Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 3 / b / i / Solution

Let $K_M=-{\partial }^2+M^2$ and let ${\Delta }_M=K_M^{-1}$ be its Euclidean quantum field theory propagator. With the source-sign convention suited to the expression in the question, definewhere $N_0$ makes $Z_0[0]=1$. Since inserting $\chi (x)$ is equivalent to acting with $-\delta /\delta J(x)$, the integral over $\chi$ gives

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