How exactly is spinor with Grassmann variables as component defined?

Question

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 $0$?

Answer 1

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.

3. For more information, see also e.g. this & this related Phys.SE posts.

References:

1. Bryce DeWitt, Supermanifolds, Cambridge Univ. Press, 1992.

2. H.J.W. Müller-Kirsten & A. Wiedemann, Intro to SUSY, 2010.