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

The inherited data

Audited field input and the packet marginal › The inherited data · lines 317–369

Em03-B@2#eq:nu

The components of this assumption have distinct source roles.

{}1.15

Component	Supplied Em03-A locator or status
Product boundary prescription and quasifreeness	State subsection and declared bulk-to-boundary input
Horizon number and anomalous covariance	`:horizonoccupation`
Infinity vacuum and independence	State subsection and `:horizonoccupation`
Complex-linear stationary scattering	`:quantumscatter`
Outgoing compact-band number covariance	`:packet`, `:numberdensity`
Packet occupation integral	`:packetlaw`
Bulk existence and Hadamard identification	Declared Dappiaggi–Moretti–Pinamonti input DMP
Weighted radiation-test completion used below	Explicit mathematical completion in Section subsec:completion; not a new bulk theorem

Hadamard regularity alone does not select these covariances.

Define

b_{h} = a_{out} (h), [b_{h}, b_{h}^{†}] = 1, N_{h} = b_{h}^{†} b_{h},

where annihilation operators are conjugate-linear in their test: $a (h) = \int h (ω) a (ω) d ω$ . Set

ν_{h} = \int_{B} ∣ h (ω) ∣^{2} n (ω) d ω, q_{h} = \frac{ν _{h}}{1 + ν _{h}} .

Then

0 \leq ν_{h} \leq n_{m a x} := (e^{β_{H} ω_{-}} - 1)^{- 1} < \infty.

Displayed formulas in this passage

Em03-B@2#eq:nu · line 360
$ν_{h} = \int_{B} ∣ h (ω) ∣^{2} n (ω) d ω, q_{h} = \frac{ν _{h}}{1 + ν _{h}} .$
unlabeled · line 352
$b_{h} = a_{out} (h), [b_{h}, b_{h}^{†}] = 1, N_{h} = b_{h}^{†} b_{h},$
unlabeled · line 366
$0 \leq ν_{h} \leq n_{m a x} := (e^{β_{H} ω_{-}} - 1)^{- 1} < \infty.$