Rigorous non-perturbative formulation of quantum field theory

Develop a mathematically rigorous and general non-perturbative formulation of quantum field theory that defines interacting models beyond perturbation theory and establishes their consistency across the full parameter space.

Background

In discussing AdS-CFT as a route to non-perturbative quantum gravity via a well-defined boundary CFT, the authors note that a fully rigorous and general non-perturbative definition of quantum field theory remains out of reach.

They reference the broader mathematical status of QFT, including its appearance among the Clay Millennium problems, and contrast QFT’s open non-perturbative formulation with the even less understood foundational principles of quantum gravity.

References

We say 'in principle', because the problem of giving a rigorous and general non-perturbative definition of quantum field theory is still open.

— Dualities in Physics  (2509.15866 - Haro et al., 19 Sep 2025) in Section 4.4 AdS-CFT and Bulk Reconstruction

The next phase of this lone journey is extending this exact master equation framework to the realistic quantum field theory of QED and QCD. However, under nonlinear many-body interactions, the memory kernel no longer derives from a simple two-point correlation function, but naturally contains higher-order many-body correlation functions. Thus, genuinely interacting requires further investigations, even though many perturbative and non-perturbative approaches, such as diagrammatic expansion and loop expansion, can be directly utilized. I leave this problem in further publications.

— From Reversible Quantum Dynamics to Statistical Probability: A dynamical solution to the origin of probability and Hilbert's sixth problem  (2609.18593 - Zhang, 16 Sep 2026) in Section Discussion and conclusion

It is also not at all clear whether they are appropriate questions which can be answered within our current knowledge.

— Geometries of Quantum Field Theories  (2609.28210 - Yamazaki, 23 Sep 2026) in Preface to the Japanese Edition

Can quantum-information-inspired microscopic models provide simple but physically faithful descriptions of interactions in quantum field theory?

— Quantum models of interaction Hamiltonian and their paradoxes  (2609.11618 - Gisin et al., 10 Sep 2026) in Section 6, “Outlook” (subsection “Quantum-field models”)