Casimir Effect in Problems with Spherical Symmetry: New Perspectives

TL;DR

This paper explores the Casimir effect in spherical geometries within quantum electrodynamics, emphasizing the importance of gauge choices and boundary conditions for accurate zero-point energy calculations.

Contribution

It introduces a detailed analysis of gauge dependence in the Casimir effect for spherical shells, highlighting the role of longitudinal and normal modes in the Lorenz gauge.

Findings

Longitudinal and normal modes are essential for reproducing known Casimir results.

Gauge choice significantly affects the eigenvalue equations in spherical geometries.

Covariant and non-covariant gauges lead to complex eigenvalue systems.

Abstract

Since the Maxwell theory of electromagnetic phenomena is a gauge theory, it is quite important to evaluate the zero-point energy of the quantized electromagnetic field by a careful assignment of boundary conditions on the potential and on the ghost fields. Recent work by the authors has shown that, for a perfectly conducting spherical shell, it is precisely the contribution of longitudinal and normal modes of the potential which enables one to reproduce the result first due to Boyer. This is obtained provided that one works with the Lorenz gauge-averaging functional, and with the help of the Feynman choice for a dimensionless gauge parameter. For arbitrary values of the gauge parameter, however, covariant and non-covariant gauges lead to an entangled system of three eigenvalue equations. Such a problem is crucial both for the foundations and for the applications of quantum field theory.