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^{3}operator with eigenvalues 1/2 and -1/2, so we say that the field has spin 1/2.

But what about other fields? Why do we say the scalar real and complex field have spin 0? I tried following the same approach and see if their solutions are eigenfunctions of that operator, but I don't know how to do it.

And side question (should I open another thread?): why is the spin related to their statistics? Everything I've read so far just shows that using commutation relations in the Dirac field quantization gives rise to a non positive-definite hamiltonian, whereas using anticommutation relations doesn't. Then they show that commutators imply Bose-Einstein statistics and anticommutators imply Dirac-Fermi statistics.

In light of this it seems that if the field has integer spin we must use commutators, and if it has half-integer spin we must use anticommutators, but nobody goes on to explain why this is.