Fermionic path integral

A fermionic path integral integrates Grassmann-valued histories. Its Gaussian integral is a determinant, in contrast with the inverse determinant produced by a bosonic Gaussian integral.

The Berezin integral is differentiation with respect to a Grassmann variable: $\int d\theta 1=0$ and $\int d\theta \theta =1$. A finite-dimensional fermionic Gaussian integral gives a determinant or Pfaffian.

For independent Grassmann variables and an invertible ordinary matrix $S$, the linear change $\eta =S\xi$ transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by $\det S$, and coefficient extraction must undo that factor. For paired variables $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$ the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.

The short-time limit rescales a Euclidean time circle toward zero size. Supersymmetry cancels paired nonzero-mode determinants, leaving an integral over constant or symmetry-fixed zero modes.