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 (*).

 

[Entropia]

it is important to approach any problem or question with a critical and logical mindset. In this case, the content provided raises several questions and assumptions that need to be addressed in order to prove the Spinor Identity in Arbitrary Dimension.

Firstly, it is necessary to define the terms used in the content. SUSY refers to supersymmetry, a theoretical framework that proposes a symmetry between particles with different spin states. It is often used in particle physics to explain the relationship between fermions (particles with half-integer spin) and bosons (particles with integer spin). The equations provided in the content are part of the SUSY algebra, which describes the transformation of fields under supersymmetry.

The Spinor Identity in Arbitrary Dimension is a mathematical equation that relates the transformation of spinors (particles with half-integer spin) under supersymmetry in any dimension. It is important to note that the dimension being referred to is not specified in the content, therefore it is assumed to be a general case for any dimension.

In order to prove the Spinor Identity, it is necessary to have a thorough understanding of the mathematical framework being used, including the properties and rules of spinors and supersymmetry in arbitrary dimensions. This may require a deep understanding of abstract algebra and differential geometry.

The content also mentions the use of two-dimensional gamma matrices and Majorana spinors. These are specific mathematical objects that have been extensively studied and used in physics, particularly in string theory and supersymmetry. However, the content does not specify the properties or relationships of these objects in the context of arbitrary dimensions, which could make it difficult to prove the Spinor Identity.

Moreover, the content mentions a helpful identity, denoted by (*), which is necessary to prove the Spinor Identity. However, the origin and properties of this identity are not provided, making it difficult to understand and use in the proof.

In order to prove the Spinor Identity in Arbitrary Dimension, it is necessary to have a solid understanding of the mathematical framework being used and the properties of spinors and supersymmetry in arbitrary dimensions. It may also be helpful to refer to existing literature or consult with experts in the field. Without further information and context, it would be challenging to provide a definitive proof of the Spinor Identity.