Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Frozen family and principal statements

Frozen family and principal statements · lines 248–304

Em01-G@5#eq:target

Let $M = R^{1 + 3}$ , with

η = diag (1, - 1, - 1, - 1), □_{η} = \partial_{t}^{2} - Δ_{x}, P_{m} = □_{η} + m^{2} .

Spacetime index conversions use $η$ . Distributional pairings are complex bilinear, with the displayed densities.

Fix

m \geq 0, λ \in R ∖ {0}, ℏ > 0, f, g \in C_{c}^{\infty} (M; R), F = E_{m}^{A} f,

a relatively compact separated coordinate patch $U \subset M^{4}$ , and

ρ \in C_{c}^{\infty} (R^{8}; R), \int ρ = 1, χ \in C_{c}^{\infty} (R^{8}), χ = 1 near f (U) .

Every datum is fixed before $δ ↓ 0$ . In particular $F$ remains the advanced root of $f$ .

The positive-scale family is

t_{δ}^{app} = i λ^{3} V (K_{ret}, ρ_{δ} * (χ C))_{U}, ρ_{δ} (H) = δ^{- 8} ρ (H / δ) .

Its arrays, source, kernels, and vertex are defined in Appendices app:inputs–app:positive. Set

K_{supp} = K_{6} + K_{8} + K_{9 d} + K_{9 m}, K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{supp} .

The weights are identically one and there is no $1/ℏ$ response prefactor.

Tests belong to $D (U; S^{*})$ , where $S = Sym^{2} ((R^{4})^{*})$ . For $L ⋐ U$ , write

∥Φ ∥_{q, L} = J, ∣ γ ∣ \leq q max L sup ∣ \partial^{γ} Φ_{J} ∣.

Compactly supported tests are extended by zero when a coordinate expression requires a fixed slightly larger neighborhood.

Displayed formulas in this passage

Em01-G@5#eq:target · line 277
$t_{δ}^{app} = i λ^{3} V (K_{ret}, ρ_{δ} * (χ C))_{U}, ρ_{δ} (H) = δ^{- 8} ρ (H / δ) .$
unlabeled · line 252
$η = diag (1, - 1, - 1, - 1), □_{η} = \partial_{t}^{2} - Δ_{x}, P_{m} = □_{η} + m^{2} .$
unlabeled · line 262
$m \geq 0, λ \in R ∖ {0}, ℏ > 0, f, g \in C_{c}^{\infty} (M; R), F = E_{m}^{A} f,$
unlabeled · line 267
$ρ \in C_{c}^{\infty} (R^{8}; R), \int ρ = 1, χ \in C_{c}^{\infty} (R^{8}), χ = 1 near f (U) .$
unlabeled · line 288
$K_{supp} = K_{6} + K_{8} + K_{9 d} + K_{9 m}, K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{supp} .$
unlabeled · line 297
$∥Φ ∥_{q, L} = J, ∣ γ ∣ \leq q max L sup ∣ \partial^{γ} Φ_{J} ∣.$