Site map
Browse QFT.org by learning path, course, subject volume, research topic, or reference collection. Expand a group to see its pages; the directory works without JavaScript.
Learning, research, and reference
Start Here
- Readiness
- Diagnose mathematical readiness
- Diagnose classical-field and relativity readiness
- Diagnose quantum-mechanics readiness
- Diagnose statistical and probability readiness
- Diagnose computational and evidence readiness
- Linear and tensor methods repair
- Fourier, distributions, and Green functions repair
- Complex and asymptotic methods repair
- Variational and classical-field repair
- Relativity, Lorentz symmetry, and spin repair
- Quantum states and operators repair
- Statistical ensembles and probability repair
- Numerical and reproducibility repair
- Core QFT
- Orientation, conventions, and study contract
- Classical fields, actions, and local dynamics
- Quantum fields, states, and observables
- Canonical quantization and the free scalar
- Functional integrals and correlators
- Fermions, spin, and anticommutation
- Vector fields and gauge redundancy
- Symmetry, currents, and Ward identities
- Perturbative expansion and Feynman rules
- LSZ and tree amplitudes
- Loops and regularization
- Renormalization and the renormalization group
- QED and Yang–Mills theory
- Effective field theory and matching
- Infrared-safe observables and synthesis
- Learning pathways
- Compressed QFT refresher
- QFT for particle and nuclear physics
- QFT for quantum matter
- QFT for gravity and cosmology
- Mathematical foundations for physicists
- Physical foundations for mathematicians
- Scattering calculations for phenomenology
- Renormalization and EFT for working researchers
- Computational field theory onboarding
- Practice
- Exercises
- Checkpoints
- Capstones
- Solutions and feedback
- Research onboarding
- Literature, sources, and claim tracing
- Scope, conventions, and status
- Reproduce and validate a result
- Uncertainty, disagreement, and negative results
- Scope a first research project
Courses
- QFT I
- QFT II
- QFT III
- Relativistic Spectrum and Oscillator Modes
- Identical Particles and Fock Space
- Nonrelativistic Field Operators
- Interacting Bose Fields and Bogoliubov Sound
- Statistics, Occupation Algebra, and Anyons
- Free Nonrelativistic Fields and Mode Hamiltonians
- Klein–Gordon Equation and Scalar Modes
- Relativistic Scalar Action and Complex Fields
- Interaction Picture and Dyson Expansion
- Time Ordering, Contact Terms, and Green Functions
- Feynman Propagator and the iε Prescription
- Wick Theorem and n-Point Functions
- Wick Contractions and Diagrammatic Expansion
- Gaussian Integrals and Classical Paths
- Source Functionals in Quantum Mechanics and QFT
- Perturbation Theory from Functional Derivatives
- Schwinger–Dyson Identities and the Classical Limit
- φ⁴ Theory and First Diagrams
- Self-Energy and the Dyson Equation
- Pole Shifts, Mass Renormalization, and 1PI Diagrams
- Causality, Commutators, and Support
- Spectral Representation and Dispersion Integrals
- Thresholds, Cuts, and Imaginary Parts
- Scattering Amplitudes and Dyson Resummation
- Pole Residue and Wavefunction Renormalization
- LSZ Reduction and Cross Sections
- Euclidean Continuation and Gaussian Path Integrals
- Statistical Mechanics and Field Theory
- Critical Behavior and Thermal Mass
- Imaginary Time and Matsubara Propagators
- Lorentz Group and Complex Coordinates
- Scalar and Vector Representations on the Mass Shell
- Spinor Representations and Gamma Matrices
- Dirac Equation and Spinor Solutions
- Dirac Field Quantization and Propagators
- Grassmann Integrals and Fermionic Wick Theorem
- Vector Fields and Gauge Redundancy
- QED Vertices and Tree Amplitudes
- One-Loop Divergences and Power Counting
- Renormalization and Running Couplings
- Effective Actions, Dimensional Estimates, and IR Physics
- Contact Scattering and Renormalization in Quantum Mechanics
- Euclidean Loop Integrals and Feynman Parameters
- Scalar Propagators and One-Loop φ⁴ Theory
- Leading Logarithms and Nested Subgraphs
- RG Equation for the Four-Point Vertex
- Running Couplings and Critical Free Energy
- Mass Tuning and Relevant Deformations
- Operator Product Expansion in Perturbation Theory
- Wilsonian RG and Operator Mixing
- Callan–Symanzik Equations and Marginal Operators
- QED as an Effective Field Theory
- Vacuum Polarization and Gauge-Invariant Counterterms
- Running Charge, Screening, and Antiscreening
- Proper Time, Determinants, and Thermal Traces
- Effective Actions in Background Fields
- QED and Yang–Mills Beta Functions
- Dimensional Transmutation and Mass Gaps
- Ward Identities and Chiral Symmetries
- Current Correlators and Polarization Tensors
- Fermi Surface and Nonrelativistic Many-Body Fields
- Fermi-Surface Instabilities and One-Dimensional Fermions
- Gauge Fields in Two Dimensions
- Schwinger Model and Gauge-Invariant Correlators
- Bosonization and Sine-Gordon–Thirring Duality
- Confinement and Screening in Two-Dimensional QED
- Nonlinear Sigma Models and Constraints
- Sigma-Model Beta Function and Asymptotic Freedom
- Large-N Saddle Point in the O(N) Model
- Antiferromagnets, Spin Chains, and Theta Terms
- Symmetry Restoration and Mermin–Wagner Physics
- Sigma-Model Instantons and Topological Charge
- Theta Angle and Instanton Corrections
- Monopoles and Confinement in Three-Dimensional Gauge Theory
- Instantons in Quantum Mechanics and Vacuum Decay
- In–Out, In–In, and Schwinger–Keldysh Functionals
- Bogoliubov Coefficients and Pair Creation
- Rindler Coordinates and Green Functions
- Unruh Temperature and Thermal Periodicity
- Spheres, de Sitter Continuation, and Outlook
- Ising Model and Graphical Expansions
- Kramers–Wannier Duality and Mean-Field Theory
- Hubbard–Stratonovich Transformation and the Continuum Field
- Critical Propagators and the Upper Critical Dimension
- Epsilon Expansion and One-Loop Scaling
- Fixed Points, Tricriticality, and Order–Disorder Variables
- Disorder Lines, Branch Cuts, and Defect Operators
- Order–Disorder Duality and Ising Fermions
- Relativistic Fields, Spinors, and the Ising Continuum Limit
- Lattice Dirac Equations and Euclidean Spinors
- Z₂ Gauge Systems, Wilson Loops, and Free Correlators
- Scaling Dimensions, Correlators, and Critical Exponents
- Callan–Symanzik Scaling and Emergent Conformal Symmetry
- Conformal Transformations in d Dimensions
- OPE, Associativity, and Conformal Correlators
- Finite Conformal Maps, Gauge Symmetry, and Free Fields
- Free Fields, Wightman Functions, and the iε Prescription
- Commutators, Currents, and Ward Identities
- Stress Tensors, Goldstone Modes, and QED Ward Identities
- Two-Dimensional QED, the Thirring Model, and Currents
- Stress-Tensor OPE and Conformal Ward Identities
- Primary Fields, Cylinder Maps, and Mode Expansions
- Virasoro Generators, Descendants, and Central Charge
- Schwarzian Derivative and the Virasoro Algebra
- Global Conformal Generators and Descendant States
- Null States and BPZ Differential Equations
- Degenerate Fields, Kac Labels, and Constraints
- Minimal Models, Fusion, and the Ising Operator Algebra
- Ising CFT, Majorana Fermions, and Tricritical Extensions
- Superconformal Currents, Maxwell Lines, and Worldline Actions
- Worldlines, Worldsheets, and Reparametrization Gauge
- Conformal Gauge, Two-Dimensional Gravity, and Vacuum Polarization
- Conformal Anomalies, Liouville Action, and Nonlocal Effective Actions
- Gauge-Invariant Observables and Asymptotic Amplitudes
- Random Lattices, Matrix Models, and Random Surfaces
- Superfluidity, Landau Criterion, and Accelerated Frames
- Compact Phases, Vortices, and Duality
- Compact Gauge Fields, Higgsing, Solitons, and Topological Defects
- Wilson Loops, Worldlines, Monopole Plasma, and Confinement
- Strings, Branes, Sigma Models, and AdS Hints
Research
- Research Fields
- Frontier Questions
- Research Methods
- Evidence Briefs
- Research Editions
- Amplitudes and Precision Scattering
- Effective Field Theory and Tests of the Standard Model
- Nonperturbative Gauge Dynamics
- Lattice and Hamiltonian Field Theory
- Conformal Field Theory and Bootstrap
- Supersymmetry and Duality
- Thermal and Nonequilibrium Field Theory
- Quantum Matter and Emergence
- Quantum Information and Entanglement in QFT
- QFT in Curved Spacetime and Semiclassical Gravity
- Holography and Quantum Gravity
- Mathematical and Constructive QFT
- Infrared-Complete Scattering Observables
- The Domain of Color–Kinematics Duality and Double Copy
- SMEFT Validity and Truncation in Global Fits
- Electroweak Vacuum Stability
- The Four-Dimensional Yang–Mills Mass Gap
- Confinement with Dynamical Matter
- Controlled Regimes of Finite-Density QCD
- Real-Time Continuum Dynamics from Regulated Computations
- From Crossing Solutions to Actual CFTs
- Certification in the Numerical Conformal Bootstrap
- Non-Supersymmetric Dualities in Three Dimensions
- Reconstructing Non-Lagrangian QFTs
- The QCD Critical Point
- The Predictive Domain of Hydrodynamic Attractors
- The Order of Deconfined Quantum Critical Transitions
- What Does Planckian Dissipation Mean?
- Operational Entanglement in Continuum QFT
- Complexity in Continuum QFT
- Robustness and Backreaction in Hawking Radiation
- Hypotheses of the Generalized Second Law
- Islands and Unitary Black-Hole Evaporation
- Bulk Reconstruction Beyond Semiclassical Code Subspaces
- Interacting Constructive QFT in Four Dimensions
- Equivalence of Axiomatic QFT Frameworks
- Multiloop Amplitudes and Resummation
- EFT Inference, Power Counting, and Truncation
- Functional Equations and Functional Renormalization Group
- Euclidean Lattice Inference and Continuum Extrapolation
- Hamiltonian Truncation, Tensor Networks, and Quantum Simulation
- Analytic and Numerical Conformal Bootstrap
- Semiclassics, Resurgence, and Transseries
- Real-Time QFT, Kinetic Theory, and Hydrodynamics
- Replica, Modular, and Operator-Algebra Methods
- Holographic Reconstruction and Gravitational Path Integrals
- Muon Anomalous Magnetic Moment
- W-Boson Mass Measurements and Fits
- Three-Dimensional Ising Bootstrap Benchmark
- Finite-Temperature QCD Equation of State
- Evidence for Hydrodynamic Attractors
- Evidence for Planckian Dissipation
- Analogue Hawking Radiation
- Scope of the Quantum Null Energy Condition
Reference
Subject volumes
Mathematical Methods
Linear, Tensor, and Graded Algebra
Analysis, Measure, and Fourier Methods
- Limits, Completeness, and Modes of Convergence
- Measures and Measurable Functions
- Lebesgue Integration and Convergence Theorems
- Product Measures, Fubini–Tonelli, and Change of Variables
- Lp Spaces, Inequalities, and Weak Convergence
- Fourier Series, Fourier Transforms, and Plancherel Theory
- Convolution, Approximate Identities, and Poisson Summation
Complex Analysis and Analytic Continuation
Distributions and Microlocal Methods
- Test-Function Spaces, Distributions, Support, and Convergence
- Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards
- Tempered Distributions and Fourier Calculus
- Distributional Kernels and Distributions on Manifolds
- Products, Scaling Degree, and Extensions of Singular Distributions
- Singular Support and Wavefront Sets
Differential Equations and Green Operators
- Linear ODEs, Evolution Operators, and Wronskians
- Sturm–Liouville Problems and Eigenfunction Expansions
- Symbols, Characteristics, and PDE Type
- Weak Solutions, Sobolev Spaces, and Well-Posedness
- Fundamental Solutions and Green Operators
- Hyperbolic Equations and Causal Propagators
- Elliptic Boundary Problems and Heat Kernels
Functional and Spectral Analysis
- Convex Cones, Separation, Conic Duality, and Semidefinite Programs
- Banach and Hilbert Spaces, Completion, and Riesz Representation
- Bounded, Compact, and Integral Operators
- Unbounded Operators, Domains, Closure, and Adjoints
- Self-Adjointness, Extensions, and Unitary Evolution
- Spectra, Resolvents, Spectral Measures, and Functional Calculus
- Trace Ideals and Fredholm Determinants
- Locally Convex, Nuclear, and Rigged Hilbert Spaces
- Operator Algebras and Positive Functionals: a Bridge
Lie Groups, Representations, and Spinors
- Groups, Actions, Quotients, and Covers
- Lie Groups, Lie Algebras, and Exponential and Adjoint Maps
- Representations, Intertwiners, Invariants, and Tensor Decomposition
- Compact Lie Groups, Roots, Weights, and Weyl Structure
- Lorentz Field Representations and Poincaré Particle Representations
- Clifford Algebras and Pin and Spin Groups
- Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities
Differential Geometry and Bundles
- Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields
- Differential Forms, Integration, Orientation, and Stokes Theorem
- Metrics, Volume Forms, Hodge Star, and Laplace Operators
- Levi–Civita Connections, Geodesics, and Riemann Curvature
- Vector, Principal, and Associated Bundles
- Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities
- Parallel Transport and Holonomy
- Spin Structures and Dirac Operators
Topology, Cohomology, and Characteristic Classes
Variational, Symplectic, and Constraint Methods
Probability and Stochastic Processes
- Probability Spaces, Random Variables, and Conditional Expectation
- Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems
- Characteristic Functions, Moments, Cumulants, and Generating Functionals
- Gaussian Vectors, Processes, Random Distributions, and Wick Structure
- Stochastic Processes and Correlation Functions
- Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics
- Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error
Asymptotic and Special-Function Methods
- Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation
- Laplace Method and Steepest Descent
- Stationary Phase, Coalescing Saddles, and Stokes Geometry
- WKB and Eikonal Methods and Turning-Point Matching
- Mellin Transforms and Scaling Asymptotics
- Special Functions from Equations and Boundary Data
- Heat Kernels, Zeta Functions, and Spectral Determinants
- Lagrangian Submanifolds, Generating Functions, and Semiclassical Phases
Categorical, Homological, and Local-to-Global Language
Foundations of Quantum Field Theory
Quantum Fields, States, and Observables
- What Is a Quantum Field Theory?
- Why Local Relativistic Quantum Theory Uses Fields
- Vacua, States, and Representations
- One-Particle States: Mass, Spin, and Relativistic Normalization
- Multiparticle States, Statistics, and Fock Organization
- Quantum Fields as Operator-Valued Distributions
- Fields, Observables, and Interpolating Operators
- Spacelike Compatibility and Local Observables
- Antiparticles and Charge-Conjugate Excitations
Classical Local Fields and Actions
Canonical Quantization and the Free Scalar
- The Klein–Gordon Field and Its Modes
- Canonical Quantization: Algebra, Representation, and State
- Quantizing the Real Scalar Field
- Scalar Propagators, Ordered Correlators, and Sources
- Fock Space, Vacuum, and Particle Number
- Complex Scalars and Conserved Charge
- Normal Ordering and Vacuum Terms
- Schrödinger Wave Functionals
- Coherent States and the Classical Limit
- Massless Scalars, Zero Modes, and Infrared Limits
Functional Integrals and Free Gaussian Fields
Free Fermion Fields
Free Spin-One Fields and Constraints
Correlators, Sources, and Effective Actions
States, Observables, and Spectra
Local Operators and Short-Distance Structure
Lorentzian, Euclidean, and In-In Formulations
- Lorentzian Boundary Conditions and the iε Prescription
- Wick Rotation and Analytic Continuation
- Euclidean Correlators and Schwinger Functions
- Reflection Positivity within Osterwalder–Schrader Reconstruction
- Retarded, Advanced, and Spectral Correlators
- In–Out versus In–In Expectation Values
- Closed-Time-Path Grammar
Structural Principles and Axiom Maps
- Poincaré Covariance and the Spectrum Condition
- Hilbert Positivity and Unitary Evolution
- Microcausality and Relativistic Compatibility
- Clustering, Vacuum Assumptions, and Long-Range Correlations
- Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks
- The Spin–Statistics Connection
- CPT: Hypotheses, Content, and Limits
- Structural Hypotheses and Failure Modes
Symmetry and Gauge Structure
Symmetry, Actions, and Redundancy
Currents, Charges, and Quantum Ward Identities
Background Fields and Gauging
Gauge Structure, Global Form, and Observables
- Gauge Fields, Redundancy, and Observable Content
- Gauge Orbits, Gauss Constraints, and Stabilizers
- Local Potentials and Global Gauge Configurations
- Gauge-Invariant and Dressed Observables
- Global Form, Matter Representations, and the Faithful Gauge Group
- Genuine Line Spectra, Discrete Theta Data, and Theory Specification
- Large Gauge Transformations and Topological Sectors
Symmetry Realization and Breaking
- Symmetry Realization and Order Parameters
- Finite Volume, Thermodynamic Limits, and Pure Phases
- Vacuum Orbits and Unbroken Subgroups
- Goldstone's Theorem: Hypotheses and Pole Argument
- Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions
- Cosets and Nonlinear Realizations
- Explicit Breaking and Pseudo-Goldstone Modes
- Elitzur's Theorem and the Gauge-Invariant Higgs Mechanism
Boundary Symmetry, Surface Charges, and Edge Modes
Gauge Fixing, BRST, and BV
- The Faddeev–Popov Construction
- Ghosts, Auxiliary Fields, and Gauge-Parameter Dependence
- Gribov Copies and the Limits of Local Gauge Fixing
- The BRST Differential and Gauge-Fixed Complex
- BRST Cohomology and Physical Observables
- Slavnov–Taylor and Zinn-Justin Identities
- BV Fields, Antifields, and the Odd Symplectic Structure
- Master Equations and BV Gauge Fixing
Anomalies, Inflow, and Matching
- What Is an Anomaly?
- Regulated Jacobians and Measure Variation
- Perturbative Chiral and Gauge Anomalies
- Consistent and Covariant Anomalies
- Wess–Zumino Consistency and Descent
- Anomaly Polynomials and Inflow
- Global and Torsion Anomalies
- Gravitational, Mixed, Discrete, and Orientation-Reversing Anomalies
- 't Hooft Anomaly Matching
Extended Operators and Defects
- Support, Codimension, and Operator Data
- Wilson Lines and Loops
- Disorder Operators and Singular Boundary Conditions
- Genuine Lines, Screening, and Charge Lattices
- Surface Defects and Codimension-Two Monodromy
- Boundaries, Interfaces, and Domain Walls
- Fusion, Junctions, and Endpoints
- Linking, Braiding, and Framing
Higher-Form and Higher-Group Symmetries
- Higher-Form Symmetry from Operators and Linking
- Higher-Form Currents, Charges, Backgrounds, and Ward Identities
- Electric and Magnetic One-Form Symmetries
- Breaking Higher-Form Symmetry and Diagnosing Phases
- Gauging a Higher-Form Symmetry
- Higher-Group Symmetry and Coupled Backgrounds
- Higher-Group Operators, Gauging, and Anomalies
Non-Invertible Symmetries
Topological Terms and Invertible Responses
Topological Gauge Theories and Symmetry TFT
Perturbative QFT and Scattering
Perturbative Expansion and Feynman Rules
- The Interaction Picture and Dyson Series
- Wick Expansion for Interacting Fields
- Diagrammatics and Symmetry Factors
- Momentum-Space Feynman Rules
- Fermion Signs and Closed Loops
- Derivative Interactions and Contact Terms
- Connected, Disconnected, and Vacuum Diagrams
- Functional Derivation of Perturbation Theory
- Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks
Asymptotic States, LSZ, and Scattering Observables
Tree Amplitudes and Gauge Consistency
S-Matrix Unitarity, Analyticity, and Resonances
On-Shell Construction
- On-Shell States and Little-Group Scaling
- Spinor-Helicity Variables
- Three-Point Amplitudes
- Massive On-Shell Variables and Little-Group Covariance
- Complex Momenta and Factorization
- BCFW Recursion
- Color Decomposition and Partial Amplitudes
- Soft Limits as On-Shell Constraints
- Constructibility, Boundary Terms, and Failure Modes
Loop Integrals and Reduction
- Anatomy of a Loop Integral
- Feynman and Schwinger Parameters
- Dimensional Regularization as an Amplitude Tool
- UV/IR Poles and the Renormalized-Amplitude Interface
- One-Loop Integral Families and Analytic Functions
- Tensor Reduction
- Expansion by Regions
- Integration-by-Parts Identities and Master Integrals
- Differential Equations for Master Integrals
- Numerical Evaluation and Validation of Loop Integrals
Singularities, Cuts, and Integrand Reconstruction
Dispersion, Positivity, and UV Constraints
Infrared Structure and Factorization
- Soft and Collinear Singularities
- Soft Theorems
- Collinear Factorization and Splitting Amplitudes
- Bloch–Nordsieck and KLN Cancellation
- Eikonal Approximation and Wilson Lines
- Hard, Jet, and Soft Factorization
- Rapidity Divergences, Glauber Exchange, and Factorization Limits
- Sudakov Logarithms and Resummation
- Dressed States and Infrared-Finite Scattering
Infrared-Safe and Precision Observables
- Measurement Functions and Inclusive Observables
- Infrared and Collinear Safety
- Real–Virtual Cancellation and Subtraction
- Phase-Space Integration and Monte Carlo Estimators
- Fixed-Order Organization and Scale Dependence
- Resummation and Fixed-Order Matching
- Jets and Event-Shape Observables
- Unstable-Particle Observables and Controlled Resonance Approximations
- Validation and Theory Uncertainties
Renormalization and Effective Field Theory
Ultraviolet Renormalization and Locality
- Ultraviolet Sensitivity and the Renormalization Problem
- Power Counting of Divergences and Perturbative Renormalizability
- QFT Regulator Families and Their Tradeoffs
- Dimensional Regularization and Minimal Subtraction
- Local Counterterms and Subdivergence Structure
- The R-Operation, Forest Formula, and Overlapping Divergences
- Renormalized Perturbation Theory and Counterterm Rules
- Symmetry Constraints and the Space of Counterterms
- Renormalization Conditions, Schemes, and Finite Parts
- Regulator Removal and Renormalized Predictions
Composite Operators and Mixing
- Renormalized Composite-Operator Insertions
- Contact Terms and Renormalized Operator Products
- Operator Mixing and Renormalization Matrices
- Operator Anomalous-Dimension Matrices
- Dual Evolution of Operators and Wilson Coefficients
- Symmetry-Protected Operators, Currents, and Improvement
- Nonperturbative Renormalization Schemes and Step Scaling
Renormalization-Group Equations and Running
- Scale Independence and the Callan–Symanzik Equation
- Beta Functions, Running Masses, and Field Anomalous Dimensions
- Running Couplings and Dimensional Transmutation
- Scheme Transformations and RG Invariants
- Multiple Couplings and Coupled RG Flows
- Large Logarithms and RG Improvement
- Local Couplings, Trace Identities, and the Local Renormalization Group
- Renormalons, OPE Ambiguities, and Power Corrections
Wilsonian and Functional Renormalization
Fixed Points, Universality, and Continuum Limits
- Fixed Points and Linearized RG Flow
- Relevant, Marginal, and Irrelevant Directions
- Critical Exponents, Scaling Relations, and Hyperscaling Caveats
- Critical Surfaces, Crossover, and Corrections to Scaling
- Universality Classes and Scaling Functions
- Gaussian and Wilson–Fisher Fixed Points
- Ultraviolet and Infrared Fixed Points: Criteria and Evidence
Effective Field Theory: Construction and Power Counting
- Effective Field Theory as a Controlled Expansion
- Scale Separation, Locality, and the Domain of an EFT
- Degrees of Freedom, Symmetry, and the Local Operator Expansion
- Power Counting and Predictive Order
- Loops, Counterterms, and Closure of an EFT Expansion
- Nonperturbative Iteration, Shallow Scales, and Power-Counting Consistency
- EFT Truncation Errors and Breakdown Diagnostics
Matching, Decoupling, and Threshold Evolution
- Integrating Out Heavy Fields
- Matching with Amplitudes, Green Functions, and Background Fields
- Tree-Level Matching and Classical Elimination
- Matching Conditions Beyond Tree Level
- Infrared Cancellation, Regulators, and Matching Consistency
- Decoupling Theorems and Threshold Corrections
- Running and Matching across Multiple Thresholds
- Nondecoupling Effects and Matching Validation
Operator Bases and Field Redefinitions
- From Operator Lists to Independent Bases
- Integration by Parts and Equation-of-Motion Redundancy
- Local Field Redefinitions and the Equivalence Theorem
- Representation and Spurion Constraints on Operator Bases
- Flavor, Hermiticity, and CP Bookkeeping
- Fierz Relations and Dimension-Specific Identities
- Evanescent Operators, Finite Renormalization, and RG Closure
- Hilbert-Series Counting and Completeness Diagnostics
- On-Shell Amplitude Bases and the Operator Correspondence
- Basis Translation, Scheme Dependence, and Reproducibility
Modes, Factorization, and Multiscale RG
- Modes, Virtualities, and EFT Scale Separation
- Multipole Expansion and Homogeneous Mode Power Counting
- Matching onto Factorized Operator Structures
- Evolution Kernels, Consistency Relations, and Resummation Architecture
- Rapidity Renormalization and Two-Scale Evolution
- Soft-Collinear Effective Theory: Architecture and Validity
- Unstable-Particle Effective Theory and the Width Expansion
Effective-Theory Architecture Atlas
- A Map of Effective-Theory Architectures
- Chiral Effective Theory and Nonlinear Symmetry
- Nuclear and Few-Body EFT Architecture
- Heavy-Particle EFT and HQET Architecture
- NRQED, NRQCD, and Potential EFT Architecture
- SMEFT and HEFT: Architecture and Domain
- Effective Field Theory of Gravity: Architecture and Power Counting
- Hydrodynamic Effective-Theory Architecture
- Open-System Effective-Theory Architecture and Consistency Conditions
Naturalness, Scale Sensitivity, and Emergence
Gauge Theories and the Standard Model
Gauge Dynamics: Charges, Scales, and Phases
Quantum Electrodynamics
- The QED Action, Charges, and Observables
- Tree-Level QED Processes
- Renormalizing the Electron and Photon
- The Ward–Takahashi Identity and Charge Renormalization
- Vacuum Polarization, Running Charge, and Screening
- Soft Photons and Infrared-Finite QED
- Lepton Magnetic Moments and the QED Form-Factor Expansion
- Bound-State QED and NRQED
Yang–Mills Theory
- The Yang–Mills Action and Gauge Self-Interaction
- Yang–Mills Equations, Constraints, and Observables
- Gauge-Fixed Yang–Mills Action and Ghost Sector
- Yang–Mills Color Algebra and Perturbative Vertices
- Non-Abelian Screening and Asymptotic Freedom
- Background-Field Yang–Mills Effective Action
- Consistency of Perturbative Yang–Mills Theory
Perturbative QCD and Partons
- QCD Fields, Scales, and the Perturbative Domain
- Short-Distance QCD Parameters, Mass Schemes, and Thresholds
- Inclusive Annihilation and the Emergence of Jets
- Deep-Inelastic Scattering and the Parton Model
- Collinear Factorization and Operator-Defined PDFs
- DGLAP Evolution and Scaling Violation
- Fragmentation Functions and Timelike Evolution
- High-Energy QCD and Small-x Evolution
- TMD Factorization, Rapidity Evolution, and Glauber Limits
- Hadron-Collider Factorization and Parton Luminosities
- QCD Radiation, Jets, and Event Shapes
- QCD Prediction and Uncertainty Ledger
Confinement and Chiral QCD
- What Confinement Means in QCD with Dynamical Quarks
- Static Sources, Center Symmetry, and String Breaking in QCD
- QCD Confinement versus the Yang–Mills Mass Gap
- Chiral Symmetry in QCD
- Chiral Order Parameters, Current Algebra, and Pions
- Chiral Lagrangians and Low-Energy QCD
- The U(1)A Problem and QCD Topology
Hadrons and Heavy Quarks
Nuclear and Few-Body EFT
Electroweak Theory and the Higgs
- Electroweak Gauge and Matter Structure
- The Higgs Doublet and Electroweak Symmetry Breaking
- Gauge-Boson Masses and Electroweak Mixing
- Charged and Neutral Weak Currents
- Yukawa Couplings and Fermion Masses
- Higgs Self-Interactions and the Scalar Potential
- The Fermi Limit of Weak Interactions
- Longitudinal Vector Bosons and the Equivalence Theorem
- Electroweak Renormalization and Input Schemes
- Higgs Interactions, Production, Decay, and Pole Observables
Quark Flavor and CP
Neutrino and Lepton Physics
Standard Model Assembly and Consistency
- The Standard Model Lagrangian and Parameter Map
- The Global Form of the Standard Model Gauge Group
- Gauge-Anomaly Cancellation and Quantum Consistency
- Accidental Symmetries and Their Violations
- Standard Model Running, Effective Potential, and Vacuum-Stability Criteria
- SMEFT and HEFT in Standard Model Observables
Precision Standard Model: Observables and Inference
Consistent Extensions and Portals
- Consistency Checklist for Standard Model Extensions
- Consistent New Matter and Gauge Sectors
- Higgs-Singlet Scalar Portals
- Extended Higgs Sectors: Alignment, Custodial Symmetry, and Decoupling
- Vector Portals and Kinetic Mixing
- Fermion and Neutrino Portals
- Axionlike and Pseudoscalar Portals
- Effective, Simplified, and Mediator Descriptions
- Search Validity and Reinterpretation for Portal Models
Nonperturbative Dynamics
Nonperturbative Questions, Vacua, and Sector Dynamics
- What Does Nonperturbative Mean?
- Local Expansions and Global Vacuum Structure
- Topological Sectors, Boundary Data, and Global Form
- Theta Parameters, Theta States, and Sector Sums
- Theta Dependence, CP, and Model-Dependent Branches
- Topological Susceptibility and Vacuum Response
- Tunneling, Superselection, and the Infinite-Volume Limit
- Anomaly and Generalized-Symmetry Constraints on Infrared Phases
Semiclassical Expansions and Integration Cycles
- Saddles, Control Parameters, and Loop Counting
- Fluctuation Operators and Determinant Ratios
- Zero Modes, Collective Coordinates, and Moduli Measures
- Negative Modes and Instability Indices
- Complex Saddles, Lefschetz Thimbles, and Integration Cycles
- Multi-Saddle Sums and Dilute Ensembles
- Stokes Jumps, Saddle Dominance, and Contour Dependence
- Renormalized Saddle Contributions and Validity Tests
Solitons, Defects, and Collective Dynamics
- Finite-Energy Boundary Data, Charge, and Stability
- Derrick Scaling, Virial Tests, and Nontopological Stability
- Kinks and Domain Walls
- Vortices, Flux Quantization, and Core Scales
- Monopoles and Dyons
- Lumps, Textures, and Skyrmions
- Q-Balls, Oscillons, and Sphalerons
- Bogomolny Bounds and First-Order Equations
- Moduli-Space Dynamics and Collective Quantization
Instantons, Fermion Zero Modes, and Tunneling
- Euclidean Tunneling Saddles and Boundary Conditions
- Quantum-Mechanical Instantons and Tunnel Splitting
- Gauge Instantons, Topological Charge, and Moduli
- Instanton Measures, Zero Modes, and Determinants
- Dilute Instanton Ensembles and Theta Dependence
- Fermion Zero Modes, Index Data, and Selection Rules
- The Instanton Size Modulus and Infrared Limitations
- Fractional Events, Calorons, and Monopole Constituents
Metastability and Vacuum Decay
- Bounce Solutions and False-Vacuum Boundary Conditions
- Decay Rates, the Negative Mode, and Prefactors
- Bounce Existence, Symmetry, and Multifield Geometry
- Thin-Wall Control and Beyond-Thin-Wall Corrections
- Gauge and Renormalization-Scale Dependence of Decay Rates
- Validity Limits and Thermal and Gravitational Handoffs
Strong-Coupling Laboratories and Dual Variables
- Sigma-Model Dynamics: Target Geometry, Dimension, and Control
- The O(N) Model as a Strong-Coupling Laboratory
- The CP(N-1) Model
- Gap Equations, Dimensional Transmutation, and Physical Mass
- Theta Terms and Topological Effects in Sigma Models
- Strong-Coupling Phases and Cross-Method Evidence
- The Principal Chiral Model and the Integrability Bridge
- The Gross–Neveu Model and Dynamical Mass Generation
- The Schwinger Model, Screening, and Bosonization
Confinement, Screening, and Mass Gaps in Controlled Regimes
- Confinement Definitions and Their Non-Equivalence
- Line Operators, Screening, and Generalized-Symmetry Diagnostics
- Static Potentials, Flux Tubes, and Effective Strings
- Compact U(1) in 2+1 Dimensions and the Monopole Plasma
- Small-Circle Abelianization, Bions, and Adiabatic Continuity
- Proposed Confinement Mechanisms and Discriminating Observables
- The Non-Abelian Mass Gap as a Spectral Statement
- Confinement Evidence, Mechanisms, and Open Problems
- Low-Dimensional Confinement and Screening Laboratories
Large-N Dynamics
- Large-N Limits, Normalizations, and Orders of Limits
- Vector Models, Auxiliary Fields, and Large-N Saddles
- Matrix Eigenvalue Saddles, Loop Equations, and Phase Transitions
- Double-Line Counting and the Topological Expansion
- Large-N Factorization and Master-Field Claims
- Planar Gauge Dynamics and String-Like Organization
- Volume Reduction, Large-N Equivalence, and Their Hypotheses
- Tensor Large N and Melonic Dominance
- Subleading Corrections, Double Scaling, and Nonuniform Limits
Continuum Functional Equations and Controlled Truncations
- Schwinger–Dyson Hierarchies and Renormalization Inputs
- Closure, Symmetry Constraints, and Branch Selection
- 2PI and nPI Effective Actions
- Bethe–Salpeter and Faddeev Bound-State Equations
- Functional-RG Applications and Cross-Checks
- Coupled Propagator, Vertex, and Bound-State Systems
- Complex-Momentum, Spectral, and Real-Time Information
- Functional-Method Validation and Error Control
- Gauge Fixing, BRST Constraints, and the Gribov Problem
Resurgence and Transseries
- Large-Order Growth and the Borel Transform
- Borel Singularities, Lateral Sums, and Stokes Data
- Transseries Sectors and Parameters
- Instantons and Large-Order Relations
- Renormalons, OPE Ambiguities, and Transseries
- Ambiguity Cancellation and Transseries Consistency
- Complex Saddles, Quasi-Zero Modes, and Hidden Phases
- Constructive Borel Summability and Its Boundaries
- Evidence and Limits for Resurgence in QFT
Integrable Relativistic QFT
- Classical and Quantum Conserved Charges
- Elasticity, Factorization, and Their Hypotheses
- The Yang–Baxter Equation
- Exact S-Matrix Bootstrap, CDD Freedom, and Completeness
- Bound-State Poles, Fusion, and Coleman–Thun Alternatives
- Bethe Quantization and Finite-Volume Spectra
- Form-Factor Bootstrap and Correlator Expansions
- Thermodynamic Bethe Ansatz and Finite-Size Ground-State Energy
- Integrable Deformations and Renormalization-Group Flows
- Integrable-QFT Casebook, Exact Data, and Limits
Lattice and Hamiltonian QFT
Lattice Regulators and Continuum Targets
- Lattice Regulators and Target Continuum Theories
- Lattice Geometry, Boundaries, and Anisotropy
- Scalar Lattice Actions and Difference Operators
- Lattice Momentum, Propagators, and Cutoff Dispersion
- Exact Symmetries, Broken Spacetime Symmetries, and Restoration
- Reflection Positivity and Transfer-Matrix Criteria
- Bare Parameters, Tuning Conditions, and Continuum Targets
Lattice Observables and Continuum Inference
- Euclidean Correlators and Spectral Information
- Operator Bases, Effective Masses, and Excited-State Control
- Three-Point Functions, Matrix Elements, and Disconnected Contributions
- Spectral Reconstruction and Ill-Posed Euclidean Inverse Problems
- Scale Setting and Dimensionless Ratios
- Nonperturbative Renormalization, Mixing, and Step Scaling
- Lattice Perturbation Theory, Symanzik Analysis, and Improvement
- Lines of Constant Physics and Continuum Extrapolation
Lattice Gauge Theory
- Links, Plaquettes, and Gauge Invariance
- The Wilson Gauge Action and Its Continuum Limit
- Gauge Fixing, Gribov Copies, and Gauge-Dependent Correlators
- Wilson and Polyakov Loops, Static Energies, and Screening Diagnostics
- Gauge Ensembles and Renormalized Observables
- Strong-Coupling and Character Expansions
- Topology and Lattice Index Diagnostics
- Gradient Flow, Flow Scales, and Renormalized Gauge Observables
Lattice Fermions and Chirality
- Naive Fermions and Species Doubling
- The Nielsen–Ninomiya Obstruction
- Wilson and Clover Fermions
- Staggered Fermions and Taste
- Ginsparg–Wilson Symmetry and the Lattice Index
- Overlap and Domain-Wall Fermions
- Fermion Determinants, Pfaffians, and Measure Positivity
- Anomalies, Ward Identities, and Chiral Diagnostics
- Chiral Gauge Theories on the Lattice
Sampling Algorithms for Lattice Fields
- Markov-Chain Sampling of Lattice Fields
- Local, Cluster, and Global Update Families
- Hybrid Monte Carlo and Symplectic Molecular Dynamics
- Pseudofermions, Determinant Ratios, and Solver Bias
- Algorithm Validation and Ensemble Provenance
- Critical Slowing Down, Topological Freezing, and Algorithmic Acceleration
Statistical Inference and Error Budgets
- Autocorrelation Times and Effective Sample Size
- Estimators, Covariance, and Resampling
- Correlated Fits, Model Selection, and Stability Tests
- Equilibration, Stationarity, and Chain Diagnostics
- Blinding, Analysis Choices, and Independent Reproduction
- Multi-Stage Analyses and Correlated Uncertainty Propagation
- Complete Lattice Error Budgets
Finite-Volume Spectra and Amplitudes
- Finite Volume as a Controlled Deformation
- Exponential Finite-Volume Effects
- Massless Fields, Long-Range Forces, and Finite-Volume QED
- Spectra from Euclidean Correlation Matrices
- Elastic Two-Body Quantization Conditions
- Moving Frames, Cubic Irreducible Representations, and Partial-Wave Mixing
- Scattering Amplitudes and Resonance Poles from Finite-Volume Spectra
- Coupled-Channel Quantization and Inference
- Finite-Volume Matrix Elements and 1→2 Transitions
- Three-Body Quantization and Decay Amplitudes
Finite Density and Sign Problems
- Chemical Potential on the Euclidean Lattice
- Anatomy and Severity of a Sign Problem
- Basis Dependence, Computational Complexity, and No Universal Cure
- Reweighting, Taylor Expansion, and Imaginary Density
- Dual Variables, Worldlines, and Tensor Reformulations
- Complex Langevin and Stochastic Correctness
- Lefschetz Thimbles and Holomorphic Contour Flow
- Canonical, Fugacity, and Density-of-States Methods
- Cross-Method Validation and Reliability Standards
Hamiltonian Lattice Field Theory
- Regulated Hamiltonian Field Theory
- Lattice Gauge Hamiltonians and Gauss’s Law
- Transfer Matrices between Euclidean and Hamiltonian QFT
- Local Hilbert-Space Regulators, Quantum Links, and Finite Gauge Groups
- Physical Gauge Hilbert Spaces and Constraint Enforcement
- Real-Time Evolution, Scattering, and Observable Extraction
- Hamiltonian Continuum Limits and Euclidean Cross-Validation
Light-Front QFT
- Light-Front Coordinates and Quantization
- Light-Front Constraints, Zero Modes, and Vacuum Structure
- Light-Front Fock Space, Wavefunctions, and Bound-State Equations
- DLCQ and Basis Light-Front Quantization
- Light-Front Regulators, Counterterms, and Symmetry Restoration
- Light-Front Observables and Continuum Validation
Hamiltonian Truncation and Variational Methods
- Hilbert-Space Truncation as a Regulator
- Basis Construction, Symmetry Sectors, and Matrix Elements
- Conformal and Hamiltonian Truncation
- Renormalizing a Truncated Hamiltonian
- Variational Principles and Field-Theory Ansätze
- Observables and Dynamics in Truncated Spaces
- Convergence, Extrapolation, and Error Certification
- Benchmark Theories for Truncation Methods
Tensor-Network QFT
- Entanglement Structure and Tensor-Network Ansätze
- Matrix Product States, Finite Entanglement, and Continuum Limits
- PEPS and Higher-Dimensional Field Theories
- Tensor Renormalization of Euclidean Path Integrals
- MERA, cMERA, and Renormalization Geometry
- Symmetric and Gauge-Invariant Tensor Networks
- Real-Time Tensor-Network Dynamics
- Contraction, Truncation, and Continuum Error Certification
Quantum Simulation of QFT
- What Counts as a Quantum Simulation of QFT?
- Encoding Fields and Truncating Local Hilbert Spaces
- Enforcing Gauge Symmetry and Constraints
- Digital Hamiltonian Simulation and Algorithmic Error
- Preparing Interacting QFT States
- Real-Time Evolution and Observable Extraction
- Verification, Error Mitigation, and Observable Certification
- Resource and Continuum Certification
- A Benchmark Ladder for Quantum Field Simulators
Conformal Field Theory and Bootstrap
Conformal Symmetry and Representations
- Conformal Geometry, Maps, and Compactification
- The Conformal Algebra and Its Generators
- Primaries, Descendants, and Conformal Multiplets
- Spin and Tensor Representations
- Unitarity Bounds and Null States
- Conserved Currents and the Stress Tensor
- Characters and Conformal Multiplet Counting
- Scale versus Conformal Invariance: Hypotheses and Counterexamples
Radial Quantization and State–Operator Correspondence
Correlators, OPE, and Conformal Blocks
- Scalar Two- and Three-Point Functions
- Spinning Correlators and Tensor Structures
- From the Local OPE to Conformal Data
- OPE Convergence, Associativity, and Domain Control
- Cross Ratios and Four-Point Kinematics
- Conformal Blocks and Casimir Equations
- Conformal Partial Waves and the Shadow Formalism
- Crossing Equations and Positivity
- Mixed Correlators and Global-Symmetry Sectors
- Momentum-Space Correlators and Conformal Ward Identities
Conformal Field Theory in One Dimension
Two-Dimensional CFT: Algebraic and Rational Core
- Complex Coordinates and Local Conformal Symmetry
- The Virasoro Algebra and the Stress Tensor
- Highest-Weight Modules, Null States, and the Kac Determinant
- Minimal Models and Fusion Rules
- Free Bosons and Vertex Operators
- Affine Current Algebras and WZW Models
- Cosets and Orbifolds
- Chiral Blocks, Sewing, and Modular Invariance
Nonunitary, Logarithmic, Noncompact, and Nonrational Two-Dimensional CFT
- Nonunitary CFTs, Effective Central Charge, and Complex Data
- Logarithmic CFT and Indecomposable Modules
- Noncompact CFTs and Continuous Spectra
- Liouville Theory and the Virasoro Bootstrap
- Irrational CFT, Fusion Kernels, and Crossing
- Distributional Correlators, Zero Modes, and Normalization
- Nonrational Modular Consistency and Spectral Densities
Higher-Dimensional CFT and Controlled Regimes
- Free and Generalized Free CFTs
- The Embedding-Space Formalism
- Spinning Operators and Blocks in Higher Dimensions
- Current and Stress-Tensor CFT Data
- Perturbative CFT Data near Free Fixed Points
- Large-N CFT Data and Vector Models
- Controlled Families of Interacting CFTs
- Large-Charge EFT and Fixed-Charge CFT Data
- Weakly Broken Higher-Spin Symmetry
- Nonunitary Sectors, Complex CFTs, and Fixed-Point Collisions
Boundaries, Defects, and Interfaces
- Conformal Boundaries and Defects
- Defect Representations, Transverse Spin, and Tensor Structures
- Bulk-to-Defect Expansion and Defect Operators
- Boundary and Defect Correlators and Blocks
- Boundary and Defect Bootstrap
- The Displacement Operator and Defect Ward Identities
- Interfaces, Folding, and Fusion
- Boundary Entropy and Defect Monotonicity
Weyl Anomalies, Deformations, and Flow Constraints
- Weyl Covariance on Curved Backgrounds
- The Trace Ward Identity and Weyl Anomaly
- Anomaly Coefficients and Central Charges
- Boundary and Defect Weyl Anomalies
- Sphere Partition Functions and Universal CFT Data
- Local RG and Weyl Consistency Conditions
- Conformal Perturbation Theory and Beta Functions
- Exactly Marginal Deformations and Conformal Manifolds
- Monotonicity Theorems and Flow Constraints
Numerical Bootstrap
- From Crossing Equations to Convex Optimization
- Linear Functionals and Positivity
- Block Approximations and Semidefinite Programs
- Automated Crossing-System Generation
- Precision, Convergence, and Numerical Error Budgets
- Solver Certificates and Independent Verification
- Numerical Boundary and Defect Bootstrap
- Single-Correlator Bounds
- Mixed-Correlator Islands
- Global Symmetry and Spinning Bootstrap Systems
- Extremal Functionals, Navigators, and Spectrum Reconstruction
- Benchmark Reproduction and Data Provenance
Analytic and Lorentzian Bootstrap
- Lorentzian Correlators and Causal Orderings
- The Lightcone OPE and Large-Spin Expansion
- Double-Twist Families and Anomalous Dimensions
- The Lorentzian Inversion Formula
- Defect Lorentzian Inversion and Lightcone Expansions
- Crossing Kernels and Conformal 6j Symbols
- The CFT Regge Limit and Boundedness
- Dispersion Relations for CFT Correlators
- Analytic Functionals and Polyakov Blocks
- Tauberian Theorems and Asymptotic Spectral Data
- Multipoint Correlators and Multilightcone Limits
- Celestial and Cosmological Correlators: Axiom and Handoff Audit
Energy Flow, Light-Ray Operators, and Conformal Colliders
- Detector Operators and Energy Flow at Null Infinity
- Averaged Null Energy and Positivity
- Conformal Collider Observables and Stress-Tensor Bounds
- Event Shapes and Energy Correlators in CFT
- Light-Ray OPE and Detector Expansions
- Bootstrap of Energy Correlators
- Energy Conditions, Causality, and Regge Consistency
Modular and Thermal Bootstrap
Superconformal Bootstrap Interfaces
Supersymmetry and Duality
SUSY Algebras and Unitary Representations
- Graded Spacetime Symmetry and the Supersymmetry Theorems
- Supersymmetry Across Dimensions, Signatures, and Reality Conditions
- The Four-Dimensional N=1 Super-Poincaré Algebra
- Extended Supersymmetry, R-Symmetry, and Central Charges
- Massive and Massless Unitary Supermultiplets
- BPS Bounds, Shortening, and Multiplet Recombination
- On-Shell Supermultiplets and Supersymmetric Ward Identities
- Superconformal Algebras, Shortening Data, and the CFT Handoff
Supersymmetric Quantum Mechanics, Cohomology, and the Witten Index
- Supercharges, Partner Hamiltonians, and Positive Energy
- Spectral Pairing, Ground States, and Supersymmetry Breaking
- Q-Cohomology, Hodge Decomposition, and Zero-Energy States
- Morse Deformation, Gradient Flow, and Semiclassical Tunneling
- The Witten Index, Vacuum Counting, and Its Failure Modes
- Continuum Spectra, Boundaries, Wall Crossing, and Ground-State Bundles
Supermultiplets, Superspace, and Off-Shell Closure
- Component Multiplets and Closure Records
- Superspace and Supertranslations
- Supercovariant Derivatives, Chirality, and Integrability
- Chiral, Vector, Linear, and Field-Strength Superfields
- Euclidean Superspace, Conjugation, and Field-Space Complexification
- Off-Shell Closure and Auxiliary Fields
- Constrained, On-Shell, and Nonlinear Superfields
- Extended Superspace Methods and Off-Shell Limits
Supersymmetric Actions, Supercurrents, and Quantum Effective Theory
- Superspace Measures, F-Terms, D-Terms, and Component Extraction
- Supersymmetric Action Principles and Component Reduction
- Wess–Zumino Models
- Kähler Sigma Models and Supersymmetric Target Geometry
- Supersymmetric Yang–Mills Actions
- Gauge–Matter Systems, F- and D-Term Potentials, and FI Data
- Supergraphs, D-Algebra, and Quantum Effective Actions
- Supercurrent Multiplets, Improvements, Anomalies, and Background Sources
Supersymmetric Vacua, Moduli Geometry, and BPS Sectors
- F- and D-Flatness and Gauge Quotients
- Gauge-Invariant Coordinates and Classical Moduli Varieties
- Kähler and Hyperkähler Quotients in Supersymmetric QFT
- Moduli-Space Metrics and Quantum Corrections
- Singular Loci and Emergent Low-Energy Degrees of Freedom
- BPS Particles and Central Charges
- BPS Solitons, Walls, Strings, Vortices, and Junctions
- Framed BPS States, Line Defects, and Defect Moduli
- Marginal Stability, Chambers, and Wall Crossing
Holomorphy, Anomalies, and Exact Quantum Constraints
- Holomorphic Couplings and Background Superfields
- Nonrenormalization Theorems: Wilsonian, 1PI, and Infrared Scope
- R-Symmetry, Anomalies, and the Holomorphic Scale
- Holomorphic and Canonical Couplings and the NSVZ Relation
- Instanton Zero Modes and Selection Rules for Exact Terms
- Nonperturbative Superpotentials and Exactness Conditions
- Holomorphic Decoupling, Integrating In and Out, and Scale Matching
- Quantum Chiral Rings and Konishi-Type Anomaly Constraints
Supersymmetry Breaking and Controlled Deformations
- F- and D-Term Breaking, Vacuum Energy, and the Goldstino
- O’Raifeartaigh and Fayet–Iliopoulos Model Laboratories
- R-Symmetry and Nelson–Seiberg-Type Criteria
- Pseudomoduli, Quantum Lifting, and Metastability
- Nonlinear Goldstino Dynamics and Constrained Effective Theory
- Soft Breaking, Spurions, and Controlled Decoupling
Four-Dimensional N=1 Gauge Dynamics
- SQCD Fields, Symmetries, Global Data, and Classical Moduli
- Pure Super-Yang–Mills Vacua and Domain Walls
- Gaugino Condensation and Discrete Vacua
- Gauge Instantons, Fermion Zero Modes, and Condensates
- The Affleck–Dine–Seiberg Superpotential
- Quantum-Modified Moduli and s-Confinement
- Infrared Phases and the Conformal Window
- Chiral Gauge Theories and Anomaly Constraints
- Dynamical Supersymmetry Breaking and Calculability
- Product-Group and Quiver Gauge Dynamics
- Compactification, Semiclassical Continuity, and Order of Limits
Field-Theory Duality: Dictionaries, Operations, and Global Data
- Duality Claims, Dictionaries, Regimes, and Evidence
- Exact, Infrared, and Emergent Equivalence Claims
- Abelian Electric–Magnetic Dualization, Charge Lattices, and Global Form
- Duality Operations: Gauging, Quotients, and Orbifolds
- Generalized Symmetries, Global Forms, and Anomalies under Duality
- Duality Defects, Walls, Interfaces, and Fusion
- Deformations, Compactification, and Duality Flows
- Duality Checks, Evidence Independence, Status, and Failure Modes
N=1 Duality, RG Fixed Points, and Protected SCFT Data
- Seiberg Duality in SQCD
- Operator Dictionaries, Chiral Rings, and Anomaly Matching
- Mass Deformations, Higgsing, and Dual RG Flows
- The Conformal Window and Interacting Fixed Points
- Accidental Symmetries and a-Maximization
- Conformal Manifolds and Duality Actions
- Protected N=1 SCFT Data and the Bootstrap Export
- Emergent and Non-Lagrangian N=1 Descriptions
N=2 Gauge Dynamics and Seiberg–Witten Geometry
- N=2 Multiplets, Lagrangians, and Vacuum Branches
- Coulomb Branches and the Abelian Low-Energy Theory
- Prepotentials and Special Kähler Geometry
- Charge Lattices, Duality Frames, and Local Systems
- Singularities, Vanishing Cycles, and Monodromies
- Seiberg–Witten Curves, Differentials, and Periods
- The Pure SU(2) Seiberg–Witten Solution
- Matter, Higher Rank, and Integrable-System Structures
- BPS Spectra, Chambers, and Wall Crossing
- Argyres–Douglas Theories, Class S, and Non-Lagrangian Interfaces
- N=1 Deformations, Monopole Condensation, and Controlled Confinement
N=4 SYM, S-Duality, and Higher-Dimensional Interfaces
- Extended Supersymmetry Gauge Dynamics: Scope and Structure
- N=4 SYM Field Content, Action, and Superconformal Data
- N=4 Moduli, BPS States, and Protected Sectors
- Finiteness, Conformality, and the Nonperturbative Evidence Ceiling
- Line Operators, Global Forms, and Discrete Theta Data
- Montonen–Olive and S-Duality
- Duality Groupoids, Walls, and Generalized-Symmetry Refinements
- Five- and Six-Dimensional Supersymmetric Fixed Points, Branches, and Status
- Six-Dimensional Origins, Compactification, and Duality Frames
Two-Dimensional Supersymmetric QFT, GLSMs, and Mirror Symmetry
- Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace
- Landau–Ginzburg Models and Chiral Rings
- Supersymmetric Sigma Models and Kähler Geometry
- GLSMs, Phases, and Quantum Kähler Moduli
- Effective Twisted Superpotentials and Quantum Vacua
- A- and B-Twists and Cohomological Observables
- Two-Dimensional Mirror Symmetry and Object-Level Dictionaries
- Elliptic Genera, Anomaly Data, and c-Extremization
- tt* Geometry and Ground-State Bundles
Three-Dimensional Supersymmetric Gauge Theory and Duality Webs
- Three-Dimensional Supersymmetry Algebras and Multiplets
- Yang–Mills and Chern–Simons–Matter Actions
- Parity Anomalies and Background Contact Terms
- Monopole Operators and Quantum Coulomb Branches
- Three-Dimensional N=4 Mirror Symmetry
- Aharony and Giveon–Kutasov-Type Dualities
- Real-Mass, FI, and Compactification Flows
- Supersymmetric Parents of Mirror, Particle–Vortex, and Bosonization Webs
Rigid Backgrounds, Topological Twists, and Localization
- Supercurrent Multiplets and Rigid-Background Compatibility
- Rigid Supersymmetry from Nondynamical Supergravity Backgrounds
- Generalized Killing Spinors and Global Spin-R Bundles
- Topological and Holomorphic Twists
- Finite-Dimensional and Equivariant Localization
- Q-Cohomology and Path-Integral Deformation
- Gauge Fixing and the Localization Deformation Complex
- Localization Loci, Zero Modes, and One-Loop Determinants
- Complex Contours, Stokes Chambers, and Regularization
- Boundaries, Gluing, and Jeffrey–Kirwan Residues
Exact Partition Functions, Indices, and Instanton Counting
- Supersymmetric Partition Functions, Counterterms, and Universal Data
- Sphere Partition Functions and Matrix Models
- F-Maximization and Accidental Symmetries
- Supersymmetric and Superconformal Indices
- Twisted Indices and Elliptic Genera
- Index Inversion, Recombination, and Protected-Spectrum Limits
- Instanton Counting and the Omega Background
- Factorization, Holomorphic Blocks, and Gluing
- Exact-Observable Identities as Duality Tests
Protected Operators, BPS Defects, and Exact Correspondences
- Protection, Q-Cohomology, Recombination, and Operator Mixing
- Chiral Rings and Exact Quantum Relations
- Protected Correlators and Operator Algebras
- Four-Dimensional Chiral-Algebra Sectors and Lost Information
- BPS Wilson, 't Hooft, and Dyonic Line Observables
- BPS Boundaries, Surface Defects, Interfaces, and Fusion
- Topological Twists and the Geometric–Langlands Interface
- AGT and Exact-Correspondence Dictionaries: Status and Limits
- Versioned Protected-Data Exports and Cross-Volume Round Trips
Thermal and Nonequilibrium QFT
Equilibrium States, Ensembles, and Thermodynamic Limits
- Statistical Ensembles and Field Configurations
- Partition Functions and Thermodynamic Response
- Euclidean Fields and Classical Statistical Systems
- Correlations, Susceptibilities, and Correlation Lengths
- Gaussian Fluctuations and Effective Free Energy
- Thermodynamic Limits, Phases, and Ensemble Equivalence
- Large Deviations and Phase Coexistence
KMS States, Imaginary Time, and Thermal Spectra
- Thermal Density Operators and the KMS Condition
- Infinite-Volume KMS States, Passivity, and Phase Multiplicity
- Imaginary Time and Matsubara Frequencies
- Thermal Boundary Conditions and Graded Traces
- Thermal Propagators and Spectral Representations
- Thermal Spectral Positivity and Sum Rules
- Exact Euclidean–Real-Time Analytic Continuation
- Noisy Spectral Reconstruction as an Inverse Problem
Finite Density and Conserved Charges
- Conserved Charges and Grand-Canonical States
- Chemical Potentials and Finite-Density Ensembles
- Generalized Gibbs Ensembles for Integrable Charges
- Noncommuting Charges and Generalized Gibbs States
- Chemical Potentials as Temporal Gauge Fields and Holonomies
- Finite-Density Correlators and Charge Susceptibilities
- Silver Blaze Behavior, Thresholds, and Density Instabilities
- Euclidean and Real-Time Access at Finite Density
Thermal Perturbation Theory and Renormalization
- Thermal Loop Expansions and Scale Power Counting
- Thermal Sum-Integrals and Vacuum Subtraction
- Thermal Counterterms and Renormalization-Group Invariance
- Free Energy and Pressure in Loop Expansion
- Thermal Self-Energies and Mass Definitions
- Damping Rates, Thermal Cuts, and Quasiparticle Widths
- Bosonic Zero Modes and Infrared Breakdown
- Gauge Dependence and Thermal Observables
Thermal EFT, Screening, and Resummation
- Thermal Modes, Matching, and EFT Power Counting
- Screening and Infrared Scale Separation
- Ring and Daisy Resummation
- Screened and Variational Thermal Expansions
- Dimensional Reduction and Static Effective Theories
- Real-Time Thermal EFT and Dissipative Matching
- Double Counting, Matching Dependence, and Breakdown
Thermal Phases, Metastability, and Nucleation
- Landau–Ginzburg Functional Landscapes and Order Parameters
- Thermal Effective Potentials and Phase Diagrams
- Convexity, Gauge Dependence, and Physical Phase Criteria
- Metastability, Spinodals, and Thermal Decay
- Thermal Bounces, Determinants, and Nucleation Rates
- Bubble Growth and Wall Friction
- Percolation, Reheating, and Transition Completion
- Anomalous Charge Violation, Baryogenesis, and Cosmological Interfaces
Real-Time Contours, Keldysh Bases, and Response
- Closed-Time-Path Generating Functionals in Practice
- Initial Density Matrices and Contour Boundary Conditions
- Unitarity, Normalization, and Largest-Time Identities
- Retarded, Advanced, and Keldysh Bases
- Causal and Statistical Propagators
- Keldysh Vertices and Causal Diagrammatics
- KMS Relations and Fluctuation–Dissipation
- Sources and Nonlinear Real-Time Response
Nonequilibrium Green Functions and Kadanoff–Baym Evolution
- Two-Time Green Functions and Self-Energies
- 2PI Effective Actions and Conserving Truncations
- Kadanoff–Baym Evolution Equations
- Initial Correlations and Boundary Terms
- Memory Kernels in Closed-System Evolution
- Wigner Transformation and Gradient Expansion
- Spectral and Statistical Evolution
- Numerical Kadanoff–Baym Evolution and Validation
Critical and Stochastic Dynamics
Kinetic Theory and Transport Equations
- Distribution Functions and Transport Equations
- From Kadanoff–Baym to Kinetic Theory
- Quasiparticle Shell Reduction and Gradient Power Counting
- Collision Kernels, Conservation, and Detailed Balance
- The H-Theorem and Kinetic Entropy Production
- Moment Hierarchies and Closure Schemes
- Linearized Kinetics and Relaxation Modes
- Matrix-Valued Kinetics and Coherent Transport
- Validity and Breakdown of Kinetic Descriptions
Hydrodynamic Variables, Frames, and Ideal Modes
Relativistic Dissipation, Transients, Stability, and Causality
- Relativistic Dissipative Hydrodynamics
- Onsager Reciprocity and Entropy Production
- Frame-Invariant Dissipative Data
- Conventional Relativistic Navier–Stokes Instability and Acausality
- Israel–Stewart, BRSSS, and DNMR Transient Hydrodynamics
- BDNK First-Order Causal Hydrodynamics
- Strong Hyperbolicity, Stability, and Causal Propagation
- Hydrodynamic Attractors and Asymptotic Gradient Expansions
Fluctuating, Generalized, and Schwinger–Keldysh Hydrodynamics
- Hydrodynamic Fluctuations and Noise
- Schwinger–Keldysh Effective Actions for Fluids
- Dynamical KMS and Topological Symmetries
- Long-Time Tails and Fluctuation Renormalization
- Charged and Anomalous Hydrodynamics
- Superfluid Hydrodynamics and Goldstone Modes
- Magnetohydrodynamics and Higher-Form Symmetries
- Generalized Hydrodynamics of Integrable Systems
- Spin Hydrodynamics, Polarization, and Pseudo-Gauge Dependence
- Hydro+ and Parametrically Slow Critical Modes
Response, Transport, and Inference
- Sources, Linear Response, and Kubo Formulae
- Contact Terms, Magnetization Currents, and Order of Limits
- Nonlinear Response and Higher-Order Kubo Relations
- Spectral Functions and Transport Peaks
- Transport Sum Rules and Ultraviolet Constraints
- Diffusion, Conductivity, and Susceptibility
- Shear and Bulk Viscosity
- Memory Functions and Slow-Mode Projection
- Transport Extraction, Inverse Problems, and Error Budgets
Thermalization, Integrability Breaking, and Quantum Chaos
- Equilibration, Thermalization, and Dephasing
- Prethermalization, Floquet Dynamics, and Generalized Ensembles
- Nonthermal Fixed Points and Wave Turbulence
- Eigenstate Thermalization Hypothesis
- Quantum Scars, Hilbert-Space Fragmentation, and ETH Exceptions
- Out-of-Time-Order Correlators and Contour Regularization
- Lyapunov Growth and Chaos Bounds
- Operator Spreading and Scrambling
- Spectral Statistics, Form Factors, and Late-Time Evidence
Open QFT and Driven Dynamics
- System–Environment Splits and Influence Functionals
- Quantum Master Equations for Fields
- Lindblad Field Dynamics and Complete Positivity
- Schwinger–Keldysh Actions for Open QFT
- Noise, Dissipation, and Fluctuation Relations
- Non-Markovian Dynamics and Memory Kernels
- Driven-Dissipative Steady States and Criticality
- Trace, Positivity, and Causal Consistency
- Renormalization of Open Effective Dynamics
Hot Gauge Theory and Plasma EFTs
- Scale Hierarchies in Hot Gauge Theories
- EQCD, MQCD, and Dimensional Reduction
- Hard-Thermal-Loop Effective Theory
- Collective Modes and Dynamical Screening
- The Magnetostatic Sector and Linde Problem
- Ultrasoft Color Dynamics and Bödeker Theory
- Gauge-Theory Effective Kinetic Theory
- Weak-Coupling Plasma Transport Coefficients
- Non-Abelian Plasma Instabilities
QCD Matter and Multistage Collision Inference
- Thermodynamics and the QCD Equation of State
- QCD Phase Structure and Crossover Thermodynamics
- Conserved-Charge Fluctuations and Criticality
- Dense QCD Regimes, Access, and Evidence Boundaries
- Initial Conditions and Pre-Equilibrium Evolution
- Hydrodynamization and the Kinetic-to-Hydrodynamic Map
- Viscous Hydrodynamics, Particlization, Afterburners, and Bulk–Flow Observables
- Hard Probes and Jet–Medium Tomography
- Electromagnetic Probes and Thermal Radiation
- Open Heavy-Flavor Transport and Hadronization
- Quarkonium In-Medium Dynamics and Regeneration
- Multistage Bayesian Global Inference and Evidence
Many-Body QFT and Quantum Matter
Nonrelativistic Fields and Low-Energy Reduction
- Galilean Fields, Scales, and Low-Energy Degrees of Freedom
- Schrödinger Symmetry, Scale Invariance, and Anomalies
- Second-Quantized Bosons and Fermions
- Coherent-State Path Integrals for Many-Body Systems
- From Microscopic Hamiltonians to Continuum Fields
- Hubbard–Stratonovich Fields and Collective Channels
- Nonrelativistic Power Counting and Universality
- Densities, Currents, and Nonrelativistic Ward Identities
- Finite-Density Goldstone Counting
Many-Body Correlators, Response, and Quasiparticles
- Quantum-Matter Correlators and Observable Conventions
- Lehmann Representations and Spectral Functions in Matter
- Dyson Equations and Self-Energies
- Diagrammatics at Finite Density
- Quasiparticle Poles, Residues, and Lifetimes
- Spectral Moments and Many-Body Sum Rules
- Polarization, the Lindhard Function, and Particle–Hole Continua
- Coulomb Screening, Dielectric Response, and RPA Validity
- Plasmons and Collective Charge Modes
- Current Vertices and Ward-Consistent Response
- Irreducible Vertices and Bethe–Salpeter Equations
- Hedin's Equations, Screened Interactions, and the GW Approximation
- Luttinger–Ward Functionals and Φ-Derivable Self-Energies
- Baym–Kadanoff Conservation Laws and Approximation Validity
Resonant Interactions and Universal Quantum Gases
- Short-Range Scattering Data as Many-Body Inputs
- Effective Range, Shallow Poles, and Universality Windows
- Two-Channel Resonance Models
- Efimov Physics and the Three-Body Parameter
- Universal Relations and Tan Contact
- Dimensional Crossover and Confinement-Induced Resonances
- Few-Body Data in the Virial Expansion
- Resonant Bose Matter and Metastable Branches
- From Few-Body Inputs to Many-Body Predictions
Bose Quantum Fluids and Lattice Bosons
- The Ideal Bose Gas and Bose–Einstein Condensation
- Condensation, Off-Diagonal Order, and Superfluidity
- The Weakly Interacting Bose Gas
- Bogoliubov Theory and Bose Quasiparticles
- Beyond Bogoliubov Theory in the Dilute Expansion
- Phase–Density EFT for Bose Superfluids
- Low-Dimensional Bose Gases and BKT Physics
- Strongly Correlated One-Dimensional Bose Fluids
- The Bose–Hubbard Model and Controlled Limits
- The Superfluid–Mott Quantum Phase Transition
- Self-Bound Quantum Droplets
Fermi Surfaces and Fermi Liquids
- The Fermi Gas and Fermi-Surface Kinematics
- Landau Fermi-Liquid Theory
- Microscopic Quasiparticles and Fermi-Liquid Self-Energy
- Luttinger's Theorem, Fermi Volume, and Failure Modes
- Fermi-Liquid Response and Zero Sound
- Fermi-Surface Patch Theory and Low-Energy Scaling
- Functional RG for Competing Fermi-Surface Instabilities
- Pomeranchuk and Density-Wave Instabilities
Correlated Lattice Fermions and Mott Physics
- From Bands and Orbitals to Effective Lattice Hamiltonians
- Hubbard Models: Symmetries and Controlled Limits
- Superexchange and the t–J Projection
- Hubbard Bands and Spectral-Weight Transfer
- Mott, Slater, Band, and Charge-Transfer Insulators
- DMFT Impurity Mapping and Self-Consistency
- Cluster DMFT and Nonlocal-Correlation Validity
- Sign-Free Quantum Monte Carlo and Hubbard-Phase Inference
- Electron–Phonon Fields and Retarded Interactions
- Multiorbital Correlations and Hund's Metals
- Competing Orders and Electronic Nematicity
- Doped Mott Matter and the Pseudogap Evidence Problem
Pairing, Superfluidity, and Superconductivity
- The Cooper Instability and Pairing Channels
- BCS Mean-Field Theory and the Gap Equation
- Nambu–Gor'kov Green Functions and Anomalous Propagators
- Bogoliubov–de Gennes Theory in Inhomogeneous Systems
- Phase, Amplitude, and Leggett Collective Modes
- Superfluid Order and Phase Stiffness
- Gauge-Invariant Meissner Response and Superfluid Weight
- Vortices and Topological Defects in Paired Matter
- Phase Winding, Flux Quantization, and Josephson Effects
- Migdal's Theorem and Vertex-Correction Validity
- Eliashberg Equations and Retarded Pairing
- The BCS–BEC Crossover
- Unconventional Pairing Symmetries and Diagnostics
- Unconventional Pairing Mechanisms and Competing Evidence
- Topological BdG Superconductors and Boundary Modes
- Majorana Platforms and Evidence Standards
Quantum Impurities, Polarons, and Kondo Matter
- Quantum Impurity Models and Local Moments
- The Anderson Impurity Model and Hybridization
- From the Anderson Model to Kondo Exchange
- Kondo Screening and Impurity RG Flow
- Strong-Coupling Screening, Phase Shifts, and the Friedel Sum Rule
- Multichannel and Overscreened Kondo Fixed Points
- NRG and CT-QMC Impurity Solvers: Validity and Error
- Quantum-Dot Transport and Impurity Spectroscopy
- Orthogonality Catastrophe and the X-Ray Edge
- Mobile Impurities and Polarons
- Kondo Lattices, RKKY Competition, and the Doniach Regime
- Heavy Fermi Liquids, Hybridization, and Large Fermi Surfaces
- Kondo Breakdown and Local-Criticality Evidence
One-Dimensional Quantum Matter
- Jordan–Wigner Maps and the Lattice–Continuum Dictionary
- The Abelian Bosonization Dictionary
- Zero Modes, Klein Factors, and Compactification
- Luttinger Liquids
- Spinful and Multicomponent Luttinger Liquids
- Sine–Gordon Perturbations, Commensurability, and Duality
- Non-Abelian Bosonization and Spin Sectors
- Spin–Charge Separation and Spectral Observables
- Impurities, Junctions, and Transport in One Dimension
Quantum Magnetism and Frustration
- Exchange Interactions and Effective Spin Hamiltonians
- Spin Coherent States and Berry Phases
- Spin Waves and Magnons
- Antiferromagnets, Sigma Models, and Theta Terms
- Quantum Spin Chains and Haldane Physics
- Frustration and Order by Disorder
- Valence-Bond Solids and Quantum Paramagnets
- Itinerant Magnetism and Spin-Fluctuation Physics
- Spin Correlations and Dynamical Structure Factors
Quantum Phase Transitions and Critical Metals
- Quantum Phase Transitions and Competing Scales
- Landau–Ginzburg–Wilson Quantum Criticality
- Quantum-Critical Fans and Finite-Temperature Scaling
- Hertz–Millis Theory and Landau Damping
- Spin-Fermion and Hot-Spot Theories
- Dangerously Irrelevant Couplings and Hyperscaling Violation
- Metallic Non-Fermi Liquids and Quasiparticle Breakdown
- Critical Fermi Surfaces and Patch Theories
- SYK Models and Local Quantum Criticality
- Strange-Metal Transport and Planckian Claims
- Evidence and Model Discrimination at Quantum Criticality
Band Geometry and Symmetry-Protected Matter
- Bloch Bands, Wannier Functions, and Effective Band Theories
- Berry Geometry and the Quantum Metric
- Polarization and Thouless Pumping
- Integer Quantum Hall Matter and Chern Insulators
- Time-Reversal-Invariant Z2 Topological Insulators
- Symmetry Classes and the Free-Fermion Periodic Table
- Crystalline and Higher-Order Topological Matter
- Interacting SPT Matter and Physical Diagnostics
- Protected Boundaries and Interfaces of Invertible Matter
- Topological Semimetals and Nodal Surface States
Fractional Quantum Hall Matter and Anyons
- Topological Order, Invertible Phases, and Matter Diagnostics
- Landau Levels, Projection, and Magnetic Translations
- Fractional Quantum Hall Fluids
- Composite Fermions and Hierarchical Hall States
- Fractional Chern Insulators and Moiré Hall Matter
- Anyons as Quasiparticles in Quantum Matter
- Abelian Topological Orders and K-Matrix Data
- Non-Abelian Topological Orders
- Chiral Edges, Interfaces, and Bulk–Boundary Tests
- Anyon Interferometry and Evidence Standards
- Topological Entanglement and Spectrum Diagnostics
Fractionalization and Emergent Gauge Fields
- Parton Constructions and Gauge Constraints
- Symmetry Fractionalization and Projective Quantum Numbers
- Emergent Z2 Gauge Fields, Visons, and Gapped Topological Order
- Compact U(1) Gauge Fields, Emergent Photons, and Monopoles
- Confinement, Higgsing, and Deconfinement in Quantum Matter
- Particle–Vortex and Matter Dualities in Quantum Matter
- Quantum Dimer Models and Height/Gauge Descriptions
- Gapped Spin Liquids and Positive Diagnostics
- Gapless Spin Liquids and Dirac Spinons
- Quantum Spin-Liquid Evidence and Competing Explanations
- Gauge-Coupled Fermi Surfaces
- Deconfined Quantum Criticality
- Deconfined Criticality: Evidence, Drift, and Pseudocriticality
- Fracton and Subsystem-Symmetric Orders
Disorder, Localization, and Glasses
- Quenched Disorder and Disorder Averaging
- Replica and Supersymmetry Methods for Disorder
- Diffusive Modes and the Disorder Nonlinear Sigma Model
- Weak Localization and Weak Antilocalization
- Interaction Corrections and Finkel'stein Scaling
- Anderson Localization and Scaling Theory
- Symmetry Classes and Topological Localization
- Dirty Bosons and the Bose Glass
- Griffiths Effects, Rare Regions, and Avalanches
- Many-Body Localization: Diagnostics and Evidence
- Mean-Field Quantum Spin Glasses and Replica Symmetry Breaking
- Finite-Dimensional Quantum Glass Dynamics and Evidence
Cold Atoms and Synthetic Matter
- Ultracold Platforms, Scales, and Traps
- Feshbach Control and Resonance Calibration
- Optical Lattices and Hubbard-Model Realizations
- Unitary Fermi-Gas Platforms and Benchmark Evidence
- Long-Range, Dipolar, and Molecular Quantum Gases
- Synthetic Gauge Fields and Spin–Orbit Coupling
- Moiré Flat-Band Quantum-Matter Platforms
- Analog-Simulation Model Realization and Observable Validation
Integrable Quantum Matter and Generalized Hydrodynamics
- Bethe-Integrable Quantum Gases and Spin Chains
- Root Densities, Macrostates, and Model Observables
- Integrable Charges and Generalized-Ensemble Model Tests
- Quench Action and Integrable Steady States
- Generalized-Hydrodynamic Riemann Problems and Trap Expansions
- Diffusive GHD Corrections as Model Benchmarks
- Weak Integrability Breaking and Hydrodynamic Crossover
- Finite-Size and Experimental Validation of Integrable Matter
Nonequilibrium, Driven, and Open Quantum Matter
- Quenches and Relaxation in Lattice and Continuum Matter
- Kibble–Zurek Scaling and Defect Production
- Nonthermal Fixed-Point Realizations in Quantum Matter
- Floquet Engineering in Quantum Matter
- Floquet Phases and Discrete Time Crystals
- Quantum-Scar Models and Exceptional Eigenstates
- Hilbert-Space Fragmentation and Constrained Dynamics
- Driven-Dissipative Condensate, Cavity, and Circuit Matter
- Dynamical Phase Transitions and Loschmidt Diagnostics
- Heating, Finite Size, and Open-System Evidence
Quantum-Matter Probes, Inference, and Evidence
- From Measured Intensity to Many-Body Claim
- Photoemission and ARPES Spectral Inference
- Tunneling Spectroscopy and Quasiparticle Interference
- Optical Conductivity and Spectral-Weight Diagnostics
- Neutron Scattering and Dynamical Structure Factors
- Raman, RIXS, and EELS Response
- NMR, μSR, and Local Magnetic Probes
- Quantum Oscillations and Thermodynamic Fermi-Surface Inference
- Cold-Atom RF, Bragg, Time-of-Flight, and Microscopy Probes
- Quantum Monte Carlo Formulations and Phase Inference
- Tensor-Network Phase Diagnostics in Quantum Matter
- Exact Diagonalization and Finite-Size Evidence
- Model Selection and Parameter Inference
- Evidence Triangulation and Reproducibility
Quantum Information in QFT
Continuum Subsystems and Local Algebras
- Regions, Causal Complements, and Nets of Observables
- Restricted States and Subregion Observables
- Why Continuum QFT Does Not Factorize Naively
- Von Neumann Factors and Type-III Local Algebras
- Standard Form, Cyclic and Separating Vectors
- The Split Property and Approximate Tensor Products
- Nuclearity, Phase-Space Bounds, and Split Distance
- Additivity, Haag Duality, and Information Completeness
- Local Preparation, State Dependence, and Operational Independence
- Reeh–Schlieder Property and Limits of Localization
- Choosing a Continuum Subsystem: Algebra, Split, or Regulator
- Local Operations, Separability, and Distillability in QFT
Regulated Entropy and Replica Methods
- Entropy of a Regulated Subregion
- Replica Trick and Branched Geometries
- Rényi Entropies and Replica Analytic Continuation
- Twist Operators, Replica Defects, and Sewing Data
- Analytic Continuation: Uniqueness and Failure Modes
- Gaussian States and Correlation-Matrix Entropy
- Entanglement Entropy of Free Fields
- Zero Modes, Boundaries, and Infrared Sensitivity
- Ultraviolet Divergences and the Area Law
- Entropy Counterterms and Renormalization Ambiguities
- Numerical Replica and Thermodynamic-Integration Estimators
- From Lattice Entropy to a Continuum Claim
- Replica and Entropy Calculation Verification Ledger
Relative Entropy, Distinguishability, and Recovery
- Relative Entropy for QFT States
- Araki Relative Entropy and Regulated Limits
- Positivity, Monotonicity, and Data Processing
- Hypothesis Testing and Asymptotic Distinguishability
- Fidelity, Chernoff Bounds, and State Overlap
- Operational Distinguishability and Continuity Bounds
- Mutual Information and Regulator-Independent Correlations
- Strong Subadditivity and Entropic Inequalities
- Conditional Mutual Information and Quantum Markov Structure
- Recovery Maps and Approximate Markovianity
- Petz, Rotated, and Universal Recovery Maps
- Infinite-Dimensional and Energy-Constrained Channel Distances
- Information-Measure Domain and Comparison Atlas
Universal, Multipartite, and Shape-Dependent Entanglement
- Universal Terms and Entangling-Surface Geometry
- Shape Dependence and Entanglement Variations
- Corners, Cusps, and Defect Data
- Interval, Sphere, and Cylinder Entanglement in CFT
- Mutual Information for Disjoint Regions
- Entanglement Negativity in QFT
- Reflected Entropy and Canonical Purifications
- Multipartite Information and Entropy Cones
- Entanglement Spectra and Modular Spectral Data
- Topological and Long-Range Entanglement Interfaces
- Finite-Temperature and Excited-State Entanglement
- Choosing an Entanglement Measure for the Physical Question
Modular Operators and Geometric Flow
- Modular Hamiltonians: Definitions and Domains
- Tomita–Takesaki Modular Operators and Flow
- Relative Modular Operators and Connes Cocycles
- Modular Conjugation, Commutants, and Standard Form
- Rindler Wedges and the Bisognano–Wichmann Theorem
- Ball Regions and Conformal Modular Hamiltonians
- Modular KMS Relations and Modular Correlators
- Modular Spectrum and Spectral Measures
- Half-Sided Modular Inclusions and Emergent Translations
- Modular Intersections and Spacetime Organization
- Nonlocal Modular Hamiltonians and Limits of Explicit Control
- Exact Modular-Flow Examples and Convention Atlas
Modular Response and Quantum Information Geometry
- First Law of Entanglement and Quadratic Corrections
- State Perturbations and Relative-Entropy Susceptibility
- Shape Deformations and Modular Perturbation Theory
- Modular Response Kernels and Stress-Tensor Insertions
- Quantum Fisher Information in QFT
- Bures, Kubo–Mori, and Monotone Metrics
- Fidelity Susceptibility and Relevant Deformations
- Modular Berry Transport and Holonomy
- Modular Analyticity and Chaos Bounds
- Noncommutative Information Geometry and Exponential Families
- Contact Terms, Operator Domains, and Perturbative Error Budgets
Information Across Scales and Renormalization
- Separating Cutoff, Region, Correlation, and RG Scales
- Information Measures Along RG Flows
- Entropic Monotones in Two Dimensions
- Sphere Entanglement and Three-Dimensional F Monotonicity
- Four-Dimensional a-Type Flow Constraints
- Renormalized Entanglement and Scheme Dependence
- Relevant Deformations and Entanglement Crossover Functions
- Relative Entropy and RG Irreversibility
- Coarse-Graining Channels and Recoverability
- Scale-Space Entanglement and Factorization Caveats
- Information Geometry on Theory Space
Symmetry, Superselection, and Information Resources
- Symmetry-Constrained Operations in QFT
- Superselection Rules and Accessible Entanglement
- Fermionic Subsystems, Parity, and Graded Tensor Products
- Reference Frames, Asymmetry, and Charged Resources
- Charge-Resolved Entanglement and Charged Moments
- Entanglement Asymmetry and Symmetry Restoration
- Gauge Constraints, Centers, and Edge Data
- Gauge-Field Entanglement and Edge Contributions
- Centers, Edge Extensions, and Distillable Entanglement
- Dressed Charges, Soft Sectors, and Infrared Information
- Generalized Symmetries and Information Diagnostics
- Covariant Channels and Symmetry-Restricted Recovery
Local Measurements, Detectors, and Instruments
- Operational Locality: Couplings, Supports, and Protocols
- Localized Probe and Detector Models
- Switching, Smearing, and Detector Regularization
- Particle Detectors versus Field Observables
- System–Probe Scattering and Measurement Models
- Local Measurement Instruments in QFT
- Completely Positive Maps and Causal Quantum Channels
- Selective Operations, Postselection, and State Update
- Measurement-Induced Energy, Noise, and Backreaction
- Joint Measurements and Spacelike Composition
- Entanglement Witnesses and Local Tomography Limits
- Detector and Instrument Validation
Causal Channels and Relativistic Communication
- Signaling, No-Signaling, and Causal Composition
- Algebraic Quantum Channels and Localized Operations
- Channel–State Correspondence in Infinite Dimensions
- Communication Through Quantum Fields
- Quantum Communication and Entanglement Distribution
- Vacuum Entanglement Harvesting
- Correlation Extraction versus Causal Exchange
- Energy-Constrained Capacities and Coding Tasks
- Wavepackets, Modes, Frames, and Localization
- Bell Nonlocality with Quantum Fields
- Relativistic Communication Protocols: Assumptions and Status
Energy, Entropy, and Localization
- Passivity, Work, and Information in QFT
- Complete Passivity, KMS Structure, and Resource Conversion
- Quantum Energy Inequalities
- Averaged Null Energy Condition
- Relative Entropy and Bekenstein-Type Bounds
- Quantum Null Energy Condition
- Quantum Interest and Negative-Energy Compensation
- Entropy Bounds, Species, and Regulator Dependence
- Energy Cost of Localization and Measurement
- Landauer Cost, Information Engines, and Field Reservoirs
- Quantum Energy Teleportation
- Energy–Information Bounds: Assumptions and Status
Entanglement Dynamics and Information Transport
- State Preparation, Quench, and Regulator Contracts
- Entanglement Growth after Quenches
- Quasiparticle Pictures of Entanglement Growth
- Membrane, Hydrodynamic, and Coarse-Grained Entanglement Laws
- Entanglement, Correlation, and Hydrodynamic Fronts
- Operator Entanglement and Channel–State Maps
- Mutual Information and Correlation Spreading
- Entanglement in Open and Monitored Field Dynamics
- Conservation Laws and Information Transport
- Integrable versus Chaotic Entanglement Dynamics
- Continuum Extrapolation and Finite-Time Windows
- Entanglement-Dynamics Diagnostic Comparison
Scrambling, Decoupling, and Recoverability
- Information Scrambling and Recovery Diagnostics
- Decoupling and Subsystem Information Loss
- OTOCs, Commutators, and Information Measures
- Tripartite Information and Multipartite Scrambling
- Operator Growth versus Recoverability
- Channel Capacities During Scrambling
- Recovery Thresholds, Access Structures, and Side Information
- Information Velocities and Causal Bounds
- Finite Size, Symmetry Sectors, and Scrambling False Positives
- Scrambling Evidence and Claim-Status Matrix
Complexity and Continuum Resource Costs
- What Task Does Complexity Answer?
- Circuit Complexity in Quantum Field Theory
- Cost Geometry, Gate Sets, and Reference States
- State, Unitary, Channel, and Operator Complexity
- Gaussian-State Complexity
- Mixed-State, Purification, and Formation Complexity
- Interacting-Field Complexity
- Path-Integral and Euclidean Preparation Complexity
- Krylov and Operator-Growth Complexity
- Complexity with Symmetry, Gauge, and Locality Constraints
- Regulator Dependence and Continuum Complexity
- Operational Preparation Cost and Energy-Constrained Bounds
- Complexity, Chaos, and Computational Claims
Quantum Error Correction in Continuum Fields
- Error Models, Codes, and Recovery Conditions
- Operator-Algebra Quantum Error Correction
- Approximate Recovery and Information–Disturbance
- Energy-Constrained Errors and Infinite-Dimensional Norms
- Erasure, Subregions, and Correctability
- Complementary Recovery and Cleaning Relations
- Locality, Code Distance, and Causal Constraints
- Symmetry, Covariance, and QEC Constraints
- Subsystem and Gauge-Code Structures in QFT
- Continuum Codes and Type-III Obstacles
- Regularization, Continuum Limits, and Code Validation
- Renormalization and Coarse Graining as Encoding
- Tensor Networks as Encoding Maps: Scope and Limits
- Reconstruction from Redundant Encodings
Inference, Certification, and Evidence
- From Field Data to Information Claims
- Entropy and Rényi Estimation Protocols
- Randomized Measurements and Classical Shadows
- Replica, Swap, and Multi-Copy Measurement Protocols
- Correlation-Matrix and Gaussian-State Reconstruction
- Detector, Channel, and Local-Parameter Tomography
- Modular-Response and Relative-Entropy Estimation
- Witnesses for Entanglement, Scrambling, and Recoverability
- Continuum Extrapolation, Bias, and Uncertainty
- Cross-Method Benchmarks and Adversarial Null Tests
- Claim–Evidence Records, Replication, and Retraction Handling
- Reproducible Reporting Standard for QFT Information
QFT in Curved Spacetime
Fields and Local Algebras on Curved Backgrounds
- Fixed-Background, Semiclassical, Gravitational-EFT, and Quantum-Gravity Regimes
- Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity
- Covariant Scalar Fields and Curvature Coupling
- Spinors, Tetrads, and Spin Connections
- Gauge Fields, Gauge Fixing, and Ghosts on Curved Backgrounds
- Differential-Form Fields and Reducible Gauge Systems
- Green Operators, Causal Propagators, and State-Dependent Two-Point Functions
- Covariant Symplectic Structure and Conserved Inner Products
- Covariant Algebraic Quantization and Fock Realizations
- Local Field Algebras, Causality, and the Time-Slice Property
- Local Covariance, Isometries, and Background Embeddings
- Relative Cauchy Evolution and Background Response
- Topology, Zero Modes, and Global Sectors
- Timelike Boundaries, Self-Adjoint Extensions, and AdS Boundary Conditions
- Conformal Transformations and Frame Changes
States, Hadamard Structure, and Microlocal Control
- Vacuum Ambiguity, Time Flow, and Observer Dependence
- Complex Structures and One-Particle Spaces
- Quasifree States and Two-Point Functions
- Bogoliubov Transformations and Unitary Implementability
- GNS Representations, Local Normality, and Local Quasiequivalence
- Hadamard Parametrix and Short-Distance Structure
- Hadamard Admissibility and the Two-Point Wavefront Criterion
- Propagation of the Hadamard Property
- Constructing Hadamard States by Deformation and Gluing
- Adiabatic States, WKB Order, and Regularity
- Ground, KMS, and Symmetry-Selected States
- States of Low Energy and Smeared-Energy Selection
- Hadamard States for Fermion and Gauge Fields
- Instantaneous Vacua, Alpha Vacua, Sorkin–Johnston States, and Failure Tests
Interacting Fields and Local Covariant Renormalization
- Why Interacting Curved-Space QFT Requires Local and Covariant Renormalization
- Interacting Correlators and Curved-Space Time-Ordered Products
- Curvature Counterterms and Composite-Operator Mixing
- Local S-Matrices and Causal Factorization
- Running Couplings and Curvature Couplings
- Background Splitting and Perturbative Agreement
- Interacting Stress Tensors and Ward Identities
- Gauge, BRST, and BV Interfaces on Curved Backgrounds
- Curved-Space OPE and Local Short-Distance Expansions
- Adiabatic Limits and Infrared Obstructions
- Interacting Scalar Theory on a Curved Background: Benchmark Ledger
Particles, Detectors, and Nonadiabatic Production
- Particles, Local Observables, and Detector Dependence
- Mode Bases, In/Out States, and Number Operators
- Detector Response Along Curved and Accelerated Worldlines
- Switching, Smearing, Finite-Time Response, and Transients
- Particle Creation in Time-Dependent Backgrounds
- Adiabaticity, Stokes Phenomena, and Production Rates
- Parametric-Oscillator and Solvable Production Benchmarks
- Unruh Effect and Uniformly Accelerated Detectors
- Tolman Redshift, KMS Structure, and Local Temperature
- Acceleration, Gravity, and the Limits of Equivalence-Principle Arguments
- Moving Mirrors and the Dynamical Casimir Effect
- Schwinger and Gravitational Production: A Controlled Comparison
Curved Channels, Communication, and Entanglement Harvesting
- Curved-Spacetime Channel Deployment Contract
- From Propagators and Response Functions to Channel Maps
- Causal Support, Signaling, and Global Geometry
- Accelerated Detector Communication Channels
- Redshift, Restricted Access, and Effective Channel Noise
- Entanglement Harvesting in Curved Spacetime
- Correlation Extraction versus Causal Exchange on Curved Backgrounds
- Horizon-Restricted Local Operations and Distillability
- Entanglement Distribution and State Transfer Through Curved Fields
- Energy-Constrained Capacity Under Redshift and Acceleration
- Relativistic Protocols, Clocks, Encoding, and Channel Tomography
- Evidence, Analogue Systems, and Astrophysical Claim Limits
Horizons and Hawking Radiation
- Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface
- Stationary Horizons, Surface Gravity, and Redshift
- Hawking Radiation from Gravitational Collapse
- Ray Tracing and the Hawking Bogoliubov Map
- Stress-Tensor Flux and Two-Dimensional Reductions
- Euclidean Periodicity, KMS Structure, and Horizon Thermality
- Boulware, Hartle–Hawking, and Unruh States
- Greybody Scattering and Flux Accounting
- Rotating and Charged Horizons, Superradiance, and Chemical Potentials
- Cosmological and Multiple-Horizon State Obstructions
- Trans-Planckian Sensitivity of the Hawking Derivation
- Modified Dispersion, Analogue Horizons, and Universality
- Evaporating Backgrounds and Adiabatic Backreaction
- Extremal, Near-Extremal, and Late-Time Limits
Local Observables, Stress Tensors, and Anomalies
- Wick Polynomials and Hadamard Point Splitting
- Renormalized Stress Tensor: Axioms and Curvature Ambiguities
- Curved-Space Renormalization Schemes: Domains and Translation
- Conservation, Local Covariance, and the Backreaction Source
- Trace Anomalies and Convention Translation
- Spin, Gauge, and Gravitational-Anomaly Responses
- Renormalized Currents and Charge Density
- Vacuum Polarization and Curved-Space Casimir Effects
- Mode-Sum and Numerical Renormalization
- Geometric Discretization and Continuum Checks
- Boundaries, Surface Counterterms, and Boundary Stress
- Null-Projected and Smeared Stress Observables
- Stress Bi-Tensors and Noise-Kernel Input
Matter Effective Actions in Curved Spacetime
- One-Loop Matter Effective Actions in Curved Space
- Proper-Time, Zeta, and Determinant Prescriptions
- Heat Kernels and the Schwinger–DeWitt Expansion
- Seeley–DeWitt Coefficients and Curvature Invariants
- Heat Kernels with Boundaries and Conical Singularities
- Worldline Methods in Curved Space
- Renormalization of Gravitational Couplings by Matter Loops
- Mass Thresholds, Decoupling, and Curvature Expansions
- Matter-Induced Nonlocal Form Factors
- Anomaly-Induced and Nonlocal Actions
- Imaginary Effective Actions and Vacuum Instability
- Effective-Action Variation, Stress Tensors, and Consistency Checks
Mean Semiclassical Backreaction
- The Semiclassical Einstein Equation
- Coupled State–Geometry Initial Data
- In-In Effective Actions and Causal Mean Backreaction
- Constraints, Conservation, and the Bianchi Identity
- Large-N, Loop, and ℏ Approximation Hierarchies
- Self-Consistent State–Geometry Solutions
- Quantum-State Evolution on Backreacted Backgrounds
- Higher-Derivative Semiclassical Initial-Value Problems
- Order Reduction and Runaway Prescriptions
- Linear Response and Semiclassical Stability
- Gauge-Invariant Response Kernels
- Cosmological Backreaction Benchmarks
- Black-Hole Evaporation and Mean Backreaction
- Numerical Self-Consistency and Error Budgets
Stochastic Gravity and Metric Fluctuations
- The Stress-Tensor Noise Kernel
- Renormalizing Stress Fluctuations and Coincident Limits
- Influence Functionals, Dissipation, and Noise
- Fluctuation–Dissipation Relations in Stationary States
- Einstein–Langevin Dynamics
- Intrinsic and Induced Metric Fluctuations
- Gauge-Invariant Stochastic Observables
- Large-N Quantum–Stochastic Correspondence
- Higher Cumulants and Non-Gaussian Noise
- Validity, Decoherence, and What Stochastic Gravity Does Not Capture
Generalized Entropy and Quantum Extremal Surfaces
- Black-Hole Thermodynamics at the QFT Interface
- Fine-Grained, Coarse-Grained, and Algebraic Entropy
- Horizon Entanglement Entropy
- Relative Entropy and Modular Horizon Laws
- Generalized Entropy and UV Renormalization
- Species, Gauge Edges, and Contact Terms
- Noether-Charge Entropy and Higher-Curvature Terms
- Replica Constructions on Fixed and Semiclassical Backgrounds
- Conical Entropy and Effective-Action Variations
- Semiclassical First Laws and Physical-Process Variations
- The Generalized Second Law: Hypotheses and Proven Scope
- Quantum Extremal Surfaces: Renormalized Semiclassical Definition
- Semiclassical Black-Hole Information: Scope and Limits
Energy Conditions, Inequalities, and Focusing
- Classical Energy Conditions and Quantum Violations
- Raychaudhuri Evolution, Null Focusing, and Renormalized Stress
- Quantum Energy Inequalities in Curved Spacetime
- QEI Sampling, State Domains, and Scheme Dependence
- Averaged Null Energy in Curved Spacetime
- Negative Energy and Quantum Interest
- QNEC on Curved Backgrounds
- The Quantum Focusing Conjecture: Dimension, EFT, Smearing, and Status
- Quantum Expansion and Covariant Entropy Bounds
- Quantum Trapped Surfaces and Semiclassical Singularity Theorems
- Wormholes, Chronology, and Superluminal-Travel Constraints
- Causal, Thermodynamic, and Stability Constraints
- Energy and Entropy Bounds: Hypotheses and Handoffs
Gravity as Effective Field Theory
- Applying EFT Power Counting to Gravity
- Curvature Operator Bases and Field Redefinitions
- Background-Field Quantization, Gauge Fixing, and Ghosts
- Local Supersymmetry and Supergravity as Low-Energy EFT
- One-Loop Graviton EFT
- Matter Contributions to Gravitational Matching
- Heavy Thresholds, Species, and the Gravitational Cutoff
- Matching Amplitudes to Background Observables
- Graviton and Matter Nonlocal Form Factors
- Long-Distance Quantum Corrections
- Relational and Gauge-Invariant Gravitational Observables
- Vacuum Energy and the Cosmological Constant
- Higher-Derivative Poles, Ghost Diagnostics, and Order Reduction
- Validity, Unitarity, and Breakdown
Quantum Fields in Cosmology
- FLRW Fields and Mode Quantization
- Conformal and Minimally Coupled Scalars in FLRW
- Spinor and Gauge Fields in Expanding Universes
- Adiabatic Particle Number in Cosmology
- Adiabatic Subtraction for Local Observables
- Vacuum Choice and Initial-State Effects
- Initial Density Matrices and Boundary EFT
- Bunch–Davies, Euclidean, and Alpha-State Diagnostics
- Trans-Planckian Initial-State Sensitivity
- Gravitational Production of Massive Relics
- Mode Matching Across Cosmological Eras
- Renormalized Stress and Backreaction in FLRW
Gravitational Vacuum Decay and Cosmological Transitions
- False-Vacuum Decay with Gravity: Regime and Observable Contract
- Euclidean Gravitational Saddles and Boundary Terms
- Coleman–De Luccia Bounces
- Hawking–Moss Transitions and the Stochastic Crossover
- Negative Modes, Determinants, and Decay Prefactors
- Real-Time Vacuum Decay and State Dependence
- Thermal Nucleation in an Expanding Universe
- Expansion History, Percolation, and Completion
- Baryogenesis Inputs and the Cosmological Yield
- Bubble Sources and Cosmological Gravitational-Wave Propagation
- Metastability, Eternal Inflation, and Claim Limits
Inflationary Correlators and Cosmological Perturbation Theory
- Inflationary Backgrounds, Broken Time Diffeomorphisms, and Decoupling
- Inflationary Perturbations and Gauge-Invariant Variables
- Mukhanov–Sasaki Scalar Modes
- Tensor Modes and Primordial Gravitons
- Constraint Solving and the Gauge-Observable Ledger
- Power Spectra, Horizon Crossing, and Freeze-Out
- In-In Cosmological Correlators
- Contours, iε, and Initial-Boundary Terms
- Effective Field Theory of Inflation and the Goldstone Mode
- Cubic Interactions and Bispectrum Shapes
- Soft Limits and Inflationary Consistency Relations
- Cosmological Loops and Renormalization
- Secular Growth, Resummation, and Dynamical RG
- Decoherence and the Quantum-to-Classical Claim
de Sitter Infrared Physics and Stochastic Inflation
- de Sitter Infrared Regimes: States, Observables, Gauges, and Limits
- Euclidean and Bunch–Davies Free Fields in de Sitter
- Massless Zero Modes and the Invariant-State Obstruction
- Interacting Infrared Logs and Secular Regimes
- Graviton Infrared Claims and Relational Observables
- Infrared Resummation and Dynamical Mass
- Stochastic Coarse-Graining and Its Derivation from QFT
- Langevin and Fokker–Planck Dynamics
- Open-System Noise and Dissipation in de Sitter
- Stationary Distributions and First-Passage Observables
- QFT–Stochastic Matching and Renormalization
- Quasi-de Sitter Validity and Evidence Status
Cosmological Analyticity and Bootstrap
- Wavefunction and Correlator Object Dictionary
- Cosmological Symmetry and Ward Identities
- Contact, Exchange, and Seed Solutions
- Total-Energy and Factorization Singularities
- Cosmological Cutting Identities and Unitarity Constraints
- Locality, Contact Ambiguities, and Field-Redefinition Equivalence
- Dispersion Relations, Subtractions, and Qualified Positivity
- Reconstruction from Symmetry, Singularities, and Cuts
- Cosmological Bootstrap: Loops, Initial States, Validity, and Handoffs
Validity, Breakdown, and Quantum-Gravity Handoffs
- Observable-Specific Validity and Error Contracts
- Semiclassical Breakdown Diagnostics
- Curvature, Coupling, Loop, Derivative, and Secular Hierarchies
- Large-N, Species, and Cutoff Hierarchies
- Cross-Regime Trans-Planckian Sensitivity and Universality
- Cauchy Horizons, Chronology Horizons, and Loss of Global Hyperbolicity
- Singularities, Initial Conditions, and Predictive Limits
- Evaporation Endpoints and the Information Handoff
- Low-Energy Constraints on UV Completion
- Analogue and Phenomenological Evidence Ceilings
- The Quantum-Gravity Handoff
Holography and Quantum Gravity
Quantum-Gravity Claims, Observables, and Evidence
- Holographic Duality: Claims, Dictionaries, and Regimes
- Observable and Regime Matrix for Quantum Gravity
- Relational, Boundary, and Asymptotic Observables
- Exact Statements, Saddle Expansions, and Conditional Derivations
- Dictionary Completeness and Global Data
- Fixed-Theory, Ensemble, and Superselection Claims
- Nonperturbative Definition and Completion Criteria
- Evidence Programs for Holographic Duality
- Evidence Independence, Circularity, and Double Counting
- Falsifiers, Negative Results, and Counterexamples
- Claim Status, Freshness, and Research Handoffs
Large N and Semiclassical Bulk Criteria
- Large-N Factorization and Classical Bulk Scaling
- Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria
- From Genus Counting to a Holographic String Regime
- Central Charge, Newton Coupling, and the Planck Scale
- Weakly Coupled Bulk Fields from Connected Correlators
- Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria
- Higher-Spin Gaps and Einstein-Regime Obstructions
- Bulk Interaction Scaling and Effective Cutoffs
- Approximate Bulk Locality from Spectral and Mellin Data
- Corrections, Nonuniform Limits, and Failure Modes
- Nonperturbative Exponential Effects and Finite-N Sectors
- Necessary, Sufficient, and Heuristic Bulk Criteria
AdS Geometry, Boundary Problems, and the Dictionary
- Anti-de Sitter Geometry and the Conformal Boundary
- Global, Poincaré, and AdS-Rindler Patches
- Timelike-Boundary Causality and Boundary-Value Problems
- Asymptotically Locally AdS Boundary Data
- Radius, Couplings, and the Parameter Map
- Bulk Fields and Boundary Operators
- Spinning Fields, Forms, and Mixed-Symmetry Operators
- The GKPW Generating-Functional Dictionary
- Currents, Stress Tensor, and Bulk Gauge and Metric Fields
- Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form
- Extended Operators, Defects, and Brane Charges
- States, Geometries, and Radial Quantization
- Euclidean Preparation and Lorentzian State Dictionaries
- Heavy States, Coherent States, and Semiclassical Geometries
- Boundary Conditions, Alternate Quantization, and Deformations
- Dictionary Normalization and Global-Data Audit
String, Brane, and Top-Down Constructions
- String Theory as a Holographic Construction Interface
- Worldsheet Sigma Models and Spacetime Consistency
- String Spectra, Scales, and Low-Energy Limits
- Decoupling Limits and the Original AdS/CFT Proposal
- D-Branes, Open-Closed Duality, and Gauge Sectors
- Near-Horizon Brane Geometries and Top-Down Dictionaries
- Flux Quantization, Compact Factors, and Kaluza–Klein Towers
- Consistent Truncations and Lower-Dimensional Effective Actions
- D3-Branes and AdS5/CFT4: Parameter-Controlled Regimes and Evidence
- M2, M5, and Higher-Dimensional Brane Examples
- D1-D5 Systems and AdS3 Top-Down Data
- Stringy and Quantum Corrections Beyond Supergravity
- Top-Down, Bottom-Up, and UV-Completion Claim Contracts
Nonperturbative String- and M-Theory Definition Proposals
- Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers
- String and M-Theory Duality Webs and Parameter Maps
- BFSS Matrix Quantum Mechanics and M-Theory Conjectures
- BMN Plane-Wave Matrix Model and Controlled Sectors
- IKKT Type-IIB Matrix Model and Emergent-Spacetime Claims
- Matrix-String Constructions and Second-Quantized Strings
- Open String Field Theory and Tachyon Dynamics
- Covariant and Closed String Field Theory
- AdS/CFT as a Conditional Nonperturbative Definition
- Background Dependence, Background Independence, and Emergence
- Numerical Matrix Evidence, Classical-Spacetime Recovery, and Limits
Holographic Renormalization and Radial Dynamics
- Asymptotically Locally AdS Fields and Fefferman–Graham Expansions
- Radial Cutoffs and Hamilton–Jacobi Flow
- Scalar Counterterms and the Renormalized On-Shell Action
- Gauge-Field and Differential-Form Counterterms
- Metric Counterterms and the Boundary Stress Tensor
- Renormalized One-Point Functions and the Variational Problem
- Ward Identities, Weyl Anomalies, and Contact Terms
- Hamiltonian Constraints and Radial Canonical Transformations
- Holographic RG Flows and Domain-Wall Geometries
- Wilsonian Cutoffs and Integrating-Out Proposals
- Holographic c-, a-, and F-Theorem Interfaces
- Finite Counterterms, Schemes, and Multi-Trace Data
Witten Diagrams and AdS Perturbation Theory
- Bulk-to-Boundary and Bulk-to-Bulk Propagators
- Contact Witten Diagrams
- Exchange Witten Diagrams and Conformal Blocks
- Spinning and Tensor Witten Diagrams
- Lorentzian Witten Diagrams and Real-Time Orderings
- AdS Integrals and D-Functions
- AdS Unitarity Cuts and Cutting Rules
- Loop Witten Diagrams and Bulk EFT Renormalization
- Loop Integral Bases and Multi-Trace Mixing
- Bulk Field Redefinitions and Contact Ambiguities
- Diagram Normalization and Reproducibility Benchmarks
AdS Scattering, Mellin Methods, and Bulk Locality
- Imported Mellin Representations and Bulk-Normalization Conventions
- Mellin Amplitudes and Penedones-Type Flat-Space Limits
- Mellin Contact Polynomials and Exchange Poles
- Bulk-EFT Information in Double-Trace Data and Large-Spin Anomalous Dimensions
- Bulk Causality from Lorentzian Inversion and Dispersive Sum Rules
- Regge Limits, Eikonal Scattering, and Causality
- Bulk-Point Singularities and Locality Diagnostics
- AdS Wavepackets and Boundary Extraction of Flat-Space Scattering
- Finite-Gap Corrections and Locality Error Budgets
- Bulk-EFT Checks from OPE, Unitarity, and Causality Data
- CFT Criteria for Approximate Bulk Locality
- CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities
Thermal Phases and AdS Black Holes
- Euclidean Saddles, Thermal States, and Hawking–Page Transitions
- Euclidean Actions, Boundary Terms, and Free-Energy Comparisons
- AdS Black Branes and Holographic Thermodynamics
- Rotating and Charged AdS Black Holes
- Two-Sided Black Holes and Thermofield-Double States
- Imported Wald Entropy in Holographic Higher-Derivative Thermodynamics
- Confinement and Deconfinement Dictionaries
- Black-Hole Instabilities and New Phases
- Ensemble Inequivalence and Microcanonical Saddles
- Finite-N Smoothing, Tunneling, and Metastability
Lorentzian, Nonequilibrium, and Chaotic Holography
- Lorentzian Holographic Correlators and Infalling Conditions
- Schwinger–Keldysh Contours and Real-Time Bulk Geometries
- Holographic Initial States and Euclidean Caps
- Holographic Influence Functionals and Open-Sector Dynamics
- Quasinormal Modes, Poles, and Spectral Response
- Bulk Quasinormal Modes and Boundary Hydrodynamic Poles
- Time-Dependent Geometries and Holographic Thermalization
- Holographic Quenches and Entanglement Growth
- Holographic Hydrodynamization, Attractors, and Gradient Asymptotics
- Shockwaves, OTOCs, and Scrambling
- Real-Time Claims, Horizons, and Validity Limits
- Finite-N Spectra, Recurrences, and the Late-Time Plateau
Holographic Matter, Transport, and Model Building
- Chemical Potential and Charged Black Branes
- Probe Branes, Flavor, and Mesonic Sectors
- Holographic Superconductors and Symmetry Breaking
- Holographic Fermions and Spectral Functions
- Holographic Models of Non-Fermi Liquids and Locally Critical Metals
- Holographic Models of Quantum Critical Matter and Strange Metals
- Hyperscaling-Violating and Lifshitz Geometries
- Kubo Formulae and Horizon Response
- Holographic Diffusion, Conductivity, and Momentum Relaxation
- Holographic Translation Breaking: Explicit and Spontaneous
- Holographic Viscosity and Higher-Derivative Hydrodynamics
- Fluid-Gravity Correspondence
- Holographic Anomalous and Magnetohydrodynamic Transport
- QCD-Like Holography and Phenomenological Limits
- Calibration, Uncertainty, and Cross-Model Inference
Bulk Reconstruction and Gravitational Dressing
- The Bulk Reconstruction Problem
- Extrapolate Dictionaries versus Interior Reconstruction
- HKLL Reconstruction for Free Bulk Fields
- Mode Completeness and Smearing-Kernel Domains
- Time-Band and Boundary-Diamond Reconstruction
- Interactions, Gravitational Dressing, and Microcausality
- Gravitational Gauss Laws and Boundary Anchoring
- Relational Bulk Observables and Dressing Choices
- Causal Wedges and Subregion Reconstruction
- Finite N, Horizons, State Dependence, and Reconstruction Limits
- Reconstruction Error Norms and Nonperturbative Precision
Holographic Entropy and Quantum Extremal Geometry
- The Ryu–Takayanagi Formula
- Covariant Extremal Surfaces and HRT
- Maximin Constructions and Extremal-Surface Existence
- Holographic Higher-Derivative Entropy Functionals
- FLM Corrections and Holographic Use of Imported Generalized Entropy and QES
- Replica Derivations and Cosmic Branes
- Rényi Entropies and Cosmic-Brane Backreaction
- Bit Threads, Multiflow, and Optimization Duality
- Tensor-Network Models of Holographic Entanglement Geometry
- Entanglement Wedges, Nesting, and Information Inequalities
- Holographic Entropy Inequalities and Entropy Cones
- Reflected Entropy and Proposed Entanglement-Wedge Cross-Section Duals
- Negativity, Purification Measures, and Proposed Mixed-State Bulk Duals
Modular Response, Relative Entropy, and Emergent-Gravity Claims
- Boundary Relative Entropy and Bulk Modular Data
- Modular Flow Reconstruction and Bulk Modular Evolution
- Entanglement First Law and Conditional Linearized Field-Equation Inference
- Canonical Energy and Second-Order Relative Entropy
- Shape Deformations and Displacement Response
- Quantum Fisher Information and Bulk Symplectic Forms
- Crossed-Product Gravitational Algebras and Generalized-Entropy Terms
- Modular Berry Transport and Bulk Connections
- Holographic ANEC, QNEC, and Focusing Arguments
- Beyond Linear Order: Conditional Emergent-Equation Claims
- Emergent-Gravity Claims: Assumptions, Nonconverses, and Status
Entanglement Wedges and Holographic Quantum Error Correction
- Entanglement-Wedge Reconstruction
- Code Subspaces, Logical Algebras, and Encoding Maps
- Exact Toy-Code Reconstruction Theorems
- Leading Semiclassical JLMS and Code-Subspace Claims
- Complementary Recovery, Area Terms, and Center Data
- Holographic Quantum Error Correction
- Approximate Finite-N Recovery, Alpha-Bits, and Error Bounds
- Holographic-QEC Algebras, Centers, and Gravitational Edge Data
- Continuum Factorization and Type-III Obstacles
- Non-Isometric Encoding Proposals
- QEC Evidence, Current Disputes, and Status
Holographic Complexity Proposals and Diagnostics
- Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets
- Complexity Equals Volume Proposals
- Complexity Equals Action Proposals
- Path-Integral and Tensor-Network Complexity
- Complexity of Formation and Time Growth
- Divergences, Counterterms, and Scheme Dependence
- Shockwaves, Switchbacks, and Scrambling Diagnostics
- Proposed Complexity Bounds and Their Counterexamples
- Operational Meaning, Nonuniqueness, and Evidence Status
AdS3/CFT2 and Three-Dimensional Gravity
- AdS3/CFT2 and the Brown–Henneaux Central Charge
- AdS3 Boundary Gravitons and Vacuum Characters
- BTZ Black Holes and Modular CFT Thermodynamics
- Virasoro Symmetry, Vacuum Blocks, and Semiclassical Gravity
- Chern–Simons Gravity and Boundary Currents
- Euclidean AdS3 Saddles and Handlebody Sums
- Wilson Lines, Geodesic Blocks, and Entanglement
- Pure AdS3 Gravity and Candidate CFT Constraints
- Conical Defects, Orbifolds, and Heavy States
- Finite-c Corrections and Nonperturbative Questions
AdS2, JT Gravity, SYK, and Random Matrices
- AdS2 Boundary Conditions and Fragmentation
- JT Gravity and the Schwarzian Boundary Mode
- Nearly AdS2 Effective Theory Beyond the Leading Schwarzian
- JT Correlators, Bilocals, and Chaos
- JT Topological Expansion and Weil–Petersson Volumes
- Near-AdS2, SYK, and the Duality Interface
- SYK Bilocal Collective Fields as Near-AdS2 Data
- SYK Conformal Regime and Schwarzian Matching
- Random Matrices, Spectral Statistics, and Ensemble Questions
- JT/SYK Spectral Form Factors and Universality Windows
- Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions
Black-Hole Microstates and Stringy Entropy
- Microscopic Black-Hole Entropy: Claim and Ensemble Contract
- BPS Indices, Absolute Degeneracies, and Wall Crossing
- D-Brane Bound States and the Strominger–Vafa Count
- D1-D5 CFT and AdS3 Microstate Data
- Attractor Mechanism and Charge-Only Entropy
- Higher-Derivative and Quantum Entropy Corrections
- Supersymmetric Localization Tests of the Quantum Entropy Function
- Microstate Geometries and Fuzzball Proposals
- Absorption, Emission, and Dynamical Tests
- Typicality, Non-BPS Extensions, and Evidence Limits
Wormholes, Gravitational Path Integrals, and Ensembles
- Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions
- Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals
- Saddles, Negative Modes, and Steepest-Descent Cycles
- Lorentzian Einstein–Rosen Bridges and Two-Boundary States
- Traversable Wormholes, Couplings, and Energy Conditions
- Euclidean Wormholes and Connected Boundary Amplitudes
- Semiclassical Gravitational Replicas
- Replica Wormholes and Saddle Competition
- Baby Universes, Alpha Parameters, and Proposed Superselection Sectors
- Factorization, Ensembles, and the Gravitational Path Integral
- Fixed-Theory Factorization and Nonperturbative Completion Tests
Black-Hole Information, Islands, and Interiors
- The Information Problem: Assumptions and Observables
- Evaporating Black Holes Coupled to Baths
- Complementarity, Firewalls, Remnants, and Final-State Proposals
- Page Curves and Entropy Bookkeeping
- Evaporating Replicas, Radiation Entropy, and Entanglement-Wedge Transitions
- Island Selection and Quantum Extremal-Surface Competition
- Hayden–Preskill Recovery and Decoding Tasks
- Algebraic Factorization of Radiation and Gravity
- Interior Reconstruction, State Dependence, Recovery, and Scrambling
- Non-Isometric Encoding in Evaporation Models
- Black-Hole Endpoints and Baby-Universe Alternatives
- What Island Calculations Establish—and What They Do Not
- Microscopic Unitarity versus Semiclassical Entropy Calculations
Higher-Spin and Vector-Model Holography
- Higher-Spin and Vector-Model Dualities
- Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions
- Vector-Model Bilocals as Higher-Spin Bulk Data
- Free and Critical Vector Models at the Higher-Spin Dictionary Interface
- Boundary Conditions, Double-Trace Flows, and Alternate Quantization
- Higher-Spin Dictionaries for Chern–Simons Matter and Parity Phases
- Higher-Spin Correlator Tests and Ward Identities
- Tensionless String Limits and Higher-Spin Enhancement
- Higher-Spin Symmetry Breaking and Einstein-Regime Limits
de Sitter and Cosmological Holography
- de Sitter and Cosmological Holography
- de Sitter Observables, States, and Horizon Patches
- Late-Time Wavefunctions and Boundary Data
- dS/CFT Dictionaries and Analytic Continuation
- Wavefunction Coefficients versus In-In Correlators
- Cosmological Bootstrap as Holographic Input
- Static-Patch Algebras and Observer Dependence
- de Sitter Entropy and Finite-Hilbert-Space Proposals
- Higher-Spin de Sitter Examples
- Matrix and Lower-Dimensional de Sitter Models
- de Sitter Holography: Dictionary Completeness, Obstructions, and Status
Flat-Space, BMS, and Celestial Holography
- Flat-Space Holography and Asymptotic Observables
- Null-Infinity Radiative Data as Candidate Boundary Data
- BMS, Memory, and Soft Sectors as Holographic Data
- Infrared-Dressed Scattering States and Boundary Dictionaries
- AdS Flat Limits as Candidate Flat-Holography Dictionaries
- Celestial CFT and Scattering Interfaces
- Celestial Bases and Boost Eigenstates as Boundary-Dictionary Inputs
- Celestial OPEs, Loop Corrections, and Infrared Factorization
- Carrollian and Other Flat-Boundary Holography Proposals: Evidence and Obstructions
- Flat and Celestial Dictionaries: Unitarity, Completeness, and Status
Quantum-Gravity Consistency and Swampland
- Quantum-Gravity Consistency Claims and Comparison Contract
- No-Global-Symmetry and Compactness Results Conditional on AdS/CFT
- Charge-Lattice and Gauge-Completeness Conjectures and Tests
- Weak-Gravity Conjectures and Their Variants
- Distance Conjectures, Towers, and Infinite-Distance Limits
- Species Bounds, Cutoffs, and Scale Separation
- Emergence Proposals for Gauge and Gravitational Couplings
- de Sitter Constraints and Vacuum-Structure Claims
- Landscape and Swampland Evidence Grammar
- Cobordism Conjecture and Topological-Sector Completeness
- Dated Status, Counterexamples, and Falsifiers
Canonical, Loop, Spin-Foam, and Group-Field Quantum Gravity
- Canonical Constraints, Dirac Observables, and Constraint Algebras
- Wheeler–DeWitt Quantization and the Problem of Time
- Ashtekar–Barbero Variables and Connection Dynamics
- Loop-Quantum-Gravity Kinematics and Spin Networks
- Kinematical Area and Volume Operators and Spectra
- Hamiltonian Constraints and Quantum Dynamics
- Spin-Foam and EPRL Amplitudes: Covariant-Dynamics Proposals
- Group-Field-Theory Fields, Feynman Diagrams, and States
- Group-Field Condensates and Cosmological Reduction
- Semiclassical States, Continuum Limits, and Classical Recovery
- Matter Coupling, Relational Observables, and Operational Predictions
- Canonical and Loop Programs: Evidence, Obstructions, and Current Status
Asymptotic Safety, Causal, and Discrete Quantum-Gravity Programs
- Asymptotic Safety and UV Fixed-Point Claims
- Functional Renormalization Group and Truncation Control
- Perturbative and Higher-Derivative Gravity Interfaces
- Causal Dynamical Triangulations: Phases and Continuum Evidence
- Causal-Set Kinematics and Dynamics
- Euclidean Dynamical Triangulations and Their Relation to CDT
- Tensor Models and Random Geometry
- Matrix-Model Lessons Beyond Two Dimensions
- Spectral Dimension and Dimensional Flow
- Emergence and Continuum-Limit Tests
- Unitarity, Reflection Positivity, and Causality Checks
- Safety, Causal, and Discrete Programs: Evidence, Obstructions, and Status
Quantum Cosmology and Singularity-Resolution Programs
- Quantum-Cosmology Observables and the Problem of Time
- Minisuperspace Reductions and Approximation Control
- Wheeler–DeWitt Cosmology: Boundary Conditions, Inner Products, and Probabilities
- No-Boundary and Tunneling Wavefunction Proposals
- Loop Quantum Cosmology and Effective Difference Dynamics
- Bounce and Singularity-Resolution Claims
- BKL, Mixmaster, and Inhomogeneous Singularities
- Quantum Geometrodynamics Beyond Minisuperspace
- Initial-State and Trans-Planckian Interfaces
- Semiclassical Recovery, Decoherence, Obstructions, and Status
Quantum-Gravity Phenomenology and Comparative Status
- Quantum-Gravity Observables and Test Taxonomy
- Lorentz-Invariance and Modified-Dispersion Tests
- Equivalence-Principle and Universality Tests
- Interferometry, Decoherence, and Entanglement Proposals
- Gravitational-Wave Propagation and Polarization Tests
- Black-Hole Ringdown, Shadows, and Compact-Object Probes
- Cosmological Background and Primordial-Spectrum Probes
- High-Energy, Cosmic-Ray, and Multimessenger Constraints
- Cross-Program Evidence and Falsifier Matrix
- Open Problems, Research Handoffs, and Update Ledger
Mathematical QFT
Theorem-First Frameworks and Claim Grammar
- Theorem-First Claim Records: Objects, Hypotheses, Conclusions, and Status
- QFT Frameworks, Object Classes, and Typed Maps
- Existence, Construction, Reconstruction, and Continuum Claims
- Equivalence, Uniqueness, and Comparison Notions
- Domains, Signatures, Supports, and Regularity Ledgers
- Positivity, Spectrum, Covariance, and Locality Hypotheses
- Counterexamples, Nonconverses, and Hypothesis Stress Tests
- Machine-Checked Theorem Records and Formal-Proof Provenance
- Source Authority, Dated Status, and Specialist Review
Wightman Fields, Reconstruction, and Structural Theorems
- Wightman Fields, Domains, and Axioms
- Wightman Functions and Spectral Support
- The Wightman Reconstruction Theorem
- Tube Domains, Complex Lorentz Covariance, and Analyticity
- Jost Points, Edge-of-the-Wedge, and Locality
- Analyticity, CPT, and Spin–Statistics
- CPT Theorem Variants and Their Hypotheses
- Spin–Statistics Theorems and Failure Modes
- Haag's Theorem and Inequivalent Representations
- Haag-Theorem Variants, Domains, and Proposed Evasions
- Wightman-Framework Models, Counterexamples, and Scope
Euclidean Fields, Reflection Positivity, and OS Reconstruction
- Euclidean Random Fields and Schwinger Hierarchies
- Osterwalder–Schrader Axioms and Reflection Positivity
- Reflection Positivity and Hilbert-Space Reconstruction
- Osterwalder–Schrader Reconstruction
- Euclidean Growth, Regularity, and Temperedness Conditions
- Analytic Continuation between Euclidean and Lorentzian Domains
- Clustering, Vacuum Uniqueness, and Mass-Gap Implications
- Fermionic, Gauge, and Lattice Reflection Positivity
- OS–Wightman Comparison Directions and Failure Modes
Particles, Scattering Theory, and Infrared Structure
- Particles, Mass-Shell Spectrum, and One-Particle Subspaces
- Haag–Ruelle Scattering-State Construction
- Wave Operators and Asymptotic Fields
- LSZ Reduction and Amputated Distributions
- Araki–Haag Detectors and Particle Weights
- Asymptotic Completeness: Definitions and Known Models
- Massless Scattering and Radiation Fields
- Infraparticles and Velocity Superselection
- Gauss-Law Charges and Infrared Sectors
- Dressed, Inclusive, and Algebraic Infrared Observables
- Scattering Analyticity, Crossing, and Rigorous Bounds
Local Nets, States, and Representations
- Haag–Kastler Nets and Locality
- Quasilocal C*-Algebras and Inductive Limits
- States, GNS Representations, and Folia
- Local Normality, Quasiequivalence, and Folia
- The Reeh–Schlieder Theorem
- Isotony, Additivity, Duality, and Primitive Causality
- From Pointlike Fields to Nets and Affiliated Operators
- Factorizing S-Matrices, Wedge-Local Fields, and Borchers Constructions
- Representation Types, Factors, and Local-Algebra Structure
Modular Theory, Nuclearity, and Split Inclusions
- Modular Theory, Nuclearity, and the Split Property
- Standard von Neumann Algebras and Tomita–Takesaki Theory
- Modular Automorphisms, Conjugations, and Standard Forms
- The Bisognano–Wichmann Theorem and Geometric Modular Action
- Phase-Space Nuclearity and Compactness Maps
- Split Inclusions, Type-I Intermediates, and Statistical Independence
- Modular Nuclearity and Wedge-Local Constructions
- Half-Sided Modular Inclusions and Spacetime Reconstruction
- Type-III Local Algebras, Entropy, and Cutoff Limits
Superselection Sectors, Statistics, and Gauge Reconstruction
- Sector Selection, Localization, and Transportability
- Superselection Sectors and DHR Reconstruction
- Endomorphisms, Intertwiners, and Tensor Products
- Conjugates, Statistics Operators, and Statistical Dimension
- Doplicher–Roberts Compact-Gauge Reconstruction
- Braided Sectors, Anyonic Statistics, and Low-Dimensional Nets
- BF Sectors, Spacelike Cones, and Massive Charges
- Gauss-Law Infrasectors and Asymptotic Charge Classes
- Solitonic, Topological, and Boundary Sectors
- Sector Completeness, Field Algebras, and Classification Limits
Locally Covariant QFT and Dynamical Locality
- Globally Hyperbolic Spacetimes and the Loc Categories
- Locally Covariant QFT as a Functor
- Natural Transformations, Fields, and Subtheory Embeddings
- Time-Slice Axiom and Relative Cauchy Evolution
- Stress–Energy Response and Background Variation
- Dynamical Locality and Kinematic–Dynamic Nets
- The Same-Physics-in-All-Spacetimes Principle
- No-Natural-State Results and Covariant State Spaces
- Local Covariance with Boundaries and Background Structures
- LCQFT, Nets, Fields, and Factorization: Typed Comparisons
Thermal AQFT, KMS States, and Nonequilibrium Structures
- C*-Dynamical Systems and the KMS Condition
- Modular Dynamics and Equilibrium Representations
- Passivity, Complete Passivity, and Ground States
- Relativistic KMS Analyticity and Spectrum
- Factorial KMS States, Phases, and Symmetry Breaking
- Thermal Nuclearity, Return to Equilibrium, and Mixing
- Nonequilibrium Steady States and Entropy Production
- Local Thermal Equilibrium and Thermal Observables
- Horizon KMS, Unruh, and Hawking Theorem Interfaces
Noncommutative Information, Measurement, and Algebraic QEC
- Von Neumann Algebra Channels and Positive Normal Maps
- Araki Relative Entropy, Monotonicity, and Data Processing
- Sufficiency, Conditional Expectations, and Petz Recovery
- Noncommutative Lp Spaces, Divergences, and Information Bounds
- Local Operations, Instruments, and AQFT Measurement
- Fewster–Verch Probe Measurements, State Updates, and Causal Composition
- Operational Independence, the Split Property, and Bell Correlations
- Algebraic Quantum Error Correction and Correctable Subalgebras
- Subfactor Index, Sector Information, and Entropy
- Type-III Information Tasks, Split-Regulated Entropy, and Energy Constraints
Constructive Euclidean QFT and Cutoff Removal
- The Constructive Program and Cutoff Removal
- Gaussian Euclidean Fields as Measures
- Interacting Measures, Stability, and Wick Ordering
- Cluster Expansions and Correlation Inequalities
- Thermodynamic Limits, Correlation Decay, and Phase Control
- Constructing P(φ)₂ and φ⁴₃ Models
- Constructive Fermionic and Yukawa Models
- Constructive Scattering, Particle Structure, and Mass Gaps
- Stochastic Quantization: Invariant Measures and Convergence Theorems
- Three-Dimensional Yang–Mills–Higgs Scaling-Limit Theorems
- From Euclidean Measures to Relativistic Models
- Constructive Existence Ledger by Model, Dimension, and Observable
Rigorous Renormalization Group and Continuum Control
- Rigorous RG as a Dynamical System
- Finite-Range Decompositions and Multiscale Integration
- Polymer Activities and Normed RG Coordinates
- Renormalized Trajectories and Counterterm Tuning
- Stable Manifolds and Relevant–Marginal Control
- Universality, Critical Manifolds, and Observable Control
- Critical φ⁴ Models and Logarithmic Corrections
- Weakly Self-Avoiding Walk and Supersymmetric RG
- Fermionic Multiscale RG and Fermi-Surface Problems
- Rigorous-RG Scheme Comparison and Continuum-Construction Status
Perturbative AQFT and Causal Renormalization
- Causal Factorization and Time-Ordered Products
- Local and Microcausal Functionals with Peierls Brackets
- Epstein–Glaser Induction
- Scaling Degree and Extension of Distributions
- Time-Ordered Products, Causal Wick Expansion, and Renormalization
- Renormalization Freedom and the Stückelberg–Petermann Group
- Perturbative AQFT and the Bogoliubov Map
- Perturbative Agreement and Background Independence
- Ward Identities and Anomalous Obstructions
- Adiabatic Limits and Algebraic Interacting Nets
- Nonperturbative Local C*-Net Developments: Construction and Scope
- Formal-Power-Series Interacting Constructions, States, and Scope
Microlocal QFT and Renormalized Local Fields
- Microlocal Calculus for Quantum Fields
- Wavefront-Set Products, Pullbacks, and Pushforwards
- Green-Hyperbolic Operators and Causal Propagators
- Propagation of Singularities for Hyperbolic Fields
- Higher-Point Microlocal Spectrum Conditions
- Hadamard States and the Wavefront-Set Characterization
- Wick Polynomials under Microlocal Conditions
- Local Covariant Wick Powers and Operator Products
- Time-Ordered Products and the Renormalized Stress Tensor
- Microlocal Renormalization Ambiguities and Their Classification
Curved-Spacetime AQFT, Stress Tensor, and Energy Inequalities
- Algebraic Free Fields on Curved Spacetimes
- Existence, Deformation, and Gluing of Hadamard States
- The Renormalized Stress Tensor: Conservation and Ambiguities
- Trace Anomalies, Local Covariance, and Scaling
- Absolute and Difference Quantum Energy Inequalities
- Worldline, Spacetime-Volume, and Null-Energy Bounds
- Averaged Null Energy and QNEI Geometric Hypotheses
- Quantum Energy Inequalities and Singularity-Theorem Interfaces
- Semiclassical Einstein Equation: Existence and Stability Results
- Interacting pAQFT on Curved Spacetimes
- Boundaries, Nonglobally Hyperbolic Spacetimes, and State Obstructions
Factorization Algebras and Local-to-Global Observables
- Prefactorization and Factorization Algebras
- Weiss Descent and Local-to-Global Observables
- Elliptic Complexes and Factorization Observables
- Classical Observables and Poisson Factorization
- BV Quantization and Obstruction–Deformation Complexes
- Quantum Observables, Renormalization, and Factorization
- Locally Constant Factorization Algebras and Eₙ Algebras
- Factorization Homology and Manifold Invariants
- Holomorphic, Topological, and Mixed Factorization Theories
- Factorization Comparison Theorems and Their Limits
- AQFT to Prefactorization on Time-Orderable Covers
- Prefactorization to AQFT: The Infinity-Categorical Converse Gap
Derived, Deformation, and Homotopical QFT
- Derived Critical Loci and Derived Gauge Quotients
- L∞ Algebras, Formal Moduli, and Field Equations
- Shifted Symplectic, Poisson, and BV Geometry
- Deformation Quantization, Formality, and Star Products
- Obstruction–Deformation Complexes and Quantization Classes
- Derived Intersections, Boundary Conditions, and Correspondences
- Higher Morita Categories and Theories as Objects
- Homotopical Renormalization and Effective-Theory Maps
- Strictification, Comparison, and Foundational Limits
Gauge Orbit Geometry, Global Obstructions, and Measures
- Gauge Configuration Groupoids and Moduli
- Orbit Spaces, Stabilizers, and Stratified Quotients
- Principal-Bundle Sectors and Large Gauge Transformations
- Local Slices, Gauge Fixing, and Faddeev–Popov Geometry
- Elliptic Gauge Complexes and Gribov Obstructions
- The Gribov–Singer Global Gauge-Fixing Obstruction
- Moduli Stacks and Derived Geometry of Gauge Fields
- Determinant Lines, Global Obstructions, and Orientations
- Nonperturbative Gauge Measures: Positivity and Configuration-Space Limits
BRST–BV Quantization and Anomalies
- Koszul–Tate Resolution and the BRST Bicomplex
- BRST Cohomology as Derived Invariants
- Local BRST Cohomology, Consistent Deformations, and Currents
- The BV Complex and Classical Master Equation
- Gauge-Fixing Fermions, Canonical Transformations, and Observables
- Quantum Master Equation and Anomaly Obstructions
- Local Anomaly Descent and Wess–Zumino Consistency
- Global Anomalies, Determinant Lines, and Eta Invariants
- Anomaly Cancellation and Positive-Hilbert-Space Existence
- Perturbative Gauge QFT: Constructions and Scope
- Chiral Gauge Perturbative Construction: Open Obligations
BV–BFV Boundaries, Corners, and Gluing
- BV–BFV Structures, Boundaries, and Gluing
- Boundary Phase Spaces, Constraints, and the BFV Charge
- Bulk–Boundary Master Equations and Anomaly Inflow
- Corners, Stratification, and Higher-Codimension Data
- Gluing, Reduction, and Composition Theorems
- Edge Modes and Extended Observables at Gauge Boundaries
- AKSZ Sigma Models as BV–BFV Examples
- Factorization Algebras with Boundaries and Defects
- Perturbative Boundary Gauge Theory: Scope and Open Problems
Conformal Nets and Operator-Algebraic CFT
- Conformal Nets and Covariance Axioms
- Circle and Interval Nets: Möbius and Diffeomorphism Covariance
- Strong Additivity, Haag Duality, and the μ-Index
- Split Property and Complete Rationality
- DHR Sectors and Modular Tensor Categories of Nets
- Subfactors, Jones Index, and Q-Systems
- Extensions, Orbifolds, Cosets, and Alpha-Induction
- Full Two-Dimensional CFT from Chiral Nets
- Nonrational, Infinite-Index, and Logarithmic Net Problems
- Conformal-Net Classification: Invariants and Limits
Vertex Operator Algebras and Direction-Specific Net Bridges
- Vertex Operator Algebras: Axioms, Grading, and Locality
- Modules, Intertwining Operators, and Tensor Categories
- C₂-Cofiniteness, Rationality, and Regularity
- Zhu Algebras, Characters, and Modular Invariance
- VOA Extensions, Orbifolds, Cosets, and Commutants
- Unitary VOAs, Energy Bounds, and Strong Locality
- Energy Bounds, Strong Locality, and Smeared Vertex Operators
- From VOAs to Conformal Nets
- From Nets to VOAs: Partial Reconstructions and Limits
- Direction-Specific VOA–Net Extension Theorems and Open Converses
Extended TQFT, Bordisms, and the Cobordism Hypothesis
- Bordism Categories and Symmetric Monoidal TQFTs
- Atiyah–Segal Functorial TQFT and Gluing
- Fully Extended TQFTs and Higher Categories
- Higher-Categorical Targets and Levels of Dualizability
- Dualizability and the Cobordism Hypothesis
- Tangential Structures: Oriented, Spin, and Framed Theories
- Invertible Field Theories and Generalized Cohomology
- Anomalies as Relative and Invertible Field Theories
- Reflection Positivity, Unitarity, and Hermitian TQFTs
- Boundaries, Defects, and Extended Operators in TQFT
- Extended-TQFT Classification: Scope and Counterexamples
Defects, Generalized Symmetry, and Categorical QFT
- Defects on Stratified Spacetimes and Higher-Categorical Composition
- Fusion Categories, Module Categories, and Bimodule Defects
- Invertible p-Form Symmetries and Topological Defects
- Noninvertible Symmetries, Fusion, and Junction Data
- Categorical Symmetries, Higher Representations, and Charges
- Gauging, Equivariantization, Orbifolds, and Condensation
- Symmetry TFTs, Anomaly Inflow, and Boundary Realizations
- Boundary Conditions and Interfaces in Functorial Field Theory
- Generalized-Symmetry Sectors, Selection Rules, and Reconstruction
- Semisimple, Nonsemisimple, and Unitarity Boundaries
- Categorical Classification: Equivalence and Completeness Limits
Existence, Continuum Limits, and Classification Frontiers
- Existence, Uniqueness, and Equivalence Claims
- Two- and Three-Dimensional Constructive Model Atlas
- Four-Dimensional Scalar QFT: Existence and Triviality
- Yang–Mills Existence and the Mass Gap
- Rigorous Mass-Gap, Scattering, and Confinement Obligations
- Chiral Gauge Theories and Standard Model Construction
- Interacting Curved-Spacetime and Gauge Existence Problems
- Continuum Limits and Universality Problems
- Lattice-to-Continuum Constructive Proof Obligations
- Classification and Comparison Problems across QFT Frameworks
- Conformal-Net, VOA, TQFT, and Categorical Classification Frontiers
- Dated Open-Problem Evidence Search and Research Handoff
Original QFT.org content:CC BY 4.0, unless an item supplies different terms. Third-party material retains its own terms.