A spin observable for a Dirac particle

We discuss the form of the spin operator in relativistic quantum mechanics. We derive the form of the spin operator in the case when the states with negative energies are admitted. It appears that for a Dirac particle the spin operator reduces to the so called mean-spin operator introduced by Foldy and Wouthuysen. We show that the spin operator transforms under Lorentz group action according to an operator Wigner rotation, analogously as a Bloch vector describing polarization of a particle in momentum representation.

► We examine the problem of a relativistic spin operator in the case of a Dirac particle.
► We show that a proper spin operator coincides for positive energies with the operator used in quantum field theory.
► This operator can be extended for negative energies.
► We show that this operator is equivalent to the so called mean-spin operator introduced by Foldy and Wouthuysen.
► The spin operator transforms under Lorentz group action according to the operator Wigner rotation.