TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--04_qft_source/sections/008-section-relative-reflection-positivity-with-fermions.tex
Original file SHA-256: 898aa879cf6a3eb6464d3180274bc2fc028408aeec9676af70a862dad9c3f4dc
Inclusive original lines: 217-254
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000217 | \begin{theorem}[Tailored OS/CAR reconstruction on \texorpdfstring{$\Padm$}{Padm}]
000218 | \label{thm:tailored-os-car-reconstruction}
000219 | Let $\{S_n^T\}_{n\ge 1}$ be the admissible Schwinger family on the Euclidean continuation of
000220 | $\mathcal U_{\mathrm{scat}}$. If reflection positivity, temperedness, graded Euclidean covariance,
000221 | graded symmetry, and clustering hold on $\Padm$, then there exist:
000222 | \begin{enumerate}[label=(\roman*), leftmargin=1.8em, topsep=3pt, itemsep=2pt]
000223 |     \item a Hilbert space $\mathcal H_{\mathrm{OS}}$ with vacuum vector $\Omega$,
000224 |     \item a strongly continuous positive-energy semigroup $T_t=e^{-tH_{\mathrm{adm}}}$ with
000225 |     self-adjoint generator $H_{\mathrm{adm}}\ge 0$,
000226 |     \item a reconstructed graded local field net on $\Padm$ whose Euclidean boundary values are the
000227 |     original admissible Schwinger functions.
000228 | \end{enumerate}
000229 | Equivalently, the admissible Euclidean family reconstructs a Wightman/CAR theory on the admissible
000230 | sector.
000231 | \end{theorem}
000232 | 
000233 | \begin{proof}
000234 | Define the OS sesquilinear form by
000235 | \[
000236 | (F,G)_{\mathrm{OS}}:=\langle \Theta F,G\rangle_{\mathrm{rel}}
000237 | \]
000238 | for positive-time admissible observables. Reflection positivity from
000239 | \TFPTxref{thm:reflection-positivity-admissible} and the fermionic quotient-form analysis of
000240 | \Cref{app:os-fermions} show that this form is positive semidefinite on the full graded admissible
000241 | algebra. Quotienting by the null space and completing gives $\mathcal H_{\mathrm{OS}}$.
000242 | 
000243 | Positive Euclidean time translation preserves the null space and acts by contractions, hence defines
000244 | a strongly continuous contraction semigroup $T_t$ on $\mathcal H_{\mathrm{OS}}$. The dense domain of
000245 | positive-time polynomial observables gives the symmetric generator domain. By Hille--Yosida, the
000246 | closure of this generator is self-adjoint and nonnegative, so $T_t=e^{-tH_{\mathrm{adm}}}$ for a
000247 | self-adjoint $H_{\mathrm{adm}}\ge 0$.
000248 | 
000249 | The temperedness, covariance, graded symmetry, and clustering established in
000250 | \Cref{lem:tempered-schwinger-bounds,lem:graded-euclidean-covariance-padm} are precisely the
000251 | remaining OS/CAR reconstruction hypotheses. Therefore analytic continuation yields the reconstructed
000252 | Wightman/CAR family with vacuum $\Omega$. Because $\Theta$, $P_\Theta$, and $\Padm$ commute, the
000253 | entire reconstruction stays on the admissible sector $\Padm$.
000254 | \end{proof}