Casimir Energy of a Spherical Shell

TL;DR

This paper recalculates the Casimir energy of a spherical shell using a corrected mode summation method, addressing previous methodological issues and confirming the numerical results without the criticized assumptions.

Contribution

It introduces a corrected mode summation approach for Casimir energy calculations that resolves prior methodological criticisms.

Findings

Numerical results remain unchanged after correction.

The method avoids ill-defined mathematical expressions.

Addresses and resolves longstanding criticisms of earlier calculations.

Abstract

The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a longstanding criticism raised against earlier calculations of the Casimir energy.