Grassmann variables and Weyl spinors

[302021895]

I just started studying supersymmetry, but I am a little bit confused with the superspace and superfield formalism. When expanding the vector superfield in components, one obtains therms of the form $\theta^{\alpha}\chi_{\alpha}$, where $\theta$ is a Grassmann number and $\chi$ is a Weyl vector.

I am aware that Grassmann numbers anticommute $\{\theta_{\alpha},\theta_{\beta}\}=0$, and that ordinary numbers commute with Grassmann variables. Do Weyl spinor components commute or anticommute with the Grassmann variables? ($\{\theta_{\alpha},\chi_{\beta}\}=0$ or $[\theta_{\alpha},\chi_{\beta}]=0$).

 

[haushofer]

Weyl spinor components are Grassman variables. As such they anticommute with other Grassman variables.

I'm not sure why you call the chi a "Weyl vector". The vector superfield V is not a vector, but it describes a vector multiplet and as such contains a field $v_{\mu}$ which is a spacetime vector.

 

[alialice]

Hi!
I have a similar problem: what is the result of a commutator/anticommutator like this?
$\left\{ \left( \gamma^{\mu} \theta \right)_{\alpha} \partial_{\mu} , \frac{\partial}{\partial \bar{\theta}^{\beta} } \right\}$
Thank you

 

[haushofer]

Use that the theta's (both the parameters and derivatives wrt them) anticommute and form an orthogonal basis for the fermionic part of superspace.