 Solution

Use signature $(-,+,+,+)$, $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$, and unit-weight antisymmetrized gamma products. The massless Rarita-Schwinger field has the gauge freedomsince the antisymmetric ${\gamma }^{\mu \nu \rho }$ annihilates ${\partial }_{\nu }{\partial }_{\rho }ϵ$. This gauge freedom is essential to remove the lower-spin components of the vector-spinor.

Put $\chi ={\gamma }^{\mu }{\psi }_{\mu }$. Contracting the field equation with ${\gamma }_{\mu }$ and using ${\gamma }_{\mu }{\gamma }^{\mu \nu \rho }=2{\gamma }^{\nu \rho }$ givesExpanding the original three-gamma product then reduces its equation toA gauge transformation changes $\chi$ by $/\partial ϵ$. Locally solve $/\partial ϵ=-\chi$ to impose gamma-trace gauge. The two equations becomeApplying $/\partial$ again gives $\square {\psi }_{\mu }=0$. Nonzero propagating Fourier modes therefore have $p^2=0$: the physical modes are massless.

For the massless Rarita-Schwinger polarization count, choose a nonzero future null momentum $p^{\mu }=(\omega ,0,0,\omega )$. The remaining gauge parameters satisfy $/pϵ=0$. Transversality gives ${\psi }_0+{\psi }_3=0$, and this residual gauge removes ${\psi }_0$, hence also ${\psi }_3$, because ${\psi }_0$ itself satisfies $/p{\psi }_0=0$. Only ${\psi }_1,{\psi }_2$ remain, withThe kernel of $/p$ on four-component spinors has complex dimension two, so there are exactly two independent polarization amplitudes.

To identify their helicities, the spinor rotation generator about the third axis in this convention is $S_3=-i{\gamma }^1{\gamma }^2/2$. Its two eigenvalues on that kernel are $h_s=\pm 1/2$. If ${\psi }_1$ is such an eigenvector, then ${\psi }_2=2i{h}_s{\psi }_1$. Its transverse vector polarization is consequently $(1,+i)$ for $h_s=+1/2$ or $(1,-i)$ for $h_s=-1/2$. These carry vector helicity $+1$ or $-1$, respectively. Adding the vector and spinor angular momenta leavesThe unwanted opposite-sign vector-spinor combinations, which would have helicity $\pm 1/2$, violate the gamma-trace constraint. For the gravitino of simple supergravity, its Majorana spinor reality condition identifies the negative-frequency antiparticle modes, giving two propagating degrees of freedom. If one instead chooses a complex Dirac vector-spinor, its independent antiparticle sector doubles the particle-plus-antiparticle count; the equation alone does not impose a Majorana condition.