# What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?

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.

• One formal difference is that the phase space formulation is an extension of classical Hamiltonian mechanics, while the path integral formulation is an extension of classical Lagrangian mechanics. Apr 22, 2015 at 16:18
• 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 $e^{S(x,v)}$ 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. Apr 22, 2015 at 16:40