Definition: Admissible polynomial kernels

The space ${\mathcal{P}}_{\mathrm{mc}}$ consists of finite polynomials in the field multiplet whose coefficient kernels are compactly supported distributions $u_n$ satisfying

The kernels have the graded symmetry appropriate to their field slots. In a purely even sector,

The quantum polynomial algebra is

with the graded Wick product determined by $\mathcal{H}$.