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

Assumption: Inherited radiation prescription

Audited field input and the packet marginal › The inherited data · lines 279–317

Em03-B@2#ass:field, Em03-B@2#eq:nomega

The horizon and past-infinity input states are independent, centered, regular, and quasifree. Their positive-Killing-frequency covariances are

⟨ a_{H} (ω)^{†} a_{H} (ω^{'})⟩ ⟨ a_{H} (ω) a_{H} (ω^{'})⟩ = n_{H} (ω) δ (ω - ω^{'}), = 0, n_{H} (ω) ⟨ a_{I} (ω)^{†} a_{I} (ω^{'})⟩ = (e^{β_{H} ω} - 1)^{- 1}, = 0.

The infinity state is its positive-frequency vacuum, hence is centered with zero anomalous covariance. Cross-channel covariances vanish.

Stationary scattering is complex linear:

a_{out} (ω) = τ_{ℓ} (ω) a_{H} (ω) + ρ_{I, ℓ} (ω) a_{I} (ω), ∣ τ_{ℓ} ∣^{2} + ∣ ρ_{I, ℓ} ∣^{2} = 1.

Write

n (ω) = \frac{Γ _{ℓ} ( ω )}{e ^{β_{H} ω} - 1}, Γ_{ℓ} = ∣ τ_{ℓ} ∣^{2}, 0 \leq Γ_{ℓ} \leq 1.

The outgoing state is therefore centered, gauge-invariant, and quasifree, with number covariance $n (ω) δ (ω - ω^{'})$ .

Displayed formulas in this passage

unlabeled · line 284
$⟨ a_{H} (ω)^{†} a_{H} (ω^{'})⟩ ⟨ a_{H} (ω) a_{H} (ω^{'})⟩ = n_{H} (ω) δ (ω - ω^{'}), = 0, n_{H} (ω) ⟨ a_{I} (ω)^{†} a_{I} (ω^{'})⟩ = (e^{β_{H} ω} - 1)^{- 1}, = 0.$
Em03-B@2#eq:nomega · line 308
$n (ω) = \frac{Γ _{ℓ} ( ω )}{e ^{β_{H} ω} - 1}, Γ_{ℓ} = ∣ τ_{ℓ} ∣^{2}, 0 \leq Γ_{ℓ} \leq 1.$
unlabeled · line 300
$a_{out} (ω) = τ_{ℓ} (ω) a_{H} (ω) + ρ_{I, ℓ} (ω) a_{I} (ω), ∣ τ_{ℓ} ∣^{2} + ∣ ρ_{I, ℓ} ∣^{2} = 1.$