For the past few months I’ve been working on writing up some ideas I’m quite excited about, and the pandemic has helped move things along by removing distractions and forcing me to mostly stay home. There’s now something written that I’d like to publicize, a draft manuscript entitled Twistor Geometry and the Standard Model in Euclidean Space, which at some point soon I’ll put on the arXiv. My long experience with both hype about unification in physics as well as theorist’s huge capacity for self-delusion on the topic of their own ideas makes me wary, but I’m very optimistic that these ideas are a significant step forward on the unification front. I believe they provide a remarkable possibility for how internal and space-time symmetries become integrated at short distances, without the usual problem of introducing a host of new degrees of freedom.

Twistor theory has a long history going back to the 1960s, and it is such a beautiful idea that there always has been a good argument that there is something very right about it. But it never seemed to have any obvious connection to the Standard Model and its pattern of internal symmetries. The main idea I’m writing about is that one can get such a connection, as long as one looks at what is happening not just in Minkowski space, but also in Euclidean space. One of the wonderful things about twistor theory is that it includes both Minkowski and Euclidean space as real slices of a complex, holomorphic, geometry. The points in these spaces are best understood as complex lines in another space, projective twistor space. It is on projective twistor space that the internal symmetries of the Standard Model become visible.

The draft paper contains the details, but I should make clear what some of the arguments are for taking this seriously:

- Unlike other ideas about unification out there, it’s beautiful. The failure of string theory unification has caused a backlash against the idea of using beauty as a criterion for judging unification proposals. I won’t repeat here my usual rant about this. As an example of what I mean about “beauty”, the fundamental spinor degree of freedom appears here tautologically: a point is by definition exactly the $\mathbf C^2$ spinor degree of freedom at that point.
- Conformal invariance is built-in. The simplest and most highly symmetric possibility for what fundamental physics does at short distances is that it’s conformally invariant. In twistor geometry, conformal invariance is a basic property, realized in a simple way, by the linear $SL(4,\mathbf C)$ group action on the twistor space $\mathbf C^4$. This is a complex group action with real forms $SU(2,2)$ (Minkowski) and $SL(2,\mathbf H)$ (Euclidean).
- The electroweak $SU(2)$ is inherently chiral. For many other ideas about unification, it’s hard to get chiral interactions. In twistor theory one problem has always been the inherent chiral nature of the theory. Here this becomes not a problem but a solution.

At the same time I should also make clear that what I’m describing here is very incomplete. Two of the main problems are:

- The degrees of freedom naturally live not on space-time but on projective twistor space $PT$, with space-time points complex projective lines in $PT$. Standard quantum field theory with fields parametrized by space-time points doesn’t apply and how to work instead on $PT$ is unclear. There has been some work on formulating QFT on $PT$ as a holomorphic Chern-Simons theory, and perhaps that work can be applied here.
- There is no idea for where generations come from. Instead of $PT$ perhaps the theory should be formulated on $S^7$ (space of unit length twistors) and other aspects of the geometry there exploited. In some sense, the incarnations of twistors as four complex number or two quaternions are getting used, but maybe the octonions are relevant.

What I think is probably most important here is that this picture gives a new and compelling idea about how internal and space-time symmetries are related. The conventional argument has always been that the Coleman-Mandula no-go theorem says you can’t combine internal and space-time symmetries in a non-trivial way. Coleman-Mandula does not seem to apply here: these symmetries live on $PT$, not space-time. To really show that this is all consistent, one needs a full theory formulated on $PT$, but I don’t see a Coleman-Mandula argument that a non-trivial such thing can’t exist.

What is most bizarre about this proposal is the way in which, by going to Euclidean space-time, you change what is a space-time and what is an internal symmetry. The argument (see a recent posting) is that, formulated in Euclidean space, the 4d Euclidean symmetry is broken to 3d Euclidean symmetry by the very definition of the theory’s state space, and one of the 4d $SU(2)$s give an internal symmetry, not just analytic continuation of the Minkowski boost symmetry. There is still a lot about how this works I don’t understand, but I don’t see anything inconsistent, i.e. any obstruction to things working out this way. If the identification of the direction of the Higgs field with a choice of imaginary time direction makes sense, perhaps a full theory will give Higgs physics in some way observably different from the usual Standard Model.

One thing not discussed in this paper is gravity. Twistor geometry can also describe curved space-times and gravitational degrees of freedom, and since the beginning, there have been attempts to use it to get a quantum theory of gravity. Perhaps the new ideas described here, including especially the Euclidean point of view with its breaking of Euclidean rotational invariance, will indicate some new way forward for a twistor-based quantum gravity.

**Bonus (but related) links:** For the last few months the CMSA at Harvard has been hosting a Math-Science Literature Lecture Series of talks. Many worth watching, but one in particular features Simon Donaldson discussing *The ADHM construction of Yang-Mills instantons* (video here, slides here). This discusses the Euclidean version of the twistor story, in the context it was used back in the late 1970s to relate solutions of the instanton equations to holomorphic bundles.

**Update**: After looking through the literature, I’ve decided to add some more comments about gravity to the draft paper. The chiral nature of twistor geometry fits naturally with a long tradition going back to Plebanski and Ashtekar of formulating gravity theories using just the self-dual part of the spin connection. For a recent discussion of the sort of gravity theory that appears naturally here, see Kirill Krasnov’s Self-Dual Gravity. For a discussion of the relation of this to twistors, see Yannick Herfray’s Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity.

How do you get around the related problems that (Penrose) twistors are inherently massless & on shell?

Warren Siegel,

This is meant as a proposal for short-distance physics, conformally-invariant physics. What’s missing is how to do gauge theory on PT and how to give dynamics to the Higgs, providing the interactions that would give you an effective field theory at longer distances that would have masses.

I donâ€™t have the background to evaluate your paper. Still, I am impressed that you have put it out to allow all those whom you have criticized to take their shots at you. Also, does your theory predict anything and how can it be experimentally verified?

John C. Rodney,

At this point in my life, the last thing I want to waste my time on is dumb arguments about these ideas, string theory, testability, etc. One could for instance argue that what I’m discussing predicts four space-time dimensions, only SU(2), SU(3) and U(1) internal symmetries, spontaneous breaking of electroweak symmetry, etc. For a truly completely convincing prediction, what you want is a calculation of one of the Standard Model parameters, or, better, some prediction from a model based on these ideas that is different than what the Standard Model predicts. There’s still a lot missing here, with perhaps the most important idea missing being an idea about where different generations of fermions come from.

String theorists have argued that it doesn’t matter how bad a unified theory string theory provides, that it has to be compared to other ideas about unification. I don’t think that there’s a serious argument that string theory unification based on the multiverse, the swampland or whatever can be compared favorably to the ideas presented here.

I’m happy to engage anyone in serious discussion of these ideas, but string theory really is off-topic.

Hi Peter,

There are many unification models out there. Even I have one (easily found on arXiv) — using category theory, it allows you to discuss fermion generations, automatically includes gravity, works only in 4D, and has potential to circumvent Coleman-Mandula. And some other stuff. So the lack of unification proposals, with various interesting properties, is not the problem.

The problem is lack of peer interest.

Namely, until someone demonstrates that one of all those models out there actually reduces the number of SM free parameters, or predicts something outside the SM, it’s all nothing more than just fascinating math. The majority of researchers will not bother to pay any attention to unification models. This is quite unfortunate, and IMO a bit sad, but I don’t see any other way to attract a bit of attention and gain interest from other researchers in the field.

Even if one explicitly demonstrates that some model reduces the number of free parameters, it is tough to convince people. A typical example is the noncommutative SM, developed by Chamseddine, Connes and their collaborators. That’s also based on some intriguing and fascinating math, and has some provocative properties regarding unification of interactions. Does the hep-th community pay any attention to it? Not much, really.

I share your point of view that proposing unification models and studying them is a worthwhile endeavour, but it appears that any such model requires a substantial number of people to drive the research in order to flesh out a convincing prediction. But attention of other researchers can apparently be obtained only if one already has a prediction. And thus you reach Catch 22, having a nontrivial manpower-problem to move the idea off the ground.

Math is always beautiful, various algebraic structures have some captivating properties, and offer tempting ways to explain various properties of nature that we do not yet understand. But the lack of collective peer interest in any such structure is a warning to all of us pursuing this research area — physics requires *more* than math.

Best, đź™‚

Marko

Since this is your (pre-?)preprint’s debut, I’m guessing the preliminary answer is “no”, but…

Have you discussed this with or gotten feedback from others with deep interest in twistors? You wrote some very nice things about Andrew Hodges a while back, and he seems to be among the outstanding proponents of Penrose’s invention, and something of a guru.

Not that anyone’s opinion matters ultimately, but what constructive assessments you receive will be of interest to readers of this blog (even those who can’t understand ~95% of the above post).

I share others’ interest in hearing more about any observational consequences of these ideas, when the time comes.

Best of luck!

LMMI

vmarko,

The main problem right now is that the SM is too good. Traditionally the way the field made progress was by focusing on explaining experimental results that contradicted the best available model. At this point all there is of that kind is arguably dark matter, where you have an extremely small amount of relevant info and a huge number of people working on it. In reaction to this difficult situation, most of the field has essentially abandoned the problem, which is a rational thing to do.

The only reason to keep working on this is if you see some possible way forward. To me there have always been aspects of the SM that don’t quite fit together (the issue of Wick rotation and spinors is an example), and new things to learn about mathematical ideas that somehow relate to the SM, giving possible new ways of thinking about it. I’ve spent 40 years doing this, learned a huge amount, enjoyed myself, but until recently never felt that I’d come upon a set of ideas that fit together in a really convincing way. I’ve now tried to write up what I do understand, there are lots of new obvious questions and directions to pursue that I don’t understand, plenty to keep me busy. I’d like to think that others will sooner or later find something in these ideas that resonates with them, we’ll see…

LMMI,

I did start writing people about this a while before making this public. Not a lot of reaction, partly I think because this is far from what people are used to thinking about, partly because it’s August. There have been a small number of extremely helpful responses, of the form: “I’m not understanding this particular point”, making it clear where I need to do more work to clarify what it going on and explain it. I”m working now on expanding parts of what I’ve written, will write more on the blog once that is done.

This feels to me like the story relating the standard model to the octonions. In both cases, there is a beautiful mathematical structure that happens to have the standard model gauge group sitting inside it, maybe even acting in a somewhat suggestive way. It feels to me like the group theory version of the Strong Law of Small Numbers, as in “there aren’t enough small Lie groups to meet the many demands made of them.”

Anonymous,

There may very well be a role for the octonions, twistor space is conventionally thought of as four complex numbers or two quaternions, perhaps the octonionic structure can be exploited.

But note that what I’m discussing is not just an algebraic framework for internal symmetries, but is based on twistor geometry, which is normally thought of as explaining space-time symmetries. It is putting internal and space-time geometry together which I think is what makes these ideas so interesting.

Peter,

As you know, putting fermions on a lattice is a mess. Does your work give any clues as to how to do that? My gut feeling is that any theory that cannot be cleanly discretized is missing something.

Have you sought out comments from John Baez? This is the sort of thing he has been publicly musing about for decades, and I would think John would either have some interesting ideas as to how to move forward or, perhaps, some ideas as to why your approach won’t work (if the latter is true, better to learn earlier than later!).

And please continue to keep us all informed: one of us may come up with some useful idea.

Best of luck!

Dave Miller in Sacramento

Dave Miller,

Actually, thinking about fermions on the lattice a very long time ago is what got me interested in this whole subject of the geometry of spinors. Euclidean lattice gauge theory is beautifully adapted to the geometry of connections and curvature, one would like something similar for spinors.

One problem with twistor methods and the lattice is that they typically very much exploit the properties of holomorphic functions, and it seems hard to do that in a discretized theory.

Someday maybe I’ll get back to thinking about the lattice and spinors. What is natural on the lattice is not spinors but differential forms, and those give you Kogut-Susskind fermions. The Euclidean twistor picture says you need to take into account at each point in 4d the fiber above it, which you can think of as projective (half)-spinors, or orthogonal complex structures. This gives you what you need to deal with spinors. Maybe there is some way to exploit this on the lattice.

As always, John Baez has been helpful…

I think there is a misprint at the end of section 4.1: surely you mean Spin(2,2) = SL(2,R) x SL(2,R), and not Spin(3,3)?

There is of course a well-known translation from twistors to octonions, which may throw up new insights. If you take the real split octonion algebra, and fix a choice of complex subalgebra, then you get SU(2,2) acting by (left, say) multiplications, and SO(2,4) acting by bi-multiplications. If you intersect with the automorphism group, you get SU(1,2), whose compact part is the EW gauge group U(2).

If I had a model like this that only covered one generation of fermions, I’d be wondering what role the choice of complex structure within the quaternions plays in all this. I take it this is what you are hinting at by suggesting extending to Spin(7) or Spin(8)? The appropriate real forms would seem to be Spin(3,4) and Spin(4,4), but I’d be worried about the group getting too big, and would wonder what could be done by restricting to Spin(3,3) instead.

“One problem with twistor methods and the lattice is that they typically very much exploit the properties of holomorphic functions, and it seems hard to do that in a discretized theory.”

That might depend on which properties of holomorphic functions you are interested in. In the last decade a theory of univariate holomorphic functions on planar graphs has been developed. This was motivated by questions from statistical mechanics, and does e.g. connect to conformal field theory.

The first steps were taken by Smirnov but now many others are working on this as well.

A survey talk by Smirnov

https://mta.videotorium.hu/en/recordings/2382/discrete-complex-analysis-and-probability

Here you can find further references

https://link.springer.com/chapter/10.1007/978-3-662-50447-5_2

Pingback: New preprint | Hidden assumptions