Positive tempered moments on the tensor test algebra

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.