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
