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Tested positive moments reconstruct closable smeared fields

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and let $(W_{n})_{n \geq 0}$ be a positive tempered moment family on $T (S)$ as in Positive tempered moments on the tensor test algebra. Write $ω$ for its linear extension, and let $P_{+}^{↑}$ , $T_{g}$ , and $α_{g}$ be the group and pullbacks of Proper-orthochronous Poincaré action and test pullback and Proper-orthochronous Poincaré pullbacks act continuously on Schwartz space. Let $(H_{ω}, D_{ω}, π_{ω}, Ω_{ω})$ be the GNS construction of GNS Hilbert reconstruction with invariant operator domain for this functional, with the first-variable- linear Hilbert pairing.

For $f \in S$ , put $X (f) : = f$ in degree one and $Φ (f) : = π_{ω} (X (f)) ∣_{D_{ω}}$ . Then $D_{ω}$ is a common invariant dense domain, every $Φ (f)$ is closable with $Φ (\overline{f}) = Φ (f)^{*} ∣_{D_{ω}}$ , and $Φ$ is a weak operator-valued tempered distribution in the exact sense of Weak operator-valued tempered distribution on a common domain. Its vacuum moments are exactly the input: $(Φ (f_{1}) \dots Φ (f_{n}) Ω_{ω}, Ω_{ω}) = ⟨ W_{n}, J_{n} (f_{1} \otimes \dots \otimes f_{n}) ⟩ .$

The following additional conclusions are conditional on additional moment identities.

If $ω \circ α_{g} = ω$ for every $g \in P_{+}^{↑}$ , then there is a strongly continuous unitary representation $g \mapsto U_{g}$ on $H_{ω}$ fixing $Ω_{ω}$ and satisfying $U_{g} Φ (f) U_{g}^{- 1} ∣_{D_{ω}} = Φ (T_{g} f) (g \in P_{+}^{↑}, f \in S) .$
Let $f, h \in D (R^{4}) = C_{c}^{\infty} (R^{4}; C)$ be the complex test functions of Test function space d of an open set, viewed as elements of $S$ under the canonical inclusion of Test function inclusion in schwartz space is continuous. In this definition, $supp f$ and $supp h$ are the closures in $R^{4}$ of their nonzero sets and hence compact. If these supports are spacelike separated, meaning $(x^{0} - y^{0})^{2} - ∣ x - y ∣^{2} < 0$ for every $x \in supp f$ , $y \in supp h$ , and $ω (b^{*} [X (f), X (h)] a) = 0 (a, b \in T (S)),$ then $[Φ (f), Φ (h)] = 0$ on $D_{ω}$ . More generally, any polynomial relation $r \in T (S)$ satisfying $ω (b^{*} r a) = 0$ for all $a, b \in T (S)$ is represented by $π_{ω} (r) = 0$ on $D_{ω}$ ; tested field equations are instances of this statement when their required identities are assumed.
For translations $τ_{z} = (I, z)$ define $D_{b, a} (z) : = ω (b^{*} α_{τ_{z}} (a)) (a, b \in T (S)) .$ If $ω$ is translation-invariant, each $D_{b, a}$ is a bounded continuous function and hence a regular tempered distribution; applying the symmetry-implementation lemma to the translation subgroup gives strongly continuous unitaries $U_{τ_{z}}$ . If, in addition, $supp {\hat{D}}_{b, a, M} \subseteq {\overline{V}}_{+}$ for every $a, b$ , with the Fourier convention of Minkowski Fourier transform and forward-cone support convention, then the reconstructed translation matrix coefficients $z ⟼ (U_{τ_{z}} π_{ω} (a) Ω_{ω}, π_{ω} (b) Ω_{ω})$ have precisely the same Fourier support. This is a matrix-coefficient spectrum condition; it does not assert joint self-adjoint energy-momentum generators or their joint spectral measure.

This theorem reconstructs a weak field only from supplied, actually positive continuous moments. It does not construct an unspecified positive functional, complete a formal perturbation series, or establish the full Wightman reconstruction theorem with assumptions not listed here.

Facts & Assumptions

Given: Countable Choice ${A C}_{ω}$ ; a positive tempered moment family $(W_{n})$ ; the proper-orthochronous Poincaré pullback of Proper-orthochronous Poincaré action and test pullback; and the GNS inputs described below. The covariance, locality, and spectrum conclusions use only the additional identities stated alongside those conclusions.

[A1]

Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is assumed for the Hilbert completion in the GNS supplier and for its cited Fourier suppliers; no full Axiom of Choice is assumed.

[F1]

Positive tempered moments on the tensor test algebra: $T (S)$ consists of finite sums of algebraic tensor powers with concatenation product and reversal/conjugation involution; each $W_{n}$ is a continuous linear functional on $S (R^{4 n})$ ; the extension satisfies $ω (1) = 1$ and $ω (a^{*} a) \geq 0$ .

[F2]

GNS Hilbert reconstruction with invariant operator domain: under Countable Choice and these positive normalized algebraic-state hypotheses, the quotient has a Hilbert completion and a common invariant dense domain $D_{ω}$ , with $π_{ω} (a) [b] = [a b]$ , $Ω_{ω} = [1]$ , and $ω (a) = (π_{ω} (a) Ω_{ω}, Ω_{ω})$ ; each represented operator is closable and its adjoint restricts to $π_{ω} (a^{*})$ on the common domain.

[F3]

Positive-state Cauchy–Schwarz and left radical: $∣ ω (b^{*} a) ∣^{2} \leq ω (a^{*} a) ω (b^{*} b)$ , including zero-norm cases.

[F4]

Weak operator-valued tempered distribution on a common domain: weak operator-valued temperedness means continuity of each first-variable-linear matrix coefficient $f \mapsto (Φ (f) u, v)$ for $u, v \in D_{ω}$ .

[F5]

Proper-orthochronous Poincaré action and test pullback and Proper-orthochronous Poincaré pullbacks act continuously on Schwartz space: $T_{g} f (x) = f (g^{- 1} x)$ ; the pullbacks are continuous on $S$ and induce a group of unital star-automorphisms $α_{g}$ on the algebraic tensor algebra, with continuous coefficient dependence for each finite tensor polynomial.

[F6]

State-preserving symmetries implemented unitarily: for a state-preserving action of a topological group, continuity at the identity of every coefficient $g \mapsto ω (b^{*} α_{g} (a))$ yields a strongly continuous unitary implementation on the GNS Hilbert space, fixing the cyclic vector and implementing the action on the common domain.

[F7]

Minkowski Fourier transform and forward-cone support convention: ${\hat{D}}_{M}$ is defined by bilinear transposition of $F_{M} ψ (z) = \int e^{- i p \cdot z} ψ (p) d^{4} p$ ; support in ${\overline{V}}_{+}$ means vanishing on every compact test supported off that cone.

[F8]

Polynomial growth functions define tempered distributions: a locally integrable function bounded by a polynomial outside compact sets defines a regular tempered distribution.

[A2]

For covariance, assume $ω \circ α_{g} = ω$ for every $g \in P_{+}^{↑}$ .

[A3]

For tested locality, assume the displayed commutator moment identity for all $a, b \in T (S)$ and all $f, h \in D (R^{4})$ with the stated spacelike-separated supports.

[A4]

For tested relations, fix $r \in T (S)$ and assume $ω (b^{*} r a) = 0$ for every $a, b \in T (S)$ .

[A5]

For the matrix-coefficient spectrum conclusion, assume translation invariance $ω \circ α_{τ_{z}} = ω$ and $supp {\hat{D}}_{b, a, M} \subseteq {\overline{V}}_{+}$ for every $a, b \in T (S)$ .

Proof

1.1F1algebra

For $f_{1}, \dots, f_{n} \in S$ , the product test map $J_{n}$ satisfies the seminorm identity in [F1]: every joint Schwartz seminorm is the product of the corresponding four-dimensional seminorms. It is therefore continuous multilinear. Since each $W_{n}$ is continuous, fixing all but one factor and evaluating $W_{n}$ on $J_{n}$ gives a continuous linear functional of the remaining test.

1.2F1F2A1given

Apply [F2] to the positive normalized functional $ω$ on the unital star algebra $T (S)$ . Under the Countable Choice premise [A1] this produces $H_{ω}$ , the common invariant dense quotient domain $D_{ω}$ , the closable representation $π_{ω}$ , and the unit cyclic vector $Ω_{ω}$ .

1.3F2A3A4algebra

Assume [A4]. For the fixed $r$ , [F2] gives $(π_{ω} (r) [a], [b]) = ω (b^{*} r a) = 0$ for every $a, b$ . The classes $[b]$ are dense in $H_{ω}$ , so $π_{ω} (r) [a] = 0$ for each $[a] \in D_{ω}$ ; hence $π_{ω} (r) = 0$ on $D_{ω}$ . Taking $r = X (f) X (h) - X (h) X (f)$ and [A3] proves the stated locality conclusion. Any tested field equation is obtained by the same substitution of its own polynomial relation, without adding an unstated identity.

2.1F1F2F4step 1.1

For $a, b \in T (S)$ , the GNS quotient pairing gives $(Φ (f) [a], [b]) = ω (b^{*} X (f) a) .$ Because $a$ and $b$ are finite sums, the right-hand side is a finite sum of distributions $W_{n}$ evaluated on product tests with $f$ in one tensor slot. Step 1.1 makes each term continuous and linear in $f$ . Thus every matrix coefficient is tempered, and [F4] shows that $Φ$ is a weak operator-valued tempered distribution on the common domain.

2.2F1F2step 1.2algebra

The degree-one map $f \mapsto X (f)$ is complex-linear and left multiplication preserves $D_{ω}$ , so $Φ : S \to Lin (D_{ω})$ is linear and has the stated common invariant domain. By [F2], each $Φ (f)$ is closable and $Φ (\overline{f}) = Φ (f)^{*} ∣_{D_{ω}}$ . Repeated application of $π_{ω} (X (f_{j})) [a] = [X (f_{j}) a]$ gives $(Φ (f_{1}) \dots Φ (f_{n}) Ω_{ω}, Ω_{ω}) = ω (X (f_{1}) \dots X (f_{n})) = ⟨ W_{n}, J_{n} (f_{1} \otimes \dots \otimes f_{n}) ⟩,$ which is the asserted moment identity for $n \geq 1$ . For $n = 0$ , the empty product is $Ω_{ω}$ and $(Ω_{ω}, Ω_{ω}) = ω (1) = W_{0} = ⟨ W_{0}, J_{0} (1) ⟩$ .

2.3F1F5step 1.1algebra

For fixed $a, b$ , expand them as finite sums of tensor monomials. Under $α_{g}$ , each test factor varies continuously in Schwartz space by [F5]; the product maps $J_{n}$ are continuous by step 1.1, so each $W_{n}$ evaluation and hence $g \mapsto ω (b^{*} α_{g} (a))$ is continuous at the identity.

3.1F5F6A2step 1.2step 2.3

Under the additional invariance hypothesis [A2], the hypotheses of [F6] hold because step 2.3 provides the required coefficient continuity. The lemma supplies strongly continuous unitaries $U_{g}$ fixing $Ω_{ω}$ and implementing $α_{g}$ on $D_{ω}$ . Since $α_{g} (X (f)) = X (T_{g} f)$ , its covariance identity gives $U_{g} Φ (f) U_{g}^{- 1} ∣_{D_{ω}} = Φ (T_{g} f)$ .

3.2F3F5F8A5step 2.3algebra

Assume [A5]. For fixed $a, b$ , [F3] and translation invariance give $∣ D_{b, a} (z) ∣^{2} = ∣ ω (b^{*} α_{τ_{z}} (a)) ∣^{2} \leq ω (b^{*} b) ω (α_{τ_{z}} (a)^{*} α_{τ_{z}} (a)) = ω (b^{*} b) ω (a^{*} a),$ uniformly in $z$ . The coefficient is continuous in $z$ by the argument of step 2.3 and the translation subgroup's topology, so it is locally integrable. Its boundedness is polynomial growth of degree zero; [F8] therefore makes it a regular tempered distribution.

3.3F6A5step 2.3given

Under translation invariance in [A5], restrict the action to the translation subgroup $R^{4}$ . Step 2.3 gives continuity at the identity of each coefficient on this subgroup; [F6] therefore supplies strongly continuous unitaries $U_{τ_{z}}$ fixing $Ω_{ω}$ and implementing $α_{τ_{z}}$ on $D_{ω}$ .

4.1F2F7A5step 3.3algebra∎

The GNS state-recovery identity and step 3.3 give $D_{b, a} (z) = (π_{ω} (α_{τ_{z}} (a)) Ω_{ω}, π_{ω} (b) Ω_{ω}) = (U_{τ_{z}} π_{ω} (a) Ω_{ω}, π_{ω} (b) Ω_{ω}) .$ Equal tempered distributions have equal transposed Fourier transforms by [F7], so the assumed support of ${\hat{D}}_{b, a, M}$ is exactly the support of the reconstructed matrix coefficient. No generator or spectral-measure assertion is used.

Tested positive moments reconstruct closable smeared fields

Statement

Facts & Assumptions

Proof

Depends on

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