Today’s Slashdot tells us that Physicists Discover Geometry Underlying Particle Physics, a story that is based on an excellent article, A Jewel at the Heart of Quantum Physics, by Natalie Wolchover at the new *Quanta Magazine* sponsored by the Simons Foundation.

As you might suspect, the Slashdot headline is simply nonsense. What’s really going on here is some new progress on computing scattering amplitudes in a very special conformally-invariant QFT, one not known to “underly particle physics”. This is a long story, one going back to Roger Penrose’s work on twistors from the late 1960s. In recent years this has been a very active and successful field of mathematical physics research, with a large group last year putting out Scattering Amplitudes and the Positive Grassmanian, which showed how to express some amplitudes to all loops in terms of volumes of geometric objects defined as subspaces of a Grassmanian. Mathematicians who want to see some speculation about the relation of this to other areas of mathematics should take a look at section 15 of that paper.

The more recent news is that Nima Arkani-Hamed and his ex-student Jaroslav Trnka now have an improvement on that calculational method, which uses the volume of a particular such geometric object they call the “Amplituhedron”. There’s no paper yet, but you can watch recent Arkani-Hamed talks about this here or here (the last from yesterday). How this ended up with the ridiculous Slashdot headline is pretty clear, as Arkani-Hamed with his trademark enthusiasm promotes this work as a road to revolutionizing physics, getting rid of locality and unitarity as fundamental principles, finding emergent space-time, maybe emergent quantum mechanics, etc (while admitting that what has been accomplished is just step 0 of step 1 of a multi-step program). From this, one gets to the rather excessive Quanta headline about a “jewel at the heart of quantum mechanics”, ensuring that the next stage of publicity (e.g. Slashdot) will launch the hype level into outer space, escaping any relationship to reality.

For the details of what this really is, the Quanta article gives a good overview, but you need to consult the long paper and recent talks to dig out a non-hyped version of what the real recent advances are. I’m nowhere near expert enough to provide this, hope that if this turns out to be as important as claimed, surely there will soon be lots of expositions of the story from various points of view. In the meantime, best perhaps to pay attention to what Witten has to say on the topic:

The field is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be.

**Update**: Here are some slides about the Amplituhedron from Trnka (hat-tip to George Ellis).

**Update**: Scott Aaronson has come up with an even more dramatic advance, the discovery of the Unitarihedron, which includes the Ampltituhedron as a special case, just “a single sparkle on an infinitely greater jewel”. See his posting The Unitarihedron: The Jewel at the Heart of Quantum Computing, where he unveils this new theory.

**Update**: See comments here by Lance Dixon and this paper for an alternative approach to computing planar amplitudes in this theory, one not using the “amplituhedron”.

**
Update**: Congratulations to Kosower, Dixon and Bern for the award of the Sakurai prize for their work on amplitudes.

**Update**: Dixon has a guest post about this topic at Sean Carroll’s blog.

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I found this article on Reddit! So I came over to see if you had smacked it around like it was a puppy slobbering all over your shoes…

Thx! My takeaway is its interesting, possibly important, but still just another angle to await further actual results..

Here’s a set of slides about it:

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

Can anyone give a brief explanation of “maximally supersymmetric Yang-Mills theory” at an undergrad level?

It sounds sorta like the spherical chicken/cow of higher dimensional quantum field theories. I got a bit excited by the Quanta Magazine article until I hit that sentence and realized it was probably still all wrapped up with supersymmetry and string theory at its core…

Lamont,

What’s being referred to is the “CFT” in the AdS/CFT correspondance, the QFT that is supposed to be dual to a certain string theory. It’s a very interesting 4d QFT, conformally invariant and the subject of a huge amount of attention, but not directly related to the QFTs of the Standard Model.

I wonder if you aren’t leaving out the most special thing about this result. In this special case, all those annoying extra guage degrees of freedom go away. They have been a nuisance to many smart people for a while.

What struck me about this article is that it seems to heavily involve twistors and yet Penrose gets no mention whatsoever. But Witten does, rather irrelevantly. One is left in no doubt which side of the Atlantic is having its curry favoured.

More destructive boosterism.

This work descends from Witten’s 2003 invention of twistor string theory.

Nasren,

You’re wrong. Witten is very relevant to this story. Penrose is relevant, too, but not as relevant as Witten for this particular line of research on scattering amplitudes. Witten’s paper

hep-th/0312171

is one of the most cited in this amplitude business, and he is the W in BCFW, which was a crucial paper.

You don’t have to be so critical! And if you’re going to be, at least get your facts straight first.

I stand corrected.

@ Lamont Granquist & others who might not know what “maximally supersymmetric Yang Mills” is:

Take a look at the rather nice explanations of Matt von Hippel:

http://arstechnica.com/science/2013/05/earning-a-phd-by-studying-a-theory-that-we-know-is-wrong/

or his rather excellent blog:

http://4gravitonsandagradstudent.wordpress.com/a-theorists-theory/

After browsing the interesting slides, I was struck by the repeated “not known to mathematicians”. I am a mathematician and I don’t have any precise idea about what exactly those amplitudes are. However, as far as I understand after a quick browsing of the slides, my understanding is the following: with a goal which I don’t grasp, they attach some “forms with logarithmic singularities” to geometric data coming from affine parametrizations of points in convex polygons in projective spaces. (No wonder that some grasmannian type spaces appear and that some geometric constraints are expressed as the positivity and invariance of parametrizations.) Then they make sense of some, impenetrable to me, calculation by reformulating it in graph rewriting terms. Maybe is not at all related, but that’s what I do as a mathematician (impenetrable physics goal aside). Affine parametrizations of such convex polygons are particular examples (yes,mathematician, sorry) of what I call emergent algebras, i.e. one-parameter deformations of idempotent right quasigroups (that’s related to Yang-Baxter, of course), and their graph rewriting version is called graphic lambda calculus. A friend noticed some days ago that the identity (13.1), p. 94 arxiv:1212.5605 looks very much alike my graphic beta move arxiv:1305.5786 (or better web tutorial), which is the graph rewriting version of the beta reduction in lambda calculus. So, maybe, with excuses for what could be a completely unrelated rant, but maybe some of it is not unknown to mathematicians.

@chorasimilarity: I think the “not known to mathematicians” line is somewhat hyperbolic. Two of the co-authors, Goncharov and Postnikov, are mathematicians, so the lower bound for “number of mathematicians who know this” is 2, if not higher. Goncharov, in particular, has done quite a bit of work dealing with polylogarithms, algebraic cycles and Feynman diagrams.

@anewcommenter:

The claim in the talk that I saw was that the amplituhedron, not the positive Grassmannian, was new to mathematicians. The latter is definitely not new. Since the paper isn’t out and we don’t know the precise definition of the amplituhedron, it’s hard to say for sure whether or not it is known to mathematicians; hearsay is the best we can do.

That being said, Nima’s mathematician friends would likely know if such objects exist in the math literature. It’s not unheard of that physicists stumble upon something that mathematician’s haven’t seen yet, e.g. mirror symmetry. Perhaps we have again stumbled upon something new . . .

Considering that this is a theory in the planar limit, and the scattering amplitudes are all on-shell, what reason is there to assume that there is no qualitative transition between Trnka’s step 1.1 and “grand vision”? (slide 2 of http://www.staff.science.uu.nl/~tonge105/igst13/Trnka.pdf )

could all this not be a mathematically elegant way to codify conformal invariance, unitarity, and energy-momentum conservation, rather than a “radical reformulation” of a generic QFT?

So as a layman on the subject, let me see if I’ve distilled the media hype to what is really going on here.

They’ve found a replacement for Feynman Diagrams? Is that the meat and potatoes of this announcement?

CIP,

I don’t think gauge symmetry is just a nuisance, it’s part of the deep structure of our best theory (and an integral part of the modern point of view on geometry). Yes, it creates all sorts of problems in a QFT, but you could also see these as opportunities. Anyway, I think I’ve gone on about this before, and should again sometime, but it’s a complicated story for which simplistic sloganeering isn’t very interesting.

lun,

Yes, that’s a real question. What people have found here is the solution to a very special problem, and from this claiming evidence for a revolutionary reworking of the foundations of physics seems very premature. There’s a huge amount to be done to understand the significance of why these specific amplitudes can be expressed in this form.

Beelzebud,

Yes, in a very specific, highly symmetric toy theory they have a way of getting scattering amplitudes without summing the usual Feynman diagrams. Quite interesting piece of mathematical physics, but I suggest taking Witten’s advice:

“it is difficult to guess what will happen or what the lessons will turn out to be.”

Just a very naive question:

If one can do everything on-shell, no virtual particles any more, what is left of q.m., e.g. the uncertainty principle ?

Thomas,

That’s not so naive. Part of Arkani-Hamed’s program refers to maybe even replacing QM, with QM “emergent”. This kind of exact formula indicates that something may be going on here like a classical approximation being exact.

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Peter,

thanks for your answer.

Having seen the video, my hunch is that something really “deep” is going on here, but I guess one has to wait and see how things develop further.

@P:

I was being a bit tongue-in-cheek in that comment. Of course the amplituhedron is a novel thing to have come out of the work, and I trust that the authors have verified that it is new, to the best of their abilities.

However, I also wanted to object to assertions like your analogy to mirror symmetry. This is quite unlike mirror symmetry – which, to my limited knowledge, appeared seemingly unbidden from physics – in that there are mathematical precursors to Arkani-Hamed et al. (as well as work done in parallel and independently, such as the work of @chorasimilarity) that are brought together in that work.

In particular, I mentioned Goncharov’s work on Feynman diagrams and polylogarithms, because there have been glimpses of computational efficiency emerging from the work of people studying that locus of ideas, cf. Kreimer and others. This has been something that some mathematicians have found to be exciting for at least the past few years now.

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Thank, you, very interesting subject that I was not aware of – Amplituhedron articles.

19 years ago, people stopped using Feynman diagrams to compute perturbative scattering amplitudes in N=4 SYM, and began using unitarity. That was also the beginning of the focus on the analytic properties of the loop integrand — the object which the amplituhedron is describing for N=4 SYM in the `planar’ limit of a large number of colors. Ironically, this year the loop integrand has become obsolete for computing at least the simplest nontrivial amplitudes in this planar N=4 SYM theory. See arXiv:1308.2276, in which the 3-loop 6-point amplitude was computed, as a function of the external data only (all loop integrals having been performed). The loop integrand was bypassed entirely. Certain boundary information necessary for the construction was supplied from a source (the operator product expansion and integrability) that can be tapped to all orders in the coupling, so the method is very robust.

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Lance thank you very much for the clarifications.

For non-experts can you also specify

a) to what extent can one be hopeful that

these results can be generalized beyond

CFT

b) I take it this goes beyond on-shell scattering amplitudes

Is that correct? How relevant are such

calculations to efforts to probe N=4 is integrable?

(Conversely if it is not integrable would

You expect such methods to fail?)

Thanks

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Lun: a) unitarity for the loop integrand works in any theory, CFT or not. It’s used in QCD calculations for the LHC, for example. The CFT in question (N=4 SYM) is very special, without a doubt, so you can go much further than in (say) QCD. The complete four-loop four-point integrand is known, for example, whereas 2 loops is the current limit for the QCD 4-point amplitude.

The limit of this CFT where the number of colors is large (planar N=4 SYM, corresponding to only drawing the planar Feynman diagrams) is even more special, however. Its amplitudes can be mapped to “polygonal Wilson loops” – which you can visualize as a polygonal rigid wire frame tiled with Feynman diagrams. Related to that, there is a symmetry that is like conformal symmetry, so that the answers only depend on a subset of the variables you thought they did. In fact the scattering of 2->2 gluons, or 2->3 gluons is totally fixed by this symmetry (up to constants). So the first nontrivial scattering is 2->4 (the 6-point amplitude). Also, integrability should only hold in the planar limit.

b) Everything I was talking about was with the aim of determining on-shell scattering amplitudes. However, they are closely related to the spectrum and other properties of the gauge-invariant composite operators of the CFT. The operator dimensions and operator product coefficients of the CFT appear as you take limits where the scattering amplitude degenerates or factorizes (where that wire frame assumes some singular limiting form where two or more edges become parallel). In fact, such limits simultaneously provide information to fix the answer for the generic, non-limiting case, and also test the source of that limiting information – which includes integrability and further assumptions. So yes, the fact that there is a consistent solution tests integrability. And if the approach failed at a given point, it could indicate a failure of integrability (or of some other assumption, so one would have to figure out which). So far, no failures though!

Just to clarify for Lun on point b), in case this was the source of the confusion: Nima’s diagrams are “on-shell” in the sense that they represent things on the inside in terms of on-shell rather than the off-shell states that Feynman diagram methods use. That’s just a feature of the particular way he sets things up, it doesn’t have anything to do with whether the incoming and outgoing particles are on-shell or off-shell.

When people talk about amplitudes, they’re generally talking about processes where the external particles are on-shell. There are other terms (correlation functions in particular) for the more general case.

Thanks Lance,

I’ve added a note to the posting with a link to the paper that you mention. If there are other sources you’d recommend for people who want to learn more about this, let me know.

http://www.youtube.com/watch?v=2rjbtsX7twc

The lyrics are going to have to changed….again!!!!

Thanks for adding that note, Peter. Our approach to planar N=4 SYM works hand in hand with the boundary data (from the OPE and integrability) provided by Benjamin Basso, Amit Sever and Pedro Vieira in their papers 1303.1396 and 1306.2058. In brief, they use the polygonal Wilson loop formulation of the problem. They solve the problem of exact, factorizable scattering for a system with one time dimension and one space dimension, which is roughly the direction along an edge of the polygon. Once they have that 2-d scattering matrix, they find a solution for a pentagonal Wilson polygon in terms of it. Then they build all other polygons (in a certain limit) out of their pentagons. The cool thing is that when they solve a certain part of the problem, they do so to all orders in perturbation theory in the ‘t Hooft coupling, thanks to the exact 2-d scattering matrix (which in turn is thanks to integrability).

If we don’t have to sum up Feynman diagrams but only have to compute volumes of the Amplituhedron, then the Feynman-diagram-inspired picture of the vacuum as a seething cauldron of virtual particles may be an artifact of the method of computation. Doesn’t this cast a different light on the alleged non-naturalness of the value of the cosmological constant?