## Wednesday, September 18, 2013 ... /////

### Amplituhedron: wonderful P.R. on new Arkani-Hamed+Trnka papers

Autom. translation to Romanian

Spacetime is doomed and its murderer has finally been spotted

Update: See this newer essay whether the amplituhedron is revolutionary
In December 2012, Nima Arkani-Hamed along with Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka released a very interesting paper
Scattering Amplitudes and the Positive Grassmannian
on a new way to calculate planar amplitudes of four-dimensional gauge theories using some generalized higher-dimensional "polytopes", more precisely "positive Grassmannians" (see previous TRF articles on that concept). This development arose from the twistor minirevolution sparked by Witten's 2003 paper although the twistors turned into a small part of a greater story with many more important mathematical structures.

The tesseract isn't quite the same as the objects talked about here (the amplituhedron should be a jewel in an infinite-dimensional space, in fact) but it's close enough and I just liked the animation.

While the locality in the four-dimensional spacetime is being obscured by the new formalism (even the existence of time: the mathematical structure is "timeless"), and so is the unitarity, in fact, the new formalism unmasks many other fascinated symmetries (associated with the integrability of such models: the Yangian) and related mathematical properties of the scattering amplitude which are behind the physicists' ability to calculate amplitudes previously uncalcalculable even with the best computers (due to their complexity) on a sheet of paper.

Nima was preparing me for a looming publication of a new, even cooler paper (now it seems that it will be several papers) about these issues that will be co-authored by him and his Czech PhD student (now a postdoc) Mr Jaroslav Trnka [Yuh-rho-sluff trrrrn-kuh; please treat the ringing sound "rrrr" as a good enough ersatz vowel – the same thing with "llll" in "Motl" – and avoid bullshit claims that the Czech language suffers from a shortage of vowels]. :-)

This paper should reorganize the previous new method to calculate the scattering amplitudes in yet another, deeper way. The publication seems imminent because Natalie Wolchover just wrote a wonderful report about the advance for the Simons Foundation:
A Jewel at the Heart of Quantum Physics
Yes, this server of Jim Simons only has the same traffic as this blog. ;-)

I recommend you to read the piece. You will learn about the technical content just a little bit – it wasn't enough for me to learn much more than what is already clear from the previous papers. (Update: I actually have learned new things from that, especially when I checked Nima's and Jaroslav's talks linked at the bottom and compared them with the Simons Foundation popular piece.)

However, you will also notice some excited comments about the simplicity and originality of the new pictures from some very well-known mathematicians and theoretical physicists including the co-authors of the previous papers mentioned above as well as Hodges, Skinner, Deligne (who helped them), Witten, and others.

So I am surely looking forward to the new paper. Unless the famous mathematicians etc. are lying, it should allow us to look at the amplitudes in yet another new way. We're being promised that in the new picture, the physical notions will follow "just" from the amplituhedron's geometry and other physical assumptions may be erased – a vowed progress relatively to the positive Grassmannians. The amplitudes should "literally" be the volume of the jewels again (but the volume form isn't constant, it has factors like the inverse distance from the faces) and the shape of the jewel should be determined by some brand new principles – a convex envelope of a sort. Faces of the jewel should correspond to the cuts.

Nima's and Jaroslav's new calculations should increase the efficiency and "covariance" relatively to the BCFW-like twistor prescriptions which had to divide the jewel to many pieces and sum them; they can sometimes find the full result without summing anything. An amplitude exact up to 6 loops was calculated as a volume of a shape only and verified to agre with laborious multipiece old calculations. They also claim to have found a "master amplituhedron" with infinitely many faces in infinitely many dimensions which should now be as important as the circle in two dimensions. ;-) Its volume counts the "total amplitude" (?) of all processes; faces of this master jewel harbor the amplitudes for processes with finite collections of particles.

To be honest, I am probably going to look at the slides of Trnka's 1-month-old talk in Utrecht, NL and Nima's talk in Italy a week later. ;-) Other links are available via the Wikipedia's entry for the amplituhedron and via Google.

#### snail feedback (35) :

The Wiki article is not useful. "It also does away with the spacetime properties of locality and unitarity." Barf.

This is definitely not some noise added by a Wiki editor – it's the key meme of this whole research that Nima in particular emphasizes all the time.

Unitary is not manifest in their formalism, all right, but it is still a necessary condition. In fact, I recall Nima himself referring to unitarity as a "sanity condition", necessary if you want to stay sane. ;-)
Of course, the validity of unitarity has been discussed and debated in the black hole context (but settled in favor of its being valid).
In the context of gauge theories, where gravity appears at best in a dual description, questioning unitarity seems downright stupid to me. In the end their formalism must be compatible with unitarity or be discarded as inconsistent.

I agree with you. Unitarity is a derived feature in their picture (and it exactly holds - because they derive the very same amplitudes we know) but it will ultimately be needed to hold exactly, the two of us think.

Nima speculates about the future deformations away from the unitarity and the locality - in the case of the unitarity, I am confident that he can't get new consistent physical theories.

Lubos, I think you'll like this:

I did like it but I had also embedded it at the top of the yesterday's blog entry.

Whoops!! Sorry about that -- not the first nor the last time I'll be late I think :)

Readers of the physics blogosphere should remember "Kea", aka Marni Sheppeard, who for many years preached that twistor polytopes were the future of physics. She did a PhD thesis on the subject in New Zealand while working as a waitress. She's still living somewhere alone and in poverty and occasionally putting a paper on vixra.org.

A courageous babe. Still, if we look for postings containing both polytope and twistor on her blog,

we find 2 hits and none of them looks like having more than 1% of the understanding that's necessary for the recent twistor uprising papers.

Well, it seems the idea works quite well for N=4 supersymmetric Yang-Mills theory (a non-physical theory). However, it would be really superb to see Nima's approach working for pure YM at small coupling. Would it be able to derive confinement? ;-)

Is Jaroslav related to Pavel Trnka? The optics of his elegant glass sculture, gives their interior an extra-dimensional quality such that they would have made interesting illustrations for this topic.

LOL, do you mean the Pilsen-born hockey player Pavel Trnka? There has also been a famous painter and sculptor Jiří Trnka – my grandfathers actually taught him at the high school.

I would guess that all these folks are unrelated by close relationships. Trnka is 174th most frequent Czech surname with almost 3,000 men carrying it:

http://prijmeni.unas.cz/

You clearly have no idea what you are talking about. "Non-physical"?! Can you keep your stupidity to yourself, please?

N=4 SYM is the "harmonic oscillator of the 21st century"

I do have a clear idea about what I am talking about. A toy theory as N=4 supersymmetric Yang-Mills describes absolutely no high-energy experiment on physics. Thus, just watch the video (http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html) at 59:40 and see how the girl's question concerning quark-quark scattering at low momenta wakes up Nima from his "Dreams of a Final Amplituhedron". Thus, please, shut up and calculate!

You Dan have absolutely no idea what a physical theory has to be. Or, do you think N=4 SYM describes any high-energy physics experiment ? So, just watch (http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html) at 59:40 and see how the girl's questions wakes up Nima from his "Dreams of a Final Amplituhedron". Thereby, make us a favor, shut up and calculate!

The idea in general looks very interesting but I do have to read the 130 pages from the 2012 paper... about the speculations related to non-unitarity and non-locality ... I think, as has been stated by others that a theory that doesn't have some property in a *manifest* form doesn't mean it doesn't have that property at all... one doesn't "give up" a property... one just makes it less visible in order to make other properties, of larger interest at a given moment, more obvious...

Thanks for your reply and explanation. One more sentence about locality if I may. We (all but) know that locality becomes approximate near black holes, fine. In a gauge theory, however, which is Lorentz invariant by construction it makes little no sense to even think about nonlocality, does it? I gather from your previous comment that this may relate to the possibility of deforming the original gauge theory to obtain a new, different (non-gauge?!) theory. Is that right?
I still find that the sentence from the Wiki I quoted above adds lots of noise and confusion to this otherwise beautiful and important subject. Am I wrong?

Dear Lubos,
I think if Nima can generalize these concepts to more general QFTs such as the ones appearing in the standard model he may be on his way to be named in the context as Einstein or Feynman.

It was "black hole", was not it? Well, probably the joke will change each decade.

Sometimes a very simple idea up ends very complex ideas. I have a question, the jewel looks like a kaleidoscope minus two facets. Ive long felt that the arrow of time does not reduce time to just the one dimension of the present (now) any more than the "arrow of space" (gravity) reduces it to just the one dimension of depth (down) but rather that just as with space where 2 dimensions reveal all (width and height) and one hides things (depth) time has two dimensions that can not be changed (past and future/cause and effect) but one that can change things ( present/connection) Is it possible the two missing facets would be filled in if a mathematical model equates the arrow of time with gravity as an arrow of space?

I am somewhat dismayed about how nonlocality and nonunitary are used in some of the discussions. These things can be very interpretation dependent. As far as amplituhedrons, I like them, and see them tied very closely to certain linear problems. I don't know how else to interpret them as modes in another class of expansion problems though.

It may seem rather abstract to me. I do not know much about this issue. Nhwung seems pretty interesting to find out.

Just an absolute wild speculation, but do you think there is a relation between the jewel an Hilbert Space?

I concur, in particular what about fermions, even massless fermions would be of interest in this direction.

The ideas in that paper made me think... a real subject for further investigation :) Again, congrats for this Blog!

Dear Luboš,

Thank you for this article. I, too, enjoyed reading the article by Natalie Wolchover. It sounds like this is a really big deal! Also, thanks to your link to the Wikipedia article on the ampli-, ample-, ... my fingers don't have practice typing that word yet ;) ... I found this article by Matt von Hippel on Ars Technica: highly recommended! Especially for a general audience! And I was pleasantly surprised by the level of the comments below that article. In fact, those software developers sound more appreciative of cutting-edge theoretical physics than many working physicists! ;)

Is this the confirmation of the old dream of Cvitanovic? "On basis of very skimpy numerical evidence, I conjecture that the gauge invariance induced cancellations are so dramatic that the growth rate of high order perturbation theory corrections to mass-shell gauge-invariant quantities is slower than Dyson's asymptotic series n! estimate."

Nice thought here Bill. It might have to be 6 dimensions.

to calculate amplitudes previously uncalcalculable even with the best computers (due to their complexity) on a sheet of paper.

Well, that sounds very exciting!

This is what the simonsfoundation link says:

How exactly are we supposed to get any usable information out of that, though?

naive question, does it apply to other QFT theories like QCD? I always heard that it is restricted to N=4SYM...