Relativistic wave equations coupled to external fields: an algebraic study of the problem of constraints

Abstract

A general matrix algebraic study is made of higher spin wave equations with minimal electromagnetic interaction, in relation to one of the basic problems, namely the problem of possible change in the number of constraints implied in the equation on introducing the interaction. Considering equations of the general form (βπ-m)ψ=0, wherein the matrix β₀ is required to have a minimal equation β₀ⁿ=β₀^n-2 to ensure uniqueness of mass, we show that when n=4 extra constraints may be generated at critical external fields, while for n=5 there may also be loss of constraints on introduction of external fields. We obtain general algebraic criteria which determine whether or not such pathologies would arise in any particular case, and verify the validity of these criteria by considering a variety of known equations.