Nonrelativistic Quantum Mechanics for Particles with Arbitrary Spin

Motivated by the difficulties which persist in the construction of a consistent description of interacting relativistic particles with spin greater than 1, we examine the analogous problem within the framework of nonrelativistic quantum mechanics and find that all of the difficulties vanish in the (Galilei-invariant) limit. It is found that a unique, first-order, Galilei-covariant wave equation describing massive, spin-$s$ particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has $6s+1$ components ($2s+1$ of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin-$s$ particles so described are found to possess only an electric charge and a magnetic dipole with a $g$ factor of $\frac{1}{s}$. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-$s$ fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schrödinger theory is presented, as well as examples of theories which violate the minimality condition.