- #1
ismaili
- 150
- 0
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!
\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!
