Supermultiplets and relativistic problems: II. The Bhabha equation of arbitrary spin and its properties

In 1945 Bhabha was probably the first to discuss the problem of a free relativistic particle with arbitrary spin in terms of a single linear equation in the four-momentum vector p(nu), but substituting the gamma(nu) matrices of Dirac by other ones. He determined the latter by requiring that their appropriate Lorentz transformations lead to their formulation in terms of the generators of the O(5) group. His program was later extensively amplified by Krajcik, Nieto and others. We returned to this problem because we had an ab-initio procedure for deriving a Lorentz-invariant equation of arbitrary spin and furthermore could express the matrices appearing in them in terms of ordinary and what we called sign spins. Our procedure was similar to that of the ordinary and isotopic spin in nuclear physics that give rise to supermultiplets, hence the appearance of this word in the title. In the ordinary and sign spin formulation it is easy to transform our equation into one linear in both the p(nu) and some of the generators of O(5). We can then obtain the matrix representation of our equation for an irrep (n(1)n(2)) of O(5) and End, through a similarity transformation, that for the irrep mentioned the particle satisfying our equation will have, in general, several spins and masses determined by a simple algorithm.