# Yang-Mills Millenium question and Dynin formalism

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I repost this question here as it went without an answer on MathOverflow.

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.

Please, note that the claim has been moved to the talk page of Wikipedia.

Has Dynin solved the problem?

No. His approach is on the whole sensible (and interesting), but there is a serious gap which I think is impossible to patch. I'll write a review of his paper, but need more time to check things.

Arnold, This is published material. So, if you find a flaw this should be widely known turning back the matter to the start.

What is the published version?

By the way, published does by no means mean correct. I have seen plenty of mistaken papers. It is the responsibility of the author(s), not of the referee(s) or the publisher, to ensure that a paper is correct.

A Gaussian partition function can be constructed for lots of covariance operators. But the correlation functions obtained by repeated differentiation satisfy the Wightman axioms only if the covariance operator has the correct properties. You need a covariance that vanishes for pairs of points at spacelike separation to satisfy the causality condition, and that is Poincare invariant to satisfy the covariance condition.  Finally, to satisfy the positivity condition you need to ensure that the Fourier transform with respect to time has a positive support. (This means that you get the ground state correlations rather than a partition function, or the latter in the limit of zero temperature.)

(Only) having shown all this you have verified the Wightman axioms for a quasifree field. For an interacting theory, you need to do much more, as the 2-point function determines the N-point functions only in the quasifree (Gaussian) case.

Actually, the covariances that satisfy the Wightman axioms have been completely classified; they are (according to the Kallen-Lehmann formula) those that can be represented as an integral over the covariances of free fields with mass m, withe respect to an arbitrary probability measure on the set of nonnegative masses. The 1-point measure gives the free fields themselves.

Existence and mass gap are not enough to solve the millennium problem. The resulting properties also count. It is explicitly stated that the verification of the Wightman axioms is part of the solution requirement.

Arnold, a trivial infrared fixed point, I mean a Gaussian partition function, is it a proof for the existence? As I know from Glimm and Jaffe that satisfies Wightman axioms even if the theory is trivial and so says practically nothing.

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I reviewed 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 here 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. Invitiation to review the claim here.

If I find more attempts I'll add them here; if I find the time, I'll review the papers.

answered Aug 10, 2014 by (13,199 points)
edited Aug 10, 2014

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 this one appeared in JHEP. I am currently applying my ideas doing computations in some processes of QCD as also you can see in this paper appeared in PRC. Due to the high reputation of these journals I would try to take a look. My question about triviality was aimed to understand how far I am from a proof of existence. Thanks.

I saw some of your papers. While they may provide partial information useful towards the same goal, I didn't see any claim of having solved either half or both parts of the millennium problem. (If I missed it, please point to paper and paragraph.) Reviewing all work that works towards the conjecture would have to include much more, and I don't have time to do this.

Do you mean they do not contain a proof of existence of the mass gap? Ok, keep on with your reviews and be happy.

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.

Your paper referenced here constructs some (classical) solutions of the YM equations from solutions of $\phi^4$ theory. But you claim an equivalence, which is not the case as most of the solutions of YM will not have this form. Thus you cover only a tiny part of the phase space.

My view is somewhat different. I interpret these classical solutions as oscillations around a non-trivial vacuum expectation value. They break BRST symmetry as found from lattice computations and provide the correct spectrum and the proper gluon propagator in the Landau gauge. So, they agree with evidence from the lattice.So, I proved the existence of a class of QFTs from Yang-Mills theory that have a mass gap. Agreement with lattice computations seems to say that also uniqueness holds and classical chaotic solutions play no role in quantization.

For every group G there is (expected to be) just one family of quantum YM theories, parameterized by the mass gap. Construction of one of these implies solvablility of the corresponding quantum initial value problem (Schroedinger equation) for all initial states.

You only provided informal evidence for the existence of some particular states of quantum YM, with a discrete spectrum. As your solutions are very special you are, even informally (i.e., on the level of typical arguments in theoretical physics), very far from an existence proof. And all your arguments being nonrigorous, they contribute nothing to the millennium problem.

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The claim in the paper is wrong, unfortunately, and the paper is fundamentally ill-concieved. This, even though it comes from a justly respected author with a prior history of valuable contributions.

There is no formalism for constructing quantum Yang-Mills theory starting from any sort of global classical solution, and there simply cannot be (it might be possible locally, at short distances, but then all the issues are in pasting together the quantum theory from the local near-classical descriptions).

Quantum Yang Mills is a well-defined computational thing, the Euclidean version can be computed on a Lattice. The continuum action is $1/g^2 \mathrm{tr} F^2$, the lattice action replaces $F^2$ by the holonomy of the link variables over each plaquette. The coupling plays the role of $\hbar$, and the $\hbar\rightarrow 0$ limit is exactly the weak coupling limit, or zero coupling limit, where the gauge group turns into a bunch of noninteracting vector bosons. It is also the small-lattice limit, because of asymptotic freedom. The continuum classical equations locally (meaning at scales which are exponentially teeny tiny) are solutions of zero coupling Yang-Mills theory. They have no relation to the behavior at longer distances, where the coupling runs stronger.

So any procedure which starts with the classical solutions and tries to do some formal jiggling to produce "solutions" to quantum Yang Mills by formally defining operators is always infinitely far away from the quantum vacuum of Yang-Mills. It is solving equations which have a fixed coupling independent of scale to describe a system which does not have a fixed coupling independent of scale. So it can never work.

But Dynin claims it does work! Why? It's because he is doing something completely different, something which has no relation to what is normally called Euclidean Yang-Mills theory, and misinterpreting this alternate construction as a quantization of Yang-Mills theory.

What he is doing is using a well-known stochastic formalism which is a known mathematician-obfuscated version of the physicist's Euclidean creation and annihilation operators, and applying it to configurations of initial data, which he identifies as classical solutions of Yang-Mills theory. The result of this is associating a creation and annihilation operator to each classical solution, producing a tower of free bosons sitting on top of each initial data. These quanta have no relation to any excitations in actual quantum Yang-Mills, they are simply a mathematical artifact of this construction. These quanta are indexed by classical solutions, not by fluctuations of the true Euclidean vacuum.

He then defines a nonsensical energy for his model, which preserves the N=0 state free-vacuum as the interacting vacuum (I made a false claim here initially). I speculate that this mistake is due to the fact that mathematicians don't have experience with simulations of Lattice Yang-Mills theory. Such simulations produce a valuable picture, first developed by Kenneth Wilson, and without it, one is lost, and can easily go down a path very distant from any correct one.

The vacuum of quantum Yang-Mills is calculated on the Lattice and it is described accurately by small perturbations to Wilson's strong coupling fixed point. The strong coupling limit is $g\rightarrow\infty$, or $\hbar\rightarrow\infty$, where the lattice action is zero. Zero action means that every link matrix is equally likely and completely independent. This is the description of Yang-Mills at long distances, as known for sure from even the very first lattice studies by Wilson. If you perturb this by introducing perturbations, slight correlations between neighboring link-matrices, these die off exponentially, and the leading approximation to a long Wilson-loop action is by the minimal surface spanning the loop. This behavior, discovered by Kenneth Wilson, is what led confinement to become universally accepted in physics.

This behavior is quite simple to understand, and the basic problem of Yang-Mills mass-gap is simply to prove rigorously that the Yang-MIlls lattice field randomizes in this way, even when at short distances, the path integral is dominated by configurations which are close to classical solutions. The problem of existence is simply to show that the limit of small lattice spacings can be taken, using a coupling which vanishes as the perturbative running of the coupling at small distances, the statistical fluctuations at long-distances converge.

This paper simply cannot do this, because the method is starting from classical solutions to classical Yang-Mills. There is no useful relation between global solutions of Yang-Mills and quantum Yang-Mills, because the classical global solutions are globally correlated--- the classical data on an initial slice uniquely determines the solution at all future times--- while the quantum system has all the correlations dying off exponentially at a distance scale of $1/\Lambda$. So the initial approximation has a structure which is infinitely far from the correct totally random structure at long distances, and the approach is fundamentally misunderstanding the classical correspondence in the confining infinitely strong interaction limit of long distances, I am sorry to say.

answered Aug 10, 2014 by (7,545 points)
edited Mar 19, 2015
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I am pretty amused by the spume on physicsoverflow  over my  papers.

I have been immersed into  functional analysis and its applications during my student years at Moscow Gelfand seminar.

My master degree paper was 1958 solution of a Gelfand problem presented  at the 1956 Mocsow conference on functional analysis. My 1961 Ph.D. dissertation prequeled Kohn-Nirenberg theory of pseudodifferential operators. It put Gelfand index problem into the framework of homotopy theory.  This framework was instrumental in the first  Atyah-Singer proof. (All four authors duly acknowledged my contribution).

In 1963 I received  the coveted   Prize of Moscow mathematical society  and a postdoc position at the Dubna Joint Institute for Nuclear Research  from  1963-1968. There I was  introduced to quantum field theory by outstanding  physicists of  Bogoliubov Laboratory.

Those years were high time for axiomatic quantum field theory. On top of it I learned Schwinger quantum action principle in the form of Bogoliubov-Shirkov quantization that I applied in my paper under discussion.

From 1977-1978 I was a visitor at IAS (Princeton), from 1978-1980 I have taught mathematics at SUNY, Stony Brook (were I was in contact with Frank Yang), and from 1980-2014 at OSU, Columbus, Ohio (where I was in contact with Robert Mills until his retirement in 1998.

Now I am a retired  professor emeritus of OSU.

I have started my YM project in 2006 as an attempt to extend the immensely successful theory of pseudo-differential operators on finite-dimensional manifolds to  cinfinite-dimensional ones in the lines of SBS quantization postulate. The development took all past years, and its progress is reflected in the series of arXiv preprints. The 2009 preprint has been certainly a rough draft but the ideas were basically right. In the tinkering   they become much more precise.

Two essential novelties in the latest incarnation arXiv:1005.3779 [math-phys]. First, by L. Faddeev's insistence, the dimensional transmutation of the running constant (though the theory is not perturbative); second,  full characterization of the quantum YM spectrum. The main difficulty has been the  infinite degeneracy of non-vacuum eigenvalues until it has been measured with respect to a von Neumann algebras with regular trace.

Certainly, it does neither address mathematical pitfalls  of  the conventional perturbation QFT, nor checking Whigtman axioms. I do not quantize classical solutions of Yang-Mills equations per se but following Bogoliubov-Shirkov I quantize the energy-mass functional on the nuclear Gelfand triple of their  constrained initial data spaces.The result is a the Shroedinger operator in the sesqui-holomorphic Fock-Kree triple.  Thus  Beothians cries (if  I may use a famous Gauss expression) on physicsoverflow are just overshooting. I recognize that my paper is not easy for non-experts. But I hope for your honesty.

This has been a very difficult unterprise that required not only developing of a powerful general framework but,  even more difficult, the concrete apllication to Yang-Mills theory. The simplicity of the gauge group is very essential.

Of course I have in mind the Clay YM problem.  However I have achieved much more in construction of a rigorous mathematical quantum field theory on a par with mathematical quantum mechanics based on Neumann axiomatics and Weyl pseudodifferential quantization.

answered Aug 21, 2014 by Alexander Dynin

Welcome to Physicsoverflow, professor Dynin. I think nobody here is questioning your professional capability as a whole, and your credentials are made clear in the main post. People here are more concerned with the truth values of the particular pieces of your work cited. Therefore this answer does not seem to contain much information directly addressing the question, it is more appropriate for a chat, but no downvote from me.

Hi prof. Dynin--- apologies for any percieved disrespectfulness. Would you be willing to share some old reviews (positive, negative, anything) you might have regarding Kohn-Nirenberg theory, or any other work? We would be happy to place any papers you wish to review in the reviews section, for example, a neglected or forgotten historical paper, a neglected reference in a classic paper, a paper with an interesting extension which is not common knowledge, etc.

Regarding the extension to $C^\infty$ manifolds of symplectic methods, this is made difficult by the unitary inequivalence of the states associated to even the smallest changes in the underlying system. This is the basis of the criticism.

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