Black hole entropy from non-proper gauge degrees of freedom: II. The charged vacuum capacitor

The question which degrees of freedom are responsible for the classical part of the Gibbons-Hawking entropy is addressed. A physical toy model sharing the same properties from the viewpoint of the linearized theory is a charged vacuum capacitor. In Maxwell's theory, the gauge sector including ghosts is a topological field theory. When computing the grand canonical partition function with a chemical potential for electric charge in the indefinite metric Hilbert space of the BRST quantized theory, the classical contribution originates from the part of the gauge sector that is no longer trivial due to the boundary conditions required by the physical set-up. More concretely, in the benchmark problem of a planar charged vacuum capacitor, we identify the degrees of freedom that, in the quantum theory, give rise to an additional contribution to the standard black body result proportional to the area of the plates, and that allow for a microscopic derivation of the thermodynamics of the charged capacitor.


Introduction
The question which degrees of freedom are responsible for the Bekenstein-Hawking entropy of black holes naturally leads one to study non-proper gauge degrees of freedom, i.e., gauge degrees of freedom that are no longer pure gauge because of non trivial boundary conditions. (i) The most direct line of reasoning is probably to consider the Hamiltonian formulation of linearized Einstein gravity. The linearized Schwarzschild solution does not involve physical degrees of freedom since the transverse-traceless parts of the spatial metric and its momenta vanish for that solution. (ii) Another argument, which holds on the non-linear level, concerns the Bekenstein-Hawking entropy of the black hole in three dimensional anti-de Sitter spacetime where there are no physical bulk gravitons to begin with. (iii) Yet another approach has to do with the type of observables that are involved: in general relativity, the ADM mass is a codimension 2 surface integral, with similar properties to electric charge in Maxwell's theory. In particular, it does not involve transverse-traceless variables. Furthermore, the classification of such observables is directly related to non-proper diffeomorphisms or large gauge transformations.
One possibility is to introduce the non-trivial boundary conditions as dynamical canonical variables in the theory, with suitable additional constraints. This idea goes back to Dirac [1] and has been used in an investigation of the definition of energy, and more generally of the Poincaré generators, in the Hamiltonian formulation of asymptotically flat general relativity [2]. In the context of Yang-Mills theory, it has been implemented for various related questions [3][4][5][6][7][8], including the infrared problem [9].
These arguments suggest to study the analogue problem in the context of the quantized electromagnetic field, where the role of the black hole is played by the Coulomb solution, the electromagnetic field created by a static point particle source with macroscopic charge Q. Besides being a physical problem in its own right where all conceptual issues are present, the linearity of the problem and the wealth of results readily available in the literature make it directly tractable.
In the first paper of this series [10], a quantum mechanical understanding has been achieved when all polarizations of the photon are quantized in an indefinite metric Hilbert space: the quantum state |0 Q corresponding to the classical Coulomb solution is a coherent state of null oscillators, made up of a linear combination of longitudinal and temporal photons. In this computation, infrared divergences occur when showing that the expectation value Q 0| π i (x)|0 Q of the electric field operator is indeed the classical field produced by a point-like source: one uses that the Fourier transform of k −2 is proportional to (4πr) −1 which really requires an infrared regularisation, (k 2 + m 2 ) −1 giving the Yukawa potential (4πr) −1 e −mr , with m → 0 + . Unlike ordinary coherent states, null coherent states have the same norm than the standard vacuum, Q 0|0 Q = 1. Furthermore, the expectation value of the energy of physical photons vanishes. It is in this sense that these states behave like different vacua of the theory.
That longitudinal and temporal photons have an important role to play in topologically non-trivial situations is in agreement with the standard interpretation of the Aharanov-Bohm effect [11] when extrapolated to the quantized electromagnetic field.
Rather than quantizing the theory for a fixed charge, what we would like to address here is the computation of the grand canonical partition function, with a precise understanding of the underlying Hilbert space and thus of the trace that is involved. Again, when trying to deal directly with the electric charge operator, in a large volume V, one has to face infrared questions since − Q is the zero mode of the longitudinal part of the electric field.
On the classical level, the role of the chemical potential is played by the constant value of A 0 = −µ at the surface of the body, while a non-vanishing electric charge requires π r = O(r −2 ). In order to take electric charge into account, non trivial fall-off or boundary conditions are thus required.
The approach we will follow here is not to introduce additional degrees of freedom besides those already contained in (A µ , π µ ). For trivial boundary conditions, standard results equivalent to those derived in the framework of reduced phase space quantization are then recovered in the indefinite metric BRST Fock space through the quartet mechanism [12] in the bulk. We will analyze in detail how these results are affected when the boundary conditions that are used in the context of the Casimir effect [13] are taken into account. For technical reasons, it is then also easier for us here to start with a vacuum capacitor consisting of 2 large parallel plates instead of a spherical vacuum capacitor, so that one may use Fourier series instead of Bessel functions [14].
The paper is organized as follows. In the next section, we start by deriving the thermodynamics of a charged vacuum capacitor in the sense of Gibbons and Hawking [29]. Section 3 points out that the gauge sector of Maxwell's theory can be understood as a topological field theory. It is not really needed for the rest of the paper, but is included in order to better understand the relation with three dimensional gravity for instance.
In section 4, we impose the boundary conditions adapted to perfectly conducting parallel plates, taken at constant z. Through a detailed Hamiltonian analysis, we show that the modes with vanishing momenta in the z direction of (A z , π z ), even though formally longitudinal, are to be considered as physical in the problem at hand. In that sense, we refer to them as non-proper gauge degrees of freedom.
In the quantum theory, we compute in section 4.3 the contribution of the non-zero modes of the non-proper gauge degrees of freedom to the standard black body result. It is proportional to the area of the plates. After turning on the chemical potential for electric charge, the proper quantum mechanical understanding of the classical thermodynamics of the vacuum capacitor follows from the contribution of the zero mode of the non-proper gauge degrees of freedom.
Additional remarks are relegated to section 5. Conventions for mode expansions adapted to the various boundary conditions are given in appendix A. In order to be selfcontained, a summary of standard material on BRST quantization as applied to Maxwell's theory is provided in appendix B and appendix C.

Thermodynamics of a charged vacuum capacitor
When making the Legendre transformation of the standard Lagrangian action S[A µ ] = − 1 4 d 4 x F µν F µν forȦ i , and after adding the boundary term, − ∂B dσ i [π i A 0 ], the first order action is Alternatively, this action may be obtained from the extended first order action after eliminating the Lagrange multiplier for the primary constraint and the momentum π 0 .
From the viewpoint of constrained Hamiltonian systems, there are two gauge invariant observables in the problem, the reduced phase space energy and also the electric charge where S is a closed 2-surface.
Consider a spherical vacuum capacitor consisting of two conducting spheres S 1 , S 2 centered at the origin with radii R 1 < R 2 and charges q, −q. Let us focus on timeindependent fields and assume that there are no sources inside the body. We will assume here that A i = 0, even though the field equations only require ∂ j F ji = 0. In this context, there are then no transverse degrees of freedom and for R 1 < r < R 2 and zero otherwise.
The thermodynamics can then be obtained from the Euclidean action evaluated onshell. Since the problem is at fixed electric charge, no improvement boundary terms are needed [30], and Using π i L = ∂ i φ and ∆φ = 0 on-shell for , I E can also be written in terms of boundary terms as where φ S = q 4πr and Q = q for the problem at hand. In this case, the entropy is Indeed, from the first law at dU = T dS + µdQ with dU = 0, it follows that ∂S ∂q = −βµ. This implies and then ln Z(β, µ) = [S(β, q) Alternatively, in order to deal directly with one supposes instead that the electric potentials at the boundary are fixed and constant, Under the additional assumptions that there are no sources inside the body, ∂ i A i = 0 and A T i = 0 = π i T , the classical solution is (2.11) In this situation, following [2], the action needs to be improved by boundary terms so that this solution is a true extremum of the variational principle, On-shell, the Euclidean action is now This leads to 14) and the saddle point approximation is For the case of the so-called exterior problem, the thermodynamics of a charged spherical shell of radius R 1 can be obtained from the above by letting R 2 → ∞ and taking φ 2 = 0.
For two parallel plates P 1 , P 2 at z = 0 and at z = L 3 , with charge densities q A and -q A , one finds under the same assumptions and in the same manner that π i = −δ i 3 q A , (when What we will study below is the quantum mechanical origin of this classical contribution.

Gauge sector of electromagnetism as topological field theory
The gauge sector including ghosts of Maxwell's theory is treated in the context of the Batalin-Fradkin-Vilkovisky Hamiltonian formalism [31][32][33]. It contains the information on the electric charge in regions where there are no sources. Not surprisingly, this sector can be identified with a Witten-type supersymmetric quantum mechanical model [34] when treating the spatial dimensions in a formal way. We follow the reviews [35], chapter 19, and [36] chapter 3, for the BFV treatment of electromagnetism and for supersymmetric quantum mechanics, respectively.
In the non-minimal BFV-BRST approach in which (A 0 , π 0 ) are among the canonical variables, the action to be used in the Hamiltonian path integral for electromagnetism is where the BRST invariant Hamiltonian is H 0 = d 3 x H 0 , H 0 is given in (2.1), and the graded Poisson brackets are determined by The BRST charge is and the gauge fixing fermion is chosen as Eliminating the auxiliary fields gives the covariant gauge fixed Faddeev-Popov action, but we will not do so here in order to keep better track of the various degrees of freedom.
Decomposing into transverse and longitudinal fields, the first order action decomposes into a transverse piece, with H ph given in (2.2), and a piece from the gauge sector (including ghosts), where includes the contribution of the longitudinal electric fields, and is explicitly given by Turning on the chemical potential for electric charge can be done through the shift In the case of a constant metric, supersymmetric quantum mechanics is described by the action The entire action is BRST exact where the BRST charge is 13) and the fundamental Poisson brackets are with all other brackets vanishing. As consequence, the BRST transformations s = {Ω, ·} are explicitly given by 14) The Hamiltonian can be written as withΩ generating the so-called anti-BRST symmetry,s = {Ω, ·}, explicitly given bȳ The gauge sector can be written as a supersymmetric quantum mechanical model with , (3.17) and provided spatial integrations by parts are allowed. Formally, DeWitt's condensed notation is used (in the sense that summation over i includes an integration over x, while δ i j includes a Dirac delta function in three dimensions).
Such a reformulation is clearly not essential for an understanding of the problem. Nevertheless, it indicates at this stage already that the explicit computation of the partition function involves the value of the exponential at the classical saddle point, the "instanton"

Planar vacuum capacitor
In this main section, the partition function for the vacuum capacitor is computed, after identifying the complete Hilbert space from a constrained Hamiltonian analysis that takes the non-trivial boundary conditions of the physical set-up into account. Notations and conventions are fixed in appendix A. In order to understand how the boundary conditions influence the result, it is instructive to first review the standard and well-known results in the case of periodic boundary conditions. This is done in appendix B and C, following [35].

Spatial boundary conditions
For conducting plates, spatial boundary conditions on the fields have to be imposed that implement n · B = 0 = n × E = 0 on the boundary. If x i = (x a , x 3 ) with a = 1, 2, this is guaranteed if the mode expansion of (A a , π a ) contains sines only, with non-vanishing Poisson brackets In order for bulk cancellations to work as in the case of periodic boundary conditions, one is forced to use Neumann conditions for (A 3 , π 3 ), so that This implies that These conditions are consistent with the boundary conditions used in the context of the Casimir effect when one works in radiation gauge A 0 = 0, ∂ i A i = 0 (see e.g. [37]). The boundary conditions on the remaining variables then follow from the Hamiltonian analysis starting from . Indeed, in order to impose the Gauss law in the bulk, (A 0 , π 0 ) should satisfy Dirichlet conditions. In turn, the same then goes for the ghost pairs (η, P), (C, ρ), and also for (A, π). Again, this is consistent with the conditions in the context of the Casimir effect (e.g. [38] where it is shown that there is a standard supersymmetric cancellation between the zero point energies of the gauge sector, and also [39], [40] [41] for related considerations).

Degrees of freedom and dynamics
When substituting the mode expansion, the canonical Hamiltonian splits into three pieces, with a standard bulk piece The piece will give rise to the secondary constraints, As expected and can be easily checked, there are no tertiary constraints.
The most interesting piece from the current perspective is In summary, we can split degrees of freedom according to whether they are k 3 zero modes or not. In the latter group, we have (A b , π b ), (A 0 , π 0 ), (η, P), (C, ρ), which all satisfy Dirichlet boundary conditions, as well as the k 3 = 0 modes of (A 3 , π 3 ) satisfying Neumann conditions. The former group contains (A C 3,ka,0 , π 3C ka,0 ), respectively the fields None of these variables is involved in any of the constraints. They are thus physical. Note that while the associated vector potential and electric fields are formally longitudinal, this is not really the case since z is restricted to the closed interval [0, L 3 ]. Note also that the Poisson brackets for these variables given in (4.4) and the Hamiltonian (4.8), which are encoded in the bulk first order action restricted to these degrees of freedom, completely determine the Lagrangian action of a massless scalar in (2+1) dimensions after integrating out the momenta, In this context, the electric charge operator, by analogy with the discussion in section 2, is taken to be the quantum version of the classical observable which Poisson commutes both with the complete Hamiltonian and all constraints.

Partition function
For the non-zero mode sector of the theory, one can then follow the analysis of the periodic case (fix the gauge, choose suitable variables). The difference is only that the modes involved are restricted to k 3 > 0. Up to details related to the standard Casimir effect (which will be addressed elsewhere), one finds that the contribution to the partition function from this sector is the standard black body result, equation (B.29).
For the new sector, we first consider the non-zero modes of the non proper gauge degrees of freedom, (A C 3,ka,0 , π 3C ka,0 ), with k a = 0. For them, one defines standard oscillator variables (4.14) The contribution to the partition function, is given by The standard approximation then leads to It follows that the contribution to the partition function, 19) of this free particle is where ∆q denotes the divergent interval of integration over q. In particular, the starting point Hamiltonian corresponds to ρ = 1 = ν, so that the classical contribution discussed in section 2 is recovered through the last term of equation (4.20).

Discussion and perspectives
We have used a Hamiltonian approach here in order to keep track of the various degrees of freedom and of their nature. It is of course possible to streamline these derivations by using finite temperature Lagrangian path integral methods combined with techniques from topological field theory and extend the considerations here to more complicated non trivial boundary conditions than the ones we have treated explicitly.
The non-trivial effect is a zero mode effect, like in the case of Bose-Einstein condensation [42]. The difference is however that in the latter both observables H and N involve the same degrees of freedom, whereas in the our case, the physical Hamiltonian H and the electric charge Q involve different degrees of freedom. The electromagnetic analog of the classical Bekenstein-Hawking entropy comes here from the zero mode of the non-proper gauge degrees of freedom, which are themselves zero-modes from the bulk perspective.
Magnetic charge can be treated in the same way when using a magnetic instead of an electric formulation. Both types of charges simultaneously can be understood in a manifestly duality invariant first order formulation [43] (see also e.g. [44]) which includes an additional quartet [45,46]. The next, in principle straightforward, step is then to generalize the result discussed here to the spherical vacuum capacitor. For linearized gravity around flat space, one can easily adapt the result of [10] and understand the Schwarzschild solution as a coherent state of unphysical gravitons. Generalizing the derivation here for the Bekenstein-Hawking entropy is also direct, and will be done in a follow-up paper. This is then what an observer at spatial infinity would see. He would however not be able to distinguish between a black hole and a star from that computation alone.
It would be interesting to fully explore the consequences of the present computation, both from a theoretical and experimental viewpoint. Also interesting would be to understand in detail from the current perspective what happens in full-fledged QED, how to resum contributions from the gauge sector and to get different charged sectors in the electromagnetic case, and similarily, to go from a flat to a black hole background in the gravitational case.
As we have tried to show in [10] and with this computation here, in order to deal consistently with charged sectors or black holes in the operator formalism, all polarizations of the four potential or of the metric should be quantized in a non-unitary Hilbert space. This is also implicitly the case in the Euclidean path integral formulation when choosing real paths for the Euclidean version of A 0 , or for the shift vectors. Since most of the questions on black hole entropy have little to do with transverse-traceless variables but rather with variables from the gauge sector, one might want to take this specific non-unitarity into account when discussing paradoxes related to black hole physics.

A Mode expansions A.1 Periodic boundary conditions
Consider first periodic boundary conditions in a box B P with sides of lengths 2L i and volume V P = 8L 1 L 2 L 3 . Note that in this case, no improvement terms are needed for the gauge fixed Hamiltonian H 0 + {Ω, K ξ }. The fields are expanded in terms of Fourier series at fixed time t, with n i ∈ Z and k i = πn i L (i) (no summation over i). Quadratic integrals are related as The canonical Poisson bracket relations that originate from the kinetic term for each canonically conjugated pair are then with all other Poisson brackets following from the middle of equation (A.2). Here σ AB is the canonical symplectic matrix obtained by combining for each canonical pair. Translating back to position space gives Alternatively, if one replaces the exponentials by sines and cosines in the z = x 3 direction, and, for k 3 > 0, . (A. 10) In this case, .11) and the Poisson brackets are and, for k 3 , k ′ 3 > 0, and all other Poisson brackets vanishing. In these terms, the periodic delta function can be written as cos k 3 x 3 cos ky 3 + sin k 3 x 3 sin k 3 y 3 ]. (A.14)

A.2 Neumann/Dirichlet boundary conditions
Imposing Neumann or Dirichlet boundary conditions on an interval of length L 3 in the z = x 3 direction can be achieved by extending the function of z ∈ [0, L 3 ] to an even respectively odd function of z ∈ [−L 3 , L 3 ]. This amounts to setting s A k 3 ,ka respectively c A ka,k 3 in (A.8) to zero, while keeping the definitions of the remaining modes in (A.9) and (A.10) unchanged. These formulas can then be expressed in terms of the real volume V = 4L 1 L 2 L 3 of the body B by the substitution V P = 2V . In the Neumann case, we now have while for the Dirichlet case, The canonical Poisson brackets now originate from kinetic terms of the form 17) which implies that the brackets of the remaining modes in (A.12), (A.13) are to be multiplied by 2, or equivalently, in these equations, V P needs to be replaced by V . In position space, one needs to replace δ P (x i , y i ) in the RHS of (A.7) by δ P (x a , y a )∆ ± (x 3 , y 3 ), with the + corresponding to the Neumann and the − to the Dirichlet case, and where (see e.g. [47], chapter 4) and also

B Partition function for periodic boundary conditions
When there is no electric potential at the surface of the body, no global electric charge and no non-trivial boundary conditions, the theory is quantized in such a way that the contribution to the partition function from the unphysical bosonic degrees of freedom (A 0 , π 0 ), (A, π) cancels the one from the ghost degrees of freedom (η, P), (C, ρ) so that only the physical degrees of freedom (A T i , π i T ) contribute. Let us briefly review these computations. As we are ultimately interested in infrared effects, we keep the volume finite and work with Fourier series including zero modes, instead of Fourier integrals.

B.1 Non-zero modes
For periodic boundary conditions in a box B P of volume V P = 8L 1 L 2 L 3 , we can adapt the change of variables from section 19.1.6 of [35] to the case of Fourier series instead of Fourier integrals. In this case, k i = πn (i) L (i) and one defines (B.5) where n ′ = n = 0 , and ω k = k · k, while {e m i, k } is an orthonormal triad, the first two vectors being transversal and the third longitudinal, k i e 1 i, k = 0 = e 2 i, k and e 3 Finally, there is an additional change of variables to null oscillators, For the non-zero modes, if a a, k , a = 1, 2 are the transverse physical oscillators, while a α Γ, k , α = 1, 2, Γ = 1, 2, are the null oscillators of the unphysical sector, with a 1 Γ, k = (a k , b k ) bosonic and a 2 Γ, k = (c k ,c k ) fermionic, the non-vanishing Poisson brackets are where indices are lowered (and raised) with δ ab , δ αβ and the indefinite metric η Γ∆ given by The canonical Poisson brackets of the fields z A are then equivalent to these non-zero modes Poisson brackets and the zero mode brackets Note that longitudinal fields A = A ′ , π = π ′ do not have zero modes, so that the commutation relations for the modes in a box imply . How zero modes for these fields may be re-introduced is briefly discussed in the next section.
With a view towards a subsequent large volume limit and a passage from Fourier series to integrals, zero modes are usually neglected. In this case, n → V . If discrete and continuous Fourier coefficients/oscillators are related by for all a a k , a α γ, k , sums over n may simply be replaced by integrals over k and Kronecker by Dirac deltas in the above expressions for the mode expansions of the fields, the Poisson brackets and quadratic expressions like the Hamiltonian or the BRST charge.

B.2 Zero modes
The piece of the BRST gauge fixed Hamiltonian (in Feynman gauge ξ = 1) When (A 0, 0 ,π 0 0 ) are quantized as anti-Hermitian operators and the zero-mode ghosts in the Schrödinger representation (cf. [35], sections 15.3.2 and 15.4.4), and after limiting the bosonic zeromode integrations to intervals ∆A µ, 0 , their contribution to the partition function would be with Z ′ (β) the partition function for the non-zero modes. Note also that the piece of the BRST charge involving zero-modes is Ω 0 = −iπ 0 0 ρ 0 . We will proceed differently however and start the analysis from the zero mode contribution to the classical Lagrangian L = − 1 There then is only the primary constraint π 0 0 ≈ 0, but no secondary constraint. Introducing the zero-mode ghost pair (C 0 , ρ 0 ), the associated BRST charge is Ω 0 given above. If one would like the theory to also include the zero modes of the other ghost pair, (η 0 , P 0 ), one can do so by adding a suitable non-minimal sector. This is done by considering the zero-mode Lagrangian as a function of the spurious P 0 , L 0 = L 0 [A µ, 0 , −P 0 ]. There then is an additional constraint −η 0 ≈ 0, for which one introduces the ghost pair (π 0 , A 0 ), unrelated to components of (A i, 0 , π i 0 ). The BRST charge including this non-minimal sector is then Choosing as gauge fixing fermion the BRST gauge fixed Hamiltonian is H 0 = H ph 0 + H gs 0 , with and H gs 0 = − 1 2 i{Ω 0 ,Ω 0 }, which is explicitly given by When proceeding in this way, the longitudinal fields (A, π) will also include zero modes.
Integrating out momenta can be done consistently including the zero modes. The same applies to the mode expansion of (3. When quantizing the unphysical zero-mode pairs (A 0, 0 , π 0 0 ), (π 0 , A 0 ), (η 0 , P 0 ), (C 0 , ρ 0 ), (B.15) in the Dirac-Fock representation, their contribution to the partition function cancels through the same mechanism, reviewed in appendix B.3 below, as for the non-zero modes of the unphysical sector. One then remains with the (infinite) contribution of three bosonic free particles encoded in (B.13), whose contribution to the partition function is

B.3 Bulk cancellations
When inserting the mode expansion reviewed above, the BRST charge is given by In Feynman gauge ξ = 1, the gauge fixed Hamiltonian is given by Here ω k = √ k i k i for the non-zero modes, ω 0 = 1 for the zero modes of the unphysical sector, a a, k , a = 1, 2 are the transverse oscillators of the physical sector, while a α Γ, k are the bosonic and fermionic null oscillators of the unphysical sector, with non-vanishing (graded) commutation relations where indices are lowered and raised with the appropriate metrics δ ab , δ αβ , η Γ∆ and their inverses.
At this stage, the difference with the partition function for a complex scalar field, and with Bose-Einstein condensation, appears clearly: the observable for which we would like to introduce a chemical potential involves different degrees of freedom than the ones of the Hamiltonian. Furthermore, such a BRST Fock space quantization guarantees that only the physical sector contributes. Indeed, since Alternatively, in the context of path integral quantization, it is convenient to introduce a collective notation a A for all the oscillators a a , a α Γ . BRST Fock quantization is implemented by using the holomorphic representation with boundary conditions that fix that creation operators at t ′ , a * A (t ′ ) = a * A and destruction operators at t, a A (t) = a A , (see e.g. [48,49], and also [50], chapter 9, [35], chapters 15, 16). In order to be able to turn on a chemical potential, we consider the coupling to a source by using The path integral representation of the kernel U j k (t ′ , t) at fixed k of the evolution operator e i(t ′ −t)H j k is then given by U j k (t ′ , t) = e iS j k | extr , where the classical action to be used is the one that has a true extremum when taking into account the boundary conditions When using that the appropriate extremum is When using a time independent source j c and t ′ − t = −iβ, this gives When evaluating the trace in the holomorphic representation, one should split into physical and unphysical oscillators. For each physical oscillators, there is a pre-factor of (1 − e −βω k ) −1 coming from an appropriate change of variables. As explicitly recalled in appendix C, these pre-factors cancel for the unphysical oscillators. This cancellation corresponds to the one between the bosonic and fermionic determinants in supersymmetric quantum mechanics. As a result, In the absence of sources, when integrating over all the modes and discarding the infinite contribution of the zero modes of the physical sector, one finds the standard black body result, Note that, if instead of the kernel of the evolution operator, one directly computes the trace, the alternating sign in the Lefschetz trace is taken into account through periodic boundary conditions in imaginary time for the ghosts (see e.g. [51] for finite temperature QED or [52] for supersymmetric quantum mechanics), so that all fields satisfy periodic boundary conditions in imaginary time.
Note also that, in real time, indefinite metric quantization is implemented in the path integral through imaginary values for the paths associated to (A 0 , π 0 ) (cf. [35] page 355). In the Euclidean approach, when one substitutes A 0 by i A 0 , these become then again real paths for A 0 . Conversely, this means that standard real paths for A 0 in the Euclidean approach correspond to using an indefinite metric Hilbert space in real time.
Turning on a chemical potential for electric charge, H gs → H gs + µQ, with Q = − V 2 (a 0 + a * 0 ) can be done in the above computation through the coupling to the source,
Their overlap is given by a * |a = e a * Γ a Γ , while the completeness relation is with fundamental integral Formulas for a pair of fermionic null oscillators, with anticommutation relations given by [ c Γ , c † ∆ ] = η Γ∆ , are the same except for the absence of (2πi) −1 in the integration measure.
Using the notation a α Γ = (a Γ , c Γ ), for α = 1, 2, let O(a * ; a) be the kernel of an operator O in the Fock space of a quartet. In this representation, the Lefschetz trace is given by  where N b = a † Γ a Γ is the number operator for the bosonic part of the quartet, i.e., for a pair of bosonic null oscillators.