Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 1 / b / Solution

Replace each occurrence of $x(t)$ in the interaction by the functional derivative that inserts it. In standard Minkowski source conventions,and henceThe factors of $i$ are adjusted together if one uses the source convention of part a directly.

This is a perturbation series, generally an asymptotic rather than convergent series because the number of Wick contractions grows factorially. For real cubic coupling the potential is also unbounded on one side, so the real-axis theory does not possess a stable nonperturbative ground state without a contour prescription or further stabilizing interactions.