Theorem: Ordered Weyl expectations and their classical limit

For arbitrary $A$,

Its distance from $e^{\mathrm{i}t\cdot o}$ is at most

For the displaced vacuum, the exact formula is

Under $ϵn_{ϵ}\to 0$, the ordered expectations converge to the classical product, uniformly for $t$ in compact subsets and for bounded input families with fixed $J$.

Proof. Repeated use of (eq:Weyl) gives the phase $-\mathrm{i}ϵ{\sum }_{i<j}{t}_i{t}_j{A}_{ij}$ and the final Weyl direction $V(t)$. Displacement gives $e^{\mathrm{i}t\cdot o}$, proving (eq:ordered-general).

Spectral calculus and Cauchy–Schwarz imply

Together with $|e^{-\mathrm{i}ϵa}-1|\le ϵ|a|$ and (eq:state-number), this proves the bound. Vacuum normal ordering proves (eq:ordered-vacuum).

On compact $t$-sets and bounded $\Vert v_j\Vert$, both $\Vert V(t)\Vert$ and $|a(t)|$ have common finite bounds. This proves uniformity.