Regulated generators and coherent correspondence

For a Schwartz coherent label $z$, let

Normal ordering gives the exact identity

This follows either by coherent contraction of (eq:B3) or by substituting the classical one-point function in the normal-ordered stress. The number estimate justifies the coherent expectations.

As $\Lambda \to \infty$, ${\chi }_{\Lambda }z\to z$ in Schwartz topology. Wave-packet estimates and classical response continuity therefore give convergence of the right side to the uncut classical pairing. This is an expectation limit, not an operator or instrument limit.

For $\Psi =\mathrm{Coh}(z)$, $\nu ={\Vert z\Vert }^2$, define

Then

and

To verify it, write $P={\Pi }_M$, $r=(I-P)\Psi$, and expand the difference as $⟨r,B_{\Lambda }\Psi ⟩+⟨P\Psi ,B_{\Lambda }r⟩$. Apply (eq:bound3). The vacuum expectation of every compression is exactly zero.