Tensor test algebra and positive tempered moment data

Use the Schwartz spaces and their seminorm topologies of Schwartz space and its seminorms and Schwartz topology and convergence. Put $S:=\mathcal{S}({\mathbb{R}}^4;\mathbb{C})$ and $S_n:=\mathcal{S}({\mathbb{R}}^{4n};\mathbb{C})$ for $n\ge 1$, and set $S_0:=\mathbb{C}$. A tempered distribution in $S_n^{\mathrm{\prime }}$ means a continuous complex-linear functional on $S_n$ as in Tempered distribution; its pairing with a test is bilinear.

For complex vector spaces $S$, let $S^{{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}n}$ be the algebraic tensor product, characterized by multilinear maps out of $S^n$, with $S^{{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}0}:=\mathbb{C}1$. Define the algebraic tensor test algebra $T(S):={⨁}_{n\ge 0}^{\mathrm{a}\mathrm{l}\mathrm{g}}{S}^{{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}n},$ where the direct sum consists of finite-support sequences. Its multiplication is concatenation of tensor factors, extended bilinearly, and its unit is $1$ in degree zero. Its conjugate-linear involution is specified on elementary tensors by $(f_1\otimes \cdots \otimes f_n{)}^{\ast }:=\overline{f_n}\otimes \cdots \otimes \overline{f_1},(c1{)}^{\ast }=\bar{c}1.$

For $f_j\in S$, the elementary tensor is evaluated against a distribution by the test function $J_n(f_1\otimes \cdots \otimes f_n)(x_1,\dots ,x_n):={\prod }_{j=1}^n{f}_j(x_j)\mathrm{.}$ This is in $S_n$: for multi-indices split into four-coordinate blocks $({\alpha }_j,{\beta }_j)$, its Schwartz seminorm is $p_{({\alpha }_j{)}_j,({\beta }_j{)}_j}(J_n(f_1\otimes \cdots \otimes f_n))={\prod }_{j=1}^n{p}_{{\alpha }_j{\beta }_j}(f_j)\mathrm{.}$ Thus $J_n$ is multilinear and continuous, and the displayed formula extends linearly to the algebraic tensor product. In degree zero set $J_0(c1):=c$.

By these definitions concatenation is associative with the degree-zero unit, and reversal together with complex conjugation is an involution satisfying $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$; thus $T(S)$ is a unital complex star algebra.

A positive tempered moment family is a sequence consisting of $W_0=1$ and $W_n\in S_n^{\mathrm{\prime }}$ for each $n\ge 1$ such that the multilinear maps $f_1,\dots ,f_n⟼⟨W_n,J_n(f_1\otimes \cdots \otimes f_n)⟩$ extend, degree by degree and then over finite sums, to a complex-linear functional $\omega :T(S)\to \mathbb{C}$ satisfying $\omega (1)=1,\omega (a^{\ast }a)\in [0,\mathrm{\infty })(a\in T(S))\mathrm{.}$ Equivalently, the family supplies an actually positive normalized functional on this algebra. Positivity is a hypothesis on the distributions; formal coefficient families do not qualify merely because they obey algebraic identities. No topology on $T(S)$, norm completion, C*-algebra, or existence claim for an interacting field is part of this definition.

Let $S=\mathcal{S}({\mathbb{R}}^4;\mathbb{C})$ with its Schwartz topology, let $H$ be a complex Hilbert space, and let $D\subseteq H$ be a dense complex linear subspace. Use the inner product linear in its first argument and conjugate-linear in its second, as in Real and complex inner-product spaces and their induced length. A weak operator-valued tempered distribution on the common domain $D$ is a complex-linear map $\mathrm{\Phi }:S⟶\mathrm{Lin}(D),$ where $\mathrm{Lin}(D)$ is the vector space of linear maps $D\to D$, such that for every $u,v\in D$ the scalar map $f⟼(\mathrm{\Phi }(f)u,v{)}_H$ is a continuous complex-linear functional on $S$, hence an element of $S^{\mathrm{\prime }}={\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^4)$ by Tempered distribution. In particular, every smeared map preserves the same domain $D$. This definition is weak in the matrix-coefficient sense; it does not define point values $\mathrm{\Phi }(x)$, operators on all of $H$, or self-adjoint closures.

Fix coordinates $x=(x^0,\mathbf{x})\in {\mathbb{R}}^4$ and the Minkowski matrix $\eta :=\mathrm{diag}(1,-1,-1,-1)\mathrm{.}$ The proper-orthochronous Lorentz group is $S{O}^{+}(1,3):=\{\mathrm{\Lambda }\in G{L}_4(\mathbb{R}):{\mathrm{\Lambda }}^{\mathsf{T}}\eta \mathrm{\Lambda }=\eta ,\det \mathrm{\Lambda }=1,{\mathrm{\Lambda }}^0{}_0\ge 1\},$ with its subspace topology from real $4\times 4$ matrices. The proper-orthochronous Poincaré group is the semidirect product $P_{+}^{\uparrow }:=S{O}^{+}(1,3)⋉{\mathbb{R}}^4,$ with multiplication $(\mathrm{\Lambda },a)(\mathrm{\Gamma },b)=(\mathrm{\Lambda }\mathrm{\Gamma },a+\mathrm{\Lambda }b)$, identity $e=(I,0)$, and affine action $gx=\mathrm{\Lambda }x+a$. Its topology is the product topology on the matrix and translation coordinates.

For $f\in S:=\mathcal{S}({\mathbb{R}}^4;\mathbb{C})$ define the pointwise pullback formula, viewed initially as a smooth function, $T_{g}f(x):=f(g^{-1}x)=f({\mathrm{\Lambda }}^{-1}(x-a))\mathrm{.}$ On smooth functions this formula obeys $T_g{T}_h=T_{gh}$ and $T_{e}f=f$, by the group law. This definition specifies the formula and conventions; it does not assert that $T_{g}f$ belongs to $S$. Translations have the convention $T_{(I,z)}f(x)=f(x-z)$.

With this signature, $p\cdot z:=p^0{z}^0-\mathbf{p}\cdot \mathbf{z}$, ${\bar{V}}_{+}:=\{p\in {\mathbb{R}}^4:p^0\ge \mathrm{\mid }\mathbf{p}\mathrm{\mid }\}$ is the closed forward cone, and $x-y$ is spacelike exactly when $(x^0-y^0{)}^2-\mathrm{\mid }\mathbf{x}-\mathbf{y}{\mathrm{\mid }}^2<0$.

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the Fourier transform suppliers. With $p\cdot z$ from Proper-orthochronous Poincaré action and test pullback, for $\psi \in \mathcal{S}({\mathbb{R}}^4)$ set ${\mathcal{F}}_M\psi (z):={\int }_{{\mathbb{R}}^4}{e}^{-ip\cdot z}\psi (p)d^{4}p\mathrm{.}$ To see that this is a continuous automorphism of Schwartz space, write ${\mathcal{F}}_{2\pi }$ for the Euclidean transform ${\mathcal{F}}_{2\pi }\psi (x)=\int e^{-2\pi ix\cdot p}\psi (p)d^{4}p$, as in Fourier transform of a tempered distribution. For $u,v\in {\mathbb{R}}^4$, write $u{\cdot }_{\mathrm{E}}v:={\sum }_{\mu =0}^3{u}^{\mu }{v}^{\mu }$. Since $p\cdot z=(\eta z){\cdot }_{\mathrm{E}}p$, the same defining integral gives ${\mathcal{F}}_M\psi (z)=({\mathcal{F}}_{2\pi }\psi )(\eta z\mathrm{/}(2\pi ))\mathrm{.}$ The map $z\mapsto \eta z\mathrm{/}(2\pi )$ is an invertible real-linear substitution, so output composition with it is a continuous automorphism of Schwartz space by Invertible linear substitutions preserve Schwartz space, and ${\mathcal{F}}_{2\pi }$ is a topological automorphism by Fourier transform is a topological automorphism of Schwartz space.

For $D\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^4)$ define its Minkowski Fourier transform by bilinear transposition, $⟨{\widehat{D}}_M,\psi ⟩:=⟨D,{\mathcal{F}}_M\psi ⟩(\psi \in \mathcal{S}({\mathbb{R}}^4))\mathrm{.}$ For a closed set $C\subseteq {\mathbb{R}}^4$, the statement $\mathrm{supp}{\widehat{D}}_M\subseteq C$ means $⟨{\widehat{D}}_M,\psi ⟩=0$ for every compactly supported smooth $\psi$ whose support is disjoint from $C$; the inclusion $C_c^{\mathrm{\infty }}({\mathbb{R}}^4)\hookrightarrow \mathcal{S}({\mathbb{R}}^4)$ is Test function inclusion in schwartz space is continuous. In particular, the forward-cone condition means $C={\bar{V}}_{+}$ with the definition in Proper-orthochronous Poincaré action and test pullback. No joint self-adjoint energy-momentum generators are included in this convention.

For every $g\in P_{+}^{\uparrow }$, the pullback $T_{g}f(x)=f(g^{-1}x)$ of Proper-orthochronous Poincaré action and test pullback maps $S=\mathcal{S}({\mathbb{R}}^4;\mathbb{C})$ continuously to itself. For each fixed $f\in S$, the map $g\mapsto T_{g}f$ from $P_{+}^{\uparrow }$ to $S$ is continuous. Consequently, ${\alpha }_g(f_1\otimes \cdots \otimes f_n):=T_g{f}_1\otimes \cdots \otimes T_g{f}_n$ extends algebraically to a unital star-automorphism of the algebraic tensor test algebra Positive tempered moments on the tensor test algebra, and ${\alpha }_g{\alpha }_h={\alpha }_{gh}$. No topology on the algebraic direct sum is asserted.

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and let $(W_n{)}_{n\ge 0}$ be a positive tempered moment family on $T(S)$ as in Positive tempered moments on the tensor test algebra. Write $\omega$ for its linear extension, and let $P_{+}^{\uparrow }$, $T_g$, and ${\alpha }_g$ be the group and pullbacks of Proper-orthochronous Poincaré action and test pullback and Proper-orthochronous Poincaré pullbacks act continuously on Schwartz space. Let $(H_{\omega },D_{\omega },{\pi }_{\omega },{\mathrm{\Omega }}_{\omega })$ 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 $\mathrm{\Phi }(f):={\pi }_{\omega }(X(f)){\mathrm{\mid }}_{D_{\omega }}$. Then $D_{\omega }$ is a common invariant dense domain, every $\mathrm{\Phi }(f)$ is closable with $\mathrm{\Phi }(\bar{f})=\mathrm{\Phi }(f{)}^{\ast }{\mathrm{\mid }}_{D_{\omega }}$, and $\mathrm{\Phi }$ 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: $(\mathrm{\Phi }(f_1)\cdots \mathrm{\Phi }(f_n){\mathrm{\Omega }}_{\omega },{\mathrm{\Omega }}_{\omega })=⟨W_n,J_n(f_1\otimes \cdots \otimes f_n)⟩\mathrm{.}$

The following additional conclusions are conditional on additional moment identities.

1. If $\omega \circ {\alpha }_g=\omega$ for every $g\in P_{+}^{\uparrow }$, then there is a strongly continuous unitary representation $g\mapsto U_g$ on $H_{\omega }$ fixing ${\mathrm{\Omega }}_{\omega }$ and satisfying $U_g\mathrm{\Phi }(f)U_g^{-1}{\mathrm{\mid }}_{D_{\omega }}=\mathrm{\Phi }(T_{g}f)(g\in P_{+}^{\uparrow },f\in S)\mathrm{.}$

2. Let $f,h\in \mathcal{D}({\mathbb{R}}^4)=C_c^{\mathrm{\infty }}({\mathbb{R}}^4;\mathbb{C})$ 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, $\mathrm{supp}f$ and $\mathrm{supp}h$ are the closures in ${\mathbb{R}}^4$ of their nonzero sets and hence compact. If these supports are spacelike separated, meaning $(x^0-y^0{)}^2-\mathrm{\mid }\mathbf{x}-\mathbf{y}{\mathrm{\mid }}^2<0$ for every $x\in \mathrm{supp}f$, $y\in \mathrm{supp}h$, and $\omega (b^{\ast }[X(f),X(h)]a)=0(a,b\in T(S)),$ then $[\mathrm{\Phi }(f),\mathrm{\Phi }(h)]=0$ on $D_{\omega }$. More generally, any polynomial relation $r\in T(S)$ satisfying $\omega (b^{\ast }ra)=0$ for all $a,b\in T(S)$ is represented by ${\pi }_{\omega }(r)=0$ on $D_{\omega }$; tested field equations are instances of this statement when their required identities are assumed.

3. For translations ${\tau }_z=(I,z)$ define $D_{b,a}(z):=\omega (b^{\ast }{\alpha }_{{\tau }_z}(a))(a,b\in T(S))\mathrm{.}$ If $\omega$ 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 $U_{{\tau }_z}$. If, in addition, $\mathrm{supp}{\widehat{D}}_{b,a,M}\subseteq {\bar{V}}_{+}$ for every $a,b$, with the Fourier convention of Minkowski Fourier transform and forward-cone support convention, then the reconstructed translation matrix coefficients $z⟼(U_{{\tau }_z}{\pi }_{\omega }(a){\mathrm{\Omega }}_{\omega },{\pi }_{\omega }(b){\mathrm{\Omega }}_{\omega })$ 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.