Construct any interacting quantum field theory in four spacetime dimensions
In plain words
Build, with full proof, one quantum field theory in 3+1 dimensions in which particles actually scatter off each other. Every rigorous example so far is either free (no interaction) or lives in fewer dimensions.
Precise statement
Construct a quantum field theory on $R^{(1,3)}$ satisfying the Wightman axioms (or Euclidean Osterwalder-Schrader axioms on $R^4$) whose truncated correlation functions are not all zero, i.e. a non-Gaussian theory with nontrivial scattering. Lattice $\phi^4$ and Ising models in $d = 4$ give only Gaussian scaling limits, so candidates are asymptotically free theories such as Yang-Mills. Answer: an explicit construction with proof of the axioms and of non-Gaussianity.
What would settle it
A refereed construction of one non-Gaussian Wightman or Osterwalder-Schrader theory in four dimensions.
Status in the literature
Aizenman and Duminil-Copin (Annals of Mathematics, 2021) proved that scaling limits of critical $4\mathrm{D}$ Ising and lattice $\phi^{4}$ models are Gaussian, removing the simplest candidate.
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