Proof of the spin-statistics theorem using (QFT) S-matrix arguments?

Question

In my lecture notes (not available online) it says in a comment that the Spin-Statistics Theorem can for example be (partially) proven by the fact that using commutators for quantising particles of half integer spin destroys the Lorentz invariance of the S-matrix (in the context of QFT).

Does in the same way Lorentz invariance of the S-matrix get destroyed when using anticommutators for quantizing bosons?

As this was just a short remark, I would like to see that proof in more technical detail to better appreciate this line of argument.

Answer 1

This is answered in detail in Weinberg's QFT book, vol. I, Chapter 5. 

Briefly, to satisfy Lorentz invariance of the S-matrix, one needs (5.1.3), hence  (by the form of $H$) that the spacelike (anti)commutators of the basic fields vanish. To construct the fields from the physical Poincare irreps, one needs to take appropriate linear combination of the creation and annihilation fields to ensure that the spacelike (anti)commutators vanish (5.1.32), and this is found to require the commutator for integral spin and the anticommutator for half-integral spin (5.7.28).

For full details see the discussion around (5.1.29-33) and (5.7.28-29).