Rarita-Schwinger particles in homogeneous magnetic fields, and inconsistencies of spin-32theories

Abstract

The Rarita-Schwinger equation for a spin-$\frac{3}{2}$ particle with minimal electromagnetic coupling is solved completely in the case when a constant homogeneous external magnetic field $\mathcal{H}$ is present. It is shown that the spectrum of energy eigenvalues includes complex values if $\mathcal{H}$ is such that $\ensuremath{\eta}\ensuremath{\equiv}(\frac{2e\mathcal{H}}{3{m}^{3}})>1$, and further that the norm of the Rarita-Schwinger wave function (i.e., the total "charge" integral defined from the Lagrangian) which is positive definite for $\ensuremath{\eta}<1$ becomes indefinite (even after taking account of the constraints) when $\ensuremath{\eta}$ exceeds unity. These results confirm that the difficulties in quantization first discovered by Johnson and Sudarshan are a reflection of the indefiniteness of the norm which appears already at the $c$-number level, and suggest that the nature of the energy spectrum (whether or not complex values are present) in the presence of very large magnetic fields would provide a quick means of predicting whether such difficulties would arise in quantization.