Locality & causality
Spacelike (anti)commutativity, causal support of real-time correlators, and the analytic structures that underwrite dispersion relations and (when asymptotic states exist) the S-matrix.
Quantum Field Theory
Quantum field theory, organized by principles and invariants — from core definitions to current research frontiers.
EncyclopedicResearch-forwardPrinciples-firstCross-linked notation
Quantum field theory (QFT) is less a single subject than a family of compatible structures: local operator algebras, correlation functions, and scale-dependent effective descriptions constrained by locality, symmetry, and unitarity. qft.org is a curated map of these structures — written to support both learning and research, with an emphasis on invariants that survive changes of variables, regularization scheme, or duality frame.
Start anywhere
Section titled “Start anywhere” Reference Atlas Definitions, identities, standard conventions, and fast “what does this symbol mean?” pages.
Concept Hubs Locality, unitarity, symmetry, anomalies, RG, OPE, defects, entanglement — each as an interlinked hub.
Research Frontiers Curated overviews of active directions: strong coupling, bootstrap, generalized symmetries, amplitudes, real-time QFT.
Learning Paths Structured routes with prerequisites and checkpoints: from free fields to gauge theory, CFT, and nonperturbative tools.
What we mean by “quantum field theory”
Section titled “What we mean by “quantum field theory””A QFT assigns local operators (\mathcal O(x)) to spacetime regions, with causal commutation relations and states that produce correlation functions. In this language, the theory is specified by operator content and operator relations (Ward identities, OPE data, anomalies), not by any particular Lagrangian.
When a functional integral exists, a QFT can be packaged into a generating functional (Z[J]) whose derivatives produce correlators. Euclidean formulations connect directly to statistical mechanics and often provide a precise definition via regularization (lattice, contour prescriptions, or constructive frameworks).
A QFT is a scale-dependent effective description: as we change the resolution (\mu), couplings and operators flow under the renormalization group. Universality, relevance/irrelevance, and matching explain why long-distance predictions can be sharp even with unknown UV completion.
The invariants that organize QFT
Section titled “The invariants that organize QFT”Unitarity & positivity
Probability conservation and positivity become quantitative constraints: bounds on spectra, constraints on correlators, and EFT sign/positivity conditions tied to causality and UV completion.
Symmetry (including generalized)
Global symmetries, gauge redundancy, higher-form symmetries, and categorical symmetry structures organize operator sectors, phases, and selection rules — and often explain “miracles” like dualities.
Anomalies
Robust obstructions and invariants that survive RG flow and frequently force infrared structure: anomaly matching, anomaly inflow, and sharp constraints on phase diagrams and defect content.
Renormalization group
Flows, fixed points, universality classes, and the logic of effective field theory. The RG is the bridge between microscopic models and continuum descriptions.
Defects, boundaries, topology
Line/surface operators, interfaces, and boundary conditions as first-class observables; fusion/braiding, topological response, and defect RG as organizing data for both gapped and gapless phases.
The “data” of a theory
Section titled “The “data” of a theory”Correlators & Ward identities
Euclidean/Lorentzian (n)-point functions, symmetry constraints, and operator relations designed to be invariant under field redefinitions and robust across dual descriptions.
Spectrum & OPE data
Scaling dimensions, spins, charges, and OPE coefficients (especially in CFT), together with consistency conditions such as crossing and reflection positivity.
On-shell observables
Scattering amplitudes (when defined), their analyticity/unitarity/factorization properties, and modern reconstructions of EFTs and gauge theories using on-shell consistency.
Phases, defects, responses
Partition functions, thermal/finite-density observables, topological response data, and defect networks that classify phases and encode universal long-distance structure.
Frontier spotlights
Section titled “Frontier spotlights” Generalized symmetries & anomalies Higher-form symmetries, anomaly constraints, defect operators, and modern phase diagnostics.
Bootstrap and consistency bounds Crossing + unitarity as a nonperturbative engine: bounds, extremal spectra, and precision CFT.
Strong coupling & confinement Confinement diagnostics, semiclassics, large‑N, and controlled corners of nonperturbative gauge dynamics.
Amplitudes, analyticity, EFT positivity On-shell methods and dispersion/positivity constraints linking causality, unitarity, and UV completion.