The question is simple: why aren't there elementary particles to be found in Nature which have an intrinsic spin of $\frac{3}{2}$? Do the laws of physics (say, quantum field theory) prohibit their existence? Why shouldn't there be an intermediate value (between spin $1$ and $2$, both of which do exist)?
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3Does this answer your question? What are the mathematical problems in introducing Spin 3/2 fermions? – AccidentalFourierTransform Mar 10 '21 at 17:47