Adiabatic Regularization for Gauge Field and the Conformal Anomaly

TL;DR

This paper develops an adiabatic regularization method for quantizing gauge fields in conformally flat spacetime, enabling the calculation of the conformal anomaly and aiding in the study of gauge theory renormalization.

Contribution

It introduces a novel adiabatic regularization scheme for gauge fields, extending the method to include ghost fields and applying it to compute the conformal anomaly.

Findings

Successfully reproduces the known conformal anomaly result.

Provides a framework for renormalizing gauge theories in curved spacetime.

Extends adiabatic regularization to gauge fields with ghost contributions.

Abstract

We construct the adiabatic regularization method for a $U(1)$ gauge field in a conformally flat spacetime by quantizing in the canonical formalism the gauge fixed $U(1)$ theory with mass terms for the gauge fields and the ghost fields. We show that the adiabatic expansion for the mode functions and the adiabatic vacuum can be defined in a similar way using WKB-type solutions as the scalar fields. As an application of the adiabatic method, we compute the trace of the energy momentum tensor and reproduces the known result for the conformal anomaly obtained by the other regularization methods. The availability of the adiabatic expansion scheme for gauge field allows one to study the renormalization of the de-Sitter space maximal superconformal Yang-Mills theory using the adiabatic regularization method.