Existence of nontrivial quantum field theories satisfying the Wightman axioms

Establish, for at least one four-dimensional quantum field theory used in particle physics (such as quantum electrodynamics or Yang–Mills theory), the existence of nontrivial solutions that satisfy the Wightman axioms by constructing the theory non-perturbatively and proving that it fulfills the full set of axioms (Lorentz invariance, locality, spectral condition, etc.).

Background

A.S. Wightman formulated a rigorous set of axioms intended to define physically acceptable quantum field theories at the non-perturbative level. These axioms imply key physical results such as CPT invariance and the spin–statistics connection, both well supported by experiment.

Despite the success of perturbation theory for practical calculations, it remains unknown whether any of the interacting four-dimensional quantum field theories used in particle physics can be constructed non-perturbatively to satisfy these axioms. This is a central problem in constructive quantum field theory.