Multidimensional cut-off technique, odd-dimensional Epstein zeta functions and Casimir energy of massless scalar fields

TL;DR

This paper introduces a multidimensional cut-off technique to accurately compute Casimir energies for massless scalar fields in high-dimensional rectangular spaces, linking quantum fluctuations to measurable forces and providing exact numerical methods.

Contribution

It develops a novel multidimensional cut-off method for Casimir energy calculation and generalizes it to arbitrary lengths, with a new interpretation of analytical and remainder parts.

Findings

Explicit formulas for multidimensional Casimir energy calculations.

Demonstration of exponential convergence of the remainder term.

Representation of Epstein zeta functions as products of one-dimensional sums.

Abstract

Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose-Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement - due to quantum fluctuations - in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in $d$-dimensional rectangular spaces with $q$ large dimensions and $d-q$ dimensions of length $L$ and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulas we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization.