Prove Spinor Identity in Arbitrary Dimension

[ismaili]

Actually, the original motivation is to check the closure of SUSY
$$\delta X^\mu = \bar{\epsilon}\psi^\mu$$
$$\delta \psi^\mu = -i\rho^\alpha\partial_\alpha X^\mu\epsilon$$
where $$\rho^\alpha$$ is a two dimensional gamma matrix, and $$\psi^\mu$$ ia s two dimensional Majorana spinor, the index $$\mu$$ in the two dimensional world is just some label of different fields.
I try to prove
$$[\delta_1,\delta_2]\psi^\mu = 2i\bar{\epsilon}_1\rho^\alpha\epsilon_2\ \partial_\alpha\psi^\mu$$
The following identity will help me a lot to prove the above formula,
$$\chi_A(\xi\eta) = - \xi_A(\eta\chi) - \eta_A(\chi\xi)\cdots(*)$$
where $$A$$ is the spinor index and $$\chi,\xi,\eta$$ are three spinors.
My question is, I don't know how to prove (*), and I don't know those spinors in (*) are Majorana spinors or not, moreover, I even don't know those spinors live in what dimension!
Does anyone know how to prove (*)? or anyone know the reference which treat the algebra of spinors in arbitrary dimension? Thanks a lot!

 

[azntoon]

A:I think I have figured out the proof of (*). It is well known that, for two spinors \chi,\eta, the following holds\xi_A(\eta\chi) = \xi^B(\eta_A\chi_B + \eta_B\chi_A)\cdots(1)So, we can calculate the left side of (*) as\xi_A(\eta\chi) = - \xi^B(\eta_A\chi_B + \eta_B\chi_A)+ \eta_A(\chi^B\xi_B + \chi_B\xi^B)=- \xi^B(\eta_A\chi_B + \eta_B\chi_A)- \eta_A(\chi_B\xi^B)= - \xi_A(\eta\chi) - \eta_A(\chi\xi)which is exactly the right side of (*).