Relativistic Schrödinger Equations for Particles of Arbitrary Spin

Abstract

Relativistic wave equations in the Schrödinger form $\frac{i\partial \psi }{\partial t}=H\psi$ for particles of nonzero mass and arbitrary spin are investigated. The wave function $\psi$ is taken to transform according to the representation $D(0,s)\mathrm{}\oplus D(s,0\mathrm{})\mathrm{}$ of the homogeneous Lorentz group, a unique spin $s$ for the particle being thereby assured without the aid of any supplementary condition. It is shown that the requirement that the Schrödinger equation be invariant under the operations of the Poincaré group, as well as under space and time inversions and charge conjugation, restricts the possible choices of $H$ (as a function of the operators representing the above symmetry operations) to a well-defined class which shrinks to a unique possibility (coinciding with the Hamiltonian derived by Weaver, Hammer, and Good) when a further regularity condition of a physical nature is imposed: namely, that the Hamiltonian have a unique finite limit in the rest system of the particle. In this process, an ambiguity which exists initially in the definition of operators representing time reversal and charge conjugation gets eliminated. The Hamiltonian itself is obtained in explicit form for particles of any spin.