ON SOME MATHEMATICAL PROBLEMS OF QUANTUM FIELD THEORY · 1958-mathphys-2

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No citable result was found that decides ordinary convergence versus divergence for every renormalized scattering-matrix element of an arbitrary renormalizable quantum field theory. The rigorous literature instead treats particular models, observables, dimensions, or weaker notions such as asymptotic expansion and Borel summability.

• Freeman J. Dyson, “Divergence of Perturbation Theory in Quantum Electrodynamics,” Physical Review 85 (1952), 631–632. DOI: 10.1103/PhysRev.85.631. Dyson argues that analyticity at zero electric charge would imply a physically unstable theory at negative coupling. This is the foundational evidence for divergence in QED, but it is not a rigorous proof for all renormalized QED scattering amplitudes and does not cover general renormalizable quantum field theories.
• Jean-Pierre Eckmann, Jacques Magnen, and Roland Sénéor, “Decay Properties and Borel Summability for the Schwinger Functions in P(φ)₂ Theories,” Communications in Mathematical Physics 39 (1975), 251–271. DOI: 10.1007/BF01705374. This proves Borel summability of perturbative Schwinger functions for a class of two-dimensional constructive models. It gives rigorous nonperturbative meaning to divergent or merely asymptotic expansions in those models, but it neither proves ordinary convergence nor treats every scattering-matrix element in general renormalizable theories.
• Jean-Pierre Eckmann and Henri Epstein, “Time-Ordered Products and Schwinger Functions,” Communications in Mathematical Physics 64 (1979), 95–130. DOI: 10.1007/BF01197509. For weakly coupled two-dimensional φ⁴ theory, the paper relates time-ordered products and Schwinger functions and proves Borel summability results including the two-particle scattering matrix. This is a direct special-case result for an S-matrix element, not a complete answer to the stated general problem.
• Jacques Magnen and Roland Sénéor, “Phase Space Cell Expansion and Borel Summability for the Euclidean φ⁴₃ Theory,” Communications in Mathematical Physics 56 (1977), 237–276. DOI: 10.1007/BF01614211. This establishes construction and Borel summability for the three-dimensional Euclidean φ⁴ model. It is strong control of a superrenormalizable special case, but Borel summability is weaker than ordinary convergence and the result does not cover general renormalizable S-matrix elements.
• Krzysztof Gawędzki and Antti Kupiainen, “Gross-Neveu Model Through Convergent Perturbation Expansions,” Communications in Mathematical Physics 102 (1985), 1–30. DOI: 10.1007/BF01208817. This constructs the two-dimensional Gross–Neveu model at weak coupling using convergent reorganized perturbative expansions. Such convergence in a particular fermionic model and expansion does not establish convergence of the standard renormalized S-matrix series in all theories.
• Vincent Rivasseau, “Constructive Field Theory and Applications: Perspectives and Open Problems,” Journal of Mathematical Physics 41 (2000), 3764–3775. DOI: 10.1063/1.533326; arXiv:math-ph/0006017. The review states that nonperturbative Green functions in many low-dimensional models are Borel sums of perturbation theory, while explicitly listing the nature of the renormalized perturbation series of four-dimensional φ⁴ theory, including the existence of renormalons, as open. This records a major remaining gap even for a specific four-dimensional renormalizable model.
• Stephen J. Summers, “A Perspective on Constructive Quantum Field Theory,” arXiv:1203.3991v2 (2016). Direct URL: https://arxiv.org/abs/1203.3991. This survey documents rigorous asymptotic, Borel-summability, and convergence results for selected low-dimensional models, while emphasizing that central four-dimensional theories such as QED, QCD, and the Standard Model still lack full mathematical constructions. Consequently, these results cannot collectively exhaust the theories and matrix elements quantified over in the statement.

Thus the strongest established progress consists of rigorous asymptotic expansions, Borel summability, or convergent reorganizations in special constructive models. What remains unresolved is the exact general question of ordinary convergence or rigorous divergence of every termwise-renormalized S-matrix perturbation series across renormalizable quantum field theory.