Unexpected equality of the quark and gluon Belinfante angular momentum and Canonical angular momentum in the light-cone gauge $A^+=0$, in a spin 1/2 nucleon

Abstract

A controversy is raging concerning the most physically meaningful definition of quark and gluon angular momentum, a question of importance in understanding the internal structure of the nucleon. There are, in my, opinion, two principal candidates: the Belinfante version, $\bm{J}_{bel}$, which is gauge invariant, and the Canonical version, $ \bm{J}_{can}$, which is non-gauge invariant, but which has a physically intuitive structure. I demonstrate the wholly unexpected result that for quarks in a spin 1/2 nucleon the expectation value of $\bm{J}_{bel}(\textrm{quark})$ is equal to the expectation value of $ \bm{J}_{can}(\textrm{quark})$ taken in the light-cone gauge $A^+=0$, with an analogous result for gluons.