Mathematical QFT

Mathematical quantum field theory asks a prior question about every calculation: what, exactly, has been defined or proved? The answer depends on the object—fields, Euclidean measures, local nets, functors, factorization algebras, derived moduli problems, vertex algebras, or bordism functors—and on the domains, positivity conditions, limit topologies, comparison maps, and counterexamples attached to it. This volume develops those distinctions at theorem level while continually returning the result to the physical problem that motivated it.

Helpful background. Banach and Hilbert spaces supplies completion, duality, and operator-domain language. Test-function spaces and distributions supplies smearing, support, and convergence. Operator algebras and positive functionals prepares states and representations. Vector, principal, and associated bundles prepares gauge geometry. Euclidean correlators and Schwinger functions and microcausality provide the main physical entry points. A reader missing one of these tools can repair it when first needed rather than postponing the entire volume.

What a mathematical-QFT claim contains

The basic unit of the volume is a directional statement

$$(\mathcal O,\mathcal D,\mathcal Q,\mathcal H;\mathcal S_v) \xrightarrow{\ \mathcal P\ } (\mathcal C,\mathcal U,\mathcal N).$$

Here $\mathcal O$ names the objects and maps, $\mathcal D$ their domains and regularity, $\mathcal Q$ the quantifiers and order of limits, $\mathcal H$ the hypotheses, and $\mathcal S_v$ the source at a definite version. The proof, construction, or comparison $\mathcal P$ licenses a conclusion $\mathcal C$, a stated uniqueness or equivalence strength $\mathcal U$, and an explicit nonconverse or failure boundary $\mathcal N$.

The notation is a checklist, not a new formalism. It prevents familiar category errors: a regulated integral is not yet a continuum model; reconstruction is not construction; agreement of selected correlators is not equivalence; local quasiequivalence is not global unitary equivalence; a formal power series is not a convergent measure; and a successful computation is not a theorem whose quantifiers were never encoded.

Lorentzian formulas in this volume use metric signature $(+---)$, while Euclidean quadratic forms are positive. Fields are operator-valued distributions until smearing and domain control justify stronger language. A limit always names its topology and order, a positivity condition names the space on which it acts, and a classification names its equivalence relation.

The first diagram shows the one-way anatomy of a valid claim. Read one horizontal path at a time: the examples are distinct object classes, not interchangeable presentations of one theory.

A theorem-level claim fixes its carrier and limit order, discharges the hypotheses, applies a named directional result, and states the conclusion’s ceiling and source version. The formalized free Gaussian branch is a real kernel-checked result, but it certifies only its encoded predicates. No horizontal path supplies a converse. The diagram is schematic and not to scale. Structured description and source data (JSON)

Choose a route by the question you have

To check a theorem or paper, start with the claimed verb: exists, reconstructs, converges, classifies, embeds, or is equivalent. Identify the primitive object and quantified variables; expose the domain and limit order; list assumptions; locate the exact theorem version; then test the conclusion and converse separately. Chapters 1–3 establish this method, and later chapters repeat it in their own frameworks.

To compare frameworks, write a source object, a target object, and a candidate map between them. Ask whether the result is a construction, faithful functor, fully faithful functor, equivalence onto an essential image, selected-observable recovery, or genuine equivalence. Wightman–Euclidean, AQFT–factorization, VOA–net, and bordism-based comparisons all have different hypotheses and incomplete directions.

To answer a physics question, enter through the relevant subject cluster and return with a bounded mathematical result:

• particle and scattering questions use Chapters 4–7, then return to the scattering model and observable;
• locality, thermality, measurement, entropy, and recovery use Chapters 5–10, then return to the thermal or information-theoretic setting;
• constructive existence and continuum questions use Chapters 3, 11, 12, and 25, then return to the regulated physical model;
• local renormalization and curved-spacetime questions use Chapters 13–15, then return to the chosen field, state, geometry, and observable;
• gauge fixing, anomalies, and boundaries use Chapters 18–20, then return to the gauge or gravitational application;
• chiral CFT, VOA, net, TQFT, defect, and generalized-symmetry questions use Chapters 21–24, then return to the relevant conformal, topological, or many-body theory.

In every route, the return step matters. A theorem about a mathematical model licenses only those physical claims whose variables and hypotheses have been matched.

Chapter sequence

The chapters form eleven coherent study arcs. Within each arc, read in the displayed order; later arcs may freely use the claim grammar and directional discipline established earlier.

1. Claims, axioms, and reconstruction: Theorem-First Frameworks and Claim Grammar; Wightman Fields, Reconstruction, and Structural Theorems; Euclidean Fields, Reflection Positivity, and OS Reconstruction.
2. Particles and local nets: Particles, Scattering Theory, and Infrared Structure; Local Nets, States, and Representations.
3. Modular structure, sectors, and covariance: Modular Theory, Nuclearity, and Split Inclusions; Superselection Sectors, Statistics, and Gauge Reconstruction; Locally Covariant QFT and Dynamical Locality.
4. Thermality, information, and measurement: Thermal AQFT, KMS States, and Nonequilibrium Structures; Noncommutative Information, Measurement, and Algebraic QEC.
5. Constructive models and rigorous flow: Constructive Euclidean QFT and Cutoff Removal; Rigorous Renormalization Group and Continuum Control.
6. Perturbative, microlocal, and curved-spacetime QFT: Perturbative AQFT and Causal Renormalization; Microlocal QFT and Renormalized Local Fields; Curved-Spacetime AQFT, Stress Tensor, and Energy Inequalities.
7. Factorization and derived structures: Factorization Algebras and Local-to-Global Observables; Derived, Deformation, and Homotopical QFT.
8. Gauge geometry, quantization, and boundaries: Gauge Orbit Geometry, Global Obstructions, and Measures; BRST–BV Quantization and Anomalies; BV–BFV Boundaries, Corners, and Gluing.
9. Conformal nets and vertex algebras: Conformal Nets and Operator-Algebraic CFT; Vertex Operator Algebras and Direction-Specific Net Bridges.
10. Extended topological and categorical QFT: Extended TQFT, Bordisms, and the Cobordism Hypothesis; Defects, Generalized Symmetry, and Categorical QFT.
11. Frontiers: Existence, Continuum Limits, and Classification Frontiers.

Framework, conclusion, and return map

The table compresses the volume without erasing its logical boundaries. “Main conclusion” means the kind of result sought in the arc, not a promise that every model satisfies it. The failure column names an upgrade that must be rejected unless a separate theorem supplies it.

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 table data (JSON) preserves the caption, scoped headers, rows, and reading order.

Evidence levels that must remain distinct

The word “rigorous” is too coarse to organize this subject. The following levels answer different questions.

• An axiom system defines an admissible class of objects. It does not show that an interacting example exists.
• A theorem proves an implication for every object satisfying explicit hypotheses. Its conclusion stops where its quantifiers stop.
• A construction produces an object. It may still leave uniqueness, classification, positivity, scattering, or equivalence open.
• A reconstruction theorem builds one kind of object from another under directional hypotheses. It is not a regulator-removal theorem for the input data.
• A formal perturbative result is coefficientwise in a formal parameter unless a separate convergence or summability theorem is proved.
• A controlled approximation or computation verifies a declared instance and tolerance. It supplies evidence or a counterexample, not unencoded universal quantifiers.
• Physical evidence can strongly constrain a conjecture while leaving mathematical existence or completeness unresolved.
• An open problem states the missing object, estimate, limit, or comparison theorem. Its partial results should be recorded without converting them into the final claim.

The failure diagram gives the practical discipline. Keep the proposed claim fixed, remove or alter one input, and identify the first conclusion that becomes unavailable.

Each dashed branch changes one decisive input. Shared two-point data do not establish framework equivalence; a plane wave, coincident product, or swapped limit can leave the theorem’s domain; ordinary positivity does not replace reflection positivity; a successful calculation or build does not prove an unencoded theorem; and local quasiequivalence does not imply global unitary equivalence. The diagram is schematic and not to scale. Structured description and source data (JSON)

A practical claim check

Before accepting a mathematical-QFT statement, answer these questions in order.

1. What is the primitive object? A field distribution, state, representation, local net, measure, functor, algebra, category, derived space, or bordism assignment cannot be replaced by a neighboring object without a map.
2. Where does it live? Name domains, test-function spaces, regularity, topology, support, signature, boundary conditions, and global sectors.
3. What is quantified? Put “for every,” “there exists,” “uniformly,” “after a subsequence,” and every regulator limit in the order used by the theorem.
4. What is assumed? Positivity, locality, covariance, spectrum, clustering, nuclearity, descent, energy bounds, dualizability, anomaly cancellation, and unitarity are logically separate.
5. What is the licensed verb? Define, construct, reconstruct, embed, compare, classify, approximate, and conjecture do not mean the same thing.
6. How unique is the result? Literal equality, unitary equivalence, quasiequivalence, natural equivalence, equivalence after localization, and equality modulo counterterms are different conclusions.
7. Which converse is known? State it separately. If it is open or false, preserve that boundary.
8. What breaks first? Change one hypothesis while keeping irrelevant data fixed, and identify the first failed map or estimate.
9. Which source version supports the wording? Use the primary theorem or construction, include corrections, and date genuinely mutable status.
10. Where does the result return to physics? Match mathematical variables to the model’s fields, observables, states, geometry, and approximation regime.

Scope, sources, and executable checks

The pages are self-contained for their declared theorem or construction scope, but they are not substitutes for the cited primary proofs. Exact locators appear beside consequential claims, and mutable existence or comparison statements carry page-local dates. The final chapter concentrates those moving frontiers so that settled definitions need not be rewritten whenever a new partial result appears.

Static diagrams and semantic tables explain dependencies and failure modes. An executable calculation would provide a different kind of evidence: a pinned environment, inputs, outputs, tolerances, a deliberate failure fixture, and an independent check. The calculation routes associated with this volume are not represented as completed theorem verification unless those materials are actually present. A passing computation can validate its encoded instance; it cannot raise the status of a broader claim.

This distinction is essential at the frontiers. Four-dimensional Yang–Mills, chiral gauge construction, scalar triviality and existence, general net–VOA converses, higher-categorical comparisons, and broad classification programs contain powerful partial theorems but also unresolved obligations. The relevant pages state the model, dimension, observables, implication direction, and cutoff date before summarizing what is known.

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.
• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory. Volumes 1–2. Cambridge: Cambridge University Press, 2017–2021. Publisher.
• Glimm, James, and Arthur Jaffe. Quantum Physics: A Functional Integral Point of View. 2nd ed. New York: Springer, 1987. DOI.
• Haag, Rudolf. Local Quantum Physics: Fields, Particles, Algebras. 2nd ed. Berlin: Springer, 1996. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. DOI; Open PDF.