How exactly is spinor with Grassmann variables as component defined?

I'm reading K. Muller's Introduction to Supersymmetry about spinor representation. He said that the components of a spinor are Grassmann variables.

I understood Grassmann variables as follows. For a vector space $$V=\rm{span}\{\theta_1,\cdots,\theta_n\}$$, the direct sum $$\Lambda(V)=\mathbb{C}\oplus V\oplus(V\wedge V)\oplus\cdots\oplus\wedge^nV$$ forms an algebra under wedge product $$\wedge$$.

On the other hand, Muller constructed Weyl spinors as elements of two representation spaces $$\psi\in F$$ and $$\bar{\psi}\in\dot{F}$$. Their components $$\psi_1,\psi_2,\bar{\psi}_{\dot{1}},\bar{\psi}_{\dot{2}}$$ are Grassmann variables. It seems that he meant these $$\psi_i\in V$$ instead of $$\psi_i\in\Lambda(V)$$.

My question is that what's the dimension of $$V$$ here $$n=?$$ At first I thought $$n=4$$ and $$V=\rm{span}\{\psi_1,\psi_2,\bar{\psi}_{\dot{1}},\bar{\psi}_{\dot{2}}\}$$, but $$(\sigma^0)_{A\dot{B}}\epsilon^{\dot{B}\dot{C}}\bar{\psi}_{\dot{C}}=\psi_A$$ seems to suggest that $$\psi_A$$ and $$\bar{\psi}_{\dot{A}}$$ are linear dependent. But if $$n=2$$, the expressions like $$(\theta\psi)(\theta\phi)$$ will simply be $$0$$?

• K. Muller?? Do you mean Harald J. W. Müller-Kirsten? Link to abstract page? On which page is $V$ defined? Jun 26, 2023 at 18:09

1. DeWitt [1] defines the set of supernumbers as the exterior algebra $$\begin{array}{rccccl} \bigwedge{}^{\bullet} V &~=~& \bigwedge{}^{\rm even} V &\oplus& \bigwedge{}^{\rm odd} V& \cr && ||| &&|||& \cr && \mathbb{C}^{1|0} && \mathbb{C}^{0|1} &\cr && ||| &&|||& \cr && \mathbb{C}_c && \mathbb{C}_a \cr && ||| &&||| &\cr &&\{c\text{-numbers}\} && \{a\text{-numbers}\}& \end{array}$$ of an infinite-dimensional vector space $$V$$. So $$n=\infty$$, cf. OP's question.

2. When considering an equation that is linear in $$\psi$$ (such as, e.g. the transformation rule for a spinor representation), it becomes agnostic to whether we treat $$\psi$$ as Grassmann-odd or Grassmann-even.

E.g. Ref. 2 flips on p. 49 the Grassmann-parity of the left- and right-handed Weyl spinor representations $$F$$ and $$\dot{F}$$ from Grassmann-even to Grassmann-odd.