Mathematical and Constructive QFT

Mathematical and constructive QFT asks which quantum field theories exist as mathematical objects, which axioms they satisfy, how observables and sectors are classified, and when different formulations can be reconstructed from one another. It includes Wightman and Euclidean fields, Haag–Kastler nets, perturbative algebraic QFT, constructive models, conformal nets and vertex algebras, and functorial approaches. “Rigorous QFT” is an alias only when the object class and theorem hypotheses are stated.

Evidence cutoff. 11 August 2026.

Required background. Theorem-first claim records supplies the object–hypothesis–conclusion discipline, QFT frameworks and typed maps prevents false equivalence, and the constructive existence map locates proved models by dimension and interaction.

Helpful background. Framework comparison supplies shared physics, operator algebras and positive functionals supplies the analytic language, BRST cohomology exposes gauge-theory complications, and conformal-net axioms gives a mature classification setting.

Objects, constructions, and comparison maps

Program	Primary object	Typical theorem	Principal frontier
Wightman/Euclidean reconstruction	distributions satisfying covariance, spectrum, locality, positivity	Hilbert-space reconstruction from correlators	interacting four-dimensional models with physical content
algebraic QFT	nets of local operator algebras and states	sector, modular, split, and curved-spacetime results	complete links to Lagrangian/constructive models and gauge observables
constructive QFT	cutoff measures or Hamiltonians with controlled removal	existence, clustering, spectrum, scattering	nontrivial four-dimensional gauge theories
perturbative algebraic QFT	local observables as formal power series	causal renormalization and local covariance	nonperturbative convergence/completion
conformal nets and vertex algebras	nets, VOAs, representation categories	classification and equivalence in special two-dimensional classes	higher-dimensional and nonrational reconstruction
functorial/topological QFT	symmetric monoidal functors or factorization algebras	classification by cobordism/local algebra data	relation to dynamical relativistic QFTs

Haag and Kastler’s net axioms shifted focus from singular point fields to local observable algebras Haag and Kastler 1964, foundational framework. Osterwalder–Schrader reflection positivity and regularity give a route from Euclidean correlators to a relativistic theory Osterwalder and Schrader 1973, reconstruction theorem. These are maps between specified object classes, not blanket equivalence of every Euclidean path integral, Wightman theory, and net.

What is constructed—and what is not

Constructive methods have established interacting scalar, fermionic, and gauge-related models primarily in two and three spacetime dimensions, with detailed control depending on the model. Glimm and Jaffe’s cutoff-free $\lambda\phi^4$ construction is a foundational concrete example Glimm and Jaffe 1968, constructive result. Four-dimensional asymptotically free gauge theories remain without the full existence and mass-gap theorem demanded by the Yang–Mills problem. Formal perturbative renormalizability is not nonperturbative existence.

Negative results are equally structural. Triviality theorems show Gaussian scaling limits for broad four-dimensional nearest-neighbor Ising and $\lambda\phi^4$ classes Aizenman and Duminil-Copin 2021, obstruction; they do not prove that every conceivable interacting four-dimensional QFT is trivial. Haag’s theorem obstructs a naive global interaction picture under exact relativistic assumptions, not regulated perturbation theory. Gauge theories complicate positivity and the identification of physical observables, so results for scalar measures do not transfer automatically.

Validation culture and benchmarks

A theorem-level status statement should name the field algebra or measure, spacetime dimension, geometry, coupling and mass regime, cutoff and limit, state class, observables constructed, and exact conclusion. “Model exists” can mean Schwinger functions, a net, Hamiltonian, scattering states, or only a formal series; these are not interchangeable.

Independent validation includes alternate constructions, reconstruction in another framework, matching exact spectra/correlators, machine-checked sublemmas, and correction tracking. Computer assistance can verify algebra or inequalities, but numerical agreement cannot replace uniform bounds needed for a continuum theorem. Benchmark classes include $P(\phi)_2$ , $\phi^4_3$ , two-dimensional integrable models, free fields on curved spacetime, rational conformal nets, and perturbative gauge observables.

A cross-formulation certification should state the map between objects, prove that it preserves the relevant products, states, and dynamics, and verify both directions on a model with independently known observables. For a boundary spectral problem, specify the operator domain and self-adjoint extension before comparing analytic and numerical spectra. For a BRST or BV construction, verify nilpotency or the master equation, identify the cohomology represented by observables, and state whether anomalies obstruct the quantum construction.

Entering the field

Choose one theorem and rewrite it as objects, hypotheses, conclusion, and exclusions; then follow every map in one reconstruction proof. The mathematical foundations for physicists and physical foundations for mathematicians supply complementary entry routes.

This guide omits purely formal mathematics without a stated QFT object or observable, and it does not label a framework superior merely because its theorems are stronger on a narrower class.

The finite search used MathSciNet-linked journal records where accessible, arXiv, Project Euclid/publisher archives, DOI registries, and citation chaining, with targeted searches for counterexamples and theorem corrections. Sources public through 11 August 2026 were eligible. Reassess when a new four-dimensional construction establishes nontriviality, a reconstruction theorem weakens material hypotheses, or a no-go result expands or contracts its model class.

Continue to interacting constructive QFT in four dimensions and equivalence of axiomatic frameworks.

References

M. Aizenman and H. Duminil-Copin, “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\lambda\phi_4^4$ Models,” Annals of Mathematics 194 (2021) 163–235. DOI.
J. Glimm and A. Jaffe, “A $\lambda\phi^4$ Quantum Field Theory without Cutoffs. I,” Physical Review 176 (1968) 1945–1951. DOI.
R. Haag and D. Kastler, “An Algebraic Approach to Quantum Field Theory,” Journal of Mathematical Physics 5 (1964) 848–861. DOI.
K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions,” Communications in Mathematical Physics 31 (1973) 83–112. DOI.