Rarita-Schwinger field (${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\psi }_{\rho }=0$)

A vector-spinor describes a spin-three-halves field after its lower-spin components are removed by constraints and, in the massless theory, gauge equivalence. The massless equation is invariant under ${\psi }_{\mu }\mapsto {\psi }_{\mu }+{\partial }_{\mu }\eta$. Gamma-trace gauge and the field equation give $\gamma \cdot \psi =\partial \cdot \psi =/\partial {\psi }_{\mu }=0$. Only helicities $\pm 3/2$ remain. A four-dimensional Majorana field has two physical polarizations; a complex field also has independent antiparticle modes. The gravitino is the supergravity realization of this field.

The gamma trace contracts the vector index of a Rarita-Schwinger field with a gamma matrix, producing an ordinary spinor. In the massive free theory it vanishes by the field equations. In the massless theory a gauge transformation changes it by ${\gamma }^{\mu }{\partial }_{\mu }ϵ$, allowing a gamma-traceless gauge for propagating modes. Together with transversality this removes the unwanted spin-one-half sector.

Gauge equivalence ${\psi }_{\mu }\sim {\psi }_{\mu }+{\partial }_{\mu }ϵ$ permits gamma-trace gauge. The Rarita-Schwinger field equation then gives transversality and a massless Dirac equation for each component. For momentum along the third axis, residual gauge removes the temporal and longitudinal components. The remaining constraint is ${\psi }_2={\gamma }^1{\gamma }^2{\psi }_1$, with ${\psi }_1$ in the two-dimensional kernel of $/p$. Spinor helicity $\pm 1/2$ is paired with vector helicity of the same sign, leaving only $\pm 3/2$. A Majorana spinor reality condition identifies antiparticles; a Dirac vector-spinor has separate antiparticle modes.

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 $\pm 1/2$.