# Grassman numbers and change of variables

1. May 14, 2012

### geoduck

Quick question about Grassman numbers and change of variables.

Suppose you have the function:

$$f=\frac{c\epsilon_{ij} }{2!} \psi_i \psi_j$$

and integrate it:

$$\int d\psi_2 d\psi_1 \frac{c\epsilon_{ij} }{2!} \psi_i \psi_j =c$$

Now change variables: $$\psi_i=J_{ik}\psi'_k$$ to get:

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

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