Connection between $\zeta$ and cutoff regularizations of Casimir energies

TL;DR

This paper investigates the relationship between zeta and cutoff regularizations of Casimir energies, revealing divergences and finite parts that differ, and applies findings to scalar fields in d-dimensional boxes.

Contribution

It establishes the connections and differences between zeta and cutoff regularizations for Casimir energies, providing explicit relationships among their coefficients.

Findings

Both regularization schemes produce divergent and finite parts that do not match.

Derived relationships among coefficients in zeta and cutoff regularizations.

Applied the analysis to scalar fields in d-dimensional periodic boxes.

Abstract

We study the connection between $\zeta $- and cutoff-regularized Casimir energies for scalar fields. We show that, in general, both regularization schemes lead to divergent contributions, and to finite parts which do not coincide. We determine the relationships among the various coefficients appearing in one approach and the other. As an application, we discuss the case of scalar fields in $d$-dimensional boxes under periodic boundary conditions.