Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

Regulated quadratic generator

Model 3: a specified sourced form and an exterior pairing › Regulated quadratic generator · lines 1195–1243

For clarity, the complete kernel prescription is recorded. Let

n_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2}, ℓ_{μ} (p) = (- ω_{p}, p), k = σ p + δ q, r = ∣ k ∣, Ω = σ ω_{p} + δ ω_{q},

and define

C_{μν}^{σ δ} (p, q) = n_{p} n_{q} [- σ δ ℓ_{(μ} (p) ℓ_{ν)} (q) + \frac{1}{2} η_{μν} (σ δ η^{α β} ℓ_{α} (p) ℓ_{β} (q) - m^{2})] .

Apply the initial-data rule of (eq:NW) to $C^{σ δ} e^{- i Ω t + ik \cdot x}$ , obtaining its trace-reversed amplitude $F^{σ δ} (t; p, q)$ . Explicitly,

A_{00} = \frac{2 C _{00}}{r ^{2}}, A_{0 i} = \frac{2 C _{0 i}}{r ^{2}}, A_{ij} = 0, B_{00} = i k_{i} A_{i 0}, B_{0 i} = B_{ij} = 0,

and

F (t) = cos (r t) A + \frac{sin ( r t )}{r} B + 2 C \int_{0}^{t} \frac{sin ( r ( t - s ))}{r} e^{- i Ω s} d s .

Define

K_{J}^{σ δ} (p, q) = \int J^{μν} (t, x) e^{ik \cdot x} (r F^{σ δ})_{μν} (t; p, q) d t d x .

Choose a real even smooth compact mask $χ_{Λ}$ , equal to one on $∣ p ∣ \leq Λ$ , and set

h_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{-+} (p, q), g_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{--} (p, q) .

Displayed formulas in this passage

unlabeled · line 1199
$n_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2}, ℓ_{μ} (p) = (- ω_{p}, p), k = σ p + δ q, r = ∣ k ∣, Ω = σ ω_{p} + δ ω_{q},$
unlabeled · line 1206
$C_{μν}^{σ δ} (p, q) = n_{p} n_{q} [- σ δ ℓ_{(μ} (p) ℓ_{ν)} (q) + \frac{1}{2} η_{μν} (σ δ η^{α β} ℓ_{α} (p) ℓ_{β} (q) - m^{2})] .$
unlabeled · line 1217
$A_{00} = \frac{2 C _{00}}{r ^{2}}, A_{0 i} = \frac{2 C _{0 i}}{r ^{2}}, A_{ij} = 0, B_{00} = i k_{i} A_{i 0}, B_{0 i} = B_{ij} = 0,$
unlabeled · line 1223
$F (t) = cos (r t) A + \frac{sin ( r t )}{r} B + 2 C \int_{0}^{t} \frac{sin ( r ( t - s ))}{r} e^{- i Ω s} d s .$
unlabeled · line 1228
$K_{J}^{σ δ} (p, q) = \int J^{μν} (t, x) e^{ik \cdot x} (r F^{σ δ})_{μν} (t; p, q) d t d x .$
unlabeled · line 1236
$h_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{-+} (p, q), g_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{--} (p, q) .$

Regulated quadratic generator

Displayed formulas in this passage

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