Position, spin and orbital angular momentum of a relativistic electron

TL;DR

This paper examines the spin and orbital angular momentum of relativistic electrons, comparing two main theoretical approaches and arguing for the more physically intuitive one based on positive-energy projection.

Contribution

It provides a detailed comparison of two approaches to defining angular momentum in relativistic electrons and advocates for the approach that naturally incorporates spin-orbit interactions.

Findings

The positive-energy projection approach effectively describes spin-orbit effects.

The Newton-Wigner-Foldy-Wouthuysen approach has limitations in physical interpretation.

The preferred approach offers a better zero-mass limit and physical insight.

Abstract

Motivated by recent interest in relativistic electron vortex states, we revisit the spin and orbital angular momentum properties of Dirac electrons. These are uniquely determined by the choice of the position operator for a relativistic electron. We overview two main approaches discussed in the literature: (i) the projection of operators onto the positive-energy subspace, which removes the zitterbewegung effects and correctly describes spin-orbit interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen operators based on the inverse Foldy-Wouthuysen transformation. We argue that the first approach [previously described in application to Dirac vortex beams in K.Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011)] has a more natural physical interpretation, including spin-orbit interactions and a nonsingular zero-mass limit, than the second one [S.M. Barnett, Phys. Rev. Lett. 118, 114802 (2017)].