Equivalence of zeta function technique and Abel-Plana formula in regularizing the Casimir energy of hyper-rectangular cavities

TL;DR

This paper proves the equivalence between zeta function regularization and Abel-Plana formula in calculating the Casimir energy for hyper-rectangular cavities, clarifying their physical and mathematical consistency.

Contribution

It provides a rigorous proof of the equivalence between the zeta function and Abel-Plana methods for regularizing Casimir energy in rectangular cavities.

Findings

Established the reflection formula of Epstein zeta function from Abel-Plana formula

Demonstrated the physical interpretation of infinite integrals in regularization

Confirmed the mathematical consistency of both regularization methods

Abstract

Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. By contrast, Abel-Plana formula method permits explicit separation of divergent terms. In regularizing the Casimir energy for a massless scalar field in a $D$-dimensional rectangular box, we give the rigorous proof of the equivalence of the two methods by deriving the reflection formula of Epstein zeta function from repeatedly application of Abel-plana formula and giving the physical interpretation of the infinite integrals. Our study may help with the confidence of choosing any regularization method at convenience among the frequently used ones, especially the zeta function method, without the doubts of physical meanings or mathematical consistency.