Light-Cone Expansion of the Dirac Sea to First Order in the External Potential

TL;DR

This paper develops a light-cone expansion method to analyze the first-order perturbation of the Dirac sea in an external potential, distinguishing between causal and non-causal contributions.

Contribution

It introduces a novel light-cone expansion approach for the Dirac sea perturbation, enabling explicit analysis of Feynman diagrams in position space.

Findings

Identifies causal and non-causal parts of the perturbation.

Constructs a formal solution to the Klein-Gordon equation using line integrals.

Method applicable to analysis of various quantum field equations.

Abstract

The perturbation of the Dirac sea to first order in the external potential is calculated in an expansion around the light cone. It is shown that the perturbation consists of a causal contribution, which describes the singular behavior of the Dirac sea on the light cone and contains bounded line integrals over the potential and its partial derivatives, and a non-causal contribution, which is a smooth function. As a preparatory step, we construct a formal solution of the inhomogeneous Klein-Gordon equation in terms of an infinite series of line integrals. More generally, the method presented can be used for an explicit analysis of Feynman diagrams of the Dirac, Klein-Gordon, and wave equations in position space.