Spin operator in the Dirac theory

Understand

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.