SUPERMULTIPLETS AND RELATIVISTIC PROBLEMS .1. THE FREE PARTICLE WITH ARBITRARY SPIN IN A MAGNETIC-FIELD

Equations for relativistic particles for arbitrary spin have been of i nterest since Dirac original work for spin 1/2, but they involved eith er bothersome constraints or start with as many Dirac equations as are required to get the derived spin from its original 1/2 value. We firs t show that it is possible to have just one equation involving n alpha 's and beta's matrices that give possibilities up to 1/2n for the spin . We then decompose the alpha's and beta's into direct products of ord inary spin matrices and a new type of them that we call sign spin. The problem reduces then to one in terms of the generators of a U(4) grou p entirely similar to the one in the spin-isospin theory of nuclear ph ysics and hence the name of supermultiplets in the title. Using then t he techniques of the latter we discuss the problem of a free particle in a magnetic field for n = 1,2 and 3 or equivalently eigenvales for s pins 0,1/2, 1 and 3/2, and the energies are given as solutions of elem entary algebraic equations.