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

An effective temporal port

The apparatus input: radiation plus participating modes › An effective temporal port · lines 687–727

Let

h_{port} = L^{2} (R, d ω) .

Its positive-frequency part carries the outgoing radiation covariance (eq:nomega). Its negative-frequency part is a new, independent auxiliary vacuum in the apparatus model. Define

n_{port} (ω) = {n (ω), 0, ω > 0, ω < 0.

The Fourier convention is

p (t) = \frac{1}{2 π} \int_{R} h (ω) e^{- i ω t} d ω, f (ω) = \frac{1}{2 π} \int_{R} f (t) e^{i ω t} d t,

with $h$ extended by zero outside its band.

Annihilators on this port satisfy

a (f) = \int_{R} f (t) b_{in} (t) d t, [b_{in} (t), b_{in} (s)^{†}] = δ (t - s),

as smeared identities. The two-sided envelope makes the Markov port canonical. It is not a claim that the physical scalar field contains independent negative-Killing-frequency annihilators.

All positive-frequency components outside the target packet retain their actual prescribed radiation covariance. In particular, no vacuum substitution is made for unselected Hawking radiation.

Displayed formulas in this passage

unlabeled · line 690
$h_{port} = L^{2} (R, d ω) .$
unlabeled · line 697
$n_{port} (ω) = {n (ω), 0, ω > 0, ω < 0.$
unlabeled · line 705
$p (t) = \frac{1}{2 π} \int_{R} h (ω) e^{- i ω t} d ω, f (ω) = \frac{1}{2 π} \int_{R} f (t) e^{i ω t} d t,$
unlabeled · line 714
$a (f) = \int_{R} f (t) b_{in} (t) d t, [b_{in} (t), b_{in} (s)^{†}] = δ (t - s),$