Vector Casimir effect for a D-dimensional sphere

TL;DR

This paper calculates the Casimir energy for vector fields inside and outside a spherical shell in D dimensions, extending previous scalar mode results to electromagnetic-like fields with TM boundary conditions.

Contribution

It provides a comprehensive calculation of the Casimir effect for vector (electromagnetic) modes in D-dimensional spheres, including interior and exterior regions, generalizing earlier scalar mode findings.

Findings

Reproduces known 3D Casimir results by Boyer.

Identifies poles in stress at positive even dimensions.

Finds logarithmic singularities at integer dimensions D ≤ 1.

Abstract

The Casimir energy or stress due to modes in a D-dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir effect for fluctuating ``electromagnetic'' (vector) fields inside and outside a spherical shell. Known results for three dimensions, first found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and cusps (logarithmic singularities) occur for integer dimensions $D\le1$. Particular attention is given the interesting case of D=2.