On the photon Green functions in curved space-time

TL;DR

This paper develops a detailed mathematical analysis of photon Green functions in curved space-time, providing asymptotic expansions and gauge dependence insights crucial for quantum electrodynamics in curved backgrounds.

Contribution

It derives the full asymptotic expansion of the photon Green function and explicitly characterizes its gauge dependence without introducing a mass term.

Findings

Asymptotic expansion of photon Green function obtained

Explicit gauge parameter dependence derived

Coincidence limits of derivatives evaluated

Abstract

Quantization of electrodynamics in curved space-time in the Lorenz gauge and with arbitrary gauge parameter makes it necessary to study Green functions of non-minimal operators with variable coefficients. Starting from the integral representation of photon Green functions, we link them to the evaluation of integrals involving Gamma functions. Eventually, the full asymptotic expansion of the Feynman photon Green function at small values of the world function, as well as its explicit dependence on the gauge parameter, are obtained without adding by hand a mass term to the Faddeev--Popov Lagrangian. Coincidence limits of second covariant derivatives of the associated Hadamard function are also evaluated, as a first step towards the energy-momentum tensor in the non-minimal case.