> Rigorous Feynman Path Integrals

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.