Electromagnetic δ -function sphere

Abstract

We develop a formalism to extend our previous work on the electromagnetic $\delta$-function plates to a spherical surface. The electric ($\lambda_e$) and magnetic ($\lambda_g$) couplings to the surface are through $\delta$-function potentials defining the dielectric permittivity and the diamagnetic permeability, with two anisotropic coupling tensors. The formalism incorporates dispersion. The electromagnetic Green's dyadic breaks up into transverse electric and transverse magnetic parts. We derive the Casimir interaction energy between two concentric $\delta$-function spheres in this formalism and show that it has the correct asymptotic flat plate limit. We systematically derive expressions for the Casimir self-energy and the total stress on a spherical shell using a $\delta$-function potential, properly regulated by temporal and spatial point-splitting, which are different from the conventional temporal point-splitting. In strong coupling, we recover the usual result for the perfectly conducting spherical shell but in addition, there is an integrated curvature-squared divergent contribution. For finite coupling, there are additional divergent contributions; in particular, there is a familiar logarithmic divergence occurring in the third order of the uniform asymptotic expansion that renders it impossible to extract a unique finite energy except in the case of an isorefractive sphere, which translates into $\lambda_g=-\lambda_e$.