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

Incoming modes and the weighted coefficient space

Geometry, signs, and canonical radiation coordinates › Incoming modes and the weighted coefficient space · lines 402–485

Em03-D@1#eq:testcoefficient, Em03-D@1#eq:Weyl, Em03-D@1#eq:Unruhnorm

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Let $f_{+} (x, ω) \sim e^{i ω x}$ at $+ \infty$ and $f_{-} (x, ω) \sim e^{- i ω x}$ at $- \infty$ be Jost solutions. Define

W (ω) = f_{-} (x, ω) f_{+}^{'} (x, ω) - f_{-}^{'} (x, ω) f_{+} (x, ω),

τ (ω) = \frac{2 i ω}{W ( ω )}, q_{H} = τ f_{+}, q_{I} = τ f_{-} .

These are the horizon-incident and infinity-incident modes. The infinity-incident mode has an outgoing infinity reflection amplitude denoted $ρ_{I} (ω)$ .

The incoming spectral transform is

(U u)_{j} (ω) = \frac{1}{2 π} \int_{R} q_{j} (x, ω) u (x) d x, j \in {H, I} .

It is understood in its unitary $L^{2}$ extension.

The canonical field expansion is

Ψ_{ϕ} (t, x) = j \sum \int_{0}^{\infty} \frac{q _{j} ( x , ω ) a _{j} ( ω ) e ^{- i ω t} + q _{j} ( x , ω ) a _{j} ( ω ) ^{†} e ^{i ω t}}{4 π ω} d ω .

Consequently a compact real reduced test has coefficient

k_{j} (f) (ω) = \frac{1}{4 π ω} \int_{R^{2}} f (t, x) q_{j} (x, ω) e^{i ω t} d t d x,

so that

ϕ (F [f]) = X (k (f)), X (k) = a (k) + a (k)^{†} .

Annihilators are conjugate-linear in their tests:

[a (k), a (l)^{†}] = ⟨ k, l ⟩ .

Define the real symplectic form and Weyl convention by

s (k, l) = 2 Im ⟨ k, l ⟩, W (k) = e^{i X (k)},

W (k) W (l) = e^{- i s (k, l) /2} W (k + l) .

The Unruh second-moment norm is

∥ k ∥_{U}^{2} = \int_{0}^{\infty} [(1 + 2 n_{H} (ω)) ∣ k_{H} (ω) ∣^{2} + ∣ k_{I} (ω) ∣^{2}] d ω, n_{H} (ω) = (e^{β ω} - 1)^{- 1} .

For real linear observables,

ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ω_{U} (W (k)) = e^{- ∥ k ∥_{U}^{2} /2} .

The norm dominates the ordinary two-channel $L^{2}$ norm.

Displayed formulas in this passage

Em03-D@1#eq:testcoefficient · line 440
$k_{j} (f) (ω) = \frac{1}{4 π ω} \int_{R^{2}} f (t, x) q_{j} (x, ω) e^{i ω t} d t d x,$
Em03-D@1#eq:Weyl · line 463
$W (k) W (l) = e^{- i s (k, l) /2} W (k + l) .$
Em03-D@1#eq:Unruhnorm · line 469
$∥ k ∥_{U}^{2} = \int_{0}^{\infty} [(1 + 2 n_{H} (ω)) ∣ k_{H} (ω) ∣^{2} + ∣ k_{I} (ω) ∣^{2}] d ω, n_{H} (ω) = (e^{β ω} - 1)^{- 1} .$
unlabeled · line 407
$W (ω) = f_{-} (x, ω) f_{+}^{'} (x, ω) - f_{-}^{'} (x, ω) f_{+} (x, ω),$
unlabeled · line 412
$τ (ω) = \frac{2 i ω}{W ( ω )}, q_{H} = τ f_{+}, q_{I} = τ f_{-} .$
unlabeled · line 422
$(U u)_{j} (ω) = \frac{1}{2 π} \int_{R} q_{j} (x, ω) u (x) d x, j \in {H, I} .$
unlabeled · line 431
$Ψ_{ϕ} (t, x) = j \sum \int_{0}^{\infty} \frac{q _{j} ( x , ω ) a _{j} ( ω ) e ^{- i ω t} + q _{j} ( x , ω ) a _{j} ( ω ) ^{†} e ^{i ω t}}{4 π ω} d ω .$
unlabeled · line 449
$ϕ (F [f]) = X (k (f)), X (k) = a (k) + a (k)^{†} .$
unlabeled · line 455
$[a (k), a (l)^{†}] = ⟨ k, l ⟩ .$
unlabeled · line 459
$s (k, l) = 2 Im ⟨ k, l ⟩, W (k) = e^{i X (k)},$
unlabeled · line 479
$ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ω_{U} (W (k)) = e^{- ∥ k ∥_{U}^{2} /2} .$