Rarita-Schwinger Fields

Rarita-Schwinger term which may be satisfied by spin-3/2 fields may be $$\mathcal{L}=\bar{\psi}_\mu \left(\frac{i}{2}\gamma^{\mu\nu\lambda}\partial_\nu - \frac{m}{4}\gamma^{\mu\lambda}\right)\psi_\lambda$$ Signs and coefficients may depend on notations.
Be aware that the product of vector index and spinor index may split into two representations: spin-3/2 and spin-1/2. $$V\otimes S = R\oplus S$$ Specifically expressed in highest-weight representations, $1\otimes\frac{1}{2}=\frac{3}{2}\oplus \frac{1}{2}$, especially in 4-dimensions, $\left(\frac{1}{2},\frac{1}{2}\right)\otimes\left(\frac{1}{2},0\right) = \left(1,\frac{1}{2}\right)\oplus \left(0,\frac{1}{2}\right)$. Spinor degrees of freedom may be found if we consider trace component $\gamma^\mu \psi_\mu$; from the Rarita-Schwinger term, if you combine $\gamma^\mu \psi_\mu$ as a spinor, the spinor may satisfies Dirac equation, so the trace may have spin-1/2 representation. To express spin-3/2 representation, we may introduce trace-less condition to the vector-spinor-indiced field. $$\gamma^\mu \psi_\mu=0$$
In $d$-dimensions, vector representation has $(d-2)$ degrees on-shell and $(d-1)$ off-shell. Dirac spinor has complex $2^{[d/2]}$ degrees off-shell and real $2^{[d/2]}$ degrees on-shell. The constraint (traceless condition) annihilates degrees as many as the spinor degrees. Off-shell degrees are $(d-1)n$ where $n$ is the spinor degrees $2^{[d/2]+1}$ in real degree. On-shell degrees are $(d-2)n-n=(d-3)n=(d-3)2^{[d/2]}$.

• http://www.ift.unesp.br/users/nastase/sugra.pdf
• Weinberg