Remarks on the Casimir Self-Entropy of a Spherical Electromagnetic $\delta$-Function Shell

TL;DR

This paper clarifies discrepancies in the calculation of the Casimir self-entropy of a spherical electromagnetic delta-function shell, confirming the existence of negative entropy regions at weak coupling and addressing issues of infrared divergences and regularization.

Contribution

It resolves conflicting results from previous studies by analyzing the role of infrared divergences and pole terms, confirming negative entropy regions and clarifying the behavior of the system.

Findings

Negative entropy exists for weak coupling.

Infrared divergences should be discarded on physical grounds.

Coupling-independent terms are artifacts of incomplete regularization.

Abstract

Recently the Casimir self-entropy of an electromagnetic $\delta$-function shell was considered by two different groups, with apparently discordant conclusions, although both had concluded that a region of negative entropy existed for sufficiently weak coupling. We had found that the entropy contained an infrared divergence, which we argued should be discarded on physical grounds. On the contrary, Bordag and Kirsten recently found a completely finite self-entropy, although they, in fact, have to remove an infrared divergence. Apart from this, the high- and low-temperature results for finite coupling agree precisely for the transverse electric mode, but there are significant discrepancies in the transverse magnetic mode. We resolve those discrepancies here. In particular, it is shown that coupling-independent terms do not occur in a consistent regulated calculation, they likely being an artefact of the omission of pole terms. The results of our previous analysis, especially, the existence of a negative entropy region for sufficiently weak coupling, are therefore confirmed. Finally, we offer some analogous remarks concerning the Casimir entropy of a thin electromagnetic sheet, where the total entropy is always positive. In that case, the origin of the analogous discrepancy can be explicitly isolated.