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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.

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

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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 $\mathrm{SL}(2,\mathbb{C})$.

Also note that there do exist Grassmann-even spinorsㅡspinor-helicity variables, twistors, Newman-Penrose dyads, etc. The book "Spinors and Space-time" by Penrose and Rindler (my favorite!) provides an excellent introduction to Grassmann-even spinors and their applications (as mentioned also by @KP99 in the comments). An easier version is Peter O'donnell's "Introduction to 2-spinors in general relativity." For the spinor-helicity stuff, I would recommend chapter 2 of Elvang & Huang's "Scattering Amplitudes in Gauge Theory and Gravity."

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