Manuscript Em06-A@1 · Em06-A-v1.tex · section · ee9923c6

The state family

The quantum family is constructed, not inferred › The state family · lines 723–757

Em06-A@1#eq:states, Em06-A@1#eq:characteristic

Define

Ψ_{ϵ} = D (z_{cl} / ϵ) Ω, ω_{ϵ} (A) = ⟨ Ψ_{ϵ}, A Ψ_{ϵ} ⟩ .

Each $Ψ_{ϵ}$ is a normalized vector. Its restriction to the represented Weyl algebra is consequently a positive normalized state. Explicitly,

ω_{ϵ} (W_{ϵ} (v)) = exp (i 2 Re ⟨ v, z_{cl} ⟩ - \frac{ϵ}{2} ∥ v ∥^{2}) .

To verify the formula, normal ordering gives

e^{i t Φ (v)} = e^{- t^{2} ∥ v ∥^{2} /2} e^{i t a^{†} (v)} e^{i t a (v)},

and therefore $⟨ Ω, e^{i t Φ (v)} Ω ⟩ = e^{- t^{2} ∥ v ∥^{2} /2}$ . Conjugating by the displacement in (eq:states) then gives (eq:characteristic). This construction is the familiar coherent-state mechanism Glauber,Hepp, with its normalization printed here.

Displayed formulas in this passage

Em06-A@1#eq:states · line 726
$Ψ_{ϵ} = D (z_{cl} / ϵ) Ω, ω_{ϵ} (A) = ⟨ Ψ_{ϵ}, A Ψ_{ϵ} ⟩ .$
Em06-A@1#eq:characteristic · line 734
$ω_{ϵ} (W_{ϵ} (v)) = exp (i 2 Re ⟨ v, z_{cl} ⟩ - \frac{ϵ}{2} ∥ v ∥^{2}) .$
unlabeled · line 745
$e^{i t Φ (v)} = e^{- t^{2} ∥ v ∥^{2} /2} e^{i t a^{†} (v)} e^{i t a (v)},$