A common invariant domain

Set

The topology on each fixed-particle summand is given by the seminorms $\Vert (1+H{)}^r\psi \Vert$; the algebraic sum has the corresponding locally convex direct-sum topology.

A local Wick field is a finite linear combination of

Its smearings use $\mathcal{S}(M)$, and its local smearings use ${C_c}^{\infty }(M)$. These are vacuum normal products, with no self-contractions inside a Wick block.

Define ${\mathcal{D}}_{\mathrm{old}}$ to be the span of the continuous joint-Schwartz vacuum-word vectors of the original field $\varphi$. This is the old domain of the benchmark just declared. It is not identified with an unspecified domain from another model.