# Green's functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges

August 01, 2017

We determine the Green's functions and Hadamard parametrices in curved
spacetimes for linearized massive and massless gauge bosons and linearized
Einstein gravity with a cosmological constant in general linear covariant
gauges. These vector and tensor parametrices are more singular than their
Feynman/de Donder-gauge counterpart, with the most singular part proportional
to $\sigma^{-2}$ and $\sigma^{-3}$, respectively. We also give explicit
recursion relations for the Hadamard coefficients, and indicate their
generalization to $n$ dimensions. Furthermore, we express the divergence and
trace of the vector and tensor Green's functions in terms of derivatives of
scalar and vector Green's functions, and show how these relations appear as
Ward identities in the free quantum theory.

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