta\gamma}{2}}.$
(1.7b)
To the best of our knowledge we still do not have a detailed description of
the exact analytical solutions to the TAE (1.1a) and TRE (1.2a) in terms of
the confluent Heun function $\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$
– the unique particular local solution of Eq. (1.6) which is regular in the
vicinity of the regular singular point $z\\!=\\!0$ and obeys the normalization
condition $\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,0)\\!=\\!1$ [14].222
In the present paper, we use the Maple-computer-package notation for the Heun
functions. Basically, this notation is borrowed from the two mile-stone papers
on modern theory of Heun’s functions by Decarreau et al. in [14]; at present
it seems to be most popular, since the Maple package is the only one for
analytical and numerical work with Heun’s functions.. Note that other
particular solutions to equation (1.6), as well as its general solution, are
not termed ”confluent Heun’s functions”, according to the accepted modern
terminology [14]. The reason is that, in general, other solutions can be
represented in a nontrivial way in terms of solutions
$\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ of the corresponding
arguments. Hence, from a computational point of view it is sufficient to study
only the Taylor series of this standard local solution and its analytical
continuation in the complex plane $\mathbb{C}_{z}$. Thus, the instrumental use
of the confluent Heun function
$\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ – the basic purpose of the
present paper, is much more advantageous than the simple fact, recognized
already in [13], that the TRE and TAE can be reduced to the confluent Heun
equation (1.6).
In the late 2006 a program for filling the above gaps in the study of the TME
was started as a natural extension of the papers [15], where a similar
approach was developed for the Regge-Wheeler equation (RWE). The first results
were quite stimulating [16], but serious difficulties came across in both
analytical and numerical studies. This is because the theory of Heun’s
functions, as well as numerical tools for calculations with them still are not
developed enough.
Here we pay special attention to the polynomial solutions of Eq. (1.6).
According to [14], the confluent Heun function
$\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ reduces to a polynomial of
degree $N\geq 0$ of the variable $z$, if and only if the following two
conditions are satisfied:
$\displaystyle{\frac{\delta}{\alpha}}+{\frac{\beta+\gamma}{2}}+N+1=0,$ (1.8a)
$\displaystyle\Delta_{N+1}(\mu)=0.$ (1.8b)
We call the first condition (1.8a) a ”$\delta_{N}$-condition”, and the second
one (1.8b) – a ”$\Delta_{N+1}$-condition”. An explicit form of the
”$\Delta_{N+1}$-condition” in form of a determinant useful for practical
calculations, as well as a novel derivation of confluent Heun’s polynomials
can be found in [17]. A recurrent procedure for calculation of $\Delta_{N+1}$
(1.8b) and its relation with Starobinsky’s constants are presented in [18].
On the other hand, the so-called algebraically special solutions to the RWE
and the TRE were discovered long time ago [19]. These are of a generalized
polynomial type, i.e. products of polynomials and simple non-polynomial
factors which are elementary functions. According to the existing literature,
these solutions describe pure incoming or pure outgoing waves. The
algebraically special solutions still are not discussed in terms of Heun’s
polynomials. To our knowledge, attempts for application of this class of
solutions to real physical problems cannot be found in the existing literature
on gravitational physics.
The only exception are the recent papers [16, 20], where one can find some
preliminary results. There, special polynomial solutions of the TAE were
considered in more detail. In particular, in the first two of the articles
[20] it was demonstrated that these singular with respect to the angular
variable $\theta$ solutions can describe collimated waves which resemble the
observed astrophysical jets. In the third of these papers the spectrum of
electromagnetic jets from Kerr black holes and naked singularities in the
Teukolsky perturbation theory was calculated for the first time, using some of
the basic results of the present paper.
Very recently the algebraically special solutions of the RWE and the TME were
proved to be relevant for the study of instabilities of different kind of some
more or less ”exotic” solutions to the Einstein equations [21]. Physical
manifestation of the instabilities of the mathematical solutions are the
explosions of the corresponding objects. Therefore, it seems natural to look
for a perturbative description of explosions in terms of solutions of the TME,
which are stable in the future and instable in the past. The confluent Heun
functions give a rigorous mathematical basis for analysis of these problems.
On the other hand, the recently found properties of confluent Heun’s function
[17] show that one can introduce a new subclass of ”$\delta_{N}$-confluent-
Heun’s-functions”, which obey only the $\delta_{N}$-condition – Eq. (1.8a). In
[18] is shown that such ”$\delta_{N}$-solutions” of the TRE and the TAE define
the most general class of solutions, for which properly generalized Teukolsky-
Starobinsky’s identities exist. Moreover, this approach reveals the existence
of Teukolsky-Starobinsky’s type of identities for Regge-Wheeler and Zirilli
equations, as well. Here we study in more detail the $\delta_{N}$-solutions to
the TRE and TAE. In particular, we show that the regular solutions to the TAE,
which are the only class of solutions to the TAE, used up to now [2, 3, 7],
are precisely nonpolynomial $\delta_{N}$-solutions. In contrast, the
polynomial solutions to the TAE of all spins are shown to be singular around
one of the poles ($\theta=0$, or $\theta=\pi$) of the unit sphere
$\mathbb{S}^{\,(2)}_{\theta,\varphi}$ and regular around the other one. This
new situation reflects the specific properties of the confluent Heun function.
It is not consistent with our experience, based on the work with
hypergeometric functions, solving the angular part of the Laplace equation in
celestial and quantum mechanics, or in electrostatics. It is well known that
in the last case solutions regular in the interval $\theta\in[0,\pi]$
(including both the poles) are polynomial.
In the limit $a\to 0$, when the Kerr metric approaches the non-rotating
Schwarzschild one, there exist a smooth transition from perturbations of the
Kerr metric to perturbations of the Schwarzschild metric in terms of the Weyl
scalars, but a simple transition from the solutions of the TME to the
solutions of the RWE (see [18, 15]) is not possible [3]. Nevertheless, the
mathematical analogy between the corresponding solutions becomes quite
transparent when the solutions are represented in terms of the confluent Heun
functions [18, 15]. The limit $a\to 0$ is traced in more detail in section
4.1.2 – for the TRE and in section 7 – for the TAE.
This way, using confluent Heun’s function, we hope to obtain a more clear
picture of the quite complicated present-day state of the arts in the
perturbation theory under consideration and its possible further developments
and new physical applications.
The main purposes of the present paper is to report some of the basic results,
obtained for a detailed description of the exact solutions of the TME in terms
of the confluent Heun function
$\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$, to introduce a large number
of new classes of such solutions, and to formulate some interesting boundary
problems for the TRE, TAE, and TME in terms of confluent Heun’s functions.
Besides the already stressed new developments, in the present paper for the
first time we introduce and study the differential invariants of the Weyl
tensor, which indicate in an invariant way both the event and Cauchy horizons
of the Kerr metric as singular points of the TRE (section 2.1), the explicit
form of 16 classes of exact solutions to the TRE in terms of confluent Heun’s
functions (section 2.2), a new classification of the solution to the TRE,
based on specific properties of confluent Heun’s functions (section 3),
especially, the class of $\delta_{N}$-radial solutions and, in particular, two
unknown infinite classes of exact solutions with equidistant complex spectra
of frequencies, two novel classes of polynomial solutions to the TRE (section
4), the explicit form of 16 classes of exact solutions to the TAE in terms of
confluent Heun’s functions (section 5), a new concomitant confluent Heun’s
function and its application to the TAE (section 5), a new classification of
the solution to the TAE, based on specific properties of confluent Heun’s
functions (section 6), especially, the class of $\delta_{N}$-angular
solutions, a novel description of the regular solutions to the TAE in terms of
confluent Heun’s functions (section 7), two classes of singular polynomial
solutions to the TAE (section 8), 256 classes of exact factorized solutions to
the TME (section 9), an explicit construction of exact bounded solutions to
the TME with spin $1/2$, using the singular kernel, built from the polynomial
solutions to the TAE (section 10), and novel general exact solutions of the
TME in the form of one-way running waves (section 10). It seems natural from
physical point of view to use these solutions for study of the relativistic
jets. Some other general conclusions and perspectives for further developments
are outlined in the concluding section 11.
## 2 Exact Solutions to the Teukolsky Radial Equation in Terms of the
Confluent Heun functions
### 2.1 Explicit form of the TRE and Geometrical Character of its
Singularities
Much like in the case of the Schwarzschild solution, for the Kerr one we have
a complicated space-time structure and a different physical meaning of the
space-time coordinates in the different domains. For example, consider, as
usual, only the real values of $r$. In the interior of the Kerr metric: $0\leq
r_{-}<r<r_{+}$ – between the zeros $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}},a\leq M$ of
the function $\Delta$ (i.e., between the Cauchy horizon $r_{-}$ and the event
horizon $r_{+}$), two of the eigenvalues: $\lambda_{t}$ and $\lambda_{r}$ of
the metric in the Boyer-Lindquist coordinates simultaneously change their
signs. Indeed, one pair of eigenvalues is
$\lambda_{\theta}=g_{\theta\theta}=r^{2}+a^{2}\cos^{2}\theta$ and
$\lambda_{r}=g_{rr}=(r^{2}+a^{2}\cos^{2}\theta)/\Delta$. The second pair of
eigenvalues is the roots $\lambda_{t},\lambda_{\phi}$ of the equation
$\lambda^{2}-(g_{tt}+g_{\phi\phi})\lambda+g_{tt}g_{\phi\phi}-g_{t\phi}^{2}$.
Their product equals $\lambda_{t}\lambda_{\phi}=-\Delta\sin^{2}\theta$. The
last expression, together with the form of $g_{rr}$ proves the simultaneous
change of the signs of the two eigenvalues $\lambda_{t},\lambda_{r}$, when the
variable $r$ crosses the horizons $r_{\pm}$, since the determinant
$g=-(r^{2}+a^{2}\cos^{2}\theta)^{2}\sin^{2}\theta$ of the metric does not
vanish there. As a result, between the two horizons $r_{\pm}$ the variable
$t_{in}=x\in(-\infty,\infty)$ plays the role of the interior time and the
variable $r_{in}=t$ is the interior radial variable. We use the following
Kerr-metric-tortoise-coordinate:
$r_{\\!*}=r+a_{+}\ln|(r-r_{+})/(r_{+}-r_{-})|-a_{-}\ln|(r-r_{-})/(r_{+}-r_{-})|\in(-\infty,\infty)$,
where $a_{\pm}={\frac{r_{+}+r_{-}}{r_{+}-r_{-}}}r_{\pm}$. It is a
straightforward generalization of the tortoise variable for the exterior
domain $r\in(r_{+},\infty)$ proposed in [2]. Since our expression is valid in
the interior domains, too, the inverse function defines $r=r(t_{in})$ when
$r\in(r_{-},r_{+})$. In the second interior domain $r<r_{-}$ the variables $r$
and $t$ restore their original meaning. For a detailed analysis of the light
cones in the Kerr geometry see [22]. This consideration is necessary for
understanding of the physical meaning of the solutions to the TME in the
different Kerr-space-time domains.
The explicit form of the TRE
$\displaystyle{\frac{d^{2}R_{\omega,E,m}}{d{r}^{2}}}+(1+s)\left({\frac{1}{r-{\it
r_{+}}}}+{\frac{1}{r-{\it r_{-}}}}\right){\frac{dR_{\omega,E,m}}{dr}}+$
$\displaystyle+\left({\frac{\Big{(}\omega\,\left({a}^{2}+{r}^{2}\right)-am\Big{)}^{2}}{\left(r-r_{+}\right)\left(r-r_{-}\right)}}-is\left({\frac{1}{r-{\it
r_{+}}}}+{\frac{1}{r-{\it
r_{-}}}}\right)\Big{(}\omega\,\left({a}^{2}+{r}^{2}\right)-am\Big{)}-\right.$
$\displaystyle\left.-E+s(s+1)-{a}^{2}{\omega}^{2}+2m\,a\omega+4\,is\omega
r\vphantom{\frac{\Big{(}a^{2}\Big{)}}{\big{(}a^{2}\big{)}}}\\!\right){\frac{R_{\omega,E,m}}{(r-r_{+})(r-r_{-})}}=0$
(2.1)
shows that it has three singular points: $r=r_{\pm}$ and $r=\infty$. In the
present paper, we consider only the non-extremal Kerr metric with real
$r_{+}>r_{-}\geq 0$. Then the first two are regular singular points, and the
third one (the physical infinity $r=\infty$) is an irregular singular point.
The symmetry of Eq. (2.1) under the interchange $r_{+}\\!\leftrightarrows
r_{-}$ is obvious. Thus, we see that the two horizons of the Kerr metric are
singularities for the TRE which are to be treated on equal footing. Do these
singularities have an invariant meaning independent of the coordinate choice?
It is well known that the algebraic invariants of the Riemann curvature tensor
${\cal R}_{ijkl}$ are not able to indicate the horizons of the Kerr black hole
and one usually considers them as pure coordinate singularities of the metric
in the Boyer-Lindquist coordinates. In contrast, the circle $r=0,\theta=\pi/2$
is a singularity of the algebraic invariants of the Riemann tensor [3]. Since
the pure algebraic invariants of the tensor ${\cal R}_{ijkl}$ do not fix
completely the geometry, their consideration is not sufficient to recover all
gauge-invariant space-time properties. For this purpose one must consider a
large enough number of high-order-differential-invariants of the Riemann
tensor [23].
It is not difficult to find differential invariants of the Riemann tensor of
the Kerr metric which are able to distinguish both the horizons $r_{\pm}$ and
the ergo-surface $g_{tt}=0$. Indeed, let us consider the following algebraic
invariants of the Weyl tensor ${\cal W}_{ijkl}$: $I_{1}={\frac{1}{48}}{\cal
W}_{ijkl}{\cal W}^{ijkl}$ – the density of the Euler characteristic class, and
$I_{2}={\frac{1}{48}}{\cal W}_{ijkl}\,{}^{*}{\cal W}^{ijkl}$ – the density of
the Chern-Pontryagin characteristic class [24]. Let us put
$(I_{1}-iI_{2})^{1/2}=\lambda=|\lambda|\exp(i\psi)$. Then
$r=\left({\frac{M}{|\lambda|}}\right)^{1/6}\cos(\psi/6)$ and
$\rho=\left({\frac{M}{|\lambda|}}\right)^{1/6}\cos(\psi/6)^{-1}$ are obviously
invariants of the Weyl tensor – nonalgebraic and nondifferential ones. In the
Boyer-Lindquist coordinates one obtains $\rho=r+{\frac{a^{2}}{r}}\cos\theta$
and $g_{tt}=1-2M/\rho$. The differential invariants of first order
$\displaystyle DI_{1}=-\left(\nabla\ln
r\right)^{2}={\frac{1}{r\rho}}\left(1-{\frac{2M}{r}}+{\frac{a^{2}}{r^{2}}}\right),$
(2.2a) $\displaystyle DI_{2}=\left(\nabla\ln\rho\right)^{2}-\left(\nabla\ln
r\right)^{2}={\frac{4}{\rho^{2}}}\left({\frac{\rho}{r}}-1\right)\left(1-{\frac{2M}{\rho}}\right)$
(2.2b)
indicate the two Kerr black hole horizons, the ergo-surface and some other
geometrical objects in the Kerr space-time. Thus, the two horizons $r_{\pm}$
of the Kerr metric are shown to be invariant objects, being singularities of
the same kind in equation (2.1).
In the limit $a\to 0$ we have $\rho\to r$ and the differential invariant in
equation (2.2b) becomes trivial: $DI_{2}\to 0$. In the same limit, the
differential invariant (2.2a) produces a nontrivial result
$\left(1-2M/r\right)/r^{2}$ for the Schwarzschild metric which is similar to
the one derived already in the third of the papers [23], as well as in the
very recent seventh one. Differential invariants similar to (2.2) are
considered in the sixth of the references [23] without any application.
### 2.2 Explicit Form of the Local Solutions to the TRE
The analytical study of the solutions to the TRE and TAE was started in [5]
and continued by different approximate methods [6, 12] without utilizing of
Heun’s functions. Using the confluent Heun function one can write down 16
exact local Frobenius type solutions to the TRE (2.1) in the form:
$\displaystyle{}_{s}R^{\pm}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})\Delta^{s/2}\\!=\\!e^{\sigma_{\alpha}{\frac{\alpha_{{}_{\pm}}z_{{}_{\pm}}}{2}}}z_{{}_{\pm}}^{\sigma_{\beta}{\frac{\beta_{{}_{\pm}}}{2}}}z_{{}_{\mp}}^{\sigma_{\gamma}{\frac{\gamma_{{}_{\pm}}}{2}}}\text{HeunC}({\sigma_{\alpha}\alpha_{{}_{\pm}},\sigma_{\beta}\beta_{{}_{\pm}},\sigma_{\gamma}\gamma_{{}_{\pm}},\delta_{{}_{\pm}},\eta_{{}_{\pm}},z_{{}_{\pm}}}),$
(2.3)
which is very similar to the form of the solutions to the RWE [18, 15].
Here333Note that the notation $z_{\pm}$ in Eq. (2.4f) is consistent with the
limits $z_{\pm}\to\pm\infty$ for $r\to\infty$. Their relation with the
notation of the parameters of the Kerr metric $r_{\pm}$ is illustrated by the
equations $z_{\pm}(r_{\mp};r_{+},r_{-})=0$. The labels $\pm$ in the notation
$R^{\pm}$ in Eq. (2.3) are related with the labels of their arguments
$z_{\pm}$, not with the labels of the parameters $r_{\pm}$.
$\displaystyle\alpha_{{}_{+}}\\!$
$\displaystyle=\\!{}_{s}\alpha_{\omega,E,m}(r_{+},r_{-})\\!=\\!2i\omega(r_{+}-r_{-})\\!=i2p\,{{\omega}/{\Omega}_{a}},$
(2.4a) $\displaystyle\beta_{{}_{+}}$
$\displaystyle=\\!{}_{s}\beta_{\omega,E,m}(r_{+},r_{-})\\!=s+i\left(m-\omega/\Omega_{-}\right)/p,$
(2.4b) $\displaystyle\gamma_{{}_{+}}$
$\displaystyle=\\!{}_{s}\gamma_{\omega,E,m}(r_{+},r_{-})\\!=s-i\left(m-\omega/\Omega_{+}\right)/p,$
(2.4c) $\displaystyle\delta_{{}_{+}}$
$\displaystyle=\\!{}_{s}\delta_{\omega,E,m}(r_{+},r_{-})\\!=\alpha_{+}\left(s-i\omega(r_{+}+r_{-})\right)\\!=\alpha_{+}\left(s-i\omega/\Omega_{g}\right),$
(2.4d) $\displaystyle\eta_{{}_{+}}\\!$
$\displaystyle=\\!{}_{s}\eta_{\omega,E,m}(r_{+},r_{-})\\!=\\!-E+s^{2}\\!+m^{2}\\!+{\frac{2m^{2}\Omega_{a}^{2}-\omega^{2}}{4p^{2}\Omega_{a}^{2}}}-{\frac{(2m\Omega_{a}-\omega)^{2}}{4p^{2}\Omega_{g}^{2}}}-{\frac{1}{2}}\left({s\\!-\\!i\,\frac{\omega\,\Omega_{+}}{\Omega_{a}\Omega_{g}}}\right)^{2};$
(2.4e) $\displaystyle z_{+}\\!$
$\displaystyle=\\!z_{+}(r;r_{+},r_{-})\\!=\\!{\frac{r\\!-\\!r_{-}}{r_{+}\\!-\\!r_{-}}},\,z_{-}\\!=\\!z_{-}(r;r_{+},r_{-})\\!=\\!{\frac{r_{+}\\!-\\!r}{r_{+}\\!-\\!r_{-}}};\,z_{+}\\!+\\!z_{-}\\!=\\!1,\,z_{+}z_{-}\\!=\\!{\frac{-\Delta}{(r_{+}\\!-\\!r_{-})^{2}}}.$
(2.4f)
The discrete parameters $\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}$ have
values $\pm 1$444Further on, $\sigma_{x}\\!=\\!\text{sign}(x)$ denotes the
sign of the real quantity $x$. The only exception is
$\sigma\\!\equiv\\!\sigma_{s}$ where we skip the index $s$.. In equations
(2.4a)-(2.4e) we use the following quantities: the angular velocity of the
event horizon
$\Omega_{+}=a/2Mr_{+}={\sqrt{r_{-}/r_{+}}}\big{/}\left(r_{+}+r_{-}\right)$,
the angular velocity of the Cauchy horizon
$\Omega_{-}=a/2Mr_{-}={\sqrt{r_{+}/r_{-}}}\big{/}\left(r_{+}+r_{-}\right)$,
the arithmetically-averaged angular velocity
$\Omega_{a}=\left(\Omega_{+}+\Omega_{-}\right)/2=1/(2a)$, the geometrically-
averaged angular velocity $\Omega_{g}=\sqrt{\Omega_{+}\Omega_{-}}=1/(2M)$, and
the new dimensionless parameter
$\displaystyle
p\\!=\\!{\frac{1}{2}}\left(\sqrt{r_{+}/r_{-}}-\sqrt{r_{-}/r_{+}}\right)\\!=\\!{\frac{1}{2}}\left(\sqrt{\Omega_{-}/\Omega_{+}}-\sqrt{\Omega_{+}/\Omega_{-}}\right)\\!=\\!\sqrt{M^{2}/a^{2}\\!-\\!1}.$
(2.5)
Note that the inverse relation
$r_{\pm}={\sqrt{\Omega_{\mp}/\Omega_{\pm}}}\big{/}\left(\Omega_{+}+\Omega_{-}\right)$
permits us to replace $r_{\pm}$ with $\Omega_{\pm}$ wherever it is necessary,
thus making transparent the duality of the parameters $r_{\pm}$ and
$\Omega_{\pm}$, as well as the behavior of the above quantities under
interchange of the two horizons: $r_{+}\leftrightarrows r_{-}$ $\Rightarrow$
$\Omega_{+}\leftrightarrows\Omega_{-}$, $p\mapsto-p$,
$\Omega_{a,g}\mapsto\Omega_{a,g}$ – invariant.
The parameters
$\alpha_{{}_{-}},\beta_{{}_{-}},\gamma_{{}_{-}},\delta_{{}_{-}},\eta_{{}_{-}}$
can be obtained by interchanging the places of the two horizons:
$r_{+}\\!\leftrightarrows r_{-}$ in (2.4a) – (2.4e). This procedure may be
substantiated using the known properties of the confluent Heun function under
changes of parameters [14]. One can check directly that this way we obtain
indeed solutions of equation (2.1).
According to equations (2.3) and (2.4f), the behavior of the solutions
${}_{s}R^{\pm}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})$
around the corresponding singular points $z=z_{\pm}(r_{\mp};r_{+},r_{-})=0$ is
defined by the dominant factor
$\left(z_{{}_{\pm}}\right)^{\sigma_{\beta}\beta_{\pm}/2}$. All other factors
in equation (2.3) are regular around these points. The same solutions are in
general singular around the corresponding singular points
$z=z_{\pm}(r_{\pm};r_{+},r_{-})=1$.
Only two of the sixteen solutions (2.3) are linearly independent.
Nevertheless, it is necessary to know all of them since for different purposes
one has to use different pairs of independent local solutions.
Using the known asymptotic expansion of the confluent Heun function [14] we
obtain two asymptotic solutions of Tomè type. These are local solutions of the
TRE around its irregular singular point $|r|=\infty$ in the complex plane
$\mathbb{C}_{r}$:
$\displaystyle{}_{s}R_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}^{\pm\infty}(r;r_{+},r_{-})\sim{{e^{i\sigma_{\\!\alpha}\,\omega\big{(}r+(r_{+}+r_{-})\ln
r\big{)}}}}\sum_{j\geq
0}a_{j}\left(\pm{{r_{+}-r_{-}}\over{r}}\right)^{j+1+(1+\sigma_{\alpha})s},\,\,\,\,a_{0}=1.$
(2.6)
The notation $\pm\infty$ in (2.6) denotes the two directions: $r\to+\infty$
and $r\to-\infty$ on the real $r$-axis for approaching the irregular singular
point $|r|=\infty$ in the complex plane $\mathbb{C}_{r}$. For the coefficients
$a_{j}=a_{j,\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$ one
has a recurrence relation [14] which shows that they increase together with
the integer $j$. Hence, the asymptotic series (2.6) is a divergent one.
As seen from (2.4),
${}_{s}R^{-}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})={}_{s}R^{+}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{-},r_{+})$.
Hence, one can introduce a new parity property of the solutions and construct
symmetric and anti-symmetric (with respect to the interchange
$r_{+}\rightleftarrows r_{-}$) solutions of the TRE:
$\displaystyle{}_{s}R^{SYM}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})={\frac{1}{2}}\left({}_{s}R^{+}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})+{}_{s}R^{-}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})\right),$
(2.7)
$\displaystyle{}_{s}R^{ASYM}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})={\frac{1}{2}}\left({}_{s}R^{+}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})-{}_{s}R^{-}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-})\right).$
Clearly, these solutions are singular at both horizons in the general case.
When one considers the two-singular-point boundary problem [14] on the
interval $[r_{-},r_{+}]$ in the Kerr black hole interior, the solutions (2.7)
may be regular at one, or at both the ends for some values of the separation
constants $\omega$ and $E$. Since this boundary problem is still not studied,
at present we are not able to make more definite statements about this case.
## 3 A New Classification of the Solutions to the TRE, Based on the
$\delta_{N}$-Condition. Novel Radial $\delta_{N}$-Solutions
For the TRE the $\delta_{N}$-condition reads:
$\displaystyle{}_{s}\omega^{\pm}_{m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}\,{\cal
L}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}=\Omega_{g}\left({\cal
M}^{\pm}_{m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}+i\,{}_{s}{\cal
N}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}\right),$ (3.1)
where
${\cal
L}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}\\!=\\!{\frac{\sigma_{\beta}\Omega_{\pm}\\!-\\!\sigma_{\gamma}\Omega_{\mp}}{\Omega_{\pm}\\!-\\!\Omega_{\mp}}}\\!-\\!\sigma_{\alpha},\,\,{\cal
M}^{\pm}_{m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}\\!=\\!m(\sigma_{\beta}\\!-\\!\sigma_{\gamma}){\frac{\Omega_{g}}{\Omega_{\pm}\\!-\\!\Omega_{\mp}}},\,\,{}_{s}{\cal
N}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}\\!=\\!N\\!+\\!1\\!+\\!\left(\sigma_{\alpha}\\!+\\!{\frac{\sigma_{\beta}\\!+\\!\sigma_{\gamma}}{2}}\right)s.$
We call radial $\delta_{N}$-solutions the solutions defined via the
$\delta_{N}$-condition (3.1).
The calculation of the values of the coefficients in equation (3.1) yields two
very different cases:
1\. In the first case ${\cal L}^{+}_{\pm,\pm,\pm}={\cal
L}^{-}_{\pm,\pm,\pm}=0$ and we see that one is not able to fix the frequencies
${}_{s}\omega^{+}_{m,\pm,\pm,\pm}$ and ${}_{s}\omega^{-}_{m,\pm,\pm,\pm}$.
Instead, choosing $\sigma_{\alpha}=\sigma_{\beta}=\sigma_{\gamma}=-\sigma$ and
using (3.1) one fixes the non-negative integer $N$ in the form
$\displaystyle{}_{s}N+1=2|s|\geq 1\,\,\,\text{for}\,\,\,|s|\geq 1/2.$ (3.2)
Thus, the degree of the polynomial $\Delta_{N+1}$-condition is fixed, too.
2\. In the second case the coefficients ${\cal
L}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$ are nonzero and one
can fix the values of the frequencies
${}_{s}\omega^{\pm}_{m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$ from
equation (3.1). Thus, one obtains two different types of exact equidistant
spectra:
a) For ${\cal L}^{+}_{\mp,\pm,\pm}={\cal L}^{-}_{\mp,\pm,\pm}=\pm 2$, ${\cal
M}^{+}_{\mp,\pm,\pm}={\cal M}^{-}_{\mp,\pm,\pm}=0$ and ${\cal
N}^{+}_{\mp,\pm,\pm}={\cal N}^{-}_{\mp,\pm,\pm}=(N+1)$ the
$\delta_{N}$-condition (3.1) produces the pure imaginary equidistant
frequencies with $N\geq 0$ – integer:
$\displaystyle{}_{s}\omega^{+}_{N,m,\mp,\pm,\pm}={}_{s}\omega^{-}_{N,m,\mp,\pm,\pm}=\pm
i\,{(N+1)/{4M}}=\pm i\,\Omega_{g}(N+1)/2.$ (3.3)
Note that these frequencies depend neither on the spin-weight $s$ and
azimuthal number $m$, nor on the rotation parameter $a$. The spectrum is not
influenced by the rotation of the waves and of the very Kerr metric. The
frequencies (3.3) are defined only by the monopole term in multipole expansion
of the metric.
b) For all other cases the coefficients ${\cal
L}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$, and ${\cal
M}^{\pm}_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$ are not fixed and
one obtains the following two similar double-equidistant spectra of
frequencies with $N\geq 0\text{ -- integer}$ and $m\,\text{ --
(half)integer}$:
$\displaystyle{}_{s}\omega^{+}_{N,m,\mp,\mp,\pm}={}_{s}\omega^{-}_{N,m,\mp,\pm,\mp}=\Omega_{+}\left(m\pm
ip(N+1\mp s)\right);$ (3.4a)
$\displaystyle{}_{s}\omega^{+}_{N,m,\pm,\mp,\pm}={}_{s}\omega^{-}_{N,m,\pm,\pm,\mp}=\Omega_{-}\left(m\pm
ip(N+1\pm s)\right).$ (3.4b)
A set of important new mathematical properties of the radial
$\delta_{N}$-solutions can be found in [17, 18]. In [18] it is shown that
these solutions define the most general class of solutions to the TRE for
which the properly generalized Teukolsky-Starobinsky identities exist. The
solutions which satisfy the relation (3.2) were studied in [2, 3] without
utilizing the Heun functions and the $\delta_{N}$-condition. The last
condition turns to be valid automatically for the solutions to the TRE studied
in [2, 3]. The infinite series of the solutions with equidistant spectra (3.3)
and (3.4) are introduced and considered for the first time in the present
paper. In the third of the papers [20] one can find an interesting and
unexpected recent application of the formulas (3.4) for fitting of the spectra
of electromagnetic jets from Kerr black holes and necked singularities.
## 4 Polynomial Solutions to the TRE
The $\delta_{N}$-condition yields the basic classification of the solutions
described in the previous section 3. As a result, one obtains two classes of
polynomial solutions to the TRE, imposing in addition the
$\Delta_{N+1}$-condition (1.8b). In what follows we will use the determinant
form of the $\Delta_{N+1}$-condition given in [17].
### 4.1 The First Class of Polynomial Solutions to the TRE:
The solutions of this class correspond to the first case in section 3 and obey
equation (3.2). The inequality ${}_{s}N=2|s|-1\geq 0$ excludes the existence
of scalar perturbations ($|s|=0$) of the first polynomial class.
#### 4.1.1 The General Case:
For brevity, we denote the solutions
${}_{s}R^{\pm}_{\omega,E,m,-\sigma,-\sigma,-\sigma}(r;r_{+},r_{-})$ as
${}_{s}R^{\pm}_{\omega,E,m}(r;r_{+},r_{-})$. For them the parameter $\mu$
takes the values
$\mu={}_{s}\mu_{\omega,k,m}^{\pm}(r_{+},r_{-}),\,\,\,k=1,\dots,2|s|$ – the
solutions of the algebraic equation (1.8b), which now takes the form:
$\Delta^{\pm}_{2|s|}(\mu)=0$. Its degree is $2|s|=1,2,3$, or $4$, depending on
the spin of the perturbations $|s|=1/2,1,3/2,2$. Making use of (1.7b), and
(2.4a)-(2.4e), we obtain for the separation constant
$E={}_{s}E_{\omega,k,m}^{\pm}(r_{+},r_{-})$, $k\\!=\\!1,\dots,2|s|$ the
expressions
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}(r_{+},r_{-})\\!=\\!{}_{s}\mu_{\omega,k,m}^{\pm}(r_{+},r_{-})+|s|(|s|-1)-a\omega(a\omega-2m)+2i\sigma(2|s|-1)\omega
r_{\mp},$ (4.1)
Applying the explicit expressions for the roots
${}_{s}\mu_{\omega,k,m}^{\pm}(r_{+},r_{-})$, we obtain:
$\displaystyle{}_{s}E_{\omega,m}^{\pm}(r_{+},r_{-})\\!=-a^{2}\omega^{2}+2a\omega
m-{\frac{1}{4}}:\,\,\,\text{for}\,\,\,|s|={\frac{1}{2}},\,m=\pm 1/2,\pm
3/2,\dots;$ (4.2)
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}(r_{+},r_{-})\\!=-a^{2}\omega^{2}+2a\omega\left(m-(-1)^{k}\sqrt{1-m/a\omega}\right)\\!:\,\,\text{for}\,\,\,k=1,2,\,|s|=1,\,\,m=\pm
1,\pm 2,\dots$ (4.3)
For the gravitational waves ($|s|=2$) one has to find the quantities
${}_{s}\mu_{\omega,k,m}^{\pm}(r_{+},r_{-})$ solving algebraic equation of the
fourth degree $\Delta^{\pm}_{4}(\mu)=0$. The explicit form of its roots is too
complicated and not necessary for the purposes of the present paper. It is
more instructive to demonstrate here the result, obtained using the Taylor
series expansion of the solutions ${}_{s}\mu_{\omega,k,m}^{\pm}(r_{+},r_{-})$
around the zero frequency $\omega=0$.
Thus, we obtain for $|s|=2,\,\,k=1,2$, and $m=\pm 2,\pm 3,\dots$ the eight
series of values:
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}\\!=\\!2\\!-\\!4\left(m\\!-\\!i(-1)^{k}{\frac{3M}{2a}}\right)a\omega\\!+\\!6\\!\left(\\!m^{2}\\!+\\!i(-1)^{k}2m\left((m^{2}\\!-\\!1){\frac{a}{M}}\\!+\\!{\frac{2M}{a}}\right)\\!+\\!{\frac{3M^{2}}{a^{2}}}\\!-\\!{\frac{7}{6}}\right)(a\omega)^{2}\\!+$
(4.4) $\displaystyle+{\cal{O}}_{3}(a\omega).$
For $|s|=2,\,\,k=3,4$, and $m=\pm 2,\pm 3,\dots$ we have other eight series of
values:
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}=i\,(-1)^{k}4\sqrt{ma\omega}\Bigg{(}1+i\,3\left(1+\left({\frac{3M^{2}}{8a^{2}}}-{\frac{2}{3}}\right){\frac{1}{m^{2}}}\right)ma\omega+\\!{\cal{O}}_{2}(a\omega)\Bigg{)}+\hskip
76.82234pt$
$\displaystyle+8ma\omega-6\left(1+\left({\frac{3M^{2}}{a^{2}}}-{\frac{5}{6}}\right){\frac{1}{m^{2}}}\right)(ma\omega)^{2}+\\!{\cal{O}}_{3}(a\omega).\hskip
14.22636pt$ (4.5)
Clearly, these series describe two kinds of solutions with a completely
different behavior around the origin $\omega=0$. In particular, the series
(4.4) and (4.1.1) have different limits: $2$ and $0$, respectively, when
$\omega\to 0$. For the solutions (4.1.1) the origin $\omega=0$ is a branching
point, etc.
The independence of the values of ${}_{s}E_{\omega,k,m}^{\pm}$ in (4.4) and
(4.1.1) on the upper labels $(\pm)$ is a result of the polynomial character of
the solutions, i.e. of the regularity of the corresponding HeunC-factor
simultaneously on both the horizons $r_{\pm}$.
For a complete solution of the problem one has to determine the frequency
$\omega$. Hence, one needs an additional relation between the parameters $E$
and $\omega$. This relation may appear when one solves the TAE (See the next
sections 5-8.).
The first class of polynomial solutions to the TRE is introduced and studied
in detail for the first time in the present paper.
#### 4.1.2 The Special Case of the Schwarzschild metric:
For the special value of the parameter $a=0$ we have $r_{-}=0$, $r_{+}=2M$.
This is the case of perturbations to the nonrotating Schwarzschild black hole
described in terms of the Weyl scalars. For simplicity, here we use units in
which $2M=1$. The parameters in the solution (2.3) acquire the limiting values
$\displaystyle\alpha_{+}=2i\omega,\beta_{+}=s,\gamma_{+}=s+2i\omega,\delta_{+}=2i\omega(s-i\omega),\eta_{+}=-E+{\frac{s^{2}}{2}};\hskip
36.98866pt$
$\displaystyle\alpha_{-}=-2i\omega,\beta_{-}=s+2i\omega,\gamma_{-}=s,\delta_{-}=-2i\omega(s-i\omega),\eta_{-}=-E+{\frac{s^{2}}{2}}+2\omega^{2}+2is\omega.$
(4.6)
These differ from the values of the parameters of confluent Heun’s functions
in the Regge-Wheeler approach to the perturbations of the Schwarzschild metric
[15].
In the limit $a\to 0$ equation (3.1) does not define the frequency $\omega$,
if $\sigma_{\alpha}=\mp\sigma_{\beta}=\pm\sigma_{\gamma}=-\sigma$, because
then one obtains ${\cal L}^{\pm}_{-\sigma,\pm\sigma,\mp\sigma}=0$. If, in
addition, $\sigma=\text{sign}(s)$, then the $\delta_{N}$-condition is
fulfilled for the special polynomial solutions of the first class denoted as
${}_{s}R^{\pm}_{\omega,E,m}(r)={}_{s}R^{\pm}_{\omega,E,m,-\sigma,\pm\sigma,\mp\sigma}(r;1,0)$.
Equation (3.1) yields the relation ${}_{s}N=|s|-1\geq 0$. Scalar perturbations
of this type do not exist.
In the case of integer spins $|s|=1,2$ the roots
$\mu={}_{s}\mu_{\omega,k,m}^{\pm},\,\,\,k=1,\dots,|s|$ of the equations
$\Delta^{\pm}_{|s|}(\mu)=0$, (1.7b), and (2.4a)-(2.4e) with $r_{+}=1$,
$r_{-}=0$ and $a=0$ produce the following simple expressions for
$E={}_{s}E_{\omega,k,m}^{\pm}$, $k=1,\dots,|s|$:
$\displaystyle{}_{s}E_{\omega,m}^{\pm}$ $\displaystyle=$ $\displaystyle
0:\,\,\,\text{for}\,\,\,|s|=1,$ (4.7)
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}$ $\displaystyle=$ $\displaystyle
1-(-1)^{k}\sqrt{1-i6\sigma\omega}:\,\,\,\text{for}\,\,\,|s|=2,\,k=1,2.$ (4.8)
For a complete solution of the problem, one needs an additional relation
between the parameters $E$ and $\omega$. This relation may be found by solving
the TAE, see sections 5-8.
The above considerations of the limit $a\to 0$ and the corresponding results
for the Schwarzschild black hole in terms of confluent Heun’s functions are
new and obtained for the first time in the present paper.
### 4.2 Second Class of Polynomial Solutions to the TRE:
According to the results of section 3, the solutions of this class originate
from the second case of the $\delta_{N}$-condition and fall into two
subclasses: a) and b). The complete definite frequencies
${}_{s}\omega^{\pm}_{N,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}$ –
formulae (3.3) and (3.4), yield algebraic equations
$\Delta^{\pm}_{N+1}(\mu)=0$ with $(N+1)$ roots
$\mu={}_{s}\mu^{\pm}_{N,n,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r_{+},r_{-})$,
$n=0,1,\dots,N$. It seems difficult to derive explicit analytic expressions
for these roots, but their numerical values can be easily obtained. Using the
values of
${}_{s}\mu^{\pm}_{N,n,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r_{+},r_{-})$
and equations (1.7b), (2.4a)-(2.4e) we obtain complete definite values for the
parameter
$E={}_{s}E_{N,n,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}^{\pm}(r_{+},r_{-})$:
a) In the case of frequencies (3.3) we obtain
$\displaystyle{}_{s}E_{N,n,m,\sigma,-\sigma,-\sigma}^{\pm}\\!=\\!{}_{s}\mu_{N,n,m,\sigma,-\sigma,-\sigma}^{\pm}+|s|(|s|-1)+a\omega(3a\omega-2m)\\!+4\,\omega^{2}r_{\mp}^{2}\\!+\\!2i\sigma\omega\big{(}2M|s|-r_{\pm}\big{)}.$
(4.9)
b) In the case of frequencies (3.4a), (3.4b) we have, respectively:
$\displaystyle{}_{s}E_{N,n,m,+,-,+}^{\pm}\\!-{}_{s}\mu_{N,n,m,+,-,+}^{\pm}={}_{s}E_{N,n,m,-,+,-}^{\mp}\\!-{}_{s}\mu_{N,n,m,+,-,+}^{\mp}=\pm
i\,{2(2a\omega-m)}/{2p}-\hskip 55.48277pt$
$\displaystyle-\Big{(}m^{2}+8m\left(1+M^{2}/a^{2}\right)a\omega+\left(1+10\,r_{\mp}/r_{\pm}+9\,(r_{\mp}/r_{\pm})^{2}-4\,(r_{\mp}/r_{\pm})^{3}\right)\omega^{2}r_{\pm}^{2}\Big{)}\big{/}{4p^{2}},\hskip
28.45274pt$ (4.10a)
$\displaystyle{}_{s}E_{N,n,m,+,+,-}^{\pm}\\!-\\!{}_{s}\mu_{N,n,m,+,+,-}^{\pm}\\!=\\!{}_{s}E_{N,n,m,-,-,+}^{\mp}\\!-\\!{}_{s}\mu_{N,n,m,-,-,+}^{\mp}\\!=\\!\pm
i\,2\Big{(}m\\!+\\!2a\omega\\!\left(1\\!-\\!2M^{2}/a^{2}\right)\\!\Big{)}\big{/}2p-\hskip
5.69046pt$
$\displaystyle-i2psa\omega-4\Big{(}m^{2}+2m\left(1\\!-\\!3M^{2}/a^{2}\right)a\omega-\left(1\\!-\\!5M^{2}/a^{2}\right)(a\omega)^{2}\Big{)}\big{/}4p^{2}.\hskip
28.45274pt$ (4.10b)
With $\omega$ and $E$ given by equations (3.3), (3.4) and (4.9), (4.10) we
have no more free parameters in the problem at hand. As a result, the
corresponding solutions to the TAE are fixed unambiguously by the designated
group of equations obtained for the second class of polynomial solutions to
the TRE. This situation is completely new, unexpected and described here for
the first time.
## 5 Exact Solutions to the Teukolsky Angular Equation in Terms of the
Confluent Heun functions
In terms of the variable $x=\cos\theta$ the TAE has three singular points. Two
of them: $x_{-}=-1$ (i.e., $\theta_{S}=\pi$ – South (S-)pole) and $x_{+}=1$
(i.e., $\theta_{N}=0$ – North (N-)pole) are regular singular points. The third
one $x_{\infty}=\infty$ is an irregular singular point. It is remarkable that
introducing the notation
$\displaystyle
z_{+}=z_{+}(\theta)=\left(\cos(\theta/2)\right)^{2},\,\,\,z_{-}=z_{-}(\theta)=\left(\sin(\theta/2)\right)^{2},\,\,\,z_{+}+z_{-}=1;$
(5.1)
and
$\displaystyle a_{\pm}=\pm 4a\omega,\,\,b_{\pm}=s\mp m,\,\,c_{\pm}=s\pm
m,\,\,d_{\pm}=\pm 4sa\omega,\,\,n_{\pm}={\frac{m^{2}+s^{2}}{2}}\mp
2sa\omega-a^{2}\omega^{2}-E.$ (5.2)
we can write down 16 local solutions of the TAE in the form
$\displaystyle{}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}=\\!e^{\sigma_{a}{\frac{a_{{}_{\pm}}z_{{}_{\pm}}}{2}}}z_{{}_{\pm}}^{\sigma_{b}{\frac{b_{{}_{\pm}}}{2}}}z_{{}_{\mp}}^{\sigma_{c}{\frac{c_{{}_{\pm}}}{2}}}\text{HeunC}(\sigma_{a}a_{{}_{\pm}},\sigma_{b}b_{{}_{\pm}},\sigma_{c}c_{{}_{\pm}},d_{{}_{\pm}},n_{{}_{\pm}},z_{{}_{\pm}})$
(5.3)
which is very similar to the form of Eq. (2.3).
Following the corresponding properties of the TAE (1.1a) [2], the solutions
(5.3) have the symmetries
$\displaystyle{}_{-s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}(\pi-\theta)$
$\displaystyle=$
$\displaystyle{}_{s}S_{\omega,E,m,-\sigma_{a},-\sigma_{b},-\sigma_{c}}^{\mp}(\theta),$
(5.4a)
$\displaystyle{}_{s}S_{-\omega,E,-m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}(\pi-\theta)$
$\displaystyle=$
$\displaystyle{}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\mp}(\theta).$
(5.4b)
Note that according to Eq. (5.3), the behavior of the solutions
${}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}$ around the
corresponding singular points $z=z_{+}(\theta_{S})=z_{-}(\theta_{N})=0$ is
defined by the dominant factor
$\left(z_{{}_{\pm}}\right)^{\sigma_{b}b_{{}_{\pm}}/2}$. All other factors in
(5.3) are regular around these points. The same solutions are in general
singular around the corresponding singular points
$z=z_{+}(\theta_{N})=z_{-}(\theta_{S})=1$. Hence, at this point we have a
complete analogy with the case of the TRE.
Only two of the sixteen solutions (5.3) are linearly independent.
Nevertheless, it is important to know all of them, since for various purposes
one can use different pairs of independent local solutions, see below. If one
chooses some two linearly independent solutions, then one can represent the
other fourteen using this basis. Unfortunately, at present the form of the
corresponding coefficients is completely unknown.
We can establish simple relations between some of the different solutions
(5.3) in proper domains of the parameters $s$ and $m$, if we divide the whole
plane $\\{s,m\\}$ into four sectors. In each of them we choose the solutions
with the same regular asymptotic behavior around the corresponding pole as
follows:
I. Sector $s\geq 0$, $|m|\leq|s|$:
$\displaystyle{}_{s}S^{+\,reg}_{\omega,E,m}(\theta)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{+}_{\omega,E,m,+++}\\!=\\!{}_{s}S^{+}_{\omega,E,m,-++}\\!=\\!{}_{s}S^{+}_{\omega,E,m,++-}\\!=\\!{}_{s}S^{+}_{\omega,E,m,-+-}\,\underset{\theta\to\pi}{\sim}\left(\cos{\frac{\theta}{2}}\right)^{s-m},$
(5.5a) $\displaystyle{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{-}_{\omega,E,m,+++}\\!=\\!{}_{s}S^{-}_{\omega,E,m,-++}\\!=\\!{}_{s}S^{-}_{\omega,E,m,++-}\\!=\\!{}_{s}S^{-}_{\omega,E,m,-+-}\,\underset{\theta\to
0}{\sim}\left(\sin{\frac{\theta}{2}}\right)^{s+m}.$ (5.5b)
II. Sector $m\leq 0$, $|s|\leq|m|$:
$\displaystyle{}_{s}S^{+\,reg}_{\omega,E,m}(\theta)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{+}_{\omega,E,m,+++}\\!=\\!{}_{s}S^{+}_{\omega,E,m,-++}\\!=\\!{}_{s}S^{+}_{\omega,E,m,++-}\\!=\\!{}_{s}S^{+}_{\omega,E,m,-+-}\,\underset{\theta\to\pi}{\sim}\left(\cos{\frac{\theta}{2}}\right)^{s-m},$
(5.6a) $\displaystyle{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{-}_{\omega,E,m,---}\\!=\\!{}_{s}S^{-}_{\omega,E,m,+--}\\!=\\!{}_{s}S^{-}_{\omega,E,m,--+}\\!=\\!{}_{s}S^{-}_{\omega,E,m,+-+}\,\underset{\theta\to
0}{\sim}\left(\sin{\frac{\theta}{2}}\right)^{-s-m}.$ (5.6b)
III. Sector $s\leq 0$, $|m|\leq|s|$:
$\displaystyle{}_{s}S^{+\,reg}_{\omega,E,m}(\theta)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{+}_{\omega,E,m,---}\\!=\\!{}_{s}S^{+}_{\omega,E,m,+--}\\!=\\!{}_{s}S^{+}_{\omega,E,m,--+}\\!=\\!{}_{s}S^{+}_{\omega,E,m,+-+}\,\underset{\theta\to\pi}{\sim}\left(\cos{\frac{\theta}{2}}\right)^{-s+m},$
(5.7a) $\displaystyle{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{-}_{\omega,E,m,---}\\!=\\!{}_{s}S^{-}_{\omega,E,m,+--}\\!=\\!{}_{s}S^{-}_{\omega,E,m,--+}\\!=\\!{}_{s}S^{-}_{\omega,E,m,+-+}\,\underset{\theta\to
0}{\sim}\left(\sin{\frac{\theta}{2}}\right)^{-s-m}.$ (5.7b)
IV. Sector $m\geq 0$, $|s|\leq|m|$:
$\displaystyle{}_{s}S^{+\,reg}_{\omega,E,m}(\theta)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{+}_{\omega,E,m,---}\\!=\\!{}_{s}S^{+}_{\omega,E,m,+--}\\!=\\!{}_{s}S^{+}_{\omega,E,m,--+}\\!=\\!{}_{s}S^{+}_{\omega,E,m,+-+}\,\underset{\theta\to\pi}{\sim}\left(\cos{\frac{\theta}{2}}\right)^{-s+m},$
(5.8a) $\displaystyle{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!{}_{s}S^{-}_{\omega,E,m,+++}\\!=\\!{}_{s}S^{-}_{\omega,E,m,-++}\\!=\\!{}_{s}S^{-}_{\omega,E,m,++-}\\!=\\!{}_{s}S^{-}_{\omega,E,m,-+-}\,\underset{\theta\to
0}{\sim}\left(\sin{\frac{\theta}{2}}\right)^{s+m}.$ (5.8b)
Note that in each sector the four solutions in the above relations of type
(a), or in the above relations of type (b) are equal, since under standard
normalization the local regular solution around any regular singular point of
the TAE is unique.
In the case of the TAE there exist an additional complication. The numbers $s$
and $m$ are simultaneously integers, or half-integers. Then
$\beta=\sigma_{b}b_{{}_{\pm}}=\sigma_{b}(s\mp m)$ is an integer and, in
particular, it may be a negative integer. However, the confluent Heun
functions $\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ are not defined
when $\beta$ is a negative integer [14]. Therefore, if
$\beta=\sigma_{b}b_{{}_{\pm}}<0$ is a negative integer, we must write down the
corresponding solutions in the form
$\displaystyle{}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}=\\!e^{\sigma_{a}{\frac{a_{{}_{\pm}}z_{{}_{\pm}}}{2}}}z_{{}_{\pm}}^{\sigma_{b}{\frac{b_{{}_{\pm}}}{2}}}z_{{}_{\mp}}^{\sigma_{c}{\frac{c_{{}_{\pm}}}{2}}}\underline{\text{HeunC}}(\sigma_{a}a_{{}_{\pm}},\sigma_{b}b_{{}_{\pm}},\sigma_{c}c_{{}_{\pm}},d_{{}_{\pm}},n_{{}_{\pm}},z_{{}_{\pm}}).$
(5.9)
For this purpose we define the concomitant confluent Heun function555Note that
for any value of the parameter $\beta$, when the confluent Heun function in
the right hand side of Eq. (5.10) is well defined, its left hand side
represents a second, linearly independent solution of the confluent Heun
equation.
$\displaystyle\underline{\text{HeunC}}(\alpha,\beta,\gamma,\delta,\eta,z)=z^{-\beta}\text{HeunC}(\alpha,-\beta,\gamma,\delta,\eta,z)\int{\frac{e^{-\alpha\zeta}\zeta^{\beta-1}(1-\zeta)^{-\gamma-1}}{\big{(}\text{HeunC}(\alpha,-\beta,\gamma,\delta,\eta,\zeta)\big{)}^{2}}}d\zeta.$
(5.10)
This function is well defined for negative integer
$\beta=\sigma_{b}b_{{}_{\pm}}<0$, together with the confluent function
$\text{HeunC}(\alpha,-\beta,\gamma,\delta,\eta,z)$. In this case, the function
$z^{-\beta}\text{HeunC}(\alpha,-\beta,\gamma,\delta,\eta,z)$ represents the
local regular solution to the confluent Heun equation (1.6) around the
singular point $z=0$ and the concomitant confluent function
$\underline{\text{HeunC}}(\alpha,\beta,\gamma,\delta,\eta,z)$ represents a
second linearly independent local solution, which is singular around this
point. We need the concomitant Heun function to construct second independent
local solution in the case $\beta=0$, too, since in this case
$z^{-\beta}\text{HeunC}(\alpha,-\beta,\gamma,\delta,\eta,z)\equiv\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$
can not be used for this purpose.
It can be shown that for negative integer $\beta$ the concomitant confluent
Heun function has the form
$\displaystyle\underline{\text{HeunC}}(\alpha,\beta,\gamma,\delta,\eta,z)=\sum_{n=1}^{|\beta|}{\frac{c_{n}}{z^{n}}}+h_{1}(z)+h_{2}(z)\ln(z),\,\,\,{\text{all}}\,\,\,c_{n}\neq
0.$ (5.11)
Here $h_{1,2}(z)$ denote two definite functions of the complex variable $z$
which are analytic in the vicinity of the point $z=0$. In the problem at hand
$|\beta|=|\beta_{\pm}|=|s\mp m|$. The logarithmic term is present in the
concomitant confluent Heun function when $|\beta|=0$, too, but then we have no
poles in the solution (5.11). For $|\beta|=0$ its form otherwise is similar to
(5.11). One can reach the last results using general analytical methods
described, for example, in [25].
The above consideration sows that in the case of the TAE we can construct only
eight local solutions (5.3) which are single-valued functions of the variable
$z$. The other eight solutions, being in the form (5.9), are infinitely-
valued, because of the logarithmic term in Eq. (5.11).
## 6 A New Classification of the Solutions to the TAE Based on the
$\delta_{N}$-Condition. Novel $\delta_{N}$-Angular Solutions
For solutions ${}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}$
(5.3) to the TAE the $\delta_{N}$-condition reads:
$\displaystyle
0=\mp\,m{\frac{\sigma_{b}-\sigma_{c}}{2}}+N+1+\left(\sigma_{a}+{\frac{\sigma_{b}+\sigma_{c}}{2}}\right)s.$
(6.1)
We call angular $\delta_{N}$-solutions the solutions defined via the
$\delta_{N}$-condition (6.1). To some extent these solutions are similar to
the radial $\delta_{N}$-solutions introduced in section 3. A set of important
new mathematical properties of the angular $\delta_{N}$-solutions can be found
in [17, 18]. In [18] it is shown that these solutions define the most general
class of solutions to the TAE for which properly generalized Teukolsky-
Starobinsky identities exist.
Comparing equation (6.1) with the corresponding one for the TRE – (3.1), we
see both essential differences and similarities. For the coefficients in
equation (6.1), which are analogous to the ones in (3.1), one obtains:
${\cal L}^{\pm}_{\sigma_{a},\sigma_{b},\sigma_{c}}\equiv 0,\,\,{\cal
M}^{\pm}_{m,\sigma_{a},\sigma_{b},\sigma_{c}}=\mp\,m(\sigma_{b}\\!-\\!\sigma_{c}){\frac{1}{2}},\,\,{}_{s}{\cal
N}^{\pm}_{\sigma_{a},\sigma_{b},\sigma_{c}}=N\\!+\\!1\\!+\\!\left(\sigma_{a}\\!+\\!{\frac{\sigma_{b}\\!+\\!\sigma_{c}}{2}}\right)s.$
Hence:
i) The coefficients ${\cal L}^{\pm}_{\sigma_{a},\sigma_{b},\sigma_{c}}$ vanish
identically, in contrast to the coefficients ${\cal L}^{\pm}$ in equation
(3.1). Consequently, there are no cases in which the condition (6.1) can fix
the frequencies $\omega$.
ii) The form of the coefficients ${\cal M}^{\pm}$ of both equations (3.1) and
(6.1) is the same only for $M/a=\sqrt{2}$.
iii) The coefficients ${\cal N}^{\pm}$ of both equations are of the same form.
We obtain two different cases depending on the coefficient
$(\sigma_{b}\\!-\\!\sigma_{c})$ in front of the azimuthal number $m$:
1\. The first class angular $\delta_{N}$-solutions with
$\sigma_{c}=\sigma_{b}$ and $\sigma_{a}=\sigma_{b}=\sigma_{c}=-\sigma$. As a
result, Eq. (6.1) fixes the degree of the second polynomial condition
$\Delta_{N+1}=0$ in the same form as equation (3.2)666The alternative case
$\sigma_{b}=\sigma_{c}=-\sigma_{a}$ leads to a uninteresting relation
$N+1=0$.:
$\displaystyle{}_{s}N+1=2|s|\geq 1\,\,\,\text{for}\,\,\,|s|\geq 1/2.$ (6.2)
In the case of the TAE the set of $\delta_{N}$-solutions consists of ones with
integer parameters $\beta=\sigma_{b}b_{{}_{\pm}}$ of both signs. According to
Eqs. (5.9) and (5.11), for negative integer $\sigma_{b}b_{{}_{\pm}}<0$ in the
solutions we have logarithmic terms. Such solutions are infinitely-valued
functions. To exclude this physically not admissible case, one must impose the
additional requirement $\sigma_{b}b_{{}_{\pm}}\geq 0$. As a result one obtains
$\sigma\sigma_{m}=\pm 1$, $\sigma_{b}b_{{}_{\pm}}=|m|-|s|\geq 0$ and
$\sigma_{c}c_{{}_{\pm}}=-|m|-|s|<0$. Thus, the single-valued angular
$\delta_{N}$-solutions of the first class with spin $|s|\geq 1/2$ correspond
to sectors II (5.6) and IV (5.8) and acquire the form
$\displaystyle{}_{s}S_{\omega,E,m}^{\pm}(z_{{}_{\pm}})=\\!e^{-2\sigma_{m}a\omega
z_{{}_{\pm}}}\left(z_{{}_{\pm}}\right)^{\frac{|m|-|s|}{2}}\left(z_{{}_{\mp}}\right)^{\frac{-|m|-|s|}{2}}\times\hskip
182.09746pt$ (6.3)
$\displaystyle\times\text{HeunC}(-4\sigma_{m}a\omega,|m|-|s|,-|m|-|s|,4\sigma_{m}a\omega|s|,{\frac{m^{2}+s^{2}}{2}}-2\sigma_{m}a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\pm}}).$
Obviously, the solutions (6.3) are regular around the singular points
$z_{{}_{\pm}}\\!=\\!0$. Their behavior around the singular points
$z_{{}_{\pm}}\\!=\\!1(\Leftrightarrow z_{{}_{\mp}}=0$) is more complicated. We
can study this behavior using the expansion of the solutions (6.3) with
respect to the basis of the two linearly independent local solutions (5.9)
around the points $z_{{}_{\mp}}\\!=\\!0$, which are well defined for
$\sigma\sigma_{m}=\pm 1$, $\sigma_{b}b_{{}_{\pm}}\\!=\\!|m|-|s|\geq 0$ and
$\sigma_{c}c_{{}_{\pm}}\\!=\\!-|m|-|s|<0$:
$\displaystyle{}_{s}S_{\omega,E,m}^{\pm}(z_{{}_{\pm}})={}_{s}\Gamma^{\pm}_{1}(\omega,E,m)\,e^{2\sigma_{m}a\omega
z_{{}_{\mp}}}\left(z_{{}_{\mp}}\right)^{\frac{-|m|-|s|}{2}}\left(z_{{}_{\pm}}\right)^{\frac{|m|-|s|}{2}}\times\hskip
113.81102pt$ (6.4)
$\displaystyle\times\underline{\text{HeunC}}(4\sigma_{m}a\omega,-|m|-|s|,|m|-|s|,-4\sigma_{m}a\omega|s|,{\frac{m^{2}+s^{2}}{2}}+2\sigma_{m}a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\mp}})+$
$\displaystyle+\,{}_{s}\Gamma^{\pm}_{2}(\omega,E,m)\,e^{2\sigma_{m}a\omega
z_{{}_{\mp}}}\left(z_{{}_{\mp}}\right)^{\frac{|m|+|s|}{2}}\left(z_{{}_{\pm}}\right)^{\frac{|m|-|s|}{2}}\times\hskip
108.12054pt$
$\displaystyle\times\text{HeunC}(4\sigma_{m}a\omega,|m|+|s|,|m|-|s|,-4\sigma_{m}a\omega|s|,{\frac{m^{2}+s^{2}}{2}}+2\sigma_{m}a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\mp}}).$
Now it is clear that in the general case, when
${}_{s}\Gamma^{\pm}_{1}(\omega,E,m)\\!\neq\\!0$, the solutions (6.3) are
singular around the corresponding points $z_{{}_{\pm}}\\!=\\!1$ and in
addition – still infinite valued, because of the poles and of the logarithmic
terms in the concomitant confluent Heun function in Eq. (6.4), as well as
because of the singular factor
$\left(z_{{}_{\mp}}\right)^{\frac{-|m|-|s|}{2}}$. One can remove at once all
these unwanted properties from solutions (6.4) imposing the condition
${}_{s}\Gamma^{\pm}_{1}(\omega,E,m)\\!=\\!0$. Unfortunately, the explicit form
of the connection constants ${}_{s}\Gamma^{\pm}_{1,2}(\omega,E,m)$ is
completely unknown. At present, this is one of the main unsolved problems in
the theory of the confluent Heun functions.
Another way to avoid the logarithmic terms in the solutions (6.3), (6.4) is to
impose the $\Delta_{N+1}$-condition, reducing this way confluent Heun’s
functions to polynomials. We consider in detail these two possibilities in the
next sections 7 and 8.
2\. The second class angular $\delta_{N}$-solutions: $\sigma_{b}=-\sigma_{c}$.
Then we obtain
$\displaystyle{}_{s}N_{m,\sigma_{a},\sigma_{b},-\sigma_{b}}+1=\pm\,m\sigma_{b}-\sigma_{a}s\geq
1.$ (6.5)
Now the additional requirement $\sigma_{b}(s\mp m)\geq 0$ and (6.5) yield the
solutions
$\displaystyle{}_{s}S_{\omega,E,m,-\sigma,\sigma,-\sigma}^{\pm}(z_{{}_{\pm}})=\\!e^{\mp
2\sigma a\omega z_{{}_{\pm}}}\left(z_{{}_{\pm}}\right)^{\frac{|s|\mp\sigma
m}{2}}\left(z_{{}_{\mp}}\right)^{\frac{-|s|\mp\sigma m}{2}}\times\hskip
182.09746pt$ (6.6) $\displaystyle\times\text{HeunC}(\mp 4\sigma
a\omega,|s|\mp\sigma m,-|s|\mp\sigma m,\pm 4\sigma
a\omega|s|,{\frac{m^{2}+s^{2}}{2}}\mp 2\sigma
a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\pm}}).$
with $|s|\geq 1/2$ and $m$ restricted in the asymmetric finite intervals
$1-|s|\leq\pm\sigma m\leq|s|$, i.e.,
$-|s|+(1\pm\sigma\sigma_{m})/2\leq|m|\leq|s|-(1\mp\sigma\sigma_{m})/2.$ These
solutions have $\sigma_{a}=-\sigma_{b}=\sigma_{c}=-\sigma$,
${}_{s}N_{m,-\sigma,\sigma,-\sigma}+1=|s|\pm\,\sigma m\geq 1$ and correspond
to pairs $\\{s,m\\}$ in sectors I (5.5) and IV (5.7). Under the above
conditions the solutions
${}_{s}S_{\omega,E,m-\sigma,\sigma,-\sigma}^{\pm}(z_{{}_{\pm}})$ (6.6) are
obviously regular around the points $z_{{}_{\pm}}=0$. Their behavior around
the second regular singular points $z_{{}_{\pm}}=1$ can be studied using the
following expansion with respect to the corresponding local basis
$\displaystyle{}_{s}S_{\omega,E,m,-\sigma,\sigma,-\sigma}^{\pm}(z_{{}_{\pm}})={}_{s}\Gamma^{\pm}_{1}(\omega,E,m,-\sigma,\sigma,-\sigma)\,e^{\pm
2\sigma a\omega z_{{}_{\mp}}}\left(z_{{}_{\mp}}\right)^{\frac{-|s|\mp\sigma
m}{2}}\left(z_{{}_{\pm}}\right)^{\frac{|s|\mp\sigma m}{2}}\times\hskip
54.06006pt$ (6.7) $\displaystyle\times\underline{\text{HeunC}}(\pm 4\sigma
a\omega,-|s|\mp\sigma m,|s|\mp\sigma m,\mp 4\sigma
a\omega|s|,{\frac{m^{2}+s^{2}}{2}}\pm 2\sigma
a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\mp}})+$
$\displaystyle+\,{}_{s}\Gamma^{\pm}_{2}(\omega,E,m,-\sigma,\sigma,-\sigma)\,e^{\pm
2\sigma a\omega z_{{}_{\mp}}}\left(z_{{}_{\mp}}\right)^{\frac{|s|\pm\sigma
m}{2}}\left(z_{{}_{\pm}}\right)^{\frac{|s|\mp\sigma m}{2}}\times\hskip 0.0pt$
$\displaystyle\times\text{HeunC}(\pm 4\sigma a\omega,|s|\pm\sigma
m,|s|\mp\sigma m,\mp 4\sigma a\omega|s|,{\frac{m^{2}+s^{2}}{2}}\pm 2\sigma
a\omega|s|-a^{2}\omega^{2}-E,z_{{}_{\mp}}).$
Since $-2|s|\leq-|s|\mp\sigma m\leq-1$ and $1\leq|s|\pm\sigma m\leq 2|s|$, the
two independent local solutions in (6.7) are well defined. The first solution
(with concomitant Heun’s function) is singular around $z_{{}_{\pm}}=1$
($\Leftrightarrow z_{{}_{\mp}}=0$) and, in addition, infinite-valued. One can
remove at once all these unwanted properties from solutions (6.7) imposing the
condition ${}_{s}\Gamma^{\pm}_{1}(\omega,E,m,-\sigma,\sigma,-\sigma)\\!=\\!0$.
Unfortunately, at present the explicit form of the connection constants
${}_{s}\Gamma^{\pm}_{1,2}(\omega,E,m,-\sigma,\sigma,-\sigma)$ is completely
unknown, too.
Another way to avoid the logarithmic terms in the solutions (6.6), (6.7) is to
impose the $\Delta_{N+1}$-condition, reducing this way confluent Heun’s
functions to polynomials. We consider in detail these two possibilities in the
next sections 7 and 8.
As seen, in the case of the TAE the only role of the $\delta_{N}$-condition is
to relate the degree $N$ of the $\Delta_{N+1}$-condition with the spin-weight
$s$ and the azimuthal number $m$ and to select the proper solutions.
Note that up to now only regular solutions to the TAE, which obey the
condition (6.2) have been studied and used in the literature [2, 3, 7]. In
section 7 we develop a new approach to the regular solutions, based on
confluent Heun’s functions. The nonregular angular $\delta_{N}$-solutions,
subject to the condition (6.2), and the infinite series of solutions, subject
to the condition (6.5), are introduced and considered for the first time in
the present paper.
## 7 Regular solutions of the TAE
The spectral conditions ${}_{s}\Gamma^{\pm}_{1}(\omega,E,m)=0$ and
${}_{s}\Gamma^{\pm}_{1}(\omega,E,m,-\sigma,\sigma,-\sigma)=0$ ensure the
regularity of the solutions (6.3) and (6.6), as seen from the formulas (6.4)
and (6.7). One is not able to use these conditions directly, since the
explicit form of the connection constants ${}_{s}\Gamma^{\pm}_{1}(\omega,E,m)$
and ${}_{s}\Gamma^{\pm}_{1}(\omega,E,m,-\sigma,\sigma,-\sigma)$ is not known.
Therefore, we are forced to use a roundabout way to find the regular solutions
to the TAE.
Suppose we have a solution ${}_{s}S^{+\,reg}_{\omega,E,m}(\theta)$ which is
regular around the S-pole ($\theta_{S}=\pi$) and another solution
${}_{s}S^{-\,reg}_{\omega,E,m}(\theta)$ which is regular around the N-pole
($\theta_{N}=0$). We will have a solution
${}_{s}S^{\,\,{}_{REG}}_{\omega,E,m}(\theta)$, regular everywhere in the
interval $\theta\in[0,\pi]$, if and only if
${}_{s}S^{+\,reg}_{\omega,E,m}(\theta)\\!=\\!\text{const}\times{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)$,
i.e., if the Wronskian vanishes:
$\text{W}\\!\left[{}_{s}S^{+\,reg}_{\omega,E,m}(\theta),{}_{s}S^{-\,reg}_{\omega,E,m}(\theta)\right]\\!=\\!0$.
This condition determines the constant $E$ in the form $E=E(a\omega,s,m,l)$,
$l$ being a (half)integer. The Wronskian will vanish for any
$\theta\in[0,\pi]$, if it is zero for some $\theta_{0}\in(0,\pi)$.
To utilize this idea for all values of the parameters $s$ and $m$, we have to
divide the whole plane $\\{s,m\\}$ into four sectors and to choose the
solutions ${}_{s}S^{\pm\,reg}_{\omega,E,m}(\theta)$ defined by Eqs.
(5.5)-(5.8).
The spectral condition makes equal the solutions of group (a) and the
solutions of group (b) in each sector. It can be written in different
equivalent forms combining in pairs one solution from the group (a) and
another one from the group (b). Below we give the simplest form of this
condition in each sector, written here for the first time in terms of
confluent Heun’s function $\mathrm{HeunC}$ and its derivative
$\mathrm{HeunC}^{\prime}$. The set of all conditions (7.1) defines the
separation constant in the whole plane $\\{s,m\\}$ in the form
$E=(m^{2}+s^{2})/2-a^{2}\omega^{2}+\varepsilon(a\omega,m,s)$. The new
parameter $\varepsilon(a\omega,m,s)$ is to be found from the following
transcendental equations:
$\displaystyle{{\mathrm{HeunC}^{\prime}(\pm 4a\omega,\,s+m,s-m,-4a\omega
s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\pm
4a\omega,\,s+m,s-m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}}\,+\hskip 93.89418pt$
(7.1a) $\displaystyle+\,{{\mathrm{HeunC}^{\prime}(\mp
4a\omega,\,s-m,s+m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\mp
4a\omega,\,s-m,s+m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}}=0\,\,\,\,\,\text{--
in sector I},\hskip 19.91684pt$ $\displaystyle{{\mathrm{HeunC}^{\prime}(\pm
4a\omega,\,-s-m,s-m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\pm
4a\omega,\,-s-m,s-m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}}\,+\hskip 93.89418pt$
(7.1b) $\displaystyle+\,{{\mathrm{HeunC}^{\prime}(\mp
4a\omega,\,s-m,-s-m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\mp
4a\omega,\,s-m,-s-m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}}=0\,\,\,\,\,\text{--
in sector II},\hskip 14.22636pt$ $\displaystyle{{\mathrm{HeunC}^{\prime}(\pm
4a\omega,\,-s-m,-s+m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\pm
4a\omega,\,-s-m,-s+m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}}\,+\hskip 93.89418pt$
(7.1c) $\displaystyle+\,{{\mathrm{HeunC}^{\prime}(\mp
4a\omega,\,-s+m,-s-m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\mp
4a\omega,\,-s+m,-s-m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}}=0\,\,\,\,\,\text{--
in sector III},\hskip 11.38092pt$ $\displaystyle{{\mathrm{HeunC}^{\prime}(\pm
4a\omega,\,s+m,-s+m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\pm
4a\omega,\,s+m,-s+m,-4a\omega s,+2\omega
as-\varepsilon,\left(\sin{{\theta}\over{2}}\right)^{2})}}\,+\hskip 93.89418pt$
(7.1d) $\displaystyle+\,{{\mathrm{HeunC}^{\prime}(\mp
4a\omega,\,-s+m,s+m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}\over{\mathrm{HeunC}(\mp
4a\omega,\,-s+m,s+m,+4a\omega s,-2a\omega
s-\varepsilon,\left(\cos{{\theta}\over{2}}\right)^{2})}}=0\,\,\,\,\,\text{--
in sector IV},\hskip 11.38092pt$
valid simultaneously for all values of $\theta\in(0,\pi)$. Thus, the two-
singular-points boundary problem for the TAE is solved. It yields a countable
set of values $E(a\omega,m,s,l)$ numbered by some (half)integer $l$: $l$ is
integer for an integer spin, or half-integer – for a half-integer spin. Due to
the symmetries (5.4) of the solutions to the TAE, the different relations
(7.1a) and (7.1c), or (7.1b) and (7.1d) give similar results. More precisely
$E(a\omega,m,-s,l)=E(a\omega,m,s,l)$ and
$E(-a\omega,-m,s,l)=E(a\omega,m,s,l)$.
An important consequence is that all the regular solutions obtained this way
are angular $\delta_{N}$-solutions with the same ${}_{s}N$ (6.2) in sectors II
and IV, or with the same ${}_{s}N_{m,\sigma_{a},\sigma_{b}}$ (6.5) – in
sectors I and III. This is because between the solutions (5.6) and (5.8) we
certainly have $\delta_{N}$-solutions: ${}_{s}S^{\pm}_{\omega,E,m,---}$, for
$s>0$, and ${}_{s}S^{\pm}_{\omega,E,m,+++}$, for $s<0$. Between the solutions
(5.5) and (5.7) $\delta_{N}$-solutions are
${}_{s}S^{-}_{\omega,E,m,\mp\pm\mp}$, for $m>0$, and
${}_{s}S^{+}_{\omega,E,m,\mp\pm\mp}$, for $m<0$. As a result of uniqueness of
the regular solutions with given values of the parameters, all regular
solutions inherit the $\delta_{N}$-property. Hence, all regular solutions of
the TAE obey the Teukolsky-Starobinsly identities [18].
Let us consider the limit $a\omega\to 0$ of the regular solutions to the TAE.
Since
$\displaystyle\text{HeunC}(0,\beta,\gamma,0,\eta,z)\\!=\\!(1\\!-\\!z)^{\beta+\gamma+1+\sqrt{\beta^{2}+\gamma^{2}+1-4\eta}}\times\hskip
199.16928pt$
$\displaystyle{}_{2}F_{1}\left({\frac{\beta\\!+\\!\gamma\\!+\\!1+\\!\sqrt{\beta^{2}\\!+\\!\gamma^{2}\\!+\\!1\\!-\\!4\eta}}{2}},{\frac{\beta\\!+\\!\gamma\\!+\\!1-\\!\sqrt{\beta^{2}\\!+\\!\gamma^{2}\\!+\\!1\\!-\\!4\eta}}{2}};\beta\\!+\\!1;z\right)\\!,$
(7.2)
in this limit the Heun functions in Eqs. (5.5)-(5.8) and (7.1) can be reduced
to the Gauss hypergeometric ones. Then, using the well-known properties of the
Gauss hypergeometric function ${}_{2}F_{1}$ one can derive from Eqs. (7.1)
with $a\omega=0$ the spectrum $E(0,s,l,m)=l(l+1)$,
$l=l(s,m,\bar{l})=\max(|m|,|s|)+\bar{l},\,\,\bar{l}=0,1,2,\dots$ The values of
the separation constant $E(0,s,l,m)$ in this case are real. The numerical
analysis of Eqs. (7.1) written directly in terms of confluent Heun’s functions
confirms this standard result for the limit $a\omega\\!=\\!0$. The
corresponding regular confluent Heun functions in
${}_{s}S^{\,{}_{REG}}_{\omega=0,l,m}(\theta)$ in this case are reduced to
Jacobi’s polynomials – in the case of an integer spin, and to their spin-
weighted generalizations – for a half-integer spin [4].
The solutions $E(a\omega,s,l,m)$ for small $a\omega$ and integer spin have
been studied many times [2, 7] in the form of Taylor’s series expansion
$E(a\omega,s,l,m)\\!=\\!l(l\\!+\\!1)\\!+\\!\sum_{j=\\!1}^{\infty}E_{j,s,l,m}(a\omega)^{j}$
without use of Eqs. (7.1) and without utilizing the Heun functions. A little
bit surprising thing is that the solutions
${}_{s}S^{\,{}_{REG}}_{\omega,l,m}(\theta)$ with $a\omega\neq 0$, regular at
both poles, are not polynomial and can be represented as an infinite series
with respect to Jacobi’s polynomials. Here we describe the regular solutions
to the TAE in terms of confluent $\delta_{N}$-Heun’s functions for the first
time.
## 8 Polynomial Solutions of the TAE
### 8.1 Singularities of the Polynomial solutions to the TAE
The polynomial solutions to the TAE are a special subclass of the angular
$\delta_{N}$-solutions studied in section 6, since both of the two conditions
(1.8) are valid for them. Being a polynomial in $z$, the HeunC-factor is
regular at both regular singular points $\theta=0,\pi$. Then the singularities
of the polynomial solutions around the poles are defined completely by the
factors $\left(z_{{}_{\pm}}\right)^{\sigma_{b}{{b_{{}_{\pm}}}/2}}$ and
$\left(z_{{}_{\mp}}\right)^{\sigma_{c}{{c_{{}_{\pm}}}/2}}$ in Eq. (5.3). Thus:
1\. In case of the first class angular $\delta_{N}$-solutions (6.3) with
$|m|\geq|s|$
we see that the singularities are defined by the factor
$\left(z_{{}_{\mp}}\right)^{\sigma_{c}{{c_{{}_{\pm}}}/2}}$ which gives
${}_{s}S_{\omega,E,m}^{+}(z_{{}_{\pm}})\sim\left(\sin{\theta\over
2}\right)^{-(|s|+|m|)}$, i.e., singularity at the N-pole $\theta=0$, and
${}_{s}S_{\omega,E,m}^{-}(z_{{}_{\pm}})\sim\left(\cos{\theta\over
2}\right)^{-(|s|+|m|)}$, i.e., singularity at the S-pole $\theta=\pi$.
2\. In case of the second class angular $\delta_{N}$-solutions (6.6) with
$-|s|+(1\pm\sigma\sigma_{m})/2\leq|m|\leq|s|-(1\mp\sigma\sigma_{m})/2$
we see that the singularities are defined by the factor
$\left(z_{{}_{\mp}}\right)^{\sigma_{c}{{c_{{}_{\pm}}}/2}}$ which gives
${}_{s}S_{\omega,E,m,-\sigma,\sigma,-\sigma}^{+}(z_{{}_{\pm}})\sim\left(\sin{\theta\over
2}\right)^{-(|s|+\sigma m)}$, i.e., singularity at the N-pole $\theta=0$, and
${}_{s}S_{\omega,E,m,-\sigma,\sigma,-\sigma}^{-}(z_{{}_{\pm}})\sim\left(\cos{\theta\over
2}\right)^{-(|s|-\sigma m)}$, i.e., singularity at the S-pole $\theta=\pi$.
As a result, we see that in any case the polynomial solutions are regular
around one of the poles and singular around the other one.
Using relations (1.7b) and (5.2) we obtain the general formula for the
constant $E$ in the form
$\displaystyle E^{\pm}\\!=\\!\mu^{\pm}\\!-\\!a\omega^{2}\mp
2\sigma_{a}\big{(}1\\!\mp\sigma_{b}m+(\sigma_{a}+\sigma_{b})s\big{)}a\omega+{\frac{\sigma_{b}\\!-\sigma_{c}}{2}}m\left(\sigma_{b}m\mp
1\right)+{\frac{\sigma_{b}\\!+\sigma_{c}}{2}}s\left(\sigma_{b}s\\!+1\right).$
(8.1)
Further analysis shows that some of the properties of the two classes of
polynomial solutions to the TAE resemble the corresponding properties of the
two classes of polynomial solutions of the TRA, but there exist also some
essential differences.
### 8.2 First Class of Polynomial Solutions to the TAE:
These are the solutions ${}_{s}S_{\omega,E,m,-\sigma,-\sigma,-\sigma}^{\pm}$
with $\Delta_{N+1}$-condition fulfilled. For them the specific requirement
$\sigma_{b}b_{{}_{\pm}}\geq 0$ yields the restriction $|m|\geq|s|$ and the
condition (6.2) is fulfilled independently of the values of the azimuthal
number $m$. As in the case of the first class polynomial solutions to the TRA
– section 4, the value $s=0$ is eliminated by (6.2). Hence, we have an
infinite series of the first class polynomial solutions to the TAE for all
admissible values of $s$ and $m$. Preserving the style accepted in the
previous sections we denote the polynomial solutions to the TAE of the first
class as
${}_{s}S_{\omega,E,m}^{\pm}={}_{s}S_{\omega,E,m,-\sigma,-\sigma,-\sigma}^{\pm}$.
For them the $\Delta_{N+1}$-condition reads $\Delta_{2|s|}(\mu)=0$ and has
$2|s|$-in-number solutions ${}_{s}\mu^{\pm}_{\omega,k,m}$. From formulae (8.1)
one obtains
$\displaystyle{}_{s}E^{\pm}_{\omega,k,m}={}_{s}\mu^{\pm}_{\omega,k,m}+|s|(|s|-1)-a\omega(a\omega-2m)\mp
2\sigma(2|s|-1)a\omega,$ (8.2)
where $k=1,\dots,2|s|$, $s=\pm 1/2,\pm 1,\pm 3/2,\pm 2$ and in addition
$|m|\geq|s|$.
Solving the $\Delta_{N+1}$-condition, we obtain for the different values of
$|s|$:
$\displaystyle{}_{s}E_{\omega,m}^{\pm}=-a^{2}\omega^{2}+2a\omega
m-{\frac{1}{4}}:\,\,\,\text{for}\,\,|s|={\frac{1}{2}},\,m=\pm 1/2,\pm
3/2,\dots;$ (8.3)
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}=-a^{2}\omega^{2}+2a\omega\left(m-(-1)^{k}\sqrt{1-m/a\omega}\right):\,\,\,\text{for}\,\,k=1,2;\,|s|=1,\,\,m=\pm
1,\pm 2,\dots$ (8.4)
The values (8.3) and (8.4) of the separation constant $E$ obtained for the
first class polynomial solutions to the TAE are the same as the corresponding
values (4.2) and (4.3) for the first class polynomial solutions to the TRE.
Important consequences of this unexpected fact are considered in a separate
paper [26].
For the gravitational waves ($|s|=2$) the quantities
${}_{s}\mu_{\omega,k,m}^{\pm}$ are solutions of the algebraic equations of the
fourth degree $\Delta^{\pm}_{4}(\mu)=0$. We do not need here the exact form of
these roots. It is quite complicated. Below we present only the form of the
separation constant $E$ for the TAE obtained making use of the Taylor series
expansions of the roots around the point $a\omega=0$.
Thus, we obtain for $|s|=2$, $k=1,2$, and $m=\pm 2,\pm 3,\dots$ the following
eight series of values:
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}=2-4ma\omega-i(-1)^{k}12\sqrt{(m-1)m(m+1)}\,(a\omega)^{3/2}+6\left(m^{2}-{\frac{7}{6}}\right)(a\omega)^{2}\\!+\\!{\cal{O}}_{5/2}(a\omega),\hskip
0.0pt$ (8.5)
and for $|s|=2$, $k=3,4$, and $m=\pm 2,\pm 3,\dots$ another eight series of
values:
$\displaystyle{}_{s}E_{\omega,k,m}^{\pm}\\!=\\!-(-1)^{k}4\sqrt{ma\omega}\left(1+\left(3m-{\frac{2}{m}}\right)a\omega+{\cal{O}}_{2}(a\omega)\right)\\!+\\!8ma\omega\\!-\\!6\\!\left(\\!m^{2}\\!-\\!{\frac{5}{6}}\\!\right)\\!(a\omega)^{2}\\!+\\!{\cal{O}}_{3}(a\omega).\hskip
14.22636pt$ (8.6)
As seen, for gravitational waves of the first polynomial class the values
(8.5) and (8.6) of the corresponding constants $E$ differ substantially from
the analogous values (4.4) and (4.1.1) of the constants $E$ obtained for the
TRE in section 4.1.1. This is in sharp contrast to the case of neutrino waves
($|s|=1/2$) of the first polynomial class and to the case of electromagnetic
waves ($|s|=1$) of this kind.
It can be shown that this phenomenon reflects the difference between the
Starobinsky constants for solutions with spin $2$ to the TAE and for solutions
with the same spin $2$ to the TRE [2, 3, 18]. The solutions to the TAE and to
the TRE with the same spin $1/2$ or $1$ have the same Starobinsky constants.
Despite the above essential difference, the first-polynomial-class-solutions
to the TAE and to the TRE with spin $2$ have similar qualitative properties,
discussed at the end of section 4.1.1.
### 8.3 Second Class of Polynomial Solutions to the TAE:
We have a finite number of second class polynomial solutions to the TAE for
which the relation $\sigma_{c}=-\sigma_{b}$ holds. For brevity, we list here
only the ones of spin 2, 1 and 1/2. For them the conditions $N\\!\geq\\!0$ and
$\sigma_{b}b_{{}_{\pm}}\\!\geq\\!0$ must be satisfied simultaneously, yielding
the requirement
$-|s|\leq-|s|+(1\pm\sigma\sigma_{m})/2\leq|m|\leq|s|-(1\mp\sigma\sigma_{m})/2\leq|s|$
– almost opposite to the analogous requirement $|m|\geq|s|$ for the polynomial
solutions of the first class. Altogether there exist only the following 32
polynomial solutions of the second class
${}_{s}S_{\omega,E,m,\mp,\pm,\mp}^{\pm}$ with spin 2, 1 and 1/2:
$\displaystyle{}_{s}S_{\omega,E,m,-,+,-}^{+}:$ $\displaystyle
s\\!=\\!+2,\,\,m\\!=\\!-1,\,\,\,\,0,\,1,\,2;\,\,s\\!=\\!+1,\,\,m\\!=\\!\,\,\,\,0,\,1;\,\,s\\!=+1/2,\,\,m\\!=+1/2,-1,2;$
(8.7) $\displaystyle{}_{s}S_{\omega,E,m,+,-,+}^{+}:$ $\displaystyle
s\\!=\\!-2,\,\,m\\!=\\!-2,-1,\,0,\,1;\,\,s\\!=\\!-1,\,\,m\\!=\\!-1,0;\,\,s\\!=-1/2,\,\,m\\!=-1/2,+1/2;$
$\displaystyle{}_{s}S_{\omega,E,m,-,+,-}^{-}:$ $\displaystyle
s\\!=\\!+2,\,\,m\\!=\\!-2,-1,\,0,\,1;\,\,s\\!=\\!+1,\,\,m\\!=\\!-1,\,0;\,\,s\\!=+1/2,\,\,m\\!=+1/2,-1/2;$
$\displaystyle{}_{s}S_{\omega,E,m,+,-,+}^{-}:$ $\displaystyle
s\\!=\\!-2,\,\,m\\!=\\!-1,\,\,\,\,0,\,1,\,2;\,\,s\\!=\\!-1,\,\,m\\!=\\!\,\,\,\,\,0,\,1,\,\,s\\!=-1/2,\,\,m\\!=-1/2,+1/2.$
The relation between the constants $E$ and $\omega$ follows from (8.1), when
$\mu$ in it is replaced by the solutions of the $\Delta_{N+1}$-condition in
the form $\Delta_{|s\pm m|}^{\pm}(\mu)=0$. Here we omit these relations.
## 9 The 256 Classes of Exact Factorized Solutions
to the Teukolsky Master Equation
Combining solutions to the TRE and to the TAE studied in the previous sections
we can construct the following 256 classes of exact factorized solutions to
the TME
$\displaystyle{}_{s}{\cal
K}^{\pm,\pm}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma},\sigma_{a},\sigma_{b},\sigma_{c}}(t,r,\theta,\varphi)=e^{-i\omega
t}e^{im\varphi}{}_{s}R^{\pm}_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}}(r;r_{+},r_{-}){}_{s}S_{\omega,E,m,\sigma_{a},\sigma_{b},\sigma_{c}}^{\pm}(\theta).$
(9.1)
For specific physical problems one has to impose specific additional
conditions, like stability conditions, boundary conditions, casuality
conditions, specific fixing of the in-out properties, regularity conditions
etc. Thus, one selects some specific combinations of solutions to the TRE and
to the TAE in Eq. (9.1) and derives the spectrum of the separation constants
$\omega$ and $E$ in the given problem.
For example, choosing solutions to the TRE which enter both the event horizon
and the 3D-space infinity we study the Kerr black holes (for $a<M$), or necked
singularities (for $a>M$) [2, 3]. If in addition we choose regular solution to
the TAE, we will obtain the standard QNM of the Kerr black holes, or necked
singularities. Using in Eq. (9.1) other solutions to the TRE and/or to the
TAE, we may hopefully describe different physical objects and phenomena, for
example, collimated jets, see in [16, 20].
The solutions (9.1) do not necessarily have a direct physical meaning, see the
Introduction. Instead, some linear combination of the specific solutions,
which obey proper boundary conditions, is to describe the Nature. In general
the solutions (9.1) have to be considered as auxiliary mathematical objects –
(maybe singular) kernels of integral representations (1.3) of the physical
solutions. The choice of the corresponding amplitudes
${}_{s}A_{\omega,E,m,\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma},\sigma_{a},\sigma_{b},\sigma_{c}}$
will fix completely the physical model and may ensure the convergence of the
integrals and discrete sums to physically acceptable solutions. Since at
present we have no rigorous mathematical treatment of this complicated issue,
we will study it in the next section using some constructive examples.
## 10 Construction of Bounded Linear Combinations of Polynomial Solutions to
the TAE
We have seen in section 8 that the polynomial solutions to the TAE are
singular and unbounded with respect to the angle $\theta$ around the N-pole,
or around S-pole. These solutions produce a singular kernel in the integral
representation (1.3). It is important to know whether it is possible to have
bounded with respect to the angle $\theta\in[0,\pi]$ solutions
${}_{s}\Psi(t,r,\theta,\varphi)$ defined by Eq. (1.3), despite the singular
character of the kernel in it. The answer to this question is a quite
nontrivial issue. Here we reach a positive answer for perturbations of spin
$1/2$ in several steps.
Let us consider the simplest case of double polynomial solutions of the first
class to the TME with spin $1/2$ and $s=\sigma/2$. For them we have an
essential simplification, since according to Eqs. (3.2) and (6.2) ${}_{s}N=0$.
Hence, the HeunC-factors in both the radial and the angular polynomial
solutions are equal to $\text{const}\equiv 1$. The value of the separation
constant $E=-a^{2}\omega^{2}+2a\omega m-{\frac{1}{4}}$ is uniquely defined in
both cases by Eqs. (4.2) and (8.3). Hence, the integration over the constant
$E$ in (1.3) produces only one term with this fixed value. As a result, the
corresponding singular kernel (9.1) is777To simplify formula (10.1), we have
omitted some constant factors in the corresponding solutions to the TRE and
TAE, which do not depend continuously on the real variables $r$ and $\theta$,
but may have different values outside the event horizon, in the domain between
the event horizon and the Cauchy horizon and inside the Cauchy horizon. This
is a legal operation, since one can include these factors in the amplitudes
${}_{s}A_{\dots}$ in the representation (1.3), considering separately the
different domains where these factors are constant.:
$\displaystyle{}_{\frac{\sigma}{2}}{\cal
K}_{\omega,E,m}(t,r,\theta,\varphi)=\delta\left(E+a^{2}\omega^{2}-2ma\omega+1/4\right)\Delta^{-\frac{1+\sigma}{4}}e^{-i\omega
T_{\sigma}}{\frac{\left(W_{\sigma}\right)^{m}}{\sqrt{\sin\theta}}},$ (10.1)
where $T_{\sigma}=t+\sigma\left(r_{*}-ia\cos\theta\right)$,
$W_{\sigma}=e^{i\phi_{\sigma}}\cot{{\theta_{\sigma}}\over 2}$,
$\phi_{\sigma}=\varphi+{\sigma\over{2p}}\ln\left|{\frac{r-r_{+}}{r-r_{-}}}\right|$,
and $\theta_{\sigma}\\!=\\!\theta$, if $\sigma\\!=\\!+1$, or
$\theta_{\sigma}\\!=\\!\pi-\theta$, if $\sigma=-1$.
The complex variable $W_{\sigma}$ defines a stereographic projection of the
two-sphere $\mathbb{S}^{(2)}_{\phi_{\sigma},\theta_{\sigma}}$ on the
compactified complex plane $\mathbb{\tilde{C}}_{W_{\sigma}}$. Its use is
critical for further analysis of the problem. Note that after the transition
from real variables $\\{\theta,\phi_{\sigma}\\}$ to the complex one
$W_{\sigma}$ one must introduce an additional phase factor
$\exp(-is\phi_{\sigma})$ in the spin-weighted spheroidal harmonics, due to the
back rotation of the basis (See the paper by Goldberg et al. in [4].). In the
case of spin $1/2$ the introduction of such factor $\exp(\mp
i\phi_{\sigma}/2)$ is equivalent to a transition in what follows from half-
integer to integer values of the azimuthal number $m$ and a replacement
$m\rightarrow m\pm 1/2$ in the factor
$\delta\left(E+a^{2}\omega^{2}-2ma\omega+1/4\right)$ in Eq. (10.1).
Taking the trivial integral on the variable $E$, one obtains from the
representation (1.3) and Eq. (10.1)
${}_{\sigma\over
2}\Psi(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}\sum_{m=-\infty}^{\infty}\left({1\over
2\pi}\int\limits_{\mathcal{L_{\omega}}}\\!\\!d\omega\,\,e^{-i\omega
T_{\sigma}}\,{}_{\sigma\over
2}A_{\omega,m}\right)\left(W_{\sigma}\right)^{m}.$ (10.2)
Since in this case we have no other restriction on the frequencies $\omega$,
different from the stability requirement $\Im(\omega)<0$, the otherwise
arbitrary integration contour ${\mathcal{L_{\omega}}}\in\mathbb{C}_{\omega}$
in (10.2) must lie in the lower complex half-plane. Suppose that the
amplitudes ${}_{\sigma\over 2}A_{\omega,m}$ and the contour
${\mathcal{L_{\omega}}}$ are chosen in such way that for all $m\in\mathbb{Z}$
there exist well defined integrals
$\displaystyle{1\over
2\pi}\int\limits_{\mathcal{L_{\omega}}}\\!\\!d\omega\,\,e^{-i\omega
T_{\sigma}}\,{}_{\sigma\over 2}A_{\omega,m}={}_{\sigma\over
2}\mathfrak{A}_{m}(T_{\sigma}).$ (10.3)
Then
${}_{\sigma\over
2}\Psi(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}\sum_{m=-\infty}^{\infty}{}_{\sigma\over
2}\mathfrak{A}_{m}\left(T_{\sigma}\right)\left(W_{\sigma}\right)^{m}.$ (10.4)
Suppose, in addition, that in some ring domain
$|W_{\sigma}|\\!\in\\!\left(|W|^{\prime},|W|^{\prime\prime}\right)$,
$0\\!<\\!|W|^{\prime}\\!<\\!|W|^{\prime\prime}\\!<\\!\infty$ the sum
$\sum_{m=-\infty}^{\infty}{}_{\sigma\over
2}\mathfrak{A}_{m}\left(T_{\sigma}\right)\left(W_{\sigma}\right)^{m}={}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ represents a convergent
Laurent series of some analytic function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$. For this purpose the
coefficients ${}_{\sigma\over 2}\mathfrak{A}_{m}\left(T_{\sigma}\right)$ in
Eq. (10.3) for $m>0$ and, independently, for $m<0$ must satisfy some of the
well-known criteria for convergence of the corresponding series. Thus, we
finally obtain a solution to the TME with spin $1/2$ which depends on an
arbitrary analytic function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ of the two variables
$T_{\sigma}$ and $W_{\sigma}$:
${}_{\sigma\over
2}\Psi(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}\,\,{}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right).$ (10.5)
Returning to the Boyer-Lindquist variables one can check directly that (10.5)
indeed gives a general solution to the TME with spin $1/2$. The explicit form
of the variable $T_{\sigma}$ shows that outside the event horizon these
solutions describe one-way-running waves: outgoing to space infinity running
waves – for $\sigma=-1$ and incoming from space infinity running waves – for
$\sigma=+1$.
Now it is easy to remove the singularities from the $z$-axis, i.e. on the
poles $\theta=0,\pi$. For example, let us choose ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)=1/\sqrt{\left(W_{\sigma}+W_{\sigma}^{-1}\right)/2}$.
Then ${}_{\sigma\over
2}\Psi(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}/{\sqrt{1-\sin^{2}\phi_{\sigma}\sin^{2}\theta_{\sigma}}}$
has no singularities on the poles $\theta_{\sigma}=0,\pi$, but this way we
have worked out two new singular lines
$\phi_{\sigma}=\varphi+{\sigma\over{2p}}\ln\left|{\frac{r-r_{+}}{r-r_{-}}}\right|=\pm\pi/2$
on the equatorial plane $\theta=\pi/2$. Hence, this way the singular line of
the solution has been only deformed and translated to a new position. The same
happens if we choose the more general function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)=1/\sqrt{\left(a(T_{\sigma})W_{\sigma}+b(T_{\sigma})W_{\sigma}^{-1}+c(T_{\sigma})\right)/2}$.
In this case, the singular $z$-axis will be deformed, translated and doubled
to the non-static singular lines
$\phi_{\sigma}=\varphi+{\sigma\over{2p}}\ln\left|{\frac{r-r_{+}}{r-r_{-}}}\right|=\phi_{1,2}=\arg(W_{1,2})$
on the (in general) moving cones
$\theta\\!=\\!\theta_{1,2}\\!=\\!\arctan\left(|W_{1,2}|^{-1}\right)$, where
$W_{1,2}$ are the two roots of the equation
$a(T_{\sigma})W_{\sigma}\\!+\\!b(T_{\sigma})W_{\sigma}^{-1}\\!+\\!c(T_{\sigma})\\!=\\!0$.
Here we have chosen a special form of the function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ which yields finite nonzero
values of the solution on the poles $\theta=0,\pi$.
It is possible to chose the function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ with the denominator which
is a sum of polynomials of higher degree with respect to variables
$W_{\sigma}$ and $W_{\sigma}^{-1}$. Then the solution (10.5) equals zero at
the N and S-poles and we can work out an arbitrary number of singular lines of
the solution related to the zeros of the denominator. At first glance, this
possibility may not seem to be interesting for the physical applications,
since on the singular lines the linear perturbation theory in use is not
applicable. We mention it here just to have a clear mathematical picture. It
is interesting to study the same situation in the whole nonlinear theory and
to know whether in it the singular lines may be replaced by regular ones. If
so, the perturbation theory under consideration indicates a possible
complicated structure of the exact radiation field on the Kerr background.
The most important question for a correct application of the linear
perturbation theory under consideration, is whether one can find a regular
analytical function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ without singularities in the
complex plane $\mathbb{C}_{W_{\sigma}}/\\{0,\infty\\}$, i.e., with the points
$W_{\sigma}=0$ and $W_{\sigma}=\infty$ punctured and which, in addition, can
remove the unbounded increase of the solutions due to the singularities of the
factor $\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}$ in (10.5). We
give a positive answer to this question constructing two explicit examples:
1\. Using the basic equality
${\sum\limits_{m=-\infty}^{\infty}}W^{m}I_{m}(z)\\!=\\!\exp\left({1\over
2}\left(W\\!+\\!W^{-1}\right)z\right)$ for the modified Bessel functions
$I_{m}(z)$ [27] we choose the coefficients in (10.4) in the specific form
${}_{\sigma\over
2}\mathfrak{A}_{m}\left(T_{\sigma}\right)=\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma}^{\,2}\right)I_{m}(\omega T_{\sigma})$, where
$\omega=\omega_{R}+i\omega_{I}$ is a fixed frequency and
$\bar{\sigma}=\text{sign}(|\omega_{R}|-|\omega_{I}|)$. Then
${}_{\sigma\over
2}\Psi_{\omega}(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}\,\,\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma}^{\,2}\right)\exp\left({1\over
2}\left(W_{\sigma}\\!+\\!W_{\sigma}^{-1}\right)\omega T_{\sigma}\right)$
(10.6)
is a stable solution, since by construction it goes to zero when
$t\to+\infty$. It is not difficult to obtain its limit when
$\theta_{\sigma}\to 0,\pi$ in the form
$\displaystyle\lim\limits_{\theta_{\sigma}\to 0,\pi}\left({}_{\sigma\over
2}\Psi_{\omega}(t,r,\theta,\varphi)\right)=\Delta(r)^{-\frac{1+\sigma}{4}}\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma;0,\pi}^{2}\right)\times\hskip 96.73918pt$
$\displaystyle\times\lim\limits_{\theta_{\sigma}\to
0,\pi}\left({1\over{\sqrt{\sin\theta}}}\exp\left({\frac{|\omega|\sqrt{(t+\sigma
r_{*})^{2}+a^{2}}}{\sin\theta}e^{i\Upsilon_{\omega\sigma;0,\pi}}}\right)\right).$
(10.7)
Here
$\displaystyle\Upsilon_{\omega,\sigma;0,\pi}=\pm\Bigg{(}\varphi+{\sigma\over{2p}}\ln\left|{\frac{r-r_{+}}{r-r_{-}}}\right|-\sigma\arctan\left({\frac{a}{t+\sigma
r_{*}}}\right)\Bigg{)}+\arg(\omega),\,\,\,\text{for}\,\,\,\theta=0,\,\,\text{or}\,\,\pi$
(10.8)
is the limit of the total phase of the term ${1\over
2}\left(W_{\sigma}\\!+\\!W_{\sigma}^{-1}\right)\omega T_{\sigma}$ and
$T_{\sigma;0,\pi}=t+\sigma(r_{*}\mp ia)$. In Eq. (10.8) the sign $(+)$
corresponds to the limit $\theta_{\sigma}\to 0$ and the sign $(-)$ – to the
limit $\theta_{\sigma}\to\pi$. Formula (10) shows that when
$\Upsilon_{\omega,\sigma;0,\pi}\in\left(-{\pi\over 2},{\pi\over 2}\right)$ the
solution ${}_{\sigma\over 2}\Psi_{\omega}(t,r,\theta,\varphi)$ is bounded
everywhere in the interval $\theta\in[0,\pi]$, since in this case
$\lim\limits_{\theta_{\sigma}\to 0,\pi}\left({}_{\sigma\over
2}\Psi_{\omega}(t,r,\theta,\varphi)\right)=0$. Otherwise this limit diverges
and the solution is singular and unbounded around the poles.
Actually, the value of the parameter $\Upsilon_{\omega,\sigma;0,\pi}$ is not
defined from a geometrical point of view, because the value of the angle
$\varphi$ is completely arbitrary on the poles $\theta_{\sigma}=0,\pi$. As a
result, we can choose any value of the parameter
$\Upsilon_{\omega,\sigma;0,\pi}$ without changing the geometrical points
associated with the N and S-poles of the sphere
$\mathbb{S}^{(2)}_{\theta,\varphi}$. Since the different values of this
parameter yield different solutions of the TAE, we see that under the boundary
conditions at hand the corresponding differential operator is not self-adjoint
[28], but its self-adjoint extensions do exist and can be fixed by suitable
fixing of the free parameter $\Upsilon_{\omega,\sigma;0,\pi}$. An analogous
phenomenon is well known for the potentials $V(x)\sim$ $1/x^{2}$, or $1/r^{2}$
in quantum mechanics [28]. Note that around the poles $\theta_{\sigma}=0,\pi$
the potential in the TAE (1.1) has precisely the same behavior:
${}_{s}W_{\omega,E,m}(\theta)\sim 1/\theta^{2}$ for $\theta\to 0$, and
${}_{s}W_{\omega,E,m}(\theta)\sim 1/(\theta-\pi)^{2}$ for $\theta\to\pi$. In
our case, the fixing of the parameter
$\Upsilon_{\omega,\sigma;0,\pi}\in\left(-{\pi\over 2},{\pi\over 2}\right)$
makes the solutions (10) to the TME for spin $1/2$ smooth and bounded
everywhere in the interval $\theta\in[0,\pi]$, i.e., physically acceptable.
2\. Another solution, which is finite everywhere in the interval
$\theta\in[0,\pi]$ but has an infinite number of bounded oscillations around
the poles $\theta=0,\pi$ can be obtained using the following equality for the
Bessel functions $J_{m}(z)$:
${\sum\limits_{m=-\infty}^{\infty}}(-1)^{m}W^{2m}\left(J_{m}(z)\right)^{2}\\!=\\!J_{0}\Big{(}\left(W\\!+\\!W^{-1}\right)z\Big{)}$
[27]. Now we choose the coefficients in (10.4) in the specific form
${}_{\sigma\over
2}\mathfrak{A}_{2m}\left(T_{\sigma}\right)=(-1)^{m}\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma}^{\,2}\right)\left(J_{2m}(\omega T_{\sigma})\right)^{2}$
and ${}_{\sigma\over 2}\mathfrak{A}_{2m+1}\left(T_{\sigma}\right)=0$ using the
same notation as in the previous example. Then
${}_{\sigma\over
2}\Psi_{\omega}(t,r,\theta,\varphi)=\Delta(r)^{-\frac{1+\sigma}{4}}\sqrt{\left({|W_{\sigma}|+|W_{\sigma}|^{-1}}\right)/2}\,\,\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma}^{\,2}\right)J_{0}\Big{(}\left(W_{\sigma}\\!+\\!W_{\sigma}^{-1}\right)\omega
T_{\sigma}\Big{)}$ (10.9)
is a stable solution to the TME with spin $1/2$. Taking into account the
asymptotic expansion of the Bessel function $J_{0}(z)\sim\sqrt{{\frac{2}{\pi
z}}}\cos(z-\pi/4)$ we obtain in the limits $\theta_{\sigma}\to 0,\pi$:
$\displaystyle\lim\limits_{\theta_{\sigma}\to 0,\pi}\left({}_{\sigma\over
2}\Psi_{\omega}(t,r,\theta,\varphi)\right)=\Delta(r)^{-\frac{1+\sigma}{4}}\exp\left(-{\bar{\sigma}\over
2}\omega^{2}T_{\sigma;0,\pi}^{\,2}\right){1\over{\sqrt{\pi\omega
T_{\sigma;0,\pi}}}}\times\hskip 71.13188pt$
$\displaystyle\times\lim\limits_{\theta_{\sigma}\to
0,\pi}\left(\cos\left({\frac{2|\omega|\sqrt{(t+\sigma
r_{*})^{2}+a^{2}}}{\sin\theta}e^{i\Upsilon_{\omega\sigma;0,\pi}}}\right)\right).$
(10.10)
As seen from Eq. (10), there exist only two choices of the free parameter:
$\Upsilon_{\omega,\sigma;0,\pi}=0,\pi$, for which the solutions (10.9) are
finite everywhere in the interval $\theta\in[0,\pi]$ – a critical property for
the use of the linear perturbation theory. Approaching these poles the
solutions oscillate infinitely many times with bounded finite amplitudes. In
this sense, the N and S-poles remain singularities of the bounded solutions
(10.9). Moreover, the gradients of the bounded solutions (10.9) are unbounded
around the poles.
Obviously, superpositions of solutions (10.6), or (10.9) with different
complex parameters $\omega$, running in some (discrete or continuous) sets in
$\mathbb{C}_{\omega}$, describe more general bounded solutions to the TAE with
spin $1/2$.
One more remark. In the case $\sigma=+1$ the solutions (10.5) are unbounded on
the horizons $r_{\pm}$ due to the factor $\Delta(r)^{-1/2}$. These stationary
singularities cannot be removed by any choice of the function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$, since it depends on the two
variables $T_{\sigma}$ and $W_{\sigma}$, not on the single one $r$. The
variables $t,r,\theta$ enter in $T_{\sigma}$ and the variables
$\varphi,r,\theta$ enter in $W_{\sigma}$ in a complex way. As a result, the
variable $r$ cannot be disentangled from the the function ${}_{\sigma\over
2}\mathfrak{A}\left(T_{\sigma},W_{\sigma}\right)$ and one is not able to
compensate the singularity due to the factor $\Delta(r)^{-1/2}$ which does not
depend neither on the time $t$, nor on the angles $\varphi$ and $\theta$.
## 11 Conclusion
In the present paper, we have demonstrated that the confluent Heun functions
are an adequate and natural tool for a unified description of linear
perturbations of gravitational field of the Kerr metric outside the event
horizon, as well as in interior domains. These functions give us an effective
tool for exact mathematical treatment of different boundary problems and the
corresponding physical phenomena. They can help us to solve old mathematical
issues, related to the Teukolsky separation of the variables, as well as to
study new physical problems. The same approach works, too, for the Regge-
Wheeler and Zerilli equations in the Schwarzschild metric [15, 18].
Large classes of exact solutions to the perturbation equations of the Kerr
metric were described here for the first time. All possible types of solutions
were classified uniformly in terms of confluent Heun’s functions and confluent
Heun’s polynomials, using their specific properties. As we saw, the variety of
the different solutions and possible spectra is much reacher than, for
example, the variety of the corresponding solutions and spectra of the
Hydrogen problem in quantum mechanics, solved exactly in terms of the
confluent hypergeometric functions [28]. Mathematically, this is obviously
caused by the presence of one more regular singular point in the confluent
Heun equation.
We have to stress especially the newly obtained singular polynomial solutions
to the Teukolsky angular equation. They differ drastically from the well-known
analogous regular polynomial solutions to the quantum Hydrogen problem. The
singular polynomial solutions to the Teukolsky angular equation present an
auxiliary mathematical construction which seems suitable for simple and
natural perturbative description of the collimation of radiated fields of all
spins $|s|>0$ in the Kerr metric, see [16, 20].
For spin $1/2$ we have proved that the singular kernels, constructed from
polynomial solutions, can produce bounded solutions of the continuous spectrum
to the TME with very interesting physical properties: These solutions describe
collimated one-way running waves in the Teukolsky perturbation theory
correctly.
For spin 1 we have also proved the existence of double polynomial solutions of
the continuous spectrum to the TME and can reach similar results in a more
complicated way, since there we meet a new physical phenomenon – the
electromagnetic superradiance [29, 2, 3]. For spin 2 the supperradiance is
known to be quite stronger, but we have no continuous spectrum of the TME and
the problem needs special treatment. We shall consider these two important
cases separately.
One can hope that the collimated one-way running waves, cropping up for the
first time in the present paper, are able to describe the real astrophysical
jets, observed at very different scales in the Universe. This still
speculative idea needs a more detailed mathematical development and a careful
confrontation with the real astrophysical observations. It indicates the
existence of a new universal mechanism for collimation of radiation of all
spins $|s|>0$ by the pure gravitational field of rotating compact
astrophysical objects of different nature.
Acknowledgments I am thankful to Kostas Kokkotas, Luciano Rezzolla and Edward
Malec for the stimulating discussion of exact solutions to the Regge-Wheeler
and Teukolsky equations and different boundary problems during the XXIV
Spanish Relativity Meeting, E.R.E. 2006, to Edward Malec for his kind
invitation to visit the Astrophysical Group of the Uniwersytet Jagiellonski,
Crakow, Poland in May 2007 and to participants of the seminar there, to
participants of the Conference ”Gravity, Astrophysics and Strings at the Black
Sea” 2007, Primorsko, Bulgaria, to Goran Djeorjevic for his kind invitation to
visit the Department of Physics, University of Nis, Serbia in December 2007
and to give a talk there, to Luciano Rezzolla – for his kind invitation to
visit the Albert Einstein Institute of Gravitational Physics in Golm, Germany
in March 2009 and to members of the Numerical Relativity Group there for
numerous discussions.
The author is thankful to Denitsa Staicova and Roumen Borissov for numerous
discussions during the preparation of the present paper, to Shahar Hod – for
his kind help in enriching the references, to Jerome Gariel for drawing
attention to the early papers by Marcilhacy G and by Blandin J, Pons R,
Marcilhacy G (see [13]) and for sending the corresponding copies, to Sam Dolan
for drawing attention to the first paper in [4] and for sending a copy of his
PhD Thesis and to and to Sigbjorn Hervik for drawing my attention to the
article by Kayll Lake in [23]..
I would like to express my special gratitude to Professor Saul Teukolsky for
his comments on the present paper and his useful suggestions, as well as to
Professor Alexey Starobinsky, to Professor George Alekseev, to Professor
Sergei Slavyanov, to Professor Irina Aref’eva and to Professor Igor Volovich
for the useful discussions and comments.
I am also thankful to the Bogolubov Laboratory of Theoretical Physics, JUNR,
Dubna, Russia for the hospitality and good working conditions during my stay
there in the summer of 2009 and in February 2010.
This paper was supported by the Foundation ”Theoretical and Computational
Physics and Astrophysics” and by the Bulgarian National Scientific Found under
contracts DO-1-872, DO-1-895 and DO-02-136.
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|
arxiv-papers
| 2009-08-28T16:16:47 |
2024-09-04T02:49:04.919675
|
{
"license": "Public Domain",
"authors": "Plamen P. Fiziev",
"submitter": "Plamen Fiziev",
"url": "https://arxiv.org/abs/0908.4234"
}
|
0908.4250
|
# Superconducting Quantum Interference Device Amplifiers with over 27 GHz of
Gain-Bandwidth Product Operated in the 4 GHz–8 GHz Frequency Range111This
paper is a contribution of the U.S. government and is not subject to U.S.
copyright.
Lafe Spietz lafe@nist.gov Kent Irwin José Aumentado National Institute of
Standards and Technology, Boulder, Colorado 80305, USA
###### Abstract
We describe the performance of amplifiers in the 4 GHz–8 GHz range using
Direct Current Superconducting Quantum Interference Devices(DC SQUIDs) in a
lumped element configuration. We have used external impedance transformers to
couple power into and out of the DC SQUIDs. By choosing appropriate values for
coupling capacitors, resonator lengths and output component values, we have
demonstrated useful gains in several frequency ranges with different
bandwidths, showing over 27 GHz of power gain-bandwidth product. In this work,
we describe our design for the 4 GHz–8 GHz range and present data
demonstrating gain, bandwidth, dynamic range, and drift characteristics.
The development of low-noise microwave amplifiers is an increasingly critical
need in low-temperature physics. One technology that has proven to have some
of the best characteristics in terms of gain-bandwidth product and overall
noise performance is the DC SQUID(Direct Current Superconducting Quantum
Interference Device)-based microwave amplifierMück et al. (1998, 2003);
Tarasov et al. (1995). The SQUID has been shown to have better noise than
similar semiconductor-based amplifiersMück et al. (2001), and better gain-
bandwidth product, ease of use and power handling capabilities than parametric
amplifiers that operate at or below the standard quantum limitCastellanos-
Beltran et al. (2008). The SQUID also has power dissipation as much as four
orders of magnitude lower than HEMT(High Electron Mobility Transistor)
amplifiers.
In recent years, the 4 GHz–8 GHz frequency range, known as the C-band, has
proven to be of great importance for a wide variety of quantum measurement
experimentsWallraff et al. (2004), and so we have chosen to focus our design
efforts primarily on that band. Based on the model of the input impedance of a
lumped-element DC SQUID described and experimentally verified elsewhereSpietz
et al. (2008), we have designed input transformers with hundreds of megahertz
of instantaneous bandwidth at frequencies in the range of 4 GHz–8 GHz. Our
amplifiers consist of a quarter-wave resonator with a coupling capacitor in
the 50 fF–100 fF range coupled to a section of transmission line varying in
length from 1.4 mm to 3.5 mm, which feeds into the input coil of the SQUID.
The output of the SQUID is then coupled to the 50 $\Omega$ transmission line
by a multipole transformer on the chip(see Fig. 1). The purpose of these
measurements was threefold. Firstly, we wanted to show that we could build
amplifiers in the 4 GHz–8 GHz range with useful gains and bandwidths.
Secondly, we wanted to show that we could tune the center frequencies of the
amplifiers’ gains by changing the length of the input resonator in a
predictable way that would allow us to design for specific frequencies.
Finally, by measuring amplifiers with a variety of input resonator lengths we
were able to characterize the imaginary component of the input impedance,
which will be necessary for future, more complex input transformer designs.
Figure 1: Diagram of 100 fF amplifier chip with three levels of magnification
(a,b and c), adapted from CAD files used for fabrication. The output
transformer consists of a set of two overlap capacitors to ground and two
series spiral inductors, designed to match over as much of the target band as
possible. Another overlap capacitor couples the input line to the quarter wave
resonator, which is terminated in the input coil of the SQUID.
Our SQUIDs, also described in reference Spietz et al. (2008), are designed to
behave as lumped-element components by keeping the physical size of the SQUID
below 200 $\upmu$m and use a slotted washer to minimize input stray
capacitance, moving parasitic self-resonances to frequencies well above the
measurement band. The SQUIDs are also designed with a second-order gradiometer
configuration to minimize magnetic pickup as well as the self inductance of
the SQUID loop (18 pH)Zimmerman and Frederick (1971); Drung et al. (2007);
Dantsker et al. (1997). All SQUIDs are fabricated by use of a Nb/AlOx/Nb
optical trilayer process with 60 $\upmu$A critical current for each junction,
shunted by 2.3 $\Omega$ resistors, and with two wiring layers and two
insulating layers of SiO2. The flux-to-voltage transfer function $\partial
V/\partial\phi$ at the flux and current bias points with useful gain was in
the range of 200 $\upmu V/\Phi_{0}$–300 $\upmu V/\Phi_{0}$. Each SQUID has a
dc flux bias coil that winds half a turn around each of the four lobes for the
dc flux bias, and a 600 pH microwave flux bias coil that winds a turn and a
half around each lobe for the microwave input coil.
Figure 2: Length of resonator as a function of frequency of best input match
for a 60 fF input capacitor amplifier. Each point represents a set of
measurements on a separate amplifier with a different length input resonator.
The continuous line is a fit to a second-order polynomial. Note that the
frequency dependence is smooth, allowing for interpolation as needed to design
for any frequency in the band. Data are displayed with frequency as the
independent variable since this is intended to be used as a design tool where
the frequency is selected based on some application, and the length is chosen
to match that frequency.
We measured the input return loss of amplifiers with several different lengths
in order to determine empirically how to design for different frequencies in
the 4 GHz–8 GHz band and in order to determine the imaginary component of the
input impedance as a function of frequency. For each length of resonator, we
measured the returns loss over a range of flux bias and current bias. For each
return loss curve, we extracted the frequency at which the most power was
absorbed, and then put that frequency into a weighted average frequency,
weighting for the depth of the resonance in linear power units. By taking such
weighted averages, we were able to compare a large quantity of data on each
amplifier by use of a single average frequency number. Fig. 2 plots the
resonator length for a given center frequency against this weighted average
matching frequency. An understanding of the quarter wave input circuit permits
us to use these data to find that the imaginary component of the input
impedance ranges from about 30 $\Omega$ to about 60 $\Omega$. We used similar
methods to determine the real component of the input impedance, which we
presented in an earlier paper Spietz et al. (2008).
In addition to the reflection measurements described above, we measure the
power gain of our amplifiers at the base temperature(approximately 40 mK) of a
dilution refrigerator. We use a cold microwave transfer switch, to bypass the
amplifier and obtain an in situ calibration. Thus, our gain measurement
determines the usable power gain from the input SMA connector of the amplifier
box to the output SMA connectoragi .
Figure 3: (a) Gains for several flux bias points, ranging over about 5 % of a
flux quantum, and fixed 150 $\upmu$A current bias of the amplifier with a 60
fF coupling capacitor. Note the trade-off between gain and bandwidth in the
different curves. (b) High bandwidth bias point(current bias of approximately
140 $\upmu$A) for 100 fF coupling capacitor amplifier. The data shown on this
plot demonstrate over 27 GHz of gain-bandwidth product (the integral of linear
power gain over the frequency range). (c) Same bias point and amplifier as
(b), showing that there are multiple gain maxima, and that feedback effects
cause surprisingly complex frequency dependence in the gain.
Fig. 3 shows several gain curves for two different amplifiers at different
flux and current bias points. The frequency dependence of the gain is a
complicated function of the bias points. We find that for amplifiers with a 60
fF input coupling capacitor, the gain curves are smooth and simple, showing
the trade-off between gain and bandwidth. While these gains and bandwidths are
enough to be useful for many applications, more bandwidth is desirable, and we
find that the bandwidths of the 100 fF input capacitor amplifiers are much
higher.
In both the gain data and the return loss data we observe frequencies and
bandwidths that differ widely from those expected from simply the frequency
dependence of the input and output transformers and the SQUID. If a resonator
coupled to a gain element such as a SQUID has some feedback mechanism, the
magnitude and phase of that feedback can have profound effects on the
frequency dependence of the overall circuit, leading to drastic suppression or
enhancement of Q as well as shifts in frequency by many line widths. Although
one might expect only moderate increases in bandwidth going from 60 fF to 100
fF, we observe increased bandwidth by as much as an order of magnitude, with
over 1 GHz of potentially useful gain for some bias points in the 100 fF
amplifier. We believe that this is due to these complex feedback effects of
the SQUID, and that greatly increased bandwidth should be possible with
further study of the SQUIDs intrinsic S-parameters similar to that carried out
with microwave transistors.
Figure 4: (a) Total microwave measurement chain drift as a function of time
overnight. Inset (b) shows dependence of maximum gain on the flux bias.
We find, via a set of drift and bias dependence measurements, that the
amplifier gain is stable over many hours. Fig. 4 shows both the way that gain
depends on flux bias in a small range of biases, and how the long-term
stability of the amplifier corresponds to possible flux drift. The drift data
were taken by recording a series of gain traces as a function of frequency at
a range of flux bias points every fifteen minutes for approximately 10 hours.
Between measurements, both the flux and current bias were turned off. Thus
they show not just the stability of the amplifier, but the repeatability of
the bias. This repeatability is important for a practical amplifier, because
that makes it possible to use the same bias point each time. It is worth
noting that this drift plot represents the entire microwave measurement chain,
including two semiconductor amplifiers (one at 4 K and one at room
temperature) and several cables and passive components, all of which could
contribute to the observed drift. These drift data also show the value of the
magnetic noise immunity provided by the second-order gradiometer configuration
of the SQUIDs. We had one layer of high permeability magnetic shielding, and
no superconducting shield. Movement of large ferrous objects near the dewar
had no observable effect on the bias point of the SQUID.
Figure 5: Gain at fixed bias point as a function of amplifier input power.
Another important figure of merit in any amplifier is its power-handling
capability, which determines the dynamic range. We have measured the gain as a
function of input power for a variety of bias points, and find that while the
power-handling capabilities fall short of HEMT amplifiers, they are high
enough to be useful. The 1 dB compression point is at approximately -110 dBm,
which should be more than sufficient for most applications where a near-
quantum-limited amplifier is required.
In conclusion, this amplifier has the instantaneous bandwidth, center
frequencies, power handling and stability required to be useful for a range of
quantum-measurement experiments. We are still measuring noise performance, but
preliminary SNR improvement measurements show that system noise can be
improved by an order of magnitude over typical measurements with HEMT
amplifiers. With careful selection of bias point and input circuit we believe
this amplifier could, in its present form, lead to improvements in dispersive
qubit readoutsSchoelkopf . The gain-bandwidth product demonstrated implies
that it should be possible to construct an amplifier with a full 4 GHz–8 GHz
range by sacrificing gain for bandwidth, then cascading two or three
amplifiers together. An amplifier with these characteristics would greatly
benefit the low-temperature physics community.
###### Acknowledgements.
We thank Konrad Lehnert, Michel Devoret, Dan Schmidt, Michael Elsbury, and Rob
Schoelkopf for useful discussions. We thank NSA for support.
## References
* Mück et al. (1998) M. Mück, M.-O. André, J. Clarke, J. Gail, and C. Heiden, Applied Physics Letters 72, 2885 (1998).
* Mück et al. (2003) M. Mück, C. Welzel, and J. Clarke, Applied Physics Letters 82, 3266 (2003).
* Tarasov et al. (1995) M. Tarasov, G. Prokopenko, V. Koshelets, I. Lapitskaya, and L. Filippenko, Applied Superconductivity, IEEE Transactions on 5, 3226 (1995).
* Mück et al. (2001) M. Mück, J. B. Kycia, and J. Clarke, Applied Physics Letters 78, 967 (2001).
* Castellanos-Beltran et al. (2008) M. Castellanos-Beltran, K. Irwin, G. Hilton, L. Vale, and K. Lehnert, Nature Physics 4, 929 (2008).
* Wallraff et al. (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature (London) 431, 162 (2004).
* Spietz et al. (2008) L. Spietz, K. Irwin, and J. Aumentado, Applied Physics Letters 93, 082506 (2008).
* Zimmerman and Frederick (1971) J. E. Zimmerman and N. V. Frederick, Applied Physics Letters 19, 16 (1971).
* Drung et al. (2007) D. Drung, C. Assmann, J. Beyer, A. Kirste, M. Peters, F. Ruede, and T. Schurig, IEEE Trans. Appl. Super. 17, 699 (2007).
* Dantsker et al. (1997) E. Dantsker, S. Tanaka, and J. Clarke, Applied Physics Letters 70, 2037 (1997).
* Mück et al. (2009) M. Mück, D. Hover, S. Sendelbach, and R. McDermott, Applied Physics Letters 94, 132509 (2009).
* (12) Agilent application note AN1287-3.
* (13) R. J. Schoelkopf, private communication.
|
arxiv-papers
| 2009-08-28T17:51:54 |
2024-09-04T02:49:04.931278
|
{
"license": "Public Domain",
"authors": "Lafe Spietz, Kent Irwin, and Jose Aumentado",
"submitter": "Lafe Spietz",
"url": "https://arxiv.org/abs/0908.4250"
}
|
0908.4439
|
# Estimates for the higher order buckling eigenvalues in the unit sphere
111This research is supported by NSFC of China (No. 10671181), Project of
Henan Provincial department of Sciences and Technology (No. 092300410143), and
NSF of Henan Provincial Education department (No. 2009A110010).
Guangyue Huang, Xingxiao Li 222The corresponding author. Email:
xxl$@$henannu.edu.cn, Xuerong Qi
Department of Mathematics, Henan Normal University
Xinxiang 453007, Henan, P.R. China
(August 12, 2009)
> Abstract. We consider the higher order buckling eigenvalues of the following
> Dirichlet poly-Laplacian in the unit sphere
> $(-\Delta)^{p}u=\Lambda(-\Delta)u$ with order $p(\geq 2)$. We obtain
> universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th
> eigenvalues independent of the domains. In particular, for $p=2$, our result
> is sharp than estimates on eigenvalues of the buckling problem obtained by
> Wang and Xia in [19].
> Keywords: eigenvalue, poly-Laplacian, buckling problem, unit sphere.
> Mathematics Subject Classification: Primary 35P15, Secondary 53C20.
## 1 Introduction
Let $\Omega$ be a connected bounded domain in an $n$-dimensional complete
Riemannian manifold $M$.
Assume that $\lambda_{i}$ is the $i$th eigenvalue of the Dirichlet poly-
Laplacian with order $p$:
$\left\\{\,\vbox{\openup 3.0pt\halign{$\displaystyle{#}$\hfil&
\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u=\lambda
u&\ \ {\rm in}\ \Omega,\\\u=\frac{\partial
u}{\partial\nu}=\cdots=\frac{\partial^{p-1}u}{\partial\nu^{p-1}}=0&\ \ {\rm
on}\ \partial\Omega,\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$ (1.1)
where $\Delta$ is the Laplacian in $M$ and $\nu$ denotes the outward unit
normal vector field of $\partial\Omega$. Let
$0<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots\rightarrow+\infty$
denote the successive eigenvalues for (1.1), where each eigenvalue is repeated
according to its multiplicity. When $p=1$, it is well known that the
eigenvalue problem (1.1) is called a fixed membrane problem and it is called a
clamped plate problem when $p=2$. For any $p$ and $M=\mathbb{R}^{n}$, Cheng-
Ichikawa-Mametsuka proved in [5] the following inequality of the type of Yang:
$\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4p(2p+n-2)}{n^{2}}\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\lambda_{i}.$
(1.2)
In particular, when $p=1$, the inequality (1.2) becomes the following
inequality of Yang in [22]:
$\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4}{n}\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\lambda_{i}.$
In an excellent paper of Cheng-Ichikawa-Mametsuka [4], by introducing
functions $a_{i}$ and $b_{i}$, they considered the eigenvalue problem (1.1)
with any order $p$ and $M=\mathbb{S}^{n}(1)$. They proved that
$\displaystyle\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq$
$\displaystyle\frac{4}{n^{2}}\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\left\\{\left(\lambda_{i}^{\frac{1}{p}}+n\right)^{p}-\lambda_{i}\right.$
(1.3)
$\displaystyle\left.+4\left(2^{p}-(p+1)\right)\lambda_{i}^{\frac{1}{p}}\left(\lambda_{i}^{\frac{1}{p}}+n\right)^{p-2}\right\\}\left(\lambda_{1}^{\frac{1}{p}}+\frac{n^{2}}{4}\right).$
(1.4)
We remark that the inequality (2.19) in [6] of Cheng-Yang and inequality
(4.16) in [18] of Wang-Xia are included in the inequality (1.3). For the
related research and important improvement in eigenvalue problem (1.1), we
refer to [1, 17, 11, 16, 7, 20, 15, 10, 14, 8, 21, 2, 3] and the references
therein.
Now assume that $\Lambda_{i}$ is the $i$th eigenvalue of the following
Dirichlet poly-Laplacian with order $p\ (\geq 2)$:
$\left\\{\,\vbox{\openup 3.0pt\halign{$\displaystyle{#}$\hfil&
\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u=\Lambda(-\Delta)u&\
\ {\rm in}\ \Omega,\\\u=\frac{\partial
u}{\partial\nu}=\cdots=\frac{\partial^{p-1}u}{\partial\nu^{p-1}}=0&\ \ {\rm
on}\ \partial\Omega.\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$ (1.5)
It is well known that this problem has a discrete spectrum
$0<\Lambda_{1}\leq\Lambda_{2}\leq\Lambda_{3}\leq\cdots\rightarrow+\infty$,
where each eigenvalue is repeated according to its multiplicity. When $p=2$,
the eigenvalue problem (1.5) is called a buckling problem. By introducing a
new method to construct nice trial functions, Cheng-Yang obtained in [9] that,
for $p=2$ and $M=\mathbb{R}^{n}$,
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4(n+2)}{n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
(1.6)
As a generalization of inequality (1.6), Huang-Li [12] considered the problem
(1.5) with any order $p$. In fact, for $M=\mathbb{R}^{n}$, they proved that
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4(p-1)(n+2p-2)}{n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}^{\frac{2p-3}{p-1}}.$
(1.7)
In 2007, Wang and Xia [19] considered this problem when $p=2$ and
$M=\mathbb{S}^{n}(1)$. They proved that, for any $\delta>0$,
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}+\frac{\delta^{2}(\Lambda_{i}-(n-2))}{4(\delta\Lambda_{i}+n-2)}\right)$
(1.8)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(1.9)
We remark that the right hand side of inequality (1.8) depends on $\delta$. In
a recent paper, by introducing a new parameter and using Cauchy inequality,
Huang-Li-Cao [13] obtain the following stronger inequality than (1.8) which is
independent of $\delta$:
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(1.10) $\displaystyle\leq$ $\displaystyle
2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{\frac{1}{2}}$
(1.11)
$\displaystyle\qquad\qquad\qquad\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{\frac{1}{2}}.$
(1.12)
Motivated by the idea used in [4], we consider in this paper the eigenvalue
problem (1.5) for any integer $p\ (\geq 2)$ when $M$ is $\mathbb{S}^{n}(1)$.
We obtain the following results:
###### Theorem 1.1.
Let $\Omega$ be a connected bounded domain in an $n$-dimensional unit sphere
$\mathbb{S}^{n}(1)$. Assume that $\Lambda_{i}$ is the $i$th eigenvalue of the
eigenvalue problem (1.5) with $p\geq 2$. Then, we have
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(1.13) $\displaystyle\leq$ $\displaystyle
2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\right\\}^{\frac{1}{2}}$
(1.14) $\displaystyle\hskip
68.28644pt\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{\frac{1}{2}},$
(1.15)
where
$\displaystyle f(\Lambda_{i},n)=$
$\displaystyle{\frac{1}{2(n-1)}}\left(\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-1}-\left(\Lambda_{i}^{\frac{1}{p-1}}-n+2\right)^{p-1}\right)$
$\displaystyle+\frac{n}{(n-1)}\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-2}-{\frac{1}{n-1}}\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}-n+2\right)^{p-2}$
$\displaystyle+2\left(2^{p-1}-p\right)\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-3}$
$\displaystyle+4(2^{p-2}-(p-1))\Lambda_{i}^{\frac{2}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-4}.$
###### Corollary 1.2.
Under the assumptions of Theorem 1.1, we have
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\sum_{i=1}^{k}$
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)$
(1.16) $\displaystyle\hskip
113.81102pt\times\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)$ (1.17)
and
$\Lambda_{k+1}\leq S_{k+1}+\sqrt{S_{k+1}^{2}-T_{k+1}},$ (1.18)
$\Lambda_{k+1}-\Lambda_{k}\leq 2\sqrt{S_{k+1}^{2}-T_{k+1}},$ (1.19)
where
$S_{k+1}=\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}+\frac{1}{2k}\sum_{i=1}^{k}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right),$
$T_{k+1}=\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}^{2}+\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
###### Remark 1.1.
When $p=2$, we have $f(\Lambda_{i},n)=\Lambda_{i}+1$ and
$f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}=\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}.$
Hence, for $p=2$, our inequality (1.13) becomes the inequality (1.10) of
Huang-Li-Cao. Moreover, the inequality (1.13) is sharp than the inequality
(1.8) of Wang and Xia in [19].
## 2 Proof of the main theorem
Let $u_{i}$ be the $i$th orthonormal eigenfunction of the problem (1.5)
corresponding to the eigenvalue $\Lambda_{i}$, that is, $u_{i}$ satisfies
$\left\\{\,\vbox{\openup 3.0pt\halign{$\displaystyle{#}$\hfil&
\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u_{i}=\Lambda_{i}(-\Delta)u_{i}&{\rm
in}\ \Omega,\\\u_{i}=\frac{\partial
u_{i}}{\partial\nu}=\cdots=\frac{\partial^{p-1}u_{i}}{\partial\nu^{p-1}}=0&{\rm
on}\ \partial\Omega,\\\\\int_{\Omega}\langle\nabla u_{i},\nabla
u_{j}\rangle=\delta_{ij}.\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$ (2.1)
Let $x_{1},x_{2},\ldots,x_{n+1}$ be the standard Euclidean coordinate
functions of $\mathbb{R}^{n+1}$. Then the unit sphere is defined by
$\mathbb{S}^{n}(1)=\left\\{(x_{1},x_{2},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\ ;\
\sum_{\alpha=1}^{n+1}x_{\alpha}^{2}=1\right\\}.$
Then by a rather long computation and a careful analysis, we are able to
derive a sequence of inequalities which can be successfully used to prove the
following key proposition of the present paper:
###### Proposition 2.1.
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}$
$\displaystyle\left(\langle\nabla x_{\alpha},\nabla
u_{i}\rangle+x_{\alpha}\Delta u_{i}\right)(-\Delta)^{p-2}\left(\langle\nabla
x_{\alpha},\nabla u_{i}\rangle+x_{\alpha}\Delta u_{i}\right)$ (2.2)
$\displaystyle\leq$
$\displaystyle{\frac{1}{2(n-1)}}\left(\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-1}-\left(\Lambda_{i}^{\frac{1}{p-1}}-n+2\right)^{p-1}\right)$
(2.3)
$\displaystyle+\frac{n}{(n-1)}\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-2}-{\frac{1}{n-1}}\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}-n+2\right)^{p-2}$
(2.4)
$\displaystyle+2\left(2^{p-1}-p\right)\Lambda_{i}^{\frac{1}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-3}$
(2.5)
$\displaystyle+4(2^{p-2}-(p-1))\Lambda_{i}^{\frac{2}{p-1}}\left(\Lambda_{i}^{\frac{1}{p-1}}+n\right)^{p-4}.$
(2.6)
We should remark that the main idea in proving Proposition 2.1 is similar to
that in reference [4]. However, here in our case, it seems a little more
complicated than in the case they considered.
For functions $f$ and $g$ defined on $\overline{\Omega}$, we define the
Dirichlet inner product $(f,g)_{D}$ by
$(f,g)_{D}=\int_{\Omega}\langle\nabla f,\nabla g\rangle$
and the Dirichlet norm of $f$ by
$\|f\|_{D}=\left((f,g)_{D}\right)^{1/2}=\left(\int_{\Omega}|\nabla
f|^{2}\right)^{\frac{1}{2}}.$
Define $H^{2}_{p}(\Omega)$ by
$H^{2}_{p}(\Omega)=\\{f\,:\ f,|\nabla f|,\ldots,|\nabla^{p}f|\,\in
L^{2}(\Omega)\\},$
where
$|\nabla^{p}f|^{2}=\sum_{i_{1},\cdots,i_{p}=1}^{n}|\nabla_{i_{1}}\nabla_{i_{2}}\cdots\nabla_{i_{p}}f|^{2}.$
Then $H^{2}_{p}(\Omega)$ is a Hilbert space with respect to the norm
$\|\cdot\|_{p}$:
$\|f\|_{p}=\left(\int_{\Omega}\left(f^{2}+|\nabla
f|^{2}+\cdots+|\nabla^{p}f|^{2}\right)\right)^{\frac{1}{2}}.$
Consider the subspace $H^{2}_{p,D}(\Omega)$ of $H^{2}_{p}(\Omega)$ defined by
$H^{2}_{p,D}(\Omega)=\left\\{f\in H^{2}_{p}(\Omega)\,:\ f=\frac{\partial
f}{\partial\nu}=\cdots=\frac{\partial^{p-1}f}{\nu^{p-1}}=0\ \ {\rm on}\
\partial\Omega\right\\}.$
Then the operator $(-\Delta)^{p}$ defines a self-adjoint operator acting on
$H^{2}_{p,D}(\Omega)$ for the eigenvalue problem (1.5) and eigenfunctions
$\\{u_{i}\\}_{i=1}^{\infty}$ defined in (2.1) form a complete orthonormal
basis for the Hilbert space $H^{2}_{p,D}(\Omega)$. For vector-valued functions
$F=(f_{1},f_{2},\ldots,f_{n+1}),\
G=(g_{1},g_{2},\ldots,g_{n+1})\,:\,\Omega\rightarrow\mathbb{R}^{n+1},$
we define the inner product $(F,G)$ by
$(F,G)=\int_{\Omega}\langle
F,G\rangle=\int_{\Omega}\sum_{\alpha=1}^{n+1}f_{\alpha}g_{\alpha}.$
The norm of $F$ is given by
$\|F\|=(F,F)^{\frac{1}{2}}=\left(\int_{\Omega}\sum_{\alpha=1}^{n+1}f_{\alpha}g_{\alpha}\right)^{\frac{1}{2}}.$
Let $\mathbf{H}^{2}_{p-1}(\Omega)$ be the Hilbert space of vector-valued
functions given by
$\displaystyle\mathbf{H}^{2}_{p-1}(\Omega)=\Big{\\{}F=(f_{1},f_{2},\ldots,f_{n+1})\,:$
$\displaystyle f_{\alpha},|\nabla
f_{\alpha}|,\ldots,|\nabla^{p-1}f_{\alpha}|\,\in L^{2}(\Omega),$
$\displaystyle{\rm for}\ \alpha=1,\ldots,n+1\Big{\\}}$
with norm
$\|F\|_{p-1}=\left\\{\|F\|^{2}+\int_{\Omega}\left(\sum_{\alpha=1}^{n+1}|\nabla
f_{\alpha}|^{2}+\cdots+\sum_{\alpha=1}^{n+1}|\nabla^{p-1}f_{\alpha}|^{2}\right)\right\\}^{\frac{1}{2}}.$
Observe that a vector field on $\Omega$ can be regarded as a vector-valued
function from $\Omega$ to $\mathbb{R}^{n+1}$. Let
$\mathbf{H}^{2}_{p-1,D}(\Omega)\subset\mathbf{H}^{2}_{p-1}(\Omega)$ be a
subspace of $\mathbf{H}^{2}_{p-1}(\Omega)$ spanned by the vector-valued
functions $\\{\nabla u_{i}\\}_{i=1}^{\infty}$ which form a complete
orthonormal basis of $\mathbf{H}^{2}_{p-1,D}(\Omega)$. For any $f\in
H^{2}_{p,D}(\Omega)$, we have $\nabla f\in\mathbf{H}^{2}_{p-1,D}(\Omega)$ and
for any $X\in\mathbf{H}^{2}_{p-1,D}(\Omega)$, there exists a function $f\in
H^{2}_{p,D}(\Omega)$ such that $X=\nabla f$.
Let $x_{1},x_{2},\ldots,x_{n+1}$ be the standard Euclidean coordinate
functions of $\mathbb{R}^{n+1}$, and $u_{i}$ be the $i$-th orthonormal
eigenfunction of the problem (1.5) corresponding to the eigenvalue
$\Lambda_{i}$ (see (2.1)). For any $\alpha=1,2,\ldots,n+1$ and each
$i=1,\ldots,k$, we decompose the vector-valued functions $x_{\alpha}\nabla
u_{i}$ as
$x_{\alpha}\nabla u_{i}=\nabla h_{\alpha i}+W_{\alpha i},$ (2.7)
where $h_{\alpha i}\in H^{2}_{p,D}(\Omega)$, $\nabla h_{\alpha i}$ is the
projection of $x_{\alpha}\nabla u_{i}$ in $\mathbf{H}^{2}_{p-1,D}(\Omega)$,
$W_{\alpha i}\perp\mathbf{H}^{2}_{p-1,D}(\Omega)$. Thus we have
$(W_{\alpha i},\nabla u)=\int_{\Omega}\langle W_{\alpha i},\nabla u\rangle=0,\
\ {\rm for\ any}\ u\in H^{2}_{p,D}(\Omega).$ (2.8)
By the denseness of $H^{2}_{p,D}(\Omega)$ in $L^{2}(\Omega)$ and
$C^{1}(\Omega)$ is dense in $L^{2}(\Omega)$, we conclude that
$(W_{\alpha i},\nabla h)=0,\ \ \ \forall\ h\in C^{1}(\Omega)\cap
L^{2}(\Omega),$ (2.9)
which implies from the divergence theorem that
$\int_{\Omega}h\,\,{\rm div}(W_{\alpha i})=0,$
where ${\rm div}(Z)$ denotes the divergence of $Z$. Consequently, we get
${\rm div}(W_{\alpha i})=0.$ (2.10)
Define $\phi_{\alpha i}$ by
$\phi_{\alpha i}=h_{\alpha i}-\sum_{j=1}^{k}b_{\alpha ij}u_{j},$ (2.11)
where
$b_{\alpha ij}=\int_{\Omega}x_{\alpha}\langle\nabla u_{i},\nabla
u_{j}\rangle=b_{\alpha ji}.$
Then we have
$\phi_{\alpha i}=\frac{\partial\phi_{\alpha
i}}{\partial\nu}=\cdots=\frac{\partial^{p-1}\phi_{\alpha
i}}{\partial\nu^{p-1}}=0$
and
$(\phi_{\alpha i},u_{j})_{D}=\int_{\Omega}\langle\nabla\phi_{\alpha i},\nabla
u_{j}\rangle=0,\ \ \ {\rm for\ any}\ j=1,\ldots,k.$ (2.12)
It follows from the Rayleigh-Ritz inequality that
$\Lambda_{k+1}\leq\frac{\int_{\Omega}\phi_{\alpha i}(-\Delta)^{p}\phi_{\alpha
i}}{\|\nabla\phi_{\alpha i}\|^{2}},$ (2.13)
where $\|f\|^{2}=\int_{\Omega}|f|^{2}$. It is easy to see from (2.11) and
(2.12) that
$\displaystyle\int_{\Omega}\phi_{\alpha i}(-\Delta)^{p}\phi_{\alpha i}=$
$\displaystyle\int_{\Omega}\phi_{\alpha i}\left((-\Delta)^{p}h_{\alpha
i}-\sum_{j=1}^{k}b_{\alpha ij}\Lambda_{j}(-\Delta)u_{j}\right)$ (2.14)
$\displaystyle=$ $\displaystyle\int_{\Omega}\phi_{\alpha
i}(-\Delta)^{p}h_{\alpha i}$ (2.15) $\displaystyle=$
$\displaystyle\int_{\Omega}\left(h_{\alpha i}-\sum_{j=1}^{k}b_{\alpha
ij}u_{j}\right)(-\Delta)^{p}h_{\alpha i}$ (2.16) $\displaystyle=$
$\displaystyle\int_{\Omega}h_{\alpha i}(-\Delta)^{p}h_{\alpha
i}-\sum_{j=1}^{k}b_{\alpha ij}\int_{\Omega}u_{j}(-\Delta)^{p}h_{\alpha i}$
(2.17) $\displaystyle=$ $\displaystyle\int_{\Omega}h_{\alpha
i}(-\Delta)^{p}h_{\alpha i}-\sum_{j=1}^{k}b_{\alpha ij}\int_{\Omega}h_{\alpha
i}(-\Delta)^{p}u_{j}$ (2.18) $\displaystyle=$
$\displaystyle\int_{\Omega}h_{\alpha i}(-\Delta)^{p}h_{\alpha
i}-\sum_{j=1}^{k}\Lambda_{j}b_{\alpha ij}^{2}.$ (2.19)
Since
$\|x_{\alpha}\nabla u_{i}\|^{2}=\int_{\Omega}x_{\alpha}^{2}|\nabla
u_{i}|^{2}=\|\nabla h_{\alpha i}\|^{2}+\|W_{\alpha i}\|^{2},$ (2.20) $\|\nabla
h_{\alpha i}\|^{2}=\|\nabla\Phi_{\alpha i}\|^{2}+\sum_{j=1}^{k}b_{\alpha
ij}^{2}.$ (2.21)
Therefore, (2.14) can be written as
$\displaystyle\int_{\Omega}\phi_{\alpha i}(-\Delta)^{p}\phi_{\alpha i}=$
$\displaystyle\int_{\Omega}h_{\alpha i}(-\Delta)^{p}h_{\alpha
i}-\Lambda_{i}\|x_{\alpha}\nabla u_{i}\|^{2}$ (2.22)
$\displaystyle+\Lambda_{i}\left(\|\nabla\phi_{\alpha i}\|^{2}+\|W_{\alpha
i}\|^{2}+\sum_{j=1}^{k}b_{\alpha
ij}^{2}\right)-\sum_{j=1}^{k}\Lambda_{j}b_{\alpha ij}^{2}.$ (2.23)
Inserting (2.22) into (2.13) yields
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})\|\nabla\Phi_{\alpha i}\|^{2}\leq$
$\displaystyle\int_{\Omega}h_{\alpha i}(-\Delta)^{p}h_{\alpha
i}-\Lambda_{i}\|x_{\alpha}\nabla u_{i}\|^{2}+\Lambda_{i}\|W_{\alpha i}\|^{2}$
(2.24) $\displaystyle+\sum_{j=1}^{k}(\Lambda_{i}-\Lambda_{j})b_{\alpha
ij}^{2}$ (2.25) $\displaystyle=$ $\displaystyle p_{\alpha i}+\|\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}+\Lambda_{i}\|W_{\alpha i}\|^{2}$ (2.26)
$\displaystyle+\sum_{j=1}^{k}(\Lambda_{i}-\Lambda_{j})b_{\alpha ij}^{2},$
(2.27)
where
$p_{\alpha i}=\int_{\Omega}h_{\alpha i}(-\Delta)^{p}h_{\alpha
i}-\Lambda_{i}\|x_{\alpha}\nabla u_{i}\|^{2}-\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}.$
###### Lemma 2.1.
[18] Let
$c_{\alpha ij}=\int_{\Omega}\langle Z_{\alpha i},u_{j}\rangle,$
where $Z_{\alpha i}=\nabla\langle x_{\alpha},\nabla
u_{i}\rangle-\frac{n-2}{2}x_{\alpha}\nabla u_{i}$. Then we have
$c_{\alpha ij}=-c_{\alpha ji}.$
Note that
$\displaystyle-2\int_{\Omega}\langle x_{\alpha}\nabla u_{i},Z_{\alpha
i}\rangle$ (2.28) $\displaystyle=$ $\displaystyle-2\int_{\Omega}\langle
x_{\alpha}\nabla u_{i},\nabla\langle x_{\alpha},\nabla
u_{i}\rangle\rangle+(n-2)\int_{\Omega}x_{\alpha}^{2}|\nabla u_{i}|^{2}$ (2.29)
$\displaystyle=$ $\displaystyle 2\int_{\Omega}\langle x_{\alpha},\nabla
u_{i}\rangle^{2}+\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta u_{i}+(n-2)\int_{\Omega}x_{\alpha}^{2}|\nabla u_{i}|^{2}.$
(2.30)
On the other hand, from (2.7), (2.9) and (2.12), we obtian
$\displaystyle-2\int_{\Omega}\langle x_{\alpha}\nabla u_{i},Z_{\alpha
i}\rangle=$ $\displaystyle-2\int_{\Omega}\langle\nabla h_{\alpha i}+W_{\alpha
i},Z_{\alpha i}\rangle$ (2.31) $\displaystyle=$
$\displaystyle-2\int_{\Omega}\langle\nabla h_{\alpha i},Z_{\alpha
i}\rangle+(n-2)\int_{\Omega}\langle W_{\alpha i},x_{\alpha}\nabla
u_{i}\rangle$ (2.32) $\displaystyle=$
$\displaystyle-2\int_{\Omega}\langle\nabla\phi_{\alpha
i}+\sum_{j=1}^{k}b_{\alpha ij}\nabla u_{j},Z_{\alpha
i}\rangle+(n-2)\int_{\Omega}\langle W_{\alpha i},x_{\alpha}\nabla
u_{i}\rangle$ (2.33) $\displaystyle=$
$\displaystyle-2\int_{\Omega}\langle\nabla\phi_{\alpha i},Z_{\alpha
i}\rangle-2\sum_{j=1}^{k}b_{\alpha ij}c_{\alpha ij}+(n-2)\|W_{\alpha i}\|^{2}$
(2.34) $\displaystyle=$ $\displaystyle-2\int_{\Omega}\langle\nabla\phi_{\alpha
i},Z_{\alpha i}-\sum_{j=1}^{k}c_{\alpha ij}\nabla
u_{j}\rangle-2\sum_{j=1}^{k}b_{\alpha ij}c_{\alpha ij}$ (2.35)
$\displaystyle+(n-2)\|W_{\alpha i}\|^{2}.$ (2.36)
From (2.28) and (2.31), we obtain
$r_{\alpha i}+2\sum_{j=1}^{k}b_{\alpha ij}c_{\alpha
ij}=-2\int_{\Omega}\langle\nabla\phi_{\alpha i},Z_{\alpha
i}-\sum_{j=1}^{k}c_{\alpha ij}\nabla u_{j}\rangle+(n-2)\|W_{\alpha i}\|^{2},$
(2.37)
where
$r_{\alpha i}=2\int_{\Omega}\langle x_{\alpha},\nabla
u_{i}\rangle^{2}+\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta u_{i}+(n-2)\int_{\Omega}x_{\alpha}^{2}|\nabla u_{i}|^{2}.$
Multiplying (2.37) by $(\Lambda_{k+1}-\Lambda_{i})^{2}$, one obtains from the
Schwarz inequality and (2.24) that
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})^{2}\left(r_{\alpha
i}+2\sum_{j=1}^{k}b_{\alpha ij}c_{\alpha ij}\right)$ (2.38) $\displaystyle=$
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})^{2}\left(-2\int_{\Omega}\left\langle\nabla\phi_{\alpha
i},Z_{\alpha i}-\sum_{j=1}^{k}c_{\alpha ij}\nabla
u_{j}\right\rangle+(n-2)\|W_{\alpha i}\|^{2}\right)$ (2.39)
$\displaystyle\leq$
$\displaystyle\delta(\Lambda_{k+1}-\Lambda_{i})^{3}\|\nabla\phi_{\alpha
i}\|^{2}+\frac{1}{\delta}(\Lambda_{k+1}-\Lambda_{i})\left\|Z_{\alpha
i}-\sum_{j=1}^{k}c_{\alpha ij}\nabla u_{j}\right\|^{2}$ (2.40)
$\displaystyle+(n-2)(\Lambda_{k+1}-\Lambda_{i})^{2}\|W_{\alpha i}\|^{2}$
(2.41) $\displaystyle\leq$
$\displaystyle\delta(\Lambda_{k+1}-\Lambda_{i})^{2}\left(p_{\alpha
i}+\|\langle\nabla x_{\alpha},\nabla u_{i}\rangle\|^{2}+\Lambda_{i}\|W_{\alpha
i}\|^{2}+\sum_{j=1}^{k}(\Lambda_{i}-\Lambda_{j})b_{\alpha ij}^{2}\right)$
(2.42)
$\displaystyle+\frac{1}{\delta}(\Lambda_{k+1}-\Lambda_{i})\left(\|Z_{\alpha
i}\|^{2}-\sum_{j=1}^{k}c_{\alpha
ij}^{2}\right)+(n-2)(\Lambda_{k+1}-\Lambda_{i})^{2}\|W_{\alpha i}\|^{2}.$
(2.43)
Since $b_{\alpha ij}=b_{\alpha ji}$ and $c_{\alpha ij}=-c_{\alpha ji}$,
summing over $i$ from 1 to $k$ for (2.38) yields
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}r_{\alpha i}$
(2.44) $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+\delta\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+(\delta\Lambda_{i}+n-2)\|W_{\alpha i}\|^{2}\Big{)}$ (2.45)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}.$ (2.46)
Let $\rho$ be a positive constant. Then we have
$\displaystyle\rho\|\langle\nabla x_{\alpha},\nabla u_{i}\rangle\|^{2}=$
$\displaystyle\rho\int_{\Omega}\langle\nabla x_{\alpha},\nabla
u_{i}\rangle^{2}$ (2.47) $\displaystyle=$
$\displaystyle-\rho\int_{\Omega}x_{\alpha}{\rm div}(\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\nabla u_{i})$ (2.48) $\displaystyle=$
$\displaystyle-\rho\int_{\Omega}\langle x_{\alpha}\nabla
u_{i},\nabla\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\rangle-\rho\int_{\Omega}\langle\nabla x_{\alpha},\nabla
u_{i}\rangle x_{\alpha}\Delta u_{i}$ (2.49) $\displaystyle=$
$\displaystyle-\rho\int_{\Omega}\langle\nabla h_{\alpha i},\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\rangle-\frac{\rho}{2}\int_{\Omega}\langle\nabla
x_{\alpha}^{2},\nabla u_{i}\rangle\Delta u_{i}$ (2.50) $\displaystyle\leq$
$\displaystyle(\delta\Lambda_{i}+n-2)\|\nabla h_{\alpha
i}\|^{2}+\frac{\rho^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.51)
$\displaystyle-\frac{\rho}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta u_{i}.$ (2.52)
Applying (2.47) to (2.44) yields
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}r_{\alpha i}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)\|W_{\alpha i}\|^{2}$ (2.53)
$\displaystyle+(\delta-\rho)\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+\rho\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}\Big{)}$ (2.54)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}$ (2.55) $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)(\|W_{\alpha i}\|^{2}+\|\nabla h_{\alpha
i}\|^{2})$ (2.56) $\displaystyle+(\delta-\rho)\|\langle\nabla
x_{\alpha},\nabla
u_{i}\rangle\|^{2}+\frac{\rho^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.57)
$\displaystyle-\frac{\rho}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta
u_{i}\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}$ (2.58) $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)\|x_{\alpha}\nabla u_{i}\|^{2}$ (2.59)
$\displaystyle+(\delta-\rho)\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+\frac{\rho^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.60)
$\displaystyle-\frac{\rho}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta
u_{i}\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}.$ (2.61)
Since
$\Delta h_{\alpha i}={\rm div}(\nabla h_{\alpha i})={\rm div}(x_{\alpha}\nabla
u_{i})=\langle\nabla x_{\alpha},\nabla u_{i}\rangle+x_{\alpha}\Delta u_{i},$
we get from Proposition 2.1 that
$\displaystyle\sum_{\alpha=1}^{n+1}p_{\alpha i}=$
$\displaystyle\sum_{\alpha=1}^{n+1}\left(\int_{\Omega}h_{\alpha
i}(-\Delta)^{p}h_{\alpha i}-\Lambda_{i}\|x_{\alpha}\nabla
u_{i}\|^{2}-\|\langle\nabla x_{\alpha},\nabla u_{i}\rangle\|^{2}\right)$
$\displaystyle=$ $\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}h_{\alpha
i}(-\Delta)^{p}h_{\alpha i}-(\Lambda_{i}+1)$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}\left(\langle\nabla
x_{\alpha},\nabla u_{i}\rangle+x_{\alpha}\Delta
u_{i}\right)(-\Delta)^{p-2}\left(\langle\nabla x_{\alpha},\nabla
u_{i}\rangle+x_{\alpha}\Delta u_{i}\right)-(\Lambda_{i}+1)$
$\displaystyle\leq$ $\displaystyle f(\Lambda_{i},n)-(\Lambda_{i}+1).$
A direct calculation yields (see (2.44), (2.45), (2.46) and (2.47) in [18])
$\displaystyle\sum_{\alpha=1}^{n+1}r_{\alpha i}=n,$
$\displaystyle\sum_{\alpha=1}^{n+1}\|x_{\alpha}\nabla
u_{i}\|^{2}=\sum_{\alpha=1}^{n+1}\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}=1,$
$\displaystyle\sum_{\alpha=1}^{n+1}\|\nabla\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}=\Lambda_{i}-(n-2),$
$\displaystyle\sum_{\alpha=1}^{n+1}\|Z_{\alpha
i}\|^{2}=\Lambda_{i}+\frac{(n-2)^{2}}{4}.$
Therefore, summing up (2.53) over $\alpha$ from 1 to $n+1$, one gets
$\displaystyle n\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta\left(f(\Lambda_{i},n)-(\Lambda_{i}+1)\right)+(\delta\Lambda_{i}+n-2)+(\delta-\rho)$
$\displaystyle+\frac{\rho^{2}}{4(\delta\Lambda_{i}+n-2)}\left(\Lambda_{i}-(n-2)\right)\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
That is,
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}$ (2.62)
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta
f(\Lambda_{i},n)-\rho+\frac{\rho^{2}}{4(\delta\Lambda_{i}+n-2)}\left(\Lambda_{i}-(n-2)\right)\right)$
(2.63)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(2.64)
Taking
$\rho=\frac{2(\delta\Lambda_{i}+n-2)}{\Lambda_{i}-(n-2)}$
in (2.62) yields
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta
f(\Lambda_{i},n)-\frac{\delta\Lambda_{i}+n-2}{\Lambda_{i}-(n-2)}\right)$
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
Hence, we obtain
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(2.65) $\displaystyle\leq$
$\displaystyle\delta\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)$
(2.66)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(2.67)
Minimizing the right hand side of (2.65) as a function of $\delta$ by choosing
$\delta=\left(\frac{\sum\limits_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)}{\sum\limits_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)}\right)^{\frac{1}{2}}$
concludes the proof of Theorem 1.1.
Proof of Corollary 1.2.
It is easy to see from (1.13) that
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\right\\}^{\frac{1}{2}}$
(2.68) $\displaystyle\hskip
73.97733pt\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{\frac{1}{2}}.$
(2.69)
One can check by induction that
$\displaystyle\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\right\\}\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\right)\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(f(\Lambda_{i},n)-\frac{\Lambda_{i}}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\},$
which together with (2.68) yields inequality (1.16).
Solving the quadratic polynomial of $\Lambda_{k+1}$ in (1.16), we obtain
inequality (1.18) and (1.19). It completes the proof of Corollary 1.2.
## References
* [1] Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Davies, E.B., Safarov, Yu(eds.) Spectral theory and geometry(Edinburgh. 1998). London Math. Soc., Lecture Notes, vol 273, pp 95-139. Cambridge University Press, Cambridge (1999)
* [2] Ashbaugh, M.S.: Universal eigenvalue bounds of Payne-Polya-Weinberger, Hile-Protter, and H. C. Yang. Proc. Indian Acad. Sci. Math. Sci. 112, 3-30 (2002)
* [3] Ashbaugh, M.S., Hermi, L.: A unified approach to universal inequalities for eigenvalues of elliptic operators. Pacific J. Math. 217, 201-219 (2004)
* [4] Cheng, Q.-M., Ichikawa, T., Mametsuka, S.: Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere. Calc. Var.
* [5] Cheng, Q.-M., Ichikawa, T., Mametsuka, S.: Inequalities for eigenvalues of Laplacian with any order. Commun. Contemp. Math. (2009)
* [6] Cheng, Q.-M., Yang, H.C.: Estimates on eigenvalues of Laplacian. Math. Ann. 331 , 445-460 (2005)
* [7] Cheng, Q.-M., Yang, H.C.: Inequalities for eigenvalues of a clamped plate problem. Trans. Amer. Math. Soc. 358, 2625-2635 (2006)
* [8] Cheng, Q.-M., Ichikawa, T., Mametsuka, S.: Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds. J. Math. Soc. Japan.
* [9] Cheng, Q.M., Yang, H.C.: Universal bounds for eigenvalues of a buckling problem. Commun. Math. Phys. 262, 663-675 (2006)
* [10] El Soufi, A., Harrell, E.M., Ilias, S.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Amer. Math. Soc. 361, 2337-2350 (2009)
* [11] Hile, G.N., Protter, M.H.: Inequalites for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29, 523-538 (1980)
* [12] Huang, G.Y., Li, X.X.: Universal inequalities for eigenvalues of Laplacian with any order. (To appear)
* [13] Huang, G.Y., Li, X.X., Cao, L.F.: Universal bounds on eigenvalues of the buckling problem on spherical domains. (To appear)
* [14] Huang, G.Y., Chen, W.Y.: Universal bounds for eigenvlaues of Laplacian operator with any order. Acta Math. Sci. Ser. B Engl. Ed. (2010)
* [15] Huang, G.Y., Li, X.X., Xu, R.W.: Extrinsic estimates for the eigenvalues of Schrödinger operator. Geom. Dedicata DOI 10.1007/s10711-009-9375-0
* [16] Hook, S.M.: Domain-independent upper bounds for eigenvalues of elliptic operators. Trans. Amer. Math. Soc. 318, 615-642 (1990)
* [17] Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289-298 (1956)
* [18] Wang, Q.L., Xia, C.Y.: Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds. J. Funct. Anal. 245 334-352 (2007)
* [19] Wang, Q.L., Xia, C.Y.: Universal inequalities for eigenvalues of the buckling problem on spherical domains. Commun. Math. Phys. 270, 759-775 (2007)
* [20] Wang, Q.L., Xia, C.Y.: Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds. Ann. Acad. Sci. Fenn. Math. 33, 319-336 (2008)
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|
arxiv-papers
| 2009-08-31T00:41:46 |
2024-09-04T02:49:04.938927
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Xingxiao Li, Xuerong Qi",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0908.4439"
}
|
0908.4575
|
# Density of first Poincaré returns, periodic orbits, and Kolmogorov-Sinai
entropy
Paulo R. F. Pinto(1), M. S. Baptista(1)(2) and Isabel S. Labouriau(1)
###### Abstract.
It is known that unstable periodic orbits of a given map give information
about the natural measure of a chaotic attractor. In this work we show how
these orbits can be used to calculate the density function of the first
Poincaré returns. The close relation between periodic orbits and the Poincaré
returns allows for estimates of relevant quantities in dynamical systems, as
the Kolmogorov-Sinai entropy, in terms of this density function. Since return
times can be trivially observed and measured, our approach to calculate this
entropy is highly oriented to the treatment of experimental systems. We also
develop a method for the numerical computation of unstable periodic orbits.
(1)CMUP - Centro de Matemática da Universidade do Porto
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
(2)Institute for Complex Systems and Mathematical Biology
King’s College, University of Aberdeen
AB24 3UE Aberdeen, UK
PACS: 05.45.–a Nonlinear dynamics and chaos; 65.40.gd Entropy
## 1\. Introduction
Knowing how often a dynamical system returns to some place in phase space is
fundamental to understand dynamics. There is a well established way to
quantify that: the first Poincaré return (FPR), which measures how much time a
trajectory of a dynamical system takes to make two consecutive returns to a
given region. Due to their stochastic behaviour, given a return time it is not
feasible to predict the future return times and for that reason one is usually
interested in calculating the frequency with which the Poincaré returns
happen, the density of the first Poncaré returns (DFP).
This work explains the existence of a strong relationship between unstable
periodic orbits (UPOs) and the first Poincaré returns in chaotic attractors.
Unstable orbits and first Poincaré returns have been usually employed as a
tool to analyse and characterise dynamical systems. With our novel approach we
can calculate how frequently returns happen by knowing only a few unstable
periodic orbits. Additionally, such relation allows us to easily estimate
other fundamental quantities of dynamical systems such as the Kolmogorov-Sinai
entropy.
Our motivation to search for a theoretical and simple way of calculating the
distribution of Poincaré return times comes from the fact that they can be
simply and quickly accessible in experiments and also due to the wide range of
complex systems that can be characterized by such a distribution. Among many
examples, in Ref. [1] the return times were used to characterize a
experimental chaotic laser, in Refs. [2, 3] they were used to characterize
extreme events, in Refs. [4, 5] they were used to characterize fluctuations in
fusion plasmas, and in Ref. [6] a series of application to complex data
analysis were described.
In addition, relevant quantifiers of low-dimensional chaotic systems may be
obtained by the statistical properties of the FPR such as the dimensions and
Lyapunov exponents [7, 8] and the extreme value laws [9]. For most of the
rigorous results concerning the FPR, in particular the form of the DFP [10],
one needs to consider very long returns to arbitrarily small regions in phase
space, a condition that imposes limitations into the real application to data
sets.
We first show how the DFP can be calculated from only a few UPOs inside a
finite region. Then, we explain how the DFP can be used to calculate
quantities as the Kolmogorov-Sinai entropy, even when only short return times
are measured in finite regions of the phase space.
Our work is organized as follows: We first introduce the work of Ref. [11],
which relates the natural measure of a chaotic attractor to the UPOs embedded
in a chaotic attractor. The measure of a chaotic attractor refers to the
frequency of visits that a trajectory makes to a portion of the phase space.
This measure is called natural when it is invariant for typical initial
conditions. This appears in Sec. 2, along with the relevant definitions. In
Sec. 3 we define $\rho(\tau,S)$ the density of first Poincaré returns for a
time $\tau$ to a subset $S$ of phase space and we study the relation between
the UPOs and this function. This can be better understood if we classify the
UPOs inside $S$ as recurrent and non-recurrent. Recurrent are those UPOs that
return more than once to the subset $S$ before completing its cycle. Non-
recurrent are UPOs that visits the subset $S$ only once in a period. While in
the calculation of the natural measure of $S$ one should consider the two
types of UPOs with a given large period inside it, for the calculation of the
DFP for a time $\tau$ one should consider only non-recurrent UPOs with a
period $\tau$. Sec. 4 is mostly dedicated to show how to calculate
$\rho(\tau,S)$ even when not all non-recurrent UPOs of a large period are
known. Such a situation typically arises when the time $\tau$ is large. We
have numerically shown that the error of our estimation becomes smaller, the
longer the period of the UPOs and the larger the number of UPOs considered.
Throughout the paper we illustrate results by presenting the calculations for
the tent map. Finally, in Sec. 6 we show numerical results on the logistic map
that support our approach. In particular, we obtain numerical estimates of the
Kolmogorov-Sinai entropy, the most successful invariant in dynamics, so far.
The estimates are obtained considering the density of only short first return
times, as discussed in Sec. 5. The UPOs of period $p$ are computed numerically
as stable periodic orbits of a system of $p$ coupled cells, a method described
in 6.5.
## 2\. Definitions and results
Consider a d-dimensional $C^{2}$ map of the form $x_{n+1}=F(x_{n})$, where
$x\in\Omega\subset R^{d}$ and $\Omega$ represents the phase space of the
system. Consider $A\subset\Omega$ to represent a chaotic attractor. By chaotic
attractor we mean an attractor that has at least one positive Lyapunov
exponent.
For a subset $S$ of the phase space and an initial condition $x_{0}$ in the
basin of attraction of $A$, we define $\mu(x_{0},S)$ as the fraction of time
the trajectory originating at $x_{0}$ spends in $S$ in the limit that the
length of the trajectory goes to infinity. So,
(1) $\mu(x_{0},S)=\lim_{n\rightarrow\infty}\frac{\sharp\\{F^{i}(x_{0})\in S,\
0\leq i\leq n\\}}{n}.$
###### Definition 2.1.
If $\mu(x_{0},S)$ has the same value for almost every $x_{0}$ (with respect to
the Lebesgue measure) in the basin of attraction of $A$, then we call the
value $\mu(S)$ the natural measure of $S$.
For now we assume that our chaotic attractor $A$ has always a natural measure
associated to it, normalized to have $\mu(A)=1$. In particular this means that
the attractor is ergodic[11].
We also assume that the chaotic attractor $A$ is mixing: given two subsets,
$B_{1}$ and $B_{2}$, in $A$, we have:
$\lim_{n\rightarrow\infty}\mu(B_{1}\cap F^{-n}(B_{2}))=\mu(B_{1})\mu(B_{2}).$
In addition, we consider $A$ to be a hyperbolic set.
The eigenvalues of the Jacobian matrix of the $n$-th iterate, $F^{n}$, at the
$j$th fixed point $x_{j}$ of $F^{n}$ are denoted by
$\lambda_{1j},\lambda_{2j},...,\lambda_{uj},\lambda_{(u+1)j},...,\lambda_{dj}$,
where we order the eigenvalues from the biggest, in magnitude, to the lowest
and the number of the unstable eigenvalues is $u$. Let $L_{j}(n)$ be the
product of absolute values of the unstable eigenvalues at $x_{j}$.
Then it was proved by Bowen in 1972 [12] and also by Grebogi, Ott and Yorke in
1988 [11] the following:
###### Theorem 2.1.
For mixing hyperbolic chaotic attractors, the natural probability measure of
some closed subset $S$ of the d-dimensional phase space is
(2) $\mu(S)=\lim_{n\rightarrow\infty}\sum_{x_{j}}L^{-1}_{j}(n),$
where the summation is taken over all the fixed points $x_{j}\in S$ of
$F^{n}$.
This formula is the representation of the natural measure in terms of the
periodic orbits embedded in the chaotic attractor. To illustrate how it works
let us take a simple example like the tent map:
###### Example 2.1.
Let us consider $F:[0,1]\rightarrow[0,1]$ such that
$F(x)=\left\\{\begin{array}[]{l}2x,\ if\ x\in[0,1/2]\\\ 2-2x,\ if\
x\in]1/2,1]\end{array}\right.$
For this map there is only one unstable direction. Since the absolute value of
the derivative is constant in $[0,1]$ we have $L_{j}(\tau)=L(\tau)=2^{\tau}$.
For the tent map, periodic points are uniformly distributed in $[0,1]$. Using
this fact together with some of the ideas of G.H. Gunaratne and I. Procaccia
[13], it is reasonable to write the natural measure of a subset $S$ of $[0,1]$
as:
(3) $\mu(S)=\lim_{\tau\rightarrow\infty}\frac{N(\tau,S)}{N(\tau)},$
where $N(\tau,S)$ is the number of fixed points of $F^{\tau}$ in $S$ and
$N(\tau)$ is the number of fixed points of $F^{\tau}$ in all space $[0,1]$.
For this particular case we have $N(\tau)=L(\tau)=L_{j}(\tau)$ and so
$\mu(S)=\lim_{\tau\rightarrow\infty}\frac{N(\tau,S)}{N(\tau)}=\lim_{\tau\rightarrow\infty}\frac{N(\tau,S)}{L(\tau)}=\lim_{\tau\rightarrow\infty}\sum_{j=1}^{N(\tau,S)}\frac{1}{L_{j}(\tau)}$
and we obtain the Grebogi, Ott and Yorke formula.
## 3\. Density of first returns and UPOs
In this section we relate the DFP, $\rho(\tau,S)$, and the UPOs of a chaotic
attractor. We show in Eq. (10) that $\rho(\tau,S)$ can also be calculated in
terms of the UPOs but one should consider in Eq. (2) only the non-recurrent
ones.
### 3.1. First Poincaré returns
Consider a map $F$ that generates a chaotic attractor $A\subset\Omega$, where
$\Omega$ is the phase space. The first Poincaré return for a given subset
$S\subset\Omega$ such that $S\cap A\neq\emptyset$ is defined as follows.
###### Definition 3.1.
A natural number $\tau$, $\tau>0$, is the first Poincaré return to $S$ of a
point $x_{0}\in S$ if $F^{\tau}(x_{0})\in S$ and there is no other
$\tau^{*}<\tau$ such that $F^{\tau^{*}}(x_{0})\in S$.
A trajectory generates an infinite sequence, $\tau_{1},\tau_{2},...,\tau_{i}$,
of first returns where $\tau_{1}=\tau$ and $\tau_{i}$ is the first Poincaré
return of $F^{n_{i}}(x_{0})$ with $n_{i}=\sum_{n=1}^{i-1}\tau_{n}$.
The subset $S^{\prime}$ of points in $S\subset\Omega$ that produce FPRs of
length $\tau$ to $S$ is given by
(4) $S^{\prime}=S^{\prime}(\tau,S)=\left(F^{-\tau}(S)\cap
S\right)-\bigcup_{0<j<\tau}\left(F^{-j}(S)\cap S\right).$
### 3.2. Density function
In this work, we are concerned with systems for which the DFP decreases
exponentially as the length of the return time goes to infinity. Such systems
have mixing properties and as a consequence we expect to find
$\rho(\tau,S)\approx\mu(S)(1-\mu(S))^{\tau-1}$, where $(1-\mu(S))^{\tau-1}$
represents the probability of a trajectory remaining $\tau-1$ iterations out
of the subset $S$. We are interested in systems for which the decay of
$\rho(\tau)$ is exponential, i.e., $\rho(\tau)\propto e^{-\alpha\tau}$.
The usual way of defining $\rho(\tau,S)$, for a given subset $S\subset\Omega$,
is by measuring the fraction of returns to $S$ that happen with a given length
$\tau$ with respect to all other possible first returns [see Eq. (27)]. It is
usually required for a density that
$\int\rho(\tau,S)d\tau=1.$
In this work, we also adopt a more appropriate definition for $\rho(\tau,S)$
in terms of the natural measure. We define the function $\rho(\tau,S)$ as the
natural measure of the set of orbits that makes a first return $\tau$ to $S$
divided by the natural measure in $S$. More rigorously, we have:
###### Definition 3.2.
The density function of the first Poincaré return $\tau$ for a particular
subset $S\subset\Omega$ such that $\mu(S)\neq 0$ is defined as
(5) $\rho(\tau,S)=\frac{\mu(S^{\prime})}{\mu(S)},$
where $S^{\prime}=S^{\prime}(\tau,S)\subset S$ is the subset of points that
produce FPRs of length $\tau$ defined in Eq. (4).
Even for a simple dynamical system as the tent map, the analytical calculation
of $\rho(\tau,S)$ is not trivial. However, an upper bound for this function
can be easily derived as in the following example:
###### Example 3.1.
Consider the tent map defined in example 2.1, for which the natural measure
coincides with the Lebesgue measure $\lambda$, and let $S\subset[0,1]$ be a
non-trivial closed interval.
To have a return to $S$ we only need to know the natural number $n^{*}$ such
that $F^{n^{*}}(S)=[0,1]$. Since $F$ is an expansion, this natural number
always exists. To find it when $\lambda(S)=\epsilon>0$, we first solve the
equation $2^{x^{*}}={1}/{\epsilon}$ and get
$x^{*}={-\log(\epsilon)}/{\log(2)}$, so we take
$n^{*}=\left[{-\log(\epsilon)}/{\log(2)}\right]+1$, where $[x]$ represents the
integer part of $x$. Then $n^{*}$ is an upper bound for $\tau_{min}$, the
shortest first return to $S$.
Most intervals $S$ of small measure have large values of $\tau_{min}$ and
$\tau_{min}\approx n^{*}$ is a good approximation. A sharper upper bound for
$\tau_{min}$ in $S$ is the lowest period of an UPO that visits it.
The set $D=F^{-n^{*}}(S)\cap S\neq\emptyset$ represents the fraction of points
in $S$ that return to $S$ (not necessarily first return) after $n^{*}$
iterations. Using Eq. (5) and since $S^{\prime}\subset D$ we have
$\rho(n^{*},S)\leq\frac{\lambda(D)}{\lambda(S)}\leq\frac{\epsilon\frac{1}{2^{n^{*}}}}{\epsilon}=2^{-n^{*}}.$
It is natural to expect that for $\tau$ of the order of $n^{*}$ and close to
$\tau_{min}$ we have $\rho(\tau,S)\leq 2^{-\tau}$.
We can write this equation as $\rho(\tau,S)\leq
e^{(-\tau\log(2))}=e^{(-\tau\lambda_{1})}$, where $\lambda_{1}=\log(2)$ is the
Lyapunov exponent for the tent map. In fact, in 1991, G. M. Zaslavsky and M.
K. Tippett
[14][15] presented one formula for the exact value of
$\rho(\tau,S)$. That result can only be valid under the same conditions that
we have used previously, i.e. $\tau\approx\tau_{min}$ and for most sets of
sufficiently small measure $\epsilon$, so that $\tau_{min}\approx n^{*}$.
### 3.3. Density function in terms of recurrent and non-recurrent UPOs
Since our chaotic attractor $A$ is mixing, the natural measure associated with
$A$ satisfies, for any subset $S$ of nonzero measure:
$\mu(S)=\lim_{\tau\rightarrow\infty}\frac{\mu(S\cap F^{-\tau}(S))}{\mu(S)}.$
We can write the right hand side of the last equation, for any positive
$\tau$, in two terms:
(6) $\frac{\mu(S\cap
F^{-\tau}(S))}{\mu(S)}=\frac{\mu(S^{\prime})}{\mu(S)}+\frac{\mu(S^{*})}{\mu(S)}$
with $S^{\prime}$ as defined in Eq. (4) and where $S^{*}=S^{*}(S,\tau)$ is the
set of points in $S$ that are mapped to $S$ after $\tau$ iterations but for
which $\tau$ is not the FPR to $S$, so $S^{\prime}\cup S^{*}=(S\cap
F^{-\tau}(S))$ and $S^{\prime}\cap S^{*}=\emptyset$.
An UPO of period $\tau$ is recurrent with respect to a set $S\subset\Omega$ if
there is a point $x_{0}\in S$ in the UPO with $F^{n}(x_{0})\in S$ for
$0<n<\tau$. In other words, its FPR is less than its period. Thus, the UPOs in
the set $S^{*}$ are all recurrent. We refer to them as the recurrent UPOs
inside $S$.
Associated with the recurrent UPOs in $S$ we define
(7) $\mu_{R}(\tau,S)=\sum_{j}\frac{1}{L_{j}^{R}(\tau)}$
and associated with the non-recurrent UPOs in $S$ we define
(8) $\mu_{NR}(\tau,S)=\sum_{j}\frac{1}{L_{j}^{NR}(\tau)}$
where $L_{j}^{R}(\tau)$ and $L_{j}^{NR}(\tau)$ refer, respectively, to the
product of the absolute values of the unstable eigenvalues of recurrent and
non-recurrent UPOs of period $\tau$ that visit $S$.
Notice that, if $\mu(S)\neq 0$,
$\lim_{\tau\rightarrow\infty}\frac{\mu(S^{*})}{\mu(S)}=\lim_{\tau\rightarrow\infty}\mu_{R}(\tau,S)$
and
(9)
$\lim_{\tau\rightarrow\infty}\frac{\mu(S^{\prime})}{\mu(S)}=\lim_{\tau\rightarrow\infty}\mu_{NR}(\tau,S)$
since $\mu(S^{*})/\mu(S)$ measures the frequency with which chaotic
trajectories that are associated with the recurrent UPOs visit $S$ and
$\mu(S^{\prime})/\mu(S)$ measures the frequency with which chaotic
trajectories that are associated with the non-recurrent UPOs visit $S$.
Comparing Eqs. (5), (6) and (9) we obtain the following:
Main Idea: For a chaotic attractor $A$ generated by a mixing uniformly
hyperbolic map $F$, for a small subset $S\subset A$, generated by a Markov
partition and such that the measure in $S$ is provided by the UPOs inside it,
we have that
(10) $\rho(\tau,S)\approx\mu_{NR}(\tau,S),$
for a sufficiently large $\tau$. Moreover,
$\mu(S)=\lim_{\tau\rightarrow\infty}[\rho(\tau,S)+\mu_{R}(\tau,S)].$
A Markov partition is a very special splitting of the phase space. For the
purpose of better justifying Eq. (10), if a region $C(\tau)$ belongs to a
Markov partition of order $\tau$, then there is a sub-interval
$\tilde{C}(\tau)$ of $C(\tau)$ that after $\tau$ iterations is mapped exactly
over $C(\tau)$. Moreover, points inside $\tilde{C}(\tau)$ make first returns
to $C(\tau)$ after $\tau$ iterations. Then, $\mu_{R}(\tau,C(\tau))$=0. As a
consequence, for sufficiently large $\tau$ we can write that
$\mu[C(\tau)]\rightarrow\rho[\tau,C(\tau)]$.
But approximation (10) remains valid for a small nonzero $\tau$. The reason
for that is the following: Notice that from the way Kac’s lemma is derived
(see Sec. 8.1), Eq. (2) can be written as
$\mu(S)=\frac{\int_{\tau_{min}}^{\infty}\rho(\tau,S)d\tau}{<\tau>},$
where $<\tau>$ represents the average of the FPRs inside $S$, since
$\int_{\tau_{min}}^{\infty}\rho(\tau,S)d\tau=1$. This equation illustrates
that any possible existing error in the calculation of $\mu(S)$ by Eq. (2) is
a summation over all errors coming from $\rho(\tau,S)$ for all values of
$\tau$ that we are considering. As shown in Ref. [11], $\mu(S)$ can be
calculated by Eq. (2) using UPOs with a small and finite period $p$. This
period is of the order of the time that the Perron-Frobenius operator
converges and thus linearization around UPOs can be used to calculate the
measure associated with them. As a consequence, if $\mu(S)$ can be well
estimated for $p\approx 30$ then $\rho(\tau,S)$ can be well estimated for
$\tau<<p$. As we will observe, considering $\tau$ small, of the order of $5$,
we get a very good estimation for $\rho(\tau,S)$.
In addition, we observe in our numerical simulation that $S$ does not need to
be a cell in a Markov partition but just a small region located in an
arbitrary location in $\Omega$.
We say that an UPO has FPRs associated with it if the UPO is non-recurrent.
See that for every UPO there is a neighborhood containing no other UPO with
the same period. If the UPO is non-recurrent then all points inside a smaller
neighborhood will produce FPRs associated with this UPO in the sense that
their FPR coincides with the UPO’s. Consider $\tau_{min}$ as the shortest
first return in $S$.
Case $\tau<2\tau_{min}$
UPOs of period $\tau$ are non-recurrent. This is illustrated in Fig. 1 (A),
where $\tau_{min}=7$, for the logistic map ($c=4$). In that picture we observe
that for $\tau\leq 14$ all FPRs are associated with UPOs. Because of this fact
$\mu(S^{*})=0$ and then all the chaotic trajectories that return to $S$ are
associated with non-recurrent UPOs. So, $\rho(\tau,S)\approx\mu(S)$ and thus,
$\rho(\tau,S)\approx\mu_{NR}(\tau,S)$.
Case $\tau\geq 2\tau_{min}$
We can have recurrent UPOs of period $\tau$, that do not have first returns
associated with them. As a consequence $\mu(S^{*})>0$ and recurrent UPOs
contribute to the measure of $S$. This is illustrated in Fig. 1 (B), when
$\tau=16$.
Figure 1. This picture shows some UPOs inside $S\subset[0,1]$ and first
Poincaré returns for the logistic map, [$x_{n+1}=4x_{n}(1-x_{n})$]. In this
example $\tau_{min}=7$. For $\tau<14$ all UPOs have FPRs associated with them.
For $\tau\geq 14$ (as in (B) for $\tau=16$) some UPOs are recurrent. Picture
(B) is a zoom of picture (A).
## 4\. How to calculate the density of first Poincaré returns
A practical issue is how to calculate $\mu_{NR}(\tau,S)$. There are two
relevant cases: All UPOs can be calculated; only a few can be calculated.
Assuming $\tau$ to be sufficiently small such that all UPOs of period $\tau$
can be calculated and sufficiently large so that Eq. (10) is reasonably valid,
$\mu_{NR}(\tau,S)$ can be exactly calculated and we can easily estimate
$\rho(\tau,S)$ from Eq.(10), using $\rho(\tau,S)\approx\mu_{NR}(\tau,S)$.
When $\tau$ is large then, typically, only a few UPOs can be calculated. For
this case, it is difficult to use Eq. (10) to estimate $\rho(\tau,S)$ since
there will be too many UPOs. In order to calculate $\rho(\tau,S)$ using
$\mu_{NR}(\tau,S)$ we do the following. First notice that
(11) $\mu(S)=\lim_{\tau\rightarrow\infty}(\mu_{NR}(\tau,S)+\mu_{R}(\tau,S)).$
Considering then $\tau$ sufficiently large we have that
$\mu(S)\approx\mu_{NR}(\tau,S)+\mu_{R}(\tau,S)$
which can be rewritten (using Eq. (10) which says that
$\rho(\tau,S)\approx\mu_{NR}(\tau,S)$, for finite $\tau$] as
(12)
$\rho(\tau,S)\approx\mu(S)-\mu_{R}(\tau,S)=\mu(S)\left(1-\frac{\mu_{R}(\tau,S)}{\mu(S)}\right).$
This equation allows us to reproduce, approximately, the function
$\rho(\tau,S)$, for any sufficiently large $\tau$, only using the estimated
value of the quotient
$\frac{\mu_{R}(\tau,S)}{\mu(S)}$
that is easy to obtain numerically, since not all UPOs should be calculated
but just a few ones with period $\tau$. We discuss this in 4.1 below.
### 4.1. How can we estimate $\mu_{R}(\tau,S)/\mu(S)$?
Considering a subset $S$ and fixing $\tau$, we calculate a number $t$ of
different UPOs with period $\tau$ (say, $t=50$) inside $S$ (It is explained in
Sec. 6.5 how to calculate numerically UPOs with any period of a given map).
These UPOs are calculated from randomly selected symbolic sequences for which
the generated UPOs visit $S$. See that, for example, in the tent map, for
$\tau=10$ and $S=[0,\frac{1}{8}]$, we may have $2^{10}/8$ UPOs inside $S$ and
so, here $50$ UPOs inside $S$ is, in fact, a very small number of UPOs.
Now, we separate all the $t$ UPOs that visit $S$ into recurrent and non-
recurrent ones and suppose that we have $r$ recurrent and $nr$ non-recurrent
such that $r+nr=t$. So, $r$ and $nr$ depend on $t$ and $S$. With these
particular $r(t,S)$ recurrent UPOs we use Eq. (7) and we obtain
$\tilde{\mu}_{R}[\tau,S,r(t,S)]=\sum_{j=1}^{r(t,S)}\frac{1}{L_{j}^{R}(\tau)}$
where $L_{j}^{R}(\tau)$ represents the product of the absolute values of the
unstable eigenvalues of the $j$th recurrent UPO within the set of $r(t,S)$
recurrent UPOs. See that this quantity is not equal to $\mu_{R}(\tau,S)$ since
we are not considering all recurrent UPOs inside $S$ but just a small number
$r(t,S)$ of them. We do the same thing with the $nr(t,S)$ non-recurrent UPOs
and obtain the quantity $\tilde{\mu}_{NR}[\tau,S,nr(t,S)]$.
Finally, we observe that, for a sufficiently large $t$, we have
$\frac{\tilde{\mu}_{R}[\tau,S,r(t,S)]}{\tilde{\mu}(\tau,S,t)}\approx\frac{\mu_{R}(\tau,S)}{\mu(S)},$
where
$\tilde{\mu}(\tau,S,t)=\tilde{\mu}_{R}[\tau,S,r(t,S)]+\tilde{\mu}_{NR}[\tau,S,nr(t,S)]$.
Therefore, with only a few UPOs inside $S$ we calculate an estimated value for
$\rho(\tau,S)$. This estimation is represented by $\rho_{M}$ and is given by
(13)
${\rho}_{M}[\tau,S,r(t,S)]=\mu(S)\left(1-\frac{\tilde{\mu}_{R}[\tau,S,r(t,S)]}{\tilde{\mu}(\tau,S,t)}\right)$
Notice that, for a large $\tau$ we will have more recurrent UPOs than non-
recurrent ones and therefore the larger $\tau$ is, the larger is the
contribution of the recurrent UPOs to the measure inside $S$.
### 4.2. Error in the estimation
To study how much our estimation in Eq. (13) depends on the number $t$ of
UPOs, we first assume that if all UPOs are known, the calculated distribution
in Eq. (10) is “exact”, or in other words it has a neglectable error as when
compared to the real distribution provided by Eq. (5).
Then, the error in Eq. (13) will depend on the deviation of the quotient
(14) $q_{1}=\frac{\tilde{\mu}_{R}[\tau,S,r(t,S)]}{\tilde{\mu}(\tau,S,t)},$
calculated when only $t$ UPOs are known, to the quotient
(15)
$q_{2}=\frac{\tilde{\mu}_{R}[\tau,S,r(t=N(\tau,S),S)]}{\tilde{\mu}(\tau,S,t=N(\tau,S))},$
calculated when all the $N(\tau,S)$ UPOs are known.
Thus, the amount of error that our estimate [Eq. (13)] has as when compared to
the “exact” value of $\rho$ (when all the UPOs are known) can be calculated by
(16) $E[\tau,S,t]=\frac{|q_{1}-q_{2}|}{q_{2}}$
which means that the quantity $E$ gives the amount of deviation, in a scale
from 0 to 1, of $\rho_{M}$ [Eq. (13)] as when compared to the “exact” value of
$\rho$ [Eq. (10)]. Notice that in Eq. (16), the quantity 100$E$ corresponds to
the percentage of error that our estimation has.
### 4.3. Uniformly distributed UPOs
There is another way to estimate the value of $\rho(\tau,S)$ in terms of the
number of UPOs in a subset $S$ of a chaotic attractor $A$. We define $N(\tau)$
as the number of fixed points of $F^{\tau}$ in $A$, $N(\tau,S)$ as the number
of fixed points of $F^{\tau}$ in $S$, $N_{R}(\tau,S)$ as the number of fixed
points of $F^{\tau}$ in $S$ whose orbit under $F$ is recurrent and
$N_{NR}(\tau,S)$ as the number of fixed points of $F^{\tau}$ in $S$ whose
orbit under $F$ is non-recurrent. Then, for a sufficiently large $\tau$ and
for a uniformly hyperbolic dynamical system for which periodic points are
uniformly distributed in $A$, we have
$\mu_{R}(\tau,S)\approx\frac{N_{R}(\tau,S)}{N(\tau)},\ \
\mu_{NR}(\tau,S)\approx\frac{N_{NR}(\tau,S)}{N(\tau)}.$
Using the previous approximations we can write
$\mu(S)\approx\frac{N_{R}(\tau,S)}{N(\tau)}+\frac{N_{NR}(\tau,S)}{N(\tau)}=\frac{N(\tau,S)}{N(\tau)}.$
By Eq. (10) we may write $\rho(\tau,S)\approx\mu_{NR}(\tau,S)$ and we have
that
(17) $\rho(\tau,S)\approx\mu(S)-\frac{N_{R}(\tau,S)}{N(\tau)}.$
which can be written as
(18)
$\rho(\tau,S)\approx\mu(S)\left(1-\frac{N_{R}(\tau,S)}{N(\tau,S)}\right).$
Again, we have an expression with a quotient
$\frac{N_{R}(\tau,S)}{N(\tau,S)}$
that is, again, easy to obtain numerically by the same technique from which
$\mu_{R}/\mu$ can be estimated and therefore we can obtain an estimation for
$\rho(\tau,S)$, represented by $\rho_{N}$, by
(19) ${\rho}_{N}[\tau,S,r(t,S)]=\mu(S)\left(1-\frac{r(t,S)}{t}\right)$
where $r(t,S)$ represents the number of recurrent UPOs out of a total of $t$
UPOs, exactly as previously defined.
## 5\. Kolmogorov-Sinai entropy
In 1958 Kolmogorov introduced the concept of entropy into ergodic theory and
this has been the most successful invariant so far[16]. In this section we
explain how to use the density of first Poincaré returns to estimate the
Kolmogorov-Sinai entropy $H_{KS}$.
The exposition here does not aim to be rigorous, only to explain how we have
arrived at the numerical estimates for the logistic map of Sec. 6. which is a
non-uniformly hyperbolic map.
It is known that[17]
(20) $N(\tau)\propto\exp(\tau H_{KS}).$
Consider $F$ as a dynamical system that has the following property:
$\frac{N_{NR}(\tau,S)}{N(\tau)}\approx\mu_{NR}(\tau,S)\approx\rho(\tau,S),$
for a sufficiently large $\tau$. For example, dynamical systems for which
periodic points are uniformly distributed on the chaotic attractor $A$ have
this property.
Considering the tent map and $S\subset[0,1]$ such that $N_{NR}(\tau,S)=1$ (if
there is more that one non-recurrent UPO of period $\tau$ inside $S$ we shrink
$S$ to have only one), we have $\rho(\tau,S)\approx\frac{1}{2^{\tau}}$ that
agrees with example 3.1, for $\tau$ close to $\tau_{min}$ and for most
intervals $S$. For other non-uniformly hyperbolic systems as the logistic the
Hénon maps, this property holds in an approximate sense and this approximation
is better the larger $\tau$ is and the closer the interval $S$ is to a Markov
partition.
Using the last approximation together with Eq. (20) we may write
$\frac{N_{NR}(\tau,S)}{\rho(\tau,S)}\approx b\exp(\tau H_{KS}),$
for some positive constant $b\in R$. So, we have that
(21)
$H_{KS}\approx\frac{1}{\tau}\log\left(\frac{N_{NR}(\tau,S)}{b\rho(\tau,S)}\right)=\frac{1}{\tau}\log\left(\frac{N_{NR}(\tau,S)}{\rho(\tau,S)}\right)-\frac{\log(b)}{\tau}.$
We define the quantity $H(\tau,S)$ as
(22)
$H(\tau,S)=\frac{1}{\tau}\log\left(\frac{N_{NR}(\tau,S)}{\rho(\tau,S)}\right)$
and then, for $b\geq 1$, it is clear that
$H_{KS}\approx\frac{1}{\tau}\log\left(\frac{N_{NR}(\tau,S)}{b\rho(\tau,S)}\right)\leq
H(\tau,S),$
so $H(\tau,S)$ is a local upper bound for the approximation of $H_{KS}$,
considering a sufficiently large $\tau$.
Supposing that there is at least one non-recurrent UPO inside $S$, then for
large $\tau$ we have $\frac{N_{NR}(\tau,S)}{\rho(\tau,S)}>\\!\\!>b$, as $b$ is
constant. Thus, the term
$\frac{1}{\tau}\log\left(\frac{N_{NR}(\tau,S)}{\rho(\tau,S)}\right)$
dominates the expression (21), for longer times.
This equation allows us to obtain an upper bound for $\rho(\tau,S)$. See that
$\rho(\tau,S)\leq N_{NR}(\tau,S)\exp(-\tau H_{KS})$ and if
$\tau\approx\tau_{min}$ then $N_{NR}(\tau,S)\approx 1$ and we obtain
$\rho(\tau,S)\leq\exp(-\tau H_{KS})$ as in example 3.1.
Equation (22) depends on the choice of the subset $S$ and is then a local
estimation for $H_{KS}$. To have a global estimate we take a finite number,
$n$, of subsets $S_{i}$ in the chaotic attractor and make a space average as
(23) $\frac{1}{\tau
n}\sum_{i=1}^{n}\log\left(\frac{N_{NR}(\tau,S_{i})}{\rho(\tau,S_{i})}\right).$
Better results are obtained taking the average over pairwise disjoint subsets
$S_{i}$ that are well distributed over $A$.
When we consider $N_{NR}(\tau,S)=1$ this means that we have only one non-
recurrent UPO, with period $\tau$, inside $S$. In general, for sufficiently
small subsets, $S_{i}$, we may have $N_{NR}(\tau,S_{i})=1\ \forall i$ and we
obtain an approximation that only depends on the density function of the first
Poincaré returns
(24) $H_{KS}\approx\frac{1}{\tau
n}\sum_{i}\log\left(\frac{1}{\rho(\tau,S_{i})}\right).$
An equation which can be trivially used from the experimental point of view
since we just need to estimate $\rho(\tau,S_{i})$ and we do not need to know
the UPOs. For practical purposes, we consider in Eqs. (22), (23) and (24) that
$\tau=\tau_{min}$.
## 6\. Numerical results
Figure 2. Density function of the FPRs, $\rho(\tau,S)$, as empty circles and
the measure of the non-recurrent periodic orbits, $\mu_{NR}(\tau,S)$, as
crosses, considering the following intervals: (A), $S=[0.3-0.05,0.3+0.05]$;
(B), $S=[0.3-0.01,0.3+0.01]$; (C), $S=[0.3-0.005,0.3+0.005]$. Figure 3. Red
empty circles represent $\rho(\tau,S)$ estimated by Eq. (12), green crosses
estimated by Eq. (18) and the black line calculated by Eq. (27). Picture (B)
is just a similar reproduction of (A) considering longer first return times.
We consider 200 UPOs inside $S=[0.1-0.001,0.1+0.001]$, for each $\tau$. Figure
4. We show the quantity $E[\tau,S,t]$ with respect to the number $t$ of UPOs
randomly chosen, for $\tau=9$ (A), $\tau=10$ (B), $\tau=11$ (C), $\tau=12$
(D), $\tau=13$ (E), $\tau=14$ (F), $\tau=15$ (G), $\tau=16$ (H), $\tau=17$
(I), $\tau=18$ (J), $\tau=19$ (K), and $\tau=20$ (L). The quantity $E$ gives
the amount of deviation, in a scale from 0 to 1, of $\rho_{M}$ [Eq. (13)] as
compared to the “exact” value of $\rho$ [Eq. (10)]. We consider an interval
positioned at $x=0.04$ with size $\epsilon=0.02$. Figure 5. (A) A bifurcation
diagram as points (light green) and the randomly chosen intervals as empty
(black) squares. (B) Lyapunov exponent as line and filled circles representing
the $H_{KS}$ entropy using Eq. (22), for the logistic family. We consider 400
values of $c$ and for each $c$ the size of the set $S$ is $\epsilon=0.002$.
Figure 6. The Lyapunov exponent $\lambda$ as line and the aproximation of
$H_{KS}$ entropy using Eqs. (23) and (24) as empty circles. (A), Eq. (23);
(B), Eq. (24). In this simulation we consider 100 values of $c$ and for each
$c$ we consider $40$ subsets $S_{i}$ each one with lenght $\epsilon=0.002$. A
subset $S_{i}$ is picked only if $\tau_{min}\in[10,14]$.
We illustrate our ideas with simulations on the logistic family
$F:[0,1]\rightarrow[0,1]$ given by
(25) $F(x)=cx(1-x),$
were $c\in R$. There are many biological motivations to study this family of
maps[18]. The maps that we obtain when the parameter $c$ is varied have
interesting mathematical properties. It is therefore of relevant use for
mathematical and biological study. Moreover, for this family it is possible to
compare the estimates made using all the UPOs to those using only some UPOs.
For most numerical simulations in this section we take $c=4$ in Eq. (25), for
which the map is chaotic and the chaotic attractor is compact.
### 6.1. Calculating $\rho$ when all UPOs are known
Figure 2 shows the function $\rho(\tau,S)$ calculated by Eq. (27) and the
values of $\mu_{NR}(\tau,S)$ calculated by Eq. (8), for some subsets $S$. See
that the DFP can be almost exactly obtained if all the non-recurrent UPOs
inside $S$ with period $\tau$ can be calculated: In Sec. 3 we concluded that
$\rho(\tau,S)\approx\mu_{NR}(\tau,S)$.
### 6.2. Calculating $\rho$ when not all UPOs are known
Figure 3 shows the approximations for $\rho(\tau,S)$ using Eqs. (13) and (19).
In (B), comparing with (A), we consider longer first return times. We only use
Eqs. (13) and (19) for $\tau>2\tau_{min}$.
### 6.3. Error of our estimation when not all UPOs are known
To numerically calculate the error [Eq. (16)] of our estimation in Eq. (13),
we only consider UPOs with a period smaller than 20. The reason is because in
order to calculate the quotient $q_{2}$ in Eq. (15), all the UPOs must be
known. Considering larger periods than 20 would be computationally demanding,
even thought the proposed method to calculate UPOs is capable of finding them
all.
It is also required that $\tau>2\tau_{min}$, once that to calculate the
quotient $q_{1}$ in Eq. (14) there has to exist at least one recurrent UPO
within the set of $t$ UPOs considered, i.e. $r\geq 1$. Therefore, we need to
choose the size of the interval such that 20-2$\tau_{min}-1$ is sufficiently
large, meaning an interval for which $\tau_{min}$ is sufficiently smaller. We
have chosen $\epsilon$=0.02.
Since the error of our estimation is proportional to a quotient between two
quantities that depend on the number $r$ of recurrent UPOs, it is advisable
that one consider intervals for which a reasonable number of recurrent UPOs
are found, even when their period is short (smaller or equal than 20). Such
interval is positioned in places were the natural measure is large. In the
case of the logistic map, these intervals are positioned either close to $x$=0
or $x=1$. Therefore, we consider an interval positioned at $x=0.04$. From the
previous considerations, we consider that the interval has a size of
$\epsilon=0.02$.
In Fig. 4(A-I), we show the quantity $E[\tau,S,t]$ with respect to the number
$t$ of UPOs randomly chosen, for $\tau=9$ (A), $\tau=10$ (B), $\tau=11$ (C),
$\tau=12$ (D), $\tau=13$ (E), $\tau=14$ (F), $\tau=15$ (G), $\tau=16$ (H),
$\tau=17$ (I), $\tau=18$ (J), $\tau=19$ (K), and $\tau=20$ (L).
The most important information from these figures is that as UPOs of longer
periods are considered [going from Fig. (A) to (L)], the error $E$ of our
estimation decreases in an average sense considering all the values of $t$.
Another relevant point is that the larger the number $t$ of UPOs considered,
the smaller the error. Notice that the total number of UPOs of period $\tau$
is given by 2τ. Therefore, looking at Fig. 4(L), one can see that even
considering only about 0.0009$\%$ of all the UPOs (10 UPOs, out of a total of
220=1048576), the error of our estimation is smaller than $14\%$ when compared
to the “exact” value of $\rho$.
### 6.4. Estimating the KS entropy
In order to know how good our estimation for $H_{KS}$ is we use Pesin’s
equality which states that $H_{KS}$ equals the sum of the positive Lyapunov
exponents, here denoted by $\lambda$. For the logistic map there is at most
one positive Lyapunov exponent.
Figure 5 shows the approximation for the quantity $H_{KS}$ using Eq. (22). See
that Eq. (22) only needs one subset $S$ on the chaotic attractor to produce
reasonable results. In this numerical simulation we vary the parameter $c$ of
the logistic family and for each $c$ we use just one subset $S(c)$ randomly
chosen [shown in Fig. 5 (A)] but satisfying $\tau_{min}\in[10,14]$ so that
$\tau$ considered in Eq. (22) is sufficiently large.
Finally, Fig. 6 shows the global estimation for $H_{KS}$, using the Eqs. (23)
and (24), considering $40$ intervals $S_{i}$ for each value of $c$. Recall
that if $\lambda<0$, then $H_{KS}=0$.
### 6.5. Numerical work to find UPOs
The analytical calculation of periodic orbits of a map is a difficult task.
Even for the logistic map it is very difficult to calculate periodic orbits
with a period as low as as four or five. In our numerical work we need to find
unstable periodic orbits and, in some cases, we need to find all different
UPOs inside a subset of the phase space, for a sufficiently large period. For
that, we use the method developed by Biham and Wenzel[19]. They suggest a way
to obtain UPOs of a dynamical system with dimension $D$ using a Hamiltonian,
associated to the map, with dimension $ND$, where $N$ is the number of UPOs
with period $p$. The extremal configurations of this Hamiltonian are the UPOs
of the map. The force ${\partial H}/{\partial t}$ directs trajectories of the
Hamiltonian to the position of a UPO.
The Hamiltonian associated with the map gives a physical interpretation of the
problem but in some cases it is impossible to know it. We propose a method
with a similar interpretation that is simpler in the sense that we do not need
to know the Hamiltonian associated with the map, just an array of $N$ coupled
systems where the linear coupling between nodes acts as the force directing
the network to possible periodic solutions of the dynamical system concerned.
For this method we just need the force associated with the $i$th node,
described by $x^{i}$, and satisfying the Euler-Lagrange (E-L) equations:
$\frac{\partial}{\partial t}\frac{\partial
L}{\partial\dot{x}^{i}}=\frac{\partial L}{\partial x^{i}},$
where $L$ is the Lagrangian associated with the map. We are interested only in
static extremum configurations of the Hamiltonian and therefore the kinetic
term will be neglected[19]. This implies
$\frac{\partial L}{\partial x^{i}}=0$
We illustrate the numerical calculation of UPOs with arbitrary length applying
it to the logistic family. Because the static (E-L) equations reproduce the
map, we have
$\frac{\partial L}{\partial x_{n}^{i}}=x_{n}^{i+1}-cx_{n}^{i}(1-x_{n}^{i}).$
The force of the $i$ node will be given by
$F_{i}=-\frac{\partial L}{\partial
x_{n}^{i}}=-x_{n}^{i+1}+cx_{n}^{i}(1-x_{n}^{i}).$
When the chain is in stable or unstable equilibrium (an extremum static
configuration of the Hamiltonian), $F_{i}=0$ for all $i$. To find a specific
extremum configuration of order $p$ of the Hamiltonian we introduce an
artificial dynamical system defined by
(26) $\frac{\partial x_{n}^{i}}{\partial t}=s_{i}F_{i},\ i=1,...,p,$
where $s_{i}=\pm 1$ represents the direction of the force with respect to the
$i$th node. This equation is solved subject to the periodic boundary condition
$x^{p+1}=x^{1}$ and when the forces in all nodes decrease to zero the
resulting structure $x^{i}$ is simultaneously an extremum static configuration
and an exact $p$-periodic orbit of the logistic map. For $c=4$, if we take
$s_{i}=-1\ \forall i$ then we obtain the trivial periodic point $x_{i}=0\
\forall i$. The different ways to write $s_{i}$ will give different UPOs. We
may look at $s_{i}$ as the representation of the orbit in a symbolic dynamics
with $\Sigma=\\{-1,1\\}$, taking the trivial partition on the logistic map,
i.e., $s_{i}=-1$ if $x_{i}\in[0,1/2]$ and $s_{i}=1$ if $x_{i}\in[1/2,1]$.
Equation (26) is in fact an equation for a network of coupled maps. The UPOs
with period $p$ embedded in the chaotic attractor can be calculated by finding
the stable periodic orbits of the following array of maps constructed with
$i=1,...,p$ nodes $x_{n}^{i}$, where every node is connected to its nearest
neighbor as in
$x_{n+1}^{i}=x_{n}^{i}-cs_{i}[x_{n}^{i+1}-F(x_{n}^{i})],$
with the periodic boundary condition $x_{n}^{p}=x_{n}^{1}$, where the term
$cs_{i}[x_{n}^{i+1}-F(x_{n}^{i})]$ represents the Lagrangian force.
## 7\. Conclusions
In this work we propose two ways to compute the density function of the first
Poincaré returns (DFP), using unstable periodic orbits (UPOs), where the first
Poincaré return (FPR) is the sequence of time intervals that a trajectory
takes to make two consecutive returns to a specific region. In the first way,
the DFP can be exactly calculated considering all UPOs of a given low period.
In the second way, the DFP is estimated considering only a few UPOs. We have
numerically shown that the error of our estimation becomes smaller, the longer
the period of the UPOs and the larger the number of UPOs considered.
The relation between DFP and UPOs allows us to compute easily an important
invariant quantity, the Kolmogorov-Sinai entropy.
For non-uniformly hyperbolic systems there exists some particular subsets for
which the UPOs that visit it are not sufficient to calculate their measure of
the chaotic attractor inside it[20, 21]. For such cases our approach works in
an approximate sense, but it still provides good estimates as we have shown in
our simulations performed in the logistic map, a non-uniformly hyperbolic
system. In addition, the approaches shown in here were applied in ref. [22] to
estimate the value of the Lyapunov exponent in the experimental Chua’s circuit
and in the Hénon map, both systems being non-hyperbolic.
Our approach offers an easy way to estimate the KS entropy in experiments,
since one does not need to calculate UPOs, but rather only to measure the DFP
of trajectories that make shortest returns, i.e. the quantity
$\rho(\tau_{min},S)$. These are the most frequent trajectories, and as a
consequence even if only a few returns are measured, one can obtain a good
estimation of $\rho(\tau_{min},S)$. More details of how to estimate the KS
entropy from experimental data can be found in Ref. [22].
## 8\. Appendix
### 8.1. Measure and density in terms of FPRs
We calculate $\rho(\tau,S)$ also in terms of a finite set of FPRs by
(27) $\rho(\tau,S)=\frac{K(\tau,S)}{L(S)}$
where $K(\tau,S)$ is the number of FPRs with a particular length $\tau$ that
occurred in region $S$ and $L(S)$ is the total number of FPRs measured in $S$
with any possible length.
We calculate $\mu(S)$ also in terms of FPRs by
(28) $\mu(S)=\frac{L(S)}{n_{L}}$
where $n_{L}$ is the number of iterations considered to measure the $L(S)$
FPRs and so $n_{L}=\sum_{n=1}^{L}\tau_{n}$ (see definition 3.1).
We define the average of the returns by
(29) $<\tau>=\frac{n_{L}}{L(S)}.$
Comparing Eqs. (28) and (29), we have that
(30) $\mu(S)=\frac{1}{<\tau>}$
also known as Kac’s lemma.
Acknowledgments: This work was supported by Fundação para a Ciência e a
Tecnologia (FCT), by Centro de Matemática da Universidade do Porto (CMUP) and
by the Mathematics Department of Oporto University.
## References
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* [2] M. S. Santhanam, H. Kantz, Return Interval Distribution of Extreme Events and Long-term Memory, Physical Review E, Vol. 78, Issue 5 (2008) 051113.
* [3] E. G. Altmann; H. Kantz, Recurrence Time Analysis, Long-term Correlations, and Extreme Events, Physical Review E, Vol. 71, Issue 5 (2005) 056106.
* [4] Z. O. Guimarães, I. L. Caldas, R. L. Viana, Recurrence Quantification Analysis of Electrostatic Fluctuations in Fusion Plasmas, Physics Letters A, Vol. 372, Issue 7 (2008) 1088–1095.
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* [6] N. Marwan, A. Facchini, M. Thiel, 20 Year of Recurrence Plots: Perspectives for a Multi-purpose Tool of Nonlinear Data Analysis, European Physical Journal-Special Topics, Vol. 164 (2008) 1–2.
* [7] J. B. Gao, Recurrence Time Statistics for Chaotic Systems and Their Applications, Phys. Rev. Lett. 83 (1999) 3178–3181.
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* [10] M. Hirata, B. Saussol, S. Vaienti, Statistics of return times: a general framework and new applications, Comm. Math. Phys. 206 (1999) 33–55.
* [11] C. Grebogi, E. Ott, J. A. Yorke, Unstable periodic orbits and the dimensions of multifractal chaotic attractors, Physical Review A 37 (1988) 1711–1724.
* [12] R Bowen, Periodic Orbits for Hyperbolic Flows, American Journal of Mathematics 94 (1972) 1–30.
* [13] G. H. Gunaratne, I. Procaccia, Organization of Chaos, Phys. Rev. Lett. 59 (1987) 1377–1380.
* [14] G. M. Zaslavsky, M. K. Tippett, Connection between Recurrence-Time Statistics and Anomalous Transport, Phys. Rev. Lett. 67 (1991) 3251–3254.
* [15] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports 371 (2002) 461-580.
* [16] Peter Walters, An Introduction to Ergodic Theory, Springer, GTM number 79 (1981).
* [17] Ya. G. Sinai, Classical dynamic systems with countably-multiple Lebesgue spectrum, Izv. Akad. Nauk SSSR, Ser. Mat. 30 1966 15–68.
* [18] J. D. Murray, Mathematical Biology, Springer, Biomathematics Texts number 19 (1993).
* [19] O. Biham, W. Wenzel, Characterization of Unstable Periodic Orbits in Chaotic attractors and Repellers, Phys. Rev. Lett. 63 (1989) 819–822.
* [20] Y.-C. Lai, Y. Nagai, C. Grebogi, Characterization of the Natural Measure by Unstable Periodic Orbits in Chaotic Attractors Phys. Rev. Lett. 79 (1997) 649–652.
* [21] M. S. Baptista, S. Kraut, C. Grebogi, Poincaré Recurrence and Measure of Hyperbolic and Nonhyperbolic Chaotic Attractors, Phys. Rev. Lett. 95 094101 (2005).
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|
arxiv-papers
| 2009-08-31T16:27:54 |
2024-09-04T02:49:04.946410
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paulo R. F. Pinto, M. S. Baptista, Isabel S. Labouriau",
"submitter": "Murilo Baptista S.",
"url": "https://arxiv.org/abs/0908.4575"
}
|
0909.0035
|
# Indices of quaternionic complexes
Oldřich Spáčil Department of Mathematics and Statistics, Masaryk University,
Kotlářská 2, 611 37 Brno, Czech Republic 151393@mail.muni.cz
###### Abstract.
Methods of parabolic geometries have been recently used to construct a class
of elliptic complexes on quaternionic manifolds, the Salamon’s complex being
the simplest case. The purpose of this paper is to describe an algorithm how
to compute their analytical indices in terms of characteristic classes. Using
this, we are able to derive some topological obstructions to existence of
quaternionic structures on manifolds.
###### Key words and phrases:
Quaternionic manifold, elliptic complex, analytical index, quaternionic
structures, integrality conditions
###### 2000 Mathematics Subject Classification:
53C15, 53C26, 57R20, 58J10
## 1\. Introduction
Quaternionic geometry has become a classical part of modern differential
geometry. The purpose of the present paper is to contribute to the study of
differential operators which are intrinsic to quaternionic manifolds and to
the study of topology of such manifolds. The connection is given via the
analytical index of elliptic complexes of differential operators.
A large class of elliptic complexes on quaternionic manifolds has been
recently constructed in a much more general setting in [9]. The point is that
quaternionic geometry is an example of the so-called _parabolic geometries_
(see [7]) and there is a well-developed machinery to construct invariant
differential operators on such geometries. These operators fit naturally into
sequences, the _curved Bernstein-Gelfand-Gelfand sequences_ , see [8].
Although a BGG-sequence does not form a complex in general, it was shown in
[9] that under some torsion-freeness like conditions it contains many
subcomplexes. For quaternionic manifolds some of them turned out to be
elliptic.
Our aim was to compute the analytical indices of these elliptic complexes on
compact manifolds. Some results in this direction were obtained in [2] for
another class of elliptic complexes in the case of hyperkähler manifolds and
in [10] for positive quaternion Kähler manifolds. However, it seems that there
is no such result for general quaternionic manifolds.
Techniques of both these articles rely heavily on a correspondence between the
quaternionic manifold and its twistor space, which is a complex manifold and
thus one can apply the Hirzebruch-Riemann-Roch theorem. Our approach is more
elementary and computational in nature, the main tool is a version of the
Atiyah-Singer index theorem for elliptic complexes associated to a
$G$-structure. The theorem provides us with a topological formula involving
several characteristic classes but to simplify this we need some ”common
language.” Luckily, this was described in [5], where a quaternionic structure
is viewed as an action of a bundle of quaternion algebras. Using this and the
method of Borel and Hirzebruch (see [4]) for computation of Chern classes, we
have developed an algorithm how to obtain a formula for the analytical index
for each given complex and in each given dimension.
The resulting formulas are given in terms of the Pontryagin classes of the
manifold and one another cohomology class which is related to the quaternionic
structure of the manifold. By taking integral linear combinations of the index
formulas it is possible to eliminate this class and derive some integrality
conditions on the existence of a quaternionic structure on a compact manifold
(and not only of an almost quaternionic structure), for an example see
Corollary 5.9.
## 2\. Quaternionic manifolds and quaternionic complexes
Let us first recall the classical definition of quaternionic manifolds via
$G$-structures. View $\mathbb{R}^{4m}$ as the space $\mathbb{H}^{m}$ of
$m$-tuples of quaternions. Then the group $\operatorname{\mathrm{Sp}}(1)$ of
unit quaternions acts on it by $a\cdot v=v\bar{a}$ and
$\mathrm{GL}(m,\mathbb{H})$ acts by left matrix multiplication. These actions
induce injections
$\operatorname{\mathrm{Sp}}(1)\rightarrow\mathrm{GL}(4m,\mathbb{R})$ and
$\mathrm{GL}(m,\mathbb{H})\rightarrow\mathrm{GL}(4m,\mathbb{R})$. We define
the group $\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$ as the
product of these two groups in $\mathrm{GL}(4m,\mathbb{R})$. More abstractly,
$\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$ is isomorphic to the
quotient
$\operatorname{\mathrm{Sp}}(1)\times_{\mathbb{Z}_{2}}\mathrm{GL}(m,\mathbb{H})$,
where a pair
$(a,A)\in\operatorname{\mathrm{Sp}}(1)\times\mathrm{GL}(m,\mathbb{H})$
represents the linear mapping $v\mapsto Av\bar{a}$. In the sequel we write
simply $G$ instead of
$\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$.
###### Definition 2.1.
A $4m$-dimensional manifold $M$, $m\geq 2$, is called _almost quaternionic_ if
it has a $G$-structure, i.e. there is a principal $G$-bundle $\mathcal{P}$ and
an isomorphism of vector bundles $\mathcal{P}\times_{G}\mathbb{R}^{4m}\cong
TM$. An almost quaternionic manifold is called _quaternionic_ if the
$G$-structure $\mathcal{P}$ admits a torsion-free connection.
Denote by $G_{1}$ the group
$\operatorname{\mathrm{Sp}}(1)\times\mathrm{GL}(m,\mathbb{H})$, which is the
double cover of $G$. The principal $G$-bundle $\mathcal{P}$ can be locally
lifted to a principal $G_{1}$-bundle $\mathcal{P}_{1}$. Then if $\rho\colon
G_{1}\to\mathrm{GL}(\mathbb{V})$ is a representation of $G_{1}$, we can
locally construct the associated vector bundle
$V=\mathcal{P}_{1}\times_{G_{1}}\mathbb{V}$. This vector bundle exists
globally if either the lift can be done globally or the action $\rho$ factors
through $G$.
Let $\mathbb{E}$ and $\mathbb{F}$ denote the standard complex
$\operatorname{\mathrm{Sp}}(1)$-module and $\mathrm{GL}(m,\mathbb{H})$-module,
respectively. These can be also viewed as $G_{1}$-modules by composing the
corresponding representations with the projections onto the two factors. Then
the $G_{1}$-module $\mathbb{E}\otimes_{\mathbb{C}}\mathbb{F}$ descends to a
$G$-module and one can show that we have the following isomorphisms of vector
bundles
(2.2)
$T^{*}M\otimes_{\mathbb{R}}\mathbb{C}\cong\mathcal{P}\times_{G}(\mathbb{E}\otimes_{\mathbb{C}}\mathbb{F})\cong\mathcal{P}_{1}\times_{G_{1}}(\mathbb{E}\otimes_{\mathbb{C}}\mathbb{F})\cong
E\otimes_{\mathbb{C}}F.$
However, note that the bundles $E$ and $F$ may not exist globally.
An easy consequence of the above decomposition is that there is a natural
subcomplex of the de Rham complex of complex-valued differential forms.
Indeed, decomposing the $G_{1}$-module
$\Lambda^{j}(\mathbb{E}\otimes\mathbb{F})$ into the sum of irreducible
$G_{1}$-modules we can find a summand $\mathbb{A}^{j}$ isomorphic to
$S^{j}\mathbb{E}\otimes\Lambda^{j}\mathbb{F}$. This is in fact a $G$-module
because the action of $G_{1}$ on $S^{j}\mathbb{E}\otimes\Lambda^{j}\mathbb{F}$
factors through $G$. Therefore, in view of (2.2) this implies that there is a
vector subbundle $A^{j}\subseteq\Lambda^{j}(T^{*}M\otimes\mathbb{C})$ such
that
(2.3) $A^{j}=\mathcal{P}\times_{G}\mathbb{A}^{j}\cong
S^{j}E\otimes\Lambda^{j}F.$
Let $d$ denote the exterior derivative and
$p_{j}\colon\Lambda^{j}(T^{*}M\otimes\mathbb{C})\to A^{j}$ the projection. If
we put $d_{j}=p_{j}\circ d$, we obtain the following sequence of differential
operators
(2.4) $0\rightarrow\Gamma A^{0}\xrightarrow{d_{1}}\Gamma
A^{1}\xrightarrow{d_{2}}\Gamma
A^{2}\xrightarrow{d_{3}}\cdots\xrightarrow{d_{2m}}\Gamma A^{2m}\rightarrow 0.$
This sequence is closely related to the (almost) quaternionic structure of the
manifold.
###### Proposition 2.5 (Salamon).
An almost quaternionic manifold $M$ is quaternionic if and only if the
sequence (2.4) is a complex. If this is the case, then the complex is
elliptic.
###### Proof.
The proof can be found in [11], Theorem 4.1. ∎
In the rest of the paper, the elliptic complex (2.4) will be called the
_Salamon’s complex_ because it was first constructed by S. Salamon in [11]
together with several other elliptic complexes, which may not exist globally.
Another class of elliptic complexes on quaternionic manifolds was described by
R. Baston in [2]. Recently, A. Čap and V. Souček applied the theory of
parabolic geometries to construct a wide class of complexes of differential
operators as subcomplexes in the so-called curved BGG-sequences, see [9], and
many of them turned out to be elliptic. We will describe these complexes in a
convenient way without giving their explicit construction. This can be found
in [9], the theory of BGG-sequences in [8] and the general theory of parabolic
geometries in [7].
According to [11] the complex representation theories of the Lie groups
$\mathrm{GL}(m,\mathbb{H})\subset\mathrm{GL}(2m,\mathbb{C})$ and
$\mathrm{U}(2m)$ are equivalent. In particular, irreducible complex
representations of $\mathrm{GL}(m,\mathbb{H})$ may be identified with highest
weights for $\mathrm{U}(2m)$. For each $k\geq 0$ put
(2.6)
$\mathbb{W}_{k}^{j}=S^{j+k}\mathbb{E}\otimes(\Lambda^{j}\mathbb{F}\otimes
S^{k}\mathbb{F}^{*})_{0}\quad\text{for
}j<2m,\quad\mathbb{W}_{k}^{2m}=S^{2(m+k)}\mathbb{E}\otimes\Lambda^{2m}\mathbb{F},$
where the zero subscript denotes the irreducible component of the tensor
product with the highest weight being the sum of the highest weights of the
respective factors. Note that the $\mathbb{W}_{k}^{j}$ are $G_{1}$ as well as
$G$-modules.
###### Proposition 2.7 (Čap, Souček).
Let $M$ be a $4m$-dimensional quaternionic manifold with a $G$-structure
$\mathcal{P}$. Denote by $W_{k}^{j}$ the associated vector bundle
$\mathcal{P}\times_{G}\mathbb{W}_{k}^{j}$. Then for each $k\geq 0$ there is an
elliptic complex
$D_{k}\colon\quad 0\to\Gamma W_{k}^{0}\xrightarrow{D}\Gamma
W_{k}^{1}\xrightarrow{D}\cdots\xrightarrow{D}\Gamma W_{k}^{2m}\to 0.$
###### Proof.
For the proof see [9]. Let us remark, however, that the existence of a
compatible torsion-free connection is essential in the construction.
Therefore, this result does not hold for manifolds which are only almost
quaternionic. ∎
The differential operators $D$ are explicitly constructed in [8] in a much
more general setting of parabolic geometries. In the case of quaternionic
manifolds they are shown to be _strongly invariant_ , which implies that the
symbol of $D$ is induced by a $G$-equivariant polynomial map
$\mathbb{R}^{4m}\to L(\mathbb{W}_{k}^{j},\mathbb{W}_{k}^{j+1})$. This will be
cleared out in the next section.
The goal of the paper is to compute the analytical indices of the elliptic
complexes $D_{k}$. Note that by setting $k=0$ we obtain precisely the
Salamon’s complex $\eqref{complex}$. Moreover, the complex $D_{1}$ may be
naturally interpreted as a deformation complex for quaternionic structures,
see [6].
## 3\. Atiyah-Singer index formula
In this section we present a version of the Atiyah-Singer index theorem for
elliptic complexes associated to $G$-structures, which is our main tool for
computation of the indices. At the end we apply this to the Salamon’s complex
on manifolds with a $\mathrm{GL}(m,\mathbb{H})$-structure.
Let $G$ be a Lie group and let $M$ be a compact oriented manifold with a
$G$-structure $\mathcal{P}$, i.e. there is a principal $G$-bundle
$\mathcal{P}\to M$ and a real oriented $G$-module $\mathbb{V}$ such that
$TM\cong\mathcal{P}\times_{G}\mathbb{V}$. Let $\mathbb{E}^{j}$, $0\leq j\leq
r$, be complex $G$-modules and put
$E^{j}=\mathcal{P}\times_{G}\mathbb{E}^{j}$. Denote by $D$ an elliptic complex
$0\rightarrow\Gamma E^{0}\xrightarrow{d_{0}}\Gamma
E^{1}\xrightarrow{d_{1}}\cdots\xrightarrow{d_{r-1}}\Gamma E^{r}\rightarrow 0$
of differential operators on $M$. Suppose that
$\varphi_{j}\colon\mathbb{V}^{*}\to L(\mathbb{E}^{j},\mathbb{E}^{j+1})$ is a
$G$-equivariant polynomial map such that for all $v\in\mathbb{V}^{*},v\neq 0$,
the sequence
(3.1)
$0\rightarrow\mathbb{E}^{0}\xrightarrow{\varphi_{0}(v)}\mathbb{E}^{1}\xrightarrow{\varphi_{1}(v)}\cdots\xrightarrow{\varphi_{r-1}(v)}\mathbb{E}^{r}\rightarrow
0$
is exact. If the symbol $\sigma_{D}$ of $D$ is induced via the isomorphisms
$T^{*}M\cong\mathcal{P}\times_{G}\mathbb{V}^{*}$ and
$E^{j}\cong\mathcal{P}\times_{G}\mathbb{E}^{j}$ from the sequence (3.1), then
we say that $\sigma_{D}$ is _associated to the $G$-structure $\mathcal{P}$_.
More explicitly, the symbol of the operator $d_{j}$ is a fibrewise polynomial
bundle map
$\sigma_{d_{j}}\colon T^{*}M\cong\mathcal{P}\times_{G}\mathbb{V}^{*}\to
L(E^{j},E^{j+1})\cong\mathcal{P}\times_{G}L(\mathbb{E}^{j},\mathbb{E}^{j+1})$
and we require that $\sigma_{d_{j}}([p,v])=[p,\varphi_{j}(v)]$.
For example, the symbol of the de Rham complex of complex-valued differential
forms at $(x,v)\in T^{*}M$ is given by the exterior product on $v$ and this is
clearly associated via the mappings $\varphi_{j}(v)=v\wedge-$ to a
$\operatorname{\mathrm{SO}}(m)$-structure of $M$ induced by a Riemannian
metric and an orientation.
As was already noted at the end of the previous section, the differential
operators forming the quaternionic complexes from Proposition 2.7 are strongly
invariant. This implies (see [8]) that their symbols are associated to the
$\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$-structure of our
quaternionic manifold $M$.
Before we state the version of the Atiyah-Singer index formula we will need,
let us recall some basic facts on classifying spaces. Let $G$ be a compact Lie
group and let $EG\to BG$ denote the universal principal $G$-bundle over the
classifying space $BG$. If $\mathcal{P}\to M$ is a principal $G$-bundle, then
there is up to homotopy a unique map $f\colon M\to BG$ such that
$\mathcal{P}\cong f^{*}EG$, the pullback bundle. In cohomology, $f$ induces a
ring homomorphism $f^{**}\colon H^{**}(BG;\mathbb{Q})\to
H^{**}(M;\mathbb{Q})$, where $H^{**}(-;\mathbb{Q})=\prod_{k\geq
0}H^{k}(-;\mathbb{Q})$. Finally, it is shown in [3] that
$H^{**}(BG;\mathbb{Q})$ is a ring of formal power series in several
indeterminates with rational coefficients, hence an integral domain. In the
following, $\operatorname{\mathrm{ch}}$ denotes the Chern character and
$\operatorname{\mathrm{td}}$ the Todd class.
###### Proposition 3.2 (Atiyah, Singer).
Let $M$ be a compact oriented manifold of dimension $2m$, $G$ a compact Lie
group and $\rho\colon G\to\mathrm{SO}(2m)$ a Lie group homomorphism. Assume
that $M$ has a $G$-structure $\mathcal{P}$, i.e. $TM$ is associated to
$\mathcal{P}$ via $\rho$. Let $\mathbb{E}^{j}$, $0\leq j\leq r$, be complex
$G$-modules and let $E^{j}$ be the corresponding associated vector bundles.
Suppose that
$0\rightarrow\Gamma E^{0}\xrightarrow{d_{0}}\Gamma
E^{1}\xrightarrow{d_{1}}\cdots\xrightarrow{d_{r-1}}\Gamma E^{r}\rightarrow 0$
is an elliptic complex with its symbol associated to the $G$-structure
$\mathcal{P}$. Let $f\colon M\to BG$ be the classifying map for the bundle
$\mathcal{P}$. Put $\widetilde{E}^{j}=EG\times_{G}\mathbb{E}^{j}$ and
$\widetilde{V}=EG\times_{\rho}\mathbb{R}^{2m}$. If the Euler class
$e(\widetilde{V})$ is nonzero, then it divides
$\sum(-1)^{j}\mathrm{ch}\,\widetilde{E}_{j}\in H^{**}(BG;\mathbb{Q})$ and the
index of the above complex is given by
$(-1)^{m}\left\\{f^{**}\left(\frac{\sum_{j=0}^{r}(-1)^{j}\mathrm{ch}\,\widetilde{E}^{j}}{e(\widetilde{V})}\right)\cdot\mathrm{td}(TM\otimes\mathbb{C})\right\\}[M].$
###### Proof.
The proof can be found in [1]. ∎
The proposition will be the main tool for our computation of the indices in
question. However, we first need to reduce the structure group of the
principal $\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$-bundle
$\mathcal{P}$ of the quaternionic manifold $M$ to a compact subgroup. But this
can be easily done by introducing a Riemannian metric on $M$ because
$(\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H}))\cap\mathrm{O}(4m)=\operatorname{\mathrm{Sp}}(1)\operatorname{\mathrm{Sp}}(m)$
is a compact subgroup of $\mathrm{SO}(4m)$. Moreover, it follows that a
quaternionic manifold is always orientable and thus Proposition 3.2 may be
applied if the universal Euler class is not zero. This will be shown in the
next section, where we study certain characteristic classes useful for the
actual calculation of the index formula.
###### Remark 3.3 (Chern classes).
For the computation of Chern classes and Chern characters we use the approach
of Borel and Hirzebruch from [4] which relates characteristic classes of
associated vector bundles to weights of the corresponding representations. Let
us outline this in a few lines.
Let $\lambda\colon G\to\mathrm{U}(\mathbb{V})\cong\mathrm{U}(n)$ be a complex
representation of a compact Lie group $G$. Let $S\subseteq G$ be a maximal
torus of $G$ such that $S$ is mapped via $\lambda$ to the maximal torus of all
diagonal matrices in $\mathrm{U}(n)$. Suppose that $x_{1},x_{2},\ldots,x_{n}$
are the weights of $\lambda$ with respect to $S$. Having a principal
$G$-bundle $\mathcal{P}\to M$, consider the associated vector bundle
$V=\mathcal{P}\times_{G}\mathbb{V}$. Then the total Chern class, the Chern
character and the Todd class of $V$ may be formally written as
$c(V)=1+c_{1}(V)+\ldots+c_{n}(V)=\prod_{j=1}^{m}(1+y_{j}),\quad\operatorname{\mathrm{ch}}(V)=\sum_{j=1}^{n}\mathrm{e}^{y_{j}},\quad\operatorname{\mathrm{td}}(V)=\prod_{j=1}^{n}\frac{y_{j}}{1-\mathrm{e}^{-y_{j}}}.$
for some two-dimensional integral cohomology classes $y_{j}$ derived from the
weights $x_{j}$.
Similarly, the weights of the $G$-module $\Lambda^{k}\mathbb{V}$ are the sums
$x_{j_{1}}+x_{j_{2}}+\ldots+x_{j_{k}}$, where $1\leq j_{1}<j_{2}<\ldots
j_{k}\leq n,$ and thus we can again write the Chern classes and the Chern
character $\Lambda^{j}(V)$ in terms of the $y_{j}$. The following formula for
the Chern character of the formal polynomial
$\Lambda_{t}(V)=\sum_{k=0}^{n}t^{k}\Lambda^{k}V$ will be useful
$\operatorname{\mathrm{ch}}(\Lambda_{t}(V))=\sum_{k=0}^{n}t^{k}\left(\sum_{1\leq
j_{1}<\ldots<j_{k}\leq
n}\mathrm{e}^{y_{j_{1}}+\ldots+y_{j_{k}}}\right)=\prod_{j=1}^{n}(1+t\mathrm{e}^{y_{j}}).$
Before we continue to study quaternionic structures from a more topological
viewpoint, let us apply Proposition 3.2 to compute the index of the Salamon’s
complex (2.4) in the special case of manifolds admitting a
$\mathrm{GL}(m,\mathbb{H})$-structure.
###### Example 3.4.
Let $M$ be a compact $4m$-dimensional manifold with a
$\mathrm{GL}(m,\mathbb{H})$-structure admitting a torsion-free connection.
Because $\mathrm{GL}(m,\mathbb{H})$ can be identified with a subgroup of
$\operatorname{\mathrm{Sp}}(1)\mathrm{GL}(m,\mathbb{H})$, the manifold $M$ is
quaternionic and so the Salamon’s complex has sense. However, in this case
both the bundles $E$ and $F$ exist globally and $E$ is trivial. Moreover, the
cotangent bundle $T^{*}M$ is isomorphic to the complex vector bundle $F$ up to
orientation – for $m$ even the orientations coincide and for $m$ odd they are
opposite. Indeed, $T^{*}M$ is oriented as a quaternionic vector bundle while
$F$ as a complex vector bundle. The $\mathrm{GL}(m,\mathbb{H})$-modules
$\mathbb{A}^{j}$ inducing the vector bundles in the Salamon’s complex now look
like $\mathbb{C}^{j+1}\otimes\Lambda^{j}\mathbb{F}$.
By introducing a Riemannian metric on $M$ we may reduce the
$\mathrm{GL}(m,\mathbb{H})$-structure of $M$ to a
$\operatorname{\mathrm{Sp}}(m)$-structure $\mathcal{P}$ (not necessarily
admitting a torsion-free connection) and then apply Proposition 3.2 with
$\rho\colon\operatorname{\mathrm{Sp}}(m)\hookrightarrow\operatorname{\mathrm{SO}}(4m)$
being the standard inclusion described in Section 2 and
$\mathbb{E}^{j}=\mathbb{A}^{j}$. The Euler class of the universal vector
bundle $\widetilde{V}=E\mathrm{Sp}(m)\times_{\rho}\mathbb{R}^{4m}$ is one of
the generators of the cohomology ring of $B\mathrm{Sp}(m)$ and thus it is
nonzero.
Consider the universal vector bundles
$\widetilde{F}=E\mathrm{Sp}(m)\times_{\operatorname{\mathrm{Sp}}(m)}\mathbb{F}$
and
$\widetilde{A}^{j}=E\mathrm{Sp}(m)\times_{\operatorname{\mathrm{Sp}}(m)}\mathbb{A}^{j}$.
Then $\widetilde{V}\cong\widetilde{F}$ up to orientation and
$\widetilde{A}^{j}\cong\mathbb{C}^{j+1}\otimes\Lambda^{j}\widetilde{F}$.
Because $\widetilde{F}$ comes from a quaternionic vector bundle, we have
$\overline{\widetilde{F}}\cong\widetilde{F}$ and altogether this gives
$e(\widetilde{V})=(-1)^{m}c_{2m}(\widetilde{F}),\quad\operatorname{\mathrm{td}}(\widetilde{V}\otimes\mathbb{C})=\operatorname{\mathrm{td}}(\widetilde{F}\oplus\overline{\widetilde{F}})=\operatorname{\mathrm{td}}(\widetilde{F})^{2}.$
Finally, let $f\colon M\to B\mathrm{Sp}(m)$ be the classifying map for
$\mathcal{P}$. Then according to Proposition 3.2 the index of the Salamon’s
complex is given by
(3.5)
$\mathrm{ind}=f^{**}\left(\frac{\sum_{j=0}^{2m}(-1)^{j}(j+1)\mathrm{ch}\,\Lambda^{j}\widetilde{F}}{(-1)^{m}c_{2m}(\widetilde{F})}\cdot\mathrm{td}(\widetilde{F})^{2}\right)[M].$
To simplify this formula we compute the Chern classes of the complex vector
bundle $\widetilde{F}$ as described in Remark 3.3. Let $S$ be the maximal
torus of $\operatorname{\mathrm{Sp}}(m)$ consisting of all diagonal matrices
with entries $\exp(2\pi\mathrm{i}x_{j})$, where $x_{j}\in\mathbb{R}$. Then the
weights of the $\operatorname{\mathrm{Sp}}(m)$-module $\mathbb{F}$ are $\pm
x_{j},1\leq j\leq m$, viewed as linear forms on the Lie algebra
$\mathfrak{s}$. It follows that the total Chern class and the Todd class of
$\widetilde{F}$ may be written in the form
$c(\widetilde{F})=1+c_{1}(\widetilde{F})+\ldots+c_{2m}(\widetilde{F})=\prod_{j=1}^{m}(1+y_{j})(1-y_{j}),\quad\operatorname{\mathrm{td}}(\widetilde{F})=\prod_{j=1}^{m}\frac{y_{j}(-y_{j})}{(1-\mathrm{e}^{-y_{j}})(1-\mathrm{e}^{y_{j}})}.$
In particular, the last Chern class equals
$c_{2m}(\widetilde{F})=\prod_{j=1}^{m}y_{j}(-y_{j})$.
The numerator of the fraction in (3.5) may be simplified as follows. Instead
of $(-1)^{j}$ write $t^{j}$ and recall the formula for
$\operatorname{\mathrm{ch}}(\Lambda_{t}(V))$ from Remark 3.3. Then
$\begin{split}&\sum_{j=0}^{2m}\,(j+1)t^{j}\operatorname{\mathrm{ch}}(\Lambda^{j}\widetilde{F})=\frac{\mathrm{d}}{\mathrm{d}t}\left(\sum_{j=0}^{2m}t^{j+1}\operatorname{\mathrm{ch}}(\Lambda^{j}\widetilde{F})\right)=\\\
&=\frac{\mathrm{d}}{\mathrm{d}t}\left(t\cdot\operatorname{\mathrm{ch}}(\Lambda_{t}(\widetilde{F})\right)=\frac{\mathrm{d}}{\mathrm{d}t}\left(t\prod_{j=1}^{m}(1+t\mathrm{e}^{y_{j}})(1+t\mathrm{e}^{-y_{j}})\right)=\\\
&=\prod_{j=1}^{m}(1+t\mathrm{e}^{y_{j}})(1+t\mathrm{e}^{-y_{j}})+t\sum_{j=1}^{m}(\mathrm{e}^{y_{j}}(1+t\mathrm{e}^{-y_{j}})+\mathrm{e}^{-y_{j}}(1+t\mathrm{e}^{y_{j}}))\prod_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{m}(1+t\mathrm{e}^{y_{k}})(1+t\mathrm{e}^{-y_{k}}).\end{split}$
Substituting $t=-1$ and collecting the terms we end up with
$\sum_{j=0}^{2m}(-1)^{j}(j+1)\mathrm{ch}\,\Lambda^{j}\widetilde{F}=(m+1)\prod_{j=1}^{m}(1-\mathrm{e}^{y_{j}})(1-\mathrm{e}^{-y_{j}})$
The interior of the bracket in (3.5) now reads as
$\frac{(m+1)\prod_{j=1}^{m}(1-\mathrm{e}^{y_{j}})(1-\mathrm{e}^{-y_{j}})}{(-1)^{m}\prod_{j=1}^{m}y_{j}(-y_{j})}\cdot\left(\prod_{j=1}^{m}\frac{y_{j}(-y_{j})}{(1-\mathrm{e}^{-y_{j}})(1-\mathrm{e}^{y_{j}})}\right)^{2}=(-1)^{m}(m+1)\operatorname{\mathrm{td}}(\widetilde{F}).$
Applying the map $f^{**}$ and evaluating on the fundamental class of $M$ we
obtain the desired index. The complex tangent bundle $T^{c}M$ of $M$ is
isomorphic to the vector bundle $F^{*}\cong
F=\mathcal{P}\times_{\operatorname{\mathrm{Sp}}(m)}\mathbb{F}$. We have thus
proved the following theorem, which is our first partial result on indices of
quaternionic complexes.
###### Theorem 3.6.
Let $M$ be a compact manifold with a $\mathrm{GL}(m,\mathbb{H})$-structure
admitting a torsion-free connection. Then the index of the Salamon’s complex
is given by
$(-1)^{m}(m+1)\mathrm{td}(T^{c}M)[M].$
Note that such a manifold is a complex manifold and the number
$\operatorname{\mathrm{td}}(T^{c}M)[M]$ is the index of the Dolbeault complex
associated to the complex tangent bundle $T^{c}M$ of $M$. In particular, it is
an integer and so the above index is an integer divisible by $m+1$.
Let us remark here that by a different method very similar results were
obtained in [2] for certain class of quaternionic complexes. However, it is
not clear to us whether the Salamon’s complex was included.
## 4\. Quaternionic structures
This section is devoted to basic topological properties of quaternionic
structures, the main reference here is [5]. In particular, we define
characteristic classes for these structures. Throughout the section, $X$
denotes a compact Hausdorff topological space.
###### Definition 4.1.
Let $\beta$ be an oriented real $3$-dimensional vector bundle over $X$ with a
positive-definite inner product $\langle-,-\rangle$. Then we define a _bundle
of quaternion algebras_ as the vector bundle
$\operatorname{\mathbb{H}_{\beta}}=\mathbb{R}\oplus\beta$ together with a
fibrewise multiplication given by
$(s,u)\cdot(t,v)=(st-\langle u,v\rangle,sv+tu+u\times v).$
Equivalently, if $\mathcal{P}\to X$ is the principal
$\operatorname{\mathrm{SO}}(3)=\mathrm{Aut}(\mathbb{H})$-bundle corresponding
to $\beta$, then we have
$\operatorname{\mathbb{H}_{\beta}}=\mathcal{P}\times_{\mathrm{Aut}(\mathbb{H})}\mathbb{H}$.
The definition says that fibrewise the bundle
$\operatorname{\mathbb{H}_{\beta}}$ carries a structure of the algebra of
quaternions, but globally it may not be the product bundle
$X\times\mathbb{H}$.
###### Definition 4.2.
A real vector bundle $V\to X$ is said to be a _right
$\operatorname{\mathbb{H}_{\beta}}$-bundle_ if it admits a right
$\operatorname{\mathbb{H}_{\beta}}$-module structure, i.e. there is a bundle
map $V\otimes_{\mathbb{R}}\operatorname{\mathbb{H}_{\beta}}\to V$ that
restricts to an $\mathbb{H}$-module structure in each fibre.
It follows from the definition that the dimension of an
$\operatorname{\mathbb{H}_{\beta}}$-bundle must be divisible by four.
Moreover, such a bundle can be canonically oriented. Indeed, to orient a fibre
$V_{x}$, choose a basis $e_{1},e_{2},\ldots,e_{m}$ of $V_{x}$ as an
$(\operatorname{\mathbb{H}_{\beta}})_{x}$-module and an oriented orthonormal
basis $\mathrm{i,j,k}$ of $\beta_{x}$. Then
$e_{1},e_{1}\mathrm{i},e_{1}\mathrm{j},e_{1}\mathrm{k},\ldots,e_{m},e_{m}\mathrm{i},e_{m}\mathrm{j},e_{m}\mathrm{k}$
is the oriented basis of $V_{x}$.
###### Proposition 4.3 ([5]).
A $4m$-dimensional real vector bundle $V$ is a right
$\operatorname{\mathbb{H}_{\beta}}$-bundle for some oriented 3-dimensional
vector bundle $\beta$ if and only if the structure group of the frame bundle
of $V$ may be reduced to the subgroup
$\mathrm{Sp}(1)\mathrm{Sp}(m)\subset\mathrm{GL}(4m,\mathbb{R})$.
###### Proof.
If the structure group of the frame bundle reduces to the subgroup
$G=\operatorname{\mathrm{Sp}}(1)\operatorname{\mathrm{Sp}}(m)$, then there is
a principal $G$-bundle $\mathcal{P}$ such that
$V\cong\mathcal{P}\times_{G}\mathbb{H}^{m}$, where we view $\mathbb{H}^{m}$ as
a real vector space. Now put
$\beta=\mathcal{P}\times_{G}\mathrm{im}\,\mathbb{H}$ with the action of $G$ on
$\mathrm{im}\,\mathbb{H}$ defined as follows: if
$(a,A)\in\operatorname{\mathrm{Sp}}(1)\times\operatorname{\mathrm{Sp}}(m)$
represents an element of $G$, then $(a,A)\cdot q=aq\bar{a}$. Then $\beta$ is
an orientable $3$-dimensional real vector bundle and the associated quaternion
algebra is
$\operatorname{\mathbb{H}_{\beta}}=\mathcal{P}\times_{G}\mathbb{H}$, where the
action of $G$ on $\mathbb{H}$ is the same as on $\mathrm{im}\,\mathbb{H}$. But
then right multiplication by quaternions is a $G$-equivariant map and so it
induces a right $\operatorname{\mathbb{H}_{\beta}}$-module structure on $V$.
For the other direction see [5]. ∎
The proposition applies, in particular, to the tangent or cotangent bundle of
a quaternionic manifold $M$ (after introducing a Riemannian metric). We can
actually describe the bundle $\beta$ as follows. Let $\mathbb{E}$ be the
standard complex $\mathrm{Sp}(1)$-module as in Section 2. If we view the
second symmetric power $S^{2}\mathbb{E}$ as a $G$-module, then the mapping
$\varphi\colon\mathrm{im}\,\mathbb{H}\to S^{2}\mathbb{E}$ defined by
$\varphi(u)=\mathrm{j}\otimes u-1\otimes u\mathrm{j}$ is a real linear
$G$-equivariant map. Moreover, the real basis
$\mathrm{i},\mathrm{j},\mathrm{k}$ of $\mathrm{im}\,\mathbb{H}$ is mapped to a
complex basis of $S^{2}\mathbb{E}$
$\mathrm{i}\mapsto(1\otimes\mathrm{j}+\mathrm{j}\otimes
1)\mathrm{i},\quad\mathrm{j}\mapsto 1\otimes
1+\mathrm{j}\otimes\mathrm{j},\quad\mathrm{k}\mapsto(1\otimes
1-\mathrm{j}\otimes\mathrm{j})\mathrm{i}.$
This implies that the complexification of $\mathrm{im}\,\mathbb{H}$ is
isomorphic to $S^{2}\mathbb{E}$ and, on the level of associated vector
bundles, the complexification of
$\beta=\mathcal{P}\times_{G}\mathrm{im}\,\mathbb{H}$ is isomorphic to
$S^{2}E=\mathcal{P}\times_{G}S^{2}\mathbb{E}$, which is a globally defined
vector bundle over $M$. In fact, the sphere bundle of $\beta$ is precisely the
Salamon’s twistor space.111Compare with [12, page 146].
Now we proceed to define characteristic classes for
$\operatorname{\mathbb{H}_{\beta}}$-bundles. Let $V\to X$ be a right
$\operatorname{\mathbb{H}_{\beta}}$-bundle of quaternionic dimension $m$, i.e.
real dimension $4m$. Then one can consider the associated projective bundle
$\pi\colon\mathbb{H}_{\beta}\mathrm{P}(V)\to X$ whose fibre over a point $x\in
X$ is the space of all quaternionic lines in the fibre $V_{x}$ in the sense of
the $\operatorname{\mathbb{H}_{\beta}}$-module structure. Futhermore, let
$L=\\{(\ell,v)\in\mathbb{H}_{\beta}\mathrm{P}(V)\times V\,|\,v\in\ell\\}$ be
the canonical $\operatorname{\mathbb{H}_{\beta}}$-line bundle over
$\mathbb{H}_{\beta}\mathrm{P}(V)$ oriented as a right
$\pi^{*}\operatorname{\mathbb{H}_{\beta}}$-bundle. The following proposition
defines characteristic classes $d^{\beta}_{j}(V)$ of the bundle $V$ as
coefficients of a certain polynomial over the ring $H^{*}(X;\mathbb{Z})$.
###### Proposition 4.4 ([5]).
For each right $\operatorname{\mathbb{H}_{\beta}}$-bundle $V\to X$ of
quaternionic dimension $m$ there are uniquely determined classes
$d^{\beta}_{j}(V)\in H^{4j}(X;\mathbb{Z}),1\leq j\leq m$, such that
$H^{*}(\mathbb{H}_{\beta}\mathrm{P}(V);\mathbb{Z})=H^{*}(X;\mathbb{Z})[t]/(t^{m}-d^{\beta}_{1}(V)t^{m-1}+\ldots+(-1)^{m}d^{\beta}_{m}(V)),$
where $t=e(L)\in H^{4}(\mathbb{H}_{\beta}\mathrm{P}(V);\mathbb{Z})$ is the
Euler class of the canonical bundle $L$.
###### Proof.
The proof is a standard application of the Leray-Hirsch theorem, see [5]. ∎
One can verify (see [5]) that the classes $d^{\beta}_{j}(V)$ have usual
properties of characteristic classes like naturality or multiplicativity.
Moreover, there is a splitting principle which implies that in calculations
with the classes $d^{\beta}_{j}(V)$ we may formally assume that there are
$y_{1},y_{2},\ldots,y_{m}\in H^{4}(X;\mathbb{Z})$ such that $d^{\beta}_{j}(V)$
is the $j$-th elementary symmetric polynomial in the $y_{k}$’s or, in short,
$d^{\beta}(V)=1+d^{\beta}_{1}(V)+\ldots+d^{\beta}_{m}(V)=\prod_{k=1}^{m}(1+y_{k})$.
The following proposition shows that the characteristic classes just defined
determine other characteristic classes of $V$ as a real vector bundle. This is
technically very useful.
###### Proposition 4.5 ([5]).
Let $V\to X$ be a canonically oriented right
$\operatorname{\mathbb{H}_{\beta}}$-bundle of quaternionic dimension $m$.
1. (a)
The Euler class $e(V)$ of $V$ equals the top-dimensional class
$d^{\beta}_{m}(V)$.
2. (b)
The rational Pontryagin classes $p_{j}(V)\in H^{4j}(X;\mathbb{Q})$ of $V$ are
given by
$1+p_{1}(V)+p_{2}(V)+\ldots+p_{2m}(V)=\prod_{j=1}^{m}((1+y_{j})^{2}+p_{1}(\beta)),$
where $d^{\beta}(V)=\prod_{k=1}^{m}(1+y_{k})$ and $p_{1}(\beta)$ is the first
Pontryagin class of $\beta$.
###### Proof.
See [5], but note that we deal with right
$\operatorname{\mathbb{H}_{\beta}}$-bundles rather than the left ones. ∎
Finally, the characteristic classes $d_{j}^{\beta}$ may be used to decribe the
cohomology ring of the classifying space of the group
$G=\operatorname{\mathrm{Sp}}(1)\operatorname{\mathrm{Sp}}(m)$. Let $EG\to BG$
be the universal principal $G$-bundle and put
$\beta=EG\times_{G}\mathrm{im}\,\mathbb{H}$ and
$\widetilde{V}=EG\times_{G}\mathbb{H}^{m}$. Then $\widetilde{V}$ is a right
$\mathbb{H}_{\beta}$-bundle as in the proof of Proposition 4.3. Let us write
simply $d_{j}$ for the characteristic classes $d^{\beta}_{j}(\widetilde{V})$
and $q_{1}$ for the first Pontryagin class $p_{1}(\beta)$ of $\beta$.
###### Proposition 4.6 ([5]).
The rational cohomology ring of $B\mathrm{Sp}(1)\mathrm{Sp}(m)$ is given by
$H^{*}(B\mathrm{Sp}(1)\mathrm{Sp}(m);\mathbb{Q})\cong\mathbb{Q}[q_{1},d_{1},d_{2},\ldots,d_{m}].$
###### Proof.
One can obtain this from the description of the integral cohomology ring of
the classifying space $B\mathrm{Sp}(1)\mathrm{Sp}(m)$, which was done in [5].
∎
According to Proposition 4.5 the universal Euler class $e(\widetilde{V})$
equals the class $d_{m}$, which is a generator of the cohomology ring and so
it is nonzero. We may therefore apply Proposition 3.2 to compute the indices
of the quaternionic complexes.
## 5\. The computations
Having all the necessary background at hand, we finally describe an algorithm
how to compute the indices of the elliptic complexes from Proposition 2.7.
This algorithm can be carried out for each given dimension of the manifold and
for each given complex $D_{k}$.
Let $M$ be a compact $4m$-dimensional quaternionic manifold. By introducing a
Riemannian metric on $M$ we may reduce the structure group of the principal
frame bundle of $M$ to the subgroup $G=\mathrm{Sp}(1)\mathrm{Sp}(m)$. Then $M$
is canonically oriented, see the preceding section. Let $\mathcal{P}\to M$ be
the principal $G$-bundle and $f\colon M\to BG$ its classifying map. Put
$\widetilde{V}=EG\times_{G}\mathbb{R}^{4m}$ and
$\widetilde{W}^{j}_{k}=EG\times_{G}\mathbb{W}^{j}_{k}$, where $W^{j}_{k}$ are
the $G$-modules inducing the quaternionic complexes in question, see
Proposition 2.7. Then by the Atiyah-Singer formula from Proposition 3.2 the
index of the quaternionic complex $D_{k}$ is given by
(5.1)
$\mathrm{ind}\,D_{k}=\left\\{f^{**}\left(\frac{\sum_{j=0}^{2m}(-1)^{j}\operatorname{\mathrm{ch}}\widetilde{W}^{j}_{k}}{e(\widetilde{V})}\right)\cdot\operatorname{\mathrm{td}}(TM\otimes\mathbb{C})\right\\}[M].$
To evaluate this expression we first have to solve the equation
(5.2) $x\cup
e(\widetilde{V})=\sum_{j=0}^{2m}(-1)^{j}\operatorname{\mathrm{ch}}\widetilde{W}^{j}_{k}$
in the cohomology ring $H^{**}(BG;\mathbb{Q})$. This task may be simplified in
two ways. First, because the cohomology groups of the compact manifold $M$
vanish above dimension $4m$, it suffices to determine $x$ up to this dimension
$4m$. Secondly, because the product group
$G_{1}=\operatorname{\mathrm{Sp}}(1)\times\operatorname{\mathrm{Sp}}(m)$ is
the double cover of $G$, the projection $\pi\colon G_{1}\to G$ induces an
isomorphism $(B\pi)^{**}\colon H^{**}(BG;\mathbb{Q})\to
H^{**}(BG_{1};\mathbb{Q})$. Therefore, we may pull back the above equation to
$BG_{1}$ and solve it in $H^{**}(BG_{1};\mathbb{Q})$. The advantage will be
clear soon.
Let $\mathbb{E}$ and $\mathbb{F}$ be the standard complex
$\operatorname{\mathrm{Sp}}(1)$ and $\operatorname{\mathrm{Sp}}(m)$-modules,
respectively, and put $\widetilde{E}=EG_{1}\times_{G_{1}}\mathbb{E}$ and
$\widetilde{F}=EG_{1}\times_{G_{1}}\mathbb{F}$. Then these are globally
defined vector bundles over $BG_{1}$ and we have
(5.3)
$(B\pi)^{*}(\widetilde{V}\otimes_{\mathbb{R}}\mathbb{C})\cong\widetilde{E}\otimes_{\mathbb{C}}\widetilde{F},$
compare with the isomorphism (2.2). Moreover, if
$\beta=EG\times_{G}\mathrm{im}\,\mathbb{H}$, then the real vector bundle
$\widetilde{V}_{1}=(B\pi)^{*}(\widetilde{V})$ is a right
$\mathbb{H}_{\beta_{1}}$-bundle for $\beta_{1}=(B\pi)^{*}(\beta)$. As in the
previous section, we may prove that $\beta_{1}\otimes\mathbb{C}\cong
S^{2}\widetilde{E}$ and hence for the first Pontryagin class of $\beta_{1}$ we
get
(5.4) $p_{1}(\beta_{1})=-c_{2}(S^{2}\widetilde{E})=-4c_{2}(\widetilde{E}).$
Applying the Chern character on (5.3) and comparing inductively the two sides
of the result, one may write the Pontryagin classes of $\widetilde{V}_{1}$ as
polynomials in the Chern classes of $\widetilde{E}$ and $\widetilde{F}$.
Altogether with (5.4) and Proposition 4.5 this implies that we are able to
translate between three sets of characteristic classes – the Chern classes of
$\widetilde{E}$ and $\widetilde{F}$, the Pontryagin classes of
$\widetilde{V}_{1}$ and $\beta_{1}$ and, finally, the classes
$d^{\beta_{1}}_{1}(\widetilde{V}_{1}),d^{\beta_{1}}_{2}(\widetilde{V}_{1}),\ldots,d^{\beta_{1}}_{m}(\widetilde{V}_{1})$
and $p_{1}(\beta_{1})$.
Now we may return to the equation (5.2) in the pulled-back version
(5.5) $(B\pi)^{*}(x)\cup
e(\widetilde{V}_{1})=\sum_{j=0}^{2m}(-1)^{j}\operatorname{\mathrm{ch}}((B\pi)^{*}(\widetilde{W}^{j}_{k})).$
We would like to compute the right-hand side in terms of the Chern classes of
$\widetilde{E}$ and $\widetilde{F}$. Once we do this, it remains to express
the result in terms of the $d^{\beta_{1}}_{l}$-classes and $p_{1}(\beta_{1})$
and divide by $e(\widetilde{V}_{1})=d^{\beta_{1}}_{m}(\widetilde{V}_{1})$ to
obtain the solution $(B\pi)^{*}(x)\in H^{*}(BG_{1};\mathbb{Q})$ and hence also
$x\in H^{*}(BG;\mathbb{Q})$, i.e. the fraction in (5.1).
Recall from the definition of the representations $\mathbb{W}^{j}_{k}$ (see
(2.6)) that
$(B\pi)^{*}(\widetilde{W}_{k}^{j})=S^{j+k}\widetilde{E}\otimes(\Lambda^{j}\widetilde{F}\otimes
S^{k}\widetilde{F}^{*})_{0}\quad\text{for
}j<2m,\quad(B\pi)^{*}(\widetilde{W}_{k}^{2m})=S^{2(m+k)}\widetilde{E}\otimes\Lambda^{2m}\widetilde{F}.$
The two factors in the tensor products are globally defined vector bundles and
so we can compute its Chern characters separately. We will again use the
approach described in Remark 3.3. As a maximal torus $S$ of
$G_{1}=\operatorname{\mathrm{Sp}}(1)\times\operatorname{\mathrm{Sp}}(m)$ take
the direct product of the standard maximal tori of
$\operatorname{\mathrm{Sp}}(1)$ and $\operatorname{\mathrm{Sp}}(m)$ – the
standard maximal torus of $\operatorname{\mathrm{Sp}}(1)$ is the set of
complex units $\exp(2\pi\mathrm{i}x)$ while the standard maximal torus of
$\operatorname{\mathrm{Sp}}(m)$ is the set of diagonal matrices with entries
$\exp(2\pi\mathrm{i}x_{l})$, where $x_{l}\in\mathbb{R}$.
Consider first the vector bundle
$\widetilde{E}=EG_{1}\times_{G_{1}}\mathbb{E}$. The weights of the
corresponding $G_{1}$-module $\mathbb{E}$ are $\pm x$ viewed as linear forms
on the Lie algebra $\mathfrak{s}$. Then the total Chern class of
$\widetilde{E}$ is given by $c(\widetilde{E})=(1+y)(1-y)$ and
$c_{1}(\widetilde{E})=0,\quad c_{2}(\widetilde{E})=-y^{2}.$
The Chern classes of the symmetric powers $S^{j}\widetilde{E}$ are now easy to
compute. Clearly, the weights of the $G_{1}$-module $S^{j}\mathbb{E}$ are the
forms $(k_{1}-k_{2})x$, $k_{1}+k_{2}=j$, and so we have
$c(S^{j}\widetilde{E})=\prod_{k_{1}+k_{2}=j}(1+(k_{1}-k_{2})y).$
This is clearly a polynomial expression in $-y^{2}=c_{2}(\widetilde{E})$.
Now turn to the vector bundle $\widetilde{F}=EG_{1}\times_{G_{1}}\mathbb{F}$.
The weights of the $G_{1}$-module $\mathbb{F}$ are precisely $\pm x_{l}$,
$1\leq l\leq m$, viewed as linear forms on the Lie algebra $\mathfrak{s}$. The
total Chern class of $\widetilde{F}$ is then given by
$c(\widetilde{F})=\prod_{l=1}^{m}(1+y_{l})(1-y_{l})=\prod_{l=1}^{m}(1-y_{l}^{2})$
and so $c_{2j}(\widetilde{F})$ is the $j$-th elementary symmetric polynomial
in the $-y_{l}^{2}$ while $c_{2j+1}(\widetilde{F})=0$.
The computation of the Chern classes of $(\Lambda^{j}\widetilde{F}\otimes
S^{k}\widetilde{F}^{*})_{0}$, $k\geq 0$, is a bit more complicated. This is
because $\mathbb{V}^{j}_{k}=(\Lambda^{j}\mathbb{F}\otimes
S^{k}\mathbb{F}^{*})_{0}$ was defined as a representation of the group
$\mathrm{U}(2m)$ corresponding to some maximal weight and we have to find its
weights with respect to the subgroup
$\operatorname{\mathrm{Sp}}(m)\subset\mathrm{U}(2m)$. This can be achieved as
follows. First, the character ring of complex representations of the group
$\mathrm{SU}(2m)$ differs from that of the group $\mathrm{U}(2m)$ only by a
one-dimensional determinantal representation on which
$\operatorname{\mathrm{Sp}}(m)$ acts trivially. Therefore, there is nothing
lost in assuming that $\mathbb{V}^{j}_{k}$ is a representation of
$\mathrm{SU}(2m)$. But $\mathrm{SU}(2m)$ is compact and simply connected and
so its representation theory is equivalent to that of the complex Lie algebra
$\mathfrak{sl}(2m,\mathbb{C})$. In particular, if we know the maximal weight
of $\mathbb{V}^{j}_{k}$, the remaining weights can be computed by standard
algorithms, see for example [13]. The maximal weight of $\mathbb{V}^{j}_{k}$
is by definition the sum of the maximal weight of $\Lambda^{j}\mathbb{F}$ and
the maximal weight of $S^{k}\mathbb{F}^{*}$ and these are easy to find – if
$z_{1},z_{2},\ldots,z_{2m}$ are the weights of $\mathbb{F}$ viewed as the
standard $\mathrm{SU}(2m)$-module, then the maximal weight of
$\Lambda^{j}\mathbb{F}$ is $z_{1}+z_{2}+\ldots+z_{j}$ while the maximal weight
of $S^{k}\mathbb{F}^{*}$ is $-k\cdot z_{2m}$. The remaining weights of
$\mathbb{V}^{j}_{k}$ are integral linear combinations of the $z_{l}$’s as
well. To obtain the weights of $\mathbb{V}^{j}_{k}$ as a
$\operatorname{\mathrm{Sp}}(m)$-module we only have to substitute
$z_{2l}=x_{l}$ and $z_{2l+1}=-x_{l}$ for $1\leq l\leq m$ – this can be seen
from the definition of the standard inclusion
$\operatorname{\mathrm{Sp}}(m)\subset\mathrm{SU}(2m)$. Finally, once we know
the weights, we get the total Chern class $c(\mathbb{V}^{j}_{k})$ and this
will be a polynomial expression symmetric in the variables $-y_{l}^{2}$.
Indeed, the set of weights of a $\mathrm{SU}(2m)$-module is invariant under
the action of the Weyl group of $\mathrm{SU}(2m)$, which is the symmetry group
on the set $\\{z_{1},z_{2},\ldots,z_{2m}\\}$. Therefore,
$c(\mathbb{V}^{j}_{k})$ can be expressed in terms of the Chern classes of
$\widetilde{F}$.
To sum up, we have seen that the Chern classes of both the vector bundles
$S^{j+k}\widetilde{E}$ and $(\Lambda^{j}\widetilde{F}\otimes
S^{k}\widetilde{F}^{*})_{0}$ may be written in terms of the Chern classes of
$\widetilde{E}$ and $\widetilde{F}$ and so the same holds true for the Chern
character of $(B\pi)^{*}(\widetilde{W}^{j}_{k})$. The right-hand side of the
equation (5.5) is thus a polynomial in the Chern classes of $\widetilde{E}$
and $\widetilde{F}$ and so it can be expressed in terms of the classes
$d^{\beta_{1}}_{l}(\widetilde{V}_{1})$ and $p_{1}(\beta_{1})$. Next, the
result will be a multiple of the Euler class
$e(\widetilde{V}_{1})=d^{\beta_{1}}_{m}(\widetilde{V}_{1})$ and by dividing we
obtain $(B\pi)^{*}(x)\in H^{*}(BG_{1};\mathbb{Q})$. To get the solution $x\in
H^{*}(BG;\mathbb{Q})$ of (5.2) it suffices to write
$d^{\beta}_{l}(\widetilde{V})$ and $p_{1}(\beta)$ instead of
$d^{\beta_{1}}_{l}(\widetilde{V}_{1})$ and $p_{1}(\beta_{1})$.
We are now at the end of the algorithm. The solution $x$, which is the
fraction in (5.1), may be expressed in terms of the Pontryagin classes of
$\widetilde{V}$ and $p_{1}(\beta)$ so that $f^{**}(x)$ will be a polynomial in
the Pontryagin classes of $TM$ and the class $p_{1}(f^{*}\beta)$. By
multiplying with the Todd class
$\operatorname{\mathrm{td}}(TM\otimes\mathbb{C})$ and evaluating the top-
dimensional part of the product on the fundamental class $[M]$ we get the
desired index.
Index formulas obtained in this way depend on the $G$-structure of $M$ via the
characteristic class $p_{1}(f^{*}\beta)$. This class may be expressed without
any reference to the classifying map $f$. Indeed, from the isomorphism
$\beta\otimes\mathbb{C}\cong S^{2}\widetilde{E}$, where $S^{2}\widetilde{E}$
is now viewed as a vector bundle over $BG$, follows that
$f^{*}(\beta\otimes\mathbb{C})\cong S^{2}E$ and this is a globally defined
complex vector bundle over $M$. Hence
$p_{1}(f^{*}\beta)=-c_{2}(f^{*}(\beta\otimes\mathbb{C}))=-c_{2}(S^{2}E).$
In general, the most difficult computational problem is to find the weights of
the $G$-modules $(\Lambda^{j}\mathbb{F}\otimes S^{k}\mathbb{F}^{*})_{0}$ and
then process these to obtain the Chern classes of the vector bundles
$(\Lambda^{j}\widetilde{F}\otimes S^{k}\widetilde{F}^{*})_{0}$. Of course, one
can make use of computer algebra systems such as LiE (see [14]) and Maple. We
have carried out some calculations for 8 and 12-dimensional manifolds arriving
at the following formulas.
###### Theorem 5.6.
Let $M$ be an $8$-dimensional compact quaternionic manifold. If we denote
$p_{1}=p_{1}(TM)$, $p_{2}=p_{2}(TM)$ and $q_{1}=-c_{2}(S^{2}E)$, then we have
$\displaystyle\textup{ind}\,D_{0}=\left(\frac{7}{1920}p_{1}^{2}-\frac{1}{24}p_{1}q_{1}-\frac{1}{480}p_{2}+\frac{1}{12}q_{1}^{2}\right)[M],$
$\displaystyle\textup{ind}\,D_{1}=\left(\frac{209}{1920}p_{1}^{2}+\frac{11}{24}p_{1}q_{1}-\frac{167}{480}p_{2}+\frac{25}{12}q_{1}^{2}\right)[M].$
###### Theorem 5.7.
Let $M$ be a $12$-dimensional compact quaternionic manifold. If we denote
$p_{1}=p_{1}(TM)$, $p_{2}=p_{2}(TM)$, $p_{3}=p_{3}(TM)$ and
$q_{1}=-c_{2}(S^{2}E)$, then we have
$\displaystyle\textup{ind}\,D_{0}=\biggl{(}\frac{31}{241920}p_{1}^{3}-\frac{7}{2304}p_{1}^{2}q_{1}-\frac{11}{60480}p_{1}p_{2}$
$\displaystyle+\frac{41}{2304}p_{1}q_{1}^{2}+$
$\displaystyle+\frac{1}{576}p_{2}q_{1}+\frac{1}{15120}p_{3}-\frac{73}{2304}q_{1}^{3}\biggr{)}[M],$
$\displaystyle\quad\textup{ind}\,D_{1}=\biggl{(}-\frac{1}{6720}p_{1}^{3}-\frac{77}{576}p_{1}^{2}q_{1}+\frac{1}{280}p_{1}p_{2}-\frac{35}{576}p_{1}q_{1}^{2}+\frac{7}{18}p_{2}q_{1}-\frac{17}{840}p_{3}-\frac{623}{576}q_{1}^{3}\biggr{)}[M].$
Recall that the analytical index is an integer and so the above formulas
evaluated on the fundamental class must give an integer as well. We will
verify this for the quaternionic projective spaces.
###### Example 5.8.
Let $M=\mathbb{H}\mathrm{P}^{m}$. In this case the vector bundles $E$ and $F$
exist globally and $E$ is precisely the tautological line bundle. The
cohomology ring $H^{*}(\mathbb{H}\mathrm{P}^{m};\mathbb{Z})$ is generated by
the class $u=-c_{2}(E)$, which also satisfies
$(u^{m})\,[\mathbb{H}\mathrm{P}^{m}]=1$. Moreover, one can show (see [4]) that
the Pontryagin classes of $T\mathbb{H}\mathrm{P}^{m}$ are given by
$p(T\mathbb{H}\mathrm{P}^{m})=(1+u)^{2m+2}(1+4u)^{-1},$
where $(1+4u)^{-1}$ is the inverse formal power series to $1+4u$. Finally, by
(5.4) we have $q_{1}=-4c_{2}(E)=4u$.
Take first $m=2$. Then the Pontryagin classes are
$p_{1}(T\mathbb{H}\mathrm{P}^{2})=2u$ and
$p_{2}(T\mathbb{H}\mathrm{P}^{2})=7u^{2}$ and inserting into the formulas from
Theorem 5.6 we obtain
$\displaystyle\mathrm{ind}\,D_{0}=1,\quad\mathrm{ind}\,D_{1}=35.$
Similarly, for $m=3$ we have $p_{1}(T\mathbb{H}\mathrm{P}^{3})=4u$,
$p_{2}(T\mathbb{H}\mathrm{P}^{3})=12u^{2}$,
$p_{3}(T\mathbb{H}\mathrm{P}^{3})=8u^{3}$ and
$\displaystyle\mathrm{ind}\,D_{0}=-1,\quad\textup{ind}\,D_{1}=-63.$
A drawback of the index formulas is that they depend on the class $q_{1}$,
which is not easy to compute. However, by taking integral linear combinations
we may try to eliminate the terms containing $q_{1}$ and thus obtain some
integrality conditions on the Pontryagin classes of a quaternionic manifold.
Consider for example the two formulas from Theorem 5.6 and the following
linear combinations
$\displaystyle 11\cdot\mathrm{ind}\,D_{0}+\mathrm{ind}\,D_{1}$
$\displaystyle=\biggl{(}\frac{143}{960}p_{1}^{2}-\frac{89}{240}p_{2}+3q_{1}^{2}\biggr{)}[M],$
$\displaystyle 50\cdot\mathrm{ind}\,D_{0}-2\cdot\mathrm{ind}\,D_{1}$
$\displaystyle=\biggl{(}-\frac{17}{480}p_{1}^{2}-3p_{1}q_{1}+\frac{71}{120}p_{2}\biggr{)}[M].$
But $p_{1}$ and $q_{1}$ are Pontryagin classes of some vector bundles and so
they come from integral cohomology. In particular, evaluation of $p_{1}q_{1}$
and $q_{1}^{2}$ on the fundamental class of $M$ gives an integer. But then by
evaluating the rest of the above formulas we must again obtain an integer (and
not only a rational number).
###### Corollary 5.9.
Let $M$ be an 8-dimensional compact quaternionic manifold. Then the following
expressions are integers
$\biggl{(}\frac{143}{960}p_{1}(TM)^{2}-\frac{89}{240}p_{2}(TM)\biggr{)}[M],\quad\biggl{(}-\frac{17}{480}p_{1}(TM)^{2}+\frac{71}{120}p_{2}(TM)\biggr{)}[M].$
Of course we may deal with other integral linear combinations
$a\cdot\mathrm{ind}\,D_{0}+b\cdot\mathrm{ind}\,D_{1}$ to derive more
integrality conditions and similarly for the $12$-dimensional manifolds.
As a final point, let us remark that for manifolds admitting a
$\mathrm{GL}(m,\mathbb{H})$-structure with a torsion-free connection (see also
Example 3.4) the formulas simplify considerably because the vector bundle $E$
is trivial and hence $q_{1}=0$. Moreover, the vector bundle $F$ is isomorphic
to the complex tangent $T^{c}M$ of $M$. Assuming $m=2$ we then easily compute
that $p_{1}(TM)=-2c_{2}(F)$ and $p_{2}(TM)=c_{2}(F)^{2}+2c_{4}(E)$ and
substituting to the formula from Theorem 5.6 for the Salamon’s complex $D_{0}$
we obtain
$\displaystyle\mathrm{ind}\,D_{0}=\left(\frac{1}{80}c_{2}(F)^{2}-\frac{1}{240}c_{4}(F)\right)[M].$
But this is exactly three times the top-dimensional part of
$\operatorname{\mathrm{td}}(F)=\operatorname{\mathrm{td}}(T^{c}M)$, which
verifies that the general formula for the Salamon’s complex from Theorem 5.6
coincides with that in Theorem 3.6.
## Acknowledgements
The paper presents the results of my diploma thesis and in this connection
thanks go to Martin Čadek, my supervisor. Furthermore, I am grateful to
Andreas Čap, Vladimír Souček and Petr Somberg for several discussions and to
Lukáš Vokřínek for his remarks and comments. The research was supported by the
grant MSM0021622409 of the Czech Ministry of Education.
## References
* [1] M. F. Atiyah, I. M. Singer, _The index of elliptic operators: III_ , Ann. of Math. 87 (1968), 546-604.
* [2] R. J. Baston, _Quaternionic complexes_ , J. Geom. Phys. 8 (1992), 29-52.
* [3] A. Borel, _Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts_ , Ann. of Math. 57 (1953), 115-207.
* [4] A. Borel, F. Hirzebruch, _Characteristic classes and homogeneous spaces, I_ , American J. of Math. 80 (1958), 458-538.
* [5] M. Čadek, M. C. Crabb, J. Vanžura, _Quaternionic structures_ , in preparation.
* [6] A. Čap, _Infinitesimal automorphisms and deformations of parabolic geometries_ , J. Eur. Math. Soc. 10, 2 (2008), 415-437.
* [7] A. Čap, J. Slovák, _Parabolic geometries I: Background and general theory_ , AMS, Providence, 2009.
* [8] A. Čap, J. Slovák, V. Souček, _Bernstein-Gelfand-Gelfand sequences_ , Ann. of Math. 154 (2001), 97-113, an extended version electronically available at `http://www.esi.ac.at`.
* [9] A. Čap, V. Souček, _Subcomplexes in curved BGG-sequences_ , to appear, ESI preprint available at `http://www.esi.ac.at`.
* [10] C. LeBrun, S. Salamon, _Strong rigidity of positive quaternion Kähler manifolds_ , Inventiones mathematicae 118 (1994), 109-132.
* [11] S. Salamon, _Differential geometry of quaternionic manifolds_ , Annales scientifiques de l’É. N. S. 19 (1986), 31-55.
* [12] S. Salamon, _Quaternionic Kähler manifolds_ , Inventiones mathematicae 67 (1982), 143-171.
* [13] H. Samelson, _Notes on Lie algebras_ , third ed., Springer-Verlag, New York, 1990.
* [14] Computer algebra system LiE, available at `http://young.sp2mi.univ-poitiers.fr/~marc/LiE/`.
|
arxiv-papers
| 2009-08-31T21:13:56 |
2024-09-04T02:49:04.957454
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oldrich Spacil",
"submitter": "Oldrich Spacil",
"url": "https://arxiv.org/abs/0909.0035"
}
|
0909.0075
|
11institutetext: Key Laboratory for Particle Astrophysics, Institute of High
Energy Physics, Beijing 100049, China 22institutetext: ICREA & Institut de
Ciències de l’Espai (IEEC-CSIC), Campus UAB, Facultat de Ciències, Torre
C5-parell, 2a planta, 08193 Barcelona, Spain 33institutetext: Theoretical
Physics Center for Science Facilities (TPCSF), CAS 44institutetext: Center for
Astrophysics,Tsinghua University, Beijing 100084, China
# _INTEGRAL_ and _Swift_ /XRT observations of IGR J19405-3016
Shu Zhang 11 Yu-Peng Chen 11 Diego F. Torres 22 Jian-Min Wang 1133 Ti-Pei Li
1144 Jun-Qiang Ge 11 szhang@mail.ihep.ac.cn
(Received / Accepted )
###### Abstract
Aims. IGR J19405-3016 is reported in the 3rd IBIS catalog as one of its lowest
significance sources ($\sim$ 4.6 $\sigma$ under an exposure of about 371 ks).
This leads to a caveat in multi-wavelength studies, although the source was
identified in the optical as a Seyfert 1. The currently available _INTEGRAL_
data on the source have increased to an exposure time of $\sim$ 1400 ks, which
stimulates us to investigate the reality of this source again by using all the
available data from _INTEGRAL_ and _Swift_ /XRT.
Methods. We analyzed all available observations carried out by the
International Gamma-Ray Astrophysics Laboratory (_INTEGRAL_) on IGR
J19405-3016. The data were processed by using the latest version OSA 7.0. In
addition, we analyzed all the available _Swift_ /XRT data on this source.
Results. We find that IGR J19405-3016 has a detection significance of $\sim$
9.4 $\sigma$ in the 20-60 keV band during the observational period between
March 2003 and March 2008. This confirms a real source detection reported
previously. The source position and error location are therefore updated. The
source is found to be constant over years at the hard X-rays. We analyzed the
_Swift_ /XRT observations on IGR J19405-3016 as well, and find that the
spectrum can be fitted with a simple power law model. Over the three XRT
observations, the source flux varied by up to 39% from the average, and the
spectrum is generally soft. The combined XRT/ISGRI spectrum is well fitted
with a simple power law model (photon index 2.11$\pm$0.03) with a column
density fixed at 8.73$\times$1020 atoms/cm2. Such a photon index is consistent
with the mean value 1.98 (dispersion 0.27) obtained from _Swift_ /BAT AGN
samples at 14-195 keV. The spectral slope of IGR J19405-3016 is softer than
the average spectral slope found elsewhere. A similar discrepancy is found
with other results regarding Seyfert 1 AGNs. A possible explanation for this
simple spectral description may be that the low level of the column density
allows for the ‘true’ spectrum to appear at soft X-rays as well.
###### Key Words.:
X-rays: individual: IGR J19405-3016
††offprints: Shu Zhang
## 1 Introduction
Four hundred twenty-one sources are listed in the 3rd IBIS/ISGRI catalog (Bird
et al. 2007). Among them, 171 are Galactic accreting systems, 122
extragalactic objects, and 113 are sources of unknown nature. Most of the
unclassified sources were identified through optical and near-infrared
spectroscopy (Masetti et al. 2004, 2006a, 2006b, 2006c, 2006d, 2008, 2009;
Chaty et al. 2008; Nespoli et al. 2008). For the newly identified sources,
about 55% are AGNs (almost equally divided into Seyfert 1s and 2s), 32% are
X-ray binaries, and 12% are CVs (Masetti et al. 2009). The optical
identifications are performed by first searching in the error location of
_INTEGRAL_ /ISGRI for a possible counterpart at the soft X-ray observations
and, accordingly, the error location can be largely improved to a level that
allows for looking into optical data for the final identification. IGR
J19405-3016 is one of the _INTEGRAL_ weakly identified sources studied by
Masetti et al. (2008) in their most recent optical identification campaign.
IGR J19405-3016 was first reported as a low-significance source in the 3rd
IBIS/ISGRI catalog (Bird et al. 2007). The source was detected at 4.6 $\sigma$
level under an exposure of $\sim$ 371 ks in the energy band 20-40 keV. The
lowest significance for a source in Bird’s catalog is 4.5 $\sigma$. Therefore,
IGR J19405-3016 is among the few sources that were reported by _INTEGRAL_
/ISGRI with the lowest significances. Bird et al. (2007) point out that the
catalog sources detected with significance above 5 $\sigma$ have a probability
of less than 1 percent of being spurious, and detections with lower
significance have a higher probability of being unreal. Accordingly, although
a Seyfert 1.2 was found as an optical counterpart around IGR J19405-3016 by
Masetti et al. (2008), a caveat was claimed therein in the sense that the hard
X-ray detection itself may be spurious, and hence the proposed optical
identification may not correspond to an actual counterpart (Masetti et al.
2008).
After the discovery of the source by _INTEGRAL_ , _Swift_ /XRT detected a
source at soft X-rays within the IBIS error location (Landi et al. 2007a). The
XRT observation locates the source at RA(J2000) = 19h 40m 15.15s and
Dec(J2000) = -30d 15m 48.5s, with a 3.5 arcsecond uncertainty at 90%
confidence level. The X-ray spectrum is modeled with an unabsorbed power law
with a photon index $\sim$ 2.1 (Landi et al. 2007a). Given that an additional
exposure will lead to a large increment in source significance and that the
currently available _INTEGRAL_ observations around IGR J19405-3016 have been
accumulated to roughly 1400 ks, it is appropriate to re-investigate the hard
X-ray properties of IGR J19405-3016. Also, after the report of Landi et al.
(2007a), further _Swift_ /XRT observations (roughly 1.6 ks more) were
available in 2008, which allow for further investigation on the source at soft
X-rays. In this paper we report the results of our _INTEGRAL_ and _Swift_ /XRT
analyses on IGR J19405-3016.
## 2 Observations and data analysis
_INTEGRAL_ (Winkler et al. 2003) is a 15 keV - 10 MeV $\gamma$-ray mission.
The main instruments are the Imager on Board the INTEGRAL Satellite (IBIS, 15
keV - 10 MeV; Ubertini et al. 2003) and the SPectrometer onboard INTEGRAL
(SPI, 20 keV - 8 MeV; Vedrenne et al. 2003). They are supplemented by the
Joint European X-ray Monitor (JEM-X, 3-35 keV) (Lund et al. 2003) and the
Optical Monitor Camera (OMC, V, 500-600 nm) (Mas-Hesse et al. 2003). At the
lower energies (15 keV - 1 MeV), the CdTe array ISGRI (Lebrun et al. 2003) of
IBIS has a better continuum sensitivity than SPI. The satellite was launched
in October 2002 into an elliptical orbit with a period of 3 days. Due to the
coded-mask design of the detectors, the satellite normally operates in
dithering mode, which suppresses the systematic effects on spatial and
temporal backgrounds.
The _INTEGRAL_ observations were carried out in the so-called individual
SCience Windows (SCWs), with a typical time duration of about 2000 seconds
each. Only IBIS/ISGRI public data were taken into account, because the source
is too weak to be detected by JEMX and SPI. The available _INTEGRAL_
observations, when IGR J19405-3016 fell into the 50$\%$ coded field of view of
ISGRI (offset angle less than 10 degrees), comprised about 566 SCWs, adding up
to a total exposure time of $\sim$ 1400 ks (until March 22, 2008). IGR
J19405-3016 therefore had roughly 1000 ks of exposure more than used in the
previous report (Bird et al. 2007). The details of the analyzed _INTEGRAL_
observations on IGR J19405-3016, including the exposure and the time periods,
are summarized in Table 1. Most of these observations were carried out in the
5x5 dithering mode. We subdivided the data into 4 groups according to the
observational sequence. The other observational modes of the selected data are
staring mode and Hexagonal mode, the sum of which comprise only 5.6% of the
whole data. Data from these two modes can just as well be used to produce a
mosaic map without introducing any systematic errors. This has been confirmed
through our consulting the INTEGRAL help desk. The data reduction was
performed by using the standard Online Science Analysis (OSA) software version
7.0, the latest released version. The results were obtained by running the
pipeline from the flowchart to the image level and the spectrum level. The
flux and the detection significance were derived in the mosaic map at the
source position revised in this work.
_Swift_ (Gehrels et al. 2004) is a $\gamma$-ray burst explorer launched
November 20, 2004. It carries three co-aligned detectors: the Burst Alert
Telescope (BAT, Barthelmy et al. 2005), the X-Ray Telescope (XRT, Burrows et
al. 2005), and the Ultraviolet/Optical Telescope (UVOT, Roming et al. 2005).
We took only _Swift_ /XRT data into account, because BAT data were not
available. The XRT uses a grazing incidence Wolter I telescope to focus X-rays
onto a state-of-the-art CCD. XRT has an effective area of 110 cm2, an FOV of
23.6 arcminutes, an angular resolution (half-power diameter) of 15 arcseconds,
and it operates in the 0.2-10 keV energy range, providing the possibility of
extending the investigation on the source to soft X-rays.
There are three _Swift_ snapshots available for IGR J19405-3016, each one with
an exposure over 1 ks. The observations were carried out in the photon-
counting mode, with exposures of 7.6 ks (ID 00036657004, on July 31, 2007),
5.9 ks (ID 00036657005, on August 7, 2007) and 1.6 ks (ID 00036657005, on June
7, 2008), respectively. The first two observations were reported in Landi et
al. (2007a). See details in Table 2. We analyzed the _Swift_ /XRT 0.3-7 keV
data by using the latest released analysis software, provided in HEAsoft
version 6.4. The XRT data reduction follows those described in Landi et al.
(2007b). Here the source events were extracted within a circular region of
radius $\sim$ 40 pixels, centered on the source position. A radius of 20
pixels (corresponds to 47 arcseconds) encloses about 90 percent of the PSF at
1.5 keV (Capalbi et al. 2005). A larger radius extracts more source counts,
hence improves the statistics in spectral fittings. The spectra were fitted
with XSPEC v12.3.1 (Dorman & Arnaud 2001) and the model parameters estimated
at 90$\%$ confidence level.
## 3 Results
### 3.1 _INTEGRAL_
The imaging analyses show that the best source detection, $\sim$ 9.4 $\sigma$
at 20-60 keV (Fig. 1), was derived in the mosaic map of all data. The source
is not detectable at lower energies by JEMX or at higher energies by SPI. Such
a detection is much more significant than the previous report of a 4.5
$\sigma$ signal with an exposure of 371 ks, at 20-40 keV (Bird et al. 2007).
The source flux is about 1.40$\pm$0.15 mCrab in the 20-60 keV band over the
period 2003-2008. Under such a high-significance detection at 20-60 keV, the
source position is improved to RA/Dec (J2000) = 295.0895∘/-30.2732∘, with a
radius of 3.4 arcminutes in error circle ( 90$\%$ confidence level), also
improved from the previous report of 5.4 arcminutes (Bird et al. 2007).
We also investigated the source detection in individual observational groups
as listed in Table 1, in the 20-60 keV band. The source is detected in Revs.
0056-0244 (MJD=52729-53291) with a significance of 4.9 $\sigma$ and a flux of
1.5$\pm$0.3 mCrab; in Revs. 0258-0371 (MJD=53332-53672) with a significance of
3.5 $\sigma$ and a flux of 1.4$\pm$0.4 mCrab and Revs. 0416-0498
(MJD=53805-54050) with a significance of 7.8 $\sigma$ and a flux of
1.5$\pm$0.2 mCrab. One sees that the source is detectable in each
observational group and the flux remains constant at about 1.5 mCrab level
over years at hard X-rays. The ISGRI data after Revs. 0498 consist of only 16
scws (Revs.543-664), which are about 17 ks exposure in total. The source was
not detected with this exposure, and we have a 2-$\sigma$ flux upper limit of
3.3 mCrab at 20-60 keV. If, during Revs. 543-664, the source kept at the same
flux level as that of the previous revolutions, it would have only been
’clearly detected’ at 1-$\sigma$ level. See details for the results in Table
1. The source is detected in the 4 adjacent energy bands of 20-25, 25-30,
30-40, and 40-60 keV, with detection significances of 4.0, 5.4, 6.1, and 4.2
$\sigma$, respectively, by combining all the ISGRI data. If there is a
constant source at hard X-rays, an exposure of 1400 ks can improve the
detection significance by a factor of 1.95 with respect to the previous report
(Bird et al. 2007), which means a detection significance of $\sim$ 9 $\sigma$,
similar to what we obtain from the summed data. In short, all results derived
by us indicate a real and steady source.
### 3.2 _Swift_ /XRT
The _Swift_ /XRT imaging analysis shows a single source detected at soft
X-rays within the error location of IGR J19405-3016. Figure 2 is the _Swift_
/XRT image produced by combining the three XRT observations to show the most
accurate location of the source at soft X-rays. The position is obtained as
RA(J2000) = 19h 40m 15.00s and Dec(J2000) = -30d 15m 48.6s, with an error
radius of 3.5 arcseconds at 90% confidence level, which is consistent with the
report in Landi et al. (2007a). Over plotted are the positions of IGR
J19405-3016, as derived in Bird et al. (2007) and in the present work, and
their error circles. It is obvious that the revised position and improved
error circle are more indicative of the correlation between the soft X-ray
source detected with XRT and the hard X-ray source IGR J19504-3016.
To look into the properties of the XRT source at soft X-rays, we carried out
the spectral analysis of the three XRT observations. We find that data from
each of the three observations can be fitted by using a simple power law
model. The data in ID 00036657004 and ID 00036657005 need an additional
component of absorption, with a column density derived as 5-6 $\times$ 1020
atoms/cm2, but this is not necessary for the data from observation ID
00036657006, probably due to a low exposure of only $\sim$ 1.6 ks. The source
was clearly detected in XRT observation of 2008 June as well. The relatively
low flux level and low exposure in this observation led to larger errors in
model parameters. The reduced $\chi^{2}$ were derived with values of $\leq$
1.0. The column density as measured directly from the XRT spectra of IGR
J19405-3016 is rather low: a value around 6$\times$1020atoms/cm2 can be even
comparable to the Galactic column density obtained with the web version
(9.15$\times$1020atoms/cm2)111http://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/W3nh/w3nh.pl and from Dickey $\&$ Lockman (1990)
(8.73$\times$1020atoms/cm2), measured $\sim$ 0.4 deg away from IGR
J19405-3016. We therefore fix the column density at 8.73$\times$1020atoms/cm2
and fit again the XRT data with a power law model. We have the fittings with
the reduced $\chi^{2}$ derived around 1.1 (see Table 2). We find that the XRT
fluxes in these snapshots can vary up to 39% with respect to the average at
soft X-rays. The overall spectrum is relatively soft irrespective of the
change in flux. The summed XRT data can be well-fitted by using a simple power
law model, with a reduced $\chi^{2}$ $\sim$ 1.13 for 198 dof222degrees of
freefom. This suggests that the spectral evolution, if any, should be not very
strong over the three XRT snapshots. See Table 2 for details of the spectral
results.
### 3.3 Combined XRT/ISGRI spectrum
The summed XRT spectrum is combined with the ISGRI spectrum extracted from the
all _INTEGRAL_ observations. The joint spectrum can be fitted with a model of
simple power law plus fixed absorption (reduced $\chi^{2}$$\sim$1.14 for 202
dof) (Fig. 3). The photon index $\Gamma$ and normalization were derived as
2.11$\pm$0.03 and 4.70$\pm$0.10$\times$10-3 ph cm-2 s-1 keV-1, respectively. A
constant is introduced to account for the difference in normalization between
_Swift_ /XRT and _INTEGRAL_ /ISGRI and the sporadic snapshots of XRT
observations, and is derived as 2.19${}^{+0.65}_{-0.60}$, which is slightly
more than unity. Such deviation can be ascribed to flux variability at soft
X-rays observed in the sporadic snapshots of XRT.
## 4 Discussion and summary
As listed in the 3rd IBIS catalog (Bird et al. 2007), IGR J19405-3016 belongs
to a group of sources with significance less than 5 $\sigma$. About 10-20
percent of them might result from false detections (Bird et al. 2007). By
taking the currently available _INTEGRAL_ observations, the exposure of which
is roughly a factor of four more than used in the 3rd IBIS catalog (Bird et
al. 2007), we find IGR J19405-3016 at a significance level of $\sim$ 9.4
$\sigma$ at 20-60 keV in the sum of the observations between March 2003 and
March 2008. The source is consistently detected at significance level $\ga$ 4
$\sigma$, in the three observational groups covering a time period of almost 5
years and in the adjacent four energy bands between 20-60 keV. We therefore
confirm the detection of a real source in a previous report (Bird et al.
2007). Hence the caveat mentioned in Masetti et al. (2008) in optical
identification of IGR J19405-3016 has disappeared. A similar result is shown
as well in Beckmann et al. (2009), where a significance of 11.8 $\sigma$ was
reported for IGR J19405-3016 in 18-60 keV. Given the distance of a redshift
z=0.052 (Masetti et al. 2008), the source luminosity is estimated as
2.5$\times$1044 ergs/s (1-100 keV, use $H_{0}$=70 km/s/Mpc, $q_{0}$=0,
$\Omega_{\lambda}$=0.73). Our analysis shows that the source was rather stable
over 5 years at energies above 20 keV. The source location and error circle
are updated accordingly, which are more indicative of a source observed by
_Swift_ /XRT at soft X-rays as its real counterpart.
At soft X-rays we find that IGR J19405-3016 has a relatively soft spectrum and
the XRT flux can vary up to 39% with respect to the average at soft X-rays.
Given the low sensitivity of ISGRI (about 4-5 mCrab for individual SCW) and
low average flux level at $>$ 20 keV ($\sim$1.5 mCrab for IGR J19405-3016),
such a variability at hard X-rays cannot be studied because of the lack of
sufficient signal-to-noise ratio in the ISGRI data, although short variability
cannot be excluded. In Table 1 one sees that the source is stable on a long
time scale at hard X-rays. It has been reported several times that, for
Seyfert 1, their fluxes can be less variable at hard X-rays than at soft
X-rays (Beckmann et al. 2007; Molina et al. 2009; Gliozzi et al. 2003). A
possible explanation for this is that the large number of the scatterings of
the soft X-rays in the corona region can let the variability be washed out at
hard X-rays. Such an idea was proposed in Gliozzi et al. (2003), but so far
the detailed modelings are not yet available.
The combined XRT/ISGRI energy spectrum is well-fitted using a model of simple
power law plus fixed absorption component. The spectral index falls into the
range of the spectral index of the AGN measured by _Swift_ /BAT at 14-195 keV
band (Tueller et al. 2008). The analysis of the first 9 months of the data of
_Swift_ /BAT survey of AGN resulted in 103 AGN being detected in 14-195 keV.
The average spectral index of these samples is 1.98, with an rms of 0.27. For
IGR J19405-3016, the spectral index of 2.11$\pm$0.03 is consistent with the
media value from BAT. About 74 BAT AGNs have the archival spectrum at soft
X-rays. A comparison of these samples between hard and soft X-rays shows that
the BAT spectral slope is in general 0.23 steeper than in the soft X-rays
(Tueller et al. 2008). A possible explanation for this could be that the
spectrum is closer to the intrinsic one at hard X-rays than at soft X-rays due
to the influence of the material local to AGN (Nandra et al. 1999; Tueller et
al. 2008). The broad-band (1-100 keV) analysis of 36 Seyfert 1 AGNs, detected
by INTEGRAL in the 20-40 keV band, presents a power law shape with an average
photon index of 1.7 and a dispersion of 0.2 (Molina et al. 2009). The photon
index of 2.11$\pm$0.03 derived for IGR J19405-3016 in $\sim$ 1-100 keV is thus
slightly more than the result of Molina et al. (2009). A similar discrepancy
is found with the results of Beckmann et al. (2009) regarding Seyfert 1 AGNs.
The column density as measured in IGR J19405-3016 is rather low, and from the
X-ray spectral data, there is no need to introduce a further absorption local
to the AGN. A possible explanation for the derived spectral properties of IGR
J19405-3016 may be that the low level of the column density allows for the
‘true’ spectrum to also be detected at soft X-rays, and thus the overall
measured spectral shape of the source is actually the true one.
In summary, the most recent analysis carried out with the latest software
releases confirms that the previous weak detection, hence the low-significance
source IGR J19405-3016 as listed in Bird et al. (2007), is not spurious. As a
result, the optical identification of a Seyfert 1.2 by Masetti et al. (2008)
is strengthened. A spectral index $\sim$ 2.11 supplemented with a low
absorption may indicate that the ‘true’ spectrum of IGR J19405-3016 extends
back to soft X-rays.
Table 1: IBIS/ISGRI observations log of IGR J19405-3016.
Revs. | Date | Expo. | SCW | Flux | Sig.
---|---|---|---|---|---
| MJD | ks | | mCrab | $\sigma$
0056-0244 | 52729-53291 | 262 | 146 | 1.5$\pm$0.3 | 4.9
0258-0371 | 53332-53672 | 218 | 129 | 1.4$\pm$0.4 | 3.5
0416-0498 | 53805-54050 | 879 | 275 | 1.5$\pm$0.2 | 7.8
0543-0664 | 54184-54547 | 17 | 16 | $<$3.3 | 0.0
0056-0664 | 52729-54547 | 1400 | 566 | 1.40$\pm$0.15 | 9.4
* Note:
The flux and the significance are presented in the energy band 20-60 keV for
the data in which the source was within an offset angle of 10 degrees. The
error bars are 1 $\sigma$ and the upper limit is 2 $\sigma$.
Table 2: _Swift_ /XRT observations log of IGR J19405-3016.
Model | Observation (date, exposure,ID)
---|---
Par | 2007-07-31 | 2007-08-07 | 2008-06-07 | 2007-2008
| 7.6 ks | 5.9 ks | 1.6 ks | 15.1 ks
| 0036657004 | 0036657005 | 0036657006 | all
$\Gamma$ | 2.07$\pm$0.05 | 2.11$\pm$0.04 | 2.19${}^{+0.14}_{-0.13}$ | 2.11$\pm$0.03
N | 3.98$\pm$0.13 | 5.95$\pm$0.17 | 3.16$\pm$0.26 | 4.70$\pm$0.10
flux | 1.69${}^{+0.05}_{-0.06}$ | 2.44$\pm$0.07 | 1.19$\pm$0.10 | 1.94$\pm$0.04
$\chi^{2}$/dof | 1.11/95 | 1.12/104 | 1.16/15 | 1.13/198
* Note:
The parameters $\Gamma$ (photon index) and N (Normalization at 1 keV, in units
of 10-3 ph cm-2 s-1 keV-1) are the fit results at 0.3-7 keV, using a power law
model. The column density is fixed at 8.73$\times$1020 atoms/cm2. The
$\chi^{2}$/dof is shown as well for each fit. The flux is calculated in the
0.3–10 keV band, in units of 10-11 erg cm-2 s-1.
Figure 1: The 20-60 keV significance map of IGR J19405-3016 as obtained from
the combined revolutions 0056 - 0664 (MJD 52729-54547). The contours start at
a significance level of 4 $\sigma$ with steps of 1 $\sigma$. Figure 2: _Swift_
/XRT sky map of IGR J19405-3016 detected in the three combined XRT
observations. The hot spot shows the soft X-ray source detected by _Swift_
/XRT and the color bar shows the scale associated with the image counts. The
open square (green) and the cross symbols (red) are the locations of IGR
J19405-3016 estimated in Bird et al. (2007) and in the current work,
respectively. The two circles are the corresponding 90% confidence level error
locations. Figure 3: Upper panel: the combined XRT/ISGRI energy spectrum of
IGR J19405-3016 obtained from the _INTEGRAL_ observations of the revolutions
0056 - 0664 and from the three combined _Swift_ /XRT observations (ID:
00036657004, 00036657005, 00036657006). The data are fitted by using a power
law shape plus an absorption component fixed at 8.73$\times$1020 atoms/cm2.
The residuals (lower panel) show the quality of the fit.
###### Acknowledgements.
This work was subsidized by the National Natural Science Foundation of China,
the CAS key Project KJCX2-YW-T03, and the 973 Program 2009CB824800. J.-M. W.
thanks the Natural Science Foundation of China for support via NSFC-10325313,
10521001, and 10733010. DFT acknowledges support by Spanish MEC grant AYA 2
006-00530.
## References
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|
arxiv-papers
| 2009-09-01T02:55:44 |
2024-09-04T02:49:04.964707
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shu Zhang, Yu-Peng Chen, Diego F. Torres, Jian-Min Wang, Ti-Pei Li,\n Jun-Qiang Ge",
"submitter": "Shu Zhang",
"url": "https://arxiv.org/abs/0909.0075"
}
|
0909.0080
|
# Generalized wave operators for a system of nonlinear wave equations in three
space dimensions
Hideo Kubo (Corresponding author)
Division of Mathematics, Graduate School of Information Sciences,
Tohoku University, Sendai 980-8579, Japan
Phone and Fax: +81-22-795-4628
E-mail: kubo@math.is.tohoku.ac.jp
Kôji Kubota
A professor emeritus from Hokkaido University,
Sapporo 060-0808, Japan
###### Abstract
This paper is concerned with the final value problem for a system of nonlinear
wave equations. The main issue is to solve the problem for the case where the
nonlinearity is of a long range type. By assuming that the solution is
spherically symmetric, we shall show global solvability of the final value
problem around a suitable final state, and hence the generalized wave operator
and long range scattering operator can be constructed.
Keywords : Final value problem, system of nonlinear wave equations,
generalized wave operator, long range scattering operator.
## 0 Introduction
This paper is a continuation of a previous one [5], in which we have studied
the global existence and asymptotic behavior of solutions to the system of
semilinear wave equations :
$\left\\{\begin{array}[]{ll}\partial_{t}^{2}u-\Delta
u=|\partial_{t}v|^{p}&\mbox{in}\quad{\mathbb{R}}^{3}\times{\mathbb{R}},\\\
\partial_{t}^{2}v-\Delta
v=|\partial_{t}u|^{q}&\mbox{in}\quad{\mathbb{R}}^{3}\times{\mathbb{R}},\end{array}\right.$
(0.1)
where $1<p\leq q$. Assuming
$q(p-1)>2,$ (0.2)
we have shown that the initial value problem for (0.1) has a global solution
if the initial data are radilally symmetric and sufficiently small (notice
that if (0.2) fails true, then the classical solution to (0.1) blows up in
finite time however small the initial data are, in general, due to Deng [1]).
In the present paper we consider a problem whether the scattering operator for
(0.1) can be defined or not.
When $p>2$, the global solution has the “free profile”. Therefore, in this
case, one can expect that the scattering operator is defined by
$S=(W_{+})^{-1}W_{-}$, following Lax and Phillips [6]. Here the wave operators
$W_{+}$ and $W_{-}$ are obtained by solving the final value problem for (0.1).
In fact, for a given final state $(u^{+},v^{+})$ which is a solution to the
system of homogeneous wave equations :
$\partial_{t}^{2}u^{+}-\Delta u^{+}=0,\quad\partial_{t}^{2}v^{+}-\Delta
v^{+}=0\quad\mbox{in}\quad{\mathbb{R}}^{3}\times{\mathbb{R}},$ (0.3)
if we found a unique solution $(u,v)$ in ${\mathbb{R}}^{3}\times[0,\infty)$ to
(0.1) satisfying
$\|u(t)-u^{+}(t)\|_{E}+\|v(t)-v^{+}(t)\|_{E}\to 0\quad\mbox{as}\quad
t\to\infty,$ (0.4)
then $W_{+}$ is defined by
$(u^{+},\partial_{t}u^{+},v^{+},\partial_{t}v^{+})(x,0)\longmapsto(u,\partial_{t}u,v,\partial_{t}v)(x,0).$
Here $\|w(t)\|_{E}$ stands for the energy norm of $w(x,t)$, i.e.,
$\|w(t)\|_{E}^{2}=\frac{1}{2}\int_{{\mathbb{R}}^{3}}(|\partial_{t}w(x,t)|^{2}+|\partial_{x}w(x,t)|^{2})dx.$
Analogously, $W_{-}$ is defined by replacing “ $t\to\infty$” in (0.4) by “
$t\to-\infty$”.
On the other hand, when $1<p\leq 2$, the nonlinearity becomes of a long range
type in the sense that the solution to the initial value problem for (0.1)
exists globally in time but does not approach to any free solution. In fact,
we have shown in [5] that the energy of the global solution is not generically
bounded for large $t>0$, so that it can not be asymptotic to any free solution
of finte energy. Nevertheless, we proved that the global solution $(u,v)$ to
the problem has a “generalized profile”. More precisely, letting
$(\bar{w},\bar{v})$ be a solution of
$\partial_{t}^{2}\bar{w}-\Delta\bar{w}=F(x,t),\quad\partial_{t}^{2}\bar{v}-\Delta\bar{v}=0\quad\mbox{in}\quad{\mathbb{R}}^{3}\times{\mathbb{R}}$
(0.5)
with a suitable $F(x,t)$, we have
$\|u(t)-\bar{w}(t)\|_{E}+\|v(t)-\bar{v}(t)\|_{E}\to 0\quad\mbox{as}\quad
t\to\infty.$ (0.6)
For instance, $F(x,t)=|\partial_{t}\bar{v}(x,t)|^{p}$ when $0\leq
2-p<q(p-1)-2$. These results imply that it is impossible to construct the wave
operators $W_{+}$ and $W_{-}$, in general, and suggest us to modify the
definition of wave operators for (0.1) when $1<p\leq 2$.
As in [5], we deal with only radially symmetric solution of (0.1) in the
present paper. For this, we write
$u(x,t)=u_{1}(|x|,t),\quad v(x,t)=u_{2}(|x|,t).$ (0.7)
Then (0.1) becomes to
$\left\\{\begin{array}[]{ll}\partial_{t}^{2}u_{1}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)u_{1}=|\partial_{t}u_{2}|^{p}&\mbox{in}\quad
r>0,\ t\in{\mathbb{R}},\\\
\partial_{t}^{2}u_{2}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)u_{2}=|\partial_{t}u_{1}|^{q}&\mbox{in}\quad
r>0,\ t\in{\mathbb{R}}.\end{array}\right.$ (0.8)
We wish to compare the nonlinear evolution under (0.8) with the linear
evolution obeying
$\left\\{\begin{array}[]{ll}\partial_{t}^{2}w-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)w=F(r,t)&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}},\\\
\partial_{t}^{2}v-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)v=0&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}}\end{array}\right.$ (0.9)
with a suitably chosen $F(r,t)$, as $t\to\pm\infty$. Actually, we are able to
realize this by assuming the following slightly stronger condition than (0.2)
:
$(p-1)^{2}(q-1)>1$ (0.10)
(for the detail, see Theorems 2.4 and 2.5 below). Since we are able to solve
the initial value problem in the same function space (see Theorem 2.6 below),
these results lead us to a construction of a long range scattering operator
for (0.1).
We remark that this kind of modification goes back to the seminal work of
Ozawa [7] for the nonlinear Schrödinger equation (see also [2, 3, 4, 8], for
instance). To our knowledge, this paper provides the first result on the wave
equation in this direction. In order to treat the system (0.8), we need to
overcome a difficulty to handle the nonlinearities with small powers $p$ which
can be close to $1$ under the assumption (0.10). For this reason, we shall
construct a generalized final state, which solves (0.9), by using an iteration
(see (2.30), (2.31)). Then we shall prove the solvability of the nonlinear
system (0.8) around the generalized final state by introduing a suitable
metric space given by (4.36).
This paper is organized as follows. In the next section we collect notation.
In the section 2 we present our main results. The section 3 is a summary of
[5, Section 4]. We refine Theorem 6 and the part (ii) of Theorem 7 in [5] so
that one can take a parameter $\gamma$ to be positive. The section 4 is
devoted to prove the main theorems.
## 1 Notation
First we introduce a class of initial data :
$\displaystyle\quad Y_{\nu}(\varepsilon)=\\{\vec{f}=(f,g)\in
C^{1}({\mathbb{R}})\times C({\mathbb{R}})~{};~{}\vec{f}(-r)=\vec{f}(r)\
(r\in{\mathbb{R}}),$ $\displaystyle\quad\quad\quad r\vec{f}(r)\in
C^{2}({\mathbb{R}})\times C^{1}({\mathbb{R}})\ \mbox{and}\
\sup_{r>0}\,(1+r)^{\nu}|\\!|\\!|\vec{f}(r)|\\!|\\!|\leq\varepsilon\\},$
where $\nu\in{\mathbb{R}}$, $\varepsilon>0$ and
$\displaystyle|\\!|\\!|\vec{f}(r)|\\!|\\!|=|f(r)|+(1+r)(|f^{\prime}(r)|+|g(r)|)+r(|f^{{}^{\prime\prime}}(r)|+|g^{\prime}(r)|).$
Next we define several function spaces and norms. Let $s=1$ or $s=2$. First of
all, we introduce a basic space of our argument :
$\displaystyle{X}^{s}=\\{u(r,t)\in
C^{s-1}({\mathbb{R}}\times[0,\infty))~{};~{}ru(r,t)\in
C^{s}({\mathbb{R}}\times[0,\infty)),$ $\displaystyle\hskip 42.67912pt\ \
u(-r,t)=u(r,t)\ \ \text{for}\ (r,t)\in{\mathbb{R}}\times[0,\infty)\\}.$
For $r>0$ and $t\geq 0$ we put
${[u(r,t)]_{2}}=|u(r,t)|+(1+r)\sum_{|\alpha|=1}|\partial^{\alpha}u(r,t)|+r\sum_{|\alpha|=2}|\partial^{\alpha}u(r,t)|$
if $u\in X^{2}$, and
${[u(r,t)]_{1}}=|u(r,t)|+r\sum_{|\alpha|=1}|\partial^{\alpha}u(r,t)|$
if $u\in X^{1}$, where $\partial=(\partial_{r},\partial_{t})$ and $\alpha$ is
a multi-index. For $\nu\in{\mathbb{R}}$, we define Banach spaces :
$\displaystyle X^{s}(\nu)=\\{u(r,t)\in
X^{s}~{};~{}\|u\|_{X^{s}(\nu)}<\infty\\},$ $\displaystyle
Z^{s}(\nu)=\\{u(r,t)\in X^{s}~{};~{}\|u\|_{Z^{s}(\nu)}<\infty\\},$
where we have set
$\displaystyle\|u\|_{X^{s}(\nu)}=\sup_{r>0,\,t\geq
0}[u(r,t)]_{s}\,(1+|r-t|)^{\nu},$ (1.1)
$\displaystyle\|u\|_{Z^{s}(\nu)}=\sup_{r>0,\,t\geq
0}[u(r,t)]_{s}\,(1+r+t)^{\nu-1}(1+|r-t|).$ (1.2)
Notice that $X^{s}(\nu)\subset Z^{s}(\nu)$ if $\nu\leq 1$, while
$Z^{s}(\nu)\subset X^{s}(\nu)$ if $\nu\geq 1$.
For notational symplicity, we shall denote $\|w(|\cdot|,t)\|_{E}$ by
$\|w(t)\|_{E}$ for a function $w(r,t)$.
## 2 Main Results
### 2.1 Existence of wave operators
When $p>2$, the evolution obeying (0.8) is well characterized by the
homogeneous wave equation. For this, we first recall the known fact about the
initial value problem for the homogeneous wave equation (see e.g. [5]) :
$\displaystyle
u_{tt}-\left(u_{rr}+\frac{2}{r}u_{r}\right)=0\quad\mbox{in}\quad(0,\infty)\times(0,\infty),$
(2.1) $\displaystyle(u,\partial_{t}u)(r,0)=\vec{f}(r)\ \quad\mbox{for}\quad
r>0.$ (2.2)
The solution of this problem is expressed by
$K[\vec{f}](r,t)=\frac{1}{2r}\left\\{\int_{r-t}^{r+t}\lambda
g(\lambda)d\lambda+\frac{\partial}{\partial t}\int_{r-t}^{r+t}\lambda
f(\lambda)d\lambda\right\\}.$ (2.3)
Moreover we have
###### Proposition 2.1
Let $\varepsilon>0$, $\nu>0$. If $\vec{f}\in Y_{\nu}(\varepsilon)$, then
$K[\vec{f}]\in X^{2}(\nu)$ and
$\|K[\vec{f}]\|_{X^{2}(\nu)}\leq C\varepsilon$ (2.4)
holds, where $C$ is a constant depending only on $\nu$.
We set
$\displaystyle\kappa_{1}=p-1,\quad\kappa_{2}=q-1$ (2.5)
for $p>2$. Then we see $\kappa_{1}$, $\kappa_{2}>1$. Our main result in this
subsection is as follows.
###### Theorem 2.2
$($Existence of a wave operator$)$ Let $1<p\leq q$. Suppose that $p>2$. Then
there is a positive number $\varepsilon_{0}$ $($depending only on $p$ and $q)$
such that one can define the wave operator $W_{+}=(W_{+}^{(1)},W_{+}^{(2)})$
from $Y_{\kappa_{1}}(\varepsilon_{0})\times Y_{\kappa_{2}}(\varepsilon_{0})$
to $Y_{\kappa_{1}}(2\varepsilon_{0})\times Y_{\kappa_{2}}(2\varepsilon_{0})$
by
$W_{+}^{(j)}[\vec{f}_{1},\vec{f}_{2}](r)=(u_{j},\partial_{t}u_{j})(r,0)\quad(j=1,2),$
(2.6)
where $(u_{1},u_{2})\in X^{2}(\kappa_{1})\times X^{2}(\kappa_{2})$ is a unique
solution of (0.8) satisfying
$\|u_{1}(t)-K[\vec{f_{1}}](t)\|_{E}+\|u_{2}(t)-K[\vec{f_{2}}](t)\|_{E}\to
0\quad\mbox{as}\quad t\to\infty$ (2.7)
for each $(\vec{f}_{1},\vec{f}_{2})\in Y_{\kappa_{1}}(\varepsilon_{0})\times
Y_{\kappa_{2}}(\varepsilon_{0})$. Moreover, we have for $r>0$
$\displaystyle|\\!|\\!|W_{+}^{(1)}[\vec{f}_{1},\vec{f}_{2}](r)-\vec{f_{1}}(r)|\\!|\\!|(1+r)^{\kappa_{1}}\leq
C\varepsilon^{p},$ (2.8)
$\displaystyle|\\!|\\!|W_{+}^{(2)}[\vec{f}_{1},\vec{f}_{2}](r)-\vec{f_{2}}(r)|\\!|\\!|(1+r)^{\kappa_{2}}\leq
C\varepsilon^{q},$ (2.9)
provided $\vec{f}_{j}\in Y_{\kappa_{j}}(\varepsilon)$ $(j=1,2)$ and
$0<\varepsilon\leq\varepsilon_{0}$, where $C$ is a constant depending only on
$p$ and $q$.
Our next step is to construct the inverse of $W_{+}$, based on the existence
result given in Theorem 1 of [5] about the initial value problem for (0.8)
with
$(u_{1},\partial_{t}u_{1})(r,0)=\vec{\varphi}_{1}(r),\
(u_{2},\partial_{t}u_{2})(r,0)=\vec{\varphi}_{2}(r)\quad\mbox{for}\quad r>0.$
(2.10)
Let $(u_{1},u_{2})\in X^{2}(\kappa_{1})\times{X^{2}(\kappa_{2})}$ be the
unique solution of the problem satisfying
$\displaystyle\|u_{1}\|_{X^{2}(\kappa_{1})}+\|u_{2}\|_{X^{2}(\kappa_{2})}\leq
2C_{0}\varepsilon$ (2.11)
with $C_{0}$ the canstant in (2.4). Note that $(u_{1},u_{2})$ satisfies the
following system of integral equations :
$u_{1}=K[\vec{\varphi}_{1}]+L(|\partial_{t}u_{2}|^{p}),\quad
u_{2}=K[\vec{\varphi}_{2}]+L(|\partial_{t}u_{1}|^{q}).$ (2.12)
(See e.g. [5]). Using the solution $(u_{1},u_{2})$, we define
$\displaystyle w=u_{1}-R(|\partial_{t}u_{2}|^{p}),\quad
v=u_{2}-R(|\partial_{t}u_{1}|^{q}),$ (2.13)
where $L$ and $R$ are the integral operators associated with the inhomogeneous
wave equation whose definition will be given in (3.3) and (3.8) below,
respectively. If we set
$\displaystyle\vec{f}_{1}(r)=(w(r,0),\partial_{t}w(r,0)),\quad\vec{f}_{2}(r)=(v(r,0),\partial_{t}v(r,0))$
(2.14)
for $r>0$, then we see that
$\displaystyle w=K[\vec{f}_{1}],\quad v=K[\vec{f}_{2}].$ (2.15)
Now we state the result for the inverse of ${W}_{+}$.
###### Theorem 2.3
$($Existence of the inverse of a wave operator$)$ Let the assumptions of
Theorem 2.2 hold. Then there exists a positive number $\varepsilon_{0}$
$($depending only on $p$ and $q)$ such that for any
$\varepsilon\in(0,\varepsilon_{0}]$, one can define $(W_{+})^{-1}$ by
$(\vec{\varphi}_{1},\vec{\varphi}_{2})\in Y_{\kappa_{1}}(\varepsilon)\times
Y_{\kappa_{2}}(\varepsilon)\longmapsto(\vec{f}_{1},\vec{f}_{2})\in
Y_{\kappa_{1}}(2\varepsilon)\times Y_{\kappa_{2}}(2\varepsilon)$ so that (2.7)
is valid. Here $(u_{1},u_{2})$ is the solution of (2.12) satisfying (2.11),
and $(\vec{f}_{1},\vec{f}_{2})$ is defined by (2.14) for
$(\vec{\varphi}_{1},\vec{\varphi}_{2})\in Y_{\kappa_{1}}(\varepsilon)\times
Y_{\kappa_{2}}(\varepsilon)$.
Moreover, we have for $r>0$
$\displaystyle|\\!|\\!|\vec{f_{1}}(r)-\vec{\varphi}_{1}(r)|\\!|\\!|(1+r)^{\kappa_{1}}\leq
C\varepsilon^{p},$ (2.16)
$\displaystyle|\\!|\\!|\vec{f_{2}}(r)-\vec{\varphi}_{2}(r)|\\!|\\!|(1+r)^{\kappa_{2}}\leq
C\varepsilon^{q},$ (2.17)
provided $\vec{\varphi}_{j}\in Y_{\kappa_{j}}(\varepsilon)$ $(j=1,2)$ and
$0<\varepsilon\leq\varepsilon_{0}$, where $C$ is a constant depending only on
$p$ and $q$.
Remark. Now we are in a position to conclude the existence of a scattering
operator for (0.1). As we have constructed $W_{+}$ in Theorem 2.2, we obtain
$W_{-}$ as well. Taking the range of $W_{-}$ to be included by that of
$W_{+}$, we are able to define the scattering operator by
$S=(W_{+})^{-1}W_{-}$.
### 2.2 Existence of generalized wave operators
In this subsection we consider the case where $1<p\leq 2$. We set
$\displaystyle\kappa_{1}=p-1,\quad\kappa_{2}=q(p-1)-1$ (2.18)
for $1<p<2$. While, when $p=2$, we take $\kappa_{1}$ and $\kappa_{2}$ in such
a way that
$0<\kappa_{1}<1<\kappa_{2}<q-1,\quad q\kappa_{1}=\kappa_{2}+1.$ (2.19)
For instance, $\kappa_{1}=(q+2)/(2q)$, $\kappa_{2}=q/2$ satisfy the above
conditions. Note that $0<\kappa_{1}<1$ and $\kappa_{2}>1$ in both cases, by
the assumption (0.2).
First of all, we present a result for a special case of Theorem 2.5 below,
because it would make easy to recognize the statement for the general case.
Namely, we assume that $1<p<2$ and the following stronger condition on $p,q$
than (0.10) :
$\kappa_{1}\kappa_{2}=(p-1)(q(p-1)-1)>1.$ (2.20)
In order to have an analogue to Theorem 2.2, we replace the final state
$(K[\vec{f_{1}}],K[\vec{f_{2}}])$ by $(w_{1},v_{0})\in Z^{2}(\kappa_{1})\times
X^{2}(\kappa_{2})$ which is the solution of the initial value problem for
$\left\\{\begin{array}[]{ll}\partial_{t}^{2}w_{1}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)w_{1}=|\partial_{t}v_{0}|^{p}&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}},\\\
\partial_{t}^{2}v_{0}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)v_{0}=0&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}}\end{array}\right.$ (2.21)
with
$(w_{1},\partial_{t}w_{1})(r,0)=\vec{f}_{1}(r),\quad(v_{0},\partial_{t}v_{0})(r,0)=\vec{f}_{2}(r)\quad\mbox{for}\quad
r>0.$ (2.22)
Actually, we have the following.
###### Theorem 2.4
$($Existence of a generalized wave operator ; a special case$)$ Let $1<p\leq
q$. Suppose that $1<p<2$ and (2.20) holds. Then there exists a positive number
$\varepsilon_{0}$ $($depending only on $p$ and $q)$ such that one can define a
generalized wave operator
$\widetilde{W}_{+}=(\widetilde{W}_{+}^{(1)},\widetilde{W}_{+}^{(2)})$ from
$Y_{\kappa_{1}}(\varepsilon_{0})\times Y_{\kappa_{2}}(\varepsilon_{0})$ to
$Y_{\kappa_{1}}(2\varepsilon_{0})\times Y_{\kappa_{2}}(2\varepsilon_{0})$ by
$\widetilde{W}_{+}^{(j)}[\vec{f}_{1},\vec{f}_{2}](r)=(u_{j},\partial_{t}u_{j})(r,0)\quad(j=1,2),$
(2.23)
where $(u_{1},u_{2})\in Z^{2}(\kappa_{1})\times X^{2}(\kappa_{2})$ is a unique
solution of (0.8) satisfying
$\|u_{1}(t)-w_{1}(t)\|_{E}+\|u_{2}(t)-v_{0}(t)\|_{E}\to 0\quad\mbox{as}\quad
t\to\infty$ (2.24)
for each $(\vec{f}_{1},\vec{f}_{2})\in Y_{\kappa_{1}}(\varepsilon_{0})\times
Y_{\kappa_{2}}(\varepsilon_{0})$. Moreover, we have for $r>0$
$|\\!|\\!|\widetilde{W}_{+}^{(1)}[\vec{f}_{1},\vec{f}_{2}](r)-\vec{f_{1}}(r)|\\!|\\!|(1+r)^{\kappa_{1}}\leq
C\varepsilon^{1+(p-1)q}(1+r)^{-\kappa_{1}(\kappa_{2}-1)},$ (2.25)
and
$|\\!|\\!|\widetilde{W}_{+}^{(2)}[\vec{f}_{1},\vec{f}_{2}](r)-\vec{f_{2}}(r)|\\!|\\!|(1+r)^{\kappa_{2}}\leq
C\varepsilon^{q},$ (2.26)
provided $\vec{f}_{j}\in Y_{\kappa_{j}}(\varepsilon)$ $(j=1,2)$ and
$0<\varepsilon\leq\varepsilon_{0}$, where $C$ is a constant depending only on
$p$ and $q$.
Remark. When $p=2$, we have only to assume $q>2$, instead of (2.20). In fact,
if we replace the right hand side of (2.25) by
$C\varepsilon^{1+q}(1+r)^{-(\kappa_{2}-\kappa_{1})}$, then the conclusions of
Theorem 2.4 remain valid (for the needed modification of the proof, see the
remark after the proof of Theorem 2.5).
Next we relax the condition (2.20) to
$\kappa_{1}\kappa_{2}>1+\kappa_{1}^{2}-\kappa_{1},$ (2.27)
which is equivalent to (0.10), while we shall keep $1<p<2$. In the previous
case, it suffices to iterate just once for getting $w_{1}$ as a final state
for $u_{1}$. However, in order to treat the general case, we need to iterate
many times for finding out a suitable final state for $u_{1}$.
First we define a sequence $\\{a_{j}\\}_{j=0}^{\infty}$ by $a_{0}=1$ and
$a_{j+1}=\kappa_{1}(a_{j}-1)+\kappa_{2}\quad\mbox{for}\ j\geq 0,$ (2.28)
explicitly we have
$a_{j}=\frac{\kappa_{2}-\kappa_{1}}{1-\kappa_{1}}-\frac{(\kappa_{2}-1)(\kappa_{1})^{j}}{1-\kappa_{1}}\quad\mbox{for}\
j\geq 0.$
Observe that $\\{a_{j}\\}_{j=0}^{\infty}$ is strictly increasing,
$a_{j}<(\kappa_{2}-\kappa_{1})/(1-\kappa_{1})$ for $j\geq 1$, and
$\lim_{j\to\infty}a_{j}=(\kappa_{2}-\kappa_{1})/(1-\kappa_{1})$. Since $p<2$
and (2.27) yield
$a_{0}=1<\frac{1}{\kappa_{1}}<\frac{\kappa_{2}-\kappa_{1}}{1-\kappa_{1}}=\lim_{j\to\infty}a_{j},$
there exists a nonnegative integer $\ell$ such that
$a_{\ell+1}>\frac{1}{\kappa_{1}},\quad a_{\ell}\leq\frac{1}{\kappa_{1}}.$
(2.29)
Next we introduce a sequence $\\{(w_{j},v_{j})\\}_{j=0}^{\ell+1}$ as follows :
For $(\vec{f}_{1},\vec{f}_{2})\in Y_{\kappa_{1}}(\varepsilon)\times
Y_{\kappa_{2}}(\varepsilon)$ we set
$\displaystyle w_{1}=w_{0}+L(|\partial_{t}v_{0}|^{p}),\quad
w_{0}=K[\vec{f}_{1}],$ $\displaystyle
v_{1}=v_{0}+R(|\partial_{t}w_{1}|^{q}),\quad v_{0}=K[\vec{f}_{2}].$
Moreover, we define
$\displaystyle
w_{j+1}=w_{j}+L(|\partial_{t}v_{j}|^{p}-|\partial_{t}v_{j-1}|^{p})),$ (2.30)
$\displaystyle
v_{j+1}=v_{j}+R(|\partial_{t}w_{j+1}|^{q}-|\partial_{t}w_{j}|^{q})$ (2.31)
for $1\leq j\leq\ell$. Here $L$ and $R$ are the integral operators given by
(3.3) and (3.8), respectively. Then we see that for $1\leq j\leq\ell+1$,
$\left\\{\begin{array}[]{ll}\partial_{t}^{2}w_{j}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)w_{j}=|\partial_{t}v_{j-1}|^{p}&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}},\\\
\partial_{t}^{2}v_{j}-\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}\right)v_{j}=|\partial_{t}w_{j}|^{q}&\mbox{for}\quad
r>0,\ t\in{\mathbb{R}}\end{array}\right.$ (2.32)
and $(w_{j},\partial_{t}w_{j})(r,0)=\vec{f}_{1}(r)$ for $r>0$.
Now, the following theorem shows that $w_{\ell+1}$ is a final state for
$u_{1}$.
###### Theorem 2.5
$($Existence of a generalized wave operator$)$ Let $1<p\leq q$. Suppose that
$1<p<2$ and (2.27). Assume $\kappa_{1}a_{\ell}<1$ in addition to (2.29). Then
there exists a positive number $\varepsilon_{0}$ $($depending only on $p$ and
$q)$ such that one can define a generalized wave operator
$\widetilde{W}_{+}=(\widetilde{W}_{+}^{(1)},\widetilde{W}_{+}^{(2)})$ from
$Y_{\kappa_{1}}(\varepsilon_{0})\times Y_{\kappa_{2}}(\varepsilon_{0})$ to
$Y_{\kappa_{1}}(2\varepsilon_{0})\times Y_{\kappa_{2}}(2\varepsilon_{0})$ by
(2.23), where $(u_{1},u_{2})\in Z^{2}(\kappa_{1})\times X^{2}(\kappa_{2})$ is
a unique solution of (0.8) satisfying
$\|u_{1}(t)-w_{\ell+1}(t)\|_{E}+\|u_{2}(t)-v_{0}(t)\|_{E}\to
0\quad\mbox{as}\quad t\to\infty$ (2.33)
for each $(\vec{f}_{1},\vec{f}_{2})\in Y_{\kappa_{1}}(\varepsilon_{0})\times
Y_{\kappa_{2}}(\varepsilon_{0})$. Moreover, for $r>0$, we have (2.26) and
$|\\!|\\!|\widetilde{W}_{+}^{(1)}[\vec{f}_{1},\vec{f}_{2}](r)-\vec{f_{1}}(r)|\\!|\\!|(1+r)^{\kappa_{1}}\leq
C\varepsilon^{B_{\ell}}(1+r)^{-\kappa_{1}(a_{\ell+1}-1)},$ (2.34)
provided $\vec{f}_{j}\in Y_{\kappa_{j}}(\varepsilon)$ $(j=1,2)$ and
$0<\varepsilon\leq\varepsilon_{0}$, where $C$ is a constant depending only on
$p$ and $q$. Here we put
$B_{\ell}=1+(p-1)(q+\ell(p+q-2)).$
Remark. If $1<p<2$ and (2.20) holds, then (2.27) is valid and
$\kappa_{1}a_{\ell}<1$ is satisfied for $\ell=0$. Therefore, Theorem 2.4
follows from Theorem 2.5.
On the one hand, suppose $\kappa_{1}a_{\ell}=1$ (notice that we have $\ell\geq
1$ in this case). Then we need to modify the statement of Theorem 2.5 a
little. Letting $\delta$ be a number satisfying
$\displaystyle
0<\delta<a_{\ell}-a_{\ell-1},\quad\kappa_{1}^{2}\delta<\kappa_{1}a_{\ell+1}-1,$
(2.35)
we define
$\displaystyle a_{\ell}^{\prime}=a_{\ell}-\delta,\quad\text{and}\quad
a_{\ell+1}^{\prime}=\kappa_{1}(a_{\ell}^{\prime}-1)+\kappa_{2}\,(=a_{\ell+1}-\kappa_{1}\delta).$
(2.36)
Observing that
$\displaystyle
a_{\ell-1}<a_{\ell}^{\prime}<a_{\ell+1}^{\prime},\quad\kappa_{1}a_{\ell}^{\prime}<1,\quad\text{and}\quad\kappa_{1}a_{\ell+1}^{\prime}>1,$
(2.37)
we can show the statement of the theorem with $a_{\ell+1}$ in (2.34) replaced
by $a_{\ell+1}^{\prime}$.
Our next step is to construct the inverse of $\widetilde{W}_{+}$, based on the
existence result in Theorem 1 of [5] for the initial value problem (0.8) and
(2.10). Let $(u_{1},u_{2})\in Z^{2}(\kappa_{1})\times{X^{2}(\kappa_{2})}$ be
the unique solution of (2.12) satisfying
$\displaystyle\|u_{1}\|_{Z^{2}(\kappa_{1})}+\|u_{2}\|_{X^{2}(\kappa_{2})}\leq
2C_{0}\varepsilon$ (2.38)
with $C_{0}$ the canstant in (2.4). Using the solution, we set
$w_{0}^{*}=K[\vec{\varphi}_{1}]$,
$v_{0}^{*}=u_{2}-R(|\partial_{t}u_{1}|^{q})$. Moreover, when $\ell\geq 1$, we
define for $1\leq j\leq\ell$
$\displaystyle w_{j}^{*}=w_{0}^{*}+L(|\partial_{t}v_{j-1}^{*}|^{p}),$ (2.39)
$\displaystyle v_{j}^{*}=v_{0}^{*}+R(|\partial_{t}w_{j}^{*}|^{q}).$ (2.40)
We furter define
$\displaystyle
w^{*}=u_{1}-R(|\partial_{t}u_{2}|^{p}-|\partial_{t}v_{\ell}^{*}|^{p}),$ (2.41)
which we wish to regard as a final state for $u_{1}$. If we set
$\displaystyle\vec{f}_{1}(r)=(w^{*}(r,0),\partial_{t}w^{*}(r,0))\quad\mbox{for}\quad
r>0,$ (2.42)
$\displaystyle\vec{f}_{2}(r)=(v_{0}^{*}(r,0),\partial_{t}v_{0}^{*}(r,0))\quad\mbox{for}\quad
r>0,$ (2.43)
then we see that $v_{0}^{*}$ and $w^{*}$ are represented as
$\displaystyle w^{*}=K[\vec{f}_{1}]+L(|\partial_{t}v_{\ell}^{*}|^{p}),\quad
v_{0}^{*}=K[\vec{f}_{2}].$ (2.44)
Now we state the result for the inverse of $\widetilde{W}_{+}$.
###### Theorem 2.6
$($Existence of the inverse of a generalized wave operator$)$ Let the
assumptions of Theorem 2.5 be fulfilled. Then there exists a positive number
$\varepsilon_{0}$ $($depending only on $p$ and $q)$ such that, for any
$\varepsilon\in(0,\varepsilon_{0}]$, one can define $(\widetilde{W}_{+})^{-1}$
by $(\vec{\varphi}_{1},\vec{\varphi}_{2})\in Y_{\kappa_{1}}(\varepsilon)\times
Y_{\kappa_{2}}(\varepsilon)\longmapsto(\vec{f}_{1},\vec{f}_{2})\in
Y_{\kappa_{1}}(2\varepsilon)\times Y_{\kappa_{2}}(2\varepsilon)$ so that
$\displaystyle\|u_{1}(t)-(K[\vec{f}_{1}]+L(|\partial_{t}v_{\ell}^{*}|^{p}))(t)\|_{E}\to
0\quad$ $\displaystyle\mbox{as}\quad t\to\infty$ (2.45)
$\displaystyle\|u_{2}(t)-K[\vec{f}_{2}](t)\|_{E}\to 0\quad$
$\displaystyle\mbox{as}\quad t\to\infty,$ (2.46)
hold. Here $(u_{1},u_{2})$ is the solution of (2.12) satisfying (2.38),
$(\vec{f}_{1},\vec{f}_{2})$ is defined by (2.42), (2.43), and $v_{\ell}^{*}$
is given by (2.40) for $(\vec{\varphi}_{1},\vec{\varphi}_{2})\in
Y_{\kappa_{1}}(\varepsilon)\times Y_{\kappa_{2}}(\varepsilon)$.
Moreover, for $r>0$, we have (2.17) and
$|\\!|\\!|\vec{f_{1}}(r)-\vec{\varphi}_{1}(r)|\\!|\\!|(1+r)^{\kappa_{1}}\leq
C\varepsilon^{B_{\ell}}(1+r)^{-\kappa_{1}(a_{\ell+1}-1)},$ (2.47)
provided $\vec{\varphi}_{j}\in Y_{\kappa_{j}}(\varepsilon)$ $(j=1,2)$ and
$0<\varepsilon\leq\varepsilon_{0}$, where $C$ is a constant depending only on
$p$ and $q$.
Remark. Now we are in a position to conclude the existence of a scattering
operator for (0.1). As we have constructed $\widetilde{W}_{+}$ in Theorem 2.5,
we obtain $\widetilde{W}_{-}$ as well. Taking the range of $\widetilde{W}_{-}$
to be included by that of $\widetilde{W}_{+}$, we are able to define the long
range scattering operator by
$\widetilde{S}=(\widetilde{W}_{+})^{-1}\widetilde{W}_{-}$.
## 3 Inhomogeneous wave equations
In this section we summarize the results of the section 4 in [5] for the case
$a=c=1$. The first one is concerned with the initial value problem for the
inhomogeneous wave equation with the zero initial data:
$\displaystyle
u_{tt}-\left(u_{rr}+\frac{2}{r}u_{r}\right)=F(r,t)\quad\mbox{in}\quad(0,\infty)\times(0,\infty),$
(3.1) $\displaystyle u(r,0)=(\partial_{t}u)(r,0)=0\quad\mbox{for}\quad r>0.$
(3.2)
The solution of this problem is given by
$L(F)(r,t)=\frac{1}{2r}\int_{0}^{t}ds\int_{r-(t-s)}^{r+(t-s)}\lambda
F(\lambda,s)d\lambda.$ (3.3)
In order to study the qualitative property of $L(F)$, we set
$\displaystyle\quad M_{0}(F)=\sup_{r>0,\,t\geq
0}|F(r,t)|r^{\alpha}(1+r)^{\beta}(1+r+t)^{\gamma}(1+|r-t|)^{\delta},$ (3.4)
$\displaystyle\quad M_{1}(F)=M_{0}(F)$ (3.5)
$\displaystyle\quad\quad+\sup_{r>0,\,t\geq
0}|\partial_{r}F(r,t)|r^{\alpha+1}(1+r)^{\beta-1}(1+r+t)^{\gamma}(1+|r-t|)^{\delta}.$
for $\alpha$, $\beta$, $\gamma$, and $\delta\in{\mathbb{R}}$. Then we have
###### Proposition 3.1
Let $F\in{X}^{1}$. Then we have $L(F)\in X^{2}$. Moreover, if $M_{s-1}(F)$
with $s=1$ or $s=2$ is finite for $\alpha<3-s$, $\beta\in{\mathbf{R}}$,
$\gamma\geq 0$, and $\delta>1$, then there exists a constant ${C}$ depending
only on $\alpha$, $\beta$, $\gamma$, and $\delta$ such that
$\displaystyle\|L(F)\|_{X^{s}(\nu)}\leq{C}M_{s-1}(F)\quad$
$\displaystyle\mbox{if}\quad\alpha+\beta+\gamma>2,$ (3.6)
$\displaystyle\|L(F)\|_{Z^{s}(\alpha+\beta+\gamma-1)}\leq{C}M_{s-1}(F)\quad$
$\displaystyle\mbox{if}\quad 1<\alpha+\beta+\gamma<2,$ (3.7)
where $\nu=\min(\alpha+\beta+\gamma-1,\delta)$.
Proof. Note that the statement follows from the case $\gamma=0$, since
$(1+\lambda+s)^{-\gamma}\leq(1+\lambda)^{-\gamma}$ when $\gamma>0$. Therefore,
applying Theorem 6 in [5] where the case $\gamma=0$ was shown, we conclude the
proof. $\square$
Next we study the operator
$R(F)(r,t)=\frac{1}{2r}\int_{t}^{\infty}ds\int_{(s-t)-r}^{(s-t)+r}\lambda
F(\lambda,s)d\lambda,$ (3.8)
related to the final value problem. Indeed, if $F\in X^{1}$ and
$\sup_{r\geq 1,\,t\geq
0}|F(r,t)|(1+r)^{\beta}(1+r+t)^{\gamma}(1+|r-t|)^{\delta}<\infty$ (3.9)
for $\beta+\gamma>2$, $\delta\in{\mathbf{R}}$, then we have $R(F)\in{X}^{1}$
and it satisfies the inhomogeneous wave equation :
$\displaystyle(\partial_{t}^{2}-\Delta)R(F)(|x|,t)=F(|x|,t)$ (3.10)
in the distributional sense on ${\mathbf{R}}^{3}\times(0,\infty)$. The
following result, which is a refinement of Theorem 7 in [5] in the sense that
one can take a parameter $\gamma$ to be positive, will play an essential role
in this paper.
###### Proposition 3.2
If $F\in X^{1}$ and $M_{s-1}(F)$ with $s=1$ or $s=2$ is finite for
$\alpha<3-s$, $\beta$, $\gamma\in{\mathbf{R}}$, and $\delta>1$ satisfying
$\alpha+\beta+\gamma>2$, then $R(F)\in X^{s}$ and there exists a constant
${C}$ depending only on $\alpha$, $\beta$, $\gamma$, and $\delta$ such that
$\|R(F)\|_{Z^{s}(\mu+\gamma)}\leq{C}M_{s-1}(F),$ (3.11)
where $\mu=\min(\alpha+\beta-1,\delta)$.
Proof. Since the statement for $\gamma\leq 0$ was shown in Theorem 7 in [5],
it suffices to prove it for $\gamma>0$. Seeing the proof, we find that
$R(F)\in X^{s}$ is valid also for $\gamma>0$. Hence it remains to show (3.11).
It follows from (3.8) that
$R(F)(r,t)=\frac{1}{2r}\int_{t}^{\infty}ds\int_{|(s-t)-r|}^{(s-t)+r}\lambda
F(\lambda,s)d\lambda,$
since $\lambda F(\lambda,s)$ is odd in $\lambda$. Observe that if
$\lambda\geq{|(s-t)-r|}$ and $s\geq t$, then we have $\lambda+s\geq r+t$, so
that $(1+\lambda+s)^{-\gamma}\leq(1+r+t)^{-\gamma}$ when $\gamma>0$.
Therefore, if $\alpha+\beta>2$, one can reduce the proof to the case
$\gamma=0$ which was already shown in [5].
Suppose, on the contrary, that $\alpha+\beta\leq 2$. We take a positive number
$\rho>0$ satisfying
$2-(\alpha+\beta)<\rho\leq\delta+1-(\alpha+\beta),\quad\rho<\gamma$ (3.12)
and set $\beta^{\prime}=\beta+\rho$, $\gamma^{\prime}=\gamma-\rho$. Then we
have $\alpha+\beta^{\prime}>2$, $\gamma^{\prime}>0$ and $\delta>1$. Applying
the result in the preceding case with $\beta$ and $\gamma$ replaced by
$\beta^{\prime}$ and $\gamma^{\prime}$ respectively, we obtain the needed
conclusion, because
$\min(\alpha+\beta^{\prime}-1,\delta)=\min(\alpha+\beta-1+\rho,\delta)=\alpha+\beta-1+\rho$
and $\mu=\min(\alpha+\beta-1,\delta)=\alpha+\beta-1$. This completes the
proof. $\square$
## 4 Proof of Main Results
### 4.1 Proof of Theorems 2.2 and 2.3
First we prove Theorem 2.2. Suppose $p>2$. Let $(\vec{f}_{1},\vec{f}_{2})\in
Y_{\kappa_{1}}(\varepsilon)\times Y_{\kappa_{2}}(\varepsilon)$ with
$0<\varepsilon\leq 1$, and set $w_{0}=K[\vec{f}_{1}]$, $v_{0}=K[\vec{f}_{2}]$.
Then it follows from (2.4) that
$\displaystyle\|w_{0}\|_{X^{2}(\kappa_{1})}+\|v_{0}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon.$ (4.1)
Recall that $\kappa_{1}=p-1>1$ and $\kappa_{2}=q-1>1$.
We shall solve the following system of integral equations :
$\displaystyle u_{1}=w_{0}+R(|\partial_{t}u_{2}|^{p}),\quad
u_{2}=v_{0}+R(|\partial_{t}u_{1}|^{q}),$ (4.2)
where $R$ is defined by (3.8). To this end, we define
$T(u_{1},u_{2})=(T^{(1)}(u_{2}),T^{(2)}(u_{1}))$ by
$\displaystyle T^{(1)}(u_{2})=w_{0}+R(|\partial_{t}u_{2}|^{p}),\quad
T^{(2)}(u_{1})=v_{0}+R(|\partial_{t}u_{1}|^{q}).$ (4.3)
For $\varepsilon>0$ we introduce a metric space
$\displaystyle D_{\varepsilon}=\\{(u_{1},u_{2})\in X^{2}\times
X^{2}\,;\,d((u_{1},u_{2}),(w_{0},v_{0}))\leq\varepsilon\\},$ (4.4)
where we have set
$d((u_{1},u_{2}),(u_{1}^{*},u_{2}^{*}))=\|u_{1}-u_{1}^{*}\|_{Z^{2}(\kappa_{1})}+\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})}.$
First we prepare the following.
###### Lemma 4.1
Let $(u_{1},u_{2})\in D_{\varepsilon}$. Then we have
$\displaystyle\|T^{(1)}(u_{2})-w_{0}\|_{Z^{2}(\kappa_{1})}\leq
C\varepsilon^{p},$ (4.5)
$\displaystyle\|T^{(2)}(u_{1})-v_{0}\|_{Z^{2}(\kappa_{2})}\leq
C\varepsilon^{q}.$ (4.6)
Moreover we have
$\displaystyle\|T^{(1)}(u_{2})-T^{(1)}(u_{2}^{*})\|_{Z^{2}(\kappa_{1})}\leq
C\varepsilon^{p-1}\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})},$ (4.7)
$\displaystyle\|T^{(2)}(u_{1})-T^{(2)}(u_{1}^{*})\|_{Z^{2}(\kappa_{2})}\leq
C\varepsilon^{q-1}\|u_{1}-u_{1}^{*}\|_{Z^{2}(\kappa_{1})}$ (4.8)
for $(u_{1},u_{2})$, $(u_{1}^{*},u_{2}^{*})\in D_{\varepsilon}$.
Proof. First we observe that if $(u_{1},u_{2})\in D_{\varepsilon}$, then we
have
$\displaystyle\|u_{1}\|_{X^{2}(\kappa_{1})}+\|u_{2}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon,$ (4.9)
due to $\kappa_{1}$, $\kappa_{2}>1$ and (4.1).
We start with the proof of (4.5). In view of (4.3), it suffices to show
$\displaystyle\|R(|\partial_{t}u_{2}|^{p})\|_{Z^{2}(\kappa_{1})}\leq
C\varepsilon^{p}.$ (4.10)
We see from (4.9) that $M_{1}(|\partial_{t}u_{2}|^{p})\leq C\varepsilon^{p}$
holds for $\alpha=\gamma=0$, $\beta=p$ and $\delta=p\kappa_{2}$, where
$M_{1}(F)$ is defined by (3.4) and (3.5). Since
$\alpha+\beta+\gamma-1=\kappa_{1}>1$, by (3.11) with $s=2$ we get (4.10),
which implies (4.5). Analogously we obtain (4.6), because $\kappa_{2}>1$.
Next we show (4.7). It follows from (4.3) that
$\displaystyle
T^{(1)}(u_{2})-T^{(1)}(u_{2}^{*})=R(|\partial_{t}u_{2}|^{p}-|\partial_{t}u_{2}^{*}|^{p}).$
(4.11)
Since $p>2$, we see from (4.9) that
$M_{1}(|\partial_{t}u_{2}|^{p}-|\partial_{t}u_{2}^{*}|^{p})\leq
C\varepsilon^{p-1}\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})}$
for $\alpha=0$, $\beta=p$, $\gamma=\kappa_{2}-1$, and
$\delta=1+(p-1)\kappa_{2}$. Since
$\alpha+\beta+\gamma-1=\kappa_{1}+\kappa_{2}-1>1$, by (3.11) with $s=2$ we
obtain
$\displaystyle\|R(|\partial_{t}u_{2}|^{p}-|\partial_{t}u_{2}^{*}|^{p})\|_{Z^{2}(\kappa_{1}+\kappa_{2}-1)}\leq
C\varepsilon^{p-1}\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})}.$
In view of (4.11), we get (4.7), since $\kappa_{2}>1$. Analogously we have
(4.8), because $q\geq p>2$. This completes the proof. $\square$
End of the proof of Theorem 2.2. We see from Lemma 4.1 that there exists a
positive number $\varepsilon_{0}$ depending only on $p$ and $q$ such that if
$0<\varepsilon\leq\varepsilon_{0}$, then we have $T(u_{1},u_{2})\in
D_{\varepsilon}$ and
$d(T(u_{1},u_{2}),T(u_{1}^{*},u_{2}^{*}))\leq
2^{-1}d((u_{1},u_{2}),(u_{1}^{*},u_{2}^{*}))$
for $(u_{1},u_{2})$, $(u_{1}^{*},u_{2}^{*})\in D_{\varepsilon}$, namely, $T$
is a contaction on $D_{\varepsilon}$. Hence we find a unique solution
$(u_{1},u_{2})\in D_{\varepsilon}$ of (4.2). Here and in what follows, we
suppose that $0<\varepsilon\leq\varepsilon_{0}$ and $(u_{1},u_{2})$ is the
solution.
Since $T^{(1)}(u_{2})=u_{1}$ and $w_{0}=K[\vec{f}_{1}]$, it follows from (4.5)
that
$\displaystyle[(u_{1}-K[\vec{f}_{1}])(r,t)]_{2}\leq
C\varepsilon^{p}(1+r+t)^{-(\kappa_{1}-1)}(1+|r-t|)^{-1}.$ (4.12)
Therefore we have
$\displaystyle\|(u_{1}-K[\vec{f}_{1}])(t)\|_{E}$ (4.13) $\displaystyle\leq$
$\displaystyle
C\varepsilon^{p}(1+t)^{-(\kappa_{1}-1)}\left(\int_{0}^{\infty}(1+|r-t|)^{-2}dr\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\varepsilon^{p}(1+t)^{-(\kappa_{1}-1)}$
for $t\geq 0$. Analogously by (4.6) we get
$\displaystyle\|(u_{2}-K[\vec{f}_{2}])(t)\|_{E}\leq
C\varepsilon^{q}(1+t)^{-(\kappa_{2}-1)}$ (4.14)
for $t\geq 0$. Hence we obtain (2.7).
Moreover, we easily get (2.8) by taking $t=0$ in (4.12). Analogously, (2.9)
follows from (4.6). Thus we prove Theorem 2.2. $\square$
Next we show Theorem 2.3. Let $(u_{1},u_{2})\in
X^{2}(\kappa_{1})\times{X^{2}(\kappa_{2})}$ be the unique solution of (2.12)
satisfying (2.11). Then we see from (2.13) and (2.15) that (2.7), (2.16) and
(2.17) follows from (4.10) and
$\displaystyle\|R(|\partial_{t}u_{1}|^{q})\|_{Z^{2}(\kappa_{2})}\leq
C\varepsilon^{q}.$ (4.15)
By virtue of (2.11), (4.10) can be shown as before. In the same way we obtain
(4.15). Thus we prove Theorem 2.3. $\square$
### 4.2 Proof of Theorems 2.5 and 2.6
We start by showing the following basic estimates.
###### Lemma 4.2
Let $v\in X^{2}(\kappa_{2})$. Then $L(|\partial_{t}v|^{p})\in
Z^{2}(\kappa_{1})$ and we have
$\displaystyle\|L(|\partial_{t}v|^{p})\|_{Z^{2}(\kappa_{1})}\leq
C\|v\|_{X^{2}(\kappa_{2})}^{p}.$ (4.16)
While, let $w\in Z^{2}(\kappa_{1})$. Then $R(|\partial_{t}w|^{q})\in
Z^{2}(\kappa_{2})$ and we have
$\displaystyle\|R(|\partial_{t}w|^{q})\|_{Z^{2}(\kappa_{2})}\leq
C\|w\|_{Z^{2}(\kappa_{1})}^{q}.$ (4.17)
Proof. For $v\in X^{2}(\kappa_{2})$, we have
$[v(r,t)]_{2}\leq\|v\|_{X^{2}(\kappa_{2})}(1+|r-t|)^{-\kappa_{2}},$
so that $M_{1}(|\partial_{t}v|^{p})\leq p\|v\|_{X^{2}(\kappa_{2})}^{p}$ holds
for $\alpha=0$, $\beta=p$, $\gamma=0$, and $\delta=p\kappa_{2}$. By
Proposition 3.1 with $s=2$ we get $L(|\partial_{t}v|^{p})\in
Z^{2}(\kappa_{1})$ and (4.16), since $\alpha+\beta+\gamma-1=\kappa_{1}<1$ and
$\delta=p\kappa_{2}>1$.
On the other hand, for $w\in Z^{2}(\kappa_{1})$, we have
$[w(r,t)]_{2}\leq\|w\|_{Z^{2}(\kappa_{1})}(1+r+t)^{1-\kappa_{1}}(1+|r-t|)^{-1},$
so that $M_{1}(|\partial_{t}w|^{q})\leq q\|w\|_{Z^{2}(\kappa_{1})}^{q}$ holds
for $\alpha=0$, $\beta=\delta=q$, and $\gamma=q\kappa_{1}-q=\kappa_{2}+1-q$.
By Proposition 3.2 with $s=2$ we get $R(|\partial_{t}w|^{q})\in
Z^{2}(\kappa_{2})$ and (4.17), since $\alpha+\beta+\gamma-1=\kappa_{2}>1$.
This completes the proof. $\square$
Next we examine the qualitative property of $\\{w_{j}\\}_{j=0}^{\ell+1}$ and
$\\{v_{j}\\}_{j=0}^{\ell+1}$ defined by (2.30) and (2.31). As a corollary of
Lemma 4.2, we derive the following estimates.
###### Corollary 4.3
Let $0\leq j\leq\ell+1$, $0<\varepsilon\leq 1$ and $\vec{f}_{i}\in
Y_{\kappa_{i}}(\varepsilon)$ with $i=1,2$. Then $w_{j}\in Z^{2}(\kappa_{1})$,
$v_{j}\in X^{2}(\kappa_{2})$, and we have
$\displaystyle\|w_{j}\|_{Z^{2}(\kappa_{1})}\leq C\varepsilon,$ (4.18)
$\displaystyle\|v_{j}\|_{X^{2}(\kappa_{2})}\leq C\varepsilon.$ (4.19)
Besides, we have
$\displaystyle\|w_{1}-w_{0}\|_{Z^{2}(\kappa_{1})}\leq C\varepsilon^{p},$
(4.20) $\displaystyle\|v_{1}-v_{0}\|_{Z^{2}(\kappa_{2})}\leq
C\varepsilon^{q}.$ (4.21)
Proof. Since $\vec{f}_{2}\in Y_{\kappa_{2}}(\varepsilon)$, by Proposition 2.1
we get $v_{0}\in X^{2}(\kappa_{2})$ and (4.19) for $j=0$. Analogously we have
$w_{0}\in X^{2}(\kappa_{1})$ and $\|w_{0}\|_{X^{2}(\kappa_{1})}\leq
C\varepsilon$. Since $0<\kappa_{1}<1$, we find $w_{0}\in Z^{2}(\kappa_{1})$
and (4.18) for $j=0$.
Next suppose that (4.18) and (4.19) hold for some $j$ with $0\leq j\leq\ell$.
Since $w_{j+1}-w_{0}=L(|\partial_{t}v_{j}|^{p})$ by (2.30), we have
$w_{j+1}-w_{0}\in Z^{2}(\kappa_{1})$ and
$\|w_{j+1}-w_{0}\|_{Z^{2}(\kappa_{1})}\leq C\varepsilon^{p}$, using (4.16) and
(4.19). Hence we get $\|w_{j+1}\|_{Z^{2}(\kappa_{1})}\leq C\varepsilon$ for
$0<\varepsilon\leq 1$ and (4.20) by taking $j=0$ in the above. Analogously we
obtain $\|v_{j+1}-v_{0}\|_{Z^{2}(\kappa_{2})}\leq C\varepsilon^{q}$ by (2.31),
(4.17) and (4.18). Therefore, $\|v_{j+1}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon$ for $0<\varepsilon\leq 1$ and (4.21) holds, because
$\kappa_{2}>1$. The proof is complete. $\square$
The following estimates are crucial in the proof of Theorem 2.5 for $\ell\geq
1$.
###### Lemma 4.4
Let $1\leq j\leq\ell$, $0<\varepsilon\leq 1$ and $\vec{f}_{i}\in
Y_{\kappa_{i}}(\varepsilon)$ with $i=1,2$. Then $w_{j+1}-w_{j}\in
Z^{2}(\kappa_{1}a_{j})$, $v_{j+1}-v_{j}\in Z^{2}(a_{j+1})$, and we have
$\displaystyle\|w_{j+1}-w_{j}\|_{Z^{1}(\kappa_{1}a_{j})}\leq
C\varepsilon^{b_{j-1}+p-1},$ (4.22)
$\displaystyle\|v_{j+1}-v_{j}\|_{Z^{1}(a_{j+1})}\leq C\varepsilon^{b_{j}},$
(4.23)
where we put $b_{k}=q+k(p+q-2)$ for a nonnegative integer $k$, and
$\displaystyle\|w_{j+1}-w_{j}\|_{Z^{2}(\kappa_{1}a_{j})}\leq
C\varepsilon^{B_{j-1}},$ (4.24)
$\displaystyle\|v_{j+1}-v_{j}\|_{Z^{2}(a_{j+1})}\leq
C\varepsilon^{B_{j-1}+q-1},$ (4.25)
where we put $B_{k}=1+(p-1)b_{k}$ for a nonnegative integer $k$.
Proof. Observe that (4.21) implies (4.23) for $j=0$, since $b_{0}=q$ and
$a_{1}=\kappa_{2}$.
First we show that if (4.23) holds for some $j$ with $0\leq j\leq\ell-1$, then
(4.22) with $j$ replaced by $j+1$ holds. It follows from (2.30) that
$\displaystyle w_{j+2}-w_{j+1}=L(G(v_{j+1},v_{j})),\quad
G(v,v^{*})=|\partial_{t}v|^{p}-|\partial_{t}v^{*}|^{p}$ (4.26)
for $0\leq j\leq\ell-1$. Note that if $v$, $v^{*}\in{X^{2}(\kappa_{2})}$ and
$v-v^{*}\in{Z^{1}(a_{j+1})}$, then we have
$\displaystyle\quad|G(v,v^{*})(r,t)|$ (4.27) $\displaystyle\leq
p|\partial_{t}(v-v^{*})|(|\partial_{t}v|+|\partial_{t}v^{*}|)^{p-1}$
$\displaystyle\leq
p\|v-v^{*}\|_{Z^{1}(a_{j+1})}(\|v\|_{X^{2}(\kappa_{2})}+\|v^{*}\|_{X^{2}(\kappa_{2})})^{p-1}$
$\displaystyle\quad\ \times
r^{-1}(1+r)^{-(p-1)}(1+r+t)^{-(a_{j+1}-1)}(1+|r-t|)^{-1-\kappa_{1}\kappa_{2}}.$
In addition, we have
$\displaystyle(1+r+t)^{-(a_{j+1}-1)}(1+|r-t|)^{-1-\kappa_{1}\kappa_{2}}$
(4.28)
$\displaystyle\leq(1+r+t)^{-\kappa_{1}(a_{j+1}-1)}(1+|r-t|)^{-\kappa_{1}-\kappa_{2}},$
since $0<\kappa_{1}<1$ and $a_{j+1}\geq\kappa_{2}$.
Applying (4.27) to $G(v_{j+1},v_{j})$ and using (4.19), (4.23) and (4.28), we
obtain
$\displaystyle M_{0}(G(v_{j+1},v_{j}))\leq C\varepsilon^{b_{j}+p-1}$
for $\alpha=1$, $\beta=\kappa_{1}$,
$\gamma=\kappa_{1}a_{j+1}-\kappa_{1}\,(>0)$, and
$\delta=\kappa_{1}+\kappa_{2}$. Since
$\alpha+\beta+\gamma-1=\kappa_{1}a_{j+1}\leq\kappa_{1}a_{\ell}$ and
$\kappa_{1}a_{\ell}<1$ from the assumption in Theorem 2.5, if we apply (3.7)
with $s=1$ to the right hand side on (4.26), then the desired estimate holds.
In particular, we have (4.22) with $j=1$, since (4.23) is valid for $j=0$.
Next we show that if (4.22) holds for some $j$ with $1\leq j\leq\ell$, then
(4.23) is valid for the same $j$. It follows from (2.31) that
$\displaystyle v_{j+1}-v_{j}=R(H(w_{j+1},w_{j})),\quad
H(w,w^{*})=|\partial_{t}w|^{q}-|\partial_{t}w^{*}|^{q}$ (4.29)
for $1\leq j\leq\ell$. Note that if $w$, $w^{*}\in{Z^{2}(\kappa_{1})}$ and
$w-w^{*}\in{Z^{1}(\kappa_{1}a_{j})}$, then we have
$\displaystyle\quad|H(w,w^{*})(r,t)|$ (4.30) $\displaystyle\leq
q|\partial_{t}(w-w^{*})|(|\partial_{t}w|+|\partial_{t}w^{*}|)^{q-1}$
$\displaystyle\leq
q\|w-w^{*}\|_{Z^{1}(\kappa_{1}a_{j})}(\|w\|_{Z^{2}(\kappa_{1})}+\|w^{*}\|_{Z^{2}(\kappa_{1})})^{q-1}$
$\displaystyle\quad\ \times
r^{-1}(1+r)^{-(q-1)}(1+r+t)^{-(q-1)(\kappa_{1}-1)-(\kappa_{1}a_{j}-1)}(1+|r-t|)^{-q}.$
Applying (4.30) to $H(w_{j+1},w_{j})$ and using (4.18), (4.22), we obtain
$\displaystyle M_{0}(H(w_{j+1},w_{j}))\leq C\varepsilon^{b_{j}}$
for $\alpha=1$, $\beta=q-1$, $\gamma=a_{j+1}+1-q$, and $\delta=q$. Since
$\alpha+\beta+\gamma-1=a_{j+1}>1$, if we apply (3.11) with $s=1$ to the right
hand side on (4.29), then the desired estimate holds. In conclusion, we have
proven (4.22) and (4.23) for $1\leq j\leq\ell$.
Next we show (4.24) and (4.25). Observe that if we put $B_{-1}=1$, then (4.25)
with $j=0$ follows from (4.21).
First we show that if (4.25) holds for some $j$ with $0\leq j\leq\ell-1$, then
it, in combination with (4.23), implies (4.24) with $j$ replaced by $j+1$.
Note that if $v$, $v^{*}\in{X^{2}(\kappa_{2})}$ and
$v-v^{*}\in{Z^{2}(a_{j+1})}$, then we have
$\displaystyle\quad\\{(1+r)|G(v,v^{*})(r,t)|+r|\partial_{r}G(v,v^{*})(r,t)|\\}$
(4.31) $\displaystyle\quad\ \times
r^{p-1}(1+r+t)^{\kappa_{1}(a_{j+1}-1)}(1+|r-t|)^{\kappa_{1}+\kappa_{2}}$
$\displaystyle\leq
2p\\{\|v-v^{*}\|_{Z^{1}(a_{j+1})}^{p-1}\|v\|_{X^{2}(\kappa_{2})}$
$\displaystyle\quad\quad+\|v-v^{*}\|_{Z^{2}(a_{j+1})}(\|v\|_{X^{2}(\kappa_{2})}+\|v^{*}\|_{X^{2}(\kappa_{2})})^{p-1}\\}.$
In fact, similarly to (4.27), we have
$\displaystyle\quad|G(v,v^{*})(r,t)|\leq
p\|v-v^{*}\|_{Z^{2}(a_{j+1})}(\|v\|_{X^{2}(\kappa_{2})}+\|v^{*}\|_{X^{2}(\kappa_{2})})^{p-1}$
$\displaystyle\quad\hskip
79.66771pt\times(1+r)^{-p}(1+r+t)^{-(a_{j+1}-1)}(1+|r-t|)^{-1-\kappa_{1}\kappa_{2}}.$
Since $1<p<2$, we obtain
$\displaystyle|\partial_{r}G(v,v^{*})(r,t)|$ $\displaystyle\leq$
$\displaystyle
2p|\partial_{t}(v-v^{*})|^{p-1}|\partial_{r}\partial_{t}v|+p|\partial_{r}\partial_{t}(v-v^{*})||\partial_{t}v^{*}|^{p-1}$
$\displaystyle\leq$ $\displaystyle
2p\|v-v^{*}\|_{Z^{1}(a_{j+1})}^{p-1}\|v\|_{X^{2}(\kappa_{2})}$
$\displaystyle\quad\times
r^{-p}(1+r+t)^{-\kappa_{1}(a_{j+1}-1)}(1+|r-t|)^{-\kappa_{1}-\kappa_{2}}$
$\displaystyle\
+p\|v-v^{*}\|_{Z^{2}(a_{j+1})}\|v^{*}\|_{X^{2}(\kappa_{2})}^{p-1}$
$\displaystyle\quad\times
r^{-1}(1+r)^{-(p-1)}(1+r+t)^{-(a_{j+1}-1)}(1+|r-t|)^{-1-\kappa_{1}\kappa_{2}}.$
By (4.28) we get (4.31).
Applying (4.31) to $G(v_{j+1},v_{j})$ and using (4.19), (4.23) and (4.25), we
obtain
$\displaystyle M_{1}(G(v_{j+1},v_{j}))\leq
C\varepsilon^{(p-1)b_{j}+1}+C\varepsilon^{B_{j-1}+p+q-2}$
for $\alpha=p-1\,(<1)$, $\beta=1$, $\gamma=\kappa_{1}a_{j+1}-\kappa_{1}$, and
$\delta=\kappa_{1}+\kappa_{2}$. It is not difficult to see that
$\displaystyle B_{j}\leq B_{j-1}+p+q-2$ (4.32)
for $0\leq j\leq\ell$ (recall $B_{-1}=1$). Therefore we have
$M_{1}(G(v_{j+1},v_{j}))\leq C\varepsilon^{B_{j}}$. Since
$\alpha+\beta+\gamma-1=\kappa_{1}a_{j+1}\leq\kappa_{1}a_{\ell}<1$, if we apply
(3.7) with $s=2$ to the right hand side on (4.26), then the desired estimate
holds. In particular, we have (4.24) with $j=1$, since (4.25) is valid for
$j=0$.
Finally we show that if (4.24) holds for some $j$ with $1\leq j\leq\ell$, then
(4.25) is valid for the same $j$. Note that if $w$,
$w^{*}\in{Z^{2}(\kappa_{1})}$ and $w-w^{*}\in{Z^{2}(\kappa_{1}a_{j})}$, then
we have
$\displaystyle\quad\\{(1+r)|H(w,w^{*})(r,t)|+r|\partial_{r}H(w,w^{*})(r,t)|\\}$
(4.33) $\displaystyle\quad\
\times(1+r)^{q-1}(1+r+t)^{(q-1)(\kappa_{1}-1)+\kappa_{1}a_{j}-1}(1+|r-t|)^{q}$
$\displaystyle\leq
q^{2}\|w-w^{*}\|_{Z^{2}(\kappa_{1}a_{j})}(\|w\|_{Z^{2}(\kappa_{1})}+\|w^{*}\|_{Z^{2}(\kappa_{1})})^{q-1},$
since $q>2$. Applying (4.33) to $H(w_{j+1},w_{j})$ and using (4.18), (4.24),
we obtain
$\displaystyle M_{1}(H(w_{j+1},w_{j}))\leq C\varepsilon^{B_{j-1}+q-1}$
for $\alpha=0$, $\beta=\delta=q$ and $\gamma=a_{j+1}+1-q$. Since
$\alpha+\beta+\gamma-1=a_{j+1}>1$, if we apply (3.11) with $s=2$ to the right
hand side on (4.29), then the desired estimate holds. In conclusion, all the
asserion of the lemma is proven. $\square$
Our next step is to solve the following system :
$\displaystyle u_{1}=w_{\ell+1}+R(G(u_{2},v_{\ell})),\quad
u_{2}=v_{\ell+1}+R(H(u_{1},w_{\ell+1})),$ (4.34)
where $G(v,v^{*})$ and $H(w,w^{*})$ are the notations from (4.26) and (4.29),
respectively. We define $T(u_{1},u_{2})=(T^{(1)}(u_{2}),T^{(2)}(u_{1}))$ by
$\displaystyle T^{(1)}(u_{2})=w_{\ell+1}+R(G(u_{2},v_{\ell})),\quad
T^{(2)}(u_{1})=v_{\ell+1}+R(H(u_{1},w_{\ell+1})).$ (4.35)
For $\varepsilon>0$ we introduce a metric space
$\displaystyle D_{\varepsilon}=\\{(u_{1},u_{2})\in X^{2}\times
X^{2}\,;\,d((u_{1},u_{2}),(w_{\ell+1},v_{\ell+1}))\leq\varepsilon^{(p-1)b_{\ell}}\\},$
(4.36)
where $b_{\ell}=q+\ell(p+q-2)$ and we have set
$d((u_{1},u_{2}),(u_{1}^{*},u_{2}^{*}))=d_{1}(u_{1},u_{1}^{*})+d_{2}(u_{2},u_{2}^{*})$
with
$\displaystyle
d_{1}(u_{1},u_{1}^{*})=\|u_{1}-u_{1}^{*}\|_{Z^{2}(\kappa_{1}a_{\ell+1})}+\|u_{1}-u_{1}^{*}\|_{Z^{1}(\kappa_{1}a_{\ell+1})}^{p-1},$
$\displaystyle
d_{2}(u_{2},u_{2}^{*})=\|u_{2}-u_{2}^{*}\|_{Z^{2}(a_{\ell+1})}+\|u_{2}-u_{2}^{*}\|_{Z^{1}(a_{\ell+1})}^{p-1}.$
We shall show that $T$ is a contraction on $D_{\varepsilon}$, provided
$\varepsilon$ is small enough.
First of all, we prepare the following.
###### Lemma 4.5
Let $(u_{1},u_{2})\in D_{\varepsilon}$ with $0<\varepsilon\leq 1$. Then we
have
$\displaystyle\|u_{1}\|_{Z^{2}(\kappa_{1})}+\|u_{2}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon$ (4.37)
and
$\displaystyle d_{1}(u_{1},w_{\ell+1})\leq\varepsilon^{(p-1)b_{\ell}},\quad
d_{2}(u_{2},v_{\ell})\leq C\varepsilon^{(p-1)b_{\ell}}.$ (4.38)
Proof. First we prove (4.37). Notice that $a_{\ell+1}\geq\kappa_{2}>1$ and
$(p-1)b_{\ell}\geq(p-1)q=\kappa_{2}+1$. Then (4.37) follows from (4.18) and
(4.19) with $j=\ell+1$.
Next we prove (4.38). The first inequality is apparent. On the other hand, in
order to get the second one, it suffices to show
$d_{2}(v_{\ell+1},v_{\ell})\leq C\varepsilon^{(p-1)b_{\ell}}$. When $\ell=0$,
it follows from (4.21) that
$d_{2}(v_{1},v_{0})\leq C(\varepsilon^{q}+\varepsilon^{(p-1)q})\leq
C\varepsilon^{(p-1)q}$
for $0<\varepsilon\leq 1$. While, when $\ell\geq 1$, it follows from (4.23)
and (4.25) with $j=\ell$ that
$d_{2}(v_{\ell+1},v_{\ell})\leq
C(\varepsilon^{B_{\ell-1}+q-1}+\varepsilon^{(p-1)b_{\ell}}).$
Since
$\displaystyle B_{\ell-1}+q-1=(p-1)b_{\ell}+(2-p)(p+q-1)>(p-1)b_{\ell}$
for $\ell\geq 1$, we obatin the needed estimate. This completes the proof.
$\square$
The following estimate will play a basic role in proving that $T$ is a
contaction on $D_{\varepsilon}$.
###### Lemma 4.6
Let $u_{2}$, $u_{2}^{*}\in X^{2}(\kappa_{2})$ satisfy $u_{2}-u_{2}^{*}\in
Z^{2}(a_{\ell+1})$ and
$\displaystyle\|u_{2}\|_{X^{2}(\kappa_{2})}+\|u_{2}^{*}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon.$ (4.39)
Then we have
$\displaystyle\|R(G(u_{2},u_{2}^{*}))\|_{Z^{1}(\kappa_{1}a_{\ell+1})}\leq
C\varepsilon^{p-1}\|u_{2}-u_{2}^{*}\|_{Z^{1}(a_{\ell+1})}$ (4.40)
$\displaystyle\|R(G(u_{2},u_{2}^{*}))\|_{Z^{2}(\kappa_{1}a_{\ell+1})}\leq
C\varepsilon^{p-1}d_{2}(u_{2},u_{2}^{*}).$ (4.41)
While, let $u_{1}$, $u_{1}^{*}\in Z^{2}(\kappa_{1})$ satisfy
$u_{1}-u_{1}^{*}\in Z^{2}(\kappa_{1}a_{\ell+1})$ and
$\displaystyle\|u_{1}\|_{Z^{2}(\kappa_{1})}+\|u_{1}^{*}\|_{Z^{2}(\kappa_{1})}\leq
C\varepsilon.$ (4.42)
Then we have
$\displaystyle\|R(H(u_{1},u_{1}^{*}))\|_{Z^{1}(a_{\ell+2})}\leq
C\varepsilon^{q-1}\|u_{1}-u_{1}^{*}\|_{Z^{1}(\kappa_{1}a_{\ell+1})},$ (4.43)
$\displaystyle\|R(H(u_{1},u_{1}^{*}))\|_{Z^{2}(a_{\ell+2})}\leq
C\varepsilon^{q-1}\|u_{1}-u_{1}^{*}\|_{Z^{2}(\kappa_{1}a_{\ell+1})}.$ (4.44)
Proof. First we prove (4.40). It follows from (4.27) with $j=\ell$, (4.28) and
(4.39) that
$\displaystyle M_{0}(G(u_{2},u_{2}^{*}))\leq
C\varepsilon^{p-1}\|u_{2}-u_{2}^{*}\|_{Z^{1}(a_{\ell+1})}$
for $\alpha=1$, $\beta=\kappa_{1}$, $\gamma=\kappa_{1}a_{\ell+1}-\kappa_{1}$,
and $\delta=\kappa_{1}+\kappa_{2}$. Applying (3.11) with $s=1$ to
$G(u_{2},u_{2}^{*})$, we get (4.40), because
$\displaystyle\alpha+\beta+\gamma-1=\kappa_{1}a_{\ell+1}>1,$ (4.45)
by virtue of (2.29).
Next we prove (4.41). It follows from (4.31) with $j=\ell$, (4.39) that
$\displaystyle M_{1}(G(u_{2},u_{2}^{*}))\leq
C\varepsilon^{p-1}d_{2}(u_{2},u_{2}^{*})$
for $\alpha=p-1$, $\beta=1$, $\gamma=\kappa_{1}a_{\ell+1}-\kappa_{1}$, and
$\delta=\kappa_{1}+\kappa_{2}$. Since (4.45) holds for these $\alpha$, $\beta$
and $\gamma$, we obtain (4.41) by (3.11) with $s=2$.
Next we prove (4.43). It follows from (4.30) with $j=\ell+1$ and (4.42) that
$\displaystyle M_{0}(H(u_{1},u_{1}^{*}))\leq
C\varepsilon^{q-1}\|u_{1}-u_{1}^{*}\|_{Z^{1}(\kappa_{1}a_{\ell+1})}$
for $\alpha=1$, $\beta=q-1$, $\gamma=a_{\ell+2}+1-q$, and $\delta=q$. Since
$\alpha+\beta+\gamma-1=a_{\ell+2}>1$, by (3.11) with $s=1$, we get (4.43).
Finally we prove (4.44). It follows from (4.33) with $j=\ell+1$ and (4.42)
that
$\displaystyle M_{1}(H(u_{1},u_{1}^{*}))\leq
C\varepsilon^{q-1}\|u_{1}-u_{1}^{*}\|_{Z^{2}(\kappa_{1}a_{\ell+1})}$
for $\alpha=0$, $\beta=\delta=q$ and $\gamma=a_{\ell+2}+1-q$. Since
$\alpha+\beta+\gamma-1=a_{\ell+2}>1$, (3.11) with $s=2$ yields (4.44). The
proof is complete. $\square$
###### Corollary 4.7
Let $(u_{1},u_{2})\in D_{\varepsilon}$ with $0<\varepsilon\leq 1$. Then we
have
$\displaystyle d_{1}(T^{(1)}(u_{2}),w_{\ell+1})\leq
C\varepsilon^{(p-1)b_{\ell}+(p-1)^{2}},$ (4.46) $\displaystyle
d_{2}(T^{(2)}(u_{1}),v_{\ell+1})\leq C\varepsilon^{(p-1)b_{\ell}+(p-1)(q-1)}.$
(4.47)
Moreover, we have
$\displaystyle d_{1}(T^{(1)}(u_{2}),T^{(1)}(u_{2}^{*}))\leq
C\varepsilon^{(p-1)^{2}}d_{2}(u_{2},u_{2}^{*}),$ (4.48) $\displaystyle
d_{2}(T^{(2)}(u_{1}),T^{(2)}(u_{1}^{*}))\leq
C\varepsilon^{(p-1)(q-1)}d_{1}(u_{1},u_{1}^{*})$ (4.49)
for $(u_{1},u_{2})$, $(u_{1}^{*},u_{2}^{*})\in D_{\varepsilon}$ with
$0<\varepsilon\leq 1$.
Proof. It follows from Lemma 4.5 that if $(u_{1},u_{2})\in D_{\varepsilon}$
and $0<\varepsilon\leq 1$, then we have (4.37) and
$\displaystyle\|u_{1}-w_{\ell+1}\|_{Z^{2}(\kappa_{1}a_{\ell+1})}+\|u_{2}-v_{\ell}\|_{Z^{2}(a_{\ell+1})}\leq
C\varepsilon^{(p-1)b_{\ell}},$ (4.50)
$\displaystyle\|u_{1}-w_{\ell+1}\|_{Z^{1}(\kappa_{1}a_{\ell+1})}+\|u_{2}-v_{\ell}\|_{Z^{1}(a_{\ell+1})}\leq
C\varepsilon^{b_{\ell}}.$ (4.51)
We start with the proof of (4.46). By (4.35) we have
$T^{(1)}(u_{2})-w_{\ell+1}=R(G(u_{2},v_{\ell}))$. Therefore, applying the
preceding lemma, we get (4.46). Similarly, since
$T^{(2)}(u_{1})-v_{\ell+1}=R(H(u_{1},w_{\ell+1}))$, we obtain (4.47).
Next we prove (4.48). By (4.35) we have
$T^{(1)}(u_{2})-T^{(1)}(u_{2}^{*})=R(G(u_{2},u_{2}^{*}))$. Since
$d_{2}(u_{2},u_{2}^{*})\leq 2\varepsilon^{(p-1)b_{\ell}}$ for $(u_{1},u_{2})$,
$(u_{1}^{*},u_{2}^{*})\in D_{\varepsilon}$, the preceding lemma shows (4.48).
Similarly, we obtain (4.49). This completes the proof. $\square$
End of the proof of Theorem 2.5. We see from Corollary 4.7 that there exists a
positive number $\varepsilon_{0}$ depending only on $p$ and $q$ such that if
$0<\varepsilon\leq\varepsilon_{0}$, then we have $T(u_{1},u_{2})\in
D_{\varepsilon}$ and
$d(T(u_{1},u_{2}),T(u_{1}^{*},u_{2}^{*}))\leq
2^{-1}d((u_{1},u_{2}),(u_{1}^{*},u_{2}^{*}))$
for $(u_{1},u_{2})$, $(u_{1}^{*},u_{2}^{*})\in D_{\varepsilon}$, namely, $T$
is a contaction on $D_{\varepsilon}$. Hence we find a unique solution
$(u_{1},u_{2})\in D_{\varepsilon}$ of (4.34). Here and in what follows, we
suppose that $0<\varepsilon\leq\varepsilon_{0}$ and $(u_{1},u_{2})$ is the
solution.
Next we prove (2.33). Since (4.19), (4.37) and (4.51) yield
$\displaystyle\|u_{2}\|_{X^{2}(\kappa_{2})}+\|v_{\ell}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon,\quad\|u_{2}-v_{\ell}\|_{Z^{1}(a_{\ell+1})}\leq
C\varepsilon^{b_{\ell}},$ (4.52)
applying (4.40), we obtain
$\displaystyle[R(G(u_{2},v_{\ell}))(r,t)]_{1}\leq
C\varepsilon^{p-1+b_{\ell}}(1+r+t)^{-(\kappa_{1}a_{\ell+1}-1)}(1+|r-t|)^{-1}.$
(4.53)
In view of (4.34), we have
$\displaystyle\|(u_{1}-w_{\ell+1})(t)\|_{E}\leq
C\varepsilon^{p-1+b_{\ell}}(1+t)^{-(\kappa_{1}a_{\ell+1}-1)}$ (4.54)
for $t\geq 0$. Similarly, by (4.18), (4.37), (4.51), and (4.43), we get
$\displaystyle\|(u_{2}-v_{\ell+1})(t)\|_{E}\leq
C\varepsilon^{q-1+b_{\ell}}(1+t)^{-(a_{\ell+2}-1)}$ (4.55)
for $t\geq 0$. Moreover, it follows from (4.21) and (4.23) that
$\displaystyle[(v_{\ell+1}-v_{0})(r,t)]_{1}\leq
C\varepsilon^{q}(1+r+t)^{-(\kappa_{2}-1)}(1+|r-t|)^{-1},$ (4.56)
so that
$\displaystyle\|(v_{\ell+1}-v_{0})(t)\|_{E}\leq
C\varepsilon^{q}(1+t)^{-(\kappa_{2}-1)}.$ (4.57)
Thus we obtain (2.33) from (4.54), (4.55) and (4.57).
Next we prove (2.34). Since
$(w_{\ell+1},\partial_{t}w_{\ell+1})(r,0)=\vec{f}_{1}(r)$, it suffices to
prove
$\displaystyle[R(G(u_{2},v_{\ell}))(r,t)]_{2}\leq
C\varepsilon^{B_{\ell}}(1+r+t)^{-(\kappa_{1}a_{\ell+1}-1)}(1+|r-t|)^{-1}.$
(4.58)
Using (4.31) and (4.52), we have
$M_{1}(G(u_{2},v_{\ell}))\leq
C(\varepsilon^{B_{\ell}}+\varepsilon^{p-1}\|u_{2}-v_{\ell}\|_{Z^{2}(a_{\ell+1})})$
for $\alpha=\kappa_{1}$, $\beta=1$, $\gamma=\kappa_{1}a_{\ell+1}-\kappa_{1}$,
and $\delta=\kappa_{1}+\kappa_{2}$. It follows from (4.25) and (4.32) that
$\varepsilon^{p-1}\|v_{\ell+1}-v_{\ell}\|_{Z^{2}(a_{\ell+1})}\leq
C\varepsilon^{B_{\ell}}$
for $0<\varepsilon\leq 1$. We see from (4.34), (4.35) and (4.47) that
$\varepsilon^{p-1}\|u_{2}-v_{\ell+1}\|_{Z^{2}(a_{\ell+1})}\leq
C\varepsilon^{(p-1)b_{\ell}+(p-1)q}.$
Since $(p-1)q=\kappa_{2}+1>2$, we therefore obtain
$M_{1}(G(u_{2},v_{\ell}))\leq C\varepsilon^{B_{\ell}}$
for $0<\varepsilon\leq 1$. Since (4.45) is satisfied for those $\alpha$,
$\beta$, $\gamma$, and $\delta$, applying (3.11) with $s=2$ we get (4.58).
Finally we show (2.26), which follows from
$\displaystyle\|u_{2}-v_{0}\|_{Z^{2}(\kappa_{2})}\leq C\varepsilon^{q}.$
(4.59)
Let $0<\varepsilon\leq 1$. By (4.33), (4.18) and (4.50) we have
$M_{1}(H(u_{1},w_{\ell+1}))\leq C\varepsilon^{q-1+(p-1)b_{\ell}}\leq
C\varepsilon^{q}$
for $\alpha=0$, $\beta=\delta=q$ and $\gamma=a_{\ell+2}+1-q$, since
$(p-1)b_{\ell}\geq(p-1)q>2$. Noting $\alpha+\beta+\gamma-1=a_{\ell+2}>1$ and
applying (3.11) with $s=2$, we get
$\displaystyle\|u_{2}-v_{\ell+1}\|_{Z^{2}(a_{\ell+2})}\leq C\varepsilon^{q},$
in view of (4.34). While, we see from (4.21) and (4.25) that
$\displaystyle\|v_{\ell+1}-v_{0}\|_{Z^{2}(\kappa_{2})}\leq C\varepsilon^{q},$
since $a_{j+1}\geq\kappa_{2}$ and $B_{j-1}\geq 1$ for $j\geq 1$. Therefore we
obtain (4.59). Thus we complete the proof of Theorem 2.5. $\square$
Remark. When $p=2$, Lemma 4.2 and Corollary 4.3 with $\ell=0$ remain valid, in
view of (2.19). We replace (4.36) by
$\displaystyle D_{\varepsilon}=\\{(u_{1},u_{2})\in X^{2}\times
X^{2}\,;\,\|u_{1}-w_{1}\|_{Z^{2}(\kappa_{2})}+\|u_{2}-v_{1}\|_{Z^{2}(\kappa_{2})}\leq\varepsilon^{q}\\}.$
Notice that we have
$\|R(G(u_{2},u_{2}^{*}))\|_{Z^{2}(\kappa_{2})}\leq
C\varepsilon\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})},$
instead of (4.41), because $M_{1}(G(u_{2},u_{2}^{*}))\leq
C\varepsilon\|u_{2}-u_{2}^{*}\|_{Z^{2}(\kappa_{2})}$ for $\alpha=0$,
$\beta=2$, $\gamma=\kappa_{2}-1$, and $\delta=1+\kappa_{2}$ (remark that
$\alpha+\beta+\gamma-1=\kappa_{2}>1$). Prodceeding as in the proof of Theorem
2.5, we find the desired conclusion stated after Theorem 2.4.
Next we prove Theorem 2.6. Similarly to the proofs of Corollary 4.3 and Lemma
4.4, one can establish the following lemma.
###### Lemma 4.8
Let $0<\varepsilon\leq 1$ and $\vec{\varphi}_{i}\in
Y_{\kappa_{i}}(\varepsilon)$ with $i=1,2$. Then $w_{j}^{*}\in
Z^{2}(\kappa_{1})$, $v_{j}^{*}\in X^{2}(\kappa_{2})$, and we have
$\displaystyle\|w_{j}^{*}\|_{Z^{2}(\kappa_{1})}+\|v_{j}^{*}\|_{X^{2}(\kappa_{2})}\leq
C\varepsilon$ (4.60)
for $0\leq j\leq\ell$. Moreover, $u_{1}-w_{j}^{*}\in Z^{2}(\kappa_{1}a_{j})$,
$u_{2}-v_{j}^{*}\in Z^{2}(a_{j+1})$, and we have
$\displaystyle\|u_{1}-w_{j}^{*}\|_{Z^{1}(\kappa_{1}a_{j})}\leq
C\varepsilon^{b_{j-1}+p-1},\quad\|u_{2}-v_{j}^{*}\|_{Z^{1}(a_{j+1})}\leq
C\varepsilon^{b_{j}},$ (4.61)
$\displaystyle\|u_{1}-w_{j}^{*}\|_{Z^{2}(\kappa_{1}a_{j})}\leq
C\varepsilon^{B_{j-1}},\quad\|u_{2}-v_{j}^{*}\|_{Z^{2}(a_{j+1})}\leq
C\varepsilon^{B_{j-1}+q-1}$ (4.62)
for $1\leq j\leq\ell$, together with
$\displaystyle\|u_{2}-v_{0}^{*}\|_{Z^{2}(\kappa_{2})}\leq C\varepsilon^{q}.$
(4.63)
Here $b_{k}$ and $B_{k}$ are defined in Lemma 4.4.
End of the proof of Theorem 2.6. Let $(u_{1},u_{2})\in
Z^{2}(\kappa_{1})\times{X^{2}(\kappa_{2})}$ be the unique solution of (2.12)
satisfying (2.38).
In order to prove (2.45), it suffices to show
$\displaystyle[R(G(u_{2},v_{\ell}^{*}))(r,t)]_{1}\leq
C\varepsilon^{p-1+b_{\ell}}(1+r+t)^{-(\kappa_{1}a_{\ell+1}-1)}(1+|r-t|)^{-1},$
in view of (2.41) and (2.44). The needed estimate can be deduced from (2.38),
(4.60), (4.61), and (4.63), similarly to (4.53).
Next we show (2.47). By (2.41) and (2.42), it is enough to prove
$\displaystyle[R(G(u_{2},v_{\ell}^{*}))(r,t)]_{2}\leq
C\varepsilon^{B_{\ell}}(1+r+t)^{-(\kappa_{1}a_{\ell+1}-1)}(1+|r-t|)^{-1}.$
Similarly to (4.58), we obtain the desired estimate from (2.38), (4.60),
(4.62), and (4.63).
Finally we show (2.46) and (2.17). Since it follows from (2.44) that
$u_{2}-K[\vec{f}_{2}]=u_{2}-v_{0}^{*}$, we see that (2.46) and (2.17) are
consequences of (4.63). This completes the proof of Theorem 2.6. $\square$
## Acknowledgement
The research of the first author was partially supported by Grant-in-Aid for
Science Research (20224013), JSPS.
## References
* [1] K. Deng, Blow-up of solutions of some nonlinear hyperbolic systems, Rocky Mountain J. Math. 29 (1999), 807–820.
* [2] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys. 151 (1993), 619–645.
* [3] N. Hayashi and T. Ozawa, Modified wave operators for the derivative nonlinear Schrödinger equation, Math. Ann. 298 (1994), 557–576.
* [4] N. Hayashi and P. I. Naumkin, Final state problem for Korteweg-de Vries type equations, J. Math. Phys. 47 (2006), 123501, 16 pp.
* [5] H. Kubo, K. Kubota and H. Sunagawa, Large time behavior of solutions to semilinear systems of wave equations, Math. Ann. 335 (2006), 435–478.
* [6] P.D. Lax and R.S. Phillips, “Scattering theory”, Academic Press, New York and London, 1967.
* [7] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys. 139 (1991), 479–493.
* [8] J. Wirth, Scattering and modified scattering for abstract wave equations with time-dependent dissipation, Adv. Differential Equations 12 (2007), 1115–1133.
|
arxiv-papers
| 2009-09-01T03:47:12 |
2024-09-04T02:49:04.970723
|
{
"license": "Public Domain",
"authors": "H. Kubo and K. Kubota",
"submitter": "Hideo Kubo",
"url": "https://arxiv.org/abs/0909.0080"
}
|
0909.0300
|
# Single-photon logic gates using minimal resources
Qing Lin qlin@mail.ustc.edu.cn College of Information Science and
Engineering, Huaqiao University (Xiamen), Xiamen 361021, China Bing He
bhe98@earthlink.net Institute for Quantum Information Science, University of
Calgary, Alberta T2N 1N4, Canada
###### Abstract
We present a simple architecture for deterministic quantum circuits operating
on single photon qubits. Few resources are necessary to implement two
elementary gates and can be recycled for computing with large numbers of
qubits. The deterministic realization of some key multi-qubit gates, such as
the Fredkin and Toffoli gate, is greatly simplified in this approach.
###### pacs:
03.67.Lx, 42.50.Ex
## I
Introduction
Quantum computing has attracted wide attention for its factoring power and
efficient simulation of quantum dynamics. Many efforts have been made in
building quantum computers with various physical systems, and optical qubits
are regarded as a prominent candidate for their robustness against
decoherence. An important theoretical breakthrough in the field was the Knill-
Laflamme-Milburn (KLM) protocol KLM , a circuit-based approach using single
photon sources, single photon detectors and linear optical elements. A two-
qubit gate could be realized in an asymptotically deterministic way, as the
number of photons forming an entangled state for teleportation in the protocol
grows to infinity KLM ; KD . It opens up the possibility of building any
quantum logic gate which can be decomposed into two-qubit and single-qubit
gates theoretically Nielsen . The prohibitively large overhead cost of a two-
qubit gate in the KLM protocol, however, necessitates various improvements.
Most progress follows in the direction of one way computation Raussendorf , an
approach imprinting circuits on a particular class of entangled states
(cluster states) through measurements. Though it is possible to create cluster
states with realistic optical methods P , the generation of such multiply
entangled states is still not efficient with available techniques, imposing a
bottleneck on the practical implementation. Beyond linear optics, a near-
deterministic CNOT gate based on weak nonlinearities Nemoto has been
proposed, and it suggests a way for deterministic quantum computation Munro .
In realistic quantum computation, however, it will still require considerable
resources to perform a gate involving more than two qubits, if one decomposes
a complicated quantum circuit into the basic CNOT and single-qubit gates.
An efficient quantum computation approach demanding fewer resources is
desirable. In this work, we propose an architecture for quantum logic gates
operating on qubits simply encoded as the linear combinations of two single
photon modes, e.g., $|0_{L}\rangle\equiv|H\rangle$ and
$|1_{L}\rangle\equiv|V\rangle$, where $H$ and $V$ are two polarization modes.
In this architecture a quantum logic gate can be deterministically realized
with a combination of two elementary gates. Only one ancilla photon and a few
coherent states, which can be recycled after implementing one elementary gate,
are necessary to compute with a large number of qubits. Because the qubits and
ancillas are in simple quantum states, the operation error of the logic gates
would be largely reduced.
Figure 1: (Color online) Schematic setup for Controlled-path gate. A PBS used
in the circuit transmits the mode $|H\rangle$ and reflects the mode
$|V\rangle$ of a single photon. Two qubus beams are coupled to the photonic
modes as indicated. The XPM phases on the qubus beams are $\theta$, and two
phase shifter $-\theta$ are applied to the qubus beams. The QND module in
dash-dotted line is used to perform number-resolving detection.
## II
C-path gate
The first ingredient in our architecture is the Controlled-path (C-path) gate
introduced in Lin . Here, as shown in Fig. 1, we propose a design with the
double cross-phase modulation (XPM) method in He to make it more efficient
and feasible. This gate performs the following operation on an initial two-
photon state $\left|\psi\right\rangle_{CT}$ ($C$ stands for the control and
$T$ the target):
$\displaystyle\left|\psi\right\rangle_{CT}=a\left|HH\right\rangle_{CT}+b\left|HV\right\rangle_{CT}+c\left|VH\right\rangle_{CT}+d\left|VV\right\rangle_{CT}$
$\displaystyle\rightarrow
a\left|HH\right\rangle_{C1}+b\left|HV\right\rangle_{C1}+c\left|VH\right\rangle_{C2}+d\left|VV\right\rangle_{C2}=\left|\phi\right\rangle,$
(1)
where the index $1$ and $2$ denotes two different paths, implementing the
control on the target qubit paths by the polarizations of the control qubit.
In Fig. 1, we first use a 50:50 beam splitter (BS) to divide the target photon
$T$ into two spatial modes 1 and 2. Then two quantum bus (qubus) beams
$\left|\alpha\right\rangle\left|\alpha\right\rangle$ are introduced, with the
first coupling to the target photon mode on path 1 and the
$\left|V\right\rangle$ mode of the control photon, while the second to the
target mode on path 2 and the $\left|H\right\rangle$ mode of the control
through Kerr media, respectively. Suppose the XPM phase shifts induced in the
processes are all $\theta$. After that, a $-\theta$ phase shifter is
respectively applied to two qubus beams. Finally, one more 50:50 BS implements
the transformation
$\left|\alpha_{1}\right\rangle\left|\alpha_{2}\right\rangle\rightarrow\left|\frac{\alpha_{1}-\alpha_{2}}{\sqrt{2}}\right\rangle\left|\frac{\alpha_{1}+\alpha_{2}}{\sqrt{2}}\right\rangle$,
realizing the state
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H\right\rangle_{C}\left(a\left|H\right\rangle_{2}+b\left|V\right\rangle_{2}\right)\left|-\beta\right\rangle+\left|V\right\rangle_{C}\left(c\left|H\right\rangle_{1}+d\left|V\right\rangle_{1}\right)\left|\beta\right\rangle\right)$
$\displaystyle\times\left|\sqrt{2}\alpha\cos\theta\right\rangle+\frac{1}{\sqrt{2}}\left|\phi\right\rangle\left|0\right\rangle\left|\sqrt{2}\alpha\right\rangle,$
(2)
where $\left|\beta\right\rangle=\left|i\sqrt{2}\alpha\sin\theta\right\rangle$.
Then, we could use the projections $\left|n\right\rangle\left\langle n\right|$
on the first qubus beam to get the proper output. If $n=0,$
$\left|\phi\right\rangle$ will be projected out; on the other hand, if $n\neq
0$, what is realized is
$e^{-in\frac{\pi}{2}}\left|H\right\rangle_{C}\left(a\left|H\right\rangle_{2}+b\left|V\right\rangle_{2}\right)+e^{in\frac{\pi}{2}}\left|V\right\rangle_{C}\left(c\left|H\right\rangle_{1}+d\left|V\right\rangle_{1}\right)$,
which can be transformed to $\left|\phi\right\rangle$ by the application of a
$\pm 1$ phase factor and the controlled switch of two paths following the
classically feed-forwarded measurement results. Since the weak nonlinearity
considered here is very small ($\theta\ll 1$), only a small portion of the
initial qubus beam $|\sqrt{2}\alpha\rangle$ is consumed by detection, the
unmeasured beam in the state $|\sqrt{2}\alpha\cos\theta\rangle$ will be used
in following elementary gates.
After a C-path gate, the target photon will be simultaneously in two different
spatial modes depending on the polarizations of the control photon. Therefore,
an operation conditioned on the control photon’s polarizations can be directly
performed on the spatial modes of the target. As we will demonstrate later,
this gate offers a way to realize the interaction between multiple photons
indirectly.
## III
Photon number-resolving detection
The projectors $\left|n\right\rangle\left\langle n\right|$ required in the
C-path gate could be well approximated by a transition edge sensor (TES), a
superconducting microbolometer that has demonstrated very high detection
efficiency (95% at $\lambda=1550$ nm) and high photon number resolution
detector . In practice, it is ideal to implement such number-resolving
detection with simple devices. Here, we apply the indirect measurement method
in He for the purpose. It is a quantum non-demolition (QND) module shown
inside the dash-dotted line in Fig. 1. The process in the QND module is as
follows:
$\left|\pm\beta\right\rangle\left|\gamma\right\rangle\left|\gamma\right\rangle\rightarrow
e^{-\left|\beta\right|^{2}/2}\overset{\infty}{\underset{n=0}{{\sum}}}\frac{\left(\pm\beta\right)^{n}}{\sqrt{n!}}\left|n\right\rangle\left|\gamma
e^{in\theta}\right\rangle\left|\gamma\right\rangle.$ (3)
With a 50:50 BS we will obtain a set of coherent-state components
$\left|\frac{\gamma
e^{in\theta}-\gamma}{\sqrt{2}}\right\rangle\left|\frac{\gamma
e^{in\theta}+\gamma}{\sqrt{2}}\right\rangle$ for $n=0,1,\cdots,\infty$. If the
amplitude $\left|\gamma\right|$ is large enough, the photon number Poisson
distributions of the states $\left|\frac{\gamma
e^{in\theta}-\gamma}{\sqrt{2}}\right\rangle$ will be separated with negligible
overlaps. If the dominant distribution for the component of $n=k$ is from
$n_{k}$ to $n_{k}^{{}^{\prime}}$, we could use a realistic detector described
by the following POVM elements to detect $\left|\frac{\gamma
e^{ik\theta}-\gamma}{\sqrt{2}}\right\rangle$:
$\displaystyle\Pi_{0}$
$\displaystyle=\overset{\infty}{\underset{n=0}{{\sum}}}\left(1-\eta\right)^{n}\left|n\right\rangle\left\langle
n\right|,$ $\displaystyle\Pi_{n_{k}}$
$\displaystyle=\overset{n_{k}^{{}^{\prime}}}{\underset{n=n_{k}}{{\sum}}}\left[1-\left(1-\eta\right)^{n}\right]\left|n\right\rangle\left\langle
n\right|,$ $\displaystyle\Pi_{E}$
$\displaystyle=I-\Pi_{0}-\sum_{k=1}^{\infty}\Pi_{n_{k}},$ (4)
where $\eta<1$ is the quantum efficiency of the detector. $\Pi_{0}$ here
corresponds to detecting no photon, $\Pi_{E}$ to the response to the
negligible overlaps, and $\Pi_{n_{k}}$ to the reaction to the $k$-th Poisson
curve, respectively. The operators $\Pi_{n_{k}}$ therefore select out the
components $\left|k\right\rangle$ in $\left|\pm\beta\right\rangle$ indirectly.
Physically, $\Pi_{n_{k}}$ are the different responses (by the induced currents
or voltages) of a number-non-resolving detector to $\left|\frac{\gamma
e^{ik\theta}-\gamma}{\sqrt{2}}\right\rangle$ of the different intensities.
Figure 2: (Color online) Schematic setup for Merging gate. The ancilla photon
in the state $|\pm\rangle$, the single photon modes on path 2 and 3 interact
with the qubus beams in the pattern indicated in Entangler. On the first
(second) qubus beam we use one XPM rotation $\theta$ to represent the coupling
to $|V\rangle$ ($|H\rangle$) mode on both path $2$ and $3$. A BS and four PBS±
divide the modes of the second photon to 4 paths, on which the QND modules in
Fig. 1 are to detect the photon. The operations $\sigma_{z}$ are performed
according to the feed-forwarded measurement results of the QNDs.
## IV
Merging gate
Since the C-Path operation splits the target photon into two different paths,
we also require a gate to merge these two paths back into a single path. To
perform the conversion deterministically, we introduce another elementary gate
called the Merging gate shown in Fig. 2, where an extra ancilla photon is
used. It implements the transformation
$\displaystyle\left|\psi\right\rangle$
$\displaystyle=a\left|HH\right\rangle_{12}+b\left|HV\right\rangle_{12}+c\left|VH\right\rangle_{13}+d\left|VV\right\rangle_{13}$
$\displaystyle\rightarrow
a\left|HH\right\rangle_{14}+b\left|HV\right\rangle_{14}+c\left|VH\right\rangle_{14}+d\left|VV\right\rangle_{14},$
(5)
i.e., the merging of the second photon modes on path 2 and 3 to path 4.
The ancilla photon is in the state
$\left|\pm\right\rangle=\frac{1}{\sqrt{2}}\left(\left|H\right\rangle\pm\left|V\right\rangle\right)$.
A total state $\left|\psi\right\rangle\left|+\right\rangle$, for example, is
first sent to Entangler in Fig. 2, where we let the photons interact with the
qubus beams (see the setup in the dashed line of Fig. 2). Similar to the
double XPM pattern in C-path gate, the total state
$\left|\psi\right\rangle\left|+\right\rangle$ will be transformed to
$\displaystyle\frac{1}{\sqrt{2}}a|H\rangle_{1}(|+\rangle_{2}+|-\rangle_{2})|H\rangle_{4}+\frac{1}{\sqrt{2}}b|H\rangle_{1}(|+\rangle_{2}-|-\rangle_{2})|V\rangle_{4}$
$\displaystyle+\frac{1}{\sqrt{2}}c|V\rangle_{1}(|+\rangle_{3}+|-\rangle_{3})|H\rangle_{4}+\frac{1}{\sqrt{2}}d|V\rangle_{1}(|+\rangle_{3}-|-\rangle_{3})|V\rangle_{4}$
(6)
with a bit flip $\sigma_{x}$ and a phase shifter $\pi$ conditioned on the
results of the number-resolving detection on a qubus beam (no action should be
taken if $n=0$). After the interference of the modes on path 2 and 3,
$|\pm\rangle_{2}\rightarrow\frac{1}{\sqrt{2}}(|\pm\rangle_{2}+|\pm\rangle_{3})$
and
$|\pm\rangle_{3}\rightarrow\frac{1}{\sqrt{2}}(|\pm\rangle_{2}-|\pm\rangle_{3})$,
through a 50:50 BS, two PBS± let the components $\left|+\right\rangle$ be
transmitted while having the components $\left|-\right\rangle$ reflected,
making the single photon run on four different paths numbered from $5$ to $8$.
We then use the QND modules, which are the same as that in Fig. 1, on each
path to determine where the single photon in the state $|\pm\rangle$ passes.
The QND detections therefore project out the outputs, and the projected out
photon on one of the paths can be used again in the next Merging gate.
## V
Two-qubit gates
Two-qubit gates such as CNOT, CZ and C-phase, which are included in the class
$|H\rangle\langle H|\otimes U_{1}+|V\rangle\langle V|\otimes U_{2}$, can be
simply constructed with these elementary gates. Any gate operation in this
form is performed by a C-path gate followed by the single qubit operations
$U_{1}$ and $U_{2}$ on the different paths of the target photon and then a
Merging gate. Compared with the qubus mediated CNOT gate in Nemoto , a CNOT
gate constructed with a pair of C-path and Merging gate uses the same amount
of resources—two elementary gates and one ancilla single photon—without
counting the QND modules for resolving the photon numbers in a qubus beam and
preserving the ancilla photon in detections. Since we apply the double XPM
method in He , the number of the conditional XPM phase rotations in each
elementary gate will be greater than that in Nemoto . In addition to recycling
the qubus beams, the advantage of the double XPM method is that a minus XPM
phase shift $-\theta$, which is impractical to realize Kok , can be avoided.
Moreover, unlike the scheme in Munro , there is no need for the displacement
operations on the qubus beams, which could be hard to implement if the
displacement amplitude is large Paris ; loss .
For an arbitrary two-qubit gate $U\in U\left(4\right)$, which is expressed as
$U=\left(A_{1}\otimes A_{2}\right)\cdot
N\left(\alpha,\beta,\gamma\right)\cdot\left(A_{3}\otimes A_{4}\right)$, where
$A_{i}\in U\left(2\right)$ and
$N\left(\alpha,\beta,\gamma\right)=\exp[i\left(\alpha\sigma_{x}\otimes\sigma_{x}+\beta\sigma_{y}\otimes\sigma_{y}+\gamma\sigma_{z}\otimes\sigma_{z}\right)]$
TG , we can diagonalize $N\left(\alpha,\beta,\gamma\right)$ to
$\displaystyle|H\rangle\langle H|\otimes
diag(e^{i\left(\alpha-\beta+\gamma\right)},e^{-i\left(\alpha-\beta-\gamma\right)})$
$\displaystyle+|V\rangle\langle V|\otimes
diag(e^{i\left(\alpha+\beta-\gamma\right)},e^{-i\left(\alpha+\beta+\gamma\right)})$
(7)
with so-called magic transformation $\mathcal{M}$ TD . The magic
transformation, which is equivalent to a CNOT and a few single qubit
operations TD , is also implementable with C-path and Merging gates.
## VI
Multi-qubit gates
It is straightforward to generalize to multiple qubit gates, as any multi-
qubit gate is decomposable to a product of two-qubit gates and single-qubit
gates Nielsen . In the framework of realizing quantum computation with the two
above-discussed elementary gates, however, the design of a multi-qubit gate
can be simplified much further. We illustrate the point with two typical
multi-qubit gates—the Fredkin gate and the Toffoli gate.
The schematic setup in Fig. 3 is a Fredkin gate which implements a swap
operation on two target photons controlled by the $\left|V\right\rangle$ of
the control photon. In other words, it performs the following transformation
of a triple photon state $\left|\Psi\right\rangle_{CT_{1}T_{2}}$:
$\displaystyle\left|\Psi\right\rangle_{CT_{1}T_{2}}$
$\displaystyle=A_{1}\left|HHH\right\rangle+A_{2}\left|HHV\right\rangle+A_{3}\left|HVH\right\rangle$
$\displaystyle+A_{4}\left|HVV\right\rangle+A_{5}\left|VHH\right\rangle+A_{6}\left|VHV\right\rangle$
$\displaystyle+A_{7}\left|VVH\right\rangle+A_{8}\left|VVV\right\rangle$
$\displaystyle\rightarrow
A_{6}\left|VVH\right\rangle+A_{7}\left|VHV\right\rangle+rest.,$ (8)
where $rest.$ denotes the unchanged terms. Here we use two C-path gates to map
the second photon to path 2 and 3, and the third photon to path 4 and 5:
$\displaystyle\left|H\right\rangle_{1}\left(A_{1}\left|HH\right\rangle+A_{2}\left|HV\right\rangle+A_{3}\left|VH\right\rangle+A_{4}\left|VV\right\rangle\right)_{24}$
$\displaystyle+\left|V\right\rangle_{1}\left(A_{5}\left|HH\right\rangle+A_{6}\left|HV\right\rangle+A_{7}\left|VH\right\rangle+A_{8}\left|VV\right\rangle\right)_{35}.$
(9)
A deterministic Fredkin gate can be therefore realized by exchanging the modes
on path 3 and 5, and using two Merging gates as the inverse operation of two
C-path gates. In the implementation of two Merging gates, only one ancilla
photon is necessary since it can be used again after QND detection. This
feature is especially useful when there are many Merging gates in computation.
Figure 3: (Color online) Schematic setup for the Fredkin gate. Two C-path and
Merging gates, together with the exchange of two path modes, are used to
realize a Fredkin gate directly. Figure 4: (Color online) Schematic setup for
Toffoli gate. The H mode of photon $C_{1}$ and the modes on path 3 after the
first C-path gate control the target photon in the second C-path gate. A bit
flip is performed on the target mode on path 5. The couplings of the qubus
beams with the relevant photonic modes in C-path-2 are illustrated in dashed
line. On the second qubus beam of the C-path-2 gate, we use one XPM rotation
$\theta$ to represent the couplings to both $|H\rangle$ mode of path 1 and 3.
The Toffoli gate illustrated in Fig. 4 is the bit flip of a target photon
conditioned on both $\left|V\right\rangle$ of two control photons, i.e., a
triple-qubit operation $(I\otimes I-|VV\rangle\langle VV|)\otimes
I+|VV\rangle\langle VV|\otimes\sigma_{x}$, where $I=|H\rangle\langle
H|+|V\rangle\langle V|$. To implement the gate, we start with a C-path gate
for the initial state $|\Psi\rangle_{C_{1}C_{2}T}$ (in the same form as
$|\Psi\rangle_{C_{1}T_{1}T_{2}}$ in Eq. (8)) to send the second photon $C_{2}$
to two different paths $2$ and $3$ under the control of the polarizations of
the first photon $C_{1}$:
$\displaystyle|\Psi\rangle_{C_{1}T_{1}T_{2}}$
$\displaystyle\rightarrow(A_{1}\left|HHH\right\rangle+A_{2}\left|HHV\right\rangle+A_{3}\left|HVH\right\rangle$
$\displaystyle+A_{4}\left|HVV\right\rangle)_{12T}+(A_{5}\left|VHH\right\rangle+A_{6}\left|VHV\right\rangle$
$\displaystyle+A_{7}\left|VVH\right\rangle+A_{8}\left|VVV\right\rangle)_{13T}=|\Psi_{1}\rangle.$
(10)
Meanwhile, as shown in dashed line of Fig. 4, a 50:50 BS divides the target
photon $T$ into two paths $4$ and $5$. Two qubus beams
$\left|\alpha\right\rangle\left|\alpha\right\rangle$ will be applied to
perform the second C-path gate with the control of the modes $|H\rangle_{3}$
and $|V\rangle_{3}$ on path $3$. There is a slight difference in this C-path
gate (denoted as C-path-2 in Fig. 4) from a standard one in Fig. 1—the second
beam is coupled not only to the mode on path $5$ and the
$\left|H\right\rangle$ mode on path $3$ but also to $|H\rangle_{1}$ of the
first control photon, while the first beam interacts with the target mode on
path $4$ and the $\left|V\right\rangle$ control mode on path $3$, as indicated
in the following transformation:
$\displaystyle|\Psi_{1}\rangle|\alpha\rangle|\alpha\rangle$
$\displaystyle\rightarrow\frac{1}{\sqrt{2}}|HH\rangle_{12}\\{(A_{1}|H\rangle_{4}+|A_{2}|V\rangle_{4})|\alpha
e^{i\theta},\alpha
e^{i\theta}\rangle+(A_{1}|H\rangle_{5}+|A_{2}|V\rangle_{5})|\alpha,\alpha
e^{i2\theta}\rangle\\}$
$\displaystyle+\frac{1}{\sqrt{2}}|HV\rangle_{12}\\{(A_{3}|H\rangle_{4}+|A_{4}|V\rangle_{4})|\alpha
e^{i\theta},\alpha
e^{i\theta}\rangle+(A_{3}|H\rangle_{5}+|A_{4}|V\rangle_{5})|\alpha,\alpha
e^{i2\theta}\rangle\\}$
$\displaystyle+\frac{1}{\sqrt{2}}|VH\rangle_{13}\\{(A_{5}|H\rangle_{4}+|A_{6}|V\rangle_{4})|\alpha
e^{i\theta},\alpha
e^{i\theta}\rangle+(A_{5}|H\rangle_{5}+|A_{6}|V\rangle_{5})|\alpha,\alpha
e^{i2\theta}\rangle\\}$
$\displaystyle+\frac{1}{\sqrt{2}}|VV\rangle_{13}\\{(A_{7}|H\rangle_{4}+|A_{8}|V\rangle_{4})|\alpha
e^{i2\theta},\alpha\rangle+(A_{7}|H\rangle_{5}+|A_{8}|V\rangle_{5})|\alpha
e^{i\theta},\alpha e^{i\theta}\rangle\\}.$ (11)
The remaining operations on two qubus beams are the same as those in a
standard C-path gate—two phase shifts $-\theta$ and a 50:50 BS. According to
the number-resolving detection results on one qubus beam, an output state,
$\displaystyle\left|H\right\rangle_{1}\left(A_{1}\left|HH\right\rangle+A_{2}\left|HV\right\rangle+A_{3}\left|VH\right\rangle+A_{4}\left|VV\right\rangle\right)_{24}$
$\displaystyle+\left|VH\right\rangle_{13}\left(A_{5}\left|H\right\rangle+A_{6}\left|V\right\rangle\right)_{4}+\left|VV\right\rangle_{13}\left(A_{7}\left|H\right\rangle+A_{8}\left|V\right\rangle\right)_{5},$
(12)
will be deterministically projected out. After that, a bit flip $\sigma_{x}$
is performed on path $5$ alone. The total operation for a deterministic
Toffoli gate will be then completed with two Merging gates for the modes on
path $4$ and $5$, and on path $2$ and $3$, respectively.
Figure 5: (Color online) Schematic setup for a triple-control Toffoli gate,
with the $|V\rangle$ modes of three photons from $C_{1}$ to $C_{3}$
controlling the bit flip $\sigma_{x}$ on the target photon $T$. As a simple
generalization of the design in Fig. 4, it applies another modified C-path
gate—C-path-3—illustrated in dashed line. On the second qubus beam of the
C-path-3 gate, we use just one XPM rotation $\theta$ to represent the
couplings to $|H\rangle$ mode of path 1, 3 and 5, respectively. Only one
ancilla photon will be necessary for the three Merging gates if we apply the
detections with QND modules.
This design can be generalized to the situation of more than three qubits,
where we could simply adopt the similar coupling patterns for the photonic
modes in the successive C-path gates. In Fig. 5 we outline a triple-control
Toffoli gate of such type, which implements the bit flip of a target photon
under the $|V\rangle$ modes of three control photons together. The simplicity
of this approach stands out as compared with the conventional method of
decomposing a quantum circuit into double-qubit and single-qubit gates. By the
conventional method, there should be at least five two-qubit gates for the
Fredkin and Toffoli gates SD . Here we deterministically realize them with
only two pairs of C-path and Merging gates, which are equivalent to two
double-qubit gates. Generally, there should be $O(n^{2})$ two-qubit gates to
simulate a multi-control gate of $n$ qubits in the decomposition approach B .
As shown in Fig. 4 and 5, however, we will only need a number of the
elementary gates, which grows linearly with the number of the involved qubits,
to realize a multi-control gate. The decomposition of a multi-qubit circuit
into two-qubit and single-qubit gates is also theoretically complicated. But
in our approach the construction of a multi-control gate follows a regular way
as from Fig. 4 to Fig. 5.
## VII
Experimental feasibility
Here we take a brief look at the feasibility of this quantum computation
approach. A core technique for realizing the elementary gates is the XPM in
Kerr media. Good candidates for weak cross-Kerr nonlinearity without self-
phase modulation effects are atomic systems working under electromagnetically
induced transparency (EIT) conditions EIT . With light-storage technique, for
example, it is possible to realize a considerable XPM phase shift at the
single photon level chen-yu . In principle, however, we only need a small XPM
phase shift which can be compensated by the large amplitude of the qubus beams
as in Nemoto ; Munro . The error probability of a detection in the QND modules
is
$\displaystyle||\sum_{n=0}^{\infty}e^{-\left|\beta\right|^{2}/2}\frac{\left(\pm\beta\right)^{n}}{\sqrt{n!}}\left|n\right\rangle(\Pi_{0}^{\frac{1}{2}}|\frac{\gamma
e^{in\theta}-\gamma}{\sqrt{2}}\rangle)||^{2}$ $\displaystyle\sim
exp\\{-2(1-e^{-\frac{1}{2}\eta\gamma^{2}\theta^{2}})\alpha^{2}\sin^{2}\theta\\},$
(13)
rendering a near-deterministic performance given
$\left|\alpha\right|\sin\theta\gg 1$. Moreover, there is no XPM phase shift
$-\theta$ and no displacement operation on the qubus beams. The design is
robust against small losses of the photonic modes in the double XPM processes
loss , and is workable with the realistic number-non-resolving detectors as we
apply the indirect detection of photon numbers.
## VIII
Summary
The architecture of a quantum computer based on two elementary
gates—Controlled-path gate and Merging gate—is relatively simple compared with
those of all other approaches. The data to be processed is directly encoded in
single photon modes, and the ancilla photon and communication beams are also
in simple quantum states so that the possibility for operational errors could
be minimized. The recyclable ancillas keep the resources required in multi-
qubit computing minimal. Such quantum computer may come into being with the
development of the techniques of cross-Kerr nonlinearity and quantum memory
for single photon qubits.
###### Acknowledgements.
B. H. thanks C. F. Wildfeuer for helpful discussion on photon number resolving
detections, and the partial support from iCORE.
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|
arxiv-papers
| 2009-09-02T00:35:26 |
2024-09-04T02:49:04.981493
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qing Lin and Bing He",
"submitter": "Lin Qing",
"url": "https://arxiv.org/abs/0909.0300"
}
|
0909.0302
|
# Interacting agegraphic quintessence dark energy in non-flat universe
A. Sheykhia,b111sheykhi@mail.uk.ac.ir, A. Bagheria and M.M.
Yazdanpanaha222myazdan@mail.uk.ac.ir aDepartment of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
bResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
###### Abstract
We suggest a correspondence between interacting agegraphic dark energy models
and the quintessence scalar field in a non-flat universe. We demonstrate that
the agegraphic evolution of the universe can be described completely by a
single quintessence scalar field. Then, we reconstruct the potential of the
interacting agegraphic quintessence dark energy as well as the dynamics of the
scalar field according to the evolution of the agegraphic dark energy.
## I Introduction
It is a general belief that our universe is currently experiencing a phase of
accelerated expansion Rie . Missing energy component with negative pressure
which is responsible for this expansion constitute a major puzzle of modern
cosmology. Despite the theoretical difficulties in understanding dark energy,
independent observational evidence for its existence is impressively robust.
Among the various candidates to explain the accelerated expansion, the
cosmological constant with $w=-1$ is located at a central position. Though, it
suffers the so-called fine-tuning and cosmic coincidence problems. In
quintessence QUINT and Chaplygin gas KMP $w$ always stays bigger than $-1$.
The phantom models of dark energy have $w<-1$ PHANT . However, following the
more accurate data analysis, a more dramatic result appears showing that the
time varying dark energy gives a better fit than a cosmological constant and
in particular, $w$ can cross $-1$ from above to below Alam .
An interesting attempt for probing the nature of dark energy within the
framework of quantum gravity is the holographic dark energy. This proposal,
that arose a lot of enthusiasm recently Coh ; Li ; Huang ; Hsu ; HDE ; Setare1
, is motivated from the holographic hypothesis Suss1 and has been tested and
constrained by various astronomical observations Xin . However there are some
difficulties in holographic dark energy model. Choosing the event horizon of
the universe as the length scale, the holographic dark energy gives the
observation value of dark energy in the universe and can drive the universe to
an accelerated expansion phase. But an obvious drawback concerning causality
appears in this proposal. Event horizon is a global concept of spacetime;
existence of event horizon of the universe depends on future evolution of the
universe; and event horizon exists only for universe with forever accelerated
expansion. In addition, more recently, it has been argued that this proposal
might be in contradiction to the age of some old high redshift objects, unless
a lower Hubble parameter is considered Wei0 . Another proposal to probe the
nature of dark energy within the framework of quantum gravity is a so-called
agegraphic dark energy (ADE). This model is based on the uncertainty relation
of quantum mechanics together with the gravitational effect in general
relativity. Following the line of quantum fluctuations of spacetime,
Karolyhazy et al. Kar1 argued that the distance $t$ in Minkowski spacetime
cannot be known to a better accuracy than $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$
where $\beta$ is a dimensionless constant of order unity. Based on Karolyhazy
relation, Maziashvili discussed that the energy density of metric fluctuations
of the Minkowski spacetime is given by Maz
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (1)
where $t_{p}$ is the reduced Planck time. We use the units $c=\hbar=k_{b}=1$
throughout this work. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$ and
$m_{p}$ are the reduced Planck length and mass, respectively. The agegraphic
dark energy model assumes that the observed dark energy comes from the
spacetime and matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1 .
Since in agegraphic dark energy model the age of the universe is chosen as the
length measure, instead of the horizon distance, the causality problem in the
holographic dark energy is avoided. The agegraphic models of dark energy have
been examined and constrained by various astronomical observations age ; shey1
; Setare2 . Although going along a fundamental theory such as quantum gravity
may provide a hopeful way towards understanding the nature of dark energy, it
is hard to believe that the physical foundation of agegraphic dark energy is
convincing enough. Indeed, it is fair to say that almost all dynamical dark
energy models are settled at the phenomenological level, neither holographic
dark energy model nor agegraphic dark energy model is exception. Though, under
such circumstances, the models of holographic and agegraphic dark energy, to
some extent, still have some advantage comparing to other dynamical dark
energy models because at least they originate from some fundamental principles
in quantum gravity. We thus may as well view that this class of models
possesses some features of an underlying theory of dark energy.
On the other hand, the scalar field model is an effective description of an
underlying theory of dark energy. Scalar fields naturally arise in particle
physics including supersymmetric field theories and string/M theory.
Therefore, scalar field is expected to reveal the dynamical mechanism and the
nature of dark energy. However, although fundamental theories such as string/M
theory do provide a number of possible candidates for scalar fields, they do
not uniquely predict its potential $V(\phi)$. Therefore it becomes meaningful
to reconstruct $V(\phi)$ from some dark energy models possessing some
significant features of the quantum gravity theory, such as holographic and
agegraphic dark energy model. The investigations on the reconstruction of the
potential $V(\phi)$ in the framework of holographic dark energy have been
carried out in Zhang . In the absence of the interaction between agegraphic
dark energy and dark matter, the quintessence reconstruction of the agegraphic
dark energy models have been established ageQ .
In this paper we intend to generalize the study to the case where both
components- the pressureless dark matter and the agegraphic dark energy- do
not conserve separately but interact with each other. Given the unknown nature
of both dark matter and dark energy there is nothing in principle against
their mutual interaction and it seems very special that these two major
components in the universe are entirely independent Setare3 ; wang1 ; shey2 .
We shall establish a correspondence between the interacting agegraphic dark
energy scenarios and the quintessence scalar field. We suggest the agegraphic
description of the quintessence dark energy in a universe with spacial
curvature and reconstruct the potential and the dynamics of the quintessence
scalar field which describe the quintessence cosmology.
This paper is outlined as follows. In the next section we demonstrate a
correspondence between the original agegraphic and quintessence dark energy
model. In section III, we establish the correspondence between the new model
of interacting agegraphic dark energy and the quintessence dark energy. The
last section is devoted to conclusions.
## II Quintessence reconstruction of ORIGINAL ADE
We assume the agegraphic quintessence dark energy is accommodated in the
Friedmann-Robertson-Walker (FRW) universe which is described by the line
element
$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(2)
where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . The corresponding Friedmann equation takes
the form
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right).$ (3)
We introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (4)
thus, the Friedmann equation can be written
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (5)
We adopt the viewpoint that the scalar field models of dark energy are
effective theories of an underlying theory of dark energy. The energy density
and pressure for the quintessence scalar field can be written as
$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (6)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi),$ (7)
Then, we can easily obtain the scalar potential and the kinetic energy term as
$\displaystyle V(\phi)=\frac{1-w_{D}}{2}\rho_{\phi},$ (8)
$\displaystyle\dot{\phi}^{2}=(1+w_{D})\rho_{\phi}.$ (9)
Now we are focussing on the reconstruction of the original agegraphic
quintessence model of dark energy. The original agegraphic dark energy density
has the form (1) where $t$ is chosen to be the age of the universe
$T=\int_{0}^{a}{\frac{da}{Ha}},$ (10)
Thus, the energy density of the original agegraphic dark energy is given by
Cai1
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (11)
where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time (since the energy density is derived for Minkowski
space-time), and so on. The dark energy density (11) has the same form as the
holographic dark energy, but the length measure is chosen to be the age of the
universe instead of the horizon radius of the universe. Thus the causality
problem in the holographic dark energy is avoided. Combining Eqs. (4) and
(11), we get
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (12)
The total energy density is $\rho=\rho_{m}+\rho_{D}$, where $\rho_{m}$ and
$\rho_{D}$ are the energy density of dark matter and dark energy,
respectively. The total energy density satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0.$ (13)
However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (14)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (15)
Here $w_{D}$ is the equation of state parameter of agegraphic dark energy and
$Q$ denotes the interaction term and can be taken as $Q=3b^{2}H\rho$ with
$b^{2}$ being a coupling constant. This expression for the interaction term
was first introduced in the study of the suitable coupling between a
quintessence scalar field and a pressureless cold dark matter field Ame ; Zim
. In the context of holographic dark energy model, this form of interaction
was derived from the choice of Hubble scale as the IR cutoff Pav1 . Taking the
derivative with respect to the cosmic time of Eq. (11) and using Eq. (12) we
get
$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{n}\rho_{D}.$ (16)
Inserting this relation into Eq. (15), we obtain the equation of state
parameter of the original agegraphic dark energy
$\displaystyle
w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(17)
Figure 1: The evolution of $w_{D}$ for original ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01.$
Figure 2: The evolution of $\Omega_{D}$ for original ADE with different
interacting parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and
$\Omega_{k}=0.01$. Figure 3: The reconstruction of the potential $V(\phi)$ for
original ADE with different model parameter $n$, where $\phi$ is in unit of
$m_{p}$ and $V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28$ and
$\Omega_{k}=0.01$. Figure 4: The reconstruction of the potential $V(\phi)$
for original ADE with different interacting parameter $b^{2}$, where $\phi$ is
in unit of $m_{p}$ and $V(\phi)$ in $\rho_{c0}$. We take here
$\Omega_{m0}=0.28$ and $\Omega_{k}=0.01$ Figure 5: The revolution of the
scalar-field $\phi(a)$ for original ADE with different interacting parameter
$b^{2}$, where $\phi$ is in unit of $m_{p}$ and we take here
$\Omega_{m0}=0.28.$ Figure 6: The revolution of the scalar-field $\phi(a)$ for
original ADE with different model parameter $n$, where $\phi$ is in unit of
$m_{p}$ and we take here $\Omega_{m0}=0.28$.
Differentiating Eq. (12) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(18)
where the dot and the prime stand, respectively, for the derivative with
respect to the cosmic time and the derivative with respect to $x=\ln{a}$.
Taking the derivative of both side of the Friedman equation (3) with respect
to the cosmic time, and using Eqs. (5), (11), (12) and (14), it is easy to
show that
$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{n}-\frac{\Omega_{k}}{2}+\frac{3}{2}b^{2}(1+\Omega_{k}).$
(19)
Substituting this relation into Eq. (18), we obtain the equation of motion of
agegraphic dark energy
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(20)
We plot in Figs. 1 and 2 the evolutions of the $w_{D}$ and $\Omega_{D}$ of the
original ADE with different interacting parameter $b^{2}$. From Fig. 1 we see
that $w_{D}$ of the agegraphic dark energy can cross the phantom divide. It
was argued Wei2 that without interaction ($b^{2}=0$) $w_{D}$ is always larger
than $-1$ and cannot cross the phantom divide. In the presence of the
interaction the situation is changed. An interesting observation from Fig. 1
is that $w_{D}$ crosses the phantom divide from $w_{D}<-1$ to $w_{D}>-1$. This
makes it distinguishable from many other dark energy models whose $w_{D}$ can
cross the phantom divide. Fig. 2 shows that at the early time
$\Omega_{D}\rightarrow 0$ while at the late time $\Omega_{D}\rightarrow 1$,
that is the ADE dominates as expected.
Now we suggest a correspondence between the original agegraphic dark energy
and quintessence scalar field namely, we identify $\rho_{\phi}$ with
$\rho_{D}$. Using relation $\rho_{\phi}=\rho_{D}={3m_{p}^{2}H^{2}}\Omega_{D}$
and Eq. (17) we can rewrite the scalar potential and kinetic energy term as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{n}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(21) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(22)
Using relation $\dot{\phi}=H{\phi^{\prime}}$, we get
$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(23)
Consequently, we can easily obtain the evolutionary form of the field by
integrating the above equation
$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(24)
where $a_{0}$ is the present value of the scale factor, and $\Omega_{D}$ is
given by Eq. (20). Therefore, we have established an interacting agegraphic
quintessence dark energy model and reconstructed the potential of the
agegraphic quintessence as well as the dynamics of scalar field.
As one can see from the above equations, the analytical form of the potential
$V=V(\phi)$ is hard to be derived due to the complexity of the equations, but
we can plot the agegraphic quintessence potential versus $a$ numerically. For
simplicity we take $\Omega_{k}\simeq 0.01$ fixed in the numerical discussion.
Besides, $\rho_{c0}=3m_{p}^{2}H^{2}_{0}$ is the present value of the critical
energy density of the universe. The reconstructed quintessence potential
$V(\phi)$ and the evolutionary form of the field are plotted in Figs. 3-6,
where we have taken $\phi(a_{0}=1)=0$. Selected curves are plotted for the
different model parameter $n$ with fixed $b^{2}$ and different $b^{2}$ with
fixed $n$, and the present fractional matter density is chosen to be
$\Omega_{m0}=0.28$. From these figures we can see the dynamics of the
potential as well as the scalar field explicitly. They also show that the
reconstructed quintessence potential is steeper in the early epoch and becomes
very flat near today. Consequently, the scalar field $\phi$ rolls down the
potential with the kinetic energy $\dot{\phi}$ gradually decreasing.
## III Quintessence reconstruction of NEW ADE
Soon after the original agegraphic dark energy model was introduced by Cai
Cai1 , a new model of agegraphic dark energy was proposed in Wei2 , while the
time scale is chosen to be the conformal time $\eta$ instead of the age of the
universe. It is worth noting that the Karolyhazy relation $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ was derived for Minkowski spacetime
$ds^{2}=dt^{2}-d\mathrm{x^{2}}$ Kar1 ; Maz . In the case of the FRW universe,
we have $ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$.
Thus, it might be more reasonable to choose the time scale in Eq. (11) to be
the conformal time $\eta$ since it is the causal time in the Penrose diagram
of the FRW universe. The new agegraphic dark energy contains some new features
different from the original agegraphic dark energy and overcome some
unsatisfactory points. For instance, the original agegraphic dark energy
suffers from the difficulty to describe the matter-dominated epoch while the
new agegraphic dark energy resolved this issue Wei2 . The energy density of
the new agegraphic dark energy can be written
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (25)
where the conformal time $\eta$ is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (26)
Figure 7: The evolution of $w_{D}$ for new ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01.$
Figure 8: The evolution of $\Omega_{D}$ for new ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01$.
Figure 9: The reconstruction of the potential $V(\phi)$ for new ADE with
different model parameter $n$, where $\phi$ is in unit of $m_{p}$ and
$V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28.$ Figure 10: The
reconstruction of the potential $V(\phi)$ for new ADE with different
interacting parameter $b^{2}$, where $\phi$ is in unit of $m_{p}$ and
$V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28.$ Figure 11: The
revolution of the scalar-field $\phi(a)$ for new ADE with different
interacting parameter $b^{2}$, where $\phi$ is in unit of $m_{p}$ and we take
here $\Omega_{m0}=0.28.$ Figure 12: The revolution of the scalar-field
$\phi(a)$ for new ADE with different model parameter $n$, where $\phi$ is in
unit of $m_{p}$ and we take here $\Omega_{m0}=0.28$.
The fractional energy density of the new agegraphic dark energy is now given
by
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (27)
Taking the derivative with respect to the cosmic time of Eq. (25) and using
Eq. (27) we get
$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{na}\rho_{D}.$ (28)
Inserting this relation into Eq. (15) we obtain the equation of state
parameter of the new agegraphic dark energy
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(29)
The evolution behavior of the new agegraphic dark energy is now given by
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(30)
We plot in Figs. 7 and 8 the evolutions of $w_{D}$ and $\Omega_{D}$ of the new
ADE with different interacting parameter $b^{2}$. From Fig. 7 we see that
$w_{D}$ of the new ADE can have a transition from $w_{D}>-1$ to $w_{D}<-1$.
This is in contrast to the original ADE model. Fig. 8 indicates that at the
late time $\Omega_{D}\rightarrow 1$, which is similar to the behaviour of the
original ADE.
Next, we reconstruct the new agegraphic quintessence dark energy model,
connecting the quintessence scalar field with the new agegraphic dark energy.
Using Eqs. (27) and (29) one can easily show that the scalar potential and
kinetic energy term take the following form
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(31) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(32)
Using Eq. (32), Eq. (31) can be reexpressed as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
3m^{2}_{p}H^{2}\Omega_{D}\left(1-\frac{\dot{\phi}^{2}}{6m^{2}_{p}H^{2}\Omega_{D}}\right).$
(33)
We can also rewrite Eq. (32) as
$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(34)
Therefore the evolution behavior of the scalar field can be obtained by
integrating the above equation
$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(35)
where $\Omega_{D}$ is now given by Eq. (30). In this way we connect the
interacting agegraphic dark energy with a quintessence scalar field and
reconstruct the potential of the agegraphic quintessence. The reconstructed
quintessence potential $V(\phi)$ and the evolutionary form of the scalar field
are plotted in Figs. 9-12 for different model parameter $n$ with fixed
interacting parameter $b^{2}$, and different $b^{2}$ with fixed $n$. Here the
present fractional matter density is chosen to be $\Omega_{m0}=0.28$.
## IV Conclusions
In this paper, we have suggested a correspondence between the interacting
agegraphic dark energy scenarios and the quintessence scalar field model in a
non-flat universe. We have demonstrated that the agegraphic evolution of the
universe can be described completely by a single quintessence scalar field. We
have adopted the viewpoint that the scalar field models of dark energy are
effective theories of an underlying theory of dark energy. If we regard the
scalar field model as an effective description of such a theory, we should be
capable of using the scalar field model to mimic the evolving behavior of the
interacting agegraphic dark energy and reconstructing this scalar field model
according to the evolutionary behavior of agegraphic dark energy. We have also
reconstructed the potential of the agegraphic scalar field as well as the
dynamics of the quintessence scalar field which describe the quintessence
cosmology.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, Iran.
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|
arxiv-papers
| 2009-09-02T01:21:17 |
2024-09-04T02:49:04.986950
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, A. Bagheri and M.M. Yazdanpanah",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0909.0302"
}
|
0909.0361
|
# Poisson structures compatible with the cluster algebra structure in
Grassmannians
M. Gekhtman Department of Mathematics, University of Notre Dame, Notre Dame,
IN 46556 mgekhtma@nd.edu , M. Shapiro Department of Mathematics, Michigan
State University, East Lansing, MI 48823 mshapiro@math.msu.edu , A. Stolin
Department of Mathematics University of Goteborg 411 24 Goteborg Sweden
mshapiro@math.msu.edu and A. Vainshtein Department of Mathematics &
Department of Computer Science, University of Haifa, Haifa, Mount Carmel
31905, Israel alek@cs.haifa.ac.il
###### Abstract.
We describe all Poisson brackets compatible with the natural cluster algebra
structure in the open Schubert cell of the Grassmannian $G_{k}(n)$ and show
that any such bracket endows $G_{k}(n)$ with a structure of a Poisson
homogeneous space with respect to the natural action of $SL_{n}$ equipped with
an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the
simplest class in the Belavin-Drinfeld classification. Moreover, every
compatible Poisson structure can be obtained this way.
###### Key words and phrases:
Grassmannian, Poisson-Lie group, cluster algebra
###### 1991 Mathematics Subject Classification:
53D17, 14M15
## 1\. Introduction
The notion of a Poisson bracket compatible with a cluster algebra structure
was introduced in [GSV1]. It was used to interpret cluster transformations and
matrix mutations in cluster algebras from a viewpoint of Poisson geometry. In
addition, it was shown in [GSV1] that if a Poisson variety
$\left(\mathcal{M},{\\{\cdot,\cdot\\}}\right)$ possesses a coordinate chart
that consists of regular functions whose logarithms have pairwise constant
Poisson brackets, then one can use this chart to define a cluster algebra
${\mathcal{A}}_{\mathcal{M}}$, which is closely related (and, under rather
mild conditions, isomorphic) to the ring of regular functions on
$\mathcal{M}$, and such that ${\\{\cdot,\cdot\\}}$ is compatible with
${\mathcal{A}}_{\mathcal{M}}$. This construction was applied to an open cell
$G^{0}_{k}(n)$ in the Grassmannian $G_{k}(n)$ viewed as a Poisson homogeneous
space under the action of $SL_{n}$ equipped with the standard Poisson-Lie
structure. The resulting cluster algebra ${\mathcal{A}}_{G^{0}_{k}(n)}$ can be
viewed as a restriction of the cluster algebra structure in the coordinate
ring of $G_{k}(n)$. This “larger” cluster algebra ${\mathcal{A}}_{G_{k}(n)}$
was described in [S] using combinatorial properties of Postnikov’s map from
the space of edge weights of a planar directed network into the Grassmannian
[P]. Poisson geometric properties of Postnikov’s map are studied in [GSV2].
One of the by-products of this study is the existence of a pencil of Poisson
structures on $G_{k}(n)$ compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. It
turnes out that every bracket in the pencil defines a Poisson homogeneous
structure with respect to a Sklyanin Poisson-Lie bracket associated with a
solution of the modified classical Yang-Baxter equation (MCYBE) of the form
$R_{t}=R_{0}+tA\pi_{0}$, where $t$ is a scalar parameter, $A$ is a certain
fixed skew-symmetric $n\times n$ matrix, $R_{0}=\pi_{+}-\pi_{-}$, and
$\pi_{\pm,0}$ are projections onto strictly upper/strictly lower/diagonal part
in $sl_{n}$ (the standard Poisson-Lie structure corresponds to $t=0$).
According to the Belavin-Drinfeld classification [BD], skew-symmetric
solutions of MCYBE are defined by two types of data: discrete data associated
with the Dynkin diagram and called the Belavin-Drinfeld triple and continuous
data associated with the Cartan subalgebra. We will say that two R-matrices
belong to the same class if the corresponding Belavin-Drinfeld triples are the
same. R-matrices $R_{t}$ mentioned above belong to the class associated with
the trivial Belavin-Drinfeld triple. The entire class consists of R-matrices
of the form $R_{S}=R_{0}+S\pi_{0}$, with $S$ arbitrary skew-symmetric. On the
other hand, the Poisson pencil described above does not exhaust all Poisson
structures compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. The main goal of
this paper is to prove
###### Theorem 1.1.
The Poisson homogeneous structure with respect to the Poisson-Lie bracket
associated with any $R_{S}$ is compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$.
Moreover, up to a scalar multiple, all Poisson brackets compatible with
${\mathcal{A}}_{G^{0}_{k}(n)}$ are obtained this way.
It should be noted that Poisson brackets compatible with the “larger” cluster
algebra ${\mathcal{A}}_{G_{k}(n)}$ are naturally associated with Poisson
structures on the Grassmann cone $\mathcal{C}(G_{k}(n))$ that can be realized
as one-dimensional extensions of corresponding structures on $G_{k}(n)$. Both
the formulation and the proof of Theorem 1.1 can be modified in a rather
straightforward way to the case of the cluster algebra
${\mathcal{A}}_{G_{k}(n)}$. A detailed description of the relationship between
${\mathcal{A}}_{G_{k}(n)}$ and ${\mathcal{A}}_{G^{0}_{k}(n)}$ this
modification relies upon is presented in Chapter 4 of the forthcoming book
[GSV3].
The paper is organized as follows. In Section 2 we recall the necessary
information on cluster algebras and compatible Poisson structures and show how
the latter can be completely described via the use of a toric action. Section
3 provides the background on Poisson-Lie groups and Sklyanin brackets.
Finally, in Section 4 we review Poisson homogeneous structures on $G_{k}(n)$
and the construction of ${\mathcal{A}}_{G^{0}_{k}(n)}$, and then proceed to
prove Theorem 1.1, which is re-stated there as Theorem 4.3.
## 2\. Cluster algebras and compatible Poisson brackets
We start with the basics on cluster algebras of geometric type. The definition
that we present below is not the most general one, see, e.g., [FZ3, BFZ] for a
detailed exposition. In what follows, we will use notation $[i,j]$ for an
interval $\\{i,i+1,\ldots,j\\}$ in $\mathbb{N}$, and write $[n]$ instead of
$[1,n]$.
The coefficient group ${\mathfrak{P}}$ is a free multiplicative abelian group
of a finite rank $m$ with generators $g_{1},\dots,g_{m}$. An ambient field is
the field ${\mathfrak{F}}$ of rational functions in $n$ independent variables
with coefficients in the field of fractions of the integer group ring
${\mathbb{Z}}{\mathfrak{P}}={\mathbb{Z}}[g_{1}^{\pm 1},\dots,g_{m}^{\pm 1}]$
(here we write $x^{\pm 1}$ instead of $x,x^{-1}$).
A seed (of geometric type) in ${\mathfrak{F}}$ is a pair $\Sigma=({\bf
x},\widetilde{B})$, where ${\bf x}=(x_{1},\dots,x_{n})$ is a transcendence
basis of ${\mathfrak{F}}$ over the field of fractions of
${\mathbb{Z}}{\mathfrak{P}}$ and $\widetilde{B}$ is an $n\times(n+m)$ integer
matrix whose principal part $B$ (that is, the $n\times n$ submatrix formed by
the columns $1,\dots,n$) is skew-symmetrizable. In this paper, we will only
deal with the case when $B$ is skew-symmetric.
The $n$-tuple ${\bf x}$ is called a cluster, and its elements
$x_{1},\dots,x_{n}$ are called cluster variables. Denote $x_{n+i}=g_{i}$ for
$i\in[m]$. We say that $\widetilde{{\bf x}}=(x_{1},\dots,x_{n+m})$ is an
extended cluster, and $x_{n+1},\dots,x_{n+m}$ are stable variables. It is
convenient to think of ${\mathfrak{F}}$ as of the field of rational functions
in $n+m$ independent variables with rational coefficients.
Given a seed as above, the adjacent cluster in direction $k\in[n]$ is defined
by
${\bf x}_{k}=({\bf x}\setminus\\{x_{k}\\})\cup\\{x^{\prime}_{k}\\},$
where the new cluster variable $x^{\prime}_{k}$ is given by the exchange
relation
$x_{k}x^{\prime}_{k}=\prod_{\begin{subarray}{c}1\leq i\leq n+m\\\
b_{ki}>0\end{subarray}}x_{i}^{b_{ki}}+\prod_{\begin{subarray}{c}1\leq i\leq
n+m\\\ b_{ki}<0\end{subarray}}x_{i}^{-b_{ki}};$
here, as usual, the product over the empty set is assumed to be equal to $1$.
We say that ${\widetilde{B}}^{\prime}$ is obtained from ${\widetilde{B}}$ by a
matrix mutation in direction $k$ and write
${\widetilde{B}}^{\prime}=\mu_{k}({\widetilde{B}})$ if
$b^{\prime}_{ij}=\begin{cases}-b_{ij},&\text{if $i=k$ or $j=k$;}\\\
b_{ij}+\displaystyle\frac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2},&\text{otherwise.}\end{cases}$
Given a seed $\Sigma=({\bf x},\widetilde{B})$, we say that a seed
$\Sigma^{\prime}=({\bf x}^{\prime},\widetilde{B}^{\prime})$ is adjacent to
$\Sigma$ (in direction $k$) if ${\bf x}^{\prime}$ is adjacent to ${\bf x}$ in
direction $k$ and $\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})$. Two seeds
are mutation equivalent if they can be connected by a sequence of pairwise
adjacent seeds.
The cluster algebra (of geometric type)
${\mathcal{A}}={\mathcal{A}}(\widetilde{B})$ associated with $\Sigma$ is the
${\mathbb{Z}}{\mathfrak{P}}$-subalgebra of ${\mathfrak{F}}$ generated by all
cluster variables in all seeds mutation equivalent to $\Sigma$.
Let ${\\{\cdot,\cdot\\}}$ be a Poisson bracket on the ambient field
${\mathfrak{F}}$ considered as the field of rational functions in $n+m$
independent variables with rational coefficients. We say that it is compatible
with the cluster algebra ${\mathcal{A}}$ if, for any extended cluster
$\widetilde{{\bf x}}=(x_{1},\dots,x_{n+m})$, one has
$\\{x_{i},x_{j}\\}=\omega_{ij}x_{i}x_{j},$
where $\omega_{ij}\in{\mathbb{Z}}$ are constants for all $i,j\in[n+m]$. The
matrix $\Omega^{\widetilde{\bf x}}=(\omega_{ij})$ is called the coefficient
matrix of ${\\{\cdot,\cdot\\}}$ (in the basis $\widetilde{\bf x}$); clearly,
$\Omega^{\widetilde{\bf x}}$ is skew-symmetric.
In what follows, we denote by $A(I,J)$ the submatrix of a matrix $A$ with a
row set $I$ and a column set $J$. Consider, along with cluster and stable
variables ${\widetilde{\bf x}}$, another $(n+m)$-tuple of rational functions
denoted $\tau=(\tau_{1},\dots,\tau_{n+m})$ and defined by
(2.1) $\tau_{j}=x_{j}^{\varkappa_{j}}\prod_{k=1}^{n+m}x_{k}^{b_{jk}},$
where $\widehat{B}=(b_{jk})_{j,k=1}^{n+m}$ is a fixed skew-symmetric matrix
such that $\widehat{B}([n],[n+m])=\widetilde{B}$, $\varkappa_{j}$ is an
integer, $\varkappa_{j}=0$ for $1\leq j\leq n$. We say that the entries
$\tau_{i}$, $i\in[n+m]$, form a _$\tau$ -cluster_. It is proved in [GSV1],
Lemma 1.1, that $\varkappa_{j}$, $n+1\leq j\leq n+m$, can be selected in such
a way that the transformation ${\widetilde{\bf x}}\mapsto\tau$ is non-
degenerate, provided $\operatorname{rank}{\widetilde{B}}=n$. We denote
$\varkappa=\operatorname{diag}(\varkappa_{i})_{i=1}^{n+m}$ and
$B_{\varkappa}=\widehat{B}+\varkappa$. Nondegeneracy of the transformation
${\widetilde{\bf x}}\mapsto\tau$ is equivalent to nondegeneracy of
$B_{\varkappa}$.
Recall that a square matrix $A$ is reducible if there exists a permutation
matrix $P$ such that $PAP^{T}$ is a block-diagonal matrix, and irreducible
otherwise. The following result is a particular case of Theorem 1.4 in [GSV1].
###### Theorem 2.1.
Assume that $\operatorname{rank}{\widetilde{B}}=n$ and the principal part of
${\widetilde{B}}$ is irreducible. Then a Poisson bracket is compatible with
${\mathcal{A}}({\widetilde{B}})$ if and only if its coefficient matrix
$\Omega^{\tau}$ in the basis $\tau$ has the following property: the submatrix
$\Omega^{\tau}([n],[n+m])$ is proportional to ${\widetilde{B}}$.
Starting with an arbitrary ${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ compatible
with ${\mathcal{A}}$, one can suggest an alternative description of all other
compatible Poisson brackets via the following construction. Let $C=(c_{ij})$
be an integral $(n+m)\times m$ matrix. Define an action of
$({\mathbb{C}}^{*})^{m}=\\{\mathbf{d}=(d_{1},\ldots,d_{m})\ :\ d_{1}\cdots
d_{r}\neq 0\\}$ on ${\widetilde{\bf x}}$ by
(2.2) $\mathbf{d}.{\widetilde{\bf
x}}=\left(x_{i}\prod_{\alpha=1}^{m}d_{\alpha}^{c_{i\alpha}}\right)_{i=1}^{n+m}.$
We say that (2.2) extends to an action of $({\mathbb{C}}^{*})^{m}$ on
${\mathcal{A}}$ if the action induced by it in any cluster in ${\mathcal{A}}$
has a form (2.2) (with possibly different coefficients $c_{i\alpha}$). Lemma
2.3 in [GSV1] claims that (2.2) extends to an action of
$({\mathbb{C}}^{*})^{m}$ on ${\mathcal{A}}$ if and only if
${\widetilde{B}}C=0$. The same condition guarantees that
$\tau_{i}(\mathbf{d}.{\widetilde{\bf x}})=\tau_{i}({\widetilde{\bf x}})$ for
$i\in[n]$. Since $B_{\varkappa}$ is invertible, any such $C$ of full rank has
a form $C=B_{\varkappa}^{-1}([n+m],[n+1,n+m])U$, where $U$ is any invertible
$m\times m$ matrix.
Next, assume that $({\mathbb{C}}^{*})^{m}$ is equipped with a Poisson
structure given by
$\\{d_{i},d_{j}\\}_{V}=v_{ij}d_{i}d_{j},$
where $V=(v_{ij})$ is a fixed skew-symmetric matrix.
###### Proposition 2.2.
Let ${\widetilde{B}}$ satisfy the assumptions of Theorem 2.1. Then for any
skew-symmetric $m\times m$ matrix $V$, there exists a Poisson structure
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ compatible with ${\mathcal{A}}$ such
that the map
$\left(({\mathbb{C}}^{*})^{m}\times{\mathcal{A}},{\\{\cdot,\cdot\\}}_{V}\times{\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}\right)\to\left({\mathcal{A}},{\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}\right)$
extended from the action $(\mathbf{d},{\widetilde{\bf
x}})\mapsto\mathbf{d}.{\widetilde{\bf x}}$ is Poisson. Moreover, every
compatible Poisson bracket on ${\mathcal{A}}$ is a scalar multiple of
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ for some $V$.
###### Proof.
Let $\Omega^{{\widetilde{\bf x}}}$ be the coefficient matrix of
${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ in the basis ${\widetilde{\bf x}}$. It
is easy to see that in the product structure
${\\{\cdot,\cdot\\}}_{V}\times{\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ on
$({\mathbb{C}}^{*})^{m}\times{\mathcal{A}}$,
$\\{(\mathbf{d}.{\widetilde{\bf x}})_{i},(\mathbf{d}.{\widetilde{\bf
x}})_{j}\\}=\left(\Omega^{{\widetilde{\bf
x}}}+CVC^{T}\right)_{ij}(\mathbf{d}.{\widetilde{\bf
x}})_{i}(\mathbf{d}.{\widetilde{\bf x}})_{j}.$
Thus, for the action $(\mathbf{d},{\widetilde{\bf
x}})\mapsto\mathbf{d}.{\widetilde{\bf x}}$ to be Poisson, one must have
$\\{x_{i},x_{j}\\}^{\mathcal{A}}_{V}=\left(\Omega^{\tilde{\bf
x}}+CVC^{T}\right)_{ij}x_{i}x_{j}$ for $i,j\in[n+m]$. Since
$\tau_{i}(\mathbf{d}.{\widetilde{\bf x}})=\tau_{i}({\widetilde{\bf x}})$ if
$i\in[n]$, and $\tau_{i}(\mathbf{d}.{\widetilde{\bf
x}})=\tau_{i}({\widetilde{\bf x}})m_{i}(\mathbf{d})$ for some monomials
$m_{i}(\mathbf{d})$ in $\mathbf{d}$ for $i\in[n+1,n+m]$, we see that
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}=\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{0}$
for $i\in[n]$, $j\in[n+m]$. Since ${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ is a
compatible Poisson bracket, Theorem 2.1 yields that
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ is compatible as well.
Now, let $\Omega^{\tau}$ be the coefficient matrix of
${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ in the basis $\tau$. Denote
$Z_{0}=\Omega^{\tau}([n+1,n+m],[n+1,n+m])$. Consider
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}$ for $i,j\in[n+1,n+m]$:
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}=z_{ij}\tau_{i}\tau_{j}$. To compute
$Z=(z_{ij})_{i,j=n+1}^{n+m}$, note that the matrix that describes
${\\{\cdot,\cdot\\}}^{\mathcal{A}}_{V}$ in coordinates $\tau$ is
$\Omega_{V}^{\tau}=B_{\varkappa}\left(\Omega^{{\widetilde{\bf
x}}}+CVC^{T}\right)B_{\varkappa}^{T}$, and thus
$Z=\Omega_{V}^{\tau}([n+1,n+m],[n+1,n+m])=Z_{0}+UVU^{-1}.$
It is clear that by varying $V$, one can make $Z$ to be equal to an arbitrary
skew-symmmetric $m\times m$ matrix. Theorem 2.1 implies that up to a scalar
multiple, the matrix block $Z$ determines a compatible Poisson structure
uniquely, and the result follows. ∎
## 3\. Poisson-Lie groups and Sklyanin brackets
We need to recall some facts about Poisson-Lie groups (see, e.g.[ReST]).
Let ${\mathcal{G}}$ be a Lie group equipped with a Poisson bracket
${\\{\cdot,\cdot\\}}$. ${\mathcal{G}}$ is called a Poisson-Lie group if the
multiplication map
${\mathfrak{m}}:{\mathcal{G}}\times{\mathcal{G}}\ni(x,y)\mapsto
xy\in{\mathcal{G}}$
is Poisson. Perhaps, the most important class of Poisson-Lie groups is the one
associated with classical R-matrices.
Let $\mathfrak{g}$ be a Lie algebra of ${\mathcal{G}}$. Assume that
$\mathfrak{g}$ is equipped with a nondegenerate invariant bilinear form $(\ ,\
)$. An element $R\in\operatorname{End}(\mathfrak{g})$ is a classical R-matrix
if it is a skew-symmetric operator that satisfies the modified classical Yang-
Baxter equation (MCYBE)
(3.1) $[R(\xi),R(\eta)]-R\left([R(\xi),\eta]\ +\
[\xi,R(\eta)]\right)=-[\xi,\eta].$
Given a classical R-matrix $R$, ${\mathcal{G}}$ can be endowed with a Poisson-
Lie structure as follows. Let $\nabla f$, $\nabla^{\prime}f$ be the right and
the left gradients for a function $f\in C^{\infty}({\mathcal{G}})$:
(3.2) $(\nabla
f(x),\xi)=\frac{d}{dt}f(\exp{(t\xi)}x)|_{t=0},\qquad(\nabla^{\prime}f(x),\xi)=\frac{d}{dt}f(x\exp{(t\xi)})|_{t=0}.$
Then the bracket given by
(3.3)
$\\{f_{1},f_{2}\\}=\frac{1}{2}(R(\nabla^{\prime}f_{1}),\nabla^{\prime}f_{2})-\frac{1}{2}(R(\nabla
f_{1}),\nabla f_{2})\ $
is a Poisson-Lie bracket on ${\mathcal{G}}$ called the Sklyanin bracket.
We are interested in the case ${\mathcal{G}}=SL_{n}$ and $\mathfrak{g}=sl_{n}$
equipped with the trace-form
$(\xi,\eta)=\operatorname{Tr}(\xi\eta).$
In this case, the right and left gradients (3.2) are
$\nabla f(x)=x\ \mbox{grad}f(x),\qquad\nabla^{\prime}f(x)=\mbox{grad}f(x)\ x,$
where
$\mbox{grad}f(x)=\left(\frac{\partial f}{\partial x_{ji}}\right)_{i,j=1}^{n},$
and the Sklyanin bracket becomes
(3.4) $\\{f_{1},f_{2}\\}_{R}(x)=\\\ \frac{1}{2}(R(\mbox{grad}f_{1}(x)\
x),\mbox{grad}f_{2}(x)\ x)-\frac{1}{2}(R(x\ \mbox{grad}f_{1}(x)),x\
\mbox{grad}f_{2}(x)).$
Every $\xi\in\mathfrak{g}$ can be uniquely decomposed as
$\xi=\pi_{-}(\xi)+\pi_{0}(\xi)+\pi_{+}(\xi),$
where $\pi_{+}(\xi)$ and $\pi_{-}(\xi)$ are strictly upper and lower
triangular and $\pi_{0}(\xi)$ is diagonal. The simplest classical R-matrix on
$sl_{n}$ is given by
(3.5)
$R_{0}(\xi)=\pi_{+}(\xi)-\pi_{-}(\xi)=\left(\operatorname{sign}(j-i)\xi_{ij}\right)_{i,j=1}^{n}.$
Substituting $R=R_{0}$ into (3.4), we find the bracket for a pair of matrix
entries:
(3.6)
$\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{0}}=\frac{1}{2}\left(\operatorname{sign}(i^{\prime}-i)+\operatorname{sign}(j^{\prime}-j)\right)x_{ij^{\prime}}x_{i^{\prime}j}.$
It is known (see [ReST]) that if $R_{0}$ is the standard R-matrix, $S$ is any
linear operator on the space of traceless diagonal matrices that is skew-
symmetric with respect to the trace-form, and $\pi_{0}$ is the natural
projection onto the subspace of diagonal matrices, then
(3.7) $R_{S}=R_{0}+S\pi_{0}$
satisfies MCYBE (3.1), and thus gives rise to a Sklyanin Poisson-Lie bracket.
The operator $S$ can be identified with an $n\times n$ skew-symmetric matrix
whose kernel contains the vector $(1,\ldots,1)$ and thus is uniquely
determined by its $(n-1)\times(n-1)$ submatrix $(s_{ij})_{i,j=1}^{n-1}$, which
we will also denote by $S$. Slightly abusing notation, we denote the remaining
elements of the above $n\times n$ skew-symmetric matrix by
$s_{in}=-\sum_{j=1}^{n-1}s_{ij},\qquad s_{nj}=-\sum_{i=1}^{n-1}s_{ij}.$
The Sklyanin bracket (3.4) that corresponds to (3.7) can be written in terms
of matrix entries as
(3.8)
$\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{S}}=\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{0}}+\frac{1}{2}\left(s_{ii^{\prime}}-s_{jj^{\prime}}\right)x_{ij}x_{i^{\prime}j^{\prime}}.$
Let $\mathcal{H}$ denote the subgroup of diagonal matrices in $SL_{n}$:
$\mathcal{H}=\\{\operatorname{diag}(d_{1},\dots,d_{n})\ :d_{1}\cdots
d_{n}=1\\}.$
For any skew-symmetric matrix $V=(v_{ij})_{i,j=1}^{n-1}$, define a Poisson
bracket ${\\{\cdot,\cdot\\}}^{\mathcal{H}}_{V}$ on $\mathcal{H}$ by
(3.9) $\\{d_{i},d_{j}\\}^{\mathcal{H}}_{V}=v_{ij}d_{i}d_{j};$
here $v_{in}$ and $v_{nj}$ have the same meaning as $s_{in}$ and $s_{nj}$
above.
In what follows, we denote the Poisson manifolds
$\left(\mathcal{H},{\\{\cdot,\cdot\\}}^{\mathcal{H}}_{V}\right)$ and
$\left(SL_{n},{\\{\cdot,\cdot\\}}_{R_{S}}\right)$ by $\mathcal{H}^{\\{V\\}}$
and $SL_{n}^{\\{S\\}}$, respectively.
Next, for $S$ defined as in (3.7), consider the direct product of Poisson
manifolds $\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}$; the product
structure we denote below simply by ${\\{\cdot,\cdot\\}}$.
###### Lemma 3.1.
The map $\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to SL_{n}^{\\{S\\}}$
given by $(D_{1},X,D_{2})\mapsto D_{1}XD_{2}$ is Poisson.
###### Proof.
Denote $D_{1}XD_{2}$ by $\widehat{X}=(\hat{x}_{ij})_{i,j=1}^{n}$. Let
$D_{k}=\operatorname{diag}(d_{kl})_{l=1}^{n}$ for $k=1,2$. Then
$\\{\hat{x}_{ij},\hat{x}_{i^{\prime}j^{\prime}}\\}=\\{\hat{x}_{ij},\hat{x}_{i^{\prime}j^{\prime}}\\}_{R_{0}}+x_{ij}x_{i^{\prime}j^{\prime}}\\{d_{1i}d_{2j},d_{1i^{\prime}}d_{2j^{\prime}}\\}$.
The second term is equal to
$\frac{1}{2}(s_{ii^{\prime}}-s_{jj^{\prime}})x_{ij}x_{i^{\prime}j^{\prime}}d_{1i}d_{2j}d_{1i^{\prime}}d_{2j^{\prime}}=\frac{1}{2}(s_{ii^{\prime}}-s_{jj^{\prime}})\hat{x}_{ij}\hat{x}_{i^{\prime}j^{\prime}},$
and the claim follows by (3.8). ∎
## 4\. Grassmannians
### 4.1.
Let ${\mathcal{P}}$ be a Lie subgroup of a Poisson-Lie group ${\mathcal{G}}$.
A Poisson structure on the homogeneous space
${\mathcal{P}}\backslash{\mathcal{G}}$ is called Poisson homogeneous (with
respect to the Poisson-Lie structure on ${\mathcal{G}}$) [D] if the action map
${\mathcal{P}}\backslash{\mathcal{G}}\times{\mathcal{G}}\to{\mathcal{P}}\backslash{\mathcal{G}}$
is Poisson. In particular, if ${\mathcal{P}}$ is a parabolic subgroup of a
simple Lie group ${\mathcal{G}}$ equipped with the standard Poisson-Lie
structure, then ${\mathcal{P}}\backslash{\mathcal{G}}$ is a Poisson
homogeneous space. We will be interested in the case when
${\mathcal{G}}=SL_{n}$ equipped with the bracket (3.4) and
${\mathcal{P}}={\mathcal{P}}_{k}=\left\\{\begin{pmatrix}A&0\\\
B&C\end{pmatrix}\ :A\in GL_{k},C\in GL_{n-k}\right\\}.$
The resulting homogeneous space is the Grassmannian $G_{k}(n)$ equipped with
what we will call the standard Poisson homogeneous structure
${\\{\cdot,\cdot\\}}_{0}^{Gr}$. We will recall an explicit expression of this
Poisson structure on the open Schubert cell $G^{0}_{k}(n)=\\{X\in
G_{k}(n):x_{[k]}\neq 0\\}$. Here we use the same notation for an element of
the Grassmannian and its matrix representative $X$, and $x_{I}$ denotes the
Plücker coordinate that corresponds to a $k$-element subset $I\subset[n]$.
Elements of $G^{0}_{k}(n)$ can be represented by matrices of the form
$\left[\mathbf{1}_{k}\ Y\right]$, and the entries of the $k\times(n-k)$ matrix
$Y$ serve as coordinates on $G^{0}_{k}(n)$. In terms of matrix elements
$y_{ij}$ of $Y$, the Poisson homogeneous bracket looks as follows [GSV1]:
(4.1)
$\\{y_{ij},y_{\alpha\beta}\\}_{0}^{Gr}=\frac{\operatorname{sign}(\alpha-i)-\operatorname{sign}(\beta-j)}{2}y_{i\beta}y_{\alpha
j}.$
We denote $G_{k}(n)$ equipped with the Poisson bracket (4.1) by
$G_{k}(n)^{\\{0\\}}$.
###### Proposition 4.1.
(i) For an arbitrary skew-symmetric operator $S$, there exists a Poisson
bracket ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ on $G_{k}(n)$, unique up to a scalar
multiple, such that the natural action $(X,D)\mapsto XD$ is a Poisson map from
$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}$ to
$G_{k}(n)^{\\{S\\}}:=\left(G_{k}(n),{\\{\cdot,\cdot\\}}^{Gr}_{S}\right)$.
(ii) The bracket ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ is a Poisson homogeneous
structure on $G_{k}(n)$ with respect to the bracket
${\\{\cdot,\cdot\\}}_{R_{S}}$ on $SL_{n}$ defined by (3.8).
###### Proof.
(i) Let $X=\left[\mathbf{1}_{k}\ Y\right]\in G^{0}_{k}(n)$,
$D=\operatorname{diag}(d_{1},\dots,d_{n})\in\mathcal{H}$ and let
$\left[\mathbf{1}_{k}\ \widetilde{Y}\right]$ be the matrix that represents the
element $XD\in G^{0}_{k}(n)$. Then $\tilde{y}_{ij}=y_{ij}{d_{j+k}}/{d_{i}}$,
and the Poisson bracket of any two Plücker coordinates $\tilde{y}_{ij}$ and
$\tilde{y}_{\alpha\beta}$ in the product structure
${\\{\cdot,\cdot\\}}^{Gr}_{0}\times{\\{\cdot,\cdot\\}}^{\mathcal{H}}_{-\frac{1}{2}S}$
is equal to
$\frac{\operatorname{sign}(\alpha-i)-\operatorname{sign}(\beta-j)}{2}\tilde{y}_{i\beta}\tilde{y}_{\alpha
j}+\frac{s_{i,\beta+k}+s_{j+k,\alpha}-s_{i,\alpha}-s_{j+k,\beta+k}}{2}\tilde{y}_{ij}\tilde{y}_{\alpha\beta}.$
Thus, the bracket defined on $G^{0}_{k}(n)$ by the formula
$\\{y_{ij},y_{\alpha\beta}\\}_{S}^{Gr}=\\{y_{ij},y_{\alpha\beta}\\}_{0}^{Gr}+\frac{s_{i,\beta+k}+s_{j+k,\alpha}-s_{i,\alpha}-s_{j+k,\beta+k}}{2}y_{ij}y_{\alpha\beta}$
is the unique, up to a scalar multiple, Poisson bracket that makes the map
$(X,D)\mapsto XD$ Poisson. Sinse $G^{0}_{k}(n)$ is an open dense subset in
$G_{k}(n)$ the claim follows.
(ii) To see that ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ is Poisson homogeneous with
respect to ${\\{\cdot,\cdot\\}}_{R_{S}}$, we need to check that the natural
action of $SL_{n}$ on $G_{k}(n)$ defines a Poisson map from
$G_{k}(n)^{\\{S\\}}\times SL_{n}^{\\{S\\}}$ to $G_{k}(n)^{\\{S\\}}$. Instead
of a straightforward calculation, we can use the fact that this is true for
$S=0$ and Lemma 3.1. Indeed, it is easy to check that both Poisson maps
$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to
G_{k}(n)^{\\{S\\}}$ given by $(X,D_{1})\mapsto XD_{1}$ and
$\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to SL_{n}^{\\{S\\}}$
given by $(D_{1},X,D_{2})\mapsto D_{1}XD_{2}$ are surjective. Therefore, we
can replace the map $G_{k}(n)^{\\{S\\}}\times SL_{n}^{\\{S\\}}\to
G_{k}(n)^{\\{S\\}}$ by the map
$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\times\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to G_{k}(n)^{\\{S\\}}$
given by $(X,D_{1},D_{2},A,D_{3})\mapsto XD_{1}D_{2}AD_{3}$. It is easy to
check that $(D_{1},D_{2})\mapsto D_{1}D_{2}$ is a Poisson map from
$\mathcal{H}^{\\{-\frac{1}{2}S\\}}\times\mathcal{H}^{\\{\frac{1}{2}S\\}}$ onto
$\mathcal{H}^{\\{0\\}}$, which, in turn, is a Poisson-Lie subgroup of
$SL_{n}^{\\{0\\}}$. We thus arrive to the Poisson map
$G_{k}(n)^{\\{0\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to G_{k}(n)^{\\{S\\}}$
given by $(X,\widetilde{A},D_{3})\mapsto X\widetilde{A}D_{3}$ with
$\widetilde{A}=D_{1}D_{2}A$. The standard Poisson homogeneous structure
ensures that the map $G_{k}(n)^{\\{0\\}}\times SL_{n}^{\\{0\\}}\to
G_{k}(n)^{\\{0\\}}$ is Poisson, and it remains to use part (i) of Proposition
4.1 to complete the proof. ∎
### 4.2.
Now, we recall the construction of the cluster algebra
${\mathcal{A}}_{G^{0}_{k}(n)}$ associated with the open cell $G^{0}_{k}(n)$ as
described in [GSV1, GSV2]. Denote $m=n-k$. For every $i\in[k]$, $j\in[m]$ put
(4.2) $I_{ij}=\begin{cases}[i+1,k]\cup[j+k,i+j+k-1],&\text{if $i\leq
m-j+1$}\\\ \left([k]\setminus[i+j-m,\ i]\right)\cup[j+k,n],&\text{if
$i>m-j+1$}.\end{cases}$
Denote the Plücker coordinate $x_{I_{ij}}$ by $x(i,j)$.
The initial cluster of ${\mathcal{A}}_{G^{0}_{k}(n)}$ is given by
(4.3) $\mathbf{x}=\mathbf{x}(k,n)=\left\\{\frac{x(i,j)}{x_{[k]}}\ :\ i\in[k],\
j\in[m]\right\\}.$
Stable variables are
${\displaystyle\frac{x(1,1)}{x_{[k]}},\ldots,\frac{x(k,1)}{x_{[k]}},\frac{x(k,2)}{x_{[k]}},\ldots,\frac{x(k,m)}{x_{[k]}}}$.
The entries of ${\widetilde{B}}$ that correspond to ${\bf x}$ are all $0$ or
$\pm 1$s. Thus it is convenient to describe ${\widetilde{B}}$ by a directed
graph $\Gamma({\widetilde{B}})$.
Figure 1. Graph $\Gamma({\widetilde{B}})$
The vertices of $\Gamma({\widetilde{B}})$ correspond to all columns of
${\widetilde{B}}$, and, since ${\widetilde{B}}$ is rectangular, the
corresponding edges are either between the cluster variables or between a
cluster variable and a stable variable. In our case, $\Gamma({\widetilde{B}})$
is a directed graph with $km$ vertices labeled by pairs of integers $(i,j)\
i\in[k],j\in[m]$. $\Gamma({\widetilde{B}})$ has edges $(i,j)\to(i,j+1)$,
$(i+1,j)\to(i,j)$ and $(i,j)\to(i+1,j-1)$ whenever both vertices defining an
edge are in the vertex set of $\Gamma({\widetilde{B}})$ (cf. Fig. 1). Each
cluster variable $x(i,j)$ is associated with (placed at) the vertex with
coordinates $(i,j)$ of the grid in Fig. 1 for $i\in[k],\ j\in[m]$. Equation
(2.1) then results in the following formulas for the $\tau$-cluster:
(4.4)
$\tau_{ij}=\frac{x(i+1,j-1)x(i,j+1)x(i-1,j)}{x(i-1,j+1)x(i,j-1)x(i+1,j)},\quad
i\in[k-1],\ j\in[2,m],$
where $x(0,j)=x(i,m+1)=1$.
###### Lemma 4.2.
Functions (4.4) are invariant under the natural action of $\mathcal{H}$ on
$G_{k}(n)$.
###### Proof.
Let $X\in G_{k}(n)$, $D=\operatorname{diag}(d_{1},\dots,d_{n})\in\mathcal{H}$
and $\widetilde{X}=XD$. For any subset $I=\\{i_{1},\ldots,i_{l}\\}\subset[n]$
denote $d^{I}=\prod_{j=1}^{l}d_{i_{j}}$. Then, using (4.2), we obtain
$\tilde{x}(i,j)=x(i,j)d^{I_{ij}}=\begin{cases}{\displaystyle
x(i,j)\frac{d^{[k]}d^{[i+j+k-1]}}{d^{[i]}d^{[j+k-1]}}},&\text{if $i\leq
m-j+1$},\\\\[6.99997pt] {\displaystyle
x(i,j)\frac{d^{[k]}d^{[n]}d^{[i+j-m-1]}}{d^{[i]}d^{[j+k-1]}}},&\text{if
$i>m-j+1$},\end{cases}$
and the equality $\tau_{ij}(\widetilde{X})=\tau_{ij}(X)$ follows from (4.4) by
trivial cancellation. ∎
Now we are ready to prove
###### Theorem 4.3.
A Poisson structure ${\\{\cdot,\cdot\\}}$ on $G_{k}(n)$ is compatible with
${\mathcal{A}}_{G^{0}_{k}(n)}$ if and only if a scalar multiple of
${\\{\cdot,\cdot\\}}$ defines a Poisson homogeneous structure with respect to
${\\{\cdot,\cdot\\}}_{R_{S}}$ for some skew-symmetric operator $S$.
###### Proof.
It follows from Theorem 5.4 in [GSV2] that ${\\{\cdot,\cdot\\}}_{0}^{Gr}$ is
compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. The number of stable variables
for ${\mathcal{A}}_{G^{0}_{k}(n)}$ is $n-1$. Since $\mathcal{H}$ is isomorphic
to $({\mathbb{C}}^{*})^{n-1}$, Lemma 4.2 guarantees that the map $(X,D)\mapsto
XD$ translates into an action of $({\mathbb{C}}^{*})^{n-1}$ on
${\mathcal{A}}_{G^{0}_{k}(n)}$ as described in Section 2. Assumptions of
Theorem 2.1 are verified in [GSV1], Section 3.3. Then Proposition 2.2 and
Proposition 4.1 imply that every compatible Poisson bracket on
${\mathcal{A}}_{G^{0}_{k}(n)}$ is a scalar multiple of
${\\{\cdot,\cdot\\}}_{S}^{Gr}$ for some skew-symmetric operator $S$ on the
space of traceless diagonal $n\times n$ matrices. Since
${\\{\cdot,\cdot\\}}_{S}^{Gr}$ is a unique Poisson homogeneous with respect to
${\\{\cdot,\cdot\\}}_{R_{S}}$ (see, e.g. [D]), the claim follows. ∎
## Acknowledgments
M. G. was supported in part by NSF Grant DMS #0801204. M. S. was supported in
part by NSF Grants DMS #0800671 and PHY #0555346. A. S. was supported in part
by KVVA. A. V. was supported in part by ISF Grant #1032/08. The authors are
grateful to A. Zelevinsky for useful comments.
## References
* [BD] A. A. Belavin and V. G. Drinfeld, Triangle equations and simple Lie algebras, Soviet Sci. Rev. Sect. C Math. Phys. Rev. 4 (1984), 93–165.
* [BFZ] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126 (2005), 1–52.
* [D] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. and Math. Phys. 95 (1993), no. 2, 524–525
* [FZ1] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999), 335–380.
* [FZ2] S. Fomin and A. Zelevinsky, Total Positivity: tests and parametrizations., Math. Inteligencer. 22 (2000), 23–33.
* [FZ3] S. Fomin and A. Zelevinsky, Cluster algebras.I. Foundations. J. Amer. Math. Soc. 15 (2002), 497–529.
* [GSV1] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry. Mosc. Math. J. 3 (2003), 899–934.
* [GSV2] M. Gekhtman, M. Shapiro, and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Mathematica 15, no. 1, 61-103.
* [GSV3] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, book in preparation.
* [P] A. Postnikov, Total positivity, Grassmannians and networks, arXiv: math/0609764.
* [ReST] A. Reyman and M. Semenov-Tian-Shansky Group-theoretical methods in the theory of finite-dimensional integrable systems Encyclopaedia of Mathematical Sciences, vol.16, Springer–Verlag, Berlin, 1994 pp. 116–225.
* [S] J. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), 345–380.
|
arxiv-papers
| 2009-09-02T09:22:30 |
2024-09-04T02:49:04.993249
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Gekhtman, Michael Shapiro, Alexander Stolin, Alek Vainshtein",
"submitter": "Alek Vainshtein",
"url": "https://arxiv.org/abs/0909.0361"
}
|
0909.0426
|
# On asymptotic structure at null infinity in five dimensions
Kentaro Tanabe Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan Norihiro Tanahashi and Tetsuya Shiromizu Department of
Physics, Kyoto University, Kyoto 606-8502, Japan
###### Abstract
We discuss the asymptotic structure of null infinity in five dimensional
space-times. Since it is known that the conformal infinity is not useful for
odd higher dimensions, we shall employ the coordinate based method such as the
Bondi coordinate first introduced in four dimensions. Then we will define the
null infinity and identify the asymptotic symmetry. We will also derive the
Bondi mass expression and show its conservation law.
## I Introduction
Inspired by superstring theory, fundamental studies of higher dimensional
space-time is important. One issue is the asymptotic structure at null
infinity. We often introduce the conformal infinity to discuss the asymptotic
structure at infinities in four dimensions Penrose:1962ij ; Wald . Therein the
space-time is compactified by conformal transformation and embedded into
another space-time. For example, Minkowski space-time is conformally embedded
into the Einstein static universe. Conformally embedded space-time has two
different infinities, i.e., spatial infinity and null infinity. In higher
dimensional space-times, the asymptotic structure at spatial infinity can be
well-defined. The asymptotic symmetry is identified with the Poincare group
and conserved quantities associated with the symmetry are constructed
Shiromizu:2004jt ; Tanabe:2009xb .
On the other hand, the asymptotic structure at null infinity is not completely
understood in higher dimensional space-times Hollands:2003ie ; Hollands:2004ac
; Ishibashi:2007kb . Indeed, the definition of null infinity is given only in
even dimensions Hollands:2003ie ; Ishibashi:2007kb . As seen later, the
difficulty in defining null infinity as compared to spatial infinity is due to
the presence of the gravitational waves at null infinity. Since there are no
gravitational waves at spatial infinity, the asymptotic structure is
“stationary” and the total mass and total angular momentum are conserved. On
the other hand, the asymptotic structure at null infinity might be disturbed
by gravitational waves. Hence, we need a stable definition of null infinity
against gravitational waves. We can give such a definition in even dimensions
if we use a conformal embedding method. But we cannot do this in odd
dimensional space-times because we cannot show the smoothness of the Einstein
equations at null infinity. This non-smoothness would be related to the facts
that in the conformal embedding method we introduce the conformal factor
$\Omega\sim 1/r$, and the behavior of gravitational waves near null infinity
are of the order ${\cal O}(\Omega^{(d-2)/2})$ in $d$ dimensional space-times.
The problem comes from the half-integer power of $\Omega$. From this
smoothness only, however, we cannot show that the boundary conditions at null
infinity for asymptotic flatness in the papers Hollands:2003ie ;
Hollands:2004ac ; Ishibashi:2007kb are the marginal and weakest conditions
which allow gravitational waves at null infinity to exist. To find such
marginal conditions, which the boundary conditions should be so, we must solve
the Einstein equations near null infinity and clarify the freedom of
gravitational waves.
In this paper, we define null infinity in five dimensional space-time. We do
not use the conformal embedding method, but instead we use the Bondi
coordinate to define null infinity, which was introduced firstly by Bondi and
Sachs in four dimensions Bondi ; Wnicour ; Sachs1 ; Sachs2 . The rest of this
paper is organized as follows. In section 2, we introduce the Bondi coordinate
in five dimensions. In section 3, we define the asymptotic flatness at null
infinity in five dimensions and show the robustness of the definition against
gravitational waves by solving Einstein equations. In this section, we define
the Bondi mass, and obtain the mass loss law in five dimensions. In section 4,
we discuss the asymptotic symmetry at null infinity associated with the
asymptotic flatness defined in section 3. We show that there are no
supertranslations in five dimensions unlike in four dimensions. Finally, in
section 5, we give a discussion and summary.
## II Bondi coordinate in five dimensions
We consider five dimensional space-time. We introduce the Bondi coordinate to
define asymptotic flatness at null infinity. Suppose there is a function
$u(x^{a})$ which satisfies the equation
$u_{,a}u_{,b}g^{ab}\,=\,0,$ (1)
where Latin indices run from $0$ to $4$ and $u_{,a}=\partial u/\partial
x^{a}$. Then $u=\text{constant}$ surfaces are null hypersurfaces, and we use
this function $u$ as retarded time to construct a coordinate system. Let
$\theta$, $\phi$, $\psi$ be angular coordinates, which are constant along
gradient $u$. The period of each of these coordinates are taken to be $\pi$,
$2\pi$, $2\pi$, respectively. For convenience, we introduce the notation
$(\theta,\phi,\psi)=x^{A}$, where capital Latin indices run from $2$ to $4$.
Now we define the function $r$ by the equation
$r^{6}\sin^{2}\theta\cos^{2}\theta\,=\,{\rm det}(g_{AB}).$ (2)
Using these coordinates (we call them the Bondi coordinates)
$x^{0}=u\,,\,x^{1}=r\,,\,x^{2}=\theta\,,\,x^{3}=\phi\,,\,x^{4}=\psi,$ (3)
we can write down metrics as
$ds^{2}\,=\,-(Ve^{B}/r^{2})du^{2}-2e^{B}dudr+r^{2}h_{AB}(dx^{A}+U^{A}du)(dx^{B}+U^{B}du),$
(4)
where
$h_{AB}\,=\,\begin{pmatrix}e^{C_{1}}&\sin\theta\sinh D_{1}&\cos\theta\sinh
D_{2}\\\ \sin\theta\sinh
D_{1}&e^{C_{2}}\sin^{2}\theta&\sin\theta\cos\theta\sinh D_{3}\\\
\cos\theta\sinh D_{2}&\sin\theta\cos\theta\sinh
D_{3}&e^{C_{3}}\cos^{2}\theta\end{pmatrix}.$ (5)
In the above, $V,B,h_{AB},U^{A},C_{1},C_{2},C_{3},D_{1},D_{2}$ and $D_{3}$ are
functions of $u,r$ and $x^{A}$. In this coordinate, null infinity is
represented by $r\rightarrow\infty$.
From Eq. (2), we have a relation between $C_{1},C_{2},C_{3},D_{1},D_{2}$ and
$D_{3}$ as
$e^{C_{3}}\,=\,\frac{1+e^{C_{2}}\sinh^{2}D_{2}+e^{C_{1}}\sinh^{2}D_{3}-2\sinh
D_{1}\sinh D_{2}\sinh D_{3}}{e^{C_{1}+C_{2}}-\sinh^{2}D_{1}}.$ (6)
As we will realize later, these five independent functions
$C_{1},C_{2},D_{1},D_{2}$ and $D_{3}$ correspond to the degrees of freedom of
gravitational waves in five dimensional space-times.
## III Einstein equation at null infinity
As stated in the introduction, the definition of null infinity should be not
disturbed by gravitational waves. The robustness of the null infinity
definition implies that the boundary conditions imposed on the metric (4)
should be compatible with Einstein equations. Hence, in order to define
asymptotic structure at null infinity and show the robustness of this
definition, we should specify the proper boundary conditions by solving
Einstein equations in the Bondi coordinate.
### III.1 Vacuum Einstein equations
Since equations are very complicated, we will not write down explicit forms
here. We will show the equations in a symbolic way in order to see only the
essential structure of equations. See Appendix A for some details of the
equations which will be used.
From $R_{rr}=0$, we have
$\frac{\partial B}{\partial r}\,=\,\frac{\partial\mathbb{C}^{2}}{\partial r},$
(7)
where $\mathbb{C}$ stands for $C_{1},C_{2},D_{1},D_{2}$ and $D_{3}$. From
$R_{rA}=0$,
$\frac{\partial}{\partial r}\left(r^{5}\frac{\partial U^{A}}{\partial
r}\right)=r^{2}\left(\mathbb{C}+\mathbb{C}^{2}+\cdots\right)$ (8)
The trace and traceless part of $R_{AB}=0$ implies
$\displaystyle\frac{\partial}{\partial
r}(r^{2}e^{-B}V)=\eta(\mathbb{C},U^{A})$ (9)
and
$\displaystyle\frac{\partial^{2}}{\partial u\partial
r}\mathbb{C}=\delta(\mathbb{C},U^{A}),$ (10)
where $\eta$ and $\delta$ are some functions of $\mathbb{C}$ and $U^{A}$.
Since we integrate the equations with respect to $r$ in solving the evolution
equations, some integration functions $f(u,x^{A})$ appear. Constraint
equations describe the evolution of such functions.
In the Bondi coordinates, after integrating the other equations, we can show
that the equation $R_{ur}=0$ would be satisfied trivially.
The evolution equations $R_{uu}=0$ and $R_{uA}=0$ have the following form
$\frac{\partial V}{\partial
u}\,=\,r^{3}(\mathbb{C}^{2}+\mathbb{C}^{3}+\cdots)+r^{2}(\mathbb{C}+\mathbb{C}^{2}+\cdots)$
(11)
and
$r\frac{\partial U^{A}}{\partial u}\,=\,\frac{\partial\mathbb{C}}{\partial
u},$ (12)
respectively.
If $\mathbb{C}$ is given on initial surface $u=u_{0}$, we can determine the
metric function $B$, $U^{A}$,$V$ from Eqs. (7), (8) and (9) except for
integration functions. The evolution of $\mathbb{C}$ and integration functions
are described by Eqs. (10), (11) and (12).
As seen later, the functions $\mathbb{C}=(C_{1},C_{2},D_{1},D_{2},D_{3})$ will
be identified with the freedom of gravitational waves, and Eq. (11) will
govern the evolution of their total mass. Thus, to obtain a stable definition
of null infinity, we should determine the asymptotic behavior of the function
$\mathbb{C}$. Then, using Eqs. (7), (8) and (9), we can obtain the asymptotic
behavior of $B$, $U^{A}$, $V$, and the robustness against gravitational waves
of these boundary conditions would be guaranteed by the evolution equation
(11) and Eq. (12).
### III.2 Asymptotic behavior of gravitational fields
From now on, we will write down all equations explicitly. When functions
$\mathbb{C}$ corresponds to gravitational waves, $\mathbb{C}$ behaves as $\sim
1/r^{3/2}$ near null infinity. This can been seen from the solutions to the
wave equation and/or the finiteness of the gravitational flux at null
infinity. Therefore, we assume the behavior of $C_{1},C_{2},C_{3},D_{1},D_{2}$
and $D_{3}$ near null infinity such that
$\displaystyle
C_{1}(u,r,x^{A})\,=\,\frac{C_{11}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (13)
$\displaystyle
C_{2}(u,r,x^{A})\,=\,\frac{C_{21}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (14)
$\displaystyle
C_{3}(u,r,x^{A})\,=\,\frac{C_{31}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (15)
$\displaystyle
D_{1}(u,r,x^{A})\,=\,\frac{D_{11}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (16)
$\displaystyle
D_{2}(u,r,x^{A})\,=\,\frac{D_{21}(u,x^{A})}{r^{3/2}}+O(1/r^{2})$ (17)
$\displaystyle
D_{3}(u,r,x^{A})\,=\,\frac{D_{31}(u,x^{A})}{r^{3/2}}+O(1/r^{2}).$ (18)
As noted around Eq. (6), $C_{31}$ can be written as
$C_{31}=-(C_{11}+C_{21}).$ (19)
Then, from Eq. (7), we see
$B=B_{1}/r^{3}+O(r^{-7/2})$ (20)
near null infinity, and we obtain a relation
$B_{1}=-\frac{(C_{11}^{2}+C_{11}C_{21}+C_{21}^{2}+D_{11}^{2}+D_{21}^{2}+D_{31}^{2})}{8}.$
(21)
Furthermore, integrating Eqs. (8), we have the relations
$\displaystyle U_{\theta 1}=\frac{2(C_{11}\cot\theta-
C_{21}\cot\theta-2C_{11}\tan\theta-
C_{21}\tan\theta+C_{11,\theta}+D_{11,\phi}\csc\theta+D_{21,\psi}\sec\theta)}{5},$
(22) $\displaystyle U_{\phi 1}\sin\theta=\frac{2(2D_{11}\cot\theta-
D_{11}\tan\theta+D_{11,\theta}+C_{21,\phi}\csc\theta+D_{31,\psi}\sec\theta)}{5}$
(23)
and
$\displaystyle U_{\psi
1}\cos\theta=\frac{2(D_{21}\cot\theta-2D_{21}\tan\theta+D_{21,\theta}+D_{31,\phi}\csc\theta-(C_{11,\psi}+C_{21,\psi})\sec\theta)}{5},$
(24)
where $U_{A1}$ are coefficients of the expansions defined as
$U^{A}=\frac{U_{A1}}{r^{5/2}}+O(1/r^{3}).$ (25)
Then, from Eq. (9), we find
$V\,=\,r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(r^{-1/2}),$ (26)
where
$V_{1}\,=\,-\frac{2}{3}\left(\frac{1}{\sin\theta\cos\theta}(U_{\theta
1}\sin\theta\cos\theta)_{,\theta}+U_{\phi 1,\phi}+U_{\psi 1,\psi}\right)$ (27)
and $M(u,x^{A})$ is an integration constant.
In Eq. (10), $C_{11,u}$, $C_{21,u}$, $D_{11,u}$, $D_{21,u}$ and $D_{31,u}$ do
not appear and then this means that we can fix them freely as initial
conditions. These freedom correspond to the degree of freedom of gravitational
waves in five dimensions.
Finally, from Eq. (11), we obtain the following formula
$\displaystyle\frac{\partial M(u,x^{A})}{\partial u}\,$ $\displaystyle=$
$\displaystyle\,-\frac{1}{3}\Biggl{(}(C_{11,u})^{2}+C_{11,u}C_{21,u}+(C_{21,u})^{2}+(D_{11,u})^{2}+(D_{21,u})^{2}+(D_{31,u})^{2}\Biggr{)}$
(28) $\displaystyle=$
$\displaystyle\,-\frac{1}{3}\Biggl{(}(C_{11,u}+C_{21,u}/2)^{2}+3(C_{21})^{2}/4+(D_{11,u})^{2}+(D_{21,u})^{2}+(D_{31,u})^{2}\Biggr{)}.$
Eq. (28) represents mass loss rate by gravitational waves, and total mass
always decreases as in four dimensions. Then $M(u,x^{A})$ describes the mass
in $u=\text{constant}$ surfaces.
### III.3 Boundary conditions
As shown in the previous subsection, in the presence of gravitational waves,
asymptotic behaviors of metric (4) in leading order should be
$\displaystyle V=r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(1/r^{1/2})$ (29)
$\displaystyle B=\frac{B_{1}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$ (30)
$\displaystyle U^{A}=\frac{U_{A1}(u,x^{A})}{r^{5/2}}+O(1/r^{3})$ (31)
$\displaystyle\mathbb{C}=\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+O(1/r^{2}).$
(32)
In four dimensions, boundary conditions at null infinity for asymptotic
flatness are determined by these leading behavior. As shown below, however, in
higher dimensions than four, further conditions are needed for asymptotic
flatness.
We consider asymptotic behavior in next-to-leading order terms. The function
$\mathbb{C}$ can be expanded as follows
$\mathbb{C}\,=\,\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+\frac{\mathbb{A}(u,x^{A})}{r^{2}}+\frac{\mathbb{C}_{2}(u,x^{A})}{r^{5/2}}+\frac{\mathbb{C}_{3}(u,x^{A})}{r^{3}}+O(1/r^{7/2}).$
(33)
The equations $R_{AB}=0$, which describes the evolution of $\mathbb{C}$,
becomes
$\frac{\mathbb{A}_{,u}}{r}+O(r^{-3/2})=0$ (34)
and then we see
$\frac{\partial\mathbb{A}}{\partial u}\,=\,0.$ (35)
At spatial infinity, asymptotic flatness requires that the Weyl tensor on
spatial infinity behave like $\sim r^{-5}$ in order for the Taub-NUT charge to
vanish Tanabe:2009xb . This implies that $\mathbb{A}$ vanishes at spatial
infinity ($u=-\infty$). From Eq. (35), $\mathbb{A}$ should vanish for
asymptotic flatness at null infinity
$\mathbb{A}\,=\,0.$ (36)
Then, from Eqs. (9), (7) and (8), we can show that other metric functions can
be expanded as follows 111$U_{A3}$ terms correspond to angular momentum. This
can be seen from the comparison with the asymptotic behavior of Myers-Perry
solutions MP .
$\displaystyle V=r^{2}+V_{1}(u,x^{A})r^{1/2}-M(u,x^{A})+O(1/r^{1/2}),$ (37)
$\displaystyle
B=\frac{B_{1}(u,x^{A})}{r^{3}}+\frac{B_{2}(u,x^{A})}{r^{4}}+\frac{B_{3}(u,x^{A})}{r^{9/2}}+O(1/r^{5}),$
(38) $\displaystyle
U^{A}=\frac{U_{A1}(u,x^{A})}{r^{5/2}}+\frac{U_{A2}(u,x^{A})}{r^{7/2}}+\frac{U_{A3}(u,x^{A})}{r^{4}}+O(1/r^{9/2}).$
(39)
Thus, boundary conditions at null infinity for asymptotic flatness are
$\mathbb{C}\,=\,\frac{\mathbb{C}_{1}(u,x^{A})}{r^{3/2}}+\frac{\mathbb{C}_{2}(u,x^{A})}{r^{5/2}}+\frac{\mathbb{C}_{3}(u,x^{A})}{r^{3}}+O(1/r^{7/2}),$
(40)
and Eqs. (37), (38), (39).
## IV Asymptotic symmetry at null infinity
In this section, we consider asymptotic symmetry at null infinity. Asymptotic
symmetry should be defined as transformations preserving the boundary
conditions (40), (37), (38) and (39). By infinitesimal transformation $\xi$,
the metric is transformed as
$\delta g_{ab}\,=\,2\nabla_{(a}\xi_{b)}.$ (41)
To preserve the boundary conditions given in the previous section, the
variation of metric, $\delta g_{ab}$, should satisfy
$\displaystyle\delta g_{rr}\,=\,0\,,\,\delta g_{rA}\,=\,0\,,\,g^{AB}\delta
g_{AB}\,=\,0,$ (42) $\displaystyle\delta g_{uu}\,=\,O(r^{-3/2})\,,\,\delta
g_{ur}\,=\,O(r^{-3})\,,\,\delta g_{uA}\,=\,O(r^{-1/2})\,,\,\delta
g_{AB}\,=\,O(r^{1/2}).$ (43)
Here, as a first step, we will consider leading order terms.
From Eq. (42), we see that the components of infinitesimal transformation
$\xi$ must take the following forms:
$\displaystyle\xi_{r}=f(u,x^{A})e^{B},$ (44)
$\displaystyle\xi_{B}g^{AB}=f^{A}(u,x^{A})-fU^{A}+\int^{\infty}_{r}dr^{\prime}e^{B}f_{,B}g^{AB},$
(45)
$\displaystyle\xi_{u}=-\frac{re^{B}}{3}\left(-\xi_{A,B}+\xi_{C}\Gamma^{C}_{AB}+\xi_{r}\Gamma^{r}_{AB}\right)g^{AB}.$
(46)
The infinitesimal transformation $\xi$ has four free parameters $f,f^{A}$. $f$
and $f^{A}$ corresponds to translations and Lorentz transformation,
respectively. The other components of metric variation become
$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle\frac{2r}{3}{\cal
D}_{A}f^{A}_{~{}~{},u}+\frac{2}{3}(3f+{\cal
D}^{2}f)_{,u}+\frac{2}{r^{1/2}}h^{(0)}_{AB}f^{A}_{~{}~{},u}U_{B1}+O(r^{-3/2}),$
(47) $\displaystyle\delta g_{ur}$ $\displaystyle=$
$\displaystyle\frac{1}{3}({\cal
D}_{A}f^{A}+3f_{,u})+\frac{1}{5r^{5/2}}h^{(1)AB}{\cal D}_{A}{\cal
D}_{B}f+O(r^{-3}),$ (48) $\displaystyle\delta g_{uA}$ $\displaystyle=$
$\displaystyle r^{2}h^{(0)}_{AB}f^{B}_{~{}~{},u}+\frac{r}{3}{\cal
D}_{A}(3f_{,u}+{\cal
D}_{B}f^{B})+r^{1/2}h^{(1)}_{AB}f^{B}_{~{}~{},u}+\frac{1}{3}{\cal D}_{A}({\cal
D}^{2}f+3f)+O(r^{-1/2}),$ (49) $\displaystyle\delta g_{AB}$ $\displaystyle=$
$\displaystyle\frac{2r^{2}}{3}(-{\cal D}_{C}f^{C}h^{(0)}_{AB}+3{\cal
D}_{(A}f_{B)})+\frac{2r}{3}(-{\cal D}^{2}fh^{(0)}_{AB}+3{\cal D}_{A}{\cal
D}_{B}f)+T_{AB}(u,x^{A})r^{1/2}+O(r^{-1/2}),$ (50)
where $X_{(AB)}:=(1/2)(X_{AB}+X_{BA})$ for some tensor $X_{AB}$ and $T_{AB}$
is some traceless tensor with respect to $h_{AB}^{(0)}$. In the above, we
expanded the metric $h_{AB}$ as
$h_{AB}\,=\,h^{(0)}_{AB}+\frac{1}{r^{3/2}}h^{(1)}_{AB}+O(r^{-5/2}),$ (51)
and $h^{(1)AB}=h^{(0)AC}h^{(0)BD}h^{(1)}_{CD}.$ In the Bondi coordinate,
$h^{(0)}_{AB}$ and $h^{(1)}_{AB}$ are
$h^{(0)}_{AB}=\begin{pmatrix}1&0&0\\\ 0&\sin^{2}\theta&0\\\
0&0&\cos^{2}\theta\end{pmatrix}$ (52)
and
$h^{(1)}_{AB}=\begin{pmatrix}C_{11}&D_{11}\sin\theta&D_{21}\cos\theta\\\
D_{11}\sin\theta&C_{21}\sin^{2}\theta&D_{31}\sin\theta\cos\theta\\\
D_{21}\cos\theta&D_{31}\sin\theta\cos\theta&-(C_{11}+C_{21})\cos^{2}\theta\end{pmatrix}.$
(53)
Note that $h^{(1)}_{AB}$ is traceless, $h^{(0)AB}h^{(1)}_{AB}=0$.
To satisfy the condition (43), we find that $f^{A}$ and $f$ should satisfy the
following equations:
$\displaystyle f^{A}_{~{}~{},u}=0,$ (54) $\displaystyle{\cal D}_{A}f_{B}+{\cal
D}_{B}f_{A}=-2\frac{\partial f}{\partial u}h_{AB}^{(0)},$ (55)
$\displaystyle{\cal D}_{A}{\cal D}_{B}f=\frac{1}{3}{\cal D}^{2}fh_{AB}^{(0)}.$
(56)
Integrating the trace part of Eq. (55) under the condition (54), we obtain
$f=-\frac{u}{3}F+\alpha(x^{A}),$ (57)
where $F:={\cal D}_{A}f^{A}$ and $\alpha(x^{A})$ is an arbitrary function of
$x^{A}$. Here we can show from Eq. (55) that $F$ satisfies
${\cal D}_{A}{\cal D}_{B}F=\frac{1}{3}{\cal D}^{2}Fh^{(0)}_{AB}$ (58)
and
${\cal D}^{2}F+3F=0.$ (59)
See Appendix B for the derivation. The general solution to these is given by
$F=E_{x}\sin\theta\cos\phi+E_{y}\sin\theta\sin\phi+E_{z}\cos\theta\cos\psi+E_{w}\cos\theta\sin\psi,$
(60)
where $E_{x},E_{y},E_{z},E_{w}$ are constants. Then, from Eqs. (56) and (57),
we see that
${\cal D}_{A}{\cal D}_{B}\alpha=\frac{1}{3}{\cal D}^{2}\alpha h^{(0)}_{AB}$
(61)
holds. The solution to this is
$\alpha=a_{u}+a_{x}\sin\theta\cos\phi+a_{y}\sin\theta\sin\phi+a_{z}\cos\theta\cos\psi+a_{w}\cos\theta\sin\psi,$
(62)
where $a_{u},a_{x},a_{y},a_{z},a_{w}$ are constants. As a summary the general
solution for $f$ is given by
$f\,=\,e_{u}+e_{x}(u)\sin\theta\cos\phi+e_{y}(u)\sin\theta\sin\phi+e_{z}(u)\cos\theta\cos\psi+e_{w}(u)\cos\theta\sin\psi,$
(63)
where $e_{u}:=a_{u}$ is constant,
$e_{x}(u):=-(u/3)E_{x}+a_{x},e_{y}(u):=-(u/3)E_{y}+a_{y},e_{z}(u):=-(u/3)E_{z}+a_{z}$
and $e_{w}(u):=-(u/3)E_{w}+a_{w}$. Now we can show that
${\cal D}_{A}({\cal D}^{2}f+3f)\,=\,0,\qquad\partial_{u}({\cal
D}^{2}f+3f)\,=\,0$ (64)
hold. Using the above equations, at a glance, we see that Eqs. (43) become
$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle O(r^{-3/2}),$
(65) $\displaystyle\delta g_{ur}$ $\displaystyle=$
$\displaystyle\frac{1}{5r^{5/2}}h^{(1)AB}{\cal D}_{A}{\cal D}_{B}f+O(r^{-3}),$
(66) $\displaystyle\delta g_{uA}$ $\displaystyle=$ $\displaystyle
O(r^{-1/2}),$ (67) $\displaystyle\delta g_{AB}$ $\displaystyle=$
$\displaystyle\frac{2r^{2}}{3}(-{\cal D}_{C}f^{C}h^{(0)}_{AB}+3{\cal
D}_{(A}f_{B)})+T_{AB}(u,x^{A})r^{1/2}+O(r^{-1/2}).$ (68)
$h^{(1)AB}{\cal D}_{A}{\cal D}_{B}f$ in Eq. (66) vanishes because of ${\cal
D}_{A}{\cal D}_{B}f\propto h_{AB}^{(0)}$ and the trace-free property of
$h^{(1)AB}$. Noting that condition of (55) is rearranged as ${\cal
D}_{A}f_{B}+{\cal D}_{B}f_{A}=(2/3){\cal D}^{C}f_{C}h_{AB}^{(0)}$, we can show
$\delta g_{AB}=O(r^{1/2})$. As a consequence, the transformation satisfying
the conditions of Eqs. (54), (55) and (56) does not disturb the boundary
condition of Eqs. (40), (37), (38) and (39) at leading order. By
straightforward calculation, it can be shown that these transformations keep
the metric satisfying the boundary conditions at next-to-leading order.
Thus, asymptotic symmetry is generated by $f$ and $f^{A}$ satisfying Eqs.
(54), (55) and (56). The part of $f$ not proportional to $u$ generates a
translation group. Since this part has only five degrees of freedom, there is
no supertranslation freedom unlike in four dimensions 222The fact that
supertranslation freedom does not exist in higher dimensions is firstly
pointed out in Hollands:2003ie by using conformal method in even dimensions.
We show that in five dimensions using the Bondi coordinates. . $f^{A}$
generates a Lorentz transformation group, so asymptotic symmetry at null
infinity in five dimensional space-time is a Poincare group which is semi-
direct of a five dimensional transformation group and Lorentz group.
## V summary and discussion
In this paper, we define asymptotic flatness at null infinity in five
dimensional space-time by using the Bondi coordinates. In the conformal
embedding method, we cannot show the smoothness of asymptotic structure at
null infinity because the gravitational waves behave like $\Omega^{3/2}\sim
r^{-3/2}$ near null infinity and the regularity of gravitational fields at
null infinity is not guaranteed in general in five dimensions. On the other
hand, in the Bondi coordinates, we can show the robustness of the asymptotic
structure at null infinity which is defined by boundary conditions of Eqs.
(40), (37), (38) and (39). Solving Einstein equations under these boundary
conditions, we find that total mass always decreases by gravitational waves as
in four dimensions. In addition, we show that the asymptotic symmetry at null
infinity would be a Poincare group in five dimensions.
In four dimensions, asymptotic symmetry at null infinity is not a Poincare
group. There are so called supertranslation freedoms, i.e., there are infinite
dimensional translations. This supertranslation freedom comes from the freedom
of the Bondi coordinates. In the Bondi coordinates, coordinate transformation
is described by the parameter $f$, $f^{A}$ in Eqs. (44), (45) and (46), which
corresponds to a translation and Lorentz transformation, respectively. If
$f^{A}=0$ (pure translation) we may expect that $f$ has only four independent
solutions. In general, however, there are conditions that $f$ should be a
functions on a $2$-sphere and then $f$ has functional freedom, i.e., infinite
dimensional degrees of freedom. This infinite dimensional set, which has an
Abelian group structure, is called a supertranslational group.
In four dimensional Minkowski space-time without any physical perturbations,
the term $O(r)$ of $\delta g_{AB}$ should vanish, and this condition reduces
infinite dimensional supertranslation to four dimensional translation. In
general, however, since gravitational waves contribute to $g_{AB}$ with $O(r)$
terms, which is the same order with metric variance, we cannot reduce
supertranslation to translation. That is, we cannot distinguish the
supertranslational ambiguity from gravitational waves. Thus the asymptotic
symmetry at null infinity in four dimensions is not a Poincare group.
On the other hand, as shown in this paper, in five dimensions, there is no
supertranslational freedoms. This is because the behavior of gravitational
waves in five dimensions is $1/r^{3/2}$ and this contributes to $g_{AB}$ with
$O(r^{1/2})$ terms. In asymptotically flat five dimensional space-time, the
term of $O(r)$ in $\delta g_{AB}$, which could be a contribution from the
supertranslation, should vanish to maintain asymptotic flatness. This
condition eliminates supertranslational freedom, and makes the asymptotic
symmetry a Poincare group. Although we have only shown this feature in five
dimensions, we expect that this feature would be common to the higher
dimensional space-time in general, because gravitational waves in $d>4$
dimensions decay $1/r^{(d-2)/2}$ near null infinity faster than
supertranslation $O(1/r)$.
In four dimensions, since there are supertranslations, we cannot choose a
preferred rotational axis. The definition of angular momentum at null infinity
does not have a precise meaning. Supertranslations and gravitational waves
both contribute to angular momentum change, and we cannot distinguish one from
another. However, since there is no supertranslation in five dimensions, we
can define total angular momentum at null infinity and observe the change of
total angular momentum by gravitational waves. Furthermore, as we show the
robustness of null infinity definition in five dimensions using the Bondi
coordinates, it may be possible for us to redefine asymptotic flatness at null
infinity using a conformal embedding method covariantly. This is left for
future issue.
Another remaining issue is the extension of our work to dimensions higher than
seven. Using the Bondi coordinates, we have to go on step by step. It will be
nice to have a systematic analysis for that. This is also our future work.
###### Acknowledgements.
KT is supported by JSPS Grant-Aid for Scientific Research. NT and TS are
partially supported by Grant-Aid for Scientific Research from Ministry of
Education, Science, Sports and Culture of Japan (Nos. 2056381, 20540258,
21111006, and 19GS0219), the Japan-U.K. Research Cooperative Programs. This
work is also supported by the Grant-in-Aid for the Global COE Program “The
Next Generation of Physics, Spun from Universality and Emergence h from the
Ministry of Education, Culture, Sports, Science and Technology (MEXT) of
Japan.
## Appendix A Einstein equation
We write the components of an Einstein equation explicitly in the expansion
form with $1/r$. If for example, we expand function $C_{1}(u,r,x^{A})$ such
that
$C_{1}\,=\,\frac{C_{11}(u,x^{A})}{r\sqrt{r}}+\frac{A(u,x^{A})}{r^{2}}+O(r^{-5/2}),$
(69)
The Einstein equation $R_{\theta\theta}=0$ becomes
$-\frac{A_{,u}}{2r}+O(r^{-3/2})=0.$ (70)
That is, $\partial A(u,x^{A})/\partial u=0$. In Ref. Tanabe:2009xb , it was
shown that from asymptotic flatness at spatial infinity,
$A(u,x^{A})|_{u=-\infty}=0$. From this fact and $\partial A(u,x^{A})/\partial
u=0$, we find that $A(u,x^{A})=0$ on null infinity. We can show in the same
way that the $O(r^{-2})$ term in $C_{2},D_{1},D_{2},D_{3}$ should also vanish.
Thus, function $\mathbb{C}$ is expanded as following
$\displaystyle
C_{1}(u,r,x^{A})\,=\,\frac{C_{11}(u,x^{A})}{r\sqrt{r}}+\frac{C_{12}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{C_{13}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(71) $\displaystyle
C_{2}(u,r,x^{A})\,=\,\frac{C_{21}(u,x^{A})}{r\sqrt{r}}+\frac{C_{22}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{C_{23}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(72) $\displaystyle
D_{1}(u,r,x^{A})\,=\,\frac{D_{11}(u,x^{A})}{r\sqrt{r}}+\frac{D_{12}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{13}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(73) $\displaystyle
D_{2}(u,r,x^{A})\,=\,\frac{D_{21}(u,x^{A})}{r\sqrt{r}}+\frac{D_{22}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{23}(u,x^{A})}{r^{3}}+O(1/r^{7/2})$
(74) $\displaystyle
D_{3}(u,r,x^{A})\,=\,\frac{D_{31}(u,x^{A})}{r\sqrt{r}}+\frac{D_{32}(u,x^{A})}{r^{2}\sqrt{r}}+\frac{D_{33}(u,x^{A})}{r^{3}}+O(1/r^{7/2}).$
(75)
Then we find from the Einstein equation that the other metric function in (4)
should be expanded as follows
$\displaystyle V\,=\,r^{2}+V_{1}(u,x^{A})\sqrt{r}+M(u,x^{A})+O(1/r^{1/2})$
(76) $\displaystyle
B\,=\,\frac{B_{1}(u,x^{A})}{r^{3}}+\frac{B_{2}(u,x^{A})}{r^{4}}+O(1/r^{9/2})$
(77) $\displaystyle
U^{A}\,=\,\frac{U_{A1}(u,x^{A})}{r^{5/2}}+\frac{U_{A2}(u,x^{A})}{r^{7/2}}+\frac{U_{A3}(u,x^{A})}{r^{4}}+O(1/r^{9/2}).$
(78)
Now we can write down each component of the Einstein equation as follows:
$R_{rr}=0$:
$\displaystyle-\frac{9}{8r^{5}}\left(8B_{1}+C_{11}^{2}+C_{11}C_{21}+C_{21}^{2}+D_{11}^{2}+D_{21}^{2}+D_{31}^{2}\right)$
$\displaystyle-\frac{3}{8r^{6}}\left(32B_{2}+10C_{11}C_{12}+5C_{12}C_{21}+5C_{11}C_{22}+10C_{21}C_{22}\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+10D_{11}D_{12}+10D_{21}D_{22}+10D_{31}D_{32}\right)+O(r^{-13/2})\,=\,0,$
(79)
$R_{r\theta}=0$:
$\displaystyle-\frac{3}{8r^{5/2}}\left(2C_{11}\cot\theta-2C_{21}\cot\theta-4C_{11}\tan\theta-2C_{21}\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5U_{\theta
1}\left.+2D_{21,\psi}\sec\theta+2D_{11,\phi}\csc\theta+2C_{11,\theta}\right)$
$\displaystyle+\frac{1}{8r^{7/2}}\left(-10C_{12}\cot\theta+10C_{22}\cot\theta+20C_{12}\tan\theta+10C_{22}\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+7U_{\theta
2}-10D_{22,\psi}\sec\theta-10D_{12,\phi}\csc\theta-10C_{12,\theta}\right)$
$\displaystyle~{}~{}~{}+O(r^{-4})\,=\,0,$ (80)
$R_{r\phi}=0$:
$\displaystyle-\frac{3}{8r^{5/2}}\left(4D_{11}\cos\theta-2D_{11}\sin\theta\tan\theta-5U_{\phi
1}\sin^{2}\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+2D_{31,\psi}\tan\theta+2C_{21,\phi}+2D_{11,\theta}\sin\theta\right)$
$\displaystyle+\frac{1}{8r^{7/2}}\left(-20D_{12}\cos\theta+10D_{12}\sin\theta\tan\theta+7U_{\phi
2}\sin^{2}\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-10D_{12,\theta}\sin\theta-10C_{22,\phi}-10D_{32,\psi}\tan\theta\right)$
$\displaystyle~{}~{}~{}~{}+O(r^{-4})=0,$ (81)
$R_{r\psi}=0$:
$\displaystyle\frac{3}{8r^{5/2}}(-2D_{21}\cot\theta\cos\theta+4D_{21}\sin\theta+5U_{\psi
1}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2C_{11,\psi}+2C_{21,\psi}-2D_{31,\phi}\cot\theta-2D_{21,\theta}\cos\theta)$
$\displaystyle+\frac{1}{8r^{7/2}}(-10D_{22}\cos\theta\cot\theta+20D_{22}\sin\theta+7U_{\psi
2}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+10C_{12,\psi}+10C_{22,\psi}-10D_{32,\phi}\cot\theta-10D_{22,\theta}\cos\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+O(r^{-9/2})=0,$ (82)
$R_{\theta\theta}=0$:
$\displaystyle\frac{1}{8r^{3/2}}\left(-13C_{11}-8U_{\theta
1}\cot\theta+8U_{\theta
1}\tan\theta-4V_{1}-8D_{21,\psi}\sec\theta\tan\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-8U_{\psi 1,\psi}-8U_{\phi
1,\phi}-4C_{11,\phi\phi}\csc^{2}\theta-4C_{11,\psi\psi}\sec^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+8D_{11,\phi}\cot\theta\csc\theta+4C_{11,\theta}\cot\theta-12C_{11,\theta}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-8C_{21,\theta}\tan\theta-12U_{\theta
1,\theta}+8D_{21,\theta\psi}\sec\theta+8D_{11,\theta\phi}\csc\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+4C_{11,\theta\theta}-8C_{21,\theta}\cot\theta-8C_{12,u}\right)$
$\displaystyle+\frac{3}{2r^{2}}\left(C_{13,u}-D_{11}D_{11,u}-D_{21}D_{21,u}\right)+O(r^{-5/2})=0,$
(83)
$R_{\theta\phi}=0$:
$\displaystyle\frac{1}{8r^{3/2}}\left(3D_{11}\sin\theta-4D_{31,\psi}-4D_{31,\psi}\tan^{2}\theta-4D_{11,\psi\psi}\sec\theta\tan\theta\right.$
$\displaystyle~{}~{}-4C_{11,\phi}\cot\theta-8C_{11,\phi}\tan\theta-4C_{21,\phi}\cot\theta-4C_{21,\phi}\tan\theta-2U_{\theta
1,\phi}$ $\displaystyle~{}~{}+4D_{21,\phi\psi}\sec\theta-2U_{\phi
1,\theta}\sin^{2}\theta+4D_{31,\theta\psi}\tan\theta+4C_{11,\theta\phi}$
$\displaystyle~{}~{}\left.+4C_{21,\theta\phi}-8D_{12,u}\sin\theta\right)$
$\displaystyle+\frac{3\sin\theta}{4r^{2}}\left(D_{11}C_{11,u}+D_{11}C_{21,u}+C_{11}D_{11,u}+C_{21}D_{11,u}\right.$
$\displaystyle~{}~{}\left.-2D_{13,u}+D_{31}D_{21,u}+D_{21}D_{31,u}\right)+O(r^{-5/2})=0,$
(84)
$R_{\theta\psi}=0$:
$\displaystyle\frac{1}{8r^{3/2}}(3D_{21}\cos\theta+4C_{11,\psi}\cot\theta-4C_{21,\psi}\cot\theta-4C_{21,\psi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-2U_{\theta
1,\psi}+4D_{31,\phi}+4D_{31,\phi}\cot^{2}\theta+4D_{11,\phi\psi}\csc\theta-4D_{21,\phi\phi}\cot\theta\csc\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-2U_{\psi
1,\theta}\cos^{2}\theta-4C_{21,\theta\psi}+4D_{31,\theta\phi}\cot\theta-8D_{22,u}\cos\theta)$
$\displaystyle+\frac{3\cos\theta}{4r^{2}}\left(D_{21}C_{21,u}-D_{31}D_{11,u}+C_{21}D_{21,u}+D_{23,u}-D_{11}D_{31,u}\right)+O(r^{-5/2})\,=\,0,$
(85)
$R_{\phi\phi}=0$:
$\displaystyle\frac{1}{8r^{3/2}}\left(-16C_{11}\sin^{2}\theta+3C_{21}\sin^{2}\theta-12U_{\theta
1}\cos\theta\sin\theta+8U_{\theta
1}\sin^{2}\theta\tan\theta-4V_{1}\sin^{2}\theta\right.$
$\displaystyle~{}~{}+8D_{21,\psi}\sin\theta-8U_{\psi
1,\psi}\sin^{2}\theta-4C_{21,\psi\psi}\tan^{2}\theta+8D_{11,\phi}\cos\theta-8D_{11,\phi}\sin\theta\tan\theta$
$\displaystyle~{}~{}-12U_{\phi
1,\phi}\sin^{2}\theta+8D_{31,\phi\psi}\tan\theta+4C_{21,\phi\phi}+8C_{11,\theta}\cos\theta\sin\theta-4C_{21,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}+4C_{21,\theta}\sin^{2}\theta\tan\theta-8U_{\theta
1,\theta}\sin^{2}\theta+8D_{11,\theta\phi}\sin\theta-4C_{21,\theta\theta}\sin^{2}\theta-8C_{22,u}\sin^{2}\theta\left.\right)$
$\displaystyle-\frac{3\sin^{2}\theta}{2r^{2}}\left(C_{23,u}-D_{11}D_{11,u}-D_{31}D_{31,u}\right)+O(r^{-5/2})=0,$
(86)
$R_{\phi\psi}=0$:
$\displaystyle\frac{1}{8r^{3/2}}(4D_{31}\cos^{2}\theta\cot\theta+11D_{31}\cos\theta\sin\theta+4D_{31}\sin^{2}\theta\tan\theta+8D_{11,\psi}\cos\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4D_{11,\psi}\sin\theta\tan\theta-2U_{\phi
1,\psi}\sin^{2}\theta-4D_{21,\phi}\cos\theta\cot\theta-8D_{21,\phi}\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-2U_{\psi
1,\phi}\cos^{2}\theta-4C_{11,\phi\psi}-4D_{31,\theta}\cos^{2}\theta+4D_{31,\theta}\sin^{2}\theta+4D_{11,\theta\psi}\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4D_{21,\theta\phi}\cos\theta-
D_{31,\theta\theta}\cos\theta\sin\theta-8D_{32,u}\cos\theta\sin\theta)$
$\displaystyle-\frac{3\cos\theta\sin\theta}{4r^{2}}\left(D_{31}C_{11,u}-D_{21}D_{11,u}-D_{11}D_{21,u}+C_{11}D_{31,u}+D_{33,u}\right)$
$\displaystyle~{}~{}~{}~{}+O(r^{-5/2})=0,$ (87)
$R_{\psi\psi}=0$:
$\displaystyle\frac{1}{8r^{3/2}}\left(-19C_{11}\cos^{2}\theta-3C_{21}\cos^{2}\theta-8U_{\theta
1}\cos^{2}\theta\cot\theta-4V_{1}\cos^{2}\theta+12U_{\theta
1}\cos\theta\sin\theta\right.$
$\displaystyle~{}~{}~{}~{}~{}+8D_{21,\psi}\cos\theta\cot\theta-8D_{21,\psi}\sin\theta-12U_{\psi
1,\psi}\cos^{2}\theta-4C_{11,\psi\psi}-4C_{21,\psi\psi}$
$\displaystyle~{}~{}~{}~{}~{}-8D_{11,\phi}\cos\theta-8U_{\phi
1,\phi}\cos^{2}\theta+8D_{31,\phi\psi}\cot\theta+4C_{11,\phi\phi}\cot^{2}\theta+4C_{21,\phi\phi}\cot^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}+4C_{11,\theta}\cos^{2}\theta\cot\theta-12C_{11,\theta}\cos\theta\sin\theta+4C_{21,\theta}\cos^{2}\theta\cot\theta-4C_{21,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}-8U_{\theta
1,\theta}\cos^{2}\theta+8D_{21,\theta\psi}\cos\theta+8C_{11,\theta\theta}\cos^{2}\theta+8(C_{12,u}+C_{22,u})\cos^{2}\theta\left.\right)$
$\displaystyle+\frac{3\cos^{2}\theta}{2r^{2}}\left(C_{13,u}+C_{23,u}-2D_{11}D_{11,u}-D_{21}D_{21,u}-D_{31}D_{31,u}\right)+O(r^{-5/2})=0,$
(88)
$R_{uu}=0$:
$\displaystyle\frac{1}{2r^{5/2}}(3V_{1,u}+2(\cot\theta-\tan\theta)U_{\theta
1,u}+2U_{\theta 1,u\theta}+2U_{\phi 1,u\phi}+2U_{\psi 1,u\psi})$
$\displaystyle-\frac{1}{2r^{3}}\left(3M_{,u}-(C_{11,u})^{2}-C_{11,u}C_{21,u}-(C_{21,u})^{2}-(D_{11,u})^{2}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-(D_{21,u})^{2}-(D_{31,u})^{2}\left.\right)+O(r^{-7/2})=0,$
(89)
$R_{u\theta}=0$:
$\displaystyle-\frac{1}{4r^{3/2}}(2C_{11,u}\cot\theta-4C_{11,u}\tan\theta+2C_{21,u}\cot\theta-2C_{21,u}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-5U_{\theta
1,u}+D_{21,u\psi}\sec\theta D_{11,u\phi}\csc\theta+2C_{11,u\theta})$
$\displaystyle+\frac{1}{8r^{5/2}}\left(-U_{\theta 1}-8U_{\psi
1,\psi}\tan\theta-4U_{\theta 1,\psi\psi}\sec^{2}\theta+8U_{\phi
1,\phi}\cot\theta\right.$ $\displaystyle~{}~{}~{}~{}~{}~{}-4U_{\theta
1,\phi\phi}\csc^{2}\theta-2V_{1,\theta}+4U_{\psi 1,\theta\psi}+4U_{\phi
1,\theta\phi}+4C_{12,u}\cot\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}-8C_{12,u}\tan\theta-4C_{22,u}\cot\theta-4C_{22,u}\tan\theta-14U_{\theta
2,u}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{22,u}\sec\theta+4D_{12,u}\csc\theta+4C_{12,u}\left.\right)+O(r^{-4})=0,$
(90)
$R_{u\phi}=0$:
$\displaystyle\frac{1}{r^{3/2}}(4D_{11,u}\cos\theta-2D_{11,u}\sin\theta\tan\theta-5U_{\phi
1,u}\sin^{2}\theta+2D_{31,u\psi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2C_{21,u\phi}+2D_{11,u\theta}\sin\theta)$
$\displaystyle~{}~{}+\frac{1}{8r^{5/2}}(15U_{\phi
1,\psi\psi}\sin^{2}\theta-4U_{\phi 1,\psi\psi}\tan^{2}\theta-4U_{\theta
1,\phi}\cot\theta-4U_{\theta 1,\phi}\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-2V_{1,\phi}+4U_{\psi 1,\phi\psi}-12U_{\phi
1,\theta}\cos\theta\sin\theta+4U_{\phi 3,\theta}\sin^{2}\theta\tan\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4U_{\theta 1,\theta\phi}-4U_{\phi
3,\theta\theta}\sin^{2}\theta+D_{12,u\theta}\sin\theta-4D_{12,u}\sin\theta\tan\theta-14U_{\phi
2,u}\sin^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{32,u\psi}\tan\theta+4C_{22,u\phi}+4D_{12,u\theta}\sin\theta)+O(r^{-3})=0,$
(91)
$R_{u\psi}=0$:
$\displaystyle\frac{1}{4r^{3/2}}(2D_{21,u}\cos\theta\cot\theta-4D_{21,u}\sin\theta-5U_{\psi
1,u}\cos^{2}\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2C_{11,u\psi}-2C_{21,u\psi}+2D_{31,u\phi}\cot\theta+2D_{21,u\theta}\cos\theta)$
$\displaystyle+\frac{1}{8r^{5/2}}\left(15U_{\psi 1}\cos^{2}\theta+4U_{\theta
1,\psi}\cot\theta+4U_{\theta 1,\psi}\tan\theta-2V_{1,\psi}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+4U_{\phi 3,\phi\psi}-4U_{\psi
1,\phi\phi}\cot^{2}\theta-4U_{\psi 1,\psi}\cos^{2}\theta\cot\theta+12U_{\psi
1,\theta}\cos\theta\sin\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+4U_{\theta
1,\theta\psi}-8D_{22,u}\sin\theta-14U_{\psi
2,u}\cos^{2}\theta-4C_{12,u\psi}-4C_{22,u\psi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4D_{32,u\phi}\cot\theta+4D_{22,u\theta}\cos\theta\left.\right)$
$\displaystyle~{}~{}~{}~{}~{}+O(r^{-3})=0.$ (92)
## Appendix B Derivation for Eqs. (58) and (59)
Here we sketch the derivation for Eqs. (58) and (59). We begin with
$\displaystyle{\cal D}_{B}{\cal D}_{A}F={\cal D}_{B}{\cal D}_{A}{\cal
D}_{C}f^{C}$ $\displaystyle=$ $\displaystyle{\cal D}_{B}({\cal D}_{C}{\cal
D}_{A}f^{C}-{}^{(3)}R_{AC}f^{C})$ (93) $\displaystyle=$ $\displaystyle-{\cal
D}_{B}{\cal D}_{C}{\cal D}^{C}f_{A}-2{\cal D}_{A}{\cal
D}_{B}\partial_{u}f-2{\cal D}_{B}f_{A},$
where ${}^{(3)}R_{AB}$ is the Ricci tensor with respect to $h_{AB}^{(0)}$. In
the second and last line, we used the definition of Riemann tensor and Eq.
(55), respectively. We also used the fact that that Riemann tensor with
respect to $h_{AB}^{(0)}$ is
${}^{(3)}R_{ABCD}=h_{AC}^{(0)}h_{BD}^{(0)}-h_{AD}^{(0)}h_{BC}^{(0)}$. Using
the definition of Riemann tensor two times, the first term in the last line of
the right-hand side becomes
$\displaystyle{\cal D}_{B}{\cal D}_{C}{\cal D}^{C}f_{A}$ $\displaystyle=$
$\displaystyle{\cal D}_{C}{\cal D}_{B}{\cal D}^{C}f_{A}-2{\cal
D}_{B}f_{A}+h_{AB}^{(0)}{\cal D}_{C}f^{C}-{\cal D}_{A}f_{B}$ (94)
$\displaystyle=$ $\displaystyle{\cal D}^{2}{\cal
D}_{B}f_{A}+2h_{AB}^{(0)}{\cal D}_{C}f^{C}-2({\cal D}_{A}f_{B}+{\cal
D}_{B}f_{A}).$
Substituting this into Eq. (93) and using the symmetry of indices $A$ and $B$,
we have
$\displaystyle{\cal D}_{B}{\cal D}_{A}F=\frac{1}{2}\Biggl{[}-{\cal
D}^{2}({\cal D}_{A}f_{B}+{\cal D}_{B}f_{A})-4h_{AB}^{(0)}F+2({\cal
D}_{A}f_{B}+{\cal D}_{B}f_{A})-4{\cal D}_{A}{\cal
D}_{B}\partial_{u}f\Biggr{]}.$ (95)
Then, using Eq. (55) and $f_{,u}=-(1/3)F$, we obtain
$\displaystyle{\cal D}_{B}{\cal D}_{A}F=-\frac{1}{3}h_{AB}^{(0)}{\cal
D}^{2}F-\frac{4}{3}h_{AB}^{(0)}F+\frac{2}{3}{\cal D}_{A}{\cal D}_{B}F.$ (96)
The trace part implies
$\displaystyle{\cal D}^{2}F+3F=0.$ (97)
Using this, Eq. (96) becomes
$\displaystyle{\cal D}_{A}{\cal D}_{B}F=\frac{1}{3}{\cal D}^{2}Fh_{AB}^{(0)}.$
(98)
## References
* (1) R. Penrose, Phys. Rev. Lett. 10, 66 (1963).
* (2) R. M. Wald General Relativity (Chicago:University of Chicago Press, 1984).
* (3) T. Shiromizu and S. Tomizawa, Phys. Rev. D 69, 104012 (2004) [arXiv:gr-qc/0401006].
* (4) K. Tanabe, N. Tanahashi and T. Shiromizu, arXiv:0902.1583 [gr-qc].
* (5) S. Hollands and A. Ishibashi, J. Math. Phys. 46, 022503 (2005) [arXiv:gr-qc/0304054].
* (6) S. Hollands and R. M. Wald, Class. Quant. Grav. 21, 5139 (2004) [arXiv:gr-qc/0407014].
* (7) A. Ishibashi, Class. Quant. Grav. 25, 165004 (2008) [arXiv:0712.4348 [gr-qc]].
* (8) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. A (London) 269, 21 (1962).
* (9) J. Winicour, J. Math. Phys. 7, 863 (1967).
* (10) R. K. Sachs, Proc. Roy. Soc. A (London) 270, 103 (1962).
* (11) R. K. Sachs, Phys. Rev. 128, 2851 (1962).
* (12) R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986).
|
arxiv-papers
| 2009-09-02T13:38:17 |
2024-09-04T02:49:04.999800
|
{
"license": "Public Domain",
"authors": "Kentaro Tanabe, Norihiro Tanahashi, Tetsuya Shiromizu",
"submitter": "Kentarou Tanabe",
"url": "https://arxiv.org/abs/0909.0426"
}
|
0909.0529
|
# Optimal capture of non-Gaussianity in weak lensing surveys : power spectrum,
bispectrum and halo counts
Joel Bergé11affiliation: Jet Propulsion Laboratory / California Institute of
Technology, 4800 Oak Grove Drive, MS 169-327, Pasadena, CA 91109, USA ;
e-mail: Joel.Berge@jpl.nasa.gov 22affiliation: NASA Postdoctoral Program (NPP)
fellow , Adam Amara33affiliation: Department of Physics, ETH Zurich, Wolfgang-
Pauli-Strasse16, CH-8093 Zurich, Switzerland and Alexandre
Réfrégier44affiliation: Laboratoire AIM, CEA/DSM - CNRS - Université Paris
Diderot, DAPNIA/SAp, 91191 Gif-sur-Yvette, France
(Draft version)
###### Abstract
We compare the efficiency of weak lensing-selected galaxy clusters counts and
of the weak lensing bispectrum at capturing non-Gaussian features in the dark
matter distribution. We use the halo model to compute the weak lensing power
spectrum, the bispectrum and the expected number of detected clusters, and
derive constraints on cosmological parameters for a large, low systematic weak
lensing survey, by focusing on the $\Omega_{m}$-$\sigma_{8}$ plane and on the
dark energy equation of state. We separate the power spectrum into the
resolved and the unresolved parts of the data, the resolved part being defined
as detected clusters, and the unresolved part as the rest of the field. We
consider four kinds of clusters counts, taking into account different amount
of information : signal-to-noise ratio peak counts; counts as a function of
clusters’ mass; counts as a function of clusters’ redshift; and counts as a
function of clusters’ mass and redshift. We show that when combined with the
power spectrum, those four kinds of counts provide similar constraints, thus
allowing one to perform the most direct counts, signal-to-noise peaks counts,
and get percent level constraints on cosmological parameters. We show that the
weak lensing bispectrum gives constraints comparable to those given by the
power spectrum and captures non-Gaussian features as well as clusters counts,
its combination with the power spectrum giving errors on cosmological
parameters that are similar to, if not marginally smaller than, those obtained
when combining the power spectrum with cluster counts. We finally note that in
order to reach its potential, the weak lensing bispectrum must be computed
using all triangle configurations, as equilateral triangles alone do not
provide useful information. The appendices summarize the halo model, and the
way the power spectrum and bispectrum are computed in this framework.
###### Subject headings:
cosmological parameters - large-scale structures - weak gravitational lensing
## 1\. Introduction
Since its first detections in the early 2000’s (Bacon, Refregier & Ellis 2000;
van Waerbeke et al 2000; Wittman et al. 2000), weak gravitational lensing, the
coherent distortion of distant galaxies by intervening dark matter, has become
a premier tool to constrain the cosmological model (for reviews, see e.g.
Mellier 1999; Bartelmann & Schneider 2001; Refregier 2003; Hoekstra & Jain
2008; Munshi et al. 2008) and has been shown to be the most promising probe of
dark energy (Albrecht et al. 2006; Peacock et al. 2006). Surveys’ optimization
and systematics minimizations in both software and hardware have been
investigated (Heymans et al. 2006; Massey et al. 2007a; Amara & Réfrégier
2007, 2008; Paulin-Henriksson et al. 2008; Amara et al. 2009; Bridle et al.
2009), favoring wide surveys, with well-controlled, stable Point Spread
Function, with comprehensive photometric redshift follow-up. Those
characteristics are shared by ambitious upcoming large area surveys, such as
the Large Synoptic Survey Telescope (LSST)111http://www.lsst.org, the
Panoramic Survey Telescope & Rapid Response System (Pan-STARRS)222http://pan-
starrs.ifa.hawaii.edu,
Euclid333http://sci.esa.int/science-e/www/area/index.cfm?fareaid=102,
http://www.euclid-imaging.net and the Joint Dark Energy Mission
(JDEM)444http://jdem.gsfc.nasa.gov. They will provide us with a large amount
of high quality imaging, well fitted to measure dark energy with weak lensing.
Anticipating a significant improvement in removing systematics, we are left
with the question of how to best extract the cosmological information out of
the data. For instance, one must decide how to optimally capture non-
Gaussianities and break the degeneracies between cosmological parameters as
constrained by the extensively studied and measured weak lensing power
spectrum. In this paper, we ignore primordial non-Gaussianities, and consider
non-Gaussianities due to the growth of structures only.
The power spectrum, the Fourier transform of the 2-point correlation function
of the shear field, has been the most used measurement so far, both from
ground (e.g. Massey et al. 2005; Van Waerbeke et al. 2005; Jarvis et al. 2006;
Hoekstra et al. 2006; Semboloni et al. 2006; Benjamin et al. 2007; Fu et al.
2008) and from space (e.g. Rhodes et al. 2004; Heymans et al. 2005; Schrabback
et al. 2007; Massey et al. 2007b). The introduction of tomography, the three-
dimensional information of the shear field, that captures structure evolution,
has tightened constraints on cosmological parameters, in particular the matter
density $\Omega_{m}$ and the amplitude of density fluctuations $\sigma_{8}$
(Massey et al. 2007b). Despite its success, the power spectrum leaves us with
well known degeneracies between parameters, that we must break by combining it
with other measurements and/or probes. The power spectrum capturing only the
Gaussian features of the field, it is natural to introduce measures of non-
Gaussianities, and combine them with the power spectrum, to break degeneracies
and tighten constraints.
Higher-order correlations measure the matter density field’s non-Gaussian
features. The lowest one, the weak lensing 3-point correlation function
(3PCF), and its Fourier transform, the bispectrum, have been given a lot of
attention in the past few years (Schneider & Lombardi 2003; Takada & Jain
2003a, b, c; Zaldarriaga & Scoccimarro 2003; Takada & Jain 2004; Schneider et
al. 2005; Benabed & Scoccimarro 2006; Semboloni et al. 2008; Joachimi et al.
2009; Vafaei et al. 2009). However, a clean measurement of the 3PCF is
difficult, thus few papers have reported observational measurements so far.
Bernardeau et al. (2003) and Jarvis et al. (2004) have measured the skewness,
but a full measurement of the shear 3PCFs, or of the convergence bispectrum,
is still to be done. Several efforts are underway, including an algorithm to
measure the convergence bispectrum directly in Fourier space (Pires et al.
2009a).
Large-scale structures such as clusters of galaxies, the result of the non-
linear evolution of density fluctuations, are the non-Gaussian features that
we want to take into account. Assessing their abundance as a function of
various parameters, such as redshift, has been known as a powerful probe and
used as this for several years (e.g. Oukbir & Blanchard 1992; Eke et al. 1998;
Wang & Steinhardt 1998; Holder et al. 2001; Refregier et al. 2002; Battye &
Weller 2003; Pierpaoli et al. 2003; Wang et al. 2004; Horellou & Berge 2005;
Marian & Bernstein 2006; Gladders et al. 2007; Mantz et al. 2008; Sahlén et
al. 2009). In particular, tight constraints can be obtained on $\Omega_{m}$
and $\sigma_{8}$. Because the weak lensing clusters selection function is
rapidly evolving with redshift, counting weak-lensing-selected clusters is
less sensitive than using catalogs of X-ray, optical, or Sunyaev-Zel’dovich
(SZ) clusters. Despite this fact, constraining cosmology is possible with
weak-lensing clusters only (Weinberg & Kamionkowski 2003; Marian & Bernstein
2006; Dietrich & Hartlap 2009; Kratochvil et al. 2009; Marian et al. 2009;
Maturi et al. 2009; Wang et al. 2009), as shown on real data by Bergé et al.
(2008). Although such counts give weaker constraints than the weak lensing
power spectrum, combining them with the power spectrum efficiently breaks
degeneracies (Pires et al. 2009b).
In this paper, we show how combining the weak lensing power spectrum with the
weak lensing bispectrum or counts of weak-lensing-selected clusters of
galaxies will allow one to tighten constraints on cosmological parameters with
upcoming weak lensing surveys. In particular, we show how we will be able to
measure the equation of state $w=P/\rho$, where $P$ is the pressure, and
$\rho$ the density of the dark energy, down to the percent level. While other
authors studied the combination of the power spectrum with cluster counts
(e.g. Takada & Bridle 2007) and with the bispectrum (e.g. Takada & Jain 2004),
we present, for the first time, a detailed and consistent comparison of both
combinations at once.
Constraints on cosmological parameters have been studied by several authors
(e.g. Cooray & Hu 2001a; Takada & Jain 2004; Takada & Bridle 2007). Most
constraints’ predictions have so far been made using a fitting function for
the non-linear power spectrum, such as Peacock & Dodds (1996); Ma & Fry (2000)
or Smith et al. (2003). However, fitting functions tuned for a $\Lambda$CDM
universe must be used with caution when one is varying $w$ (Joudaki et al.
2009). We use the halo model to compute the weak lensing statistics. Despite
its simplicity, it is well suited to cosmological parameters prospects, since
it allows us to vary the dark energy without extrapolating fitting functions
out of $w=-1$. Appendix A summarizes the halo model description that we use in
this paper. The halo model allows us to define the power spectrum as the
separated contributions of the resolved and unresolved parts of the weak
lensing field. Such a separation, since it gets rid of highly non-Gaussian
features, can help improve constraints on cosmological parameters. We consider
four different types of cluster counts, with more or less intrinsic
cosmological information : counts as a function of shear signal-to-noise ratio
(S/N) only, counts as a function of mass, counts as a function of redshift,
and counts as a function of mass and redshit. We show that when combined with
the power spectrum, counting clusters just as a function of their S/N gives
constraints similar to those obtained when combining the power spectrum with
counts of clusters taking the full mass and redshift information. We then show
how the weak lensing bispectrum captures non-Gaussian features and breaks the
power spectrum degeneracies as well as clusters counts. We assume a Euclid-
like survey, 20,000 deg2 wide, with $n_{g}=40~{}\mbox{galaxies arcmin}^{-2}$
with median redshift $z_{m}=1$, without any external (e.g. from CMB) priors,
so as to show what can be done with a weak lensing survey alone.
Section 2 summarizes the methods we employ to constrain cosmological
parameters with different weak lensing statistics. Section 3 presents our
results. We conclude in section 4. Our halo model code will be made part of
the public icosmo package555http://www.icosmo.org (Refregier et al. 2008).
## 2\. Method
### 2.1. Weak lensing selection function
The weak lensing selection function for clusters of galaxies has already been
extensively studied, by using different kinds of smoothing functions or
matched-filters (e.g. Weinberg & Kamionkowski 2002; Hamana et al. 2004;
Hennawi & Spergel 2005; Maturi et al. 2005; Marian & Bernstein 2006). We
derive a simplified, ideal selection function, based on a matched-filter
approach for the signal-to-noise ratio created by a halo. We neglect
projection effects and intrinsic alignments, which have been investigated e.g.
by Maturi et al. (2005), Marian & Bernstein (2006) and Fan (2007). That
selection function has been introduced, without the mathematical justification
that follows, and tested on real data in Bergé et al. (2008).
We consider a spherically symmetric weak lens, decoupled from its surrounding
and alone along the line of sight, characterized by its convergence
$\kappa_{\rm obs}(\theta)$, where $\theta$ is the distance from its center
(the convergence can be replaced by the shear without any loss of generality).
The number density of background sources is $n_{g}$. We assume that the noise
$n(\theta)$, originating from Poisson and intrinsic shape noise only, averages
to 0 in circular shells and that its variance in the circular shell $i$ is
given by $<n_{i}^{2}>=\sigma_{\gamma}^{2}/N_{i}$, where $\sigma_{\gamma}$ is
the r.m.s error on the shape measurement and $N_{i}$ is the number of lensed
galaxies in shell $i$. We fit a theoretical model $\kappa(\theta)$ to the
observable $\kappa_{\rm obs}(\theta)=\kappa(\theta)+n(\theta)$.
The signal-to-noise ratio created by the lens is defined as the ratio of an
estimator of $\kappa(\theta)$ to its associated error,
$\nu=\left<\hat{K}_{w}\right>/\sigma\left(\left<\hat{K}_{w}\right>\right)$.
The estimator is defined by $\left<\hat{K}_{w}\right>=\int{\rm
d}^{2}\theta\kappa_{\rm obs}(\theta)w(\theta)$, where $w(\theta)$ is a weight
function to be adjusted so as to optimize the S/N. This is the case when
$w=\kappa$, giving the optimized S/N as
Figure 1.— Weak lensing selection function for clusters of galaxies, in the
redshift - mass plane. We assume a survey with $n_{g}=40$ galaxies per
arcmin2, distributed along equation 4. Contours denote the S/N of galaxy
clusters.
$\nu=\frac{\sqrt{n_{g}}}{\sigma_{\gamma}}\sqrt{\int{\rm
d}^{2}\theta\kappa^{2}(\theta)},$ (1)
which is consistent with previously published expressions.
We will consider NFW halos (Navarro et al. 1996) only, the S/N of which is
expressed as (appendix B)
$\nu=2\sqrt{2\pi}\left<Z\right>\frac{\sqrt{n_{g}}}{\sigma_{\gamma}}\frac{\rho_{s}r_{s}^{2}}{D_{\rm
d}\Sigma_{\rm crit,\infty}}\sqrt{G(c)}$ (2)
where $G(c)=\int_{0}^{c}{\rm d}x~{}xg(x)^{2}$ is a function of the halo’s
concentration $c$ only, well fitted by
$G(c)\approx\frac{0.131}{c^{2}}-\frac{0.375}{c}+0.388-5\times
10^{-4}c-2.8\times 10^{-7}c^{2}$ (3)
for $1\leqslant c\leqslant 200$, and the function $g(x)$ is defined by Eq.
(B2). The quantity $\left<Z\right>$ in Eq. (2) describes the effect of the
distribution of galaxy sources. In the above equations, $D_{\rm d}$ is the
angular-diameter distance to the lens, $\rho_{s}$ and $r_{s}$ parametrize the
NFW halo (see Appendix A), and $\Sigma_{\rm crit,\infty}$ is defined in
Appendix B.
We parametrize the redshift distribution of source galaxies by (Smail et al.
1994)
$n(z)=z^{\alpha}\exp\left[-\left(\frac{z}{z_{0}}\right)^{\beta}\right]$ (4)
with $\alpha=2$, $\beta=1.5$, and $z_{0}\approx z_{\rm m}/1.412$, where
$z_{\rm m}$ is the median redshift of the survey.
Figure 1 shows our fiducial survey’s selection function, in the mass-redshift
plane. Contours originate from the S/N of halos in that plane.
### 2.2. Weak lensing statistics
#### 2.2.1 Power spectrum
The halo model allows us to decompose the full three-dimensional matter power
spectrum along
$P_{\delta}(k)=P_{\delta}^{\rm res}(k)+P_{\delta}^{\rm unr}(k),$ (5)
where $P_{\delta}^{\rm unr}(k)$ is computed on the unresolved part of the
halos distribution, defined as the ensemble of halos with S/N smaller than a
certain threshold $\nu_{\rm th}$; $P_{\delta}^{\rm res}$ is computed on the
resolved part of the halos distribution, the ensemble of halos with
$\nu>\nu_{\rm th}$. Those names come naturally from the separation made on a
mass map between peaks (defined as resolved structures with $\nu>\nu_{\rm
th}$) and the rest, unresolved part of the map. The resolved part can be seen
as the most non-Gaussian features of the field.
Figure 2.— Weak lensing convergence power spectrum, when one single redshift
bin is used to measure it. The black curve shows the power spectrum when the
contribution of all halos is taken into account. Other curves make use of
halos whose signal-to-noise is less than a given threshold (the unresolved
part) : $\nu_{\rm th}=3$ (dashed), $\nu_{\rm th}=7$ (dash-dot) and $\nu_{\rm
th}=11$ (dash-dot-dot-dot).
The full weak lensing power spectrum $P_{\kappa}(\ell)$ follows the same
decomposition. Figure 2 shows the full power spectrum together with power
spectra computed on unresolved parts defined by the thresholds $\nu_{\rm
th}=3$, $\nu_{\rm th}=7$ and $\nu_{\rm th}=11$. The effect of removing
significant clusters is mostly apparent on intermediate scales. It is less
pronounced than when cutting the power spectrum with respect to halo masses,
as shown by Takada & Bridle (2007), because of the weak lensing selection
function, that mixes masses when integrating the matter power spectrum on
redshift.
The power spectrum is a measure of the Gaussian field only, and one must rely
on direct measures of non-Gaussianity to complement it, like the bispectrum or
cluster counts.
#### 2.2.2 Bispectrum
Three point statistics are the lowest statistics to capture non-Gaussian
features in a statistical field. The dark matter and the weak lensing
bispectra have been the object of numerous studies (e.g. Takada & Jain 2004),
most commonly based on the perturbation theory on large scales and on the
hyper-extended perturbation theory (HEPT) on smaller scales (e.g. Bernardeau
et al. 2002 for a review). Here, we compare and combine the constraints from
the weak lensing power spectrum and bispectrum evaluated in the same halo
model. This fills a gap in the literature, where such constraints come from
the HEPT approximation only.
The halo model allows us to separate the contribution from the unresolved and
the resolved parts of the field when computing the bispectrum, like we did for
the power spectrum. However, discarding the unresolved part from the
bispectrum removes too much useful signal, and thus suppresses much power in
the bispectrum. Therefore, we will always evaluate the bispectrum on the
entire (resolved and unresolved) data field.
#### 2.2.3 Weak-lensing-selected halo counts
Clusters of galaxies, tracing dark matter halos, have emerged from the
primordial Gaussian field through non-linear gravitational clustering, and are
those non-Gaussianities that one can detect with higher order statistics. Or
one can simply estimate their abundance to take non-Gaussianities into
account. In this paper, we concentrate on how to break the degeneracies from
the power spectrum by combining it with non-Gaussian features. As a result, we
will not focus on the constraints that halo counts alone can bring, but on how
they improve the constraints from the power spectrum.
Weak-lensing-selected clusters are detected on a mass map derived from the
data used to measure the power spectrum and the bispectrum. Hence, they come
at no extra cost, contrary to X-ray or optically-selected clusters. Defined as
peaks with S/N greater than a certain threshold $\nu_{\rm th}$, they are the
resolved part introduced above.
Once halos are selected, one can add information to their single S/N
distribution, e.g. by introducing their redshift and mass, and thus measure
their abundance as a function of those quantities. Hereafter, we investigate
the constraints on cosmological parameters provided by four kinds of halo
counts. With increasing information, we look at (1) counts as a function of
halo’s S/N only ; (2) counts as a function of halo’s mass ; (3) counts as a
function of halo’s redshift ; and (4) counts as a function of halo’s redshift
and mass.
In the remainder of this paper, we assume that the weak lensing selection
function is perfectly known. We neglect the nuisance parameters commonly used
to account for the imperfect knowledge of the relation between an observable
and a cluster’s mass (e.g. Hu 2003; Lima & Hu 2004; Majumdar & Mohr 2004).
### 2.3. Constraints on cosmological parameters
Fisher matrices allow one to characterize how a set of observables
$\mathbf{x}$ is able to constrain a set of parameters $\mathbf{p}$ around a
fiducial model. The associated Fisher matrix is defined by (e.g. Hu & Tegmark
1999)
$F_{\alpha\beta}=-\left<\frac{\partial^{2}\ln L}{\partial p_{\alpha}\partial
p_{\beta}}\right>.$ (6)
where $L(\mathbf{x},\mathbf{p})$ is the associated likelihood.
Given a fiducial model, the inverse of the Fisher matrix $\mathbf{F}^{-1}$
estimated in its neighborhood provides a lower limit to the parameters’
covariance matrix. It thus quantifies the best statistical error that can be
reached on the parameters,
$\sigma(p_{\alpha})\geqslant\sqrt{(\mathbf{F}^{-1})_{\alpha\alpha}}$, where
$\sigma(p_{\alpha})$ is the 1$\sigma$ error on parameter $p_{\alpha}$
marginalized over other parameters $p_{\beta}$.
The Fisher matrices for the weak lensing power spectrum, bispectrum and
cluster counts can be found e.g. in Cooray & Hu (2001b); Lima & Hu (2004); Hu
& Jain (2004); Takada & Jain (2004) and Takada & Bridle (2007).
We aim to determine how to best capture non-Gaussianities in order to break
the degeneracies from the power spectrum alone. We thus need to combine the
power spectrum with the bispectrum or the halo counts. To combine two
observables ${\rm D_{1}}$ and ${\rm D_{2}}$, we create the data vector ${\rm
D}=\\{{\rm D_{1}},{\rm D_{2}}\\}$. The associated Fisher matrix is
$F_{\alpha\beta}^{({\rm(1)+(2)})}={\rm D}_{,\alpha}^{T}\left[{\rm
C^{((1)+(2))}}\right]^{-1}{\rm D}_{,\beta}$.
The rigorous estimation of the combined observables’ covariance matrix,
${\rm C^{((1)+(2))}}=\left(\begin{array}[]{cc}{\rm C^{(1)}}&{\rm
C^{(1),(2)}}\\\ {\rm C^{(1),(2)}}&{\rm C^{(2)}}\end{array}\right),$ (7)
requires the knowledge of the cross-covariance ${\rm C^{(1),(2)}}$.
The weak lensing power spectrum is dominated by the most massive halos’
contribution, those which are the most likely to be detected and taken into
account in cluster counts. Hence, the power spectrum and halo counts are not
independent observables : their cross-covariance is non-zero, and must be
accounted for. Nonetheless, Takada & Bridle (2007) showed that neglecting it
changes the errors by only a few percent.
Here, we take a different approach, by separating the contribution of
different halos to the power spectrum (Eq. 5). We combine the counts of
clusters with S/N greater than the threshold $\nu_{\rm th}$ (the resolved
part) with the power spectrum created by all halos with S/N smaller than the
same threshold $\nu_{\rm th}$ (the unresolved part). If we neglect the
clustering between halos, the power spectrum in the unresolved part and the
resolved clusters are uncorrelated : the covariance between cluster counts and
the power spectrum vanishes, and we can simply add the Fisher matrices,
$F_{\alpha\beta}^{({\rm c+ps^{\rm unr}})}=F_{\alpha\beta}^{\rm
c}+F_{\alpha\beta}^{\rm ps^{\rm unr}},$ where the superscript ${}^{\rm ps^{\rm
unr}}$ stands for the power spectrum evaluated on the unresolved part.
As shown by Takada & Jain (2004), the cross-covariance between the power
spectrum and the bispectrum has no Gaussian feature, but arises from the five-
point correlation function of the shear field. In such a case, it is safe to
approximate the Fisher matrix of the combination between the power spectrum
and the bispectrum by the sum of the individual Fisher matrices,
$F^{\rm(ps+bisp)}_{\alpha\beta}\approx F^{\rm ps}_{\alpha\beta}+F^{\rm
bisp}_{\alpha\beta}.$
## 3\. Results
Figure 3.— Marginalized errors on cosmological parameters, as a function of
the threshold between resolved and unresolved parts. Top-left: $\Omega_{m}$.
Top-right : $\sigma_{8}$. Bottom-left : $w_{0}$. Bottom-right : $w_{a}$.
Different colors and line styles label different measurements : power spectrum
on all the data (dotted black), power spectrum on the unresolved part (dashed
red), bispectrum (dash-dotted yellow), and combinations of the power spectrum
on the unresolved part with clusters counts as a function of redshift and mass
(solid green), redshift only (solid blue), mass only (solid cyan), S/N only
(solid purple) and with the bispectrum (long-dashed orange). Note that the
$y$-scale is not the same for top and bottom panels.
In this section, we compute and compare the errors on cosmological parameters,
that can be reached by measuring the power spectrum alone, and when combining
it with either counts or the bispectrum. We assume a fiducial cosmology
described by the parameters
$(\Omega_{m},~{}\Omega_{\Lambda},~{}\Omega_{b},~{}\sigma_{8},~{}h,~{}n,~{}w_{0},~{}w_{a})=(0.3,0.7,0.04,0.9,0.7,1,-1,0)$.
The Universe’s curvature is a free parameter, set by the combination of
$\Omega_{m}$ and $\Omega_{\Lambda}$. The evolution of the dark energy equation
of state is parametrized by $w(a)=w_{0}+(1-a)w_{a}$, $a$ being the scale
factor. We assume a Euclid-like survey, 20,000 deg2 wide, with
$n_{g}=40~{}\mbox{galaxies arcmin}^{-2}$ with median redshift $z_{m}=1$. We
assume that the intrinsic shape r.m.s is $\sigma_{\rm int}=0.3$.
To count halos, we use 20 linear redshift bins, spanning the entire interval
accessible to our fiducial survey ($0<z<5$) - the highest bins being empty -
and 15 logarithmic mass bins ($10^{10}{\rm h}^{-1}{\rm
M}_{\odot}<M<10^{16}{\rm h}^{-1}{\rm M}_{\odot}$ \- where the lower bound is
small enough so that all clusters detectable by their weak lensing signal are
accounted for) to compute halo counts as a function of redshift and/or mass,
and 15 linear S/N bins for S/N peak counts ($2<S/N<20$). We estimate the power
spectrum in 10 redshift bins ($0<z<5$). For numerical reasons, we cannot use
more than 3 redshift bins to compute the tomography bispectrum. This is not
problematic, since Takada & Jain (2004) showed that the tomography bispectrum
S/N quickly converges to its maximum value, and almost reaches it for three
redshift bins.
Before reporting our results, we would like to refer to the work of Vallisneri
(2008) in which he shows the limitations of the Fisher matrix formalism. In
particular, one must pay particular attention to ill-conditioned Fisher
matrices, the inversion of which is likely to be wrong. Here, we check the
condition number of all of our Fisher matrices, and consider as good only
those with a small enough condition number so that their inversion can be
trusted. We find that the Fisher matrices for counts as a function of S/N, of
mass and of redshift, are all ill-conditionned. Therefore, such halo counts
cannot give reliable Fisher constraints by themselves. This is true for our
particular set of parameters though, and could not be true on other parameter
spaces. Looking for such spaces is beyond the scope of this paper, and we will
give constraints from halo counts only when combined with the power spectrum,
for which the condition number is low enough to be safe.
Table 1Marginalized absolute errors on cosmological parameters. | $P_{\kappa}$ | $P_{\kappa}^{\rm unr}$ | $B_{\kappa}$ | $n(M,z)$ | $P^{\rm unr}_{\kappa}+n(M,z)$ | $P^{\rm unr}_{\kappa}+n(z)$ | $P^{\rm unr}_{\kappa}+n(M)$ | $P^{\rm unr}_{\kappa}+n(\nu)$ | $P^{\rm unr}_{\kappa}+B_{\kappa}$ | $P_{\kappa}+B_{\kappa}$
---|---|---|---|---|---|---|---|---|---|---
$\Omega_{m}$ (0.3) | 0.0049 | 0.0027 | 0.0047 | 0.070 | 0.0024 | 0.0024 | 0.0026 | 0.0026 | 0.0021 | 0.0023
$\Omega_{\Lambda}$ (0.7) | 0.024 | 0.012 | 0.019 | 0.254 | 0.0099 | 0.010 | 0.011 | 0.011 | 0.0087 | 0.0094
$\Omega_{b}$ (0.04) | 0.015 | 0.015 | 0.016 | 0.656 | 0.013 | 0.014 | 0.014 | 0.014 | 0.008 | 0.008
$\sigma_{8}$ (0.9) | 0.0086 | 0.0039 | 0.0065 | 0.112 | 0.0032 | 0.0034 | 0.0036 | 0.0036 | 0.0029 | 0.0033
$h$ (0.7) | 0.091 | 0.086 | 0.122 | 3.59 | 0.077 | 0.079 | 0.079 | 0.085 | 0.048 | 0.051
$w_{0}$ (-1) | 0.040 | 0.041 | 0.031 | 0.283 | 0.023 | 0.024 | 0.025 | 0.025 | 0.016 | 0.016
$w_{a}$ (0) | 0.147 | 0.133 | 0.156 | 2.179 | 0.062 | 0.063 | 0.064 | 0.064 | 0.055 | 0.063
$w_{p}$ | 0.022 | 0.015 | 0.013 | 0.252 | 0.014 | 0.014 | 0.015 | 0.015 | 0.0049 | 0.0045
$n$ (1) | 0.020 | 0.021 | 0.038 | 1.00 | 0.018 | 0.018 | 0.019 | 0.020 | 0.011 | 0.012
The S/N threshold between the resolved and unresolved parts is set to $\nu_{\rm th}=6$. |
The central value for the parameter in our fiducial model is given into parenthesis in the first column. |
We ignore combining the power spectrum with the bispectrum and cluster counts at a time (see main text). |
The goal of this paper is to compare how cluster counts and the bispectrum
capture non-Gaussianities and break degeneracies left over by the power
spectrum. Consequently, we ignore combining cluster counts with the
bispectrum, as well as combining the three observables (power spectrum,
bispectrum and cluster counts) at a time.
We evaluate Fisher matrices for power spectra estimated both on all the data
and on the unresolved parts defined by different $\nu_{\rm th}$, for the
corresponding halo counts (halos with $\nu>\nu_{\rm th}$), and for the
bispectrum, and the combinations introduced in section 2.3. Figure 3 shows the
expected errors that they provide, marginalized over all eight parameters, as
a function of the S/N threshold $\nu_{\rm th}$. The power spectrum
$P_{\kappa}(<\nu_{\rm th})$ is estimated on the unresolved part (dashed red),
then combined with clusters counts of clusters (made as a function of mass and
redshift - solid green-, redshift only -solid blue-, mass only -solid cyan-
and S/N only -solid purple). The dotted black flat line shows the errors
brought by the measurement of the power spectrum on the entire data set.
Discarding the contribution of the most massive halos (i.e. the most non-
Gaussian features) from the power spectrum improves the errors on $\Omega_{m}$
and $\sigma_{8}$, the errors being smallest for an optimal threshold $\nu_{\rm
th,opt}\approx 6$. Although the counter-intuitive increase in the performance
that we observe when discarding the resolved part of the data is not so
surprising, since the impact of non-Gaussianities on the power spectrum is
lowered (Shaw et al. 2009 observe a similar trend for Sunyaev-Zel’dovich
clusters), we should note here that it could be exacerbated if our analysis is
close to the Fisher matrices formalism’s limits. Moreover, there may be an
optimal weighting scheme that would both allow for the signal in the resolved
part and minimize its non-Gaussian errors. Smaller errors on the cosmological
parameters would then be expected, since more information (with minimal noise)
would be taken into account than that we use here with the unresolved part.
Investigating these issues is beyond the scope of this paper, and we defer it
to a later study. The dark energy equation of state parameters $w_{0}$ and
$w_{a}$ are less affected by the separation of the power spectrum into the
resolved and the unresolved parts.
Although cluster counts do not provide strong constraints by themselves (see
below, Table 1), combining them with the power spectrum improves the errors it
provides, in particular when low S/N clusters are taken into account. When the
clusters considered are significant enough, all four kinds of counts perform
as well at capturing non-Gaussianities : the power spectrum provides most of
the Fisher information, and clusters help by breaking parameters degeneracies,
independently of the way they are considered. Figure 3 shows that this is true
for $\nu_{\rm th}\gtrapprox 6$, a safe threshold to discard false detections
(Pace et al. 2007). Therefore, that makes S/N peaks counts a direct and
efficient probe to combine with the power spectrum, as shown e.g. by Pires et
al. (2009b).
The dash-dotted yellow line on Fig. 3 shows the errors given by the
bispectrum, and the long-dashed orange line shows those given when combining
it with the power spectrum. At low S/N thresholds, when combined with the
power spectrum, the bispectrum and clusters counts give similar constraints on
the four parameters $\Omega_{m}$, $\sigma_{8}$, $w_{0}$ and $w_{a}$. At higher
thresholds the bispectrum gives better constraints than clusters counts on the
dark energy parameters, yet errors remain comparable for $\Omega_{m}$ and
$\sigma_{8}$.
Table 1 lists the marginalized errors on all of our eight parameters, as well
as those of the dark energy equation of state $w_{p}$ at the pivot point
(section 3.1), for the measurements and combinations considered, with
$\nu_{\rm th}=6$. Halo counts (excepted counts as a function of redshift and
mass) are plagued by ill-conditioned Fisher matrices, and we do not consider
them alone. Nevertheless, since counts as a function of S/N, of mass alone and
redshift alone contain less information than counts as a function of mass and
redshift, their expected constraints are weaker than those listed in table 1
for counts as a function of mass and redshift. We also list in the table the
errors provided when combining the bispectrum with the power spectrum
estimated on all the data. Although this combination is not explicitly shown
on figure 3, it can be drawn from the high S/N threshold-limit of the
combination of the bispectrum with the power spectrum estimated on the
unresolved part (long-dashed orange line). The table emphasizes the
conclusions given by figure 3. The four combinations of clusters counts with
the power spectrum give similar constraints, that are comparable to those
provided by the combination of the bispectrum and the power spectrum.
Furthermore, although the bispectrum and the power spectrum give comparable
errors, combining them breaks degeneracies and lowers errors by a factor
$\approx 2$.
Pires et al. (2009b) looked at how several statistics (clusters counts,
bispectrum, skewness, kurtosis) break the $\Omega_{m}$-$\sigma_{8}$
degeneracy, and reported the high efficiency of cluster counts. They found
that the skewness could also discriminate against models, but that the
bispectrum, when using only equilateral triangles, gave poor results. We
expand their work by taking into account all triangle configuration when
computing the bispectrum, and add it to clusters counts as an efficient way to
break degeneracies (see section 3.2.2). In particular, Fig. 3 shows that this
conclusion is true not only for the $\Omega_{m}$-$\sigma_{8}$ degeneracy, but
also for the dark energy equation of state.
### 3.1. Dark energy plane
Figure 4.— Dark energy FoM, as defined by Eq. (8), as a function of the
threshold between the resolved and the unresolved parts, for the same
measurements as in Fig. 3, labeled with the same colors and line styles. Thick
lines denote measurements made with the power spectrum estimated on the
unresolved part, and thin lines take the entire information into account in
the power spectrum estimation.
We now specialize the discussion to the dark energy figure of merit, as
defined e.g. by Albrecht et al. (2009),
$\mathcal{F}_{\rm de}=\frac{1}{\sigma(w_{p})\sigma(w_{a})}$ (8)
where $w_{p}$ is the dark energy equation of state at the pivot redshift,
where the dark energy is best constrained. The errors on $w_{p}$ are listed in
Table 1.
Figure 4 shows the dependence of this figure of merit as a function of the S/N
threshold $\nu_{\rm th}$ between the resolved and the unresolved parts of the
data. The color code and line styles are the same as those used for the
previous figure. Thick lines show the FoMs when the power spectrum is
estimated on the unresolved part, and thin lines show the FoMs when it is
estimated on all the data (we neglect the covariance between the power
spectrum and cluster counts).
Considering the power spectrum on the unresolved part only (dashed red)
slightly improves the FoM. This holds true when combining clusters with
$\nu\geqslant 5$. The bispectrum gives a higher figure of merit than cluster
counts, and depends very weakly on $\nu_{\rm th}$. Only at low S/N do the
figures of merit compare.
Clusters counts, when the selection function and the contamination are
controlled well enough to allow one to consider a very low S/N threshold, and
the bispectrum are comparable at extracting non-Gaussian aspects in a weak
lensing survey and at constraining dark energy. On the other hand, if we use a
safe threshold for clusters detection ($\nu_{\rm th}\approx 6$), the
bispectrum becomes better than clusters counts to break degeneracies and to
measure dark energy.
### 3.2. Comparison with other works
Figure 5.— Constraints on the $\Omega_{m}$-$\sigma_{8}$ plane, when all other
parameters are kept constant, for a 20,000 deg2 survey, and no tomography.
Filled ellipses show constraints from the power spectrum (cyan), S/N peak
counts (grey), cluster counts as a function of mass and redshift (green) and
bispectrum when all triangle configuration are taken into account (yellow);
the constraints obtained from the bispectrum when taking into account
equilateral triangles only get out of the frame. Open ellipses are constraints
from combining the power spectrum with the bispectrum (all triangle
configurations - blue), with peak counts (black) and cluster counts (red).
#### 3.2.1 Peak counting
In two recent papers, Dietrich & Hartlap (2009) and Kratochvil et al. (2009)
show with numerical simulations how powerful weak lensing peaks counting is at
constraining $\Omega_{m}$ and $\sigma_{8}$, and $w$, respectively. These
appear to contradict our results, in which we saw that if indeed, counting
peaks efficiently captures non-Gaussianity and breaks power spectrum
degeneracies, it does it only when combined with the power spectrum, but gives
poor constraints when used alone (Table 1). A concern expressed by the
aforementioned authors is that they varied only their parameters of interest
($\Omega_{m}$ and $\sigma_{8}$ or $w$), and defer marginalizing on other
parameters for later works. In our analysis, we indeed marginalized on eight
parameters. To show that this considerably lowers the constraints, we make the
same kind of analysis, by varying $\Omega_{m}$ and $\sigma_{8}$ only, as done
by Dietrich & Hartlap (2009). For simplicity, we do not separate the power
spectrum into resolved and unresolved parts, but consider the total power
spectrum only, without tomography. Since our analytical model cannot fully
reproduce their taking into account cosmologically significant false
detections (peaks that do not correspond to a single, localized, virialized
cluster, but that are instead due to mass alignments along the line of sight),
we use counts as a function of S/N as a proxy of their measure. We use a
detection threshold $\nu_{\rm th}=3$. Figure 5 shows how counting peaks (grey
filled ellipse) compares to the power spectrum (cyan filled ellipse) in that
case. The black open ellipse shows the combination of the two. As shown by
Dietrich & Hartlap (2009), peak counts indeed is an efficient cosmological
tool when varying $\Omega_{m}$ and $\sigma_{8}$ only. Marginalizing on other
parameters hampers its ability to perform, as we showed above, unless combined
with the power spectrum.
However, since our model does not catch filaments and significant false
detections, as Dietrich & Hartlap (2009) and Kratochvil et al. (2009) do, our
conclusion about cluster counts alone may be over-pessimistic. Therefore, it
will be valuable to adapt their simulations to higher dimensional parameter
estimations.
#### 3.2.2 Bispectrum : the need for all triangle configurations
Looking for the best statistics to discriminate models, Pires et al. (2009b)
conclude that peak counting is the best way of breaking the
$\Omega_{m}-\sigma_{8}$ degeneracy, with other third-order statistics (such
the skewness) also being able to break degeneracies. In that work the
bispectrum in particular was found to be weak at breaking degeneracies. This
apparent discrepancy with our findings here, where we find that combining the
bispectrum with the power spectrum is comparable to combining peak counting
with the power spectrum, comes from the fact that the bispectrum presented
here uses all triangle configurations, while the bispectrum in Pires et al.
(2009b) used only equilateral triangles.
To emphasize the importance of considering all triangle configurations when
computing the bispectrum, we compute the constraints expected on $\Omega_{m}$
and $\sigma_{8}$, while keeping all other parameters fixed, as in Pires et al.
(2009b), from the power spectrum, the bispectrum with all triangles
information or only equilateral triangles, and cluster counts. We use one
redshift bin. Our analytical model does not allow us to compute Pires et al.
(2009b)’s Wavelet Peak Counts (counts of clusters in different wavelet
scales), so we use the counts as a function of mass and redshift as a proxy.
Figure 5 shows how constraints from these measurements compare. The cyan
filled ellipse shows constraints from the power spectrum, the green one,
constraints from cluster counts, and the red open ellipse their combination.
As shown by Pires et al. (2009b), cluster counts are very efficient at
grabbing non-Gaussianities and breaking the power spectrum degeneracies in
that particular case. The yellow filled ellipse shows constraints from the
bispectrum, when the information from all triangle configurations is taken
into account, and the blue open ellipse shows constraints from combining this
bispectrum with the power spectrum. As mentioned above, the bispectrum (with
information from all triangle configurations) provides tight constraints, and
is as efficient as cluster counts at capturing non-Gaussian features. However,
we find that when taking into account only equilateral triangles when
computing constraints from the bispectrum, the errors on the parameters reach
100%, making such a measurement extremely low efficiency. The frame of Figure
5 is much too small to show the associated ellipse. Consequently, we concur
with Pires et al. (2009b) when claiming that the bispectrum measured with
equilateral triangles only does not allow one to break the power spectrum
degeneracy between $\Omega_{m}$ and $\sigma_{8}$.
Coming back to our 8-parameters analysis, we notice the same behavior as that
mentioned in the above paragraph. To capture non-Gaussian features and
efficiently break degeneracies left over by the power spectrum, one has to
take into account all triangle configurations when measuring the weak lensing
bispectrum. Equilateral triangles only do not provide any cosmological
constraints.
## 4\. Conclusion
We investigated how to best capture non-Gaussianities in weak lensing surveys
so as to break degeneracies left over by the power spectrum in weak lensing
analysis, by means of combining it with galaxy clusters counts and with the
weak lensing bispectrum. To this end, we computed, in the halo model
framework, the power spectrum, the bispectrum and four kinds of halo counts
(as a function of S/N, mass, redshift, and mass and redshift), in a Euclid-
like survey. We have computed the errors on cosmological parameters brought by
the combination of the power spectrum with the bispectrum, and of the power
spectrum with clusters counts. We have emphasized our discussion on the
varying dark energy equation of state, the mass density parameter $\Omega_{m}$
and the amplitude of density fluctuations $\sigma_{8}$, marginalized on a
8-parameter cosmological model. We did not take any prior from outside weak
lensing (e.g. CMB priors), so as to show how a weak lensing survey alone can
allow us to capture non-Gaussianities, and what constraints we can expect from
it on cosmological parameters.
When computing the power spectrum, we have separated the contribution of the
resolved and the unresolved parts of the data, delimited by a S/N threshold
$\nu_{\rm th}$. Doing so, we noted an improvement in errors on cosmological
parameters.
Although peak counts alone are powerful at constraining $\Omega_{m}$ and
$\sigma_{8}$ when other parameters are kept constant (as shown by Dietrich &
Hartlap 2009), we have seen that marginalizing on a 8-parameter space make
them provide only weak constraints, unless combined with the power spectrum.
We have indeed shown that when combined with the power spectrum, cluster
counts provide percent level marginalized errors on cosmological parameters.
Furthermore, provided that the threshold is high enough ($\nu_{\rm th}\approx
6$), all four kinds of cluster counts we consider give similar constraints.
Hence, combining the power spectrum with S/N peak counts gives constraints
nearly as tight as combining the power spectrum with clusters counted as a
function of redshift and mass. Galaxy clusters provide the same non-Gaussian
information, independently of the way they are taken into account, as long as
they are combined with the power spectrum : different cluster counts break
degeneracies in a similar way. This will allow us to put stringent constraints
on cosmological parameters from weak lensing alone, by combining S/N peaks
with the power spectrum, without any extra cost and requirement of follow-up
for clusters’ redshift and mass measurement.
We have shown that the weak lensing bispectrum efficiently captures non-
Gaussianities, and is marginally more efficient than cluster counts at
breaking degeneracies left over by the power spectrum. Furthermore, the
bispectrum alone provides constraints comparable to those given by the power
spectrum. We have expanded Pires et al. (2009b)’s conclusions by adding the
bispectrum to clusters counts as efficient tools to capture non-Gaussianities.
We also noted that equilateral triangles alone do not bring significant
information. To provide percent level constraints, all triangle configurations
must be accounted for when measuring the weak lensing bispectrum.
All our results are based on a Euclid-like survey, and may not be directly
applicable to current surveys such as the Canada-France-Hawaii Telescope
Legacy Survey (CFHTLS) or the Cosmic Evolution Survey (COSMOS). In particular,
on current surveys, the cosmic variance could play an important role in
lowering the ability of the bispectrum to be more efficient than S/N peak
counts. How those two measures compare to capture non-Gaussian features could
then depend on the survey’s characteristics. This question is beyond the scope
of this paper, and will be addressed in a subsequent work.
We have assumed that all systematics were well accounted for so as not to bias
our results. Although that can seem over-optimistic, the amount of efforts
currently underway to correct for various systematics gives us good reasons to
think that our assumption is likely to be met when a Euclid-like survey is
undertaken. Moreover, allowing for systematics requires a more sophisticated
analysis than the Fisher matrix (Amara & Réfrégier 2008). We defer it to a
future paper, looking at how systematics enter in the correlations between
different statistics.
In light of our results, it appears that weak lensing surveys alone will be
able to reach the percent accuracy on the dark energy equation of state
parameters. Moreover, it provides us with ways to cross-check the parameters’
measurements. For instance, after measuring the power spectrum, counting S/N
peaks provides an easy and fast way to optimally capture non-Gaussianities and
tighten constraints. Then, the bispectrum and its combination with the power
spectrum should give consistent parameters’ estimation and similar
constraints. However, measuring the bispectrum on half the sky will constitute
a real challenge. Along with already existing robust and fast algorithms
(Jarvis et al. 2004; Zhang & Pen 2005), new ones are being developed (e.g.
Bergé et al in prep and Semboloni et al in prep) to measure either the 3PCF in
real space or the bispectrum in Fourier space (as proposed by Pires et al.
2009a). In the meantime, more and more precise numerical simulations will be a
needed tool to test and tune those algorithms, and current surveys such as the
CFHTLS and COSMOS are ideal benchmark to test codes in the real world.
We want to thank Sandrine Pires, Benjamin Joachimi, Xun Shi and Masahiro
Takada for fruitful discussions and for their help in comparing bispectrum
codes, as well as Jochen Weller. We also thank Jason Rhodes and Sedona Price
for useful comments on this manuscript. JB is supported by the NASA
Postdoctoral Program, administered by Oak Ridge Associated Universities
through a contract with NASA. This work was carried out at Jet Propulsion
Laboratory, California Institute of Technology, under a contract with NASA. We
thank the anonymous referee for useful comments.
## Appendix A A - Halo model
Presented as an alternative to fitting functions based on numerical
simulations, the halo model for large-scale structures describes structures as
spherical haloes - see Cooray & Sheth (2002) for a review. In this framework,
the large-scale structures statistics can be described by correlations within
a same halo (the profile of which affects them) and between different halos
(the clustering of halos is thus naturally taken into account). Although being
simplistic, the model agrees fairly well with common numerical simulations.
Here, we summarize its key ingredients : the mass function, the profile of
halos and the halo bias. We also give the expressions for the 3D dark matter
power spectrum and bispectrum.
### A.1. Mass function
The comoving number of virialized halos, whose mass ranges between $M$ and
$M+{\rm d}M$ and redshift ranges between $z$ and $z+{\rm d}z$, is given by
(Press & Schechter 1974) :
$n(M,z){\rm d}M=\frac{\rho_{0}}{M}\frac{{\rm d}\nu(M,z)}{{\rm d}M}f(\nu){\rm
d}M$ (A1)
where $\rho_{0}$ is the Universe’s current background density and
$\nu(M,z)=\delta_{c}(z)/\sigma(M)$. $\delta_{c}(z)$ is the non-linear
overdensity of a halo collapsing at redshift $z$. In a dark energy universe,
$\delta_{c}(z)$ depends weakly on cosmology, especially on the dark energy’s
equation of state $w$. We use Weinberg & Kamionkowski (2003)’s fitting
function to compute $\delta_{c}(z;w)$. This fitting function is valid for
constant $w$ only, therefore we compute it using an effective equation of
state $w_{\rm eff}(a)=w_{0}+(1-a)w_{a}$. The r.m.s of the mass fluctuations in
a sphere containing a mass $M$ at redshift $z$ is defined by
$\sigma^{2}(M)=\frac{1}{(2\pi)^{3}}\int{\rm d}^{3}kP^{\rm lin}(k)|W(kR)|^{2}$
(A2)
where $P^{\rm lin}(k)$ is the three-dimension linear matter power spectrum,
$R=(3M/4\pi\bar{\rho})^{1/3}$ is the radius of the considered sphere, and
$\bar{\rho}$ is the mean background density. The window function
$W(x)=(3/x^{3})\left[\sin(x)-x\cos(x)\right]$.
A couple of mass functions have been described in the literature (Press &
Schechter 1974; Sheth & Tormen 1999; Jenkins et al. 2001; Tinker et al. 2008).
We use that of Jenkins et al. (2001),
$f(\sigma)=a_{j}\exp(-|\ln\sigma^{-1}+b_{j}|^{c_{j}})$, where $a_{j}=0.315$,
$b_{j}=0.61$ and $c_{j}=3.8$.
### A.2. Profile
The top-hat collapse model for structure formation describes the matter infall
on the gravitational wells until virialization. At this stage, non-linear
physics takes place, that further modifies the virialized halo profile, and
that must be assessed through numerical simulation. Navarro et al. (1996)
showed that the mass profile of such virialized halos can be described by a
“universal” profile,
$\rho(r|M)=\frac{\rho_{s}}{(r/r_{s})^{\alpha}(1+r/r_{s})^{\beta}},$ (A3)
where $r_{s}$ and $\rho_{s}$ correspond to a characteristic radius and
density, respectively, and ($\alpha,~{}\beta$)=(1,2) for the usual NFW
profile, that we use in this paper.
The mass of a NFW halo is then given by $M=\int_{0}^{r_{\rm vir}}{\rm d}r4\pi
r^{2}\rho(r|M)$, where $r_{\rm vir}$ is the halo’s virial radius. In theory, a
NFW is infinite, but we assume here that it is truncated at $r_{\rm vir}$. The
virial and characteristic radius are linked by $r_{\rm vir}=cr_{s}$, where $c$
is the halo’s concentration, that can be parametrized, following Bullock et
al. (2001) and Dolag et al. (2004), by
$c(M,z)=\frac{c_{0}}{1+z}\left(\frac{M}{M_{*}(z=0)}\right)^{-\beta(z)}.$ (A4)
In this equation, $M_{*}(z=0)$ is the non-linear mass scale, defined as
$\nu(M_{*},z=0)=1$. In contrast to earlier works (Bullock et al. 2001), we add
an extra redshift-dependence in the parameter $\beta$. Based on Zhao et al.
(2003, 2008); Duffy et al. (2008) and Gao et al. (2008), it allows us to avoid
the catastrophic drop in concentration observed on simulations for high
redshift, massive halos, as parametrized by Bullock et al. (2001) with a
constant $\beta$. We find that $c_{0}=8$ and $\beta(z)=0.13-0.05z$, besides
agreeing with Duffy et al. (2008); Gao et al. (2008); Zhao et al. (2008), and
with Bullock et al. (2001) at low redshift, gives us the best agreement
between the halo model and the Smith et al. (2003)’s power spectrum and
bispectrum. One should note that the concentration has a significant impact on
the 1-halo term of the halo model power spectrum and bispectrum, as shown by
Huffenberger & Seljak (2003), and thus a thorough description is needed.
### A.3. Bias
The bias describes how halos cluster with respect to the matter distribution.
It is well described at first order by fitting functions of the form (Mo &
White 1996; Sheth et al. 2001),
$b(M,z)=1-\frac{q\nu-1}{\delta_{c}(z)}+\frac{2p/\delta_{c}(z)}{1+(q\nu)^{p}}$
(A5)
with $q=0.707$ and $p=0.3$. We neglect the higher order terms of the bias.
### A.4. Matter power spectrum and bispectrum
The halo model describes the correlation functions of the density field, in
real-space, as the sum of the correlation between points belonging to a same
halo and of the correlation between points in different halos. The same
contributions appear in Fourier space, when defining the power spectrum and
bispectrum, which are thus the sum of a 1-halo term and a 2-halo term (and
3-halo term for the bispectrum) (e.g. Cooray & Hu 2001a),
$P_{\delta}(k)=P_{\delta}^{1h}(k)+P_{\delta}^{2h}(k).$
The 1-halo component is given by
$P_{\delta}^{1h}(k)=\int{\rm
d}mn(m)\left(\frac{m}{\bar{\rho}}\right)^{2}|u(k|m)|^{2}$ (A6)
and the 2-halo term is given by
$P_{\delta}^{2h}(k)=\int{\rm
d}m_{1}n(m_{1})\left(\frac{m_{1}}{\bar{\rho}}\right)u(k|m_{1})\int{\rm
d}m_{2}n(m_{2})\left(\frac{m_{2}}{\bar{\rho}}\right)u(k|m_{2})P_{hh}(k|m_{1}m_{2}).$
(A7)
The function
$u(k|m)=\int_{0}^{r_{\rm vir}}{\rm d}r4\pi r^{2}\frac{\sin
kr}{kr}\frac{\rho(r|m)}{m}$ (A8)
is the Fourier transform of the dark matter density profile. The term
$P_{hh}(k|m_{1}m_{2})$ represents the power spectrum of halos of mass $m_{1}$
and $m_{2}$ and is approximated by
$P_{hh}(k|m_{1}m_{2})\approx\Pi_{i=1}^{2}b_{i}(m_{i})P_{\delta}^{\rm lin}(k)$,
where $b_{i}(m_{i})$ is halo $i$’s bias and $P_{\delta}^{\rm lin}(k)$ is the
matter linear power spectrum. We use the Eisenstein & Hu (1998)’s transfert
function to evaluate $P_{\delta}^{\rm lin}(k)$.
The 3D matter bispectrum can be decomposed into terms coming from one, two and
three halos :
$B_{\delta}=B_{\delta}^{1h}+B_{\delta}^{2h}+B_{\delta}^{3h},$ (A9)
with
$B_{\delta}^{1h}(k_{1},k_{2},k_{3})=I_{3}^{0}(k_{1},k_{2},k_{3}),$ (A10)
$B_{\delta}^{2h}(k_{1},k_{2},k_{3})=I_{2}^{1}(k_{1},k_{2})I_{1}^{1}(k_{3})P_{\delta}(k_{3})+C.P.,$
(A11)
and
$B_{\delta}^{3h}(k_{1},k_{2},k_{3})=[2J(k_{1},k_{2},k_{3})I_{1}^{1}(k_{3})+I_{1}^{2}(k_{3})]\\\
\times
I_{1}^{1}(k_{1})I_{1}^{1}(k_{2})P_{\delta}(k_{1})P_{\delta}(k_{2})+C.P.,$
(A12)
where $C.P.$ denotes circular permutations. The $J$ function is given by the
second-order perturbation theory (Bernardeau et al. 2002) :
$J(k_{1},k_{2},k_{3})=1-\frac{2}{7}\Omega_{m}^{-2/63}+\left(\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{2k_{1}k_{2}}\right)^{2}\\\
\times\left[\frac{k_{1}^{2}+k_{2}^{2}}{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}+\frac{2}{7}\Omega_{m}^{-2/63}\right].$
(A13)
The function $I_{\mu}^{\beta}$ is defined through the Fourier transform of the
halo profile $u(k|m)$ (Eq. A8) :
$I_{\mu}^{\beta}(k_{1},\dots,k_{\mu};~{}z)=\int
dm\left(\frac{m}{\rho_{b}}\right)^{\mu}n(m,z)b_{\beta}(m)\\\ \times
u(k_{1},m)\dots u(k_{\mu},m)$ (A14)
with $b_{0}=1$.
## Appendix B B - Signal-to-noise ratio for a NFW halo
Here, we specialize the discussion of section 2.1 to an NFW halo. To evaluate
equation (1) in this case, we first need to derive the convergence of such a
halo. The mass density, projected along the line-of-sight, of a NFW halo at
redshift $z$, with concentration $c$, is given by :
$\Sigma(x)=\int_{-\sqrt{c^{2}-x^{2}}}^{\sqrt{c^{2}-x^{2}}}{\rm
d}z\rho(x,z)=2\rho_{s}r_{s}g(x)$ (B1)
where $x=r/r_{s}$. The function $g$ depends only on the distance to the halo’s
center, and is given by (as long as $c\geqslant 1$) (Wright & Brainerd, 2000)
$g(x)=\left\\{\begin{array}[]{l}-\frac{\sqrt{c^{2}-x^{2}}}{(1-x^{2})(1+c)}+\frac{1}{(1-x^{2})^{3/2}}{\rm
arccosh}\frac{x^{2}+c}{x(1+c)},\\\ \hskip 133.72795pt(x<1)\\\
\frac{\sqrt{c^{2}-1}}{3(1+c)}\left(1+\frac{1}{1+c}\right),\\\ \hskip
133.72795pt(x=1)\\\
-\frac{\sqrt{c^{2}-c^{2}}}{(1-x^{2})(1+c)}-\frac{1}{(x^{2}-1)^{3/2}}{\rm
arccos}\frac{x^{2}+c}{x(1+c)},\\\ \hskip 116.6563pt(1<x\leqslant c)\\\ 0\hskip
125.19212pt(x>c).\end{array}\right.$ (B2)
The halo’s convergence is then, if we assume that all sources are at the same
redshift $z_{s}$,
$\kappa(x,z_{s})=\frac{\Sigma(x)}{\Sigma_{\rm
crit}}=2\frac{\rho_{s}r_{s}}{\Sigma_{\rm crit}}g(x),$ (B3)
where the critical density is defined as
$\Sigma_{\rm crit}=\frac{c^{2}}{4\pi G}\frac{D_{\rm s}}{D_{\rm d}D_{\rm ds}}.$
(B4)
In equation (B4), $c$ is the speed of light, $G$ the gravitation constant, and
$D_{\rm s}$, $D_{\rm d}$ and $D_{\rm ds}$ are the angular-diameter distances
of the source, of the lens, and between the lens and the source, respectively.
To take the redshift distribution $p_{z}(z_{s})$ of source galaxies, we follow
Seitz & Schneider (1997), Bartelmann & Schneider (2001) and Weinberg &
Kamionkowski (2002) and introduce the function
$Z(z_{s};z_{d})\equiv\frac{{\rm lim}_{z_{s}\rightarrow\infty}\Sigma_{\rm
crit}(z_{d};z_{s})}{\Sigma_{\rm crit}(z_{d};z_{s})}=\frac{\Sigma_{{\rm
crit},\infty}}{\Sigma_{\rm crit}(z_{d};z_{s})},$ (B5)
where $z_{d}$ is the lens’ redshift, and $\Sigma_{{\rm crit},\infty}$ is the
critical density for a source at infinite redshift. This function allows us to
project sources with a certain redshift distribution on a single redshift
$z_{s}$ verifying $Z(z_{s})=\left<Z\right>$, with $\left<Z\right>=\int{\rm
d}z_{s}p_{z}(z_{s})Z(z_{s};z_{d})$. One can then link the halo’s convergence
with a source at $z_{s}$ to that with infinite redshift source,
$\kappa(x,z_{s})=\kappa(x)Z(z_{s};z_{d})$ (Seitz & Schneider 1997). The
convergence given by a source population distributed in redshift along
$p_{z}(z_{s})$ is then given by
$\kappa(x,p_{z}(z_{s})=\left<Z\right>\kappa(x)$.
The signal-to-noise ratio (Eq. 1), for a NFW halo is then
$\nu_{\rm
NFW}=2\sqrt{2\pi}\left<Z\right>\frac{\sqrt{n_{g}}}{\sigma_{\gamma}}\frac{\rho_{s}r_{s}^{2}}{D_{\rm
d}\Sigma_{\rm crit,\infty}}\sqrt{G(c)}$ (B6)
where $G(c)=\int_{0}^{c}{\rm d}x~{}xg(x)^{2}$ is a function of the halo’s
concentration only, and is well fitted, for $1\leqslant c\leqslant 200$ (the
maximum deviation being of order 0.5%), by
$G(c)\approx\frac{0.131}{c^{2}}-\frac{0.375}{c}+0.388-5\times
10^{-4}c-2.8\times 10^{-7}c^{2}.$ (B7)
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|
arxiv-papers
| 2009-09-02T20:42:53 |
2024-09-04T02:49:05.007489
|
{
"license": "Public Domain",
"authors": "Joel Berge, Adam Amara, Alexandre Refregier",
"submitter": "Joel Berg\\'e",
"url": "https://arxiv.org/abs/0909.0529"
}
|
0909.0535
|
# Mode Bifurcation and Fold Points of Complex Dispersion Curves for the
Metamaterial Goubau Line
P. L. Overfelt Klaus Halterman klaus.halterman@navy.mil Simin Feng D. R.
Bowling Research and Intelligence Department, Physics Division, Naval Air
Warfare Center, China Lake, California 93555, USA
###### Abstract
In this paper the complex dispersion curves of the four lowest-order
transverse magnetic modes of a dielectric Goubau line ($\epsilon>0,\mu>0$) are
compared with those of a dispersive metamaterial Goubau line. The vastly
different dispersion curve structure for the metamaterial Goubau line is
characterized by unusual features such as mode bifurcation, complex fold
points, both proper and improper complex modes, and merging of complex and
real modes.
The Goubau line (G-Line) has been known and studied since Sommerfeld and later
Goubau considered applications using non-radiating surface waves on
transmission lines sommer ; gball . While Sommerfeld analyzed a long
cylindrical metallic wire as the transmission line of interest, Goubau
realized that adding a dielectric outer sheath to the wire reduced the radial
extent of the electromagnetic (EM) field and thus the dimensions of the
associated excitation device. Interestingly, while the Sommerfeld wave can
exist only on a conductor of finite conductivity, the Goubau wave can exist
even when the inner conductor is assumed to have perfect conductivity. The
G-Line has been investigated since the first part of the twentieth century,
and its guided modes are well known sommer ; gball ; strat ; wald ; semen . As
with all open waveguide structures, the G-Line spectrum consists of a finite
discrete set of guided modes with purely real longitudinal propagation
constants and an infinite continuum of radiation modes. Also present on open
lossless structures are leaky waves marc ; marc2 characterized by discrete
complex longitudinal propagation constant solutions to the dispersion
equation, but which are improper solutions of Maxwell’s equations in that
these solutions decay longitudinally but do not obey the transverse radiation
condition and thus may only be used in restricted regions of space. Improper
waves are not considered part of an open waveguide spectrum and are often
referred to as “nonmodal” or “nonspectral” marc . Despite this fact, leaky
waves have found great usefulness in certain applications, particularly those
related to leaky wave antennas gold . Some authors have referred to the
complex solutions of the circular dielectric rod and the standard G-Line
dispersion equations as leaky modes and have considered them on a more or less
equal footing with the guided modes kim ; kim2 . The leaky waves of even the
standard G-Line are still not well characterized (only the transverse magnetic
(TM) solutions have been considered in detail kim ) but on that structure, all
complex leaky wave solutions of the characteristic equation have EM field
components that diverge as the radial coordinate increases to infinity and are
thus improper.
Figure 1: Schematic of the Goubau line geometry. The metal core of radius $a$
is a perfect conductor.
In this Letter the G-Line geometry (see Fig. 1) is used with a negative index
of refraction dispersive metamaterial (NIM) elef ; shad replacing the usual
dielectric layer. Under these circumstances the metamaterial G-Line spectrum
consists of guided modes, radiation modes, improper complex waves, and proper
complex modes. In the following we consider only the symmetric transverse
magnetic (${\rm TM}_{0n}$) solutions of the metamaterial G-Line (MM G-line).
The transverse electric (${\rm TE}_{0n}$) and hybrid modes of the MM G-line
will be considered subsequently.
Using the geometry in Fig. 1, the characteristic equation for the G-Line with
perfectly conducting inner conductor and assuming a $\theta,z,t$ dependence of
$e^{-i(m\theta+\gamma z-\omega t)}$, can be written in the general form,
$\displaystyle\overline{\gamma}^{2}m^{2}\Bigl{(}\frac{1}{s^{2}}-\frac{1}{w^{2}}\Bigr{)}^{2}{\cal
Z}{\cal Y}+{\cal S}{\cal T}=0,$ (1)
where $\overline{\gamma}=\overline{\beta}+i\overline{\alpha}$ is the
normalized complex longitudinal propagation constant. The longitudinal phase,
$\overline{\beta}$, and attenuation $\overline{\alpha}$ are both normalized by
$k_{0}\equiv\omega/c$, the free space wavenumber. Considering the $m=0$ case,
Eq. (1) reduces to ${\cal T}=0$ for TM modes, where,
$\displaystyle{\cal T}=\frac{\epsilon_{2}{\cal W}}{w}-\epsilon_{3}{\cal
Z}g=0,$ (2)
${\cal S}={\mu_{2}{\cal X}}/{w}-\mu_{3}{\cal Y}g$, and $g\equiv-
H_{1}^{(2)}(s)/(sH_{0}^{(2)}(s))$. We also define, ${\cal
Z}=J_{0}(w)Y_{0}(v)-J_{0}(v)Y_{0}(w)$, ${\cal
Y}=-J_{0}(w)Y_{1}(v)+J_{1}(v)Y_{0}(w)$, ${\cal
W}=-J_{1}(w)Y_{0}(v)+J_{0}(v)Y_{1}(w)$, and ${\cal
X}=J_{1}(w)Y_{1}(v)-J_{1}(v)Y_{1}(w)$, with the $J$’s, $Y$’s and $H$’s the
usual Bessel functions of the first and second kind, and the Hankel functions
of the second kind, respectively. The dimensionless wavenumbers are,
$v=k_{2}a$, $w=k_{2}b$, and $s=k_{3}b$. The transverse propagation wave
numbers are given by
$k_{j}=k_{0}\sqrt{\epsilon_{j}\mu_{j}-\overline{\gamma}^{2}}$, for $j=2,3$.
For these ${\rm TM}_{0n}$ modes, the EM fields in the $j$th region are
generally written as, ${\bm{H}}_{j}=H_{j,\theta}\hat{\bm{\theta}}e^{-i(\gamma
z-\omega t)}$, and
${\bm{E}}_{j}=(E_{j,\rho}\hat{\bm{\rho}}+E_{j,z}\hat{\bm{z}})e^{-i(\gamma
z-\omega t)}$, for propagation in the positive $z$ direction. Maxwell’s
equations and the cylindrical symmetry result in the following transverse
fields, $H_{j,\theta}=-i\epsilon_{j}k_{0}/\kappa_{j}^{2}\partial
E_{j,z}/\partial\rho$, and $E_{j,\rho}=-i\gamma/\kappa_{j}^{2}\partial
E_{j,z}/\partial\rho$. The forms of (1) - (2) above are most suitable for
emphasizing material parameter changes in both $\epsilon$ and $\mu$ of Regions
2 and 3, although they have been derived previously in alternate forms semen .
Figure 2: Dispersion for standard G-line
We solve Eq. (2) numerically for its complex zeros as a function of frequency
by employing a recently developed complex root finding algorithm. Fig. 2 shows
the four lowest-order ${\rm TM}_{0n}$ ($n=0,1,2,3$) modes as functions of
frequency (in GHz) for the standard G-Line. Part (a) shows the normalized
longitudinal phase vs. frequency while Part (b) shows normalized longitudinal
attenuation vs. frequency. In Fig. 2 the parameters $\epsilon_{2}=5$,
$\mu_{2}=1$, $a=5$ mm, $b=10$ mm were used to compare our results with those
in Ref. kim, . In particular, note the lowest-order ${\rm TM}_{00}$ mode
(black curve) that has no cutoff frequency and $\alpha=0$. Except for this
mode, our results are consistent with Ref. kim, . We have assumed that the
outside region is air and thus $\epsilon_{3}=\mu_{3}=1$. At a given frequency,
when the imaginary part of a root is zero, that root represents a guided or
bound solution. When the imaginary part is nonzero, that root represents a
complex leaky wave solution of (2). Upon calculating the EM field components
associated with certain leaky wave solutions in Fig. 2, all discrete complex
solutions of the standard G-line (with Region 2 index of refraction greater
than one) are improper marc ; kim . Because the outside region is air, cutoff
occurs when $\overline{\gamma}=1$ or $s=0$. When
$1<\overline{\beta}<\sqrt{5}$, and $\overline{\alpha}=0$, the well-known
guided modes result. When $\overline{\beta}<1$, $\overline{\alpha}\neq 0$, the
waves are leaky. Also note that an infinite number of higher order leaky waves
can occur at lower frequencies These leaky waves have high frequency cutoffs
but no low frequency cutoffs. Generally these waves have
$|\overline{\alpha}|\gg 1$ and thus are not so far found to be particularly
useful. In Fig. 2, the real and imaginary parts of the longitudinal
propagation constant are correlated using color. Thus the lowest-order mode
(black) has zero attenuation and is purely guided [Fig. 2(a)], the second
lowest-order mode (magenta) is leaky at the lower frequencies with its
attenuation magnitude increasing as the frequency decreases (see Fig. 2(a) and
(b)). For this mode as the frequency increases up to cutoff at about 14.9 GHz,
the associated attenuation decreases in magnitude until this mode becomes a
guided mode once cutoff is passed (magenta $\rightarrow$ blue). The turquoise
blue and red curves show the next higher-order modes which for much of the
frequency range are leaky with $|\alpha|\gg 1$.
If the dielectric layer of Region 2 is replaced by a negative index of
refraction MM, the TM mode structure is vastly different. The MM is
characterized by the following dispersion formulas,
$\mu_{2}(f)=\mu_{l}+Ff^{2}/(f_{0}^{2}-f^{2}-i\Gamma f)$, and,
$\epsilon_{2}(f)=\epsilon_{h}+f_{p}^{2}/(f_{r}^{2}-f^{2}-i\Gamma f)$. We take
the filling factor, $F$, to be $0.25$, and the damping factor, $\Gamma$, to be
zero. The low and high frequency limits are $\mu_{l}=-1$ and
$\epsilon_{h}=-5$, respectively. The resonance frequencies are, $f_{0}=0.75$
GHz, $f_{r}=0.5$ GHz, and $f_{p}=0.25$ GHz.
Figure 3: Dispersion for the MM G-line. The curves in (c) and (d) give a
magnified view of $\overline{\beta}$ and $\overline{\alpha}$, respectively,
near the bifurcation point (Point 4).
Figure 3 contains the four lowest-order TM modes assuming a metamaterial
G-Line with both $\epsilon_{2}$ and $\mu_{2}$ negative over the frequency
range of interest. Fig. 3(a) shows the normalized longitudinal phase vs.
frequency, part (b) shows the normalized longitudinal attenuation vs.
frequency, where $|\overline{\alpha}|$ is small, and also shows the normalized
longitudinal attenuation on a larger scale. In Fig. 3, $\epsilon_{2}\approx-5$
and $\mu_{2}\approx-1$. The radii, $a$ and $b$, are the same as for the
standard G-line in Fig. 2. Referring to the various mode “paths” or “branches”
in Fig. 3, note that the green curve now has a leaky part (below about 3 GHz)
and the portion of the green curve with $\overline{\beta}>1$ now has
$\overline{\alpha}\neq 0$. As the frequency decreases, the attenuation peaks
in magnitude between 4 and 5 GHz (green curve) and then starts to decrease.
Thus the green path is a wave with a complex $\overline{\gamma}$. This path is
also improper (until it merges with the black curves at much higher
frequency). Thus the fundamental guided mode of the standard G-line is
replaced by a complex mode that is improper. The mode represented by the
magenta path is also complex with $\overline{\alpha}$ nonzero until point 4 is
reached. However this path is proper. At point 4 ($f\approx 10.025$ GHz),
there is a mode bifurcation [see Fig. 3(c) and (d)], as has been observed in
catastrophe and bifurcation theories. Catastrophe theory attempts to study how
the qualitative nature of solutions of equations depends on the parameters
appearing in those equations hanson ; poston . These techniques are invaluable
in analyzing the qualitative changes in system dynamical characteristics by
small perturbations. From this standpoint, the dispersion relationship, ${\cal
T}=0$, is cast as a nonlinear equation in $\overline{\beta}$,
$\overline{\alpha}$, and $f$ with parameters $\epsilon_{1}$, $\epsilon_{2}$,
$\mu_{1}$, $\mu_{2}$, $a$, and $b$. Using catastrophe theory, areas of unusual
qualitative interest such as the fold point at Point 4 can be determined.
Using numerical techniques alone to solve the dispersion relation over a
frequency range is difficult due to its “dense” set of zeros at each
frequency. Point 4 is a complex fold point (a point where, because no material
losses in the line have been assumed as yet, $\overline{\gamma}$,
$-\overline{\gamma}$, $\overline{\gamma}^{*}$, and $-\overline{\gamma}^{*}$
all meet at the same frequency) yak ; hanson ; poch ; poston . As the
frequency increases past the fold point, $\overline{\alpha}\rightarrow 0$
along both upper and lower black paths. The black paths near points 2 and 3
are therefore both guided modes immediately after bifurcation.
Thus, in Fig. 3 there are an improper complex set of roots, a proper complex
set of roots, and a mode bifurcation at a fold point in (a) near $f\approx 10$
GHz past which there are two guided modes (i.e., $\overline{\alpha}=0$ on both
black paths after 10 GHz). By considering the two guided paths (2 and 3) where
$\overline{\alpha}=0$, we find that the black upper path always has a slope
opposite in sign to the lower path until the first cutoff frequency, $f\approx
14.9$ GHz, and thus is backward traveling in this frequency region. At cutoff
the lower path near point 3 has zero slope and after cutoff, the slope becomes
positive. Also after cutoff, this path becomes complex with
$|\overline{\alpha}|\ll 1$, as shown by the blue curve in Fig. 3(b). The red
and turquoise blue curves are quite similar to those in Fig. 2 over the given
frequency range but they, too, exhibit the bifurcation and fold point
phenomena at higher frequencies None of the above situations occur for the
standard G-line.
Figure 4: Normalized power (see text) as a function of frequency for a few of
the dispersion curves in Fig. 3, and using the same color scheme. In (a), the
radial power flux increases, consistent with the attenuation growth for the
mode. In (b), the longitudinal power flux in the MM region declines rapidly
with frequency. The modes in (c) and (d) are strictly guided, with the EM
energy flowing in the MM mainly for those modes in the upper branch of the
dispersion curve of Fig. 3.
Certain effects similar to those in Fig. 3 have been seen in both grounded
metamaterial slab geometries shu ; lovat and in complicated grounded
dielectric-loaded, open geometries yak ; bac ; lamp ; maj ; tamir . In the
case of the MM G-line, we have a transition from a complex proper mode
(magneta curve in Fig. 3(a)) to a pair of guided modes at higher frequencies.
We see in both the normalized phase and attenuation the characteristic
intersection of a parabola and a straight line [Fig. 3(c) and (d)]. The phase
has a straight line to parabola transition characteristic of a fold point in
catastrophe theory yak ; hanson ; poch ; poston . Note that the attenuation
for the MM G-line in Fig. 3(b) has changed sign for the red, magenta, and
turquoise blue curves when compared to the standard G-line. while the
attenuation shows a parabola to straight line transition.
The guided and leaky modes of the G-line and MM G-line can be described as
slow modes and fast modes, respectively. The guided modes are slow in that
they occur in the spectral region
$\sqrt{\epsilon_{3}\mu_{3}}<\overline{\beta}<\sqrt{\epsilon_{2}\mu_{2}}$.
Since $\overline{\beta}>1$, $c\beta/\omega>1$ but $\omega/(c\beta)<1$, thus
their phase velocity is less than the speed of light. In contrast the leaky
waves are fast waves. Note that with reference to both Fig. 2 and Fig. 3 there
is a point at the low frequency end of each graph where (although we cut the
graphs off at $\sqrt{\epsilon_{2}\mu_{2}}$) there are complex solutions of the
dispersion relation that have $\overline{\beta}>\sqrt{\epsilon_{2}\mu_{2}}$,
and $\overline{\alpha}\neq 0$. For the standard G-line this region is a
“forbidden” region. For the MM G-line of Fig. 3,
$\overline{\beta}>\sqrt{\epsilon_{2}\mu_{2}}$ is no longer considered
forbidden, and “superslow modes” can arise shad .
The total power through the various regions can be described by the normalized
power in the radial and longitudinal directions,
$\overline{P}_{j,\sigma}=P_{j,\sigma}/P_{\rm tot}$, for $\sigma=\rho,z$, and
$j=2$ or $3$. The total power, $P_{\rm tot}$ is the sum of power flux, defined
as, $P_{\rm tot}\equiv\sum_{j,\sigma}|P_{j,\sigma}|$. The power flux through
the surface of a cylinder of radius $R$ and length $L$ encompassing a segment
of the structure is determined from the spatially integrated time-averaged
Poynting vector ${\bm{S}}_{j}$, with components,
${S}_{j,z}(\rho,z)=(c/8\pi)e^{2\alpha z}{\rm Re}[E_{j,\rho}H_{j,\theta}^{*}]$,
or ${S}_{3,\rho}(\rho,z)=-(c/8\pi)e^{2\alpha z}{\rm
Re}[E_{3,z}H_{3,\theta}^{*}]$. Thus, the longitudinal power transmitted
through the two circular regions is, $P_{j,z}|_{z=L}=2\pi e^{2\alpha
L}\int_{0}^{R}\rho d\rho S_{j,z}(\rho,0)$, and
$P_{j,z}|_{z=0}=-2\pi\int_{0}^{R}\rho d\rho S_{j,z}(\rho,0)$. For the surface
normal to $\hat{\bm{\rho}}$, integration gives, $P_{3,\rho}=2\pi R\,e^{\alpha
L}(1/\alpha)\sinh(\alpha L)S_{3,\rho}(R,0)$. We take $R/b=10$. In Fig. 4, we
illustrate the normalized power as a function of frequency for several of the
modes shown in Fig. 3. In panel (a), power flows radially, with maxima at
frequencies correlated to when $|\alpha|$ peaks for that mode (Fig. 3(b)).
Although the power flow is predominately in the air region, is is evident that
the counter-directed longitudinal power flux in the MM region declines rapidly
towards zero with increased frequency [panel (b)]. This indicates that power
emission from the waveguide takes place at a slight angle, $\vartheta$ from
the $z$ axis over a relatively small bandwidth. As for the proper mode
spectra, $|\overline{P}_{j,z}|=1/2$ for both regions, but counter-directed. At
the bifurcation point ($f\approx 10$ GHz), the leaky wave transforms into a
strictly guided mode at higher frequencies, where the two counter propagating
regions are clearly identified by their sign [Fig. 4(c) and (d)]. The lower
branch (see Fig. 3) in the guided mode dispersion clearly maps to power flow
that is predominately in the air region. This is in contrast to modes
occupying the upper branch in Fig. 3, where again in Fig. 4(c) and (d) we see
the power distribution shift from both regions equally to reside completely in
the MM at higher frequencies. This also verifies that the MM G-line has the
remarkable property that certain guided modes can propagate below the first
waveguide cutoff and can be of the backward as well as of the forward wave
type hrab ; hrab2 ; lub .
In conclusion, dispersion curve behavior for a dielectric versus MM G-Line has
been shown to be vastly different. The MM G-line dispersion is characterized
by unusual mode bifurcations, complex fold points a la catastrophe theory, and
both proper and improper complex modes.
The authors acknowledge G. A. Lindsay, Z. Sechrist, and G. Ostrom for valuable
discussions and the support from ONR, as well as NAVAIR’s ILIR program from
ONR. This work is also supported in part by a grant of HPC resources as part
of the DOD HPCMP.
## References
* (1) A. Sommerfeld, Ann. Physik 15, 673 (1904); G. Goubau, J. Appl. Phys. 21, 1119 (1950).
* (2) G. Goubau, IRE Trans. Antennas Propag. 7, S140 (1959).
* (3) J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
* (4) R. A. Waldron, Theory of Guided Electromagnetic Waves (Van Nostrand Reinhold, London, 1969).
* (5) N. A. Semenov, Radio Eng. Electr. Phys. 9, 989 (1964); J. G. Fikioris and J. A. Roumeliotis, IEEE Trans. Microwave Theory Tech. 27, 570 (1979); K. Sakina and J. Chiba, IEICE Trans. Electron.76, 657 (1993).
* (6) N. Marcuvitz, IRE Trans. Antennas Propag. 4,192 (1956); T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).
* (7) D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, Orlando, FL, 1974); A. W. Snyder, Appl. Phys. 4, 273 (1974).
* (8) L. O. Goldstone and A. A. Oliner, IRE Trans. Antennas Propag. XX, 307 (1958 or 1959); R. E. Collin and F. J. Zucker, Antenna Theory, Part 2 (McGraw-Hill Book Company, New York, 1969); R. C. Johnson, Antenna Engineering Handbook (McGraw-Hill Book Company, New York, 1993) Third Edition, Chapter 10; R. E. Collin, Field Theory of Guided Waves, 2nd ed. (New York, IEEE Press, 1991); A. Alu, et al., IEEE Trans. Antennas Propag. 55, 1698 (2007).
* (9) K. Y. Kim, Microwave Opt. Tech. Lett. 50, 523 (2008);
* (10) K. Y. Kim, et al., Electr. Lett. 39, 61 (2003);K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides, PhD Thesis (2004).
* (11) G. V Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials (IEEE Press, Hoboken, NJ, 2005); N. Engheta and R. W. Ziolkowski, Metamaterials (IEEE Press, Hoboken, NJ, 2006).
* (12) I. V. Shadrivov, et al., Phys. Rev. E67,057602 (2003); K. Halterman, et al., Phys. Rev. A76, 013834 (2007).
* (13) A. B. Yakovlev and G. W. Hanson, IEEE Trans. Microwave Theory Tech. 45, 87 (1997).
* (14) G. W. Hanson and A. B. Yakovlev, Radio Science 33, 803 (1998).
* (15) I. E. Pochanina, et al., Radiofizica 32, 1000 (1989).
* (16) T. Poston and I. Stewart, Catastrophe Theory and Its Applications (London, Pitman, 1978); R. Gilmore, Catastrophe Theory for Scientists and Engineers (New York, Wiley, 1981).
* (17) W. Shu and J. M. Song, PIER 65, 103 (2006).
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* (22) T. Tamir and F. Y. Kou, IEEE J. Quantum Electr. 22, 544 (1986).
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|
arxiv-papers
| 2009-09-02T21:46:25 |
2024-09-04T02:49:05.015684
|
{
"license": "Public Domain",
"authors": "P. L. Overfelt, Klaus Halterman, Simin Feng, D. R. Bowling",
"submitter": "Klaus Halterman",
"url": "https://arxiv.org/abs/0909.0535"
}
|
0909.0669
|
# Soliton and periodic wave solutions to the osmosis K(2, 2) equation
Jiangbo Zhou Corresponding author. zhoujiangbo@yahoo.cn Lixin Tian Xinghua
Fan Nonlinear Scientific Research Center, Faculty of Science, Jiangsu
University, Zhenjiang, Jiangsu 212013, China
###### Abstract
In this paper, two types of traveling wave solutions to the osmosis K(2, 2)
equation
$u_{t}+(u^{2})_{x}-(u^{2})_{xxx}=0$
are investigated. They are characterized by two parameters. The expresssions
for the soliton and periodic wave solutions are obtained.
###### keywords:
osmosis K(2, 2) equation , soliton, periodic wave solution
###### MSC:
35G25 , 35G30 , 35L05
, ,
## 1 Introduction
In 1993, Rosenau and Hyman [1] introduced a genuinely nonlinear dispersive
equation, a special type of KdV equation, of the form
$u_{t}+a(u^{n})_{x}+(u^{n})_{xxx}=0,n>1,$ (1.1)
where $a$ is a constant and both the convection term $(u^{n})_{x}$ and the
dispersion effect term $(u^{n})_{xxx}$ are nonlinear. These equations arise in
the process of understanding the role of nonlinear dispersion in the formation
of structures like liquid drops. Rosenau and Hyman derived solutions called
compactons to Eq.(1.1) and showed that while compactons are the essence of the
focusing branch where $a>0$, spikes, peaks, and cusps are the hallmark of the
defocusing branch where $a<0$ which also supports the motion of kinks.
Further, the negative branch, where $a<0$, was found to give rise to solitary
patterns having cusps or infinite slopes. The focusing branch and the
defocusing branch represent two different models, each leading to a different
physical structure. Many powerful methods were applied to construct the exact
solutions to Eq.(1.1), such as Adomain method [2], homotopy perturbation
method [3], Exp-function method [4], variational iteration method [5],
variational method [6, 7]. In [8], Wazwaz studied a generalized forms of the
Eq.(1.1), that is $mK(n,n)$ equations and defined by
$u^{n-1}u_{t}+a(u^{n})_{x}+b(u^{n})_{xxx}=0,n>1,$ (1.2)
where $a,b$ are constants. He showed how to construct compact and noncompact
solutions to Eq.(1.2) and discussed it in higher dimensional spaces in [9].
Chen et al. [10] showed how to construct the general solutions and some
special exact solutions to Eq.(1.2) in higher dimensional spatial domains. He
et al. [11] considered the bifurcation behavior of travelling wave solutions
to Eq.(1.2). Under different parametric conditions, smooth and non-smooth
periodic wave solutions, solitary wave solutions and kink and anti-kink wave
solutions were obtained. Yan [12] further extended Eq.(1.2) to be a more
general form
$u^{m-1}u_{t}+a(u^{n})_{x}+b(u^{k})_{xxx}=0,nk\neq 1,$ (1.3)
And using some direct ansatze, some abundant new compacton solutions, solitary
wave solutions and periodic wave solutions to Eq.(1.3) were obtained. By using
some transformations, Yan [13] obtained some Jacobi elliptic function
solutions to Eq.(1.3). Biswas [14] obtained 1-soliton solution of equation
with the generalized evolution term
$(u^{l})_{t}+a(u^{m})u_{x}+b(u^{n})_{xxx}=0,$ (1.4)
where $a,b$ are constants, while $l,m$ and $n$ are positive integers. Zhu et
al. [15] applied the decomposition method and symbolic computation system to
develop some new exact solitary wave solutions to the $K(2,2,1)$ equation
$u_{t}+(u^{2})_{x}-(u^{2})_{xxx}+u_{xxxxx}=0,$ (1.5)
and the $K(3,3,1)$ equation
$u_{t}+(u^{3})_{x}-(u^{3})_{xxx}+u_{xxxxx}=0.$ (1.6)
Recently, Xu and Tian [16] introduced the osmosis $K(2,2)$ equation
$u_{t}+(u^{2})_{x}-(u^{2})_{xxx}=0,$ (1.7)
were the positive convection term $(u^{2})_{x}$ means the convection moves
along the motion direction, and the negative dispersive term $(u^{2})_{xxx}$
denotes the contracting dispersion. They obtained the peaked solitary wave
solution and the periodic cusp wave solution to Eq.(1.7). In [17], the authors
obtained the smooth soliton solutions to Eq.(1.7). In this paper, following
Vakhnenko and Parkes’s strategy [18, 19] we continue to investigate the
traveling wave solutions to Eq.(1.7) and obtain soliton and periodic wave
solutions. Our work in this paper covers and extends the results in [16, 17]
and may help people to know deeply the described physical process and possible
applications of the osmosis K(2, 2) equation.
The remainder of this paper is organized as follows. In Section 2, for
completeness and readability, we repeat Appendix A in [19], which discuss the
solutions to a first-order ordinary differential equaion. In Section 3, we
show that, for travelng wave solutions, Eq.(1.7) may be reduced to a first-
order ordinary differential equation involving two arbitrary integration
constants $a$ and $b$. We show that there are four distinct periodic solutions
corresponding to four different ranges of values of $a$ and restricted ranges
of values of $b$. A short conclusion is given in Section 4.
## 2 Solutions to a first-order ordinary differential equaion
This section is due to Vakhnenko and Parkes (see Appendix A in [19]). For
completeness and readability, we state it in the following.
Consider solutions to the following ordinary differential equation
$(\varphi\varphi_{\xi})^{2}=\varepsilon^{2}f(\varphi),$ (2.1)
where
$f(\varphi)=(\varphi-\varphi_{1})(\varphi-\varphi_{2})(\varphi_{3}-\varphi)(\varphi_{4}-\varphi),$
(2.2)
and $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\varphi_{4}$ are chosen to
be real constants with
$\varphi_{1}\leq\varphi_{2}\leq\varphi\leq\varphi_{3}\leq\varphi_{4}$.
Following [20] we introduce $\zeta$ defined by
$\frac{d\xi}{d\zeta}=\frac{\varphi}{\varepsilon},$ (2.3)
so that Eq.(2.1) becomes
$(\varphi_{\zeta})^{2}=f(\varphi).$ (2.4)
Eq.(2.4) has two possible forms of solution. The first form is found using
result 254.00 in [21]. Its parametric form is
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{1}+(\varphi_{2}-\varphi_{1})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.5)
with $w$ as the parameter, where
$m=\frac{(\varphi_{3}-\varphi_{2})(\varphi_{4}-\varphi_{1})}{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},p=\frac{1}{2}\sqrt{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},w=p\zeta,$
(2.6)
and
$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{3}-\varphi_{1}}.$ (2.7)
In (2.5) $\mathrm{sn}(w|m)$ is a Jacobian elliptic function, where the
notation is as used in Chapter 16 of [22]. $\Pi(n;w|m)$ is the elliptic
integral of the third kind and the notation is as used in Section 17.2.15 of
[22].
The solution to Eq.(2.1) is given in parametric form by (2.5) with $w$ as the
parameter. With respect to $w$, $\varphi$ in (2.5) is periodic with period
$2K(m)$, where $K(m)$ is the complete elliptic integral of the first kind. It
follows from (2.5) that the wavelength $\lambda$ of the solution to (2.1) is
$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{1}K(m)+(\varphi_{2}-\varphi_{1})\Pi(n|m)|.$ (2.8)
where $\Pi(n|m)$ is the complete elliptic integral of the third kind.
When $\varphi_{3}=\varphi_{4}$, $m=1$, (2.5) becomes
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{3}}{\textstyle
p}-2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.9)
The second form of the solution to Eq.(2.4) is found using result 255.00 in
[21]. Its parametric form is
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{4}-(\varphi_{4}-\varphi_{3})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.10)
where $m,p,w$ are as in (2.6), and
$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{4}-\varphi_{2}}.$ (2.11)
The solution to Eq.(2.1) is given in parametric form by (2.10) with $w$ as the
parameter. The wavelength $\lambda$ of the solution to (2.1) is
$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{4}K(m)-(\varphi_{4}-\varphi_{3})\Pi(n|m)|.$ (2.12)
When $\varphi_{1}=\varphi_{2}$, $m=1$, (2.10) becomes
$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{2}}{\textstyle
p}+2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.13)
## 3 Solitary and periodic wave solutions to Eq.(1.7)
Eq.(1.7) can also be written in the form
$u_{t}+2uu_{x}-6u_{x}u_{xx}-2uu_{xxx}=0.$ (3.1)
Let $u=\varphi(\xi)+c$ with $\xi=x-ct$ be a traveling wave solution to
Eq.(3.1), then it follows that
$-c\varphi_{\xi}+2\varphi\varphi_{\xi}-6\varphi_{\xi}\varphi_{\xi\xi}-2\varphi\varphi_{\xi\xi\xi}=0,$
(3.2)
where $\varphi_{\xi}$ is the derivative of function $\varphi$ with respect to
$\xi$.
Integrating (3.2) twice with respect to $\xi$ yields
$(\varphi\varphi_{\xi})^{2}=\frac{1}{4}(\varphi^{4}-\frac{4c}{3}\varphi^{3}+a\varphi^{2}+b),$
(3.3)
where $a$ and $b$ are two arbitrary integration constants.
Eq.(3.3) is in the form of Eq.(2.1) with $\varepsilon=\frac{\textstyle
1}{\textstyle 2}$ and $f(\varphi)=(\varphi^{4}-\frac{\textstyle 4c}{\textstyle
3}\varphi^{3}+a\varphi^{2}+b)$. For convenience we define $g(\varphi)$ and
$h(\varphi)$ by
$f(\varphi)=\varphi^{2}g(\varphi)+b,\ \mbox{where}\
g(\varphi)=\varphi^{2}-\frac{4c}{3}\varphi+a,$ (3.4)
$f^{\prime}(\varphi)=2\varphi h(\varphi),\ \mbox{where}\
h(\varphi)=2\varphi^{2}-2c\varphi+a,$ (3.5)
and define $\varphi_{L}$, $\varphi_{R}$, $b_{L}$, and $b_{R}$ by
$\varphi_{L}=\frac{\textstyle 1}{\textstyle
2}(c-\sqrt{c^{2}-2a}),\varphi_{R}=\frac{\textstyle 1}{\textstyle
2}(c+\sqrt{c^{2}-2a}),$ (3.6)
$b_{L}=-\varphi_{L}^{2}g(\varphi_{L})=\frac{\textstyle a^{2}}{\textstyle
4}-\frac{\textstyle 1}{\textstyle 2}c^{2}a+\frac{\textstyle c^{4}}{\textstyle
6}-\frac{\textstyle 1}{\textstyle 6}(c^{3}-2ac)\sqrt{c^{2}-2a},$ (3.7)
$b_{R}=-\varphi_{L}^{2}g(\varphi_{L})=\frac{\textstyle a^{2}}{\textstyle
4}-\frac{\textstyle 1}{\textstyle 2}c^{2}a+\frac{\textstyle c^{4}}{\textstyle
6}+\frac{\textstyle 1}{\textstyle 6}(c^{3}-2ac)\sqrt{c^{2}-2a}.$ (3.8)
Obviously, $\varphi_{L}$, $\varphi_{R}$ are the roots of $h(\varphi)=0$.
Without loss of generality, we suppose the wave speed $c>0$. In the following,
suppose that $a<\frac{\textstyle c^{2}}{\textstyle 2}$ and $a\neq 0$ for each
value $c>0$, such that $f(\varphi)$ has three distinct stationary points:
$\varphi_{L}$, $\varphi_{R}$, $0$ and comprise two minimums separated by a
maximum. Under this assumption, Eq.(1.7) has periodic and solitary wave
solutions that have different analytical forms depending on the values of $a$
and $b$ as follows:
(1) $a<0$
In this case $\varphi_{L}<0<\varphi_{R}$ and $f(\varphi_{L})>f(\varphi_{R})$.
For each value $a<0$ and $0<b<b_{L}$ (a corresponding curve of $f(\varphi)$ is
shown in Fig.1(a)), there are periodic loop-like solutions to Eq.(3.3) given
by (2.10) so that $0<m<1$, and with wavelength given by (2.12). See Fig.2(a)
for an example.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 1: The curve of $f(\varphi)$ for the wave speed $c=2$. (a) $a=-40$,
$b=200$; (b) $a=-40$, $b=226.0424$; (c) $a=1.5$, $b=-0.05$; (d) $a=1.5$,
$b=0$; (e) $a=\frac{\textstyle 16}{\textstyle 9}$, $b=-0.1$; (f)
$a=\frac{\textstyle 16}{\textstyle 9}$, $b=0$; (g) $a=\frac{\textstyle
17}{\textstyle 9}$, $b=-0.24$; (h) $a=\frac{\textstyle 17}{\textstyle 9}$,
$b=-0.1842$.
The case $a<0$ and $b=b_{L}$ (a corresponding curve of $f(\varphi)$ is shown
in Fig.1(b)) corresponds to the limit $\varphi_{1}=\varphi_{2}=\varphi_{L}$ so
that $m=1$, and then the solution is a loop-like solitary wave given by (2.13)
with $\varphi_{2}\leq\varphi<\varphi_{R}$ and
$\varphi_{3}=\frac{\textstyle 1}{\textstyle 2}\sqrt{c^{2}-2a}+\frac{\textstyle
c}{\textstyle 6}-\frac{\textstyle 1}{\textstyle 3}\sqrt{c^{2}+3c\sqrt{4-2a}},$
(3.9) $\varphi_{4}=\frac{\textstyle 1}{\textstyle
2}\sqrt{c^{2}-2a}+\frac{\textstyle c}{\textstyle 6}+\frac{\textstyle
1}{\textstyle 3}\sqrt{c^{2}+3c\sqrt{4-2a}}.$ (3.10)
See Fig.3(a) for an example.
(2) $0<a<\frac{\textstyle 4c^{2}}{\textstyle 9}$
In this case $0<\varphi_{L}<\varphi_{R}$ and $f(\varphi_{R})<f(0)$. For each
value $0<a<\frac{\textstyle 4c^{2}}{\textstyle 9}$ and $b_{L}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(c)), there are periodic
valley-like solutions to Eq.(3.3) given by (2.10) so that $0<m<1$, and with
wavelength given by (2.12). See Fig.2(b) for an example.
The case $0<a<\frac{\textstyle 4c^{2}}{\textstyle 9}$ and $b=0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(d)) corresponds to the
limit $\varphi_{1}=\varphi_{2}=0$ so that $m=1$, and then the solution can be
given by (2.13) with $\varphi_{3}$ and $\varphi_{4}$ given by the roots of
$g(\varphi)=0$, namely
$\varphi_{3}=\frac{\textstyle 2c}{\textstyle 3}-\sqrt{\frac{\textstyle
4c^{2}}{\textstyle 9}-a},\varphi_{4}=\frac{\textstyle 2c}{\textstyle
3}+\sqrt{\frac{\textstyle 4c^{2}}{\textstyle 9}-a}.$ (3.11)
In this case we obtain a weak solution, namely the periodic downward-cusp wave
$\varphi=\varphi(\xi-2j\xi_{m}),(2j-1)\xi_{m}<\xi<(2j+1)\xi_{m},\ j=0,\pm
1,\pm 2,\cdots,$ (3.12)
where
$\varphi(\xi)=(\varphi_{3}-\varphi_{4}\tanh^{2}(\xi/4))\cosh^{2}(\xi/4),$
(3.13)
and
$\xi_{m}=4\tanh^{-1}\sqrt{\frac{\varphi_{3}}{\varphi_{4}}}.$ (3.14)
See Fig.3(b) for an example.
(a)
(b)
(c)
(d)
Figure 2: Periodic solutions to Eq.(3.3) with $0<m<1$ and the wave speed
$c=2$. (a) $a=-40$, $b=200$ so $m=0.8978$; (b) $a=1.5$, $b=-0.05$ so
$m=0.6893$; (c) $a=\frac{\textstyle 16}{\textstyle 9}$, $b=-0.1$ so
$m=0.8254$; (d) $a=\frac{\textstyle 17}{\textstyle 9}$, $b=-0.24$ so
$m=0.8412$.
(a)
(b)
(c)
(d)
Figure 3: Solutions to Eq.(3.3) with $m=1$ and the wave speed $c=2$. (a)
$a=-40$, $b=226.0424$; (b) $a=1.5$, $b=0$; (c) $a=\frac{\textstyle
16}{\textstyle 9}$, $b=0$; (d) $a=\frac{\textstyle 17}{\textstyle 9}$,
$b=-0.1842$.
(3) $a=\frac{\textstyle 4c^{2}}{\textstyle 9}$
In this case $0<\varphi_{L}<\varphi_{R}$ and $f(\varphi_{R})=f(0)$. For
$a=\frac{\textstyle 4c^{2}}{\textstyle 9}$ and each value $b_{L}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(e)), there are periodic
valley-like solutions to Eq.(3.3) given by (2.5) so that $0<m<1$, and with
wavelength given by (2.8). See Fig.2(c) for an example.
The case $a=\frac{\textstyle 4c^{2}}{\textstyle 9}$ and $b=0$ (a corresponding
curve of $f(\varphi)$ is shown in Fig.1(f)) corresponds to the limit
$\varphi_{3}=\varphi_{4}=\varphi_{R}=\frac{\textstyle 2c}{\textstyle 3}$ and
$\varphi_{1}=\varphi_{2}=0$ so that $m=1$. In this case neither (2.9) nor
(2.13) is appropriate. Instead we consider Eq.(3.3) with
$f(\varphi)=\frac{\textstyle 1}{\textstyle
4}\varphi^{2}(\varphi-\frac{\textstyle 2c}{\textstyle 3})^{2}$ and note that
the bound solution has $0<\varphi<\frac{\textstyle 2c}{\textstyle 3}$. On
integrating Eq.(3.3) and setting $\varphi=0$ at $\xi=0$ we obtain a weak
solution
$\varphi=-\frac{\textstyle 2c}{\textstyle 3}\exp{(-\frac{\textstyle
1}{\textstyle 2}|\xi|)}+\frac{\textstyle 2c}{\textstyle 3},$ (3.15)
i.e. a single valley-like peaked solution with amplitude $\frac{\textstyle
2c}{\textstyle 3}$. See Fig.3(c) for an example.
(4) $\frac{\textstyle 4c^{2}}{\textstyle 9}<a<\frac{\textstyle
c^{2}}{\textstyle 2}$
In this case $0<\varphi_{L}<\varphi_{R}$ and $f(\varphi_{R})>f(0)$. For each
value $\frac{\textstyle 4c^{2}}{\textstyle 9}<a<\frac{\textstyle
c^{2}}{\textstyle 2}$ and $b_{L}<b<b_{R}$ (a corresponding curve of
$f(\varphi)$ is shown in Fig.1(g)), there are periodic valley-like solutions
to Eq.(3.3) given by (2.5) so that $0<m<1$, and with wavelength given by
(2.8). See Fig.2(d) for an example.
The case $\frac{\textstyle 4c^{2}}{\textstyle 9}<a<\frac{\textstyle
c^{2}}{\textstyle 2}$ and $b=b_{R}$ (a corresponding curve of $f(\varphi)$ is
shown in Fig.1(h)) corresponds to the limit
$\varphi_{3}=\varphi_{4}=\varphi_{R}$ so that $m=1$, and then the solution is
a velley-like solitary wave given by (2.10) with
$\varphi_{L}<\varphi\leq\varphi_{3}$ and
$\varphi_{1}=\frac{\textstyle c}{\textstyle 6}-\frac{\textstyle 1}{\textstyle
2}\sqrt{c^{2}-2a}-\frac{\textstyle 1}{\textstyle
3}\sqrt{c^{2}-3c\sqrt{c^{2}-2a}},$ (3.16) $\varphi_{2}=\frac{\textstyle
c}{\textstyle 6}-\frac{\textstyle 1}{\textstyle
2}\sqrt{c^{2}-2a}+\frac{\textstyle 1}{\textstyle
3}\sqrt{c^{2}-3c\sqrt{c^{2}-2a}}.$ (3.17)
See Fig.3(d) for an example.
## 4 Conclusion
In this paper, we have found expressions for two types of traveling wave
solutions to the osmosis K(2, 2) equation, that is, the soliton and periodic
wave solutions. These solutions depend, in effect, on two parameters $a$ and
$m$. For $m=1$, there are loop-like ($a<0$), peakon ($a=\frac{\textstyle
4c^{2}}{\textstyle 9}$) and smooth ($\frac{\textstyle 4c^{2}}{\textstyle
9}<a<\frac{\textstyle c^{2}}{\textstyle 2}$) soliton solutions. For
$m=1,0<a<\frac{\textstyle 4c^{2}}{\textstyle 9}$ or $0<m<1,a<\frac{\textstyle
c^{2}}{\textstyle 2}$ and $a\neq 0$, there are periodic wave solutions.
## References
* [1] P. Rosenau, J. M. Hyman, Compactons: solitons with finite wavelengths, Phys. Rev. Lett. 70 (1993) 564-567.
* [2] A. M. Wazwaz, Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl. Math. Comput. 138 (2003) 309-319.
* [3] J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J Nonlinear Sci. Numer. Simulat. 6 (2005) 207-208.
* [4] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (2006) 700-708.
* [5] J. H. He, X. H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals 29 (2006) 108-113.
* [6] J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J Modern Phys. B 20 (2006) 1141-1199.
* [7] L. Xu, Variational approach to solitons of nonlinear dispersive equations, Chaos, Solitons and Fractals 37 (2008) 137-143.
* [8] A. M. Wazwaz, General compacts solitary patterns solutions for modified nonlinear dispersive equation in higher dimensional spaces, Math. Comput. Simulat. 59 (2002) 519-531.
* [9] A. M. Wazwaz, Compact and noncompact structures for a variant of KdV equation in higher dimensions, Appl. Math. Comput. 132 (2002) 29-45.
* [10] Y. Chen, B. Li, H. Q. Zhang, New exact solutions for modified nonlinear dispersive equations in higher dimensions spaces, Math. Comput. Simul. 64 (2004) 549-559.
* [11] B. He, Q. Meng, W. Rui, Y. Long, Bifurcations of travelling wave solutions for the equation, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 2114-2123.
* [12] Z. Y. Yan, Modified nonlinearly dispersive $mK(m,n,k)$ equations: I. New compacton solutions and solitary pattern solutions, Comput. Phys. Commun. 152 (2003) 25-33.
* [13] Z. Y. Yan, Modified nonlinearly dispersive equations: II. Jacobi elliptic function solutions, Comput. Phys. Commun. 153 (2003) 1-16.
* [14] A. Biswas, 1-soliton solution of the equation with generalized evolution, Phys. Lett. A 372 (2008) 4601-4602.
* [15] Y. G. Zhu, K. Tong, T. C. Lu, New exact solitary-wave solutions for the $K(2,2,1)$ and $K(3,3,1)$ equations, Chaos, Solitons and Fractals 33 (2007) 1411-1416.
* [16] C. H. Xu, L. X. Tian, The bifurcation and peakon for $K(2,2)$ equation with osmosis dispersion, Chaos, Solitons and Fractals 40 (2009) 893-901.
* [17] J. B. Zhou, L. X. Tian, Soliton solution of the osmosis K(2, 2) equation, Phys. Lett. A 372 (2008) 6232-6234.
* [18] V. O. Vakhnenko, E. J. Parkes, Explicit solutions of the Camassa-Holm equation, Chaos, Solitons and Fractals 26 (2005) 1309-1316.
* [19] V. O. Vakhnenko, E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos, Solitons and Fractals 20 (2004) 1059-1073.
* [20] E. J. Parkes, The stability of solutions of Vakhnenko’s equation, J. Phys. A Math. Gen. 26 (1993) 6469-75.
* [21] P. F. Byrd, M. D. Friedman, Handbook of elliptic integrals for engineers and scientists, Springer, Berlin, 1971.
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|
arxiv-papers
| 2009-09-03T14:43:03 |
2024-09-04T02:49:05.022784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangbo Zhou, Lixin Tian, Xinghua Fan",
"submitter": "Jiangbo Zhou",
"url": "https://arxiv.org/abs/0909.0669"
}
|
0909.0687
|
To the problem of cross-bridge tension in steady muscle shortening and
lengthening
.
By Valery B. Kokshenev
.
Submitted to the Journal of Biomechanics 14 April 2009, BM-D_09-00317
.
Departamento de Física, Universidade Federal de Minas Gerais, Instituto de
Ciências Exatas, Caixa Postal 702, CEP 30123-970, Belo Horizonte, Brazil,
valery@fisica.ufmg.br
.
Abstract. Despite the great success of the Huxley sliding filament model
proposed half a century ago for actin-myosin linkages (cross-bridges), it
fails to explain the force-velocity behavior of stretching skeletal muscles.
Huxley’s two-state kinetic equation for cross-bridge proportions is therefore
reconsidered and a new solution to the problem of steady muscle eccentric and
concentric contractions is reported. Instead of numerical modeling the
contractive-force data by appropriate choice of the seemingly arbitrary
heterogeneity of attachment and detachment rates of myosin heads to actin
filament cites, Huxley’s idea on mechanical equilibrium is probed into
thermodynamic equilibrium in the whole overlapped actin-myosin zone. When the
second law of statistical thermodynamics is applied to cross-bridge
proportions, the weakly bound states appear to be correlated to the strongly
bound states via structural and kinetic intrinsic muscle characteristics. A
consequent substantial reduction of the number of free parameters in cross-
bridge proportions is also due to the overall self-consistency (normalization)
of attachment-detachment stochastic events. The explicit force-velocity curve
is found to be generic when applied to the reduced tension in a single cross
bridge, sarcomere, fiber, or muscle as a whole during its active shortening or
lengthening. This universal curve fits the empirical tension-velocity data on
frog muscle shortening using only one adjustable parameter, while the Huxley
model employed four parameters. The established normally distributed cross-
bridges, detaching slowly near equilibrated states in steady lengthening
muscle and quickly in shortening muscle, are in qualitative agreement with
recent data on the force enhancement following muscle stretching.
.
1\. Introduction
The early studies of muscle fibers under the light microscope revealed cross-
striations running normal to the fiber axis. It was observed that during
either concentric or eccentric contractions, the length changes occurred via
an increase or decrease in the extent of the I-band with the A-band remaining
unchanged. Two groups laid the foundations for the cross-bridge (CB)__ theory
when they simultaneously suggested that muscle contractions occur due to the
relative sliding of the thick myosin filaments past the thin actin filaments,
mediated by the ATP-dependent actin-myosin linkages (H. E. Huxley and Hanson,
1954) working as independent force generators (A. F. Huxley and Niedergerke,
1954).
A. F. Huxley (1957) evaluated muscle tension caused by shortening, in fact,
based on the idea of the existence of _mechanical_ equilibrium of myosin heads
at the regular sites of actin filaments. His famous two-state (bound-unbound)
sliding filament model was determined on the basis of the simplest standard
kinetic equation controlled by the velocity-independent rates of attachment
and detachment of myosin heads. In spite of the great success in illuminating
the force generation and power liberation during muscle shortening, the Huxley
approach generally failed to explain the ascending branch of the
phenomenologically established tension-velocity equation (see e.g. Harry et
al., 1990 and references therein).
Exploring the fact that the sliding filament model leaves a free choice of the
attachment and detachment rate functions, many researchers successfully
simulated a range of muscle properties in lengthening by fitting the empirical
data by linear functions and constants suggested by Huxley (1957) for the CB
proportions or by bilinear and exponential functions. Likewise, considerable
efforts have been made to modify the rate functions (Zahalak, 1981; Harry et
al., 1990; Ma and Zahalak, 1991; Cole et al., 1996) or to find an exact
_numerical_ solution to Huxley’s model (Wu et al., 1997). Very recently,
controversies surrounding Huxley’s approach were brought forth by Mehta and
Herzog (2008) in their careful studies of force exposed by a single CB during
lengthening.
The theoretical problem of self-consistency in the two-state sliding filament
models was thoughtfully discussed by Hill and co-workers (Hill et al., 1975).
Considering the conditions of CB _thermodynamic_ equilibrium besides the
minimum of mechanical energy (Huxley, 1957), they demonstrated that the
original Huxley model has a low efficiency in comparison to its modified
versions. Moreover, it was noted by Eisenberg et al. (1980) that ”there is no
in vitro evidence for … the basic (Huxley’s) assumption that the cross-bridge
detaches slowly”. The lacking data were provided by Mehta and Herzog (2008).
In this study, I develop a statistical approach to the filament sliding
mechanism and show that only linear functions for the attachment-detachment
rates are compatible with the concept of thermodynamic equilibrium. A new
analytical solution to Huxley’s two-state kinetic equation is proposed and
verified using the available from the literature data on tension in steady
muscle lengthening and shortening.
.
2\. Methods
2.1. Model by Huxley (1957) revisited
At a fixed muscle _contraction velocity_ $V$, the number of bound CB states
$N_{V}$ combining myosin filament with actin filament of the total number of
sites $N_{0A}$ obeys the common ”balance” kinetic equation
$\frac{d}{dt}N_{V}(x,t)=\frac{\partial N_{V}}{\partial t}+\frac{\partial
N_{V}}{\partial x}\cdot\frac{dx}{dt}=\text{ }f(x)(N_{0A}-N_{V})-g(x)N_{V}.$
(1)
Here $f$ and $g$ are _attachment_ and _detachment rates_ of the corresponding
unbound and bound states located in time $t$ at a distance $x$ estimated from
the nearest site $x=0$. The steady process determined by late times $t\gg
f^{-1},$ $g^{-1}$ providing $\partial N_{V}(x,\infty)/\partial t=0$ in Eq.
(1), reduces Eq. (1) to
$-\frac{V}{2}\frac{d}{dx}n_{V}(x)=\text{ }f(x)(1-n_{V})-g(x)n_{V}\text{,}$ (2)
i.e., to Huxley’s Eq. (4) where the _proportion_
$n_{V}(x)=N_{V}(x,\infty)/N_{0A}$ of CBs during _steady shortening_. According
to Huxley, the force output $F(x)=kx$ is produced when $x$ decreases at a
positive velocity of sliding of the actin filament $V_{A}$ and a negative
velocity of myosin filament $V_{M}$, i.e., $V_{A}=-V_{M}=-dx/dt>0$. The
contraction velocity per one-half sarcomere $V/2$ determines the contraction
_velocity_ $V$__ of the muscle __ as a whole, when modeled by
$V=V_{A}-V_{M}=2V_{A}$. As can be derived from Huxley’s Eq. (6) with the
preservation in part its notations, the overall generated force
$F_{V}^{(total)}=\lim_{L\rightarrow\infty}\frac{sN_{0M}}{L}\int_{-L}^{+L}F(x)n_{V}(x)\frac{dx}{2l_{A}}\text{,}$
(3)
was evaluated via the force $F(x)$ per one _myosin site_ , as one actin site
is carried past it. Here $s$ is the sarcomere length, $l_{A}$ is the _trial_
distance between the nearest sites in the actin filament _evaluated_ in
Huxley’s Eq. (15), and $N_{0M}$ is the number of sites in the thick filament
in the overlapping zone of length $L$.
The solution to Eq. (2) for the shortening regime (hereafter distinguished by
index $1$) was found as a combination of the _localized_ (short-domain) and
_delocalized_ (large-domain) spatially correlated (bound) states. The
corresponding proportions reproduced exactly from Huxley’s Eqs. (7) and (8)
are
$n_{V}^{(loc)}(x)=n_{01}\left(1-\exp\left[\frac{V_{1}}{V}\left(\frac{x^{2}}{h^{2}}-1\right)\right]\right)\text{,
}0\leq x\leq h$ (4)
and
$n_{V}^{(deloc)}(x)=n_{01}\left[1-\exp\left(-\frac{V_{1}}{V}\right)\right]\exp\left(2x\frac{g_{1}^{\prime}}{V}\right)\text{,
}-\infty<x\leq 0\text{,}$ (5)
though parameterized here by
$n_{01}=\frac{f_{1}}{f_{1}+g_{1}}\text{ and }V_{1}=h(f_{1}+g_{1})\text{.}$ (6)
In turn, this description of the two CB states follows from the rates
_postulated_ by linear functions, namely
$f(x)=f_{1}\frac{x}{h}\text{ and }g(x)=g_{1}\frac{x}{h}\text{, for }0\leq
x\leq h\text{,}$ (7)
and two constants $f^{\prime}(x)=0$, $g^{\prime}(x)=g_{1}^{\prime}$, for
$-\infty<x<0$. The muscle concentric steady tension $P_{V}$ reduced to the
model isometric tension $P_{0}$ found on the basis of Eqs. (3)-(7), namely
$\frac{P_{V}^{(short)}}{P_{0}}=1-\frac{V}{V_{1}}\left[1-\exp\left(-\frac{V_{1}}{V}\right)\right]\left(1+\frac{VV_{1}}{2h^{2}g_{1}^{\prime
2}}\right)\text{, for }V>0\text{,}$ (8)
was fitted by the widely cited four model parameters: $f_{1}=43.3$ $s^{-1}$
and $g_{1}=10.0$ $s^{-1}$, indicating slow detachment of the localized CBs,
and $f_{1}^{\prime}=0$ with $g_{1}^{\prime}=209$ $s^{-1}$, for delocalized
states. In addition, two more adjustable parameters $h\thickapprox 15$ $nm$
and $V_{1}=V_{\max}^{(\exp)}/4$, where $V_{\max}^{(\exp)}$ is the empirical
maximum shortening velocity, were indirectly employed when tested by Hill’s
empirical equation (see Chapter IV in Huxley, 1957). It is noteworthy that the
nearest-site distance in the actin filament treated as a free parameter was
estimated as $l\thickapprox h$, i.e. close to the known nearest-molecular
distance in the myosin filament $l_{M}=14.5$ $nm$ (Craig and Woodhead, 2006).
However, the ratio $P_{-\infty}/P_{0}=(f_{1}+g_{1})/g_{1}=5.33$ reported by
Huxley (1957) for the muscle lengthening regime, contrasts to the observed
ratios falling between $1.8$ and $2.0$ (e.g. Harry et al., 1990).
.
2.2. A new solution to Huxley’s steady equation
Beyond any specific suggestions, the formal solution to the steady-state Eq.
(2)
$n_{V}(x)=n_{0}(x)+\Delta
n_{V}(x)=n_{0}+(1-n_{0})c_{V}\exp\left(-\frac{1}{\overset{\cdot}{x}}\int_{0}^{x}[f(x^{\prime})+g(x^{\prime})]dx^{\prime}\right),\text{
}\overset{\cdot}{x}\equiv\frac{dx}{dt}=\mp\frac{V}{2}\text{,}$ (9)
is valid for any contraction shortening velocity $V$ ($=-$
$2\overset{\cdot}{x}>0$) and lengthening velocity $V$
($=2\overset{\cdot}{x}<0$), leaving an arbitrary choice of the rate functions
$f(x)$ and $g(x)$. A differential equation of the first order possesses as
common only one free constant, denoted by $c_{V}$, whereas
$n_{0}(x)=\frac{f(x)}{f(x)+g(x)}$ (10)
straightforwardly following from Eq. (2) taken at $V=0$, describes maximal CB
proportions limited by intrinsic rates. In the Huxley model, the constant
$c_{V}=-n_{01}(1-n_{01})^{-1}\exp(-V_{1}/V)$ in Eq. (9) results from his
_boundary condition_ $n_{V}(h)=0$ providing the _non-Gaussian_ proportion (4)
for CB localized states during muscle shortening.
Besides the boundary conditions considered below, let us employ the property
of periodicity in the overlapping part of the actin filament of length $Nd$
having $N$ _occupied_ _cells_. Since the Huxley proportion $n_{V}(x)$ in Eq.
(9) plays the role of the late-time _probability_ of finding one of the two
myosin heads attached at a position $x$ between two nearest equivalent sites
(see also Hill et al. 1975, p. 346), the total force output in a _finite_
overlapped zone is
$F_{V}^{(zone)}=NF_{V}=\int_{-Nd}^{+Nd}F(x^{\prime})n_{V}(x^{\prime})\frac{dx^{\prime}}{2d}=N\int_{-d}^{d}F(x)n_{V}(x)\frac{dx}{2d}\text{,
where }x^{\prime}=xN\text{.}$ (11)
Here $F(x)$ is the active force _per one_ _actin site_ , substituting that per
one myosin cite in Eq. (3). Such a consideration suggests the statistical
equivalence of all the occupied cells in the actin filament treated as a one-
dimensional crystal of lattice constant $d$ ($=36$ $nm$, e.g. Hill et al.,
1975). In order to be consistent with Eqs. (11) and (3), the normalization
conditions for the CB distributions (proportions)
$\int_{-d}^{+d}n_{V}(x)\frac{dx}{2d}=\int_{0}^{d}n_{V}(x)\frac{dx}{d}=\int_{-d}^{0}n_{V}(x)\frac{dx}{d}=1\text{
}$ (12)
must be taken into consideration.
Furthermore, extending Huxley’s idea on the minimum of mechanical energy at
$x=0$ over the minimum of Gibbs energy (Eisenberg et al., 1980), the CB state
with $F(0)=0$ is treated as the locally equilibrated state, having _maximum
configurational entropy_ at $x=0$. As the consequence of one of the most
general principle (second law) of statistical thermodynamics, the distribution
of bound states $n_{V}(x)$ given in Eq. (9) must have Gaussian form centered
at $x=0$ (see e.g. Chapter 12 in Landau and Lifshitz, 1989). One can see from
Eq. (9), that the thermodynamical principle ensured by the _sign requirement_
$\overset{\cdot}{x}x>0$ can be satisfied solely by the linear parameterization
of the rate functions, namely
$\displaystyle f(x)$ $\displaystyle=f_{m}\frac{x}{x_{m}}\text{,
}g(x)=g_{m}\frac{x}{x_{m}}\text{,}$ $\displaystyle\text{for }x_{m}$
$\displaystyle=x_{+}\geq x\geq 0\text{ or }x_{m}=-x_{-}\leq x\leq 0\text{,}$
(13)
Consequently, the normalization constant
$c_{V}=\frac{d}{|x_{m}|}\frac{2}{\sqrt{\pi
v}}\operatorname{erf}\left(\frac{1}{\sqrt{v}}\right)^{-1}\text{,
}v=\frac{V}{V_{m}}=\frac{2\overset{\cdot}{x}}{(f_{m}+g_{m})x_{m}}>0\text{,}$
(14)
readily follows from the normalization conditions (12), where the standard
_error function_
$\operatorname{erf}(y)=(2/\sqrt{\pi})\int_{0}^{y}\exp(-t^{2})dt$, lying
between $0$ [$=\operatorname{erf}(0)$] and $1$
[$=\operatorname{erf}(\infty)$], is employed.
.
3\. Results
3.1. CB proportions in steady muscle shortening and lengthening
As seen in Eq. (9), a given CB is characterized by the equilibrated velocity-
independent _ground_ state and the _excited_ non-equilibrated state described,
respectively, by uniform proportion $n_{0}=f_{m}/(f_{m}+g_{m})$ and non-
uniform, heterogenous stochastic proportion $\Delta n_{V}(x)$, having the
meaning of probabilities normalized in Eq. (12). The requirement of signs (
$\overset{\cdot}{x}x>0$), while in particular ensuring the self-consistency
with the ground state (when $x\rightarrow 0$, $\Delta n_{V}(x)\rightarrow 0$),
also constrains possible domains for both CB states, as shown in Eq. (13).
Indeed, both kinds of domains ($0\leq x\leq x_{0}$ and $-x_{0}\leq x\leq 0$,
otherwise $n_{0}(x)=0$) are generally possible for the ground state, whereas
only the negative domain, $-x_{-}\leq x\leq 0$, satisfies the requirement of
self-consistency of the solution given in Eq. (9). In this way, the CB
boundary conditions are not postulated as in Eqs. (4)-(7), but result from the
minimum of Gibbs energy, also giving rise to the CB _mechanical constraints_ ,
consistent with the simultaneous observations of directions of both the force
output and contraction velocity, as illustrated in Fig. 1.
.
Place Fig. 1
.
The analysis in Fig. 1 specifies domains of the _short-domain_ CB states
which, being incorporated in the trial Eq. (9) with the help of Eq. (14),
yield
$n_{V}(x)=n_{0}\Theta_{0}(x)+(1-n_{0}\frac{x_{0}}{d})\frac{d}{x_{-}}\frac{2}{\sqrt{\pi
v}}\frac{\exp\left(-\frac{x^{2}}{vx_{m}^{2}}\right)}{\operatorname{erf}\left(\frac{1}{\sqrt{v}}\right)}\Theta_{V}(x)\text{.}$
(15)
Here, the auxiliary functions $\Theta_{0}(x)=\Theta(\pm x)-\Theta(x\mp x_{0})$
and $\Theta_{V}(x)\equiv\Theta(-x)-\Theta(x+x_{-})$ are introduced by the
standard Heaviside (step) function $\Theta(y)$, which is one for $y\geq 0$ and
zero for $y<0$.
.
3.2. Muscle tension in steady shortening and lengthening
The mean force output $F_{V}$ generated by a single cell of the actin filament
is evaluated using Eqs. (11) and (15), namely
$F_{V}=F_{0}+\Delta
F_{V}=k\int_{-d}^{+d}xn_{V}(x)\frac{dx}{2d}=F_{0}-k\frac{x_{-}}{2}(1-n_{0}\frac{x_{0}}{d})\Phi(v)\text{,
}F_{0}=\pm\frac{kx_{0}^{2}}{2d}n_{0}\text{,}$ (16)
via the CB _stiffness_ $k=F(x)/x$ (Huxley, 1957; Huxley and Simmons, 1971),
shown to be a velocity-independent intrinsic muscle quantity (e.g. Lombardi
and Piazzesi, 1990), where
$\Phi(v)=2\sqrt{\frac{v}{\pi}}\frac{1-\exp(-\frac{1}{v})}{\operatorname{erf}(\frac{1}{\sqrt{v}})}\text{;
with }\Phi(0)=0\text{, }\Phi(1)=0.846\text{, and }\Phi(\infty)=1\text{.}$ (17)
The upper and lower signs in the velocity-independent limiting steady force
$F_{0}$ (16) correspond to shortening and lengthening (see Fig. 1). In this
way, Huxley’s Eq. (8) is transformed into a unique equation
$\frac{P_{V}}{P_{0}}=\frac{F_{V}}{F_{0}}=1\mp\sigma_{m}\Phi\left(v\right)\text{,
with }\sigma_{m}=\frac{(d-n_{0}x_{0})x_{-}}{n_{0}x_{0}^{2}}\text{, }$ (18)
for the reduced CB tension and force output in both concentric and eccentric
muscle contractions conducted at positive and negative steady velocities
$V=vV_{m}$ (14), respectively.
The _one-parameter_ fitting analysis of the proposed theory is conducted on
the basis of Eq. (18) and the available experimental data. In Fig. 2, the
muscle shortening is described by
$\displaystyle\text{ }\frac{P_{V}^{(short)}}{P_{01}}$
$\displaystyle=1-\frac{\Phi\left(\frac{\lambda
V}{V_{\max}}\right)}{\Phi(\lambda)}\text{, }0\leq V\leq V_{\max}\text{, }$
$\displaystyle V_{\max}$ $\displaystyle=\lambda V_{m1}\text{,
}V_{m1}=x_{m1}(f_{m1}+g_{m1})\text{,}$ (19)
where $\lambda$ is an adjustable parameter.
.
Place Fig. 2
.
.
Place Fig. 3
.
The steady muscle lengthening is fitted in Fig. 3 by
$\displaystyle\frac{F_{V}^{(stret)}}{F_{02}}$
$\displaystyle=1+\Phi\left(\frac{V}{V_{m2}}\right)\text{, }-\infty<V\leq
0\text{, }$ $\displaystyle V_{m2}$
$\displaystyle=-x_{m2}(f_{m2}+g_{m2})<0\text{.}$ (20)
using the characteristic velocity $V_{m2}$ as a free parameter. Other
parameters describing two distinct regimes are specified as $x_{-}=x_{m1}$,
$f_{m}=f_{m1}$, $g_{m}=g_{m1}$, for shortening, and $x_{-}=x_{m2}$,
$f_{m}=f_{m2}$, $g_{m}=g_{m2}$, for lengthening. One can see that $V_{\max}$
plays the role of the maximum shortening velocity at which
$P_{V}^{(short)}=0$, and $V_{m2}$ is a characteristic velocity separating slow
and fast lengthening. Also, the limiting tension in the fastest steady
lengthening is $P_{-\infty}^{(stret)}=2P_{0}$.
.
3.3. CB domains
The force-velocity fitting analysis alone does not provide details on the CB
attachment-detachment rates or the domains. Physically, these domains follow
from the conditions of realization of thermodynamic stability described by a
minimum of Gibbs energy (Hill et al., 1975). Nevertheless, the overall curve
conditions of observation can be established here by the inequalities
$\sigma_{m1}^{(\exp)}>1\geq\sigma_{m2}^{(\exp)}$, resulting from Eqs. (19) and
(20), where the fitting parameter $\sigma_{m1}^{(\exp)}=\Phi(0.85)^{-1}=1.22$
is found for muscle shortening and $\sigma_{m2}^{(\exp)}$, generally lying
between $0.8$ and $1.0$, for lengthening.
Alternatively, the observation conditions of the predicted branches of the
master curve can be reformulated in terms of the CB rigor state proportions
$n_{01}<x_{m1}dx_{01}^{-1}(x_{01}+x_{m1})^{-1}$ and $n_{02}\geq
x_{m2}dx_{02}^{-1}(x_{02}+x_{m2})^{-1}$, obtained with the help of Eq. (18).
This finding can be improved when the CB geometrical constraints shown in Fig.
1 are taken into account. Indeed, since the tail of the myosin molecule is
longer than heads, one should expect a geometrical constraint $x_{m2}>x_{m1}$,
providing $n_{02}>n_{01}$. Under the simplified requirements of periodicity
($x_{01}+x_{m1}=d$ and $x_{02}=x_{m2}=d$), the CB proportions underlying the
observation of the master curve are specified in the insets in Figs. 2 and 3.
.
4. Discussion
Huxley’s model of the establishment of mechanical equilibrium of myosin heads
near actin-filament sites is based on the simplest kinetic equation
determining a balance between unbound and bound actin-myosin states. In a
muscle contracting at constant velocity $V$, these two states are described by
the proportions $1-n_{V}(x)$ and $n_{V}(x)$, satisfying the steady-state
kinetic equation (2) at generally arbitrary rates $f(x)$ and $g(x)$. Such a
property, following evidently from the solution $n_{V}(x)$ found for a general
case in Eq. (9), implies that the kinetic equation accounts for the most
general features of muscle relaxation, regardless of details underlying the
attachment-detachment mechanism of myosin heads. Consequently, theoretical
studies exploring an arbitrary choice of the functional form of the
attachment-detachment rates, which involved increasing number of numerical
parameters, guarantee good fit to phenomenological data, but do not shed light
on the muscle intrinsic characteristics.
After work by Rayment et al. (1993) on the structural study of force
generators in contracting muscles, the observations of catalytic domains of
myosin being initially weakly attached to actin are commonly associated with
the _weakly bound_ CB states, and the following structural changes resulting
in tight binding of actin-myosin linkages are associated with _strongly bound_
CB states. Since the steady-state equation (2) is a late-time part of more
general kinetic equation (1), the proportions $n_{V}(x)$ are also part of the
non-steady solutions, as demonstrated by Lombardi and Piazzesi (1990) and
recently by Walcott and Herzog (2008) employing Huxley’s Eqs. (4) and (5). It
seems therefore plausible to associate Huxley’s short-domain proportion
$n_{V}^{(loc)}(x)$ and large-domain proportion $n_{V}^{(deloc)}(x)$ with
respectively weak and strong late-time CB states. In this study, the actin-
myosin bound state is composed of the equilibrated and excited states
distributed by Gaussian function dictated by the second law of thermodynamics.
Within the proposed framework of stochastic approach to the attachment-
detachment events of myosin heads, a common requirement of normalization of
the random proportion $n_{V}(x)$ specifies the heterogeneity of the CB
distribution via the correlated intrinsic structural ($x_{m}$, $d$), kinetic
($f_{m}$, $g_{m}$) and dynamic ($V_{m}$) muscle characteristics [see e.g. Eq.
(14)], that decreases the number of free parameters. Moreover, the trend of
weakly bound myosin heads to achieve maximum structural-domain entropy in the
vicinity of actin-filament sites ($x\approx 0$) requires a correlation in
signs between the head displacements ($x$) and velocities
($\overset{\cdot}{x}$). Consequently, the conceivable CB domains for both
bound states schematically shown in Fig. 1 are eventually described by
Heaviside functions in Eq. (15). A geometrical selection of the main
components of the force resulting in the power stroke in a direction
consistent with the vector of contraction velocity are also shown in Fig.1.
The explicit solution (9) to Huxley’s kinetic equation for CB proportions
$n_{0}(x)$ and $\Delta n_{V}(x)$, distributing respectively strongly and
weakly bound states over the actin filament cells, results in the velocity-
independent (isometric) force $F_{0}$ and contractive force $\Delta F_{V}$,
components of the CB force output $F_{V}$ (16). In Fig. 2, famous Huxley’s
comparative analysis with Hill’s data on muscle concentric tension (Huxley,
1957, p. 287) is revisited. The high-velocity wing of the tension curve above
$V/V_{\max}=0.5$ controlled mostly by weak CB states is well fitted by both
non-Gaussian (5) and Gaussian (15) proportions. It is not the case of the low
velocity region $0.2<V/V_{\max}<0.5$, where a discrepancy between Huxley’s
curve (8) and the data indicate a disadvantage of the short-domain ($x<h<d$)
weak CBs exerting negative force $\Delta F_{V}$ and eventually reducing the
total produced tension. In contrast to the postulated retarded detachment
($g_{1}<f_{1}$) discussed in Eqs. (4) and (7), the Gaussian strong and weak
CBs require faster detachment than attachment ($g_{m1}>f_{m1}$) in muscle
shortening, as derived from Hill’s data and shown in the inset in Fig. 2. The
regular deviation of Gaussian CBs from the data at very low velocities is
associated with a simplified modeling of the channel of relaxation of weak
states to strong states. Indeed, the fit analyses can be improved when the
proportion of a new weak-to-strong transient Gaussian CB spreads its domain
symmetrically within the range $-\delta\leq x\leq\delta$, extending the weak
CB state in the vicinity of $x\approx 0$, as shown by the dotted line in the
inset in Fig. 2 for the case $V/V_{\max}=0.1$.
In Fig. 3, the upper branch of the force-velocity curve (18) drawn at a single
adjustable parameter ($V_{m2}=-288$ $nm/s$) fits well the empirical data on CB
force in muscle stretching. Similar to shortening, the fitting analysis could
be improved at low stretch velocities when a transient bound state is
additionally introduced, as independently proposed by Mehta and Herzog (2008,
Fig. 3). These authors also raised the central question on the existence of
Huxley’s proportions favoring myosin head attachment events at large distances
with an increase in contraction velocity. One therefore infers that although
non-Gaussian large-domain CBs (5) numerically fit the empirical data (Fig. 2),
they do favor neither thermodynamic equilibrium in the overlapped zone nor
high cycle efficiency (Hill et al., 1975).
The microscopic structures of the short-domain bound states are provided above
via observation conditions of the generic curve (18) equally applied to the
reduced tension in a single CB, sarcomere, fiber, or muscle as a whole during
its steady shortening or lengthening. It is also demonstrated (inset in Fig.
3) how the two-state muscle cycle _duty ratio_ $\beta$ derived from real
experiments can be helpful in a characterization of the Gaussian CB _rate
ratio_ $\alpha=f_{m}/g_{m}$ [$=(1-\beta)/\beta$] and strong bound state
proportion $n_{0}$ ($=1-\beta$). To summarize a comparative analysis of
structural and kinetic characteristics of CBs, one can see that _steady_
muscle eccentric and concentric contractions are well distinguished via the
attachment-detachment rate rations, with $\alpha_{stret}>1>\alpha_{short}$,
the strongly-bound occupation numbers, with$\
n_{0}^{(stret)}>n_{0}^{(short)}$, supported by the directly observable cycle
duty ratios $\beta_{stret}<\beta_{short}$ (Mehta and Herzog, 2008). Within
this context, the working hypothesis by Mehta and Herzog (2008) ”that a
stretched cross-bridge might remain attached longer than a cross-bridge that
had been shortened while attached” combined with the main finding by Lombardi
and Piazzesi (1990) that ”reattachment (in steady lengthening is)… faster than
attachment in the isometric condition or during shortening … in the same
domain of $x$” results in the predictions $\alpha_{stret}>1$,
$n_{0}^{(stret)}>1/2$, and $\beta_{stret}<1/2$, which are generally consistent
with the CB parameters derived in the insets in Figs. 2 and 3.
To conclude, the provided statistical thermodynamic analysis of the
attachment-detachment CB process, modifying Huxley’s mechanical sliding
filament model, can also be figured out as an two-headed steady walking of
synchronous myosin molecules over periodical sites of actin filaments with
multiple $36$-$nm$ steps, as directly observed by Sakamoto et al. (2008). The
fluctuating steps are statistically scattered by the normal distribution,
having the zero mean and variance linear with muscle contraction velocity. The
proposed steady contraction dynamics is universally observable through the two
branches of the force-velocity curve generic for steady shortening and
lengthening of a muscle as a whole or its counterparts. The microscopic
structural muscle characteristics appear to be strongly correlated to kinetic
and dynamic characteristics distinguished by the force output directions
generated in distinct muscle regimes. At a macroscopic level, similar kind of
correlations driven by maximum generated force were revealed via the primary
muscle functions well distinguished though the muscle structure adapted to
efficient eccentric, isometric, or concentric contractions (Kokshenev, 2008).
.
Acknowledgements
.
The author thanks Scott Medler for helpful comments. The financial support by
CNPq is also acknowledged.
.
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Figure Legends
Figure 1. Mechanical scheme of the force generation by combining myosin heads
with periodic actin filament. Each of the two heads of the effective CB may be
attached to actin filament either in equilibrated ground state (shown by the
open circle) with the uniform probability $n_{0}$, within the domains
$x\leq\pm x_{0}$, or in the non-equilibrium, excited state (closed circle)
with the probability $\Delta n_{V}(x)$, within the domains $x\leq-
x_{1},-x_{2}$. The _arrows_ indicate the directions of the sliding velocity of
the actin filament $V_{A}$ and the myosin filament $V_{M}$. During concentric
muscle contraction with a _positive_ velocity $V$, the velocity-independent
portion of the generated force $F_{0}$ is also positive, whereas the ATP
hydrolysis results in the negative portion of the contractive force $\Delta
F_{V}$ . During eccentric contractions commonly associated with the _negative_
direction of velocity $V$, both the forces are also negative.
Figure 2. Analysis of the theoretically predicted tension-velocity curve using
available data on the reduced tension during steady muscle shortening. The
_points_ and _dashed-point curve_ are the famous data by Hill (1938) for
isolated frog muscles modeled by Huxley (1957), drawn respectively by the
phenomenological equation $P^{(\exp)}/P_{0}=a(1-V/V_{\max})/(a+V/V_{\max})$,
with $a=0.25$ and Eq. (8), fitted by Huxley’s parameters listed above. The
_solid line_ is Eq. (19) taken at $\lambda=0.85$. _Inset:_ The attachment-
detachment rates and CB proportions predicted in Eq. (15) within the CB
domains at distinct shortening velocities reduced to the maximum velocity. The
CB structure discussed in the Results is exemplified by the model parameters
$x_{01}^{(\operatorname{mod})}=2d/3$ and $x_{m1}^{(\operatorname{mod})}=-d/3$,
as well as by $n_{01}^{(\operatorname{mod})}=0.47$, providing the _rate ratio_
$\alpha_{1}=f_{m1}/g_{m1}=0.88$. The _dotted line_ shows a proportion for the
modeled transient CB state schematically drawn for $V/V_{\max}=0.1$.
Figure 3. Steady force induced by one cross-bridge versus the stretching
velocity. The _open circles_ are the mean datapoints of the forces (re-scaled
by $|F_{0}|=1.95$ $pN$ ) measured by Lombardi and Piazzesi (1990, Fig. 7) in
frog muscle fibers at stretch velocities lying between $75$ and $1030$ $nm/s$
and scaled here by $d=36$ $nm$. The _closed square_ indicates the force per
one CB reported by Mehta and Herzog (2008) for unspecified velocities. The
theoretical curve is drawn based on Eq. (20) with
$V_{m2}^{(\operatorname{mod})}=-8$ $d/s$. _Inset:_ The attachment-detachment
rates within the CB domains and CB proportions predicted in Eq. (15) at three
distinct velocities reduced to the found $V_{m2}^{(\operatorname{mod})}$. They
are exemplified by model parameters $x_{02}^{(\operatorname{mod})}=$
$x_{m2}^{(\operatorname{mod})}=-d$, consistent with the observation conditions
discussed in the Results, as well as by
$n_{02}^{(\exp)}=1-\beta_{2}^{(\exp)}=0.93$, where the stretch _cycle_ _duty
ratio_ $\beta_{2}^{(\exp)}=7.35\%$ (the time of attachment $f_{m}^{-1}$
related to total CB cycling time $f_{m}^{-1}+g_{m}^{-1}$, i.e.,
$\beta=1-n_{0}$) studied by Mehta and Herzog (2008) is employed. Moreover, the
relation
$|V_{m2}^{(\operatorname{mod})}|=f_{m2}^{(\exp)}x_{m2}^{(\operatorname{mod})}/[1-\beta_{2}^{(\exp)}]$
derived from Eq. (20) provides a crude model estimate for the CB domain
$x_{m2}^{(\operatorname{mod})}\thickapprox 1.2$ $d$, if their characteristic
attachment time $[f_{2}^{(\exp)}]^{-1}=$ $0.167$ $s$ is also employed. The
rates (shown by _dashed lines_) are determined by the ratio
$f_{m2}^{(\exp)}/g_{m2}^{(\operatorname{mod})}=13$, corresponding to the model
estimate $[g_{m2}^{(\operatorname{mod})}]^{-1}=$ $2.2$ $s$. The _dotted line_
shows a proportion for the assumed transient CB state schematically drawn for
$V/V_{m2}^{(\operatorname{mod})}=0.1$.
|
arxiv-papers
| 2009-09-03T16:01:10 |
2024-09-04T02:49:05.027739
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valery B. Kokshenev",
"submitter": "Valery Kokshenev B.",
"url": "https://arxiv.org/abs/0909.0687"
}
|
0909.0712
|
In [1] Beilinson obtained a formula relating the special
value of the $L$-function of $H^2$ of a product of modular curves to
the regulator of an element of a motivic cohomology group - thus
providing evidence for his general conjectures on special values of
$L$-functions. In this paper we prove a similar formula for the
$L$-function of the product of two Drinfeld modular curves providing
evidence for an analogous conjecture in the case of function fields.
MSC classification: 11F52, 11G40
§ INTRODUCTION
§.§ Beilinson's conjectures and a function field analogue
The algebraic $K$-theory of a smooth projective variety over a
field has a finite, increasing filtration called the Adams
filtration. For a variety over a number field, in [1]
Beilinson formulated conjectures which relate the graded pieces of
this filtration, the motivic cohomology groups $H^\ast_\mathcal M$,
to special values of the Hasse-Weil $L$-function of a cohomology
group of the variety.
The conjectures are of the following nature: corresponding to the
motivic cohomology group $H^*_{\M}$ there is a real vector space
$H^*_{\D}$, called the real Deligne cohomology, whose dimension is
the order of the pole, at a specific point, of the Archimedean
factor of the $L$-function.
Beilinson defined a regulator map from the $H^*_{\M}$ to $H^*_{\D}$
and conjectured that its image determines a $\Q$-structure on the
$H^*_{\D}$. $H^*_{\D}$ has another $\Q$-structure induced by de Rham
and Betti cohomology groups. Beilinson conjectured further that the
determinant of the change of basis between these two $\Q$ structures
is, up to a non-zero rational number, the first non-zero term in the
Taylor expansion of the $L$-function at a specific point. More
details can be found in the book [14] or in the paper
Beilinson's conjectures have been proved only in a few special
cases. In [1], he proved them for the product of two
modular curves and as a result for the product of two non-isogenous
elliptic curves over $\Q$. It is these results that we generalize to
the function field case.
Since the conjectures deal with the transcendental part of the value
of the $L$-function and involve the Archimedean $L$-factor they can
be viewed as conjectures for the Archimedean place. It is natural to
ask whether one can formulate a similar question for the other
finite places.
In [15], we formulated a function field analogue of the
Beilinson conjectures. In particular we defined a group which, at a
finite place, plays the role of the real Deligne cohomology.
This group, called the $\nu$-adic Deligne cohomology, is a rational
vector space whose dimension was shown by Consani [5],
assuming some standard conjectures, to coincide with the order of
the pole, at a certain integer, of the local $L$-factor at the place
In [15] we defined a regulator map $r_{ \D,\nu}$ from the
motivic cohomology to the $\nu$-adic Deligne cohomology and, in
analogy with the Beilinson conjectures, conjectured that the image
is a full lattice. Finally, in some cases, we made a conjecture on
the special value of the $L$-function.
One such case is that of the L-function of a surface at the integer
$s=1$. It is a generalization of the Tate conjecture for a variety
over a function field. The precise statement of this conjecture is
as follows
Let $X$ be a smooth proper surface over a function
field $K$ and $\XX$ a semi-stable model of $X$ over $A$, its ring of
integers. Let $\Lambda(H^2(\bar{X},\Q_\ell),s)$ be the completed
$L$-function of $H^2$, namely the product of the local $L$-factors
at all places of $K$, where $\bar{X}=X \times Spec(\CI)$. Then,
there is a `thickened' regulator map $R_{\D}=\bigoplus_{\nu}
r_{\D,\nu} \oplus cl$
$$R_{\D}:H^3_{\M}(X,\Q(2))\oplus B^1(X) \longrightarrow \bigoplus_{\nu}
where $B^1(X)=CH^1(X)/CH^1_{hom}$ and $PCH^1(X_{\nu})$ is a subgroup
of the Chow group of the special fibre at $v$, which provides an
integral structure on the $\nu$-adic Deligne cohomology
$H^3_{\D}(X_{/\nu},\Q(2))$ defined below. We conjecture $R_\D$
satisfies the following properties:
A. $R_{\D}$ is a pseudo-isomorphism - namely it has a
finite kernel and co-kernel.
B.(Tate's conjecture) $-\ord_{s=2} \Lambda(H^2(\bar{X},\Q_\ell),s)=\dim_{\Q}
B^1(X)\otimes \Q$
$$\Lambda^*(H^2(\bar{X},\Q_\ell),1)=\pm \frac{|\text{coker}(R_{\D})|}{|\text{ker}(R_{\D})|}
\cdot \log(q)^{\ord_{s=1} \Lambda(H^2(\bar{X}),s)}$$
where $\Lambda^*$ denotes the first non-zero value in the Laurent
expansion and $|\;|$ of a finite set denotes its cardinality.
In other words, the conjecture asserts that the regulator map
provides an isomorphism of the rational motivic cohomology with the
sum of all the $\nu$-adic Deligne cohomology groups. The special
value then measures the obstruction to this map being an isomorphism
of integral structures.
This conjecture comes from the localization sequence for motivic cohomology which relates the motivic cohomologies of $X, \XX$ and $\XX_{\nu}$. The regulator map is the boundary map in the localization sequence. We stated the conjecture for surfaces - for points, when $X=Spec(K)$, this is simply a combination of the function field class number formula and units theorem – the special value conjecture in this case implies the well known formula
$$ \Lambda^* (H^0(Spec(X),0))=-\frac{h_K}{(q-1)\log(q)}$$
where $h_K$ is the class number and $(q-1)$ is the number of roots of unity which can be interpreted as the orders of the kernel and cokernel of the regulator map respectively and the power of $\log(q)$ that appears corresponds to the well known fact that the zeta function has a simple pole at $s=1$.
Beilinson [1] theorem follows from a formula relating the
cohomological $L$-function of $ h^1(M_f) \otimes h^1(M_g)$, where
$h^1(M_f)$ and $h^1(M_g)$ are the motives of eigenforms of weight
two and some level $N$, to the regulator of an element of a certain
motivic cohomology group evaluated on the $(1,1)$-form
d\bar{z}_2-d\bar{z}_1 \otimes dz_2)$. We show an analogous formula
in the Drinfeld modular case with the Archimedean place being
replaced by the prime $\infty$. More precisely, since our
$L$-functions essentially take rational values, we have an exact
formula for the value analogous to the main theorem of [4].
Our main result is the following –
Let $I$ be a square-free element of $\F_{q}[T]$ and
$\Gamma_0(I)$ the congruence subgroup of level $I$. Let $f$ and $g$
be Hecke eigenforms for $\Gamma_0(I)$ and $\Lambda(h^1(M_f)\otimes
h^1(M_g),s)$ denote the completed, that is, with the $L$-factor at
$\infty$ included, $L$-function of the motive $h^1(M_f)\otimes
h^1(M_g)$.Then one has
\begin{equation}
\Lambda(h^1(M_f)\otimes h^1(M_g),1)=\frac{q}{ 2 (q-1) \kappa}
(r_{\D,\infty} (\Xi_0(I)),\Z_{f,g})
\end{equation}
where $\Xi_0(I)$ is an element of motivic cohomology group
$H^3_{\M}(X_0(I) \times X_0(I),\Q(2))$, $r_{\D,\infty}$ is the
$\infty$-adic regulator map, $\kappa$ is an explicit integer
constant and $\Z_{f,g}$ is a special cycle in the special fibre at
$\infty$ and $(,)$ denotes the intersection pairing the Chow group
of the special fibre.
§.§ Outline of the paper
In the first few sections we introduce some of the background on
Drinfeld modular curves. This is perhaps well known to people
working with function fields, but perhaps not so well known to
people working in the area of algebraic cycles, hence it has been
We then study the analytic side of the problem, namely the special
value of the $L$-function. We use the Drinfeld uniformization and an
analogue of the Rankin-Selberg method to get an integral formula for
the $L$-function. We also formulate and prove an analogue of
Kronecker's first limit formula and use it to get an integral
formula for the special value at $1$ of the $L$-function.
Following that we study the algebraic side of the problem. We
introduce the motivic cohomology group of interest to us and define a
regulator map on it. This regulator map is the boundary map in a
localization sequence relating the motivic cohomology groups of the
generic fibre and special fibre. The result is that the regulator of
an element of our motivic cohomology group is a certain $1$-cycle on
the special fibre.
We then construct an explicit element in this motivic cohomology
group using analogues of the classical modular units and
compute its regulator. The regulator of this element is then
related to our integral formula using the relation between
components of the associated reduction of the Drinfeld modular
curve and vertices on the Bruhat-Tits tree.
In the classical case the regulator is a current on $(1,1)$-forms
and one obtains the special value by evaluating this current on a
specific form. Here, the regulator is a $1$-cycle and one obtains
the special value by computing the intersection pairing with a
specific cycle supported on the special fibre. Finally we relate
our formula with the conjecture made above.
Curiously, the formulae are almost identical to the number field
case, though the objects involved are quite different. It suggests,
however, that there should be some underlying structure on which all
these results case be proved and the case of number field and
function fields arise by specializing to the case of $\ZZ$ or
Acknowledgements: I would like to thank S. Bloch, C.Consani,
J. Korman, S. Kondo, M. Papikian, A Prasad and M. Sundara for their
help and comments on earlier versions of this manuscript. I would also like to thank
the referee for his comments.
I would like to thank the University of Toronto, Max-Planck-Institute, Bonn
and the TIFR Centre for Applicable Mathematics in Bangalore for proving me an
excellent atmosphere in which to work in. Finally I would like to dedicate this paper
to the memory of my mother, Ratna Sreekantan.
§ NOTATION
Throughout this paper we use the following notation
* $\F_q$: the finite field with $q=p^{n}$ elements, where $p$ is a prime number.
* $A=\F_{q}[T]$: the polynomial ring in one variable.
* $K=\F_q(T)$: the quotient field of $A$.
* $\pi_{\infty}=T^{-1}$: a uniformizer at the infinite place
* $K_{\infty} = \F_q((\pi_{\infty}))$: the completion of $K$ at
* $K_{\infty}^{sep}$: the separable closure of
* $K_{\infty}^{ur}$: the maximal unramified extension of
* $\CI$: the completed algebraic closure of $K_{\infty}$.
* $\ord_{\infty} = - \deg$: the negative value of the
usual degree function.
* $\OO_{\infty} = \F_q[[\pi_{\infty}]]$: the $\infty$-adic integers.
* $|\cdot|$: the $\infty$-adic absolute value on $\KI$,
extended to $\CI$.
* $|\cdot|_{i}$: the `imaginary part' of $|\cdot|$:
$|z|_{i}=\text{inf}_{x \in K_{\infty}}\{|z-x|\}$
* $G$: the group scheme $GL_2$.
* $B$: the Borel subgroup of $G$.
* $Z$: the center of $G$.
* $\K = G(\OO_{\infty})$.
* $\I=\left\{ \begin{pmatrix} a & b\\c & d \end{pmatrix} \in \K \text{ such that } c \equiv
0 \text{ mod } \infty \right\}$
* $\T$: the Bruhat-Tits tree of $PGL_2(K_{\infty})$.
* $V({\mathfrak G})$: the set of vertices of a graph ${\mathfrak G}$.
* $Y({\mathfrak G})$: the set of oriented edges of an oriented graph ${\mathfrak G}$: if $e$ is
an edge, $o(e)$ and $t(e)$ denote the origin and terminus of the
* $\m$: a divisor of $K$ with degree $\deg(\m)$
(this is different from $\deg(m)=-\ord_{\infty}(m)$ for $m \in K$).
§ PRELIMINARIES ON DRINFELD MODULAR CURVES
In the function field setting, there are two analogues of the
complex upper half-plane: the Bruhat-Tits tree and the Drinfeld
upper half-plane. These sets capture different aspects of the
classical upper half-plane. The Bruhat-Tits tree has a transitive
group action, but does not have a manifold structure, whereas the
Drinfeld upper half-plane has the structure of a rigid analytic
manifold, but no transitive group action. These two sets are related
by means of the building map. We first describe the Bruhat-Tits
tree. We refer to the paper [10] for further details.
§.§ The Bruhat-Tits tree
The Bruhat-Tits tree $\T$ of $PGL_2(K_{\infty})$ is an oriented
graph. It has the following description.
§.§.§ Vertices and Ends of $\T$.
The vertices of $\T$ consist of similarity classes $[L]$, where $L$
is a $\OO_\infty$-lattice in $(K_{\infty})^2$. Recall that a
lattice $L$ is said to be similar to $L'$ ($L \equiv L'$) if and
only if there exists an element $c \in K_{\infty}^*$ such that
$L=cL'$. Two vertices $[L]$ and $[L']$ are joined by an edge if
they are represented by lattices $L$ and $L'$ with $L
\subset L'$ and $\dim_{\F_{q}}(L'/L)=1$. Each vertex $v$ has
exactly $(q+1)$-adjacent vertices and this set is in bijection
with $\CP^1(\F_{q})$. More generally, the set of vertices of $\T$
which are adjacent to a fixed vertex $[L]$ by at most $k$
edges is in bijection with $\CP^1(L/\pi_{\infty}^kL)$. This makes
$\T$ in to a $(q+1)$-regular tree.
A half-line is an infinite sequence of adjacent non-repeating
vertices $\{v_i\}$ starting with an initial vertex $v_0$. Two
half-lines are said to be equivalent if the symmetric difference of
the two sets of vertices is a finite set. An end is an equivalence
class of half lines.
Let $\partial \T$ be the set of the ends of $\T$. There is a
bijection (independent of $L$)
\[
\partial \T \stackrel{\simeq} {\longrightarrow} \varprojlim_{k}
\CP^1(L/\pi_{\infty}^kL) \simeq \CP^1(\OO_{\infty}) =
\CP^1(K_{\infty}).
\]
The left-action of $G(K_{\infty})$ on $\T$ extends to an action on
$\partial \T$ which agrees with the action of $G(K_{\infty})$ on
$\CP(K_{\infty})$ by fractional linear transformations.
§.§.§ Orbit Spaces.
For $i\in\mathbb Z$, let $v_i \in V(\T)$ be the vertex
$[\pi_{\infty}^{-i}\OO_{\infty} \oplus \OO_{\infty}]$. As the vertex
$v_0$ has stabilizer $\K\cdot Z(\KI)$ in $G(\KI)$, one obtains the
following identification
\[
G(K_{\infty})/\K\cdot Z(\KI) \stackrel{\sim}{\rightarrow}
V(\T)\qquad g \mapsto g(v_0).
\]
Similarly, let $e_i$ be the edge $\overrightarrow{v_{i}v_{i+1}}$
(i.e $o(e_i) = v_i$, $t(e_i) = v_{i+1}$) then
\[
G(K_{\infty})/\I\cdot Z(\KI) \stackrel{\sim}{\rightarrow}
Y(\T)\qquad g \mapsto g(e_0).
\]
These identifications allow one to consider functions on vertices
and on edges of $\T$ as equivariant functions on matrices.
Let $w = \begin{pmatrix} 0&1\\1&0
\end{pmatrix}$. We set
\[
S_{V}=\left \{\begin{pmatrix}\pi_{\infty}^k & u \\0 & 1
\end{pmatrix}|~k \in \ZZ,~u \in \KI,~u~\text{mod}
\;\;\pi_{\infty}^k\OO_{\infty}\right \}
\]
\[
S_U=\left \{w\begin{pmatrix}1 &0 \\c & 1\end{pmatrix} |~c \in
\F_{q}\right \} \cup \{1\},\quad S_Y=\left \{gh~|~g \in S_V, h
\in S_U \right \}.
\]
Then, $S_V$ is a system of representatives for $V(\T)$ and $S_Y$ is
a system of representatives for $Y(\T)$ [12]. We will use
these systems to define functions on the vertices and the edges of
the tree.
§.§.§ Orientation.
The choice of an end $\infty$ representing the equivalence class
of the half line $\{v_0,v_1,\ldots\}$, where $v_i$ are as above,
defines an orientation on $\T$ in the following manner. If $e=w_0w_1
$ is an edge, $e$ is said to be positively oriented if there is a
half line in the equivalence class of $\infty$ starting with initial
vertex $w_0$ and subsequent vertex $w_1$ and negatively oriented if
the half line has initial vertex $w_1$ and subsequent vertex $w_0$.
For a positively oriented edge, $e=w_0w_1$, let $o(e)=w_0$ denote
the origin of $e$ and $t(e)=w_1$ denote the terminus. This
determines a decomposition $Y(\T)=Y(\T)^{+}\cup Y(\T)^{-}$. We say
that $\sgn(e)=+1$ if $e \in Y(\T)^+$ and $\sgn(e)=-1$ if $e \in
At a vertex $v$ there is precisely one positively oriented edge with
origin $v$ and there are $q$ positively oriented edges with
terminus $v$. That determines a bijection of $S_V$ with the set of
positively oriented edges $Y(\T)^+$. We will use the notation
$v(k,u)$ and $e(k,u)$ to denote the vertex and the positively
oriented edge represented by the matrix
$\begin{pmatrix} \pi_{\infty}^k & u \\ 0 & 1
\end{pmatrix}$
respectively. The edge $e(k,u)$ has origin $o(e)=v(k,u)$ and
terminus $t(e)=v(k-1,u)$.
§.§.§ Realizations and norms.
The realization $\T(\R)$ of the unoriented tree $\T$ is a
topological space consisting of a real unit interval for every
unoriented edge of $\T$, glued together at the end points according
to the incidence relations on $\T$. If $e$ is an edge, we denote
by $e(\R)$ the corresponding interval on the realization. Let
$\T(\ZZ)$ denote the points on $\T(\R)$ corresponding to the
vertices of $\T$. The set of points $\{t[L]+(1-t)[L']~|~t \in \Q\}$
lying on edges $([L],[L'])$ will be denoted by $\T(\Q)$.
A norm on a $\KI$-vector space $W$ is a function $\nu:W
\rightarrow \R$ satisfying the following properties
- $\nu(v) \geq 0;~\nu(v)=0 \Leftrightarrow v=0$
- $\nu(xv)=|x|\nu(v),~\forall~x \in \KI$
- $\nu(v+w)\leq \text{max}\{\nu(v),\nu(w)\},~\forall~v,w
\in W$.
Two norms $\nu_1$ and $\nu_2$ are said to be similar if there
exist non-zero real constants $c_1$ and $c_2$ such that
$$c_1 v_1 \leq v_2 \leq c_2 v_1$$
The right action of $GL(W)$ on $W$ induces an action on the set of
norms as
\[
\gamma(\nu)(v)=\nu(v\gamma).
\]
This action descends to similarity classes. The following theorem
relates norms to the realization of the tree.
There is a canonical $G(\KI)$-equivariant bijection $b$ between
the set $\T(\R)$ and the set of similarity classes of norms on
This bijection is defined as follows. To a vertex $[L]$ in
$\T(\ZZ)=V(\T)$ we associate $b([L])$, the class of the norm
$\nu_{L}$ defined by
\[
\nu_{L}(v) = \text{inf}\{|x|~:~x \in \KI, v \in xL\}.
\]
This norm makes $L$ a unit ball. If $P$ is a point of $\T(\R)$
which lies on the edge $([L],[L'])$ with $\pi_{\infty} L' \subset L
\subset L'$ and $P=(1-t)[L]+t[L']$, then $b(P)$ is the class of the
norm defined by
\[
\nu_P(v) = \text{sup}\{\nu_{L}(v),q^t\nu_{L'}(v)\}.
\]
§.§ Drinfeld's upper half-plane and the building map
The set $\Omega=\CP^1(\CI) - \CP^1(\KI)=\CI - \KI$ is called the
Drinfeld upper half-plane. This space has the structure of a rigid
analytic space over $\KI$. There is a canonical $G(\KI)$-equivariant
\begin{equation}\label{bm}
\lambda:\Omega \longrightarrow \T(\R)
\end{equation}
called the building map. It is defined as follows. To $z \in
\Omega$, we associate the similarity class of the norm $\nu_{z}$
on $(\KI)^2$ defined by
\[
\nu_{z}((u,v)) = |uz+v|.
\]
Since $|\cdot|$ takes values in $q^{\Q}$, the image of $\lambda$
in contained in $\T(\Q)$ and in fact one shows that
§.§ The pure covering and its associated reduction
The Drinfeld upper half space is a rigid analytic
space. We need to study its reduction at the prime $\infty$.
However, there is no canonical reduction but there is a natural one
obtained by the associated analytic reduction of a certain pure
cover of $\Omega$. This is described in detail in [10], pg
33-34 and in fact, we essentially copy from there.
The pure cover is described as follows. For $n\in \ZZ$, let $D_n$
denote the subset of $\C_{\infty}$ defined by
* $1.\; D_{n}=\{z \in \C_{\infty}:\;|\pi_{\infty}|^{n+1} \leq |z| \leq
* $2.\; |z-c\pi_{\infty}^n|\geq |\pi_{\infty}|^n,
|z-c\pi_{\infty}^{n+1}| \geq |\pi_{\infty}^{n+1}|$ for all $c\in
\F_{q}^{*} \subset K_{\infty}.$
* Equivalently $2'.\; |z|=|z|_i$
Condition $2'$ shows $D_{n} \subset \Omega$ and is independent of
the choice of $\pi_{\infty}$. This is an affinoid space over
$K_{\infty}$ with ring of holomorphic functions
\in \F_{q}^{*}>$$
which is the algebra of `strictly convergent power series' in
$\pi_{\infty}^{-n}z$. This allows one to define the canonical
reduction $(D_n)_{\infty}$ and this is isomorphic to the union of
two projective lines meeting at an $\F_{q}$-rational point, with all
other rational points deleted.
For $\bi=(n,x),n\in \ZZ,x \in K_{\infty}$ let
$D_{\bi}=D_{(n,x)}=x+D_n$. Then one can see that, if $\bi'=(n',x')$,
$$D_{\bi}=D_{\bi}' \Leftrightarrow n=n' \text{ and } |x-x'|\leq
So if $I=\{(n,x)|n\in \ZZ, x\in K_{\infty}/\pi_{\infty}^{n+1}
\OO_{\infty}\}$, where, for each $n$, $x$ runs through a set of
representatives, then
$$\Omega=\bigcup_{\bi \in I} D_{\bi}$$
is a pure covering of $\Omega$. For any $\bi$, there are only
finitely many $\bi'$ such that $D_{\bi} \cap D_{\bi'}\neq \phi$.
With respect to this covering one has an associated analytic
$$R:\Omega \longrightarrow \Omega_{\infty}$$
where $\Omega_{\infty}$ consists of a union of $\CP^1_{\F_{q}}$'s
each of which meets $q+1$ other ones at $\F_q$ rational points.
Conversely, any $\F_q$ rational point $s$ of a component $M$
determines a component $M'$ such that $M'\cap M=\{s\}$. For adjacent
$M$ and $M'$ let $M^*=M-M(\F_q)$ and $(M \cup M')^*=M\cup
M'-(M(\F_q) \cup M'(\F_q)) \cup (M \cap M')$. Then, there exits
$\bi,\bi'$ such that
$$R^{-1}(M^*)=D_{\bi} \cap D_{\bi'} \text{ and } R^{-1}((M \cup
The intersection graph of $\Omega_{\infty}$ is the graph whose
vertices are the components $M$ of $\Omega_{\infty}$. Two vertices
$M$ and $M'$ are joined by an oriented edge if and only if $M$ and
$M'$ are adjacent components of $\Omega_{\infty}$, that is, if
$M\cap M'\neq \phi$. The map $\lambda$ in (<ref>) determines a
canonical identification of this graph with the Bruhat-Tits tree:
Given a component $M$ there exists a unique $[L] \in \T(\ZZ)$ such
and this association is compatible with the group actions and
identifies the two graphs. We will use this identification rather
crucially in the final step of the proof.
§.§ Drinfeld modular curves of level $I$
For a monic polynomial $I$ in $A$ let
\Gamma_0(I)=\left\{\begin{pmatrix} a & b \\
c & d \end{pmatrix}\in \Gamma~|~c \equiv 0 \;\; \text{mod}
\;\;I\right \}.
$\Gamma_0(I)$ acts discretely on $\Omega$ via Möbius
transformations: For $z \in \Omega$ and $\gamma=\begin{pmatrix} a& b
\\c & d \end{pmatrix} \in \Gamma_0(I)$, define
$$\gamma z=\frac{az+b}{cz+d}.$$
The Drinfeld modular curve $X_0(I)$ of level $I$ is a smooth,
proper, irreducible algebraic curve, defined over $K$, such that
its $\CI$ points have the structure of a rigid analytic space and
there is a canonical isomorphism of analytic spaces over $\CI$
X_0(I)(\CI) \simeq \Gamma_0(I)\backslash \Omega\cup
\{\text{cusps}\}.
where the cusps are finitely many points in bijection with
$\Gamma_0(I)\backslash \CP^1(K)$. Entirely analogous to the
classical construction over a number field, the Drinfeld modular
curve $X_0(I)$ parameterizes Drinfeld modules of rank two with level
$I$ structure.
$$\T_0(I)=\Gamma_0(I)\backslash \T$$
denote the corresponding quotient of the Bruhat-Tits tree by the
left action of $\Gamma_0(I)$. Let $X(\T_0(I))$ and $Y(\T_0(I))$
denote the vertices and edges of the graph $\T_0(I)$ respectively.
$\T_0(I)$ is an infinite graph consisting of a finite graph
$\T_0(I)^0$ and a finite number of ends corresponding to the
finitely many cusps ([10],Section 2.6).
The curve $X_0(I)$ is a totally split curve over $K_{\infty}$. The
pure covering of $\Omega$ induces a pure covering of $X_0(I)$ and
the associated analytic reduction $R$ is a scheme $X_0(I)_{\infty}$
over $\F_q$ which is a finite union of $\CP^1_{\F_q}$'s intersecting
at $\F_q$ rational points. The intersection graph of this scheme is
the finite part $\T_0(I)^0$ of the graph $\T_0(I)$.
§.§ Harmonic cochains on the Bruhat-Tits tree
In the function field setting there are two notions of modular
forms – corresponding to the two analogues of the complex upper half
plane. One notion deals with certain equivariant functions on the
Drinfeld upper half plane while the other refers to certain
invariant harmonic co-chains on the Bruhat-Tits' tree. The latter
are sometimes called automorphic forms of Jacquet-Langlands-Drinfeld
(JLD) type. It is these that we will be concerned with and will
review their definition and properties in this section.
If $R$ is a commutative ring, an $R$-valued harmonic co-chain on
$Y(\T)$ is a map $\phi:Y(\T) \longrightarrow R$ satisfying the
harmonic conditions:
- $\phi(e) + \phi(\overline{e})=0$
- $\displaystyle{\sum_{t(e)=v}} \phi(e)=0$
where, for $e$ in $Y(\T)$, $\overline{e}$ denotes the same edge
with the opposite orientation. The second condition can also be
stated as follows - First, notice that there is precisely one edge
$e_0$ with $t(e_0)=v$ and $sgn(e_0)=-1$. The second condition is
then equivalent to
\[
\phi(e_0)=\sum_{t(e)=v\atop sgn(e)=1} \phi(e).
\]
If $\Gamma$ is a subgroup of $G(A)$ we will consider co-chains
satisfying the further condition of $\Gamma$-invariance, namely,
- $\phi(\gamma e)=\phi(e),\quad \forall \gamma \in
\Gamma.$
The group of $\Gamma$-invariant, $R$-valued harmonic co-chains
on the edges of $\T$ is denoted by $H(Y(\T),R)^{\Gamma}$. The
harmonic functions on the edges of $\T$ are the analogues of
classical cusp forms of weight $2$. In fact, if $\ell \neq p$ is
a prime number, the $\Gamma$-invariant harmonic co-chains detect
`half' of the étale cohomology group
$H^1_{\text{\'et}}(X(\Gamma),\Q_{\ell})$ ([16], pg. 272).
An R-valued harmonic cochain $f$ is said to be of level $I$ if
$f \in H(Y(\T),R)^{\Gamma_0(I)}$. If $f$ has finite support as
a function on $\Gamma_0(I)\backslash Y(\T)$, it is called a cusp
form or said to be cuspidal. Usually we will deal with $\ZZ$,$\C$ or
$\Q_{\ell}$ valued functions. In analogy with the classical case,
we sometimes will use the word `form' to denote these functions.
§.§.§ Fourier expansions.
A harmonic function on the set of positively oriented edges
$Y(\T)^+$ which is invariant under the action of the group
\[
\Gamma_{\infty}=\left \{\begin{pmatrix} a & b \\0 & d \end{pmatrix}
\in G(A)\right \}
\]
has a Fourier expansion. This statement is a consequence of the
general theory of Fourier analysis on adèle groups. Details can be
founds in [8]. This expansion has the following
description. Let $\eta:\KI \rightarrow {\mathbb C}^*$ be the
character defined as
\[
\eta\left (\sum_j a_j \pi_{\infty}^j\right ) = \exp\left (\frac{2
\pi i {\mathrm Tr}(a_1)}{p}\right )
\]
where ${\mathrm Tr}$ is the trace map from $\F_q$ to $\F_p$. Then,
the Fourier expansion of a $\Gamma_{\infty}$-invariant function $f$
on $Y(\T)^+$ is given by
\[
f\left (\begin{pmatrix} \pi_{\infty}^k & u \\0 & 1
\end{pmatrix}\right ) = c_0(f,\pi_{\infty}^k) + \sum_{0 \neq m \in A \atop \deg(m)
\leq k-2} c(f,\div(m)\cdot \infty^{k-2}) \eta(mu).
\]
The constant Fourier coefficient $c_0(f,\pi_{\infty}^k)$ is the
function of $k \in \ZZ$ given by
\[
c_0(f,\pi_{\infty}^k) = \begin{cases} f\left (\begin{pmatrix}
\pi_{\infty}^k & 0
\\0 & 1
\end{pmatrix}\right) & {\mathrm if} \; k\leq 1 \\ q^{1-k} \displaystyle{\sum_{u \in
(\pi_{\infty})/(\pi_{\infty}^k)} f\left(\begin{pmatrix}
\pi_{\infty}^k & u \\0 & 1
\end{pmatrix}\right)} & {\mathrm if} \; k \geq 1. \end{cases}
\]
For a non-negative divisor $\m$ on $K$, with $\m=\div(m) \cdot
\infty^{\deg(\m)}$, the non-constant Fourier coefficient is
\[
c(f,\m)=q^{-1-\deg(\m)} \sum_{u \in
f\left(\begin{pmatrix} \pi_{\infty}^{2+\deg(\m)}& u \\0 & 1 \end{pmatrix}\right )
\eta(-mu).
\]
§.§.§ Petersson inner product.
There is an analogue of the Petersson inner
product for invariant functions on the tree $\T$.
If $f$ and $g$ are complex valued harmonic co-chains for
$\Gamma_0(I)$, one of which is cuspidal, define
\[
\delta(f,g)(e) =f(e)\overline{g(e)}d\mu(e) \qquad \text { for } e\in
\]
where $\mu(\cdot)$ is the Haar measure on the discrete set
$Y(\T_0(I))$ defined by $\mu(e) =
\frac{q-1}{2}|\text{Stab}_{\Gamma_0(I)}(e)|^{-1}$, where
$|\text{Stab}_{\Gamma_0(I)}(e)|$ is the cardinality of the
stabilizer of $e\in Y(\T_0(I))$. The Petersson inner product of $f$
and $g$ is defined as
\[
<f,g> = \int_{Y(\T_0(I))} \delta(f,g)= \int_{Y(\T_0(I))} f(e)
\overline{g(e)}d\mu(e).
\]
§.§.§ Hecke operators and Hecke eigenforms.
Let $\PP$ be a prime and $I$ a fixed level. The Hecke operator
$T_{\PP}$ is the operator on $H(Y(\T),\C)^{\Gamma_0(I)}$ defined by
\[
T_{\PP}(f)(e)=\begin{cases} f\left(e \begin{pmatrix} \PP & 0 \\ 0
& 1
\end{pmatrix} \right ) + \displaystyle{\sum_{r\text{ mod } \PP}} f\left (e \begin{pmatrix} 1 & r
\\ 0 & \PP \end{pmatrix} \right ) & if\;\;\PP \not{|}~I\\
\displaystyle{\sum_{r\text{mod}~\PP}} f\left(e \begin{pmatrix} 1 & r \\
0 & \PP
\end{pmatrix} \right) & if \;\;\PP~|~I. \end{cases}
\]
A Hecke eigenform $f$ is a harmonic co-chain of level $I$ which is
an eigenfunction of all the Hecke operators $T_{\PP}$. $f$ is
called a newform if in addition it lies in the orthogonal
complement, with respect to the Petersson inner product, of the
space generated by all cusp forms of level $I'$ for all levels $I'$
properly dividing $I$.
If $f$ is a non-zero newform, then the coefficient $c(f,1)$ in the
Fourier expansion is not zero. The form $f$ is said to be normalized
if one further assumes that $c(f,1)=1$. Let $\lambda_{\PP}$ denote
the eigenvalue of the Hecke operator $T_{\PP}$. The Fourier
coefficients of a cuspidal, normalized newform $f$ have the
following special properties:
- $c_0(f,\pi_{\infty}^k)=0,\quad \forall k \in \ZZ$
- $c(f,1)=1$
- $c(f,\m)c(f,{\mathfrak n}) = c(f,\m {\mathfrak n})$,
whenever $\m$ and ${\mathfrak n}$ are relatively prime
- $c(f,\PP^{n-1})-\lambda_{\PP} c(f,\PP^n) +
|\PP|c(f,\PP^{n+1})=0$, if $\PP \not{|}~I \cdot \infty$
- $c(f,\PP^{n+1})-\lambda_{\PP} c(f,\PP^n)=0$, if
$\PP~|~ I$
- $c(f,\infty^{n-1})=q^{-n+1}$, if $n \geq 1.$
If $f$ and $g$ are normalized Hecke eigenforms and $f \neq g$,
then $<f,g>~= 0$. Further, since the Hecke operators are self
adjoint, $f=\bar{f}$.
§.§.§ Logarithms and the logarithmic derivative.
Let $f$ be a $\C_{\infty}$-valued invertible function on $\Omega$. There is a notion of
the logarithm of $|f|$ defined as follows. Let $v$ be a vertex of
$\T$ and $\tau_{v} \in \Omega$ an element of $\lambda^{-1}(v)$
where $\lambda$ is the building map defined in (<ref>). From Section <ref> one can see that
the function $|\cdot|$ factors through the building map, so the quantity $|f|$ depends only on $v$ and not on the choice of $\tau_v$.
\begin{equation}\label{thelog}
\log|f|(v)=\log_{q} |f(\tau_v)|
\end{equation}
This function takes values in $\ZZ$.
If $g$ is a function on the vertices of the tree $\T$, then the
derivative of $g$ is a function on the edges of $\T$ defined to be
\begin{equation}
\label{log}
\partial g(e) = g(t(e))-g(o(e)).
\end{equation}
The logarithmic derivative of an invertible function $f$ on $\Omega$
is the composite of these two maps, namely
\begin{equation}
\label{dlog}
\partial \log|f| (e) = \log|f|(t(e))-\log|f|(o(e)).
\end{equation}
§.§.§ The cohomology of a Drinfeld modular curve.
The cohomology of a Drinfeld modular curve has a decomposition, due
to Drinfeld, which is analogous to the classical decomposition of
the cohomology of a modular curve into eigenspaces of modular forms
of weight two.
Let $\ell$ be a prime, $\ell \neq p$. There is a two dimensional $\ell$-adic representation $\SSP_\ell$ of
the Galois group $Gal(K_{\infty}^{sep}/K_{\infty})$, called the
special representation, which acts through a quotient isomorphic to
$\hat{\ZZ}\ltimes \ZZ_{\ell}(1)$. The group $\hat{\ZZ}$ is
isomorphic to $Gal(K_\infty^{ur}/K_\infty)$. The canonical generator
of $\hat\ZZ$ corresponds to $F_\infty$, the Frobenius automorphism
of $K_\infty^{ur}/K_\infty$. The group $\ZZ_\ell(1)$ is isomorphic
to $Gal(E_\ell/K_\infty^{ur})$, where $E_\ell/K_\infty^{ur}$ is the
field extension obtained by adjoining all the $\ell^r$-th roots of
the uniformizer $\pi_{\infty}$ to $K_\infty^{ur}$. The action of
$F_\infty = 1 \in \hat\ZZ$ on $\ZZ_\ell(1)$ is given by $F_\infty u
F_\infty^{-1} = u^q$, for $u\in\ZZ_\ell(1)$. Choose an isomorphism
$\ZZ_\ell(1) \cong \ZZ_\ell$, then
\begin{equation*}
\SSP_\ell: Gal(K_{\infty}^{sep}/K_\infty) \twoheadrightarrow \hat\ZZ
\ltimes \ZZ_\ell \to Gl(2,\mathbb Q_\ell)
\end{equation*}
where the right hand-side arrow is defined as
\begin{equation}
(1,0) = F_\infty \mapsto \begin{pmatrix}1 & 0 \\0 &
q^{-1}\end{pmatrix};\quad (0,1) \mapsto
\begin{pmatrix} 1 & 1 \\0 & 1\end{pmatrix}. \label{frob}
\end{equation}
We recall the following theorem of Drinfeld [10].
Let $X_0(I)$ be a Drinfeld modular curve
with level $I$ structure and let $\bar{X}_0(I)=X_0(I) \times
Spec(\CI)$. Then
\begin{equation}
H^1_{\text{\'et}}(\bar{X}_0(I),\Q_{\ell}) \cong
H(Y(\T),\Q_{\ell})^{\Gamma_0(I)} \otimes \SSP_\ell. \label{drinisom}
\end{equation}
This isomorphism is compatible with the action of the local Galois
group $Gal(K_{\infty}^{sep}/K_{\infty})$ and the action of the Hecke
A consequence of this theorem and Eichler-Shimura type relations
[10](4.13.2) is the decomposition of the $L$-function of a
Drinfeld modular curve into a product of $L$-functions of Hecke
\begin{equation}
L(H^1_{\text{\'et}}(\bar{X}_0(I)),s)=\prod L(h^1(M_f),s),\qquad
s\in\C. \label{dec0}
\end{equation}
where $L(h^1(M_f),s)=L(f,s)$ is the $L$-function of the motive
$h^1(M_f)$ corresponding to the Hecke eigenform $f$ ([12], pg
In this paper, we focus on the study of the $L$-function of
$H^2(\bar{X}_0(I) \times \bar{X}_0(I),\mathbb Q_\ell)$. Applying the
Künneth formula we get the following decomposition
\begin{equation}\label{dec}
\bar{X}_0(I)),s)=L(H^2_{\text{\'et}}(\bar{X}_0(I)),s)^2
\end{equation}
The incomplete ( here we omit the local factor at $\infty$ )
$L$-function of $H^2_{\text{\'et}}(\bar{X}_0(I))$ is
$\zeta_A(s-1)=\frac{1}{1-q^{2-s}}$. Under the assumption that the
level is square-free and using the decomposition above (<ref>)
the $L$-function of the last factor in (<ref>) can be expressed
as a product
$$L(H^1(\bar{X}_0(I))\otimes H^1(\bar{X}_0(I)),s) = \zeta_A(2s)^{-1} \prod_{f,g} L_{f,g}(s)$$
where $f$ and $g$ are normalized newforms of JLD type and level $I$.
$L_{f,g}(s)$ is the Rankin-Selberg convolution $L$-function defined
in the next section. It is essentially the $L$-function of the
tensor product of the motives $h^1(M_f) \otimes h^1(M_g)$.
§ THE RANKIN-SELBERG CONVOLUTION
The main goal of this section is the computation of a special value
of the convolution $L$-function of two automorphic forms of
JLD-type verifying certain prescribed conditions.
We begin by studying certain Eisenstein series on the Bruhat-Tits
tree $\T$. The classical Eisenstein-Kronecker-Lerch series are real
analytic functions on the upper half-plane, invariant under the
action of a congruence subgroup $\Gamma\subset SL_2(\ZZ)$. They are
related to logarithms of modular units on the associated modular
curve via the Kronecker Limit formulas.
There are function field analogues of these series, as well as an
analogue of Kronecker's First Limit formula. These results follow
from the work of Gekeler [8] and they are the crucial steps
in the process of relating the regulators of elements in $K$-theory
to special values of $L$-functions.
§.§ Eisenstein series
The real analytic Eisenstein series for $\Gamma_0(I)$ is defined as
\[
E_I(\tau,s)= \displaystyle{\sum_{\gamma \in
\Gamma_\infty\backslash\Gamma_0(I)}} |\gamma(\tau)|_{i}^{s},\qquad
\tau \in \Omega,~s\in\C.
\]
This series converges absolutely for $Re(s) \gg 0$ and from the
definition one can see that it is $\Gamma_0(I)$ invariant.
The `imaginary part' function $|z|_{i}=\text{inf}\{|z-x|;~x \in
K_{\infty}\}$ factors through the building map, so the Eisenstein
series can be thought of as a function defined on the vertices of
the Bruhat-Tits tree. In terms of the matrix representatives
$S_{V}$, $E_I(\tau,s)$ can be expressed as follows.
Let $m,n \in A,~(m,n)\neq (0,0)$ and let $v \in V(\T)$ be a vertex
represented by $v=\begin{pmatrix} \pi^k & u\\0 & 1
\end{pmatrix}$. For $\omega=\ord_{\infty}(mu+n)$ and $s\in\C$, define
\begin{equation}\label{thephi}
\phi_{m,n}^{s}(v)=\phi^s_{m,n} \begin{pmatrix} \pi^k & u \\ 0 & 1
\end{pmatrix} =
\begin{cases} q^{(k-2\deg(m))s} & if\;\;\ \omega \geq k-\deg(m)\\
q^{(2\omega-k)s} & if \;\; \omega < k-\deg(m). \end{cases}
\end{equation}
Then, using an explicit set of representatives for
$\Gamma_\infty\backslash\Gamma_0(I)$, we have ( see [12],
section 4 for details)
\begin{equation*}
\label{E-I} E_{I}(v,s)=q^{-ks}+\sum_{{m\in A\atop m~\text{ monic
}}\atop m\equiv 0~\text{mod }I}\sum_{n \in A,\atop (m,n)=1}
\phi^{s}_{m,n}(v).
\end{equation*}
Let $E(v,s)=E_{1}(v,s)$. In [12], it is shown that
$E_I(v,s)$ has an analytic continuation to a meromorphic function on
the entire complex plane, with a simple pole at $s=1$. Lemma 3.4 of
[12] relates the two series $E$ and $E_I$ through the formula
\begin{equation}\label{relat}
\zeta_I(2s)E_I(v,s)=\frac{\zeta(2s)}{|I|^s} \sum_{d|I\atop d\text{
monic }} \frac{\mu(d)}{|d|^s} E((I/d)v,s)
\end{equation}
where $\zeta(s)=\frac{1}{1-q^{1-s}}$ is the zeta function of $A$, $\zeta_I(s) =
\displaystyle{\prod_{\PP \nmid I}} (1-|\PP|^{-s})^{-1}$,
$\mu(\cdot)$ is the Möbius function of $A$, defined entirely
analogously to the usual Möbius function using the monic prime
factorization of an element of $A$, and $(I/d)v$ denotes the action
of the matrix $\begin{pmatrix} I/d &0\\0 & 1
\end{pmatrix}$ on $v$.
Notice that a function $F$ on the vertices of $\T$ can be
considered as a function on the edges of the tree by defining
$F(e)=F(o(e))$. In particular, if we define
\begin{equation}\label{Eisfunct}
E_I(e,s)=E_I(o(e),s),\quad e\in Y(\T)
\end{equation}
we recover the definition given in section 3 of [12].
§.§.§ Functional equation.
The Eisenstein series $E(e,s)$ satisfies a functional equation
analogous to that satisfied by the classical
Eisenstein-Kronecker-Lerch series. The analogue of the `archimedean
factor' of the zeta function of $A$ is
\begin{equation}
\label{lfactorinfty}
\end{equation}
We recall the following result
Define $\Lambda(e,s) = -L_{\infty}(s) E(e,s)$. Then,
$\Lambda(e,s)$ has a simple pole at $s=1$ with residue $-(\log q)^{-1}$ and satisfies the functional equation
\[
\Lambda(e,s)=-\Lambda(e,1-s).
\]
Theorem 3.3.
§.§ The Rankin-Selberg convolution
In [12], M. Papikian describes a function field analogue of
the Rankin-Selberg formula. In this section we will apply this
result by using the interpretation of the Eisenstein series
$E_{I}(v,s)$ as an automorphic form on the edges of the Bruhat-Tits
tree $\T$.
Let $f$ and $g$ be two automorphic forms of level $I$ on $\T$.
Consider the series
\[
L_{f,g}(s) = \zeta_I(2s) \sum_{\m \; \text{effective divisors}\atop
(\m,\infty)=1} \frac{c(f,\m) \bar c(g,\m)}{|\m|^{s-1}}
\]
If $f$ and $g$ are normalized newforms, using the decomposition
$\m=\m_{fin}\infty^d$ ($d \geq 0$) and the last property of the
Fourier coefficients listed in section <ref>, namely
$$c(f,\m)=c(f,\m_{fin}\cdot \infty^d)=c(f,\m_{fin})q^{-d}$$
so we can pull out the Euler factor at $\infty$ and we have
\[
\zeta_I(2s) \sum_{\m~\text{effective divisors}} \frac{c(f,\m)
\bar c(g,\m)}{|\m|^{s-1}}=L_{\infty}(s+1) L_{f,g}(s).
\]
Let $f$ and $g$ be two cusp forms of level $I$. Then
\[
\zeta_{I}(2s) <f~E_I(e,s),~g>~=~\zeta_I(2s) \int_{Y(\T_0(I))}
E_I(e,s)f(e)\overline{g(e)} d\mu(e) =
\]
[12], section 4.
We set $\Phi(s) = \Phi_{f,g}(s)$ where
\begin{equation}\label{thefunct}
\Phi_{f,g}(s)
E_I(e,s) f(e) \overline{ g(e)} d\mu(e).
\end{equation}
It follows from the proposition above, (<ref>),
(<ref>) and Theorem <ref> that $\Phi(s)$ has the
following description
\begin{align}\label{thefunct1}
\Phi(s) &= \sum_{d|I\atop d\text{ monic }} \frac{\mu(d)}{|d|^s}
\int_{Y(\T_0(I))} \Lambda ((I/d)e,s)f(e)\overline {g(e)}d\mu(e) \\
\end{align}
$\Phi(s-1)$ is the completed, namely with the factors at
$\infty$ included, $L$-function of the motive $h^1(M_f) \otimes
For future use we recall the following result
$L_{f,g}(s)$ has a simple pole at $s=1$ with residue a non-zero
multiple of $<f,g>$, the Petersson inner product of $f$ and $g$. In
particular, if $f$ and $g$ are normalized newforms and $f \neq g$,
then $<f,g>=0$, so $L_{f,g}(s)$ does not have a pole at $s=1$.
It follows that if $f$ and $g$ are normalized newforms and $f\neq
g$, then $\Phi(1) = \frac{q|I|L_{f,g}(1)}{(1-q^2)}$. To compute the
value $\Phi(0)$ we will make use of the following theorem.
Let $f$ and $g$ be two cuspidal eigenforms of square-free levels
$I_1,I_2$ respectively, with $I_1$ and $I_2$ co-prime monic
polynomials. Let $I = I_1I_2$. Then, the function $\Phi(s)$ as
defined in (<ref>) satisfies the functional equation
\begin{equation*}
\Phi(s) = -\Phi(1-s).
\end{equation*}
The method of the proof is similar to that of Ogg [11],
section 4. Using the Atkin-Lehner operators, we can simplify the
integral in (<ref>). Let $\PP$ be a monic prime element of
$A$ such that $\PP|I$, say $\PP|I_1$. The Atkin-Lehner operator
$W_{\PP}$ corresponding to $\PP$ is represented by
\[
\beta=\begin{pmatrix} a\PP & -b \\I & \PP \end{pmatrix}, \;
\det(\beta) = \PP,\quad \beta \in \Gamma_0(I/\PP) \begin{pmatrix}
\PP & 0 \\0 & 1 \end{pmatrix},\qquad \text{for some}~a,b\in A.
\]
Let $d$ be a monic divisor of $I/\PP$, then
\[
\begin{pmatrix} I / \PP d & 0 \\ 0 & 1
\end{pmatrix} \beta \begin{pmatrix} I/d & 0 \\ 0 & 1
\end{pmatrix}^{-1} \in \Gamma.
\]
As $\Lambda(e,s)$ of Theorem <ref> is $\Gamma$-invariant we
\begin{equation*}
\Lambda((I/\PP d)\beta e,s)=\Lambda((I/d)e,s)
\end{equation*}
where $(I/\PP d) = \begin{pmatrix} I / \PP d & 0 \\ 0 & 1
\end{pmatrix}$. Since $\beta$ normalizes $\Gamma_0(I_1)$ and $f$ is a
newform, we obtain
\[
\]
where $c(f,\PP)=\pm 1$. Further, if $h=g|_{\beta}$, then
$h|_{\beta}=g|_{\beta^2}=g$. We then have
\begin{align}\label{int}
\int_{Y(\T_0(I))} \Lambda((I/\PP d)e,s)\delta(f,g) &=
\int_{\beta^{-1}(Y(\T_0(I)))}
\Lambda((I/d)e,s)c(f,\PP)\delta(f,g|_{\beta}) \\
&= \int_{Y(\T_0(I))} \Lambda((I/d) e,s) c(f,\PP)\delta(f,h)\nonumber
\end{align}
as $\beta^{-1}(Y(\T_0(I)))$ is a fundamental domain for $\beta^{-1}
\Gamma_0(I) \beta=\Gamma_0(I)$. We deduce that
\begin{equation}
\label{fe1} \Phi(s)= \sum_{d|(I/\PP)\atop d\text{ monic
}}\frac{\mu(d)}{|d|^s} \int_{Y(\T_0(I))} \Lambda((I/d)e,s)(
\delta(f,g)+c(f,\PP)\delta(f,h)).
\end{equation}
Now, we repeat this process with $h$ in the place of $g$. It follows
from the Fourier expansion that
\[
\]
Substituting this expression in (<ref>), we have
\begin{align}\label{fe2}
c(f,\PP)|\PP|^{-s}\Phi(s) &= \sum_{d|(I/\PP)\atop d\text{ monic }}
\int_{Y(\T_0(I))}
(\Lambda((I/d)e,s)-|\PP|^{-s}\Lambda((I/\PP d) e,s) ) \delta(f,h) \\
&= \sum_{d|(I/\PP)\atop d\text{ monic }} \frac{\mu(d)}{|d|^{s}}
\int_{Y(\T_0(I))} \Lambda((I/d)e,s) \left(
\delta(f,h)+\frac{c(f,\PP)}{|\PP|^{s}}\delta(f,g) \right). \nonumber
\end{align}
We denote by ${\mathfrak S}$ the sum of the terms involving
$\delta(f,h)$. Comparing (<ref>) and (<ref>), we obtain
$$ {\mathfrak S}(\frac{c(f,\PP)}{|\PP|^{s}}-1)=0.$$
The only way this equation can hold for all $s$ in $\C$ is if
${\mathfrak S}=0$. Hence we obtain
\[
\Phi(s)=\sum_{d|(I/\PP)\atop d\text{ monic }} \frac{\mu(d)}{|d|^{s}}
\int_{Y(\T_0(I))} \Lambda((I/d)e,s) \delta(f,g).
\]
Repeating this process for all primes $\PP$ dividing $I$, keeping
in mind the assumption that a prime divides $I_1$ or $I_2$ but not
both. We get
\begin{equation*} \Phi(s)=\int_{Y(\T_0(I))}
\Lambda((I)e,s)\delta(f,g).
\end{equation*}
As $\Lambda((I)e,s)$ satisfies the functional equation
$\Lambda((I)e,s)= -\Lambda((I)e,1-s)$, we finally obtain
\[
\Phi(s)=-\Phi(1-s).
\]
§.§ Kronecker's limit formula and the Delta function
For the computation of the value $\Phi(0)$ we introduce the Drinfeld
discriminant function $\Delta$.
§.§.§ The discriminant function and the Drinfeld modular unit.
Let $\tau$ be a coordinate function on $\Omega$ and let
$\Lambda_{\tau} = <1,\tau>$ be the rank two free $A$-submodule of
$\CI$ generated by $1$ and $\tau$. Consider the following product
\[
e_{\Lambda_{\tau}}(z)= z \prod_{\lambda \in \Lambda_{\tau}
\backslash \{0\}} \left(1-\frac{z}{\lambda}\right) = z
\displaystyle{\prod_{a,b \in A\atop (a,b) \neq (0,0)}}
\left(1-\frac{z}{a \tau +b}\right).
\]
This product converges to give an entire, $\F_{q}$-linear,
surjective, $\Lambda_{\tau}$-periodic function
$e_{\Lambda_{\tau}}:\CI \rightarrow \CI$ called the Carlitz
exponential function attached to $\Lambda_\tau$. This is the
function field analogue of the classical $\wp$-function and it
provides the structure of a Drinfeld $A$-module to the additive
group scheme $\CI/\Lambda_{\tau}$.
The discriminant function $\Delta:\Omega \rightarrow \CI$ is the
analytic function defined by
\[
\Delta(\tau) = \mathop{\prod_{\alpha,\beta \in
T^{-1}A/A}}_{(\alpha,\beta) \neq (0,0)} e_{\Lambda_{\tau}}(\alpha z
+ \beta).
\]
This is a modular form of weight $q^2-1$. For $I\neq 1$ a monic
polynomial in $A$ , let $\Delta_I$ be the function
\begin{equation}\label{dmu}
\Delta_{I}(\tau) := \prod_{d|I\atop d\text{ monic }}
\Delta((I/d)\tau)^{\mu(d)}.
\end{equation}
Here $\mu(\cdot)$ denotes the Möbius function on $A$. Since
$$\sum_{d|I\atop d\text{ monic }} \mu(d)=0$$
one has that the weight of $\Delta_I$ is $0$ so it is in fact a
$\Gamma_0(I)$-invariant function on $\Omega$. As we will see later
in equation (<ref>), it is a modular unit, that is, its
divisor is supported on the cusps and defined over $K$ . We call
this the Drinfeld modular unit for $\Gamma_0(I)$.
§.§.§ The Kronecker Limit Formula.
The classical Kronecker limit formula links the Eisenstein series to
the logarithm of the discriminant function $\Delta$. In this
section, we prove an analogue of this result in the function field
We first compute the constant term $a_0(v)$ in the Taylor expansion
of $E(v,s)=E_1(v,s)$ around $s=1$. We have
\begin{equation}
\label{eisensteinone}
\end{equation}
where $a_{-1}$ is a constant independent of $v$. To compute
explicitly the coefficient function $a_0(v)$ we differentiate `with
respect to $v$', namely we apply the $\partial$ operator defined in
section 3.4.4 and then evaluate the result at $s=1$. This
computation gives
\[
\partial E(\cdot,s)|_{s=1}=\partial a_0 (\cdot).
\]
It follows from (<ref>) that
\[
a_0(v)=\int_{v_0}^{v} \partial E(\cdot,s)(e)|_{s=1} d\mu(e)+C
\]
where $v_0$ is any vertex on the tree and $C$ is a constant. For
definiteness we can choose $v_0$ to be the vertex corresponding to
the lattice $[\OO_{\infty} \oplus \OO_{\infty}]$. The integration is
to be understood as the weighted sum of the value of the function
on the edges lying on the unique path joining $v_0$ and $v$.
The function $\partial E(\cdot,s)$ on the edges in $Y(\T_0)$ is
related to the logarithmic derivative of the discriminant function
$\Delta$ through an improper Eisenstein series studied by Gekeler in
We first define Gekeler's series [8]. For $e=e(k,u) \in
Y(\T)$, $s\in\C$, let $\psi^s(e)= \text{sgn}(e) q^{-ks}$. Consider
the following Eisenstein series
\[
F(e,s)=\sum_{\gamma \in \Gamma_{\infty}\backslash \Gamma}
\psi^s(\gamma(e)).
\]
This series converges for $Re(s)\gg 0$. Let $m,n\in A$, $(m,n) \neq
(0,0)$. For $\omega = \text{ord}_\infty(mu+n)$ let
\[
\psi^s_{m,n}(e) = \psi_{m,n}^{s}(e(k,u)) =
\psi^s_{m,n}\left(\begin{pmatrix} \pi^k & u
\\ 0 & 1 \end{pmatrix}\right) =
\begin{cases} -q^{(k-2\deg(m)-1)s} & if\;\;\ \omega \geq k-\deg(m)\\
q^{(2\omega-k)s} & if \;\; \omega < k-\deg(m) \end{cases}
\]
we have
\[
F(e,s)=\psi^s(e)+ \sum_{{m \in A\atop m\text{ monic }}}\sum_{n\in
A\atop (m,n)=1} \psi^{s}_{m,n}(e).
\]
One can consider the limit as $s\rightarrow 1$ but the resulting
function $F(e,1)$, while $G(A)$ invariant, is not harmonic. The
series for $F(e,1)$ does not converge.
Gekeler [8](4.4) defines a conditionally convergent
improper Eisenstein series $\tilde{F}(e)$ as follows.
\begin{equation}
\tilde{F}(e)=\sum_{m \in A \atop \text{monic}} \sum_{n \in A \atop
(m,n)=1} \psi_{m,n}(e) + \psi(e) \label{impeis}
\end{equation}
This function is harmonic, but not $G(A)$ invariant. The relation
between these two functions is given as follows
[8](Corollary 7.11)
\begin{equation}
\label{releis} F(e,1)=\tilde{F}(e)-\sgn(e)\frac{q+1}{2q}
\end{equation}
This equation helps us relate the functions $E(\cdot,1)$ and
$\log|\Delta|$ as they are connected to $F$ and $\tilde{F}$ through
their derivatives. Precisely, we have the following
Let $\tilde{F}$ be as above. We have
\begin{equation}
\partial \log |\Delta|(e)=(1-q) \tilde{F}(e)
\label{gekeis}
\end{equation}
See [9],
Corollary 2.8.
Moreover the following lemma describes a relation between the
series $E(v,s)$ and $F(e,s)$.
Let $E(v,s)$ be the series in equation (<ref>) and let
$F(e,s)$ be the series defined in equation (<ref>). Then
\begin{equation}
\partial E(\cdot,s)(e)=(q^s-1)F(e,s).
\label{partial}
\end{equation}
It follows from the definition of the derivative of a function given
in (<ref>) that
\[
\partial E(\cdot,s)(e) = E(t(e),s) - E(o(e), s).
\]
Set $e=e(k,u)=\vec{v(k,u)v(k-1,u)}$. Let $\phi^s_{m,n}(v)$ be the
function defined in (<ref>). There are four cases to
Case 0. For $e=e(k,u)$
\begin{align*}
\phi^s(t(e))-\phi^s(o(e))
&= q^{(k-1)s}-q^{-ks} \\
&= (q^{s}-1)q^{-ks} \nonumber\\
&= (q^s- 1)\psi^s(e) .\nonumber
\end{align*}
Case 1. If $\omega > k-1-\deg(m)$, then
\begin{align*}
\phi^s_{m,n}(t(e))-\phi^s_{m,n}(o(e))
&= q^{(k-1-2\deg(m))s}-q^{(k-2\deg(m))s} \\
&= (1-q^{s})q^{(k-1-2\deg(m))s} \nonumber\\
&= (q^s- 1)\psi^s_{m,n}(e).\nonumber
\end{align*}
Case 2. If $\omega < k-1-\deg(m)$, then
\begin{align*}
\phi^s_{m,n}(t(e))-\phi^s_{m,n}(o(e)) &=
&= (q^s-1)(q^{(2\omega-k)s}\nonumber \\
&= (q^s-1)\psi^s_{m,n}(e).\nonumber
\end{align*}
Case 3. If $\omega= k-1-\deg(m)$, so
$2\omega-k=2k-2-2\deg(m)$, then
\begin{align*}
\phi^s_{m,n}(t(e))-\phi^s_{m,n}(o(e))
&= q^{(k-2\deg(m)-1)s}-q^{(2\omega -k)s}\\
&= q^{(k-1-2\deg(m))s}-q^{(2k-2-2\deg(m))s}\nonumber \\
&= (q^s-1)(q^{(2\omega-k)s}\nonumber\\
&= (q^s-1)\psi^s_{m,n}(e).\nonumber
\end{align*}
These computations show that
\[
\partial E(\cdot,s)(e)=(q^s-1)F(e,s).
\]
Taking the limit as $s\rightarrow 1$ we have
$$\partial E(\cdot,1)(e)=(q-1)F(e,1)$$
Let $\Lambda(v,s)=-L_{\infty}(s)E(v,s)=\frac{q}{q-1} E(v,s)$ be the
function defined in Theorem <ref>. We have the following
The function $\Lambda(v,s)$ has an expansion around $s=1$ of the
\begin{equation}
\Lambda(v,s)=\frac{b_{-1}}{s-1}+ \frac{q}{1-q}
\log|\Delta|(v)-\frac{q-1}{2} \log |\cdot|_i(v) + C +
b_1(v)(s-1)+\ldots \label{klf1}
\end{equation}
where $b_{-1}$ and $C$ are constants independent of $v$.
It follows from Lemma <ref> that
\[
\partial \Lambda(\cdot,1)(e)=q F(e,1).
\]
Using (<ref>) we obtain
\[
\partial\Lambda(\cdot,1)(e)=q(\tilde{F}(e)-\frac{q+1}{2q} \sgn(e))
\]
To obtain the constant term in the Laurent expansion we integrate
the right hand side of this expression from $v_0$ to $v$. From
(<ref>) we have that the first term is
$$q\tilde{F}(e)=\frac{q}{1-q} \partial log|\Delta|(e)$$
so its integral is $\log|\Delta|(v)-\log|\Delta|(v_0)$. The integral
of the second term is
$$\int_{v_0}^{v} \frac{q+1}{2q} \sgn(e) d\mu(e)=\frac{q+1}{2q}
k(v)=\frac{q+1}{2q}(-\log |\cdot|_i(v))$$
where $|\cdot|_i$ is the `imaginary part', namely the distance from
$K_{\infty}$ of any element $\tau_v$ in $\lambda^{-1}(v)$, which
descends to a function on the tree. This follows from the discussion
on pp 371-372 of [8].
Combining these two expressions we get the theorem.
Observe that each term that appears here is analogous to a term
which appears in the classical Kronecker Limit formula.
§.§ A special value of the $L$-function
Using the functional equation for $\Lambda(v,s)$ as stated in
Theorem <ref> and the expansion (<ref>), we obtain the
following result
Let $f$ and $g$ be two newforms of level $I_1$ and $I_2$
respectively with $I_1$ and $I_2$ relatively prime polynomials in
$A$. Let $I=I_1I_2$. Then
\begin{equation*}
\Phi_{f,g}(0)= \frac{q}{1-q} \int_{Y(\T_0(I)}
\log|\Delta_I|\delta(f,g)
\end{equation*}
Here, $\log|\Delta_I|$ is interpreted as a function on $Y(\T)$ by
$\log|\Delta_I|(e)=\log|\Delta_I|(o(e))$ and $\delta(f,g)$ is as in
From the description of $\Phi(s)$ given in (<ref>) we have
\[
\Phi(0)= \lim_{s \rightarrow 0} \sum_{d|I\atop d\text{ monic }}
\frac{\mu(d)}{|d|^{s}}\int_{Y(\T_0(I))}
\Lambda((I/d)e,s)\delta(f,g).
\]
From the functional equation of $\Lambda(e,s)$ and the Limit
Formula (<ref>), we have
\begin{equation*}
\Lambda(e,s)=\frac{b_{-1}}{s}+\frac{q}{1-q}\log|\Delta|(e)
-\frac{q^2-1}{2} \log |\cdot|_i(e) + C + \text{h.o.t.(s)}
\end{equation*}
where $C$ is a constant and h.o.t.(s) denotes higher order terms in
$s$. Since
$$\sum_{d|I\atop d\text{ monic }} \mu(d)=0 \text{ and } <f,g>=0$$
we have
$$\sum_{d|I\atop d\text{ monic }}
\Lambda((I/d)e,0)=\frac{q}{1-q}\log|\Delta_I(e)|$$
as the sum of the residues of the poles is $0$ , the constant term
$C$ gets multiplied by $0$ and finally
$$\sum_{d|I\atop d\text{ monic }} \mu(d)\log|\cdot|_i((I/d)e)=0$$
as $|(I/d)e|_i=|(I/d)||e|_i$ and one can easily check that
$\prod_{d|I\atop d\text{ monic }} (I/d)^{\mu(d)}=(1)$, so its log is
$0$. It follows that
\begin{equation*}
\Phi_{f,g}(0)=\Phi(0)=-\frac{q}{q-1} \int_{Y(\T_0(I))}
\log|\Delta_I|(o(e))\delta(f,g).
\end{equation*}
Since $\delta(f,g)$ is orientation invariant, we can replace the
usual integration on edges by integration over positively oriented
edges to get
\begin{equation}
\Phi_{f,g}(0)=\Phi(0)=-\frac{q}{q-1} \int_{Y^{+}(\T_0(I))} \left(
\log|\Delta_I|(o(e)) + \log|\Delta_I|(t(e)) \right)
\delta^{+}(f,g). \label{specialvalue}
\end{equation}
Here $\delta^{+}(f,g)=f(e)g(e)d\mu^{+}(e)$ and $\mu^{+}$ denotes the
Haar measure on the positively oriented edges:
§ ELEMENTS IN $K$-THEORY
§.§ The group $H^3_{\M}(X,\Q(2))$
Let $X$ be an algebraic surface over a field $F$. The second graded
piece of the Adams filtration on $K_1(X)\otimes \Q$ is usually
denoted by $H^3_{\M}(X,\Q(2))$. It has the following description in
terms of generators and relations.
The elements of this group are represented by finite formal sums
\[
\sum_{i} (C_i,f_i)
\]
where $C_i$ are curves on $X$ and $f_i$ are $F$-valued rational
functions on $C_i$ satisfying the cocycle condition
\begin{equation}\label{coco}
\sum_i \div(f_i)=0.
\end{equation}
Relations in this group are given by the tame symbol of
functions. Precisely, if $C$ is a curve on $X$ and $f$ and $g$ are
two functions on $X$, the tame symbol of $f$ and $g$ at $C$ is
defined by
\[
T_C(f,g) = (-1)^{\ord(g) \ord(f)}
\frac{f^{\ord(g)}}{g^{\ord(f)}},\qquad \ord(\cdot) = \ord_C(\cdot).
\]
Elements of the form $\displaystyle{\sum_C} (C, T_C(f,g))$ are said
to be zero in $H^3_{\M}(X,\Q(2))$.
§.§ The regulator map on surfaces
We define the regulator as the boundary map in a localization
sequence. We use the formalism of Consani [5].
Let $\Lambda$ be a henselian discrete valuation ring with fraction
field $F$ and let $X$ be a smooth, proper surface defined over $F$.
We set $\bar{X}=X \times Spec(\bar{F})$ for $\bar{F}$ an algebraic
closure of $F$. By a semi-stable model of $X$ (or semi-stable
fibration) we mean a flat, proper morphism $\XX \to Spec(\Lambda)$
of finite type over $\Lambda$, with generic fibre $\XX_{\eta} \cong
X$ and special fibre $\XX_{\nu}=Y$, a reduced divisor with normal
crossings in $\XX$. $\eta$ and $\nu$ denote respectively the generic
and closed points of $Spec(\Lambda)$. The scheme $\XX$ is assumed to
be non-singular and the residue field at $v$ is assumed to be
The scheme $Y$ is a finite union of irreducible components:
$Y=\cup_{i=1}^{r} Y_i$, with $Y_i$ smooth, proper, irreducible
surfaces. Let $J$ be a subset of $\{1,2,\ldots,r\}$ whose
cardinality is denoted by $|J|$. We set $Y_J=\cap_{j \in J} Y_j$ and
$$Y^{(j)}=\begin{cases} \XX & \text{ if } j=0 \\ \displaystyle{\coprod_{|J|=j}} Y_{J}& \text{
if } 1 \leq j \leq 3 \\
\emptyset & \text { if } j>3. \end{cases}$$
Let $\iota:Y \rightarrow \XX$ denote the subscheme inclusion map.
$\iota$ induces a push-forward homomorphism $\iota_\ast:
CH_1(Y^{(1)}) \to CH_1(\XX)$ and a pullback map $\iota^\ast:
CH^2(\XX) \to CH^2(Y^{(1)})$. Let $J=\{j_1,j_2\}$, with $j_1 < j_2$
and $I = J-\{j_t\}$, for $t\in\{1,2\}$. Then, the inclusions
$\delta_t: Y_J \rightarrow Y_{I}$ induce push-forward maps
$\delta_{t\ast}$ on the Chow homology groups. The Gysin morphism
$\gamma: CH_1(Y^{(2)}) \to CH_1(Y^{(1)})$ is defined by $\gamma =
\sum_{t=1}^2(-1)^{t-1}\delta_{t\ast}$.
$$PCH^1(Y)=\frac{ker [\iota^*{\iota}_{*}:CH_{1}(Y^{(1)})
\rightarrow CH^2(Y^{(1)})]}{im [\gamma:CH_{1}(Y^{(2)}) \rightarrow
and define the $\nu$-adic Deligne cohomology to be
$$H^3_{\D}(X_{/\nu},\Q(2))=PCH^1(Y)\otimes \Q$$
If certain `standard conjectures' are satisfied, it follows from
Theorem 3.5 of [5] that
$$\dim_{\Q} H^3_{\D}(X_{/\nu},\Q(2))=-{\ord_{s=1}} L_{\nu}(H^2(\bar{X},\Q_\ell),s),\qquad s\in\C$$
where $L_{\nu}(H^2(\bar{X}),s)$ is the local Euler factor at ${\nu}$
of the Hasse-Weil $L$-function of $X$.
There is a localization sequence which relates the motivic
cohomology of $\XX,X$ and $Y=\XX_{\nu}$ [3]. When
specialized to our case it is as follows:
$$\dots \longrightarrow H^3_{\M}(\XX,\Q(2))\longrightarrow H^3_{\M}(X,\Q(2))
\stackrel{\partial}{\longrightarrow} H^3_{\D}(X_{/\nu},\Q(2))
\longrightarrow H^4_{\M}(\XX,\Q(2))\longrightarrow \dots
The $\nu$-adic regulator map $r_{\D,\nu}$ is defined to be the
boundary map $\partial$ in this localization sequence. If $\sum_i
(C_i,f_i)$ is an element of $H^3_{\M}(X,\Q(2))$ then
$$r_{\D, \nu}\left(\sum_i (C_i,f_i)\right)=\sum_i \div(\bar{f_i})$$
where $\bar{f_i}$ is the function $f_i$ extended to the Zariski
closure of $C$ in $\XX$. The condition $\sum \div(f_i)=0$ shows that
the `horizontal' divisors cancel each other out and so the image of
the regulator map is supported in the special fibre $\XX_v$.
Explicitly, one has the following formula for the regulator
\begin{equation}
\label{explicitregulator} r_{\D, \nu}\left(\sum_i (C_i,f_i)\right)=\sum_i
\sum_{Y} \ord_Y(f_i) Y
\end{equation}
where $Y$ runs through the components of the reduction of the
Zariski closure of the curves $C_i$.
This regulator map clearly depends on the choice of model. However,
Consani's work shows that the dimension of the target space does not
depend on the model since the local $L$-factor does not. Since the regulator map
is simply the boundary map of a localization sequence it satisfies the expected functoriality
While all our calculations below are with respect to a particular model,
perhaps the correct framework to work with the non-Archimedean
Arakelov theory of Bloch-Gillet-Soule [2].
§.§ The case of products of Drinfeld modular curves
We now apply the results of the previous section to the case of the
self product of a Drinfeld modular curve $X_0(I)$ and the prime
$\infty$. In [16] page 280, Teitelbaum describes how to construct a
model $\XX_0(I)$ of the curve $X_0(I)$ over $\OO_{\infty}$. This
model has semi-stable reduction at $\infty$ and he describes a covering by affinoids
which have a canonical reduction. The special fibres of the affinoids covering
$\XX_0(I)$ are made up of two types of components – either of the type $(T_i \cup T_j)$
where the $T_i$ and $T_j$ are isomorphic to $\CP^1_{\F_q}$ with all but one rational point deleted and
meet at that point $T_{ij}=T_i \cap T_j$, or of the form $T_i$ where $T_i$ is isomorphic to $\CP^1_{\F_q}$
with all but one rational point deleted.
The self-product $\XX_0(I) \times \XX_0(I)$ has, therefore, a covering by products of the affinoids covering $\XX_0(I)$ so there are four possibilities for the special fibre :
* (i) $(T_1 \cup T_2) \times T_3$
* (ii) $T_1 \times (T_3 \cup T_4) $
* (iii) $T_1 \times T_3$
* (iv) $(T_1 \cup T_2) \times (T_3 \cup T_4)$
depending on whether the reduction is of the first or second type above. Therefore it is made up of components of the form
$$T_1 \times T_4 \hspace{1in} T_1\times T_3$$
$$T_2 \times T_4 \hspace{1in} T_2 \times T_3$$
One has the following schematic representation –
\[
\xy
(0,0)*{}; (20,0)*{} **\dir{-};
(0,0)*{}; (-20,0)*{} **\dir{-};
(-20,0)*{}; (-20,-20)*{} **\dir{-};
(10,10)*{T_1 \times T_3};
(10,-10)*{T_2 \times T_3};
(-10,10)*{T_1 \times T_4};
(-10,-10)*{T_2 \times T_4};
(2,2)*{{\mathbf P}};
\endxy
\]
$$\mathrm {Figure \;\;1.}$$
This reduction, however is not semi-stable. In the first three cases there is no problem but in case (iv) above there are four components and all of them meet at the point ${\bf P}=(T_{12},T_{34})$, hence it is not semi-stable.
However, if we blow up $\XX_0(I) \times \XX_0(I)$ at this point the
special fibre of the blow-up is locally normal crossings. Locally
the special fibre consists of five components, $Y_1,\dots, Y_5$,
$$Y_3=\widetilde{T_1 \times T_4} \hspace{1in} Y_1=\widetilde{T_1 \times T_3}$$
$$Y_4=\widetilde{T_2 \times T_4} \hspace{1in} Y_2=\widetilde{T_2 \times T_3}$$
are the strict transforms of the components $T_i \times T_j$ above
and $Y_5\simeq \CP^1 \times \CP^1$ is the exceptional fibre
[6], (Lemma 4.1).
We label it in this curious way as it is important in what follows. If one
thinks of the point of intersection as the origin, then $Y_1$ is the
strict transform of the first quadrant, $Y_2$ of the one below it,
$Y_3$ of the quadrant to the left of $Y_1$ and $Y_4$ the strict
transform of the quadrant to the left of $Y_2$. The diagram below is a schematic representation of the situation –
\[
\xy
% Dots..
\endxy
\]
$$\mathrm{Figure \;\;2.}$$
Recall that this is the local picture – to obtain the semi-stable model we have to repeat this procedure
for every point of intersection of the components $T_i \times T_j$ - namely
at the points denoted by $\circ$ in Figure 1.
So the components of special fibre consist of the the strict
transforms of the $T_i \times T_j$ with all the four corners being
blown up.
Observe that the labeling $Y_i$ above is with respect to which
corner of the $T_i \times T_j$ is being considered – so for example $T_i \times T_j$ will be labelled $Y_1$ if the South-Western corner is blown up but will be labelled $Y_4$ if the North-Eastern corner is blown up, $Y_2$ if the North-Western corner and $Y_3$ if the South-Eastern corner is blown up.
Let $Y_{ij}$ denote the cycle $Y_{i} \cap Y_j$, if it exists. For example, one has cycles $Y_{15}, Y_{12}, Y_{13}$ but no cycle $Y_{14}$ as $Y_1$ and $Y_4$ do not intersect. Similarly, let $Y_{ijk}$ denote the cycle $Y_i \cap Y_j \cap Y_k$, if it exists. From the diagram one can see that for any such cycle at least one of the $i, j$ or $k$ has to be $5$, say $k=5$ and the cycle is $Y_{ij5}= Y_{i5} \cap Y_{j5}$. Further, one does not have cycles $Y_{145}$ and $Y_{235}$. Since the cycles $Y_{i5}$ and $Y_{j5}$ are rulings on $Y_5 \simeq \CP^1 \times \CP^1$ their intersection number is either $1$ or $0$ and so the cycles $Y_{ij5}$ have support on one point with multiplicity one.
In the group $H^3_{\D}((X_0(I) \times X_0(I))_{/\infty},\Q(2))$ one has
cycles coming from the restriction of the generic cycles as well as
certain cycles supported in the exceptional fibres. Locally, the
restriction of horizontal and vertical components give the cycles
$Y_{12}+Y_{34}+(Y_{15}-Y_{45}+Y_{25}-Y_{35})$ and
$Y_{13}+Y_{24}+(Y_{15}-Y_{45}-Y_{25}+Y_{35})$ [6], (Lemma
4.1). In the exceptional fibre $Y_5$ over $\mathbf{P}$ one also has
the cycle $\Z_{\bf{P}}=Y_{15}+Y_{45}-Y_{25}-Y_{35}$. Computing the
intersection with the other cycles show that this is not the
restriction of a generic cycle – in fact, it is orthogonal to them and the cycles $Y_{12}, Y_{13}, Y_{24}$ and $Y_{34}$ as well.
There are relations in this group coming from the image of the Gysin
map $\gamma$. For example, the difference of the image of the cycles
$Y_{15}$ in $CH^1(Y_1)$ and $CH^1(Y_5)$ lies in the image of the
Gysin map, so is $0$ in $H^3_{\D}((X_0(I) \times
X_0(I))_{/\infty},\Q(2))$. So there is a well defined $Y_{15}$ in $H^3_{\D}((X_0(I) \times
X_0(I))_{/\infty},\Q(2))$. Similarly, the cycles $Y_{ij}$, which lie in both $Y_i$ and $Y_j, i, j \in \{1,\dots 5\}$, are well defined. Further, the cycle which is $Y_{12}$ with respect to ${\bf P}$ is $Y_{34}$
with respect to the point ${\bf P'}$ to the right of ${\bf P}$ and so is counted only once in
$H^3_{\D}((X_0(I) \times X_0(I))_{/\infty},\Q(2))$, and similarly for the others. So the local cycles described above coming from the restriction of the horizontal and vertical cycles patch up to give global cycles in $H^3_{\D}((X_0(I) \times X_0(I))_{/\infty},\Q(2))$.
§.§.§ A description in terms of the graph.
Using the relation between the Bruhat-Tits tree and the special
fibre described at the end of Section <ref> or in [16] one can also
express this local picture in terms of the graph. Recall that components of the special fibre of $\Omega$ correspond to vertices on the tree and two components intersect at an edge. From that we have that the graph $\T_0(I)$ consists of a finite graph $\T_0(I)^0$ and finitely many ends. $\T_0(I)^0$ is the dual graph of the intersection graph
of the special fibre of $\XX_0(I)$. The situation where the canonical reduction of an affinoid has
two components corresponds to an edge $e$ with vertices $o(e)$ and $t(e)$, both of which are $\T_0(I)^0$.
The situation when the canonical reduction has only one component corresponds to an edge $e$ with a distinguished
vertex which is in $\T_0(I)^0$.
The special fibre of the product then has the following local description - it corresponds to either two, one or four pairs of vertices depending on whether we have case (i) or (ii), (iii) or (iv) above. In case (iv), the four pairs of vertices correspond to the four pairs of components and the point ${\mathbf P}=(T_{12},T_{34})$ corresponds to a pair of
edges $(e_{12},e_{34})$ . So we can re-label the cycle $\Z_{{\mathbf P}}$ as
$\Z_{(e_{12},e_{34})}$ where $(e_{12},e_{34})$ is the point being blown up.
The regulator of an element supported on curves uniformized by the
Drinfeld upper half plane lying on $X_0(I) \times X_0(I)$ can also
be expressed in terms of the graph. Since components in the special
fibre correspond to vertices of the graph on can rewrite the
regulator in terms of vertices. Let $Y_v$ denote the component
corresponding to a vertex $v$. From the definition of $\log|\cdot|$ one
has $\ord_{Y_v}(f)=\log|f|(v)$. So one can rewrite the expression
(<ref>) as
\begin{equation}
r_{\D,\infty}\left(\sum_i (C_i,f_i)\right)=\sum_i \sum_{v} \log|f_i|(v) Y_v
\label{logregulator}
\end{equation}
where $v$ runs through the vertices of the Bruhat-Tits graphs of
$C_i$. In Section <ref>, the element we construct will be
supported on curves isomorphic to $X_0(I)$ so we can express its
regulator using (<ref>).
§.§ The special cycle $\Z_{f,g}$
As mentioned before, the motivic cohomology group of the surface
$X_0(I) \times X_0(I)$ can be decomposed with respect to eigenspaces
for pairs of cusp forms $(f,g)$ and this results in a decomposition
of the $\infty$-adic Deligne cohomology group as well. We denote
these groups by $H^3_{\D}(h^1(M_f) \otimes
h^1(M_g)_{/\infty},\Q(2))$. The local $L$-factor at $\infty$
(<ref>) and Consani's theorem [5], Theorem
3.5, shows that this space is $1$ dimensional.
There is a special cycle in this group which plays the role played
by the $(1,1)$-form
$$\omega_{f,g}=f(z_1)\overline{g(z_2)}(dz_1\otimes d\bar{z}_2- d\bar{z}_1\otimes dz_2)$$
in the classical case. While $\omega_{f,g}$ is not represented by an
algebraic cycle, in our case there is a special cycle, supported in
the special fibre, which represents it. It is defined as follows.
For $f,g$ two cuspidal automorphic forms of JLD type and
$\Z_{(e,e')}$ as above, we define
$$\Z_{f,g} = \sum_{e,e' \in Y(\T_0(I)^0)} f(e)\overline{g(e')} \Z_{(e,e')}$$
in $H^3_{\D}(h^1(M_f) \otimes h^1(M_g)_{/\infty},\Q(2))$. The action of the Hecke correspondence
is through its action on $f$ and $g$ and so that shows that this cycle lies in the $(f,g)$
component with respect to the Hecke action.
Note this this cycle is orientation invariant as
$(\bar{e},\bar{e}')=(e,e')$ and
$f(\bar{e})\bar{g}(\bar{e}')=f(e)\bar{g}(e')$. Further, as it is
composed of the cycles $\Z_{(e,e')}$ it is orthogonal to the cycles
which come by restriction from the generic Neron-Severi group.
§.§ A special element in the motivic cohomology group
In this section we will use the Drinfeld modular unit $\Delta_I$
defined in (<ref>) on the diagonal $D_0(I)$ of $X_0(I)$ to
construct a canonical element $\Xi_0(I)$ in the motivic cohomology
$H^3_\M(X_0(I) \times X_0(I),\Q(2))$ of the self-product $X$ of the
Drinfeld curve $X_0(I)$. The trick is to `cancel out' the zeroes and
the poles of (a power of) $\Delta_I$ using certain functions
supported on the vertical and horizontal fibres of $X$. The
existence of these functions is a consequence of the function field
analogue of the Manin-Drinfeld theorem proved by Gekeler in
[9]. Theorem <ref> provides a more explicit
description of them. As a corollary, we get an effective version of
the Manin-Drinfeld theorem in the function field case.
§.§.§ Cusps.
Let $I\in A$ be a monic, square-free polynomial. We first compute
the divisor of the function $\Delta_I$ explicitly. For this we need
to work with an explicit description of the set of the cusps of
$X_0(I)$. It is well known that the set of these points is in
bijection with the set
\[
\Gamma_0(I)\backslash\Gamma/\Gamma_\infty
\stackrel{\simeq}{\to}\{[a:d]~:~d~|I,~a \in
(A/tA)^{*},~t=(d,I/d),~~a,d ~\text{monic, coprime}\}/\mathbb
\]
We will denote the cusp corresponding to $[a:d]$ by $P^{a}_{d}$.
Since $I$ is square-free, the cusps are of the form $P_d=P^1_{d}$,
where $d$ is a monic divisor of $I$. For a function $F = F(\tau)$ on
$\Omega$ and $f \in A$ let $F(f)$ denote the function $F(f\tau)$.
For $a,b \in A$, let $(a,b) = \text{g.c.d}\{a,b\}$ and $[a,b] =
\text{l.c.m}\{a,b\}$. As $A$ is a P.I.D. they are both elements of
$A$. If $J$ is an element of $A$, the symbol $|J|$ denotes the
cardinality of the set $A/(J)$, where $(J)$ is the ideal generated
by $J$.
Let $I \in A$ be square-free and monic and assume that $I'$ and $d$
are monic divisors of $I$. Then
\begin{equation*}
\ord_{P_d} \Delta(I') = |I|~\frac{|(d,I')|}{|[d,I']|}
\end{equation*}
where the order at a cusp is computed in terms of a local
uniformizer as in [10] section 2.7.
It follows from [9] section 3 that
\[
\ord_{P_d}\Delta= |(I/d)|,\qquad \ord_{P_d} \Delta(I)= |d|.
\]
To obtain an explicit description of the divisor of $\Delta(I')$ on
$X_0(I)$ we need to compute the ramification index of $P_d$ over
$P_{d'}$, where $d' = \text{g.c.d}\{d,I'\}$. It follows from op.cit, Lemma 3.8 that
\[
\text{ram}^{P_d}_{P_{d'}} = \frac{ |I||(d,I')|}{|d||I'|}.
\]
Therefore, one gets
\begin{align*}
\ord_{P_d} \Delta(I') &= \text{ram}^{P_d}_{P_{d'}}\cdot\ord_{
P_{d'}} \Delta(I') =
\frac{|I||(d,I')|}{|d||I'|} \cdot |(d,I')| \\
&= |I| \frac{|(d,I')|}{|[d,I']|}.
\end{align*}
It follows from the definition of the function $\Delta_I$ in
(<ref>) and Lemma <ref> that
\begin{equation}
\div(\Delta_I)=\sum_{d|I\atop d\text{ monic }}
\mu(d)\div(\Delta(I/d))=\prod_{f|I \atop f \text{ prime
}}(1-|f|)(\sum_{d|I\atop d\text{ monic }} \mu(d) P_d).
\label{drindiv}
\end{equation}
A simple modular unit is a (Drinfeld) modular unit whose divisor is
of the form $k(P-Q)$, where $P$ and $Q$ are cusps of $X_0(I)$
and $k\in \mathbb Z$. The following theorem shows that there exists
$\kappa \in \mathbb N$ such that the function $\Delta_I^{\kappa}$
can be decomposed into a product of such units.
Let $I$ be a square-free, monic element of $A$ and
let $I=\prod_{i=0}^{r} f_i$ be the prime factorization of $I$, with
the $f_i's$ monic elements of $A$. Let $\kappa=\prod_{i=0}^{r}
(1+|f_i|)$. Then
\[
\Delta_I^{\kappa}=\prod_{a|(I/f_0)\atop a\text{ monic }} F_{a}
\]
where the functions $F_a$ are simple modular units and
\[
\div(F_a)=\prod_{i=0}^{r}(1-|f_i|^2) \mu(a)(P_a-P_{f_0 a}).
\]
The proof will follow from the following lemmas.
Let $P_a$ be a cusp of $X_0(I)$. Then, the divisor
of the form
$$D_a=\prod_{d|I\atop d\text{ monic }} \Delta(d)^{\mu(I/d)
$$\div(D_a)=\prod_{f|I\;\atop f\;\text{ monic },\;\text{ prime }} (1-|f|^2) \mu(a) P_a.$$
Let $P_b$ be a cusp of $X_0(I)$. Then, it follows from
lemma <ref> that
$$\ord_{P_b} (D_a)=\sum_{d|I\atop d\text{ monic }}
\mu(I/d)|I|^2\frac{|(a,d)(b,d)|}{|[a,d][b,d]|}.$$
We consider the following cases
Case 1 ($a\neq b$). In this case there is a prime
element $f\in A$ dividing $a$ but not $b$ (or vice versa). Assume
that $f|a$ and $\text{g.c.d.}\{f,b\}=1$. Then
$$\ord_{P_b}(D_a)= \sum_{d|(I/f)\atop d\text{ monic }}
\mu(I/d)|I|^2\left(\frac{|(a,d)(b,d)|}{|[a,d][b,d]|} -
\frac{|(a,fd)(b,fd)|}{|[a,fd][b,fd]|} \right ).$$
Since $f|a$, $(a,fd)=f(a,d)$ and $[a,fd]=[a,d]$. Further, $(f,b)=1$,
$(b,fd)=(b,d)$ and $[b,fd]=f[b,d]$. Therefore
$$\frac{|(a,d)(b,d)|}{|[a,d][b,d]|} -
\frac{|(a,fd)(b,fd)|}{|[a,fd][b,fd]|}=0.$$
so we have $\ord_{P_b}(D_a)=0$.
Case 2 ($a=b$). In this case we have to show that
\begin{equation}\label{form}
\sum_{d|I\atop d\text{ monic }}
\mu(I/d)|I|^2\frac{|(a,d)|^2}{|[a,d]|^2}=\mu(a)\prod_{f|I\atop
f\text{ monic, prime }}(1-|f|^2).
\end{equation}
The proof is by induction on $a$. If $a=1$, then the left hand side
of (<ref>) is
$$\sum_{d|I\atop d\text{ monic }} \mu(I/d)|I|^2\frac{|(1,d)|^2}{|[1,d]|^2}=
\sum_{d|I\atop d\text{ monic }}
\mu(I/d)\left(\frac{|I|}{|d|}\right)^2=\prod_{f|I\atop f\text{
monic, prime }}(1-|f|^2)$$
and the lemma follows.
Now, we assume that (<ref>) holds for some $a|I$. Let $f$ be a
monic prime of $A$ such that $f|I$ and $(f,a)=1$. We will show that
(<ref>) holds for $fa$. The left hand side of (<ref>) is
$$\sum_{d|I\atop d\text{ monic }} \mu(I/d)|I|^2\frac{|(fa,d)|^2}{|[fa,d]|^2}=\sum_{d|(I/f)\atop d\text{ monic }}
\mu(I/d)|I|^2
If $d|(I/f)$, we have $(fa,d)=(a,d)$, $[fa,d]=f[a,d]$ and
$(fa,fd)=f(a,d)$,$[fa,fd]=(f)[a,d]$. So
$$\sum_{d|I\atop d\text{ monic }} \mu(I/d)|I|^2\frac{|(fa,d)|^2}{|[fa,d]|^2}=\sum_{d|(I/f)\atop d\text{ monic }}
\mu(I/d)|I|^2
\left(\frac{1}{|f|^2}-1\right)\left(\frac{|(a,d)|^2}{|[a,d]^2}\right)=$$
$$=-(1-|f|^2) \sum_{d|(I/f)\atop d\text{ monic }}\mu(I/fd)|(I/f)|^2
\frac{|(a,d)|^2}{|[a,d]|^2}.$$
By induction, this is
$$-(1-|f|^2)\mu(a)\prod_{g|(I/f)\atop g\text{ monic, prime }}(1-|g|^2)=\mu(fa)\prod_{g|I\atop g\text{ monic, prime }}(1-|g|^2).$$
This concludes the proof of the lemma.
Let $f_0$ be a prime element of $A$ dividing $I$. For $a|(I/f_0)$,
we set
$$F_a=D_a D_{f_0a}$$
where the functions $D_a$ and $D_{f_0a}$ are defined as in
lemma <ref>.
Then, by applying that lemma we have
$$\div(F_a)=\prod_{f|I\atop f \text{ monic, prime }}
So $F_a$ is a simple modular unit. The statements of the theorem
will follow by applying the next lemma
Under the same hypotheses of Theorem <ref>, we
$$\prod_{a|(I/f_0)\atop a\text{ monic }} F_a=\prod_{d|(I/f_0)\atop d\text{ monic }}\prod_{a|(I/f_0)\atop a\text{ monic }} \left(
\frac{\Delta(d)}{\Delta(f_0d)} \right)^{\mu(I/d)|I|
From the definition of $F_a$ we have
\[
\prod_{a|(I/f_0)\atop a\text{ monic }} F_a=\prod_{a|(I/f_0)\atop
a\text{ monic }} \prod_{d|I\atop d\text{ monic }}
\Delta(d)^{\mu(I/d) |I| \left( \frac{|(a,d)|}{|[a,d]|} +
\frac{|(f_0a,d)|}{|[f_0a,d]|} \right)}.
\]
If $(d,f_0)=1$, then $(f_0a,d) = (a,d)$ and $[f_0a,d] = f_0[a,d]$.
So we get
$$ \frac{|(a,d)|}{|[a,d]|}+ \frac{ |(f_0 a,d)|}{|[f_0 a,d]|}=
\frac{|(a,d)|}{|[a,d]|} \left( 1+\frac{1}{|f_0|} \right) = \frac{|(a,f_0d)|}{|[a,f_0d]|}+ \frac{ |(f_0 a,f_0d)|}{|[f_0 a,f_0d]|}.$$
Collecting together the terms with the same $d$, we obtain
$$\prod_{a|(I/f_0)\atop a\text{ monic }} F_a= \prod_{d|(I/f_0)\atop d\text{ monic }}
\left( \frac{\Delta(d)}{\Delta(f_0d)} \right)^{\mu(I/d)|I|
(1+\frac{1}{|f_0|})\left(\sum_{a|(I/f_0)\atop a\text{ monic }}
\frac{|(a,d)|}{|[a,d]|} \right) }.$$
Using an induction argument similar to the one used in the proof of
Lemma <ref>, we have
$$\sum_{a|(I/f_0)\atop a\text{ monic }} \frac{|(a,d)|}{|[a,d]|} =\prod_{f|(I/f_0)\atop f\text{ prime }}
To finish the proof of the theorem, we notice that
$$\Delta_I=\prod_{d|I\atop d\text{ monic }} \Delta(d)^{\mu(I/d)}=\prod_{d|(I/f_0)\atop d\text{ monic }}
\left( \frac{\Delta(d)}{\Delta(f_0d)} \right) ^{\mu(I/d)}.$$
Let $\kappa=\prod_{i=0}^{r} (1+|f_i|)$. Then, it follows from
lemma <ref> that
∏_a|(I/f_0)a monic F_a= Δ_I^κ.
As a corollary of Lemma <ref>, we obtain the following
result of independent interest.
Let $I=\prod_{i=0}^{r} f_i$ be the monic, prime factorization of a
square free, monic polynomial $I$ in $A$. Then, the cuspidal divisor
class group is finite and its order divides
If $a$ and $a'$ are two cusps of $X_0(I)$, then it follows from
lemma <ref> that the function
$$F_{a,a'}=\frac{D_a}{ D_{a'}^{\frac{\mu(a)}{\mu(a')}}}$$
has divisor
$$\div(F_{a,a'}) = \prod_{i=0}^{r}(1-|f_i|^2)
\mu(a)(P_a-P_{a'}).$$
§.§ An element in $H^3_{\M}(X_0(I) \times
Using the factorization in Theorem <ref>, we can construct an
element of the motivic cohomology group as follows:
Let $D_0(I)$ denote the diagonal on $X_0(I)\times X_0(I)$ and let
$I=\prod_{i=0}^{r} f_i$ be the monic prime factorization of $I$.
Let $\kappa=\prod_{i=0}^{r} (1+|f_i|)$. Let $F_d = D_dD_{f_0d}$ as
in Lemma <ref>. Consider the element
\begin{equation}\label{el}
\Xi_0(I) = (D_0(I),\Delta_I^{\kappa})-\left( \sum_{d|(I/f_0)} (P_{d}
\times X_0(I),P_d \times F_d) + (X_0(I) \times P_{f_0d}, F_{d}
\times P_{f_0d}) \right).
\end{equation}
It follows from Theorem <ref> that this element satisfies the
cocycle condition (<ref>), as the sum of the divisors of the
functions is a sum of multiples of terms of the form
\[ (P_d,P_d)-(P_{f_0d},P_{f_0d}) - (P_d,P_d) + (P_{d},P_{f_0d}) +
(P_{f_0d},P_{f_0d}) - (P_{d},P_{f_0d}).
\]
Hence $\Xi_0(I)$ determines an element of $H^3_{\M}(X_0(I) \times
§.§.§ The regulator of $\Xi_0(I)$.
From the formula give in (<ref>), the regulator of our
element of $H^3_{\M}(X_0(I) \times X_0(I),\Q(2))$ is given by the
\begin{equation}
r_{\D,\infty}(\Xi_0(I))= \sum_{v \in X(D_0(I))}
\log|\Delta_I^{\kappa}|(v)Y_{v} \label{regform}
\end{equation}
+ \sum_{d|(I/f_0)} \left( \sum_{v
\in X((P_{d} \times X_0(I)))} \log|P_d \times F_d|(v)Y_v + \sum_{v
\in X((X_0(I) \times P_{f_0d}))} \log|F_{d} \times P_{f_0d}|(v)Y_v
\right)$$
§.§ The final result
We have the following theorem which relates the special value of the
L-function with the intersection pairing of certain cycles. This
intersection pairing is the intersection pairing on the group
$PCH^1(Y)$ obtained as the sum of the intersection pairings on the
Chow groups of the components. It is well defined as it vanishes on
the image of the Gysin map.
Let $f$ and $g$ be Hecke eigenforms for $\Gamma_0(I)$ and
$\Phi_{f,g}$ the completed Rankin-Selberg $L$-function. Then one has
\begin{equation}
\Phi_{f,g}(0)=\frac{q}{ 2 (q-1) \kappa} (r_{\D,\infty}
\end{equation}
where $\Xi_0(I)$ is the element of the higher chow group constructed
above, $r_{\D,\infty}$ is the regulator map and $\Z_{f,g}$ is the
special cycle described above.
We first compute the pairing of the regulator of $\Xi_0(I)$ with
$\Z_{f,g}$. For this we have to compute the pairing of special fibre of the total
transform of the diagonal $D_0(I)$ with $\Z_{f,g}$ as well as the
pairing of the vertical and horizontal components with $\Z_{f,g}$. Since the pairing
is the sum of all the pairings of the components one can compute it
locally - around a point ${\bf P}=(e,e')$ which is being blown up as in Section <ref> .
Recall that $\Z_{\BP}=Y_{15}+Y_{45}-Y_{25}-Y_{35}$. We have the following intersection numbers of $\Z_{{\BP}}$ with the various cycles $Y_{ij}$ –
* $(\Z_{\BP},Y_{12})=(\Z_{\BP},Y_{13})=(\Z_{\BP},Y_{24})=(\Z_{\BP},Y_{34})=0$
* $(\Z_{\BP},Y_{15})=(\Z_{\BP},Y_{45})=-2$
* $(\Z_{\BP},Y_{25})=(\Z_{\BP},Y_{35})=2$
These can easily be computed using the fact $\Z_{\BP}$ is the difference of rulings on $Y_{5}$.
Locally, $D_0(I)$ is the blow-up of the diagonal in $(T_1 \cup T_3) \times (T_2 \cup T_4)$, where $T_i$ are as in Section <ref>. The part of the diagonal which passes through ${\BP}$ is the sum of the diagonals in $T_1 \times T_3$ and $T_2 \times T_4$. Let $\Delta_1$ and $\Delta_4$ denote the strict transforms of these diagonals in $Y_1$ and $Y_4$. The total transform is
as the blow up of the diagonal in $T_1 \times T_3$ has exceptional fibre $Y_{15}$ and similarly for the other diagonal. One has $(\Z_{\BP},\Delta_i)=0$ since $\Z_{\BP}$ is supported in the exceptional fibre.
For vertical or horizontal components the total transform is [6], Lemma 4.1,
$$Y_{13}+(Y_{15}-Y_{35}) + Y_{24}+(Y_{25}-Y_{45})$$
respectively. Hence, using the intersection numbers computed above, we have
* $(\Z_{\BP},Y_{13}+(Y_{15}-Y_{35}) + Y_{24}+(Y_{25}-Y_{45}))=0$
* $(\Z_{\BP},Y_{12}+(Y_{15}-Y_{25})+Y_{34}+(Y_{35}-Y_{45}))=0$
The regulator of $\Xi_0(I)$ is
$$\sum_{v \in X(D_0(I))} \log|\Delta_I^{\kappa}|(v)Y_{v} + $$
$$+ \sum_{d|(I/f_0)} \left( \sum_{v \in X((P_{d} \times X_0(I)))}
\log|P_d \times F_d|(v)Y_v + \sum_{v \in X((X_0(I) \times
P_{f_0d}))} \log|F_{d} \times P_{f_0d}|(v)Y_v \right).$$
From above we can see that the vertical and horizontal components have intersection number $0$ with $\Z_{f,g}$, so it suffices to compute the intersection number of the diagonal component of the regulator with $\Z_{f,g}$.
Locally, at the picture corresponding to the point $(e,e)$, the diagonal components
appear with multiplicities
* $\kappa \log|\Delta_0(I)|(o(e))$ for $\Delta_4$ and $Y_{45}$
* $\kappa \log|\Delta_0(I)(t(e))$ for $\Delta_1$ and $Y_{15}$.
as the vertex $o(e)$ corresponds to the component $\Delta_4$ and the vertex $t(e)$
corresponds to the component $\Delta_1$ of the diagonal. Hence the diagonal component is a sum of terms of the type
$$\kappa \log|\Delta_0(I)(o(e)) (\Delta_4+Y_{45}) + \kappa \log|\Delta_0(I)(t(e)) (\Delta_1+Y_{15}).$$
Using the fact that $t(e)=o(\bar{e})$ and the calculations above, we get –
$$(r_{\D,\infty}(\Xi_0(I)),\Z_{f,g})=(-2\kappa)\int_{e \in Y^{+}_0(I)}
\left( \log|\Delta_I|(o(e))+\log|\Delta_I|(t(e))\right) f(e)g(e)
This is a finite sum as $f$ and $g$ have finite support.
Comparing this with (<ref>) gives us our final result.
\begin{equation}
\Phi_{f,g}(0)=\frac{q}{ 2 (q-1) \kappa} (r_{\D,\infty}
(\Xi_0(I)),\Z_{f,g}) \label{finalresult}
\end{equation}
As $\Phi_{f,q}(s-1)=\Lambda(h^1(M_f) \otimes h^1(M_g),s)$, we get
Theorem <ref>.
§.§.§ An application to elliptic curves.
Theorem <ref> provides some evidence for
Conjecture <ref> in the case of a product of two non-isogenous
elliptic curves over $K$.
If $E$ is a non-isotrivial (that is, $j_E \notin \F_q$),
semi-stable elliptic curve over $K$ with conductor
$I_{E}=I\cdot\infty$ and split multiplicative reduction at
$\infty$, by the work of Deligne [7], Drinfeld, Zarhin and
eventually Gekeler-Reversat [10] we have that $E$ is modular.
This means that the Hasse-Weil $L$-function $L(E,s)$ is equal to
the $L$-function of an automorphic form $f$ of JLD-type with
rational fourier coefficients
\[
L(E,s)=L(f,s)=\sum_{\m \;\text{pos.div}} \frac{c(f,\m)}{|\m|^{s-1}}.
\]
Furthermore, there exists a Drinfeld modular curve $X_0(I)$ of level
$I$ and a dominant morphism (the modular parametrization)
\begin{equation}\label{param}
\pi_f:X_0(I) \longrightarrow E.
\end{equation}
Now, let $E$ and $E'$ be two such modular elliptic curves
with corresponding automorphic forms $f$ and $g$ of levels
$I_1$ and $I_2$. Assume that $(I_1,I_2)=1$ and that $I=I_1 I_2$ is
square-free . Then, the $L$-function of $H^2(\bar{E} \times
\bar{E}',\Q_\ell)$ can be expressed in terms of the $L$-function of
the Rankin-Selberg convolution of $f$ and $g$. Künneth's theorem
gives the decomposition
\[
L(H^2(\bar{E}\times \bar{E}'),s) =
H^1(\bar{E}'),s)=\zeta_A(s-1)^2 L(H^1(\bar{E})\otimes
\]
The completed $L$-function of $H^1(\bar{E})\otimes H^1(\bar{E})$ is
the function $\Phi(s-1)=\Phi_{f,g}(s-1)$ of (<ref>). We
\begin{equation}\label{clf}
\Lambda_{E,E'}(s)=L_{\infty}(s-1)^2 \zeta_A(s-1)^2 \Phi(s-1).
\end{equation}
Then $\Lambda_{E,E'}(s)$ is the completed $L$-function of
$H^2(\bar{E} \times \bar{E}',\Q_\ell)$.
The following result is an application of Theorem <ref>.
Let $E$ and $E'$ be elliptic curves over $K$ satisfying the above
conditions. Then, there is an element $\Xi \in H^3_{\M}(E\times
E',\Q(2))$ such that
\begin{equation}\label{spv2}
\Lambda_{E,E'}^*(1)=\frac{q \deg(\Pi)^2}{2 \kappa (1-q)^3
\log_e(q)^{2}} \left(r_{\D,\infty}(\Xi),\Z_{E,E'}\right)
\end{equation}
where $\Pi$ is the restriction of the product of the modular
parameterizations of $E$ and $E'$ to the diagonal $D_0(I)$ of
$X_0(I)$ and $\Lambda_{E,E'}^*(1)$ is the first non-zero value in
the Laurent expansion at $s=1$.
Let $\pi_f \times \pi_g: X_0(I) \times X_0(I) \rightarrow E \times
E'$ be the product of the modular parameterizations $\pi_f$ and
$\pi_g$. Let $\Xi = (\pi_f \times \pi_g)_*(\Xi_0(I)) \in H^3_{\M}(E
\times E',\Q(2))$ be the push-forward cocycle in motivic cohomology,
where $\Xi_0(I)$ is the class defined in (<ref>). Let $\Z_{E,E'}
= (\pi_f \times \pi_g)_*(\Z_{f,g})$ be the push-forward cycle in the
Chow group where $\Z_{f,g}$ is the 1-cycle considered in
theorem <ref>. The two push-forward maps contribute a factor
$\deg(\Pi)^2$ to the equation. Moreover, the residue at $s=1$ of the
archimedean factor in (<ref>) is ${\log_e(q)}^{-2}$. The result
then follows from Theorem <ref>.
For a self-product of elliptic curves of the type considered in
Theorem <ref>, part C. of Conjecture <ref> asserts that
$$\Lambda_{E,E'}^*(1)=\frac{|coker(R_{\D})|} {|ker(R_{\D})|} \log_e(q)^{-2}.$$
Note that (<ref>) contains the correct power of $\log_e(q)$.
Further, one has that the intersection number $( r_{\D,\infty}(\Xi),
Z_{E,E'} )$ divides $coker(R_{\D})$. Finally, the power $(1-q)^3$ in
the denominator of (<ref>) can be partly explained in terms of
the kernel of the regulator map $R_\D$. The group $H^3_{\M}(E \times
E',\Q(2))$ contains certain elements coming from $H^2_{\M}(E \times
E',\Q(1)) \otimes H^1_\M(E \times E',\Q(1))$ called decomposable
elements. Note that $H^2_{\M}(E \times E',\Q(1)) \cong Pic(E \times
E')$ and $H^1_{\M}(E \times E',\Q(1)) \cong K^*$. Elements of the
form $D \otimes u$, for $D \in NS(E \times E')$ and $u$ a torsion
element in $K^*$, belong to $ker(R_\D)$. There are $(q-1)$ elements
$u$ coming from $\F_{q}^{*}$ and there are two independent elements
$D$ of $NS(E \times E')$ providing $(q-1)^2$ such elements.
§ FINAL REMARKS
Many of the arguments can be carried out in much greater generality
- for example, the ground field could be any local field. The
assumption $(I_1,I_2) = 1$ in Theorem <ref> is not that
essential. Along the lines of the arguments in [4], we can
prove a similar result under the weaker assumption that $I_1$ and
$I_2$ have some common factors, but are not identical.
As suggested by the referee, another direction in which this work can be generalized is that of higher weight forms. Scholl generalized the work of Beilinson's for forms of weight $> 2$ – however, while in our case the analogue of weight 2 forms are the $\Q_{\ell}$-valued harmonic cochains on the tree, it is not clear to me what the analogue of higher weight forms is. One might expect that perhaps harmonic cochains with values in local systems might play the role.
As remarked earlier, since all the factors appearing are analogous to factors appearing in the classical number
field case it would be interesting to know if there was some common underlying field over which the conjecture can be formulated and proved, for which the above work and the classical theorems are special cases.
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D}$-conjecture for products of elliptic curves.
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Ramesh Sreekantan
Indian Statistical Institute
$8^{th}$ Mile, Mysore Road
Jnana Bharathi
Bangalore, 560 059 India
Email: rameshsreekantan@gmail.com
|
arxiv-papers
| 2009-09-03T17:02:56 |
2024-09-04T02:49:05.034921
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ramesh Sreekantan",
"submitter": "Ramesh Sreekantan",
"url": "https://arxiv.org/abs/0909.0712"
}
|
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