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Grassmann variables and Weyl spinors

  1. Feb 15, 2012 #1
    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 [itex]\theta^{\alpha}\chi_{\alpha}[/itex], where [itex]\theta[/itex] is a Grassmann number and [itex]\chi[/itex] is a Weyl vector.

    I am aware that Grassmann numbers anticommute [itex]\{\theta_{\alpha},\theta_{\beta}\}=0[/itex], and that ordinary numbers commute with Grassmann variables. Do Weyl spinor components commute or anticommute with the Grassmann variables? ([itex]\{\theta_{\alpha},\chi_{\beta}\}=0[/itex] or [itex][\theta_{\alpha},\chi_{\beta}]=0[/itex]).
  2. jcsd
  3. Feb 16, 2012 #2


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    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 [itex]v_{\mu}[/itex] which is a spacetime vector.
  4. Jul 9, 2012 #3
    I have a similar problem: what is the result of a commutator/anticommutator like this?
    [itex]\left\{ \left( \gamma^{\mu} \theta \right)_{\alpha} \partial_{\mu} , \frac{\partial}{\partial \bar{\theta}^{\beta} } \right\}[/itex]
    Thank you
  5. Jul 9, 2012 #4


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