In the Standard Model of particle physics, the strong nuclear force is “mediated” by gluons, which are usually described by the 8 Gell-Mann matrices, that generate the 8-dimensional Lie algebra su(3). Fifty years ago, Gunaydin and Gursey had the idea that, since the Lie algebra u(1) is associated to electromagnetism and the complex numbers, while the Lie algebra su(2) is associated to the weak nuclear force and the quaternions, wouldn’t it be nice if the Lie algebra su(3) and the strong force could be associated to the octonions? Fifty years on, some people are still pursuing this idea, although the mainstream abandoned it long ago.
What is the ongoing attraction of this idea, when all the evidence is that it fails, and that it failed catastrophically in the 1970s, beyond all hope of resurrection?
Let us go back to the beginning, and the classification by Hurwitz of the real division algebras. These are what they say on the tin: they are real algebras, in which you can divide by anything except zero. There are only four of them: the real numbers, the complex numbers, the quaternions and the octonions, which have dimensions 1, 2, 4 and 8 respectively. Geometrically, they are Euclidean spaces, so have orthogonal symmetry groups SO(1), SO(2), SO(4) and SO(8) respectively. Of these, SO(1) is the trivial group, while SO(2) is isomorphic to U(1), so that complex numbers work perfectly to describe electromagnetism, with a “gauge group” U(1) and its associated Lie algebra u(1). So far so good.
At the quaternion level, things get more interesting, because so(4) is a 6-dimensional algebra, but su(2) is only 3-dimensional. There are two completely different ways to embed su(2) in so(4), which are best described in terms of the symmetry groups: one corresponds to SO(3) inside SO(4), and the other corresponds to SU(2) inside SO(4). Essentially it was worked out very early on that the latter is the correct way to describe the weak force. That is, we need the embedding in which there are no fixed points in the 4-dimensional space of quaternions. Remember that: no fixed points.
Now what happens in the octonions? Now we have that so(8) is 28-dimensional, while su(3) is only 8-dimensional. It turns out that there are again just two different ways to embed su(3) in so(8). This follows for example from Dynkin’s complete classification of (maximal) embeddings of complex (simple) Lie algebras in each other, completed in the early 1950s. One of them is an embedding via so(6), or su(4), or g_2, and has fixed points. The other is a maximal embedding, with no intermediate stops, and with no fixed points. Which one do you think we need? Which one do you think they chose?
I think we need the one with no fixed points, don’t you? Why would you want fixed points? We want to be able to mix up all the 8 gluons, we don’t want any of them to be fixed. Surely? Isn’t it obvious this is the one we need?
They chose the other one. No wonder they failed. And in the ensuing 50 years, it has never occurred to anyone that they have got the wrong copy of su(3). But they have. And they still carry on trying to force their way through this route, which doesn’t go anywhere. They took a wrong turning on day one, and have never looked back. And still they keep going, after 40 years in the wilderness: the triumph of hope over experience. I could help them if they would listen to me. But they don’t, and they won’t. So they are condemned to keep on wandering in the wilderness.
Hidden assumptions 
January 2, 2024 at 7:40 pm |
I forget whether I mentioned this, but there was a paper a few months back, proposing to work with exactly that other embedding of SU(3) into SO(8). They obtained the other SU(3) as the automorphism group of a different eight-dimensional algebra, the Okubo algebra or “okubonions”. https://arxiv.org/abs/2309.17435
January 3, 2024 at 12:10 am |
Can I just ask please if you’re aware of the following “anomalies” with the strange quark (electric charge -1/3) and the omega minus (triplet of strange quarks) which has is normally swept under the carpet but has enormous implications (I’ll try to keep this brief and clear).
1. Fermi’s point theory of beta decay says a muon decays into an electron, and strange quarks decay into upquarks.
But when the W- propagator was added to Fermi’s theory, an anomaly emerged: if a muon decays into an electron, it has to become a W- boson (briefly) en route, then the corresponding Feynman diagram for beta decay of a strange QUARK decaying via a W- boson turns the strange quark into an electron! There you have quark-lepton unification. (Diagram: https://vixra.org/pdf/1111.0111v1.pdf at Fig 34 in the middle of p44.)
2. Fractional “electric charges” of quarks are artifacts like emergent from the very large vacuum polarization shielding of a pair or triplet of quarks and this is proved by the Omega Minus, which should be viewed as the Rosetta Stone for understanding everything: it is a triplet of strange quarks with the electric charge -1, so the strange quarks are all -1/3. This makes it understandably simple!
Look at the maths in this: the you have in close proximity THREE similar electric charges, which must physically produce a vacuum polarization (pairs of charged virtual fermions which align to shield the core charge within) THREE times stronger than a single charge would produce. It’s like wearing three pairs of sunglasses at once, you get three times as much filtering. Hence, the strange quarks hypothetical isolated electric charge is 3 x (-1/3) = -1, the same as the electron.
So there’s your quark-lepton unification.
The missing “shielded” energy, can be easily caculated here: 2/3 of the -1 electric charge per quark is shielded, so you see an apparent total omega minus charge of -1. The virtual particles acquire it this energy, adding to their survival time beyond the Heisenberg’s t = ℏ/E (virtual fermions only exist between UV and IR cutoff energies, which translate into distances out to ~33 fm). So this acquired electric field energy allows them to briefly behave like real particles, obeying Pauli’s exclusion principle and thus gaining a quasi nuclear shell structure (near the UV cutoff where virtual quark pairs exist) and a quasi electron structure further out (nearer the IR cutoff where electron-positron pair production occurs). Simple calculations prove this predicts particle masses: Table 1 in https://vixra.org/pdf/1408.0151v1.pdf (which is merely based on the nuclear shell model magic numbers analogy). It should also be possible to make more detailed calculations by calculating statistically average mass of the polarized vacuum particles using the easily deduced omega minus shielded electric field energy.
January 3, 2024 at 9:10 am |
[…] I tried to shorten and improve that argument: […]
January 3, 2024 at 12:59 pm |
Identification and correction of errors pertaining to efforts to try to model “fundamental particles” (muon beta decay compared to quark decay “anomaly” in standard model, omega minus strange quarks as Rosetta stone to start correcting errors): https://nige.wordpress.com/2024/01/03/unification-of-particles-and-fundamental-forces/ (it’s not possible to put it into words in a brief comment, diagrams needed, so I put post up about this). If you accept as “data” a mainstream mixup of empirical facts and flawed analysis, you’re —-ed from day one, crackpot or no crackpot.
January 3, 2024 at 1:04 pm |
[…] This post is a response to https://robwilson1.wordpress.com/2024/01/02/gluons-and-octonions/ […]