Yang-Mills millenium problem solved?


I am an avid reader of Wikipedia as there is always a lot to be learned. Surfing around I have found the article Yang-Mills existence and mass gap and the corresponding discussion page. Well, someone put out my name but this is not the real matter. A Russian mathematician, Alexander Dynin, presently at Ohio State University, was doing self-promotion on Wikipedia at his paper claiming to have found a solution to the problem. This is not published material, so Wiki Admins promptly removed it and started a discussion. By my side, I tried to make aware the right person for this and presently no answer come out. I cannot say if the proof is correct so far but, coming from a colleague, it would be a real pity not to take a look. Waiting for more significant judgments, I will take some time to read it.


Arnold passed away


Latest of fathers of KAM theorem, Vladimir Arnold, passed away yesterday. He was 72. He has been one of the most influential mathematical-physicists of XX century. I studied analytical mechanics on his book and I have learned a lot about differential equations from other fine texts of him. It is with great sadness for me to apprehend this piece of news that is going around all the World, giving the exact stature of this man. The reason for my deep sadness relies on the fact that I built on his very strong shoulders, being him a giant (see here).  KAM theorem has been a key result in dynamical systems opening the way toward a deep understanding of chaos in conservative systems and extending beyond to more general classes of Hamiltonian systems. Personally, I lived the struggling to understand plasma heating under the spell of this theorem but applications range everywhere in all fields of physics. I think that the best way to remember a scientist is through his works and I think that a simple post is not enough to embrace all the contributions due to Arnold. He is also remembered for being a pioneer in catastrophe theory and solved XIII Hilbert problem. He has been awarded a Fields medal but Soviet government impeded him to accept the prize.  He will live forever in the work of people of our community and a lot more will be the fruits coming from his ideas.

And you are calling it a gluon yet…


One of the more questionable points I have discussed so far is: What are QCD asymptotic states at very low  momenta? This question is not trivial at all. If you will speak with experts in this matter, a common point they will share is that gluons carry color charge and so must form bound states. A claim like this has a strong implication indeed. The implication is that Yang-Mills Hamiltonian must display the same asymptotic states at both ends of the energy range. But the problem is exactly in the self-interaction of the theory that, at very low momenta, becomes increasingly large and gluons, asymptotic states of Yang-Mills theory in the asymptotic freedom regime, are no more good to describe physics. So, what are good states at low energies? I have already answered to this question a lot of times (recently here) and more and more confirmations are around. I would like just to cite a very nice paper I have seen recently on arxiv (see here) by Stanley Brodsky, Guy de Teramond and Alexandre Deur. These authors have nicely exploited AdS/CFT symmetry obatining striking results in the understanding of low-energy QCD. I would like to cite again the work of these authors as their soft-wall model is indeed a strong support to my view. It would be really interesting to get them working out a pure Yang-Mills model obtaining beta function and all that.

What one has at low end of momenta is a new set of states, glue states or glueballs if you prefer, that permits strong interactions. These states have already been seen in most laboratories around the World and belong to the open question of the understanding of the lower part of the hadronic spectrum.

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