On the determination of the relativistic wave equations associated with a given representation of SL (2,C)

Abstract

Straightforward algebraic techniques are presented and used to determine the structure of wave equations whose relativistic covariance is governed by two representations of SL (2,C), S₀(Λ)=(1,½)⊕(½,1)⊕(½,0)⊕(0,½) and S₁(Λ)=S₀(Λ)⊕(½,0)⊕(0,½), subject to the requirements that the equations should be parity preserving, admit an invariant Hermitian bilinear form realized by a numerical matrix η, and that they should describe a particle with a unique mass and spin. It is shown that S₀(Λ) leads to a unique algebraic structure, that of the Rarita-Schwinger equation, whereas S₁(Λ) leads either to a trivial extension of the former case or to a family of equations whose matrices have a minimal algebra with degree one higher than that of the former case. One such example reproduces the equation presented by Glass. When, contrary to custom, a singular η matrix is considered, it is shown that S₁(Λ) allows for equations whose coefficient matrices are reducible but indecomposable. These equations are completely equivalent to the Rarita-Schwinger equation in the free case, but the added components may enter the dynamics in the presence of certain interactions. The present examples serve to illustrate techniques which may be applied in the study of any relativistic wave equation.