OP-3081OpenMathematical physics

Existence or triviality of continuum QED₄

Either construct nontrivial four-dimensional continuum QED satisfying specified relativistic QFT axioms, or prove that removing the ultraviolet cutoff forces its renormalized interaction to vanish; control the charge and infrared sectors in either formulation.

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Background

QED is the most experimentally precise theory in physics, yet it has no nonperturbative mathematical construction. The central worry is 'triviality': the Landau pole suggests the renormalized charge is driven to zero as the ultraviolet cutoff is removed, so a nontrivial continuum interacting QED may not exist.

What is known: Scalar triviality theorems are important comparisons, not QED theorems. Aizenman and Fröhlich established results above four dimensions; Aizenman–Duminil-Copin proved Gaussianity of scaling limits for specified critical four-dimensional Ising and $\phi^4$ models. These do not establish triviality for a gauge–fermion theory or for the Higgs sector coupled to gauge fields. The perturbative Landau-pole argument motivates the question but is not a nonperturbative exclusion proof.

So the open problem has two faces: either construct a nontrivial 4D continuum QED (controlling the Landau pole, infrared/soft-photon sectors, and charge renormalization), or rigorously establish its triviality and characterize QED only as an effective theory with a cutoff. Either resolution is a major programme. Related: QF7 (infrared).

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

References

[1]Aizenman & Duminil-Copin, 'Marginal triviality of the scaling limits of critical 4D Ising and φ⁴ models', Ann. of Math. 194 (2021) link
[2]Aizenman, 'Proof of the Triviality of φ⁴ Field Theory and Some Mean-Field Features of Ising Models for d > 4', Phys. Rev. Lett. 47, 1 (1981) link

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