TFPT scientific source excerpt
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000365 | \label{def:relative-generating-functional}
000366 | The relative Euclidean generating functional is normalized as
000367 | \[
000368 | Z_{\mathrm{rel}}[J,\eta,\bar\eta]
000369 | :=
000370 | \frac{Z_{\mathrm{adm}}[J,\eta,\bar\eta]}{Z_{\mathrm{ref}}[0,0,0]}.
000371 | \]
000372 | The reference sector is therefore a fixed normalizing denominator rather than a second fluctuating measure that would mix with admissible positivity.
000373 | \end{definition}
000374 | 
000375 | \begin{lemma}[Admissible reflection positivity]
000376 | \label{lem:admissible-reflection-positivity}
000377 | Assume
000378 | \[
000379 | [\Theta,P_\Theta]=0,
000380 | \qquad
000381 | \Theta\Padm=\Padm\Theta,
000382 | \]
000383 | the admissible bosonic kernel satisfies the Bernstein-positivity remark above and is
000384 | reflection-stable on the admissible sector, and the
000385 | admissible fermionic CAR kernel is reflection positive on that same sector. Then every
000386 | admissible observable $O$ supported in $\tau>0$ satisfies
000387 | \[
000388 | \langle \Theta O,O\rangle_{\mathrm{rel}}
000389 | =
000390 | \frac{1}{Z_{\mathrm{ref}}[0,0,0]}
000391 | \langle \Theta O,O\rangle_{\mathrm{adm}}
000392 | \ge 0.
000393 | \]
000394 | \end{lemma}