TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex
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000302 | \begin{definition}[Primitive Euclidean action]
000303 | For any declared spectral test profile $f$, the primitive Euclidean action is the relative
000304 | spectral seed
000305 | \[
000306 | S_{\mathrm{prim}}[\chiseed]
000307 | :=
000308 | \tr f\!\left(\frac{D_{\mathrm{rel}}}{\chiseed}\right)
000309 | -
000310 | \tr f\!\left(\frac{D_{\mathrm{ref}}}{\chiseed}\right)
000311 | +\frac{i\pi}{2}\Delta\eta_\Sigma.
000312 | \]
000313 | Gauge fixing, ghosts, and carrier-compatible bosonic fluctuations are introduced only after the
000314 | hard carrier theorem and the canonical admissibility projector are in place. Whenever convenient,
000315 | the compressed expression $\Padm A\Padm$ may be used as shorthand for matrix elements on the
000316 | physical admissible sector, but it is not the definition of a local operator net.
000317 | \end{definition}
000318 | 
000319 | \begin{definition}[Physical admissible local net]
000320 | \label{def:physical-admissible-local-net}
000321 | Let $\mathfrak A_E(\mathcal O)$ be the Euclidean source algebra generated by local field insertions
000322 | with test-function support in $\mathcal O$. After the later Osterwalder--Schrader reconstruction,
000323 | let $\mathfrak A_M(\mathcal O)$ denote the corresponding Minkowski local algebra and let
000324 | $\mathcal H_{\mathrm{OS}}$ be the reconstructed Hilbert space. Define the physical admissible
000325 | Hilbert space by
000326 | \[
000327 | \mathcal H_{\mathrm{adm}}:=\operatorname{Ran}(\Padm)\subset \mathcal H_{\mathrm{OS}}.
000328 | \]
000329 | Define the physical admissible local net by
000330 | \[
000331 | \mathfrak A_{\mathrm{adm}}(\mathcal O)
000332 | :=
000333 | \left\{
000334 | A|_{\mathcal H_{\mathrm{adm}}}\;:\;
000335 | A\in\mathfrak A_M(\mathcal O),\;
000336 | A\mathcal H_{\mathrm{adm}}\subset \mathcal H_{\mathrm{adm}},\;
000337 | A^\ast\mathcal H_{\mathrm{adm}}\subset \mathcal H_{\mathrm{adm}}
000338 | \right\}.
000339 | \]
000340 | Thus $\Padm$ acts as the selector of the physical state space, while the local net is defined by
000341 | restriction to the invariant admissible subspace rather than by compression with a global
000342 | projector.