My joint paper with Corinne Manogue and Tevian Dray, entitled “Octions: An E_8 description of the Standard Model,” was published online today, 08-24-2022, in Journal of Mathematical Physics (Vol.63, Issue 8). It may be accessed via the link below:

DOI: 10.1063/5.0095484

It represents one view of how E8 can contain the Standard Model of Particle Physics, and incorporate three generations of fundamental fermions. It does not contain any of the more radical proposals that I have made to alter the Standard Model rather than extend it.

23 Responses to “Octions”

  1. Mitchell Porter Says:

    Did you get much feedback from the reviewers?

    • Robert A. Wilson Says:

      Generally very positive. They asked for a bit more detail on chirality (i.e how do we refute the Distler-Garibaldi arguments) and the three generations of fermions, as well as comments on non-compactness of the gauge groups, but that was about it.

  2. mitchellporter Says:

    I’ve looked again at this paper, just to get some idea of what it is saying, and also to walk the line between openness to any new ideas it introduces, and awareness of where it may be running into trouble.

    I take the essential idea to be, “division of an exceptional Lie algebra into adjoint and spinor representations” of its maximal subalgebra, followed by further division of this subalgebra into a space-time symmetry algebra and a particle symmetry algebra. And this program can be pursued for several such algebras (see Table II); the paper just happens to focus on e8(-24) as phenomenologically promising.

    So, we’re getting space-time, gauge bosons, and fermions, all from the same object. At this point I can state two standard objections to this. Of course, the Coleman-Mandula theorem is supposed to tell us that space-time symmetry and internal symmetries don’t “mix”; and the other piece of conventional wisdom is that you can’t have bosons and fermions in the same multiplet of particles, unless the symmetry is described by a “superalgebra”.

    To the extent that these obstructions really are theorems, they have life within a particular context, e.g. Coleman-Mandula pertains to the scattering matrix of a relativistic quantum field theory. Meanwhile, the studied decomposition of these Lie algebras is a mathematical fact, it’s just the labeling of the parts as space-time, boson, and fermion which is contentious.

    The challenge for the advocate (of octions and related ideas), therefore, is to find some alternative realization of these physical concepts which is different enough that the logic of the standard objections no longer applies, but also still tethered enough to the essence of the concepts, that it has some chance of applying to physical reality.

    In this regard, it might be useful to consider the *simplest* examples of “space-time + bosons + fermions” from an single Lie algebra. These would be too simple to describe our reality, but they might illustrate with maximum clarity, the anticipated difficulties that arise when one wishes to turn this into physics, as well as any opportunities to circumvent the theoretical obstructions.

    That’s my main comment.

    Also, one technical point: so(12,4) was briefly considered in part 6 of https://arxiv.org/abs/hep-th/9703123, a paper written within the framework of conventional quantum field theory, and an associated superalgebra is said to cause problems for construction of Yang-Mills theory, that don’t exist e.g. for so(11,3).

    • Robert A. Wilson Says:

      We have, of course, looked long and hard at the standard objections to such models, and we believe we have dealt with them all satisfactorily. That doesn’t mean I think it is a good model – I don’t! But it is a model that contains some interesting ideas that cannot be immediately rejected as “wrong”.

      I looked briefly at the so(11,3) paper, but I really haven’t got the time or the expertise to look at super-duper-hyper-mega-symmetric models (which seems to be the technical term in string theory for a group of order 24).

      The “obstructions” are, as you say, all a matter of how you assume the mathematics and the physics are related. I examine these assumptions carefully, and reject the ones that cause obstructions.

      I am hoping there will be a sequel to this paper, in which we are a bit more adventurous. We’ll see.

      • mitchellporter Says:

        Although Coleman-Mandula gets all the attention, to me it’s the attempt to put bosons and fermions in the same multiplet (but without using supersymmetry), that is most clearly problematic.

        For example: suppose you have two bosons in the same state, and then you act on this multi-particle state with a group element that maps them to some species of fermion, but leaves other properties unchanged. Now you have two fermions in the same state – but that violates the Pauli exclusion principle (or Fermi statistics, or the antisymmetry of fermionic wavefunctions).

        I simply don’t see a way to do this, except by removing that group element from the physical part of the theory; similar to how “graviGUT” theorists may sometimes say, regarding Coleman-Mandula, that in practice their unified symmetry is broken to a product of space-time and internal symmetries, after all.

        In the case of the “octions” paper, that could mean that the true physical symmetry is something like, SO(3,1) space-time symmetry, times SO(4) x SO(3,3) internal symmetry.

        In that case you would no longer have problems with Coleman-Mandula or with boson-vs-fermion incompatible statistics. But a critic might say (and they do say this, to graviGUT theorists who avail themselves of this loophole): in what sense does your theory actually possess the larger symmetry that you wanted?

        In the case e.g. of electroweak unification, the theory has a high-temperature phase in which the Higgs field is in a thermal state rather than a condensate, and the full electroweak symmetry is manifest. Analogously, some graviGUT theorists have claimed that their theory has a “topological” phase in which the full symmetry applies, and the breaking to a Coleman-Mandula-compliant product, is associated with the emergence of a metric. Perhaps one could also have an unbroken phase in which all particles have parameter-dependent parastatistics that can range between bosonic and fermionic, but which settles into one or the other when the big symmetry is broken. But I have never seen this possibility explored, or even suggested.

    • Robert A. Wilson Says:

      Oh, no, I think I may have confused super-duper-hyper-mega-symmetriosous with super-cali-fragilistic-expialidocious. Anyway, it doesn’t matter, both of them are fairy stories for children.

    • Robert A. Wilson Says:

      These are all valid points, that apply to all GUTs if you believe that the symmetry group is physically “real”. I don’t have any answer to them beyond the standard ones if this is the class of models you believe in. But I don’t believe these large symmetry groups are actual symmetry groups of physics. They cannot be, because quantum physics does not contain enough information to support them.

      So the question is, what is the real symmetry group, that gives the illusion of E8 symmetry? I believe it is the binary icosahedral group, and it is the McKay correspondence that is responsible for creating the illusion. I have written two papers (both on the arXiv) explaining how the binary icosahedral model works, and am writing another explaining how it relates to E8, and why the E8 symmetries are fictitious.

    • Robert A. Wilson Says:

      Ultimately, it is the hidden variable problem, that goes back to EPR in 1935. Continuous hidden variables cannot explain what quantum mechanics predicts, or what experiment reveals. But discrete hidden variables can.

    • Robert A. Wilson Says:

      I have been worried that the new version of the E8 model I’ve been working on does have a symmetry group that mixes bosons and fermions. But these “bosons” and “fermions” aren’t actually particles. The “bosons” are another name for symmetries of spacetime, and the “fermions” are another name for the vacuum. So that there is no contradiction with spin statistics. The model was built in order to explain parity violation of the weak force, which I believe it does perfectly adequately.

      But what I have now discovered is that it also contains CP-violation, in the form of a CP operator that converts the vacuum into neutrino/antineutrino pairs. As far as I can see, it does not create any other particles, only neutrinos and antineutrinos. In other words, whenever spacetime stretches and bends, according to General Relativity, what happens in quantum mechanics is the vacuum generates some neutrinos and antineutrinos.

      This fact has been obvious to me for physical reasons for several years. What is new is that I can now prove it mathematically as well. And I can confirm that the symmetry group in General Relativity is wrong – it is not SL(4,R) or GL(4,R) as is generally supposed, it is SU(2,2), as Peter Woit proposed last year. He has the correct approach to quantum gravity. It is just a pity that he won’t listen to me, because I can do the unification as well.

      Because these neutrinos are generated by the CP operator, they also cause the CP-violation of neutral kaon decay. As I have also demonstrated on physical grounds on many previous occasions. The energy for the creation of these neutrino/antineutrino pairs comes, of course, from the gravitational field, because where else could it come from? There is no “vacuum energy”, 120 orders of magnitude greater than is physically possible in the universe as a whole. It is gravitational energy. Obviously. I mean, OBVIOUSLY.

    • Robert A. Wilson Says:

      There’s a new prediction of this new E8 model – it is not compatible with neutrinoless double beta decay, so it predicts that this hypothetical reaction does not take place in our universe.

    • Robert A. Wilson Says:

      This model is totally amazing. I never expected it give so much, but it has everything. Absolutely everything. I started writing out all the particle eigenstates explicitly, for all three generations of particles and antiparticles. Just from the neutrinos, straightaway I got neutrino oscillations as a direct and immediate consequence of wave-particle duality.

      From the electrons, straightaway I got a dependence of mass on rotations of the frame of reference – again as a direct and immediate consequence of wave-particle duality. This is what got me started on this road in January 2015, when I noticed a strange coincidence between electron mass (relative to the difference between neutron and proton) and the (sine of the) angle of tilt of the Earth’s axis. Now it is no longer a strange coincidence: it is an inevitable consequence of the E8 model.

      Quantum gravity, unification, explanation of the unexplained parameters, all the anomalies – muon g-2, CP violation, W/Z mass ratio – everything. This may not be the ultimate theory I wanted, but it’s pretty damned close.

      The secret is to bang the neutrinos together.

  3. Mark Thomas Says:

    Excuse me for being an ignorant layman and maybe I should just keep my mouth shut. I know the E8 and Coleman-Mandula issue has been beaten blue. Besides a trivial mixing of internal and external spacetime there might actually be case(s) where the two could mix meaningfully in a conformal theory or is this wrong. That might mean a higher energy expression and I haven’t a clue whether the E8 could be expressed in such a conformal theory.

    • Robert A. Wilson Says:

      Mixing of internal and external spacetime symmetries is a huge issue, which as far as I can see is a no-go area for physicists. But it is really the only issue I am interested in. I don’t understand all the theoretical objections to such mixing, but I do understand the experimental evidence, which makes it abundantly clear that this mixing does actually happen in the real world, whatever the theorists say.

      • Mark Thomas Says:

        I agree. The idea of say isospin symmetry and spacetime symmetry working together is a beautiful idea. At some point how could it not be true.

    • Robert A. Wilson Says:

      Indeed. I would like to say that it is “obvious” that isospin symmetry and spacetime symmetry work together – but I have to admit it took me a few years before I saw it.

    • Robert A. Wilson Says:

      You know that story about the maths lecturer who said something was “obvious”, and started to think about it, walked out of the lecture room, and came back half an hour later and said: “yes, it is obvious”? That’s rather how I feel. It is “obvious” that isospin symmetry and spacetime symmetry are correlated – just let me think about it for ten years.

      • Mark Thomas Says:

        When a certain size star collapses and ends becoming a neutron star is that not the influence of gravity whereby protonic matter uud converts to neutronic matter udd. That looks like a instep correlation.

    • Robert A. Wilson Says:

      I would certainly agree. I don’t see how it is possible to consider the weak force and (quantum) gravity separately from each other.

    • Robert A. Wilson Says:

      Experimental evidence for the mixing of the weak gauge group with space(-time) symmetries comes from the recently reported W boson mass anomaly. There are two mixing angles, since the group is SO(3), but the standard model simply adds them together, because it insists on working in complex numbers instead of quaternions. These mixing angles occur in spacetime as (a) the angle of tilt of the Earth’s axis, and (b) the angle of inclination of the Moon’s orbit. The latter changes sufficiently that the mixing angle as measured by experiments varies by an amount significantly greater than the reported accuracy of the experiment. Hence the anomaly.

      The physical mechanism behind this is that rotating bodies lose energy in the form of neutrinos, and that these neutrinos interfere with the experiment by giving mass to (or taking mass away from) the particles being measured.
      At least, that’s how I see it. Mainstream physicists disagree, but then they can’t explain the anomaly and I can.

  4. Robert A. Wilson Says:

    I’ve been forced to learn a bit more quantum field theory (not too much, I hasten to add) in order to match this new model to the Standard Model. I already knew that physicists measure spin in the z direction of space, and was vaguely aware that they measure charge in the y direction of space. Why do they do that? I mean, really, why??

    It’s to do with chirality, of course. Any physicist will say that confidently, without necessarily knowing what it means. I don’t pretend to know what it means, but I suspect it doesn’t actually mean anything. The E8 model is big enough that we can choose the z direction for spin *independently* of the y direction for charge. That is why we can model three generations of electrons instead of one.

    You see, there is no intrinsic relationship between spin direction and charge direction. It is only the gravitational field that makes it appear so. And it makes it appear that they are almost but not quite orthogonal. Why? Because the mass of the electron is almost but not quite zero. All the mathematics of spin and charge is the mathematics of the electron and the proton. It does not accommodate the muon (“who ordered that?”).

    The E8 model does. And it makes it clear that the spin direction is not related to the three perpendicular charge directions. Any apparent relationship is entirely due to the non-inertial motion of the laboratory in which these measurements are made. It is the spinning of the laboratory that matters, not the spinning of the elementary particles. And it is the gravitational field that causes the high-energy muons to fall back to the lowest energy (electron) state. Not the weak force, as the Standard Model would have us believe.

    The (left-handed) weak force causes neutrons to decay into electrons and protons and (anti)neutrinos. The (right-handed) gravitational force causes muons and tau particles to decay to electrons plus various neutrinos and antineutrinos. It is only because particle physicists cannot distinguish their left hands from their anti-right hands that they don’t understand this.

  5. Robert A. Wilson Says:

    One of mitchellporter’s links recently sent me back to a paper written in 1937, in which the existence of four types of particles (Weyl spinors) was first pointed out. The reference that sent me back there was a paper written in 2000, that moans “Why has this fact, known since 1937, been largely ignored?” I echo this moan, as loud as I can. Only two types of Weyl spinors are recognised in the standard model, called left-handed and right-handed. These are not enough to describe the elementary particles.

    The other two should perhaps be called left-footed and right-footed. It is the feet that keep the particles on the ground, if you see what I mean – they are what give them mass and therefore weight (or do I mean give them weight and therefore mass?). The neutrinos don’t have feet, only hands (or should I call them wings?).

    The problem with the standard model is that a rotation through 180 degrees takes the right hand to the left foot, and the left hand to the right foot. Quantum mechanics thinks this is a rotation through 360 degrees, and therefore thinks that the right hand *is* the left foot. What utter balderdash. I explained all this in a paper 18 months ago, that was instantly rejected by everybody. But the result is that the entire theory is topsy-turvy, with feet in the air and hands on the ground.

    Well, I suppose if you have been taught since you were a baby that a right hand is the same as a left foot then you might as well suck your toes as suck your thumb. Grow up!

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