# Amplituhedron: newly discovered mathematical object

tiny-tim
Homework Helper
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 …

… 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)

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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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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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.

Ben Niehoff
Gold Member
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

Why isn't he wearing socks?
Because he is the Qweesatz-Haderach

and this thing... this is a sign....

Demystifier
Gold Member
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

Any idea if this offers a "loophole" for Bell analysis or does it fundamentally preempt all that?

1 person
atyy
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?

So does this thing have more than 3 dimensions or what I still don't get what's all the hassle about?

tiny-tim
Homework Helper
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

atyy
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
The paper doesn't seem to use the word "amplituhedron". Natalie Wolchover's article https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/ seems to indicate it's a different thing from the positive Grassmannian.

"The physicists hoped that the amplitude of a scattering process would emerge purely and inevitably from geometry, but locality and unitarity were dictating which pieces of the positive Grassmannian to add together to get it. They wondered whether the amplitude was “the answer to some particular mathematical question,” said Trnka, a post-doctoral researcher at the California Institute of Technology. “And it is,” he said."

"Arkani-Hamed and Trnka discovered that the scattering amplitude equals the volume of a brand-new mathematical object — the amplituhedron."

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
Ah thanks, there are some words I'll translate in order to fully understand but I get the general principle

mfb
Mentor
Any idea if this offers a "loophole" for Bell analysis or does it fundamentally preempt all that?
It is just a mathematical tool to calculate known processes in quantum field theory. It is a new way to get the same results (probably with less effort). It is not related to Bell experiments in any way.

Any idea if this offers a "loophole" for Bell analysis or does it fundamentally preempt all that?
Yes!! I want the Ansible too!

How is this different than the twistor path integral program?

atyy
How is this different than the twistor path integral program?
It's a development of it. In http://arxiv.org/abs/1212.5605 which is their paper before the amplituhedron, they mention twistors and momentum twistor many times.

I just now noticed they put a nice picture on p154.

As Natalie Wolchover wrote in
‘A Jewel at the Heart of Quantum Physics’:
“Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron
directly in some cases, without using twistor diagrams to compute the volumes of its pieces.
They have also found a ““master amplituhedron”” with an infinite number of facets, analogous
to a circle in 2-D, which has an infinite number of sides. Its volume represents, in theory, the
total amplitude of all physical processes. Lower-dimensional amplituhedra, which correspond
to interactions between finite numbers of particles, live on the faces of this master structure.”

The ampli’hedron (easier to pronounce) as a mathematical structure that encodes probability
amplitudes seems to me to be an effective idea perhaps connected to biology and even to
religion.

Suppose biological uniqueness is regarded as a probability amplitude as physically
expressed by an individual living thing. Then, since biological uniqueness is encoded
molecularly by the base sequences that bind together two polynucleotide chains twisted around
one another, the possible number of distinct encodings is large enough to allow all seven billion
of us to be unique individuals. And, I guess, for unique individuals to comprise all life that has
ever existed here or elsewhere in the universe. The master ampli’hedron that in a probabilistic
sense could represent all life is then effectively infinitely faceted

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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
If people are looking for a demo where they can "grab the knobs to rotate this and get a feeling for the shape" and if the resulting projection has to be into an even number of dimensions then it seems that these are very restrictive conditions.

If it was "gluing together four three-simplicies and adding the knobs" then that is easy. If each edge needs, what appears to my simple mind, added sine curves and even some rainbow color palette that don't really seem to have anything to do with the underlying mathematics of the structure then this would probably just take days of fiddling with all the Mathematica buttons and knobs to dress up the simplicies to turn this into art, maybe ten times more work, but still doable.

If this were going to be projected into four dimensions, then projected down into three dimensions and using the color palette to show to show the fourth dimension and being able to rotate the four dimensional structure around in three dimensions then that is tedious and requires a good deal of concentration to get it right, maybe a hundred times more work, but still doable.

I think a starting point is a very simple specific concrete example of onr infinite dimensional object with specific coordinates for all the verticies with a very simple specific projection into two or four dimensions. I know I cannot produce this, but if someone else can supply that then perhaps we can look at the feasibility of throwing that at Mathematica.

Natalie Wolchover's article https://www.simonsfoundation.org/qua...antum-physics/ [Broken] seems to indicate it's a different thing from the positive Grassmannian.
yet they seem closely related as she goes on to say:

....The pieces of the positive Grassmannian that were being calculated with twistor diagrams and then added together by hand were building blocks that fit together inside this jewel, just as triangles fit together to form a polygon.
Like the twistor diagrams, the Feynman diagrams are another way of computing the volume of the amplituhedron piece by piece, but they are much less efficient. .....Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron directly in some cases, without using twistor diagrams to compute the volumes of its pieces.

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MTd2
Gold Member
Planar limit in this case means that N->inf (number of colors of the gluons).

And the reason "planar" is used is (http://www.nworbmot.org/physics/thesis.pdf)

"Strongly-coupled gauge theories are not easily accessible either perturbatively or analytically; one must discretise spacetime on a lattice and use computers to approximate the path integral, using methods due to Wilson [1]. An alternative approach due to ’t Hooft [2] is to allow the number of colours of the gauge group SU(N) to become large and then to expand in 1/N. The gauge theory simpliﬁes and exhibits string-like behaviour. The Feynman diagrams organise themselves into an expansion in topologies of the twodimensional surfaces on which the diagrams can be written. The genus expansion is ordered by powers of 1/N2h−2 according to the number of handles h of the 2d surface, just like a string genus expansion."

So, planar means the plane of a string worldsheet (not necessarily of string theory's), which is general a very complicated topology.

So, this is kind of funny. Sending the number of colors to infinity, you get rid of complications due the finite number of colors.

Color is a quantum number, and getting rid of it by taking a limit, it is like taking a kind of classical limit. It seems a correct procedure since, that is sending the number of colors, to 0 is merely destroying the theory, but erasing any kind of charge.

So, is there a procedure to recover the correct result by re-quantizing the result to the correct number, after a desirable precision is obtained?

It is just a mathematical tool to calculate known processes in quantum field theory. It is a new way to get the same results (probably with less effort). It is not related to Bell experiments in any way.
I think the questioner is asking whether a "hedron-based" theory could provide a hidden-variables theory. Note, not a local hidden variables theory, because (we are being told) locality isn't fundamental here, it's emergent. So it's a logical question.

My general response would be to first remind everyone that the quantities you can calculate from this object are about the probability of having specified particles come in "from infinity" and having other specified particles leave "to infinity". And then to point out that in the new formalism, these probabilities come from calculating the volume of segments of the hedron - a calculation which is similar enough to evaluating an ordinary Feynman diagram, which will involve integration over a parameter space. It may even be a defensible proposition (I'm not sure yet) that the marvel of the amplituhedron can be reduced entirely to statements about integration domains for different path integrals - how they join up, and how they degenerate in singular limits.

One thing about the scattering picture (particles from infinity, to infinity) is that it doesn't say what happens in between. (Or it says that what happens in between is a superposition of all possibilities, if you take the path integral literally as a picture of reality.) Another thing is that it gets used in the real world, even though the scatterings only occur across finite space-time intervals, because of the "adiabatic hypothesis" that the particles (e.g. in colliders) start far enough apart that they can be regarded as "at infinity" - not yet interacting. And finally, that the scattering process in the hedron case is for a special sort of theory - first, it's conformal, and second, maximally supersymmetric.

It seems that these hedron calculations give us the scattering probabilities without the path integral picture. That's certainly interesting, it is suggestive of quantum mechanics coming from something else. But one of the philosophical challenges of the S-matrix picture of the universe has always been, where do we fit in? We don't live at past infinity or future infinity, we live in real time, in the middle of the process somewhere.

I can think of basically two ways that relativistic quantum field theory reaches into this "real time". One is through real-time formalisms like Schwinger-Keldysh formalism. The other is via the focus on "resonances", transient objects which show up as poles in the S-matrix. In neither case is there a simple picture like nonrelativistic QM, where you just have a state that evolves in time. (Incidentally, the latter picture, of poles, is mentioned by Susskind in his "anthropic landscape" paper, where he ponders how to get a metastable de-Sitter universe from string theory.)

At this point, fans of the Copenhagen interpretation can still regard the amplituhedron the way they have also regarded state vectors in Hilbert space - just as mathematical constructs. The real world is the world of observables in a space-time, all these other objects are just mathematics. So that's one way to "interpret" it.

I am also expecting that some Everett fans will say that it is the shape of the multiverse. Whether that sort of claim can be turned into anything more than rhetoric remains to be seen.

One problem in extracting an interpretation or an ontological theory from the amplituhedron is that its volumes translate to probabilities. But I still think that the main problem is just the limitations of this first example. It doesn't even describe the whole of N=4 super-Yang-Mills, just the sector of Feynman diagrams without crossings. We will need to know how to get theories other than CFTs - though there is reason to hope there, in that generic QFTs can be understood as flows between CFTs, so it may be possible to stitch together amplituhedra from different CFTs, in order to describe a non-CFT. Here my main concern is whether we can move away from susy - I have no idea how integral susy is to the current construction, or indeed in what way it might be integral.

And there is also still the question of how to get something more than the from-infinity-to-infinity perspective; how to say something about what happens in the middle. It wouldn't surprise me if gravity was relevant here - if the way you get a gravitational theory is by stitching together "finite-time amplituhedra" for finite-time scattering events.

See also this paper of Jaroslav Trnka :

http://www.staff.science.uu.nl/~tonge105/igst13/Trnka.pdf [Broken]

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.Scott
Homework Helper
I think the geometric drawing, beautiful as they are, are distracting most people from understanding what is being done.

The "jewels" themselves are not that complicated - although the arithmetic that follows them is. If you have three non-colinear points, you can draw a triangle. If you have four non-coplanar points, you can draw a tetrahedron. If you have five points that do not all fall with a three dimensional space, you can draw the corresponding four-dimensional object. And so on.

Now, for two dimensions, fit a bunch of those triangles together so that they all share a common vertice, adjacent ones share a common side, and all have their own non-overlapping areas. Your done.

For any higher dimension, you build in a corresponding way. The number of dimensions you need will dependent on the number of particles. If you are doing the whole universe you will need an "infinite" number of dimensions. Of course, it's not really infinite, but why quibble.

Why is the planar limit important? I don't really get how or why it is needed. Could someone explain?

MathematicalPhysicist
Gold Member
I think the geometric drawing, beautiful as they are, are distracting most people from understanding what is being done.

The "jewels" themselves are not that complicated - although the arithmetic that follows them is. If you have three non-colinear points, you can draw a triangle. If you have four non-coplanar points, you can draw a tetrahedron. If you have five points that do not all fall with a three dimensional space, you can draw the corresponding four-dimensional object. And so on.

Now, for two dimensions, fit a bunch of those triangles together so that they all share a common vertice, adjacent ones share a common side, and all have their own non-overlapping areas. Your done.

For any higher dimension, you build in a corresponding way. The number of dimensions you need will dependent on the number of particles. If you are doing the whole universe you will need an "infinite" number of dimensions. Of course, it's not really infinite, but why quibble.
How do you know that it's finite?