Local momentum space and the vector field

TL;DR

This paper develops a local momentum space expansion for the vector field's Green function, including curvature effects, and derives heat kernel coefficients, with applications to the Maxwell field's stress-energy tensor.

Contribution

It introduces a curvature-inclusive expansion for the vector field's Green function using Riemann normal coordinates, applicable to non-minimal operators and gauge fixing.

Findings

Derived the first three untraced heat kernel coefficients including curvature terms.

Re-examined the anomalous trace of the Maxwell stress-energy tensor and discussed gauge dependence.

Provided a general framework valid in arbitrary spacetime dimensions.

Abstract

The local momentum space expansion for the real vector field is considered. Using Riemann normal coordinates we obtain an expansion of the Feynman Green function up and including terms that are quadratic in the curvature. The results are valid for a non-minimal operator such as that arising from a general Feynman type gauge fixing condition. The result is used to derive the first three terms in the asymptotic expansion for the coincidence limit of the heat kernel without taking the trace, thus obtaining the untraced heat kernel coefficients. The spacetime dimension is kept general before specializing to four dimensions for comparison with previously known results. As a further application we re-examine the anomalous trace of the stress-energy-momentum tensor for the Maxwell field and comment on the gauge dependence.