The role of Pryce's spin and coordinate operators in the theory of massive Dirac fermions

TL;DR

This paper explores Pryce's spin and coordinate operators in massive Dirac fermions, revealing their conserved nature, relation to isometry generators, and clarifying the separation of spin and orbital angular momentum.

Contribution

It demonstrates that Pryce's operators form SU(2) generators, are conserved, and relate to isometry generators, providing a natural solution to separating spin and orbital angular momentum.

Findings

Pryce's spin operators are SU(2) generators for Dirac solutions.

New conserved polarization operators are defined for various polarizations.

The Pryce mass-center operator becomes a dipole one-particle operator after quantization.

Abstract

It is shown that the components of Pryce's spin operator of Dirac's theory are $SU(2)$ generators of a representation carried by the space of Pauli's spinors determining the polarization of the plane wave solutions of Dirac's equation. These operators are conserved via Noether theorem such that new conserved polarization operators can be defined for various polarizations. The corresponding one-particle operators of quantum theory are derived showing how these are related to the isometry generators of the massive Dirac fermions of any polarization, including momentum-dependent ones. In this manner, the problem of separating conserved spin and orbital angular momentum operators is solved naturally. Moreover, the operator proposed by Pryce as mass-center coordinate is studied showing that after quantization this becomes in fact the dipole one-particle operator. As an example, the quantities determining the principal one-particle operators are derived for the first time in momentum-helicity basis.