The Amplituhedron and Other Excellently Silly Words

Nima Arkani-Hamed recently gave a talk at the Simons Center on the topic of what he and Jaroslav Trnka are calling the Amplituhedron.

There’s an article on it in Quanta Magazine. The article starts out a bit hype-y for my taste (too much language of importance, essentially), but it has several very solid descriptions of the history of the situation. I particularly like how the author concisely describes the Feynman diagram picture in the space of a single paragraph, and I would recommend reading that part even if you don’t have time to read the whole article. In general it’s worth it to get a picture of what’s going on.

That said, I obviously think I can clear a few things up, otherwise I wouldn’t be writing about it, so here I go!

“The” Amplituhedron

Nima’s new construction, the Amplituhedron, encodes amplitudes (building blocks of probabilities in particle physics) in N=4 super Yang-Mills as the “area” of a multi-dimensional analog of a polyhedron (hence, Amplitu-hedron).

Now, I’m a big supporter of silly-sounding words with amplitu- at the beginning (amplitudeologist, anyone?), and this is no exception. Anyway, the word Amplitu-hedron isn’t what’s confusing people. What’s confusing people is the word the.

When the Quanta article says that Nima has found “the” Amplituhedron, it makes it sound like he has discovered one central formula that somehow contains the whole universe. If you read the comments, many readers went away with that impression.

In case you needed me to say it, that’s not what is going on. The problem is in the use of the word “the”.

Suppose it was 1886, and I told you that a fellow named Carl Benz had invented “the Automobile”, a marvelous machine that can get everyone to work on time (as well as become the dominant form of life on Long Island).

My use of “the” might make you imagine that Benz invented some single, giant machine that would roam across the country, picking people up and somehow transporting everyone to work. You’d be skeptical of this, of course, expecting that long queues to use this gigantic, wondrous machine would swiftly ruin any speed advantage it might possess…

The Automobile, here to take you to work.

Or, you could view “the” in another light, as indicating a type of thing.

Much like “the Automobile” is a concept, manifested in many different cars and trucks across the country, “the Amplituhedron” is a concept, manifested in many different amplituhedra, each corresponding to a particular calculation that we might attempt.


Each amplituhedron has to do with an amplitude involving a specific number of particles, with a particular number of internal loops. (The Quanta article has a pretty good explanation of loops, here’s mine if you’d rather read that). Based on the problem you’re trying to solve, there are a set of rules that you use to construct the particular amplituhedron you need. The “area” of this amplituhedron (in quotation marks because I mean the area in an abstract, mathematical sense) is the amplitude for the process, which lets you calculate the probability that whatever particle physics situation you’re describing will happen.

Now, we already have many methods to calculate these probabilities. The amplituhedron’s advantage is that it makes these calculations much simpler. What was once quite a laborious and complicated four-loop calculation, Nima claims can be done by hand using amplituhedra. I didn’t get a chance to ask whether the same efficiency improvement holds true at six loops, but Nima’s description made it sound like it would at least speed things up.

[Edit: Some of my fellow amplitudeologists have reminded me of two things. First, that paper I linked above paved the way to more modern methods for calculating these things, which also let you do the four-loop calculation by hand. (You need only six or so diagrams). Second, even back then the calculation wasn't exactly "laborious", there were some pretty slick tricks that sped things up. With that in mind, I'm not sure Nima's method is faster per se. But it is a fast method that has the other advantages described below.]

The amplituhedron has another, more sociological advantage. By describing the amplitude in terms of a geometrical object rather than in terms of our usual terminology, we phrase things in a way that mathematicians are more likely to understand. By making things more accessible to mathematicians (and the more math-headed physicists), we invite them to help us solve our problems, so that together we can come up with more powerful methods of calculation.

Nima and the Quanta article both make a big deal about how the amplituhedron gets rid of the principles of locality and unitarity, two foundational principles of quantum field theory. I’m a bit more impressed by this than Woit is. The fine distinction that needs to be made here is that the amplituhedron isn’t simply “throwing out” locality and unitarity. Rather, it’s written in such a way that it doesn’t need locality and unitarity to function. In the end, the formulas it computes still obey both principles. Nima’s hope is that, now that we are able to write amplitudes without needing locality and unitarity, if we end up having to throw out either of those principles to make a new theory we will be able to do so. That’s legitimately quite a handy advantage to have, it just doesn’t mean that locality and unitarity must be thrown out right now.

…and Disadvantages

It’s important to remember that this whole story is limited to N=4 super Yang-Mills. Nima doesn’t know how to apply it to other theories, and nobody else seems to have any good ideas either. In addition, this only applies to the planar part of the theory. I’m not going to explain what that term means here; for now just be aware that while there are tricks that let you “square” a calculation in super Yang-Mills to get a similar calculation in quantum gravity, those tricks rely on having non-planar data, or information beyond the planar part of the theory. So at this point, this doesn’t give us any new hints about quantum gravity. It’s conceivable that physicists will find ways around both of these limits, but for now this result, though impressive, is quite limited.

Nima hasn’t found some sort of singular “jewel at the heart of physics”. Rather, he’s found a very slick, very elegant, quite efficient way to make calculations within one particular theory. This is profound, because it expresses things in terms that mathematicians can address, and because it shows that we can write down formulas without relying on what are traditionally some of the most fundamental principles of quantum field theory. Only time will tell whether Nima or others can generalize this picture, taking it beyond planar N=4 super Yang-Mills and into the tougher theories that still await this sort of understanding.

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14 thoughts on “The Amplituhedron and Other Excellently Silly Words

  1. David Horgan

    Hi really enjoyed this post. Very clear and balanced description of the amplituhedron story and its significance. I notice you are doing research in an area of physics I’m interested in so I’ll be reading your other posts and following this blog. By own research work is looking at the quantum tetrahedron at the moment by research blog is at:

  2. Miguel Martínez

    Actually there is a “The”… it is The Master Amplituhedron. “With an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides. Its volume represents, in theory, the total amplitude of all physical processes.”

      1. 4gravitonsandagradstudent Post author

        Ah yes, I was wondering if someone was going to ask about this.

        First of all, the “Master Amplituhedron” is more speculative. Nima didn’t make a big point of it in his talk.

        Second, He did clarify in response to a question that his method is inherently order-by-order in loops and that it’s unclear how it could be generalized to the limit of infinite loops. It’s important to understand the difference here between an amplituhedron that corresponds to an infinite number of particles, and one that corresponds to an infinite number of loops. Even with a “Master Amplituhedron” that gives you a process with infinite numbers of particles, you still have to calculate one of these for every additional loop of precision you need.

        My guess (and again, this is a guess, since Nima didn’t say much about the Master Amplituhedron in his talk) is that the Master Amplituhedron refers to the fact that, since you can get from an amplitude with a higher number of particles to an amplitude with a lower number of particles by a fairly straightforward procedure, you can imagine deriving a chain of related amplitudes with different numbers of particles from a single, infinite, “Master” amplitude/amplituhedron. I don’t think that this means that Nima has a practical handle on how to actually use this object, but again, I may be mistaken. I would think that if it was that solid he would have said more about it.

        Edit: As a way to clarify: this also wouldn’t be some sort of “object that contains the whole universe”. What you might be imagining is that this would contain the lower-particle amplitudes in that it would tell you what happens first, what happens second, etc., on to the whole history of the universe. That’s not what this object would do. Rather, it still only answers the question “Given x particles coming in, what is the probability that y particles come out, to a specific level of precision?” It just would relate those probabilities to eachother, in a way that would let you think of them as being results of the same overall formula.

  3. JollyJoker

    “He did clarify in response to a question that his method is inherently order-by-order in loops and that it’s unclear how it could be generalized to the limit of infinite loops.”

    Is the lack of explicit unitarity and locality relevant here? I assume only the final all loops result has to obey these two. I remember Nima in some video saying that the requirement for unitarity in intermediate steps brought a lot of unnecessary complexity to (calculations using) Feynman diagrams.

    1. 4gravitonsandagradstudent Post author

      No actually, each order in loops ends up obeying locality and unitarity in the end. (Basically, you can think of a given order in loops as a result of setting the coupling constant to a particular (small) value and specifying how good your detectors are. Neither of those should change basic principles like locality.) The main point is that in Nima’s approach only the sum of all diagrams at any particular loop order obeys unitarity+locality, while in the Feynman diagram approach each individual diagram has to obey those principles.

      1. JollyJoker

        Ok, so intermediate steps have locality and unitarity as long as the steps are loop-by-loop. (Even with strong coupling?) Still, if you look at it completely generally rather than just this particular way, only the final result needs to obey them?

      2. 4gravitonsandagradstudent Post author

        “Even with strong coupling” doesn’t mean anything in this context. If you’ve got strong coupling, it means that you have no reason to think that higher loops will give smaller contributions than lower loops, so you don’t end up thinking about things in terms of loops at all. (The coupling is just a number you multiply each loop result by, that gets higher powers for higher loops. So if it’s much less than 1, you’ve got a series that might converge, but if it’s greater than 1 you’re in trouble and need other methods.)

        But I think I see what you’re saying. If you’ve got a situation where the coupling is strong, and you can’t expand in loops, then it’s the final result that ought to be unitary and local, and intermediate steps in general wouldn’t have to be.

      3. JollyJoker

        “Even with strong coupling” was in reference to “Basically, you can think of a given order in loops as a result of setting the coupling constant to a particular (small) value”

        Heh, I know a bit too little to come up with understandable questions :) But if loop-by-loop isn’t doable when you have strong coupling it would seem the amplituhedron can’t be “fundamental” (generic?) in some sense?

      4. 4gravitonsandagradstudent Post author

        Ok, I get ya.

        This is actually a pretty deep problem for quantum field theory in general (and often string theory too). We’re at our best when the coupling is small and we can compute things loop by loop. If we see the same sorts of structures at each loop, we can hypothesize that similar structures are relevant even when the coupling is large and the loop approach doesn’t make sense. But it’s always an open question whether new things might crop up when the coupling becomes large, and there are plenty of examples where they do!

        There are things that can be computed at strong coupling, but it’s generally harder and has to make use of more indirect techniques/tricks. Of course, what’s best is when a result can be understood both at strong coupling and at weak coupling, or when we can compare strong and weak coupling predictions and see if they can be smoothly related. Not to toot my own horn, but I’ve been involved in an example of the latter (plots on page 73 and 75 of that made heavy use of an example of the former (

  4. david

    I don’t know why you worried about hype .if this progress in physics and math can’t get you excited you chose the Wrong Career for yourself .i think article from psmag show why public and legendary physicists like edward witten are excited over this.and i’m not a physicist and didn’t think Amplituhedron contain whole universe like you write but get excited about this progress.

    1. 4gravitonsandagradstudent Post author

      There’s a difference between hype and excitement. I think these are fascinating, interesting results that deserve appreciation by the public (otherwise, why would I blog about them?), but they ought to be put into their proper context. Essentially, it’s the “language of importance” issue I talk about here:

      That article you link is an interesting example of that problem, actually. The interviewer keeps going off on tangents about how scary it is living in a world without space and time etc., while Bourjaily keeps trying to (pretty competently) draw things back to the real nature of the results and why they’re interesting for their own merits. (Though he does repeat the claim that these methods are more efficient, which I think just means more efficient than Feynman diagrams but just makes me more intent on cornering one of these guys to ask whether it’s more efficient than the dual conformal ansatz method used here:

  5. telescoper

    Reblogged this on In the Dark and commented:
    No time for a proper post between meetings, so I thought I’d take the opportunity to reblog one of many interesting posts I’ve seen recently about the Amplituhedron. “The what?” I hear you say. Well, read on. And, better still, perhaps you can pass an opinion on whether it is more than hype…


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