For a <scalar field> with classical <action> $S[\phi]$, the Euclidean source convention used throughout this question gives
$$
Z[J]=\int\mathcal D\phi\,
\exp\left[-\frac1\hbar\left(S[\phi]+\int d^dx\,J(x)\phi(x)\right)\right].
$$
The <generating functional> produces <correlation functions> through <functional derivatives>. For example,
$$
\boxed{\langle\phi(x)\phi(y)\rangle
=\left.\frac{\hbar^2}{Z[0]}
\frac{\delta^2Z[J]}{\delta J(x)\delta J(y)}\right|_{J=0}.}
$$
More generally, each derivative brings down $-\phi/\hbar$, so an $n$-point function carries $(-\hbar)^n/Z[0]$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-304/1/a/solution