Abstract
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.
Highlights
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
After turning on the chemical potential for electric charge, a quantum mechanical understanding of the classical thermodynamics of the vacuum capacitor follows from the contribution of the zero mode of the nonproper gauge degrees of freedom
Summary
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. 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 j0iQ 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 Qh0jπiðxÞj0iQ of the electric field operator is the classical field. After turning on the chemical potential for electric charge, a quantum mechanical understanding of the classical thermodynamics of the vacuum capacitor follows from the contribution of the zero mode of the nonproper gauge degrees of freedom. 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
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