Gauge covariance of the fermion Schwinger-Dyson equation in QED

Any practical application of the Schwinger-Dyson equations to the study of n-point Green's functions in a strong coupling field theory requires truncations. In the case of QED, the gauge covariance, governed by the Landau-Khalatnikov-Fradkin transformations (LKFT), provides a unique constraint on such truncation. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the LKFT are linear operations on the spectral densities. 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.