Front Matter
# Notation
Every symbol used in the book, and where it is introduced.
Algebras and states
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| --- | --- | --- |
| $\mathfrak{A}(\mathcal{O})$ | The von Neumann algebra of observables localised in the spacetime region $\mathcal{O}$. Type III$_1$ in any reasonable QFT. | §3.1 |
| $\mathcal{O}_1\subset\subset\mathcal{O}_2$ | Strict inclusion: the closure of $\mathcal{O}_1$ lies in the interior of $\mathcal{O}_2$. | §4.1 |
| $\mathcal{N}$ | The intermediate type I factor supplied by the split property. This is the subject. | §4.2 |
| $\mathcal{N}'$ | The commutant of $\mathcal{N}$ — everything that is not this subject. | §4.2 |
| $\omega,\;\rho$ | A state on an algebra; $\rho$ the density matrix it induces on $\mathcal{N}$. | §4.2 |
| $\Omega$ | The vacuum vector; cyclic and separating by Reeh–Schlieder. | §3.2 |
| $\varepsilon,\;\xi$ | Split collar thickness; Ginzburg–Landau healing length. Identified in §4.5. | §4.3 |
Modular theory
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| --- | --- | --- |
| $S=J\Delta^{1/2}$ | The Tomita operator and its polar decomposition. | §5.1 |
| $\Delta$ | Modular operator. Generates the intrinsic flow. | §5.1 |
| $J$ | Modular conjugation. Fixes the subject/world cut. | §5.1 |
| $K=-\ln\Delta$ | Modular Hamiltonian. | §5.1 |
| $\sigma^{\omega}_{s}$ | Modular automorphism group; $s$ is phenomenal time. | §5.2 |
| $\beta_{\mathrm{eff}}$ | Inverse modular temperature; $t=\hbar\beta_{\mathrm{eff}}s$ converts to physical time. | §5.3 |
Valence and its ingredients
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| --- | --- | --- |
| $\mu_\Psi$ | Spectral measure of the state $\Psi$ with respect to the modular flow. | §6.1 |
| $\mathcal{A}$ | Coherence. Squared mass of the atoms of $\mu_\Psi$; the inverse participation ratio. | (6.1) |
| $\kappa(x)$ | Consonance kernel; a mollified Thomae function, weighted $(pq)^{-\sigma}$. | (7.2) |
| $\mathcal{C}$ | Consonance. The $\kappa$-weighted spectral self-overlap; $\mathcal{C}\ge\mathcal{A}$. | (7.1) |
| $q_{ab}$ | Overlap between replicas $a$ and $b$. | §8.2 |
| $P(q)$ | Parisi overlap distribution. | (8.1) |
| $\mathcal{D}$ | Frustration. $\mathrm{Var}_{P}(q)$; zero iff replica symmetry is unbroken. | (8.1) |
| $\mathfrak{V}$ | Valence. $\mathcal{C}\,(1-2\mathcal{D}/\mathcal{D}_{\max})$, taking values in $[-\mathcal{C},+\mathcal{C}]$. | (8.2) |
| $S_2$ | Rényi-2 entropy, $-\ln\mathrm{Tr}\rho^2$. Felt intensity. | (9.2) |
| $N_{\mathrm{eff}}$ | Effective participation number, $1/\mathcal{A}$. | §9.2 |
Geometry and topology
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| --- | --- | --- |
| $g^{\mathrm{B}}$ | Bures metric — the quantum Fisher information metric on state space. | (10.1) |
| $U_\gamma$ | Uhlmann holonomy around a loop $\gamma$; proposed carrier of qualitative character. | §10.4 |
| $\psi$ | Coarse-grained order parameter of the electromagnetic field, $\psi=\|\psi\|e^{i\theta}$. | §4.5 |
| $\pi_n(\mathcal{T})$ | Homotopy groups of the order-parameter target; classify topological pockets. | §4.4 |
| $\mathfrak{Q}$ | The qualia functor: $(\mathcal{O},\omega)\mapsto$ the phenomenal object of (2.2). | §2.2 |
Conventions
- Spacetime dimension $d=4$ unless stated. Signature is irrelevant to every argument here.
- $\hbar$ and $k_B$ are kept explicit wherever a number is computed, and set to 1 in formal passages.
- "Region" always means a bounded open region with non-empty causal complement, so that Reeh–Schlieder applies.
- "Subject" is reserved for the technical sense of Axiom 4.1 and never used loosely.
For agents
GET /api/library/spectral-panpsychism/notation.md