Vacuum energy via dimensional reduction of functional determinants

TL;DR

This paper introduces a dimensional reduction method using a generalized Gelfand-Yaglom theorem to evaluate vacuum energy, providing an alternative Casimir energy representation and new approximation schemes for complex backgrounds.

Contribution

It develops a novel dimensional reduction approach for functional determinants, offering an alternative Casimir energy formulation and new approximation techniques.

Findings

Reduced problem yields valuable insights into vacuum energy.

New approximation scheme enhances computational efficiency.

Alternative representation simplifies complex background evaluations.

Abstract

We apply a `dimensional reduction' mechanism to the evaluation of the functional integral for the vacuum energy of a real scalar field in the presence of non-trivial backgrounds, in d+1 dimensions. The reduction is implemented by applying a generalized version of Gelfand-Yaglom's theorem to the corresponding functional determinant. The main outcome of that procedure is an alternative representation for the Casimir energy, which involves one spatial dimension less than the original problem. We show that, for some configurations, important information about the reduced problem can be obtained. We also show that the reduced problem allows for the introduction of an approximation scheme which is novel within this context.