^{3}.

The semiclassical contribution to the partition function is obtained by evaluating the Euclidean action improved through suitable boundary terms. We address the question of which degrees of freedom are responsible for this contribution. A physical toy model for the gravitational problem 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 Becchi-Rouet-Stora-Tyutin quantized theory, the classical contribution to the partition function originates from the part of the gauge sector that is no longer trivial due to the boundary conditions required by the physical setup. More concretely, for a planar charged vacuum capacitor with perfectly conducting plates, we identify the degrees of freedom that, in the quantum theory, give rise to additional contributions to the standard blackbody result proportional to the area of the plates and that allow for a microscopic derivation of the thermodynamics of the charged capacitor.

The question of which degrees of freedom are responsible for the Bekenstein-Hawking entropy of black holes naturally leads one to study nonproper gauge degrees of freedom, i.e., gauge degrees of freedom that are no longer pure gauge because of nontrivial 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 nonlinear 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-two 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 nonproper diffeomorphisms or large gauge transformations.

One possibility is to introduce the nontrivial boundary conditions as dynamical canonical variables in the theory, with suitable additional constraints. This idea goes back to Dirac

These arguments suggest studying 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

In the first paper of this series

Unlike ordinary coherent states, null coherent states have the same norm than the standard vacuum,

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,

On the classical level, the role of the chemical potential is played by the constant value of

That longitudinal and temporal photons have an important role to play in topologically nontrivial situations is in agreement with the standard interpretation of the Aharanov-Bohm effect

Recent work on infrared physics has been driven by new connections in the field summarized in

The paper is organized as follows. In the next section, we start by discussing the thermodynamics of a charged vacuum capacitor following the method developed by Gibbons and Hawking

In Sec.

In Sec.

In the quantum theory, we compute in Sec.

Additional remarks are relegated to Sec.

When making the Legendre transformation of the standard Lagrangian action

From the viewpoint of constrained Hamiltonian systems, there are two gauge invariant observables in the problem, the reduced phase space energy

Consider a spherical vacuum capacitor consisting of two conducting spheres

The thermodynamics can then be obtained from the Euclidean action evaluated on-shell. Since the problem is at fixed electric charge, no improvement boundary terms are needed

The conditions under which some of these terms can be neglected will be discussed elsewhere.

This implies thatAlternatively, in order to deal directly with

For the case of the so-called exterior problem, the thermodynamics of a charged spherical shell of radius

For two parallel plates

What we will study below is the quantum mechanical origin of the semiclassical contribution to the partition function, together with the additional one-loop contributions.

The gauge sector of Maxwell’s theory is treated in the context of the Batalin-Fradkin-Vilkovisky Hamiltonian formalism

We follow the reviews

In the nonminimal BFV-BRST approach in which

Decomposing into transverse and longitudinal fields,

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 gauge sector can be written as a supersymmetric quantum mechanical model with

This formulation of the gauge sector can be turned into a local topological field theory with a BRST exact Hamiltonian when using the potential

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” solution

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 nontrivial boundary conditions of the physical setup into account. Notations and conventions are fixed in Appendix

For conducting plates, spatial boundary conditions on the fields have to be imposed that implement

When substituting the mode expansion, the canonical Hamiltonian splits into three pieces,

The most interesting piece from the current perspective is

In summary, we can split degrees of freedom according to whether they are

The former group contains

For the nonzero-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

For the new sector, we first consider the nonzero modes of the nonproper gauge degrees of freedom,

For the zero mode of the nonproper gauge degrees of freedom, the variables

The starting point Hamiltonian corresponds to

We have used a Hamiltonian approach here in order to keep track of the various degrees of freedom and of their nature. It should be 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 nontrivial boundary conditions than the ones we have treated explicitly.

The nontrivial effect is a zero-mode effect, like in the case of Bose-Einstein condensation

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

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

The analysis in this paper in terms of a detailed mode expansions is possible because boundary conditions at both

It would be interesting to study in more detail how the quantization of the electromagnetic field in this topologically nontrivial setup appears from the viewpoint of large gauge symmetries. In Chern-Simons theories for instance, large gauge symmetries become gobal symmetries of the Wess-Zumino or Liouville theories that describe the residual dynamics in the presence of boundaries. This role is played here by the massless scalar theory, which does indeed possess an infinite number of global symmetries.

The consequences of the present computation, both from a theoretical and an experimental viewpoint should be fully explored. One would need to understand from the current perspective what happens in an interacting theory like QED for instance, how to resum contributions from the gauge sector and to get different charged sectors in the electromagnetic case, and similarly, to go from a flat to a black hole background in the gravitational case.

As we have tried to show in

This work is supported by the F. R. S.-FNRS Belgium, convention FRFC PDR T.1025.14 and convention IISN 4.4503.15. Part of the work was done at the Kavli Institute for Theoretical Physics China during the program “Quantum Gravity, Black Holes and Strings 2014.” Another part has been completed while visiting the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation. The author is grateful to Cédric Troessaert, Hernán González, Marc Geiller, Laurent Freidel, and Marc Henneaux for useful discussions.

Consider first periodic boundary conditions in a box

The canonical Poisson bracket relations that originate from the kinetic term

Alternatively, if one replaces the exponentials by sines and cosines in the

Imposing Neumann or Dirichlet boundary conditions on an interval of length

The canonical Poisson brackets now originate from kinetic terms of the form

When there is no electric potential at the surface of the body, no global electric charge and no nontrivial boundary conditions, the theory is quantized in such a way that the contribution to the partition function from the unphysical bosonic degrees of freedom

For periodic boundary conditions in a box

Finally, there is an additional change of variables to null oscillators,

For the nonzero modes, if

The canonical Poisson brackets of the fields

Note that longitudinal fields

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,

The piece of the BRST gauge fixed Hamiltonian (in Feynman gauge

We will proceed differently however and start the analysis from the zero-mode contribution to the classical Lagrangian

When quantizing the unphysical zero-mode pairs,

When inserting the mode expansion reviewed above, the BRST charge is given by

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

Note also that, in real time, indefinite metric quantization is implemented in the path integral through imaginary values for the paths associated to

Turning on a chemical potential for electric charge,

To a pair of bosonic null oscillators,

Formulas for a pair of fermionic null oscillators, with anticommutation relations given by

Using the notation

It also follows that