Footnote: Haag's theorem only invalidates naive perturbative approximation. But that's not what physicists actually use; they use it for motivation and then s

Footnote: Haag's theorem only invalidates naive perturbative approximation. But that's not what physicists actually use; they use it for motivation and then switch to a renormalized perturbative approximation, which is compatible with Haag's theorem. Your point stands though: mathematicians are indeed able to cope with path integrals. But several of the models -- qed, Higgs, electroweak, basically everything but qcd -- used in particle theory suffer from a fatal flaw, the Landau pole. They can only exist as incomplete approximations to something else.