Spinor index notation craziness

[TriTertButoxy]

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

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?

 

[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.

 

[Entropia]

I would first verify the calculations to ensure they are correct. If they are indeed correct, then this discrepancy could be due to the properties of Grassman variables in spinor index notation.

It is important to note that Grassman variables are anticommutative, meaning that their multiplication follows the rule of \xi^a \xi^b = -\xi^b \xi^a. This property may explain the different signs in the two derivatives, as the position of the spinor index changes the order of multiplication.

In addition, spinor index notation is a powerful tool for handling spinors and their derivatives, but it is also known to be a complex and non-intuitive notation. It is possible that this discrepancy is a result of the complexity of the notation and not necessarily a contradiction.

Furthermore, it is important to consider the physical implications of this discrepancy. Does it affect the overall results or predictions in the scientific study? If not, then it may not be a significant issue. However, if it does affect the results, it may be necessary to re-examine the calculations and find a more consistent approach.

In conclusion, while this discrepancy may seem confusing and inconsistent, it is important to thoroughly analyze the calculations and consider the properties of Grassman variables and spinor index notation. If the discrepancy persists and has significant implications, it may be necessary to seek alternative approaches or consult with other experts in the field.