Gauge Invariant Noether's Theorem and The Proton Spin Crisis

TL;DR

This paper develops a gauge invariant Noether's theorem to clarify the definitions of spin and orbital angular momentum in gauge theories, revealing limitations in existing definitions and the non-necessity of the Belinfante-Rosenfeld tensor.

Contribution

It introduces a combined Lorentz and gauge transformation approach to derive gauge invariant conserved quantities, challenging previous notions about electromagnetic spin angular momentum.

Findings

Gauge invariant spin angular momentum of electromagnetic field does not exist in Dirac-Maxwell theory.

Existing gauge invariant definitions of electromagnetic spin angular momentum are incorrect due to non-vanishing surface terms.

The Belinfante-Rosenfeld tensor is not necessary for obtaining symmetric, gauge invariant energy-momentum tensors.

Abstract

Due to proton spin crisis it is necessary to understand the gauge invariant definition of the spin and orbital angular momentum of the quark and gluon from first principle. In this paper we derive the gauge invariant Noether's theorem by using combined Lorentz transformation plus local gauge transformation. We find that the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electromagnetic field does not exist in Dirac-Maxwell theory although the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electron exists. We find that the gauge invariant definition of the spin angular momentum of the electromagnetic field in the literature is not correct because of the non-vanishing surface term in Dirac-Maxwell theory although the corresponding surface term vanishes for linear momentum. We also show that the Belinfante-Rosenfeld tensor is not required to obtain symmetric and gauge invariant energy momentum tensor of the electron and the electromagnetic field in Dirac-Maxwell theory.