= Vanishing odd correlators of a free Gaussian field
{title2=$\langle\widehat f_1\cdots\widehat f_{2n+1}\rangle=0$}
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.
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