Thursday, December 27, 2012

Amplitudes, permutations, Grassmannians

Back in July, I mentioned some new highly intriguing results by a group of physicists and mathematicians including Nima Arkani-Hamed and others:
Permutations join twistor minirevolution
That report echoed a talk at Strings 2012. Finally, the paper is out.

One of the 263 figures in the paper.

On the 154 pages of their new article,
Scattering Amplitudes and the Positive Grassmannian (PDF),
Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka expose the power of their new formalism based on the positive Grassmannians (a Grassmannian in this sense is a space of \(k\)-dimensional hyperplans in an \(n\)-dimensional hyperspace; the positivity condition means that all minors i.e. sign-corrected subdeterminants or minors have the positive sign).

The physical building blocks are UV and IR divergences and they are composed into magnificent structures by new Grassmannian-valued variables whose treatment may be described in terms of polytopes, permutations, and otherwise. Conformal theories are mostly analyzed in twistor variables because it is "wise" to do so. However, I feel that twistors have been downgraded to secondary technical tools in this research.

Their methods apply to 2-dimensional integrable systems, the 3-dimensional ABJM theory of the membrane minirevolution, the 4-dimensional maximally supersymmetric gauge theory, and others. They make the full Yangian (or related) symmetries manifest while obscuring locality. So far, I don't quite understand whether the appearance of these very different theories in one papers is just a symptom of some superficial mathematical similarity in the tools that can be used – this is what mathematicians often market as unification but I would never buy it – or whether the unifying picture is more profound.

It's a long paper and the length itself will probably discourage many people from reading it, especially people who resemble your humble correspondent by the expectation that a deeper understanding of a system should ultimately make it possible to write more concise explanations of theories, but a quick reading makes it self-evident that it's a paper that badly deserves to be read. So please try to look at it.

Update: I decided to read the paper rather systematically. It's beautifully written, many things that were only vaguely clear to me or that I had to "guess" are clearly articulated. I will probably postpone the evaluation of my potential far-reaching corollaries and interpretations (that I've spent quite some time thinking about) to a future date, after I am confident that I understand the known stuff properly.


  1. This paper has to do with what Nima spoke briefly about at the end of a lecture that I watched/listen to. It got me very interested.

    However, I'll leave the reading and digesting of it to you, Lubos! ;-)
    But I promise to read some of your wizardly reflections and conclusions, which I'll promise to try to taste best I can. :-)

  2. Just reading the abstract gives me goosebumps...

  3. Yeah, I alway like Lumo's translations of cool papers into an even for me readable form too :-)

  4. I rather like his poems because of their scientific content. I Also think he wrote the paper because of its artistic beauty.

  5. Hi Lubos,
    why no one has proposed a 11-dimensional string theory whose fundamental objects are 3D-branes?

  6. You may "propose" it - at the level of writing the sentence you just did - but if you actually scientifically study whether such a theory may exist and be consistent, you will find out that the answer is No.

    If the fundamental objects are 3-branes, it's not really string theory because the fundamental objects in string theory are 1-branes, also known as strings. This is not an arbitrary choice. The 1-dimensional character of strings - and the related 2D conformal invariance - is needed to get rid of unphysical degrees of freedom and various UV divergences etc.

    String theory is also known to possess higher-dimensional objects that are, at strong coupling and from a more general viewpoint, equally fundamental as the "fundamental" strings. However, the map of the possible backgrounds and objects in string theory shows that in the 11D vacua - "M-theory" - there are no 3-branes. The "fundamental" objects in M-theory are M2-branes and M5-branes.

    All these things are completely determined by maths. There isn't a hair of room in string theory to manipulate or "adjust according to subjective criteria". Consistency completely dictates every detail of the theory - that's really a necessary part of the fact that string theory may be called a "theory of everything".

  7. Hello Lubos, hope all is well.

    On your post "On-shell N=4 SYM: recursively solved to all orders à la BCFW" you said you don't think that the off-shell physics can be extended from the on-shell results of the Arkani-Hamed et al paper.

    However on the new paper they point out the following:

    "We should emphasize that we are not merely using
    on-shell information to determine scattering amplitudes, but rather seeing that the amplitudes can be directly computed in terms of fully on-shell processes. The off-shell, virtual particles familiar from Feynman diagrams are replaced by internal,
    on-shell particles (with generally complex momenta)."


    "For scattering amplitudes, we have seen that off-shell notions like virtual loop integration variables can be fully understood in on-shell terms"

    Do you feel convinced by these claims?

  8. Dear Pablo, I am absolutely convinced that these new claims are right. In other words, the off-shell loop integrals are completely replaced by other integrals that only deal with on-shell data, so that the dependence of the scattering amplitudes on on-shell data is manifest.

    But this is a totally different question than the question in my post you referred to. Indeed, there is still no indication on the market that these methods may be extended to off-shell correlation functions. In some sense, you could say that the quotes at the bottom of your comments are suggesting exactly the opposite - namely that this formalism is inseparable from the on-shell conditions not only on the external particles (gluons) but, in fact, all gluons that ever appear in the calculations.

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  11. Sorry off topic (or maybe not that much ...?):

    Dear Lumo,

    could you not write something nice here if you find time for it?

    I somehow agreed with Qmechanic that people who want to write an answer could ping him, but now it picks me again and I'd really like to read a nice answer to this :-)

  12. Dear Dilaton, dualities are obviously important and unify several seemingly different descriptions. This is by definition of dualities. In this most general sense, they are analogous to the wave-particle dualism and unification of pictures in quantum mechanics and perhaps other things (unification of electricity and magnetism is substantially different).

    However, the question is so incredibly vague, loaded, and arguably unavoidably vacuous that I wouldn't know what to do with it, and I would probably also vote for it to be closed. In particular, the examples that are shown by the author of the question are trivial as well as misleading. The quantum particle is the same thing as the object displaying both wave and particle properties, so the "two" concepts related by the arrow on that line are really the same concept, and the whole relationship claim is vacuous.

    In the same way, the matrix and wave mechanics may be unified but the unification is nothing else than the Dirac formalism for quantum mechanism. We already have this description for dualities in string theory, sort of. One may discuss physics in the description-invariant way. The problem is that we don't have a universal definition of the "Hamiltonian" or "action" but we may still write the general equations with *a* Hamiltonian or an action that is duality-invariant.

    So while the wording of the question is similar to some potential discoveries that many people including me would like to be made, I think that the question in the particular wording that was chosen was nonsensical and I would probably vote to close it, too.

  13. Dear Lumo,

    thanks for these very nice explanations and clarifications, this is what I would have liked to read as an answer over there too :-). I just wanted to know what an expert (as you) thinks and has to say about it.
    (Sometimes "crapily" formulated questions can lead to very nice answers explaining what is wrong and how things really work and have to be thought about...)