Manuscript Em06-B@1 · Em06-B-v1.tex · section · 61e38859

One-particle map and classical preparation

Independent classical audit and compact-test estimates › One-particle map and classical preparation · lines 482–525

Em06-B@1#eq:k, Em06-B@1#eq:v, Em06-B@1#eq:phi-z, Em06-B@1#eq:classical-symbol

Use the unitary spatial Fourier convention

F (p) = (2 π)^{- 3/2} \int_{R^{3}} F (x) e^{- i p \cdot x} d^{3} x .

Let

h = L^{2} (R^{3}, d^{3} p), ⟨ v, z ⟩ = \int v (p) z (p) d^{3} p, ω_{p} = ∣ p ∣^{2} + m^{2} .

The inner product is linear in its second argument. For a real compact smooth spacetime test $F$ , define

k_{F} (p) = \frac{1}{( 2 π ) ^{3/2} 2 ω _{p}} \int_{M} F (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x .

Write

v (f, h, χ) = k_{q_{f, h, χ}} .

For $z \in S (R^{3}; C)$ , define the real smooth free scalar

ϕ_{z} (t, x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} \frac{z ( p ) e ^{- i ω_{p} t + i p \cdot x} + z ( p ) e ^{i ω_{p} t - i p \cdot x}}{2 ω _{p}} d^{3} p .

Direct integration gives

O_{f, h, χ} [ϕ_{z}] = 2 Re ⟨ v (f, h, χ), z ⟩ =: o (f, h, χ, z) .

The target therefore depends on the scalar preparation as well as the prescribed metric coefficient. Writing $o [g_{cl}]$ without naming the preparation would suppress an essential input.

Displayed formulas in this passage

Em06-B@1#eq:k · line 496
$k_{F} (p) = \frac{1}{( 2 π ) ^{3/2} 2 ω _{p}} \int_{M} F (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x .$
Em06-B@1#eq:v · line 502
$v (f, h, χ) = k_{q_{f, h, χ}} .$
Em06-B@1#eq:phi-z · line 508
$ϕ_{z} (t, x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} \frac{z ( p ) e ^{- i ω_{p} t + i p \cdot x} + z ( p ) e ^{i ω_{p} t - i p \cdot x}}{2 ω _{p}} d^{3} p .$
Em06-B@1#eq:classical-symbol · line 516
$O_{f, h, χ} [ϕ_{z}] = 2 Re ⟨ v (f, h, χ), z ⟩ =: o (f, h, χ, z) .$
unlabeled · line 485
$F (p) = (2 π)^{- 3/2} \int_{R^{3}} F (x) e^{- i p \cdot x} d^{3} x .$
unlabeled · line 489
$h = L^{2} (R^{3}, d^{3} p), ⟨ v, z ⟩ = \int v (p) z (p) d^{3} p, ω_{p} = ∣ p ∣^{2} + m^{2} .$

One-particle map and classical preparation

Displayed formulas in this passage

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