A note on the Casimir energy of a massive scalar field in positive curvature space

TL;DR

This paper re-evaluates the Casimir energy of a massive scalar field in a positively curved space using zeta function techniques, confirming the equivalence with high temperature regularization methods and exploring analytic continuation approaches.

Contribution

It provides a detailed analysis of the Casimir energy calculation for a massive scalar field in curved space, comparing different regularization and analytic continuation methods.

Findings

Zeta function regularization is equivalent to high temperature regularization for this problem.

Two valid methods of analytic continuation are described and compared.

Results clarify the computation of vacuum energy in curved spacetime.

Abstract

We re-evaluate the zero point Casimir energy for the case of a massive scalar field in $\mathbf{R}^{1}\times\mathbf{S}^{3}$ space, allowing also for deviations from the standard conformal value $\xi =1/6$, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularization of the vacuum energy, as conjectured in a previous publication. Two different, albeit equally valid, ways of doing the analytic continuation are described.