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000144 | \begin{definition}[Admissible Schwinger functions]
000145 | \label{def:admissible-schwinger-functions}
000146 | Let
000147 | \[
000148 | W_{\mathrm{rel}}[J,\eta,\bar\eta]:=\log Z_{\mathrm{rel}}[J,\eta,\bar\eta].
000149 | \]
000150 | For local source insertions supported in the Euclidean continuation of
000151 | $\mathcal U_{\mathrm{scat}}$, define the truncated admissible Schwinger distributions by
000152 | \[
000153 | S_n^T(x_1,\ldots,x_n)
000154 | :=
000155 | \left.
000156 | \frac{\delta^n W_{\mathrm{rel}}}{\delta\mathcal J(x_1)\cdots\delta\mathcal J(x_n)}
000157 | \right|_{\mathcal J=0},
000158 | \qquad
000159 | \mathcal J:=(J,\eta,\bar\eta).
000160 | \]
000161 | \end{definition}
000162 | 
000163 | \begin{theorem}[\textnormal{[M]} Full admissible Osterwalder--Schrader closure]
000164 | \label{thm:full-admissible-os-closure}
000165 | On the Euclidean continuation of $\mathcal U_{\mathrm{scat}}$, the family
000166 | $\{S_n^T\}_{n\ge 1}$ satisfies the local Osterwalder--Schrader axioms: regularity, Euclidean
000167 | covariance, reflection positivity, symmetry, and clustering. Hence there exists a reconstructed
000168 | Wightman theory
000169 | \[
000170 | (\mathcal H_{\mathrm{OS}},U(a,\Lambda),\Omega,\{\Phi_i\})
000171 | \]
000172 | whose Schwinger functions are the boundary values of the admissible Euclidean family
000173 | $\{S_n^T\}_{n\ge 1}$.
000174 | \end{theorem}
000175 | 
000176 | \begin{proof}
000177 | Regularity and moment bounds are recorded in
000178 | \Cref{lem:tempered-schwinger-bounds}, while graded Euclidean covariance and symmetry on $\Padm$ are
000179 | recorded in \Cref{lem:graded-euclidean-covariance-padm}. Reflection positivity is exactly
000180 | \TFPTxref{thm:reflection-positivity-admissible}, and clustering is the exponential clustering statement
000181 | of \TFPTxref{thm:nonperturbative-admissible-qft}. Therefore the admissible Schwinger family satisfies
000182 | the hypotheses of the tailored OS/CAR reconstruction theorem
000183 | \Cref{thm:tailored-os-car-reconstruction}, which yields the claimed reconstructed Wightman theory.
000184 | \end{proof}