Three scalar one-particle vectors

Declare the plane Poisson bracket by $\{q,\pi \}=1$, the scalar bracket by Lemma lem:scalarCCR, and mixed brackets zero. This is a specified initial-data bracket, not the reduced bracket of a coupled Einstein–scalar theory. The coefficient quantization

satisfies

Indeed, the surviving terms contain only one scalar CCR; brackets between two correction terms carry ${\kappa }^2$.