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A news has recently became viral that physicists have discovered a geometrical object that simplifies a lot our models quantum physics. For an outsider like me, it is difficult to actually understand the significance of that finding.

Is it actually a new model that literaly makes every quantum physics book obsolete - or is it just a tool for a very specific calculation or effect that barely change everything else on the field?

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@dj_mummy: hep-th/1212.5605 is the latest (comprehensive) paper put out by the people working on these Grassmannian methods. I've seen a talk where Arkani-Hamed claimed to have made some progress since then but it's not published yet. –  Vibert Sep 18 '13 at 6:08
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For a readable description try simonsfoundation.org/quanta/… –  anna v Sep 19 '13 at 15:46
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Amplituhedron on Wikipedia and Mathoverflow. –  Qmechanic Sep 24 '13 at 14:50
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Related: What is the amplituhedron? and The amplituhedron minus the physics at MathOverflow. –  Emilio Pisanty Sep 27 '13 at 23:16
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Nima's talk @ SUSY2013 on Youtube. –  Qmechanic Jan 4 at 22:08
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5 Answers 5

There was a presentation of the idea at SUSY2013 by Nima Arkani-Hamed which is available on video at http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html

The amplituhedron is a buzzword for a description of a way to solve maximally supersymmetric (i.e. N=4) Yang-Mills theory in 4 dimensions. Ordinary Yang-Mills theory is a generalization of quantum gauge field theories which include electrodynamics and quantum chromodynamics. The supersymmetric extensions have not been found in nature so far.

The usual way to calculate scattering amplitudes in quantum field theory is by adding together the effects of many Feynman diagrams, but the number and complexity of diagrams increases rapidly as the number of loops increases and if the coupling is strong the sum converges slowly making it difficult to do accurate calculations.

The new solution for Super Yang-Mills uses the observation that the theory has a superconformal invariance in space-time and another dual superconformal invariance in momentum space. This constrains the form that the scattering amplitudes can take since they must be a representation of these symmetries. There are further constraints imposed by requirements of locality and unitarity and all these constraints together are sufficient to construct the scattering amplitudes in the planar limit without doing the sum over Feynman diagrams. The mathematical tools needed are twistors and grassmanians. The answer for each scattering amplitude takes the form of a volume of a high dimensional polytope defined by the positivity of grassmanians, hence the name amplituhedron.

The first thing to say about this is that so far it is only applicable to the planar limit of one specific quantum field theory and it is not one encountered in nature. It is therefore very premature to say that this makes conventional quantum field theory obsolete. Some parts of the theory can be generalised to more physical models such as QCD but only for the tree diagrams and the planar limit. There is some hope that the ideas can be broadened beyond the planar limit but that may be a long way off.

On its own the theory is very interesting but of limited use. The real excitement is in the idea that it extends in some way to theories which could be physical. Some progress has been made towards applying it in maximal supergravity theories, i.e. N=8 sugra in four dimensions. This is possible because of the observation that this theory is in some sense the square of the N=4 super Yang Mills theory. At one time (about 1980) N=8 SUGRA was considered a candidate theory of everything until it was noticed that its gauge group is too small and it does not have chiral fermions or room for symmetry breaking. Now it is just considered to be another toy model, albeit a very sophisticated one with gravity, gauge fields and matter in 4 dimensions. If it can be solved in terms of something like an amplituhedron it would be an even bigger breakthrough but it would still be unphysical.

The bigger hope then is that superstring theory also has enough supersymmetry for a similar idea to work. This would presumably require superstring theory to have the same dual superconformal symmetry as super Yang Mills, or some other even more elaborate infinite dimensional symmetry. Nothing like that is currently known.

Part of the story of the amplituhedron is the idea that space, time, locality and unitarity are emergent. This is exciting because people have always speculated that some of these things may be emergent in theories of quantum gravity. In my opinion it is too strong to call this emergence. Emergence of space-time implies that space and time are approximate and there are places such as a black hole singularity where they cease to be a smooth manifold. The amplituhedron does not give you this. I think it is more accurate to say that with the amplituhedron space-time is derived rather than emergent. It is possible that true emergence may be a feature in a wider generalisation of the theory especially if it can be applied to quantum gravity where emergence is expected to be a feature. Having space-time and unitarity as a derived concept may be a step towards emergence but it is not the same thing.

For what my opinion is worth I do think that this new way of looking at quantum field theories will turn out to something that generalises to something that is really part of nature. I have advocated the idea that string theory has very large symmetries in the form of necklace algebras so these ideas seem on the right track to me. However I do think that many more advances will be required to work up from super yang mills to sugra and then string theory. They will have to find a way to go beyond the planar limit, generalise to higher dimensions, include gravity and identify the relevant symmetries for string theory. Then there is just the little issue of relating the result to reality. It could be a long road.

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Plato said it 2500 years ago (tongue in cheek) en.wikipedia.org/wiki/List_of_Greek_phrases ἀεὶ ὁ θεὸς γεωμετρεῖ Aei ho theos geōmetreî. "God always geometrizes", Plato –  anna v Sep 19 '13 at 5:03
    
@Philip Gibbs: A very good summary. I have two questions: Is the amplituhedron about a perturbative calculation? Does the perturbative expansion converge for N=4 YM theory? –  Xiao-Gang Wen Sep 19 '13 at 8:29
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My main sources of info are the latest 130 minute talk at scgp.stonybrook.edu/archives/8698 and Motl's blog motls.blogspot.co.uk/2013/09/… –  Philip Gibbs Sep 19 '13 at 9:45
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Neither of the two options offered by the OP is the correct answer, and your option isn't either. That is why more information is given. The answers here are meant to be useful to a wider audience than just the OP, so some technical details are fine. You will find that more technical answers here get more upvotes and this at least avoided equations. –  Philip Gibbs Oct 6 '13 at 18:14
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I rolled back a change made by "community" because I did not like it. I dont mind edits that correct typos but if my answers are going to have more substantial edits then I dont want to write answers for stackexchange. –  Philip Gibbs May 13 at 6:38
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The publication of the paper by Arkani-Hamed and Trnka is imminent. It should offer a completely new way to calculate the probability amplitudes in quantum field theories – relatively to the usual Feynman diagrams – but the observable results are finally the same. See

http://motls.blogspot.com/2013/09/amplituhedron-wonderful-pr-on-new.html?m=1

for some basic info, links, and the big picture. The resulting observable quantities are the same as calculated by other methods so you don't have to throw away the usual textbooks on quantum mechanics and quantum field theory; they haven't been invalidated.

However, the calculation using the shape in an infinite-dimensional space, the amplituhedron, should provide us with completely new perspectives how to look at the dynamics – perspective that is timeless, obscures the location of objects and events in the space and time, and obscures the unitarity (the requirement that the total quantum-calculated probability of all possibilities remains 100%), but it unmasks some other key structures that dictate what the probabilities should be, structures we were largely ignorant about.

If the new picture becomes sufficiently generalized, you could perhaps throw away the old books because you will get an entirely new framework to compute these things and to think about all these things. But once again, you don't have to throw them away because the physical results are the same.

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The new paper will still only be about ${\cal N} = 4$, right? (Apart from the remarks made in the paper that appeared around Christmas.) –  Vibert Sep 18 '13 at 19:15
    
Probably only $N=4$ but during the SUSY 2013 talk, Nima indicated he has an idea what it means to describe non-scale-invariant theories - a whole new region that can produce singularities away from the main "jewel". –  Luboš Motl Sep 18 '13 at 20:20
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Most of the success in this whole area is in N=4 super yang mills seems great. At least at tree level the scattering amplitude for tree level is the same as the scattering amplitude for non-supersymmetric theories or QCD for gluon scattering. However, it seems to me that all the progress has been made in the context of integrand. Now integrands are not trivial things but an amplitude is much more difficult. More importantly, one needs to regularize. Is it always clear that when one simplifies these integrands, we do not lose the information when one simplify these terms out? For example for pure Yang Mills there are all plus(or minus) helicity amplitude, at tree level they all vanish. However, at one loop level they have a finite contribution. In supersymmetric theory these all plus helicity one loop amplitude tend to vanish and they do not contribute. However, is it plausible that we lose certain information when we simplify these integrands in terms of these very interesting expression?

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Certainly texts that address the probabilities of different particle interactions will be incomplete without mentioning this overall connection to geometry. Perhaps the biggest text book change would be the loss of "unitarity", the notion that the sum off the probabilities of each possible interaction should add up to one.

Beyond that, this new tool is expected to make further investigations into particle physics easier.

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"Locality is the notion that particles can interact only from adjoining positions in space and time."

True, except for the time part. Time is derived from expanding space. And, there's also that little thing of 'low pass filtered expanding space' and 'particle locality'. Particle is only as much local as its leading and trailing edges (in terms of 2D visualization of information propagation in a low pass filtered system/space)... interacting with edges of another particle (which is why electron size differs depending on which other particle is used to measure it).

Is space works as it seems to be working, every particle in existence has an information 'tail' that extends outwards by as much as 99.9% of the particle's true size. There's a whole sea of invisible information in the space out there, visibly interacting only when combined edges of interacting particles reach the threshold value needed for visible interaction. The rest of the time, that information appears as 'virtual particles'. In other words, who needs 'virtual photons' when electrons themselves can act on much, much larger scales than their apparent size?

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