TFPT scientific source excerpt
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000339 | \section{Relative reflection positivity with fermions}
000340 | \label{app:os-fermions}
000341 | 
000342 | \subsection{Normalized quotient form}
000343 | 
000344 | The relative Euclidean theory used in the main closure section is not interpreted as a difference of measures. Instead it is organized as the normalized quotient
000345 | \[
000346 | Z_{\mathrm{rel}}[J,\eta,\bar\eta]
000347 | =
000348 | \frac{Z_{\mathrm{adm}}[J,\eta,\bar\eta]}{Z_{\mathrm{ref}}[0,0,0]}.
000349 | \]
000350 | Because the denominator is source independent on the admissible sector, the reference theory acts only as a fixed normalizer and does not interfere with reflection positivity.
000351 | 
000352 | \subsection{Bosonic and fermionic positivity on \texorpdfstring{$\Padm$}{Padm}}
000353 | 
000354 | The commutation relations
000355 | \[
000356 | [\Theta,P_\Theta]=0,
000357 | \qquad
000358 | \Theta\Padm=\Padm\Theta
000359 | \]
000360 | ensure that reflection preserves the admissible subspace. The bosonic part then follows the standard Markov / reflection-stable Osterwalder--Schrader argument on admissible observables. For fermions, the admissible CAR kernel may be written as a reflected quadratic form; equivalently one may express the same condition as positivity of the associated Pfaffian structure on the admissible sector. Hence for every admissible observable $O$ supported in $\tau>0$,
000361 | \[
000362 | \langle \Theta O,O\rangle_{\mathrm{rel}}
000363 | =
000364 | \frac{1}{Z_{\mathrm{ref}}[0,0,0]}
000365 | \langle \Theta O,O\rangle_{\mathrm{adm}}
000366 | \ge 0.
000367 | \]
000368 | This appendix section is the mechanism behind \Cref{lem:admissible-reflection-positivity,thm:reflection-positivity-admissible,thm:lorentzian-reconstruction}; it is recorded here so that the main-text reconstruction theorem is not supported only by a one-line axiom call.