Statement copy in the gpt 6 astra pro collection (Phys-6)

Non-Perturbative Construction and Non-Triviality of 4D Euclidean Phi4 Quantum Field Theory

Definitions and mathematical statement

A Euclidean scalar field is a probability law on generalized functions on ${\mathbb{R}}^4$, or equivalently a compatible collection of Schwinger correlation distributions. The formal quartic action is \[ S(\phi)=\int_{{\mathbb{R}}^4}\left(\tfrac12|\nabla\phi|^2+\tfrac12m^2\phi^2+\lambda\phi^4\right)dx. \] This expression by itself does not define the continuum measure. A construction must specify regularization, renormalization, and a limit satisfying the Osterwalder--Schrader conditions, including Euclidean invariance, reflection positivity, symmetry, and regularity. Non-Gaussian means that the correlations are not all determined by the two-point function through Wick's rule. Source-fit warning: the cited Aizenman--Duminil-Copin theorem proves Gaussian limits for its four-dimensional ferromagnetic lattice Ising/$\phi^4$ class. The requested non-Gaussian construction cannot use that class of limits; the catalogue does not specify an alternative class. The record is retained, not certified as a well-posed currently open construction problem.

Formulation and concept references: [S1], [E1].

Short English statement

Can the proposed four-dimensional quartic scalar theory be constructed as a genuinely non-Gaussian continuum field, outside the class already ruled out by the cited triviality theorem?

Sources

• S1 M. Aizenman, H. Duminil-Copin, Ann. of Math. 194$1$:163-235, 2021. https://doi.org/10.4007/annals.2021.194.1.3. Catalogue formulation source.
• S2 Marginal triviality of the scaling limits of critical 4D Ising and phi4 models. https://annals.math.princeton.edu/2021/194-1/p03. Catalogue research/status reference; not a proof endorsement.
• S3 The non triviality of a Phi4 model, III, the Osterwalder-Schrader Positivity. https://arxiv.org/abs/2101.11980. Catalogue research/status reference; not a proof endorsement.
• S4 Triviality in a Non-Perturbative Second-Order Mean-Field Theory for phi4. https://arxiv.org/abs/2608.05998. Catalogue research/status reference; not a proof endorsement.
• E1 Michael Aizenman and Hugo Duminil-Copin, Marginal triviality of the scaling limits of critical 4D Ising and phi4 models, Annals of Mathematics 194 $2021$, 163--235; arXiv:1912.07973. https://arxiv.org/abs/1912.07973. Added primary source: Source-fit warning and proved triviality for the paper's model class. Consulted 24 September 2026.
• E2 Romain Panis, Triviality of the scaling limits of critical Ising and $\phi^4$ models with effective dimension at least four, arXiv:2309.05797; Annals of Probability, DOI 10.1214/25-AOP1782. Assumptions $A1$--$A6$, Theorems 1.12--1.13 and Corollary 1.14. https://arxiv.org/abs/2309.05797. Added research source: Quantitative triviality for the specified long-range model class; not a no-go theorem for arbitrary continuum constructions. Consulted 24 September 2026.
• E3 Francesca Arici, Daniel Becker, Chris Ripken, Frank Saueressig and Walter D. van Suijlekom, Reflection positivity in higher derivative scalar theories, arXiv:1712.04308 $2017$, Theorem 3.7; DOI 10.1063/1.5027231. https://arxiv.org/abs/1712.04308. Added research source: Rational-covariance reflection-positivity criterion. The negative reflection form used here is also derived explicitly in the attempt. Consulted 24 September 2026.
• E4 Majdouline Borji, Triviality in a Non-Perturbative Second-Order Mean-Field Theory for $\phi^4$, arXiv:2608.05998 (August 2026), concluding discussion of the ultraviolet remainder. https://arxiv.org/abs/2608.05998. Added research source: Mean-field result and its stated limitation; no assertion of full-theory triviality is imported. Consulted 24 September 2026.
• E5 Marietta Manolessou, The "non triviality" of a $\Phi^4_4$ model, III, the "Osterwalder--Schrader Positivity", arXiv:2101.11980 $2021$, Theorem 3.2. https://arxiv.org/abs/2101.11980. Added research source: Claimed positivity result. The preceding hierarchy construction and the full construction claim were not independently verified in this notebook; not a proof endorsement. Consulted 24 September 2026.