Exact embedding and the ordering interface

Map each XI-B linear generator to its linear functional in ${\mathcal{E}}_H$. The equations and commutators agree, so the map extends to a homomorphism

It is injective. Indeed, PBW gives the polynomial presentation of the old central graded relations. Independent old test classes are separated by smooth solutions of the reduced wave equations. On any finite-dimensional span this gives the ordinary symmetric/exterior polynomial algebra as the classical limit. No nonzero old polynomial can acquire a zero leading symbol; the same argument, coefficientwise in $\hslash$, proves injectivity.

Here is the precise finite-rank interface. Put

On regular finite-rank kernels, $S$ is graded symmetric and

intertwines the causal-Weyl and $H$ products. Thus an old Weyl-symbol polynomial $F$ has image ${\alpha }_{S}F$. Equivalently, the old $H$-normally ordered polynomial ${{\alpha }_S}^{-1}F$ has image $F$.

Admission below uses these $H$-normal symbol coordinates. It agrees exactly with old inclusion in that presentation and has the required unchanged classical symbol. We do not assert the stronger, generally divergent claim that ${\alpha }_S$ extends unchanged to every singular kernel. That stronger statement would confuse normal ordering with an extension of raw causal multiplication.