The scalar Wick algebra

Fix one scalar Hadamard bisolution ${\Delta }_{+}$ on $\hat{M}\times \hat{M}$, with

Its wavefront set has the Hadamard null-relation form: both slot covectors are nonzero, and the first lies in one fixed half of the null cone Radzikowski. No state on the conical product is constructed by this choice.

Include zero covectors in the closed Minkowski causal cones ${\overline{V}}_{\pm }$, and define

Use the polynomial microcausal free presentation: a polynomial functional has compact distributional derivative kernels satisfying

for every positive derivative order.

Denote the corresponding off-shell scalar algebra by ${\mathfrak{A}}_{\mathrm{off}}$. Its Wick product is

Each polynomial pair has a finite sum. The usual microcausal contraction theorem supplies this free product and its associativity FR. No interacting product theorem is imported from it.

Let ${\mathfrak{J}}_0$ be the pointwise ideal generated by

Because ${\Delta }_{+}$ is a bisolution,

The same observation applied inside a pointwise multiple shows that ${\mathfrak{J}}_0$ is a star ideal. Define

In particular,

The off-shell insertion domain and the output sector are

Zero is included in this homogeneous degree-one sector. Its topology is the quotient test topology on $\mathcal{D}/P_m\mathcal{D}$. Formal $\hslash$-series are treated coefficientwise.