# What anticommutes in the two-component formalism for spinors?

I'm studying the 2-component (alias Pauli) formalism for spinors, and I'm confused with the anticommutativity of the objects it uses and describes.
For "objects" i mean the spinors $$\chi_\alpha$$, the antispinors $$\bar{\chi}_\dot{\alpha}$$, the transformation matrices $$M^\alpha_\beta,{M^*}^\dot{\alpha}_\dot{\beta}\in SL(2,\mathbb{C})$$, the invariant tensors $$\epsilon^{\alpha\beta},\epsilon^{\dot\alpha\dot\beta}$$, and the generalized Pauli matrices $$\sigma^m_{\alpha\dot\alpha}$$.
I know that spinors anticommute $$\chi_\alpha \psi_\beta=-\psi_\beta \chi_\alpha$$ and that the indices in the invariant tensors do too $$\epsilon^{\alpha\beta}=-\epsilon^{\beta\alpha}$$ but what about any other combination? Some examples to show what I mean: $$\chi_\alpha \epsilon^{\alpha\beta}\stackrel{?}{=}\pm\epsilon^{\alpha\beta}\chi_\alpha$$ $$\epsilon^{\alpha\beta}\epsilon^{\rho\sigma}\stackrel{?}{=}\pm\epsilon^{\rho\sigma}\epsilon^{\alpha\beta}$$ $$\epsilon^{\alpha\beta}\sigma^m_{\alpha\dot\alpha}\stackrel{?}{=}\pm\sigma^m_{\alpha\dot\alpha}\epsilon^{\alpha\beta}$$

If this is of any help, I'm trying to understand this to derive the Fierz identities.

• Hi Mauro Giliberti. What is the Grassmann parity of your spinors? Odd or even? Feb 8, 2022 at 16:49
• @Qmechanic I guess odd, how could spinors be even? Wouldn't they just be vectors? I may know even less than what I think. Feb 8, 2022 at 16:53
• You should check the book "Spinors and Space-time" Volume-I by Penrose and Rindler
– KP99
Feb 8, 2022 at 17:24

The epsilon tensors $$\epsilon^{\alpha\beta}$$, $$\bar{\epsilon}^{\dot{\alpha}\dot{\beta}}$$ and the sigma symbols $$(\sigma^m)_{\alpha\dot{\beta}}$$, $$(\bar{\sigma}^m)^{\dot{\alpha}\beta}$$ are Grassmann-even. Hence, $$\chi_\gamma \epsilon^{\alpha\beta} = \epsilon^{\alpha\beta}\chi_\gamma \,,\quad \epsilon^{\alpha\beta}\epsilon^{\gamma\delta}=\epsilon^{\gamma\delta}\epsilon^{\alpha\beta} \,,\quad \epsilon^{\alpha\beta}(\sigma^m)_{\gamma\dot\gamma}=(\sigma^m)_{\gamma\dot\gamma}\epsilon^{\alpha\beta} \,,\quad \text{etc.}$$ In general, the "invariant tensors" intertwining between representations are Grassmann-even. The $$\mathrm{SL}(2,\mathbb{C})$$ matrices $$L_\alpha{}^\beta$$ and $$\bar{R}^{\dot{\alpha}}{}_{\dot{\beta}}$$ are also Grassmann-even because they value in $$\mathrm{SL}(2,\mathbb{C})$$.