My long-promised paper on chirality and E8 has today appeared on the arXiv, and you can find it at https://arxiv.org/abs/2210.06029. Of course, they downgraded it from hep-th (where I submitted it) to gen-ph, although it is clearly a sequel to https://arxiv.org/abs/2204.05310, which is in hep-ph (where I am not allowed to submit). But they did post it immediately, without keeping it on hold for a day or two, which I suppose is progress. Anyway, the paper shows that by making making small but fundamental changes to the model in the earlier paper, it is possible to remove a number of technical objections to the model.

First, it is possible to have compact gauge groups: compactness is a basic assumption of all gauge theories of particle physics, and although one can argue that this assumption is not necessary, it is much easier to get a theory accepted if the gauge group is compact.

Second, the chirality of the model is explicit, and fundamental. It has long been considered that chirality is a major stumbling block for E8 models of fundamental physics, but I show that this is based on an incorrect mathematical definition of chirality. The correct mathematical definition has been known to physicists since 1937, but has been almost completely ignored. Unfortunately, the basic assumption of physicists, that they do not need mathematical rigour, is false.

Other features of the model are perhaps even more important:

Third, the model allows for non-inertial motion of the laboratory and/or the observer, by allowing for an 8-parameter family of copies of the Lorentz group. This opens the door to a uniform explanation of many different anomalies, including the muon g-2 anomaly, the W mass anomaly and various kaon and B-meson anomalies, that are not currently explained, as well as old anomalies such as neutrino and kaon oscillations (CP violation), that are generally considered to be adequately explained, but in my opinion are not. Four of the parameters are dimensionless, and correspond to the four fundamental dimensionless parameters of the non-inertial motion of the laboratory, that I have expounded many times and in many places (the number of days in a month, and in a year, and the angles of inclination of the Earth’s axis and the Moon’s orbit).

Fourth, the Dirac equation appears in two forms (a differential equation and a momentum-space equation) which are not mathematically equivalent (unlike in the Standard Model), and therefore the model permits a distinction between gravitational and inertial mass which I have argued extensively is both necessary and experimentally confirmed, if one cares to look at the evidence in an unbiased fashion.

Fifth, the infinitesimal version of the Dirac equation is the same for both particles and anti-particles, which implies that anti-particles have positive energy and positive mass (as is experimentally confirmed) rather than negative energy/mass (as Dirac’s original equation, still in use today, implies).

Sixth, there is an explicit restriction of this model to my discrete model, based on the binary icosahedral group, which permits a reduction of quantum field theory to a (potentially, but not necessarily, completely deterministic) model in which the ultraviolet catastrophe is avoided, and singularities such as black holes and the Big Bang do not occur. Now that the James Webb Space Telescope is producing strong evidence that the `early universe’ does not look the way the Big Bang Theory predicts, the necessity for replacing the Big Bang by something else is becoming urgent.

October 13, 2022 at 8:07 pm |

Alright! Thanks for the acknowledgement. Now it’s time to figure out what’s really going on here. 🙂

If I skim the paper without trying too hard, in section 7, I see we have a grand unified SU(5), conformal SO(2,4) ~ SU(2,2), and I know we have an extra SU(2) from elsewhere that is to give us gravity, as in Ashtekar and Woit. OK.

The first point of confusion for the casual reader might be, how we obtain 3 generations rather than just 1. Section 7 lists states for one generation. There seems to be a generation somehow associated with each of I, J, K, but – unless I missed something – the generations aren’t associated with completely independent degrees of freedom?

How this works must be tied up with the “weak SO(4)” that acts on l, I, J, K, from which the usual weak SU(2) is, I think, extracted. But SU(2)_L in the standard model acts within generations (e.g. exchanging left-handed up quark with left-handed down quark), not across generations. In terms of the standard model lagrangian, the mixing of generations ultimately arises from the yukawa terms. These terms *do* couple Weyl spinors from different generations, and in a way that takes account of their “weak isospin” charge, but the weak isospin *gauge field* does not appear in these couplings, just the Higgs field (which itself has weak isospin charge).

From a symmetry perspective, one may certainly have generation symmetries (often called family symmetries), e.g. U(3). And the family symmetry can even be unified with SU(2)_L; but normally one supposes that these two symmetries commute. It doesn’t look like that’s the case here.

These are just quick initial comments. On further reading, of this and the “octions” paper, I may understand more of what’s intended, but I make these comments now, in case there’s some obvious clarification.

October 13, 2022 at 8:09 pm |

I’ll quickly add that of course the “L” in “SU(2)_L” is not the L in your notation, it’s just “L for left” as in the usual standard model.

October 15, 2022 at 8:14 am |

The three generations are implemented in much the same way as in the octions paper. Thus the choice of one generation is effectively a choice of complex structure (I,J,K) in the quaternions. I believe, therefore, that a better model can be obtained by using this quaternion structure more effectively. That is in a separate paper that I am now working on, where I use the quaternion structure to combine the SU(2) structure of e/p/n spin with the SU(2) structure of the three generations of electron into an Sp(4) = SO(5) structure for “all” electro-weak interactions. This allows me to make a more explicit connection to the “icosion” model, in which Sp(4) acts on the Dirac spinors, and all the representations are either real or quaternionic, never complex.

The current paper, however, breaks the generation symmetry, so that the group SU(5) x SU(2,3) takes centre stage. To my mind the interesting bit is SU(2,3), which extends the Penrose SU(2,2) in a chiral way, and thus gives an intrinsic chirality to the spacetime vacuum. That seems to be why there are no right-handed neutrinos…. but I can’t say I really understand what the model is telling us here.

The splitting between fermions and bosons breaks the icosahedral symmetry to dihedral of order 6, which is the same splitting that appears in the icosion model to split neutrinos from electrons from quarks. I have no idea what this means, but it seems to be something really fundamental.

October 16, 2022 at 10:17 pm

In the octions paper it says:

“it is well understood how to combine spin eigenstates along one axis in order to get spin eigenstates along another… Here, we have the analogous ability to combine weak eigenstates of one “type” in order to get weak eigenstates of another “type” … We propose interpreting these different types as generations.”

Does that mean that e.g. third-generation states are to be regarded as superpositions of states from the first and second generations? That seems problematic from an empirical standpoint. There are bound to be situations in which you need to be able to talk about a superposition of two flavors (e.g. up and charm), as something distinct from the third flavor (in this case, top).

If we took that example literally – that |t> = a |u> + b |c> for some values of a and b – it would run up against the fact that the top quark decays *very* quickly. It would mean that a mesonic superposition like |u ubar> + |u cbar> would be equivalent to |u tbar>, and would therefore decay orders of magnitude faster than actual charm mesons do…

October 17, 2022 at 7:39 am |

I suspect my answer here is different from that of my co-authors. In my view, a “superposition” of states is a shorthand for a complicated relationship with the environment, and is not an intrinsic property of a particle. As far as mesons are concerned, I have been at some pains to point this out, for example in the case of kaons, in which I claim the supposed superposition of decay mode states is nothing of the sort, but an expression of the changing relationship with the gravitational environment.

Superposition of quarks is too complicated to discuss in this way, because the environment includes the other quarks bound into a hadron, and is essentially unknowable. Superposition of electrons is easier to discuss, because it basically does not occur. So now can you tell me *why* it doesn’t occur?

It doesn’t occur, because the symmetry is discrete, not continuous. No E8 model can ever explain that, I think, unless perhaps one chooses once and for all a complete set of commuting operators, and uses the Weyl group to describe the discrete symmetries. But in that case one might as well start with the discrete symmetries, and use the representation theory of the finite group to build the model.

This is what I do in several of my papers (e.g. the Integer Yang-Mills paper), and the present paper is intended as much as anything as an advertisement for a better model. A step on the road from the Standard Model to the Ultimate Theory of Everything.

So I might as well come clean at this point. The model in my new paper is clearly wrong. But analysing where and why it is wrong will help me to build a better model. In particular, the model does *not* translate accurately into the discrete model that I am advocating. Initially, I found this puzzling, but I think I now understand it: the structure of E8 mimics a lot of the structure of fundamental particles, but is not completely accurate. The search for an E8 “theory of everything” is based on a mirage, that hides the grains of sand that describe the fundamental structure of the quantum desert.

October 19, 2022 at 6:43 am

https://arxiv.org/abs/0706.1216 (pages 6-7) argues that coherent superpositions of electron and muon are produced in W-boson decay, but that they decohere after traveling about 10^-10 meters, so it would be difficult to test this.

You seem to be saying that this kind of superposition will not occur in some post-E8 model, because “the symmetry is discrete”. I don’t know which symmetry that would be, or why the discreteness has this effect. But meanwhile let me ask this. Will this model allow the kinds of superposition that *are* regularly created and manipulated, and which involve particles like electron, photon, proton, neutron?

October 19, 2022 at 9:04 am |

I have thought again about your comments, as they do appear to have some real force, and deserve to be considered seriously. Any real E8 model that avoids the naive counting argument (15 Weyl spinors per generation, = 45 in total, where E8 has only 32) must either use SU(2) for generation symmetry, or reduce the symmetry groups of the Standard Model. Thus the current paper and the octions paper, and the Chester-Marrani-Rios model https://arxiv.org/abs/2002.02391, all use SU(2) for generation symmetry. Therefore they use 30 Weyl spinors, so have two left over.

Your argument implies we must use SU(3) – or more likely SL(3,R) – as a generation symmetry group, at least for massive fermions. If we allow (at least one of) the neutrinos to be massless, then the superpositions of neutrino generations that undoubtedly do occur in reality can be described using SU(2) rather than SU(3). The other thing we then have to do is use colour confinement as a reason to say that the colour symmetry group is not SU(3), but only SU(2). The only reason I haven’t done this (actually I have, elsewhere) is that it sets off people’s crackpot alarms immediately, because it throws QCD out of the window.

So it disagrees with accepted theory, but it does not disagree with any experiment that I know of, or can even imagine. It uses 2 Weyl spinors for three generations of neutrinos, 6 for electrons, and 3x2x2x2=24 for quarks, making 32 in all, exactly the right number for an E8 model. At this point we have some real work to do, because we’ve got to re-write the whole of QCD. That is too difficult for me, but even at the level of symmetries and basic concepts it has some startling things to say.

First of all, the neutrino generation symmetry SU(2) and the quark colour symmetry SU(2) are *the same group*. Actually, I would allow left-handed and right-handed SU(2)s to be independent, so that the colour symmetry group is actually SO(4). This implies there are six linearly independent gluons (as in the octions paper) rather than eight (as in the SM). But it also means that the gluons (adjoint SU(3)) have been split apart into LH and RH vector representations of SU(3), and therefore into neutrino/antineutrino pairs. Hence the strong force is effectively mediated by virtual neutrinos, without the necessity for combining them into gluons.

What this does is unify the weak and strong forces in a gauge group SO(4) – or possibly Spin(4) – giving an overall splitting of SO(12,4) into SO(4) x SL(3,R) x SO(3,1) x SO(2). I suppose I am now going to have to write a paper on this model! One can put SO(4) into compact G2 and SL(3,R) into split G2, so that both are simultaneously symmetries of spinors and vectors – an essential property for the existence of particles. Then SO(3,1) x SO(2) lies outside G2 x G2, so cannot consist of particles. Instead it describes classical (non-quantised) electromagnetism. This decomposition, incidentally, is exactly the one described in the octions paper – it is only the interpretations that are different!

Now we are missing QED, so we have to get it back by mixing weak-strong SO(4) with Lorentz SO(3,1). There is an SO(3) inside compact G2 that maps onto both an SO(3) in SO(4) and an SO(3) in SO(3,1). This will be needed for electro-weak unification. There is no canonical way of mixing in the boosts of SO(3,1), or the SO(2)=U(1). Therefore if we do try to extend the model to a Lorentz-invariant model we find that the mixing parameters vary with the energy scale (as is known from experiment).

What do you think? Is it worth a shot?

October 19, 2022 at 9:28 pm

If we were doing ordinary quantum field theory, then tinkering with the symmetries would be no problem methodologically, you could just calculate the consequences and compare with experiment. It’s a lot less clear to me how to proceed, when the dynamical framework itself is unspecified.

One could choose from among the known candidates for a subquantum theory, like Bohm’s mechanics or Adler’s trace dynamics. And on the other hand, one could try to reason like a 1950s-1960s particle physicist, the era in which gauge theory was not yet figured out, and one used all kinds of heuristics in order to reason about particles.

October 19, 2022 at 10:15 pm |

Yes, I don’t really see any strong reason to invest effort in this direction at the moment. No-one will take it seriously, so I might as well put my effort into something I really believe in instead.

October 22, 2022 at 8:26 pm |

Thinking about it some more, I am not sure that your comments about superpositions are actually relevant. The “superpositions” in the oction model, and in the new model, are not simple superpositions of two well-defined particles in different generations. To get any third-generation quark out of the first two generations, you have to take all four of the first two generation quarks. Hence your argument does not directly apply to either of these models.

Part of the problem, at least, is that the mass eigenstates are not compatible with the charge eigenstates, so that it is not possible to consider the generations of one particular charge separately from the other charges.

October 23, 2022 at 12:05 am

“To get any third-generation quark out of the first two generations, you have to take all four of the first two generation quarks.”

Could you state the mathematical proposition to which this corresponds?

This equivalence is still potentially problematic, because e.g. if |b> = α |u> + β |d> + γ |s> + δ |c>, for some choice of coefficients, then electric charge isn’t conserved (since a bottom quark then has some probability of behaving like up or charm quark, with different charge).

October 23, 2022 at 8:43 am |

The point is that the space of spinors generated by the three generations of up quark is the space of all quark spinors. The reason for this (in my model, at least, maybe not in the octions model) is that there is no universal charge operator, but only separate operators for each generation.

It is easier to see what is going on for the leptons: here the *difference* between the electron and the neutrino is independent of the generation. This means you get the muon as the sum of an electron, a neutrino and an antineutrino. There is no superposition happening here at all, it is just a straightforward description of muon decay.

In my opinion, superpositions of distinct quarks are hypothetical theoretical constructions that have no experimental validity.

What happens in my model is that the three generations of a particular particle are discrete quantum states, that are distinguished *only* by the (quantised) gravitational field. The “masses” are simply a code for how the different particles interact with the particular gravitational field we happen to find ourselves in.

October 23, 2022 at 4:58 pm

But does not gravity scale in relation to the vacuum. Although the vacuum always seems to have a hovering vev of zero it is more or less energetic in scaling action. Also, any change in that action always maintains a false vacuum because of a mass gap. Therefore, there may be a relation between the mass gap and gravity and this changes with scale. Gravity has to be part of the SM or all is nought. QFT ignores gravity and just ‘switches it off’. This is an ongoing problem.

October 23, 2022 at 6:15 pm |

Mark Thomas,

Yes, absolutely, this is the problem. QFT “switches off” gravity – but in the real universe it is *impossible* to switch off gravity. I have said this thousands of times over many years, until I am blue in the face – to no avail. Physicists divide into two camps: those who ignore gravity, and those who ignore everything except gravity. Someone has to get these two opposing camps talking to each other, somehow. God knows how.

October 23, 2022 at 8:05 pm |

Leptons and gravity seems like a promising topic of discussion. We can study how your model deals with mass and gravity, without the complications of color.

As a way to orient myself in this discussion, I’m going to take the first paragraph of section 7 as my foundation. To a first approximation, this model is about the representations 5×10+10×5, of SU(5) x SU(2,3), where SU(5) is a grand unification group and SU(2,3) is an extension of the space-time conformal group. It’s standard grand unification to get one generation from “5-bar” + 10 of SU(5), so there’s certainly room for one set of leptons in those representations. The challenge for the reader is to understand how the new ideas, like dependence of “generation” on gravitational context, are implemented…

October 23, 2022 at 9:25 pm |

Yes, that is the heart of the matter, I think. I’ve left out almost all discussion of gravity from the paper, because as soon as I talk about quantum gravity, people label me a crackpot. But the quantum gravity is still there, behind the scenes. The challenge is to understand why the three generations of electrons have completely different masses (spanning 3.5 orders of magnitude), while the neutrino masses (in all three generations) are experimentally (if not theoretically) indistinguishable from zero.

The neutrinos, to me, have the same relationship to mass that the photon has to charge – they *communicate* the information of where the mass/charge is, without having any mass/charge of their own. There are two “polarisations” of photon, which are necessary in order to communicate the information of whether the charge is positive or negative. Similarly, there are three “generations” of neutrino, in order to communicate the information of which direction in the “mass plane” the particle lies. It isn’t a “plane” of course, it is a sphere, but never mind.

October 24, 2022 at 8:21 am

You may be aware that there is a long history of trying to get gravity from neutrinos. Dean Rickles’s history of early quantum gravity, “Covered in Deep Mist”, has a section on proposals that the missing energy in beta decay might be going into gravitational waves. A physicist called Gleb Wataghin is named as proposing in the 1930s that the graviton might be the neutrino. In chapter 2 of his “Lectures on Gravitation”, Feynman also considers the possibility of trying to get a gravitational potential from neutrinos. He says no for various reasons, but he concludes with a remark about a three-body force, in which the third body is much further away, mediated by spin-1 objects that might be pairs of neutrinos.

These days it is considered proven that a graviton, being a perturbation of the metric tensor, must be spin-2. So neutrino gravity is a lot rarer. Bob McElrath proposed (in “Emergent Electroweak Gravity”) that gravity comes from a neutrino condensate; this paper provided the occasion for a memorable blog post by Lubos Motl, summarizing all the reasons why it shouldn’t work. Marni Sheppeard also believed in a neutrino/gravity correspondence, as you would know. I’ll also add that in supersymmetry, the spin 1/2 goldstino can give mass to the spin 3/2 gravitino, superpartner of the graviton.

October 27, 2022 at 5:02 pm |

Yes, I have never understood why the “spin 2 graviton” is supposed to be a “fact”, without any experimental evidence whatsoever, and in the face of incontrovertible evidence that the theory that gave rise to it (general relativity) does not hold in the universe we actually live in.

October 27, 2022 at 7:24 pm |

If Lubos Motl ever wrote such a rant against my work, I should consider it a badge of honour, and a sure sign that I was on the right track.

October 27, 2022 at 8:09 pm |

Having got this paper out, and submitted to a journal, it is more or less inevitable that I have more or less immediately found something that I think is better. Not that either of them is a correct model of physics, of course, but the new one has additional features that make it more attractive.

The basic idea of the paper already submitted is that one should split E8 as A4+A4, rather than one of the more traditional splittings as D4+D4, D5+D3, E6+A2 and so on. Looking at the list of other possible splittings, there is one that stands out like a sore thumb, namely A5+A2+A1. This is because A2+A1 is just a different notation for SU(3) x SU(2), that is the gauge group of the nuclear forces.

So, this means A5 is available for a really rich theory of the structure of spacetime, with 35 dimensions compared to Lorentz’s 6, Poincare’s 10, Einstein’s 15, Penrose’s different 15, and so on. Of course, I don’t mean that spacetime itself has such large dimension – all of Lorentz, Poincare, Einstein and Penrose are quite clear that spacetime has four dimensions and that’s that. It’s a question of how you model the dynamics of spacetime. As far as I can see, 15 dimensions are not enough to model a dynamical spacetime, they are only enough to model a static spacetime. The other 20 dimensions are what is called the Riemann Curvature Tensor in General Relativity. But that is the wrong tensor to describe what is actually happening. It ignores the fact that matter *moves*.

Anyway, the particular version of A5 that is available in semi-split E_8 is SU(2,4), which contains Penrose SU(2,2) plus an SU(2) for generation symmetry, and a U(1) for charge/hypercharge or whatever, so contains Lorentz SO(1,3) plus a mass gauge SO(1,1) and everything else you could possibly desire. Without going into the strong force, colour symmetries and quarks, we have four fundamental fermions (proton, electron, muon and tau particle), exactly enough for all of nuclear/atomic/electro/magnetic/gravitational physics. All we are missing is the internal structure of the proton, which, to be honest, I’m not too worried about, since we know that protons do not decay, so we’re not actually going to encounter any quarks in nature. Oh, and the neutrinos – which obviously should not be classified as “matter”, although they usually are. They are, of course, not matter, but a force.

November 14, 2022 at 6:35 pm |

Today the paper has been rejected by the journal I sent it to. That is only to be expected. I do not object to journals rejecting my papers. What I object to is them rejecting them for specious reasons. The referee in this case did not read my paper, but read the conclusion, and claimed that I mis-represented the Banff workshop – which I was at, and the referee obviously wasn’t, because they relied on a google search for a journalist’s report for their information. I was there, I know what happened. The referee was not there, and does not know what happened. The editor apparently rejected my paper because the referee’s information via google trumps my information from personal participation. Ha! Trump! Now there’s a word with meaning for today.