Vanishing odd correlators of a free Gaussian field ($⟨{\hat{f}}_1\cdots {\hat{f}}_{2n+1}⟩=0$)

A centered free field in a Gaussian vacuum has only pair contractions, so every odd correlator vanishes by Wick theorem. Equivalently, number parity reverses the field while preserving the vacuum. This statement applies to smeared or regulated fields; it does not assert that unregulated coincident products exist. Interactions or a nonzero field mean can invalidate it.