Casimir Energy of a Ball and Cylinder in the Zeta Function Technique

TL;DR

This paper introduces a straightforward zeta function method for calculating the electromagnetic Casimir energy of spherical and cylindrical geometries, avoiding divergences and renormalization, and providing rigorous mathematical justification for known contributions.

Contribution

It develops a simple, divergence-free approach using contour integrals and asymptotic expansions to compute Casimir energies for spheres and cylinders with boundary conditions.

Findings

Spectral zeta functions constructed without divergences

Method applied to material and perfect conductor cases

Mathematically rigorous justification for logarithmic contributions

Abstract

A simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder. When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed. The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used. After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms. The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in an uniform endless medium under the condition that the velocity of light does not change when crossing the interface. As a special case, perfectly conducting spherical and cylindrical shells are also considered in the same line. In this approach one succeeds, specifically, in justifying, in mathematically rigorous way, the appearance of the contribution to the Casimir energy for cylinder which is proportional to $\ln (2\pi)$.