Today it is appeared my contribution to the proceedings in arxiv (see here). I am happy to hear your comments if any. I prefer to write this kind of papers rather soon, when the memory of the talk is fresh enough to convey the same contents, if possible. As you can see, there has been a lot of maturation of these ideas since their inception and now I am really satisfied on the way the scenario of mass generation and low-energy behavior of Yang-Mills theory is formed. It is also striking to see how all this matter fits well inside the current understanding that is forming in the community as I discussed in the preceding posts. Mass generation is a key matter today, mostly due to the Standard Model and the expectations from LHC (e.g. see here). My view is that our work will be pivotal and it is possible that similar mechanisms are at work here both for Standard Model and QCD. If this is so, Nature will prove to be mind-boggling again.

## The many faces of QCD (3)

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Congratulations! I am quite impressed though still have to digest the paper. It has been an experts opinion that the YM mass gap problem is beyond perturbation theory.

Interestingly, the leading term of your asymptotics of quantum YM energy spectrum is a harmonic oscillator spectrum. This echoes with the spectrum estimate from below given in Dynin, A.,

“Energy-mass spectrum of Yang-Mills bosons is infinite and discrete”, arXiv:math- ph/09034727. That paper purported a mathematically rigorous (also Higgsless) solution of the Clay Institute YM gap problem.

The paper was submitted to Journal of Mathematical

Physics in May 2009 but withdrawn after 18 months of their indecision. Currently a purified version is in preparation for a mathematical journal.

Dear Alexander,

Thank you a lot for your comment. As you know, I tried to do some ads to your work but without too much success so far (see my post here). Contrarily to what I used to think, physics community responds more promptly also for the reason that there are other ways to check if something does work or not without too much demand for rigor.

About the question of experts and perturbation theory, you should know that it was an idea (let me say a prejudice) of both mathematicians and physicists that the only perturbation theory available is that with a small parameter. But, as I have shown, a problem has two sides. All started on 1992 but the most important paper of the series is this one published on Physical Review A. One can always gets a perturbation series for a differential equation when a parameter goes to zero but also when this same parameter goes to infinity. The series are dual each other in the sense that a series has a development parameter being the inverse of the other. From this I was able to manage problems in almost all areas of physics and QCD is the last of the series, the reason why, twenty years ago, I started all this.

I have looked at your paper and the fact that you get an estimation of the spectrum being the one of a harmonic oscillator implies that your arguments are sound as this conclusion was drawn in d=2+1 from lattice computations and seems in agreement with the spectrum of the theory computed in d=3+1. I have asked to my colleagues to check all this through their computations on the lattice of the Green functions and I hope the answer should not delay too much.

Sorry for you not so good conclusion with JMP. What appears a mystery to me is why some papers get immediate hype and others, like yours, do not while coming from respectable colleagues. I hope these few lines will help you to go through.

Regards,

Marco

Dear Marco,

I was a student of great I. Gelfand, who, universal as he was, had a special predilection for mathematical physics, an important subject at his celebrated seminar in Moscow. In particular, he invented path integral independently from Feynman but, unfortunately rejected by caustic L. Landau and his cohort of physicists.

50 years ago in my PH. D. dissertation I made important inroads to a solution of Gelfand Index Problem. The work got the prize of Moscow Mathematical Society and, more importantly, used by Atiyah and Singer in their first solution of the Gelfand

problem. Certainly, an immature student had no chance in competition with the grandmasters, but afterwards my younger friend S. Novikov (the great topologist and mathematical physicist) regretted that he did not pay more attention to my questions during

graduate school time. Otherwise the famous Atiyah-Singer index theorem might have different names.

Gelfand influence is apparent in my YM paper. Actually the paper applies a rather non-conventional

but rigorous mathematical QFT based on Gelfand triples from 55 years ago as well as on Hida white noise calculus. Most of my difficulties with (math) physicists are due to the conflict with their paradigms. Just as in the Gelfand-Landau case!

Regards,

Alexander

Dear Alexander,

As I can see you have a fine list of masters on your education. I should say the same from the physics side as I have had very good physicists in Rome that taught me so much that it was impossible not to do good research.

The question you put forward says to me that somehow my idea that mathematicians could be more fair than physicists is plainly wrong. Both communities share identical problems. My idea arises from the fact that in mathematics there is no other way than right or wrong while physicists can live for a lot of years in a middle earth where something “could be surely proved right in the future and we can take it as a paradigm”. Most of this relies on experiments that the current technology cannot yet afford. This means that some people can build an entire career with something never proved right in his/her lifetime. This is the reason why I chose research arguments that could be proved right or wrong in a very few time if someone should take the time to take a look at them. This is a short path and when it works it works in the finest way.

For my results on Yang-Mills equations I have had the experience of the intervention of a famous mathematician. His intervention helped me to give a right form both to the result and its proof. Of course now he is on a harsh position. He agreed with my latest proof (in the way you can read here) but he will not support my work at all. Of course, I am a physicist and this is not so important. I care more for the support of my community that I have always had. Besides, we have stronger ways to see if something is right and wrong because this can be read from experiments. This is without any appeal. The point is that, if I am right, behaviors like that will go into the spotlight and remembered forever. But Nature may prove me wrong and we can forget it all.

Finally, I hope that your results will be supported by your community as I am convinced that you are supporting mine and mathematics displays different ways to arrive at the same goal.

Regards,

Marco

Dear Marco,

May I have references to those lattice computations?

Thanking,

Alexander

Dear Alexander,

About d=2+1 there is this paper by Bruce McKellar and Jesse Carlsson. You should check, by the same authors, also this. These authors do calculations using a lattice theory and arrive at the conclusion that the spectrum is that of a harmonic oscillator. You can also read other papers about from them here. So far, no general consensus has been reached about this, mostly because a harmonic oscillator spectrum implies a trivial theory and people, for a lot of years, believed that Yang-Mills theory was not trivial when a mass gap emerges. Here “trivial” means that the theory is free in this low energy limit. Of course, one should see at this the other way round as a trivial limit at lower energies grants the existence of asymptotic states to produce a meaningful quantum field theory at increasing energies and doing analytical calculations to compare with experiments. This is something physicists are very eager to find because there is a lot to be understood yet at low energies in what is seen in accelerator facilities.

For d=3+1 there are a couple of classical papers on lattice computations performed on computers. This one and this one but the first gives the mass spectrum through pure numbers and you can easily check by yourself that mass ratios go approximately like the series 3,5,7,… of a harmonic oscillator. Both groups miss the ground state at n=1 because they worked with too small volumes and such states are lost in the noise.

There are other theoretical approaches supporting this conclusion in d=2+1 that also appeared in archival journals. They do not give a proof of the mass gap but just guess the vacuum functional and produce from it the spectrum that is given by the zeros of Bessl functions. Indeed, these zeros have the sequence proper to a harmonic oscillator in the limit considered. If you need these refs I will give you the links.

Finally, there are the lattice computations of the Green functions of Yang-Mills theory. But you can read about this in my latest posts here or, if you have more time, in my oldest ones. These researchers show inequivocally that the Green function for the Yang-Mills field reaches a finite non-zero value lowering momenta and the running coupling goes to zero in the same limit. These are the chrisms of triviality and you are left with a harmonic oscillator spectrum for the theory.

All this agrees perfectly well with my mathematics. I think you should be delighted.

Let me know if you need more.

Regards,

Marco

Dear Marco,

Certainly, because of the YM self-interaction the YM spectrum cannot be additive (unlike with harmonic oscillator). However your result is asymptotic, and my is an estimate of the spectrum from below.

An intrigue here is that non-additivity of the YM spectrum suggests an inner structure of YM bosons!

Best,

Alexander

Dear Alexander,

Yes, I am at the trivial fixed point. This is the reason why I have a harmonic oscillator spectrum. Your lower bound just says this.

I agree that an inner structure could explain this (asymptotic) spectrum: A stringy one?

Best,

Marco

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