I'm differentiating with respect to Grassman variables, and I'm getting something very inconsistent:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose [itex]\xi[/itex] and [itex]\chi[/itex] are two-component, left-handed, grassman-valued spinors. Now, I take derivatives with of the product, [itex]\xi^a \chi_a[/itex], respect to [itex]\xi[/itex] two different ways, and denote their results by [itex]\Pi[/tex] :

[tex]1. \hspace{5mm} \Pi_b=\frac{\partial}{\partial\xi^b}(\xi^a\chi_a)=\delta_b^{\phantom b a}\chi_a=\chi_b[/tex]

[tex]2. \hspace{5mm} \Pi^b=\frac{\partial}{\partial\xi_b}(\xi^a\chi_a)=\frac{\partial}{\partial\xi_b}(\epsilon^{ac}\xi_c\chi_a)=\epsilon^{ac}\delta_c^{\phantom c b}\chi_a=\epsilon^{ab}\chi_a=-\epsilon^{ba}\chi_a=-\chi^b[/tex]

I would have expected 1 and 2 to come out with the same sign (with the only difference being the position of the spinor index). Apparently, they are not coming out with the same sign. If I did the math correctly, how am I supposed to interpret this?

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# Spinor index notation craziness

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