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000123 | \subsection{Admissible Schwinger functions, Minkowski locality, and scattering closure}
000124 | 
000125 | \begin{definition}[Laboratory scattering sector]
000126 | \label{def:laboratory-scattering-sector}
000127 | Let $\mathcal U_{\mathrm{scat}}\subset M$ denote a low-curvature branch region on which
000128 | \[
000129 | \|g_{\mu\nu}-\eta_{\mu\nu}\|_{C^2(\mathcal U_{\mathrm{scat}})}\ll 1,
000130 | \qquad
000131 | \|\nabla\chigeo\|_{C^0(\mathcal U_{\mathrm{scat}})}\ll m_*\chigeozero,
000132 | \]
000133 | and
000134 | \[
000135 | \left\|
000136 | \Phi-\frac{1}{\sqrt2}\binom{0}{\vgeo}
000137 | \right\|_{C^0(\mathcal U_{\mathrm{scat}})}
000138 | \ll
000139 | \vgeo.
000140 | \]
000141 | All scattering statements below are formulated on $\mathcal U_{\mathrm{scat}}$.
000142 | \end{definition}
000143 | 
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}
000185 | 
000186 | \begin{theorem}[Admissible local Minkowski net and microcausality]
000187 | \label{thm:admissible-local-minkowski-net}
000188 | Let $\mathfrak A_{\mathrm{adm}}(\mathcal O)$ be the physical admissible local net of
000189 | \TFPTxref{def:physical-admissible-local-net}. Then:
000190 | \begin{enumerate}[label=\textup{(\roman*)}, leftmargin=1.8em, topsep=3pt, itemsep=2pt]
000191 |     \item if $\mathcal O_1\subset\mathcal O_2$, then
000192 |     \[
000193 |     \mathfrak A_{\mathrm{adm}}(\mathcal O_1)\subset \mathfrak A_{\mathrm{adm}}(\mathcal O_2);
000194 |     \]
000195 |     \item if $\mathcal O_1\perp\mathcal O_2$, then
000196 |     \[
000197 |     [A_1,A_2]=0
000198 |     \qquad
000199 |     (A_1\in\mathfrak A_{\mathrm{adm}}(\mathcal O_1),\ A_2\in\mathfrak A_{\mathrm{adm}}(\mathcal O_2)).
000200 |     \]
000201 | \end{enumerate}
000202 | Equivalently, the admissible branch reconstructs a causal Minkowski local net on
000203 | $\mathcal H_{\mathrm{adm}}$.
000204 | \end{theorem}
000205 | 
000206 | \begin{proof}
000207 | Isotony is immediate from the support definition of the unreduced local net
000208 | $\mathfrak A_M(\mathcal O)$. For locality, consider the Wick-rotated quadratic operators on the
000209 | low-curvature branch. Scalar and bosonic modes are normally hyperbolic, fermions are
000210 | Dirac-hyperbolic, and gauge fields become hyperbolic after covariant gauge fixing and BRST
000211 | reduction. Hence the advanced and retarded fundamental solutions $E^\pm$ exist and satisfy
000212 | \[
000213 | \operatorname{supp}(E^\pm)\subset J^\pm.
000214 | \]
000215 | Therefore the Pauli--Jordan propagator
000216 | \[
000217 | E:=E^+-E^-
000218 | \]
000219 | has support in the causal cone, so for spacelike separated test-function supports one has
000220 | \[
000221 | [\Phi_i(f),\Phi_j(g)]_{\mp}=0.
000222 | \]
000223 | Now let $A_1$ and $A_2$ be local operators preserving $\mathcal H_{\mathrm{adm}}$. Since they
000224 | commute on $\mathcal H_{\mathrm{OS}}$ for spacelike separated supports, their restrictions to the
000225 | invariant subspace $\mathcal H_{\mathrm{adm}}$ also commute:
000226 | \[
000227 | [A_1|_{\mathcal H_{\mathrm{adm}}},A_2|_{\mathcal H_{\mathrm{adm}}}]=0.
000228 | \]
000229 | Thus locality survives admissibility because admissibility is implemented as a state-space
000230 | restriction, not as a nonlocal compression defining the algebra itself.
000231 | \end{proof}
000232 | 
000233 | \begin{figure}[t]
000234 | \centering
000235 | \scriptsize
000236 | \begin{tikzpicture}[
000237 |     >=Latex,
000238 |     selector/.style={draw=blue!60!black, fill=blue!6, rounded corners, text width=2.65cm, minimum height=1.0cm, align=center},
000239 |     dyn/.style={draw=orange!75!black, fill=orange!8, rounded corners, text width=2.65cm, minimum height=1.0cm, align=center},
000240 |     result/.style={draw=green!50!black, fill=green!8, rounded corners, text width=2.9cm, minimum height=1.0cm, align=center},
000241 |     note/.style={font=\footnotesize\itshape, text=black!60, align=center}
000242 | ]
000243 | \node[selector] (padm) at (-4.8,1.15) {$\Padm$\\state-space selector};
000244 | \node[dyn] (zrel) at (-1.6,1.15) {$Z_{\mathrm{rel}},\ W_{\mathrm{rel}}$\\relative dynamics};
000245 | \node[dyn] (sch) at (1.6,1.15) {$\{S_n^T\}$\\Schwinger family};
000246 | \node[dyn] (os) at (4.8,1.15) {$(\mathcal H_{\mathrm{OS}},\mathfrak A_M)$\\OS/Wightman theory};
000247 | 
000248 | \node[selector] (hadm) at (-1.6,-1.2) {$\mathcal H_{\mathrm{adm}}=\operatorname{Ran}(\Padm)$\\physical sector};
000249 | \node[result] (aadm) at (2.6,-1.2) {$\mathfrak A_{\mathrm{adm}}(\mathcal O)$\\local net by restriction};
000250 | 
000251 | \draw[->, thick] (zrel) -- (sch);
000252 | \draw[->, thick] (sch) -- (os);
000253 | \draw[->, thick, dashed] (padm) -- (hadm);
000254 | \draw[->, thick] (os) -- (aadm);
000255 | \draw[->, thick] (hadm) -- (aadm);
000256 | 
000257 | \node[note] at (0,-2.35) {Locality is inherited by restriction to $\mathcal H_{\mathrm{adm}}$, not
000258 | by defining the net through a global compression $\Padm A \Padm$.};
000259 | \end{tikzpicture}
000260 | \caption{Selector versus dynamics on the admissible branch. The projector $\Padm$ selects the