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 am not so sure. I have to be a little careful since it is a long time since I looked at that paper, always on the lookout for a common space in which to express both classical and quantum physics. But here are my current thoughts after a quick rereading.
The paper consists of two more or less independent parts. The first part is a reformulation of classical mechanics in terms of complex variables (which resemble the classical counterparts of creation and annihilation operators). I've seen that done in other places, but maybe Strocchi was the first. I don't think there is anything wrong with it. The four canonical transformations themselves are treated extensively in Ch. 8 of Goldstein's "Classical Mechanics", it is not too hard to translate between the two formulations.
The second part, starting at section IV seems rather unrelated to the first. As far as I understand he needed the classical complex formulation first to make a connection with quantum mechanics where complex numbers are mandatory. It is not too hard to work through the equations for, for instance, a two level system (until ResearchGate implements mathjax I won't do it here, however), and it is indeed possible to derive the equations just above section V. As far as I can see there is no connection between the first and the second part, how could there be: there is no classical limit for a spin system. The idea of mapping finite systems (qubits) to classical oscillators was pursued further by Briggs and Eisfeld in PHYSICAL REVIEW A 85, 052111 (2012), and a number of followup papers, who also note that Dirac had the same idea as Strocchi.
So at the end of the paper only unresolved questions remain. Is it indeed suitable for a classical limit? That interesting question is not treated in this paper, and I could not find a later paper by Strocchi on that topic. In that case you would need to take the same system for both classical and quantum, let's say a quantum oscillator, construct the infinite dimensional "classical complex" phase space going with that, see if the two dimensional classical phase space (p,q) is somehow part of it, and that the limit h\to 0 indeed projects on that subspace. Or something like that. This would only work if the spaces have common properties, and if limits are properly defined. Is it a good space for quantum statistical mechanics? I don't know, I could not find the paper referred to at the bottom of p.40 on Web of Science. My gut feeling is that the two phase spaces will still be different in very fundamental ways. I forgot, or at least cannot find notes about it, if I checked for instance if the Jacobi identity is still satisfied (I assume it could be), and what the Liouville equation would look like in both spaces. In the classical case that equation has a Poisson bracket, in the quantum case a commutator, I do not see how to take an average of the second to get the first. Or, equally important, how the average of the quantum density operator would give the classical phase space density.
So in the end I do not agree. I also noted that the paper is quoted often, WoS gives 84 citations. I scanned through the list and looked at a few, but did not see anything along the lines of the statements made in the abstract.
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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