Manuscript V@1 · V.tex · section · e7ddf37d

Fully Wick form and joint cutoff limit

Contrast: the Newton–wave cubic form › Fully Wick form and joint cutoff limit · lines 3455–3505

V@1#eq:pnorm, V@1#eq:cubicsector

On the full symmetric scalar Schwartz domain $D_{ϕ}$ , define

s_{n} (ψ_{n}) = p_{1}, \dots, p_{n} sup (j = 1 \prod^{n} ⟨ p_{j} ⟩^{30}) ∣ ψ_{n} ∣, s_{0} (z) = ∣ z ∣,

I_{*} = \int_{R^{3}} ⟨ p ⟩^{- 60} d^{3} p, p (Ψ) = n \geq 0 \sum (1 + I_{*})^{n} (n + 3)^{3} s_{n} (Ψ_{n}) .

This is a continuous norm on the algebraic direct-sum Schwartz domain, stronger than the Fock norm.

Full cubic normal ordering is prescribed before cutoff removal. For $r + s = 3$ , let $k_{r, s} = K_{f}^{+^{r} -^{s}}$ . Its sector pairing, with $n \geq s$ and $m = n + r - s$ , is

C_{r, s} (Ψ_{m}, X_{n}) = (r 3) \frac{n ! m !}{( n - s )!} \int Ψ_{m} (p_{1}, \dots, p_{r}, z) k_{r, s} (p; q) X_{n} (q_{1}, \dots, q_{s}, z) d^{3 r} p d^{3 s} q d^{3 (n - s)} z .

Sum these finitely many occupied-sector pairings to define $C_{f}^{(3)}$ .

The cutoffs are bounded smooth profiles:

θ_{Λ} (p) = 1 (∣ p ∣ \leq Λ), θ_{Λ} (p) = 0 (∣ p ∣ \geq 2Λ),

c_{δ} (q) = 0 (∣ q ∣ \leq δ), c_{δ} (q) = 1 (∣ q ∣ \geq 2 δ),

and an edge profile $e_{ϵ} (s)$ , zero for $s - t_{0} \leq ϵ$ and one for $s - t_{0} \geq 2 ϵ$ . In each ordered summand of (eq:cubickernel), multiply by $\prod_{a} θ_{Λ} (p_{a}) c_{δ} (q)$ and by $e_{ϵ} (s)$ in the outer response integral. The metric Cauchy data remain fixed.

Displayed formulas in this passage

V@1#eq:cubicsector · line 3476
$C_{r, s} (Ψ_{m}, X_{n}) = (r 3) \frac{n ! m !}{( n - s )!} \int Ψ_{m} (p_{1}, \dots, p_{r}, z) k_{r, s} (p; q) X_{n} (q_{1}, \dots, q_{s}, z) d^{3 r} p d^{3 s} q d^{3 (n - s)} z .$
unlabeled · line 3459
$s_{n} (ψ_{n}) = p_{1}, \dots, p_{n} sup (j = 1 \prod^{n} ⟨ p_{j} ⟩^{30}) ∣ ψ_{n} ∣, s_{0} (z) = ∣ z ∣,$
V@1#eq:pnorm · line 3465
$I_{*} = \int_{R^{3}} ⟨ p ⟩^{- 60} d^{3} p, p (Ψ) = n \geq 0 \sum (1 + I_{*})^{n} (n + 3)^{3} s_{n} (Ψ_{n}) .$
unlabeled · line 3491
$θ_{Λ} (p) = 1 (∣ p ∣ \leq Λ), θ_{Λ} (p) = 0 (∣ p ∣ \geq 2Λ),$
unlabeled · line 3495
$c_{δ} (q) = 0 (∣ q ∣ \leq δ), c_{δ} (q) = 1 (∣ q ∣ \geq 2 δ),$

Fully Wick form and joint cutoff limit

Displayed formulas in this passage

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