Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Regulator placement and numerical triangle

Fixed data and inherited inputs › Regulator placement and numerical triangle · lines 314–362

IX-B@2#eq:T, IX-B@2#eq:G

As in VII-D, the same fixed radial mollifier is used on both slots of each insertion, and the middle observable is unsmeared VIID (sc:placement, prop:positive). Specifically, $ρ \in C_{c}^{\infty} (R^{4})$ is real, nonnegative, even, Euclidean radial, and normalized by $\int ρ = 1$ . Set

ρ_{ε} (z) = ε^{- 4} ρ (z / ε), S_{ε} = ρ_{ε} *, B_{f, ε} = B_{f} [S_{ε} h, S_{ε} ϕ] .

With the Fourier convention $v (k) = \int e^{- ik \cdot z} v (z) d z$ , define

ϑ_{ρ} (u) = ρ (- u, u n), M_{ε} (λ) = ϑ_{ρ} (ε λ^{2} + m^{2} /2), n \in S^{2} .

Radiality makes the profile independent of $n$ . Let $K_{ε}$ be (eq:K) with the multiplier $ϑ_{ρ} (ε ∣ p ∣)$ , and set

D_{ε} (z) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) z^{0} + i q \cdot z} .

VII-D identifies the exact numerical family as

T_{ε}^{ρ} (x, y, x^{'}) = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .

It is smooth for every $ε > 0$ VIID (prop:positive, eq:Texact). On real product tests, its pairing is the single VII-C coefficient

G_{ε} (f, g, f^{'}) := ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) = ⟨ T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ .

The joint-test distribution is complex-linear; conjugation in (eq:T) conjugates the kernel, not the test.

Displayed formulas in this passage

IX-B@2#eq:T · line 345
$T_{ε}^{ρ} (x, y, x^{'}) = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .$
IX-B@2#eq:G · line 353
$G_{ε} (f, g, f^{'}) := ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) = ⟨ T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ .$
unlabeled · line 322
$ρ_{ε} (z) = ε^{- 4} ρ (z / ε), S_{ε} = ρ_{ε} *, B_{f, ε} = B_{f} [S_{ε} h, S_{ε} ϕ] .$
unlabeled · line 329
$ϑ_{ρ} (u) = ρ (- u, u n), M_{ε} (λ) = ϑ_{ρ} (ε λ^{2} + m^{2} /2), n \in S^{2} .$
unlabeled · line 338
$D_{ε} (z) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) z^{0} + i q \cdot z} .$