Gauge-invariant momentum and angular momentum operators in quantum electrodynamics and chromodynamics

TL;DR

This paper demonstrates that gauge-invariant momentum and angular momentum operators in quantum electrodynamics and chromodynamics can be explicitly constructed in the Coulomb gauge, simplifying quantization while maintaining gauge invariance.

Contribution

It provides explicit expressions for gauge-invariant momentum, spin, and orbital angular momentum operators in the Coulomb gauge, showing their properties and relation to physical transverse photons and gluons.

Findings

Gauge-invariant operators are explicitly constructed in Coulomb gauge.

Quantization in Coulomb gauge involves only physical transverse modes.

Operators are more complex than their quantum mechanical counterparts.

Abstract

Differences between vector potentials in different gauges contain no dynamics in both classical and quantum electrodynamics and chromodynamics. Consequently, once gauge invariance is established, results calculated in non-covariant gauges can be expected to agree with results obtained in covariant gauges in all Lorentz frames. We show in particular that canonical quantization in the Coulomb gauge can be used without giving up explicit gauge invariance. Quantization in the Coulomb gauge is particularly simple because it involves only the two transverse photons/gluons present in all gauges. These transverse photons/gluons reside on a 2-dimensional physical plane in momentum space perpendicular to the photon/gluon momentum {\bf k}. Explicit expressions are given for the basic momentum, spin and orbital angular momentum field operators of photons/gluons in the Coulomb gauge. Their properties are discussed in some detail. In particular, these field operators are shown to be more complicated than the corresponding operators in quantum mechanics which they also contain.