In a 1966 paper, Strocchi shows that there exists formal correspondence between Poisson brackets and commutator brackets using complex coordinate. His paper was published in Rev. Mod. Phys. 38, 36 (URL: http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.38.36). His abstract goes as follows: "By introducing complex canonical coordinates, classical and quantum mechanics may be embedded in the same formulation. In such a way, the connection between Poisson brackets and commutators, canonical transformations and unitary transformations, etc., become apparent. This formulation is also particularly suitable for discussing the classical limit of quantum mechanics and for quantum-statistical mechanics."
So do you agree that there is formal correspondence between Poisson brackets and commutators under complex coordinate?
I had a quick look at the paper now and admittedly did not see why the complexification helps for taking the classical limit. I'm not saying that it doesn't help, but I didn't see it in the rush. But I wanted to make one comment about the relation between Poisson bracket and commutator: If you use the (Moyal-Weyl) star product to represent your products of phase space "operators" after quantization by a deformed-product (the star-product) of functions on classical phase space, then you get an hbar-expansion of the quantum commutator. And at lowest order in hbar, this agrees precisely with the Poisson-bracket. That's my preferred way to see the relation between commutator and Poisson bracket.
Dear Gert and Sebastian. Thank you for your kind answer. In a recent paper, Strocchi explored the same idea of correspondence between CM and QM using Lie-Rinehart algebra. You can find it in arxiv. Best wishes
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