Ordered Grassmann Gaussian contractions (source code)

= Ordered Grassmann Gaussian contractions
{title2=$\langle\theta_i\bar\theta_j\theta_k\bar\theta_l\rangle=C_{ij}C_{kl}-C_{il}C_{kj},\quad C=B^{-1}$}

Use the <Berezin integral> orientation $\int\prod_i d\bar\theta_i d\theta_i\,\theta_1\bar\theta_1\cdots\theta_n\bar\theta_n=1$. For an invertible <matrix> $B$, the normalized weight $e^{-\bar\theta B\theta}$ has $\langle\theta_i\bar\theta_j\rangle=(B^{-1})_{ij}$. Differentiate its <determinant> with respect to $B_{ji}$ to obtain this formula; two derivatives give the displayed four-point formula. The second pairing carries a <fermionic sign>. These are ordered contractions, so exchanging two odd factors changes their sign.