76. Chaos in quantum field theory

Clearly, the topic of interplay between confinement and chaos in classical YM got some interest, so let me continue. Contrary to what the title says, I shall not mention “confinement” this time, focusing on “chaos” instead.

Our debate got heated up quite a bit in the comments to the last post Marco is explaining on his blog that (citing his words):

1. Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.
2. A quantum field theory does not exist having as building classical solutions just chaotic solutions.
3. In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

Let me start with the point 3 – I believe, it is actually wrong. Let us again take a hamiltonian classical system with self-interaction. To get the intermittency (i.e., periodic orbits becoming chaotic trajectories and vise versa), it is enough to fix initial conditions and than vary the coupling constant, as I have explained in the previous post. Since one has running coupling in a QFT without sweating, one will have intermittency as well in the Schwinger-Dyson equations for Keldysh Green functions. So, strictly speaking, one can only get rid of chaos at the RG fixed point which corresponds to CFT anyway (that is — no particles, nor quasiparticles, just unparticles ).

The point 2 – this is actually Marco’s hypothesis as he explains. Since he needs it to prove why Smilga mapping is important for determining correct confining propagator, it is his work to prove it, not mine – to disprove it, as I have explained in comments. But I think I’ll try at least to say something about it in a couple of posts.

The point 1:  this is indeed so – nobody really knows how to quantize a classical  solution with chaotic behaviour or at least, how to start quantization from such a solution (for interested readers ready for some technicalities – see this intoduction to quantum chaos theory). The opposite argument was actually used by old Einstein against Niels Bohr in their famous dispute: classical systems often admit quantum behaviour, but the Bohr-Sommerfeld quantization rule only knows about periodic orbits. So, where is chaos in quantum mechanics, where is it encoded?

Since it does not look like the theory of quantum chaos is in a good shape currently, let us try to develop it a bit My impression is that chaos is encoded a) in the spectrum of the system, and b) in quantum phases of modes.

Let us take some hamiltonian system with selfinteraction potential (with appropriate form to allow chaos – basically, a billiard with negative curvature walls), its strength being managed by the coupling g. In the limit the system is free and there is no chaos in the classical limit, so Fourier modes are nice eigenmodes of the Hamiltonian in the sense that particles corresponding to them have infinite lifetime.

We now make coupling small but non-zero. My modes still do fine. At the classical level some trajectories (for some choice of initial conditions) may become chaotic, but almost all classical trajectories are periodic ones and they carry most information about the system. There is a new (long) time scale that appears in the system, mean free time that shows how long my particles live between collisions.

Finally, let us proceed to the regime of the strong coupling. The time scale is very short, basically, , where is the level splitting in the spectrum of the Hamiltonian. My Fourier modes do not look like a good basis anymore. On the other hand, at the classical level chaos is strong: almost all classical trajectories (i.e., trajectories corresponding to almost arbitrary choice of initial conditions) are chaotic. Of course, if we are lucky enough, then we can introduce quasiparticles – long living excitations with long mean free time. This is, I think, guaranteed if our strongly coupled theory can be mapped to a weakly coupled theory – the latter than shows what are long living eigenstates.

Now, quantum phases of modes are also important. Basically, one has to also consider anomalous correlators of the form

to be nonzero an learn how they behave as functions of time ( is the quantum phase of the mode with momentum k). If quantum phases randomize quickly due to, say, collisions of particles, then again our basis is good. But if they are not getting randomized, we interpret it as strong coherence effect.

To be continued.