75. Chaos in YM and confinement
Save This Post as PDFI think we are currently having a somewhat fruitful discussion with Marco Frasca on his blog. The question is how relevant is chaotic behavior of classical solutions of Yang-Mills equations of motion for the quantum theory (or, more precisely, for YM at strong coupling).
Marco argues that it is absolutely irrelevant, and strong couping behavior of YM (that is, confinement) is completely determined by classical solutions without chaos, so called Smilga choices. If this is so, then, he says, strongly coupled Yang-Mills can be reduced to 
scalar field theory, and the strong coupling behavior of the latter is known.
Honestly, at this point I don’t see how it might happen. Before I proceed to the explanation, let me recall how classical YM behaves. First of all, equations of motion of the YM field are non-linear and therefore their solutions admit chaotic behavior. Are all the solutions of these equations of motion chaotic? The answer is of course negative: depending on the coupling strength and initial conditions, one can get whole sets of classical solutions without chaos, which we will call Smilga choices, following Marco. Suppose that we fix coupling and continuously change initial conditions for the YM equations of motion – as a result of this variation, we will first get, say, a chaotic solution, than a solution without chaos, than again a chaotic soltion, etc.
Now let us fix initial conditions and vary the coupling. With its variation my non-chaotic classical solutions may become chaotic and vise versa. This phenomenon is rather typical for non-linear systems and is called intermittency.
Are chaotic classical solutions important for the quantum YM or not? Well, let us consider a non-equilibrium formulation of YM (action is defined on the Schwinger-Keldysh contour in the complex plane of 
) and follow the dynamics of Schwinger-Keldysh Green functions. Since the EOM for YM admit chaos, this chaos will be also inevitably imprinted in the behavior of Keldysh Green functions. Namely, for a one initial condition we will get a solution of Dyson equations that admits chaos, for another choice of initial condition – a solution without chaos. It is hardly believable that Green functions do not carry any information about this chaotic behavior and intermittency. (Note, that if chaos is irrelevant, there would be a good chance that YM is exactly solvable, wouldn’t it? First hint of the quantum integrability is usually classical integrability.
But, Marco says, the hadron spectrum consists of quasiparticles (hadrons) – how this can be possible for the system with chaotic behavior? Well, hadrons are a) composite particles, and b) there are whole Regge trajectories of them. I doubt that the latter can be explained by mapping YM to a scalar field theory. But let me forget about this issue for a moment and rephrase the Marco’s question as follows: where the YM chaos is hidden at the quantum level?
I think, it is encoded in the behavior of the flux and might be actually responsible for confinement. According to the old idea by Olesen, necessary and sufficient condition for confinement (in the large 
limit) is that there should exist a flux independently distributed over the minimal surface spanned by the Wilson loop.
This criterion for confinement is really easy to check for the confining 1+1 Schwinger model, which is exactly solvable. Namely, one has
,
where 
is the electric flux through the contour of the electric flux. The area law for the Wilson line is automatically guaranteed if the flux is independently distributed over the surface spanned by the contour 
.
Going back to YM, chaos is encoded in the distribution function for the flux (which happens to be very simple in the 
limit). If the discussion will get warmed up, I shall write about it more.
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Hi Dmitry,
I think you did not answer to the two main questions I put forward to you:
1) Classical chaotic behavior is at odd with experiments
and, at best, you have nothing to compare with.
2) How do you reproduce a well defined spectrum in QFT when a Fourier decomposition in plane waves does not exist?
The recurring question in these latter years about physics is that we have lost the fundamental pragmatic lesson that comes from very far in time: We need numbers to compare with experiments. But a lot of physicists are producing useless theories believing they have reached an understanding. If one has nothing to compare with experiments a theory is nothing. But we are accepting this metaphysical approach today with all the consequences this implies.
Of course, this has nothing to do with you, Dmitry. I do not share your point of view but only experiments (or lattice) can say. Your analysis about non-equilibrium may be as well wrong but I am just asking you to answer my questions for the pure theory. After you have answered these you can extend your analysis to more complex situations. It is a fact that a Smilga’s choice is just a map that brings a Yang-Mills action into a scalar field action. This is true, being a mathematical truth, and independent on any take one may like. Of course, as any kind of mathematics, nature may chooses it or not. Let us wait and see.
Ciao,
Marco
Dear Marco
Thanks again for keeping discussion up!
> 1) Classical chaotic behavior is at odd with experiments
> and, at best, you have nothing to compare with.
I think the question is what is the measured quantity. Are experimentalists able to measure, say, the VEV of the Wilson line for a given contour?
> 2) How do you reproduce a well defined spectrum in QFT
> when a Fourier decomposition in plane waves does not
> exist?
I think, it is actually funny but old Einstein has put the opposite argument against quantum mechanics to Bohr: classical systems admit chaotic behavior, while, say, Bohr-Sommerfeld quantization condition knows only about periodic, integrable, orbits. So, where is chaos in quantum systems?
The known answer, I think, is that the spectrum carries information about chaos, in particular, it is not descrete.
Take a hamiltonian system with interaction. When the coupling constant is small, Fourier modes are nice eigenmodes of the Hamiltonian in the sense that mode mixing is weak and the corresponding quantum states are long living. On the other hand, chaos is also weak in the regime of weak coupling: for almost any choice of initial conditions classical trajectories are periodic.
In the strong coupling the situation might be the following: Fourier modes are no good anymore, since the life time of the corresponding particles is shorter than inverse \Delta E. Chaos is also strong, i.e., almost all classical trajectories are chaotic and just few of them are periodic.
Now, if you are lucky enough, you can get quasi-particles. The question is how long the life time of these quasiparticles if g is very large? If the life time is very long, then, I think, it is almost automatically guaranteed that you have a weak coupling dual for your strongly coupled theory. Is there any for QCD? Nobody is really sure of that.
> Of course, this has nothing to do with you, Dmitry.
Of course
Cheers,
Dmitry,
My bias is to say that chaos in quantum Yang-Mills theories ought to play an essential role in understanding the physics of infrared QCD. It is most likely, in my view, that there is decoherence induced by unsupressed radiative corrections in the strong coupling sector of QCD. It is in this region where non-equilibrium dynamics and manifest criticality takes over and one can no longer rely on traditional equilibrium field theory and its perturbative methods to correctly describe phenomena. Decoherence and inherent statistical fluctuations on all scales is prone to motivate the transition from quantum to classical behavior.
There is undisputable numerical and analytic proof put forward by Pascalutsa, Plessas and others (about two or three years ago) showing the presence of quantum chaos in the spectrum of hadron energy spacings. This finding reinforces the need to look for truly non-perturbative approaches to deal with infrared QCD. It also hints that nonlinear dynamics and chaos must be the drivers of field evolution in this regime.
Ervin
Dear Ervin
Thanks a lot for the comment, I do appreciate you dropped by. Do you expect that the strong coupling dynamics is classical but stochastic?
Could you point me to some references where decoherence might be discussed in the YM context?
Cheers,
Dmitry.
Dmitry,
In response to your questions:
1) I expect a dynamics regime that is strongly coupled to itself or to its environment to be well approximated in terms of classical fields and stochastic methods. If there are a large number of fluctuations constantly interfering with the internal degrees of freedom of the system,that system is well described by a classical statistical ensemble (actually Frasca himself has proved this conjecture, you can read about this on his blog. Jaffe also proved this conjecture.)
2)There is a large body of papers dealing with transition to classicality as a result of decoherence in quantum physics. You can do a Google search using “decoherence, Yang-Mills” and you get a large number of entries. For example, there is an excellent book by Hans-Thomas Elze called “Decoherence and Entropy in Complex Systems” with contributions from people like Biro, Matynian and Muller.
The bottom line is that there are too many “die-hards” quantum field theorists that refuse to recognize the fact that ignoring nonlinearity, emergent dynamics and non-equilibrium physics continue to lead us to more and more anomalies!(look how many puzzles remain unsolved in the Standard Model for particle physics….)
Best regards,
Ervin
Hi again Ervin
Thanks, I actually knew the book by Biro and Matinyan but wanted to be sure whether something new came to the market.
Cheers
Thats a neat and interesting paper on the IR and Ghost propagators for the gluon.
So, I gather in the IR that the Smilga map (for quantum systems) is throwing out some amount of information, but that it should be very small, at least in so far as perturbation series can see!
Its a little weird that it seems to become trivial, rather than hitting a fixed point though. That’s a little shocking.
So yea, how exactly are we supposed to identify the huge spectrum of hadrons in this setup?
Dear Haelfix
Although you clearly expect a reply from Marco, I shall also add my 5 cents into the discussion
>So, I gather in the IR that the Smilga map (for quantum
>systems) is throwing out some amount of information, but
>that it should be very small, at least in so far as >perturbation series can see!
I think the “perturbation series” argument is flawed (see also my reply to Marco above). The reason is that we have a free field theory at g=0, with classical EOMs that of course do not have any chaotic solutions.
What happens, say, in turbulence with intermittency regime is the following. When the coupling g becomes nonzero but small, most solutions (i.e., solutions corresponding to most choices of initial conditions) remain periodic, and only a small fraction of them becomes chaotic. That’s the essence of the perturbation theory argument — in the limit g\to 0 periodic solutions indeed carry almost all the information about the system.
Not so when g becomes large (the regime we are primarily interested in): the situation at large couplings is the opposite — almost all solutions are chaotic, and a small fraction of them — periodic.
Cheers,
Dmitry.
Hi Dmitry,
Of course I cannot see any clear answer to my questions and I can conclude that you are simply evading them. We are doing particle physics and the spectrum are the particles themselves and you cannot get around this claiming that Einstein said blah,blah,blah… It seems to me that you are missing some mundane aspects of quantum field theory (absit iniuria verbis). So, I will keep on discussing with you about this matter again when you will solve Frasca’s conjecture:
A quantum field theory does not exist having as building classical solutions just chaotic solutions.
Please, give me a counterargument and I will be able to appreciate your criticisms. Claiming flaws in other work requires a strong support that you are blatantly missing here. Indeed, your arguments seem deprived of any foundations and as I see no concrete or serious questions against my work I will stop my comments here. People interested can read my latest post
http://marcofrasca.wordpress.c.....-argument/
where Dmitry seems to miss another mundane argument about quantum field theory.
Haelfix:
The question of the running coupling in QCD is not mine. With my approach I was able to get back other results. You can look at the work of Boucaud et al.
http://arxiv.org/abs/hep-ph/0212192
published in JHEP. This group obtains a coupling going to zero as the fourth power of momentum. This is in agreement with the scaling of my propagator if you analyze it with a Callan-Symanzik equation. Recently, analysis of experimental data due to Prosperi’s group at University of Milano gets back a similar result. You can see the following paper published on PRL:
http://arxiv.org/abs/0705.0329
So, it seems like the running coupling reaches a trivial fixed point also in the infrared exactly as happens to a massles scalar field theory. Both theories seem to share triviality.
Marco
Hi Marco
Thanks again for visiting.
>We are doing particle physics and the spectrum are the >particles themselves and you cannot get around this >claiming that Einstein said blah,blah,blah?
But we are (well, were) just discussing physics, aren’t we? It seems to me that you neglect the fact that your particles are composite objects and there is a whole Regge trajectory of them. That’s why some people mention word “strings” when they discuss strongly coupled QCD.
> It seems to me that you are missing some mundane
> aspects of quantum field theory (absit iniuria verbis)
Whatever.
> So, I will keep on discussing with you about this
> matter again when you will solve Frasca?s conjecture:
> A quantum field theory does not exist having as
> building classical solutions just chaotic solutions.
I am afraid this is conjecture necessary to prove that Smilga choices give correct confining propagator for strongly coupled YM. It is therefore your work to prove it, not mine to disprove it.
> Indeed, your arguments seem deprived of any foundations > and as I see no concrete or serious questions against
> my work I will stop my comments here.
That’s of course your right. I was interested in our work and wanted to discuss it with you. You have started with making somewhat funny statements like “quantum theory requires special initial conditions to be selfconsistent” and ended discussing my personality instead of physics.
> where Dmitry seems to miss another mundane argument
> about quantum field theory.
Oh, Marco
Cheers,
“I am afraid this is conjecture necessary to prove that Smilga choices give correct confining propagator for strongly coupled YM. It is therefore your work to prove it, not mine to disprove it.”
You are wrong and mostly incorrect here. You claimed that classical chaotic solutions are relevant for Y-M quantum field theory not me and the burden of the proof relies on you. My idea is that nobody is able to do that because no quantum field theory can be built on classical chaotic solutions. This is all the point and you must prove that you are not wrong as you firstly put this argument forward. Full stop.
Marco
> You claimed that classical chaotic solutions are
> relevant for Y-M quantum field theory
No, I did not claim that, take a whole discussion step by step. I have no idea how much more periodic or chaotic orbits important for the quantum YM and confinement. It was you who said in the paper that Smilga choices = periodic classical orbits are responsible for the IR structure of the theory at strong coupling. That is the same as if you have said that chaotic orbits are absolutely irrelevant — what I’ve tried to question.
My basis (why I think chaos should be somehow relevant for confinement) is the Olesen theorem as I have discussed in this post. Biro, Mueller and Matinyan have tried to make sense of chaos in YM. There is also old hypothesis (due to Kirzhnits, I think) that confinement is due to the Anderson localization on classical chaotic trajectories.
In other words, you are making strong statement (chaos is ABSOLUTELY irrelevant), and I have also heard other statements on the market. Since you are the author of idea, you are selling the idea, I thought that you are supposed to convince the readers why you are right.
As for your hypothesis, of course, I can neither prove it nor disprove it. In essence, you are asking me to develop quantum chaos theory, the problem as hard as to prove the confinement, I think.
I have also mentioned large N, which you somehow completely preferred to ignore. If the YM can be fully mapped anyhow to a single scalar field, it is in the regime of large N – recall the discussion by Witten (Witten’s master field).
Cheers,
“In other words, you are making strong statement (chaos is ABSOLUTELY irrelevant), and I have also heard other statements on the market. Since you are the author of idea, you are selling the idea, I thought that you are supposed to convince the readers why you are right.”
The only way I have to convince of the goodness of my theory is experiment (or lattice computations) as I do strong predictions from it. That is all. I have already said this before (…time will say…glueball spectrum…and so on) and if this is all our conclusion we could have arrived it before. I do not pretend to have the truth in my pocket. I am a physicist like you trying to make sense out of a simple but not simpler world.
Best,
Marco
Hi Marco
I think I did appreciate your point about relation to experiment when you put it forward. But it is natural for me, since I am a theorist (as you are), that I just want to test the theory as much as I can before we proceed to experiments.
Cheers and thanks that you stayed.
Dmitry.
Dear Dmitry and Marco,
I have to excuse myself for rudely interfering in your conversation but I believe you are boldly debating one of the most difficult and controversial topics of contemporary field theory: is chaotic behavior of classical solutions relevant to the corresponding quantum sector at strong coupling?. It is rather natural to encounter opposite viewpoints on such a highly nontrivial matter!
Let me suggest that the ONLY way to settle this lively debate is to come up with models that match reality (and not only lattice simulations). I know that this is a challenging proposition, but a convincing argument would be derivation on the spectrum of lowest lying hadrons from first principles.
I wish to let you know that I intend to publish in the upcoming months a relevant paper on this important topic. Here I present analytic evidence that the distribution of hadron masses follows from the universal route to chaos in nonlinear dynamical systems. In particular, I show that both meson and baryon spectra obey a scaling hierarchy dependent on Feigenbaum?s constant, with critical exponents organized in natural progression. I find that numerical predictions are in close agreement with experimental data.
This work is a sequel to a paper I published this year, namely:
http://dx.doi.org/10.1209/0295-5075/82/11001
My best regards to both,
Ervin