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Integrable derivatives yield scattering vectors

Statement

Let $D_{σ}$ and the direction-labelled packet families $Ψ_{t}^{η} (f) \in H$ be as in the massive scattering approximant contract, with $η \in {+ 1, - 1}$ , $f \in D_{σ}$ , and $t \geq 1$ . Assume that for each fixed $η, f$ , the map $t \mapsto Ψ_{t}^{η} (f)$ is continuously differentiable in Hilbert-space norm and $∥ \frac{d}{d t} Ψ_{t}^{η} (f) ∥ \leq C_{η, f} (1 + t)^{- 1 - ϵ_{η, f}}$ for constants $C_{η, f} \geq 0$ and $ϵ_{η, f} > 0$ . Then the norm limit $W_{η} f = \lim_{t \to \infty} Ψ_{t}^{η} (f)$ exists and $∥ Ψ_{t}^{η} (f) - W_{η} f ∥ \leq \frac{C_{η, f}}{ϵ_{η, f}} (1 + t)^{- ϵ_{η, f}} .$ In particular, each approximant family is uniformly bounded in norm. Write $(f, g)_{F, 1} : = ⟨ g, f ⟩_{F, 2}$ for the first-variable-linear pairing corresponding to the QFT Fock definition's second-variable-linear pairing. If $f \mapsto Ψ_{t}^{η} (f)$ is linear on $D_{σ}$ and its pairwise first-variable inner products converge to $(f, g)_{F, 1}$ , then $W_{η}$ is an isometry on $D_{σ}$ and extends uniquely to an isometry on its Hilbert-space closure. The derivative estimate alone does not assert this inner-product convergence or that the range is all of $H$ .

Statement references

Massive scattering packets and Cook approximant contract

Schwartz/distribution spaces and Hilbert Fock conventions

Facts & Assumptions

Given: A complete complex Hilbert space $H$ , a packet domain $D_{σ}$ , a sign $η \in {+ 1, - 1}$ , continuously differentiable maps $Ψ_{t}^{η} : D_{σ} \to H$ for $t \geq 1$ that are linear in their packet label, and constants $C_{η, f} \geq 0$ , $ϵ_{η, f} > 0$ satisfying the displayed derivative bound for every $f \in D_{σ}$ . For the isometry conclusion, also assume the stated pairwise inner-product limit.

[F1]

Massive scattering packets and Cook approximant contract: the candidate families are Hilbert-space-valued norm- $C^{1}$ functions on each time half-line; their one-particle limits, smooth packet labels, and symmetric or antisymmetric packet-domain inner product are part of the construction data. The derivative and Fock inner-product estimates remain hypotheses of this theorem. The contract assumes Countable Choice.

[F2]

Schwartz/distribution spaces and Hilbert Fock conventions: the bosonic and fermionic Fock spaces are square-summable direct sums of the symmetric and antisymmetric completed tensor sectors, with a second-variable-linear componentwise pairing. Its first-variable-linear counterpart is $(f, g)_{F, 1} = ⟨ g, f ⟩_{F, 2}$ .

[F3]

Fundamental theorem of calculus for Banach-valued continuous curves: under Countable Choice, for real $s < t$ , a real or complex Banach space $X$ , and a continuous curve $φ : [s, t] \to X$ that is differentiable on $(s, t)$ with derivative extending continuously to $[s, t]$ , the Bochner integral with respect to Lebesgue measure satisfies $φ (t) - φ (s) = \int_{s}^{t} φ^{'} (r) d r$ . The contract's Countable Choice assumption in [F1] supplies this hypothesis; a complex Hilbert space is a Banach space, and the contract's norm- $C^{1}$ condition supplies the required curve and derivative regularity on each compact interval.

[F4]

Bochner integral norm inequality: for a Bochner-integrable Banach-valued function $g$ on a measurable set $E$ , $∥ \int_{E} g ∥ \leq \int_{E} ∥ g ∥$ . The continuous derivative in [F3] is Bochner integrable on $[s, t]$ .

[A1]

For all $f, g \in D_{σ}$ , $\lim_{t \to \infty} (Ψ_{t}^{η} (f), Ψ_{t}^{η} (g)) = (f, g)_{F, 1}$ .

Proof

1.1F1F3F4given

Fix $1 \leq s \leq t$ . If $s = t$ the increment inequality is immediate. If $s < t$ , restrict $r \mapsto Ψ_{r}^{η} (f)$ to $[s, t]$ . By [F1] this is a continuous curve into the Banach space $H$ , differentiable on $(s, t)$ with derivative extending continuously to $[s, t]$ . The contract's Countable Choice premise satisfies the hypothesis of [F3], so the Banach-valued fundamental theorem gives $Ψ_{t}^{η} (f) - Ψ_{s}^{η} (f) = \int_{s}^{t} \partial_{r} Ψ_{r}^{η} (f) d r$ . The derivative is continuous and hence Bochner integrable on this compact interval; [F4] then gives $∥ Ψ_{t}^{η} (f) - Ψ_{s}^{η} (f) ∥ \leq \int_{s}^{t} ∥ \partial_{r} Ψ_{r}^{η} (f) ∥ d r$ .

2.1F1step 1.1algebra

Apply the derivative bound in [F1] and integrate the scalar power: $∥ Ψ_{t}^{η} (f) - Ψ_{s}^{η} (f) ∥ \leq \frac{C_{η, f}}{ϵ_{η, f}} ((1 + s)^{- ϵ_{η, f}} - (1 + t)^{- ϵ_{η, f}}) \leq \frac{C_{η, f}}{ϵ_{η, f}} (1 + s)^{- ϵ_{η, f}} .$ As $s \to \infty$ , the last expression tends to zero uniformly for $t \geq s$ . Thus $Ψ_{t}^{η} (f)$ is Cauchy as $t \to \infty$ ; completeness gives a limit $W_{η} f \in H$ . Taking $t \to \infty$ in the first inequality gives the claimed rate, and taking $s = 1$ gives $∥ Ψ_{t}^{η} (f) ∥ \leq ∥ Ψ_{1}^{η} (f) ∥ + C_{η, f} / ϵ_{η, f}$ for every $t \geq 1$ .

3.1F1step 2.1algebra

For $f, g \in D_{σ}$ , linearity of each $Ψ_{t}^{η}$ and norm convergence from 2.1 imply $W_{η} (a f + b g) = a W_{η} f + b W_{η} g$ for all scalars $a, b \in C$ : apply the triangle inequality to the finite linear combination and pass to the limit. Hence $W_{η}$ is linear on $D_{σ}$ .

4.1A1F2step 3.1algebra

By continuity of the Hilbert inner product and the assumed limit [A1], $(W_{η} f, W_{η} g) = \lim_{t \to \infty} (Ψ_{t}^{η} (f), Ψ_{t}^{η} (g)) = (f, g)_{F, 1}$ . Setting $g = f$ proves preservation of norm, so $W_{η}$ is an isometry on $D_{σ}$ .

5.1F1F2step 4.1algebra∎

If $(f_{n})$ in $D_{σ}$ converges in its Hilbert-space closure, then $(W_{η} f_{n})$ is Cauchy because $∥ W_{η} f_{n} - W_{η} f_{k} ∥ = ∥ f_{n} - f_{k} ∥$ . Completeness of $H$ defines a limit independent of the approximating sequence; the same equality passes to the limit. This is the unique isometric extension to $\overline{D_{σ}}$ . No surjectivity onto $H$ follows.

Integrable derivatives yield scattering vectors

Statement

Statement references

Facts & Assumptions

Proof

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