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

Understand

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.