# Amplituhedron?

I recently ran across a rather interesting result while browsing the Internet:

Physics Discover Geometry Underlying Particle Physics

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.

Unfortunately, I'm still at the point in my education where my best response to new physics is "cache for later," and the fact that it claims to eliminate locality/unitarity seems decidedly *odd* to my mostly-untrained mind. I notice I am confused, and that LessWrong has a rather large number of trained physicists.

## Comments (15)

Best*21 points [-]Here is an attempt to create a roadmap to the amplituhedron work. My relevant background and disclaimers: I am a mathematician with interests in particle physics who has been trying to learn about Arkani-Hamed and collaborators' ideas for the last two years. The specific result which is getting press now is one that has not been public for most of that time; my goal had been to understand the story of scattering amplitudes as described in his prior 154 page paper. I have been meeting regularly with a group of mathematicians and physicists here at the University of Michigan in pursuit of this goal.

So, what should you learn first:

You should be completely comfortable with quantum mechanics and special relativity. I would point out that Less Wrong will give you great ideas about the philosophy of QM but is very short on computing any actual examples; you should understand how to actually use QM to solve problems.

Mathematically, I found my familiarity with representation theory and Lie groups extremely useful. However, a lot of the physicists in our group didn't have this background and compensated for it with strengths of their own.

You should understand the material of a first graduate course in Quantum Field Theory, through the computation of tree-level amplitudes. To learn this, I audited a course taught out of Srednicki's book, and also read on my own in Peskin-Schroeder and Zee. I can't claim to have a great understanding of this material, and if anyone has advice as to how to learn it better, I'd love to hear some. However, I feel confident in saying that, had I been enrolled in that class, I would have gotten an A, and I think you should at least be at that level. A second course in QFT certainly wouldn't hurt -- the fact that I had never worked through any loop integrals in detail handicapped me -- but I am managing without it.

If you get this far, I strongly recommend you next read Henriette Elvang and Yu-Tin Huang's notes on scattering amplitudes http://arxiv.org/abs/1308.1697 . As the abstract says, "The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes." I have found this extremely helpful. (Of course, being able to knock on Henriette's door and get her to explain something to me is even more valuable :).)

After that, I'd look at "Scattering Amplitudes and the Positive Grassmannian" http://arxiv.org/abs/1212.5605 . This is long and hard, but has the advantage that it is written down in full detail, unlike the current subject which only exists in lecture notes.

At this point, you will have caught up to me, so I'm not sure I can advise you how to go further. However, I will suggest that I find Arkani-Hamed's co-author, Trnka, much more understandable than Arkani-Hamed. These lecture notes http://wwwth.mpp.mpg.de/members/strings/amplitudes2013/amplitudes_files/program/Talks/WE/Trnka.pdf are the clearest presentation of the amplituhedron material I have found yet.

Thanks! I'm a busy undergrad, so this'll take me a few years to work through, but it's always good to have more things to read :P.

The article is confusing on this point. IIUC, the amplituhedron does not eliminate either unitarity or locality. What it does is to avoid taking them as axioms, as the Feynman approach does. Then it later turns out that they

fall naturally out of the math, which is much nicer.I figured as much. That still seems mildly odd, but that's probably because I'm used to thinking of them as axiom-ish principles and not as derivative results.

Where did you get that from? Did you read the primary, or is there some actually decent exposition of this somewhere?

Comments on HN. Sorry! But I read it on the Internet, so it must be true.

FWIW I've seen the same thing said by other people elsewhere, though of course it's possible that they and the HN comments derive from a common source. Also, my understanding is that the amplituhedron approach produces the exact same numbers as enumerating Feynman diagrams does, just quicker and in something more like a closed form, so it has to be as local and unitary as any other way of doing QFT.

You can watch/listen to Arkani-Hamed's recent talk at SUSY 2013. At around 2:00, he says:

At around 6:00, a written slide describes his strategy:

He goes on to discuss this subject in more detail.

Also, (somewhat technical) slides from his former student have a section called "Emergent Locality and Unitarity".

Scott Aaronson's take on the amplitudihedron.

I couldn't figure out the tone of this post.

Is he trying to poke fun at it? Pointing out there's a more general version? What?

He's saying it's alright but nothing special. He's saying it's over-hyped because it has a snappy name. If only he'd called BQP the Unitarihedron!

It also has a pretty picture.

An article about the underlying math with a very accessible introduction (using lots of illustrative graphs) can be found on Arxiv:

"Scattering Amplitudes and the Positive Grassmannian" by N. Arkani-Hameda at al http://arxiv.org/pdf/1212.5605v1.pdf

Thanks!

I posted about this here.