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 interest since Dirac original work for spin 1/2, but they involved either bothersome constraints or start with as many Dirac equations as are required to get the derived spin from its original 1/2 value. We first 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 ordinary 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) group entirely similar to the one in the spin-isospin theory of nuclear physics and hence the name of supermultiplets in the title. Using then the 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 spins 0,1/2, 1 and 3/2, and the energies are given as solutions of elementary algebraic equations.