# Exact Solutions to the Fermion Propagator Schwinger-Dyson Equation in Minkowski space with on-shell Renormalization for Quenched QED

## TL;DR

This paper derives explicit analytic solutions for the fermion propagator in Minkowski space within quenched QED, using spectral representation and on-shell renormalization, providing insights into the propagator's analytic structure.

## Contribution

It introduces a spectral representation approach to solve the fermion propagator Schwinger-Dyson equation analytically in Minkowski space with on-shell renormalization in quenched QED.

## Key findings

- Explicit fermion propagator spectral functions obtained
- Solutions exhibit correct analytic structure
- Padé approximation applied to spectral functions

## Abstract

With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic structure. The Pad\'{e} approximation for the spectral functions is also investigated.