Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue

TL;DR

This paper calculates the multiplicative anomaly for zeta-function regularized determinants of Laplace-type operators on compact manifolds, revealing its dependence on dimension and applying results to quantum field theory models.

Contribution

It provides explicit calculations of the multiplicative anomaly using Wodzicki's residue, highlighting cases where the anomaly vanishes and applying findings to scalar field theories.

Findings

Anomaly vanishes for odd dimensions and D=2

Explicit formulas for the anomaly using Wodzicki residue

Application to one-loop effective potential in scalar models

Abstract

The multiplicative anomaly associated with the zeta-function regularized determinant is computed for the Laplace-type operators $L_1=-\lap+V_1$ and $L_2=-\lap+V_2$, with $V_1$, $V_2$ constant, in a D-dimensional compact smooth manifold $ M_D$, making use of several results due to Wodzicki and by direct calculations in some explicit examples. It is found that the multiplicative anomaly is vanishing for $D$ odd and for D=2. An application to the one-loop effective potential of the O(2) self-interacting scalar model is outlined.