Tag Archives: Nima Arkani-Hamed

So You Want to Prove String Theory, Part II: How Can QCD Be a String Theory?

A couple weeks back, I had a post about Nima Arkani-Hamed’s talk at Strings 2016. Nima and his collaborators were trying to find what sorts of scattering amplitudes (formulas that calculate the chance that particles scatter off each other) are allowed in a theory of quantum gravity. Their goal was to show that, with certain assumptions, string theory gives the only consistent answer.

At the time, my old advisor Michael Douglas suggested that I might find Zohar Komargodski’s talk more interesting. Now that I’ve finally gotten around to watching it, I agree. The story is cleaner, more conclusive…and it gives me an excuse to say something else I’ve been meaning to talk about.

Zohar Komargodski has a track record of deriving interesting results that are true not just for the sorts of toy models we like to work with but for realistic theories as well. He’s collaborating with amplitudes miracle-worker Simon Caron-Huot (who I’ve collaborated with recently), Amit Sever (one of the integrability wizards who came up with the POPE program) and Alexander Zhiboedov, whose name seems to show up all over the place. Overall, the team is 100% hot young talent, which tends to be a recipe for success.

While Nima’s calculation focuses on gravity, Zohar and company are asking a broader question. They’re looking at any theory with particles of high spin and nonzero mass. Like Nima, they’re looking at scattering amplitudes, in the limit that the forces involved are weak. Unlike Nima, they’re focusing on a particular limit: rather than trying to fix the full form of the amplitude, they’re interested in how it behaves for extreme, unphysical values for the particles’ momenta. Despite being unphysical, this limit can reveal something about how the theory works.

What they figured out is that, for the sorts of theories they’re looking at, the amplitude has to take a particular form in their unphysical limit. In particular, it takes a form that indicates the presence of strings.

What sort of theories are they looking at? What theories have “particles of high spin and nonzero mass”? Well, some are string theories. Others are Yang-Mills theories … theories similar to QCD.

For the experts, I encourage you to watch Zohar’s talk or read the paper for more detail. It’s a fun story that showcases how very general constraints on scattering amplitudes can translate into quite specific statements.

For the non-experts, though, there’s something that may already be confusing. When I’ve talked about Yang-Mills theories before, I’ve talked about them in terms of particles of spin 1. Where did these “higher spin” particles come from? And where are the strings? How can there be strings in a theory that I’ve described as “similar to QCD”?

If I just stuck to the higher spin particles, things could almost stay familiar. The fundamental particles of Yang-Mills theories have spin 1, but these particles can combine into composite particles, which can have higher spin and higher mass. That should be intuitive: in some sense, it’s just like protons, neutrons, and electrons combining to form atoms.

What about the strings? I’ve actually talked about that before, but I’d like to try out a new analogy. Have you ever heard of Conway’s Game of Life?


Not this one!


This one!

Conway’s Game of Life starts with a grid of black and white squares, and evolves in steps, with each square’s color determined by the color of adjacent squares in the last step. “Fundamentally”, the game is just those rules. In practice, though, structure can emerge: a zoo of self-propagating creatures that dance across the screen.

The strings that can show up in Yang-Mills theories are like this. They aren’t introduced directly in the definition of the theory. Instead, they’re consequences: structures that form when you let the rules evolve and see what they create. They’re another description of the theory, one with its own advantages.

When I tell people I’m a theoretical physicist, they inevitably ask me “Have any of your theories been tested?” They’re operating from one idea of what a theoretical physicist does: propose new theories to describe the world, based on available evidence. Lots of theorists do that, they’re called phenomenologists, but it’s not what I do, or what most theorists I interact with day-to-day do.

So I describe what I do, how I test new mathematical techniques to make particle physics calculations faster. And in general, that’s pretty easy for people to understand. Just as they can imagine people out there testing theories, they can imagine people who work to support the others, making tools to make their work easier. But while that’s what I do, it’s not the best description of what most of my colleagues do.

What most theorists I know do is like finding new animals in Conway’s game of life. They start with theories for which we know the rules: well-tested theories like QCD, or well-studied proposals like string theory. They ask themselves, not how they can change the rules, but what results the rules have. They look for structures, and in doing so find new perspectives, learning to see the animals that live on Conway’s black and white grid. (This is something I’ve gestured at before, but this seems like a cleaner framing.)

Doing that, theorists have seen strings in the structure of QCD-like theories. And now Zohar and collaborators have a clean argument that the structures others have seen should show up, not only there, but in a broader class of theories.

This isn’t about whether the world is fundamentally described by string theory, ten dimensions and all. That’s an entirely different topic. What it is is a question about what sorts of structures emerge when we try to describe the world. What it does is show that strings are, in some sense (and, as for Nima, [with some conditions]) inevitable, that they come out of our rules even if we don’t expect them to.

So You Want to Prove String Theory (Or: Nima Did Something Cool Again)

Nima Arkani-Hamed, of Amplituhedron fame, has been making noises recently about proving string theory.

Now, I can already hear the smartarses in the comments correcting me here. You can’t prove a scientific theory, you can only provide evidence for it.

Well, in this case I don’t mean “provide evidence”. (Direct evidence for string theory is quite unlikely at the moment given the high energies at which it becomes relevant and large number of consistent solutions, but an indirect approach might yet work.) I actually mean “prove”.

See, there are two ways to think about the problem of quantum gravity. One is as an experimental problem: at high enough energies for quantum gravity to be relevant, what actually happens? Since it’s going to be a very long time before we can probe those energies, though, in practice we instead have a technical problem: can we write down a theory that looks like gravity in familiar situations, while avoiding the pesky infinities that come with naive attempts at quantum gravity?

If you can prove that string theory is the only theory that does that, then you’ve proven string theory. If you can prove that string theory is the only theory that does that [with certain conditions] then you’ve proven string theory [with certain conditions].

That, in broad terms, is what Nima has been edging towards. At this year’s Strings conference, he unveiled some progress towards that goal. And since I just recently got around to watching his talk, you get to hear my take on it.

 Nima has been working with Yu-tin Huang, an amplitudeologist who tends to show up everywhere, and one of his students. Working in parallel, an all-star cast has been doing a similar calculation for Yang-Mills theory. The Yang-Mills story is cool, and probably worth a post in its own right, but I think you guys are more interested in the quantum gravity one.

What is Nima doing here?

Nima is looking at scattering amplitudes, probabilities for particles to scatter off of each other. In this case, the particles are gravitons, the particle form of gravitational waves.

Normally, the problems with quantum gravity show up when your scattering amplitudes have loops. Here, Nima is looking at amplitudes without loops, the most important contributions when the force in question is weak (the “weakly coupled” in Nima’s title).

Even for these amplitudes you can gain insight into quantum gravity by seeing what happens at high energies (the “UV” in the title). String amplitudes have nice behavior at high energies, naive gravity amplitudes do not. The question then becomes, are there other amplitudes that preserve this nice behavior, while still obeying the rules of physics? Or is string theory truly unique, the only theory that can do this?

The team that asked a similar question about Yang-Mills theory found that string theory was unique, that every theory that obeyed their conditions was in some sense “stringy”. That makes it even more surprising that, for quantum gravity, the answer was no: the string theory amplitude is not unique. In fact, Nima and his collaborators found an infinite set of amplitudes that met their conditions, related by a parameter they could vary freely.

What are these other amplitudes, then?

Nima thinks they can’t be part of a consistent theory, and he’s probably right. They have a number of tests they haven’t done: in particular, they’ve only been looking at amplitudes involving two gravitons scattering off each other, but a real theory should have consistent answers for any number of gravitons interacting, and it’s doesn’t look like these “alternate” amplitudes can be generalized to work for that.

That said, at this point it’s still possible that these other amplitudes are part of some sort of sensible theory. And that would be incredibly interesting, because we’ve never seen anything like that before.

There are approaches to quantum gravity besides string theory, sure. But common to all of them is an inability to actually calculate scattering amplitudes. If there really were a theory that generated these “alternate” amplitudes, it wouldn’t correspond to any existing quantum gravity proposal.

(Incidentally, this is also why this sort of “proof” of string theory might not convince everyone. Non-string quantum gravity approaches tend to talk about things fairly far removed from scattering amplitudes, so some would see this kind of thing as apples and oranges.)

I’d be fascinated to see where this goes. Either we have a new set of gravity scattering amplitudes to work with, or string theory turns out to be unique in a more rigorous and specific way than we’ve previously known. No matter what, something interesting is going to happen.

After the talk David Gross drew on his experience of the origin of string theory to question whether this work is just retreading the path to an old dead end. String theory arose from an attempt to find a scattering amplitude with nice properties, but it was only by understanding this amplitude physically in terms of vibrating strings that it was able to make real progress.

I generally agree with Nima’s answer, but to re-frame it in my own words: in the amplitudes sub-field, there’s something of a cycle. We try to impose general rules, until by using those rules we have a new calculation technique. We then do a bunch of calculations with the new technique. Finally, we look at the results of those calculations, try to find new general rules, and start the cycle again.

String theory is the result of people applying general rules to scattering amplitudes and learning enough to discover not just a new calculation technique, but a new physical theory. Now, we’ve done quite a lot of string theory calculations, and quite a lot more quantum field theory calculations as well. We have a lot of “data”.

And when you have a lot of data, it becomes much more productive to look for patterns. Now, if we start trying to apply general rules, we have a much better idea of what we’re looking for. This lets us get a lot further than people did the first time through the cycle. It’s what let Nima find the Amplituhedron, and it’s something Yu-tin has a pretty good track record of as well.

So in general, I’m optimistic. As a community, we’re poised to find out some very interesting things about what gravity scattering amplitudes can look like. Maybe, we’ll even prove string theory. [With certain conditions, of course. 😉 ]

Where do you get all those mathematical toys?

I’m at a conference at Caltech this week, so it’s going to be a shorter post than usual.

The conference is on something call the Positive Grassmannian, a precursor to Nima Arkani-Hamed’s much-hyped Amplituhedron. Both are variants of a central idea: take complicated calculations in physics and express them in terms of clean, well-defined mathematical objects.

Because of this, this conference is attended not just by physicists, but by mathematicians as well, and it’s been interesting watching how the two groups interact.

From a physics perspective, mathematicians are great because they give us so many useful tools! Many significant advances in my field happened because a physicist talked to a mathematician and learned that a problem that had stymied the physics world had already been solved in the math community.

This tends to lead to certain expectations among physicists. If a mathematician gives a talk at a physics conference, we expect them to present something we can use. Our ideal math talk is like when Q presents the gadgets at the beginning of a Bond movie: a ton of new toys with just enough explanation for us to use them to save the day in the second act.

Pictured: Mathematicians, through Physicist eyes

You may see the beginning of a problem here, once you realize that physicists are the James Bond in this analogy.

Physicists like to see themselves as the protagonists of their own stories. That’s true of every field, though, to some degree or another. And it’s certainly true of mathematicians.

Mathematicians don’t go to physics conferences just to be someone’s supporting cast. They do it because physics problems are interesting to them: by hearing what physicists are working on they hope to get inspiration for new mathematical structures, concepts jury-rigged together by physicists that represent corners that mathematics hasn’t yet explored. Their goal is to take home an idea that they can turn into something productive, gaining glory among their fellow mathematicians. And if that sounds familiar…

Pictured: Physicists, through Mathematician eyes

While it’s amusing to watch the different expectations go head-to-head, the best collaborations between physicists and mathematicians are those where both sides respect that the other is the protagonist of their own story. Allow for give-and-take, paying attention not just to what you find interesting but to what the other person does, without assuming a tired old movie script, and it’s possible to make great progress.

Of course, that’s true of life in general as well.

What Can Replace Space-Time?

Nima Arkani-Hamed is famous for believing that space-time is doomed, that as physicists we will have to abandon the concepts of space and time if we want to find the ultimate theory of the universe. He’s joked that this is what motivates him to get up in the morning. He tends to bring it up often in talks, both for physicists and for the general public.

The latter especially tend to be baffled by this idea. I’ve heard a lot of questions like “if space-time is doomed, what could replace it?”

In the past, Nima and I both tended to answer this question with a shrug. (Though a more elaborate shrug in his case.) This is the honest answer: we don’t know what replaces space-time, we’re still looking for a good solution. Nima’s Amplituhedron may eventually provide an answer, but it’s still not clear what that answer will look like. I’ve recently realized, though, that this way of responding to the question misses its real thrust.

When people ask me “what could replace space-time?” they’re not asking “what will replace space-time?” Rather, they’re asking “what could possibly replace space-time?” It’s not that they want to know the answer before we’ve found it, it’s that they don’t understand how any reasonable answer could possibly exist.

I don’t think this concern has been addressed much by physicists, and it’s a pity, because it’s not very hard to answer. You don’t even need advanced physics. All you need is some fairly old philosophy. Specifically we’ll use concepts from metaphysics, the branch of philosophy that deals with categories of being.

Think about your day yesterday. Maybe you had breakfast at home, drove to work, had a meeting, then went home and watched TV.

Each of those steps can be thought of as an event. Each event is something that happened that we want to pay attention to. You having breakfast was an event, as was you arriving at work.

These events are connected by relations. Here, each relation specifies the connection between two events. There might be a relation of cause-and-effect, for example, between you arriving at work late and meeting with your boss later in the day.

Space and time, then, can be seen as additional types of relations. Your breakfast is related to you arriving at work: it is before it in time, and some distance from it in space. Before and after, distant in one direction or another, these are all relations between the two events.

Using these relations, we can infer other relations between the events. For example, if we know the distance relating your breakfast and arriving at work, we can make a decent guess at another relation, the difference in amount of gas in your car.

This way of viewing the world, events connected by relations, is already quite common in physics. With Einstein’s theory of relativity, it’s hard to say exactly when or where an event happened, but the overall relationship between two events (distance in space and time taken together) can be thought of much more precisely. As I’ve mentioned before, the curved space-time necessary for Einstein’s theory of gravity can be thought of equally well as a change in the way you measure distances between two points.

So if space and time are relations between events, what would it mean for space-time to be doomed?

The key thing to realize here is that space and time are very specific relations between events, with very specific properties. Some of those properties are what cause problems for quantum gravity, problems which prompt people to suggest that space-time is doomed.

One of those properties is the fact that, when you multiply two distances together, it doesn’t matter which order you do it in. This probably sounds obvious, because you’re used to multiplying normal numbers, for which this is always true anyway. But even slightly more complicated mathematical objects, like matrices, don’t always obey this rule. If distances were this sort of mathematical object, then multiplying them in different orders could give slightly different results. If the difference were small enough, we wouldn’t be able to tell that it was happening in everyday life: distance would have given way to some more complicated concept, but it would still act like distance for us.

That specific idea isn’t generally suggested as a solution to the problems of space and time, but it’s a useful toy model that physicists have used to solve other problems.

It’s the general principle I want to get across: if you want to replace space and time, you need a relation between events. That relation should behave like space and time on the scales we’re used to, but it can be different on very small scales (Big Bang, inside of Black Holes) and on very large scales (long-term fate of the universe).

Space-time is doomed, and we don’t know yet what’s going to replace it. But whatever it is, whatever form it takes, we do know one thing: it’s going to be a relation between events.