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.

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.