Rarita-Schwinger term which may be satisfied by spin-3/2 fields may be
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.
Specifically expressed in highest-weight representations,
, especially in 4-dimensions, .
Spinor degrees of freedom may be found if we consider trace component ; from the Rarita-Schwinger term, if you combine 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.
In -dimensions, vector representation has degrees on-shell and off-shell. Dirac spinor has complex degrees off-shell and real degrees on-shell. The constraint (traceless condition) annihilates degrees as many as the spinor degrees. Off-shell degrees are where is the spinor degrees in real degree. On-shell degrees are .
- http://www.ift.unesp.br/users/nastase/sugra.pdf
- Weinberg