Tag Archives: DoingScience

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.

Post-Scarcity Academia

Anyone will tell you that academia is broken.

The why varies, of course: some blame publication pressure, or greedy journals. Some think it’s the fault of grant committees, or tenure committees, or grad admission committees. Some argue we’re driving away the wrong people, others that we’re letting in the wrong people. Some place the fault with the media, or administrators, or the government, or the researchers themselves. Some believe the problem is just a small group of upstarts, others want to tear the whole system down.

If there’s one common theme to every “academia is broken” take, it’s limited resources. There are only so many people who can make a living doing research. Academia has to pick and choose who these people are and what they get to do, and anyone who thinks the system is broken thinks those choices could be made better.

As I was writing my version of the take, I started wondering. What if we didn’t have to choose? What would academia look like in a world without limited resources, where no-one needed to work for a living? Can we imagine what that world might look like?

Then I realized I didn’t need to imagine it. I’d already seen it.


And it was glorious

Let me tell you a bit about Dungeons and Dragons.

Dungeons and Dragons doesn’t have “pro gamers”, nobody makes money playing it. It isn’t even really the kind of game you can win or lose. It’s collaborative storytelling, backed up with a pile of dice and rulebooks. Nonetheless, Dungeons and Dragons has an active community dedicated to thinking about the game. They call themselves “optimizers”, and they focus on figuring out the best way the rules allow to do what they want to do.

Sometimes, the goal is practical: “what’s the best archer I can make?” “how can I make a character that has something useful to do no matter what?” Sometimes it’s more farfetched: “can I deal infinite damage?” “how can I make a god at level one?” Optimizing for these goals requires seeking out obscure rules, debating loopholes and the meaning of the text, and calculating probabilities.

I like to joke that Dungeons and Dragons was my first academic community, and that isn’t too far from the truth. These are people obsessed with understanding a complex system, who “publish” their research in forum posts , who collaborate and compete and care about finding the truth. While these people do have day jobs, that wasn’t a real limit. Dungeons and Dragons, I am forced to admit, is easier than theoretical physics. Even with day jobs or school, most of the D&D optimization community had plenty of time to do all the “research” they wanted. In a very real sense, they’re a glimpse at a post-scarcity academia.

There’s another parallel, one relevant to the current situation in theoretical physics. When I was most active in optimization, we played an edition of the game that was out of print. Normally there’s a sort of feedback between game designers and optimizers. As new expansions and errata are released, debates in the optimization community get resolved or re-ignited. With an out of print edition though, that feedback isn’t available. The optimization community was left by itself, examining whatever evidence it already had. This feels a lot like the current situation in physics, when so many experiments are just confirming the Standard Model. Without much feedback, the community has to evolve on its own.


So what did post-scarcity academia look like?

First, the good: this was a community highly invested in education. The best way to gain status wasn’t to build the strongest character, or discover a new trick. Instead, the most respected members of the community were the handbook writers, people who wrote long, clearly written forum posts summarizing optimization knowledge for newer players. I’m still not at the point where I read physics textbooks for fun, but back when I was active I would absolutely read optimization handbooks for fun. For those who wanted to get involved, the learning curve was about as well-signposted as it could be.

It was a community that could display breathtaking creativity, as well as extreme diligence. Some optimization was off-the-cuff and easy, but a lot of it took real work or real insight, and it showed. People would write short stories about the characters they made, or spend weeks cataloging every book that mentioned a particular rule. Despite not having to do their “research” for a living, motivation was never in short supply.

All that said, I think people yearning for a post-scarcity academia would be disappointed. If you think people do derivative, unoriginal work just because of academic careers, then I regret to inform you that a lot of optimization was unoriginal. There were a lot of posts that were just remixes of old ideas, packaged into a “new” build. There were also plenty of repetitive, pointless arguments, to the point that we’d joke about “Monkday” and “Wizard Wednesday”.

There was also a lot of attention-seeking behavior. There’s no optimization media, no optimization jobs that look for famous candidates, but people still cared about being heard, and pitched their work accordingly. We’d get a lot of overblown posts: “A Fighter that can beat any Wizard!” (because he’s been transformed by a spell into an all-powerful shapeshifter), “A Sorceror that can beat any Wizard!” (using houserules which change every time someone points out a flaw in the idea).

(Wizards, as you may be noticing, were kind of the String Theory of that community.)


Some problems in academia are caused by bad incentives, by the structure of academic careers. Some, though, are caused because academics are human beings. If we didn’t have to work for a living, academics would probably have different priorities, and we might work on a wider range of projects. But I suspect we’d still have good days and bad, that we’d still puff ourselves up for attention and make up dubious solutions to famous problems.

Of course, Dungeons and Dragons optimizers aren’t the only example of “post-scarcity academia”, or even a perfect example. They’ve got their own pressures, due to the structure of the community, that shape them in particular ways. I’d be interested to learn about other “amateur academics”, and how they handle things. My guess is that the groups whose work is closer to “real academia” (for example, the Society for Creative Anachronism) are more limited by their day jobs, but otherwise might be more informative. If there’s a “post-scarcity academia” you’re familiar with, mention it in the comments!

The Quantum Kids

I gave a pair of public talks at the Niels Bohr International Academy this week on “The Quest for Quantum Gravity” as part of their “News from the NBIA” lecture series. The content should be familiar to long-time readers of this blog: I talked about renormalization, and gravitons, and the whole story leading up to them.

(I wanted to title the talk “How I Learned to Stop Worrying and Love Quantum Gravity”, like my blog post, but was told Danes might not get the Doctor Strangelove reference.)

I also managed to work in some history, which made its way into the talk after Poul Damgaard, the director of the NBIA, told me I should ask the Niels Bohr Archive about Gamow’s Thought Experiment Device.

“What’s a Thought Experiment Device?”


This, apparently

If you’ve heard of George Gamow, you’ve probably heard of the Alpher-Bethe-Gamow paper, his work with grad student Ralph Alpher on the origin of atomic elements in the Big Bang, where he added Hans Bethe to the paper purely for an alpha-beta-gamma pun.

As I would learn, Gamow’s sense of humor was prominent quite early on. As a research fellow at the Niels Bohr Institute (essentially a postdoc) he played with Bohr’s kids, drew physics cartoons…and made Thought Experiment Devices. These devices were essentially toy experiments, apparatuses that couldn’t actually work but that symbolized some physical argument. The one I used in my talk, pictured above, commemorated Bohr’s triumph over one of Einstein’s objections to quantum theory.

Learning more about the history of the institute, I kept noticing the young researchers, the postdocs and grad students.


Lev Landau, George Gamow, Edward Teller. The kids are Aage and Ernest Bohr. Picture from the Niels Bohr Archive.

We don’t usually think about historical physicists as grad students. The only exception I can think of is Feynman, with his stories about picking locks at the Manhattan project. But in some sense, Feynman was always a grad student.

This was different. This was Lev Landau, patriarch of Russian physics, crowning name in a dozen fields and author of a series of textbooks of legendary rigor…goofing off with Gamow. This was Edward Teller, father of the Hydrogen Bomb, skiing on the institute lawn.

These were the children of the quantum era. They came of age when the laws of physics were being rewritten, when everything was new. Starting there, they could do anything, from Gamow’s cosmology to Landau’s superconductivity, spinning off whole fields in the new reality.

On one level, I envy them. It’s possible they were the last generation to be on the ground floor of a change quite that vast, a shift that touched all of physics, the opportunity to each become gods of their own academic realms.

I’m glad to know about them too, though, to see them as rambunctious grad students. It’s all too easy to feel like there’s an unbridgeable gap between postdocs and professors, to worry that the only people who make it through seem to have always been professors at heart. Seeing Gamow and Landau and Teller as “quantum kids” dispels that: these are all-too-familiar grad students and postdocs, joking around in all-too-familiar ways, who somehow matured into some of the greatest physicists of their era.

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.

Visiting Uppsala

I’ve been in Uppsala this week, visiting Henrik Johansson‘s group.


The Ångström Laboratory here is substantially larger than an ångström, a clear example of false advertising.

As such, I haven’t had time to write a long post about the recent announcement by the LIGO and VIRGO collaborations. Luckily, Matt Strassler has written one of his currently all-too-rare posts on the subject, so if you’re curious you should check out what he has to say.

Looking at the map of black hole collisions in that post, I’m struck by how quickly things have improved. The four old detections are broad slashes across the sky, the newest is a small patch. Now that there are enough detectors to triangulate, all detections will be located that precisely, or better. A future map might be dotted with precise locations of black hole collisions, but it would still be marred by those four slashes: relics of the brief time when only two machines in the world could detect gravitational waves.

On the Care and Feeding of Ideas

I read Zen and the Art of Motorcycle Maintenance in high school. It’s got a reputation for being obnoxiously mystical, but one of its points seemed pretty reasonable: the claim that the hard part of science, and the part we understand the least, is coming up with hypotheses.

In some sense, theoretical physics is all about hypotheses. By this I don’t mean that we just say “what if?” all the time. I mean that in theoretical physics most of the work is figuring out the right way to ask a question. Phrase your question in the right way and the answer becomes obvious (or at least, obvious after a straightforward calculation). Because our questions are mathematical, the right question can logically imply its own solution.

From the point of view of “Zen and the Art”, as well as most non-scientists I’ve met, this part is utterly mysterious. The ideas you need here seem like they can’t come from hard work or careful observation. In order to ask the right questions, you just need to be “smart”.

In practice, I’ve noticed there’s more to it than that. We can’t just sit around and wait for an idea to show up. Instead, as physicists we develop a library of tricks, often unstated, that let us work towards the ideas we need.

Sometimes, this involves finding simpler cases, working with them until we understand the right questions to ask. Sometimes it involves doing numerics, or using crude guesses, not because either method will give the final answer but because it will show what the answer should look like. Sometimes we need to rephrase the problem many times, in many different contexts, before we happen on one that works. Most of this doesn’t end up in published papers, so in the end we usually have to pick it up from experience.

Along the way, we often find tricks to help us think better. Mostly this is straightforward stuff: reminders to keep us on-task, keeping our notes organized and our code commented so we have a good idea of what we were doing when we need to go back to it. Everyone has their own personal combination of these things in the background, and they’re rarely discussed.

The upshot is that coming up with ideas is hard work. We need to be smart, sure, but that’s not enough by itself: there are a lot of smart people who aren’t physicists after all.

With all that said, some geniuses really do seem to come up with ideas out of thin air. It’s not the majority of the field: we’re not the idiosyncratic Sheldon Coopers everyone seems to imagine. But for a few people, it really does feel like there’s something magical about where they get their ideas. I’ve had the privilege of working with a couple people like this, and the way they think sometimes seems qualitatively different from our usual way of building ideas. I can’t see any of the standard trappings, the legacy of partial results and tricks of thought, that would lead to where they end up. That doesn’t mean they don’t use tricks just like the rest of us, in the end. But I think genius, if it means anything at all, is thinking in a novel enough way that from the outside it looks like magic.

Most of the time, though, we just need to hone our craft. We build our methods and shape our minds as best we can, and we get better and better at the central mystery of science: asking the right questions.