Scope, the release theorem, and the domain correction

The proposed calculation involves two logically separate steps: constructing a retarded polynomial functional and admitting that functional as an element of a quantum algebra. The first step does not require a state. The second cannot be inferred solely from the existence of a causal inverse.

XI-B's Definition of the free causal polynomial algebra uses compact interior smearings of linear fields and their finite products. Its construction expressly does not extend to all local or microcausal polynomial functionals. The local vertex and the nonlocal bilinear insertion in the present problem therefore require more than the algebraic operations already handed off.

We retain that limitation rather than silently replace ${\mathcal{A}}_0$ by a Wick algebra. The additional space used below, ${\mathcal{P}}_{\mathrm{mc}}$, is a space of classical polynomial symbols. No quantum product on that space is selected. The usual distinction between regular polynomial generators, local polynomials, and microcausal functionals is relevant here FR,HRV.