Equivalence of the adiabatic expansion and Hadamard renormalization for a charged scalar field

Abstract

We examine the relationship between three approaches (Hadamard, DeWitt-Schwinger and adiabatic) to the renormalization of expectation values of field operators acting on a charged quantum scalar field. First, we demonstrate that the DeWitt-Schwinger representation of the Feynman Green's function is a particular case of the Hadamard representation. Next, we restrict attention to a spatially flat Friedmann-Lemaitre-Robertson-Walker universe with time-dependent, purely electric, background electromagnetic field, considering two, three and four-dimensional space-times. Working to the order required for the renormalization of the stress-energy tensor (SET), we find the adiabatic and DeWitt-Schwinger expansions of the Green's function when the space-time points are spatially separated. In two and four dimensions, the resulting DeWitt-Schwinger and adiabatic expansions are identical. In three dimensions, the DeWitt-Schwinger expansion contains terms of adiabatic order four which are not necessary for the renormalization of the SET and hence absent in the adiabatic expansion. The equivalence of the DeWitt-Schwinger and adiabatic approaches to renormalization in the scenario considered is thereby demonstrated up to well-known renormalization ambiguities in three space-time dimensions.