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Yes, but what the hell is it? A polyhedron of some sort, I hope…
I haven’t looked into this much – and that’s easy because the paper that causes all this excitement hasn’t even been posted anywhere yet!
But, yes, from what I have seen and from e.g. Trnka’s slide 11-12 the idea is precisely this:
Certain very nicely behaved classes of amplitudes in certain very nicely behaved gauge field theories seem to want to be integrals of certain natural volume forms over certain polyhedra.
Slide 13 in the above set of slides (which gets repeated a few times through the whole set of slides, so I guess the author wants us to stare at that) means to tell us what specifically the “amplituhedron” is. I’d say the slide is lacking a bit of context, but I gather the upshot is that the author is claiming that for those MHV amplitudes in super-Yang-Mills theory that people have been looking at so much in recent years, he has a concrete definition of which polyhedron to use.
Of course without saying which differential form to put on that this seems to be an empty information. The “definition” of the differential forms appears from slide 27 on, I guess the general definition for the above polyhedron is that on slide 28 and following.
That’s what I know for the moment. I don’t feel super-motivated to look more into this at the moment. To me the broad statement currently is that in some highly symmetric field theory some highly symmetric scattering amplitudes happen to have some semi-nice re-expression as integrals over polyhedra. Right now I am inclined to think that of course the highly symmetric amplitudes in a highly symmetric field theory will satisfy some special properties that a random class of amplitudes in a random field theory does not. But that it only gets interesting once one has a conceptual understanding of the nice rewriting of the amplitude. Until then, I find the hype-to-result ratio here too large.
Maybe I am missing something. But when the last article that this is based on came out, I couldn’t help but noticing (here) that on just the first one and a half pages of the introduction, the reader was subjected to the following advertizement
” […] amazing physical and mathematical structures […] This has been seen dramatically […] astonishingly simple and beautiful properties […] closely tied to remarkable new structures […] It is remarkable […] extraordinary progress […] a huge amount of data has been generated […] Remarkable strategies have also been presented […] a strong resonance with the explosion of progress in the last decade […] the spectacular solution […] powerful new mathematical structures underlying the extraordinary properties […] beautiful and active areas of current research […] vigorous interactions […] involves NEW areas of mathematics […] It is both startling and exciting […] “
(Warning: possibly ill-informed opinion ahead)
Thinking about this result for a little bit, and comparing it to work of people on motivic approaches to Feynman sums etc, it seems not that surprising. In slightly more highbrow language, the amplitudes have been expressed as certain periods in some nice moduli spaces generalising flag varieties. But we suspect already that we get results of these sorts from field theories corresponding to the standard model, up to knowing that certain multizeta functions are periods.
One can also find the 1-hour talk by Nima Arkani-Hamed at SUSY 2013 conference at youtube: http://www.youtube.com/watch?v=By27M9ommJc
The paper The Amplituhedron by Nima Arkani-Hamed and Jaroslav Trnka has now been published. I’ve added that to the references.
What should we make of Fig. 1 of Strassler’s post showing the maximally supersymmetric end, where the amplituhedron lives, as involving the closest relation to advanced mathematics, along with the least close relationship to the world?
Thanks for adding the pointers!
Concering “relation to mathematics”: it’s just the familiar fact that clean mathematical structure is visible in the foundations more than in the messy realizations.
Already Newton had to make the idealization assumption that we live in a vacuum to justify his first law. In the actually observed macroscopic world, objects do not keep moving at constant speed and are not described by clean mathematics. Instead one has to assume that the clean maths of Newton’s laws holds for tiny building blocks of matter and then deduce a messy macroscopic world from that. Somebody argued that had we been intelligent octopuses instead of intelligent apes, we would have never found the mathematical laws of physics, since friction under water is so immensely higher that this idealization step for Newton’s first law would never have been guessed by anyone.
Same here with YM gauge theory. The highly supersymmetric version are more idealized and hence more amenable to mathematical description. The standard model of particle physics is already quite a bit of a mess, compared to that.
Is there a difference here though in that we’re not expecting the maximally supersymmetric end to be like “the clean maths of Newton’s laws”, i.e., true in some ideal sense? I think Strassler suggests a more subtle sense of how the more supersymmetric end can help us, via (2) analogies and (3) new features.
In sum, aren’t there slightly different idealizations going on? Like with the Ising model for magnetism, there’s no sense in which it accurately represents all inter-atomic interactions, whatever conditions I place my magnet in, whereas if I place two bodies far enough away from any others, then Newton’s Laws hold almost exactly.
The Outlook of The Amplituhedron is full of ambition!
Yes, the Ising model is an idealization which is not actually realized anywhere exactly, in contrast to Newton’s first law.
Super Yang-Mills is at least like the Ising model. But there are also arguments/speculation that it is more. If the perspective of the “IKKT matrix model” or of strings in AdS/CFT should turn out to be phenomenologically viable, then super Yang-Mills would actually be at work at the foundations of observable reality. We don’t know, but there are some arguments that it might be.
Given our recent discussions about understand gauge equivalence via groupoids, I wonder what we should make of
We have given a formulation for planar N = 4 SYM scattering amplitudes with no reference to space-time or Hilbert space, no Hamiltonians, Lagrangians or gauge redundancies, no path integrals or Feynman diagrams, no mention of “cuts”, “factorization channels”, or recursion relations.
If locality was also a reason to adopt action groupoids, then is it just the extreme degree of supersymmetry which allows
while in the usual formulation of field theory, locality and unitarity are in tension with each other, necessitating the introduction of the familiar redundancies to accommodate both, in the new picture they emerge together from positive geometry?
Hm, I should look at the article to see what exactly they mean there. Can you give me the precise page where you take the last quote from (I am sort of busy elswhere, would help me to have a page number…)
I’m sure you have better things to do, but it was p. 29.
That’s them summing up. It seems to be section 11 Locality and Unitarity from Positivity (p. 21) where they do the work.
Locality and unitarity are encoded in the positive geometry of the amplituhedron in a beautiful way. As is well-known, locality and unitarity are directly reflected in the singularity structure of the integrand for scattering amplitudes.
They develop the line in the Outlook section that just as we see with hindsight the determinism of classical mechanics emerge as derived from the picture of the least action principle understood quantum mechanically, so we need some new formulation which will allow us to see locality and unitarity emerge as derived.
But perhaps this all part of dealing with a very special case, as you once said.
I have edited expanded the entry amplituhedron a little, trying to make it just a tad more informative. It is still not anywhere close to explaining anything, but at least now semi-experts might get away with a rough idea of what’s going on (or not).
Concerning those summary statements on p. 29 of arXiv:1312.2007:
So the upshot of the “amplituhedron” story is that scattering amplitudes in SYM turn out to have a more efficient expression than Feynman’s, by fewer integrals. From what I understand, to say that “no Hamiltonians, no Lagrangians” etc. is involved is a bit misleading, for we wouldn’t be considering integrals over cells in the “amplituhedron” if we didn’t know how these relate to the Lagrangian of SYM. This seems to me like, say, pointing to the Balmer series and saying: “look, no Hamiltonian operator!” This would be pointless as the Balmer series is an efficient encoding of what fundamentally is the spectrum of a certain Hamiltonian.
Also it would seem to me that the fact that summing Feynman diagrams is intrinsically less concise than the full answer should be is not surprising or mysterious. This is almost the definition of “perturbation theory”. We know that in principle there is a single well defined non-perturbative scattering amplitude and that Feynman diagrams are away to approximate this by summing up a vast number of tiny contributions.
That by looking at this one can find ways to reorganize the perturbative expansion such as to more manifestly look like a non-perturbative result is nice, of course, but currently I don’t see why it should shatter our understanding of the foundations of QFT.
But that’s just me. Experts who read this are kindly asked to set me straight, where necessary.
The recent post
by one of the inventors of the general approach may serve to put the “amplituhedron” thing into perspective. (I have added a pointer to the entry.)
Concerning putting the hype into perspectve, Peter Woit kindly extracted (here) the following quote of Arkani-Hamed from a video of his recent talk:
So, usually I’ll get up when I talk about scattering amplitudes and give a long introduction about how spacetime is doomed, we have to find some way of thinking about quantum field theory without local evolution in space time and maybe even without a Hilbert space and blah-blah-blah. This is all very high-falutin stuff, this is stuff that Lance [ Dixon ] wouldn’t be get caught dead saying. I think none of these guys [ Dixon’s collaborators ] would ever say something that sounds so pretentious, but I have to say it, you know I have to say it, because this is the only way I can get up in the morning, and like “I suck again, OK, here we go, I’m doing it because spacetime is doomed, I swear to God, right”.
Dixon’s post is very interesting. He doesn’t mention Kreimer’s work with Connes on Hopf algebras of Feynman diagrams. Has that turned out to be unimportant, I wonder? Broadhurst is mentioned who did work with Kreimer, but this was after the 1993 paper Dixon links to.
First I suppose in these circles the Connes-Kreimer re-formulation of BPHZ-renormalization is regarded as just that: a re-formulation of something known. But probably more importantly, Connes-Kreimer’s work is all about handling Feynman diagrams (their Hopf algebra is one formedof Feynman diagrams, of course) and the work of Dixon et al. is all about not using Feynman diagrams but other methods for expressing perturbative scattering amplitudes.
I wonder whether the amplituhedron (jeez, what an ungainly term) can be obtained by some natural series of blowings up, as e.g. occurs with associahedra, or Fulton-MacPherson compactifications, etc.
Logan Maingi’s exposition is the clearest I’ve seen.
Thanks, David. Would you or anyone else be able to amplify what “tree-level” is supposed to signify (intuitively)?
In Feynman diagrams the loop order measures the number of loops, so trees are diagrams with no loops, loop order 0. But given this approach is supposed to be avoiding Feynman diagrams, I’m not sure how loops appear.
Added Arkani-Hamed and Trnka’s latest preprint, The Amplituhedron, arxiv/1312.2007 to references.
Ovidiu: it is indicative that the author does not even use LaTeX. I consider it a spam.
Arkani-Hamed and Trnka in their article refer to their topic as “remarkable” 13 times. The stock market article does it only 7 times.
What is it about a positive Grassmannian that makes it a “remarkable structure”? Is it more remarkable, than, say, a Weyl alcove, the fundamental domain in the upper half plane, or, say, associahedron? Maths is full of geometric shapes. Grassmannian spaces are taught to freshmen. If you approach a mathematician and boast that you know what a Grassmannian space is, you won’t have the intended effect.
So why is it that for discussioon of positive Grassmannians in scattering amplitudes it is emphasized so much how remarkable it all is. If one removed this advertizing, might people not recognize the result as remarkable in itself anymore?
(In fact, this is once what somebody on G+suggested: that Arkani-Hamed is including all the hyperbole in his articles as a service to his lowly readers, so that they have a chance to appreciate how great it all is ).
Ovidiu, why don’t you make yourself useful and write up some clear explanations of the amplituhedron on the nLab that manage to avoid hyperbole?
I for one would read it, if you were to write a clear mathematical explanation. It might be nice to contemplate as a purely mathematical topic, and it would be nice to compare it if possible to another known shapes such as Fulton-MacPherson compactifications of various configuration or moduli spaces.
added pointer to today’s:
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