Spin operator in the Dirac theory

TL;DR

This paper identifies all spin operators for a Dirac particle satisfying specific physical and symmetry conditions, revealing four such operators and highlighting the unique one consistent with non-relativistic limits and known spin operators.

Contribution

The paper systematically derives all spin operators for Dirac particles under broad conditions, clarifying their algebraic properties and physical relevance, and establishing the equivalence of the unique operator with known spin definitions.

Findings

Four spin operators satisfy the conditions.

Only one operator has a proper non-relativistic limit.

This operator is equivalent to the Newton-Wigner and Foldy-Wouthuysen spin operators.

Abstract

We find all spin operators for a Dirac particle satisfying the following very general conditions: (i) spin does not convert positive (negative) energy states into negative (positive) energy states, (ii) spin is a pseudo-vector, and (iii) eigenvalues of the projection of a spin operator on an arbitrary direction are independent of this direction (isotropy condition). We show that there are four such operators and all of them fulfill the standard su(2) Lie algebra commutation relations. Nevertheless, only one of them has a proper non-relativistic limit and acts in the same way on negative and positive energy states. We show also that this operator is equivalent to the Newton-Wigner spin operator and Foldy-Wouthuysen mean-spin operator. We also discuss another operators proposed in the literature.