The Role of the Pauli-Lubanski Vector for the Dirac, Weyl, Proca, Maxwell, and Fierz-Pauli Equations

TL;DR

This paper investigates fundamental relativistic wave equations for various classical fields through the lens of the Pauli-Lubanski vector and Poincare group Casimir operators, revealing their overdetermined structures and spin-related consistency conditions.

Contribution

It provides a group-theoretical analysis of key relativistic wave equations, highlighting their overdetermined forms and the role of the Pauli-Lubanski vector in their structure.

Findings

Wave equations can be derived from overdetermined systems reducible to standard forms.

The Pauli-Lubanski vector links particle spin to the consistency of wave equations.

Massless cases show a special connection between spin and equation overdetermination.

Abstract

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-Lubanski vector and the Casimir operators of the Poincare group. 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.