Status of the lower spins in the Rarita-Schwinger four-vector spinor $\psi_\mu$ within the method of the combined Lorentz- and Poincar\'e invariant projectors

摘要

We investigate the status of the lower spin-1/2 companions to spin-3/2 withinthe four-vector spinor, $\psi_\mu$. According to its reducibility,$\psi_\mu\longrightarrow \left[(1/2,1)\oplus (1,1/2)\right]\oplus[(1/2,0)\oplus (0,1/2)]$ this representation space contains two spin-1/2sectors, the first one transforming as a genuine Dirac-spinor, $(1/2,0)\oplus(0,1/2)$, and the second as the companion to spin-3/2 in $(1/2,1)\oplus(1,1/2)$. In order to correctly identify the covariant spin-1/2 degrees offreedom in the Rarita-Schwinger field of interest we exploit the properties ofthe Casimir invariants of the Lorentz algebra to distinguish between theirreducible Dirac- and $(1/2,1)\oplus (1,1/2)$ representation spaces andconstruct corresponding momentum-independent (static) projectors which we thencombine with a dynamical spin-1/2 Poincar\'e covariant projector. In so doingwe obtain two spin-1/2 wave equation, and prove them to be causal within anelectromagnetic field. We furthermore calculate Compton scattering off each oneof the above states, and find that the amplitudes corresponding to the firstspin-1/2 are identical to those of a Dirac particle and conclude on theobservability of this state. Also for the second spin-1/2 we find finite crosssections in all directions in the ultra-relativistic limit, and conclude thatits observability is not excluded neither by causality of propagation within anelectromagnetic environment, nor by unitarity of the Compton scatteringamplitudes in the ultraviolet. Finally, we notice that the method of thecombined Lorentz- and Poincar\'e invariant projectors could be instrumental inopening a new avenue toward the consistent description of any spin by means ofsecond order Lagrangians written in terms of sufficiently large reduciblerepresentation spaces equipped with separate Lorentz-- and Dirac indices.