# Proofs and Insight

Hearing us talking about the Amplituhedron, the professor across the table chimed in.

“The problem with you amplitudes people, I never know what’s a conjecture and what’s proven. The Amplituhedron, is that still a conjecture?”

The Amplituhedron, indeed, is still a conjecture (although a pretty well-supported one at this point). After clearing that up, we got to talking about the role proofs play in theoretical physics.

The professor was worried that we weren’t being direct enough in stating which ideas in amplitudes had been proven. While I agreed that we should be clearer, one of his points stood out to me: he argued that one benefit of clearly labeling conjectures is that it motivates people to go back and prove things. That’s a good thing to do in general, to be sure that your conjecture is really true, but often it has an added benefit: even if you’re pretty sure your conjecture is true, proving it can show you why it’s true, leading to new and valuable insight.

There’s a long history of important physics only becoming clear when someone took the time to work out a proof. But in amplitudes right now, I don’t think our lack of proofs is leading to a lack of insight. That’s because the kinds of things we’d like to prove often require novel insight themselves.

It’s not clear what it would take to prove the Amplituhedron. Even if you’ve got a perfectly clear, mathematically nice definition for it, you’d still need to prove that it does what it’s supposed to do: that it really calculates scattering amplitudes in N=4 super Yang-Mills. In order to do that, you’d need a very complete understanding of how those calculations work. You’d need to be able to see how known methods give rise to something like the Amplituhedron, or to find the Amplituhedron buried deep in the structure of the theory.

If you had that kind of insight? Then yeah, you could prove the Amplituhedron, and accomplish remarkable things along the way. But more than that, if you had that sort of insight, you would prove the Amplituhedron. Even if you didn’t know about the Amplituhedron to begin with, or weren’t sure whether or not it was a conjecture, once you had that kind of insight proving something like the Amplituhedron would be the inevitable next step. The signpost, “this is a conjecture” is helpful for other reasons, but it doesn’t change circumstances here: either you have what you need, or you don’t.

This contrasts with how progress works in other parts of physics, and how it has worked at other times. Sometimes, a field is moving so fast that conjectures get left by the wayside, even when they’re provable. You get situations where everyone busily assumes something is true and builds off it, and no-one takes the time to work out why. In that sort of field, it can be really valuable to clearly point out conjectures, so that someone gets motivated to work out the proof (and to hopefully discover something along the way).

I don’t think amplitudes is in that position though. It’s still worthwhile to signal our conjectures, to make clear what needs a proof and what doesn’t. But our big conjectures, like the Amplituhedron, aren’t the kind of thing someone can prove just by taking some time off and working on it. They require new, powerful insight. Because of that, our time is typically best served looking for that insight, finding novel examples and unusual perspectives that clear up what’s really going on. That’s a fair bit broader an activity than just working out a proof.

# An Elliptical Workout

I study scattering amplitudes, probabilities that particles scatter off each other.

In particular, I’ve studied them using polylogarithmic functions. Polylogarithmic functions can be taken apart into “logs”, which obey identities much like logarithms do. They’re convenient and nice, and for my favorite theory of N=4 super Yang-Mills they’re almost all you need.

Well, until ten particles get involved, anyway.

That’s when you start needing elliptic integrals, and elliptic polylogarithms. These integrals substitute one of the “logs” of a polylogarithm with an integration over an elliptic curve.

And with Jacob Bourjaily, Andrew McLeod, Marcus Spradlin, and Matthias Wilhelm, I’ve now computed one.

This one, to be specific

Our paper, The Elliptic Double-Box Integral, went up on the arXiv last night.

The last few weeks have been a frenzy of work, finishing up our calculations and writing the paper. It’s the fastest I’ve ever gotten a paper out, which has been a unique experience.

Computing this integral required new, so far unpublished tricks by Jake Bourjaily, as well as some rather powerful software and Mark Spradlin’s extensive expertise in simplifying polylogarithms. In the end, we got the integral into a “canonical” form, one other papers had proposed as the right way to represent it, with the elliptic curve in a form standardized by Weierstrass.

One of the advantages of fixing a “canonical” form is that it should make identities obvious. If two integrals are actually the same, then writing them according to the same canonical rules should make that clear. This is one of the nice things about polylogarithms, where these identities are really just identities between logs and the right form is comparatively easy to find.

Surprisingly, the form we found doesn’t do this. We can write down an integral in our “canonical” form that looks different, but really is the same as our original integral. The form other papers had suggested, while handy, can’t be the final canonical form.

What the final form should be, we don’t yet know. We have some ideas, but we’re also curious what other groups are thinking. We’re relatively new to elliptic integrals, and there are other groups with much more experience with them, some with papers coming out soon. As far as we know they’re calculating slightly different integrals, ones more relevant for the real world than for N=4 super Yang-Mills. It’s going to be interesting seeing what they come up with. So if you want to follow this topic, don’t just watch for our names on the arXiv: look for Claude Duhr and Falko Dulat, Luise Adams and Stefan Weinzierl. In the elliptic world, big things are coming.

# One, Two, Infinity

Physicists and mathematicians count one, two, infinity.

We start with the simplest case, as a proof of principle. We take a stripped down toy model or simple calculation and show that our idea works. We count “one”, and we publish.

Next, we let things get a bit more complicated. In the next toy model, or the next calculation, new interactions can arise. We figure out how to deal with those new interactions, our count goes from “one” to “two”, and once again we publish.

By this point, hopefully, we understand the pattern. We know what happens in the simplest case, and we know what happens when the different pieces start to interact. If all goes well, that’s enough: we can extrapolate our knowledge to understand not just case “three”, but any case: any model, any calculation. We publish the general case, the general method. We’ve counted one, two, infinity.

Once we’ve counted “infinity”, we don’t have to do any more cases. And so “infinity” becomes the new “zero”, and the next type of calculation you don’t know how to do becomes “one”. It’s like going from addition to multiplication, from multiplication to exponentiation, from exponentials up into the wilds of up-arrow notation. Each time, once you understand the general rules you can jump ahead to an entirely new world with new capabilities…and repeat the same process again, on a new scale. You don’t need to count one, two, three, four, on and on and on.

Of course, research doesn’t always work out this way. My last few papers counted three, four, five, with six on the way. (One and two were already known.) Unlike the ideal cases that go one, two, infinity, here “two” doesn’t give all the pieces you need to keep going. You need to go a few numbers more to get novel insights. That said, we are thinking about “infinity” now, so look forward to a future post that says something about that.

A lot of frustration in physics comes from situations when “infinity” remains stubbornly out of reach. When people complain about all the models for supersymmetry, or inflation, in some sense they’re complaining about fields that haven’t taken that “infinity” step. One or two models of inflation are nice, but by the time the count reaches ten you start hoping that someone will describe all possible models of inflation in one paper, and see if they can make any predictions from that.

(In particle physics, there’s an extent to which people can actually do this. There are methods to describe all possible modifications of the Standard Model in terms of what sort of effects they can have on observations of known particles. There’s a group at NBI who work on this sort of thing.)

The gold standard, though, is one, two, infinity. Our ability to step back, stop working case-by-case, and move on to the next level is not just a cute trick: it’s a foundation for exponential progress. If we can count one, two, infinity, then there’s nowhere we can’t reach.

# Thoughts on Polchinski’s Memoir

I didn’t get a chance to meet Joseph Polchinski when I was visiting Santa Barbara last spring. At the time, I heard his health was a bit better, but he still wasn’t feeling well enough to come in to campus. Now that I’ve read his memoir, I almost feel like I have met him. There’s a sense of humor, a diffidence, and a passion for physics that shines through the pages.

The following are some scattered thoughts inspired by the memoir:

A friend of mine once complained to me that in her field grad students all brag about the colleges they went to. I mentioned that in my field your undergrad never comes up…unless it was Caltech. For some reason, everyone I’ve met who went to Caltech is full of stories about the place, and Polchinski is no exception. Speaking as someone who didn’t go there, it seems like Caltech has a profound effect on its students that other places don’t.

Polchinski mentions hearing stories about geniuses of the past, and how those stories helped temper some of his youthful arrogance. There’s an opposite effect that’s also valuable: hearing stories like Polchinski’s, his descriptions of struggling with anxiety and barely publishing and “not really accomplishing anything” till age 40, can be a major comfort to those of us who worry we’ve fallen behind in the academic race. That said, it’s important not to take these things too far: times have changed, you’re not Polchinski, and much like his door-stealing trick at Caltech getting a postdoc without any publications is something you shouldn’t try at home. Even Witten’s students need at least one.

Last week I was a bit puzzled by nueww’s comment, a quote from Polchinski’s memoir which distinguishes “math of the equations” from “math of the solutions”, attributing the former to physicists and the latter to mathematicians. Reading the context in the memoir and the phrase’s origin in a remark by Susskind cleared up a bit, but still left me uneasy. I only figured out why after Lubos Motl posted about it: it doesn’t match my experience of mathematicians at all!

If anything, I think physicists usually care more about the “solutions” than mathematicians do. In my field, often a mathematician will construct some handy basis of functions and then frustrate everyone by providing no examples of how to use them. In the wider math community I’ve met graph theorists who are happy to prove something is true for all graphs of size $10^{10^10}$ and larger, not worrying about the vast number of graphs where it fails because it’s just a finite number of special cases. And I don’t think this is just my experience: a common genre of jokes revolve around mathematicians proving a solution exists and then not bothering to do anything with it (for example, see the joke with the hotel fire here).

I do think there’s a meaningful sense in which mathematicians care about details that we’re happy to ignore, but “solutions” versus “equations” isn’t really the right axis. It’s something more like “rigor” versus “principles”. Mathematicians will often begin a talk by defining a series of maps between different spaces, carefully describing where they are and aren’t valid. A physicist might just write down a function. That sort of thing is dangerous in mathematics: there are always special, pathological cases that make careful definitions necessary. In physics, those cases rarely come up, and when they do there’s often a clear physical problem that brings them to the forefront. We have a pretty good sense of when we need rigor, and when we don’t we’re happy to lay things out without filling in the details, putting a higher priority on moving forward and figuring out the basic principles underlying reality.

Polchinski talks a fair bit about his role in the idea of the multiverse, from hearing about Weinberg’s anthropic argument to coming to terms with the string landscape. One thing his account makes clear is how horrifying the concept seemed at first: how the idea that the parameters of our universe might just be random could kill science and discourage experimentalists. This touches on something that I think gets lost in arguments about the multiverse: even the people most involved in promoting the multiverse in public aren’t happy about it.

It also sharpened my thinking about the multiverse a bit. I’ve talked before about how I don’t think the popularity of the multiverse is actually going to hurt theoretical physics as a field. Polchinski’s worries made me think about the experimental side of the equation: why do experiments if the world might just be random? I think I have a clearer answer to this now, but it’s a bit long, so I’ll save it for a future post.

One nice thing about these long-term accounts is you get to see how much people shift between fields over time. Polchinski didn’t start out working in string theory, and most of the big names in my field, like Lance Dixon and David Kosower, didn’t start out in scattering amplitudes. Academic careers are long, and however specialized we feel at any one time we can still get swept off in a new direction.

I’m grateful for this opportunity to “meet” Polchinski, if only through his writing. His is a window on the world of theoretical physics that is all too rare, and valuable as a result.

# Bootstrapping in the Real World

I’ll be at Amplitudes, my subfield’s big yearly conference, next week, so I don’t have a lot to talk about. That said, I wanted to give a shout-out to my collaborator and future colleague Andrew McLeod, who is a co-author (along with Øyvind Almelid, Claude Duhr, Einan Gardi, and Chris White) on a rather cool paper that went up on arXiv this week.

Andrew and I work on “bootstrapping” calculations in quantum field theory. In particular, we start with a guess for what the result will be based on a specific set of mathematical functions (in my case, “hexagon functions” involving interactions of six particles). We then narrow things down, using other calculations that by themselves only predict part of the result, until we know the right answer. The metaphor here is that we’re “pulling ourselves up by our own bootstraps”, skipping a long calculation by essentially just guessing the answer.

This method has worked pretty well…in a toy model anyway. The calculations I’ve done with it use N=4 super Yang-Mills, a simpler cousin of the theories that describe the real world. There, fewer functions can show up, so our guess is much less unwieldy than it would be otherwise.

What’s impressive about Andrew and co.’s new paper is that they apply this method, not to N=4 super Yang-Mills, but to QCD, the theory that describes quarks and gluons in the real world. This is exactly the sort of thing I’ve been hoping to see more of, these methods built into something that can help with real, useful calculations.

Currently, what they can do is still fairly limited. For the particular problem they’re looking at, the functions required ended up being relatively simple, involving interactions between at most four particles. So far, they’ve just reproduced a calculation done by other means. Going further (more “loops”) would involve interactions between more particles, as well as mixing different types of functions (different “transcendental weight”), either of which make the problem much more complicated.

That said, the simplicity of their current calculation is also a reason to be optimistic.  Their starting “guess” had just thirteen parameters, while the one Andrew and I are working on right now (in N=4 super Yang-Mills) has over a thousand. Even if things get a lot more complicated for them at the next loop, we’ve shown that “a lot more complicated” can still be quite doable.

So overall, I’m excited. It looks like there are contexts in which one really can “bootstrap” up calculations in a realistic theory, and that’s a method that could end up really useful.

# You Can’t Smooth the Big Bang

As a kid, I was fascinated by cosmology. I wanted to know how the universe began, possibly disproving gods along the way, and I gobbled up anything that hinted at the answer.

At the time, I had to be content with vague slogans. As I learned more, I could match the slogans to the physics, to see what phrases like “the Big Bang” actually meant. A large part of why I went into string theory was to figure out what all those documentaries are actually about.

In the end, I didn’t end up working on cosmology due my ignorance of a few key facts while in college (mostly, who Vilenkin was). Thus, while I could match some of the old popularization stories to the science, there were a few I never really understood. In particular, there were two claims I never quite saw fleshed out: “The universe emerged from nothing via quantum tunneling” and “According to Hawking, the big bang was not a singularity, but a smooth change with no true beginning.”

As a result, I’m delighted that I’ve recently learned the physics behind these claims, in the context of a spirited take-down of both by Perimeter’s Director Neil Turok.

My boss

Neil held a surprise string group meeting this week to discuss the paper I linked above, “No smooth beginning for spacetime” with Job Feldbrugge and Jean-Luc Lehners, as well as earlier work with Steffen Gielen. In it, he talked about problems in the two proposals I mentioned: Hawking’s suggestion that the big bang was smooth with no true beginning (really, the Hartle-Hawking no boundary proposal) and the idea that the universe emerged from nothing via quantum tunneling (really, Vilenkin’s tunneling from nothing proposal).

In popularization-speak, these two proposals sound completely different. In reality, though, they’re quite similar (and as Neil argues, they end up amounting to the same thing). I’ll steal a picture from his paper to illustrate:

The picture on the left depicts the universe under the Hartle-Hawking proposal, with time increasing upwards on the page. As the universe gets older, it looks like the expanding (de Sitter) universe we live in. At the beginning, though, there’s a cap, one on which time ends up being treated not in the usual way (Lorentzian space) but on the same footing as the other dimensions (Euclidean space). This lets space be smooth, rather than bunching up in a big bang singularity. After treating time in this way the result is reinterpreted (via a quantum field theory trick called Wick rotation) as part of normal space-time.

What’s the connection to Vilenkin’s tunneling picture? Well, when we talk about quantum tunneling, we also end up describing it with Euclidean space. Saying that the universe tunneled from nothing and saying it has a Euclidean “cap” then end up being closely related claims.

Before Neil’s work these two proposals weren’t thought of as the same because they were thought to give different results. What Neil is arguing is that this is due to a fundamental mistake on Hartle and Hawking’s part. Specifically, Neil is arguing that the Wick rotation trick that Hartle and Hawking used doesn’t work in this context, when you’re trying to calculate small quantum corrections for gravity. In normal quantum field theory, it’s often easier to go to Euclidean space and use Wick rotation, but for quantum gravity Neil is arguing that this technique stops being rigorous. Instead, you should stay in Lorentzian space, and use a more powerful mathematical technique called Picard-Lefschetz theory.

Using this technique, Neil found that Hartle and Hawking’s nicely behaved result was mistaken, and the real result of what Hartle and Hawking were proposing looks more like Vilenkin’s tunneling proposal.

Neil then tried to see what happens when there’s some small perturbation from a perfect de Sitter universe. In general in physics if you want to trust a result it ought to be stable: small changes should stay small. Otherwise, you’re not really starting from the right point, and you should instead be looking at wherever the changes end up taking you. What Neil found was that the Hartle-Hawking and Vilenkin proposals weren’t stable. If you start with a small wiggle in your no-boundary universe you get, not the purple middle drawing with small wiggles, but the red one with wiggles that rapidly grow unstable. The implication is that the Hartle-Hawking and Vilenkin proposals aren’t just secretly the same, they also both can’t be the stable state of the universe.

Neil argues that this problem is quite general, and happens under the following conditions:

1. A universe that begins smoothly and semi-classically (where quantum corrections are small) with no sharp boundary,
2. with a positive cosmological constant (the de Sitter universe mentioned earlier),
3. under which the universe expands many times, allowing the small fluctuations to grow large.

If the universe avoids one of those conditions (maybe the cosmological constant changes in the future and the universe stops expanding, for example) then you might be able to avoid Neil’s argument. But if not, you can’t have a smooth semi-classical beginning and still have a stable universe.

Now, no debate in physics ends just like that. Hartle (and collaborators) don’t disagree with Neil’s insistence on Picard-Lefschetz theory, but they argue there’s still a way to make their proposal work. Neil mentioned at the group meeting that he thinks even the new version of Hartle’s proposal doesn’t solve the problem, he’s been working out the calculation with his collaborators to make sure.

Often, one hears about an idea from science popularization and then it never gets mentioned again. The public hears about a zoo of proposals without ever knowing which ones worked out. I think child-me would appreciate hearing what happened to Hawking’s proposal for a universe with no boundary, and to Vilenkin’s proposal for a universe emerging from nothing. Adult-me certainly does. I hope you do too.

# KITP Conference Retrospective

I’m back from the conference in Santa Barbara, and I thought I’d share a few things I found interesting. (For my non-physicist readers: I know it’s been a bit more technical than usual recently, I promise I’ll get back to some general audience stuff soon!)

James Drummond talked about efforts to extend the hexagon function method I work on to amplitudes with seven (or more) particles. In general, the method involves starting with a guess for what an amplitude should look like, and honing that guess based on behavior in special cases where it’s easier to calculate. In one of those special cases (called the multi-Regge limit), I had thought it would be quite difficult to calculate for more than six particles, but James clarified for me that there’s really only one additional piece needed, and they’re pretty close to having a complete understanding of it.

There were a few talks about ways to think about amplitudes in quantum field theory as the output of a string theory-like setup. There’s been progress pushing to higher quantum-ness, and in understanding the weird web of interconnected theories this setup gives rise to. In the comments, Thoglu asked about one part of this web of theories called Z theory.

Z theory is weird. Most of the theories that come out of this “web” come from a consistent sort of logic: just like you can “square” Yang-Mills to get gravity, you can “square” other theories to get more unusual things. In possibly the oldest known example, you can “square” the part of string theory that looks like Yang-Mills at low energy (open strings) to get the part that looks like gravity (closed strings). Z theory asks: could the open string also come from “multiplying” two theories together? Weirdly enough, the answer is yes: it comes from “multiplying” normal Yang-Mills with a part that takes care of the “stringiness”, a part which Oliver Schlotterer is calling “Z theory”. It’s not clear whether this Z theory makes sense as a theory on its own (for the experts: it may not even be unitary) but it is somewhat surprising that you can isolate a “building block” that just takes care of stringiness.

Peter Young in the comments asked about the Correlahedron. Scattering amplitudes ask a specific sort of question: if some particles come in from very far away, what’s the chance they scatter off each other and some other particles end up very far away? Correlators ask a more general question, about the relationships of quantum fields at different places and times, of which amplitudes are a special case. Just as the Amplituhedron is a geometrical object that specifies scattering amplitudes (in a particular theory), the Correlahedron is supposed to represent correlators (in the same theory). In some sense (different from the sense above) it’s the “square” of the Amplituhedron, and the process that gets you from it to the Amplituhedron is a geometrical version of the process that gets you from the correlator to the amplitude.

For the Amplituhedron, there’s a reasonably smooth story of how to get the amplitude. News articles tended to say the amplitude was the “volume” of the Amplituhedron, but that’s not quite correct. In fact, to find the amplitude you need to add up, not the inside of the Amplituhedron, but something that goes infinite at the Amplituhedron’s boundaries. Finding this “something” can be done on a case by case basis, but it get tricky in more complicated cases.

For the Correlahedron, this part of the story is missing: they don’t know how to define this “something”, the old recipe doesn’t work. Oddly enough, this actually makes me optimistic. This part of the story is something that people working on the Amplituhedron have been trying to avoid for a while, to find a shape where they can more honestly just take the volume. The fact that the old story doesn’t work for the Correlahedron suggests that it might provide some insight into how to build the Amplituhedron in a different way, that bypasses this problem.

There were several more talks by mathematicians trying to understand various aspects of the Amplituhedron. One of them was by Hugh Thomas, who as a fun coincidence actually went to high school with Nima Arkani-Hamed, one of the Amplituhedron’s inventors. He’s now teamed up with Nima and Jaroslav Trnka to try to understand what it means to be inside the Amplituhedron. In the original setup, they had a recipe to generate points inside the Amplituhedron, but they didn’t have a fully geometrical picture of what put them “inside”. Unlike with a normal shape, with the Amplituhedron you can’t just check which side of the wall you’re on. Instead, they can flatten the Amplituhedron, and observe that for points “inside” the Amplituhedron winds around them a specific number of times (hence “Unwinding the Amplituhedron“). Flatten it down to a line and you can read this off from the list of flips over your point, an on-off sequence like binary. If you’ve ever heard the buzzword “scattering amplitudes as binary code”, this is where that comes from.

They also have a better understanding of how supersymmetry shows up in the Amplituhedron, which Song He talked about in his talk. Previously, supersymmetry looked to be quite central, part of the basic geometric shape. Now, they can instead understand it in a different way, with the supersymmetric part coming from derivatives (for the specialists: differential forms) of the part in normal space and time. The encouraging thing is that you can include these sorts of derivatives even if your theory isn’t supersymmetric, to keep track of the various types of particles, and Song provided a few examples in his talk. This is important, because it opens up the possibility that something Amplituhedron-like could be found for a non-supersymmetric theory. Along those lines, Nima talked about ways that aspects of the “nice” description of space and time we use for the Amplituhedron can be generalized to other messier theories.

While he didn’t talk about it at the conference, Jake Bourjaily has a new paper out about a refinement of the generalized unitarity technique I talked about a few weeks back. Generalized unitarity involves matching a “cut up” version of an amplitude to a guess. What Jake is proposing is that in at least some cases you can start with a guess that’s as easy to work with as possible, where each piece of the guess matches up to just one of the “cuts” that you’re checking.  Think about it like a game of twenty questions where you’ve divided all possible answers into twenty individual boxes: for each box, you can just ask “is it in this box”?

Finally, I’ve already talked about the highlight of the conference, so I can direct you to that post for more details. I’ll just mention here that there’s still a fair bit of work to do for Zvi Bern and collaborators to get their result into a form they can check, since the initial output of their setup is quite messy. It’s led to worries about whether they’ll have enough computer power at higher loops, but I’m confident that they still have a few tricks up their sleeves.