# 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. 😉 ]

## 17 thoughts on “So You Want to Prove String Theory (Or: Nima Did Something Cool Again)”

Haven’t gotten a chance to watch it, but I could believe that. Amit Sever gave a talk on it (the YM side of the story) at Amplitudes, and it seemed a lot cleaner (which is probably part of why they’ve published and Nima and Yu-tin haven’t yet).

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1. Giotis

I’ve only watched bits and pieces; I see he bypassed Higher Spin Gravity quickly which is the only competitor I can think of at least in AdS/dS.

The interested reader can check this classic review aimed to the non expert where the various no-go theorems for HS Gravity are confronted even in Minkowski space-time:

http://arxiv.org/abs/1007.0435v3

That being said I never thought that HS Gravity could be a standalone theory, all indications point to the fact that is some subsector of String theory.

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2. Wyrd Smythe

Trying to prove a theory is the only possible one seems like trying to prove a black swan you found is the only black swan in existence. I’m not sure how one can claim an exhaustive search with regard to any discovery, so that part confused me.

(I’ll add the video to my watch list, but right now I’m trying to work through Leonard Susskind’s General Relativity lecture series at Stanford… but I think I’m gonna have to go take a high-level calc class… tensors still aren’t gelling with me. I’m pretty lost on the whole covarient and contravarient thing. Fun challenge, though.)

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Think of it less like an exhaustive search, and more like a derivation. The idea is not to show that every possible theory is wrong except string theory. Rather, it’s to show that the answers string theory gives to certain questions (here, scattering amplitudes) are dictated not just by string theory but by the nature of the problem we’re trying to solve, no matter how you approach it.

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1. Wyrd Smythe

It’s the “no matter how you approach it” part that requires an “exhaustive search” — that is, a conclusion the theory does, indeed, cover all approaches. Seems like demonstrating that is quite a trick.

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So, the reason [with certain conditions] is there is that indeed, we can’t rule out everything. In particular, we can’t rule out an out-of-the-box approach that changes the nature of the problem.

What Nima&co are doing here, rather, is finding a sub-problem. Unlike the broad physics question, the sub-problem is purely mathematical: it’s just asking “are there functions that obey this set of conditions, and if so which ones?” Those sorts of questions sometimes do have unique solutions, and even when they don’t you can often find a way to list all the options. (I don’t know if you did complex analysis back in college, but analytic continuations are an example of that kind of problem, where there’s one unique function that can solve it.)

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1. Wyrd Smythe

Ah, okay, got it, thanks.

(Formally, I only got as high as Calc I and II. I’ve been chewing on the edges of higher maths ever since, but I find it tough stuff. Love it, though!)

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3. Jan Reimers

I found this talk much harder to follow than many of is amplitude/grassmanian talks. Thanks for the link to the “all star” paper, maybe I can get more out of that. One problem is when he talks about s I never know if that s is spin or the Mandelstam invariant. He used to use a script s for spin … but apparently not here. Do you know why it is necessary to mention high spin gravity? I thought all spins > 2 (for fundamental particles) were well known to be off the allowed menu … or is that not a well accepted result?

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As far as I can tell he’s using the same convention here, it’s just that his script s occasionally looks a little too much like his non-script s. But the main thing is, if he’s assigning values to s, it’s spin, so it should be easier to tell if you expect that.

Higher spin gravity isn’t just adding higher interacting spins (which are indeed off the menu), it’s a specific avenue of research using specific loopholes. I don’t think I can do it justice with a summary here, but if you’re up for watching a lecture a friend of mine at Perimeter did a few last year.

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1. Jan Reimers

Thanks for the link I will have a look at the talk. I also saw the link provided by Goitis after I posted which has a whole section on getting around the HS no-go theorem. It would be interesting to tally all the no-go theorems over years and how many of them been eventually worked around … and if any of the workarounds are actually realized/observed in nature.

I have a condensed matter background but I find the recent amplitudes progress fascinating to follow for a number reasons. This blog and that of Edward Hughs are a great help in providing enough pedagogical material to allow a non-specialist to try and follow the field. SO thanks for that.

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2. Giotis

Caution, massive higher spins are included in the spectrum of String theory and there aren’t any no-go theorems for them.

The theory of massless higher spins on the other hand is what we refer to when we talk about HS Gravity.

For these indeed there are no-go theorems in flat space but they work whenever a cosmological constant is present (AdS/dS space); it is known as Vasiliev’s type HS Gravity.

But even in flat space we have loopholes, check the paper I referred to in my comment for more information.

Broadly speaking, the Higher Spin symmetry is broken and massless higher spins with spin > 2 acquire mass via some Higgs like mechanism (dynamic breaking also considered) and thus only the plain Gravity we are familiar to remains.

In the tensionless limit of String theory where massive higher spins become massless, HS Gravity is believed to be a subsector of String theory.

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4. Tom Andersen

With QM firmly in hand, John von Neumann, one of the greatest mathematicians of the twentieth century, claimed he had proved that Einstein’s dream of a deterministic completion or reinterpretation of quantum theory was mathematically impossible. (von Neumann 1932, p. 325)

“It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics — the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.”

Yet his proof was wrong.

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5. Zack Yezek

Doesn’t this entire approach assume gravitons in some QFT version of gravity actually exist, and thus if they don’t it won’t prove anything? Our current understanding of gravity is that it isn’t caused by scattering particles, but rather by a warping of the very geometry, the environment such scatterings occur in.

I get that you could use fictitious mathematical gravitons to do certain calculations with gravity at the quantum scale, even if they aren’t real particles. That sort of trick is already used for some computations w/ Feynman diagrams. The rest of String Theory has also produced math that’s been useful in things like condensed matter physics too. None of it has proved or can prove that the universe is really made of strings, or that gravitons really exist. At most Nima might prove that certain assumptions about quantum gravity mathematically imply some version of string theory, but in fact he appears to have shown the opposite.

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It’s pretty difficult to have a quantum theory of gravity without gravitons. Once you’ve got gravitational waves then you need to quantize them somehow, and that’s going to result in something like the graviton. That doesn’t mean that gravity isn’t the environment in which scatterings happen, the two pictures describe the same physics, in much the same way that “electromagnetic fields are the environment electron scattering happens in” ends up requiring the existence of photons. I don’t know your level of expertise, but I’ve got a general audience level post on this here.

That said, I don’t know if a consistent formula for graviton scattering is absolutely inevitable in any theory of quantum gravity. The few Loop Quantum Gravity people I’ve talked to about this have given me conflicting answers, so I don’t know whether they ultimately expect there to be some sensible formula there.

Nima is pretty open that his goal here is to prove string-like behavior given certain assumptions, and one of those assumptions is that graviton scattering makes sense. There are other assumptions as well. But even with those assumptions you’d be able to constrain some non-string models.

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