I just sent a note to the Millennium editors at Claymath, informing them that their Yang-Mills problem has no solution, beyond jettisoning Yang-Mills. Jaffe and Witten, the sponsors of this particular problem, think that the Yang-Mills Theory requires more math to fully explain it. This is the current solution to everything: pile more new math on top of piles of old math, until you can no longer smell the dead mouse at the bottom of the heap. At Claymath* it says,
This “mass gap” is the dead mouse, and it is the same furry creature rotting at the bottom of renormalization. This mouse has been stinking up the entire history of math and science, not only since Newton but since Archimedes. We will get to that in moment, but the other keyword in that paragraph is the word “successful.” If Yang-Mills were successful, we wouldn't have a problem here, would we? The mass gap occurs because we have a mismatch between the math, the data, and the theory. This must mean that Yang-Mills is unsuccessful, which means that the statement of the problem at Claymath is propaganda. Jaffe, Witten, and the rest want to find a solution to this “lack of understanding from a theoretical view,” but they cannot admit that it is really a mathematical failure. It is not the theory that is failing, since there is no mechanical theory under QED or QCD. It is a straight mathematical failure, a failure to contain non-zero masses. But, as you see, the promoters try to pass off Yang-Mills as “successful” from the first sentence, from the very second word. Propaganda.
The mass gap cannot be understood using the Yang-Mills theory, because Yang-Mills theory is not a mechanical theory, it is just math. It is not a theory at all. Like Faraday's theory or Maxwell's, it is no more than heuristics: math created to explain a limited set of data. Since the math is wrong, it is uncorrectable. You don't correct bad math and bad postulates by piling more math on top of the old, you correct it by starting over with good postulates and good math, both based on mechanics. What this paragraph from Claymath is really saying is that the mathematical models we now have can't explain the data we now see, but we want to keep the models anyway because they have made everyone famous. A lot of people would have to give back their Nobel Prizes if it were proved that the models utterly failed, so we prefer to continue to try to patch them up. We will even give you a million dollars if you will patch them up for us.
Well, it might be that I could do so, but I am too scrupulous to attempt it, even with such a bribe. Unlike the rest, I prefer the truth to any amount of cash or prize, and the truth is, there is no strong force. If there is no strong force, there can hardly be any math or theory to represent it. I know this is not what these people want to hear, but I am not interested in their career desires. At some point they will have to face their failures. Everything that has come in contact with the old E/M theory is fatally corrupt, and that includes all of QED, QCD, and string theory.
To be specific, the mass gap cannot be bridged because it doesn't exist. Their models predict that particles moving at c cannot have mass, because they have given the photon a zero mass. If the waves move at c, they should be massless. But the entire problem is with their math. They think the photon must be a point particle, and they think this because according to their math, c is a limit. At a limit like this, mass must either go to infinity or zero. Why? Because this is what the calculus tells them. This is how the calculus works according to Newton, and no one has ever questioned it.
But I have shown that the calculus has been misinterpreted. Modern mathematicians think that the mass must go to zero at the limit for the same reason they think that they can find an instantaneous velocity: they believe in the existence of the zero differential and the singularity. In other words, they still have not fully understood what is happening when you approach a limit, when you talk of infinitesimals, or when you reach Newton's ultimate interval. They don't understand this because they don't understand how or why the calculus really works.
Just as with General Relativity, QED is still infested with the point. Einstein was still writing equations for mass points, and QED is still trying to write equations for point particles. But mass at a point is simply a contradiction in terms. There is and cannot be any mass at a point, by definition. Contemporary physicists read this upside down: they think it means that a point particle can't have any mass. But that is wrong. What the sentence means is that anything that exists must have mass. Any proposed particle, to be a particle, must have extension, and therefore mass. You cannot propose a point particle, because that is logically equivalent to proposing a tall midget or a short giant. You cannot propose a contradiction as a postulate.
The photon is useful only as a field particle, and to have any mechanical use it must have mass and a radius. I have shown that it does indeed have both mass and radius. Therefore, if you have created a math that can't contain this fact, then you first have to get rid of that math. If your math is telling you that the photon must have a zero mass, then your math must be wrong. It is that simple. You don't correct that math, you jettison it.
This doesn't mean you jettison all of calculus, of course, but you must jettison the modern interpretation that tells us that numbers can exist at points. Numbers cannot exist at points, whether those numbers apply to masses or velocities or lengths or volumes or times or anything else. Any number above zero represents some extension, and you cannot have extension at a point. “Number” and “point” are mutually exclusive. Points are ordinals, but numbers in physics must be cardinals.
Even if you fix this foundational problem, you cannot save Yang-Mills. This is because Yang-Mills is a gauge theory, and all gauge theory is physically intrusive. As I have already shown, gauge theory is too much math for the job, and I mean for all of physics, not just QED or QCD. It is too much math because it already contains symmetries. It is gauged, which means it has immovable structure even before you apply it to nature. But this is not what we want. What we want is ungauged math, math that can contain nature without forcing her into any pre-defined shapes. All gauged math has become a problem, because physicists have begun trying to fit nature to the math, rather than trying to fit math to nature. Nature will always rebel under such pressure.
Yang-Mills, like most modern math, should be sent back to the math department. It is a nuisance. That is the solution to the Millennium problem.
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