Amplituhedron: newly discovered mathematical object

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  • #51
atyy
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Why is the planar limit important? I don't really get how or why it is needed. Could someone explain?
4 gravitons and a grad student gives his take on the amplitudhedron http://4gravitonsandagradstudent.wordpress.com/2013/09/20/the-amplituhedron-and-other-excellently-silly-words/ [Broken] and explains the planar limit http://4gravitonsandagradstudent.wordpress.com/2013/09/27/planar-vs-non-planar-a-colorful-story/ [Broken].
 
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  • #52
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4 gravitons and a grad student gives his take on the amplitudhedron http://4gravitonsandagradstudent.wordpress.com/2013/09/20/the-amplituhedron-and-other-excellently-silly-words/ [Broken] and explains the planar limit http://4gravitonsandagradstudent.wordpress.com/2013/09/27/planar-vs-non-planar-a-colorful-story/ [Broken].
Thanks for the links. I read them and it said focusing on planar graphs makes things simpler, but why?
 
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  • #53
atyy
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Thanks for the links. I read them and it said focusing on planar graphs makes things simpler, but why?
What I understood from the 4 gravitons and a grad student's posts is that in non-planar theories you have to consider two sorts of graphs, but in planar theories, you only have to consider planar graphs.
 
  • #54
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Planar theories produce strings. 't Hooft discovered this forty years ago. For example, a quark and an antiquark with many gluons shuttling between them, along a line of flux. The internal history of this quark-antiquark string is described by planar Feynman diagrams.
 
  • #56
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I think the basic meaning of N->infinity is that it is an asymptotic statement about theories with large but finite number of colors: the more colors the theory has, the more accurate a planar approximation will be. So the significance for low N, like N=3, might be that there are significant corrections to, or deviations from, the string / worldsheet picture, in the real world case of SU(3) QCD.
 
  • #57
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I think the real magic of all this is BCFW recursion. The Amplitudehedron interprets this as a triangulation. My question is how generalizable is BCFW, i.e. in what cases does it apply?
 
  • #58
MTd2
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So the significance for low N, like N=3, might be that there are significant corrections to, or deviations from, the string / worldsheet picture, in the real world case of SU(3) QCD.
Yes, but that's part of the observation before my question, but with a caveat, that the corrections would be in terms of that N->inf. But what correction would be? That kind of limit resembles a classic limit. So, what I mean is, quantizing the perturbative expansion in terms of N=3.

I am not sure if this makes sense, but given that the planar limit is a string, wouldn't that mean attaching it to an orbifold of 3 branes and calculating the combinatorics of attaching incoming and outcoming strings?
 
  • #59
Physics Monkey
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I just now noticed they put a nice picture on p154.
Reminds me of entanglement ...
 
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atyy
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  • #61
Physics Monkey
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I have no idea :), but why not tensor networks for scattering amplitudes?
 
  • #62
atyy
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It'd be neat. I have to understand more what they are doing. I'd kind of assumed it doesn't generalize beyond such a high symmetric theory, unlike AdS/CFT. But maybe that's wrong.
 
  • #63
Physics Monkey
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I have a similar feeling. In fact, in the case of AdS/CFT I do think it generalizes as you know, but here I'm not so sure.
 
  • #64
atyy
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I have no idea :), but why not tensor networks for scattering amplitudes?
Like http://arxiv.org/abs/0907.0151 and http://arxiv.org/abs/1209.3304 ?

But I'm not sure if Arkani-Hamed and collaborators mean amplitude in the same way, I remember it's just the integrand or something like that. But conceptually it seems like it should be related to the normal meaning of "amplitude".
 
  • #65
Physics Monkey
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Yes, I think they focus on the integrand of the multi-loop scatting amplitude. It still has to integrated over internal momenta (which I think are "on shell" but have been complexified). I'm not exactly sure what the full procedure is.

But if they're somehow considering twistors or something similar, perhaps we can build a twistor tensor network so that the sum over the internal variables is like a sum over loop momenta.
 
  • #66
.Scott
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How do you know that it's finite?
As I understand it, the number of dimensions you need for the "gem" is proportional to the number of particles involved in the interaction. So how many particles are there in the universe? Or more precisely, what is the cardinality of particles in the universe?
 
  • #67
MathematicalPhysicist
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As I understand it, the number of dimensions you need for the "gem" is proportional to the number of particles involved in the interaction. So how many particles are there in the universe? Or more precisely, what is the cardinality of particles in the universe?
How do you know that there's a finite number of particles in the universe?
 
  • #68
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Lemme' ask you this hameed, you code? I mean am I the only one in PF that actually likes coding in Mathematica for fun? Keep in mind who ever does this can never be forgotten even in death cus' all we have to do is google them and bam! There it is: first person in the world to create a nice 3D realistic, interactive image of the amplituhedron.

But that's ok, that's alright, no big deal if no one is interested. I got plenty other stuff to do.
I can do that and i would love to do that :) Just a couple of days of coding, that's the easy part. The hardest part is to understand Amplituhedron's geometrical properties and how it actually works.
 
  • #69
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Lemme' ask you this hameed, you code? I mean am I the only one in PF that actually likes coding in Mathematica for fun? ...
I do also.

However, I don't see what's so special about the amplituhedron. It looks like it could be a good graphical method, like Young diagrams, but I think that it may be too early to say.
 

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