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.
 A: The connection has been provided explicitly repeatedly, best by P Sharan (1978). In words, essentially, the time-evolution kernels of the Wigner function from each phase-space point to all other such points is computed, and then concatenated with kernels for a subsequent move, and integrated over all intermediate points. Concatenation of an infinity of such successive time evolutions for infinitesimal time intervals and elimination of the momentum integrals produces the Feynman path integral with its infinite ordered variables of integration; while the reverse process yields the *-evolution operator describing propagation in the phase-space formulation. 
For more connections, you might try Ref. 1.  
Indeed, the three equivalent formulations: Hilbert space, Path Integral, and Phase space are joined at the hip (phase space). There are logically independent functor bridges between the first two; and, detailed in Ref. 1: between the 1st & 3rd and, 
your question, between the 2nd & 3rd. The most expeditious bridges go through paths in phase space, as shown in Sharan's paper, but your can find several other paths, if that one is not to your liking, in Ref. 1.
References:


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*Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos,  A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.  

*M. S. Marinov,  Phys. Lett. A 153, 5 (1991), A new type of phase-space path integral.
