Theorem: Em06-A scalar-only classical-limit theorem

Fix Specification spec:data, $h=h_{\mathrm{cl}}$, and the preparation $z_{\mathrm{cl}}$ of (eq:classicalfield). For the observable (eq:observable), the quantum section (eq:quantumobservable), and the states (eq:states), the following statements hold.

1. Mean. For every $0<ϵ\le 1$,

   $$\boxed{{\omega }_{ϵ}({\hat{\mathcal{O}}}_{f,ϵ})={\mathcal{O}}_f[g_{\mathrm{cl}};{\varphi }_{\mathrm{cl}}].}$$

   In particular, the mean converges in the ordinary real topology to the named classical value.

2. Variance. For every $0<ϵ\le 1$,

   $$\boxed{{\mathrm{Var}}_{{\omega }_{ϵ}}({\hat{\mathcal{O}}}_{f,ϵ})=ϵ\Vert k_{q_{f,h_{\mathrm{cl}}}}{\Vert }^2⟶0.}$$

3. Spectral topology. Let ${\mu }_{f,ϵ}$ be the spectral probability law of ${\hat{\mathcal{O}}}_{f,ϵ}$ in ${\Psi }_{ϵ}$, and put $o_f={\mathcal{O}}_f[g_{\mathrm{cl}};{\varphi }_{\mathrm{cl}}]$. Then

   $${\mu }_{f,ϵ}=\mathcal{N}(o_f,ϵ\Vert v_f{\Vert }^2),$$

   with the point-mass interpretation if $v_f=0$, and

   $$W_2({\mu }_{f,ϵ},{\delta }_{o_f})=\sqrt{ϵ}\Vert v_f\Vert .$$

   Thus the spectral laws converge in the quadratic Wasserstein metric, not merely by a cutoff estimate.

4. Restricted gauge independence. Replacing $h_{\mathrm{cl}}$ by $h_{\mathrm{cl}}+K\xi$, $\xi \in {\mathfrak{g}}_U$, leaves the represented observable, its classical value, and its spectral law unchanged.

5. Fixed-support uniformity. For any bounded subset $\mathcal{B}\subset {C_K}^{\infty }((0,T)\times {\mathbb{R}}^3;\mathbb{R})$,

   $$\underset{f\in \mathcal{B}}{\sup }{\mathrm{Var}}_{{\omega }_{ϵ}}({\hat{\mathcal{O}}}_{f,ϵ})\le ϵC_{\mathcal{B}},\underset{f\in \mathcal{B}}{\sup }{W}_2({\mu }_{f,ϵ},{\delta }_{o_f})\le \sqrt{ϵC_{\mathcal{B}}},$$

   where $C_{\mathcal{B}}<\infty$ is independent of $ϵ$.

The metric jet and causal structure are prescribed inputs. The theorem asserts neither metric reconstruction nor a derivation of Einstein dynamics.

Proof. By (eq:displacement),

The vacuum mean of $\Phi (v_f)$ is zero. Equations (eq:classicalpairing) and (eq:representation) identify the constant in (eq:shiftedoperator) with $o_f$. This proves the mean statement separately.

For the variance, center (eq:shiftedoperator). Since $\Phi (v_f)\Omega =a^{†}(v_f)\Omega$,

Lemma lem:k proves that the norm is finite. This proves the vanishing-variance statement without inferring it from the mean.

The characteristic function of the spectral law, obtained from (eq:characteristic) with $v=t{v}_f$, is

This identifies the stated Gaussian law. Every coupling to the point mass ${\delta }_{o_f}$ has second coordinate $o_f$, so

For gauge independence, Proposition prop:testgauge and Lemma lem:k give

Hence the quantum representatives agree as operators. Classical agreement is Proposition prop:plateau.

Finally, the continuous linear maps of Proposition prop:test and Lemma lem:k send a bounded fixed-support test set to a bounded subset of $\mathfrak{h}$. Taking

proves both uniform estimates.