Manuscript Em03-D@1 · Em03-D-v1.tex · section · 450d878a

The outgoing target and a controlled state class

The outgoing target and a controlled state class · lines 624–667

Em03-D@1#eq:packetcoefficient, Em03-D@1#eq:Hawkingbaseline

Fix a normalized

h \in C_{c}^{\infty} ((0, \infty)), supp h \subset [ω_{-}, ω_{+}], 0 < ω_{-} < ω_{+} .

Stationary scattering gives

a_{out} (h) = a_{H} (τ h) + a_{I} (ρ_{I} h) .

Thus

k_{*} = \frac{ϑ}{2} (τ h, ρ_{I} h), ϑ Q_{h} = X (k_{*}) .

This is the coefficient of the outgoing scattering observable in the bulk incoming coordinates. It is not a finite-radius pointwise limit of an unsmeared quantum field.

Unitarity gives

∥ k_{*} ∥_{0}^{2} = ϑ^{2} /2, ∥ k_{*} ∥_{U}^{2} = ϑ^{2} (ν_{h} + 1/2) .

Consequently

ω_{U} (E_{h, ϑ}) = \frac{1}{2} [1 + e^{- ϑ^{2} (2 ν_{h} + 1) /4}] .

There is no packet-count rate factor $1/ (2 π)$ in this formula.

Displayed formulas in this passage

Em03-D@1#eq:packetcoefficient · line 640
$k_{*} = \frac{ϑ}{2} (τ h, ρ_{I} h), ϑ Q_{h} = X (k_{*}) .$
Em03-D@1#eq:Hawkingbaseline · line 658
$ω_{U} (E_{h, ϑ}) = \frac{1}{2} [1 + e^{- ϑ^{2} (2 ν_{h} + 1) /4}] .$
unlabeled · line 628
$h \in C_{c}^{\infty} ((0, \infty)), supp h \subset [ω_{-}, ω_{+}], 0 < ω_{-} < ω_{+} .$
unlabeled · line 634
$a_{out} (h) = a_{H} (τ h) + a_{I} (ρ_{I} h) .$
unlabeled · line 652
$∥ k_{*} ∥_{0}^{2} = ϑ^{2} /2, ∥ k_{*} ∥_{U}^{2} = ϑ^{2} (ν_{h} + 1/2) .$

The outgoing target and a controlled state class

Displayed formulas in this passage

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