You probably know that at the beginning of the Millennium the Clay Institute set seven mathematical challenges, and offered prizes of a million dollars each. One of these problems is called the Yang-Mills mass gap problem, and I’m about to tell you how I’m (not) going to win the prize for this problem. Of course, it’s a very technical problem, but I can explain it in simple terms, provided you understand the difference between a straight line and a circle. You do understand the difference, right? I have to ask, because, unfortunately, a lot of physicists don’t. Straight lines go on and on, circles go round and round.

So to explain. In technical terms, a straight line is called “split” (it splits the plane into two pieces, right?) and a circle is “compact” (it encloses a compact area of the plane, right?) They are called “groups” because you can add angles in a circle, and add distances along a straight line, and they are called “gauge groups” because you need some “gauge”, i.e. a measuring device like a protractor or a ruler. Yang-Mills theory, in a nutshell, is that all the gauge groups of particle physics are compact – i.e. all circles, no straight lines.

Now how do you measure mass with a compact gauge group? How do you make sure you don’t run off the end of the scale and get back to zero again at some large mass? Well, you can’t. You have to use a split gauge group to measure mass. Especially if, in the current model of elementary particles, there is a factor of about a trillion between the lightest mass (neutrino) and the heaviest (top quark). In other words, mass *cannot* be described by a Yang-Mills theory. Never, ever, in a trillion years. Do you see what I mean when I say some physicists can’t tell the difference between circles and straight lines?

In other words, the problem, even if it can be solved, has no relevance to physics.

I’d better explain what “mass gap” means, I suppose, in case anyone here doesn’t understand the word “gap”. It means that there is a definite gap between the lightest mass (neutrino) and zero. The Millennium Problem is to prove that there is a definite mass gap in any Yang-Mills theory. Besides the fundamental problem that Yang-Mills theories cannot describe mass, there are two further problems with this, one experimental and one theoretical. The experimental problem is that this gap is so small they can’t measure it. The theoretical one is that straight lines don’t have gaps.

I’ve explained countless times what you have to do about these problems. I have explained, for example, that the apparent “mass gap” is caused by doing all your experiments on Earth. The size of the mass gap depends on the geometry of the gravitational field, and the geometry of the gravitational field can change to reduce the mass gap to be as small as you like. In other words, there is no mass gap to explain. I have also explained how you can get masses of elementary particles in split or semi-split extension of Yang-Mills theories, rather than in Yang-Mills theories themselves.

I’m going to get a bit more technical now, and for that I have to explain what a clock is. Oh, you already know what a clock is? A clock is a compact gauge group for measuring time, which unfortunately is not compact. Never mind. All I need you to understand is the difference between a 12-hour clock and a 24-hour clock. The technical term for this difference in the theory of gauge groups is “double cover”: the 24-hour clock is a double cover of a 12-hour clock, because for every time registered on the 12-hour clock there are two possible times on the 24-hour clock. So if you are using a 12-hour clock, you have to specify a.m. or p.m. to distinguish them. This happens all over quantum physics, where such distinctions as spin up versus spin down are made by using double covers (24-hour clocks) so that it all happens automatically and you don’t have to worry about it.

Now to understand particle physics, all you need to understand is how to combine a clock with a calendar. You do understand that, right? I have to ask, because a lot of physicists think a calendar is a compact gauge group, whereas I am sure you understand that time goes on and on, not round and round, so that a calendar is a split gauge group. The Standard Model combines the two compact gauges (one day and one year) into a compact gauge group, which is called SO(3) in its 12-hour version, and SU(2) in its 24-hour version. My model combines one compact gauge and one split gauge into a split gauge group, which is called SO(2,1) in its 12-hour version, and SL(2,R) in its 24-hour version. Got it?

Oh, you want to know why I am using clocks and calendars to measure mass? Ask a particle physicist, they *always* use clocks to measure mass, because mass is energy, and energy is dual to time. Anyway, my version has two straight lines and a circle, where the standard version has three circles. My version therefore has two masses, which are the masses of the proton and the neutron. These masses are very similar, and if you use a calendar you can’t tell the difference. To tell the difference, you need a clock. A 12-hour clock. The difference is in fact almost exactly the difference between a year of 365 days, and a year of 365 days and 12 hours. Not quite exactly, which is interesting – and also means that they shout at me and say “BLAH! BLAH! BLAH! YOU ARE MAD! WE ARE RIGHT!” (See previous post)

Well, they can shout and ignore me all they like. Posterity will judge. I can explain the mass difference between proton and neutron to 3 significant figures, and they can’t even predict the order of magnitude. That is because they are trying to use a Yang-Mills theory to describe mass – which forces them to measure mass with a clock instead of a calendar. It is as though they haven’t noticed that the year has seasons, and that they are getting older every year. Anyway, this post is getting longer every minute so I’d better stop.

I have explained how to put mass into the weak interaction by extending a circle (SO(2)) to three dimensions (SO(2,1), not SO(3)). Next I’ll explain how to put mass into the strong interaction by extending a sphere (SO(3)) to four dimensions (SO(3,1), not SO(4), and most definitely not SU(3)), and how to do both at once, and create a not-very-grand unified theory by extending SO(5) to SO(5,1). Yang-Mills theory instead extends the circle to SU(2), and the sphere to SU(3), and Georgi-Glashow theory combines the two into SU(5), which then predicts proton decay and is therefore falsified by experiment. As I have explained, the error comes much earlier than this, in the 1950s, not the 1970s.

August 7, 2022 at 8:58 am |

I think the reason I don’t get the proton/neutron mass ratio exactly right is that my clock is running a little fast. It seems to be gaining about 8 minutes per day compared to the Standard Model atomic clocks. Not bad, considering I made it myself and it runs on gravity like an old-fashioned grandfather clock. And considering that the length of a day actually does vary by about 15 minutes over the course of a year.

August 7, 2022 at 10:44 am |

The other possibility is it is a systematic error in my calculation, which does not actually use the eccentricity of the Earth’s orbit. I count only the 365 wobbles due to the rotation of the Earth, and not the 2 (much larger) wobbles due to the eccentric orbit or the 12 (smaller) wobbles due to the Moon. I don’t know how to correct for these other motions, so my figure is equivalent to a circular orbit of 362.8 days, without a Moon. The eccentricity could easily introduce a correction of up to +/- 1.7%, or +/- 6 days.

August 7, 2022 at 9:21 am |

Did you notice the epicycles? The Yang-Mills extension of one circle (SO(2)) to many circles (a sphere, or SO(3)) is just epicycles. My extension goes from circles to ellipses: SO(2,1) is just a 2-dimensional Lorentz group which allows stretching and squeezing a circle into an ellipse. So I am not saying anything new here. The problem with the Standard Model of Particle Physics was first pointed out by Kepler, 400 years ago! There is nothing new under the Sun. Those who do not remember the past are condemned to repeat it. The secret is to bang the rocks together, guys.

No, the secret is to use ellipses. The absurd thing is, that Dirac actually put those ellipses into the model in 1928. But no-one, not even Dirac, has ever understood what they are there for.

August 7, 2022 at 10:25 am |

Maybe that’s a bit unfair on Dirac. The Dirac equation does actually explain how to get mass from those ellipses. What Dirac didn’t understand, and what no-one even today understands, is that those ellipses are the

sameellipses that Kepler drew attention to.August 7, 2022 at 11:34 am |

The double covers are needed because circles don’t just go round, they go round and round.

August 7, 2022 at 11:39 am |

The real “problem of mass” is the fact that straight lines aren’t straight. All theories of gravity from General Relativity onwards take this into account, but they all have different ideas of what “straight” means – if it doesn’t mean “straight”.

August 7, 2022 at 12:56 pm |

Spoiler alert.

SO(5,1) allows 5-fold symmetry of mass equations, and because an element of order 5 fixes a 2-dimensional subspace of the 6-space, there are two independent equations with a 5-fold symmetry. I’ve told you what they both are, but you may have forgotten.

In one case there are 5 neutrons, and they add up to three protons plus an electron from each generation. In the other case, there are five strange quarks, and if you add a charm quark to get the charge to be an integer, while staying in the second quark generation, you get a charge of -1 and the mass of a tau particle.

To be more precise, you get a value of 1750 +/- 35, compared to an experimental measurement of 1776.86+/-.12. Statistically, this corresponds to a “confidence level” of 50%: which in colloquial language, means a confidence level of zero, i.e. a complete lack of evidence in either direction, and you might as well just toss a coin.

Except that there is a good mathematical reason to think these numbers must be equal, and there is no such good reason to think they must be different. You don’t need to toss that coin.