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Grassman numbers and change of variables

  1. May 14, 2012 #1
    Quick question about Grassman numbers and change of variables.

    Suppose you have the function:

    [tex]f=\frac{c\epsilon_{ij} }{2!} \psi_i \psi_j[/tex]

    and integrate it:

    [tex]\int d\psi_2 d\psi_1 \frac{c\epsilon_{ij} }{2!} \psi_i \psi_j =c [/tex]

    Now change variables: [tex]\psi_i=J_{ik}\psi'_k [/tex] to get:

    [tex]\int d\psi_2 d\psi_1 \frac{c\epsilon_{ij} }{2!} \psi_i \psi_j=
    \int J_{2r} J_{1s}d\psi'_r d\psi'_s \frac{c\epsilon_{ij} }{2!} J_{im}\psi'_m J_{jn} \psi'_n
    =
    \int det[J]d\psi'_2 d\psi'_1 \frac{c\epsilon_{mn} }{2!} det[J] \psi'_m \psi'_n
    =c [/tex]

    Doesn't this imply that det[J]2 has to equal one though? That can't be right.
     
  2. jcsd
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