Correspondence between quantum and classical integrability I'm looking for connections between quantum and classical integrability. I know quantum integrability is not well-defined, but let us just take one of the popular definitions by promoting the Poisson brackets in the definition of classical integrability to commutators.
Let us focus on the case of $N$ particles with well-defined classical analog (i.e., exclude quantum systems with finite dimension, which does not have a well-defined classical limit). Then here are our classical and quantum definitions of integrability, respectively.
Classical integrability
$$\{H,I_n\}=0$$
$$\{I_n,I_m\}=0.$$
Quantum integrability
$$[H,I_n]=0$$ and
$$[I_n,I_m]=0.$$
My questions are: With above definitions and focusing on $N$-particle with well-defined classical analog

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*(i) Does classical integrability imply quantum integrability defined above?
i.e., if one promotes the Poisson brackets among the integrals of motions into commutators, will they still commute?


*(ii) Does the quantum integrability defined above imply classical integrability? Say, one demotes the commutators among the integrals of motion for your quantum mechanical model into Poisson brackets, will they vanish?
I think both (i) and (ii) may be right as long as we focus on say $N$-particle system with well-defined classical limit. Good examples are the Calogero model and Toda lattice. Both of them are classically and quantum mechanically integrable.  I don't know if there is any counter-example against these two claims. Or someone has found some rigorous proof that (i) and (ii) are true for $N$-particle system with well-defined classical limit
 A: I am not aware of "natural" superintegrable systems failing to translate across the quantization map and certainly not the classical limit. (Actually, these are all described by classical & quantum Nambu brackets which demolishes bogus "proofs"  that NBs could not be quantized. This is why I barely listen to proofs for the non-existence of fish as I'm fishing.)
It is extremely hard to not get confused about quantization to commutators, unless one uses their isomorphs in phase space, namely Moyal Brackets, whose classical limit is PBs, if a classical limit is defined. So, then, the answer to your (ii) is "yes, of course". QM includes CM.
The pioneer fleshing out the systematics of (i), extension/correction of PBs to MBs by suitable ℏ corrections, for integrable systems, is Hietarinta, J Math Phys 25 1833 (1984), and subsequent citers. The key-point is that quantization allows for an infinity of options, and the ones sought and chosen are the ones preserving the symmetry/integrability structure of the classical theory.
Well-meaning finite-dimensional systems do follow suit, as a rule, and I have never seen plausible simple counterexamples in ordinary QM. For strings and recondite QFTs, you may always effectively conceal  ambiguities in their definitions; you appreciate the requirement of good faith here.
