Remarks on the Casimir self-entropy of a spherical electromagnetic δ -function shell

Abstract

Recently the Casimir self-entropy of an electromagnetic δ-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. * kmilton@ou.edu †

Negativity of the Casimir Self-Entropy in Spherical Geometries

“…[19] and Figure 1 of Ref. [20] (see Figure 1 here). Evidently, the entropy, the negative derivative of the free energy with respect to temperature, becomes negative for sufficiently weak coupling (large ξ), as is seen from the analytic limiting behavior:…”

“…We showed in Refs. [19,20] that the leading behaviors for high temperature of the TM free energy and entropy are…”

“…(An alternative derivation is given in Appendix A of ref. [ 20 ].) This function is plotted in Figure 3 of ref.…”

“…[ 19 ] and Figure 1 of ref. [ 20 ] (see Figure 1 here). Evidently, the entropy, the negative derivative of the free energy with respect to temperature, becomes negative for sufficiently weak coupling (large ), as is seen from the analytic limiting behavior: The TE contribution to the entropy is always negative, so the total entropy turns negative for a sufficiently small coupling.…”

“…In the limit of , we recover the entropy for a perfectly conducting sphere first obtained by Balian and Duplantier [ 18 ]. However, for a sufficiently weak coupling, even at high temperatures, we found that the entropy could turn negative [ 19 , 20 ]. (The results found there largely agreed with those found subsequently by Bordag and Kirsten [ 21 , 22 ].)…”