Scattering Amplitude Workshop at SCGP – the Amplituhedron!

I’ve just attended a workshop on Scattering Amplitudes at the Simons center (posting here: http://scgp.stonybrook.edu/archives/7136, and check out the video here), so I thought I would put a few thoughts about the most interesting recent result in this direction – the Amplituhedron. Naturally, since this is the Simons center you need to translate “Scattering Amplitudes” to mean “scattering of strings and super-things”, but the techniques come from good old QFT.

Also, since this is not my direct area of expertise, the content of this post will be from the “interested outsider”…anyway…

The point of all this focus on scattering amplitudes is that although Feynman diagrams provide us a nice way of organizing diagrams for complicated QFT calculations, in the end we always end up calculating things which look like

$\int d^4l \frac{\mathcal{N}}{l^2(l+p_1)^2(l+p_1+p_2)^2(l-p_1)^2}$

(specifically, this is the scattering of 4 particles with momentum $p_1, p_2, p_3$, and $p_4=-(p_1+p_2+p_3)$ at 1-loop with momentum $l$). The numerator $\mathcal{N}$ might be some complicated thing, but when these particles are all massless, all the interesting information is contained at the poles (where $l=p_1$, for example). There is nothing too new in wanting to understand these kinds of integrals – this was the point of “The S-matrix Program”, which lead to all kinds of interesting work in the 1960s and 70s, but has died out a little since then. The revival has occured because apparently when one restricts to N=4 super Yang-Mills, enough simplifications occur so that further progress can be made.

This further progress has lead to “the amptliduhedron”, which has been propagandized as “the end of locality and unitarity in physics!” by Quanta magazine (link) – although is it worth it to note that Quanta is an “editorially independent division of the Simons Foundation”? Anyway, it’s really discussed much better at Sean Carrol’s blog (link). Nearly everything is discussed much better there. This was topic of much debate during this workshop, but last week I got the chance to see one of the original workers on the subject (Nima Arkani-Hamed) give a review of it. Having seen him talk, it’s easy to see why everyone is so excited – he is a fantastic speaker, very passionate and energetic. He also has some “colorful metaphors” to describe how he thinks and works, so it’s not surprising that the blogosphere has attached themselves to him to deliver us from these 19th century notations that the universe should make sense.

So I will try and relay some of his talk to describe this beast (so this next part is not mine). Essentially, look at the integral above. If you were dumb, you might say “the integrand is a product of $d(log l^2)$s”. If you were a little smarter, you would notice that not only does the numerator screw up that nice description, but also you can’t take the log of a dimensional number. However, if you were *a lot* smarter, you would see there is a change of coordinates (which is very similar to the duality transformations that Feynman came up with back in the day) which takes the integrand to the form

$\frac{d\alpha}{\alpha}\frac{d\beta}{\beta}\frac{d\gamma}{\gamma}\frac{d\rho}{\rho}$

(This is the part that only seems to work for N=4 super-YM). The amplituhedron is the geometric shape which describes a form with log-singularities on its boundaries, and thus completely encodes all the information about this scattering amplitude. The sweeping claims made by Arkani-Hamed and others is that *this describes everything*, so the universe can be reduced to a bunch of amplitudhedrons, which do not require locality and unitarity because these transformations are independent of them.

(just a note that unitary is still encoded when you actually *do* the integral, but my understanding is that if they can make this integrand into an honest volume form, unitary will not be needed either).

So the work is very interesting and sounds totally reasonable – but what about the claim that “this describes everything”? Well, if you believe that QFT describes everything (seems reasonable – I guess you have to assume we have no souls), that scattering amplitudes in QFT describe everything (which they don’t – various topological properties are not detected by them), and further that supersymmetry is real (not my bag, but I think 80% of HEP physicists would agree), then the amplitudhedron should “describe everything”. At least, it provides a method to construct any interaction by referring only to a specific geometric object. A rather fascinating idea, but don’t we already have such an object (at least for the standard model), called a fibre bundle?

Anyway, more questions remain, but my impression is that this is the most productive area of S-matrix work – since they have actually been able to analyze a scattering amplitude in terms of some very concrete geometry.  I expect more interesting work and grand claims in the near future!