 Fermionic path integral

A fermionic path integral integrates histories valued in Grassmann variables. Products and integration order carry signs, so the measure convention forms part of the definition. Its finite-dimensional building block is the Berezin integral.

For independent complex Grassmann pairs ${\theta }_i,{\bar{\theta }}_i$, choose the measure convention for which the one-pair integral of $e^{-\bar{\theta }a\theta }$ is $a$. ThenExpanding the exponential function terminates because each Grassmann variable squares to zero. Extracting the term containing every variable antisymmetrizes the products of matrix entries, producing the determinant.

This determinant formula concerns paired complex variables. For an even number of real Grassmann variables with an antisymmetric matrix, the corresponding Gaussian gives a Pfaffian. For comparison, the ordinary complex Gaussian integral with positive Hermitian matrix $M$ is ${\pi }^n/\det M$. Thus the precise determinant power depends on whether variables are real, complex or paired; it should be specified before passing to infinitely many field modes.