Casimir Energy in the Axial Gauge

TL;DR

This paper calculates the Casimir energy for a conducting spherical shell using the axial gauge, employing path-integral methods, and confirms consistency with previous Lorenz gauge results and Boyer's value.

Contribution

It provides an exact solution for the Casimir energy in the axial gauge, demonstrating gauge invariance and extending previous analyses.

Findings

Exact solution obtained via Green-function method

Complete agreement with Lorenz gauge results

Validation of Boyer's Casimir energy value

Abstract

The zero-point energy of a conducting spherical shell is studied by imposing the axial gauge via path-integral methods, with boundary conditions on the electromagnetic potential and ghost fields. The coupled modes are then found to be the temporal and longitudinal modes for the Maxwell field. The resulting system can be decoupled by studying a fourth-order differential equation with boundary conditions on longitudinal modes and their second derivatives. The exact solution of such equation is found by using a Green-function method, and is obtained from Bessel functions and definite integrals involving Bessel functions. Complete agreement with a previous path-integral analysis in the Lorenz gauge, and with Boyer's value, is proved in detail.