CCR scaling and the observable section

On $\mathcal{H}={\mathcal{F}}_{\mathrm{s}}(\mathfrak{h})$, let $a(v)$ be antilinear in $v$, with

Define

For $0<ϵ\le 1$, the scaled scalar field is

Equivalently, on the underlying real space of $\mathfrak{h}$, set

Its Weyl relation is

Thus $ϵ$ is a dimensionless effective Planck parameter, with a reference unit of action fixed. It is not $\kappa$, not an edge width, and not a cutoff.

Fix the prescribed $h_{\mathrm{cl}}$, and abbreviate

The quantum representative of the named observable is the section

The classical symbol ${\mathcal{O}}_f[h_{\mathrm{cl}};\cdot ]$ is fixed while its CCR representation varies. This is the states-or-representations interpretation of the Em06 acceptance pair.