From tracial anomalies to anomalies in Quantum Field Theory

TL;DR

This paper explores how tracial anomalies arising from zeta-regularized traces in pseudo-differential operators can cause unexpected phenomena in quantum field theory, demonstrated through two specific examples.

Contribution

It demonstrates, with two examples, how non-cyclic and non-commuting properties of zeta-regularized traces lead to anomalies in quantum field theory.

Findings

Tracial anomalies can induce anomalous phenomena in quantum field theory.

Zeta-regularized traces are not cyclic and do not commute with differentiation.

Examples illustrate the impact of these anomalies on quantum field theoretical models.

Abstract

zeta-regularized traces, resp. super-traces, are defined on a classical pseudo-differential operator A by: tr^Q(A):= f.p.tr(A Q^{-z})_{|_{z=0}}, resp. str^Q(A):= f.p.str(A Q^{-z})_{|_{z=0}}, where f.p. refers to the finite part and Q is an (invertible and admissible) elliptic reference operator with positive order. They are widly used in quantum field theory in spite of the fact that, unlike ordinary traces on matrices, they are neither cyclic nor do they commute with exterior differentiation, thus giving rise to tracial anomalies. The purpose of this article is to show, on two examples, how tracial anomalies can lead to anomalous phenomena in quantum field theory.