Multiplicative Anomaly matches Casimir Energy for GJMS Operators on Spheres

TL;DR

This paper derives a formula for the multiplicative anomaly of zeta-regularized products, applies it to spheres and finite-temperature universes, and reveals that Casimir energy equals the accumulated multiplicative anomaly for GJMS operators, offering new physical insights.

Contribution

It introduces a generalized explicit formula for the multiplicative anomaly and demonstrates its equivalence to Casimir energy for GJMS operators on spheres and Einstein universes.

Findings

Casimir energy matches the accumulated multiplicative anomaly.

The formula generalizes Shintani-Mizuno results.

An improved Casimir energy accounting for the anomaly is proposed.

Abstract

An explicit formula to compute the multiplicative anomaly or defect of $\zeta$-regularized products of linear factors is derived, by using a Feynman parametrization, generalizing Shintani-Mizuno formulas. Firstly, this is applied on $n$-spheres, reproducing known results in the literature. Then, this framework is applied to a closed Einstein universe at finite temperature, namely $S^1_{\beta}\times S^{n-1}$. In doing so, it is shown that the standard Casimir energy for GJMS operators coincides with the accumulated multiplicative anomaly for the shifted Laplacians that build them up. This equivalence between Casimir energy and multiplicative anomaly, unnoticed so far to our knowledge, brings about a new turn regarding the physical significance of the multiplicative anomaly, putting both now on equal footing. An emergent improved Casimir energy, that takes into account the multiplicative anomaly among the building Laplacians, is also discussed.