TFPT scientific source excerpt
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000143 | \subsection{Normalized quotient form}
000144 | 
000145 | 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
000146 | \[
000147 | Z_{\mathrm{rel}}[J,\eta,\bar\eta]
000148 | =
000149 | \frac{Z_{\mathrm{adm}}[J,\eta,\bar\eta]}{Z_{\mathrm{ref}}[0,0,0]}.
000150 | \]
000151 | 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.
000152 | 
000153 | \subsection{Bosonic and fermionic positivity on \texorpdfstring{$\Padm$}{Padm}}
000154 | 
000155 | The commutation relations
000156 | \[
000157 | [\Theta,P_\Theta]=0,
000158 | \qquad
000159 | \Theta\Padm=\Padm\Theta
000160 | \]
000161 | 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 $\tau>0$,
000162 | \[
000163 | \langle \Theta O,O\rangle_{\mathrm{rel}}
000164 | =
000165 | \frac{1}{Z_{\mathrm{ref}}[0,0,0]}
000166 | \langle \Theta O,O\rangle_{\mathrm{adm}}
000167 | \ge 0.
000168 | \]
000169 | This appendix section is the mechanism behind \TFPTxref{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.