Zero-point energies and the multiplicative anomaly

TL;DR

This paper investigates the presence of the multiplicative anomaly in the one-loop effective action of a relativistic scalar field at finite temperature, finding no anomaly when using full determinants and explaining its high-temperature behavior.

Contribution

It provides an exact calculation of the effective action using zeta-function regularisation and clarifies the conditions under which the multiplicative anomaly appears or vanishes.

Findings

No multiplicative anomaly for the full fourth order determinant.

The anomaly at high temperature is explained as a difference in zero-point energies.

The anomaly is linked to non-normal ordered charge operators.

Abstract

For the case of a relativistic scalar field at finite temperature with a chemical potential, we calculate an exact expression for the one-loop effective action using the full fourth order determinant and zeta-function regularisation. We find that it agrees with the exact expression for the factored operator and thus there appears to be no mulitplicative anomaly. The appearance of the anomaly for the fourth order operator in the high temperature limit is explained and we show that the multiplicative anomaly can be calculated as the difference between two zeta-regularised zero-point energies. This difference is a result of using a charge operator in the Hamiltonian which has not been normal ordered.