# How is a half-odd integer spin field defined in the Wightman axioms?

Can anyone give me a proper definition of 'half-odd integer spin fields' in terms of the Wightman axioms? I'm trying to prove the anti-unitarity of the PCT operator in 'PCT, Spin and Statistics, and all that.' by Streater and Wightman, but for that I need to know what they mean by this and I cannot find a proper definition anywhere.

• A field has half-integer iff the matrix $S$ in eq.3-4 (pg. 99) is a spinorial representation of the Lorentz group. Example c) on that page concerns a Dirac field. – AccidentalFourierTransform Jun 9 '18 at 19:54
• and what is a spinorial representation of the Lorentz group? – user353840 Jun 9 '18 at 19:57
• In four dimensions, it is such that $S_x^2+S_y^2+S_z^2=j(j+1)$, with $j$ a half-integer. In an arbitrary number of spacetime dimensions, the definition is slightly trickier (essentially, those representations of $\mathrm{Spin}(d)$ that are not representations of $\mathrm{SO}(d)$; this is easy to tell given the Dynkin labels of the $\mathfrak{so}(d)_{\mathbb C}$ representation). – AccidentalFourierTransform Jun 9 '18 at 20:03
• Hmm, that doesn't really help me much. The thing I'm stuck with is that I have no information on this power 'F' in equation 3-66 on page 131. Is there any condition? For example, if F would be even, it would solve my problem.. – user353840 Jun 9 '18 at 20:08
• Essentially, a field $\varphi_{(\alpha)(\dot\beta)}$ is spinorial iff $|(\alpha)|+|(\dot\beta)|\neq 0\ \mathrm{mod}\ 2$. From this, you should be able to determine $F$. – AccidentalFourierTransform Jun 9 '18 at 20:15