Manuscript IV@1 · IV.tex · section · 351cb2d7

Explicit scalar kernels

Model 3: scalar stress form and exterior record › Explicit scalar kernels · lines 2733–2797

IV@1#eq:Cstress, IV@1#eq:transfer3

For $σ, τ \in {+ 1, - 1}$ , let $u_{p}^{+} = u_{p}, u_{p}^{-} = u ˉ_{p}$ , and define

ℓ_{μ} (p) = (- ω_{p}, p), k = σ p + τ q, r = ∣ k ∣, Ω_{σ τ} = σ ω_{p} + τ ω_{q} .

The polarized stress amplitude is

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

It multiplies $e^{- i Ω_{σ τ} t + ik \cdot x}$ .

For $r > 0$ , the Newton–wave amplitude has initial tensors

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 evolution

F_{μν}^{σ τ} (t; p, q) = cos (r t) A_{μν} + \frac{sin ( r t )}{r} B_{μν} + 2 C_{μν}^{σ τ} \int_{0}^{t} \frac{sin ( r ( t - s ))}{r} e^{- i Ω_{σ τ} s} d s .

Define

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

Plane waves are used here to compute a kernel, not as members of the original Schwartz source class. Local integrability is proved below.

Choose a real even smooth compact mask $χ_{Λ} (p) = χ (p /Λ)$ , equal to one on $∣ p ∣ \leq Λ$ and zero on $∣ p ∣ \geq 2Λ$ . Apply it to each scalar field leg. The cutoff field still solves $P ϕ_{Λ} = 0$ , so its normal-ordered stress is conserved. 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

IV@1#eq:Cstress · line 2743
$C_{μν}^{σ τ} (p, q) = n_{p} n_{q} [- σ τ ℓ_{(μ} (p) ℓ_{ν)} (q) + \frac{1}{2} η_{μν} (σ τ η^{α β} ℓ_{α} (p) ℓ_{β} (q) - m^{2})] .$
IV@1#eq:transfer3 · line 2764
$F_{μν}^{σ τ} (t; p, q) = cos (r t) A_{μν} + \frac{sin ( r t )}{r} B_{μν} + 2 C_{μν}^{σ τ} \int_{0}^{t} \frac{sin ( r ( t - s ))}{r} e^{- i Ω_{σ τ} s} d s .$
unlabeled · line 2737
$ℓ_{μ} (p) = (- ω_{p}, p), k = σ p + τ q, r = ∣ k ∣, Ω_{σ τ} = σ ω_{p} + τ ω_{q} .$
unlabeled · line 2756
$A_{00} = \frac{2 C _{00}}{r ^{2}}, A_{0 i} = \frac{2 C _{0 i}}{r ^{2}}, A_{ij} = 0,$
unlabeled · line 2760
$B_{00} = i k_{i} A_{i 0}, B_{0 i} = B_{ij} = 0,$
unlabeled · line 2774
$K_{J}^{σ τ} (p, q) = \int J^{μν} (t, x) e^{ik \cdot x} (r F^{σ τ})_{μν} (t; p, q) d t d x .$
unlabeled · line 2789
$h_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{-+} (p, q),$
unlabeled · line 2793
$g_{Λ} (p, q) = 2 χ_{Λ} (p) χ_{Λ} (q) K_{J}^{--} (p, q) .$

Explicit scalar kernels

Displayed formulas in this passage

Nearby in this section