TFPT scientific source excerpt
Original repository path: documents/archive--paper-latex--tfpt-42/sections/022-section-closure-theorems-from-the-admissibility-complex.tex
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000270 | \[
000271 | [\Theta,P_\Theta]=0,
000272 | \qquad
000273 | \Theta\Padm=\Padm\Theta,
000274 | \]
000275 | the admissible bosonic kernel satisfies the Bernstein-positivity remark above and is
000276 | reflection-stable on the admissible sector, and the
000277 | admissible fermionic CAR kernel is reflection positive on that same sector. Then every
000278 | admissible observable $O$ supported in $\tau>0$ satisfies
000279 | \[
000280 | \langle \Theta O,O\rangle_{\mathrm{rel}}
000281 | =
000282 | \frac{1}{Z_{\mathrm{ref}}[0,0,0]}
000283 | \langle \Theta O,O\rangle_{\mathrm{adm}}
000284 | \ge 0.
000285 | \]
000286 | \end{lemma}
000287 | 
000288 | \begin{proof}
000289 | By definition,
000290 | \[
000291 | \langle \Theta O,O\rangle_{\mathrm{rel}}
000292 | =
000293 | \frac{1}{Z_{\mathrm{ref}}[0,0,0]}\,
000294 | \langle \Theta O,O\rangle_{\mathrm{adm}}.
000295 | \]
000296 | The denominator is a fixed positive normalizing constant, so the sign is determined entirely by the
000297 | admissible numerator.
000298 | 
000299 | Split $O$ into its bosonic and fermionic parts on the retained admissible sector. For the bosonic
000300 | part, the Bernstein representation of the admissible kernel writes the Euclidean covariance as a
000301 | positive superposition of reflection-stable heat kernels. Reflection across $\tau=0$ therefore sends
000302 | every positive-time bosonic observable to a positive quadratic form, so the bosonic contribution to
000303 | \[
000304 | \langle \Theta O,O\rangle_{\mathrm{adm}}
000305 | \]
000306 | is nonnegative.
000307 | 
000308 | For the fermionic part, the CAR reflection-positivity hypothesis on the admissible subspace implies
000309 | that the reflected two-point kernel is positive in the standard fermionic sense; equivalently, the
000310 | corresponding Pfaffian / determinant form on positive-time test spinors is nonnegative. Because the
000311 | bosonic and fermionic admissible factors commute inside the normalized expectation, their product
000312 | remains nonnegative. Dividing by the fixed reference normalization preserves that sign. Hence
000313 | \[
000314 | \langle \Theta O,O\rangle_{\mathrm{rel}}\ge 0.
000315 | \]
000316 | \end{proof}
000317 |