Hadamard parametrix of the Feynman Green’s function of a five-dimensional charged scalar field

Abstract

The Hadamard parametrix is a representation of the short-distance singularity structure of the Feynman Green’s function for a quantum field on a curved spacetime background. Subtracting these divergent terms regularizes the Feynman Green’s function and enables the computation of renormalized expectation values of observables. We study the Hadamard parametrix for a charged, massive, complex scalar field in five spacetime dimensions. Even in Minkowski spacetime, it is not possible to write the Feynman Green’s function for a charged scalar field exactly in closed form. We, therefore, present covariant Taylor series expansions for the biscalars arising in the Hadamard parametrix. On a general spacetime background, we explicitly state the expansion coefficients up to the order required for the computation of the renormalized scalar field current. These coefficients become increasingly lengthy as the order of the expansion increases, so we give the higher-order terms required for the calculation of the renormalized stress-energy tensor in Minkowski spacetime only.

Quantization of a charged scalar field on a charged black hole background

“…Work to date on this question has largely focused on a neutral quantum scalar field (apart from the recent work considering a charged scalar field in [57,58]). A general formalism for the Hadamard renormalization of expectation values for a charged quantum scalar field was developed in [61,85] (see also [86,87] for earlier work based on DeWitt-Schwinger renormalization). Using this approach, renormalized expectation values of the current have been computed on an RN-de Sitter black hole [57,58] for a charged scalar field in the Unruh state.…”