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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.

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  • $\begingroup$ 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. $\endgroup$
    – jabirali
    Commented Apr 22, 2015 at 16:18
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    $\begingroup$ 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. $\endgroup$
    – Matta
    Commented Apr 22, 2015 at 16:40

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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:

  1. 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.

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

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