Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 4 / ii / Solution

Write each free field as creation and annihilation parts, ${\varphi }_i={\varphi }_i^{-}+{\varphi }_i^{+}$. Moving every annihilation part to the right produces commutators $[{\varphi }_i^{+},{\varphi }_j^{-}]$, which are precisely the contractions. Applying the $n=2$ identity once and then normal ordering the remaining field givesNo double contraction is possible for three fields. This is exactly Wick theorem at $n=3$.