In the most recent post I wrote a short and simple summary of my most recent findings mainly in Einstein’s theories, but in this summary I needed to refer to an example how to build a gauge field. I constructed a gauge field, a Yang-Mills field in a paper I wrote in 2010
https://arxiv.org/abs/1011.3962
So there is an explicit example. Then I thought that maybe the readers doubt if I actually can do this kind of mathematics correctly. People doubt such things, I can understand it, but actually I do have some books on quantum field theory, whether the paper looks like that or not. So I decided to add this post looking at the question whether this paper has perhaps been shown incorrect. These seems to be two people in 8 years who read and checked the calculations in this paper. Both found the solution to the classical Yang-Mills equations correct, so you can believe it is correct. One of the persons questions my way of quantization, but quantization is not so clearly defined and understood. The second person thinks my solution is correct.
My paper solves one of the Clay Millennium problems. I offered it to two journals but the editor in both cases refused to send it to reviewers. I also sent my solution to the authors of the problem statement. They did not answer. It was useless, I got tired of trying to find a publisher or an expert to comment the paper.
Let me remind what happened to my solution to the Navier-Stokes problem, the one that I after several efforts got published in a peer reviewed journal: the original problem statement depended on an old theorem that was believed by the community to be true but I found it was faulty. Thus the problem statement allowed a simple and nonphysical solution that answered the original problem statement. I had hard time to get the paper accepted as reviewers did not want to believe that the old theorem was incorrect. When my paper was already accepted, the CMI secretly modified (but I saw it as I managed to follow closed discussion board and there was discussion on it: in a search google shows pieces of content that you do not have access to) the problem statement so that it did not any more depend on the theorem.
And then these people claimed that they have always known that the theorem is wrong and that the solution must be physical. However, the CMI problems were not physical problems, they were mathematical problems and in a mathematical problem you must give all assumptions you need. As they were not correctly given, I found a non-physical solution satisfying all required properties. Then some web commenters claim that my solution is not physical, but I already in the paper explained very clearly that it is not a physical solution and not a solution to what should be the Navier-Stokes problem, it was a solution to the CMI Millennium Prize problem, which was stated poorly.
The Navier-Stokes paper also has a rather general solution to the Navier-Stokes equations in a series form. I do not check if and in what environment the series converges (it converges in some environment, but I could not make the paper too long). By the way, those equations are quite difficult to solve, nonlinear partial differential equations with many unknowns. Not so trivial as it may appear.
Finally, the CMI ignored my solution. So, I was not too interested in getting my other solutions published as after such an effort the paper would still be ignored. But some time ago I noticed that a blogger Zulfahmed had read my solution to the Yang-Mills fields.
In that paper I construct an explicit solution to Yang-Mills field equations. I first solve the classical equations, construct the solution so that there cannot be a mass gap and then use one of the possible quantization approaches. There is no mass gap. Let me explain this a bit more. The problem asked for proving something for quantum Yang-Mills fields. This is a mathematical problem statement. It is not a statement that this is an interesting field of mathematical physics, so if you write a good and interesting paper we will give you a prize. A mathematical problem statement is assumed to state precisely what is to be solved, and you are assumed to solve exactly what is asked.
In quantum field theory quantum Yang-Mills fields are much studied. They are gauge fields with SU(N) symmetry. The Lagrangian of Yang-Mills theories has three parts: Yang-Mills term, Dirac term and a stream term. A quantum field theory starts with Schrodinger’s equation (or usually Dirac’s) and moves to the occupation number formalism. Then the Lagrangian has the Hamiltonian surrounded by creation and annihilation operators for position (which look like wave functions but are operators). Here you can try to insert Yang-Mills fields, like the Maxwell equations can be inserted in a covariant form as an operator involving a vector potential. The usual way in quantum field theory is as I mentioned: add a free Dirac spinor field, Noether stream and the Yang-Mills term.
However, when I read the problem setting, it said nothing of this. What I explained was what a theoretical physicist would like to see solved: find the solution to all physical forces. But this is a mathematical problem. There are many possible ways to create a theory of this type in theoretical physics, but this problem should be: solve this particular problem and do not complicate it with such things that are not even mentioned in the problem statement. So, drop the Dirac spinor field and such things. Keep only the Lagrangian of Yang-Mills. Then the field can be just as well be classical.
So I concluded that I have to follow the problem setting. Yang-Mills fields can be taken as classical, let us take them as classical, solve them and then quantize the result. That must be what the problem statement says. (It is a very hazy problem statement formulated with differential forms. I case you know something of quantum field theory and expect that you should see perturbation series, Feynman diagrams, Dirac equations, canonical or path integral quantization, take a look e.g. at Freed-Uhlenbeck, Instatons and Four-Manifolds to see how Freedman and Donaldson treat Yang-Mills fields as connections on fibre bundles with differential topology. Considering the problem statement and the authors this seems to be the intended setting of the problem, and then you understand why I made an extremely concrete and simple counterexample by using a totally elementary constructions. A proof would need to be by differential topology, but a counterexample can be concrete and elementary.)
Zulmahmed apparently did check my solution to the classical equations, so did Simone Farinelli who I mention below, and the solution is correct. Yang-Mills field equations are also quite difficult equations. You can try, not so trivial.
Simone Farinelli wrote a paper especially studying my solution to the Yang-Mills fields:
http://inspirehep.net/record/1300860?ln=en
Four Dimensional Quantum Yang-Mills Theory and Mass Gap by Simone Farinelli, Jun 16, 2014, e-Print: arXiv:1406.4177 [math.DG] | PDF Abstract (arXiv)
We prove that Jormakka’s classical solution of the Yang-Mills equations for the Minkowskian R1+3 can be quantized to field maps satisfying Wightman’s axioms of Constructive Quantum Field Theory and that the spectrum of the corresponding Hamilton operator is positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level.
In the later version of the paper the abstract does not say it analyses the solution from my paper. I did not know of Farinelli’s paper (I did not look at this problem later as those people were not willing to respond to me) but it makes a different quantization than mine and gives a mass-gap. My paper shows that there is no mass-gap and I still consider my result as the correct one.I just noticed Farinelli’s paper yesterday and have not read it, and am unlikely to read papers of a field where the people of the field refuse to read my papers, but I read the text referring to my paper. Farelli writes that Jormakka first solves the classical Yang-Mills equations and the quantizes, while they first quantize and then solve the equations. In this way, by quantizing in their chosen way they do not get my solution that has no mass gap. They get their solution that has a mass gap.
Farinelli claims their quantization is the correct way. It may be more in line with the quantization that physicists think should be the correct one, but the Clay problem is a mathematical problem. Whatever physicists might have been thinking should be the problem setting, for a mathematical problem the problems setting is what is given in the problem statement. The problem setting in the paper is hazy, especially with quantization. I sent my solution to the authors of the problem statement. They had all the possibility to explain that they meant the statement in a different way. They did not answer. This being so, I as a mathematician have the right to interpret the problem statement in the way any mathematician would do, that is, trying to understand what the hazy person writing it thought it should be, and my quantization is fully legal in the scope of the problem setting.
The blogger Zulfahmed thinks I am correct:
ZULF THINKS THERE IS NO MASS GAP IN R4 QUANTUM YANG-MILLS
ZULF THINKS THERE IS NO MASS GAP IN R4 QUANTUM YANG-MILLS November 1, 2017 by zulfahmed This is a Millenium Problem, to show a mass gap in quantum Yang-Mills for nonabelian gauge group. There are two serious attempts. One by Jormakka provides explicit solutions of classical Yang-Mills with arbitrarily low energy and then argues that the corresponding quantum Yang-Mills also has arbitrarily low energy and no mass gap. The other is from work of Alexander Dynin which shows a mass gap for a quantum Yang-Mills operator that is self-adjoint on an infinite dimensional space with respect to a Gaussian measure that has discrete spectrum. Refinements of the second solution is done by Simone Farinelli including showing consistency with Wightman axioms. To me it seems still that this the second approach is too complicated to be right. So Jorma’s construction is criticised on the step to quantization. I think he’s right and there is something wrong with the more complicated constructions. This is because I believe that experimentally observed mass gap is happening for an entirely different reason that quantum phenomena are results of a fixed static compact geometry of space where mass gap is trivial essentially because self-adjoint operators on Hilbert spaces of functions have discrete spectra.
Nevertheless, all of this has no physical relevance.
These ways of quantization of gauge fields are not physical. What I more or less showed in the papers i just put to vixra is that tn the physical reality the field particles are finite size and field interaction particles (bosons) are simply Lagrange multipliers needed for giving the Langangian symmetry of a chosen symmetry group. That is, the field is created by finite size space elements, which are constructions in the 3-space and keep the state of the field (any of the four interaction fields). Then you take the propagator of the interaction, like the Dirac equation for fermions, and make a Lagrangian. Then you select a symmetry group and require that the Lagrangian is invariant under this symmetry group. You will have to add some terms to the Lagrangian so that it becomes invariant. These terms actually are Lagrangian multipliers and the interaction particles of the field are these Lagrangian multipliers. Quantization of photons, from which all this started, is caused by the finite size space volume elements. Therefore the bosons (the interaction particles) are not actually particles, they are waves, and quantization is not done as these people think, it what we want is a physically realistic model.
Max Planck and Albert Einstein pushed through the false paradigm of a photon as a particle, the space as a 4-dimensional Lorentz space, the principle of relativity and seeing the interactions as geometry. This is not the case. The space is a 3-space, there are no additional dimensions, the volume elements are of finite size as that is the only way to realize gauge fields without adding dimensions. Quantization is a result of the finite size of field particle (field particles do not move with the speed c, do not confuse them with field interaction particle). This is the reason Planck measured that the energy of a photon is quantized: it gets its energy from the field and a discrete field can only give or absorb energy in quanta.
When I wrote solution attempts to the seven Clay problems I made a trial publication campaign. It was a test and i expected it to show the result which it showed: it is not possible to get papers checked through any way one can initiate. Only the powers that be can decide to create publicity and this allows them to select the people who are allowed to solve famous problems.
I know this would likely be the result of the publicity trial (notice that was almost totally ignored by the media and totally ignored by the academia). I knew it because from 1987 to 2000 I tried to get my proof of the Poincare Conjecture reviewed. Later I put the paper to the web, but it is essentially the same as the paper from 1987. I did rewrite is a couple of times, but the basic idea did not change. I did not find an error in 13 years and none of the about 10 experts (including Siebenmann) and 5 journals whom I asked to read the paper pointed to any errors. So, I think it is correct. I put the final version of my paper to a discussion board in 2000. It was not long after that that another solution of the Poincare Conjecture was accepted. It was strange: this person divided his paper to four papers and put them to arxiv. Nothing in the papers indicated that they solve the Poincare Conjecture. Immediately many experts were reading these papers. Another guy solver Fermat’s last theorem. He made a seminar without telling what the topic was, top experts came to the seminar. In 2010 I put seven papers to the web. I received only one answer, from the Hodge conjecture that I did not manage to solve. I explained my P not NP paper in a seminar in the university. They did not find errors and did not get a bit interested. I wonder it to experts would have come to a seminar as they did not even respond to my emails. Yes, I was a professor at that time. This is the reality, so how did he manage so well? His solution uses Ricci flows. My solution used the Morse theory. The Morse theory is 2-surfaces in a 3-manifold, they are moved as flows. I moved the surfaces and ended to a contradiction, but in the Morse theory you have to imagine how these surfaces move and see in your mind what happens to a loop. That is done so in geometric topology, it is a valid method of proof, but naturally if somebody claims that he cannot imagine it, then you cannot do anything. Ricci flows are metric versions of these Morse flows and can be expressed as functions, so it is easier to convince people. But I did not read his proof. I was not interested as I already knew that the Poincare Conjecture is true. But I was interested in seeing how he got his arxiv papers checked by experts so easily. That is why I made the publicity trial.
This time I will make no such publicity trial, but maybe I will send a link to this post to the blogger Zulfahmed. Why not, nice people are not easy to find among physicists. But let us be realists, nothing positive is to be expected.
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