Canonical quantization of the electromagnetic field in arbitrary $\xi$-gauge

TL;DR

This paper performs the canonical quantization of the electromagnetic field in any $\xi$-gauge, clarifies the propagator and $i\epsilon$ prescription, and demonstrates gauge parameter independence of physical results.

Contribution

It provides a comprehensive canonical quantization framework for the electromagnetic field in arbitrary $\xi$-gauge, resolving ambiguities and clarifying gauge independence.

Findings

Derived the propagator in arbitrary $\xi$-gauge

Clarified the $i\\epsilon$ prescription for the propagator

Proved gauge independence of physical processes

Abstract

We carry out the canonical quantization of the electromagnetic field in arbitrary $\xi$-gauge and compute its propagator. In this way we fill a gap in the literature and clarify some existing confusion about Feynman $i\epsilon$ prescription for the propagator of the electromagnetic field. We also discuss the BRST quantization and investigate the apparent singularities present in the theory when the gauge parameter $\xi$ takes the value -1. We find that this is a mere artifact due to the choice of basic modes and show that in the appropriate basis the commutation relations and the BRST transformation are, in fact, independent of the gauge parameter. The latter only appears as the coefficient of a BRST exact term in the Hamiltonian, which constitutes an extremely simple proof of the independence of any physical process on the gauge parameter $\xi$.