Hadamard renormalization for a charged scalar field

TL;DR

This paper develops explicit Hadamard renormalization techniques for a charged scalar field in various dimensions, enabling precise calculations of quantum effects near charged black holes.

Contribution

It provides explicit Hadamard coefficients and a formalism for renormalizing key expectation values for charged scalar fields in curved space-times.

Findings

Explicit Hadamard coefficients for 2, 3, and 4 dimensions.

Formalism for renormalizing scalar condensate, current, and stress-energy tensor.

Application potential for quantum stability analysis of black hole horizons.

Abstract

The Hadamard representation of the Green's function of a quantum field on a curved space-time is a powerful tool for computations of renormalized expectation values. We study the Hadamard form of the Feynman Green's function for a massive charged complex scalar field in an arbitrary number of space-time dimensions. Explicit expressions for the coefficients in the Hadamard parametrix are given for two, three and four space-time dimensions. We then develop the formalism for the Hadamard renormalization of the expectation values of the scalar field condensate, current and stress-energy tensor. These results will have applications in the computation of renormalized expectation values for a charged quantum scalar field on a charged black hole space-time, and hence in addressing issues such as the quantum stability of the inner horizon.