Euclidean Schwinger hierarchies and controlled limit data

Fix an integer $d\ge 1$, write Euclidean space as $E_d=\mathbb{R}\times {\mathbb{R}}^{d-1}$ with coordinates $x=(t,\mathbf{x})$, and put $S_d=\mathcal{S}(E_d;\mathbb{C})$ and $S_{d,n}=\mathcal{S}(E_d^n;\mathbb{C})$ for $n\ge 1$, with $S_{d,0}=\mathbb{C}$. For $d=1$, ${\mathbb{R}}^0$ is the one-point vector space with unit measure. A Schwinger hierarchy is a sequence $S=(S_n{)}_{n\ge 0}$ with $S_0=1$ and each $S_n\in S_{d,n}^{\mathrm{\prime }}$; its moments on elementary tests are $⟨S_n,f_1\otimes \cdots \otimes f_n⟩$. The associated unital algebraic test algebra is $T_d={⨁}_{n\ge 0}^{\mathrm{a}\mathrm{l}\mathrm{g}}{S}_d^{{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}n}$ with concatenation multiplication, unit $1\in S_{d,0}=\mathbb{C}$, and involution $(f_1\otimes \cdots \otimes f_n{)}^{\ast }={\bar{f}}_n\otimes \cdots \otimes {\bar{f}}_1$ and $(c1{)}^{\ast }=\bar{c}1$. The moments define ${\omega }_S:T_d\to \mathbb{C}$ by linear extension. At $d=4$ this is the same four-dimensional algebra and moment convention used by the positive-tempered-moment definition.

Ordinary moment positivity means ${\omega }_S(1)=1$ and ${\omega }_S(a^{\ast }a)\ge 0$ for every $a\in T_d$. It is distinct from reflection positivity. Let $\vartheta (t,\mathbf{x})=(-t,\mathbf{x})$ and $E_{d,+}=\{(t,\mathbf{x}):t>0\}$. The positive-time polynomial algebra $T_{d,+}$ is the unital star subalgebra generated by tests in $C_c^{\mathrm{\infty }}(E_{d,+};\mathbb{C})$. Define the antilinear reflection $\mathrm{\Theta }$ by $\mathrm{\Theta }f=\overline{f\circ \vartheta }$ on degree one, $\mathrm{\Theta }(c1)=\bar{c}1$, and $\mathrm{\Theta }(ab)=\mathrm{\Theta }(b)\mathrm{\Theta }(a)$. Reflection positivity is ${\omega }_S(\mathrm{\Theta }aa)\in [0,\mathrm{\infty })$ for every $a\in T_{d,+}$. For the finite-site version, fix a finite set $J$, a permutation $\rho :J\to J$ with ${\rho }^2=\mathrm{id}$, and a specified subset $J_{+}\subseteq J$. Let $A_J=\mathbb{C}[z_j:j\in J]$ be the commutative polynomial algebra with $z_j^{\ast }=z_j$ and conjugation of coefficients, and let $A_{J,+}$ be its unital subalgebra generated by $z_j$ for $j\in J_{+}$. Define ${\mathrm{\Theta }}_J$ by conjugating coefficients and replacing every $z_j$ by $z_{\rho (j)}$. For a specified complex-linear functional $\omega :A_J\to \mathbb{C}$ with $\omega (1)=1$, ordinary positivity means $\omega (a^{\ast }a)\in [0,\mathrm{\infty })$ for all $a\in A_J$, and reflection positivity means $\omega (({\mathrm{\Theta }}_{J}a)a)\in [0,\mathrm{\infty })$ for all $a\in A_{J,+}$. One may also use $Q=A_J\mathrm{/}I$ for an ideal $I\subseteq A_J$ invariant under both $\ast$ and ${\mathrm{\Theta }}_J$, provided $I\subseteq \ker \omega :=\{a\in A_J:\omega (a)=0\}$. Writing $[a]=a+I$, define $[a{]}^{\ast }=[a^{\ast }]$, ${\bar{\mathrm{\Theta }}}_J[a]=[{\mathrm{\Theta }}_{J}a]$, the positive-site algebra $Q_{+}=\{[a]:a\in A_{J,+}\}$, and the normalized complex-linear functional $\bar{\omega }([a])=\omega (a)$. The kernel condition makes this value independent of the representative: if $a-b\in I$, then $\omega (a)-\omega (b)=\omega (a-b)=0$. On $Q$, ordinary positivity means $\bar{\omega }(u^{\ast }u)\in [0,\mathrm{\infty })$ for every $u\in Q$, and reflection positivity means $\bar{\omega }(({\bar{\mathrm{\Theta }}}_{J}u)u)\in [0,\mathrm{\infty })$ for every $u\in Q_{+}$. Equivalently, one may specify a normalized complex-linear functional $\bar{\omega }$ on $Q$ first and set $\omega (a)=\bar{\omega }([a])$; this automatically satisfies the kernel condition. An exchanging pair of sites is $J=\{-,+\}$ with $\rho (-)=+$, $\rho (+)=-$, and $J_{+}=\{+\}$.

A regulated hierarchy is a family $S^{\lambda }=(S_n^{\lambda }{)}_{n\ge 0}$ indexed by a nonempty directed set $(\mathrm{\Lambda },⪯)$, where directed means every two indices have a common upper bound. A distributional continuum limit is a hierarchy $S$ such that for every fixed $n$ and every $\mathrm{\Phi }\in S_{d,n}$, $⟨S_n^{\lambda },\mathrm{\Phi }⟩\to ⟨S_n,\mathrm{\Phi }⟩$ as $\lambda$ advances in $\mathrm{\Lambda }$. A limit contract records these pairings, the regulator family, and any additional uniform estimates; pointwise-in-$n$ convergence by itself asserts no uniform bound as $n$ varies.

For explicit comparison, define the Schwartz seminorm $P_k(\mathrm{\Phi })={\max }_{\mathrm{\mid }\alpha \mathrm{\mid },\mathrm{\mid }\beta \mathrm{\mid }\le k}{\sup }_{x\in E_d^n}\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }\mathrm{\Phi }(x)\mathrm{\mid }$ on $S_{d,n}$. A per-order uniform continuity record is a choice, for each fixed $n$, of $r_n\in {\mathbb{N}}_0$ and $C_n\ge 0$ such that $\mathrm{\mid }⟨S_n^{\lambda },\mathrm{\Phi }⟩\mathrm{\mid }\le C_n{P}_{r_n}(\mathrm{\Phi })$ for every $\lambda \in \mathrm{\Lambda }$ and $\mathrm{\Phi }\in S_{d,n}$. Separately, a uniform-order bound on the limit hierarchy is the explicit cross-order condition that there exist fixed $s\in \mathbb{N}$ and $A,B,q>0$ with $\mathrm{\mid }⟨S_n,\mathrm{\Phi }⟩\mathrm{\mid }\le A{B}^n(n!{)}^q{P}_{sn}(\mathrm{\Phi })$ for every $n\ge 1$ and $\mathrm{\Phi }\in S_{d,n}$. This bound is recorded separately and is not inferred from distributional convergence or asserted equivalent to a corrected Osterwalder–Schrader regularity hypothesis.

For $\gamma (x)=Rx+a$ with $R\in O(d)$ and $a\in E_d$, define $(U_{\gamma }\mathrm{\Phi })(x_1,\dots ,x_n)=\mathrm{\Phi }({\gamma }^{-1}{x}_1,\dots ,{\gamma }^{-1}{x}_n)$. Euclidean invariance means $⟨S_n,U_{\gamma }\mathrm{\Phi }⟩=⟨S_n,\mathrm{\Phi }⟩$ for every such $\gamma$ and $\mathrm{\Phi }\in S_{d,n}$. Permutation symmetry means invariance of the pairing under each permutation of its $n$ argument blocks. Clustering means that for every $n,m\ge 1$, $f\in S_{d,n}$ and $g\in S_{d,m}$,

where $e_0=(1,\mathbf{0})$ and $({\tau }_{a{e}_0}g)(x_1,\dots ,x_m)=g(x_1-a{e}_0,\dots ,x_m-a{e}_0)$. This contract definition asserts neither existence of a regulated limit nor any of these additional properties.

Positive tempered moments on the tensor test algebra

Schwartz space and its seminorms

Tempered distribution

The reflection form, the ordinary positive-moment form, and the clustering limit are three distinct interfaces. Fixed-order distributional convergence can pass a reflection quadratic form; it does not by itself establish clustering or a uniform-order growth bound. The locally defined uniform-order bound above is not asserted equivalent to an Osterwalder–Schrader hypothesis. Any later reconstruction claim must state the exact corrected theorem version and prove its test-space and order-growth hypotheses. This definition does not invoke a reconstruction theorem or assert construction of a null-quotient Hilbert space, positive-time translation semigroup, self-adjoint generator, analytic continuation, locality, or spectrum condition.