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Homework Help: Fierz Rearrengement Identity

  1. Apr 29, 2015 #1
    1. The problem statement, all variables and given/known data

    I have to prove the Fierz rearrengement identity for Weyl Fermions. Eq 2.20 in Martin's supersymmetry primer:


    2. Relevant equations
    We have that the antisimetric tensor raises and lowers indices.

    3. The attempt at a solution

    Now I need to prove that [itex](\xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta)=-\delta^\beta_\alpha(\xi\eta)[/itex]. Can anyone help me with this? Thanks a lot!
  2. jcsd
  3. May 1, 2015 #2


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    Generally the way to prove these formulae is to manipulate indices so that you can write an expression as a sum of terms involving just epsilons and deltas multiplying the same expression of fermions. For example:

    $$\begin{split} \xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta +\delta^\beta_\alpha(\xi\eta) & = \epsilon_{\alpha\gamma} \xi^\gamma \eta^\beta + \epsilon_{\alpha\gamma} \eta^\gamma \xi^\beta + \delta^\beta_\alpha \epsilon_{\gamma\delta} \xi^\gamma\eta^\delta \\
    &= (\epsilon_{\alpha\gamma} \delta^\beta_\delta - \epsilon_{\alpha\delta} \delta^\beta_\gamma + \delta^\beta_\alpha \epsilon_{\gamma\delta} ) \xi^\gamma\eta^\delta.\end{split}$$

    We can then manipulate the coefficent to see that it vanishes due to the Jacobi identity

    $$ \epsilon_{\alpha\beta} \epsilon_{\gamma\delta} + \epsilon_{\alpha\gamma} \epsilon_{\delta\beta} + \epsilon_{\alpha\delta} \epsilon_{\beta\gamma} =0.$$

    We could have directly applied this method to the original Fierz formula.
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