= 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 $a$. Then
$$
\int D\bar\theta\,D\theta\,e^{-\bar\theta_iM_{ij}\theta_j}=\det M.
$$
Expanding 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.
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