The one-loop effective action and trace anomaly in four dimensions

TL;DR

This paper computes the one-loop effective action for quantum fields in four dimensions as a nonlocal curvature expansion, and derives the trace anomaly directly from this effective action using covariant methods and heat kernel techniques.

Contribution

It provides a third-order nonlocal curvature expansion of the one-loop effective action and a direct derivation of the trace anomaly in four dimensions.

Findings

Effective action expanded to third order in curvature.

Trace anomaly obtained directly from the effective action.

Nonlocal terms involve complex functions of three operators.

Abstract

The one-loop effective action for a generic set of quantum fields is calculared as a nonlocal expansion in powers of the curvatures (field strengths). This expansion is obtained to third order in the curvature. It is stressed that the covariant vertices are finite. The trace anomaly in four dimensions is obtained directly by varying the effective action. The nonlocal terms in the action, producing the anomaly, contain non-trivial functions of three operator arguments. The trace anomaly is derived also by making the conformal transformation in the heat kernel.