# Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot
Groups

E. Bou Dagher  Esther Bou Dagher:
Department of Mathematics
Imperial College London
180 Queen’s Gate, London SW7 2AZ
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

In the setting of Carnot groups, we prove the $q-$Logarithmic Sobolev
inequality for probability measures as a function of the Carnot-Carathéodory
distance. As an application, we use the Hamilton-Jacobi equation in the
setting of Carnot groups to prove the $p-$Talagrand inequality and
hypercontractivity.

###### Contents

  1. 1 Introduction
  2. 2 The Carnot Group and the Hamilton-Jacobi Equation
  3. 3 $q-$Logarithmic Sobolev Inequality
  4. 4 $p-$Talagrand inequality for ${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$
  5. 5  Hypercontractivty of Hamilton-Jacobi solutions for ${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$

###### Key words and phrases:

Logarithmic Sobolev inequality, Talagrand inequality, hypercontractivity,
Hamilton-Jacobi equation, Carnot-Carathéodory distance, Carnot groups

##  1\. Introduction

In [18], L. Gross obtained the Logarithmic Sobolev inequality

(1.1)
$\int_{\mathbb{R}^{n}}f^{2}log\left(\frac{f^{2}}{\int_{\mathbb{R}^{n}}f^{2}d\mu}\right)d\mu\leq
2\int_{\mathbb{R}^{n}}|\triangledown f|^{2}d\mu,$

where ${\displaystyle\triangledown}$ is the standard gradient on
$\mathbb{R}^{n}$ and ${\displaystyle
d\mu=\frac{e^{-\frac{|x|^{2}}{2}}}{Z}d\lambda}$ is the Gaussian measure. He
proved that if $\mathcal{L}$ is the non-positive self-adjoint operator on
$L^{2}\left(\mu\right)$ such that
${\displaystyle\left(-\mathcal{L}f,f\right)_{L^{2}\left(\mu\right)}=\int_{\mathbb{R}^{n}}|\triangledown
f|^{2}d\mu,}$ then (1.1) is equivalent to the fact that the semigroup
${\displaystyle P_{t}=e^{t\mathcal{L}}}$ generated by $\mathcal{L}$ is
hypercontractive: i.e. for $q\left(t\right)\leq 1+\left(q-1\right)e^{2t}$ with
$q>1$, we have ${\displaystyle\parallel
P_{t}f\parallel_{q\left(t\right)}\leq\parallel f\parallel_{q}}$ for all $f\in
L^{q}\left(\mu\right).$

In [2], D. Bakry and M. Emery extended the Logarithmic Sobolev inequality for
a larger class of probability measures defined on manifolds under an important
Curvature-Dimension condition. More generally, if $\left(\Omega,F,\mu\right)$
is a probability space, and $\mathcal{L}$ is a non-positive self-adjoint
operator acting on $L^{2}\left(\mu\right),$ we say that the measure $\mu$
satisfies a Logarithmic Sobolev inequality if there is a constant $c$ such
that, for $f\in D\left(\mathcal{L}\right),$

$\int f^{2}log\frac{f^{2}}{\int f^{2}d\mu}d\mu\leq c\int
f\left(-\mathcal{L}f\right)d\mu.$

F. Otto and C. Villani showed in their celebrated paper [29] that in the
setting of manifolds under D. Bakry and M. Emery’s Curvature-Dimension
condition, the Logarithmic Sobolev inequality implies the Talagrand
transportation cost inequality. Their proof relied on PDE’s methods and Otto
calclulus in Wasserstein space [28]. The Talagrand transportation cost
inequality was first introduced in [31] by M. Talagrand:

(1.2) $T_{w}(\mu,\nu)\leq 2\int log(f)d\mu,$

where $\mu$ is a measure on $\mathbb{R}^{N}$ absolutely continuous with
respect to the Gaussian measure $\nu,$ $f={\displaystyle\frac{d\mu}{d\nu}}$ is
the relative density, and
$w(x,y)={\displaystyle\sum_{i=1}^{N}(x_{i}-y_{i})^{2}}$ is the cost of moving
a unit mass from $x$ to $y.$ $T_{w}(\mu,\nu)$ is the transportation cost
measuring how much effort is required to transport a mass distributed
according to $\mu$ to a mass distributed according to $\nu$, i.e.

$T_{w}(\mu,\nu)={\displaystyle\underset{\pi\in\Pi(\mu,\nu)}{inf}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}}w(x,y)d\pi(x,y),$

where $\Pi(\mu,\nu)$ is the set of probability measures on
$\mathbb{R}^{N}\times\mathbb{R}^{N}$ with $\mu$ the first marginal and $\nu$
the second marginal. M. Talagrand showed how the Talagrand transportation cost
inequality (1.2) implies the concentration of measure phenomenon, and his
approach was motivated by the work of K. Marton [26, 27]. Later on, many works
[1, 5, 6, 15, 16, 23, 24, 29] studied links between the Logarithmic Sobolev
inequality, the Talagrand inequality, and the concentration of measure
phenomenon, which is one of the main tools in probability theory, statistical
mechanics, quantum information theory, stochastic dynamics, etc.

In this note, in the setting of Carnot groups, we are first interested in
proving the $q-$Logarithmic Sobolev inequality (see [7, 8]) for
$q\in\left(1,\infty\right)$,

(1.3) $\int f^{q}log\frac{f^{q}}{\int f^{q}d\mu}d\mu\leq
c\int|\triangledown_{\mathbb{}}f|^{q}d\mu,$

for measures of the form ${\displaystyle
d\mu=\frac{e^{-U\left(d\right)}}{Z}d\lambda}$ . $U(d)$ is a function having a
suitable growth at infinity, $\lambda$ is a natural measure like the Lebesgue
measure for instance, and $d$ is a metric related to the sub-gradient
${\displaystyle\triangledown=\left(X_{1},\ldots,X_{n}\right),}$ where
$X_{i}^{\prime}s$ are the generators of the group’s lie algebra. Since the
Laplacian is of Hörmander type and has some degeneracy, D. Bakry and M.
Emery’s Curvature-Dimension condition in [2] will no longer hold true. In
[19], W. Hebisch and B. Zegarliński developed a method of studying coercive
inequalities on general metric spaces that does not require a bound on the
curvature of space. Working on a general metric space equipped with non-
commuting vector fields $\\{X_{1},\ldots,X_{n}\\},$ their method is based on
U-bounds, where U stands for the potential, which are inequalities of the
form:

$\int f^{q}\eta\left(d\right)d\mu\leq C\int|\triangledown f|^{q}d\mu+D\int
f^{q}d\mu,$

where $\eta(d)$ is a function having a suitable growth at infinity. Those
methods were applied, for example, in the works [10, 12, 11] in the setting of
Carnot groups. In [19], W. Hebisch and B. Zegarliński proved the
$q-$Logarithmic Sobolev inequality for the measure ${\displaystyle
d\mu=\frac{e^{-\alpha d^{p}}}{Z}d\lambda}$, where $p$ is the finite index
conjugate of $q$ and $\alpha>0.$ In this note, we enlarge the class of
measures that satisfy the $q-$logarithmic Sobolev inequality to include
measures of the form ${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda},$ under
some growth conditions for $U(d)$, and get examples such as
$U(d)=(d+1)^{p}log(d+1)$ and $U(d)=sinh(d).$ We would like to apply the
$q-$Logarithmic Sobolev inequality to get hypercontractivity and to obtain the
$p-$Talagrand inequality on $(X,d,\mu)$ with a constant $K:$

(1.4)
$W_{p}(\mu,\nu)^{p}\leq\frac{1}{K}Ent_{\mu}\left(\frac{d\nu}{d\mu}\right),$

with $p$ finite index conjugate of $q.$ The $p-$Wasserstein distance between
two probability measures on $X$ is defined as ${\displaystyle
W_{p}(\mu,\nu)^{p}={\displaystyle\underset{\pi\in\Pi(\mu,\nu)}{inf}\int_{X\times
X}}d(x,y)^{p}d\pi(x,y),}$ where $\Pi(\mu,\nu)$ is the set of probability
measures on $X\times X$ with $\mu$ the first marginal and $\nu$ the second
marginal.

$Ent_{\mu}{\displaystyle\left(\frac{d\nu}{d\mu}\right)=\int\frac{d\nu}{d\mu}log\left(\frac{d\nu}{d\mu}\right)d\mu}$

is the entropy functional such that $\nu$ is a probability measure absolutely
continuous with respect to $\mu.$ We note that for $p=2,$ (1.2) is a special
case of (1.4).

The $p-$Talagrand inequality was introduced by Z.M. Balogh et al. in [4] who
proved that it was implied by the $q-$Logarithmic Sobolev inequality for $p$
the finite index conjugate of $q$; their work generalises the result by J.
Lott and C. Villani [25] who proved the implication for the quadratic case
$p=q=2$. J. Lott and C. Villani [25] used the Hamilton-Jacobi infimum
convolution operator under the assumption where the space $(X,d,\mu)$ supports
local Poincaré inequality and the measure $\mu$ is a doubling measure i.e. the
measure of any open ball is positive and finite and there exists a constant
$c_{d}\geq 1$ such that for all $x\in X$ and $r>0,$

(1.5) ${\displaystyle\mu(B(x,2r))\leq c_{d}\mu(B(x,r))}.$

This idea of using the Hamilton-Jacobi infimum convolution operator was first
used by S. Bobkov et al. in [5] who explored the equivalence between the
Logarithmic Sobolev inequality with a constant $\rho$ in $\mathbb{R}^{n}$ and
hypercontractivity of the quadratic Hamilton-Jacobi semigroup $Q_{t}$ i.e.

${\displaystyle||e^{Q_{t}f}||_{a+\rho t}\leq}||e^{f}||_{a},$

for every bounded measureable function $f$ on $\mathbb{R}^{n},$ $t\geq 0,$
$a\in\mathbb{R},$ and $||.||_{p}$ the $L^{p}-$norm with respect to $\mu.$

In the setting of the Heisenberg group $\mathbb{H}$, Z.M. Balogh et al. in [4]
used W. Hebisch and B. Zegarliński’s [19] $q-$Logarithmic Sobolev inequality
for the measure ${\displaystyle d\mu=\frac{e^{-\alpha d^{p}}}{Z}d\lambda},$
where $d$ is the Carnot-Carathéodory distance, to obtain the $p-$Talagrand
inequality. To prove the above implication, Z.M. Balogh et al. observed that
by Pansu’s differentiability theorem [17], for Lipschitz continuous functions
$f:\mathbb{H}\rightarrow\mathbb{R},$ the norm of the sub-Riemannian gradient
$|\triangledown
f(x)|=\left({\displaystyle\sum_{i=1}^{N}|X_{i}f|^{2}}\right)^{\frac{1}{2}}$used
in [19] coincides with the metric subgradient in [4] for $\mu_{p}$ almost
every $x$ for which $|\triangledown f(x)|>0.$

In section 2, we introduce the Carnot group and the Hamilton-Jacobi equation
in that setting ([14, 3]). In section 3, we extend W. Hebisch and B.
Zegarliński’s results in [19] by enlarging the class of measures that satisfy
the $q-$logarithmic Sobolev inequality to include measures of the form
${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda},$ under some growth
conditions for $U(d)$, and get examples such as $U(d)=(d+1)^{p}log(d+1)$ and
$U(d)=sinh(d).$ As an application, we adapt Z.M. Balogh et al.’s Theorem 4.1
[4] to get the $p-$Talagrand inequality (section 4), and we adapt S. Bobkov et
al.’s Theorem 2.1 in [5] to get hypercontractivity (section 5). However,
instead, we use the Hamilton-Jacobi equation in the setting of Carnot groups
by F. Dragoni [14]. The advantage of doing so is that the restriction (1.5) to
have $\mu$ a doubling measure is no longer required!

##  2\. The Carnot Group and the Hamilton-Jacobi Equation

Carnot groups are geodesic metric spaces that appear in many mathematical
contexts like harmonic analysis in the study of hypoelliptic differential
operators ([13, 30]) and in geometric measure theory (see extensive reference
list in the survey paper [22]). The following series of definitions are from
[9]:

######  Definition 1.

We say that a Lie group on $\mathbb{R}^{N}$ ,
$\mathbb{G=}(\mathbb{R}^{N},\circ)$ is a (homogeneous) Carnot group if the
following properties hold:

(C.1) $\mathbb{R}^{N}$ can be split as
$\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times...\times\mathbb{R}^{N_{r}},$ and the
dilation $\delta_{\lambda}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$

$\delta_{\lambda}(x)=\delta_{\lambda}(x^{(1)},...,x^{(r)})=(\lambda
x^{(1)},\lambda^{2}x^{(2)},...,\lambda^{r}x^{(r)}),\;\;\;\;\;\;\;\;\;x^{(i)}\in\mathbb{R}^{N_{i}},$

is an automorphism of the group $\mathbb{G}$ for every $\lambda>0.$ Then
$\mathbb{}(\mathbb{R}^{N},\circ,\delta_{\lambda})$ is a homogeneous Lie group
on $\mathbb{R}^{N}$ . Moreover, the following condition holds:

(C.2) If $N_{1}$ is as above, let $Z_{1},...,Z_{N_{1}}$ be the left invariant
vector fields on $\mathbb{G}$ such that $Z_{j}(0)=\partial/\partial
x_{j}|_{0}$ for $j=1,...,N_{1}$. Then

$rank(Lie{\\{Z_{1},...,Z_{N_{1}}\\}}(x))=N\;\;\;\;\;\;\forall
x\in\mathbb{R}^{N}.$

If (C.1) and (C.2) are satisfied, we shall say that the triple
$\mathbb{G}=(\mathbb{R}^{N},\circ,\delta_{\lambda})$ is a homogeneous Carnot
group. We also say that $\mathbb{G}$ has step $r$ and $N_{1}$ generators. The
vector fields $Z_{1},...,Z_{N_{1}}$ will be called the (Jacobian) generators
of $\mathbb{G}$.

######  Definition 2.

The vector valued operator $\triangledown:=(Z_{1},Z_{2},...,Z_{N_{1}})$ is
called the sub-gradient on $\mathbb{G},$ and
$\triangle={\displaystyle\sum_{i=1}^{N_{1}}Z_{i}^{2}}$ is called the sub-
Laplacian on $\mathbb{G}.$

######  Definition 3.

We say that $\gamma$ is horizontal if there exist measurable functions
$a_{1},\ldots,a_{N_{1}}:\left[0,1\right]\rightarrow\mathbb{R}$ such that

$\gamma^{\prime}\left(t\right)=\sum_{i=1}^{N_{1}}a_{i}\left(t\right)Z_{i}\left(\gamma\left(t\right)\right)$

for almost all $t\in\left[0,1\right]$ i.e. $\leavevmode\nobreak\
\gamma^{{}^{\prime}}\left(t\right)\in
Span\left\\{Z_{1}\left(\gamma\left(t\right)\right),\ldots,Z_{N_{1}}\left(\gamma\left(t\right)\right)\right\\}$
almost everywhere. For such a horizontal curve $\gamma,$ we define the length
of $\gamma$ to be

$|\gamma|=\int_{0}^{1}\left(\sum_{i=1}^{N_{1}}a_{i}^{2}\left(t\right)\right)^{\frac{1}{2}}dt.$

######  Definition 4.

The Carnot-Carathéodory distance or the control distance between two points
$x$ and $y$ is defined by

$d\left(x,y\right)=inf\left\\{t|\gamma:\left[0,t\right]\rightarrow\mathbb{G},\leavevmode\nobreak\
\gamma\left(0\right)=x,\leavevmode\nobreak\
\gamma\left(t\right)=y\leavevmode\nobreak\ |\gamma^{\prime}\left(s\right)|\leq
1\;\;\forall s\in\left[0,t\right]\right\\},$

where $\gamma:\left[0,1\right]\rightarrow\mathbb{G}$ is an absolutely
continuous horizontal path on $\left[0,1\right],$ i.e. if for every
$\varepsilon>0$ there exists a $\delta>0$ such that when a finite number of
pairwise disjoint subintervals $\left[x_{k},y_{k}\right]$ of
$\left[0,1\right]$ satisfy $\sum_{k}|y_{k}-x_{k}|<\delta,$ then
$\sum_{k}d\left(\gamma\left(y_{k}\right),\gamma\left(x_{k}\right)\right)<\varepsilon.$

It can be shown, proof found in [20], that $d$ is associated to the sub-
gradient through:

$|\triangledown f\left(x\right)|=\underset{d\left(x,y\right)\rightarrow
0}{limsup}\frac{|f\left(x\right)-f\left(y\right)|}{d\left(x,y\right)}.$

We will be using F. Dragoni’s Theorem 4 of [14] to get hypercontractivity and
$p-$Talagrand inequality from the $q-$Logarithmic sobolev inequality.

######  Theorem 5 (F. Dragoni [14]).

Let $\phi:[0,\infty)\rightarrow[0,\infty)$ be a continous, non-decreasing, and
convex function with $\phi(0)=0.$ Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$
a lower semicontinuous function such that there exists $C>0:$

$f(x)\geq-C(1+d(x)).$

Then a viscosity solution for the Hamilton-Jacobi problem in the Carnot Group
$\mathbb{G}$

$\begin{cases}u_{t}(x,t)+\phi(|\mathbb{\triangledown}u(x,t)|)=0\text{
}&(x,t)\in\mathbb{G}\times(0,\infty)\\\ u(x,0)=f(x)&x\in\mathbb{G}\end{cases}$
is given by the Hopf-Lax formula

$Q_{t}f(x):=\underset{y\in\mathbb{G}\mathbb{}}{inf}\left\\{t\phi^{*}\left(\frac{d(x,y)}{t}\right)+f(y)\right\\},$

where $d(x,y)$ is the Carnot-Carathéodory distance as in Definition 4,
$\triangledown$ is the sub-gradient as in Definition 2, and $\phi^{*}$ is the
Legendre transform of $\phi.$

In this note, we will be considering $f(x)$ to be a continuous and bounded
function and we will be choosing the function
${\displaystyle\phi(s)=\frac{s^{q}}{q},}$ where $1<q\leq 2.$

Thus, the viscosity solution for the Hamilton-Jacobi problem in the Carnot
group $\mathbb{G}$

(2.1) $\begin{cases}u_{t}(x,t)+\frac{|\triangledown u(x,t)|^{q}}{q}=0\text{
}&(x,t)\in\mathbb{}\mathbb{G}\times(0,\infty)\\\
u(x,0)=f(x)&x\in\mathbb{}\mathbb{G}\end{cases}$

is given by the Hopf-Lax formula

$Q_{t}f(x)=\underset{y\in\mathbb{G}}{inf}\left\\{\frac{d(x,y)^{p}}{t^{p-1}}+f(y)\right\\},$

where ${\displaystyle\frac{1}{p}+\frac{1}{q}=1.}$

##  3\. $q-$Logarithmic Sobolev Inequality

Given the probability measure ${\displaystyle d\mu=\frac{e^{-U}}{Z}d\lambda},$
where $U$ is an increasing unbounded function and $Z$ is the normalization
constant. To obtain the $q-$Logarithmic Sobolev inequality, we will need the
following theorems stated in the beginning of this section by W. Hebisch and
B. Zegarliński in [19].

Let $\lambda$ be a measure satisfying the q-Poincaré inequality for every ball
${\displaystyle B_{R}=\\{x:N\left(x\right)<R\\},\leavevmode\nobreak\ }$ i.e.
there exists a constant $C_{R}\in\left(0,\infty\right)$ such that

$\frac{1}{|B_{R}|}\int_{B_{R}}\left|f-\frac{1}{|B_{R}|}\int_{B_{R}}f\right|^{q}d\lambda\leq
C_{R}\frac{1}{|B_{R}|}\int_{B_{R}}|\triangledown f|^{q}d\lambda,$

where $1\leq q<\infty.$

Note that we have this Poincaré inequality on balls in the setting of
Nilpotent lie groups thanks to J. Jerison’s celebrated paper [21].

######  Theorem 6 (W. Hebisch, B. Zegarliński [19]).

Let $\mu$ be a probability measure on $\mathbb{R}^{n}$ which is absolutely
continuous with respect to the measure $\lambda$ and such that

(3.1) $\int f^{q}\eta d\mu\leq C\int|\triangledown f|^{q}d\mu+D\int f^{q}d\mu$

with some non-negative function $\eta$ and some constants
$C,D\in\left(0,\infty\right)$ independent of a function $f.$ If for any
$L\in\left(0,\infty\right)$ there is a constant $A_{L}$ such that
${\displaystyle\frac{1}{A_{L}}\leq\frac{d\mu}{d\lambda}\leq A_{L}}$ on the set
$\left\\{\eta<L\right\\}$ and, for some $R\in\left(0,\infty\right)$ (depending
on L), we have $\left\\{\eta<L\right\\}\subset B_{R},$ then $\mu$ satisfies
the q-Poincaré inequality

$\mu|f-\mu f|^{q}\leq c\mu|\triangledown f|^{q}$

with some constant $c\in\left(0,\infty\right)$ independent of $f.$

######  Theorem 7 (W. Hebisch, B. Zegarliński [19]).

Suppose the following Sobolev inequality is satisfied

$\left(\int|f|^{q+\varepsilon}d\lambda\right)^{\frac{q}{q+\varepsilon}}\leq
a\int|\triangledown f|^{q}d\lambda+b\int|f|^{q}d\lambda,$

and the following bound is true

(3.2) $\mu\left(|f|^{q}\left[|\triangledown U|^{q}+U\right]\right)\leq
C^{\prime}\mu|\triangledown f|^{q}+D^{\prime}\mu|f|^{q}.$

Then, the following inequality is true

$\mu\left(f^{q}log\frac{f^{q}}{\mu f^{q}}\right)\leq C\mu|\triangledown
f|^{q}+D\mu|f|^{q}.$

Moreover, if $q\in\left(1,2\right]$ and the following $q-$Poincaré inequality
holds $\mu|f-\mu f|^{q}\leq\frac{1}{M}\mu|\triangledown f|^{q},$ then one has

(3.3) $\mu\left(f^{q}log\frac{f^{q}}{\mu f^{q}}\right)\leq c\mu|\triangledown
f|^{q}$

with some constant $c\in\left(0,\infty\right)$ independent of $f.$

Under two assumptions on $d,$ outside the unit ball
$B\equiv\left\\{d\left(x\right)<1\right\\},$

$\Delta d\leq K+\beta\varepsilon d^{p-1},$

and

$\frac{1}{\sigma^{2}}\leq|\triangledown d|\leq 1,$

where
${\displaystyle\varepsilon\in\left(0,\frac{1}{\sigma}\right),\>\beta\in\left(0,\infty\right),and\>\sigma\in\left[1,\infty\right),}$
W. Hebisch and B. Zegarliński proved that the measure

$d\mu_{p}=\frac{e^{-d^{p}}}{Z}d\lambda$

where $p>1$ satisfies both Poincaré and Logarithmic Sobolev inequalities. To
get the Poincaré inequality from Theorem 6, W. Hebisch and B. Zegarliński
proved a U-bound inequality (3.1) with ${\displaystyle\eta=d^{p-1}}$ (Theorem
2.1 of [19]). However, the Logarithmic Sobolev inequality requires an
additional U-bound inequality with ${\displaystyle U=d^{p}}$ (Theorem 2.4 of
[19]). The additional U-bound inequality is obtained from a more general
inequality of the form:

$\int d^{q\left(p-1\right)}f^{q}d\mu_{p}\leq C_{q}\int|\triangledown
f|^{q}d\mu_{p}+D_{q}\int|f|^{q}d\mu_{p}$

which for a given $p\in\left(1,\infty\right)$ is valid for all
$q\in\left[1,\infty\right)$ (Theorem 2.3 of [19]). In their proof of the
Logarithmic Sobolev inequality, since ${\displaystyle U=d^{p},}$ and
${\displaystyle\frac{1}{p}+\frac{1}{q}=1,}$ then

$|\triangledown U|=|pd^{p-1}\triangledown d|\leq pd^{p-1}|\triangledown d|\leq
pd^{p-1},$

which implies that

$|\triangledown U|^{q}+U\leq
p^{q}d^{\left(p-1\right)q}+d^{p}=p^{q}d^{p}+d^{p}.$

Therefore, the second U-bound allows condition (3.2) to be satisfied in
Theorem 7. So, they obtain the Logarithmic Sobolev inequality for $p$ the
finite conjugate of $q$.

In the following, we present a generalised version of Theorems 2.1, 2.3, and
2.4 of [19]: In place of the function $U(d)=d^{p},$ we let
$U:\left[0,\infty\right)\rightarrow\left[0,\infty\right)$ be a twice
differentiable and increasing function; and let ${\displaystyle
d\mu_{U}=\frac{e^{-U\left(d\right)}d\lambda}{Z}}$ be a probability measure
defined in terms of the function $U\left(d\right),$ where $Z$ is the
normalization constant.

######  Theorem 8.

Assume that outside the open unit ball $B=\left\\{d\left(x\right)<1\right\\},$
the metric $d$ satisfies the following: $|\triangledown d|$ is bounded, say
$|\triangledown d|\leq 1,$ and there exist finite positive constants $K$ and
$c_{0}$ such that

(3.4) $\Delta d\leq K+U^{\prime}\left(d\right)\left(|\triangledown
d|^{2}-c_{0}\right).$

(i) If $U^{\prime\prime}\leq\beta U^{{}^{\prime}}$ for some positive constant
$\beta,$ outside $B$, then for any $q\in\left(1,\infty\right),$ there exist
constants $C_{q},D_{q},$ independent of $f$, such that

$\int|f|^{q}|U^{\prime}\left(d\right)|^{q}d\mu_{U}\leq C_{q}\int|\triangledown
f|^{q}d\mu_{U}+D_{q}\int|f|^{q}d\mu_{U}.$

(ii) If, in addition, $U\leq\gamma U^{\prime q}$ for some positive constant
$\gamma$ and some $q>1,$ outside $B,$ then

$\int|f|^{q}U\left(d\right)d\mu_{U}\leq C_{q}\int|\triangledown
f|^{q}d\mu_{U}+D_{q}\int|f|^{q}d\mu_{U}.$

In the proof, we adapt, with simplification, the general lines of proof in
[6].

###### Proof.

Using integration by parts and condition (3.4),

$\int\left(\triangledown
d\right)\cdot\triangledown\left(fe^{-U\left(d\right)}\right)d\lambda=-\int\Delta
d\left(fe^{-U\left(d\right)}\right)d\lambda$ (3.5) $\geq-K\int
fe^{-U\left(d\right)}d\lambda-\int U^{\prime}\left(d\right)|\triangledown
d|^{2}fe^{-U\left(d\right)}d\lambda+c_{0}\int
U^{\prime}\left(d\right)fe^{-U\left(d\right)}d\lambda.$

Computing

$\triangledown\left(fe^{-U\left(d\right)}\right)=\triangledown
fe^{-U\left(d\right)}-U^{\prime}\left(d\right)e^{-U\left(d\right)}\triangledown
d,$

and taking the dot product with $\triangledown d,$

$\left(\triangledown
d\right)\cdot\triangledown\left(fe^{-U\left(d\right)}\right)=\left(\triangledown
f\right)\cdot\left(\triangledown
d\right)e^{-U\left(d\right)}-U^{\prime}\left(d\right)f|\triangledown
d|^{2}e^{-U\left(d\right)},$

and replacing in (3.5) to get

$c_{0}\int U^{\prime}\left(d\right)fe^{-U\left(d\right)}d\lambda\leq K\int
fe^{-U\left(d\right)}d\lambda+\int|\triangledown f||\triangledown
d|e^{-U\left(d\right)}d\lambda.$

Using the fact that $|\triangledown d|\leq 1$ ($|\triangledown d|$ bounded is
enough), we get

(3.6) $\int U^{\prime}\left(d\right)fd\mu_{U}\leq C\int
fd\mu_{U}+D\int|\triangledown f|d\mu_{U}.$

To prove (i), we replace $f$ in (3.6) by $f^{q}$ to get

$\int
f^{q}\left(U^{\prime}\left(d\right)\right)^{q}d\mu_{U}=\int\left[f^{q}U^{\prime}\left(d\right)^{q-1}\right]U^{\prime}\left(d\right)d\mu_{U}$
$\leq
C\int\left[f^{q}U^{\prime}\left(d\right)^{q-1}\right]d\mu_{U}+D\int\left|\triangledown\left[f^{q}U^{\prime}\left(d\right)^{q-1}\right]\right|d\mu_{U}$
$=C\int ff^{q-1}U^{\prime}\left(d\right)^{q-1}d\mu_{U}+D\int
qf^{q-1}|\triangledown f|U^{\prime}\left(d\right)^{q-1}d\mu_{U}$ (3.7)
$+D\int\left(q-1\right)ff^{q-1}U^{\prime}\left(d\right)^{q-2}U^{\prime\prime}\left(d\right)|\triangledown
d|d\mu_{U}.$

Using Young’s inequality with $\alpha,$ i.e. ${\displaystyle ab\leq\alpha
a^{p}+\frac{b^{q}}{q\left(\alpha p\right)^{q/p}}}$ where $p$ and $q$ are
conjugate finite indices in each of the three terms on the right-hand side of
(3.7), in addition to using the conditions $|\triangledown d|\leq 1$ and
$U^{\prime\prime}<\beta U^{{}^{\prime}}$ in the last term, we get

$\int f^{q}U^{\prime}\left(d\right)^{q}d\mu_{U}\leq\frac{C}{q\left(\alpha
p\right)^{\frac{q}{p}}}\int f^{q}d\mu_{U}+C\alpha\int
f^{q}U^{\prime}\left(d\right)^{q}d\mu_{U}+\frac{D}{\left(\alpha
p\right)^{\frac{q}{p}}}\int|\triangledown f|^{q}d\mu_{U}$ $+Dq\alpha\int
f^{q}U^{\prime}\left(d\right)^{q}d\mu_{U}+\frac{D\beta\left(q-1\right)}{q\left(\alpha
p\right)^{\frac{q}{p}}}\int f^{q}d\mu_{U}+D\left(q-1\right)\alpha\int
f^{q}U^{\prime}\left(d\right)^{q}d\mu_{U}.$

Choosing $\alpha$ small enough so that
$C\alpha+Dq\alpha+D\left(q-1\right)\alpha\leq 1,$ we get

$\int|f|^{q}U^{\prime}\left(d\right)^{q}d\mu_{U}\leq C_{q}\int|\triangledown
f|^{q}d\mu_{U}+D_{q}\int|f|^{q}d\mu_{U}.$

∎

######  Corollary 9.

If the positive twice differentiable and increasing function $U(d)$ satisfies
the two inequalities $U^{\prime\prime}<\beta U^{{}^{\prime}}$ and $U\leq\gamma
U^{\prime q},$ where $\beta$ and $\gamma$ are finite positive constants, then
the measure

$d\mu_{U}=\frac{e^{-U\left(d\right)}d\lambda}{Z}$

satisfies a q-Poincaré and a q-Logarithmic Sobolev inequality.

###### Proof.

To get the Logarithmic Sobolev inequality, we verify inequality (3.2) from
Theorem 7

$|\triangledown
U\left(d\right)|^{q}+U\left(d\right)=|U^{\prime}\left(d\right)\triangledown
d|^{q}+U\left(d\right)\leq\left(1+\gamma\right)U^{{}^{\prime}q}\left(d\right)$

Since $U^{\prime\prime}\left(d\right)<\beta U^{\prime}\left(d\right)$ by
assumption, we can use (i) of Theorem 8, and the result follows. ∎

######  Remark 10.

In Corollary 9, it is enough to assume that the inequalities

$U^{{}^{\prime\prime}}<\beta U^{{}^{\prime}}\>and\>{}U\leq\gamma U^{\prime q}$

hold almost everywhere for $\beta$ and $\gamma$ finite positive constants.

######  Remark 11.

To get Theorem 2.3 of [19], take $U\left(d\right)=d^{p},$ then

$U^{\prime}\left(d\right)=pd^{p-1}\>and\>U^{\prime\prime}\left(d\right)=p\left(p-1\right)d^{p-2}.$

Outside the open unit ball $B=\left\\{d\left(x\right)<1\right\\},$

$\frac{U^{\prime\prime}\left(d\right)}{U^{\prime}\left(d\right)}=\frac{p-1}{d}\leq
p-1.$

Therefore, (i) from Theorem 8 holds, and the statement of Theorem 2.3 of [19]
is recovered.

Suppose $q$ is the finte conjugate index of $p,$ then,

$\frac{U\left(d\right)}{U^{{}^{\prime}q}\left(d\right)}=\frac{d^{p}}{p^{q}d^{\left(p-1\right)q}}=\frac{1}{p^{q}}.$

Therefore, (ii) of Theorem 8 holds, and the statement of Theorem 2.4 of [19]
is recovered.

######  Example 12.

The q-Poincaré and a q-Logarithmic Sobolev inequality are satisfied for the
measure

$d\mu_{U}=\frac{e^{-\left(d+1\right)^{p}log\left(d+1\right)}d\lambda}{Z}$

for $q\geq\beta$, where $\beta$ is the finite index conjugate to $p.$

###### Proof.

Given $U\left(d\right)=\left(d+1\right)^{p}log\left(d+1\right).$Then,

$U^{\prime}\left(d\right)=\left(d+1\right)^{p-1}\left(plog\left(d+1\right)+1\right),$

and

$U^{\prime\prime}\left(d\right)=\left(d+1\right)^{p-2}\left(p\left(p-1\right)log\left(d+1\right)+2p-1\right).$

So, outside the open unit ball $B=\left\\{d\left(x\right)<1\right\\},$

$\frac{U^{\prime\prime}\left(d\right)}{U^{\prime}\left(d\right)}=\frac{\left(p\left(p-1\right)log\left(d+1\right)+2p-1\right)}{d+1}\leq
p\left(p-1\right)+\frac{2p-1}{d+1}\leq p^{2}-\frac{1}{2}.$

In addition,

$\frac{U\left(d\right)}{U^{{}^{\prime}q}\left(d\right)}=\frac{\left(d+1\right)^{p}log\left(d+1\right)}{\left(d+1\right)^{\left(p-1\right)q}\left(plog\left(d+1\right)+1\right)^{q}}\leq\frac{\left(d+1\right)^{p}log\left(d+1\right)}{\left(d+1\right)^{\left(p-1\right)\beta}\left(plog\left(d+1\right)+1\right)^{q}}\leq
1.$

So, by Corollary 9, the q-Poincaré and a q-Logarithmic Sobolev inequality are
satisfied for the measure ${\displaystyle
d\mu_{U}=\frac{e^{-\left(d+1\right)^{p}log\left(d+1\right)}}{Z}d\lambda}$ for
$q\geq\beta,$ where $\beta$ is the finite index conjugate to $p.$ ∎

######  Example 13.

For $U\left(d\right)=sinh\left(d\right),$
$U\left(d\right)=U^{{}^{\prime\prime}}\left(d\right)\leq
cosh\left(d\right)=U^{{}^{\prime}}\left(d\right).$

So, by Corollary 9, the q-Poincaré and q-Logarithmic Sobolev inequalities hold
true for the measure ${\displaystyle
d\mu_{U}=\frac{e^{-sinh\left(d\right)}}{Z}d\lambda}$ for all $q\geq 1.$

##  4\. $p-$Talagrand inequality for ${\displaystyle
d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$

To prove that the $q-$Logarithmic Sobolev inequality implies the $p-$Talagrand
inequality, we are inspired by the proof of Theorem 4.1 in [4]. However,
instead of using the Hamilton-Jacobi equation as in that paper, we use the
following Hamilton-Jacobi equation as in Theorem 4 in the paper by F. Dragoni
[14]. In the setting of Carnot groups, the restriction for using a doubling
measure $\mu$ is no longer needed.

Let $d$ be the Carnot-Carathéodory distance and ${\displaystyle
d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$ be a measure where $Z$ is a normalization
constant, $\lambda$ is the Lebesgue measure, and $U(d)$ satisfies the
conditions of Theorem 8.

######  Theorem 14.

Let $1<q\leq 2,$ and $p\geq 2$ be its finite index conjugate, so that
${\displaystyle\frac{1}{p}+\frac{1}{q}=1.}$

If $(\mathbb{G},d,\mu)$ satisfies the q-Logarithmic Sobolev 3.3 inequality
with constant $c={\displaystyle(q-1)\left(\frac{q}{K}\right)^{q-1}}$ for some
constant $K>0,$ then it also satisfies the p-Talagrand inequality with the
same constant $K$.

For every continuous bounded function $f$ where $Qf=Q_{1}f,$ we will be
proving the dual formulation of the Talagrand inequality

(4.1) $\int_{\mathbb{G}}e^{KQf}d\mu\leq e^{K\int_{\mathbb{G}}fd\mu},$

which is shown equivalent to the $p-$Talagrand inequality (1.4) in [6].

###### Proof.

For some $n\geq 1,$ define

$\phi(t)=\frac{1}{Kt^{n}}log\left(\int_{\mathbb{G}}e^{kt^{n}Q_{t}f}d\mu\right).$

Using the fact that $f$ is bounded and the dominated convergence theorem,
Balogh et al. prove in Theorem 4.1 of [4] that:

$\underset{t\rightarrow 0^{+}}{lim}\phi(t)=\int_{\mathbb{G}}fd\mu.$

The main goal is to prove that $\phi(t)$ is non-increasing in $t$, which
implies that $\phi(1)\leq\underset{t\rightarrow 0^{+}}{lim}\phi(t).$ In other
words, replacing:

$\phi(1)=\frac{1}{K}log\left(\int_{\mathbb{G}}e^{kQf}d\mu\right)\leq\underset{t\rightarrow
0^{+}}{lim}\phi(t)=\int_{\mathbb{G}}fd\mu.$

Rearranging the terms and exponentiating the last inequality, we get the dual
Talagrand inequality (4.1). We now show that $\phi(t)$ is non-increasing. The
two main ingredients to prove that are the q-Logarithmic Sobolev inequality
and the solution to the Hamilton-Jacobi problem (2.1).

Fix $t\in(0,1].$ For $s>0,$ we have

$\underset{s\rightarrow
0^{+}}{lim}\frac{\phi(t+s)-\phi(t)}{s}=\underset{s\rightarrow
0^{+}}{lim}\frac{1}{s}\left(\frac{1}{K(t+s)^{n}}-\frac{1}{Kt^{n}}\right)log\int_{\mathbb{G}}e^{K(t+s)^{n}Q_{t+s}f}d\mu$

(4.2) $+\underset{s\rightarrow
0^{+}}{lim}\frac{1}{Kt^{n}s}\left(log\int_{\mathbb{G}}e^{K(t+s)^{n}Q_{t+s}f}d\mu-
log\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu\right).$

As $s\rightarrow 0^{+},$ the first term on the right-hand side of (4.2)
converges to

$-\frac{n}{Kt^{n+1}}log\left(\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu\right).$

The limit of the second term of (4.2) is

$\frac{1}{Kt^{n}}\frac{1}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}\underset{s\rightarrow
0^{+}}{lim}\left[\frac{1}{s}\left(\int_{\mathbb{G}}e^{K(t+s)^{n}Q_{t+s}f}d\mu-\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu\right)\right]$

rewriting, we get

$=\frac{1}{Kt^{n}}\frac{1}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}\underset{s\rightarrow
0^{+}}{lim}\left[\int_{\mathbb{G}}\frac{e^{K(t+s)^{n}Q_{t+s}f}-e^{Kt^{n}Q_{t+s}f}}{s}d\mu+\int_{\mathbb{G}}\frac{e^{Kt^{n}Q_{t+s}f}-e^{Kt^{n}Q_{t}f}}{s}d\mu\right]$

so, using the dominated convergence theorem for the first term, we get

(4.3)
$=\frac{1}{Kt^{n}}\frac{\int_{\mathbb{G}}Knt^{n-1}Q_{t}fe^{Kt^{n}Q_{t}f}d\mu}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}+\frac{1}{Kt^{n}}\frac{1}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}\underset{s\rightarrow
0^{+}}{lim}\left[\int_{\mathbb{G}}\frac{e^{Kt^{n}Q_{t+s}f}-e^{Kt^{n}Q_{t}f}}{s}d\mu\right].$

In order to compute the limit in (4.3), we use the Hamilton-Jacobi equation in
Theorem 5, which has the solution $Q_{t}f,$ so

$\frac{\partial Q_{t}f}{\partial_{t}}=-\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q},$

so,

(4.4) $\underset{s\rightarrow
0^{+}}{lim}\left(\frac{e^{Kt^{n}Q_{t+s}f}-e^{Kt^{n}Q_{t}f}}{s}\right)=Kt^{n}e^{Kt^{n}Q_{t}f}\frac{\partial
Q_{t}f}{\partial_{t}}=-Kt^{n}e^{Kt^{n}Q_{t}f}\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q}.$

Since $Q_{(.)}f(.)$ is Lipschitz on $\mathbb{G}\times\mathbb{R}_{+},$
$Q_{t+s}f=Q_{t}f+O(s)$ holds uniformly on $\mathbb{G}.$ Since $Q_{t}f(x)$ is
uniformly bounded in $x,$ we can use the dominated convergence theorem and
(4.4), to find the limit in (4.3):

$\underset{s\rightarrow
0^{+}}{lim}\left[\int_{\mathbb{G}}\frac{e^{Kt^{n}Q_{t+s}f}-e^{Kt^{n}Q_{t}f}}{s}d\mu\right]=-Kt^{n}\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q}d\mu.$

In other words, equation (4.3) becomes:

$\underset{s\rightarrow
0^{+}}{lim}\frac{1}{Kt^{n}s}\left(log\int_{\mathbb{G}}e^{K(t+s)^{n}Q_{t+s}f}d\mu-
log\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu\right)$
$=\frac{1}{Kt^{n+1}}\frac{1}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}\left[\int_{\mathbb{G}}nKt^{n}Q_{t}fe^{Kt^{n}Q_{t}f}d\mu-\int_{\mathbb{G}}Kt^{n+1}\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q}e^{Kt^{n}Q_{t}f}d\mu\right].$

Replacing the last equation in (4.2), we get:

$\underset{s\rightarrow
0^{+}}{lim}\frac{\phi(t+s)-\phi(t)}{s}=\frac{1}{Kt^{n+1}}\frac{1}{\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu}$
$\times[-nlog\left(\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu\right)\int_{\mathbb{G}}e^{Kt^{n}Q_{t}f}d\mu+\int_{\mathbb{G}}nKt^{n}Q_{t}fe^{Kt^{n}Q_{t}f}d\mu$
$-\int_{\mathbb{G}}Kt^{n+1}\frac{\left|\triangledown_{\mathbb{}}Q_{t}f\right|^{q}}{q}e^{Kt^{n}Q_{t}f}d\mu].$

Let ${\displaystyle g=e^{\frac{1}{q}Kt^{n}Q_{t}f}},$ and ${\displaystyle
n=\frac{1}{q-1},}$ we obtain:

$\underset{s\rightarrow
0^{+}}{lim}\frac{\phi(t+s)-\phi(t)}{s}=\frac{1}{Kt^{\frac{q}{q-1}}}\frac{1}{\int_{\mathbb{G}}g^{q}d\mu}$

$\times[-\frac{1}{q-1}log\left(\int_{\mathbb{G}}g^{q}d\mu\right)\int_{\mathbb{G}}g^{q}d\mu+\frac{1}{q-1}\int_{\mathbb{G}}g^{q}log(g^{q})d\mu$
(4.5) $-\int_{\mathbb{G}}Kt^{\frac{q}{q-1}}\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q}g^{q}d\mu].$

Using the q-Logarithmic Sobolev inquality

$Ent_{\mu}(g^{q})=\int_{\mathbb{G}}g^{q}log(g^{q})d\mu-
log\left(\int_{\mathbb{G}}g^{q}d\mu\right)\int_{\mathbb{G}}g^{q}d\mu$
$\leq(q-1)\left(\frac{q}{K}\right)^{q-1}\int_{\mathbb{G}}|\triangledown
g|^{q}d\mu$

in (4.5):

$\underset{s\rightarrow
0^{+}}{lim}\frac{\phi(t+s)-\phi(t)}{s}\leq\frac{1}{Kt^{\frac{q}{q-1}}}\frac{1}{\int_{\mathbb{G}}g^{q}d\mu}$

$\times\left[\left(\frac{q}{K}\right)^{q-1}\int_{\mathbb{G}}|\triangledown
g|^{q}d\mu-\int_{\mathbb{G}}Kt^{\frac{q}{q-1}}\frac{\left|\triangledown_{\mathbb{}}Q_{t}f\right|^{q}}{q}g^{q}d\mu\right]$
$=\frac{1}{Kt^{\frac{q}{q-1}}}\frac{1}{\int_{\mathbb{G}}g^{q}d\mu}\left[\left(\frac{q}{K}\right)^{q-1}\int_{\mathbb{G}}|\triangledown{\displaystyle(e^{\frac{1}{q}Kt^{n}Q_{t}f})}|^{q}d\mu-\int_{\mathbb{G}}Kt^{\frac{q}{q-1}}\frac{\left|\triangledown
Q_{t}f\right|^{q}}{q}g^{q}d\mu\right]=0.$

Hence, $\phi(t)$ is non-increasing, and we obtain the dual Talagrand
inequality . ∎

##  5\.  Hypercontractivty of Hamilton-Jacobi solutions for ${\displaystyle
d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$

Denote $||.||_{p},$ $p\in\mathbb{R},$ the $L^{p}-$norms with respect to $\mu,$
and ${\displaystyle||f||_{0}=e^{\int log|f|d\mu}}$ whenever $log|f|$ is
$\mu-$integrable. Also, consider the Hamilton-Jacobi problem on a Carnot group
$\mathbb{G}$ as in Theorem 5. The following Hypercontractivity theorem is
inspired by Theorem 2.1 by Bobkov et al. in [5], but with the measure
${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$ and the Euclidean gradient
replaced by the sub-gradient $\triangledown.$

######  Theorem 15.

Assume we have the following 2-Logarithmic Sobolev inequality with the measure
${\displaystyle d\mu=\frac{e^{-U(d)}}{Z}d\lambda}$, and in the setting of the
Carnot group:

$\rho Ent_{\mu}(f^{2})=\rho\int_{\mathbb{G}}f^{2}log(f^{2})d\mu-\rho
log\left(\int_{\mathbb{G}}f^{2}d\mu\right)\int_{\mathbb{G}}f^{2}d\mu\leq
2\int_{\mathbb{G}}|\triangledown f|^{2}d\mu.$

Then, for every bounded measurable function $f$ on $\mathbb{G},$ every $t\geq
0,$ and every $a\in\mathbb{R},$

$||e^{Q_{t}f}||_{a+\rho t}\leq||e^{f}||_{a}.$

The following proof adapts the proof of Theorem 2.1 done by Bobkov et al. in
[5] in the Euclidean setting, but uses the Hamilton-Jacobi problem in the
Carnot group setting (Theorem 5).

###### Proof.

Let ${\displaystyle F(t)=||e^{Q_{t}f}||_{\lambda(t)},}$ with
$\lambda(t)=a+\rho t,$ $t>0.$

For all $t>0$ and almost every $x,$ the partial derivatives
${\displaystyle\frac{\partial}{\partial_{t}}Q_{t}f(x)}$ exist, and by the
Hamilton-Jacobi problem (Theorem 5),

(5.1)
${\displaystyle\frac{\partial}{\partial_{t}}Q_{t}f(x)}=-\frac{1}{2}|\triangledown
Q_{t}f(x)|^{2}.$

So, $F$ is differentiable at every point $t>0$ where $\lambda(t)\neq 0.$

Computing,

(5.2) $\left(F(t)^{\lambda(t)}\right)^{\prime}=\rho
F(t)^{\lambda(t)}log(F(t))+\lambda(t)F(t)^{\lambda(t)-1}F^{\prime}(t).$

Also,

$\left(F(t)^{\lambda(t)}\right)^{\prime}=\frac{\partial}{\partial_{t}}\left(\int_{\mathbb{G}}e^{\lambda(t)Q_{t}f}d\mu\right)$
$=\int_{\mathbb{G}}\left(\rho
Q_{t}f+\lambda(t)\frac{\partial}{\partial_{t}}Q_{t}f\right)e^{\lambda(t)Q_{t}f}d\mu$

using (5.1),

$=\frac{\rho}{\lambda(t)}Ent_{\mu}(e^{\lambda(t)Q_{t}f})+\frac{\rho}{\lambda(t)}log\left(\int_{\mathbb{G}}e^{\lambda(t)Q_{t}f}d\mu\right)\int_{\mathbb{G}}e^{\lambda(t)Q_{t}f}d\mu-\frac{\lambda(t)}{2}\int_{\mathbb{G}}|\triangledown
Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu$
$=\frac{\rho}{\lambda(t)}Ent_{\mu}(e^{\lambda(t)Q_{t}f})+\rho
F(t)^{\lambda(t)}log\left(F(t)\right)-\frac{\lambda(t)}{2}\int_{\mathbb{G}}|\triangledown
Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu.$

Combining this with (5.2), we get

$\lambda(t)^{2}F(t)^{\lambda(t)-1}F^{\prime}(t)=\rho
Ent_{\mu}(e^{\frac{1}{2}\lambda(t)Q_{t}f})^{2}-\frac{\lambda(t)^{2}}{2}\int_{\mathbb{G}}|\triangledown
Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu$

using the 2-Logarithmic Sobolev inequality,

$\leq\int_{\mathbb{G}}|\triangledown_{\mathbb{G}}e^{\frac{\lambda(t)Q_{t}f}{2}}|^{2}d\mu-\frac{\lambda(t)^{2}}{2}\int_{\mathbb{G}}|\triangledown
Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu$
$=\frac{\lambda(t)^{2}}{4}\int_{\mathbb{G}}|\triangledown_{\mathbb{G}}Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu-\frac{\lambda(t)^{2}}{2}\int_{\mathbb{G}}|\triangledown
Q_{t}f(x)|^{2}e^{\lambda(t)Q_{t}f}d\mu$ $\leq 0,$

for all $t>0.$ Thus, $F^{\prime}(t)\leq 0$ for all $t>0.$ Since $F$ is
continuous, then it is non-increasing. Hence, $F(t)\leq F(0),$ for all $t>0.$

$||e^{Q_{t}f}||_{\lambda(t)}\leq||e^{Q_{0}f}||_{\lambda(0)}.$

So,

$||e^{Q_{t}f}||_{a+\rho t}\leq||e^{f}||_{a}.$

∎

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#  Riiid! Answer Correctness Prediction Kaggle Challenge:
4th Place Solution Summary

Duc Kinh Le Tran
Axance Technology, France
<EMAIL_ADDRESS>
###### Abstract

This paper presents my solution to the challenge ”Riiid! Answer Correctness
Prediction” on Kaggle hosted by Riiid Labs111https://www.kaggle.com/c/riiid-
test-answer-prediction/, which scores 0.817 (AUC) and ranks 4th on the final
private leaderboard222Code to reproduce the solution is available at
https://github.com/dkletran/riiid-challenge-4th-place. It is a single
transformer-based model heavily inspired from previous works such as SAKT,
SAINT and SAINT+. Novel ingredients that I believed to have made a difference
are the time-aware attention mechanism, the concatenation of the embeddings of
the input sequences and the embedding of continuous features.

## Introduction

Knowledge Tracing has been a very interesting problem from both the
application and the academic point of view. The objective of Knowledge Tracing
is to keep track of a student’s _knowledge state_ on an online learning
platform (?). The problem of Knowledge Tracing can be formulated like this:
giving the history of learning activities of a student, predict if he/she
answers correctly the question he/she is facing. This problem has been tackled
by many methods from collaborative filtering to deep learning with recurrent
networks. Recently, transformers (?), originally designed for language
understanding, have been proven to outperform other aproaches. Some of these
transformer-based models are SAKT (?), SAINT (?) and SAINT+ (?)).

In this competition (?), with data the host made available (?), we have for
each student the list of his/her questions in the past, along with his/her
answers, the relative timestamp, time spent for each question and whether
he/she viewed the explanation after answering the question. We also have a
list of lectures watched by users as they progress in their education. The
metadata of questions and lectures can also be helpful in predicting the
correctness of students’ answers.

Like many other competitors I decided to use transformers for this challenge
believing there is still room for improvement in existing transformer-based
models. The other reason for using transformers in this competition is that
the input data are much richer that those used in previous works. Effective
integration of all potentially useful data on the inputs of the transformer’s
encoders and decoders could be the key to boost the performance.

## Data Preparation

To feed the transformer, raw train data in the form one activity per row, need
to be preprocessed. The first steps is to encode categorical features into
integer indices for embedding layers of the transformer. This step is also
applied to the metadata tables (question metadata and lecture metadata). On
the train table, I also added _time lag_ \- time interval from the last
question bundle - which has been proven useful in the SAINT+ (?) (not exactly
the same version of _time lag_ here but still very similar). Since all
question of the same bundle (same container) share a timestamp, they also
share one _time lag_.

Two statistics on the questions - _question popularity_ and _question
difficulty_ \- are also computed from the whole train table. Adding these two
features was proven (experimentally) to be helpful, at least in terms of
convergence speed of the training process.

The train data table is then grouped by _user id_ into one row per student so
that we have sequences representing the history of activities for each
student. These sequences will be the inputs of the transformers’ embedding
layers.

## Model Architecture

Figure 1: Model architecture

.

As shown in Figure 1, the model is designed to take into account all available
input data that have been prepared from the previous step. Below is a
walkthrough of the model architecture with details on its key features.

### Embedding Layers

Each input sequence goes through an embedding layer with the same _embedding
size_.

Embeddings of the categorical features (_answer correctly_ , _user answer_ ,
_question had explanation_) are normal embedding which is basically a lookup
on the weight matrix by the indices in of the input sequences.

Embedding of the content (questions and lectures) is computed using the
_ContentEmbeddingLayer_. The layer aims to encode all metadata of questions
and lectures into the output embedding. Each element of the metadata of
questions and lectures (for example tags, part, type) has a weight matrix and
the embedding weight matrix of the layer is the concatenation of theses weight
matrices. Two computed statistics of questions (_question popularity_ and
_question difficulty_) also contribute to the layer weight matrix. The output
embedding is the result of the lookup on this concatenated matrix by the
content index.

Embeddings of continuous features (_time lag, question elapsed time, question
difficulty, question popularity_) are computed using a _ContinuousEmbedding_
layer. The computation of this layer is illustrated Figure 2. The main idea is
to sum up (with weighted windowed sum) a number of consecutive embedding
vectors of the embedding weight matrix. The _window size_ is a hyper-parameter
of the layer. The output of the layer is a ”smooth” embedding vector of the
continuous feature, in the sense that 2 values very close together should have
similar embeddings. In the experimentation, I observed that this method of
embedding gave the model a little edge in performance in comparison with
traditional _categorical embedding_ (where continuous features are descretized
then gone through normal a embedding layer).

### Encoder and Decoder Input

Embeddings of all input sequences are concatenated and linear-transformed to
feed the first layer of the encoder and that of the decoder of the
transformer. As shown in the schema, input of the first encoder layer contains
embeddings of all input elements and input of the decoder doesn’t not contain
user answer related elements (only _content embeddings_ and _time lag
embeddings_). This is an intuitive choice justified by the following:

  * •

The encoder input value of a position represents the knowledge state of the
student up to that position. We need all information (content, time lag,
his/her answer, answer correctness, time spent on the question and whether
he/she watched the explanation after answer the question).

  * •

The decoder input (query) at a position represents the exercise (question) and
available context information at that position. We only have information about
the content itself and _time lag_ from the previous exercises.

  * •

To predict the user answer (its correctness) at a position we need to know
his/her ”knowledge” and the current question and context. That is why the both
the encoder output and the decoder query are combined in the decoder layer to
form the output of the model.

  * •

Attention and self-attention of the encoder and decoder layers help to look
back and take into account not not only the most recent events but the whole
history of students’ activities.

Figure 2: Embeddings of continuous features

.

### Attention And Self-Attention Mechanism

Encoder and decoder layers are almost identical to those in the transformer
original paper (?). The differences reside in the multi-head attention (and
self-attention) part. Fisrt of all, to prevent the current position from
seeing future inputs, we must apply causal masks to the self-attention of
encoder and decoder layers. This causal mask is computed from the timestamp
sequence: a position at timestamp $t_{0}$ only attends to positions with
timestamp $t\geq t_{0}$ (which means it can attend to itself and other
interactions of the same container). Attention mask of the encoder output to
the decoder is a little different because the decoder at a position cannot see
user answer related inputs at that position. This means that the encoder
output at a timestamp $t_{0}$ can only attend to decoder positions that are
strictly in the future $t>t_{0}$ (i.e not including the current timestamp).

The other feature that gives the model some boost in performance is the _Time-
Weighted Multihead Attention_ , an improvement of the original _Multihead
Attention_ based on the observation that events in a distant past should
attend less to the current position than recent events do. The idea is to
decay the attention coefficients by a factor of $T_{ij}^{-w}$ where
$T_{ij}=t_{j}-t_{i}$ is the difference in timestamp of a position $i$ and the
position $j$ it attends to and $w$ is a non-negative trainable parameter. This
is implemented by modifying the ”Scaled Dot-Product Attention” of the encoders
and decoders. In the original paper this ”Scaled Dot-Product Attention” is:

$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}({QK^{T}\over\sqrt{d_{k}}})V$ (1)

Since the dot product $QK^{T}\over\sqrt{d_{k}}$ is in logarithmic scale with
respect to the attention coefficients (the softmax of the dot prodcut),
decaying the attention coefficients by a factor of $T_{ij}^{-w}$ is equivalent
to substracting $w\log{T_{ij}}$ from the dot product. In other words, the
”Scaled Dot-Product Attention” can be rewritten as the following to take into
account the time decay factor:

$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}({QK^{T}\over\sqrt{d_{k}}}-w\log
T)V$ (2)

where $T$ is the timestamp difference matrix ($T_{ij}=t_{j}-t_{i}$ with
$t_{i},t_{j}$ is the timestamp at the position $i$ and $j$ respectively). In
practice, $\log T$ is computed once and then used as an input to all the
encoder and decoder layers.

As stated earlier, $w$ is a trainable parameter constrained to be non-
negative. We have one parameter per attention head. When tuning the model, I
found it works best when $w$ is randomly initialized in the interval (0,1).

## Training

The output of the transformer goes through a dense layer with _sigmoid_
activation after which we get the predicted probability of the correctness of
the user answer to the question at each position. _Binary Cross-Entropy Loss_
is used to train the model parameters. Positions where the content is a
lecture are masked out to be excluded from the loss function.

Input sequences are cut and padded to have the same length. To put it more
concretely, short sequences are padded to have the required length and long
sequences are randomly cut into smaller parts, each of which is kept and
padded if necessary to the required length. In training, the sequence length
is set to 1024.

The model was implemented in Tensorflow and trained on Colab Pro TPU. To
monitor the model performance locally and also for hyper-parameter tuning, I
used the validation strategy by ? (?). This is basically a train-validation
split which leaves 2.5% of the train set (around 2.5M iteractions) for
validation. It is a very reliable validation strategy thanks to the good
quality of the data. Throughout the competition, all improvements measured on
the validation set lead to improvement on the leader board.

The final version of the model has embedding size of 128, model size of 512, 4
encoder layers and 4 decoder layers. Adam optimizer is used with
$\beta_{1}=0.9,\beta_{2}=0.98,\epsilon=1e-9$. Batch size is set to 64 and
dropout rate is 0.0. Learning rate, initially set to $2.5e-4$ is scheduled to
have a warm up stage of 4000 steps followed by cosine decay stage of around
30000 steps. Training took around 4 hours. The AUC on the validation set is
around 0.818 and the accuracy is 75.35% at the end of training.

The model used for my final submission (scored 0.817 AUC) was trained on the
whole train dataset with the same configuration as above.

## Inference

Due to resource limitation (9 hours of execution with GPU to make prediction
for 2.5M rows), in the inference kernel, the sequence length is reduced to
512. Prediction in inference kernel, constrained by the time-series API, is
not entirely in the same condition as in training and validation : we only
predict the correctness of students’ answers at the latest timestamp (the last
batch of questions). For each student, history in a window of 512 latest
activities are kept in the memory (in form of numpy arrays). In each
iteraction, these activity histories are updated and fed into the model to
predict the correctness of the student’s answer to the last question (or a few
questions in the last batch). Inference on the whole test set took around 8.5
hours, estimated from the moment the kernel get submitted and the moment its
score is available on the leaderboard.

## Observation and Discussion

The most challenging part of training the transformer is to avoid overfitting,
which comes after 1-2 epochs. Various methods like dropout, label smoothing,
partial label masking have been tried out but all led to no obvious
improvement or worse results.

It turns out that reducing the embedding size and concatenating the embeddings
of different input elements work much better than having the embedding size
equal to the model dimension and summing the embeddings. Reducing the
embedding size obviously has some kind of regularization effect. Also, in the
final version of the model, I added a _layer normalization_ layer (?) after
the content embedding layer. It helps regularize the layer weights and give a
small improvement in performance.

Two other features proposed in the model (_Time-Weighted Atttention_ and
embedding of continuous features also contribute to improve AUC. Without the
_Time-Weighted Atttention_ the model scores 0.816 AUC on the validation set
and without the proposed embedding layer for continuous features (using
categorical embedding instead), the model’s AUC is 0.817 on the validation
set. As mentioned earlier, the model’s AUC on the validation set is around
0.818 when both of the proposed features are activated. This is not a big
improvement but still significant in the context of this competition.

Longer sequence length also helps. Due to resource limitation I ended up
training the model with the size of 1024 and reduce it to 512 in the
submission kernel. What is interesting is that training with sequence length
1024 then doing inference with 512 works better than training with 512 and
then doing inference with 512 (same length) (the gap is around 0.002 AUC).

My choice of model size (model dimension 512 and and number of layers 4) was
influenced by previous work (SAINT, SAINT+), but it turns out this is a good
choice in terms of compromising the performance and computational resources.
With this configuration, the model has just enough time (within 9h limitation)
to do inference on the entire test set (with the sequence length of 512) in
the submission kernel so there’s no need to increase it further.

## References

  * [Ba, Kiros, and Hinton 2016] Ba, J. L.; Kiros, J. R.; and Hinton, G. E.  2016\.  Layer normalization.
  * [Choi et al. 2020a] Choi, Y.; Lee, Y.; Cho, J.; Baek, J.; Kim, B.; Cha, Y.; Shin, D.; Bae, C.; and Heo, J.  2020a.  Towards an appropriate query, key, and value computation for knowledge tracing.
  * [Choi et al. 2020b] Choi, Y.; Lee, Y.; Shin, D.; Cho, J.; Park, S.; Lee, S.; Baek, J.; Bae, C.; Kim, B.; and Heo, J.  2020b.  Ednet: A large-scale hierarchical dataset in education.  In International Conference on Artificial Intelligence in Education, 69–73.  Springer.
  * [Kaggle 2020] Kaggle, R. L.  2020\.  Riiid! answer correctness prediction.  https://www.kaggle.com/c/riiid-test-answer-prediction/.  Accessed: 2021-01-01.
  * [Pandey and Karypis 2019] Pandey, S., and Karypis, G.  2019\.  A self-attentive model for knowledge tracing.
  * [Shin et al. 2020] Shin, D.; Shim, Y.; Yu, H.; Lee, S.; Kim, B.; and Choi, Y.  2020\.  Saint+: Integrating temporal features for ednet correctness prediction.
  * [Tito 2020] Tito, I.  2020\.  Cv strategy.  https://www.kaggle.com/its7171/cv-strategy/.  Accessed: 2021-01-01.
  * [Vaswani et al. 2017] Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A. N.; Kaiser, L.; and Polosukhin, I.  2017\.  Attention is all you need.


# Compton scattering in TFD formalism

D. S. Cabral<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this paper, the cross section for the Compton scattering process at finite
temperature is calculated. Temperature effects are introduced using the
Thermofield Dynamics (TFD) formalism. It is a real-time finite temperature
quantum field theory. Our result shows that thermal effects become relevant as
the temperature increases. A comparison between the TFD and closed-time path
results is presented.

##  I Introduction

Compton scattering is a very important process with regard to Quantum
Electrodynamics (QED), mainly because it brings with it a historical fact: one
of the particular phenomena that brought about the birth of quantum mechanics
[1]. However, despite being a topic discussed in all studies of basic physics,
its formulation is made, in almost all cases, from the classical fundamentals
to finite temperature. To make a more accurate and complete description of
this scattering process, it is necessary to use some QED concepts, starting
from the theory of the Standard Model [2, 3, 4]. Furthermore, it is inherent
to the environment that temperature and, consequently, thermal effects on
physical systems are fundamental and natural characteristics presents in any
real physical process. Thus, it is necessary and interesting to analyze the
Compton scattering subject to finite temperature, looking at the consequences
and for the asymptotic cases.

To discuss quantum field theory at finite temperature, in particular QED
theory, there are some formalisms that introduce these thermal effects into
the usual zero-temperature theories. These formalisms are based on ideas that
use real-time or imaginary-time transformations as a basis [5, 6, 7]. For
example, Matsubara approach is an imaginary-time formalism, which is based on
replacing the time, $\displaystyle t$, with a complex time, $\displaystyle
i\tau$ [8, 9]. In this formalism, the propagators have a non-trivial
dependence on temperature. For the real-time formalism there are two
approaches: the closed-time path formalism [10] and Thermofield Dynamics (TFD)
formalism [11, 12, 13, 14, 15]. The closed-time path formalism acts on the
complex-time plane in such a way that the contour goes along the real axis and
then returns. The TFD formalism, on the other hand, consists of two
ingredients: the doubling of the Fock space and the Bogoliubov transformation.
Here the TFD formalism is chosen, as its procedures and calculations are
analogous to those performed in zero-temperature theories.

The difficulty encountered in these formalisms is the fact that the relevant
thermal effects, to be observed experimentally, are expected to arise when the
temperatures involved are of an astronomical order, as in the center of stars,
for example. Although there are several experimental results involving Compton
scattering [16, 17, 18, 19], these measurements are obtained in laboratories
in which the experiments are carried out as close as possible to the ideal
situation of zero temperature. Based on this fact, the present work describes
the Compton scattering at finite temperature, using the QED formalism and
introducing the thermal effects via TFD formalism. Although the temperature
effects for Compton scattering have already been calculated using the closed-
time path formalism [20], in this paper such a study is developed following a
different approach. It is important to note that TFD and closed-time path are
real-time approaches with different procedures and tools. There are
similarities between these formalisms, however, these analogies should not go
too far. The structure of the propagator in the closed-time path formalism and
in the TFD formalism is not the same. In TFD the fields of the duplicated
Hilbert space commute or anti-commute, while in the closed-time formalism the
operators on the path above the real axis do not commute or anti-commute with
those on the path below the real axis [14, 21, 22]. Here the result for a
scattering process is considered and it is compared in both approaches. The
differential cross section at high temperatures, at least for the tree-level,
exhibits different behavior.

This paper is organized as follows. In section II, a brief introduction to the
TFD formalism is presented. In section III, Compton scattering process is
introduced. Feynman diagrams characterizing this scattering are shown. The
probability amplitude at finite temperature is obtained. In section IV, the
differential cross section for this problem is calculated. The usual result at
zero temperature is recovered at the proper limit. Our result is compared with
solutions obtained from the closed-time path approach. Finally, in the Section
V, some final considerations are discussed.

##  II Thermofield Dynamics

TFD is a formalism that introduces temperature into quantum fields from ideas
of symmetries and Lie groups, based on concepts of statistical mechanics and
thermodynamics [12, 13, 14]. It is done by doubling the Hilbert space so that
from this doubling, one can write a thermal Hilbert space
$\displaystyle\mathbb{H}_{T}=\mathbb{H}\otimes\widetilde{\mathbb{H}}$, where
$\displaystyle\mathbb{H}$ deals with the usual Hilbert space, while
$\displaystyle\widetilde{\mathbb{H}}$ is obtained from the former by the tilde
conjugation rules. For a dynamical operator $\displaystyle A$, these rules are
defined as

$\begin{split}(A_{i}A_{j})^{\widetilde{}}&=\widetilde{A}_{i}\widetilde{A}_{j};\\\
(cA_{i}+A_{j})^{\widetilde{}}&=c^{*}\widetilde{A}_{i}+\widetilde{A}_{j};\\\
(A_{i}^{\dagger})^{\widetilde{}}&=(\widetilde{A}_{i})^{\dagger};\\\
(\widetilde{A}_{i})^{\widetilde{}}&=\pm A_{i};\\\
[A_{i},\tilde{A}_{j}]&=0.\end{split}$ (1)

Here the positive (negative) sign represents the bosonic (fermionic)
quantization. Note that $\displaystyle A$ acts on $\displaystyle\mathbb{H}$
while $\displaystyle\widetilde{A}$ acts on
$\displaystyle\widetilde{\mathbb{H}}$.

In this approach, the expected value of a dynamical operator $\displaystyle A$
is given by $\displaystyle\langle A\rangle=\bra{0(\beta)}A\ket{0(\beta)}$,
where $\displaystyle\ket{0(\beta)}$ is the thermal vacuum state defined as

$\ket{0(\beta)}=\frac{1}{\sqrt{Z(\beta)}}\sum_{n}e^{-\frac{1}{2}\beta
E_{n}}\ket{n,\widetilde{n}},$ (2)

and $\displaystyle Z(\beta)$ the canonical partition function, with
$\displaystyle\beta=1/k_{B}T$, being $\displaystyle k_{B}$ the Boltzmann
constant and $\displaystyle T$ the temperature. The state
$\displaystyle\ket{n,\widetilde{n}}$ is in the thermal space
$\displaystyle\mathbb{H}_{T}$, such that a generic state in it can be written
as

$\ket{m,\widetilde{n}}=\frac{1}{\sqrt{m!}\sqrt{n!}}(a^{\dagger})^{m}(\widetilde{a}^{\dagger})^{n}\ket{0,\tilde{0}}$
(3)

with $\displaystyle\ket{0,\tilde{0}}=\ket{0}\otimes\ket{\tilde{0}}$, where
this first is the vacuum state of $\displaystyle\mathbb{H}$, while the second
one has a similar role in $\displaystyle\widetilde{\mathbb{H}}$. In this
approach, the operators $\displaystyle a^{\dagger}$ and
$\displaystyle\widetilde{a}^{\dagger}$, along with $\displaystyle a$ and
$\displaystyle\widetilde{a}$ have the role of creation and annihilation
operators in their respective spaces.

Furthermore, this formalism also brings the Bogoliubov transformations. These
are unitary and act by connecting thermal and non-thermal operators through a
rotation in the double space. This implies that the thermal creation and
annihilation operators act on $\displaystyle\mathbb{H}_{T}$ as

$\ket{m(\beta),\tilde{n}(\beta)}=\frac{1}{\sqrt{m!}}\frac{1}{\sqrt{n!}}(a^{\dagger}_{\beta})^{m}(\tilde{a}^{\dagger}_{\beta})^{n}\ket{0(\beta)},$
(4)

where the $\displaystyle\beta$ index represents the temperature dependence of
the operators.

For fermions, whose creation and annihilation operators are $\displaystyle
a^{\dagger}(p)$ and $\displaystyle a(p)$, respectively, there are the
following transformations,

$a^{\dagger}(p)=U(\beta)a_{\beta}^{\dagger}(p)+V(\beta)\tilde{a}_{\beta}(p);\quad\quad
a(p)=U(\beta)a_{\beta}(p)+V(\beta)\tilde{a}_{\beta}^{\dagger}(p),$ (5)

where

$U=\frac{1}{\sqrt{1+e^{-\beta E}}};\quad\quad V=\frac{1}{\sqrt{1+e^{\beta
E}}}.$ (6)

The bosons follow similar thermal transformations, i.e.

$d^{\dagger}(p)=U^{\prime}(\beta)d_{\beta}^{\dagger}(k)+V^{\prime}(\beta)\tilde{d}_{\beta}(k);\quad\quad
d(p)=U^{\prime}(\beta)d_{\beta}(k)+V^{\prime}(\beta)\tilde{d}_{\beta}^{\dagger}(k),$
(7)

in such a way that

$U^{\prime}=\frac{1}{\sqrt{1-e^{-\beta E}}};\quad\quad
V^{\prime}=\frac{e^{-\beta E/2}}{\sqrt{1-e^{-\beta E}}}.$ (8)

With these definitions, the thermal operators of creation and annihilation of
bosons (fermions) obey the following commutation (anti-commutation) relations
in equal time

$[d_{\beta}(k,\lambda),d_{\beta}^{\dagger}(k^{\prime},\lambda^{\prime})]=\frac{1}{N_{k}}\delta_{\lambda,\lambda^{\prime}}\delta^{3}(\vec{k}-\vec{k}^{\prime}),\quad\quad\\{a_{\beta}(p,s),a_{\beta}^{\dagger}(p^{\prime},s^{\prime})\\}=\frac{1}{N_{p}}\delta_{s,s^{\prime}}\delta^{3}(\vec{p}-\vec{p}^{\prime}),$
(9)

with $\displaystyle N_{p}$ and $\displaystyle N_{k}$ being normalization
factors.

The formalism presented here will be used in the next sections to calculate
the differential cross section for Compton scattering at finite temperature.

##  III Compton scattering

In this section, the main objective is to calculate the probability amplitude
at finite temperature for Compton scattering. The Lagrangian that describes
this process is given as

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi,$
(10)

where $\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
is the electromagnetic tensor, $\displaystyle\psi$ is the fermion field,
$\displaystyle\gamma^{\mu}$ are the Dirac matrices, $\displaystyle m$ is the
fermion mass and $\displaystyle D_{\mu}=\partial_{\mu}-ieA_{\mu}$ with
$\displaystyle e$ being the electron charge and $\displaystyle A_{\mu}$ the
photon field. For the interest proposed here, the main part is the interaction
Lagrangian written as

$\mathcal{L}_{\text{int}}=e\bar{\psi}\gamma^{\mu}\psi A_{\mu}.$ (11)

As a consequence, the interaction Hamiltonian is

$\mathcal{H}_{\text{int}}=-\mathcal{L}_{\text{int}}=-e\bar{\psi}\gamma^{\mu}\psi
A_{\mu}.$ (12)

To deal with a theory at finite temperature, the workspace is doubled, i.e.
generating the tilde space. And from this, one can write the operators that
generate the symmetry, as the $\displaystyle\hat{\mathcal{H}}$ which is
related to the time evolution. This operator is defined as

$\hat{\mathcal{H}}=\mathcal{H}-\tilde{\mathcal{H}},$ (13)

where $\displaystyle\mathcal{H}$ and $\displaystyle\tilde{\mathcal{H}}$ are
operators in standard Hilbert space and tilde space, respectively.

Compton scattering is the reaction between an electron and a photon through a
virtual electron, according to the Feynman diagrams shown in Figure 1.

Figure 1: Feynman diagrams of the Compton scattering. Adapted from Ref. [23].

For this process, in the TFD formalism, the initial and final states are

$\ket{i}=\ket{e_{i},\gamma_{i}}=a^{\dagger}_{\beta}(p)d^{\dagger}_{\beta}(k)\ket{0(\beta)};\quad\quad\ket{f}=\ket{e_{f},\gamma_{f}}=a^{\dagger}_{\beta}(p^{\prime})d^{\dagger}_{\beta}(k^{\prime})\ket{0(\beta)},$
(14)

where $\displaystyle e_{i}$ ($\displaystyle e_{f}$) and
$\displaystyle\gamma_{i}$ ($\displaystyle\gamma_{f}$) represent the initial
(final) states of the electron and photon, respectively.

In quantum electrodynamics, the second order term of the scattering matrix is
given by

$\hat{S}^{(2)}=\frac{(-i)^{2}}{2}\int
d^{4}xd^{4}y:\hat{\mathcal{H}}_{\text{int}}(x)\hat{\mathcal{H}}_{\text{int}}(y):\;\;=\frac{(-i)^{2}}{2}\int
d^{4}xd^{4}y:\mathcal{H}_{\text{int}}(x)\mathcal{H}_{\text{int}}(y):,$ (15)

with $\displaystyle:(\cdots):$ being the normal ordering operator, which
introduces the correct order for the operators to be applied, in order to
avoid divergences. In addition, to the last term in (15), only the non-tilde
operators are considered, since these are actually the observables of the
theory.

For this scattering process, the probability amplitude is

$\mathcal{M}=\bra{f}\hat{S}^{(2)}\ket{i}=-\frac{e^{2}}{2}\int
d^{4}xd^{4}y\bra{f}:\bar{\psi}(x)\gamma^{\mu}\psi(x)\bar{\psi}(y)\gamma^{\nu}\psi(y)A_{\mu}(x)A_{\nu}(y):\ket{i},$
(16)

so that, after an analysis of the field operators, it can be seen that only
the terms described by the Feynman diagrams will contribute to the scattering.
Then

$\begin{split}\mathcal{M}&=-e^{2}\int
d^{4}xd^{4}y\bra{f}:\bar{\psi}^{-}(x)\gamma^{\mu}[\psi(x)\bar{\psi}(y)]\gamma^{\nu}\psi^{+}(y)A_{\mu}^{-}(x)A_{\nu}^{+}(y):\ket{i}\\\
&-e^{2}\int
d^{4}xd^{4}y\bra{f}:\bar{\psi}^{-}(x)\gamma^{\mu}[\psi(x)\bar{\psi}(y)]\gamma^{\nu}\psi^{+}(y)A_{\mu}^{+}(x)A_{\nu}^{-}(y):\ket{i}\equiv\mathcal{M}_{a}+\mathcal{M}_{b},\end{split}$
(17)

where $\displaystyle\mathcal{M}_{a}$, $\displaystyle\mathcal{M}_{b}$
corresponding to the diagrams $\displaystyle(a)$ and $\displaystyle(b)$ in
Figure 1, respectively, and the positive (negative) index represents the part
of field operators with positive (negative) frequencies, in other words

$\psi^{+}(x)=\int
d^{3}pN_{p}\sum_{s}a(p,s)u(p,s)e^{-ipx};\quad\quad\bar{\psi}^{-}(x)=\int
d^{3}pN_{p}\sum_{s}a^{\dagger}(p,s)\bar{u}(p,s)e^{ipx}$ (18)

for the fermionic field, and

$A^{+}_{\mu}(x)=\int
d^{3}kM_{k}\sum_{\lambda}d(k,\lambda)\epsilon_{\mu}(k,\lambda)e^{-ikx};\quad\quad
A^{-}_{\mu}(x)=\int
d^{3}kM_{k}\sum_{\lambda}d^{\dagger}(k,\lambda)\epsilon^{\dagger}_{\mu}(k,\lambda)e^{ikx},$
(19)

for the photonic field. Note that the four-vector product $\displaystyle
px=\xi t-\vec{p}\cdot\vec{x}$ leads to
$\displaystyle\xi^{2}=\vec{p}^{2}+m^{2}$, where $\displaystyle\xi$ is the
energy and $\displaystyle\vec{p}$ is the 3-momentum.

Therefore, from Eq. (17) suppressing the spin and polarization indices, taking
into account the Bogoliubov transformations Eq. (5) and Eq. (7) for the
operators, the probability amplitude to the first diagram becomes

$\begin{split}\mathcal{M}_{a}&=-e^{2}\int
d^{4}xd^{4}y\int\frac{d^{4}q}{(2\pi)^{4}}(U^{\prime})^{2}U^{2}\left[\bar{u}(p^{\prime})\gamma^{\mu}e^{i(p^{\prime}+k^{\prime}-q)x}S(q)e^{-i(p+k-q)y}\gamma^{\nu}u(p)\epsilon_{\mu}^{\dagger}(k^{\prime})\epsilon_{\nu}(k)\right],\end{split}$
(20)

where $\displaystyle S(q)=S^{(0)}(q)+S^{(\beta)}(q)$ is the electron
propagator in the momentum space, with

$S^{(0)}(q)=\frac{\not{q}+m}{q^{2}-m^{2}},$ (21)

being the zero temperature part, and

$S^{(\beta)}(q)=\frac{\pi i}{(e^{\beta
q^{0}}+1)}\left[\frac{(\gamma^{0}\xi-\gamma^{i}q_{i}+m)}{2\xi}\Delta_{1}\delta(q^{0}-\xi)+\frac{(\gamma^{0}\xi+\gamma^{i}q_{i}-m)}{2\xi}\Delta_{2}\delta(q^{0}+\xi)\right]$
(22)

is the temperature dependent part [24]. The terms $\displaystyle\Delta_{1}$
and $\displaystyle\Delta_{2}$ arise from the doubled space and are thermal
matrices $\displaystyle 2\times 2$ given by

$\Delta_{1}=\begin{pmatrix}1&e^{\beta q_{0}/2}\\\ e^{\beta
q_{0}/2}&-1\end{pmatrix};\quad\quad\Delta_{2}=\begin{pmatrix}-1&e^{\beta
q_{0}/2}\\\ e^{\beta q_{0}/2}&1\end{pmatrix}.$ (23)

From the thermal functions Eqs. (6) and (8), derived from the Bogoliubov
transformations, we get

$(U^{\prime})^{2}U^{2}=\frac{1}{1-e^{-2\beta E}}=\frac{1+\coth\beta E}{2}$
(24)

and using the definition of the Dirac delta function

$\int
d^{4}xd^{4}ye^{i(p^{\prime}+k^{\prime}-q)x}e^{-i(p+k-q)y}=\delta^{4}(p^{\prime}+k^{\prime}-q)\delta^{4}(p+k-q),$
(25)

the probability amplitude for the diagram $\displaystyle(a)$ becomes

$\mathcal{M}_{a}=-e^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\left[\bar{u}(p^{\prime})\gamma^{\mu}\delta^{4}(p^{\prime}+k^{\prime}-q)S(q)\delta^{4}(p+k-q)\gamma^{\nu}u(p)\epsilon_{\mu}^{\dagger}(k^{\prime})\epsilon_{\nu}(k)\right]\left(\frac{1+\coth{\beta
E}}{2}\right),$ (26)

in such a way that, performing the integration over the momentum space of the
electron in the middle, remembering that the Dirac delta function imposes
momentum conservation, we obtain

$\mathcal{M}_{a}=-e^{2}\bar{u}(p^{\prime})\gamma^{\mu}S(p+k)\gamma^{\nu}u(p)\epsilon_{\mu}^{\dagger}(k^{\prime})\epsilon_{\nu}(k)\frac{1+\coth{\beta
E}}{2}.$ (27)

Analogously, for the diagram $\displaystyle(b)$ the probability amplitude is

$\mathcal{M}_{b}=-e^{2}\bar{u}(p^{\prime})\gamma^{\mu}S(p-k^{\prime})\gamma^{\nu}u(p)\epsilon_{\mu}(k)\epsilon_{\nu}^{\dagger}(k^{\prime})\frac{1+\coth{\beta
E}}{2}.$ (28)

In the next section, these results are used to calculate the differential
cross section for the Compton scattering at finite temperature in the TFD
approach.

##  IV Differential Cross section

Here the differential cross section at finite temperature is obtained for
Compton scattering. To achieve this objective, the main quantity to be
calculated is the probability density. Since the scattering amplitude Eq. (17)
is given by $\displaystyle\mathcal{M}=\mathcal{M}_{a}+\mathcal{M}_{b}$, the
probability density is given as

$|\mathcal{M}|^{2}=|\mathcal{M}_{a}|^{2}+|\mathcal{M}_{b}|^{2}+2\Re{\mathcal{M}_{a}^{\dagger}\mathcal{M}_{b}}.$
(29)

Taking into account the unpolarized initial and final beams, it is necessary
to calculate the average over all spins and polarizations. Then

$\langle|\mathcal{M}|^{2}\rangle=\frac{1}{4}\sum_{\begin{subarray}{c}\lambda,\lambda^{\prime}\\\
s,s^{\prime}\end{subarray}}|\mathcal{M}|^{2}=\langle|\mathcal{M}_{a}|^{2}\rangle+\langle|\mathcal{M}_{b}|^{2}\rangle+\left\langle
2\Re{\mathcal{M}_{a}^{\dagger}\mathcal{M}_{b}}\right\rangle.$ (30)

The average of the diagram $\displaystyle(a)$ leads to

$\displaystyle\displaystyle\langle|\mathcal{M}_{a}|^{2}\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{4}\sum_{\begin{subarray}{c}\lambda,\lambda^{\prime}\\\
s,s^{\prime}\end{subarray}}e^{4}\left[\bar{u}(p^{\prime})\gamma^{\mu}S(p+k)\gamma^{\nu}u(p)\epsilon_{\mu}^{\dagger}(k^{\prime})\epsilon_{\nu}(k)\right]$
(31)
$\displaystyle\displaystyle\times\left[\epsilon^{\dagger}_{\omega}(k)\epsilon_{\alpha}(k^{\prime})\bar{u}(p)\gamma^{\omega}S^{\dagger}(p+k)\gamma^{\alpha}u(p^{\prime})\right]\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.$

Using the completeness relations

$\sum_{s}u(p,s)\bar{u}(p,s)={\not{p}+m};\quad\quad\sum_{\lambda}\epsilon_{\mu}(k,\lambda)\epsilon^{\dagger}_{\alpha}(k,\lambda)=g_{\mu\alpha},$
(32)

and the relation

$[\bar{u}(p)\gamma^{\mu}\gamma^{\nu}u(p^{\prime})\bar{u}(p^{\prime})\gamma_{\mu}\gamma_{\nu}u(p)]=\Tr[u(p)\bar{u}(p)\gamma^{\mu}\gamma^{\nu}u(p^{\prime})\bar{u}(p^{\prime})\gamma_{\mu}\gamma_{\nu}],$
(33)

as well as the following property of Dirac matrices

$\gamma^{\mu}\not{p}\gamma_{\mu}=-2\not{p},$ (34)

Eq. (31) reads

$\langle|\mathcal{M}_{a}|^{2}\rangle={e^{4}}\Tr{\not{p}^{\prime}S(p+k)\not{p}S^{\dagger}(p+k)}\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.$ (35)

Splitting the propagator into its zero and finite temperature parts, we have

$\displaystyle\displaystyle\langle|\mathcal{M}_{a}|^{2}\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle{e^{4}}\left\\{\Tr[\not{p}^{\prime}S^{(0)}(p+k)\not{p}S^{(0)}(p+k)]+\Tr[\not{p}^{\prime}S^{(0)}(p+k)\not{p}S^{(\beta)\dagger}(p+k)]\right.$
(36)
$\displaystyle\displaystyle\left.+\Tr[\not{p}^{\prime}S^{(\beta)}(p+k)\not{p}S^{(0)}(p+k)]+\Tr[\not{p}^{\prime}S^{(\beta)}(p+k)\not{p}S^{(\beta)\dagger}(p+k)]\right\\}$
$\displaystyle\displaystyle\times\left(\frac{1+\coth{\beta E}}{2}\right)^{2}.$

Considering the centre of mass frame, in which the four momenta are given by

$\displaystyle\displaystyle p$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle(E,0,0,-\omega);\quad k=(\omega,0,0,\omega);$
$\displaystyle\displaystyle p^{\prime}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle(E,-\omega\sin{\theta},0,-\omega\cos{\theta});\quad
k^{\prime}=(\omega,\omega\sin{\theta},0,\omega\cos{\theta}),$ (37)

with $\displaystyle\theta$ being the scattering angle, and also the high
energy approximation $\displaystyle m\to 0$ (which implies $\displaystyle
E\to\omega$), we get

$\Tr[\not{p}^{\prime}S^{(0)}(p+k)\not{p}S^{(0)}(p+k)]=\frac{8}{(p+k)^{4}}(p^{\prime}\cdot
k)(p\cdot k).$ (38)

The same result has been obtained by [23], for the analysis of the scattering
at zero temperature.

The other terms of Eq. (36) are all null, because for $\displaystyle
p+k=(E+\omega,0,0,0)$, in the propagator at the centre of mass frame subject
to high energies approximation, we have $\displaystyle(p+k)_{i}=0$ and,
consequently, $\displaystyle\xi=0$. Therefore, the average over the first
diagram is

$\begin{split}\langle|\mathcal{M}_{a}|^{2}\rangle&=e^{4}(\cos{\theta}+1)\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.\end{split}$ (39)

In a similar way, the average of the diagram $\displaystyle(b)$, where the
momentum conservation is $\displaystyle
p-k^{\prime}=(E-\omega,\vec{p}+\vec{p}^{\prime})$, is

$\displaystyle\displaystyle\langle|\mathcal{M}_{b}|^{2}\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle{e^{4}}\left\\{\Tr[\not{p}^{\prime}S^{(0)}(p-k^{\prime})\not{p}S^{(0)}(p-k^{\prime})]+\Tr[\not{p}^{\prime}S^{(0)}(p-k^{\prime})\not{p}S^{(\beta)\dagger}(p-k^{\prime})]\right.$
(40)
$\displaystyle\displaystyle\left.+\Tr[\not{p}^{\prime}S^{(\beta)}(p-k^{\prime})\not{p}S^{(0)}(p-k^{\prime})]+\Tr[\not{p}^{\prime}S^{(\beta)}(p-k^{\prime})\not{p}S^{(\beta)\dagger}(p-k^{\prime})]\right\\}$
$\displaystyle\displaystyle\times\left(\frac{1+\coth{\beta E}}{2}\right)^{2},$

where

$\Tr[\not{p}^{\prime}S^{(0)}(p-k^{\prime})\not{p}S^{(0)}(p-k^{\prime})]=\frac{8}{(p-k^{\prime})^{4}}(p^{\prime}\cdot
k^{\prime})(k^{\prime}\cdot p),$ (41)

as in the case of zero temperature [23].

The next terms in Eq. (40) imply that

$\Tr[\not{p}^{\prime}S^{(0)}(p-k^{\prime})\not{p}S^{(\beta)\dagger}(p-k^{\prime})]=-\Tr[\not{p}^{\prime}S^{(\beta)}(p-k^{\prime})\not{p}S^{(0)}(p-k^{\prime})],$
(42)

and, finally

$\displaystyle\displaystyle\Tr[\not{p}^{\prime}S^{(\beta)}(p-k^{\prime})\not{p}S^{(\beta)\dagger}(p-k^{\prime})]$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{8\pi^{2}\omega^{2}}{(e^{\beta
q^{0}}+1)^{2}4\xi_{b}^{2}}\Bigl{\\{}\cos{\left(\frac{\theta}{2}\right)}\bigl{[}(2\omega-\xi_{b})^{2}\delta^{2}(q^{0}-\xi_{b})$
(43)
$\displaystyle\displaystyle+(2\omega+\xi_{b})^{2}\delta^{2}(q^{0}+\xi_{b})\bigl{]}$
$\displaystyle\displaystyle+\left[4\omega^{2}(\cos{\theta}+1)-2\xi_{b}^{2}\cos{\theta}\right]\delta(q^{0}-\xi_{b})\delta(q^{0}+\xi_{b})\Bigl{\\}},$

with $\displaystyle q_{0}=E-\omega$,
$\displaystyle\xi_{b}=2\omega\cos{(\theta/2)}$ and $\displaystyle E\to\omega$
at the high energy limit. In such a way that

$\displaystyle\displaystyle\langle|\mathcal{M}_{b}|^{2}\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Biggl{\\{}\frac{4e^{4}}{\cos{\theta}+1}+\frac{\pi^{2}e^{2}\omega^{2}}{2[e^{\beta(E-\omega)}+1]^{2}}\sec{\left(\frac{\theta}{2}\right)}\left[10\cos{\theta}+\cos{2\theta}+17\right]$
(44)
$\displaystyle\displaystyle\times\delta^{2}\left(2\omega\cos{\frac{\theta}{2}}\right)\Biggl{\\}}\left(\frac{1+\coth{\beta
E}}{2}\right)^{2},$

where only the first element of the resulting thermal matrix was taken, since
only terms of the non-tilde space are related to the observables. For the
mixing term, we get

$\left\langle 2\Re{\mathcal{M}_{a}^{\dagger}\mathcal{M}_{b}}\right\rangle=0.$
(45)

The Mandelstam variables have been used

$s=2p\cdot k=2p^{\prime}\cdot k^{\prime};\quad\quad t=-2p\cdot
p^{\prime}=-2k\cdot k^{\prime};\quad\quad u=-2p\cdot
k^{\prime}=-2p^{\prime}\cdot k;$ (46)

or in terms of the scattering angle of the centre of mass frame

$s=(2\omega)^{2};\quad\quad t=-2\omega^{2}(1-\cos{\theta});\quad\quad
u=-2\omega^{2}(1+\cos{\theta}).$ (47)

With Eqs. (39), (44) and (45), Eq. (30) becomes

$\begin{split}\langle|\mathcal{M}|^{2}\rangle&=2e^{4}\left(\frac{\cos{\theta}+1}{2}+\frac{2}{\cos{\theta}+1}\right)\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}\\\
&+\frac{\pi^{2}e^{2}\omega^{2}\sec{\left(\frac{\theta}{2}\right)}\left[10\cos{\theta}+\cos{2\theta}+17\right]}{2[e^{\beta(E-\omega)}+1]^{2}}\delta^{2}\left(2\omega\cos{\frac{\theta}{2}}\right)\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.\end{split}$ (48)

Thus, using the definition of the differential cross section for any
scattering process, i.e.,

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}s}\langle|\mathcal{M}|^{2}\rangle,$
(49)

for Compton scattering at finite temperature, we get

$\begin{split}\left(\frac{d\sigma}{d\Omega}\right)_{\beta}&=\frac{2e^{4}}{64\pi^{2}(2\omega)^{2}}\left(\frac{\cos{\theta}+1}{2}+\frac{2}{\cos{\theta}+1}\right)\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}\\\
&+\frac{\pi^{2}e^{2}\omega^{2}\sec{\left(\frac{\theta}{2}\right)}\left[10\cos{\theta}+\cos{2\theta}+17\right]}{64\pi^{2}(2\omega)^{2}2[e^{\beta(E-\omega)}+1]^{2}}\delta^{2}\left(2\omega\cos{\frac{\theta}{2}}\right)\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.\end{split}$ (50)

It should be noted that for the low temperature limit $\displaystyle T\to 0$,
it leads to $\displaystyle\beta\to\infty$, such that

$\lim_{\beta\to\infty}\coth{\beta
E}=1;\quad\quad\lim_{\beta\to\infty}\frac{1}{[e^{\beta(E-\omega)}+1]^{2}}=0,$
(51)

therefore

$\lim_{T\to
0}\left(\frac{d\sigma}{d\Omega}\right)_{\beta}=\frac{2e^{4}}{64\pi^{2}(2\omega)^{2}}\left(\frac{\cos{\theta}+1}{2}+\frac{2}{\cos{\theta}+1}\right)=\left(\frac{d\sigma}{d\Omega}\right).$
(52)

This is the differential cross section for Compton scattering at zero
temperature, as obtained by [23].

For the high temperatures limit, $\displaystyle\beta\to 0$, we have
$\displaystyle[e^{\beta(E-\omega)}+1]^{2}\to 1$, but
$\displaystyle\coth{\beta}\to\infty$. Then the temperature contribution
becomes a leading term and the thermal effects are more evident and relevant.
Furthermore, it has no problem dealing with a product of delta functions with
identical arguments. In other words, there are regularized forms to work with
these functions. Which means that their values exists and are finite [7].

In terms of the differential cross section at zero temperature, one can
rewrite Eq. (50) as

$\displaystyle\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(\frac{d\sigma}{d\Omega}\right)\left[1+\frac{\pi^{2}\omega^{2}}{4[e^{\beta(E-\omega)}+1]^{2}}\frac{2\sec{\left(\frac{\theta}{2}\right)}\left[55\cos{\theta}+12\cos{2\theta}+\cos{3\theta}+44\right]}{4\cos{\theta}+\cos{2\theta}+11}\delta^{2}\left(2\omega\cos{\frac{\theta}{2}}\right)\right]$
(53) $\displaystyle\displaystyle\times\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.$

In order to compare our result with the result obtained using the closed-time
path (CTP) formalism [20], for simplicity, will be considered

$\delta^{2}\left(2\omega\cos{\frac{\theta}{2}}\right)=\frac{2}{\pi^{2}\omega^{2}}\frac{4\cos{\theta}+\cos{2\theta}+11}{\sec{\left(\frac{\theta}{2}\right)}\left[55\cos{\theta}+12\cos{2\theta}+\cos{3\theta}+44\right]}.$
(54)

Then, one can write
$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta}=F^{2}_{\text{TFD}}(\beta)\left(\frac{d\sigma}{d\Omega}\right)$,
with

$F^{2}_{\text{TFD}}(\beta)\equiv\left\\{1+\frac{1}{[e^{\beta(E-\omega)}+1]^{2}}\right\\}\left(\frac{1+\coth{\beta
E}}{2}\right)^{2}.$ (55)

A similar result using the closed-time path formalism was obtained by [20],
i.e.
$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta}=F^{2}_{\text{CTP}}(\beta)\left(\frac{d\sigma}{d\Omega}\right)$,
where

$F^{2}_{\text{CTP}}=\left(\frac{1+\coth{\beta E}}{2}\right)^{2}\coth^{2}{\beta
E}.$ (56)

The functions (55) and (56) are represented in Figure 2.

Figure 2: Thermal differential cross section in terms of the cross section at
zero temperature, for the closed-time path formalism (black solid line), TFD
with $\displaystyle E-\omega=0.1E$ (red dashed line) and TFD with
$\displaystyle E-\omega=0.3E$ (blue dotted line). The horizontal axis is
normalized by an fixed energy.

It is important to note that, although both formalisms have similar behavior,
the functions (55) and (56) are different, then at high temperature the
behavior is different as shown in Figure 2. This difference arises because the
procedures and tools used in these formalisms are completely different. The
main difference is due to the form in which the average of an operator is
calculated. In the TFD formalism, the main idea is to interpret the
statistical average of an arbitrary operator as its expected value in a
thermal vacuum $\displaystyle|0(\beta)\rangle$. Two elements are needed to
construct this thermal state: (i) doubling of the original Hilbert space and
(ii) Bogoliubov transformations. In the closed-time path approach, the
statistical average is calculated using the density matrix $\displaystyle\rho$
[20]. Therefore, it is important to emphasize that the result obtained here
uses the Bogoliubov transformation to introduce temperature effects, while in
the closed-time path approach the thermal effects are introduced in a
different way. In addition, as stated in reference [20] in the imaginary-time
formalism such an analysis is not possible, since the Matsubara formalism
cannot give a first principles formulation of thermal scattering because it is
impossible to define the transition probability, and hence the cross-section.
Then, there is no general formulation to define a thermal scattering process,
although TFD and closed-time path present the same functional behavior as
shown in Figure 2. For more details related to the similarities and
differences between these formalisms, see the references [14, 21].

##  V Conclusion

It is known that there are two real-time approaches, i.e. TFD and closed-time
path, to introducing temperature effects into a quantum field theory. However,
solving finite temperature field theory problems using the TFD formalism gives
us calculations very similar to those present in zero temperature theories.
This is the main advantage over using the closed-time path formalism. The
results obtained here, although not the same, are similar. It can be noticed
that, for very high temperatures, i.e. very low values of
$\displaystyle\beta$, the thermal effects will be dominant. In this case, the
differential cross section at finite temperature will be many times greater
than at zero temperature. Quantitatively speaking, at centre of mass energies
on the order of $\displaystyle 1KeV$, thermal effects would be detected at
temperatures on the order of $\displaystyle 10^{7}K$. This corresponds to
situations in the center of stars, such as the Sun, for example.

From Figure 2 it can be seen that the solution obtained by the closed-time
path formalism grows faster when $\displaystyle\beta\to 0$. Therefore, the
thermal effects at high temperatures, in this approach, are more expressive
than those obtained by the TFD formalism. This can be understood from Eqs.
(55) and (56), which shows that $\displaystyle
F^{2}_{\text{CTP}}>F^{2}_{\text{TFD}}$ when very high values of $\displaystyle
T$ are assumed. Still in Figure 2, the TFD formalism for this specific
scattering strongly depends on the difference $\displaystyle E-\omega$, due to
the presence of the exponential in the first term of (55). What comes from the
centre of mass frame in approaching high energies. This fact implies a slower
decay in the asymptotic limit $\displaystyle\beta\to\infty$ of the TFD
function compared to that of closed-time path. Therefore, despite the
different contributions, it can be noted that both approaches have the same
functional behavior.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2.

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