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Spinor index notation craziness

  1. Mar 8, 2009 #1
    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?
    Last edited: Mar 8, 2009
  2. jcsd
  3. Mar 10, 2009 #2


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    Science Advisor

    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.
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