Manuscript En02-E@2 · En02-E-v2.tex · section · c2769986

Normalization and outgoing covariance

Specified stationary radiation model › Normalization and outgoing covariance · lines 292–356

En02-E@2#eq:outcov

The current convention is

j (u) = - i \int_{0}^{\infty} \frac{ω}{4 π} (a (ω) e^{- i ω u} - a (ω)^{†} e^{i ω u}) d ω, [a (ω), a (ω^{'})^{†}] = δ (ω - ω^{'}) .

The affine vacuum current covariance pulls back to

C_{κ} (s) = - \frac{κ ^{2}}{16 π} csch^{2} (\frac{κ ( s - i 0 )}{2}), C_{κ} (ν) = \frac{ν}{2 ( 1 - e ^{- β ν} )} .

These are the stationary boundary identities imported from Em03-A. The positive- and negative-frequency current coefficients are $ω (1 + n_{H}) / (4 π)$ and $ω n_{H} / (4 π)$ , respectively; Fourier transformation contributes $2 π$ . Hence

n_{H} (ω) = \frac{1}{e ^{β ω} - 1} .

In particular, a normalized packet covariance has no extra $1/ (2 π)$ factor.

Let $S_{ℓ} (ω)$ be the unitary, flux-normalized two-channel scattering matrix of (eq:radial). Write its horizon-to-infinity transmission amplitude as $τ_{ℓ}$ , and put $Γ_{ℓ} = ∣ τ_{ℓ} ∣^{2}$ . For $ω > 0$ , $0 < Γ_{ℓ} \leq 1$ . The outgoing annihilator is

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

The incoming channels are independent and have zero means. Their number covariances give

⟨ a_{out} (ω)^{†} a_{out} (ω^{'})⟩ q_{out} (ω) ⟨ a_{out} (ω) a_{out} (ω^{'})⟩ = q_{out} (ω) δ (ω - ω^{'}), = \frac{Γ _{ℓ} ( ω )}{e ^{β ω} - 1}, = 0.

Quasifreeness is part of the input. Number covariance alone would not determine a general non-Gaussian state.

The associated time–frequency mode-window convention is

N_{B, T} = \frac{T}{2 π} \int_{B} q_{out} (ω) d ω .

It is distinct from a normalized packet occupation and from a specified detector instrument.

Hawking's fixed-background calculation motivates this model Hawking. A collapse scaling-limit analysis FredenhagenHaag, renormalized backreaction HollandsWald, and a dynamical mass-loss model are separate programmes.

Displayed formulas in this passage

En02-E@2#eq:outcov · line 331
$⟨ a_{out} (ω)^{†} a_{out} (ω^{'})⟩ q_{out} (ω) ⟨ a_{out} (ω) a_{out} (ω^{'})⟩ = q_{out} (ω) δ (ω - ω^{'}), = \frac{Γ _{ℓ} ( ω )}{e ^{β ω} - 1}, = 0.$
unlabeled · line 295
$j (u) = - i \int_{0}^{\infty} \frac{ω}{4 π} (a (ω) e^{- i ω u} - a (ω)^{†} e^{i ω u}) d ω, [a (ω), a (ω^{'})^{†}] = δ (ω - ω^{'}) .$
unlabeled · line 303
$C_{κ} (s) = - \frac{κ ^{2}}{16 π} csch^{2} (\frac{κ ( s - i 0 )}{2}), C_{κ} (ν) = \frac{ν}{2 ( 1 - e ^{- β ν} )} .$
unlabeled · line 313
$n_{H} (ω) = \frac{1}{e ^{β ω} - 1} .$
unlabeled · line 324
$a_{out} (ω) = τ_{ℓ} (ω) a_{H} (ω) + ρ_{I, ℓ} (ω) a_{I} (ω) .$
unlabeled · line 344
$N_{B, T} = \frac{T}{2 π} \int_{B} q_{out} (ω) d ω .$