TFPT scientific source excerpt
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000171 | \subsection{OS and CAR reconstruction on \texorpdfstring{$\Padm$}{Padm}}
000172 | 
000173 | \begin{theorem}[Tailored OS/CAR reconstruction on \texorpdfstring{$\Padm$}{Padm}]
000174 | \label{thm:tailored-os-car-reconstruction}
000175 | Let $\{S_n^T\}_{n\ge 1}$ be the admissible Schwinger family on the Euclidean continuation of
000176 | $\mathcal U_{\mathrm{scat}}$. If reflection positivity, temperedness, graded Euclidean covariance,
000177 | graded symmetry, and clustering hold on $\Padm$, then there exist:
000178 | \begin{enumerate}[label=(\roman*), leftmargin=1.8em, topsep=3pt, itemsep=2pt]
000179 |     \item a Hilbert space $\mathcal H_{\mathrm{OS}}$ with vacuum vector $\Omega$,
000180 |     \item a strongly continuous positive-energy semigroup $T_t=e^{-tH_{\mathrm{adm}}}$ with
000181 |     self-adjoint generator $H_{\mathrm{adm}}\ge 0$,
000182 |     \item a reconstructed graded local field net on $\Padm$ whose Euclidean boundary values are the
000183 |     original admissible Schwinger functions.
000184 | \end{enumerate}
000185 | Equivalently, the admissible Euclidean family reconstructs a Wightman/CAR theory on the admissible
000186 | sector.
000187 | \end{theorem}
000188 | 
000189 | \begin{proof}
000190 | Define the OS sesquilinear form by
000191 | \[
000192 | (F,G)_{\mathrm{OS}}:=\langle \Theta F,G\rangle_{\mathrm{rel}}
000193 | \]
000194 | for positive-time admissible observables. Reflection positivity from
000195 | \TFPTxref{thm:reflection-positivity-admissible} and the fermionic quotient-form analysis of
000196 | \Cref{app:os-fermions} show that this form is positive semidefinite on the full graded admissible
000197 | algebra. Quotienting by the null space and completing gives $\mathcal H_{\mathrm{OS}}$.
000198 | 
000199 | Positive Euclidean time translation preserves the null space and acts by contractions, hence defines
000200 | a strongly continuous contraction semigroup $T_t$ on $\mathcal H_{\mathrm{OS}}$. The dense domain of
000201 | positive-time polynomial observables gives the symmetric generator domain. By Hille--Yosida, the
000202 | closure of this generator is self-adjoint and nonnegative, so $T_t=e^{-tH_{\mathrm{adm}}}$ for a
000203 | self-adjoint $H_{\mathrm{adm}}\ge 0$.
000204 | 
000205 | The temperedness, covariance, graded symmetry, and clustering established in
000206 | \Cref{lem:tempered-schwinger-bounds,lem:graded-euclidean-covariance-padm} are precisely the
000207 | remaining OS/CAR reconstruction hypotheses. Therefore analytic continuation yields the reconstructed
000208 | Wightman/CAR family with vacuum $\Omega$. Because $\Theta$, $P_\Theta$, and $\Padm$ commute, the
000209 | entire reconstruction stays on the admissible sector $\Padm$.
000210 | \end{proof}