Status of the lower spins in the Rarita-Schwinger four-vector spinor $ψ_μ$ within the method of the combined Lorentz- and Poincaré invariant projectors

Abstract

We investigate the status of the lower spin-1/2 companions to spin-3/2 within the four-vector spinor, $ψ_μ$. According to its reducibility, $ψ_μ\longrightarrow \left[(1/2,1)\oplus (1,1/2)\right]\oplus [(1/2,0)\oplus (0,1/2)]$ this representation space contains two spin-1/2 sectors, 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 of freedom in the Rarita-Schwinger field of interest we exploit the properties of the Casimir invariants of the Lorentz algebra to distinguish between the irreducible Dirac- and $(1/2,1)\oplus (1,1/2)$ representation spaces and construct corresponding momentum-independent (static) projectors which we then combine with a dynamical spin-1/2 Poincaré covariant projector. In so doing we obtain two spin-1/2 wave equation, and prove them to be causal within an electromagnetic field. We furthermore calculate Compton scattering off each one of the above states, and find that the amplitudes corresponding to the first spin-1/2 are identical to those of a Dirac particle and conclude on the observability of this state. Also for the second spin-1/2 we find finite cross sections in all directions in the ultra-relativistic limit, and conclude that its observability is not excluded neither by causality of propagation within an electromagnetic environment, nor by unitarity of the Compton scattering amplitudes in the ultraviolet. Finally, we notice that the method of the combined Lorentz- and Poincaré invariant projectors could be instrumental in opening a new avenue toward the consistent description of any spin by means of second order Lagrangians written in terms of sufficiently large reducible representation spaces equipped with separate Lorentz-- and Dirac indices.