Formulating Schwinger--Dyson equations for QED propagators in Minkowski space

TL;DR

This paper develops a truncation scheme for Schwinger--Dyson equations of QED propagators in Minkowski space, aiming to preserve gauge principles and enable nonperturbative solutions.

Contribution

It introduces a novel truncation approach combining dimensional regularization and spectral representation to solve QED SDEs in Minkowski space.

Findings

Truncation scheme preserves gauge invariance principles.

Solutions obtainable directly in Minkowski space.

Potential application to nonperturbative QFT studies.

Abstract

The Schwinger--Dyson equations (SDEs) are coupled integral equations for the Green's functions of a quantum field theory (QFT). The SDE approach is the analytic nonperturbative method for solving strongly coupled QFTs. When applied to QCD, this approach, also based on the first principle, is the analytic alternative to lattice QCD. However, the SDEs for the n-point Green's functions involves (n+1)-point Green's functions (sometimes (n+2)-point functions as well). Therefore any practical method for solving this infinitely coupled system of equations requires a truncation scheme. When considering strongly coupled QED as a modeling of QCD, naive truncation schemes violate various principles of the gauge theory. These principles include gauge invariance, gauge covariance, and multiplicative renormalizability. The combination of dimensional regularization with the spectral representation of propagators results in a tractable formulation of a truncation scheme for the SDEs of QED propagators, which has the potential to preserve the aforementioned principles and renders solutions obtainable in the Minkowski space.