TriTertButoxy
Mar8-09, 12:40 AM
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
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
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?
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
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
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?