# TFD formalism: applications to the scalar field in a Lorentz-violating
theory

L. H. A. R. Ferreira<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  Faqir C. Khanna 111Professor Emeritus
- Physics Department, Theoretical Physics Institute, University of Alberta
Edmonton, Alberta, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Victoria,
3800 Finnerty Road, Victoria BC V8P 5C2, Canada

###### Abstract

The Thermofield Dynamics (TFD) formalism is considered. In this context, a
Lorentz-breaking scalar field theory is introduced. In contrast to the
Matsubara formalism, the best-known approach to introducing the temperature
effect, TFD is a real-time formalism and is a topological field theory. While
in Matsubara the temperature effects are introduced as a consequence of a
compactification of the field in a finite interval on the time axis, in the
TFD this effect emerges through a condensed state related to the Bogoliubov
transformation. An advantage of the TFD formalism is that different
topologies, which lead to different effects, can be chosen. Here, three
different topologies are considered. Then the Stefan-Boltzmann law and Casimir
effect at zero and non-zero temperature are calculated. This is a unique
feature of TFD, which allows us to treat different phenomena in the same way.

##  I Introduction

Temperature is a natural ingredient in all phenomena described by nature.
However, to study the effects of temperature, a thermal theory is needed. One
of the best-known theories for introducing temperature is the Matsubara
formalism, also known as the imaginary time formalism [1]. This approach
implies a replacement of the coordinate of time by imaginary time. Another
thermal formalism is the real-time formalism which allows to analyze the
temporal evolution of the system and temperature effects together. It is
divided into two approaches: (i) the closed-time path formulation which
consists of the doubling of degrees of freedom that modifies the Green
functions [2], and (ii) the Thermofield Dynamics (TFD) formalism proposed by
Takahashi and Umezawa [3, 4, 5, 6, 7, 8, 9]. In this approach, a thermal
vacuum state is constructed. This implies that the vacuum expectation value of
an operator is equal to its statistical average. Here, the TFD formalism is
used.

To build the TFD formalism two fundamental elements are required: the
duplication of the Hilbert space and the Bogoliubov transformation. The
expanded Hilbert space is composed of the standard Hilbert space
$\displaystyle{\cal S}$ and the tilde space $\displaystyle\tilde{\cal S}$,
i.e. $\displaystyle{\cal S}_{T}={\cal S}\otimes\tilde{\cal S}$. These two
spaces are related through the tilde conjugation rules. Then an operator in
the original Hilbert space is associated with two operators in the thermal or
double Hilbert space. The other key ingredient is the Bogoliubov
transformation, which is a rotation that involves the tilde and non-tilde
operators. From this, the effects of temperature are introduced. Furthermore,
this formalism is a topological field theory described by
$\displaystyle\Gamma_{D}^{d}=(\mathbb{S}^{1})^{d}\times\mathbb{R}^{D-d}$,
where $\displaystyle D$ are the space-time dimensions and $\displaystyle d$ is
the number of compactified dimensions. This implies that any number of
dimensions of the manifold can be compactified into the hyper-torus
$\displaystyle(\mathbb{S}^{1})^{d}$. Then both effects, due to thermal and
spatial compactifications, can be investigated. In this paper, the scalar
field is considered in three different topologies: (i) the time coordinate is
compactified into a circle of length $\displaystyle 1/T$, where $\displaystyle
T$ is the temperature; (ii) the $\displaystyle z$ coordinate is compactified
into a circumference of length $\displaystyle L$, and (iii) both coordinates,
i.e. $\displaystyle t$ and $\displaystyle z$ are compactified. It is
interesting to note that this formalism allows us to explore some different
phenomena, such as the Stefan-Boltzmann law and the Casimir effect, in the
same way. Using this theory, let us study the Casimir effect at zero and
finite temperature for the scalar field in a Lorentz-violating theory.

Lorentz invariance emerges as a non-exact symmetry in models that describe a
fundamental theory that arises at very high energy scales [10, 11]. A small
violation of Lorentz symmetry leads to a new physics beyond the standard
model. To study the effects due to the violation of Lorentz symmetry, an
effective field theory, called Standard Model Extension (SME), has been
constructed [12, 13, 14]. The SME is composed of general relativity, the
standard model, and all terms that violate Lorentz. Terms that belong to the
SME are classified as CPT-odd and CPT-even. The CPT-odd term violates Lorentz
symmetry and CPT (Charge conjugation, Parity and Time reversal) symmetry,
while the CPT-even term violates Lorentz symmetry, but CPT symmetry is
maintained. Here, the main interest is the CPT-even aether-like term. This
term was first introduced in a five-dimensional context and its consequences
for different fields have been analyzed [15]. Some studies with such a
Lorentz-violating term have been explored. For example: stability conditions
have been investigated [16], phenomenological properties of this theory have
been studied [17], the generation of this term in three, four and five space-
time dimensions has been done [18], a non-abelian version in four dimensions
has been constructed [19], among others. In this context, the aether-like
Lorentz-breaking scalar field theory is considered. Then, some corrections due
to Lorentz symmetry breaking are calculated for the Casimir effect.

The Casimir effect, proposed in 1948 by H. Casimir [20], is a quantum
phenomenon that arises due to the vacuum fluctuations of any field that
exhibits a quantum nature. This effect is a consequence of boundary conditions
or topology under the quantum fields. Its experimental confirmation takes
place a few years after theoretical construction [21]. Over the years its
accuracy has greatly increased [22, 23]. Our main objective is to calculate
the Casimir effect for the scalar field in four dimensions with the aether-
like term at zero and finite temperature. There are some studies that
investigate the Casimir effect for a scalar field at zero and finite
temperature in a theory where the Lorentz invariance is broken due to the
aether term [24, 25, 26, 27, 28, 29]. In each of these works, different
techniques and different analyzes are used. However, these applications do not
exhaust the possibilities of studying this effect using different tools. Here,
the Casimir effect is calculated using the TFD formalism. Through different
topologies, it is possible to analyze this effect at zero and non-zero
temperature. Furthermore, the calculations are carried out in a natural
covariant way. This aspect represents ease of calculation, such that the
results are direct and accurate. Another advantage of the TFD is that the
Stefan-Boltzmann law associated with the field can be calculated in the same
way, just by choosing an appropriate topology.

This paper is organized as follows. In section II, the TFD formalism is
introduced. In section III, the model that describes a massive scalar field
with an aether-like term that violates the Lorentz symmetry is presented. The
energy-momentum tensor associated with this theory in the TFD approach is
calculated. In section IV, some finite temperature and Lorentz violation
applications are analyzed. Considering a time-like and space-like constant
vector, the Stefan-Boltzmann law and the Casimir effect at zero and non-zero
temperatures are obtained. In section V, some concluding remarks are
discussed.

##  II TFD formalism

TFD formalism is a thermal quantum field theory that allows interpreting the
statistical average of an arbitrary operator as its vacuum expectation value
in a thermal ground state [3, 4, 5, 6, 7, 8, 9]. In addition, TFD is
classified as real-time formalism. Then, the effects of temperature and
temporal evolution of a system are investigated together.

In order to build a thermal vacuum, the degrees of freedom are doubled and, as
a consequence, the Fock space is extended, that is, the thermal Fock space
$\displaystyle{\cal S}_{T}$ is composed of the original Fock
$\displaystyle{\cal S}$ and the tilde or dual space $\displaystyle\tilde{\cal
S}$, i.e. $\displaystyle{\cal S}_{T}={\cal S}\otimes\tilde{\cal S}$. There is
a mapping between the two spaces defined by the tilde conjugation rules which
are given as

$\displaystyle\displaystyle({\cal A}_{i}{\cal A}_{j})^{\thicksim}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\tilde{{\cal
A}_{i}}\tilde{{\cal A}_{j}},$ $\displaystyle\displaystyle(c{\cal A}_{i}+{\cal
A}_{j})^{\thicksim}$ $\displaystyle\displaystyle=$ $\displaystyle\displaystyle
c^{*}\tilde{{\cal A}_{i}}+\tilde{{\cal A}_{j}},$
$\displaystyle\displaystyle({\cal A}_{i}^{\dagger})^{\thicksim}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\tilde{{\cal
A}_{i}}^{\dagger},$ $\displaystyle\displaystyle(\tilde{{\cal
A}_{i}})^{\thicksim}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\xi{\cal A}_{i},$ (1)

with $\displaystyle\xi=-1$ for bosons and $\displaystyle\xi=+1$ for fermions
and $\displaystyle{\cal A}$ is an arbitrary operator.

In addition to the duplication of Fock space, another tool is required in this
formalism. The Bogoliubov transformation consists of a rotation, between the
tilde and non-tilde operators, which introduces the dependency of a new
parameter on that operator. This parameter, called the compactification
parameter, is defined as
$\displaystyle\alpha=(\alpha_{0},\alpha_{1},\cdots\alpha_{D-1})$. The effects
of temperature are introduced by taking $\displaystyle\alpha_{0}=\beta=1/T$
and all other components equal to zero.

As an example of applying the Bogoliubov transformation, consider two
arbitrary operators $\displaystyle{\cal O}$ and $\displaystyle\tilde{\cal O}$
in $\displaystyle{\cal S}$ and $\displaystyle\tilde{\cal S}$, respectively.
Then

$\displaystyle\displaystyle\left(\begin{array}[]{cc}{\cal O}(k,\alpha)\\\
\xi\tilde{\cal O}^{\dagger}(k,\alpha)\end{array}\right)={\cal
U}(\alpha)\left(\begin{array}[]{cc}{\cal O}(k)\\\ \xi\tilde{\cal
O}^{\dagger}(k)\end{array}\right),$ (6)

where $\displaystyle{\cal U}(\alpha)$ is defined as

$\displaystyle\displaystyle{\cal
U}(\alpha)=\left(\begin{array}[]{cc}u(\alpha)&-w(\alpha)\\\ \xi
w(\alpha)&u(\alpha)\end{array}\right),$ (9)

with $\displaystyle u(\alpha)$ and $\displaystyle v(\alpha)$ related to the
Bose distribution, i.e. $\displaystyle
v^{2}(\alpha)=\frac{1}{e^{\alpha\omega}-1}$ and $\displaystyle
u^{2}(\alpha)=1+v^{2}(\alpha)$.

For the calculations that are developed in the next section, the modifications
caused by the introduction of the $\displaystyle\alpha$ parameter in the
propagator of the scalar field are investigated. In the TFD formalism, this
propagator is written as

$\displaystyle\displaystyle G_{0}^{(ab)}(x-x^{\prime};\alpha)=i\langle
0,\tilde{0}|\tau[\phi^{a}(x;\alpha)\phi^{b}(x^{\prime};\alpha)]|0,\tilde{0}\rangle,$
(10)

where $\displaystyle a,b$ $\displaystyle=1,2$, $\displaystyle\tau$ is the time
ordering operator and $\displaystyle\phi(x;\alpha)={\cal
U}(\alpha)\phi(x){\cal U}^{-1}(\alpha)$. In the thermal vacuum
$\displaystyle|0(\alpha)\rangle={\cal U}(\alpha)|0,\tilde{0}\rangle$, the
propagator is

$\displaystyle\displaystyle
G_{0}^{(ab)}(x-x^{\prime};\alpha)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik(x-x^{\prime})}G_{0}^{(ab)}(k;\alpha),$
(11)

with $\displaystyle G_{0}^{(ab)}(k;\alpha)={\cal
U}(\alpha)G_{0}^{(ab)}(k){\cal U}^{-1}(\alpha).$ It is important to note that,
the physical Green function is given by the non-tilde part, i.e.

$\displaystyle\displaystyle
G_{0}^{(11)}(k;\alpha)=G_{0}(k)+v^{2}(k;\alpha)[G^{*}_{0}(k)-G_{0}(k)],$ (12)

where $\displaystyle v^{2}(k;\alpha)$ is the generalized Bogoliubov
transformation [9] defined as

$\displaystyle\displaystyle v^{2}(k;\alpha)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{s=1}^{p}\sum_{\\{\sigma_{s}\\}}2^{s-1}\sum_{l_{\sigma_{1}},...,l_{\sigma_{s}}=1}^{\infty}(-\xi)^{s+\sum_{r=1}^{s}l_{\sigma_{r}}}\exp\left[{-\sum_{j=1}^{s}\alpha_{\sigma_{j}}l_{\sigma_{j}}k^{\sigma_{j}}}\right],$
(13)

with $\displaystyle p$ being the number of compactified dimensions,
$\displaystyle\\{\sigma_{s}\\}$ denotes the set of all combinations with
$\displaystyle s$ elements and $\displaystyle k$ is the 4-momentum.

##  III The energy-momentum tensor in the TFD formalism

Here, the main idea is to obtain the energy-momentum tensor for the scalar
field within a Lorentz-violating theory. The term that leads to invariance
breaking is the aether-like term. In the TFD formalism, the energy-momentum
tensor is thermalized and, as a consequence, some different physical
quantities are calculated at finite temperature.

The complete Lagrangian in the TFD approach, which describes the scalar field
in the presence of the aether-like term, is given as

$\displaystyle\displaystyle\widehat{{\cal L}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\cal L}-\widetilde{{\cal L}}$ (14)
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\lambda}{2}(u\cdot\partial\phi)^{2}+\frac{1}{2}m^{2}\phi^{2}-\frac{1}{2}\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}+\frac{\lambda}{2}(u\cdot\partial\tilde{\phi})^{2}+\frac{1}{2}m^{2}\tilde{\phi}^{2},$

where $\displaystyle{\cal L}$ and $\displaystyle\widetilde{\cal L}$ are the
Lagrangians in the original and tilde spaces, respectively. The violation of
Lorentz symmetry is introduced through the term
$\displaystyle\lambda(u\cdot\partial\phi)^{2}$, with $\displaystyle\lambda$
being a very small dimensionless parameter which determines the effects of
symmetry breaking in the theory and $\displaystyle u^{\mu}$ is a constant
vector that acts as a background field.

In order to obtain the energy-momentum tensor, only the non-tilde variables in
the Lagrangian are considered. However, a similar development is performed
with the tilde part. Then, using the definition

$\displaystyle\displaystyle T^{\mu\nu}=\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-g^{\mu\nu}{\cal L},$ (15)

the energy-momentum tensor associated with the Lorentz-violating scalar field
is

$\displaystyle\displaystyle
T^{\mu\nu}=\partial^{\mu}\phi\partial^{\nu}\phi+\lambda
u^{\mu}\partial^{\nu}\phi(u\cdot\partial\phi)-\frac{1}{2}g^{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi-\frac{\lambda}{2}g^{\mu\nu}(u\cdot\partial\phi)^{2}-\frac{1}{2}g^{\mu\nu}m^{2}\phi^{2}.$
(16)

It is important to note that the product of the field operators at the same
space-time points should be avoided. To solve this problem, the energy-
momentum tensor is written as

$\displaystyle\displaystyle T^{\mu\nu}(x)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\prime}\to
x}\tau{\bigg{\\{}}\partial^{\mu}\phi(x)\partial^{\prime\nu}\phi(x^{\prime})+\lambda
u^{\mu}\partial^{\nu}\phi(x)\left(g_{\rho\epsilon}u^{\epsilon}\partial^{\prime\rho}\phi(x^{\prime})\right)-\frac{1}{2}g^{\mu\nu}g_{\gamma\rho}\partial^{\gamma}\phi(x)\partial^{\prime\rho}\phi(x^{\prime})$
(17) $\displaystyle\displaystyle-$
$\displaystyle\displaystyle\frac{\lambda}{2}g^{\mu\nu}g_{\sigma\omega}g_{\lambda\theta}u^{\omega}u^{\theta}\partial^{\sigma}\phi(x)\partial^{\prime\lambda}\phi(x^{\prime})-\frac{1}{2}g^{\mu\nu}m^{2}\phi(x)\phi(x^{\prime}){\bigg{\\}}},$

where $\displaystyle x^{\prime}$ and $\displaystyle x$ represent different
points of space-time. Using the commutation relation

$\displaystyle\displaystyle[\phi(x),\partial^{\prime\mu}\phi^{\dagger}(x^{\prime})]=in^{\mu}_{0}\delta({\bf
x-x^{\prime}}),$ (18)

with $\displaystyle n^{\mu}_{0}=(1,0,0,0)$ being a time-like vector, eq. (17)
becomes

$\displaystyle\displaystyle T^{\mu\nu}(x)=\lim_{x^{\prime}\to
x}{\bigg{\\{}}\Gamma^{\mu\nu}\tau[\phi(x)\phi(x^{\prime})]+I^{\mu\nu}\delta({\bf
x-x^{\prime}}){\bigg{\\}}},$ (19)

where

$\displaystyle\displaystyle\Gamma^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial^{\mu}\partial^{\prime\nu}+\lambda
g_{\alpha\epsilon}u^{\mu}u^{\epsilon}\partial^{\nu}\partial^{\prime\alpha}-\frac{1}{2}g^{\mu\nu}g_{\gamma\rho}\partial^{\gamma}\partial^{\prime\rho}-\frac{\lambda}{2}g^{\mu\nu}g_{\sigma\omega}g_{\lambda\theta}u^{\omega}u^{\theta}\partial^{\sigma}\partial^{\prime\lambda}-\frac{1}{2}g^{\mu\nu}m^{2},$
(20) $\displaystyle\displaystyle I^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle in^{\mu}_{0}n^{\nu}_{0}+i\lambda
g_{\alpha\epsilon}u^{\mu}u^{\epsilon}n^{\nu}_{0}n^{\alpha}_{0}-\frac{i}{2}g^{\mu\nu}g_{\gamma\rho}n^{\gamma}_{0}n^{\rho}_{0}-\frac{\lambda}{2}ig^{\mu\nu}g_{\sigma\omega}g_{\lambda\theta}u^{\omega}u^{\theta}n^{\sigma}_{0}n^{\lambda}_{0}.$
(21)

Now, let us take the vacuum expectation value of the energy-momentum tensor.
Then

$\displaystyle\displaystyle\bigl{\langle}T^{\mu\nu}(x)\bigl{\rangle}$
$\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\bigl{\langle}0|T^{\mu\nu}|0\bigl{\rangle}$ (22)
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{x^{\prime}\to
x}\bigg{\\{}\Gamma^{\mu\nu}\bigl{\langle}0|\tau[\phi(x)\phi(x^{\prime})]|0\bigl{\rangle}+I^{\mu\nu}\delta({\bf
x-x^{\prime}})\bigl{\langle}0|0\bigl{\rangle}\bigg{\\}},$

with

$\displaystyle\displaystyle\bigl{\langle}0|\tau[\phi(x)\phi(x^{\prime})]|0\bigl{\rangle}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-G_{0}(x-x^{\prime})$
(23) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{im}{4\pi^{2}}\frac{K_{1}(m\sqrt{-(x-x^{\prime})^{2}})}{\sqrt{-(x-x^{\prime})^{2}}}$

being the definition of the massive scalar field propagator. Here
$\displaystyle K_{n}(x)$ is the modified Bessel function.

Introducing the TFD formalism notation as presented in the previous section,
eq. (22) reads

$\displaystyle\displaystyle\bigl{\langle}T^{\mu\nu(ab)}(x;\alpha)\bigl{\rangle}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{x^{\prime}\to
x}\bigg{\\{}\Gamma^{\mu\nu}G_{0}^{ab}(x-x^{\prime};\alpha)+I^{\mu\nu}\delta({\bf
x-x^{\prime}})\bigg{\\}}.$ (24)

There is an important observation that must be made in the vacuum expectation
value. This quantity is divergent for both cases, with or without the
$\displaystyle\alpha$ parameter. Then a renormalization procedure is required.
This prescription consists of

$\displaystyle\displaystyle{\cal T}^{\mu\nu(ab)}(x;\alpha)$
$\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\bigl{\langle}T^{\mu\nu(ab)}(x;\alpha)\bigl{\rangle}-\bigl{\langle}T^{\mu\nu(ab)}(x)\bigl{\rangle}$
(25) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\prime}\to
x}{\bigg{\\{}}i\Gamma^{\mu\nu}\overline{G}^{(ab)}_{0}(x-x^{\prime};\alpha){\bigg{\\}}},$

where

$\displaystyle\displaystyle\overline{G}^{(ab)}_{0}(x-x^{\prime};\alpha)=G^{(ab)}_{0}(x-x^{\prime};\alpha)-G^{(ab)}_{0}(x-x^{\prime}).$
(26)

In the next section, eq. (25) is used to investigate some finite temperature
applications with different choices for the constant vector $\displaystyle
u^{\mu}$ which leads to a violation of the Lorentz symmetry.

##  IV Some Applications: TFD and Lorentz violation

Here, applications for different choices of the compactification parameter
$\displaystyle\alpha$ and different directions of the constant vector
$\displaystyle u^{\mu}$ are considered.

###  IV.1 Stefan-Boltzmann law for massive scalar field

As a first application, the topology in which one spacetime dimension is
compactified is considered, i.e. $\displaystyle\alpha=(\beta,0,0,0)$. Here,
the time-axis is compactified into a circumference $\displaystyle\beta$. In
this case, the Bogoliubov transformation Eq. (13) and the Green function Eq.
(26) are given, respectively, as

$\displaystyle\displaystyle v^{2}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{l_{0}=1}^{\infty}e^{-\beta k^{0}l_{0}},$ (27)
$\displaystyle\displaystyle\overline{G}_{0}(x-x^{\prime},\beta)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\sum_{l_{0}=1}^{\infty}G_{0}(x-x^{\prime}-i\beta l_{0}n_{0}),$ (28)

where $\displaystyle n_{0}^{\mu}=(1,0,0,0)$.

Now, let us analyze the energy-momentum tensor for different choices of the
constant vector $\displaystyle u^{\mu}$.

####  IV.1.1 Time-like constant vector

In this case, the constant vector is chosen as $\displaystyle
u^{\mu}=(1,0,0,0)$. For $\displaystyle\mu=\nu=0$, Eq. (25) becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta)=i\lim_{x^{\prime}\to
x}\sum_{l_{0}=1}^{\infty}{\bigg{\\{}}(1+\lambda)\partial^{0}\partial^{\prime
0}+\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime 3}-m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-i\beta
l_{0}n_{0}).$ (29)

Performing the derivatives and some simplifications, this equation leads to

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta)=\frac{3m^{2}}{4\pi^{2}\beta^{2}}\sum_{l_{0}=1}^{\infty}\frac{1}{l_{0}}\left[(2+\lambda)K_{2}(m\beta
l_{0})+\frac{4\lambda m\beta l_{0}}{3}K_{1}(m\beta l_{0})\right].$ (30)

This expression is the Stefan-Boltzmann law associated with the massive scalar
field with corrections due to the aether term that breaks the Lorentz
symmetry. Let us consider the small mass limit, i.e. $\displaystyle(m\to 0)$.
In this limit, the Bessel function is given by

$\displaystyle\displaystyle K_{z}(\nu)\approx\Gamma(\nu)2^{\nu-1}z^{-\nu}.$
(31)

Then Eq. (30) reads

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta)=\dfrac{\pi^{2}(2+\lambda)}{60\beta^{4}}.$ (32)

This is the Stefan-Boltzmann law for the massless scalar field. It is observed
that the parameter, that violates the Lorentz symmetry, contributes to
increase the energy density.

####  IV.1.2 Space-like constant vector

Here, the following cases are investigated: (i) $\displaystyle
u^{\mu}=(0,1,0,0)$, (ii) $\displaystyle u^{\mu}=(0,0,1,0)$ and (iii)
$\displaystyle u^{\mu}=(0,0,0,1)$. First, let us consider $\displaystyle
u^{\mu}=(0,1,0,0)$. For this case, the energy-momentum tensor, Eq. (25),
becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta)=i\lim_{x^{\prime}\to
x}\sum_{l_{0}=1}^{\infty}{\bigg{\\{}}\partial^{0}\partial^{\prime
0}+(1-\lambda)\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime 3}-m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-i\beta
l_{0}n_{0}),$ (33)

Taking the component $\displaystyle\mu=\nu=0$, it turns out that

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta)=\sum_{l_{0}=1}^{\infty}\frac{1}{l_{0}^{2}}\left[\frac{m^{2}(6-\lambda)K_{2}(m\beta
l_{0})}{4\pi^{2}\beta^{2}}\right].$ (34)

This is the Stefan-Boltzmann law for a space-like constant vector
$\displaystyle u^{\mu}$. If the small mass limit is taken, this expression
becomes

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta)=\frac{\pi^{2}(6-\lambda)}{180\beta^{4}}.$ (35)

It is observed that the other choices for the constant vector, which are
$\displaystyle u^{\mu}=(0,0,1,0)$ and $\displaystyle u^{\mu}=(0,0,0,1)$ lead
to the same results, i.e. Eq. (34) and Eq. (35). Furthermore, it is
interesting to note that the violation of Lorentz symmetry changes the Stefan-
Boltzmann law. However, this modification depends on the direction of the
constant vector $\displaystyle u^{\mu}$. In other words, a time-like constant
vector contributes in a different way compared to the contribution due to a
space-like constant vector.

###  IV.2 Casimir effect for the massive scalar field at zero temperature

In order to calculate the Casimir effect at zero temperature, the
compactification parameter is chosen as $\displaystyle\alpha=(0,0,0,i2d)$.
With such a choice, the described topology consists of a compactification
along the $\displaystyle z$ coordinate, i.e. the topological theory is defined
as $\displaystyle\Gamma^{1}_{4}=\mathbb{S}^{1}\times\mathbb{R}^{3}$ with
$\displaystyle\mathbb{S}^{1}$ being a circumference of length $\displaystyle
L=2d$, where $\displaystyle d$ is the distance between the two conducting
plates. This spatial compactification leads to the Bogoliubov transformation

$\displaystyle\displaystyle
v^{2}(d)=\sum_{l_{3}=1}^{\infty}e^{-i2dk^{3}l_{3}},$ (36)

and the Green function

$\displaystyle\displaystyle\overline{G}_{0}(x-x^{\prime};d)=2\sum_{l_{3}=1}^{\infty}G_{0}(x-x^{\prime}-2dl_{3}n_{3})$
(37)

with $\displaystyle n_{3}^{\mu}=(0,0,0,1)$. With these ingredients and for
different choices of the constant vector $\displaystyle u^{\mu}$, the Casimir
effect at zero temperature is calculated.

####  IV.2.1 Time-like constant vector

Assuming that $\displaystyle u^{\mu}=(1,0,0,0)$ and $\displaystyle\mu=\nu=0$,
the energy-momentum tensor reads

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)=i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}(1+\lambda)\partial^{0}\partial^{\prime
0}+\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime
3}-m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}),$ (38)

that provides us

$\displaystyle\displaystyle{\cal
T}^{00(11)}(d)=-\frac{m^{2}}{8\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\left[\frac{(2+\lambda)}{2}K_{2}(2mdl_{3})+2mdl_{3}K_{1}(2mdl_{3})\right].$
(39)

This is the Casimir energy for the massive scalar field with Lorentz symmetry
violation. Similarly, the Casimir pressure is obtained for
$\displaystyle\mu=\nu=3$. Then

$\displaystyle\displaystyle{\cal T}^{33(11)}(d)=i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}(1+\lambda)\partial^{0}\partial^{\prime
0}-\partial^{1}\partial^{\prime 1}-\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime
3}+m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}).$ (40)

This equation leads to

$\displaystyle\displaystyle{\cal
T}^{33(11)}(d)=-\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\left[\frac{m^{2}(6+\lambda)K_{2}(2mdl_{3})}{16d^{2}\pi^{2}}\right].$
(41)

In the small mass limit, i.e. $\displaystyle m\to 0$, the Casimir energy and
pressure at zero temperature become

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\pi^{2}(2+\lambda)}{2880d^{4}},$ (42)
$\displaystyle\displaystyle{\cal T}^{33(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\pi^{2}(6+\lambda)}{2880d^{4}}.$ (43)

These expressions show that, although the aether term that violates Lorentz
symmetry changes the Casimir effect, it does not modify the attractive nature
of this phenomenon. Instead, even being very small, it helps to increase the
Casimir effect associated with a massive scalar field.

####  IV.2.2 Space-like constant vector

First, let us assume $\displaystyle u^{\mu}=(0,1,0,0)$. Then the component
$\displaystyle\mu=\nu=0$ of the energy-momentum tensor reads

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)=i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\partial^{0}\partial^{\prime
0}+(1-\lambda)\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime
3}-m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}).$ (44)

This equation gives us the Casimir energy which is given as

$\displaystyle\displaystyle{\cal
T}^{00(11)}(d)=-\frac{m^{2}}{8d^{2}\pi^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\left[\frac{(2+\lambda)}{2}K_{2}(2mdl_{3})+2mdl_{3}K_{1}(2mdl_{3})\right].$
(45)

And for $\displaystyle\mu=\nu=3$ we get

$\displaystyle\displaystyle{\cal T}^{33(11)}(d)=i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\partial^{0}\partial^{\prime
0}-(1-\lambda)\partial^{1}\partial^{\prime 1}-\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime
3}+m^{2}G_{0}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}),$ (46)

which results in the Casimir pressure, i.e.

$\displaystyle\displaystyle{\cal
T}^{33(11)}(d)=-\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\left[\frac{m^{2}(6-\lambda)K_{2}(2mdl_{3})}{16\pi^{2}d^{2}}\right].$
(47)

Taking the limit $\displaystyle m\to 0$, Eq. (45) and Eq. (47) are given as

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\pi^{2}(2+\lambda)}{2880d^{4}},$ (48)
$\displaystyle\displaystyle{\cal T}^{33(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{\pi^{2}(6-\lambda)}{2880d^{4}}.$ (49)

In this case, the Lorentz-violating parameter acts against the usual Casimir
pressure, i.e., the Casimir force provided by the Lorentz symmetry violation
is repulsive. The same results are obtained for the case $\displaystyle
u^{\mu}=(0,0,1,0)$.

Now, let’s look at the constant vector $\displaystyle u^{\mu}=(0,0,0,1)$. In
this case, the components $\displaystyle\mu=\nu=0$ and
$\displaystyle\mu=\nu=3$ of the energy-momentum tensor are, respectively

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\partial^{0}\partial^{\prime
0}+\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+(1-\lambda)\partial^{3}\partial^{\prime
3}-m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}),$ (50)
$\displaystyle\displaystyle{\cal T}^{33(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle i\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\partial^{0}\partial^{\prime
0}-\partial^{1}\partial^{\prime 1}-\partial^{2}\partial^{\prime
2}+(1-\lambda)\partial^{3}\partial^{\prime
3}+m^{2}{\bigg{\\}}}G_{0}(x-x^{\prime}-2dl_{3}n_{3}).$ (51)

Performing some calculations, the Casimir energy and pressure are given,
respectively, as

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{m^{2}}{8\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\left[\frac{(2-3\lambda)}{2}K_{2}(2mdl_{3})+(2-\lambda)mdl_{3}K_{1}(2mdl_{3})\right],$
(52) $\displaystyle\displaystyle{\cal T}^{33(11)}(d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{m^{2}}{16\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l^{2}_{3}}\Bigl{[}(6-3\lambda)K_{2}(2mdl_{3})-2\lambda
mdl_{3}K_{1}(2mdl_{3})\Bigl{]}.$ (53)

Following the same steps as in the previous case, the results obtained for the
case $\displaystyle m\to 0$ are

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$
$\displaystyle\displaystyle=-\frac{\pi^{2}(2-3\lambda)}{2880d^{4}},$ (54)
$\displaystyle\displaystyle{\cal T}^{33(11)}(d)$
$\displaystyle\displaystyle=-\frac{\pi^{2}(2-\lambda)}{960d^{4}}.$ (55)

Therefore, the corrections for the Casimir effect at zero temperature due to
Lorentz symmetry violation depend on the direction of the constant vector
$\displaystyle u^{\mu}$.

###  IV.3 Casimir effect for the massive scalar field at finite temperature

In this section, the main point of our discussion is calculated, i.e. the
Casimir effect for the massive scalar field at finite temperature in the
presence of the aether term that introduces violation of Lorentz symmetry. To
achieve this goal, two compactifications are needed. Then the compactification
parameter is chosen as $\displaystyle\alpha=(\beta,0,0,i2d)$. For this case,
the Bogoliubov transformation is given as

$\displaystyle\displaystyle v^{2}(\beta,d)=\sum_{l_{0}=1}^{\infty}e^{\beta
k^{0}l_{0}}+\sum_{l_{3}=1}^{\infty}e^{-i2dk^{3}l_{3}}+2\sum_{l_{0},l_{3}=1}^{\infty}e^{-\beta
k^{0}l_{0}-i2dk^{3}l_{3}}.$ (56)

The first two terms correspond to the previous cases, while the third term
represents the effect of two compactifications together. The Green function
for the last term is

$\displaystyle\displaystyle\overline{G}(x-x^{\prime};\beta,d)=4\sum_{l_{0},l_{3}=1}^{\infty}G_{0}(x-x^{\prime}-i\beta
l_{0}n_{0}-2dl_{3}n_{3}).$ (57)

Now, let us analyze the energy-momentum tensor considering the constant vector
$\displaystyle u^{\mu}$ as a time-like vector in the first analysis and in the
sequence as a space-like vector.

####  IV.3.1 Time-like constant vector

Considering $\displaystyle u^{\mu}=(1,0,0,0)$, the third term of Eq. (56) and
Eq. (57), the energy-momentum tensor with $\displaystyle\mu=\nu=0$ becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2i\lim_{x^{\prime}\to
x}\sum_{l_{0},l_{3}=1}^{\infty}{\bigg{\\{}}(1+\lambda)\partial^{0}\partial^{\prime
0}+\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime 3}-m^{2}{\bigg{\\}}}$ (58)
$\displaystyle\displaystyle\times$ $\displaystyle\displaystyle
G_{0}(x-x^{\prime}-i\beta l_{0}n_{0}-2dl_{3}n_{3}).$

From this equation, the Casimir energy at finite temperature is obtained as

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{\left[(2dl_{3})^{2}-3(\beta
l_{0})^{2}\right]}{\left[(2dl_{3})^{2}+(\beta
l_{0})^{2}\right]^{2}}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})\left(\frac{2+\lambda}{2}\right)$ (59)
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\left(\frac{2(2dl_{3})^{2}+\lambda(\beta
l_{0})^{2}}{2}\right)\frac{mK_{1}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})}{\left[(2dl_{3})^{2}+(\beta
l_{0})^{2}\right]^{3/2}}{\Bigg{\\}}}.$

This result shows that the Lorentz violation alters the phenomenon in both
situations, at zero and finite temperature.

In order to get a more detailed analysis, some interesting limits are
considered. As a first attempt, let us take just the first term in the sum of
$\displaystyle l_{3}$ in Eq. (59). Then

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0}=1}^{\infty}{\Bigg{\\{}}\frac{\left(4-3\beta^{2}l_{0}^{2}/d^{2}\right)}{d^{2}\left(4+\beta^{2}l_{0}^{2}/d^{2}\right)}K_{2}(md\sqrt{4+\beta^{2}l_{0}^{2}/d^{2}})\left(\frac{2+\lambda}{2}\right)$
$\displaystyle\displaystyle+\left(8+\frac{\lambda\beta^{2}l_{0}^{2}}{d^{2}}\right)\frac{mK_{1}(md\sqrt{4+\beta^{2}l_{0}^{2}/d^{2}})}{2d\left(4+\beta^{2}l_{0}^{2}/d^{2}\right)^{3/2}}{\Bigg{\\}}}.$
(60)

Taking the large mass limit ($\displaystyle md\gg 1$), the asymptotic
expression for the modified Bessel function is given as

$\displaystyle\displaystyle K_{\nu}(z)=\sqrt{\frac{\pi}{2z}}e^{-z}.$ (61)

Defining $\displaystyle\alpha_{l_{0}}=\frac{\beta l_{0}}{d}$, Eq. (60) is
written as

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=\frac{m^{3/2}}{\sqrt{2}\pi^{3/2}d^{3/2}}\sum_{l_{0}=1}^{\infty}{\bigg{\\{}}\frac{(4-3\alpha^{2}_{l_{0}})(2+\lambda)+md(8+\lambda\alpha^{2}_{l_{0}})(4+\alpha^{2}_{l_{0}})^{1/2}}{2d(4+\alpha^{2}_{l_{0}})^{9/4}}{\bigg{\\}}}e^{-md\sqrt{4+\alpha^{2}_{l_{0}}}}.$
(62)

For low temperatures, i.e. $\displaystyle\beta/d\gg 1$, the last equation
becomes

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{\sqrt{m^{5}}\lambda}{2\sqrt{2\pi^{3}}}\frac{1}{\beta^{3/2}}\sum_{l_{0}=1}^{\infty}\frac{e^{-m\beta
l_{0}}}{l_{0}^{3/2}}.$ (63)

Therefore, at this limit the Casimir energy decays exponentially with
$\displaystyle\beta$.

Now the massless case, i.e. $\displaystyle(md=0)$, is investigated. Using the
asymptotic expression for the modified Bessel function Eq. (31), Eq. (59) is
written in the form

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{(2+\lambda)}{\pi^{2}d^{4}}\sum_{l_{0},l_{3}=1}^{\infty}\frac{(4l_{3}^{2}-3\alpha^{2}_{l_{0}})}{(4l_{3}^{2}+\alpha^{2}_{l_{0}})}.$
(64)

Performing the sum in $\displaystyle l_{3}$, it gives

$\displaystyle\displaystyle\sum_{l_{3}=1}^{\infty}\frac{(4l_{3}^{2}-3\alpha^{2}_{l_{0}})}{(4l_{3}^{2}+\alpha^{2}_{l_{0}})^{3}}$
$\displaystyle\displaystyle=\frac{1}{16\alpha^{4}_{l_{0}}}{\bigg{[}}24-4\pi\alpha_{l_{0}}\coth\left(\frac{\pi\alpha_{l_{0}}}{2}\right)-2\pi^{2}\alpha^{2}_{l_{0}}\mbox{csch}^{2}\left(\frac{\pi\alpha_{l_{0}}}{2}\right)$
$\displaystyle\displaystyle-\pi^{3}\alpha^{3}_{l_{0}}\coth\left(\frac{\pi\alpha_{l_{0}}}{2}\right)\mbox{csch}^{2}\left(\frac{\pi\alpha_{l_{0}}}{2}\right){\bigg{]}}.$
(65)

In the low temperatures limit, the hyperbolic functions assume the values
$\displaystyle\mbox{csch}\left(\frac{\pi\alpha_{l_{0}}}{2}\right)\to 0$ and
$\displaystyle\mbox{coth}\left(\frac{\pi\alpha_{l_{0}}}{2}\right)\to 1$. Then

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{(2+\lambda)}{16\pi^{2}d^{4}}\sum_{l_{0}=1}^{\infty}\frac{1}{\alpha^{4}_{l_{0}}}\left[24-4\pi\alpha_{l_{0}}\right].$
(66)

carrying out the sum, the Casimir energy at such limit reads

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{(2+\lambda)}{60\pi\beta^{4}}\left[\pi^{3}-15\frac{\beta\zeta(3)}{d}\right],$
(67)

where $\displaystyle\zeta(3)$ is the Riemann zeta function.

The general expression for the Casimir pressure is given as

$\displaystyle\displaystyle{\cal T}^{33(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{[3(2dl_{3})^{2}-(\beta
l_{0})^{2}]}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})$
$\displaystyle\displaystyle+\frac{[(2dl_{3})^{2}-3(\beta
l_{0})^{2}]}{2[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{2}}\lambda
K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2})}$
$\displaystyle\displaystyle-\frac{(\beta
l_{0})^{2}(2+\lambda)}{2[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3/2}}mK_{1}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}}){\Bigg{\\}}}.$ (68)

Similar limits can be analyzed for this result. The main result is that the
aether term modifies the Casimir effect at all limits, large mass, no mass, or
low temperatures.

####  IV.3.2 Space-like constant vector

Assuming that $\displaystyle u^{\mu}=(0,1,0,0)$ is a space-like vector, the
Casimir energy at finite temperature with contribution due to the aether term
is

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{[(2dl_{3})^{2}-3(\beta
l_{0})^{2}]}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})$
$\displaystyle\displaystyle+\frac{\lambda K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})}{2[(2dl_{3})^{2}+(\beta
l_{0})^{2}]}+\frac{m(2dl_{3})^{2}K_{1}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})}{2[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{3/2}}{\Bigg{\\}}}.$ (69)

In order to compare the results, the same limits discussed in the previous
subsection are investigated for this case. First, let us consider the large
mass limit $\displaystyle md\gg 1$. Then Eq. (69) becomes

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=\frac{m^{3/2}}{\sqrt{2}\pi^{3/2}d^{3/2}}\sum_{l_{0}=1}^{\infty}{\Bigg{\\{}}\frac{8-6\alpha_{l_{0}}^{2}+\lambda(4+\alpha_{l_{0}}^{2})+8md(4+\alpha_{l_{0}}^{2})^{1/2}}{2d^{2}(4+\alpha_{l_{0}}^{2})^{9/4}}{\Bigg{\\}}}e^{-md\sqrt{4+\alpha_{l_{0}}^{2}}}.$
(70)

At the limit of low temperatures, we get

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{\sqrt{m^{3}}e^{-m\beta}}{2\sqrt{2\pi^{3}}\beta^{5/2}}(\lambda-6).$
(71)

And for the massless and low temperatures case, the Casimir energy is given as

$\displaystyle\displaystyle{\cal
T}^{00(11)}(\beta,d)=-\frac{1}{30\pi\beta^{4}}\left[\frac{\pi^{3}(\lambda-6)}{6}+\frac{15\beta(\lambda-4)}{4d}\zeta(3)\right].$
(72)

The Casimir pressure at finite temperature is obtained as

$\displaystyle\displaystyle{\cal T}^{33(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{[3(2dl_{3})^{2}-(\beta
l_{0})^{2}]}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{2}}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})$
$\displaystyle\displaystyle-\frac{\lambda K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})}{2((2dl_{3})^{2}+(\beta l_{0})^{2})}-\frac{m\lambda(\beta
l_{0})^{2}K_{1}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}})}{[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{3/2}}{\Bigg{\\}}}.$ (73)

Again, all previously applied limits can be used for this pressure. The same
results are obtained for the space-like vector $\displaystyle
u^{\mu}=(0,0,1,0)$.

As a last case, let us consider $\displaystyle u^{\mu}=(0,0,0,1)$. Performing
similar calculations, the Casimir energy and pressure are given, respectively,
as

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{[(2dl_{3})^{2}-3(\beta
l_{0})^{2}]}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{2}}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})$
$\displaystyle\displaystyle-\frac{[3(2dl_{3})^{2}-(\beta
l_{0})^{2}]}{2[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{2}}\lambda
K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})$
$\displaystyle\displaystyle+\frac{2m(dl_{3})^{2}(2-\lambda)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3/2}}K_{1}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}}){\Bigg{\\}}}$
(74)

and

$\displaystyle\displaystyle{\cal T}^{33(11)}(\beta,d)=$
$\displaystyle\displaystyle-\frac{m^{2}}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}{\Bigg{\\{}}\frac{[3(2dl_{3})^{2}-(\beta
l_{0})]}{2[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3}}K_{2}(m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}})(2-\lambda)$
(75) $\displaystyle\displaystyle-\frac{[2\lambda(dl_{3})^{2}+(\beta
l_{0})^{2}]}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3/2}}mK_{1}(m\sqrt{(2dl_{3})^{2}+(\beta
l_{0})^{2}}){\Bigg{\\}}}.$

In order to avoid repetitions, limits are not applied as the analyzes are very
similar to the previous cases. The main observation that must be considered is
the fact that the modifications caused by the aether term depend on the
direction of the constant vector which is coupled with the derivative of the
scalar field.

##  V Conclusions

The TFD formalism, which is a real-time quantum field theory at finite
temperature, allows us to study some applications by choosing different
topologies. Here, three different cases are considered. The first case
consists of the investigation of effects resulting from the compactification
of the time axis. The second case uses a similar topology, however, the
compactification takes place along the $\displaystyle z$ coordinate. In the
latter case, both compactifications are considered at the same time. Based on
this topological theory, different phenomena such as the Stefan-Boltzmann law
and the Casimir effect can be considered on an equal footing. In order to
analyze these effects, a massive scalar field in a Lorentz-violating theory is
introduced. The term that introduces the violation of Lorentz symmetry is the
CPT-even aether-like term. Then our results show how the Lorentz symmetry
violation modifies the energy density associated with the scalar field and the
Casimir effect at zero and non-zero temperature. The aether-like term, which
breaks Lorentz, is constructed as the coupling between the derivative of the
field and a constant vector $\displaystyle u^{\mu}$. It is verified that the
modifications due to this Lorentz-violating term depend on whether the
constant vector is a time-like or a space-like vector. Therefore, the aether
term changes both, the Stefan-Boltzmann law and the Casimir effect at zero and
finite temperature. In addition, the limits of large mass, small mass, and low
temperature are discussed. Although some results discussed here have been
calculated in other works, similar analyzes using the TFD formalism have never
been done in the literature. In this way, the results obtained are direct and
exact, without any simplification or approximation. Within this formalism, the
Casimir effect is calculated at zero and finite temperature just by choosing a
different topology, and then the same proceeding is considered. It is a unique
feature of the TFD formalism. It is important to note that our results at
finite temperature, i.e. Eqs. (59), (68), (69), (73), (74) and (75), are
consistent with the results obtained in [25] where the Matsubara approach has
been used. Furthermore, although the study developed here is theoretical, if
an experimental apparatus is possible, the effects of temperature can help
improve constraints on the Lorentz-violating parameter.

## Acknowledgments

This work by A. F. S. is supported by National Council for Scientific and
Technological Development - CNPq projects 430194/2018-8 and 313400/2020-2. L.
H. A. R. Ferreira thanks CAPES for financial support.

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