The lower bound on the minimal degree of the matrices in first-order relativistic wave equations

Abstract

A proof is given, on the basis of a theorem due to Garding (1944), for a lower bound on the degree (l+2) of the minimal equation of matrices beta ^mu in first-order unique-mass relativistic wave equations. It is not necessary, for the applicability of the theorem, that a hermitising operator should exist or that the equation be irreducible; and the generalisation of the bound to multimass equations is also straightforward. The bound is not, in general, linked to the physical spin or spins s allowed by the wave equation or the maximum spin j_m contained in the wavefunction. However, in the physically important case of irreducible equations which admit a hermitising operator, the bound becomes (l+2)>or= (2j_m+1), which is stronger than the bound (2s+1) suggested in the recent literature.