I read Peter Woit's reply to an objection, and it really didn't help either. It feels circular, basically, "angular momentum mathematically looks like this if y

I read Peter Woit's reply to an objection, and it really didn't help either. It feels circular, basically, "angular momentum mathematically looks like this if you start from the infinitesimal generators"...okay, so that means the spinor components of the dirac field transform like they're spinning...but why man, why must that be the case? God didn't give you the dirac states and the gamma matrices and say "all fermions behave as so! " Arguably, you can say the opposite: people use spinor valued fields because they describe spin 1/2 particles, which electrons and positrons are observed to be experimentally. I once read a comment on how the lorentz invariance "falls out" of the Lagrangian and thus it's not mysterious, this is such a particle theorist perspective (if it's not clear I'm being derisive) who just copied the notes in class but doesn't understand theory or the history of the situation. The Lagrangians you use and then quantize in QFT are chosen to be Lorentz invariant because experiment says physics is Lorentz invariant, not the other way around. Similarly, Zeeman effect and other such things informed Pauli's theory of spin 1/2 and all that followed. And thus, since such facts (constant speed of light, intrinsic spin) are experimental facts, it is natural and fine to wonder if there is a more fundamental reason underneath the theory. The thing is despite his reputation, I expect better from someone as smart as Peter Woit. This is kind of disappointing.