Proposition: Model 3 metric-valued form

Equation (eq:jform) is a well-defined sesquilinear form on ${\mathcal{D}}_3$, symmetric for real $J$. Its unsmeared smooth matrix elements satisfy

with the explicitly prescribed Newton–wave initial data. The form is continuous for an algebraic Schwartz Fock topology.

Proof. The first assertions follow from Lemma lem:stress3 and Proposition prop:response3. Here is an explicit meaning of the final assertion. For an integer $L$, let

Write $e_n(u_1,\dots ,u_n)=a^{†}(u_1)\cdots a^{†}(u_n)\Omega /\sqrt{n!}$, and define

These are finite seminorms dominating the Hilbert norm. The annihilation formula

shows that any fixed source seminorm is bounded by $C{p}_L(\Psi )p_L(\mathrm{X})$, with sufficiently large $L$. There are at most two field factors, and the factors $(n+2{)}^2$ absorb the finite annihilation choices. Momentum integrations by parts control the uncontracted wave packets. Continuity of the classical response on a compact set containing $\mathrm{supp}J$ therefore gives