One way to see the expression vanishes (maybe the one you already know how to do?) is to use the fact that there can not exist a 2*2*2 totally antisymmetric matrix. First, write everything using indices:

$$\begin{align}
A &= \chi_\alpha (\xi \eta) + \xi_\alpha (\eta \chi) + \eta_\alpha (\chi \xi) \\
&= \chi_\alpha \xi_\beta\eta_\gamma\varepsilon^{\beta\gamma} + \xi_\alpha\eta_\beta\chi_\gamma\varepsilon^{\beta\gamma} + \eta_\alpha\chi_\beta\xi_\gamma\varepsilon^{\beta\gamma} \\
&= \varepsilon^{\beta\gamma}(\chi_\alpha\xi_\beta\eta_\gamma +\xi_\alpha\eta_\beta\chi_\gamma+\eta_\alpha\chi_\beta\xi_\gamma)
\end{align}$$
Then use the antisymmetry of the epsilon tensor and the anticommutivity of spinors to get
$$\begin{align}
A_\alpha &= \tfrac12\varepsilon^{\beta\gamma}(
\chi_\alpha\xi_\beta\eta_\gamma - \chi_\alpha\xi_\gamma\eta_\beta
+\xi_\alpha\eta_\beta\chi_\gamma-\xi_\alpha\eta_\gamma\chi_\beta
+\eta_\alpha\chi_\beta\xi_\gamma-\eta_\alpha\chi_\gamma\xi_\beta)
= \tfrac12\varepsilon^{\beta\gamma}(
\chi_\alpha\xi_\beta\eta_\gamma - \chi_\alpha\xi_\gamma\eta_\beta
+\chi_\gamma\xi_\alpha\eta_\beta-\chi_\beta\xi_\alpha\eta_\gamma
+\chi_\beta\xi_\gamma\eta_\alpha-\chi_\gamma\xi_\beta\eta_\alpha)
\end{align}$$
This can be written as
$$ A_\alpha = \varepsilon^{\beta\gamma}A_{\alpha\beta\gamma}$$
where
$A_{\alpha\beta\gamma}=A_{[\alpha\beta\gamma]}
=3\chi_{[\alpha}\xi_\beta\eta_{\gamma]}$
is totally antisymmetric and thus must be zero.

This can generalize to other spinors, but the expressions scale as the dimensions!

Maybe the more algorithmic way is to use Clifford algebra (sigma / gamma matrix) identities to get all of the terms into a standard form. For example, move the free index onto the $\chi$ spinor. You can do this with 2-component spinors using the identity
$$\psi_\alpha \chi_\beta = \tfrac12 \varepsilon_{\alpha\beta} (\psi\chi)
- \tfrac12 \sigma^{ab}_{\alpha\beta}(\psi\sigma_{ab}\chi) $$
A couple of simplifications then reduce the expression for $A_\alpha$ to zero.

Once again, similar things work with other spinors, but as they are higher dimensional spin representations, the expressions are in general larger.

I talked a bit more about Fierz identites and Majorana Fierz identities in some older questions. Just as in those posts I'm using the conventions in Kuzenko and Buchbinder and recommend you have a look at Generalized Fierz Identities and references within.

This post imported from StackExchange Physics at 2015-06-15 18:11 (UTC), posted by SE-user Simon