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 gamm a(nu) matrices of Dirac by other ones. He determined the latter by req uiring that their appropriate Lorentz transformations lead to their fo rmulation in terms of the generators of the O(5) group. His program wa s later extensively amplified by Krajcik, Nieto and others. We returne d to this problem because we had an ab-initio procedure for deriving a Lorentz-invariant equation of arbitrary spin and furthermore could ex press the matrices appearing in them in terms of ordinary and what we called sign spins. Our procedure was similar to that of the ordinary a nd isotopic spin in nuclear physics that give rise to supermultiplets, hence the appearance of this word in the title. In the ordinary and s ign spin formulation it is easy to transform our equation into one lin ear 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 gen eral, several spins and masses determined by a simple algorithm.