The Amplituhedron and Twistors

Yesterday Nima Arkani-Hamed was here at Columbia, giving a theory seminar on the topic of the Amplituhedron, which is a characterization of the integration region in a calculation of scattering amplitudes by integrating over regions in the so-called positive Grassmannian. This is a modest advance in mathematical physics, one that for some reason a few months ago garnered a lot of hype (see here for more about this).

As seems to often be the case, the Arkani-Hamed talk was a bit bizarre as an event. Scheduled to start at noon, people soon settled in with the sandwiches provided by this seminar, and he started talking about 12:15. About an hour and a half into the talk, people were reminded that he doesn’t mind if they leave while he’s speaking. Two hours into the talk, soon after he said he was only a quarter of the way through his material, I had to leave in order to do some other things. I don’t know how long the talk actually went on. It’s too bad I didn’t get a chance to stay until the end, since he promised to then explain what the current state of progress on these calculations is.

What is being calculated are scattering amplitudes in a conformally invariant theory, with the simplest example the planar limit of tree-level amplitudes of N=4 super Yang-Mills. One wants to extend these methods to loops, to higher order terms in 1/(number of colors), and to non-conformally invariant theories like ordinary Yang-Mills (at the tree level, ordinary and N=4 super YM give the same results).

As usual, Arkani-Hamed was a clear and very engaging speaker. Also as usual though, it’s unclear why he thinks it’s a good idea to not bother trying to fit his talk into a conventional length, but just keep talking. One reason for the length was the extensive motivation section at the beginning, which had basically no connection at all to the topic of the talk. There was a lot about quantum gravity of an extremely vague sort. In a recent talk I wrote about here, he explains one reason why he does this, that he’s describing the motivation he needs to keep doing this kind of mathematical physics. I suspect another related reason is that this kind of vague argument about quantum gravity and getting rid of space-time is all the rage, so if you’re not working on the firewall paradox, you have to justify that somehow.

Once he got beyond the motivational stuff (and a complaint about BRST: “almost anytime you hear BRST, there is something formal and complicated going on”) the talk was worthwhile and I learned a fair amount. The main thing that struck me was just how much the whole story has to do with Penrose’s twistor program. Penrose developed twistors also with a quantum gravity motivation: they provide a very different set of basic variables, with usual space-time points not the fundamental objects. Of course I was aware of some of the twistor part of the amplitudes story (see for instance here), but I was unaware of the important role played by Andrew Hodges, of Penrose’s twistor group at Oxford, in these recent developments. Hodges, besides writing a fantastic biography of Alan Turing, has worked on twistor theory for about forty years, and some of his innovations have been crucial for the recent advances on gauge theory amplitudes. One example is his “twistor diagrams”, and for more about this and how other work of his has contributed to the emerging story, see his up-to-date Twistor Diagrams website. Hodges is a wonderful story of someone who didn’t follow fashion, but stuck to pursuing something truly worthwhile, it’s great that he has now been getting attention for this, as his work has become useful for more popular research programs.

For those who keep asking about interesting, promising ideas about fundamental physics to work on, twistors are something they definitely should look into. The recent amplitudes work is one specific application of thinking in twistor variables, but the whole question of how to do quantum field theory in twistor space seems to me to still be wide open. Twistor theory involves some wonderfully different ways of thinking about four-dimensional geometry, and these seem far more likely to play some role in future advances in the direction of unification than any of the tired ones (GUTs, SUSY, string theory) that have dominated the field for so long.

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5 Responses to The Amplituhedron and Twistors

  1. OMF says:


    Are you implying that a lot of what is fashionable is not very worthwhile? (I am asking a very serious question here.)

  2. J.S. says:

    I find peculiar that according to you what Hodges does is awesome but not so what Nima does, after explaining how related both things are.

  3. Peter Woit says:

    I was writing here about what I learned from Arkani-Hamed’s talk. He was the one who credited Hodges with coming up with the important ideas he was discussing (with three examples “twistor diagrams”, dual conformal symmetry, and the idea that the amplitude was a volume of a polytope). For more details of who did what, see this page at Hodges’s web-site
    which tells the story in a way quite consistent with Arkani-Hamed’s version in his talk.

    So, if you want to simplify this to “what Hodges does is awesome”, you can, but that’s not “according to me”, it’s a report of the story according to Arkani-Hamed.

    Yes, a lot of what is fashionable is not worthwhile. This is true to some extent in all subjects, certainly is true in theoretical physics. One can get into an argument about what of such currently fashionable ideas are the ones that are not worthwhile, but if you want an example, I’ve made it very clear what I think of the multiverse. No matter what your views on that, I think if one goes back and looks at the topics fashionable at any given moment in HEP theory, most of the time you’ll find that some of the fashionable topics didn’t turn out to be worthwhile, while people doing some very unfashionable things in obscurity turned out to be doing something much more worthwhile. Hodges is a good example of this.

  4. Peter Woit says:

    Thinking about both this talk and the Tegmark story, an odd parallel occurred to me. Tegmark early in his career downplayed the multiverse part of his interests, in order to make sure people took him seriously as a scientist. These days, it’s somewhat the reverse, with Arkani-Hamed using, if not the multiverse, quantum gravity research at the boundaries of science, in order to get people to take seriously his work on what they otherwise might decide was just an obscure corner of mathematical physics.

  5. J.S. says:

    To say that the multiverse mania is fashionable you have to provide evidence: what fraction of the vacua contain string theorists? and in what fraction of those are they actively discussing the multiverse?

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