Amplituhedron

Physicists have discovered a jewel-like geometric object that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.

“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.

The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

> Physicists have discovered a jewel-like geometric object that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.

(Checks date; not April 1).

> “removing two deeply rooted principles of physics: locality and unitarity. … unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one”.

Yes. I’ve actually been waiting for this. When probabilities don’t add up to 1, rather strange things happen. It’s difficult to envisage the case where probabilities don’t add up to 1. For example have a certain experiment of tossing a coin – the results could be that it lands heads, or tails, or on its edge. The possibility that the coin vanishes from the universe with finite probability is not normally taken into account.

Douglas Adams had a similar idea of violating unitarity when he invented Bistromath.

> “is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry”

In the Causal Dynamical Triangulation TOE, space and time are not fundamental but arise as a consequence of the geometry, so that bears some resemblance to this, but even in Causal Dynamical Triangulation probabilities sum to 1.

> “1980s … Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula. Then, (they) guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler.

I hadn’t heard of that.

>> “1980s … Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula. Then, (they) guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler.

> I hadn’t heard of that.

Looking further, I find that I had. The problem appears in Chapter 7.4 of A. Zee “Quantum field theory in a nutshell” where is is solved by the relatively simple technique of combining many Feynman diagrams into much fewer “generalized” Feynman diagrams (using notation that Feynman never envisaged). This simplification is an essential part of all analytic solutions to quantum chromodynamic problems.

> BCFW recursion relations, named for Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. The BCFW relations are best couched in terms of strange variables called “twistors,” and particle interactions can be captured in a handful of associated twistor diagrams.

See Twistor theory
I know it vaguely in its original form from 1967, where it was proposed as an alternative description of general relativity. It’s still an important topic in theoretical physics, Google scholar gives 11,800 scientific papers on twistors.

> “In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3+1 space-time (Minkowski space) into geometric objects in a 4 dimensional space with metric signature (2,2). This space is called twistor space, and its complex valued coordinates are called twistors.”

This is equivalent to having a 4-D space-time with two spacelike coordinates and two timelike coordinates.

> “the positive Grassmannian is the slightly more grown-up cousin of the inside of a triangle”. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in an N-dimensional space bounded by intersecting planes. (N is the number of particles involved in a scattering process.)

I call that a “simplex”. Or is is more complicated than a simplex?

> Even without unitarity and locality, the amplituhedron formulation of quantum field theory does not yet incorporate gravity.

That’s not a major concern, yet. Far more pressing is the need for a proof that quantum chromodynamics is self-consistent (i.e. not self-contradictory). Perhaps the theory will help with that, or perhaps the non-unitarity means that quantum chromodynamics really is self-contradictory when it assumes unitarity.

> maximally supersymmetric Yang-Mills theory

;-0 It’s far more usual to use MINIMALLY supersymmetric Yang-Mills theory.

The mathematics of this is on the borders of what I can understand, if not beyond it, and before now I’ve only said that about string theory. Whereas string theory is based on topology, this new structure is based on geometry, and I like geometry.

>> “the positive Grassmannian is the slightly more grown-up cousin of the inside of a triangle”. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in an N-dimensional space bounded by intersecting planes. (N is the number of particles involved in a scattering process.)

> I call that a “simplex”. Or is is more complicated than a simplex?

Looks like I was right. 22 minutes into a one hour presentation on the “amplituhedron” (OK, I fast forwarded through the presentation) there’s a direct comparison between the positive Grassmannian and the simplex . The differences are negligible.

The positive Grassmannian is a useful way for calculating results in quantum mechanics without Feynmann diagrams. The simplex is the simplest possible polyhedron in N-dimensions.

In the talk, a particular slide appears briefly, giving the plan for future action.

After 6 years he has finally got to the end of Step 1.0, to be followed by other parts of Step 1 and then Step 2 etc.

Step Task
1.0. Planar N=4 SYM (ie. 4-D space representing four interacting particles, supersymmetry)
1.1. Non-planar
1.2. Non-supersymmetric
1.3. Gravity
1.4. Perturbative Strings
1.5. AdS/CFT
…

He’s got a baby model working in four dimensions, the final version of the amplituhedron will be infinite dimensional.

“That object is known as “Bong”,
there’s the problem, they have smoked too many bongs.

Have sent an email to Arkani-Hamed asking for more information on this. Many years ago I did some some playing around with polyhedrons (more correctly, polytopes) in 4-D.

This whole scheme looks like a possible winner for a TOE.

mollwollfumble said:

Have sent an email to Arkani-Hamed asking for more information on this. Many years ago I did some some playing around with polyhedrons (more correctly, polytopes) in 4-D.

This whole scheme looks like a possible winner for a TOE.

This “amplituhedron” thingy seems rather cool and I guess it implies a level of structure that has been previously overlooked, but does it really imply a new TOE?

I suppose that it adds credibility to the twistor approach and simplifies twistor-based calculations to a degree, but twistors and the so-called positive Grassmannian are hardly new – QCD calculations would be insane without them.

Trying to get actual numbers out of QCD using QED-style Feynman diagrams is impractical, if not downright impossible: you need to use ridiculous numbers of diagrams and the convergence is so slow that the calculation process takes ages, is error-prone, and requires very high precision arithmetic if you don’t want to get swamped by rounding errors. And even using twistors, it’s not a walk in the park: it took Wilczek months to calculate a rough value for the proton mass from first principles using a supercomputer.

So if this “amplituhedron” speeds up such calculations I’m very pleased, but I remain skeptical of the theoretical implications. As you said earlier “He’s got a baby model working in four dimensions, the final version of the amplituhedron will be infinite dimensional”. From 4 to infinity is a rather large leap. :) And as I’ve often said, I’m not a huge fan of supersymmetry. The mathematician in me thinks it’s very neat, the physicist in me thinks it’s too neat, and smells slightly of phlogiston. :)

OTOH, it’s great to see twistor theory getting the attention it deserves. FWIW, Andrew Hodges (mentioned at the start of the linked article) maintains http://www.twistordiagrams.org.uk/ ; he did his PhD under Penrose. Hodges also wrote the best biography of A M Turing, and maintains http://www.turing.org.uk/ .

To finish this post, I’d like to quote from the currently rather short xkcd thread on the Amplituhedron

SU3SU2U1 » Sat Sep 21, 2013 1:59 am UTC
This has actually been around for a few years now (although this is a heck of a marketing coup), but it only works for SUSY Yang-Mills theories, and it seems pretty clear that breaking SUSY and adding quarks that aren’t paired with the gluons via SUSY will probably break the nice geometric regularities.

A few years ago, Nima was having lots of trouble getting grad students/postdocs to work on it, because they (probably correctly) perceived spending years hammering at this would be a career killer. I imagine thats part of the reason for this marketing push- they can now say “See its hot, people will care about it, come work on it.”

If the behaviour of a particle is geometry based to create spacetime, would it not make sense that gravity would be a result of interaction between particles and vacuum? ie, the most obvious interaction between vacuum and a particle is that the vacuum provides the particle with freedom of range for it’s geometry, – a particle will seek equilibrium of pressure. To do this it provides the particle with a means to express internal energy as kinetic momentum.

If the energy being expressed by a particle is not c, so that the particle is travelling at luminal velocity, the momentum of the particle may be imposed upon it’s immediate vacuum’s virtual particle behaviour, creating a negative time space reflection. The virtual momentum reflected on the vacuum cannot be expressed other than through particle momentum. This would fundamentally govern the velocity of the universal vacuum to essentially be “faster than a subluminal particle and travelling in the opposite direction”.

??

Riff-in-Thyme said:

If the behaviour of a particle is geometry based to create spacetime, would it not make sense that gravity would be a result of interaction between particles and vacuum? ie, the most obvious interaction between vacuum and a particle is that the vacuum provides the particle with freedom of range for it’s geometry, – a particle will seek equilibrium of pressure. To do this it provides the particle with a means to express internal energy as kinetic momentum.

If the energy being expressed by a particle is not c, so that the particle is travelling at luminal velocity, the momentum of the particle may be imposed upon it’s immediate vacuum’s virtual particle behaviour, creating a negative time space reflection. The virtual momentum reflected on the vacuum cannot be expressed other than through particle momentum. This would fundamentally govern the velocity of the universal vacuum to essentially be “faster than a subluminal particle and travelling in the opposite direction”.

??

bumped this for you pm. The last line doesn’t sum up what I am exactly but this ‘geometric object’ thing has just about got me convinced that gravitation is an effect of vacuum on particles rather than directly between particles. If you don’t have any thoughts I will just get back to it if this picture in my head makes logic when put into words.

Riff-in-Thyme said:

‘thinking exactly’