Today I gave a talk via Zoom at the Algebra, Particles and Quantum Theory seminar series organized by Nichol Furey. The slides from the talk are here (I gather the talk was recorded and video might be available at some point).
This talk emphasized explaining the twistor geometry, integrating some of what I’ve learned over the last few months thinking about the “twistor $P^1$” (see here). For instance, one way to think of the basic object of Euclidean twistor theory is as $\mathbf {CP}^3$, together with a different real structure (the twistor real structure) than the usual one given by conjugation of complex coordinates. One thing that struck me while writing up these slides is that the Euclidean twistor story gets a lot of mileage out of identifying $\mathbf C^2$ and $\mathbf H$, together with taking as fundamental $\mathbf H^2$. It has always seemed possible that the octonions might have a role to play here; one way into that might be to think about identifying $\mathbf H^2$ with $\mathbf O$ in some analogous way to the $\mathbf C,\mathbf H$ story.
There’s nothing new here about any of the many open questions of how to use this geometrical framework to get a fully worked out dynamics that would include the Standard Model and gravity. After a detour into number theory and hyper-Kähler geometry for several months, I’m now getting back to thinking about those questions.
Update: Video of the talk is now available here.
“one way into that might be to think about identifying H^2 with O in some analogous way to the C, H story.”
People have thought about this before, and it seems that the end result is always some variant of the Cayley-Dickson construction.
Very nice, your speculations seem noticeably stronger than in your earlier public writings on the topic. Also nice to see Octonions making a debut in your work, was Prof Baez an influence there?
How was the talk received?
Aula,
More specifically what I’m wondering about is more geometrical, whether there’s an octonionic analog of the relation between the twistor P^1 and the quaternions. That one is identifying C^4 with H^2 gives the twistor real structure on CP^3, what extra structure on CP^3 does one get from identifying H^2 with the octonions?
Bertie,
The more I think about this framework for thinking about fundamental physics, the more convincing I find it. You can interpret that as a reason others should take this seriously, or as evidence that I’m becoming increasingly delusional…
The reaction to the talk seemed to me quite positive, there were good questions.
John Baez’s article on octonions in the Bulletin of the AMS is one of the few things I’ve read about them that gave me some insight especially into the geometry. I plan to soon reread it, see if there’s anything there or elsewhere about the question I mentioned in the previous comment.
Speculations on twistors and fundamental theoretical physics reminds me this lectures by Atiyah: https://arxiv.org/pdf/1009.4827.pdf
On octonions, I am also reminded of this informal lecture given by Atiyah some years ago: https://www.youtube.com/watch?v=lp2cXnNt0Xs
I wonder if he ever published anything more substantial about this…
Jackiw-Teitelboim,
Thanks for those references, which certainly resonate with me, for instance the story of the twistor P^1 fits perfectly with Atiyah’s
“In the big picture, physics is at infinity, and number theory at the finite
points.”
Atiyah was always a huge inspiration to me, and what he has to say about his speculative dreams relating number theory and physics is fascinating. It’s frustrating not to have more detail from him about this put down to paper. Also frustrating is what happened in the last few years of his life, when he apparently lost the ability to see needed details. The last time I saw him (2016 in Heidelberg), at one point he was talking about how the octonions were central for understanding gravity, but he didn’t seem to have a really coherent argument one could follow (at least I couldn’t). Likely what’s in the video you link to may be the best source for trying to understand what he had in mind.
I watched Atiyah’s 2010 video in its entirety, and I would like, if I may, to comment on what he says about octonions, since I am in fact an acknowledged expert on octonions. If you don’t like it, then please ask Nichol Furey or John Baez, who are also acknowledged experts on the octonions, and will back me up. Atiyah did no work on octonions until his mid-70s, and he simply doesn’t understand what he is talking about. It is clear in the work of Nichol Furey, and Geoffrey Dixon, and Corinne Manogue and Tevian Dray, and many others, that EM is complex, the weak force is quaternionic, and the strong force is octonionic. These are all people who have worked seriously, rigorously and productively in the area, which Atiyah never did. Atiyah links EM to real numbers, the weak force to complex numbers, the strong force to quaternions, and gravity to octonions. This is just nonsense.
Robert A. Wilson,
I just took a longer look at the Atiyah video and have to agree that on gravity/octonions there’s not much at all there, just like some years later in Heidelberg there didn’t seem to be much to the idea, beyond a vague “hunch”. He does admit that the idea is “vague and philosophical”. I agree that the connections strong interactions/octonions studied by Nichol Furey and others seem much more substantive. Beyond these, one might somehow speculate that triality has something to do with three generations, but I don’t know that anyone has anything specific about that.
The explanation Atiyah gave in that talk of the Tits-Freudenthal magic square was really nice, but there’s no clear connection to physics.
On this kind of thing, Baez has written extensively and in detail, I recommend his series of blog posts that start here
https://golem.ph.utexas.edu/category/2020/07/octonions_and_the_standard_mod.html
One aspect of the octonions that has often showed up in speculations about relations to physics is that the Lie algebra sl(2,O) in some sense is so(9,1), the Lie algebra of the 10d Lorentz rotations. This is of some encouragement to the idea of relating octonions to the 10d superstring, but I’ve never seen anything much coming out of that.
The Tits-Freudenthal magic square is undoubtedly very beautiful, but I wouldn’t call Atiyah’s remarks an “explanation”. What is magic about the square is that it is symmetric, and that is a very deep and difficult theorem of Tits. For an explanation, you can’t do better than the Barton-Sudbery paper, but for deep insight, in a vaguely unintelligible way, I think Vinberg has the edge. Atiyah’s remarks do not seem to me to amount to much more than the observation that there are four dimensions of space, four fundamental forces, and four rows and columns in the magic square, but perhaps I am missing something. I agree with him that there is something very special about the number four, but I don’t find anything deep in what he says here – although in other parts of his talk I think there is real depth.
The idea that triality is connected to the three generations is a key part of Garrett Lisi’s infamous “Exceptionally simple theory of everything”, that few people take seriously these days. It is a seductive idea, but it hasn’t worked out. You can make your own decision whether to work on it, but I lost interest in it many years ago. I don’t know why you qualify the isomorphism between sl(2,O) and so(9,1) with the words “in some sense”: it is a perfectly valid and rigorous mathematical result, well understood in the 1930/40s by von Neumann, Jordan, Schafer, Chevalley, etc. But they quickly abandoned the idea that it had anything to do with physics, and I think this conclusion is amply supported by your suggestion that it might have something to do with string theory!
Robert A. Wilson,
I would like to take issue with “It is clear in the work of Nichol Furey, and Geoffrey Dixon, and Corinne Manogue and Tevian Dray, and many others, that […] the strong force is octonionic” [my italics].
In section 5 of the nice article “Quark structure and octonions”, by Günaydin and Gürsey (J. Math. Phys. 14, 1973) it reads: “Since G2 has only real representations, only real representations of SU(3) can occur in the representations of G2.”
So, for one thing, the G2 automorphism group of the octonions, cannot naturally explain, say, the (physical) difference between the quark triplets [3] and [3*].
John Fredsted
Günaydin and Gürsey also demonstrated that the complexified octonions (S = C ⊗ O) give rise to an SU(3) with respect to which S itself (which is a spinor space – see for example, Conway and Sloane Sphere Packings, if you are dismissive of the 4 names listed above) transforms as a singlet, anti singlet, triplet, and anti triplet. This barely scratches the surface.
Cheers, Geoffrey Dixon
John Fredsted,
I happen to agree with you, but I was not discussing my own opinions, I was discussing the work of others, so that *if* you believe the octonions are useful in physics, *then* they must show themselves in the strong force. If they do not show themselves there, as you seem to be saying, then I don’t see how you can believe in the usefulness of octonions in quantum gravity – which seems to have been Atiyah’s point of view, and still has some currency today. One can argue about lots of this – but not on Peter Woit’s blog – but I agree with Peter that both the weak force and gravity are quaternionic, quite contrary to what Atiyah was saying.
All,
Enough about the octonions and gravity. From all I can tell, there wasn’t a serious Atiyah proposal about gravity. About SU(3) and octonions, I’m just starting to think about whether this fits into the twistor geometry I’m studying. Pointers to previous work on this continue to be appreciated and I’ll look into them.
Perimeter hosted some online talks about “Octonions and the Standard Model” last year; the recordings are on PIRSA.
If you want to relate octonions directly to twistors, you need the split octonions in order to get the right symmetry groups. Then you get SO(4,2) and SO(3,3) and the corresponding spin groups in the isotopy groups, and SU(2,1) and SL(3,R) as the possible real forms of SU(3) in the automorphism group. Physicists tell me they are not interested in non-compact real forms. Various of the people already mentioned have their various ways of dealing with this.
If you are interested in links to number theory, you’ll want discrete versions of octonions, which are discussed in the Conway/Sloane book to some extent, but you probably want to look at Conway/Smith “On quaternions and octonions” for more. There is also a section of my book “The finite simple groups” that goes beyond what I have seen elsewhere. Incidentally, ‘t Hooft’s latest paper arxiv:2202.05367 asks for integer versions of SU(3) for describing physics at the Planck scale. A list of all such groups is in Blichfeldt’s “Finite collineation groups” (1917), and an argument for a specific choice is in arxiv:2202.08263.
I don’t know anything about the connection of exceptional groups to physics, although some claims on E8 have been debunked by Distler and Garibaldi:
Distler, Jacques; Garibaldi, Skip
There is no “theory of everything” inside E8.
Comm. Math. Phys. 298 (2010), no. 2, 419–436.
The symmetry of the magic square is not difficult from the point of view of
Deligne, Pierre; Gross, Benedict
On the exceptional series, and its descendants.
C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 877–881.
which also extends the magic square of Lie algebras to a magic triangle of Lie groups.
… a glass of pure wine to facilitate the debate…
– Octonions are non-associative (or more exactly anti-associative)
and so far nobody knows physics with such property.
– Complex Quaternions, (a.k.a. Weyl spinors) C x H — isomorphic but not the same as C^4 or H^2 — are exactly what one needs to include SM and gravity under one roof. Twistors are basically spinors enriched by projective geometry point of view.
– E8 ~ SU(5) x SU(5) ~ [SU(2) x SU(3)] x [SU(2) x SU(3)] — doubly GUT 🙂 — has to be also part of the final solution, as spinor geometry is 8D, and E8 is the most exceptional Lie group describing “the-most-ever-conceivable-still-admissible-eight-dimensional-something”.
It is simply not restricted just to Octonions.
(Distler, Jacques; Garibaldi, Skip’s argumentation is correct only in respect to Lisi’s “theory of everything” but not universally).