How can the Fierz Rearrangement Identity be proven for Weyl Fermions?

[fuchini]

Homework Statement

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

\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\chi\xi)

Homework Equations

We have that the antisimetric tensor raises and lowers indices.

The Attempt at a Solution

\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\chi\xi)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\xi\chi)
\chi_\alpha(\xi\eta)=-\xi_\alpha(\eta^\beta\chi_\beta)-\eta_\alpha(\xi^\beta\chi_\beta)
\chi_\alpha(\xi\eta)=-\chi_\beta(\xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta)

Now I need to prove that (\xi_\alpha\eta^\beta+\eta_\alpha\xi^\beta)=-\delta^\beta_\alpha(\xi\eta). Can anyone help me with this? Thanks a lot!

 

[fzero]

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 coefficient 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.