Modular invariance in finite temperature Casimir effect

The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series.


INTRODUCTION
When studying the Casimir effect [1] at finite temperature [2,3], one cannot fail to be intrigued by the temperature inversion symmetry of the partition function, originally derived using an image-source construction of the Green's function [4]. This result has been rediscussed from various points of view including a derivation in terms of Euclidean path integrals and Epstein zeta functions, while the Casimir energy has been related to the thermodynamic potentials of the system [5][6][7][8][9][10][11].
A natural question is then whether one may include another observable in the partition function so as to enhance the temperature inversion symmetry to transformations under the full modular group and complete the parallel to the massless free boson on the torus with momentum operator included. Whereas for two-dimensional conformal field theories on the torus, the central charge has been related to the Casimir energy of a field theory with boundary conditions in one spatial dimension [12][13][14][15], we show here how techniques developed in that context can be applied in the original setting of the Casmir effect in order to produce exact results.
Our motivation for studying this question originated from an attempt to understand the contribution of specific degrees of freedom related to non-trivial boundary conditions to partition functions. More precisely, a pair of perfectly conducting plates at constant x 3 requires Neumann conditions for the third component of the electric field and the vector potential. This gives rise to an additional polarization at zero value of the quantized transverse momentum whose dynamics is governed by a free massless scalar field in 2+1 dimensions. Its contribution to the partition function scales with the area of the plates and provides the leading correction at low temperature to the zero temperature Casimir result for the free energy [16,17]. Just as the contribution of the zero mode of a free boson on a torus is essential to achieve modular invariance of the partition function [18,19], so is the contribution of this lower-dimensional massless scalar field in the current context.

CAPACITOR PARTITION FUNCTION
The partition function for perfectly conducting parallel plates of area L 2 separated by a distance d, such that L ≫ d, is given by [2][3][4] (see e.g. [20,21] for reviews) and or, equivalently, when using a Sommerfeld-Watson transformation, The next step [8,10] is to extend the sum over all integers (l, m) except for (0, 0) and to recognize the relevant Epstein zeta function, in order to write the result as or, in terms of the free energy, The expression for the partition function in (1) is in line with the discussion in [4], but differs from the corresponding result for the free energy in [20,21] by the last term in (1), which is absent in the latter references. The reason is that the latter approach includes the subtraction of the full free energy of the black body in empty space, whereas this subtraction is limited to the (divergent) zero temperature part in the former approach. This can be done because the thermal part of the free energy is convergent. It means that the system that one considers contains, in addition to the photons between the plates, also those in the large box that surrounds the plates [22,23].
As pointed out in [10], an advantage of the expression as given in (1) is that the inversion symmetry, f (1/t) = t −4 f (t), established in [4], extends to the Epstein zeta function, Z(2; 1/t 2 , 1) = t 4 Z(2; t 2 , 1), and thus turns into a symmetry of the full partition function and the free energy, Up to exponentially suppressed terms, the high temper- where the leading piece is the black body contribution V π 2 45 β −3 while the sub-leading temperature-independent contribution scales like the area; the low temperature expansion t ≫ 1 in turn is given by where the second term is due to the lower-dimensional scalar as described above.

BOUNDARY CONDITIONS: E AND H MODES
Let us now provide some details on electromagnetism with Casimir boundary conditions needed for our purpose. We work in radiation gauge A 0 = 0 = ∇ · A and implement from the outset the constraints π 0 = 0 = ∇· π. Perfectly conducting large parallel plates of sides L at x 3 = 0 and x 3 = d with unit normal n require E × n = 0 = B · n on the plates, and periodic boundary conditions in the x a directions. Let i = 1, 2, 3, a = 1, 2, Following for instance [24] (see also e.g. [25,26] for closely related discussions), one introduces satisfying Neumann, respectively Dirichlet, conditions as well as the Helmholtz equations with λ = (E, H). In these terms, the mode expansion of the canonical pair ( A, π) and the associated electric and magnetic fields E = − π, B = ∇ × A is given by the sum of E or transverse magnetic modes, where ǫ ab is skew-symmetric with ǫ 12 = 1, and H or transverse electric modes, with the understanding that a H ka,0 = 0. In these variables, the first order electromagnetic action is given by with λ = (E, H). In particular, Poisson brackets are read off from the kinetic term and oscillators do have the usual time dependence. Action (16) coincides with the mode expansion of an action for two massless scalar fields, one with Neumann and one with Dirichlet conditions. For later purposes, it is useful to introduce an equivalent formulation in terms of a single free massless scalar field, which satisfies periodic boundary conditions in x, y in intervals of length L and in x 3 of length 2d, where V P = 2dL 2 with n 3 ∈ Z as well, and , n 3 = 0, a ka,0 = a E ka,0 .
With these definitions, the first order scalar field action agrees with the first order electromagnetic action (15) because its expression in terms of modes is given by the RHS of (16).

THE OBSERVABLE
In the equivalent scalar field formulation, we will show below that the correct observable to be turned on in order to produce a real part for the modular parameter τ and to consider full modular transformations is linear momentum in the x 3 direction, which, in terms of oscillators, is given by The action of this observable in electromagnetic terms can be inferred from with similar relations holding for π λ by using (13) and (14). On the electromagnetic E and H vector potentials, electric and magnetic fields, it acts like the curl followed by an application of (−∂ 3 )(−∆) − 1 2 and an exchange of E and H: with the transformations of the H fields obtained by exchanging E and H in the above. Similarly, one may show by direct computation that the observable is given by Up to multiplication of each term in momentum space by k, this observable is the component in the x 3 direction of spin angular momentum, since one may show that

EXTENDED PARTITION FUNCTION AND MODULAR PROPERTIES
For the computation of the extended partition function, the fastest way for our purpose is to follow [27] and to start from the Hamiltonian path integral representation where the sum is over periodic phase space path of period β, and the Euclidean action is After integration over the momenta, this leads to with Following [28] (see also e.g. [25,29] for earlier connected work and [30] for a review), the evaluation of this path integral is done by zeta function techniques. Except for the replacement ∂ 4 → ∂ 4 + iµ∂ 3 , the operator in the action is the Laplacian in 4 dimensions with periodic boundary conditions in all directions. Since the eigenfunctions are e i(kj x j + 2πn 4 β x 4 ) , the eigenvalues are where A = 1, . . . 4, the zeta function of the capacitor is where the prime means that the term with n A = 0 is excluded. The next step is to take the limit of large plate size L, and hence to turn the sum over the transverse directions turns into integrals. As in two-dimensional conformal field theories on the torus and also in the context of QCD (see e.g. [31-33]), one now uses a purely imaginary chemical potential µ = iν with ν real. After introducing the complex parameter and doing the integral in polar coordinates, the zeta function becomes When n 3 = 0 = n 4 the integral is regulated through an infrared cut-off ǫ, ∞ ǫ dyy −2s+1 = − ǫ 2−2s 2−2s . This expression together with its derivative both vanish in the limit at s = 0 in the limit ǫ → 0 and can thus be discarded. After performing the integral for the remaining terms, the result may be written in terms of a real analytic Eisenstein series (see e.g. [34] for a recent review), Using the inversion formula, then yields The behaviour of the partition function under modular transformations where a, b, c, d ∈ Z, then follow directly from modular invariance of f 2 (τ ), ln Z(τ ′ ) = |cτ + d| 2 ln Z(τ ).
The previous result (5) corresponds to vanishing chemical potential ν = 0 = Re(τ ) in which case the temperature inversion formula is recovered for a = 0 = d, b = −c = 1.

DISCUSSION
Identifying the correct observable is straightforward in the scalar field formulation. The corresponding expression in terms of electromagnetic fields is somewhat harder to guess directly. More details on a direct derivation in the operator formalism and on an underlying infinitedimensional symmetry algebra will appear elsewhere. Other formulations making gauge invariance manifest will also be explored. At this stage let us just point out that, after having identified the modular parameter, one may write in the standard way the contribution to the partition function that corresponds to the vacuum energy, i.e., the first term in the RHS of (9), L 2 π 2 β d 3 720 = ln (qq) −c/24 = πIm(τ )c 6 , q = e 2πiτ , (42) provided that the central charge of the planar capacitor is taken as