Gamma trace of a vector-spinor ($\chi ={\gamma }^{\mu }{\Psi }_{\mu }$)

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.