Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

A specified occupation baseline

E3: controlled semiclassical comparison › A specified occupation baseline · lines 1116–1158

Em03-E@1#eq:Hbaseline, Em03-E@1#eq:calibration

Choose two orthonormal positive-frequency packet tests $h_{1}, h_{2}$ in the Em03-A–C scalar model, with disjoint frequency supports contained in compact bands away from zero. The stationary outgoing prescription then has vanishing off-diagonal occupation covariance.

Its selected state is the product of geometric densities

ρ_{H} = i = 1 ⨂^{2} [(1 - q_{i}) n = 0 \sum^{\infty} q_{i}^{n} ∣ n ⟩ ⟨ n ∣],

where

ν_{i} = \int_{0}^{\infty} ∣ h_{i} (ω) ∣^{2} \frac{Γ _{ℓ} ( ω )}{e ^{8 π GM ω} - 1} d ω, q_{i} = \frac{ν _{i}}{1 + ν _{i}} .

These are predecessor inputs, with state selection retained. They are not calculated from the finite Hamiltonians of this paper.

For the occupation emissions choose

w_{i, n} = {(1 - q_{i}) q_{i}^{n}, q_{i}^{J_{i}}, 0 \leq n < J_{i}, n = J_{i} .

We use $0^{0} = 1$ . This is a normalized probability vector.

Equation (eq:calibration) is an explicit calibration of interaction amplitudes to a clipped packet law. It is not a derivation of the Hawking law from the finite dynamics.

Displayed formulas in this passage

Em03-E@1#eq:Hbaseline · line 1125
$ρ_{H} = i = 1 ⨂^{2} [(1 - q_{i}) n = 0 \sum^{\infty} q_{i}^{n} ∣ n ⟩ ⟨ n ∣],$
Em03-E@1#eq:calibration · line 1143
$w_{i, n} = {(1 - q_{i}) q_{i}^{n}, q_{i}^{J_{i}}, 0 \leq n < J_{i}, n = J_{i} .$
unlabeled · line 1132
$ν_{i} = \int_{0}^{\infty} ∣ h_{i} (ω) ∣^{2} \frac{Γ _{ℓ} ( ω )}{e ^{8 π GM ω} - 1} d ω, q_{i} = \frac{ν _{i}}{1 + ν _{i}} .$