In this discussion: http://chat.stackexchange.com/rooms/4243/discussion-between-arnold-neumaier-and-ron-maimon Arnold Neumaier suggested that there might be a close link between classical and quantum integrability, while I think there are many more classically integrable systems than quantum integrable ones.
The reason is that classically integrable systems are easy to make up--- you make up an infinite number of action and angle variables, and change canonical coordinates in some complicated way to x,p pairs, and say this x-p version is your system of interest. But quantum systems don't admit the same canonical transformation structure as classical systems, so there might be systems which have an integrable classical limit, but no real sign of integrability outside of the classical limit.
But I don't know any examples! Most of the 1+1d integrable stuff is for cases where the classical and quantum integrability are linked up, for the obvious reason that people are interested in finding integrable systems, not examples where they are not. The reason I think finding an example is not trivial is because the classical integrability guarantees that the motion is not classically chaotic, and that the asymptotic quantum energy states are pretty regular. So I don't think one can look for a counterexample in finite dimensions, where all high enough energy states are permanently semi-classical.
But consider a field theory on a lattice in 2+1 dimensions (continuous time). The lattice is so that the dynamics can be arbitrary, no continuum limit, no renormalization. Even if you have an integrable classical dynamics for the field theory, the energy can still dissipate over larger volumes (this isn't 1+1 d), and eventually the classical field will be weak enough that the classical limit is no longer valid, and you see the quanta. This allows the possibility that every finite energy state to eventually leave the semi-classical domain, and turns quantum, and then the integrability is lost.
So is there a 2+1 (or 3+1) dimensional lattice scalar field theory where the classical dynamics is integrable, but the quantum mechanical system is not?
By saying that the quantum system is not integrable, I mean:
- the many particle S-matrix doesn't factorize or simplify in any significant way (aside from the weak asymptotic relations implied by having a classical integrable limit)
- there are only a finite number of quantum conserved currents (but an infinite number of conserved currents in the classical limit).