Manuscript Em03-C@1 · Em03-C-v1.tex · section · 47dd71ec

The effective port and its domain

Joint temporal modes and participating radiation › The effective port and its domain · lines 841–877

As in Em03-B, the effective port has one-particle space $L^{2} (R, d ω)$ . Its positive-frequency sector carries (eq:radiationdensity); its negative-frequency sector is a new independent auxiliary vacuum.

This two-sided Markov envelope is model data. It is not a physical negative-energy sector of the Schwarzschild scalar. Unselected positive-frequency radiation is not replaced by vacuum.

Use

p_{i} (t) = \frac{1}{2 π} \int h_{i} (ω) e^{- i ω t} d ω, f (ω) = \frac{1}{2 π} \int f (t) e^{i ω t} d t .

The input annihilator $a (f)$ is the temporal smearing $\int f (t) b_{in} (t) d t$ .

Finite restrictions are defined on the weighted domain

\int_{R} (1 + n_{port} (ω)) ∣ v (ω) ∣^{2} d ω < \infty,

where $n_{port} = n$ on positive frequencies and zero on negative frequencies. Compact positive-frequency tests are dense in the positive-frequency weighted space. Covariances and Gaussian Weyl positivity extend by continuity. Every finite canonical subspace with finite occupation matrix has a normal density by the same diagonalization proof as Theorem thm:gaussian.

This extension is a radiation-test construction, not a new local bulk characteristic-restriction theorem.

Displayed formulas in this passage

unlabeled · line 853
$p_{i} (t) = \frac{1}{2 π} \int h_{i} (ω) e^{- i ω t} d ω, f (ω) = \frac{1}{2 π} \int f (t) e^{i ω t} d t .$
unlabeled · line 863
$\int_{R} (1 + n_{port} (ω)) ∣ v (ω) ∣^{2} d ω < \infty,$