# Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I think his should represent respectable work. The question is that I am a physicist and I have not the right knowledge to approach Dynin's work. Please, could you give me some hints and references about so I can make an idea by myself of these techniques? My aim is to get a comparison with the work currently pursued in the area of theoretical physics about this same problem.

Thanks a lot beforehand.

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I am not familiar with this work, but It seems the wiki page has been edited by the author of the paper himself, which seems to violate the no original research policy of wikipedia. Usually discussions of claims of validity of papers proposing solutions to famous problems are not encouraged on mathoverflow. – Thomas Rot Jul 13 '14 at 18:53
@ThomasRot Yes, the claim has been inserted by Dynin himself. My question is rather different and, I think, in the framework of MO. This author's claim implies the use of a kind of technique that I am not aware of, being a physicist, and I would like to know more about. – Jon Jul 13 '14 at 19:56
It's more complicated than that (Dynin wasn't the first to add something about the paper). But I have moved two paras off the page and onto the talk page so that they can be discussed. – Charles Matthews Jul 14 '14 at 9:58
@CharlesMatthews But you have left the link to Dynin's paper. Is it ok? – Jon Jul 14 '14 at 10:01
@Jon Well, maybe - thanks for pointing that out. The policy on external links should probably be applied at some later time. – Charles Matthews Jul 14 '14 at 12:33

The paper is currently (and will be at least for a few days) under discussion at

http://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete

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As soon as your discussion with Dynin is completed, I will accept this as the answer to my question. Thank you again a lot for your time and sorry for having considered you a theoretical physicist rather than a mathematician as you are. – Jon Aug 14 '14 at 15:42
@Jon: I polished my review in view of online and offline discussions, and strengthened my conclusions there. With this, I consider the discussion of the paper now as being completed. – Arnold Neumaier Aug 17 '14 at 14:52

I reviewed at

http://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete?show=21846#a21846

four nearly identical unpublished papers by Dynin on the Clay millennium problem. The most recent paper claims at the beginning of Section 1:

A mathematically rigorous solution is given for both parts of the
7th Millennium problem of Clay Mathematics Institute''

As I discuss in my review, his claim is wrong. Neither are the explicit requirements of the problem definition satisfied (no discussion of Poincare invariance and causality), nor is the paper mathematically rigorous in a crucial part of the construction (it is not proved that there is an operator with the anti-normal symbol specified in the construction).

The main criticism also applies to the published paper

Alexander Dynin,
Quantum Yang-Mills-Weyl Dynamics in Schroedinger paradigm,
Russian Journal of Mathematical Physics 21 (2014),No.2,169-188.
http://arxiv.org/abs/1005.3779

which wrongly claims to give a construction of massless QED.

Note that there are other attempts in the literature to settle this millennium problem or variants of it.

Simone Farinelli, Four Dimensional Quantum Yang-Mills Theory and Mass Gap I: Quantization of the Solution of the Classical Equation, claimed to prove a mass gap given existence of a quantum Yang-Mills theory. This claim was reviewed at http://www.physicsoverflow.org/21788 and also found wanting.

Agostino Prastaro, Quantum Extended Crystal Super Pde's, claimed to have quantized a super-Yang-Mills theory with mass gap; see Theorem 3.28. An invitiation to review the claim is at http://www.physicsoverflow.org/21787 .

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Dear Arnold, As you could notice by the citations in some of these papers, I am somewhat involved in this strange race. Also Terry Tao entered to comment on one of my papers finally agreeing on the fix he asked for in one of them. So, if you would like to make a complete list, my papers should be in. I think the last of the series is arxiv.org/abs/1309.3966 appeared in JHEP. I am currently applying my ideas doing computations in some processes of QCD as also you can see in arxiv.org/abs/1105.5274 appeared in Physical Review C. Thanks. – Jon Aug 10 '14 at 12:47
@Jon: None of your arguments is mathematically rigorous, hence what you do can only be a ''proof'' on the usual relaxed level of rigor of theoretical physics, which in mathematical terms is only a (perhaps persuasive) informal argument, but without any logical force. For a proof in mathematical terms, you need to show that the exact spectrum of the theory is bounded away from zero. This requires to find rigorous error bounds for your approximations and a proof that the approximation errors do not change your informal conclusion. – Arnold Neumaier Aug 11 '14 at 11:14
So, why Terry Tao asked for a fix to a theorem of mine? Of course, I am a theoretical physicist so I appreciate a lot your being more mathematical oriented that helped a lot in this quest on Dynin's work. In some sense, your answer is some I expected and I think that Tao's intervention was motivated by arxiv.org/abs/0709.2042, my Physics Letters B paper. In this paper I work with theorems and is the most rigorous I have written about this matter. After that my interest diverted and I was mostly interested in applications to be compared with experiments. At the end, I am a physicist. – Jon Aug 11 '14 at 15:43
@CharlesMatthews: I polished my review in view of online and offline discussions, and strengthened my conclusions there. With this, I consider the discussion of the paper now as being completed. – Arnold Neumaier Aug 17 '14 at 14:53

Here is the answer by Alexander Dynin to the preceding criticisms that I post on his behalf. I cannot post it as a comment being too long. A point of view from mathematicians would be helpful at this point.

The Schroedinger paradigm in QFT is a quantization of functionals on the initial data. Under certain conditions the latter parametrize the solutions of classical YM equations but not the latter are quantized. Functionals may be non-linear but the Shroedinger operator is linear of course. No disentanglement is required.

Certainly, my procedure is not relativistic but the energy-mass component of the classical relativistic energy-momentum vector is not either. That is very important. In particular, Poincare generators have no role here even when an action functional is relativistic. The results are the same qualitatively in all Lorentz coordinates.

The energy-mass functional is a tame polynomial in infinite dimensions. As such it is a symbol of a tame operator in the Gelfand-Kree triple and defines an unbounded self-adjoint operator in the corresponding Fock space. As such the quantum Yang-Mills operator does exist. My latest main theorem about the structure of its spectrum achieves much more than a solution of the YM mass gap problem.

Alexander

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