added pointer to today’s:

- Livia Ferro, Tomasz Lukowski,
*Amplituhedra, and Beyond*, Topical Review invited by Journal of Physics A: Mathematical and Theoretical (arXiv:2007.04342 )

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.

]]>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?

]]>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: it is indicative that the author does not even use LaTeX. I consider it a spam.

]]>Added Arkani-Hamed and Trnka’s latest preprint, *The Amplituhedron*, arxiv/1312.2007 to references.

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.

]]>Thanks, David. Would you or anyone else be able to amplify what “tree-level” is supposed to signify (intuitively)?

]]>Logan Maingi’s exposition is the clearest I’ve seen.

]]>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.

]]>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.

]]>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.

]]>The recent post

- Lance Dixon,
*Calculating Amplitudes*, December 2013 (web)

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”.

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.

]]>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.

]]>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…)

]]>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?

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.

]]>