Thermal Casimir effect with general boundary conditions

In this paper we study the system of a quantum field confined between two plane parallel plates at finite non-zero temperature and thermal equilibrium. We will represent the plates by the most general boundary conditions enabled by the principles of quantum field theory, assuming they are isotropic and homogeneous. Under these assumptions we will be able to compute the thermal correction to the quantum vacuum energy as a function of the free parameters enclosed in the boundary condition and the temperature. The latter will enable to obtain similar results for the pressure between plates and the quantum thermal correction to the entropy.


Introduction
The so-called Casimir effect between two dielectric plates is a long-standing phenomenon profusely studied, both theoretically [1][2][3][4][5] and experimentally [6,7]. In physical grounds, the Casimir energy is the interaction energy of the quantum fluctuations of the electromagnetic field with the charged current fluctuations of the plates [8,9]. For separation distances much larger than any other length scale which determines the electric response of the plates, only the long-wavelength transverse modes of the electromagnetic field are relevant to the interaction, and they can be mimicked by the normal modes of a scalar field [3,10,11]. In the last years, motivated by its applications to the design of electronic devices at a e-mail: jose.munoz.castaneda@uva.es b e-mail: lucia.santamaria@uva.es c e-mail: manuel.donaire@uva.es d e-mail:marcos.tello@uva.es nanometer scales [12], a renewal interest in the problem has arisen at finite temperature. To this respect, from a theoretical perspective, two are the main issues which efforts have focused on. In the first place, there exists an intense debate on the appropriate description of the dielectric response of metallic plates in terms of dissipative or lossless models. In particular, whereas the optical response of any metal at low frequency needs of the finite conductivity of Drude's model, the results of experiments on the Casimir pressure are better described by the lossless plasma model, which has come to be referred to as the Drude-plasma puzzle [13,14]. On the other hand, directly motivated by technological applications of the Casimir effect, and indirectly motivated by Casimir-like effects in cosmological models [15], there exists an interest on the possibility of finetuning the sign of the Casimir energy and the direction of the Casimir pressure with the temperature.
In this article, we address the problem of the Casimir effect at finite temperature from a mathematical perspective. Eventually, our results might find physical realisation in some realistic setups in which the electromagnetic response of the plates be well approximated by uniform and constant Robin's boundary conditions. In particular, we compute the free energy of a scalar field confined between two homogeneous parallel plates in thermal equilibrium, subject to the most general type of dispersionless boundary conditions. Physically, the normal modes of the scalar field are intended to mimic the transverse modes of the electromagnetic field. In regards to the dispersionless boundary conditions, they are intended to mimic the permittivities and permeabilities of lossless plates which, for the range of wavelengths relevant to the problem, behave as effective constants. Obviously, the aforementioned discussion on the Drude-plasma puzzle for metallic plates lies out-side the scope of the present article. Nonetheless, some general lessons can be learned out of our simplified model in regards to the role of boundary conditions and its interplay with thermal field fluctuations. To this end, we extend the results of the works of Asorey-Ibort-Marmo and Asorey-Munoz-Castaneda on a massless scalar field at zero temperature to the thermal environment [16,17].
The quantum vacuum energy of a massless scalar field confined between two parallel plates with general boundary conditions was studied in [17], using the theory of selfadjoint extensions for the Laplace-Beltrami operator developed in [16]. The most remarkable result of Ref. [17] is the computation of the quantum vacuum energy for a scalar field confined between two homogeneous isotropic plates as a function over the space of those selfajoint extensions that are allowed in quantum field theory. As a consequence the authors were able to characterise those selfadjoint extensions that give rise to attractive, repulsive or null Casimir force between plates. The aim of this paper, is to extend the results obtained in [17] to the same quantum field theory but at finite temperature by computing the thermal correction to the free energy of the system. In addition the formalism developed will enable us to compute any other thermodynamical quantities such as the entropy and the Casimir pressure at finite temperature as functions of the free parameters of the boundary conditions.
The article is organised as follows. In Section 2 we make a compilation of basic formulas and previous results needed to obtain the main calculations of the paper. In Section 3 we proceed to the calculation of the Helmholtz free energy, and entropy for quantum massless scalar fields confined between two plates with general boundary conditions. Afterwards in Section 4 we use the previous section's formulas to compute the Casimir quantum pressure between plates at finite temperature. To conclude, in Section 5 we present our conclusions.
Throughout the paper we will use a system of units such that = c = k B = 1, being the Plank constant, c the speed of light, and k B the Boltzman constant. In this paper we will study a massless scalar field confined between two homogenous parallel plates mimicked by the most general type of boundary conditions. There is a highly dependence of the quantum vacuum state and the vacuum energy on the geometry of the physical space and the physical properties of the boundaries that interact with the quantum field, that are encoded in the boundary conditions [2,3,5,[18][19][20][21].
We consider a free massless complex scalar field φ confined in a domain Ω ∈ R 3 bounded by two parallel homogenous two-dimensional plates orthogonal to the x-axis and placed at x = 0, L, i. e.. Ω = [0, L] × R 2 . In this situation the classical action for the massless scalar field that gives rise to local equations of motion is given by After a standard canonical second quantization the equation for the modes of the scalar quantum field is given by the non-relativistic Schrödinger eigenvalue problem Splitting the spatial coordinate as x = (x, y ) with y = (y, z) ∈ R 2 and x ∈ [0, L] we can separate variables in the mode equation above by writing the modes of the quantum field as φ ω (x) = ψ k (y )g k (x). Under these assumptions the total Laplace operator can be splitted as Assuming that the two plates are isotropic and homogeneous ∆ is nothing but the Laplace-Beltrami operator in R 2 . Hence the only nontrivial equation eigenvalue equation that is leftover is the one corresponding to the OX 3 axis: The Laplace operator ∆ over Ω = [0, L] × R 3 is not essentially selfadjoint, so it admits an infinite set of selfadjoint extensions. To respect the unitarity principle of quantum field theory we must only take into account those selfadjoint extensions of the Laplace operator that give rise to non-negative selfadjoint operators for all L ∈ (0, ∞). The set of selfadjoint extensions of ∆ in Ω has been widely studied. From a physical point of view the most meaningful way to determine the set of selfadjoint extensions is given in Ref. [16]. Under our assumptions of homogeneity and isotropy of the plates the set of selfadjoint extensions of the Laplacian ∆ over Ω is in one-to-one correspondence with the set of selfadjoint extensions of the operator −d 2 /dx 2 over [0, L] which are given by the group U (2) (see Ref. [17]). The domain D U that defines the selfadjoint extension ∆ U is given in terms of the matrix U ∈ U (2) (see [17,22]) by where n = (n 1 , n 2 , n 3 ) is an unitary vector, σ are the Pauli matrices, the angles α, θ are such that α ∈ [0, π] and θ ∈ [−π/2, π/2] and a is a fundamental length scale related to the electromagnetic response of the plates. Note that, except for the trivial choice a = 0, in which case Eq.(4) reduces to Dirichlet's boundary conditions, Eq.(4) leaves us with two independent length scales at our disposal, in addition to the temperature. From a physical perspective, considering φ as the effective field of transverse electromagnetic modes, and noting that a relates field values and field derivatives, the length a can be associated to the relationship between the electric and magnetic response of the plates. The space boundary conditions M F that give rise to non-negative sefladjoint extensions ∆ U of the Laplacian operator is 1 We can characterize the non-zero part of the spectrum for any selfadjoint extension ∆ U ∈ M F including multiplicities of eigenvalues throughout the secular equation obtained in [17] h U (k) = sin (kL)[(k 2 a 2 − 1) cos θ + (k 2 a 2 + 1) cos α] − 2ka cos (kL) sin α − 2ka n 1 sin θ, where, by assumption, we consider that the boundary condition parameters do not depend on k and are uniform on the plates. In addition, since they are temperature independent, Eq. (7) is equivalent to the spectrum of normal modes obtained in Refs. [17,22]. Note that since f U contains terms in different powers of the dimensionless quantities kL and ka, the spectral function varies with respect to a and L in an independent manner. Hereafter, in order to simplify matters we will disregard the trivial case a = 0 and consider without loss of generality a = 1, unless stated otherwise. Thus, the separation length L and the inverse of the temperature T −1 will be expressed in units of a in most of the reminder of this paper.
The vacuum energy is given by the sum of the eigenvalues of This sum is ultraviolet divergent due to the contributions of the energy density of the field theory in the bulk and the surface energy density associated to the plates but there are finite volume corrections to the vacuum energy that give rise to a finite neat Casimir effect. These divergencies can be subtracted to obtain a finite result. Following [17,23] we can write the zero temperature finite Casimir energy per unit area of the plates in three dimensions as where, A is the area of the plates and h and defining the polynomial 2.2 Free energy and thermodynamics.
When considering thermal excitations of an essemble of particles, the statistical behaviour of such ensemble is characterized by a temperature T and a probability distribution once the equilibrium is reached. In our case a quantum scalar field between plates is nothing but an infinite collection of harmonic oscillators that do not interact between each other. The system is characterised by the grand canonical partition function, Z(T ) which will be computed in the next section. The Helmholtz free energy of the system in thermal equilibrium at a temperature T is given in terms of the partition function as where β = 1/T is the inverse temperature. We can split the free energy (12) in two parts [5]: The first one corresponds to the quantum vacuum energy at zero temperature, E vac 0 ≡ F 0 T =0 . The second part, ∆ T F 0 corresponds to a purely thermal term and it encloses all the temperature dependence of F. It is the first part the one which carries all the divergences; thus, it must be renormalized [5]. The divergences and regularization methods for E vac 0 have been largely studied (see e.g. Refs. [2,3,5]), and in our case the finite quantum vacuum energy will be given by (9) following Refs. [17,23]. Once we have computed the free energy, other thermodynamic quantities can be computed easily. In particular, we will focus our attention in the entropy (S) and the force between plates per unit of area of the plates, i. e. the pressure P The main aim of this work is to calculate the thermal correction ∆ T F 0 to the vacuum energy E vac 0 , the entropy, and the quantum vacuum pressure of the system for arbitrary temperature and arbitrary boundary conditions allowed by the principles of quantum field theory.

Free energy and entropy at finite temperature
The fact that the boundary condition parameters are dispersionless and hence do not depend on k in Eq. (7), as well as the restriction of the spectrum to real modes, allow us to simplify greatly the expression of the free energy. Indeed, we take into account that in our case the Hamiltonian of the quantum field theory can be written as a formal summation over the normal modes of the quantum field as where σ(∆ U ) is the spectrum of the corresponding selfadjoint extension ∆ U . Therefore we can simply write Thus, we can treat the system as an ensemble of noninteracting oscillators with energy levels where ω 2 is an eigenvalue of the selfadjoint extension of ∆ U with boundary condition given by a certain matrix U ∈ M F , i. e. the non-zero eigenvalues {ω 2 } are given by the zeros of f U in Eq. (7). The fact that the ensemble of harmonic oscillators do not interact enables to write the partition function of the quantum field theory as an infinite product of harmonic oscillator canonical partition functions, one for each frequency ω (ω 2 ∈ σ(∆ U )): It is a well known result that the canonical partition function for a single harmonic oscillator of frequency ω can be written as [24] Z os (T ) = and the corresponding free energy is Hence from this expression and using Eqs. (12) and (19) after some straightforward manipulations we obtain for the total Helmholtz free energy the well known result The first term in the last equality corresponds to the quantum vacuum energy at zero temperature and it has ultraviolet divergences that need to be regularised, and the second term is the temperature dependent part of the free energy that is finite. After regularization and dropping all the divergences E 0 (T = 0) is given by formula (9). The temperature dependent part of the free energy can be computed as the sum of the Boltzman factors B(ω, T ) over the quantum field modes that form the spectrum The summation over the field modes ω can be separated into the summation over the parallel modes which is an integration over k , and the discrete summation over the transverse modes k.

Summation over the parallel modes modes
Starting from Eq. (23) and taking into account that the frequencies of the field modes are given by ω = k 2 + k 2 , being k the 2-dimensional parallel momenta and k the discrete orthogonal momenta (which can be obtained from the non-null zeroes of the spectral function, i.e., Z * (h U )). Hence the summation over the whole spectrum σ(∆ U ) when ∆ U does not have zero modes transforms into being A the area of the plates. The integration over the parallel momenta can be commuted with the summation over the discrete transverse momenta. Doing so the integration over the parallel momenta reads This integration can be performed analytically using Mathematica, to obtain where Li s (z) denotes the polylogarithmic function of order s [25]. Now it is only left the summation over transverse momenta, i. e. the zeros of h U (k) different from k = 0. At this point we need to distinguish between those boundary conditions that give rise to a selfadjoint extension that does not have zero modes, and those that do. The subset M (0) F ∈ M F of selfadjoint extensions that admit zero modes was characterised in Ref. [22]. Furthermore in Ref. [22] it was demonstrated that any ∆ U ∈ M The summation over transverse momenta is equivalent the zeros of h U (k) different from k = 0. As was explained in Ref. [22] when U ∈ M F − M (0) F , the spectral function h U (k) from Eq. (7) needs to be replaced by in order to be able to write the summation over the discrete transverse momenta as a contour integral [26,27] avoiding the possible problems at k = 0. Hence if we take into account that summing over the spectrum of discrete transverse momenta is equal to sum over the zeros of the secular function f U (k), the final formula for the temperature dependent part of the free energy is when being Z(f U ) the set of zeros of f U (k). The sum over the zeros of the secular equation f U (k) can be written down by using a complex contour integral as where Γ is the contour shown in Fig. 1, which encloses all the zeroes of f U (k) when R → ∞. The integral (29) is well defined because f U (k) is a holomorphic function on k. When R → ∞ in the contour of in Fig. 1 the integration over the circumference arc z = Re iµ , with µ ∈ [−γ, γ], goes to zero. Hence taking the limit R → ∞, integrating over the whole contour in Fig. 1 leaves us with the integration over the two straight lines z = ξe ±iγ being γ a constant angle and ξ ∈ [0, ∞): The residue theorem ensures that the result of this integration does not depend on the angle γ taken in the contour because all the zeros enclosed lie on R + since F has no zero modes and it is a definite positive selfadjoint operator [17]. It is of note that when we take γ = π/2, and Cauchy Principal Values we obtain the well known Matsubara formula after integrating by parts in ξ [5,8,13,28]. This is shown in Refs. [29,30].   F with Mathematica to obtain ∆ T F(U ). In addition, formula (9) from Refs. [17,23] enables to obtain the zero temperature energy which summed to ∆ T F(U ) gives the total Helmholtz free energy as a function of the temperature T and the parameters of the most general boundary condition allowed by the principles of quantum field theory.
Low temperature behaviour of the Helmholtz energy. In Figs. 2-4 we can observe the numerical results for the free energy at low temperatures (T = 0.35). In each figure we can see the quantum vacuum energy at T = 0 (left plots) computed with formula (9), the thermal correction ∆ T F(α, θ, n 1 ) (central plots) and the total free energy F(α, θ, n 1 ; T ) (right plots) as functions of the parameters {α, θ, n 1 } defining the boundary condition. It can be seen that although the thermal correction ∆ T F is definite negative for any boundary condition, the total free energy can be positive negative or zero, in a similar way as it happens for the quantum vacuum energy at zero temperature [17]. Nevertheless, unlike it happens for T = 0 where the quantum vacuum energy behaves with the distance between plates as for any arbitrary finite T > 0 positive or negative total Helmholtz energy does not ensure attractive or repulsive quantum vacuum thermal force. The quantum vacuum thermal force is discussed in the next section.
High temperature behaviour of the Helmholtz energy.
The thermal correction ∆ T F is negative and monotone decreasing with T , hence as the temperature grows it dominates the total Hemholtz free energy. As can be seen in Fig. 5 when T > 1 the temperature dependent part of the Helmholtz free energy ∆ T F is negative and when compared with the left plots in Figs. 2-4. Therefore at high temperature the total Helmholtz free energy is always negative.
On the critical temperature T F c . From the numerical results discussed above, we infer that for a fixed length L there should be a critical temperature T F c such that for any T > T F c there are no boundary conditions giving rise to positive total Helmholtz energy. On the other hand whenever T < T F c there will be boundary conditions for which the total Helmholtz energy F is positive. From Ref. [17] we know that the boundary condition for which the quantum vacuum energy at T = 0 reaches its maximum is given by anti-periodic boundary conditions: Moreover, the quantum vacuum energy for anti-periodic boundary conditions can be obtained analytically: Since ∆ T F is a monotone decreasing function of T , we can ensure that the situation in which the possibility of having positive total Helmholtz free energy at a given T disappears whenever The condition (34) gives an equation in T and L than enables to obtain numerically T F c . The analytical solution can not be obtained. Nevertheless in Fig. 6 it is shown T F c as a function of the length L. It is of note that the critical temperature T F c does not separate the regimes in which quantum vacuum force is fully repulsive and the case in which it can be repulsive, attractive or zero.
The entropy. The entropy arising from Casimir selfenergies has been a field of a lot of activity in recent years since it was noticed in Ref. [31] that the quantum vacuum energy at finite temperature can give rise to negative corrections to the entropy. The existence  of negative entropy is interpreted as a hint of possible instabilities in this kind of systems [32]. Therefore, the calculation of the entropy for the system we are studying is of great interest to infer if there are boundary conditions that can give rise to negative entropy corrections. By using formula (14) we can compute numerically the entropy as a function of the parameters that determine the boundary condition for any arbitrary temperature. In Figs. 7-8 we show the results of the behaviour of the entropy as a function of the parameters of the boundary condition for low and high temperature. As can be seen the entropy is positive definite for any {α, θ, n 1 } and is monotone increasing with T . This ensures that there is no boundary condition giving rise to negative entropy corrections, and therefore all the boundary conditions are stable. Moreover the maximum entropy is reached for Neumann boundary conditions (α = θ = 0) for any T as it will be clarified in the next section. In addition de minimum entropy occurs for Dirichlet boundary conditions (α = 2π, θ = 0) for any T > 0.

Quantum field theories with zero modes:
The boundary conditions in M F that give rise to a quantum field theory where there are zero-modes was studied in detail in Ref. [22]. Concerning these boundary conditions the two most important results from Ref. [22] are: • There are boundary conditions in the space M F for which ∆ U has at most one constant zero-mode: the space F can be characterised in terms of the parameters of the unitary matrices that determine the boundary condition as: In this situation, the spectral function that must be used is given by [22] f (0) Plugging n 1 = ±1, θ = −n 1 α into the equation above and taking into account Eq.(7) we obtain k cos(α) sin(kL) + sin(α)(1 − cos(kL)) k 2 It is of note that the zeros of f U ), characterises the non-zero spectrum of the discrete transverse momenta. Therefore the whole spectrum of transverse momenta when U ∈ M Hence in this case the summation over the spectrum σ(∆ U ) splits into two terms: The integration accounts for all the field modes characterised by frequencies ω = k 2 + 0 2 , that are not included when we perform the summation over the zeroes of f (0) U (k). The integral (39) can be obtained analytically Hence the temperature dependent part of the Helmholtz free energy for the case in which there are zero modes reads: The extra term arising due to the existence of transverse zero-modes will not contribute to the force between plates since it does not depend on the distance between plates. Nevertheless, it contributes to the dominant term of the free energy at high temperature since The second term in (41) can be again computed using (30) replacing f U (z) by f F . Periodic boundary condition (α = π/2) corresponds to the minimum of F| M0 for any T < 1 (see Fig 9). In contrast, when the temperature is sufficiently high (see Fig. 10) F| M0 becomes a monotone increasing function of α, so the minimum of F| M0 occurs at Neumann boundary condition (α = 0).

Finite temperature Casimir force. Attraction, repulsion and no-force boundary conditions
The finite temperature quantum vacuum force per unit area of the plates, i. e. the quantum vacuum pressure, can be obtained from (15). Since only those terms that depend on the distance between plates contribute to the pressure in this case the general formula does not need to make any distinction between boundary conditions in M (0) F and the rest of boundary conditions in M F because the zero mode contribution in (41) does not depend on the distance L. Taking this into account the general formula for the temperature dependent part of the quantum vacuum pressure reads: In addition the zero-temperature pressure can be easily obtained from formula (9) if we take into account that being c U a coefficient that does not depend on the distance between plates L [17]. Hence the zero temperature quantum vacuum pressure can be obtained as Hence, putting together formulas (43) and (45) we finally obtain the quantum vacuum pressure for any temperature and any boundary condition U ∈ M F P U (T ) = P U (T = 0) + ∆ T P U .
It is of note that since ∆ T P U does not behave with the distance between plates as L −4 the regions where the quantum vacuum force becomes attractive, repulsive or zero do not match the regions when the total quantum vacuum energy F is negative, positive, or zero. In Fig.11 we show the numerical results for the quantum vacuum pressure at finite temperature computed using formulas (43), (45), and (46). As can be seen the quantum vacuum pressure still gives rise to attraction, repulsion or no-force regimes when the temperature is low enough. Furthermore the minimum pressure occurs for periodic boundary conditions (see plot n 1 = 1 lefthand-side corner in Fig.11) and the maximum pressure happens for anti-periodic boundary conditions (see plot n 1 = 1 right-hand-side corner in Fig.11) as it happens for T = 0 (see Ref. [17]). The biggest temperature effect at low temperature happens when n 1 = 0 where the attractive regime almost disappear with respect the T = 0 case (see plot n 1 = 0 in Fig.11 and its analogue for T = 0 from Ref. [17]).

Critical temperature: the fully repulsive regime
As it expected from the previous results, as the temperature increases the quantum vacuum pressure will be dominated by the thermal fluctuations, which tend to produce a repulsive force. Hence the temperature at which the minimum quantum vacuum pressure is equal to zero defines a critical temperature T P c that separates the thermal-dominated regime (if T > T P c there is only repulsion due to the thermal fluctuations for any U ∈ M F ) and the quantum vacuum fluctuations dominated regime (if T > T P c there can be attractive, repulsive or null quantum vacuum pressure). Since the minimum quantum vacuum pressure occurs for periodic boundary conditions the equation that enables to obtain the critical temperature T P c for a given distance between plates L is being P Up (T = 0) the Casimir pressure at zero temperature, given by [17] The Eq. (47) can not be solved analytically, so we have to proceed by using numerical methods. In Fig. 12 we show the numerical results for T P c . As can be seen the temperature at which the possibility of having attractive quantum vacuum pressure disappears (T P c ) is higher that the critical temperature T F c at which the total quantum vacuum energy F becomes definite negative.

Conclusions and further comments
In this paper we considered a massless scalar field confined between two plates mimicked by the most general boundary conditions enabled by the principles of the theory of quantum fields at finite temperature different from zero. We have been able to compute and study the most important thermodynamic magnitudes as functions of the free parameters entering the boundary condition and the temperature: the Helmholtz free energy, the entropy and the pressure.
Concerning the Helmholtz free energy the main result obtained is the existence of possible sign change in the energy for temperatures under a certain critical temperature T F c (see Fig. 6) as can be seen in Figs. 2-4. As it happens for T = 0 (see Ref. [17]) for T < T F c the maximum of the free energy occurs for anti-periodic boundary conditions, and the minimum is reached for periodic boundary conditions (see Fig. 3). Nevertheless since the finite temperature correction to the free energy ∆ T F has its minimum for Neumann boundary conditions 2 (see Figs. 2-5 9 and 10) and its maximum at Dirichlet boundary conditions 3 when the temperature is high enough the maximum and the minimum of F are Dirichlet and Neumann boundary conditions respectively.
Regarding the entropy as a function of the free parameters mimicking the plates and the temperature we find out that the one-loop quantum correction to the entropy is positive definite for any temperature and any U ∈ M F . Taking into account recent research where there has been found negative one-loop quantum corrections to the entropy in Casimir-like systems [31,[33][34][35][36] and that this phenomenon suggests certain instabilities of the quantum system [32], we can conclude that the quantum system of a scalar field confined between two plates mimicked by the most general boundary conditions is always stable. Moreover, for any temperature the maximum entropy is reached for Neumann boundary conditions meanwhile the minimum happens for Dirichlet boundary conditions (see Figs. 7 and 8). It is remarkable that the extrema of the entropy for any temperature occur for the most unstable boundary renormalization group flow fixed point (Dirichlet boundary condition) and the most stable boundary renormalization group flow fixed point (Neumann boundary condition) [37].
For the Casimir pressure at finite temperature we have obtained the critical temperature T P c that separates the thermal fluctuation dominated regime (T > T P c ), and the quantum vacuum fluctuation dominated regime (T < T P c ) (see Fig. 12). On the one hand when T < T P c the effect of the behaviour of the quantum vacuum fluctuations at zero temperature still dominates the pressure behaviour of the system, therefore there are boundary conditions that produce attractive, repulsive or null force between plates (see Fig. 11). On the other had for T > T P c the thermal fluctuations become dominant giving rise to a repulsive force between plates for any boundary condition U ∈ M F . In addition, as it happens at zero temperature, when T < T P c the maximum and minimum of the quantum vacuum force occur for anti-periodic and periodic boundary conditions respectively. As a consequence, we can conclude that the theorem of Kenneth and Klich that states the opposites attract (see Refs. [38,39]) only holds for T < T P c . The simplest way to see this is the Casimir pressure as a function of the temperature for Dirichlet or Neumann boundary conditions shown in Fig. 13. As can be seen attraction between plates is not maintained for any temperature.
It is of note that our results can easily be generalised to arbitrary dimension. If the physical space with boundary in which the scalar field lives is given by [0, L] × R D , then all our arguments and formulas to compute the free energy, the entropy and the pressure remain valid by just changing I 3 defined in Eq. (25) by This integral can be computed in terms of more complicated combinations of polylogarithms of higher or- der. Nevertheless since in any case the arguments of the polylogarithms will be e −k/T it is ensured that non of them will go through the branch cut when performing the integration in k to sum over the orthogonal modes. For the boundary conditions U ∈ M (0) F the dominant contribution of the zero-mode to the free energy will be given by