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

With left functional derivatives for the Grassmann sources, replace fields in the interaction by $\varphi \mapsto i^{-1}\delta /\delta J$, $\psi \mapsto i^{-1}\delta /\delta \bar{\eta }$, and $\bar{\psi }\mapsto -i^{-1}\delta /\delta \eta$. ThusThe ordering displayed fixes the Grassmann signs and makes this an explicit source functional with no dynamical fields.