Abstract
It is known that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with ε μ = 1 , a so-called isorefractive or diaphanous ball. Here we re-examine that example and attempt to extend it to an electromagnetic δ -function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.
Highlights
It is clear that Casimir energies between distinct rigid bodies are finite, even though they arise formally from a summation of changes in zero-point field energies by material bodies, that finiteness fails for the self-energy of a single body
Brevik and Einevoll [13] include dispersion in the coupling, but this leads to linearly divergent terms that are regulated by insertion of an arbitrary parameter
Casimir self energies of bodies are divergent, so it is of interest to study examples where unambiguous finite self energies can be extracted. Such is the case of a perfectly conducting spherical shell, or the diaphanous ball discussed in the previous section, which reduces to the former in the ξ → 1 limit. Another generalization of the spherical shell that would seem to admit finiteness is an electromagnetic δ-function sphere, with equal and opposite electric and magnetic couplings; it was observed in [20] that the catastrophic divergence in the third order in the coupling cancels, because only even powers in the coupling appear in the uniform asymptotic expansion of the energy integrand
Summary
It is clear that Casimir energies between distinct rigid bodies are finite, even though they arise formally from a summation of changes in zero-point field energies by material bodies, that finiteness fails for the self-energy of a single body. Much earlier, Brevik and collaborators realized that, in the special case eμ = 1, that is, when the speed of light is the same both inside and outside the sphere, the divergences cancel, and a completely unique finite self-energy can be found [10,11,12,13,14,15,16]. One could note analogous results for diaphanous cylinders [17,18,19], which yield a vanishing energy in the dilute limit, in contrast to the nonvanishing energy of a diaphanous ball Another generalization was explored more recently—that of an electromagnetic δ-function shell [20].
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