@ACuriousMind so my first question is, whether a path integral formulation of classical problems would be a better candidate than the quantization procedure based on the Hamiltonian picture of classical mechanics, in order to avoid the consistency issues with the axioms of the latter.
My second question is regarding Witten's volume formula for moduli space of flat connections. I can follow but not deeply understand his proof. But I recently learned about evaluating stationary phase integrals (I believe this has some similarity with path integral formulation of a classical Lagrangian),