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

+ 3 like - 0 dislike
177 views

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.

+ 3 like - 0 dislike

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).

answered Apr 16, 2015 by (13,189 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.