TFPT scientific source excerpt
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000217 | \begin{definition}[Carrier polarization on the finite block]
000218 | \label{def:carrier-polarization-involution}
000219 | Let $\mathcal H_{\mathrm{car}}$ denote the finite-dimensional carrier block selected after
000220 | primitive admissibility compression. Define
000221 | \[
000222 | \epscar:=\iC|_{\mathcal H_{\mathrm{car}}},
000223 | \qquad
000224 | E_-:=\ker(\epscar+1),
000225 | \qquad
000226 | E_+:=\ker(\epscar-1).
000227 | \]
000228 | Then the finite carrier decomposition is
000229 | \[
000230 | E=E_-\oplus E_+.
000231 | \]
000232 | \end{definition}
000233 | 
000234 | \begin{remark}[Forward reference to the full admissibility complex]
000235 | \label{rem:forward-Padm}
000236 | The full admissibility projector is obtained by upgrading $\Pprim$ with the color center
000237 | projector $P_{\mathrm{sing}}$ and the determinant involution projector $P_\Theta$, which are
000238 | available only after the carrier theorem of \Cref{sec:minimal-carrier-proof} identifies the
000239 | color factor $\mathcal H_c$ and after the compact bosonic index of \Cref{sec:closure-theorems}
000240 | fixes the determinant involution $Q_\Theta$. The full complex and the explicit upgrade are
000241 | stated in \Cref{def:full-admissibility-complex,thm:padm-upgrade}; nothing in the present
000242 | primitive subsection depends on those inputs.
000243 | \end{remark}
000244 | 
000245 | \begin{definition}[Primitive Euclidean action]
000246 | For any declared spectral test profile $f$, the primitive Euclidean action is the relative
000247 | spectral seed
000248 | \[
000249 | S_{\mathrm{prim}}[\chiseed]
000250 | :=
000251 | \tr f\!\left(\frac{D_{\mathrm{rel}}}{\chiseed}\right)
000252 | -
000253 | \tr f\!\left(\frac{D_{\mathrm{ref}}}{\chiseed}\right)
000254 | +\frac{i\pi}{2}\Delta\eta_\Sigma.
000255 | \]
000256 | Gauge fixing, ghosts, and carrier-compatible bosonic fluctuations are introduced only after the
000257 | hard carrier theorem and the canonical admissibility projector are in place. Whenever convenient,
000258 | the compressed expression $\Padm A\Padm$ may be used as shorthand for matrix elements on the
000259 | physical admissible sector, but it is not the definition of a local operator net.
000260 | \end{definition}
000261 | 
000262 | \begin{definition}[Physical admissible local net]
000263 | \label{def:physical-admissible-local-net}
000264 | Let $\mathfrak A_E(\mathcal O)$ be the Euclidean source algebra generated by local field insertions
000265 | with test-function support in $\mathcal O$. After the later Osterwalder--Schrader reconstruction,
000266 | let $\mathfrak A_M(\mathcal O)$ denote the corresponding Minkowski local algebra and let
000267 | $\mathcal H_{\mathrm{OS}}$ be the reconstructed Hilbert space. Define the physical admissible
000268 | Hilbert space by
000269 | \[
000270 | \mathcal H_{\mathrm{adm}}:=\operatorname{Ran}(\Padm)\subset \mathcal H_{\mathrm{OS}}.
000271 | \]
000272 | Define the physical admissible local net by
000273 | \[
000274 | \mathfrak A_{\mathrm{adm}}(\mathcal O)
000275 | :=
000276 | \left\{
000277 | A|_{\mathcal H_{\mathrm{adm}}}\;:\;
000278 | A\in\mathfrak A_M(\mathcal O),\;
000279 | A\mathcal H_{\mathrm{adm}}\subset \mathcal H_{\mathrm{adm}},\;
000280 | A^\ast\mathcal H_{\mathrm{adm}}\subset \mathcal H_{\mathrm{adm}}
000281 | \right\}.
000282 | \]
000283 | Thus $\Padm$ acts as the selector of the physical state space, while the local net is defined by
000284 | restriction to the invariant admissible subspace rather than by compression with a global
000285 | projector.
000286 | \end{definition}
000287 | 
000288 | \begin{remark}[Bernstein realizations as analytic choices]
000289 | \label{ass:bernstein-positivity}
000290 | Whenever a Bernstein representation of a spectral test profile is used, it is treated only as an