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

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-$3\mathrm{/}2$ $\in (3\mathrm{/}2,0)\oplus (0,3\mathrm{/}2)$. 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, ${\mathcal{P}}^{(3\mathrm{/}2,0)}$, 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 $(3\mathrm{/}2,0)\oplus (0,3\mathrm{/}2)$ 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 $(3\mathrm{/}2,0)\oplus (0,3\mathrm{/}2)$ 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, $p^\mu \left[ w^{(3/2,0)}\right]{\mu\nu}=0$,supplementedbytheadditionalcondition,$\gamma^\mu\gamma^\nu \left[ w^{(3/2,0)}\right]_{\mu \nu} =0$, a constraint that invokes the Majorana sector.