Applications in physics of the multiplicative anomaly formula involving some basic differential operators

TL;DR

This paper proves the validity of the multiplicative anomaly formula for zeta-function regularisation on non-compact manifolds, with detailed analysis of Dirac operators and harmonic oscillators, highlighting its physical significance.

Contribution

It demonstrates the validity of the multiplicative anomaly formula in non-compact manifolds and provides explicit analytical expressions for the anomaly in physical operators.

Findings

The anomaly can be expressed by a simple analytical formula.

The anomaly is non-zero for Dirac operators and harmonic oscillators.

The anomaly impacts the calculation of determinants of differential operators in physics.

Abstract

In the framework leading to the multiplicative anomaly formula ---which is here proven to be valid even in cases of known spectrum but non-compact manifold (very important in Physics)--- zeta-function regularisation techniques are shown to be extremely efficient. Dirac like operators and harmonic oscillators are investigated in detail, in any number of space dimensions. They yield a non-zero anomaly which, on the other hand, can always be expressed by means of a simple analytical formula. These results are used in several physical examples, where the determinant of a product of differential operators is not equal to the product of the corresponding functional determinants. The simplicity of the Hamiltonian operators chosen is aimed at showing that such situation may be quite widespread in mathematical physics. However, the consequences of the existence of the determinant anomaly have often been overlooked.