Black hole entropy from nonproper gauge degrees of freedom: The charged vacuum capacitor

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.