Symbol calculus and zeta--function regularized determinants

TL;DR

This paper develops a method using symbol calculus and semigroup integrals to evaluate zeta-function regularized determinants, especially for non-positive operators like the Dirac operator, aiding in quantum effective action analysis.

Contribution

It introduces a novel approach combining semigroup integrals and Weyl symbol calculus to compute determinants for complex operators in quantum field theory.

Findings

Effective evaluation of determinants for Dirac and bosonic operators

Application of derivative expansion to quantum effective actions

Enhanced understanding of kinetic and potential terms in quantum theories

Abstract

In this work, we use semigroup integral to evaluate zeta-function regularized determinants. This is especially powerful for non--positive operators such as the Dirac operator. In order to understand fully the quantum effective action one should know not only the potential term but also the leading kinetic term. In this purpose we use the Weyl type of symbol calculus to evaluate the determinant as a derivative expansion. The technique is applied both to a spin--0 bosonic operator and to the Dirac operator coupled to a scalar field.