Covariant non-local action for massless QED and the curvature expansion

TL;DR

This paper develops a covariant non-local action approach for massless QED in curved space, enabling the study of gravitational couplings and anomalies through a novel non-linear completion method.

Contribution

It introduces a non-linear completion technique for constructing non-local actions in curved space, providing a new way to analyze QED trace anomalies.

Findings

The non-linear completion matches diagrammatic expansion of the effective action.

The approach clarifies the curved space generalization of ln clow, including the ln clow term.

The method identifies the non-local terms responsible for the QED trace anomaly in 4D.

Abstract

We explore the properties of non-local effective actions which include gravitational couplings. Non-local functions originally defined in flat space can not be easily generalized to curved space. The problem is made worse by the calculational impossibility of providing closed form expressions in a general metric. The technique of covariant perturbation theory (CPT) has been pioneered by Vilkovisky, Barvinsky and collaborators whereby the effective action is displayed as an expansion in the generalized curvatures similar to the Schwinger-De Witt local expansion. We present an alternative procedure to construct the non-local action which we call {\em non-linear completion}. Our approach is in one-to-one correspondence with the more familiar diagrammatic expansion of the effective action. This technique moreover enables us to decide on the appropriate non-local action that generates the QED trace anomaly in 4$D$. In particular we discuss carefully the curved space generalization of $\ln \Box$, and show that the anomaly requires both the anomalous logarithm as well as $1/\Box$ term where the latter is related to the Riegert anomaly action.