A criterion for reducibility of a relativistic wave equation

Abstract

In general when one writes a relativistic wave equation of the form (−iΓ·∂+m)ψ(x)=0, that transforms covariantly under some representation Λ→T (Λ) of SL(2,C), it is nontrivial to determine whether or not the equation is irreducible or to avoid ending up with a reducible equation; especially if T (Λ) contains repeating irreducible representations. In this paper a simple (st) criterion is given by which one can determine whether or not an equation is irreducible. It is shwon that if Λμ have any invariant subspace at all, then that subspace must be a representation space of some combination of SL(2,C) representations in T (Λ). Knowing this it is shown that a wave equaiton is reducible if an only if there exists some idempotent projector P~ such that (1−P) Λ₀P=0 other than P=O or I. A method for constructing all possible addmissable P's is given. A simple example of the technique is given. A simple example of the technique is also given.