Spin and orbital angular momentum in gauge theories (I): QED and determination of the angular momentum density

TL;DR

This paper resolves the gauge invariance issue of angular momentum in QED, showing how to define and compute gauge-invariant spin and orbital angular momentum for electrons and photons, and correcting a common formula for electromagnetic angular momentum density.

Contribution

It introduces gauge-invariant definitions of angular momentum in QED and demonstrates their practical computation using Coulomb gauge, correcting longstanding misconceptions.

Findings

Gauge-invariant spin and orbital angular momentum can be defined for electrons and photons.

These quantities can be computed using canonical operators in Coulomb gauge.

The traditional formula for electromagnetic angular momentum density is incorrect.

Abstract

This two-paper series addresses and fixes the long-standing gauge invariance problem of angular momentum in gauge theories. This QED part reveals: 1) The spin and orbital angular momenta of electrons and photons can all be consistently defined gauge invariantly. 2) These gauge-invariant quantities can be conveniently computed via the canonical, gauge-dependent operators (e.g, $\psi ^\dagger \vec x \times\frac 1i \vec \nabla \psi$) in the Coulomb gauge, which is in fact what people (unconsciously) do in atomic physics. 3) The renowned formula $\vec x\times(\vec E\times\vec B)$ is a wrong density for the electromagnetic angular momentum. The angular distribution of angular-momentum flow in polarized atomic radiation is properly described not by this formula, but by the gauge invariant quantities defined here. The QCD paper [arXiv:0907.1284] will give a non-trivial generalization to non-Abelian gauge theories, and discuss the connection to nucleon spin structure.