Interacting Constructive QFT in Four Dimensions

The question is: Which nontrivial interacting four-dimensional continuum QFTs have rigorous existence results, and which obstructions remain? With “construction” taken to mean a cutoff-independent, infinite-volume local theory satisfying a standard reconstruction or algebraic axiom system and demonstrably non-Gaussian interaction, no phenomenologically standard four-dimensional model presently has the complete package. There are decisive negative results for important scalar universality classes and deep partial constructions for gauge theory; neither should be misstated as a universal no-go theorem or as a completed Yang–Mills construction.

Evidence cutoff. 11 August 2026.

Required background. Four-dimensional scalar QFT existence and triviality supplies the precise scalar problem. Constructive existence by model, dimension, and observable distinguishes a measure bound, Schwinger functions, and a reconstructed Lorentzian theory.

Helpful background. Regulators, cutoffs, and continuum limits supplies the limits that must commute. Rigorous construction status and open problems prevents numerical and perturbative evidence from being relabeled as existence. Ultraviolet and infrared fixed points supplies the RG distinction between asymptotic freedom, asymptotic safety, and an actually constructed continuum measure.

What counts as a four-dimensional construction

A Euclidean route begins with regulated Schwinger functions $S_n^{(a,L)}$ and must control both ultraviolet and infrared limits,

$$S_n=\lim_{L\to\infty}\lim_{a\to0}S_n^{(a,L)},$$

or a jointly controlled limit. The limiting functions must satisfy the Osterwalder–Schrader hypotheses needed for reconstruction. Interaction must then survive the limit—for example through a nonzero connected higher correlation or nontrivial scattering—rather than being present only in the cutoff action.

Coordinate	Required evidence
Dimension and spacetime	Four Euclidean dimensions with reconstruction to 3+1 Lorentzian dimensions, or a direct Lorentzian construction
Cutoffs	Ultraviolet regulator removed and infinite-volume/infrared limit controlled
Axioms	Reflection positivity plus OS regularity, or a stated Wightman/Haag–Kastler alternative
Gauge structure	Gauge-invariant observables, positivity, and control of gauge fixing/Gribov or lattice-to-continuum issues
Nontriviality	A theorem excluding a Gaussian/generalized-free limit in the claimed observable class
Dynamics	Locality, covariance, spectrum/energy positivity, and preferably a mass gap or scattering sector where claimed
Non-example	A finite lattice, a formal power series, a UV bound with a fixed IR cutoff, or a continuum correlator inferred only numerically

Established facts and their reach

Scalar nearest-neighbor universality class. Aizenman and Duminil-Copin prove that scaling limits of critical four-dimensional Ising-type models with nearest-neighbor ferromagnetic interactions are Gaussian, and obtain the corresponding result for lattice-cutoff $\lambda\phi^4_4$ fields Aizenman and Duminil-Copin 2021. This is a rigorous triviality theorem for a major class, not a theorem that every imaginable four-dimensional scalar QFT is Gaussian.

Gauge-theory progress. Balaban’s renormalization-group program establishes strong ultraviolet stability and effective-action bounds for lattice gauge theories, including four-dimensional steps such as coupling renormalization Balaban 1987. Magnen, Rivasseau, and Sénéor construct four-dimensional pure $SU(2)$ Yang–Mills Schwinger functions with no ultraviolet cutoff but with a fixed infrared cutoff in a regularized axial gauge and verify nonperturbative Slavnov identities Magnen, Rivasseau, and Sénéor 1993. The fixed infrared cutoff and restricted setting are exactly why this is a major partial result rather than the full infinite-volume theory with mass gap demanded by the Yang–Mills existence problem.

Assumptions behind extrapolation. Perturbative asymptotic freedom, lattice universality, numerical mass spectra, and continuum extrapolations strongly support four-dimensional Yang–Mills as a physical QFT. They do not replace uniform bounds proving the existence, positivity, locality, and infinite-volume properties of the limiting theory.

Interpretation. The shortage of completed constructions reflects marginal couplings, simultaneous ultraviolet and infrared control, gauge redundancy, and the need to retain interaction at the critical limit. It is not evidence that QFT calculations in four dimensions are internally worthless; it identifies a stricter mathematical question than order-by-order renormalization or numerical prediction.

Candidate families and obstructions

Candidate	Positive evidence	Negative result or missing step
Nearest-neighbor ferromagnetic $\phi^4_4$/Ising scaling limit	Rigorous control of the critical scaling limit.	The limit is Gaussian in the proved class, so it does not furnish an interacting construction. Aizenman and Duminil-Copin 2021
Pure non-Abelian Yang–Mills	Asymptotic freedom, constructive UV bounds, IR-cutoff Schwinger functions, and extensive lattice evidence.	No proof simultaneously removes cutoffs, reconstructs the full local theory on $\mathbb R^4$, establishes interaction, and proves the mass gap. The official problem remains open. Clay Mathematics Institute, accessed 11 August 2026
Four-dimensional supersymmetric or conformal theories	Exact protected data, dualities, localization, and bootstrap constraints can be exceptionally strong.	These determine sectors or observables under physical assumptions; a full OS/Wightman construction for the interacting theory generally does not follow.
Perturbative algebraic QFT	Local interacting observables and renormalization can be defined covariantly to all formal orders.	A formal series need not converge or define a positive Hilbert-space theory at finite coupling.
Noncommutative or otherwise modified models	Some evade ordinary scalar triviality mechanisms and admit constructive control.	Modified locality, spacetime symmetry, or observable content places them outside the ordinary local-QFT question unless that scope is declared.

Method limits and common overclaims

Uniform boundedness of a regulated partition function is not enough: correlation functions can converge to a free theory. Borel summability of perturbation theory around a cutoff model is not enough unless the summed objects survive all required limits and satisfy positivity. A lattice continuum extrapolation with shrinking statistical and discretization errors is evidence for a limit, but not a proof of tightness and uniqueness over all scales.

Likewise, “asymptotically free” means the renormalized coupling weakens at short distance in a controlled expansion; it does not itself construct the probability measure or Hilbert space. The infrared is especially consequential for Yang–Mills because confinement and the mass gap are precisely the nonperturbative properties a full construction must recover.

Status — no complete standard interacting construction; decisive partial results. A major scalar class is rigorously Gaussian in the scaling limit. Four-dimensional non-Abelian gauge theory has powerful ultraviolet and IR-regulated constructions but not the complete cutoff-free, infinite-volume, positive local theory with mass gap. Other four-dimensional frameworks give exact or formal sectors without yet meeting the full criterion used here.

Concrete resolution criteria

A positive construction should provide:

1. a regulator family and uniform estimates that permit ultraviolet removal and infinite volume in a declared order or joint limit;
2. limiting gauge-invariant correlation functions or local algebras satisfying a complete named axiom set;
3. a reconstruction of a positive-energy Lorentzian theory;
4. a proof of non-Gaussianity or another invariant interaction criterion; and
5. for Yang–Mills, a mass gap and enough uniqueness/universality to connect the construction to the intended continuum theory.

A negative resolution would require a theorem whose hypotheses cover the proposed universality class and whose conclusion rules out all non-Gaussian limits—not an extrapolation from the nearest-neighbor scalar theorem.

Research connections

The field context is mathematical and constructive QFT. Euclidean lattice and continuum extrapolation supplies the regulator-to-continuum route, while functional equations and functional RG organizes candidate fixed points but does not by itself prove existence. Any constructive claim should identify a finite-cutoff measure, prove the uniform bounds or tightness needed for the continuum and infinite-volume limits, verify the reconstruction axioms, and demonstrate nontrivial correlators. Comparisons must use the same existence notion rather than placing partial measures, formal fixed points, and fully reconstructed theories in one undifferentiated class.

Literature selection and cutoff

The finite source set was selected through targeted journal, institutional, and citation searches for a sharp four-dimensional scalar theorem, representative constructive gauge-theory bounds, the strongest IR-regulated Yang–Mills construction, and the official full-problem statement available through 11 August 2026. Reviews and numerical studies informed terminology but did not determine the rigorous status. The selection is not exhaustive.

References

• Aizenman, M., and Duminil-Copin, H. (2021). “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi^4_4$ Models.” Annals of Mathematics 194, 163–235. DOI; arXiv:1912.07973.
• Balaban, T. (1987). “Renormalization Group Approach to Lattice Gauge Field Theories. I. Generation of Effective Actions in a Small Field Approximation and a Coupling Constant Renormalization in Four Dimensions.” Communications in Mathematical Physics 109, 249–301. DOI.
• Clay Mathematics Institute. “Yang–Mills and Mass Gap.” Millennium Prize Problems. Official problem page.
• Magnen, J., Rivasseau, V., and Sénéor, R. (1993). “Construction of $YM_4$ with an Infrared Cutoff.” Communications in Mathematical Physics 155, 325–383. DOI.