A. F. Santos, Faqir C. Khanna
The energy-momentum tensor for the gravitoelectromagnetism (GEM) theory in the real-time finite temperature field theory formalism is presented. Expressions for the Casimir energy and pressure at zero and finite temperature are obtained. An analysis of the Casimir effect for the GEM field is developed.
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Gravitons are described by the propagator in Teleparallel gravity in nearly flat space-time. Finite temperature is introduced by using Thermo Field Dynamics formalism. The Gravitational Casimir effect and Stefan-Boltzmann law are calculated as a function of temperature. Then an equation of state for gravitons is determined.
A detailed investigation is presented of the energy-momentum tensor approach to the evaluation of the force acting on a rigid Casimir cavity in a weak gravitational field. Such a force turns out to have opposite direction with respect to the gravitational acceleration. The order of magnitude for a multi-layer cavity configuration is derived and experimental feasibility is discussed, taking into account current technological resources.
We calculate the gravitational self-energy of vacuum quantum field fluctuations using a Casimir approach. We find that the Casimir gravitational self-energy density can account for the measured dark energy density when the SUSY-breaking energy is approximately 5 TeV, in good agreement with current estimates. Furthermore, the Casimir gravitational self-energy appears to provide a quantum mechanism for the well-know geometric relation between the Planck, SUSY and cosmological constant energy scales.
We show through Schwinger's approach that in a static weak gravitational background, the Casimir Energy for a real massless scalar field obeying Dirichlet boundary conditions on rectangular plates is unaltered from it's flat space-time value. The result is obtained considering an rather general class of backgrounds, adding further generality and consistency test for the previous work. The proposed result has direct consequences on earlier works in the literature that found gravity related corrections for similar setups.
This paper derives a set of general relativistic Cardinal Equations for the equilibrium of an extended body in a uniform gravitational field. These equations are essential for a proper understanding of the mechanics of suspended relativistic systems. As an example, the prototypical case of a suspended vessel filled with radiation is discussed. The mechanics of Casimir apparatuses at rest in the gravitational field of the Earth is then considered. Starting from an expression for the Casimir energy-momentum tensor in a weak gravitational field recently derived by the authors, it is here shown th…
The influence of the gravity acceleration on the regularized energy-momentum tensor of the quantized electromagnetic field between two plane parallel conducting plates is derived. We use Fermi coordinates and work to first order in the constant acceleration parameter. A new simple formula for the trace anomaly is found to first order in the constant acceleration, and a more systematic derivation is therefore obtained of the theoretical prediction according to which the Casimir device in a weak gravitational field will experience a tiny push in the upwards direction.
Recent work in the literature had evaluated the energy-momentum tensor of a Casimir apparatus in a weak gravitational field, for an electromagnetic field subject to perfect conductor boundary conditions on parallel plates. The Casimir apparatus was then predicted to experience a tiny push in the upwards direction, and the regularized energy-momentum tensor was found to have a trace anomaly. The latter, unexpected property made it compelling to assess what happens in a simpler case. For this purpose, the present paper studies a free, real massless scalar field subject to homogeneous Dirichlet c…
We show that the statement in F. Sorge [Class. Quant. Grav. 36, no.23, 235006 (2019)] that the Casimir effect receives second order corrections due to gravity is not consistent. We remark especially on the tracing of the proper time Hamiltonian, where the correct procedure is to use the eigenfunctions and eigenvalues of the covariant DAlembertian. After some cancellations we find that the value of the functional W [0] is the same as obtained by Sorge. However, we argue that the proper vacuum energy density carries extra space-time volume terms that cancel over the gravitational correction, ret…
How does Casimir energy fall in $κ$-deformed space-time?
本文在 κ 变形时空中用 κ 变形 Rindler 坐标和标量场计算弱引力场下的 Casimir 能量，证明其像质量一样加速，从而验证质能等价原理在时空非对易背景下依然成立。
The discussion of vacuum energy is currently a subject of great theoretical importance, specially concerning the cosmological constant problem in General Relativity. From Quantum Field Theory, it is stated that vacuum states subject to boundary conditions may generate tensions on these boundaries related to a measurable non-zero renormalized vacuum energy: the Casimir Effect. As such, investigating how these vacuum states and energy behave in curved backgrounds is just natural and might provide important results in the near future. In this paper we revisit a model of the Casimir Effect in weak…
We review and assess a part of the recent work on Casimir apparatuses in the weak gravitational field of the Earth. For a free, real massless scalar field subject to Dirichlet or Neumann boundary conditions on the parallel plates, the resulting regularized and renormalized energy-momentum tensor is covariantly conserved, while the trace anomaly vanishes if the massless field is conformally coupled to gravity. Conformal coupling also ensures a finite Casimir energy and finite values of the pressure upon parallel plates. These results have been extended to an electromagnetic field subject to per…
We consider a Casimir apparatus consisting of two perfectly conducting parallel plates, subject to the weak gravitational field of the Earth. The aim of this paper is the calculation of the energy-momentum tensor of this system for a free, real massless scalar field satisfying Neumann boundary conditions on the plates. The small gravity acceleration (here considered as not varying between the two plates) allows us to perform all calculations to first order in this parameter. Some interesting results are found: a correction, depending on the gravity acceleration, to the well-known Casimir energ…
This paper studies two perfectly conducting parallel plates in the weak gravitational field on the surface of the Earth. Since the appropriate line element, to first order in the constant gravity acceleration g, is precisely of the Rindler type, we can exploit the formalism for studying Feynman Green functions in Rindler spacetime. Our analysis does not reduce the electromagnetic potential to the transverse part before quantization. It is instead fully covariant and well suited for obtaining all components of the regularized and renormalized energy-momentum tensor to arbitrary order in the gra…
We calculate the renormalized vacuum energy density for a massless scalar field confined between two nearby parallel plates formed by ideal uncharged conductors, placed very close to the surface of a rotating spherical gravitational source with mass $M$, radius $R$ and momentum angular $J$, at the equatorial region. We consider that the source rotates slowly and that the gravitational field is weak. Corrections to the Casimir energy density induced by the gravitational field generated by this source are calculated up to $M/R^2$ order. The obtained results show us that there is an important mod…
We study the finite temperature Casimir interactions on two parallel boundaries in the anti-de Sitter spacetime AdS$_{D+1}$ induced by the vacuum fluctuations of a massive fermionic field with MIT bag boundary conditions. As in the Minkowski spacetime, the Casimir interaction always tends to attract the two boundaries to each other at any temperature and for any mass. For bosonic fields, it is well known that the high temperature leading term of the Casimir interaction is linear in temperature. However, for fermionic fields, the Casimir interaction decays exponentially at high temperature due …
We compute the Casimir energy between an unusual pair of parallel plates at finite temperature, namely, a perfectely conducting plate ($ε\to\infty$) and an infinitely permeable one ($μ\to\infty$) by applying the generalized zeta function method. We also compute the Casimir pressure and discuss the high and the low temperature limits.
We study the zero and finite temperature Casimir force acting on a perfectly conducting piston with arbitrary cross section moving inside a closed cylinder with infinitely permeable walls. We show that at any temperature, the Casimir force always tends to move the piston away from the walls and towards its equilibrium position. In the case of rectangular piston, exact expressions for the Casimir force are derived. In the high temperature regime, we show that the leading term of the Casimir force is linear in temperature and therefore the Casimir force has a classical limit. Due to duality, all…
We proceed with the study of the Casimir force for parallel plates at finite temperature in the Horava-Lifshitz (HL) theory. We find that the HL exponent can not be chosen as an integer, or the Casimir energy will be a constant and further the Casimir force between two parallel plates will vanish. The higher temperature makes the attractive Casimir force weaker, which is consistent with the original consequence confirmed theoretically and experimentally. We can select the HL factor adequately to lead the thermally revised Casimir force to be similar to the standard results for the parallel pla…
In this article, we consider the finite temperature Casimir effect in Kaluza-Klein spacetime due the the vacuum fluctuation of massless scalar field with Dirichlet boundary conditions. We consider the general case where the extra dimensions (internal space) can be any compact connected manifold or orbifold without boundaries. Using piston analysis, we show that the Casimir force is always attractive at any temperature, regardless of the geometry of the internal space. Moreover, the magnitude of the Casimir force increases as the size of the internal space increases and it reduces to the Casimi…
In this paper the Casimir effect for parallel plates at finite temperature in the presence of compactified universal extra dimensions is analyzed. We show the thermal corrections to the effect in detail. We investigate the Casimir effect for different size of universal extra dimensions.
Casimir Effect and Gravitational Balance: a Search for Stable Configurations
本文研究了Casimir力在对抗引力收缩中的作用，探讨了在弱场极限下实现稳定平衡配置的可能性。
Through the Casimir effect, we obtain a temperature correction to the ground state mass of the Bosonic string. Accordingly, we show that higher than the temperature $\approx 1/2π$, the Tachyon attain real mass, and also we find massive Graviton for nonzero temperatures. As another consequence, we demonstrate that for temperatures higher than the Hagedorn temperature, the string would find again a Tachyonic state.
The influence of the gravity acceleration on the regularized energy-momentum tensor of the quantized electromagnetic field between two plane parallel conducting plates is derived. We use Fermi coordinates and work to first order in the constant acceleration parameter. A perturbative expansion, to this order, of the Green functions involved and of the energy-momentum tensor is derived by means of the covariant geodesic point splitting procedure. In correspondence to the Green functions satisfying mixed and gauge-invariant boundary conditions, and Ward identities, the energy-momentum tensor is c…
The influence of the gravity acceleration on the regularized energy-momentum tensor of the quantized electromagnetic field between two plane parallel conducting plates is derived. A perturbative expansion, to first order in the constant acceleration parameter, of the Green functions involved and of the energy-momentum tensor is derived by means of the covariant geodesic point splitting procedure. The energy-momentum tensor is covariantly conserved and satisfies the expected relation between gauge-breaking and ghost parts.
The role of bulk matter quantum effects (via the corresponding effective potential discussed on the example of conformal scalar) and of boundary matter quantum effects (via the conformal anomaly induced effective action) is considered in brane-world cosmology. Scenario is used where brane tension is not free parameter, and the initial bulk-brane classical action is defined by some considerations. The effective bulk-brane equations of motion are analyzed. The quantum creation of 4d de Sitter or Anti-de Sitter(AdS) brane Universe living in 5d AdS space is possible when quantum bulk and (or) bran…
We discuss the Casimir effect for massless scalar fields subject to the Dirichlet boundary conditions on the parallel plates at finite temperature in the presence of one fractal extra compactified dimension. We obtain the Casimir energy density with the help of the regularization of multiple zeta function with one arbitrary exponent and further the renormalized Casimir energy density involving the thermal corrections. It is found that when the temperature is sufficiently high, the sign of the Casimir energy remains negative no matter how great the scale dimension $δ$ is within its allowed regi…
In this paper, the finite temperature Casimir force acting on a two-dimensional Casimir piston due to electromagnetic field is computed. It was found that if mixed boundary conditions are assumed on the piston and its opposite wall, then the Casimir force always tends to restore the piston towards the equilibrium position, regardless of the boundary conditions assumed on the walls transverse to the piston. In contrary, if pure boundary conditions are assumed on the piston and the opposite wall, then the Casimir force always tend to pull the piston towards the closer wall and away from the equi…
We study the Casimir effect for parallel plates with massless Majorana fermions obeying the bag boundary conditions at finite temperature. The thermal influence will modify the effect. It is found that the sign of the Casimir energy keeps negative if the product of plate distance and the temperature is larger than a special value or the energy will change to be positive. The Casimir energy rises with the stronger thermal influence. We show that the attractive Casimir force between two parallel plates becomes greater with the increasing temperature. In the case of piston system involving the sa…
The Casimir effect for parallel plates satisfying the Dirichlet boundary condition in the context of effective QED coming from a six-dimensional Nielsen-Olesen vortex solution of the Abelian Higgs model with fermions coupled to gravity is studied at finite temperature. We find that the sign of the Casimir energy remains negative under the thermal influence. It is also shown that the Casimir force between plates will be weaker in the higher-temperature surroundings while keeps attractive. This Casimir effect involving the thermal influence is still inconsistent with the known experiments. We fi…
The Casimir effect for Dirac as well as for scalar charged particles is influenced by external magnetic fields. It is also influenced by finite temperature. Here we consider the Casimir effect for a charged scalar field under the combined influence of an external magnetic field and finite temperature. The free energy for such a system is computed using Schwinger's method for the calculation of the effective action in the imaginary time formalism. We consider both the limits of strong and weak magnetic field in which we compute the Casimir free energy and pressure.
We consider the finite temperature Casimir effect of a massive fermionic field confined between two parallel plates, with MIT bag boundary conditions on the plates. The background spacetime is $M^{p+1}\times T^q$ which has $q$ dimensions compactified to a torus. On the compact dimensions, the field is assumed to satisfy periodicity boundary conditions with arbitrary phases. Both the high temperature and the low temperature expansions of the Casimir free energy and the force are derived explicitly. It is found that the Casimir force acting on the plates is always attractive at any temperature r…
In this article, we study the finite temperature Casimir effect for scalar field with Robin boundary conditions on two parallel plates in a background spacetime that has a compact internal manifold with arbitrary geometry. The finite temperature Casimir force acting on the plates is derived using the piston approach, with which we show that the divergences of the Casimir forces from the region between the plates and the region outside the plates cancel each other. The nature and asymptotic behaviors of the Casimir force at different limits such as small and large plate separation, low and high…
We consider the finite temperature Casimir interaction between two Dirichlet spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the term with zero Matsubara frequency, and it is known as the classical term since this term is independent of the Planck constant $\hbar$. Explicit expression of the classical term is derived and it is computed exactly using appropriate similarity transforms of matric…
In a recent paper [arXiv:0904.2904] using a conjecture it is shown how one can calculate the effect of a weak stationary gravitational field on vacuum energy in the context of Casimir effect in an external gravitational field treated in 1+3 formulation of spacetime decomposition.. In this article, employing quntum field theory in curved spacetime, we explicitly calculate the effect of a weak static gravitational field on virtual massless scalar particles in a Casimir apparatus. It is shown that, as expected from the proposed conjecture, both the frequency and renormalized energy of the virtual…
The Galilean covariance, formulated in 5-dimensions space, describes the non-relativistic physics in a way similar to quantum field theory. Using a non-relativistic approach the Stefan-Boltzmann law and the Casimir effect at finite temperature for a particle with spin zero and $1/2$ are calculated. The Thermo Field Dynamics is used to calculate the temperature effect.
In this paper we use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzman law and Casimir effect both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) and the quantities are analyzed once projected in the coordinates space.
A spatially flat Friedmann-Robertson-Walker background with a general scale factor is considered. In this space-time, the energy-momentum tensor of the scalar field with a general curvature coupling parameter is obtained. Using the Thermo Field Dynamics (TFD) formalism the Stefan-Boltzmann law and the Casimir effect at finite temperature are calculated. The Casimir effect at zero temperature is also considered. The expansion of the universe changes these effects. A discussion of these modifications is presented.
In this letter, we derive the explicit exact formulas for the finite temperature Casimir force acting on a pair of parallel plates in the presence of extra compactified dimensions within the framework of Kaluza-Klein theory. Using the piston analysis, we show that at any temperature, the Casimir force is always attractive and the effect of extra dimensions becomes stronger when the size or number of the extra dimensions increases. These properties are not affected by the explicit geometry and topology of the Kaluza-Klein space.
The Casimir effect for a three-parallel-plate system at finite temperature within the frame of five-dimensional Randall-Sundrum models is studied. In the case of Randall-Sundrum model involving two branes we find that the Casimir force depends on the plates distance and temperature after one outer plate has been moved to the distant place. Further we discover that the sign of the reduced force is negative as the plate and piston locate very close, but the reduced force nature becomes repulsive when the plates distance is not very tiny and finally the repulsive force vanishes with extremely lar…
We consider the Casimir force acting on a $d$-dimensional rectangular piston due to massless scalar field with periodic, Dirichlet and Neumann boundary conditions and electromagnetic field with perfect electric conductor and perfect magnetic conductor boundary conditions. It is verified analytically that at any temperature, the Casimir force acting on the piston is always an attractive force pulling the piston towards the interior region, and the magnitude of the force gets larger as the separation $a$ gets smaller. Explicit exact expressions for the Casimir force for small and large plate sep…
The free energy due to the vacuum fluctuations of matter fields on a classical gravitational background is discussed. It is shown explicitly how this energy is calculated for a non-minimally coupled scalar field in an arbitrary gravitational background, using the heat kernel method. The treatment of (self-)interacting fields of higher spin is outlined, using a meanfield approximation to the gaugefield when treating the gauge boson self interaction and the fermion-gauge boson interaction.
This paper studies the Casimir effect due to fractional massless Klein-Gordon field confined to parallel plates. A new kind of boundary condition called fractional Neumann condition which involves vanishing fractional derivatives of the field is introduced. The fractional Neumann condition allows the interpolation of Dirichlet and Neumann conditions imposed on the two plates. There exists a transition value in the difference between the orders of the fractional Neumann conditions for which the Casimir force changes from attractive to repulsive. Low and high temperature limits of Casimir energy…
We find the free energy of the string by applying the known Matsubara formalism. Then through the Casimir effect we offer a temperature correction to the tachyon mass of the string. We see that for the fermionic part the temperature correction is precisely the opposite of that of bosonic part, so the quantum ground state of the superstring would remain massless, as expected.
In this paper, a massless scalar field coupled to gravity is considered. Then the Casimir effect at finite temperature is calculated. Such development is carried out in the Thermo Field Dynamics formalism. This approach presents a topological structure that allows for investigating the effects of temperature and the size effect in a similar way. These effects are calculated considering Gödel-type solutions as a gravitational background. The Stefan-Boltzmann law and its consistency are analyzed for both causal and non-causal Gödel-type regions. In this space-time and for any region, the Casimir…
Currently, the discussions and investigations for the vacuum energy is drawing great both theoretical and experimental attention. The vacuum states of variety of fields, subject to special boundary conditions, may contribute to non-trivial macroscopic vacuum energy, i.e. the Casimir effect, which become an interdisciplinary subject and plays an important role in a variety of fields of physics. We adopt Schwinger's source theory and study the Casimir effect due to the quantization of gravitation, i.e. the gravitational Casimir effect (GCE), in inspiraling neutron star (NS) binaries with wide se…
Fermionic Casimir effect at finite temperature in Horava-Lifshitz theories
本研究针对Horava-Lifshitz理论中违反洛伦兹不变性的无质量费米子场，采用广义ζ函数技术研究满足MIT袋边界条件的平行平板间的有限温度卡西米尔效应，得到了不同参数与温度极限下自由能和卡西米尔压强的解析表达式。
By the thermofield dynamics (TFD) formalism we obtain the energy-momentum tensor for the Electromagnetism with Lorentz Breaking Even term of the Standard Model Extended (SME) Sector in a topology $S^{1}\times S^{1}\times R^{2}$. We carry out the compactification by a generalized TFD-Bogoliubov transformation that is used to define a renormalized energy-momentum tensor, and the Casimir energy and pressure at finite temperature are then derived. A comparative analysis with the electromagnetic case is developed, and we remark the influence of the background in the traditional Casimir effect.
The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.
The standard model and general relativity are local Lorentz invariants. However it is possible that at Planck scale there may be a breakdown of Lorentz symmetry. Models with Lorentz violation are constructed using Standard Model Extension (SME). Here gravitational sector of the SME is considered to analyze the Lorentz violation in Gravitoelectromagnetism (GEM). Using the energy-momentum tensor, the Stefan-Boltzmann law and Casimir effect are calculated at finite temperature to ascertain the level of local Lorentz violation. Thermo Field Dynamics (TFD) formalism is used to introduce temperature…
Recently a great deal of interest in field theories formulated in a Lorentz violating framework has been developed. Here the Very Special Relativity (VSR) is considered. The main aspect in the VSR proposal is that laws of physics are invariant under the subgroups of the Poincaré group preserving the basic elements of special relativity. An important point is that the photon acquires a mass. In this context, the energy-momentum tensor for the electromagnetic field is calculated. From this, the Stefan-Boltzmann law and Casimir effect at finite temperature in VSR are obtained. The effects of temp…
The Casimir effect at finite temperature is investigated for a dilute dielectric ball; i.e., the relevant internal and free energies are calculated. The starting point in this study is a rigorous general expression for the internal energy of a system of noninteracting oscillators in terms of the sum over the Matsubara frequencies. Summation over the angular momentum values is accomplished in a closed form by making use of the addition theorem for the relevant Bessel functions. For removing the divergences the renormalization procedure is applied that has been developed in the calculation of th…
The one-loop correction to the spectrum of Kaluza-Klein system for the $ϕ^3$ model on $R^{1,d}\times (T_θ^2)^L$ is evaluated in the high temperature limit, where the $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the $L$ extra two-dimensional tori are chosen to be the noncommutative torus with noncommutativity $θ$. The corrections to the Kaluza-Klein mass formula are evaluated and used to compute the Casimir energy with the help of the Schwinger perturbative formula in the zeta-function regularization method. The results show that the one-loop Casimir energy is independent of …
The survey summarizes briefly the results obtained recently in the Casimir effect studies considering the following subjects: i) account of the material characteristics of the media and their influence on the vacuum energy (for example, dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to gain some insight, specifically, in the geometrical origin of the divergences that enter the vacuum energy and to develop the relevant renormalization procedure; iii) universal method for calculating the high tempera…
A non-Abelian gauge theory describes the strong interactions among particles with the commutator of generators are non-zero. An $SU(3)$ gauge theory describes the interactions that lead to nuclear forces among particles. The Lagrangian density refers to fermions with colour and flavour and the gauge field quanta implying gluons. The gauge theory is treated at finite temperature using the Thermo Field Dynamics (TFD). Using self interaction of gluons the Stefan-Boltzmann law and the Casimir effect are calculated at finite temperature. An appendix is attached to give a response of a massless quar…
We use the thermofield dynamics (TFD) formalism to obtain the energy-momentum tensor for the Kalb-Ramond (KR) field in a topology $% S^{1}\times S^{1}\times R^{2}$. The compactification is carried out by a generalized TFD-Bogoliubov transformation that is used to define a renormalized energy-momentum tensor. The expressions for the Casimir energy and pressure at finite temperature are then derived. A comparative analysis with the electromagnetic case is developed, and the results may be important for applications, as in cuprate superconductivity, for instance.
Hot Casimir wormholes in Einstein-Gauss-Bonnet gravity
本文在高维EGB引力中引入经热修正的Casimir能量，推导虫洞形状函数并验证形成条件，揭示热修正对喉部曲率与可穿越性的影响。
The survey summarizes briefly the results obtained recently in the Casimir effect studies considering the following subjects: i) account of the material characteristics of the media and their influence on the vacuum energy (for example, dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to gain some insight, specifically, in the geometrical origin of the divergences that enter the vacuum energy and to develop the relevant renormalization procedure; iii) universal method for calculating the high tempera…
We calculate the finite vacuum energy density of the scalar and electromagnetic fields inside a Casimir apparatus made up of two conducting parallel plates in a general weak gravitational field. The metric of the weak gravitational field has a small deviation from flat spacetime inside the apparatus and we find it by expanding the metric in terms of small parameters of the weak background. We show that the found metric can be transformed via a gauge transformation to the Fermi metric. We solve the Klein-Gordon equation exactly and find mode frequencies in Fermi spacetime. Using the fact that t…
I investigate the finite temperature Casimir effect for a charged and massless scalar field satisfying mixed (Dirichlet-Neumann) boundary conditions on a pair of plane parallel plates of infinite size. The effect of a uniform magnetic field, perpendicular to the plates, on the Helmholtz free energy and Casimir pressure is studied. The zeta-function regularization technique is used to obtain finite results. Simple analytic expressions are obtained for the zeta function and the free energy, in the limits of small plate distance, high temperature and strong magnetic field. The Casimir pressure is…
We use a functional approach to evaluate the Casimir free energy for a self-interacting scalar field in $d+1$ dimensions, satisfying Dirichlet boundary conditions on two parallel planes. When the interaction is turned off, exact results for the free energy in some particular cases may be found, as well as low and high temperature expansions based on a duality relation that involves the inverse temperature $β$ and the distance between the mirrors, $a$. For the interacting theory, we derive and implement two different approaches. The first one is a perturbative expansion built with a thermal pro…
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