Inductive proof of the scalar Wick theorem

Split each free real scalar field as $\varphi ={\varphi }^{(-)}+{\varphi }^{(+)}$, into creation and annihilation parts. For chronologically ordered arguments, the first field has the latest time. Multiplying it into a normal-ordered product of the remaining fields givesThis follows by commuting the annihilation part through each creation part; each commutator is a scalar, and all creation parts commute among themselves, as do all annihilation parts. The scalar commutator is the Feynman propagator for the chronological pair. Induction on the number of fields therefore lists every partial pairing exactly once: either field 1 remains unpaired or it pairs with one $j$. This proves the Wick theorem for free bosonic fields, including the empty pairing. Vacuum expectation retains only complete pairings. The identities are understood for regulated or smeared operator-valued distributions.