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

We determine the retarded and advanced 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 contain additional singular terms compared with their Feynman/de Donder-gauge counterpart. 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.