Euclidean Schwinger hierarchy in two dimensions

Let $E(2)=O(2)⋉{\mathbb{R}}^2$ be the group of Euclidean isometries $g(x)=Rx+a$ of ${\mathbb{R}}^2$, with $R\in O(2)$ and $a\in {\mathbb{R}}^2$. For each integer $n\ge 1$, let $\mathcal{S}(({\mathbb{R}}^2{)}^n)$ be the Schwartz space of smooth functions whose derivatives decrease faster than every polynomial, with the seminorms given by polynomially weighted derivatives. A scalar Euclidean Schwinger hierarchy is the data $S_0=1,S_n\in {\mathcal{S}}^{\mathrm{\prime }}(({\mathbb{R}}^2{)}^n)(n\ge 1),$ where each $S_n$ is a continuous complex-linear functional on that Schwartz space.

For $g\in E(2)$ define the continuous pullback $T_{g,n}$ by $(T_{g,n}f)(x_1,\dots ,x_n)=f(g^{-1}{x}_1,\dots ,g^{-1}{x}_n)\mathrm{.}$ For $\pi \in {\mathfrak{S}}_n$, define the block permutation pullback by $(T_{\pi ,n}f)(x_1,\dots ,x_n)=f(x_{{\pi }^{-1}(1)},\dots ,x_{{\pi }^{-1}(n)})\mathrm{.}$ The pullbacks above are continuous Schwartz-space automorphisms. The hierarchy is Euclidean-covariant and boson-symmetric when $S_n(T_{g,n}f)=S_n(f),S_n(T_{\pi ,n}f)=S_n(f)$ for every $n\ge 1$, $g\in E(2)$, $\pi \in {\mathfrak{S}}_n$, and $f\in \mathcal{S}(({\mathbb{R}}^2{)}^n)$. Here the pairing is bilinear in the test function, with the bilinear test-function convention.

This definition imposes no reflection positivity, cluster condition, growth bound, probability-measure representation, or reconstruction conclusion. In particular, it does not assert that $S_n$ is a moment $\mathbb{E}[{\varphi }^n]$ of a random distribution.

Schwartz space and its seminorms

Tempered distribution

Euclidean and block-permutation pullbacks preserve Schwartz space