Massless Rarita-Schwinger field from a divergenceless anti-symmetric-tensor spinor of pure spin-$3/2$

Abstract

We construct the Rarita-Schwinger basis vectors, $U^\mu$, spanning the direct product space, $U^\mu:=A^\mu\otimes u_M$, of a massless four-vector, $ A^\mu $, with massless Majorana spinors, $u_M$, together with the associated field-strength tensor, ${\mathcal T}^{\mu\nu}:=p^\mu U^\nu -p^\nu U^\mu$. The ${\mathcal T}^{\mu\nu}$ space is reducible and contains one massless subspace of a pure spin-$3/2$ $\in (3/2,0)\oplus (0,3/2)$. We show how to single out the latter in a unique way by acting on ${\mathcal T}^{\mu\nu}$ with an earlier derived momentum independent projector, ${\mathcal P}^{(3/2,0)}$, properly constructed from one of the Casimir operators of the algebra $so(1,3)$ of the homogeneous Lorentz group. In this way it becomes possible to describe the irreducible massless $(3/2,0)\oplus (0,3/2)$ carrier space by means of the anti-symmetric-tensor of second rank with Majorana spinor components, defined as $\left[ w^{(3/2,0) }\right]^{\mu\nu}:=\left[{\mathcal P}^{(3/2,0)}\right]^{\mu \nu}\,\,_{\gamma\delta}{\mathcal T}^{\gamma \delta }$. The conclusion is that the $(3/2,0)\oplus (0,3/2)$ bi-vector spinor field can play the same role with respect to a $U^\mu$ gauge field as the bi-vector, $(1,0)\oplus (0,1)$, associated with the electromagnetic field-strength tensor, $F_{\mu\nu}$, plays for the Maxwell gauge field, $A_\mu$. Correspondingly, we find the free electromagnetic field equation, $p^\mu F_{\mu\nu}=0$, is paralleled by the free massless Rarita-Schwinger field equation, $p^\mu \left[ w^{(3/2,0)}\right]_{\mu\nu}=0$, supplemented by the additional condition, $\gamma^\mu\gamma^\nu \left[ w^{(3/2,0)}\right]_{\mu \nu} =0$, a constraint that invokes the Majorana sector.