Algebraic study of a class of relativistic wave equations

Abstract

First-order relativistic wave equations are considered whose irreducible matrix coefficients satisfy the simplest (except for the Dirac algebra) unique mass condition, (β·p)³=p²(β·p), which is also sufficient to guarantee causality in a minimally coupled external electromagnetic field. All of the associated representations of SL(2, ©) are classified and studied up to and including those which are the direct sum of four irreducible components, (n, m), with either n or m less than two. A large number of families of representations are found which permit the algebraic condition to be satisfied. These are tabulated according to whether a Hermitian choice for β⁰ is possible and their spin content is given. If a unique spin is described, then the only possible representations are (1) (n,0)⊕(n−½, ½) (2) (n,0)⊕(n+½, ½) (3) (n+½, ½)⊕(n,0)⊕(n−½, ½) (4) (1,0)⊕(½, ½)⊕(0,1) and their conjugates. If, in addition, the representation is assumed to be self-conjugate, then only the Dirac and Petiau-Duffin-Kemmer equations survive.