There's been a lot of progress nonperturbatively defining quantum mechanics and quantum field theory using resurgence lately. See, for example, the very recent https://arxiv.org/abs/1802.10441
Re:Lost Causes of Theoretical Physics (2003)
by auntienomen on Friday March 02, 2018 @08:21PM
(#16507241)
I think it's incorrect to say that Feynman diagrams are as rigorous as standard perturbation theory. Perturbation theory in QM has a completely rigorous description. It's controlled approximations to known mathematical objects. By contrast, the derivation of perturbation theory in quantum field theory involves applying reasonable well-understood transformations to mathematical objects which are largely _unknown_. Nothing protects you from garbage-in, garbage out. (Perturbation theory turns out perfectly sensible looking answers for theories like 6 dimensional scalar field theory, which doesn't actually exist.)
That said, Feynman integrals are something mathematical physicists have a reasonable level of comfort with these days. There's a well-developed theory of Euclidean path integrals which works just the way Feynman integrals are supposed to. The rigorously constructed examples are lower dimensional field theories, like scalar field theory in 3d, but they don't really differ in kind from the 4d theories used in particle physics. It's just that the analysis gets significantly harder in 4d.
It depends on how rigorous you want the treatment to be.
The whole machinery of Feynman diagrams is derived from Path Integrals and Wick's Theorem, so they are rigorous, at least as rigorous as standard perturbation theory using operatorial quantum mechanics.
A mathematical physicist won't be amused, though, that's correct. But I wouldn't say that Path Integrals are less rigorous than the average technique used in Theoretical Physics, and particularly in many-body theory.