On the admissibility of repeated irreducible representations in barnacle-free unique-mass relativistic wave equations

Abstract

Considering relativistic wave equations in the first-order form, wherein the transformation of the wavefunction under the proper Lorentz group involves a certain number of inequivalent irreducible representations (IIRS) repeated an arbitrary number of times, the authors note some of the restrictions (on the skeleton matrix of the matrices beta ^mu occurring in the equation and on the spin blocks of beta ⁰) which arise from the requirements that the equation be barnacle free, and that there be solutions corresponding to a single spin and single mass only (without any degeneracy). If the number of IIRS is just two, these restrictions permit only two types of equations with no repeated IRS in either case. The authors also consider equations involving three IIRS with arbitrary multiplicity, carry out a reduction of the skeleton matrix, and analyse the implications of the mentioned requirements with regard to the possible existence of equations in which the multiplicity of one of the IRS is the sum of the multiplicities of the other two. Nothing is assumed about the specific IRS involved, except that they are linked.