Amplituhedron: newly discovered mathematical object

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  • #1
Khashishi
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Is this anything to get excited about? I have no idea what it's about.
https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/

https://www.simonsfoundation.org/wp-content/uploads/2013/09/amplutihedron_span.jpg [Broken]

Artist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated — the probabilities of outcomes of particle interactions.
 
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  • #3
arildno
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It must exist because it is so beautiful! :approve:
 
  • #4
Aero51
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I guess the next question is "where did this structure come from in the first place?"
 
  • #5
kreil
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This is amazing! Great article link, thanks.
 
  • #6
korialstasz
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You can watch Arkani Hamed give a talk at the SUSY 2013 conference about the Ampllituhedron and how their research led them to its discovery here. It's pretty involved as far as having an in-depth understanding of particle physics and QFT, but it's engaging regardless.

It's a pretty fantastic discovery, that all of the information about scattering amplitudes can be encoded in one geometric object, with no reference to space or time, locality or unitarity (which come out as emergent properties of the geometry). Granted, keep in mind that this model is built on maximally supersymmetric Yang-Mills Theory, and it will take a lot of work to generalize it to more complex QFTs (especially since we have found no evidence to support SUSY), but it looks pretty promising just in the beauty of the theory.
 
  • #7
reenmachine
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Don't know what it is , but it's beautiful.
 
  • #9
atyy
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Why isn't he wearing socks?
 
  • #10
Enigman
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Why isn't he wearing socks?

He's copying Albert.
 
  • #11
atyy
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He's copying Albert.

What a wannabe. But Albert would be pleased if he got rid of unitarity.
 
  • #12
jackmell
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I hate to be disagreeable but I feel I must add some sense of sensibility to this dog and pony show.

That picture does nothing for me. It's just some rainbow curvy-looking thing that appears to be missing a piece. Is the actual object a higher-dimensional object and if so then what is the picture representing and I bet a dollar the colors are really not part of the object but just added to make it look pretty.

I request a better explanation of what exactly the geometric object is, and a better illustration of what exactly it looks like. Hold the pretty colors please.
 
  • #13
Enigman
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  • #14
jackmell
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Ok Enigman. Thank you. That is in my opinion much, much better and I can actually see what it looks like in 3D although a nice 3D model that I could rotate would be better. And I did read the link above but I guess I missed that diagram. Sorry.

Also, while we're at it, I don't think it would be too hard to actually create a 3D model of that figure in Mathematica that we could rotate and if so, then we could say, "hey, anybody if you want more than a pretty flat picture of what's happening, PF's got it cus' we know what's happening." So I say we move this thread to the Math and Science software forum and ask someone to do that. No I don't want to do it because I do that stuff all the time and would like someone else to.

Thank you.
 
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  • #15
ABD EL HAMEED
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There's so many ways to look at that thing...
 
  • #16
jackmell
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Oh guys don't you see, don't you see? Someone here has a marvelous opportunity to contribute to humanity. What I would like is for someone to create a Wolfram Demonstration Project that not only draws the amplituhedron accurately in 3D, (with colors if you want) but also gives the user the opportunity to adjust the dimensions of each face interactively (if that is an allowable transformation). And that person would, I suggest, be the first person in the world to write a very nice 3D demonstration on this wonderful, newly created mathematical object.

Just sayin' that's all. :)

Haven't checked the demonstration project but I doubt seriously it's already in there.
 
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  • #17
ABD EL HAMEED
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How can it be a mathematical object?
 
  • #18
jackmell
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How can it be a mathematical object?

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.
 
  • #19
ABD EL HAMEED
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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.

No I don't code but I will start learning how to do so when I start collage but I still don't get your point, are you trying to draw this shape using coding in mathematica?
 
  • #20
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I doubt that you can accurately represent that thing in just 3 dimensions. Feel free to prove me wrong ;).
 
  • #21
kreil
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I doubt that you can accurately represent that thing in just 3 dimensions. Feel free to prove me wrong ;).

It would just look silly. Like describing a cow with a single point.

However, we can take 3D cross sections. That's where the steak is:

https://www.simonsfoundation.org/wp-content/uploads/2013/09/amplituhedron-drawing_web-271x300.jpg [Broken]
 
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  • #22
korialstasz
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Is the actual object a higher-dimensional object and if so then what is the picture representing

The actual object itself would prepresent every possible particle interaction, and is infinite dimensional. The hand-drawn geometric object with the subtitle "8-gluon particle interaction" is actually only one face of a 4-dimensional object which fully describes the 8-gluon interaction. The 3-D face describes one possible permutation, so to speak, of the possible ways that 8 gluons can interact, as written underneath [1- 2+ 3+ 4+ 5+ 6+ 7- 8-].
 
  • #23
jackmell
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No I don't code but I will start learning how to do so when I start collage but I still don't get your point, are you trying to draw this shape using coding in mathematica?

Yes hameed, just a nice 3D plot of that hand-drawn figure posted above. It's not rocket science guys. Jesus, why you guys trying to make it so complicated?
 
  • #24
arivero
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Epps, ¿moved to General Discussion?
 
  • #25
dipole
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So this is just a re-formulation of an existing theory, which its self is just a toy-model.

So it simplifies and already over-simplified theory?

I don't see the excitement.
 
  • #26
tiny-tim
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at about 16:00, arkani-hamed says "pictures of space-time on-shell processes are in 1-to-1 correspondence with cells of the positive Grassmanian which in turn are in 1-to-1 correspondence with a new way of thinking about permutations"

he presents black-or-white dot-diagrams that represent permutations

since even a ten-year-old could understand them (after a little thought), let's look at a couple of diagrams …
https://www.simonsfoundation.org/wp-content/uploads/2013/09/twistor-diagrams-300x200.png [Broken]

… the rule is that you start at one of the numbers round the outside, you turn LEFT at a white dot, and RIGHT at a black dot, and you end at another number on the outside

eg in the first diagram, 1 goes to 3, 3 goes to 5, 5 goes back to 1; 2 goes to 4, 4 goes to 6, 6 goes back to 4, so it represents the permutation (135) (246)

similarly the second diagram represents (153) (264) :smile:

EDIT: if you're only interested in the amplituhedron itself, you can start at about 20:30,

where he indeed says "I could have skipped the whole of the first part of the talk, which is physics-motivated, and started at this point, and this can be explained to a smart junior-high-school student"​
 
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  • #27
jackmell
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Compute the volume of amplituhedron. I don't think that would be too hard. Of course you have to draw it first (define the planes algebraically) to do that.

Is micromass still doing those challenge problems?
 

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  • #28
korialstasz
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Compute the volume of amplituhedron. I don't think that would be too hard. Of course you have to draw it first (define the planes algebraically) to do that.

I believe they use a triangulation method to compute the cell volumes. There was also mention of a special volume form defined for the amplituhedron to make calculuations easier. In the talk he says that he was able to compute the scattering amplitude for the diagram included in your post with pen and paper on the flight over.
 
  • #29
Ben Niehoff
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The funny rainbow picture is just the artist's imaginative illustration after hearing the physicists attempt to describe what they've done in simple language.

The original paper is here:

http://arxiv.org/abs/1212.5605
 
  • #30
sanman
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Why isn't he wearing socks?

Because he is the Qweesatz-Haderach

and this thing... this is a sign....
 
  • #31
Demystifier
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As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
Albert Einstein

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
Albert Einstein
 
  • #32
bahamagreen
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Any idea if this offers a "loophole" for Bell analysis or does it fundamentally preempt all that?
 
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  • #33
atyy
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Granted, keep in mind that this model is built on maximally supersymmetric Yang-Mills Theory, and it will take a lot of work to generalize it to more complex QFTs (especially since we have found no evidence to support SUSY), but it looks pretty promising just in the beauty of the theory.

Why should one think it's generalizable? Could it be that this works only because the system is "integrable" and highly symmetric?
 
  • #34
ABD EL HAMEED
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So does this thing have more than 3 dimensions or what I still don't get what's all the hassle about?
 
  • #35
tiny-tim
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it has an infinite number of dimensions

but we only use projections of it (like a 2D plan of a 3D building), and those projections are into a finite even number of dimensions

each projection is a "polyhedron" in 2n dimensions, and the 2n-dimensionsal "volume" of each polyhedron gives us the value of a coefficient of an interaction

we calculate the volume by dividing it into 2n-dimensional "pyramids" whose volumes are easy to calculate (like finding the area of a 2D polygon by dividing it into 2D triangles)

the number of dimensions (2n) depends on the accuracy with which we want to calculate our coefficient :smile:
 

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