Simple method for calculating the Casimir energy for sphere

TL;DR

This paper introduces a straightforward method for calculating the Casimir energy of a sphere using mode summation and complex plane integration, demonstrated on perfect conductors and scalar fields with boundary conditions.

Contribution

It presents a novel, simple approach based solely on classical eigenfrequency equations, streamlining Casimir energy calculations for spherical geometries.

Findings

Efficient calculation of Casimir energy for spherical shells.

Application to perfect conductors and scalar fields with boundary conditions.

Discussion on renormalization of energy and sphere radius.

Abstract

A simple method for calculating the Casimir energy for a sphere is developed which is based on a direct mode summation and counter integration in a complex plane of eigenfrequencies. The method uses only classical equations determining the eigenfrequencies of the quantum field under consideration. Efficiency of this approach is demonstrated by calculation of the Casimir energy for a perfectly conducting spherical shell and for a massless scalar field obeying the Dirichlet and Neumann boundary conditions on sphere. The possibility of rationalizing the removal of divergences in this problem as a renormalization of both the energy and the radius of the sphere is discussed.