= Real-complex compatibility of Grassmann Gaussian integrals
{title2=$\operatorname{Pf}(\operatorname{diag}(M,M))=\det M$}
For an even-dimensional <antisymmetric matrix> $M$, put $\theta=(u+iv)/\sqrt2$ and $\bar\theta=(u-iv)/\sqrt2$, with independent <Grassmann variables> $u,v$. Anticommutation cancels the mixed terms, giving $\bar\theta^TM\theta=(u^TMu+v^TMv)/2$. The constant <Grassmann change-of-variables formula> changes only the normalization. The two real <Pfaffians> multiply to $\operatorname{Pf}(M)^2=\det M$, agreeing with the complex <Grassmann Gaussian integral>. In odd dimension $\det M=0$ and both top-degree integrals vanish.
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