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.
