Soft degrees of freedom, Gibbons-Hawking contribution and entropy from Casimir effect

TL;DR

This paper reviews the role of non proper-gauge degrees of freedom in the Casimir effect, analyzing their contributions to entropy and partition functions, and connecting these findings with the Gibbons-Hawking method and classical thermodynamics.

Contribution

It provides a first-principles derivation of the dynamics of additional modes in the Casimir effect and links their thermodynamic contributions to established methods like Gibbons-Hawking.

Findings

Additional modes contribute to entropy proportional to the plate area.

Zero modes in charged plates affect the partition function without entropy contribution.

The results connect Casimir effect phenomena with Gibbons-Hawking and classical thermodynamics.

Abstract

Recent work on non proper-gauge degrees of freedom in the context of the Casimir effect is reviewed. In his original paper, Casimir starts by pointing out that, when the electromagnetic field is confined between two perfectly conducting parallel plates, there is an additional physical polarization of the electromagnetic field at zero value for the discretized longitudinal momentum besides the standard two transverse polarizations at non-zero values. In this review, the dynamics of these additional modes is obtained from first principles. At finite temperature, their contribution to the entropy is proportional to the area of the plates and corresponds to the contribution of a massless scalar field in 2+1 dimensions. When the plates are charged, there is a further contribution to the partition function from the zero mode of this additional scalar that scales with the area but does not contribute to the entropy. It reproduces the result obtained when the Gibbons-Hawking method is applied to the vacuum capacitor. For completeness, a brief discussion of the classical thermodynamics of such a capacitor is included.