quantum field theory - Under what conditions is a vector-spinor gamma trace free

I came upon the concept of irreducible vector-spinors while trying to simplify an expression involving the gravitino field.

It is claimed that an irredicible vector-spinor is gamma-traceless, i.e.

$$\Gamma^M \psi_{\alpha M} = 0.$$

Are there conditions on the irreducibility besides the above equation? When can I use the relation? Does it follow from more general group-theoretic relations or is it a rehash of the equations of motion?

Answer

Yes, it follows from a simple group theoretical consideration. A gravitino is basically a spinor $\times$ a gauge field. However $\psi^{\alpha}_{\mu}$ is not an irreducible representation so you need to take the gamma traceless part away.

To see it, just look at spinor $\times$ a gauge field decomposition which looks as $1\otimes \frac{1}{2}= \frac{3}{2}\oplus\frac{1}{2}$, the $1/2$ part is precisely $\gamma^\mu\psi_\mu$ since it transforms to itself. So you need to impose the condition $\gamma^\mu\psi_\mu=0$ to get the irreducible $3/2$ part as required.