Manuscript Em03-B@2 · Em03-B-v2.tex · theorem · c1a0977d

Theorem: Normal geometric packet marginal

Audited field input and the packet marginal › B0: complete one-mode identification · lines 372–426 · proof included or linked

Em03-B@2#thm:marginal, Em03-B@2#eq:rhoh

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Under Assumption ass:field, the selected packet state is represented normally on $H_{h} = F_{s} (C) ≅ ℓ^{2} (N_{0})$ by

ρ_{h} = (1 - q_{h}) n = 0 \sum^{\infty} q_{h}^{n} ∣ n ⟩ ⟨ n ∣ .

For $ν_{h} = 0$ , it is the vacuum. Its count law has mean $ν_{h}$ , variance $ν_{h} (1 + ν_{h})$ , and generating function

E s^{N_{h}} = \frac{1 - q _{h}}{1 - q _{h} s}, ∣ s ∣ < q_{h}^{- 1}

when $q_{h} > 0$ .

Proof. Complex-linear scattering and restriction preserve centered quasifreeness. The packet has $⟨ b_{h}^{2} ⟩ = 0$ and $⟨ b_{h}^{†} b_{h} ⟩ = ν_{h}$ . Thus its displacement characteristic function is

⟨ e^{z b_{h}^{†} - z b_{h}} ⟩ = e^{- (ν_{h} + 1/2) ∣ z ∣^{2}} .

For $ν_{h} > 0$ , construct the trace-class coherent-state mixture

σ = \int_{C} \frac{e ^{- ∣ α ∣^{2} / ν_{h}}}{π ν _{h}} ∣ α ⟩ ⟨ α ∣ d^{2} α .

The measure has total mass one. Its characteristic function is the displayed Gaussian, since

⟨ α ∣ e^{z b_{h}^{†} - z b_{h}} ∣ α ⟩ = e^{- ∣ z ∣^{2} /2 + z α - z α} .

Angular and radial integration give

⟨ n ∣ σ ∣ m ⟩ = δ_{nm} \frac{1}{ν _{h} n !} \int_{0}^{\infty} e^{- (1 + ν_{h}^{- 1}) t} t^{n} d t = δ_{nm} \frac{ν _{h} ^{n}}{( 1 + ν _{h} ) ^{n + 1}} .

This is (eq:rhoh). Equality of characteristic functions identifies the states on the Weyl algebra, because its dense linear span consists of Weyl operators. The vacuum case follows directly. Summing the geometric series proves the moment and generating-function statements.

Displayed formulas in this passage

Em03-B@2#eq:rhoh · line 377
$ρ_{h} = (1 - q_{h}) n = 0 \sum^{\infty} q_{h}^{n} ∣ n ⟩ ⟨ n ∣ .$
unlabeled · line 385
$E s^{N_{h}} = \frac{1 - q _{h}}{1 - q _{h} s}, ∣ s ∣ < q_{h}^{- 1}$
unlabeled · line 398
$⟨ e^{z b_{h}^{†} - z b_{h}} ⟩ = e^{- (ν_{h} + 1/2) ∣ z ∣^{2}} .$
unlabeled · line 403
$σ = \int_{C} \frac{e ^{- ∣ α ∣^{2} / ν_{h}}}{π ν _{h}} ∣ α ⟩ ⟨ α ∣ d^{2} α .$
unlabeled · line 410
$⟨ α ∣ e^{z b_{h}^{†} - z b_{h}} ∣ α ⟩ = e^{- ∣ z ∣^{2} /2 + z α - z α} .$
unlabeled · line 415
$⟨ n ∣ σ ∣ m ⟩ = δ_{nm} \frac{1}{ν _{h} n !} \int_{0}^{\infty} e^{- (1 + ν_{h}^{- 1}) t} t^{n} d t = δ_{nm} \frac{ν _{h} ^{n}}{( 1 + ν _{h} ) ^{n + 1}} .$

Theorem: Normal geometric packet marginal

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