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.

18 pages. Updated to journal version

References in corpus (3)

• Elementary particles with continuous spin
• Bosonic and fermionic Weinberg-Joos (j,0)+ (0,j) states of arbitrary spins as Lorentz-tensors or tensor-spinors and second order theory
• Problems and Progress in Covariant High Spin Description

Cited by in corpus (1)

• Ghost and Tachyon Free Propagation up to spin-3 in Lorentz Invariant Field Theories