Grassman numbers and change of variables

[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:

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

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

 

[azntoon]

What am I missing here?The mistake you are making is that when you change the variables, you must also change the Grassman numbers accordingly. In this case, the appropriate change is to replace εij with εrs det[J]εmn. This will ensure that the integral remains equal to c.