The role of the Pauli-Lubański vector for the Dirac, Weyl, Proca, Maxwell and Fierz-Pauli equations

We analyze basic relativistic wave equations for the classical fields, such as Dirac's equation, Weyl's two-component equation for massless neutrinos and the Proca, Maxwell and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubański vector and the Casimir operators of the Poincarégroup. In general, in this group-theoretical approach, the above wave equations arise in certain overdetermined forms, which can be reduced to the conventional ones by a Gaussian elimination. A connection between the spin of a particle/field and consistency of the corresponding overdetermined system is emphasized in the massless case.