We are all familiar with the version of Quantum Mechanics based on state space, operators, Schrodinger equation etc. This allows us to successfully compute relevant physical quantities such as expectation values of operators in certain states and then compare with experiment.
However, it is often claimed that the path integral is an "equivalent" way to do all of this. To me, the "equivalent" part is a bit vague. I understand that the Feynman path integral allows you compute the propagator $\langle x | e^{-iHt} |x' \rangle $ by just using the classical Lagrangian of the system. Then any expectation value in a state can be computed by two resolutions of the identity to get an integral over this propagator. This shows that the path integral is a way to compute a quantity that's very useful, but not much more than that, since we still need the concept of operators, and their representation in position space, as well as position space wave functions, all of these objects along with their usual interpretations.
Ultimately, regardless of how you compute things, QM will still be based on probabilities and thus wave functions, however my question is, is there anything analogous to the axioms of Quantum mechanics usually mentioned in textbooks that are based on the path integral?
The path integral if seen as an independent object gives us the propagator, correlation functions, and the partition function (and maybe other objects which I'm not aware of). Are all these sufficient to give us the same information that quantum mechanics based on Hilbert space and operators give us? I would really appreciate if someone can make these connections precise.
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