Manuscript IX-B-ST@1 · IX-B-ST.tex · section · f6af1523

Regulator placement

Chart formulae and the exact dependency ledger › Regulator placement · lines 550–594

IX-B-ST@1#eq:Deps, IX-B-ST@1#eq:triangle

Let $ρ \in C_{c}^{\infty} (R^{4})$ be real, nonnegative, even, Euclidean radial, and normalized by $\int ρ = 1$ . Use the same $ρ$ on both original field slots of each insertion:

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

The middle observable $W_{g}$ is not smoothed.

Use the coordinate Fourier pairing $ρ (k) = \int e^{- ik \cdot z} ρ (z) d^{4} z$ . For $n \in S^{2}$ , put

ϑ_{ρ} (v) = ρ (- v, v n), M_{ε} (λ) = ϑ_{ρ} (ε λ^{2} + m^{2} /2) .

Radiality makes $ϑ_{ρ}$ independent of $n$ . Let $K_{ε}$ be (eq:K) with multiplier $ϑ_{ρ} (εP)$ , 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} .

The exact numerical family under this placement is

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

Its positive-cutoff smoothness and its identification with the old-domain coefficient are included in the following analytic input.

Displayed formulas in this passage

IX-B-ST@1#eq:Deps · line 577
$D_{ε} (z) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) z^{0} + i q \cdot z} .$
IX-B-ST@1#eq:triangle · line 585
$T_{ε}^{ρ} (x, y, x^{'}) = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .$
unlabeled · line 556
$ρ_{ε} (z) = ε^{- 4} ρ (z / ε), S_{ε} = ρ_{ε} *, B_{f, ε} [h, ϕ] = B_{f} [S_{ε} h, S_{ε} ϕ] .$
unlabeled · line 568
$ϑ_{ρ} (v) = ρ (- v, v n), M_{ε} (λ) = ϑ_{ρ} (ε λ^{2} + m^{2} /2) .$