Supermultiplets and relativistic problems: III. The non-relativistic limit for a particle of arbitrary spin in an external field

In previous papers, with the same series title, an ab-initio procedure was developed for deriving a Lorentz invariant equation with arbitrary spins. This equation is linear in the four momentum p(nu), and its coefficients are matrices that can be expressed in terms of ordinary spin and what we called sign spin. In the present paper we consider this equation in an external field A(nu) which implies just replacing p(nu) by Pi(nu) = p(nu) - A(nu) and discuss the cases when A(0) = 1/2(r(2)/a(2)) (a = root mc(2)/(H) over bar Omega, Omega being the frequency of the oscillator), A = 0 and A(0) = 0, A = 1/2 (r x H) corresponding respectively to harmonic oscillator potential and a constant magnetic field. By using an appropriate complete set of states, with part of them characterized by the irreps of the chain of groups SU(4) superset of SUs(2) x SUt (2) where the subscripts s and t respectively stand for the ordinary and sign spin, the problem can be formulated in a matrix representation whose diagonalization gives the energy spectrum. For simplicity we shall only consider the symmetric representation {n} of SU(4) for which s = t, and our interest is focussed on the case when the external field is weak, which gives the non-relativistic limit, and where a perturbation analysis can be applied. We show that the expected non-relativistic result can be obtained only when the sign spin projection takes its maximum value, i.e, when all individual states contributing to the final one correspond to positive energies. In the case of constant magnetic field, we obtain the gyromagnetic ratio 1/s consistent with other derivations.