Casimir Energy for Spherical boundaries

TL;DR

This paper critically examines methods for calculating Casimir energy in spherical geometries, identifying errors in stress tensor approaches and demonstrating an improved, equivalent calculation method based on mode summation.

Contribution

It highlights errors in previous stress tensor-based calculations and presents a corrected approach that aligns with mode summation results, clarifying the proper methodology.

Findings

Previous stress tensor methods contain serious errors.

Proper calculations align with mode summation results.

An improved stress tensor approach is demonstrated.

Abstract

Calculations of the Casimir energy for spherical geometries which are based on integrations of the stress tensor are critically examined. It is shown that despite their apparent agreement with numerical results obtained from mode summation methods, they contain a number of serious errors. Specifically, these include (1) an improper application of the stress tensor to spherical boundaries, (2) the neglect of pole terms in contour integrations, and (3) the imposition of inappropriate boundary conditions upon the relevant propagators. A calculation which is based on the stress tensor and which avoids such problems is shown to be possible. It is, however, equivalent to the mode summation method and does not therefore constitute an independent calculation of the Casimir energy.