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As for the question itself, I'd like to point out that I often think about high spin fields when we discuss any type of theory and so this problem seemed really interesting to me. My friend is playing with Weinberg and found a problem asking him to do this and I thought it would be fun, but we both found it impossibly difficult to start, and I was wondering if anyone could explain how.

My thoughts were to use PS (the most familiar QFT text to me) and just redo the steps in section 3.1-2 but for spin-3/2 instead of 1/2, but this didn't work and I'll outline why: we know the the dimension of our Lorentz group must be dimension ##n = 2s+1## where ##s## is the spin. For spin-1/2 this gives ##n=2## and for spin-3/2, ##n=4##. They nicely show that the Pauli matrices satisfy the 2D representation needed for spin-1/2, but then go to Minkowski spacetime and inexplicably jump to 4D representations of the Dirac matrices. So then when it comes to spin-3/2, I don't see how to generalize this: it needs to start with a 4D representation, but we've already done that for spin-1/2, so would it not just be the same?

My last problem was trying to just use their definition $$ \Lambda_{1/2} = \exp(-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}) $$ in eq 3.30 but with the spin-3/2 representation of SU(2) (found here, or alternatively somewhere in Georgi's book) but this led me to nothing concrete. My goal, ultimately, is to end up at the Rarita-Schwinger equation using similar principles to those used to find the Dirac equation, and I would appreciate it if anyone had any insight.