Replace each occurrence of $x(t)$ in the interaction by the <functional derivative> that inserts it. In standard Minkowski source conventions,
$$
Z[J]
=\exp\left[
-\frac{i\lambda}{3!}\int dt
\left(\frac1i\frac{\delta}{\delta J(t)}\right)^3
\right]Z_0[J]
$$
and hence
$$
Z[J]
=\sum_{n=0}^\infty\frac1{n!}
\left[-\frac{i\lambda}{3!}\int dt
\left(\frac1i\frac{\delta}{\delta J(t)}\right)^3\right]^n
Z_0[J].
$$
The 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 theorem>[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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-304/1/b/solution