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.

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.

For a Witten index,the path integral of supersymmetric quantum mechanics has periodic fermionic boundary conditions. They differ from the antiperiodic conditions for an ordinary thermal partition function. Under suitable trace and regulator assumptions, positive-energy bosonic and fermionic partners cancel, permitting a short-time evaluation from the surviving zero modes. A symmetry insertion instead imposes the twisted boundary conditions of an equivariant supertrace.

For the supersymmetric oscillator with bosonic energies $E_n^B=n$ and fermionic energies $E_n^F=n+1$ in units with unit oscillator gap,Pairing before taking $\beta \to 0$ preserves the single unpaired ground state. Taking separate short-time limits of the two traces gives divergent quantities and obscures that cancellation. A continuum spectrum or ill-defined trace requires additional analysis; supersymmetry alone does not justify exchanging all limits and integrals.