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Find scoped QFT concepts and the volume pages that define them, with ambiguity and preparation boundaries kept visible.

This index is a concise projection, not an independent explanation. Each record keeps the scope needed for use and links to its source page.

Coverage notes
  • Each entry identifies a concept and the page that develops its primary definition; it is not a standalone dictionary definition or alias list.
  • The projection displays the owner question and bounded scope as orientation and never manufactures a dictionary definition from a page heading.

Coverage note: records included through 2026-08-12

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  • Term

    Analytic Continuation: Uniqueness and Failure Modes

    Which replica continuations are nonunique, nonuniform, branch-sensitive, or obstructed by phase transitions, and how should uncertainty be reported?

    Principal question
    Which replica continuations are nonunique, nonuniform, branch-sensitive, or obstructed by phase transitions, and how should uncertainty be reported?
    Boundary
    It does not dismiss every continuation as impossible or hide a selected analytic ansatz behind an exact-looking entropy value.
    Scope
    This page owns uniqueness hypotheses, branch points, replica-symmetry breaking, saddle exchange, finite-n sampling, and ill-conditioned n → 1 extrapolation.
    Assumptions
    Rényi Entropies and Replica Analytic Continuation
  • Term

    Anatomy and Severity of a Sign Problem

    When does a negative or complex weight create a sign problem, how is its severity quantified, and why are variance, overlap, and correctness distinct?

    Principal question
    When does a negative or complex weight create a sign problem, how is its severity quantified, and why are variance, overlap, and correctness distinct?
    Boundary
    Generic probability belongs to Volume 1, complexity qualifications to the next page, dense-matter physics to Volume 11, and method-specific responses to later pages.
    Scope
    Phase reweighting identity, average sign or phase, free-energy scaling, overlap problem, cancellations, observable dependence, volume and temperature severity, basis dependence as a bridge, effective sample complexity, and distinction from generic instability.
    Assumptions
    Chemical Potential on the Euclidean Lattice
  • Term

    Anomaly and Generalized-Symmetry Constraints on Infrared Phases

    Which infrared vacuum or phase options are ruled in or out by anomaly and generalized-symmetry data without solving the dynamics?

    Principal question
    Which infrared vacuum or phase options are ruled in or out by anomaly and generalized-symmetry data without solving the dynamics?
    Boundary
    Definitions and anomaly technology remain in Volume 3, QCD realization in Volume 6, and theorem-level status in Volume 16.
    Scope
    A concrete constraint workflow from ultraviolet symmetry and anomaly data to the allowed infrared alternatives: symmetry preservation, gaplessness, spontaneous breaking, or topological order.
    Assumptions
    't Hooft Anomaly Matching; Breaking Higher-Form Symmetry and Diagnosing Phases; Theta Terms, Periodicity, and Vacuum Sectors
  • Term

    Anti-de Sitter Geometry and the Conformal Boundary

    How do AdS curvature, causal structure, isometries, conformal compactification, and its timelike boundary set the geometric stage for a holographic dictionary?

    Principal question
    How do AdS curvature, causal structure, isometries, conformal compactification, and its timelike boundary set the geometric stage for a holographic dictionary?
    Boundary
    Volume I owns differential-geometric machinery, Volume IX owns conformal symmetry, Volume XIV owns generic timelike-boundary QFT, and this chapter owns the AdS deployment.
    Scope
    The physical AdS geometry ledger in global and embedded descriptions, including radius, boundary conformal class, causal travel time, volume element, and sign conventions.
    Assumptions
    Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields
  • Term

    Approximate Bulk Locality from Spectral and Mellin Data

    How do a large gap, Mellin analyticity, polynomial boundedness, and controlled contact ambiguities support only approximate bulk locality?

    Principal question
    How do a large gap, Mellin analyticity, polynomial boundedness, and controlled contact ambiguities support only approximate bulk locality?
    Boundary
    Volume IX owns Mellin correlators and CFT-side locality tests; Chapter 8 owns detailed pole, Regge, bulk-point, and dispersive analyses; Chapter 12 owns operator locality.
    Scope
    A preliminary locality inference connecting imported spectral and Mellin data to an AdS derivative expansion, with finite-gap, finite-N, and Regge-domain qualifications.
    Assumptions
    Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria; Bulk Interaction Scaling and Effective Cutoffs; Mellin-Space CFT Correlators
  • Term

    Background Fields versus Dynamical Gauging

    Which new data and sums distinguish probing a symmetry with a background field from gauging that symmetry to construct a new theory?

    Principal question
    Which new data and sums distinguish probing a symmetry with a background field from gauging that symmetry to construct a new theory?
    Boundary
    Specific Yang-Mills or matter dynamics belong to Gauge Theories; lattice implementations belong to Lattice and Hamiltonian QFT.
    Scope
    The conceptual transition from fixed background to dynamical integration or bundle summation, including measure, action, counterterm, global-sector, and gauge-redundancy data.
    Assumptions
    Coupling to Background Gauge Fields and Bundles; Symmetry, Gauge Redundancy, and Duality
  • Term

    Basis Dependence, Computational Complexity, and No Universal Cure

    Why is a sign problem representation dependent, what do stoquasticity and complexity results actually constrain, and why is there no universal cure?

    Principal question
    Why is a sign problem representation dependent, what do stoquasticity and complexity results actually constrain, and why is there no universal cure?
    Boundary
    Computational-complexity foundations belong to Volume 13, rigorous complexity theorems to Volume 16, specific dual variables to a later page, and algorithm capability status to Research.
    Scope
    Basis and variable dependence, unitary or dual transformations, stoquastic and nonstoquastic distinctions at the physics level, complexity-hardness statements with hypotheses, local versus global transformations, model-family versus instance claims, and no-go overclaim prevention.
    Assumptions
    Anatomy and Severity of a Sign Problem
  • Term

    Bilinear and Hermitian Forms, Adjoints, and Isometries

    How do bilinear or Hermitian forms define adjoints, orthogonality, signatures, and norm-preserving maps?

    Principal question
    How do bilinear or Hermitian forms define adjoints, orthogonality, signatures, and norm-preserving maps?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Bilinear and Hermitian Forms, Adjoints, and Isometries at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Vector Spaces, Duals, Linear Maps, and Bases
  • Term

    Charges, Screening, and Long-Range Forces

    How does Gauss law constrain charged sectors and distinguish screening from long-range flux?

    Principal question
    How does Gauss law constrain charged sectors and distinguish screening from long-range flux?
    Boundary
    The general dressing taxonomy, edge observables, and generalized-symmetry structure belong to Volume 3; DHR and algebraic superselection rigor belongs to Volume 16.
    Scope
    Charge-flux relations, screening tests, minimal dressing consequences, and a physical orientation to charged sectors and carefully qualified superselection language.
    Assumptions
    Dynamical Gauge Fields and Matter; Gauge Orbits, Gauss Constraints, and Stabilizers
  • Term

    Chiral Symmetry in QCD

    What is the chiral flavor symmetry of massless QCD, and how do quark masses and anomalies modify it?

    Principal question
    What is the chiral flavor symmetry of massless QCD, and how do quark masses and anomalies modify it?
    Boundary
    Electroweak chirality; Full flavor phenomenology; Primary anomaly derivation
    Canonical treatment
    Chiral Symmetry in QCD
    Scope
    Left/right flavor groups, vector and axial currents, quark-mass breaking, anomaly of U(1)A, discrete remnants orientation, and order parameters
    Assumptions
    QCD Fields, Scales, and the Perturbative Domain; Quantum Currents, Improvements, and Conservation
  • Term

    Collinear Factorization and Operator-Defined PDFs

    How does collinear QCD factorization separate a hard process from universal parton distributions, and what operator, scheme, scale, and power-correction data make the statement testable?

    Principal question
    How does collinear QCD factorization separate a hard process from universal parton distributions, and what operator, scheme, scale, and power-correction data make the statement testable?
    Boundary
    Generic factorization proofs and mode architecture belong to Volumes 4 and 5; fragmentation, TMD/rapidity, and small-x regimes have separate pages, while current PDF releases remain versioned external inputs.
    Scope
    Leading-power collinear factorization and PDF operator definitions, hard/PDF scheme cancellation, support and sum rules, universality tests, and power-suppressed failure terms.
    Assumptions
    Deep-Inelastic Scattering and the Parton Model; Collinear Factorization and Splitting Amplitudes
  • Term

    Commutators and Operator Exponentials

    How do commutators control ordering, conjugation, and the composition of exponentials?

    Principal question
    How do commutators control ordering, conjugation, and the composition of exponentials?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Commutators and Operator Exponentials at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Vector Spaces, Duals, Linear Maps, and Bases
  • Term

    Completeness and the Operator Basis

    Which completeness statement justifies inserting conformal families between operator products, and when can operator bases fail to be discrete or ordinary?

    Principal question
    Which completeness statement justifies inserting conformal families between operator products, and when can operator bases fail to be discrete or ordinary?
    Boundary
    General Euclidean axioms, reflection positivity, and operator definitions remain in Volume 2; thermal cylinder physics remains in Volume 11; executable spectral checks require separately supplied code and data.
    Scope
    Resolution of the identity on spherical slices, sums over conformal families and descendants, sector decomposition, convergence conditions, continuous spectra, and the distinction between completeness and a chosen truncation.
    Assumptions
    Descendant States and Gram Matrices
  • Term

    Condensation, Off-Diagonal Order, and Superfluidity

    How are Bose condensation, off-diagonal long-range order, spontaneous symmetry breaking, and superfluid response related without being identified?

    Principal question
    How are Bose condensation, off-diagonal long-range order, spontaneous symmetry breaking, and superfluid response related without being identified?
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    One-body density matrix, condensate fraction, quasi-long-range order, symmetry-breaking limits, helicity modulus, superfluid density, and counterexamples separating the notions.
    Assumptions
    The Ideal Bose Gas and Bose–Einstein Condensation; Symmetry Realization and Order Parameters
  • Term

    Conformal Geometry, Maps, and Compactification

    Which transformations are conformal locally, and when do signature, compactification, or global topology obstruct extending them to honest global symmetries?

    Principal question
    Which transformations are conformal locally, and when do signature, compactification, or global topology obstruct extending them to honest global symmetries?
    Boundary
    General Lie theory and Lorentz representations remain in Volume 1; general global-symmetry, Ward-identity, and anomaly machinery remains in Volume 3; RG criteria for fixed points remain in Volume 5; theorem-level edge cases remain in Volume 16.
    Scope
    Conformal maps in Euclidean and Lorentzian signature, inversions, compactified Minkowski and Euclidean spaces, connected components, singular loci, and the local-versus-global symmetry distinction.
    Assumptions
    Lie Groups, Lie Algebras, and Exponential and Adjoint Maps; Lorentz Field Representations and Poincaré Particle Representations
  • Term

    Conjugation and Reflection Positivity

    What is Hermitian conjugation in radial quantization, and how does Euclidean reflection positivity become positivity of the CFT Hilbert space?

    Principal question
    What is Hermitian conjugation in radial quantization, and how does Euclidean reflection positivity become positivity of the CFT Hilbert space?
    Boundary
    General Euclidean axioms, reflection positivity, and operator definitions remain in Volume 2; thermal cylinder physics remains in Volume 11; executable spectral checks require separately supplied code and data.
    Scope
    Radial inversion as conjugation, bra insertions at infinity, operator reality, reflection-positive pairings, norm positivity, and the boundary between unitary and nonunitary CFT.
    Assumptions
    The State–Operator Correspondence; Reflection Positivity within Osterwalder–Schrader Reconstruction
  • Term

    Connected, Disconnected, and Vacuum Diagrams

    Which classes of diagrams contribute to normalized correlators and scattering amplitudes?

    Principal question
    Which classes of diagrams contribute to normalized correlators and scattering amplitudes?
    Boundary
    Cluster decomposition as a structural theorem; Cumulants as a primary definition
    Scope
    Vacuum bubbles, connected correlators, disconnected external processes, exponentiation of vacuum diagrams, and linked-cluster logic
    Assumptions
    Diagrammatics and Symmetry Factors
  • Term

    Conserved Charges and Grand-Canonical States

    Which conserved charges may enter an equilibrium statistical operator, and what commutation, boundedness, and superselection conditions make the construction meaningful?

    Principal question
    Which conserved charges may enter an equilibrium statistical operator, and what commutation, boundedness, and superselection conditions make the construction meaningful?
    Boundary
    This chapter owns generic finite-density state and correlator structure; Volume 3 retains symmetry and background-gauge definitions, Volume 8 owns lattice sign-problem methods, and Chapter 18 owns current QCD evidence.
    Scope
    Grand-canonical generators, commuting conserved charges, generalized thermodynamic potentials, charge sectors, stability of the exponent, neutral observables, and equilibrium time evolution by the appropriate modular Hamiltonian.
    Assumptions
    Thermal Density Operators and the KMS Condition; Continuous Symmetries, Generators, and Charges
  • Term

    Coulomb, Higgs, and Confining Regimes

    Which gauge-invariant observables distinguish Coulomb, Higgs, and confining regimes, and when can they be continuously connected?

    Principal question
    Which gauge-invariant observables distinguish Coulomb, Higgs, and confining regimes, and when can they be continuously connected?
    Boundary
    A gauge-variant order parameter; Complete phase diagrams; Microscopic confinement mechanisms
    Scope
    Long-range forces, mass gaps, screening, flux tubes, Higgs and confinement diagnostics, complementarity orientation, and matter-representation dependence
    Assumptions
    Charges, Screening, and Long-Range Forces; Elitzur's Theorem and the Gauge-Invariant Higgs Mechanism
  • Term

    Coupling to Background Gauge Fields and Bundles

    How does a nondynamical background connection and its global bundle data make an ordinary symmetry Ward identity local and covariant?

    Principal question
    How does a nondynamical background connection and its global bundle data make an ordinary symmetry Ward identity local and covariant?
    Boundary
    Dynamical integration over fields is deferred to the gauging pages; background-field gauge fixing and gauge dynamics belong to Gauge Theories.
    Scope
    Nondynamical background connections, bundle sectors, covariant source coupling, background gauge transformations, and covariant Ward identities.
    Assumptions
    Current Sources and Generating Functionals; Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities
  • Term

    Covariant Algebraic Quantization and Fock Realizations

    How are curved-space solution spaces promoted to CCR or CAR algebras, and when does an additional complex structure produce a Fock representation rather than canonical physics?

    Principal question
    How are curved-space solution spaces promoted to CCR or CAR algebras, and when does an additional complex structure produce a Fock representation rather than canonical physics?
    Boundary
    Volume II owns general canonical and free-field quantization, Chapter 2 owns state and representation selection, and Volume XVI owns theorem-first CCR, CAR, and functorial constructions.
    Scope
    The algebraic quantization of free curved-background fields, commutator or anticommutator kernels, positivity requirements, representation choice, and the noncanonical status of Fock realizations.
    Assumptions
    Covariant Symplectic Structure and Conserved Inner Products; Green Operators, Causal Propagators, and State-Dependent Two-Point Functions; Quantizing the Real Scalar Field
  • Term

    Crossing Equations and Positivity

    How do OPE associativity and unitarity become crossing equations with positive spectral weights?

    Principal question
    How do OPE associativity and unitarity become crossing equations with positive spectral weights?
    Boundary
    The local Wilson OPE is defined in Volume 2; general distribution and tensor analysis remain in Volume 1; amplitude observables remain in Volume 4; block-generation code and benchmark data require separately supplied code and data and Reference.
    Scope
    Channel equality, reduced crossing equations, positivity for identical Hermitian operators, matrix positivity for degeneracies, gap assumptions, identity isolation, and the limits of positivity in nonunitary or spinning systems.
    Assumptions
    OPE Convergence, Associativity, and Domain Control; Conformal Blocks and Casimir Equations; Conjugation and Reflection Positivity
  • Term

    Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity

    How do Lorentzian causal structure, Cauchy surfaces, domains of dependence, and global hyperbolicity determine whether a curved-spacetime field problem has predictive evolution?

    Principal question
    How do Lorentzian causal structure, Cauchy surfaces, domains of dependence, and global hyperbolicity determine whether a curved-spacetime field problem has predictive evolution?
    Boundary
    Volume I owns Lorentzian-geometry and hyperbolic-PDE theorems, blackhole.org owns deep horizon geometry, and Volume XVI owns proof-first propagation and algebraic consequences.
    Scope
    The physical geometry contract for QFT on a prescribed spacetime, including time orientation, causal diamonds, Cauchy development, global hyperbolicity, and explicit failure outside that class.
    Assumptions
    Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields; Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities
  • Term

    Derivative Interactions and Contact Terms

    How do derivatives, identical fields, and local contact terms modify momentum-space vertices?

    Principal question
    How do derivatives, identical fields, and local contact terms modify momentum-space vertices?
    Boundary
    EFT operator-basis reduction; Renormalized contact operators; Gauge-theory seagull catalogs
    Scope
    Momentum factors, integration by parts, identical-field combinatorics, time-derivative cautions, contact terms, and convention checks
    Assumptions
    Momentum-Space Feynman Rules
  • Term

    Diagrammatics and Symmetry Factors

    How do contractions become graphs, and how are graph multiplicities converted into symmetry factors?

    Principal question
    How do contractions become graphs, and how are graph multiplicities converted into symmetry factors?
    Boundary
    Graph-polynomial methods; Package syntax; Renormalization forests
    Scope
    Vertices, internal and external lines, automorphisms, labeled contractions, connected components, and scalar examples
    Assumptions
    Wick Expansion for Interacting Fields
  • Term

    Direct Sums, Tensor Products, and Index Structure

    How do direct sums and tensor products encode alternatives, composites, multilinear maps, and index symmetries?

    Principal question
    How do direct sums and tensor products encode alternatives, composites, multilinear maps, and index symmetries?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Direct Sums, Tensor Products, and Index Structure at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Vector Spaces, Duals, Linear Maps, and Bases
  • Term

    Effective Field Theory as a Controlled Expansion

    What makes a low-energy field theory systematically improvable even when its ultraviolet completion is unknown?

    Principal question
    What makes a low-energy field theory systematically improvable even when its ultraviolet completion is unknown?
    Boundary
    A catalog of specific EFT frameworks; Claims beyond the declared breakdown scale
    Scope
    Separation of scales, low-energy degrees of freedom, locality, symmetry, cutoff or matching scale, operator expansion, Wilson coefficients, predictive truncation, and top-down versus bottom-up viewpoints
    Assumptions
    Regulator Removal and Renormalized Predictions; Local and Composite Operator Insertions
  • Term

    Efimov Physics and the Three-Body Parameter

    Why does resonant three-body physics require an additional parameter, and how does discrete scale invariance enter many-body predictions?

    Principal question
    Why does resonant three-body physics require an additional parameter, and how does discrete scale invariance enter many-body predictions?
    Boundary
    Volumes 4 and 5 own scattering and few-body EFT machinery. This chapter owns the use of matched two- and three-body data in quantum gases; current resonance and loss records belong to Research.
    Scope
    Efimov spectrum, limit-cycle scaling, three-body parameter, atom–dimer thresholds, recombination interfaces, and separation of few-body input from medium effects.
    Assumptions
    Effective Range, Shallow Poles, and Universality Windows
  • Term

    Entropy of a Regulated Subregion

    Which regulator, region algebra, state, boundary prescription, and order of limits define a subregion entropy before continuum interpretation?

    Principal question
    Which regulator, region algebra, state, boundary prescription, and order of limits define a subregion entropy before continuum interpretation?
    Boundary
    It does not call a divergent bare entropy a continuum observable or compare values from different center and boundary conventions without matching them.
    Scope
    This page owns the regulated density operator or covariance restriction, von Neumann entropy, geometric region, cutoff prescription, and nonuniversality of the raw value.
    Assumptions
    Direct Sums, Tensor Products, and Index Structure; Why Continuum QFT Does Not Factorize Naively
  • Term

    Euclidean Fields and Classical Statistical Systems

    When can a Euclidean field functional be interpreted as a classical statistical system, and which quantum or real-time information is lost in that correspondence?

    Principal question
    When can a Euclidean field functional be interpreted as a classical statistical system, and which quantum or real-time information is lost in that correspondence?
    Boundary
    Volume 1 retains probability, measure, asymptotic, and convex-analysis foundations; Volume 2 retains vacuum functional-integral definitions; rigorous infinite-system existence belongs to Volume 16.
    Scope
    The Boltzmann-weight dictionary, transfer and dimensional interpretations, ultraviolet regulator and measure, reflection-positivity caveat, classical critical observables, and explicit limits of Euclidean-to-dynamical inference.
    Assumptions
    Wick Rotation and Analytic Continuation; Statistical Ensembles and Field Configurations
  • Term

    Euclidean Random Fields and Schwinger Hierarchies

    How does a probability measure on a distribution space define Euclidean fields and a compatible hierarchy of Schwinger functions?

    Principal question
    How does a probability measure on a distribution space define Euclidean fields and a compatible hierarchy of Schwinger functions?
    Boundary
    Volume II owns regulated functional-integral intuition and Chapter 11 owns interacting measure construction; this page defines the Euclidean random-field object before reconstruction.
    Scope
    Cylinder measures, characteristic functionals, random tempered distributions, moments and cumulants, Euclidean covariance, symmetry of Schwinger functions, and the distinction between a hierarchy and an underlying measure.
    Assumptions
    Existence, Construction, Reconstruction, and Continuum Claims; Probability Spaces, Random Variables, and Conditional Expectation; Gaussian Vectors, Processes, Random Distributions, and Wick Structure
  • Term

    Euclidean Superspace, Conjugation, and Field-Space Complexification

    Why do Euclidean superspace and localized path integrals require explicit complexification, independent conjugate fields, and a separately chosen reality contour?

    Principal question
    Why do Euclidean superspace and localized path integrals require explicit complexification, independent conjugate fields, and a separately chosen reality contour?
    Boundary
    Chapter 1 owns physical-state representations; Volume 1 retains Grassmann and spinor mathematics; Volume 3 retains gauge redundancy; dynamical actions begin in Chapter 4.
    Scope
    Wick rotation of spinors and supercharges, Euclidean real forms, independent chiral variables, reflection operations, complexified field space, integration cycles, and limits of Lorentzian conjugation notation.
    Assumptions
    Supercovariant Derivatives, Chirality, and Integrability; Euclidean Correlators and Schwinger Functions
  • Term

    Existence, Construction, Reconstruction, and Continuum Claims

    How do existence, explicit construction, reconstruction from correlation data, regulator removal, thermodynamic limit, and continuum limit differ as quantified mathematical claims?

    Principal question
    How do existence, explicit construction, reconstruction from correlation data, regulator removal, thermodynamic limit, and continuum limit differ as quantified mathematical claims?
    Boundary
    Constructive models belong to Chapters 11–12, lattice procedures to Volume VIII, and physical renormalization to Volume V; this page supplies their common logical vocabulary.
    Scope
    The implication ledger separating finite-regulator definitions, compatible families, limiting objects, nontriviality, uniqueness, reconstruction, and recovery of selected observables.
    Assumptions
    Theorem-First Claim Records: Objects, Hypotheses, Conclusions, and Status; Limits, Completeness, and Modes of Convergence
  • Term

    Extended Supersymmetry, R-Symmetry, and Central Charges

    How do extended supercharges, R-symmetries, central charges, and tensorial charges fit into a consistent super-Poincaré algebra?

    Principal question
    How do extended supercharges, R-symmetries, central charges, and tensorial charges fit into a consistent super-Poincaré algebra?
    Boundary
    Volume 1 retains spinor and representation mathematics, Volume 3 retains ordinary symmetry and charge machinery, Volume 9 owns conformal crossing, and Volume 4 owns amplitude methods.
    Scope
    Extended-supercharge indices, automorphism groups, central and p-form extensions, their Lorentz and R representations, Hermiticity properties, Jacobi constraints, and physical charge interpretations.
    Assumptions
    The Four-Dimensional N=1 Super-Poincaré Algebra; Quantum Implementations, Projective Actions, and Central Extensions
  • Term

    Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration

    How do grading and anticommutation organize fermionic variables, signs, determinants, and Pfaffians?

    Principal question
    How do grading and anticommutation organize fermionic variables, signs, determinants, and Pfaffians?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Direct Sums, Tensor Products, and Index Structure
  • Term

    Fermion Signs and Closed Loops

    Where do the signs of fermionic diagrams come from, and how can they be checked reliably?

    Principal question
    Where do the signs of fermionic diagrams come from, and how can they be checked reliably?
    Boundary
    Anomaly traces; Automated fermion-flow packages; Supersymmetric sign systems
    Scope
    Grassmann reordering, external fermion ordering, closed-loop signs, arrow flow, Majorana caution, and explicit low-order checks
    Assumptions
    Momentum-Space Feynman Rules; The Fermion Propagator
  • Term

    Fields, Observables, and Interpolating Operators

    How can a local field create or detect particle content without itself being identical to a particle or necessarily being observable?

    Principal question
    How can a local field create or detect particle content without itself being identical to a particle or necessarily being observable?
    Boundary
    Asymptotic reduction belongs to Scattering; gauge-invariant observable construction belongs to Symmetry and Gauge Theories; local algebraic criteria belong to Mathematical QFT.
    Scope
    The physical distinctions among field coordinatizations, observables, sources, interpolating operators, and their matrix elements.
    Assumptions
    One-Particle States: Mass, Spin, and Relativistic Normalization
  • Term

    Finite Volume as a Controlled Deformation

    When is finite spatial volume a controlled infrared deformation, and which scales and orders of limits determine its effect?

    Principal question
    When is finite spatial volume a controlled infrared deformation, and which scales and orders of limits determine its effect?
    Boundary
    Thermal compactification belongs to Volume 11, general boundary-state foundations to Volume 2, finite-volume scattering theory to later pages, and executable volume studies must be accompanied by code, inputs, and validation checks.
    Scope
    Spatial box geometry, momentum quantization, boundary conditions, interaction range, mass-gap control, symmetry reduction, exponential versus power-law effects, temporal-wrap separation, and the volume-continuum-mass order of limits.
    Assumptions
    Euclidean Correlators and Spectral Information; Lattice Geometry, Boundaries, and Anisotropy
  • Term

    Fixed Points and Linearized RG Flow

    How does linearizing an RG transformation near a fixed point reveal scaling operators and critical exponents?

    Principal question
    How does linearizing an RG transformation near a fixed point reveal scaling operators and critical exponents?
    Boundary
    Conformal representation theory and bootstrap data owned by Volume 9; Current model-specific exponent tables
    Scope
    Fixed points, stability matrices, eigenoperators, scaling exponents, anomalous dimensions, coordinate dependence, discrete versus continuous RG, and nonlinear qualifications
    Assumptions
    Wilsonian Coarse Graining and Theory Space
  • Term

    Fixed-Theory, Ensemble, and Superselection Claims

    When does a bulk calculation describe one fixed boundary theory, an average over theories or couplings, or a superselection sector, and how can those possibilities be told apart?

    Principal question
    When does a bulk calculation describe one fixed boundary theory, an average over theories or couplings, or a superselection sector, and how can those possibilities be told apart?
    Boundary
    Volume XI owns statistical ensembles, Chapter 18 owns JT/SYK realizations, Chapter 20 owns gravitational topology sums and baby universes, and Research owns live factorization disputes.
    Scope
    An ensemble-status contract specifying what is integrated, averaged, conditioned, or held fixed, together with factorization, variance, and sector-resolving diagnostics.
    Assumptions
    Observable and Regime Matrix for Quantum Gravity; Dictionary Completeness and Global Data
  • Term

    Fragmentation Functions and Timelike Evolution

    How do fragmentation functions describe identified final-state hadrons, evolve timelike, satisfy momentum sum rules, and differ from PDFs in universality and factorization limits?

    Principal question
    How do fragmentation functions describe identified final-state hadrons, evolve timelike, satisfy momentum sum rules, and differ from PDFs in universality and factorization limits?
    Boundary
    Hadronization models, current fragmentation fits, transverse-momentum fragmentation, and nonperturbative determinations remain with specialist or versioned evidence sources.
    Scope
    Fragmentation-function operator and factorization definitions, timelike DGLAP evolution, momentum sum rules, identified-hadron observables, scheme dependence, and universality tests.
    Assumptions
    DGLAP Evolution and Scaling Violation
  • Term

    Framed BPS States, Line Defects, and Defect Moduli

    How do a supersymmetric line defect and its boundary conditions define framed BPS sectors and defect moduli rather than an ordinary bulk-particle spectrum?

    Principal question
    How do a supersymmetric line defect and its boundary conditions define framed BPS sectors and defect moduli rather than an ordinary bulk-particle spectrum?
    Boundary
    Volume 1 retains quotient geometry, Volume 3 retains generic spontaneous symmetry breaking, Volume 7 retains generic solitons and zero modes, and dimension-specific dynamics stay in later chapters.
    Scope
    Preserved subalgebras, defect boundary conditions, core charges, framed charge sectors, halo configurations, defect moduli, framed indices, chamber dependence, and fusion with bulk BPS states.
    Assumptions
    BPS Solitons, Walls, Strings, Vortices, and Junctions; Disorder Operators and Singular Boundary Conditions
  • Term

    From the Local OPE to Conformal Data

    How does the general local OPE specialize at a conformal fixed point to spectrum, representation, and three-point-coefficient data?

    Principal question
    How does the general local OPE specialize at a conformal fixed point to spectrum, representation, and three-point-coefficient data?
    Boundary
    The local Wilson OPE is defined in Volume 2; general distribution and tensor analysis remain in Volume 1; amplitude observables remain in Volume 4; block-generation code and benchmark data require separately supplied code and data and Reference.
    Scope
    The conformal-data package, descendant coefficients fixed by symmetry, two-point metric, OPE-basis transformations, selection rules, and the distinction between defining the OPE and exploiting conformal covariance.
    Assumptions
    Free-Field OPE Preview and Ownership Map; Scalar Two- and Three-Point Functions
  • Term

    Galilean Fields, Scales, and Low-Energy Degrees of Freedom

    How do Galilean symmetry, particle number, and scale separation determine the field content of a nonrelativistic many-body theory?

    Principal question
    How do Galilean symmetry, particle number, and scale separation determine the field content of a nonrelativistic many-body theory?
    Boundary
    Volume 2 owns Fock-space and path-integral foundations, Volume 3 owns symmetry and Ward identities, and Volume 5 owns generic EFT construction; this chapter owns their nonrelativistic many-body realization and matching.
    Scope
    Galilean transformations, mass central charge, particle-number symmetry, dispersion, dynamical exponent, microscopic versus collective scales, and the continuum limit.
    Assumptions
    Lie Groups, Lie Algebras, and Exponential and Adjoint Maps; The Action Principle and Field Equations; One-Particle States: Mass, Spin, and Relativistic Normalization
  • Term

    Generalized Gibbs Ensembles for Integrable Charges

    When does a commuting family of local or quasilocal integrable charges define a charge-complete generalized Gibbs ensemble and its thermodynamic macrostate?

    Principal question
    When does a commuting family of local or quasilocal integrable charges define a charge-complete generalized Gibbs ensemble and its thermodynamic macrostate?
    Boundary
    This page consumes Volume 7 integrability, conserved-charge, and thermodynamic-Bethe-ansatz machinery rather than re-owning it. Genuinely noncommuting constraints belong to the sibling noncommuting-charge page; dynamical approach and prethermal plateaus belong to Chapter 15; Euler-scale generalized hydrodynamics belongs to Chapter 13; and model- or platform-specific realizations belong to Volume 12.
    Scope
    Commuting local and quasilocal integrable charges, maximum-entropy generalized Gibbs ensembles, generalized chemical potentials, charge completeness, truncated-GGE diagnostics, representative-state equivalence, and the interface from charge data to a thermodynamic-Bethe-ansatz macrostate.
    Assumptions
    Conserved Charges and Grand-Canonical States; Classical and Quantum Conserved Charges; Thermodynamic Bethe Ansatz and Finite-Size Ground-State Energy
  • Term

    Ginsparg–Wilson Symmetry and the Lattice Index

    How does the Ginsparg-Wilson relation define an exact modified lattice chiral symmetry and a finite-spacing index?

    Principal question
    How does the Ginsparg-Wilson relation define an exact modified lattice chiral symmetry and a finite-spacing index?
    Boundary
    Overlap and domain-wall realizations belong to the next page, general anomalies to Volume 3, functional-analytic and index rigor to Volume 16, and sign-function algorithms must be accompanied by code, inputs, and validation checks.
    Scope
    The Ginsparg-Wilson relation, modified chiral transformation, Jacobian, lattice index trace, spectral circle, locality requirement, anomaly reproduction, and its logical relation to the no-go assumptions.
    Assumptions
    The Nielsen–Ninomiya Obstruction
  • Term

    GNS Representations, Local Normality, and Local Quasiequivalence

    How do GNS representations, local normality, and local quasiequivalence separate physically comparable curved-space states from globally inequivalent Fock descriptions?

    Principal question
    How do GNS representations, local normality, and local quasiequivalence separate physically comparable curved-space states from globally inequivalent Fock descriptions?
    Boundary
    Volume XIII owns continuum-subsystem consequences, Volume I owns operator-algebra prerequisites, and Volume XVI owns type classification and quasiequivalence proofs.
    Scope
    The operational comparison of state representations on local algebras, including GNS cyclic data, normal state transfer, local folia, and limitations of global particle language.
    Assumptions
    Local Field Algebras, Causality, and the Time-Slice Property; Quasifree States and Two-Point Functions; Operator Algebras and Positive Functionals: a Bridge
  • Term

    Haag–Kastler Nets and Locality

    How does assigning an observable algebra to each spacetime region encode isotony, Einstein causality, covariance, vacuum structure, and spectral positivity?

    Principal question
    How does assigning an observable algebra to each spacetime region encode isotony, Einstein causality, covariance, vacuum structure, and spectral positivity?
    Boundary
    Volume XIII owns operational subsystem consequences and Volume XIV owns curved deployment; this page states the local-net framework rather than deriving it from every field theory.
    Scope
    The Haag–Kastler net object and axiom variants on double cones, wedges, and causally complete regions, including the distinction between C*-nets and represented von Neumann nets.
    Assumptions
    Positivity, Spectrum, Covariance, and Locality Hypotheses; Wightman Fields, Domains, and Axioms; The Wightman Reconstruction Theorem
  • Term

    Hadamard Admissibility and the Two-Point Wavefront Criterion

    How does the wavefront set of a two-point distribution encode Hadamard ultraviolet admissibility without selecting a preferred state or vacuum?

    Principal question
    How does the wavefront set of a two-point distribution encode Hadamard ultraviolet admissibility without selecting a preferred state or vacuum?
    Boundary
    Volume I owns wavefront-set calculus, Volume XVI owns propagation and equivalence proofs, and this page does not promote microlocal admissibility into a vacuum-selection rule.
    Scope
    The physical two-point wavefront criterion, its future-directed null covector pairing, relation to local singularity form, admissible distribution products, and declared free-field hypotheses.
    Assumptions
    Hadamard Parametrix and Short-Distance Structure; Singular Support and Wavefront Sets; Products, Scaling Degree, and Extensions of Singular Distributions
  • Term

    Higher-Spin Gaps and Einstein-Regime Obstructions

    Why does a low higher-spin tower obstruct a local weakly curved Einstein regime even when large-N factorization and many other holographic features persist?

    Principal question
    Why does a low higher-spin tower obstruct a local weakly curved Einstein regime even when large-N factorization and many other holographic features persist?
    Boundary
    Volume IX owns weakly broken higher-spin CFT data, Chapter 22 owns higher-spin dualities, and Chapter 8 owns Regge and causality bounds on finite-gap corrections.
    Scope
    The higher-spin-gap criterion, its relation to string scale and derivative interactions, vector-model counterexamples, and the distinction between holography and Einstein-like locality.
    Assumptions
    Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria
  • Term

    Holomorphic Couplings and Background Superfields

    How does promoting masses and couplings to background chiral superfields turn holomorphy and spurionic symmetries into controlled constraints?

    Principal question
    How does promoting masses and couplings to background chiral superfields turn holomorphy and spurionic symmetries into controlled constraints?
    Boundary
    Volume 3 retains anomaly foundations, Volume 5 retains scheme and RG machinery, Volume 7 retains generic instanton zero modes, and Chapters 8 and 10 own theory-specific exact dynamics.
    Scope
    Background chiral couplings, spurions, holomorphic Wilsonian data, charge and dimension tables, anomalous transformations, regularity, decoupling limits, and the distinction between a constraint and a unique answer.
    Assumptions
    Wess–Zumino Models; Coupling to Background Gauge Fields and Bundles
  • Term

    In and Out States

    Under which assumptions can interacting states be compared with free particle states in the distant past and future?

    Principal question
    Under which assumptions can interacting states be compared with free particle states in the distant past and future?
    Boundary
    Haag–Ruelle proof; Dressed-state construction; Unstable external particles
    Canonical treatment
    In and Out States
    Scope
    Asymptotic conditions, wave packets, stable particles, Møller operators as physical orientation, and limitations from massless or confining dynamics
    Assumptions
    One-Particle States: Mass, Spin, and Relativistic Normalization; Multiparticle States, Statistics, and Fock Organization
  • Term

    Infraparticles and Velocity Superselection

    How do long-range gauge fields replace a charged particle mass shell by an infraparticle spectrum and produce disjoint representations labeled by asymptotic velocity data?

    Principal question
    How do long-range gauge fields replace a charged particle mass shell by an infraparticle spectrum and produce disjoint representations labeled by asymptotic velocity data?
    Boundary
    Volume IV owns perturbative infrared resummation and inclusive observables, Volume VI owns QED phenomenology, and this page does not claim a universal nonperturbative charged-sector construction.
    Scope
    The spectral signature of infraparticles, absence of sharp charged mass eigenstates, soft photon clouds, velocity superselection, and model-dependent rigorous results.
    Assumptions
    Particles, Mass-Shell Spectrum, and One-Particle Subspaces; Araki–Haag Detectors and Particle Weights; Massless Scattering and Radiation Fields
  • Term

    Internal, Spacetime, Discrete, and Antiunitary Symmetries

    How do internal, spacetime, continuous, discrete, unitary, antiunitary, and orientation-reversing symmetries differ operationally?

    Principal question
    How do internal, spacetime, continuous, discrete, unitary, antiunitary, and orientation-reversing symmetries differ operationally?
    Boundary
    CPT proofs remain in Foundations and Mathematical QFT; conformal symmetry belongs to CFT; curved-background isometries and diffeomorphisms belong to Curved Spacetime.
    Scope
    A convention-aware taxonomy of ordinary symmetries and their actions on amplitudes, operators, spacetime arguments, complex coefficients, and Euclidean backgrounds.
    Assumptions
    What Is a Symmetry of a QFT?
  • Term

    Kähler and Hyperkähler Quotients in Supersymmetric QFT

    When do supersymmetric vacuum spaces inherit Kähler or hyperkähler quotient structures, and which metric and complex structures are protected?

    Principal question
    When do supersymmetric vacuum spaces inherit Kähler or hyperkähler quotient structures, and which metric and complex structures are protected?
    Boundary
    Volume 1 retains quotient geometry, Volume 3 retains generic spontaneous symmetry breaking, Volume 7 retains generic solitons and zero modes, and dimension-specific dynamics stay in later chapters.
    Scope
    Kähler reduction, hyperkähler moment-map triples, FI levels, complex structures, quotient metrics, dimensional requirements, supersymmetry constraints, and the distinction between algebraic variety and metric data.
    Assumptions
    F- and D-Flatness and Gauge Quotients; Hamiltonian Group Actions and Moment Maps
  • Term

    Landau Fermi-Liquid Theory

    How does an interacting Fermi system retain quasiparticles while encoding interactions in a functional of the distribution?

    Principal question
    How does an interacting Fermi system retain quasiparticles while encoding interactions in a functional of the distribution?
    Boundary
    Volume 5 owns RG construction and Volume 11 owns generic transport. This chapter owns Fermi-surface kinematics, Landau theory, Luttinger hypotheses, and matter-specific instabilities.
    Canonical treatment
    Landau Fermi-Liquid Theory
    Scope
    Adiabatic quasiparticles, Landau interaction function and parameters, thermodynamics, stability conditions, effective mass, susceptibilities, and validity assumptions.
    Assumptions
    The Fermi Gas and Fermi-Surface Kinematics; Quasiparticle Poles, Residues, and Lifetimes
  • Term

    Large Deviations and Phase Coexistence

    How do large-deviation rate functions encode rare macroscopic fluctuations, phase coexistence, interface costs, and finite-volume rounding?

    Principal question
    How do large-deviation rate functions encode rare macroscopic fluctuations, phase coexistence, interface costs, and finite-volume rounding?
    Boundary
    Volume 1 retains probability, measure, asymptotic, and convex-analysis foundations; Volume 2 retains vacuum functional-integral definitions; rigorous infinite-system existence belongs to Volume 16.
    Scope
    Scaled cumulant-generating functions, rate functions, Legendre–Fenchel duality, nonconvex constrained free energies, Maxwell construction, coexistence weights, interface suppression, and limits of asymptotic probability claims.
    Assumptions
    Partition Functions and Thermodynamic Response; Thermodynamic Limits, Phases, and Ensemble Equivalence
  • Term

    Lattice Geometry, Boundaries, and Anisotropy

    How do lattice geometry, boundary conditions, anisotropy, and the order of limits alter the regulated theory and its observables?

    Principal question
    How do lattice geometry, boundary conditions, anisotropy, and the order of limits alter the regulated theory and its observables?
    Boundary
    General boundary-state preparation belongs to Volume 2, thermal compactification and phase structure to Volume 11, and geometric discretizations on curved backgrounds to Volume 14.
    Scope
    Hypercubic and bounded geometries, spatial and temporal extents, periodic, antiperiodic, open, and twisted boundary conditions, anisotropic spacings, lattice cells and orientations, and the finite-size versus cutoff hierarchy.
    Assumptions
    Lattice Regulators and Target Continuum Theories
  • Term

    Lattice Perturbation Theory, Symanzik Analysis, and Improvement

    How are lattice Feynman rules, compact-Brillouin-zone loop integrals, and lattice-side matching organized, and how does the Symanzik expansion turn their coefficients into controlled improvement predictions?

    Principal question
    How are lattice Feynman rules, compact-Brillouin-zone loop integrals, and lattice-side matching organized, and how does the Symanzik expansion turn their coefficients into controlled improvement predictions?
    Boundary
    Continuum Dyson-Wick diagrammatics and general loop-integration machinery belong to Volume 4; general renormalization schemes, EFT basis construction, matching logic, and RG evolution belong to Volume 5; nonperturbative renormalization, mixing, and step scaling belong to the preceding page; formulation-specific gauge and fermion improvement belongs to Chapters 3 and 4; executable loop notebooks require separate implementations.
    Scope
    Lattice perturbation theory from a declared regulated action and operator basis: lattice propagators and vertices, compact-Brillouin-zone measures and loop integrals, gauge-fixing and measure terms when applicable, lattice-side action and operator coefficient matching, lower-dimensional and power-divergent mixing diagnostics, the Symanzik action and operator expansion, regulator-symmetry classification, on-shell versus off-shell improvement, redundant operators, anisotropy and boundary terms, coefficient tuning, and residual scaling tests.
    Assumptions
    Lattice Momentum, Propagators, and Cutoff Dispersion; Bare Parameters, Tuning Conditions, and Continuum Targets; Relevant, Marginal, and Irrelevant Directions
  • Term

    Lattice Regulators and Target Continuum Theories

    Which regulator data and limiting claim turn a continuum QFT target into a well-specified spacetime-lattice theory?

    Principal question
    Which regulator data and limiting claim turn a continuum QFT target into a well-specified spacetime-lattice theory?
    Boundary
    General regulator taxonomy, renormalization-group flow, and regulator removal belong to Volume 5; formal and constructed continuum definitions belong to Volumes 2 and 16; executable lattice construction requires a separate implementation.
    Scope
    The lattice-specific regulator contract: dimension, geometry, spacings, extents, boundary data, bare variables, measure, observables, symmetry losses, and the distinction between a finite lattice model and its target continuum QFT.
    Assumptions
    Regulators, Cutoffs, and Continuum Limits
  • Term

    Limits, Completeness, and Modes of Convergence

    Which notion of convergence is being used, what does it preserve, and when may limits be interchanged?

    Principal question
    Which notion of convergence is being used, what does it preserve, and when may limits be interchanged?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Metric-space limits, Cauchy completeness, pointwise and uniform convergence, and counterexamples to invalid limit interchange.
  • Term

    Local Covariance, Isometries, and Background Embeddings

    How should a curved-space QFT respond to causal background embeddings, isometries, boundary choices, and changes of presentation while preserving local physical content?

    Principal question
    How should a curved-space QFT respond to causal background embeddings, isometries, boundary choices, and changes of presentation while preserving local physical content?
    Boundary
    Volume XVI owns the categorical functor, natural-transformation, and theorem-first formulation; this page retains physical interpretation, examples, and failure diagnostics.
    Scope
    The operational local-covariance contract for embedding one admissible background into another, transporting observables, distinguishing covariance from symmetry, and recording boundary dependence.
    Assumptions
    Local Field Algebras, Causality, and the Time-Slice Property; Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity
  • Term

    Local Expansions and Global Vacuum Structure

    Which vacuum data can change nonanalytically or remain invisible around any one perturbative saddle?

    Principal question
    Which vacuum data can change nonanalytically or remain invisible around any one perturbative saddle?
    Boundary
    Spontaneous-symmetry-breaking primary treatment; Gauge-theory phase catalog; Constructive vacuum existence. Those subjects remain with their canonical volume or Research owner.
    Scope
    Vacuum energy, multiple extrema, order parameters, disconnected sectors, nonanalytic parameter dependence, finite-volume uniqueness, and infinite-volume limits, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Vacua, States, and Representations
  • Term

    Local Field Algebras, Causality, and the Time-Slice Property

    How do local algebras assigned to spacetime regions encode causality, isotony, equations of motion, and the time-slice property without assuming Hilbert-space factorization?

    Principal question
    How do local algebras assigned to spacetime regions encode causality, isotony, equations of motion, and the time-slice property without assuming Hilbert-space factorization?
    Boundary
    Volume XIII owns continuum-subsystem and information-theoretic consequences, while Volume XVI owns Haag–Kastler, locally covariant, and time-slice theorem proofs.
    Scope
    The physical net-of-observables construction for free curved fields, including region assignment, causal commutation, quotient by equations of motion, and Cauchy-neighborhood completeness.
    Assumptions
    Covariant Algebraic Quantization and Fock Realizations; Microcausality and Relativistic Compatibility
  • Term

    Local Operations, Separability, and Distillability in QFT

    How must local operations, separability, PPT tests, and distillability be formulated for algebraic QFT bipartitions and their split realizations?

    Principal question
    How must local operations, separability, PPT tests, and distillability be formulated for algebraic QFT bipartitions and their split realizations?
    Boundary
    It does not identify abstract vacuum distillability with a finite-rate laboratory protocol, and it hands theorem-first classification to Volume XVI and horizon-dependent extraction to Volume XIV.
    Scope
    This page owns algebraic bipartitions, split-property tensor realizations, separable and PPT state classes, LOCC assumptions, vacuum and thermal distillability, and finite extractable resource.
    Assumptions
    Choosing a Continuum Subsystem: Algebra, Split, or Regulator; The Split Property and Approximate Tensor Products
  • Term

    Measures and Measurable Functions

    What structure makes size, almost-everywhere statements, and measurable observables precise?

    Principal question
    What structure makes size, almost-everywhere statements, and measurable observables precise?
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Measures and Measurable Functions at reusable, hypothesis-aware mathematical-methods depth.
  • Term

    Moduli-Space Metrics and Quantum Corrections

    Which moduli-space structures receive perturbative or nonperturbative quantum corrections, and which supersymmetry or holomorphy arguments constrain them?

    Principal question
    Which moduli-space structures receive perturbative or nonperturbative quantum corrections, and which supersymmetry or holomorphy arguments constrain them?
    Boundary
    Volume 1 retains quotient geometry, Volume 3 retains generic spontaneous symmetry breaking, Volume 7 retains generic solitons and zero modes, and dimension-specific dynamics stay in later chapters.
    Scope
    Low-energy sigma-model metrics, Kähler potentials versus complex structures, wavefunction renormalization, singular metrics, higher-derivative terms, nonrenormalization scope, and dimension-dependent protection.
    Assumptions
    Kähler and Hyperkähler Quotients in Supersymmetric QFT; Kähler Sigma Models and Supersymmetric Target Geometry
  • Term

    Momentum-Space Feynman Rules

    How are propagators, vertices, momentum conservation, and integration measures read from a local action?

    Principal question
    How are propagators, vertices, momentum conservation, and integration measures read from a local action?
    Boundary
    Model-specific rule catalogs; Counterterm vertices; Gauge-fixing derivations
    Canonical treatment
    Momentum-Space Feynman Rules
    Scope
    Fourier transform, all-momenta-incoming convention, propagators, vertex factors, delta functions, internal integrations, and overall signs
    Assumptions
    Diagrammatics and Symmetry Factors
  • Term

    Multiparticle States, Statistics, and Fock Organization

    How are identical-particle sectors organized, and which parts of that organization precede a particular Fock representation?

    Principal question
    How are identical-particle sectors organized, and which parts of that organization precede a particular Fock representation?
    Boundary
    Spin-statistics hypotheses are audited later in this volume; braided statistics and generalized charge sectors belong to Symmetry, CFT, and Mathematical QFT.
    Scope
    Symmetric and antisymmetric multiparticle kinematics, normalization, and the distinction between sector organization and a chosen Fock representation.
    Assumptions
    One-Particle States: Mass, Spin, and Relativistic Normalization
  • Term

    Noncommuting Charges and Generalized Gibbs States

    What maximum-entropy state is licensed by genuinely noncommuting or non-Abelian charge constraints, and which conclusions depend on representation, reference frame, or preparation protocol?

    Principal question
    What maximum-entropy state is licensed by genuinely noncommuting or non-Abelian charge constraints, and which conclusions depend on representation, reference frame, or preparation protocol?
    Boundary
    This chapter owns generic finite-density state and correlator structure; Volume 3 retains symmetry and background-gauge definitions, Volume 8 owns lattice sign-problem methods, and Chapter 18 owns current QCD evidence.
    Scope
    Maximum-entropy exponentials with noncommuting Lie-algebra charges, non-Abelian thermal states, modular generators, residual symmetry and symmetry breaking by charge potentials, reference-frame and preparation dependence, and limits on simultaneous sharp charge assignment.
    Assumptions
    Conserved Charges and Grand-Canonical States
  • Term

    Nonperturbative Exponential Effects and Finite-N Sectors

    What physics is invisible to every order in 1/N or perturbative genus expansion, and how can exponentially suppressed finite-N sectors affect exact observables?

    Principal question
    What physics is invisible to every order in 1/N or perturbative genus expansion, and how can exponentially suppressed finite-N sectors affect exact observables?
    Boundary
    Chapter 4 owns brane and string origins, Chapter 5 owns definition proposals, Chapters 18 and 20 own topology and ensemble realizations, and Research owns unsettled interpretations.
    Scope
    A taxonomy of exp(-N), exp(-N squared), brane, instanton, topology-changing, and level-discreteness effects, with observable-dependent estimates and no claim of universal completion.
    Assumptions
    From Genus Counting to a Holographic String Regime; Corrections, Nonuniform Limits, and Failure Modes
  • Term

    Normal Forms, Spectra, and Projectors

    Which finite-dimensional operators admit useful normal forms and spectral projectors, and what fails for non-normal maps?

    Principal question
    Which finite-dimensional operators admit useful normal forms and spectral projectors, and what fails for non-normal maps?
    Boundary
    Infinite-dimensional spectra and spectral measures belong to Functional and Spectral Analysis.
    Scope
    Eigenvalues, algebraic versus geometric multiplicity, diagonalization, finite spectral theorem, Jordan form, SVD orientation, and projectors.
    Assumptions
    Vector Spaces, Duals, Linear Maps, and Bases
  • Term

    One-Particle States: Mass, Spin, and Relativistic Normalization

    How do Poincaré quantum numbers, little-group data, and invariant normalization define a relativistic one-particle state?

    Principal question
    How do Poincaré quantum numbers, little-group data, and invariant normalization define a relativistic one-particle state?
    Boundary
    Representation-theory machinery belongs to Mathematical Methods; interacting asymptotic states and scattering normalizations belong to Scattering.
    Scope
    The physical Wigner classification, invariant state normalization, and the meanings of mass, spin, and helicity.
  • Term

    Osterwalder–Schrader Axioms and Reflection Positivity

    Which Euclidean covariance, symmetry, regularity, clustering, and reflection-positivity conditions form an Osterwalder–Schrader input suitable for relativistic reconstruction?

    Principal question
    Which Euclidean covariance, symmetry, regularity, clustering, and reflection-positivity conditions form an Osterwalder–Schrader input suitable for relativistic reconstruction?
    Boundary
    Lattice transfer-matrix practice remains in Volume VIII and radial-quantization positivity in Volume IX; this page states the continuum reconstruction hypotheses without conflating those settings.
    Scope
    The OS axiom ledger with a fixed time reflection, positive-time test algebra, reflection-positive quadratic form, Euclidean action, permutation symmetry, growth assumptions, and clustering variants.
    Assumptions
    Euclidean Random Fields and Schwinger Hierarchies; Positivity, Spectrum, Covariance, and Locality Hypotheses
  • Term

    Particles, Mass-Shell Spectrum, and One-Particle Subspaces

    How are stable particles defined by isolated mass hyperboloids in the joint energy–momentum spectrum rather than by a chosen free-field expansion?

    Principal question
    How are stable particles defined by isolated mass hyperboloids in the joint energy–momentum spectrum rather than by a chosen free-field expansion?
    Boundary
    Volume II owns particle intuition and Volume IV owns observable scattering kinematics; this page supplies the spectral theorem assumptions needed for rigorous asymptotic states.
    Scope
    Spectral projections, mass operators, isolated shells, one-particle irreducible Poincaré representations, multiplicity, stability, and the distinction between poles, resonances, and infraparticles.
    Assumptions
    Positivity, Spectrum, Covariance, and Locality Hypotheses; Wightman Functions and Spectral Support; The Wightman Reconstruction Theorem
  • Term

    Primaries, Descendants, and Conformal Multiplets

    How are local operators organized into irreducible conformal families, and which algebraic data distinguish primaries from descendants?

    Principal question
    How are local operators organized into irreducible conformal families, and which algebraic data distinguish primaries from descendants?
    Boundary
    General Lie theory and Lorentz representations remain in Volume 1; general global-symmetry, Ward-identity, and anomaly machinery remains in Volume 3; RG criteria for fixed points remain in Volume 5; theorem-level edge cases remain in Volume 16.
    Scope
    Primary conditions at the origin, descendant generation, scaling dimension, spin labels, multiplet modules, shortening, null descendants, and operator-state language before imposing unitarity.
    Assumptions
    The Conformal Algebra and Its Generators; Local and Composite Operator Insertions
  • Term

    Pure Super-Yang–Mills Vacua and Domain Walls

    How do anomaly-free discrete R symmetry, vacuum labels, line data, and domain-wall charges organize pure N=1 super-Yang–Mills?

    Principal question
    How do anomaly-free discrete R symmetry, vacuum labels, line data, and domain-wall charges organize pure N=1 super-Yang–Mills?
    Boundary
    Volume 6 retains generic gauge phases and confinement definitions, Volume 7 retains generic nonperturbative mechanisms, Chapter 10 owns duality, and claims about nonsupersymmetric QCD remain outside this chapter.
    Scope
    Pure-SYM global theory data, anomalous U(1)R remnant, candidate discrete vacua, gaugino order parameter, theta dependence, domain walls, one-form symmetry, mixed anomalies, and distinctions among exact, inferred, and semiclassical statements.
    Assumptions
    Supersymmetric Yang–Mills Actions; R-Symmetry, Anomalies, and the Holomorphic Scale; BPS Solitons, Walls, Strings, Vortices, and Junctions
  • Term

    Quantum Currents, Improvements, and Conservation

    When does a classical Noether current define a meaningful conserved quantum operator and charge?

    Principal question
    When does a classical Noether current define a meaningful conserved quantum operator and charge?
    Boundary
    Operator mixing, anomalous dimensions, and algebraic renormalization belong to Renormalization and EFT; curved-space stress tensors belong to Curved Spacetime.
    Scope
    Local current operators, conservation equations, improvements, surface dependence, charge existence, and the structural need for renormalized composite definitions.
    Assumptions
    Continuous Symmetries, Generators, and Charges; Classical Symmetries, Currents, and Stress Tensors
  • Term

    Quantum Fields as Operator-Valued Distributions

    Why must a quantum field be smeared with test functions, and what statements remain meaningful before choosing a rigorous axiom system?

    Principal question
    Why must a quantum field be smeared with test functions, and what statements remain meaningful before choosing a rigorous axiom system?
    Boundary
    Common invariant domains, Wightman axioms, microlocal spectrum conditions, and operator-algebraic constructions belong to Mathematical QFT.
    Scope
    The physical need for smearing, domains, distributional kernels, and the distinction between point notation and defined smeared operators.
  • Term

    Quantum Implementations, Projective Actions, and Central Extensions

    When is a physical symmetry implemented projectively, and what information is carried by its cocycle or central extension?

    Principal question
    When is a physical symmetry implemented projectively, and what information is carried by its cocycle or central extension?
    Boundary
    Full group-cohomology classification belongs to Mathematical Methods or Mathematical QFT; supersymmetry central charges belong to Supersymmetry and Duality; anomalies remain distinct.
    Scope
    Projective Hilbert-space actions, two-cocycles, central extensions, ray representations, and controlled relativistic or nonrelativistic examples.
    Assumptions
    What Is a Symmetry of a QFT?; Representations, Intertwiners, Invariants, and Tensor Decomposition
  • Term

    Quasifree States and Two-Point Functions

    Which positivity, commutator, field-equation, reality, and continuity conditions make a curved-space two-point distribution define a quasifree state?

    Principal question
    Which positivity, commutator, field-equation, reality, and continuity conditions make a curved-space two-point distribution define a quasifree state?
    Boundary
    Volume II owns Gaussian-state basics, Volume XIII owns information measures for Gaussian fields, and Volume XVI owns C-star algebraic existence and continuity proofs.
    Scope
    Quasifree-state reconstruction from two-point data, symmetric covariance versus antisymmetric causal kernel, Wick factorization, purity criteria, and state-dependent expectation values.
    Assumptions
    Covariant Algebraic Quantization and Fock Realizations; Complex Structures and One-Particle Spaces
  • Term

    Quasiparticle Poles, Residues, and Lifetimes

    When does a pole or narrow spectral feature define a quasiparticle, and how are residue, spectral lifetime, and transport lifetime kept distinct?

    Principal question
    When does a pole or narrow spectral feature define a quasiparticle, and how are residue, spectral lifetime, and transport lifetime kept distinct?
    Boundary
    Volumes 2 and 11 own the universal correlator, spectral, equilibrium, and response grammar; Volume 7 owns reusable functional methods. This chapter owns finite-density quantum-matter observables and approximation audits.
    Scope
    Pole equations, quasiparticle residue, dispersion, damping width, spectral versus transport lifetime, incoherent weight, and breakdown criteria; general spectral moments now belong to the next page.
    Assumptions
    Lehmann Representations and Spectral Functions in Matter; Dyson Equations and Self-Energies
  • Term

    Radial Time and Quantization on Spheres

    How does radius become Euclidean time, and what Hilbert-space data are defined on concentric spheres?

    Principal question
    How does radius become Euclidean time, and what Hilbert-space data are defined on concentric spheres?
    Boundary
    General Euclidean axioms, reflection positivity, and operator definitions remain in Volume 2; thermal cylinder physics remains in Volume 11; executable spectral checks require separately supplied code and data.
    Scope
    Radial ordering, spherical slices, the dilatation Hamiltonian, inside/outside states, path-integral preparation, and the relation between radial evolution and scale transformations.
    Assumptions
    Conformal Geometry, Maps, and Compactification; Euclidean Correlators and Schwinger Functions
  • Term

    Regions, Causal Complements, and Nets of Observables

    Which isotony, locality, covariance, and time-slice assumptions make a region-to-algebra assignment a physically usable continuum subsystem?

    Principal question
    Which isotony, locality, covariance, and time-slice assumptions make a region-to-algebra assignment a physically usable continuum subsystem?
    Boundary
    It does not own theorem-first Haag–Kastler reconstruction or any entropy formula that silently replaces a local algebra by a Hilbert-space tensor factor.
    Scope
    This page owns the net assignment O ↦ A(O), inclusion arrows, causal complements, commutants, representation dependence, and the operational meaning of local observables.
    Assumptions
    Operator Algebras and Positive Functionals: a Bridge; Spacelike Compatibility and Local Observables; Microcausality and Relativistic Compatibility
  • Term

    Relational, Boundary, and Asymptotic Observables

    How do relational bulk quantities, timelike-boundary data, and asymptotic observables evade the absence of local gauge-invariant gravitational operators while retaining operational meaning?

    Principal question
    How do relational bulk quantities, timelike-boundary data, and asymptotic observables evade the absence of local gauge-invariant gravitational operators while retaining operational meaning?
    Boundary
    Volume XIV owns low-energy relational observables, Volume IV owns S-matrix definitions, Chapter 12 owns explicit AdS reconstruction and dressing, and Volume XVI owns rigorous observable-algebra formulations.
    Scope
    A comparison of gravitationally dressed relational observables, AdS boundary observables, and asymptotic scattering data, including their anchors, algebras, causal domains, and perturbative precision.
    Assumptions
    Observable and Regime Matrix for Quantum Gravity
  • Term

    Relativistic Scattering Kinematics

    How are masses, channels, thresholds, and scattering angles encoded in Lorentz-invariant variables?

    Principal question
    How are masses, channels, thresholds, and scattering angles encoded in Lorentz-invariant variables?
    Boundary
    Analytic crossing theorem; Multi-Regge and high-energy resummation limits
    Scope
    Mandelstam invariants, center-of-mass frame, thresholds, Gram constraints, two-body angles, and crossing-region orientation
    Assumptions
    S-Matrix and T-Matrix Normalization
  • Term

    Relevant, Marginal, and Irrelevant Directions

    What makes a deformation relevant, marginal, or irrelevant, and why can marginality require higher-order analysis?

    Principal question
    What makes a deformation relevant, marginal, or irrelevant, and why can marginality require higher-order analysis?
    Boundary
    Naive classification by engineering dimension away from controlled regimes; Operator-basis construction owned by Chapter 8
    Scope
    Scaling dimensions, RG eigenvalues, relevant and irrelevant perturbations, marginally relevant and irrelevant flows, dangerously irrelevant variables, redundant operators, and convention translation
    Assumptions
    Fixed Points and Linearized RG Flow
  • Term

    Renormalized Composite-Operator Insertions

    Why does a local composite operator require its own renormalization beyond fields and couplings?

    Principal question
    Why does a local composite operator require its own renormalization beyond fields and couplings?
    Boundary
    Primary definitions of local operators and OPE owned by Volume 2; Model-specific matrix elements
    Scope
    Sources for composite insertions, short-distance singularities, operator counterterms, renormalized matrix elements, vacuum subtraction, normalization conditions, and insertion-dependent contact terms
    Assumptions
    Renormalization Conditions, Schemes, and Finite Parts; Local and Composite Operator Insertions
  • Term

    Renormalons, OPE Ambiguities, and Power Corrections

    How do factorial perturbative growth and Borel ambiguities communicate with operator-product power corrections without becoming a proof of a nonperturbative effect?

    Principal question
    How do factorial perturbative growth and Borel ambiguities communicate with operator-product power corrections without becoming a proof of a nonperturbative effect?
    Boundary
    Resurgence and condensate dynamics belong to Volume 7, model-specific estimates to Volume 6 and rigorous asymptotic analysis to Volume 16.
    Scope
    Factorial perturbative growth, Borel singularities and ambiguities, and their cancellation against OPE or power-suppressed ambiguities at the structural level.
    Assumptions
    Running Couplings and Dimensional Transmutation; Large Logarithms and RG Improvement; Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation; Free-Field OPE Preview and Ownership Map
  • Term

    Restricted States and Subregion Observables

    What exactly is restricted when a global QFT state is viewed on a local algebra, and which observables remain available to distinguish restrictions?

    Principal question
    What exactly is restricted when a global QFT state is viewed on a local algebra, and which observables remain available to distinguish restrictions?
    Boundary
    It does not identify every restriction with a trace-class density operator or import finite-dimensional tomography without specifying the represented local algebra.
    Scope
    This page owns algebraic state restriction, normality in a chosen representation, local expectation values, reduced descriptions, and the distinction between restriction and partial trace.
    Assumptions
    Operator Algebras and Positive Functionals: a Bridge; Operators, Observables, and Matrix Elements; Regions, Causal Complements, and Nets of Observables
  • Term

    S-Matrix and T-Matrix Normalization

    How are the identity contribution, momentum-conserving delta function, and invariant amplitude separated in the S-matrix?

    Principal question
    How are the identity contribution, momentum-conserving delta function, and invariant amplitude separated in the S-matrix?
    Boundary
    Unitarity cuts; Cross-section formulas; Alternative finite-volume normalization
    Scope
    S = 1 + iT convention, connected matrix elements, invariant amplitude, normalization factors, and dimensional checks
    Assumptions
    In and Out States
  • Term

    Scale versus Conformal Invariance: Hypotheses and Counterexamples

    Under which hypotheses does scale invariance enhance to conformal invariance, and where do counterexamples or unresolved cases evade those hypotheses?

    Principal question
    Under which hypotheses does scale invariance enhance to conformal invariance, and where do counterexamples or unresolved cases evade those hypotheses?
    Boundary
    General Lie theory and Lorentz representations remain in Volume 1; general global-symmetry, Ward-identity, and anomaly machinery remains in Volume 3; RG criteria for fixed points remain in Volume 5; theorem-level edge cases remain in Volume 16.
    Scope
    Virial currents, stress-tensor improvement, dimension, unitarity, discreteness, locality and spectrum assumptions, representative theorems, counterexample taxonomy, and calibrated claim language.
    Assumptions
    Conserved Currents and the Stress Tensor; Local Couplings, Trace Identities, and the Local Renormalization Group
  • Term

    Schrödinger Symmetry, Scale Invariance, and Anomalies

    When does a Galilean theory enlarge to Schrödinger symmetry, and how do contact interactions, renormalization, and dimensional transmutation obstruct scale invariance?

    Principal question
    When does a Galilean theory enlarge to Schrödinger symmetry, and how do contact interactions, renormalization, and dimensional transmutation obstruct scale invariance?
    Boundary
    Volume 2 owns Fock-space and path-integral foundations, Volume 3 owns symmetry and Ward identities, and Volume 5 owns generic EFT construction; this chapter owns their nonrelativistic many-body realization and matching.
    Scope
    The Schrödinger algebra, primary operators, nonrelativistic state–operator map, scale Ward identity, contact anomalies, and hypothesis ledger for exact versus emergent symmetry.
    Assumptions
    Galilean Fields, Scales, and Low-Energy Degrees of Freedom
  • Term

    Second-Quantized Bosons and Fermions

    How are indistinguishable nonrelativistic particles encoded as bosonic or fermionic fields and Fock-space operators?

    Principal question
    How are indistinguishable nonrelativistic particles encoded as bosonic or fermionic fields and Fock-space operators?
    Boundary
    Volume 2 owns Fock-space and path-integral foundations, Volume 3 owns symmetry and Ward identities, and Volume 5 owns generic EFT construction; this chapter owns their nonrelativistic many-body realization and matching.
    Scope
    Creation and annihilation fields, equal-time (anti)commutators, number and Hamiltonian operators, continuum and lattice bases, normal ordering, and observables.
    Assumptions
    Galilean Fields, Scales, and Low-Energy Degrees of Freedom; Canonical Quantization: Algebra, Representation, and State; Fock Space, Vacuum, and Particle Number
  • Term

    Sector Selection, Localization, and Transportability

    Which representations count as superselection sectors relative to the vacuum, and how do localization and transportability criteria encode physically movable charges?

    Principal question
    Which representations count as superselection sectors relative to the vacuum, and how do localization and transportability criteria encode physically movable charges?
    Boundary
    Volume III owns physical charges and generalized symmetries, while Volume XIII owns accessible information under superselection; this page fixes the algebraic sector object.
    Scope
    Vacuum-relative selection criteria, double-cone and cone localization, unitary equivalence outside a localization region, charge transporters, irreducibility, and finite-statistics qualifications.
    Assumptions
    Haag–Kastler Nets and Locality; States, GNS Representations, and Folia; Isotony, Additivity, Duality, and Primitive Causality
  • Term

    Silver Blaze Behavior, Thresholds, and Density Instabilities

    Why can zero-temperature observables remain independent of chemical potential below threshold, and how do onset, condensation, pairing, or inhomogeneous instabilities invalidate that Silver Blaze regime?

    Principal question
    Why can zero-temperature observables remain independent of chemical potential below threshold, and how do onset, condensation, pairing, or inhomogeneous instabilities invalidate that Silver Blaze regime?
    Boundary
    This chapter owns generic finite-density state and correlator structure; Volume 3 retains symmetry and background-gauge definitions, Volume 8 owns lattice sign-problem methods, and Chapter 18 owns current QCD evidence.
    Scope
    Ground-state charge thresholds, contour-shift explanation, onset nonanalyticity, Bose condensation, Fermi surfaces, pairing channels, negative susceptibilities, inhomogeneous response, and regulator/order-of-limits requirements.
    Assumptions
    Finite-Density Correlators and Charge Susceptibilities; Chemical Potentials and Finite-Density Ensembles
  • Term

    Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria

    Which large-N operator sectors behave like elementary bulk fields, multiparticle states, or collective variables, and where does that identification become basis dependent?

    Principal question
    Which large-N operator sectors behave like elementary bulk fields, multiparticle states, or collective variables, and where does that identification become basis dependent?
    Boundary
    Volume IX owns CFT operator and OPE data, Volume VII owns regulated collective constructions, and Chapters 3 and 7 own the AdS dictionary and interactions.
    Scope
    A bulk-facing operator taxonomy relating normalized single traces, multi-traces, collective bilocals, mixing matrices, and particle-number interpretations at specified orders in 1/N.
    Assumptions
    Large-N Factorization and Classical Bulk Scaling
  • Term

    Spacelike Compatibility and Local Observables

    Which graded commutators vanish at spacelike separation, and how does that statement differ for fields, gauge-fixed variables, and physical observables?

    Principal question
    Which graded commutators vanish at spacelike separation, and how does that statement differ for fields, gauge-fixed variables, and physical observables?
    Boundary
    The structural microcausality principle appears later; local nets and gauge-field locality subtleties belong to Mathematical QFT and Symmetry.
    Scope
    The first physical locality test, including graded locality and the field-versus-observable qualification.
    Assumptions
    Fields, Observables, and Interpolating Operators
  • Term

    Spacetime Currents, Stress Tensors, and Charge Algebras

    How do translations and Lorentz transformations lead to stress-tensor currents, angular-momentum charges, improvements, and charge algebras?

    Principal question
    How do translations and Lorentz transformations lead to stress-tensor currents, angular-momentum charges, improvements, and charge algebras?
    Boundary
    Foundations owns free-model derivations; Curved Spacetime owns metric stress tensors and renormalization; CFT owns conformal improvements and current algebras.
    Scope
    General physical spacetime-current structure, canonical versus improved currents, hypersurface generators, and bounded central-extension cautions.
    Assumptions
    Continuous Symmetries, Generators, and Charges; Classical Symmetries, Currents, and Stress Tensors
  • Term

    Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria

    Which spectral sparsity and operator-dimension gaps are needed for a semiclassical bulk with finitely many light fields and a parametrically separated cutoff?

    Principal question
    Which spectral sparsity and operator-dimension gaps are needed for a semiclassical bulk with finitely many light fields and a parametrically separated cutoff?
    Boundary
    Volume IX owns measurement and bootstrap bounds on CFT spectra; this page owns their bulk interpretation, while Chapter 8 owns quantitative locality diagnostics.
    Scope
    A criterion ledger distinguishing low-dimension sparsity, single-trace gaps, higher-spin gaps, central-charge scaling, and their separate implications for bulk field content and locality.
    Assumptions
    Central Charge, Newton Coupling, and the Planck Scale; Weakly Coupled Bulk Fields from Connected Correlators; Large-N and Sparse-Spectrum CFT Data
  • Term

    Standard Form, Cyclic and Separating Vectors

    How do cyclic and separating vectors place a local algebra, its commutant, and modular operators in standard form for information-theoretic use?

    Principal question
    How do cyclic and separating vectors place a local algebra, its commutant, and modular operators in standard form for information-theoretic use?
    Boundary
    It does not assert that every vector is cyclic and separating or replace domain analysis of antilinear operators with finite-matrix intuition.
    Scope
    This page owns cyclicity, separatingness, standard representation, natural cones, and the domain assumptions needed to define Tomita operators for a state-algebra pair.
    Assumptions
    Von Neumann Factors and Type-III Local Algebras
  • Term

    Statistical Ensembles and Field Configurations

    How does a probability measure on field configurations encode microcanonical, canonical, grand-canonical, and constrained equilibrium descriptions without confusing the ensemble with a single configuration?

    Principal question
    How does a probability measure on field configurations encode microcanonical, canonical, grand-canonical, and constrained equilibrium descriptions without confusing the ensemble with a single configuration?
    Boundary
    Volume 1 retains probability, measure, asymptotic, and convex-analysis foundations; Volume 2 retains vacuum functional-integral definitions; rigorous infinite-system existence belongs to Volume 16.
    Scope
    Classical and Euclidean field configuration spaces, density functionals, ensemble constraints, normalization, expectation values, conserved extensive data, and the operational distinction between microscopic states, macrostates, and ensemble measures.
    Assumptions
    Probability Spaces, Random Variables, and Conditional Expectation; Hamiltonian Initial Data and Phase Space
  • Term

    Superspace and Supertranslations

    How does superspace realize supertranslations geometrically, and how do its coordinates reproduce the super-Poincaré composition law?

    Principal question
    How does superspace realize supertranslations geometrically, and how do its coordinates reproduce the super-Poincaré composition law?
    Boundary
    Chapter 1 owns physical-state representations; Volume 1 retains Grassmann and spinor mathematics; Volume 3 retains gauge redundancy; dynamical actions begin in Chapter 4.
    Scope
    Bosonic and Grassmann coordinates, supertranslation group multiplication, left and right actions, differential supercharges, invariant one-forms, chiral coordinates, and finite transformations.
    Assumptions
    The Four-Dimensional N=1 Super-Poincaré Algebra; Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration
  • Term

    Symmetry, Gauge Redundancy, and Duality

    Which operational tests distinguish a physical global symmetry, a gauge redundancy, a duality between descriptions, and a spurionic covariance?

    Principal question
    Which operational tests distinguish a physical global symmetry, a gauge redundancy, a duality between descriptions, and a spurionic covariance?
    Boundary
    Gauge orbit mechanics is developed in Chapter 5; duality dynamics belongs to Supersymmetry and Duality; RG-emergent equivalences belong to Renormalization and EFT.
    Scope
    The diagnostic distinction among transformations acting on physical data, transformations quotiented as redundancy, equivalences between presentations or theories, and transformations of couplings or sources.
    Assumptions
    What Is a Symmetry of a QFT?
  • Term

    The Superfluid–Mott Quantum Phase Transition

    Which universality class governs the Bose–Hubbard superfluid–Mott transition at a lobe tip versus a density-driven edge?

    Principal question
    Which universality class governs the Bose–Hubbard superfluid–Mott transition at a lobe tip versus a density-driven edge?
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    Commensurate rotor theory, particle–hole symmetry at the tip, dilute-boson edges, dynamical exponents, scaling variables, dangerously irrelevant effects, and finite-size diagnostics.
    Assumptions
    The Bose–Hubbard Model and Controlled Limits; Universality Classes and Scaling Functions
  • Term

    The U(1)A Problem and QCD Topology

    How do the axial anomaly and topological fluctuations resolve the U(1)A puzzle in the hadron spectrum?

    Principal question
    How do the axial anomaly and topological fluctuations resolve the U(1)A puzzle in the hadron spectrum?
    Boundary
    Strong-CP solution mechanisms; Full instanton calculus; Current lattice determination
    Scope
    Singlet axial current, anomaly, topological charge and susceptibility, eta-prime mass orientation, large-N relation bridge, zero modes, and scheme cautions
    Assumptions
    Chiral Symmetry in QCD; Regulated Jacobians and Measure Variation; Theta Dependence in Yang–Mills and QCD
  • Term

    Thermal Boundary Conditions and Graded Traces

    How do ordinary thermal traces, chemical twists, and graded index traces differ in boundary conditions, positivity, and physical interpretation?

    Principal question
    How do ordinary thermal traces, chemical twists, and graded index traces differ in boundary conditions, positivity, and physical interpretation?
    Boundary
    This chapter owns thermal-state, KMS, Matsubara, and spectral dictionaries; Volume 8 retains regulator-specific Euclidean reconstruction machinery, Volume 9 owns thermal-CFT specializations, and Volume 16 owns theorem-first operator-algebraic status.
    Scope
    Trace insertions, spin-statistics boundary conditions, twists by conserved charges, fermion parity, supertraces, index versus thermal partition-function semantics, and admissible analytic continuations of holonomies.
    Assumptions
    Imaginary Time and Matsubara Frequencies; Continuous Symmetries, Generators, and Charges
  • Term

    Theta Dependence in Yang–Mills and QCD

    How does a theta parameter weight topological sectors and alter the vacuum structure of a gauge theory?

    Principal question
    How does a theta parameter weight topological sectors and alter the vacuum structure of a gauge theory?
    Boundary
    Instanton calculus; Strong-CP phenomenology; Complete nonperturbative vacuum solution
    Scope
    Topological sectors, theta weighting and periodicity, large transformations, vacuum branches, discrete-symmetry points, and observable consequences orientation
    Assumptions
    Theta Terms, Periodicity, and Vacuum Sectors
  • Term

    Theta Dependence, CP, and Model-Dependent Branches

    How can periodic theta dependence coexist with multiple vacuum branches and special CP behavior?

    Principal question
    How can periodic theta dependence coexist with multiple vacuum branches and special CP behavior?
    Boundary
    Universal claim of CP breaking at theta equals pi; QCD parameter bounds; Supersymmetric exact vacuum formulas. Those subjects remain with their canonical volume or Research owner.
    Scope
    Vacuum-energy periodicity, branch exchange, CP at special theta values, cusps and level crossings, anomaly constraints as inputs, and dimension/model dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Theta Parameters, Theta States, and Sector Sums
  • Term

    Theta Parameters, Theta States, and Sector Sums

    How do large gauge transformations and sector phases produce a theta-labeled quantum vacuum?

    Principal question
    How do large gauge transformations and sector phases produce a theta-labeled quantum vacuum?
    Boundary
    Primary gauge-global-form treatment; Instanton evaluation of tunneling; Strong-CP phenomenology. Those subjects remain with their canonical volume or Research owner.
    Scope
    Large-transformation action, sector superposition, theta character, periodicity, canonical and functional viewpoints, and convention dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Topological Sectors, Boundary Data, and Global Form
  • Term

    Topological Sectors, Boundary Data, and Global Form

    When does finite action split a configuration space into sectors labeled by a topological charge?

    Principal question
    When does finite action split a configuration space into sectors labeled by a topological charge?
    Boundary
    Theta coupling as a primary definition; Lattice topology algorithms; Classification for every gauge group and dimension. Those subjects remain with their canonical volume or Research owner.
    Scope
    Boundary conditions, homotopy classification, integer topological charge, sector-restricted functional integrals, and normalization, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Theta Terms, Periodicity, and Vacuum Sectors
  • Term

    Topological Susceptibility and Vacuum Response

    How are derivatives of the theta-dependent vacuum energy related to topological-charge cumulants?

    Principal question
    How are derivatives of the theta-dependent vacuum energy related to topological-charge cumulants?
    Boundary
    Lattice estimator implementation; Current numerical values; Axion phenomenology. Those subjects remain with their canonical volume or Research owner.
    Scope
    Generating relation, connected correlators, susceptibility, higher cumulants, contact-term and sign cautions, and finite-volume definition, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Theta Dependence, CP, and Model-Dependent Branches; Connected Correlators and Cumulants
  • Term

    Tunneling, Superselection, and the Infinite-Volume Limit

    When does tunneling mix candidate vacua, and when do they become distinct superselection sectors?

    Principal question
    When does tunneling mix candidate vacua, and when do they become distinct superselection sectors?
    Boundary
    Instanton prefactor calculation; Rigorous superselection theory; Finite-volume numerical spectroscopy. Those subjects remain with their canonical volume or Research owner.
    Scope
    Finite-volume level splitting, infinite-volume suppression, boundary conditions, cluster properties, domain walls, and order-of-limits cautions, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Local Expansions and Global Vacuum Structure
  • Term

    Twist Operators, Replica Defects, and Sewing Data

    How do twist operators or codimension-two replica defects encode cyclic sewing, and which defect data determine entanglement observables?

    Principal question
    How do twist operators or codimension-two replica defects encode cyclic sewing, and which defect data determine entanglement observables?
    Boundary
    It does not treat every entangling surface as a local primary insertion or import defect-CFT results beyond their dimension and symmetry assumptions.
    Scope
    This page owns permutation monodromy, twist scaling data, defect expansions, orientation, normalization, and the link from defect correlators to replicas.
    Assumptions
    Replica Trick and Branched Geometries
  • Term

    Ultraviolet Sensitivity and the Renormalization Problem

    Why do short-distance singularities require new parameters without destroying the predictive content of QFT?

    Principal question
    Why do short-distance singularities require new parameters without destroying the predictive content of QFT?
    Boundary
    Generic loop-integration machinery owned by Volume 4; Theorem-first construction of interacting QFT owned by Volume 16
    Scope
    Bare and renormalized parameters, ultraviolet sensitivity, local divergences, counterterm logic, renormalization conditions, predictive inputs, and regulator independence as a goal
    Assumptions
    Coincident Products and Contact Terms
  • Term

    Universality Classes and Scaling Functions

    Why can microscopically different theories share exponents and scaling functions near the same RG fixed point?

    Principal question
    Why can microscopically different theories share exponents and scaling functions near the same RG fixed point?
    Boundary
    Many-body phase catalogs owned by Volume 12; Conformal data as a replacement treatment
    Scope
    Universality classes, relevant data, scaling hypotheses, amplitudes and universal ratios, scaling functions, corrections, boundaries of universality, and emergent symmetry cautions
    Assumptions
    Critical Surfaces, Crossover, and Corrections to Scaling
  • Term

    Vacua, States, and Representations

    How are a state, a vacuum, an operator algebra, and a Hilbert-space representation related without assuming one universal Fock space?

    Principal question
    How are a state, a vacuum, an operator algebra, and a Hilbert-space representation related without assuming one universal Fock space?
    Boundary
    GNS theory, folia, superselection theory, and inequivalent-representation theorems belong to Mathematical QFT.
    Scope
    The physical distinction among states, vacua, algebras, representations, and sectors at introductory QFT depth.
  • Term

    Vacuum Ambiguity, Time Flow, and Observer Dependence

    Why does a generic curved spacetime lack a preferred vacuum or particle splitting, and which time flow, observer, asymptotic region, or operational choice supplies additional structure?

    Principal question
    Why does a generic curved spacetime lack a preferred vacuum or particle splitting, and which time flow, observer, asymptotic region, or operational choice supplies additional structure?
    Boundary
    Volume II owns general vacuum and excitation language, Chapter 4 owns detector and particle applications, and Volume XVI owns representation-theoretic state theorems.
    Scope
    The distinction among algebra, state, vacuum, particle interpretation, and observer response, including stationary selection, asymptotic choices, and the limits of instantaneous diagonalization.
    Assumptions
    Covariant Algebraic Quantization and Fock Realizations; Vacua, States, and Representations
  • Term

    Vector Spaces, Duals, Linear Maps, and Bases

    Which statements about vectors, covectors, and linear maps remain meaningful when a basis changes?

    Principal question
    Which statements about vectors, covectors, and linear maps remain meaningful when a basis changes?
    Boundary
    Topological duals and unbounded maps belong to Functional and Spectral Analysis.
    Scope
    Finite-dimensional spaces, duals, linear maps, kernels, images, rank, bases, and change of basis.
  • Term

    What Confinement Means in QCD with Dynamical Quarks

    How is confinement formulated in QCD with dynamical quarks when strings can break and center symmetry is not an exact criterion?

    Principal question
    How is confinement formulated in QCD with dynamical quarks when strings can break and center symmetry is not an exact criterion?
    Boundary
    A cross-theory definition or universal confinement order parameter; Proof of confinement in four dimensions; Generic confinement mechanisms owned by Volume 7
    Scope
    Gauge-invariant physical spectrum, absence of isolated colored asymptotic states, string breaking, static-source limits, and the contrast between quenched or pure-gauge diagnostics and dynamical-quark QCD
    Assumptions
    Coulomb, Higgs, and Confining Regimes; QCD Fields, Scales, and the Perturbative Domain
  • Term

    What Does Nonperturbative Mean?

    Relative to which expansion, formulation, observable, and limiting operation is a contribution nonperturbative, and what controls it?

    Principal question
    Relative to which expansion, formulation, observable, and limiting operation is a contribution nonperturbative, and what controls it?
    Boundary
    General asymptotic analysis belongs to Volume 1, interacting-QFT definitions to Volume 2, and renormalization-group or EFT control to Volume 5.
    Scope
    A control-first taxonomy of exponential sectors, strong coupling, reorganized expansions, exact special sectors, and regulated numerical formulations, with the expansion variable, observable, remainder, and limiting operation stated.
    Assumptions
    Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation; Regulators, Cutoffs, and Continuum Limits
  • Term

    What Is a Quantum Field Theory?

    Which data, observables, states, limits, and consistency conditions are required before a formal expression deserves to be called a quantum field theory?

    Principal question
    Which data, observables, states, limits, and consistency conditions are required before a formal expression deserves to be called a quantum field theory?
    Boundary
    Theorem-level comparisons and constructions belong to Mathematical QFT; renormalized continuum construction belongs to Renormalization, Lattice, and Mathematical QFT as appropriate.
    Scope
    A physical orientation to QFT as a structure of states, observables, correlations, dynamics, and controlled limits, together with a map of common formulations.
  • Term

    What Is a Symmetry of a QFT?

    What physical data must a transformation preserve, and how is the faithful symmetry obtained after quotienting transformations that act trivially?

    Principal question
    What physical data must a transformation preserve, and how is the faithful symmetry obtained after quotienting transformations that act trivially?
    Boundary
    Specific representations belong to model volumes; local-algebraic automorphisms and superselection reconstruction belong to Mathematical QFT.
    Canonical treatment
    What Is a Symmetry of a QFT?
    Scope
    Operational global-symmetry data on states, observables, operators, sectors, and correlators, including faithful kernels and the distinction between exact theory symmetry and Lagrangian presentation.
    Assumptions
    Groups, Actions, Quotients, and Covers
  • Term

    Why Continuum QFT Does Not Factorize Naively

    Why does the continuum local-algebra pair generally fail to realize H = H_A ⊗ H_B, and which regulated substitutes remain legitimate?

    Principal question
    Why does the continuum local-algebra pair generally fail to realize H = H_A ⊗ H_B, and which regulated substitutes remain legitimate?
    Boundary
    It does not forbid all approximate tensor products, nor may a cutoff factorization be promoted to a regulator-independent continuum identity.
    Scope
    This page owns the type-III, ultraviolet, and causal reasons naive spatial tensor factorization fails, together with a dictionary of lattice, split, and mode-factorized substitutes.
    Assumptions
    Direct Sums, Tensor Products, and Index Structure; Regions, Causal Complements, and Nets of Observables
  • Term

    Why Local Relativistic Quantum Theory Uses Fields

    Why are local quantum fields the natural language for relativistic systems with particle creation, locality, and many degrees of freedom?

    Principal question
    Why are local quantum fields the natural language for relativistic systems with particle creation, locality, and many degrees of freedom?
    Boundary
    Rigorous localization, local algebras, and no-go theorems belong to Mathematical QFT; nonrelativistic fields belong to Many-Body QFT and Quantum Matter.
    Scope
    The physical motivation for local fields without claiming that relativity alone logically forces one unique formalism.
  • Term

    Wightman Fields, Domains, and Axioms

    How are relativistic quantum fields defined as operator-valued tempered distributions on a common invariant domain, and what exactly do the Wightman axioms require?

    Principal question
    How are relativistic quantum fields defined as operator-valued tempered distributions on a common invariant domain, and what exactly do the Wightman axioms require?
    Boundary
    Volume II owns the physical field interpretation and free-field calculations; this chapter owns the axiom system and does not assume an interacting model exists merely because the axioms are consistent.
    Scope
    The field-domain statement of covariance, vacuum cyclicity, spectrum condition, locality, adjoint compatibility, temperedness, and common-domain invariance with all unbounded-operator qualifications visible.
    Assumptions
    Domains, Signatures, Supports, and Regularity Ledgers; Positivity, Spectrum, Covariance, and Locality Hypotheses; Test-Function Spaces, Distributions, Support, and Convergence; Unbounded Operators, Domains, Closure, and Adjoints
  • Term

    Wilsonian Coarse Graining and Theory Space

    How does integrating out short-distance degrees of freedom define a flow through theory space?

    Principal question
    How does integrating out short-distance degrees of freedom define a flow through theory space?
    Boundary
    Lattice blocking algorithms as executable methods; Subject-specific many-body coarse graining owned by Volume 12
    Scope
    Wilsonian effective actions, cutoff scale, coarse graining, theory space, quasi-local operators, semigroup structure, dimensionless variables, and exact versus truncated flow
    Assumptions
    Scale Independence and the Callan–Symanzik Equation; Gaussian Fields and Sources