With left <functional derivative>[functional derivatives] for the Grassmann sources, replace fields in the interaction by $\phi\mapsto i^{-1}\delta/\delta J$, $\psi\mapsto i^{-1}\delta/\delta\bar\eta$, and $\bar\psi\mapsto-i^{-1}\delta/\delta\eta$. Thus
$$
Z[\bar\eta,\eta,J]=
\exp\left[-ig\int d^4x
\left(-\frac1i\frac\delta{\delta\eta(x)}\right)
\left(\frac1i\frac\delta{\delta J(x)}\right)
\left(\frac1i\frac\delta{\delta\bar\eta(x)}\right)
\right]Z_{0,f}[\bar\eta,\eta]Z_{0,b}[J].
$$
The ordering displayed fixes the Grassmann signs and makes this an explicit source functional with no dynamical fields.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-304/2/c/solution