The analogous <Grassmann Gaussian integral>, normalized by $Z_0[0,0]=1$, gives
$$
Z_0[\bar\eta,\eta]=\exp\left[-\int d^4x,d^4y\,
\bar\eta(x)S_F(x-y)\eta(y)\right],
$$
with the free <Dirac propagator>
$$
S_F(x-y)=\int\frac{d^4p}{(2\pi)^4}
\frac{i(\not p+m)e^{-ip\cdot(x-y)}}{p^2-m^2+i\epsilon}.
$$
Indeed $(i\not\partial_x-m)S_F(x-y)=i\delta^{(4)}(x-y)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 2 / b / Solution

A Grassmann Gaussian integral satisfies $\int D\bar\psi D\psi\,e^{-\bar\psi A\psi}=\det A$. Adding Grassmann sources and completing the square produces an exponent containing $A^{-1}$.


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 Incoming links: Grassmann Gaussian integral