Proposition: Joint products, old-domain inclusion, and locality

The following statements hold.

1. ${\mathcal{D}}_E$ is Hilbert-dense and invariant under every smeared Wick field and its formal adjoint.

2. Every finite ordered list of Wick fields defines continuous joint-Schwartz matrix-element distributions on ${\mathcal{D}}_E$. The corresponding joint vacuum-vector maps are continuous.

3. ${\mathcal{D}}_{\mathrm{old}}\subseteq {\mathcal{D}}_E$, and ${\mathcal{D}}_{\mathrm{old}}$ is Hilbert-dense.

4. All these Wick fields are closable on ${\mathcal{D}}_E$. Spacelike separated smearings commute on ${\mathcal{D}}_E$.

Proof. Energy cutoffs in every finite-particle sector prove Hilbert density. Lemma lem:wickest, for every $s$, proves invariance.

Iterating the same lemma gives continuous multilinear maps in the individual Schwartz smearings, with bounds by products of finitely many Schwartz seminorms. Nuclearity identifies the completed projective tensor product of the Schwartz spaces with the joint Schwartz space. Hence the maps extend uniquely to arbitrary joint Schwartz kernels. The same estimates give continuous vacuum-vector maps.

For old vacuum words, the Fourier variables creating particles have positive energies. A joint Schwartz kernel therefore gives rapid decay in their total energy, and hence finite moments of every order. This proves ${\mathcal{D}}_{\mathrm{old}}\subseteq {\mathcal{D}}_E$. Old field vacuum words generate the scalar Fock representation: restrictions of spacetime Schwartz Fourier transforms to the positive mass shell are dense in $\mathfrak{h}$, and their symmetric tensor products are dense in each particle sector. Thus ${\mathcal{D}}_{\mathrm{old}}$ is Hilbert-dense.

The adjoint pairing of a normal product is the normal product of the adjoint factors, with the reversed operator order for an ordered word. It is defined on ${\mathcal{D}}_E$. A densely defined formal adjoint therefore gives closability.

Finally, Wick's expansion of a commutator between spacelike separated blocks contains commutators of the elementary fields or their derivatives. Each vanishes by (eq:CCR). The support-preserving joint-test extension preserves this identity.