I am trying to conceptually connect the two formulations of quantum mechanics.
The phase space formulation deals with Wigner quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space.
I see how they both lead to non-classical physics but how do they relate? Either conceptually or formally.
The thing that motivates me is the idea that the Lagrangian, via the action, is a map from the tangent bundle of the configuration space to the reals. The Wigner function is a map from the cotangent bundle (phase space) of the configuration space to the reals. To get expectation values out both W(x,p) and eS(x,v) act as weightings in an integral (S=action, W=Wigner function). I would like to get from one to the other without using Hilbert space as an intermediary.
https://en.wikipedia.org/wiki/Phase_space_formulation
https://en.wikipedia.org/wiki/Path_integral_formulation
https://en.wikipedia.org/wiki/Configuration_space
strangebeautiful.com/papers/curiel-geom-ele.pdf
https://en.wikipedia.org/wiki/Phase_space
The paths integral leads to classical wave mechanics. The result is not a transition probability, but a wave intensity that can be identified to a probabilty through born's interpretation. The action is the phase difference of the wave integrated along a path, and can be thought of as a distance in some space.
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