One real symplectic plane and one oscillator

For this section assume the declared real TT plane is symplectic for

and choose its basis with ${\sigma }_{\mathrm{TT}}(e,r)=1$. An arbitrary independent plane need not be symplectic; the restricted-current theorem did not need this extra assumption. The example with data $(a,0)$ and $(0,a)$, normalized by $\int a_{ij}{a}^{ij}=1$, is symplectic.

A real Darboux plane has one canonical oscillator. Let

and use

The mixed core is

Let $N=N_g+N_{\varphi }$.

The oscillator-valued metric is

It commutes with the scalar algebra, but

This smooth commutator is not generally supported on causally related pairs. The model is not a local relativistic metric field and does not support an unrestricted local-field causal induction.

The product constructed here has only one metric factor. Its commutator with the scalar insertion inherits causal support from the scalar commutator, so (eq:nonlocalcomm) does not enter that calculation.