Light-front gauge propagator reexamined-II

TL;DR

This paper reexamines the light-front gauge propagator, proposing a new method with two Lagrange multipliers to eliminate residual gauge freedom and produce a well-defined, Lorentz non-invariant propagator.

Contribution

It introduces a novel gauge fixing approach using two Lagrange multipliers, resulting in a complete and exact light-front propagator free of residual gauge ambiguities.

Findings

Achieves a well-defined light-front propagator without residual gauge freedom

Provides an exact propagator with explicit gauge fixing terms

Results in a Lorentz non-invariant but consistent propagator

Abstract

Gauge fields are special in the sense that they are invariant under gauge transformations and \emph{``ipso facto''} they lead to problems when we try quantizing them straightforwardly. To circumvent this problem we need to specify a gauge condition to fix the gauge so that the fields that are connected by gauge invariance are not overcounted in the process of quantization. The usual way we do this in the light-front is through the introduction of a Lagrange multiplier, $(n\cdot A)^{2}$, where $n_{\mu}$ is the external light-like vector, i.e., $n^{2}=0$, and $A_{\mu}$ is the vector potential. This leads to the usual light-front propagator with all the ensuing characteristics such as the prominent $(k\cdot n)^{-1}$ pole which has been the subject of much research. However, it has been for long recognized that this procedure is incomplete in that there remains a residual gauge freedom still to be fixed by some ``ad hoc'' prescription, and this is normally worked out to remedy some unwieldy aspect that emerges along the way. In this work we propose \emph{two} Lagrange multipliers with distinct coefficients for the light-front gauge that leads to the correctly defined propagator with no residual gauge freedom left. This is accomplished via $(n\cdot A)^2+(\partial \cdot A)^2$ terms in the Lagrangian density. These lead to a well-defined and exact though Lorentz non invariant propagator.