On the degree of the minimal equation of the matrices in first-order relativistic wave equations

Abstract

Starting with an analysis of the general structure of the matrix beta ⁰ entering in first-order relativistic wave equations, the authors show that the degree of the minimal equation of beta ⁰ is determined by the size and nature of the various spin blocks of the 'skeleton matrix' of the theory. Since it is the numbers of Lorentz irreducible representations contributing to particular spins which determine the sizes of the spin blocks (and the value j_m of the maximum spin contained in psi has no direct bearing on these), the reason for the failure of the Umezawa-Visconti rule and its extrapolation by Chandrasekaran et al. (1972) becomes clear. The authors obtain some general results concerning the minimal degree in certain types of theories and on certain procedures whereby the minimal degree can be raised without altering j_m, and finally analyse a few interesting examples.