Gauge covariance of the fermion Schwinger-Dyson equation in QED

TL;DR

This paper demonstrates that gauge covariance constraints in QED's fermion Schwinger-Dyson equations can be represented as linear operations on spectral densities, aiding in testing truncation validity.

Contribution

It introduces a spectral representation approach that formalizes gauge covariance constraints as linear group operations, simplifying the validation of Schwinger-Dyson truncations in QED.

Findings

Constraints are linear operations on spectral densities.

Group operations can test gauge covariance of truncations.

Provides practical examples for validation.

Abstract

Any practical application of the Schwinger-Dyson equations to the study of $n$-point Green's functions of a field theory requires truncations, the best known being finite order perturbation theory. Strong coupling studies require a different approach. In the case of QED, gauge covariance is a powerful constraint. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the Landau-Khalatnikov-Fradkin transformations are linear operations on the spectral densities. Here we formally define these group operations and show with a couple of examples how in practice they provide a straightforward way to test the gauge covariance of any viable truncation of the Schwinger-Dyson equation for the fermion 2-point function.