Mathematical QFT

Mathematical QFT asks when fields, observables, states, and limits define a quantum theory, and which conclusions follow from that definition. It turns questions about locality, particles, thermal behavior, gauge symmetry, and existence into precise statements with testable hypotheses.

The central distinction is between assuming a framework, constructing a model, and proving a property of that model. For a free scalar field, for example, a differential equation helps construct an observable algebra. Choosing a state then produces a Hilbert-space representation. Neither step by itself constructs an interacting theory. This volume develops these connections across analytic, operator-algebraic, geometric, and categorical approaches.

Where to begin

• First theorem-level study: start with Theorem-First Frameworks and Claim Grammar, then Wightman Fields, Reconstruction, and Structural Theorems. Learn to identify the objects, hypotheses, and exact conclusion before using a structural theorem.
• For Euclidean or regulated calculations: enter Euclidean Fields, Reflection Positivity, and OS Reconstruction, then Constructive Euclidean QFT and Cutoff Removal. Separate reconstructing a theory from suitable correlators from proving that a regulator limit supplies those correlators.
• For local observables or curved spacetime: begin with Local Nets, States, and Representations, then choose modular theory, thermal states, or local covariance according to the physical question.

Helpful background. Hilbert spaces, distributions, and operator algebras prepare the analytic routes. Bundles support gauge geometry; Schwinger functions and microcausality connect the mathematics to physical correlators and locality. Consult these tools when your chosen route needs them.

Chapters

1. Theorem-First Frameworks and Claim Grammar — State objects, domains, hypotheses, implication directions, and the strength of a conclusion.
2. Wightman Fields, Reconstruction, and Structural Theorems — Connect field distributions and vacuum correlators to relativistic structural results.
3. Euclidean Fields, Reflection Positivity, and OS Reconstruction — Determine when Euclidean data reconstruct a positive-energy quantum theory.
4. Particles, Scattering Theory, and Infrared Structure — Specify the assumptions needed for particle states, scattering limits, and infrared alternatives.
5. Local Nets, States, and Representations — Build a theory from regional observable algebras and their states.
6. Modular Theory, Nuclearity, and Split Inclusions — Relate state-dependent algebraic structure to localization and phase-space restrictions.
7. Superselection Sectors, Statistics, and Gauge Reconstruction — Recover charges, statistics, and gauge structure under stated localization assumptions.
8. Locally Covariant QFT and Dynamical Locality — Compare observables consistently across spacetimes and metric perturbations.
9. Thermal AQFT, KMS States, and Nonequilibrium Structures — Formulate equilibrium relative to a specified dynamics and study departures from it.
10. Noncommutative Information, Measurement, and Algebraic QEC — Treat entropy, measurement, and recovery using observable algebras.
11. Constructive Euclidean QFT and Cutoff Removal — Prove existence through controlled measures and regulator limits in specified models.
12. Rigorous Renormalization Group and Continuum Control — Use estimates on scale-dependent transformations to control continuum behavior.
13. Perturbative AQFT and Causal Renormalization — Construct formal interacting observables from causality and distribution extension.
14. Microlocal QFT and Renormalized Local Fields — Control singularities, distribution products, and local composite fields.
15. Curved-Spacetime AQFT, Stress Tensor, and Energy Inequalities — Relate geometry and state conditions to stress-energy and its bounds.
16. Factorization Algebras and Local-to-Global Observables — Assemble compatible observables on larger regions from local data.
17. Derived, Deformation, and Homotopical QFT — Organize deformation, equivalence, and quantization using complexes and homotopy.
18. Gauge Orbit Geometry, Global Obstructions, and Measures — Identify the geometric obstacles to global gauge fixing and integration.
19. BRST–BV Quantization and Anomalies — Express gauge reduction and quantization obstructions through cohomology and master equations.
20. BV–BFV Boundaries, Corners, and Gluing — Track boundary data and compatibility when cutting and gluing a theory.
21. Conformal Nets and Operator-Algebraic CFT — Study chiral theories through interval algebras, sectors, and extensions.
22. Vertex Operator Algebras and Direction-Specific Net Bridges — Compare vertex-algebra and conformal-net constructions under their actual hypotheses.
23. Extended TQFT, Bordisms, and the Cobordism Hypothesis — Classify specified extended topological theories through dualizability and bordism data.
24. Defects, Generalized Symmetry, and Categorical QFT — Formulate defect composition, gauging, and categorical symmetry with explicit obstructions.
25. Existence, Continuum Limits, and Classification Frontiers — Identify the missing construction, estimate, or comparison theorem in a frontier claim.

How the ideas connect

Consider the real free Klein–Gordon field of mass $m>0$ on a globally hyperbolic spacetime without boundary. Global hyperbolicity supplies a well-posed causal problem and advanced and retarded Green operators. Their difference determines a symplectic form on the appropriate space of solutions. Quantizing that space gives a Weyl algebra: an algebra of bounded exponentials of smeared fields, encoding the canonical commutation relations.

This construction can be made compatible with spacetime embeddings that preserve the relevant geometry and causal structure. For the free field it gives a locally covariant theory satisfying causality and the time-slice property: observables near a Cauchy surface determine the full algebra Brunetti, Fredenhagen, and Verch 2003, §2.3, Theorem 2.2, preprint pp. 8–9 (PDF).

An algebra still needs a state to give expectation values. A positive normalized functional $\omega$ supplies the inner product $\langle[A],[B]\rangle=\omega(A^*B)$ after quotienting out zero-norm elements. Completion gives the GNS Hilbert space and its representation of the algebra. Different states can yield inequivalent representations; on curved spacetime the algebraic construction itself need not choose a vacuum. The states and representations page develops these steps and their uniqueness statement.

Three different questions now become visible. Reconstruction starts with suitable data and builds a new description; Euclidean reconstruction, for example, requires reflection positivity and further regularity and covariance hypotheses on the full hierarchy of correlators. Construction must produce data satisfying those hypotheses, often while removing cutoffs with uniform estimates. Comparison asks whether two descriptions are related by a specified map and whether a converse exists. Matching a two-point function, or writing a formal perturbation series, does not answer all three questions.

Keep the limitation attached to the result. A perturbative construction can be exact coefficient by coefficient without establishing convergence. A theorem for extended topological theories does not classify arbitrary dynamical QFTs. A finite computation checks its encoded instance. Similarly, a Hilbert-space reconstruction does not establish asymptotic completeness of scattering states without further arguments.

The comparison of representative claims shows how these distinctions work in concrete frameworks, with the hypotheses, conclusions, and failed converses kept together.

For a broader comparison, the following optional table groups the chapters into eleven study arcs. Each row identifies the mathematical objects, decisive assumptions, permissible conclusions, and physical applications.

Compare the frameworks, hypotheses, and limitations

Framework objects, licensed conclusion types, failure boundaries, and physical return routes across eleven study arcs

Study arc	Primitive objects	Main conclusion type	Decisive hypotheses	Excluded upgrade	Physical return
Claims, Wightman fields, and Euclidean reconstruction	Typed theorem records, operator-valued distributions, Wightman functions, Schwinger hierarchies	Axiom consequences and directional Hilbert-space reconstruction	Common domains, spectral support, locality, full positivity hierarchy, Euclidean growth and regularity	Matching low-point functions or Euclidean covariance alone does not establish existence or equivalence	Relativistic fields, correlators, and foundational structural claims
Particles, scattering, and local nets	Local observables, asymptotic fields, detector limits, quasilocal algebras, states and representations	Particle construction, scattering limits, net construction, or representation comparison	Mass isolation or infrared substitute, locality, stability, clustering, selection criterion, controlled asymptotic limit	LSZ notation does not imply Haag–Ruelle hypotheses or asymptotic completeness; an abstract net need not recover point fields	Scattering amplitudes, infrared observables, and physical particle interpretation
Modular theory, sectors, and local covariance	Von Neumann algebras, modular objects, localized endomorphisms, tensor categories, spacetime functors	Geometric modular action, split inclusion, gauge reconstruction, or natural dynamics	Faithful cyclic separating states, nuclearity bounds, localization and conjugates, typed spacetime morphisms, time-slice property	The split property is not sharp tensor factorization; DHR reconstruction and dynamical locality do not apply outside their stated domains	Thermal structure, charge sectors, gauge symmetry, and curved-spacetime dynamics
Thermality, information, and measurement	C*-dynamical systems, KMS states, weights, normal maps, instruments, recovery channels, correctable algebras	Equilibrium characterization, passivity, entropy inequality, local measurement, or recovery theorem	Named automorphism group, state normality, modular or split assumptions, locality and causal ordering, exact recovery criterion	A KMS identity is not detector thermality, and a sharp type-III local algebra has no canonical density matrix	Thermal, nonequilibrium, measurement, and quantum-information applications
Constructive QFT and rigorous renormalization	Cutoff measures, Schwinger functions, polymer activities, Banach-space RG trajectories, scaling limits	Measure construction, cutoff removal, controlled flow, universality, or continuum limit	Stability, uniform estimates, tightness, tuned relevant directions, contraction, limit topology, retained observables	A finite regulated model, numerical flow, or perturbative beta function is not a nonperturbative continuum construction	Renormalization, lattice, and nonperturbative model claims
Causal, microlocal, and curved-spacetime QFT	Microcausal functionals, time-ordered products, wavefront sets, Hadamard states, local fields and stress tensors	Formal interacting construction, distribution extension, local-covariant renormalization, or energy inequality	Causal factorization, formal coefficient ring, scaling-degree and wavefront conditions, local covariance, state and sampling class	Formal construction does not imply convergence; a quantum energy inequality does not automatically imply ANEC or solve backreaction	Renormalized observables, curved-spacetime fields, horizons, and semiclassical gravity
Factorization and derived QFT	Prefactorization algebras, Weiss cosheaves, elliptic complexes, derived critical loci, deformation and obstruction complexes	Descent, local-to-global computation, formal quantization, or direction-specific framework comparison	Cover class, target category, additivity and time-slice assumptions, gradings, completions, obstruction vanishing	One comparison functor is not an unrestricted AQFT equivalence, and a derived formal moduli problem is not a nonperturbative Hilbert-space theory	Local observables, effective field theory, and geometric or topological sectors
Gauge geometry, BRST–BV, and BV–BFV	Orbit groupoids, bundle sectors, resolution complexes, BV actions, anomaly classes, boundary phase spaces and relative states	Local slice, cohomological resolution, perturbative quantization, anomaly cancellation, or boundary gluing	Global sector data, properness or derived reduction, master equations, renormalization, Lagrangian boundary data, residual modes and corners	Faddeev–Popov notation does not solve the global orbit problem; topological gluing formulas do not construct four-dimensional Yang–Mills	Gauge theory, anomalies, edge modes, and bounded-region field theory
Conformal nets and vertex operator algebras	Interval nets, positive-energy representations, subfactors, Q-systems, graded vertex algebras and modules	Sector or extension classification, tensor-category structure, VOA-to-net construction, or restricted converse	Covariance and positivity, finite-index or rationality assumptions, unitarity, energy bounds, strong locality, cofiniteness and completion	Neither rational examples nor strong locality establish a universal equivalence between conformal nets and VOAs	Two-dimensional CFT, chiral algebras, modular data, and conformal constructions
Extended TQFT, defects, and categorical symmetry	Bordism categories, higher targets, fully dualizable objects, defect data, fusion and module categories, symmetry TFTs	Functorial construction, extended classification, gauging, condensation, or anomaly realization	Dimension, tangential structure, extension depth, target category, dualizability, positivity, semisimplicity or its replacement, anomaly cancellation	The cobordism hypothesis does not classify arbitrary physical QFTs, and fusion data alone do not prove completeness of defects or charges	Topological phases, generalized symmetry, conformal defects, and many-body applications
Existence and classification frontiers	Model families, regulator systems, continuum limits, comparison functors, proposed classification data	Dated existence, mass-gap, universality, equivalence, or classification statement	Exact dimension and fields, observables, topology of limits, nontriviality, positivity or reconstruction, source version, contrary results	Physical evidence, perturbation theory, a partial model theorem, or a classification invariant cannot close the remaining proof obligations	The physical model together with a dated research record

Structured comparison data (JSON) contains the same headers and rows.

Where to go next

Return to Foundations of QFT for the physical interpretation of states, correlators, and particles. Use QFT in Curved Spacetime for state selection and geometric applications, or Nonperturbative Dynamics for controlled analytic methods and their observables. The mathematical and constructive QFT research field tracks developing existence and comparison questions.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Field Theory.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF, arXiv:math-ph/0112041v1. Locators above refer to the December 2001 preprint.