Lemma: Conditional regular polynomial realization

Suppose normalized free contraction data $W,E$ on the required graded test space have been supplied. If a linear and quadratic observable have smooth compact coefficient kernels $l,k$, then all finite Wick contractions needed for that regular polynomial are defined.

For

one bosonic Wick convention is

Its finite polynomial star products are defined by the corresponding finite contraction sums.

Proof. Every contraction pairs a distribution or a tensor product of distributions with a smooth compact joint test kernel. The tensor products exist, and the pairings are finite by distributional continuity. A polynomial has only finitely many nonzero functional derivatives, so the contraction sums terminate. The displayed subtraction removes the unique self-contraction of the quadratic term in that convention.