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larger volumes (this isn't 1+1 d), and eventually the classical field will be weak enough that the classical limit is no longer valid, and you see the quanta.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).
  • 2
    $\begingroup$ Just to read along, could you elaborate more on the meaning of a quantum integrable system? And what are the infinite currents (in the classical system), they are not independend objects are they? $\endgroup$
    – Nikolaj-K
    Jul 27, 2012 at 8:30
  • $\begingroup$ One possibility is that in a quantum system one can have discrete symmetries which lead to quantum integrability, and these discrete symmetries would never be sufficient to give classical integrability. For example, see: prl.aps.org/abstract/PRL/v107/i10/e100401 $\endgroup$
    – genneth
    Jul 27, 2012 at 10:10
  • $\begingroup$ Your ''many'' classical systems are just copies of the same in different coordiantes. If you have a quantum integrable system made from action and angle variables, you can similarly apply an arbitrary unitary transformation and get another quantum integrable system. In this sense, there are ''as many'' quantum integrable systems as classical ones, though the kind of transformations you apply to each system to make equivalent copies of them is different. $\endgroup$ Jul 27, 2012 at 11:32
  • $\begingroup$ @genneth: This is just the opposite of what Ron wants to have! $\endgroup$ Jul 27, 2012 at 11:36
  • $\begingroup$ On first glance it would seems that the quantum case should be more often integrable because of linearity. $\endgroup$
    – MBN
    Jul 27, 2012 at 11:40

3 Answers 3


I think one set of classically integrable models that are not generically quantum integrable are the sigma models on symmetric spaces. For an example of anomalies of the $CP^n$ model see, eg., this paper

  • $\begingroup$ Can't the anomaly be cancelled by a Wess-Zumino term? sciencedirect.com/science/article/pii/0370269385911414 $\endgroup$ Jul 27, 2012 at 18:07
  • $\begingroup$ @Olof: I can't read it right now, but I think this is the 1+1 d model, these do have quantum scaling anomaly which I always think of Friedan style, in terms of the Ricci flow on the group manifold. I can see how this would break some classical currents, but I didn't know that these 1d group models are classically integrable. Is this a complete example for sure? Is there an arxiv review somewhere? (I'll upvote as soon as I read the linked paper). $\endgroup$
    – Ron Maimon
    Jul 28, 2012 at 7:06
  • $\begingroup$ I accepted my example, because it's exactly what I was looking for, but I would still like to see if CPn models are the same way. $\endgroup$
    – Ron Maimon
    Jul 30, 2012 at 14:15
  • $\begingroup$ @Arnold: The paper you link to discusses fermions coupled to the sigma model, while I'm talking about a purely bosonic sigma model, so I'm not sure it's directly applicable. $\endgroup$
    – Olof
    Jul 30, 2012 at 21:00
  • $\begingroup$ @RonMaimon: Sorry, I missed that you wanted an example in 2+1 dim. For a discussion on the classical integrability of symmetric space sigma models see eg the recent paper arxiv.org/abs/1207.3325 and references in it. The result of anomalies for the CP^n case are well cited, but I can't find any kind of recent review. Similar, more general, results are also given in arxiv.org/abs/hep-th/0003264 (but I haven't read that paper in any detail). $\endgroup$
    – Olof
    Jul 30, 2012 at 21:28

First quantized string theory is precisely such an example. See the question here. The classical theory has n noninteracting strings, each of which has integrable dynamics. The quantum theory allows strings to split and merge, and is hence no longer integrable.

  • $\begingroup$ I think this example is not very good, (and smart-alecky). The quantum theory is integrable at any order, on any genus worldsheet. This means that string theory to any finite order is integrable in a trivial sense (you can calculate it by higher genus free fields) and it only gives nonintegrable dynamics in the resummation of infinite genus. The "classical dynamics" in this sense is the supergravity and is non-integrable. $\endgroup$
    – Ron Maimon
    Jul 29, 2012 at 17:12
  • $\begingroup$ I said stupid things before, but this answer keeps haunting me. Ok you are right--- the no loop tree level superstring is free on the worldsheet, it has a classical interacting limit, so you can calculate the n-vertex scattering amplitude to any tree order with an arbitrary number of lines, but there is no way that the classical 10d supergravity is integrable in any meaningful sense classically. So all that is going on here is that the n-th order classical perturbation theory of the model is given by certain simple free-field correlators, but the classical theory itself is an all orders sum. $\endgroup$
    – Ron Maimon
    Jul 31, 2012 at 1:31

I came up with a model of the sort I wanted:

consider a lattice field theory in 2 dimensions of space and 1 continuous time whose action is:

$$ \sum_i {1\over 2} \dot{\phi_i}^2 - \sum_{\langle i,j\rangle} {1\over 2}(\phi_i - \phi_j)^2 $$

Where $i$ and $j$ are nearest lattice neighbors. This is a massless lattice free theory, and hence integrable. The canonical momentum is

$$ \Pi_i = \dot{\phi_i}$$

and the Hamiltonian is

$$ H = \sum_i {1\over 2} \Pi_i^2 + \sum_{\langle i,j\rangle} (\phi_i -\phi_j)^2 $$

The Poisson brackets/commutators are the obvious ones.

Consider the classical invertible regular change of variables

$$ \phi_i = u_i + {\lambda\over 3} u_i^3 $$

You can invert this to find $u$ in terms of $\phi$ as an infinite series in $\lambda$. It has an unilluminating closed form too.

If you perform this change of variables in the classical Hamiltonian, you get a new canonical momentum:

$$ \Pi'_i = \Pi_i (1 + \lambda u_i^2) $$

One can check that the symplectic form is the sum over $i$ of $du_i\wedge d\Pi'_i$. This change of variables in the classical equations of motion gives a transformed classical Hamiltonian,

$$ H = \sum_i \frac{\Pi'^2_i}{2(1+\lambda u_i^2 )^2}+ \sum_{\langle i,j\rangle} (u_i - u_j + {\lambda\over 3}(u_i^3 - u_j^3))^2$$

which is derived from the sigma-model like transformed classical Lagrangian obtained by substituting $u$ for $\phi$:

$$ L = \sum_i {1\over 2} \dot{u_i}^2 (1+\lambda u_i^2)^2 - V(u)$$

Where $V(u)$ is the second complicated potential term in the Hamiltonian. Now I will consider this action as defining a quantum system. The classical limit is integrable by construction, by a classical change of variables.

The quantum theory consists of variables $u_i$ at each lattice site, with a conjugate momentum $\Pi'_i$ obeying canonical commutation relations with $u$

$$ [\Pi'_i,u_j] = i \delta_{ij}$$

and with a quantum Hamiltonian equal to one appropriate interpretion of the classical expression above:

$$ H = {1\over 2} \sum_i \Pi'_i {1\over(1+\lambda u_i^2)^2} \Pi'_i + V(u) $$

This is a well defined quantum system, and the new action is not quantum equivalent by change of variables to the free field theory. The reason is that to do the change of variables in the path integral properly, you would need to introduce a further measure factor at each site into the Lagrangian:

$$ \sum_i \log(1 + \lambda\phi_i^2)$$

And I did not introduce this measure factor in the new action, so the interactions are nonzero for the quantum variables. This system is a 2 space dimension lattice field theory of sigma-model type, and it's not going to have any special integrability properties in the quantum domain (aside from those derived from the integrability of the classical limit, which isn't saying much). So it will have wild quantum scattering which will conspire to make integrable free-field behavior at large coherent waves made up of large numbers of excitations per wavelength, degenerating to a random mess when the waves disperse to low enough amplitude that the semiclassical approximation is no good.


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