Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Ordered Weyl identity and bound

Static concentration and normalized fluctuations › Ordered expectations for noncommuting families · lines 737–801 · proof included or linked

Psy02-A@1#thm:ordered, Psy02-A@1#eq:ordered, Psy02-A@1#eq:ordered-bound, Psy02-A@1#eq:ordered-vacuum

Back to manuscript

For arbitrary real $t = (t_{1}, \dots, t_{J})$ ,

Tr (ρ_{ϵ} j = 1 \prod^{J} e^{i t_{j} O_{j, ϵ}}) = e^{i t \cdot o - i ϵ a (t)} Tr (ρ_{ϵ} e^{i ϵ Φ (V (t))}), \abs Tr (ρ_{ϵ} j \prod e^{i t_{j} O_{j, ϵ}}) - e^{i t \cdot o} \leq ϵ \abs a (t) + ϵ C_{ϵ} \norm V (t) .

For residual vacuum,

Tr (ρ_{ϵ} j \prod e^{i t_{j} O_{j, ϵ}}) = exp (i t \cdot o - \frac{ϵ}{2} \norm V (t)^{2} - i ϵ a (t)) .

Under the bounded-family assumptions of Corollary cor:uniform, convergence to $e^{i t \cdot o}$ is uniform on compact $t$ -sets.

Proof. Repeated application of (eq:Weyl) produces the phase $- i ϵ a (t)$ and the final direction $ϵ V (t)$ . Conjugation by $U_{ϵ}$ gives the target phase.

For self-adjoint $B$ , spectral calculus gives

\abs Tr (ρ (e^{i B} - 1)) \leq Tr (ρ B^{2}) .

Apply this with $B = ϵ Φ (V (t))$ , use Lemma lem:number, and combine with $\abs e^{- i ϵ a} - 1 \leq ϵ \abs a$ . Vacuum normal ordering gives $⟨ 0 ∣ W (v) ∣ 0 ⟩ = e^{- \norm v^{2} /2}$ . On compact $t$ -sets, the vector norms and symplectic phase have common bounds.

Displayed formulas in this passage

Psy02-A@1#eq:ordered · line 740
$Tr (ρ_{ϵ} j = 1 \prod^{J} e^{i t_{j} O_{j, ϵ}}) = e^{i t \cdot o - i ϵ a (t)} Tr (ρ_{ϵ} e^{i ϵ Φ (V (t))}), \abs Tr (ρ_{ϵ} j \prod e^{i t_{j} O_{j, ϵ}}) - e^{i t \cdot o} \leq ϵ \abs a (t) + ϵ C_{ϵ} \norm V (t) .$
Psy02-A@1#eq:ordered-bound · line 740
$Tr (ρ_{ϵ} j = 1 \prod^{J} e^{i t_{j} O_{j, ϵ}}) = e^{i t \cdot o - i ϵ a (t)} Tr (ρ_{ϵ} e^{i ϵ Φ (V (t))}), \abs Tr (ρ_{ϵ} j \prod e^{i t_{j} O_{j, ϵ}}) - e^{i t \cdot o} \leq ϵ \abs a (t) + ϵ C_{ϵ} \norm V (t) .$
Psy02-A@1#eq:ordered-vacuum · line 765
$Tr (ρ_{ϵ} j \prod e^{i t_{j} O_{j, ϵ}}) = exp (i t \cdot o - \frac{ϵ}{2} \norm V (t)^{2} - i ϵ a (t)) .$
unlabeled · line 789
$\abs Tr (ρ (e^{i B} - 1)) \leq Tr (ρ B^{2}) .$

Theorem: Ordered Weyl identity and bound

Displayed formulas in this passage

Nearby in this section