Green Functions for Classical Euclidean Maxwell Theory

TL;DR

This paper investigates the classical Euclidean Green functions for Maxwell theory, demonstrating the existence of solutions that satisfy various gauge conditions, including axial and Coulomb gauges, on bounded flat Euclidean space.

Contribution

It provides a detailed proof of the existence of Euclidean Green functions that satisfy specific gauge conditions in classical Maxwell theory.

Findings

Existence of Euclidean Green functions for second- and fourth-order operators.

Admissibility of axial and Coulomb gauges at the classical level.

Green functions ensure gauge conditions on bounded Euclidean regions.

Abstract

Recent work on the quantization of Maxwell theory has used a non-covariant class of gauge-averaging functionals which include explicitly the effects of the extrinsic-curvature tensor of the boundary, or covariant gauges which, unlike the Lorentz case, are invariant under conformal rescalings of the background four-metric. This paper studies in detail the admissibility of such gauges at the classical level. It is proved that Euclidean Green functions of a second- or fourth-order operator exist which ensure the fulfillment of such gauges at the classical level, i.e. on a portion of flat Euclidean four-space bounded by three-dimensional surfaces. The admissibility of the axial and Coulomb gauges is also proved.