Functional Determinant of the Massive Laplace Operator and the Multiplicative Anomaly

TL;DR

This paper explores the zeta function regularization and multiplicative anomaly, applying these methods to compute functional determinants of massive Laplacians on spheres across various dimensions, providing explicit formulas for specific cases.

Contribution

It presents explicit formulas for functional determinants of massive Laplacians on spheres in multiple dimensions, extending the understanding of zeta regularization and anomalies in these contexts.

Findings

Explicit formulas for Laplace operators on spheres in 1-4 dimensions

Formulas for vector and tensor Laplacians on S^4

Application of zeta regularization to quantum field theory

Abstract

After a brief survey of zeta function regularization issues and of the related multiplicative anomaly, illustrated with a couple of basic examples, namely the harmonic oscillator and quantum field theory at finite temperature, an application of these methods to the computation of functional determinants corresponding to massive Laplacians on spheres in arbitrary dimensions is presented. Explicit formulas are provided for the Laplace operator on spheres in $N=1,2,3,4$ dimensions and for `vector' and `tensor' Laplacians on the unitary sphere $S^4$.