Local momentum space and the vector field

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.