Berezin integral

Summary

The Berezin integral is the notion of integration over anticommuting variables, the odd coordinates of a supermanifold. Over one odd variable $\theta$ it sends $a + b\theta$ to $b$; over several it picks out the coefficient of the product of all odd variables. It is translation invariant and kills derivatives, like the usual integral, but it changes under linear substitutions by the determinant instead of its inverse. It is the basic tool of fermionic path integrals in quantum field theory, where Gaussian integrals give Pfaffians and determinants.