Proposition: Physical L in the declared benchmark

The field

has mixed ordered products with every original scalar field, every adjoint, and every Wick polynomial on ${\mathcal{D}}_E\supseteq {\mathcal{D}}_{\mathrm{old}}$. It is local, and every real compact smearing has a self-adjoint affiliated realization in the original Weyl net.

Complex compact smearings have compatible closed affiliated realizations whose action and adjoint action agree with the corresponding Wick insertions on ${\mathcal{D}}_E$.

Proof. Mixed products, locality, and domain invariance follow from Proposition prop:domain. Real affiliated realizations follow from Lemma lem:localapprox and multiplication by $-1/2$.

For a complex smearing, write $f=f_1+\mathrm{i}{f}_2$ with real $f_1,f_2$. Let $A,B$ be the affiliated self-adjoint realizations of $L(f_1),L(f_2)$. On $\mathrm{Dom}A\cap \mathrm{Dom}B$, the sum $A+\mathrm{i}B$ is densely defined and closable: its adjoint contains $A-\mathrm{i}B$ on the same dense intersection. Every unitary in the commutant of the local algebra preserves both domains and commutes with both summands. It consequently preserves the closure of the sum. That closure is affiliated, agrees with $L(f)$ on ${\mathcal{D}}_E$, and its Hilbert adjoint agrees there with $L(\overline{f})$.