About two years ago I posted a paper on the arxiv in which I argued that the correct foundation for particle physics is the real group algebra of the binary tetrahedral group, of order 24. The numerology is correct, as the standard model contains 12 elementary bosons, splitting 1+3+8, and 12 elementary fermions, that is 6 leptons and 6 quarks. The group algebra naturally splits into components which individually provide the gauge groups of the standard model, including the Lorentz group, but with one fatal flaw – the gauge group for the strong force appears in the split real form SL(3,R) instead of the compact real form SU(3). At the time, this was considered by my critics to be a deal-breaker.

But times have changed. The octions paper published last year also uses SL(3,R) for the strong force, and explains how it is possible to use a copy of U(1) to complexify the representations, but not the group itself, and argues that this is sufficient to do everything that the standard model requires. If so, then I can return to that original model, and forget all about the various other models I have worked on in the past two years. The new lamps may be bright and shiny, but they don’t contain the old magic. So let us summon the djinni and make a wish: I wish for a quantum theory of everything.

No sooner said than done. Unimaginable riches pour out of this poor little old lamp. In particular, the group that emerges is the direct product of U(1), SU(2), SL(3,R) and (U(1) x SL(2,C))/Z_2. The group that is used in the octions paper is exactly the same, except without the separate copy of U(1). So I can mix the two copies of U(1) together, and then follow the recipe in the octions paper, and the whole standard model drops out.

But why would I want to do that, and use all the ghastly complications of E_8 and its 248 dimensions, when everything I need is already in the 24-dimensional group algebra? Simplify, simplify, simplify. The fermionic part of the group algebra consists of two Dirac spinors acted on by SL(2,C), plus one weak isospinor acted on by SU(2). The Dirac spinors represent elementary particles (fermions) in the standard model, but the isospinors do not. Why not? As soon as you put any spinor or isospinor into space, you tensor with the 3-dimensional space representation, and you get the sum of a Dirac spinor and an isospinor. What is this isospinor doing? Why isn’t it a particle?

Good question. The answer, I have discovered, is because it is a neutrino. The neutrinos are the fermionic manifestation of the weak force. They appear whenever the weak force indulges itself in destroying mass in radioactive frenzy. Therefore the fermionic representation associated to the weak gauge group SU(2) cannot be anything other than a neutrino. A neutrino, therefore, is not a Dirac particle. Nor is it a Majorana particle. Nor is it a left-handed Weyl particle. It is something else entirely.

Dirac particles participate in electromagnetic interactions. That is what the Dirac spinors, Dirac algebra and quantum field theory are for. The neutrinos do not participate in electromagnetic interactions. They do not belong in the Dirac spinors, they are not acted on by the Dirac algebra, they do not participate in quantum field theory. They are something else entirely. The standard model identifies the neutrino “isospinor” representation with the left-handed part of the Dirac spinor. This confuses two different representations of the binary tetrahedral group, and is WRONG.

Undo this ghastly error, and the whole beautiful elegant and unbelievably simple theory of everything just pops out, pretty much as I explained it two years ago. Of course, I understand it much better now, so I’ll try and explain it again, without so many mistakes, and without so many irrelevant distractions. For example, the change from SU(3) to SL(3,R) is not a bug but a feature – the split form permits the introduction of five apparently independent masses for elementary particles. These five masses live in a spin 2 representation – but it is frankly absurd to interpret this representation as a spin 2 graviton, or even as a set of five gluons. Again, the separation of the real isospinor from the complex spinor entails a separation of a real scalar from a complex scalar, which implies the separation of a real (gravitational) mass from a complex (electromagnetic) mass.

What we lose is special (and general) relativity. Extraordinarily we find that there is no mixing of space with time in the quantum world. Absolutely none. Space is space and time is time and ne’er the twain shall meet. What we gain is a quantum gravity that (unlike general relativity) is consistent with the broad features of galactic and cosmological dynamics. Are you prepared to give up special relativity for this prize? You should be. You are allowed to keep the observational consequences of special relativity, including time dilation, length contraction, and mass increases. But you are not allowed to keep the group SO(3,1). You must instead use the group GL(3,R) for these purposes. This is because SO(3,1), like SU(3), is NOT a subgroup of the group algebra.

You must therefore give up all the theories of fundamental physics developed during the 20th century. ALL of them. It is not that they are especially “wrong” in practice – they describe the universe pretty well. But they are based on fundamentally wrong-headed physical principles. The first fundamental principle (of special relativity) is that time and space are different aspects of the same thing. The second fundamental principle (of quantum mechanics) is that neutrinos and matter are different aspects of the same thing. The third fundamental principle (of general relativity) is that inertia and gravity are different aspects of the same thing.

All these fundamental principles are wrong. Time is not space, and the two do not mix. Neutrinos are not matter, and the two do not mix. Inertia is not gravity, and the two do not mix.