The Epstein-Glaser causal approach to the Light-Front QED$_{4}$. I: Free theory

TL;DR

This paper applies the Epstein-Glaser causal approach to light-front QED, providing a rigorous mathematical treatment of singular propagators without prescriptions, and deriving consistent (anti)commutation relations and propagator structures.

Contribution

It introduces a distribution-based, prescription-free method to handle singular light-front propagators using Epstein-Glaser causality in free QED.

Findings

Derived analytic representations for light-front propagators.

Established equal-time (anti)commutation relations for fields.

Reproduced known prescriptions under specific conditions.

Abstract

In this work we present the study of light-front field theories in the realm of axiomatic theory. It is known that when one uses the light-cone gauge pathological poles $(k^{+}) ^{-n}$ arises, demanding a prescription to be employed in order to tame these ill-defined poles and to have correct Feynman integrals due to the lack of Wick rotation in such theories. In order to shed a new light on this long standing problem we present here a discussion based on the use rigorous mathematical machinery of distributions combined with physical concepts, such as causality, to show how to deal with these singular propagators in a general fashion without making use of any prescription. The first step of our development will consist in showing how analytic representation for propagators arises by requiring general physical properties in the framework of Wightman's formalism. From that we shall determine the equal-time (anti)commutation relations in the light-front form for the scalar, fermionic fields and for the dynamical components of the electromagnetic field. In conclusion, we introduce the Epstein-Glaser causal method in order to have a mathematical rigorous treatment of the free propagators of the theory, allowing us to discuss the general treatment for propagators of the type $(k^{+}) ^{-n}$. Moreover, we show that at given conditions our results reproduce known prescriptions in the literature.