Massive Rarita-Schwinger constraints ($\gamma \cdot \psi =\partial \cdot \psi =0$)

For ${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\psi }_{\rho }-m{\gamma }^{\mu \rho }{\psi }_{\rho }=0$ with $m\ne 0$, divergence gives ${\gamma }^{\mu \rho }{\partial }_{\mu }{\psi }_{\rho }=0$. Gamma contraction then forces $\gamma \cdot \psi =0$, hence $\partial \cdot \psi =0$. The remaining equation is $(/\partial +m){\psi }_{\mu }=0$ in a mostly-plus convention. In the rest frame the temporal component vanishes, leaving a spatial vector times an on-shell two-component spinor; its gamma trace removes the spin-one-half sector. The remaining spin-three-halves representation has four physical states, including longitudinal helicities $\pm 1/2$.