Spinor index notation craziness

[TriTertButoxy]

I'm differentiating with respect to Grassman variables, and I'm getting something very inconsistent:

Suppose $\xi$ and $\chi$ are two-component, left-handed, grassman-valued spinors. Now, I take derivatives with of the product, $\xi^a \chi_a$, respect to $\xi$ two different ways, and denote their results by [itex]\Pi[/tex] :$$1. \hspace{5mm} \Pi_b=\frac{\partial}{\partial\xi^b}(\xi^a\chi_a)=\delta_b^{\phantom b a}\chi_a=\chi_b$$<br /> <br /> $$2. \hspace{5mm} \Pi^b=\frac{\partial}{\partial\xi_b}(\xi^a\chi_a)=\frac{\partial}{\partial\xi_b}(\epsilon^{ac}\xi_c\chi_a)=\epsilon^{ac}\delta_c^{\phantom c b}\chi_a=\epsilon^{ab}\chi_a=-\epsilon^{ba}\chi_a=-\chi^b$$<br /> <br /> 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?[/itex]

 

[Avodyne]

Well, your math is right. It must mean that, for consistency, the rule for raising and lowering a spinor index on a derivative must carry an opposite sign to the rule for rule for raising and lowering a spinor index on the field itself.