Exact solutions of Yang-Mills theory: The situation

Some time passed by since Terry Tao was so kind to take a look to my work. His concern about a main theorem in my paper, the so called mapping theorem, was motivated by the fact that no proof exists that there are common solutions between Yang-Mills equations and the one of the quartic scalar field. This point is quite crucial as, if such solutions do not exist, I cannot do any claim about Yang-Mills theory.

Some people are in confusion yet about this matter and I find occasionally someone, e.g. the Czech guy, claiming that my paper is false also after I have proved that such solutions exist.

Of course, Terry meant to point out a weakness in the proof given in my paper as I gave no evidence whatsoever of the form of these solutions and so the proof is, at least, incomplete. My next preprint proved that such solutions indeed exist and my argument is true already at level of perturbation theory. The conclusion is straightforward: Smilga’s choice select a class of common solutions between Yang-Mills equations and a quartic scalar field. I have not presented them explicitly in my paper and this is the reason why all this arguing was started. Terry’s suggestion was to complete the proof  and this I have done.

Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration.  I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here.

Dmitry Podolsky (see his blog here) hit correctly the point when asked for the fate of chaotic solutions in the infrared quantum field theory. Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.

Finally, this gives the current situation about this matter. My paper that started all this is correct and in agreement with current lattice results. People’s mood about lattice computations range from fully convinced to skeptical.  My view is that they represent correctly the infrared physics at hand but I am a supporter of these people working on lattice computations and so, my judgement should not be counted.


21 Responses to Exact solutions of Yang-Mills theory: The situation

  1. Daniel de França MTd2 says:

    “Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.”

    You shouldn’t be so clueless. t’Hoof has beeing trying that for 10 years:


    If you want more inspiration, for anything, look at here:


  2. mfrasca says:

    Hi Daniel,

    Thank you a lot for the links to ‘t Hooft’s works. I have had the chance to listen him to a talk about this matter about three years ago. This is a nice line of research and my hope is that will produce the deeper understanding the author is aiming to. I am also aware of ‘t Hooft’s homepage but I think that a list of his publications is plain history of physics and so a lot of inspiration, not only for me, but for several generations to follow.


  3. Daniel de França MTd2 says:

    I forgot to say. Look for the citations, he is not the only one working in his theory! There is a lot of stuff! 🙂

  4. mfrasca says:

    Yes, I think it is a serious mistake to ignore ‘t Hooft. 🙂

  5. Luboš Motl says:

    Dear Marco, Terry is actually telling you exactly the same things as I do, only in a less transparent way that allows you to pretend that you don’t hear him – he is just a polite and nearly invisible Fields medal winner, after all.

    In fact, I am convinced that you know very well that he is telling you that (and why) your approach is fundamentally flawed and there is no chance to get something out of it.

  6. rafael says:

    Hi Marco,
    Which people are skeptical about lattice computations?
    Cheers, R.

  7. mfrasca says:


    Silence is telling you something? Your natural paranoia is working for you?

    As Newton said “Hypotheses non fingo” and I am convinced that Newton was smarter than you, just by looking at your scientific contributions.

    I am just aware of the fact that you tried to vandalize his blog and he removed your comment. I am also convinced that you are wrong as happens quite frequently.


  8. mfrasca says:

    Hi Rafael,

    Nice to hear from you again. Just check this thread on physicsforums:




  9. rafael says:

    Dear Marco,

    comments such as that from “Haelfix” are surely from people that are not part of lattice/SD/FRG community.
    Would I recommend: http://books.google.com/books?id=V48ddclvbioC&printsec=frontcover&dq=lattice+methods as an excelent book on the topic?
    After reading one will understand why Lattice calculations are the only “ab initio” method able to rigorously plug QCD-action to experiments at all energy scales.


    • mfrasca says:

      Dear Rafael,

      I have that book. Thank you for your suggestion. Of course, I trust lattice work that is also a strong support to my work.



    • Haelfix says:

      Rafael, I am not skeptical of lattice computations, indeed I find them very reliable, even without being part of the lattice community (though I have worked with lattice calculations before as a graduate student)

      Otoh, just like I said in that forum post (and as someone else pointed out also), there are and has been conflicts with standard field theory notions involving technicalities with the Gribov ambiguity which a quick survey of Hep-lat will convince you off.

      You can take two points of view on the question. Either the standard field theory prescription is incomplete, or the ‘naive’ lattice calculation is incomplete, or perhaps both. There are indeed some subtle issues going on with the gauge fixing procedure and there has been debate about that in the expert circles.

      From the experience i’ve had with lattice calculations in the past, you do in fact have to tinker with the lattice in order to get good results. For instance with fermions or supersymmetry, one needs to get creative so that the computer doesn’t return junk. I personally suspect this is the case with the gribov ambiguity, but im sure experts out there might have a more illuminated opinion.

  10. Luboš Motl says:

    Dear Marco, there has been no silence. Terry told you very explicitly why your work is wrong and why your work is wrong. He just doesn’t have the time and guts to repeat himself.

    • mfrasca says:

      Terry was wrong about that argument as I proved. As can be read in wikipedia, he pretended that common solutions between Yang-Mills equations and quartic scalar field theory do not exist. This is plainly wrong. To satisfy Terry’s requirement all is needed is to show, by solving equations, that this common set exists and this is done straightforwardly in my paper (if Smilga’s book was not enough). Indeed, he was not even sure such solutions were in Smilga’s book. Interested people should read the discussion here


      A silence cannot be evidence against a mathematical truth. One can remain skeptical about the rest of the paper sharing others position. But, about the specific argument Terry put forward I am right.

      So, there remains only your personal interpretation about his silence.


  11. carlbrannen says:

    Okay, that was worth reading. The Kasner metric is fascinating, particularly in 5 dimensions, where you can put p for the three usual spatial dimensions as 1/2 and p for the extra spatial dimension as -1/2. One then ends up with a vacuum GR solution where the speed of light, coordinate speed that is, slows down as $tex c = \sqrt(1/t)$. Reminds me of Louise Riofrio’s cosmology.

  12. Kea says:

    Ahh! Excellent observation, Carl. The next time some crusty relativist tries to tell me that c cannot change, I will know exactly what to say.

  13. carlbrannen says:

    Marco, I ran into this article on complex Bohmian trajectories and it seemed to echo some of the methods you’re using to me: quant-ph/0604150

  14. mfrasca says:

    Carl, thanks for the link. I am not a supporter of Bohmian mechanics and there are some people arguing about experimental evidence against it. E.g. see



  15. carlbrannen says:

    Marco, yes I’ve seen that literature. Of course the Bohmianists are arguing back, here’s papers on both sides.

    My own feeling is that Bohmian mechanics is too complicated to be ontologically correct, but that its results match standard QM. The proof is rather simple and seems fool-proof.

    But I prefer a variant of Bohmian mechanics where the measurement process is about the relationship between the time of the event as compared to the time of the observer (as an event). That is, if the observer sees the event in the past, then it’s a particle, and if he sees it in the future, it’s a wave. So the same situation does include a wave function and a particle path, which exists depends on the relationship of the observer to the event. Either he sees the event in his past or in his future.

    To do this ontologically, you have to apply two time coordinates to each event in spacetime. This implies that rather than wave functions, events in spacetime need to be described by density operators. Density operators automatically include two copies of the time coordinate, \rho(x,t) = \psi(x,t)\psi^*(x',t'). We usually assume that both time coordinates t and t’ refer to wave functions, but we don’t have to do this.

    This is part of a paper I’m busily writing up, but the main topic is applications to the quarks and leptons.

  16. Daniel de França MTd2 says:

    Speaking of Bohmian mechanics, here is a QFT version of it.


    QFT as pilot-wave theory of particle creation and destruction
    Authors: H. Nikolic
    (Submitted on 15 Apr 2009)

    Abstract: States in quantum field theory (QFT) are represented by many-particle wave functions, such that a state describing n particles depends on n spacetime positions. Since a general state is a superposition of states with different numbers of particles, the wave function lives in the configuration space identified with a product of an infinite number of 4-dimensional Minkowski spacetimes. The squared absolute value of the wave function is interpreted as the probability density in the configuration space, from which the standard probabilistic predictions of QFT can be recovered. Such a formulation and probabilistic interpretation of QFT allows to interpret the wave function as a pilot wave that describes deterministic particle trajectories, which automatically includes a deterministic and continuous description of particle creation and destruction. In particular, when the conditional wave function associated with a quantum measurement ceases to depend on one of the spacetime coordinates, then the 4-velocity of the corresponding particle vanishes, describing a trajectory that stops at a particular point in spacetime.

  17. mfrasca says:

    Hi Daniel,

    Thanks for the link. I was aware of this. The author talked about that at DICE 2006 in Piombino, a conference I attended. At the same conference there was also the main researcher that pursued experiments to falsify this approach. My view is that, at least, this way to look at quantum mechanics is exceptionally awkward and sensibly useless to work with. There must be serious reasons to switch to that but presently I cannot see them.



  18. Kea says:

    I view the Bohmian approach as a clumsy (and ontologically incorrect) way of properly accounting for both classical and quantum information, which works much better in the category theory formulation now common in quantum information theory.

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