Tag Archives: DoingScience

PSI Winter School

I’m at the Perimeter Scholars International Winter School this week. Perimeter Scholars International is Perimeter’s one-of-a-kind master’s program in theoretical physics, that jams the basics of theoretical physics into a one-year curriculum. We’ve got students from all over the world, including plenty of places that don’t get any snow at all. As such, it was decided that the students need to spend a week somewhere with even more snow than Waterloo: Musoka, Ontario.

A place that occasionally manages to be this photogenic

This isn’t really a break for them, though, which is where I come in. The students have been organized into groups, and each group is working on a project. My group’s project is related to the work of integrability master Pedro Vieira. He and his collaborators came up with a way to calculate scattering amplitudes in N=4 super Yang-Mills without the usual process of loop-by-loop approximations. However, this method comes at a price: a new approximation, this time to low energy. This approximation is step-by-step, like loops, but in a different direction. It’s called the Pentagon Operator Product Expansion, or POPE for short.

Approach the POPE, and receive a blessing

What we’re trying to do is go back and add up all of the step-by-step terms in the approximation, to see if we can match to the old expansion in loops. One of Pedro’s students recently managed to do this for the first approximation (“tree” diagrams), and the group here at the Winter School is trying to use her (still unpublished) work as a jumping-off point to get to the first loop. Time will tell whether we’ll succeed…but we’re making progress, and the students are learning a lot.

Trust Your Notation as Far as You Can Prove It

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Calculus contains one of the most famous examples of physicists doing something silly that irritates mathematicians. See, there are two different ways to write down a derivative, both dating back to the invention of calculus: Newton’s method, and Leibniz’s method.

Newton cared a lot about rigor (enough that he actually published his major physics results without calculus because he didn’t think calculus was rigorous enough, despite inventing it himself). His notation is direct and to the point: if you want to take the derivative of a function f of x, you write,

$f'(x)$

Leibniz cared a lot less about rigor, and a lot more about the scientific community. He wanted his notation to be useful and intuitive, to be the sort of thing that people would pick up and run with. To write a derivative in Leibniz notation, you write,

$\frac{df}{dx}$

This looks like a fraction. It’s really, really tempting to treat it like a fraction. And that’s the point: it’s to tell you that treating it like a fraction is often the right thing to do. In particular, you can do something like this,

$y=\frac{df}{dx}$

$y dx=df$

$\int y dx=\int df$

and what you did actually makes a certain amount of sense.

The tricky thing here is that it doesn’t always make sense. You can do these sorts of tricks up to a point, but you need to be aware that they really are just tricks. Take the notation too seriously, and you end up doing things you aren’t really allowed to do. It’s always important to stay aware of what you’re really doing.

There are a lot of examples of this kind of thing in physics. In quantum field theory, we use path integrals. These aren’t really integrals…but a lot of the time, we can treat them as such. Operators in quantum mechanics can be treated like numbers and multiplied…up to a point. A friend of mine was recently getting confused by operator product expansions, where similar issues crop up.

I’ve found two ways to clear up this kind of confusion. One is to unpack your notation: go back to the definitions, and make sure that what you’re doing really makes sense. This can be tedious, but you can be confident that you’re getting the right answer.

The other option is to stop treating your notation like the familiar thing it resembles, and start treating it like uncharted territory. You’re using this sort of notation to remind you of certain operations you can do, certain rules you need to follow. If you take those rules as basic, you can think about what you’re doing in terms of axioms rather than in terms of the suggestions made by your notation. Follow the right axioms, and you’ll stay within the bounds of what you’re actually allowed to do.

Either way, familiar-looking notation can help your intuition, making calculations more fluid. Just don’t trust it farther than you can prove it.

Amplitudes for the New Year

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Ah, the new year, time of new year’s resolutions. While some people resolve to go to the gym or take up online dating, physicists resolve to finally get that paper out.

At least, that’s the impression I get, given the number of papers posted to arXiv in the last month. Since a lot of them were amplitudes-related, I figured I’d go over some highlights.

Everyone once in a while people ask me for the latest news on the amplituhedron. While I don’t know what Nima is working on right now, I can point to what others have been doing. Zvi Bern, Jaroslav Trnka, and collaborators have continued to make progress towards generalizing the amplituhedron to non-planar amplitudes. Meanwhile, a group in Europe has been working on solving an issue I’ve glossed over to some extent. While the amplituhedron is often described as calculating an amplitude as the volume of a geometrical object, in fact there is a somewhat more indirect procedure involved in going from the geometrical object to the amplitude. It would be much simpler if the amplitude was actually the volume of some (different) geometrical object, and that’s what these folks are working towards. Finally, Daniele Galloni has made progress on solving a technical issue: the amplituhedron gives a mathematical recipe for the amplitude, but it doesn’t tell you how to carry out that recipe, and Galloni provides an algorithm for part of this process.

With this new algorithm, is the amplituhedron finally as efficient as older methods? Typically, the way to show that is to do a calculation with the amplituhedron that wasn’t possible before. It doesn’t look like that’s happening soon though, as Jake Bourjaily and collaborators compute an eight-loop integrand using one of the more successful of the older methods. Their paper provides a good answer to the perennial question, “why more loops?” What they find is that some of the assumptions that people made at lower loops fail to hold at this high loop order, and it becomes increasingly important to keep track of exactly how far your symmetries can take you.

Back when I visited Brown, I talked to folks there about some ongoing work. Now that they’ve published, I can talk about it. A while back, Juan Maldacena resurrected an old technique of Landau’s to solve a problem in AdS/CFT. In that paper, he suggested that Landau’s trick might help prove some of the impressive simplifications in N=4 super Yang-Mills that underlie my work and the work of those at Brown. In their new paper, the Brown group finds that, while useful, Landau’s trick doesn’t seem to fully explain the simplicity they’ve discovered. To get a little partisan, I have to say that this was largely the result I expected, and that it felt a bit condescending for Maldacena to assume that an old trick like that from the Feynman diagram era could really be enough to explain one of the big discoveries of amplitudeology.

There was also a paper by Freddy Cachazo and collaborators on an interesting trick to extend their CHY string to one-loop, and one by Bo Feng and collaborators on an intriguing new method called Q-cuts that I will probably say more about in future, but I’ll sign off for now. I’ve got my own new years’ physics resolutions, and I ought to get back to work!

Who Needs Non-Empirical Confirmation?

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I’ve figured out what was bugging me about Dawid’s workshop on non-empirical theory confirmation.

It’s not the concept itself that bothers me. While you might think of science as entirely based on observations of the real world, in practice we can’t test everything. Inevitably, we have to add in other sorts of evidence: judgments based on precedent, philosophical considerations, or sociological factors.

It’s Dawid’s examples that annoy me: string theory, inflation, and the multiverse. Misleading popularizations aside, none of these ideas involve non-empirical confirmation. In particular, string theory doesn’t need non-empirical confirmation, inflation doesn’t want it, and the multiverse, as of yet, doesn’t merit it.

In order for non-empirical confirmation to matter, it needs to affect how people do science. Public statements aren’t very relevant from a philosophy of science perspective; they ebb and flow based on how people promote themselves. Rather, we should care about what scientists assume in the course of their work. If people are basing new work on assumptions that haven’t been established experimentally, then we need to make sure their confidence isn’t misplaced.

String theory hasn’t been established experimentally…but it fails the other side of this test: almost no-one is assuming string theory is true.

I’ve talked before about theorists who study theories that aren’t true. String theory isn’t quite in that category, it’s still quite possible that it describes the real world. Nonetheless, for most string theorists, the distinction is irrelevant: string theory is a way to relate different quantum field theories together, and to formulate novel ones with interesting properties. That sort of research doesn’t rely on string theory being true, often it doesn’t directly involve strings at all. Rather, it relies on string theory’s mathematical abundance, its versatility and power as a lens to look at the world.

There are string theorists who are more directly interested in describing the world with string theory, though they’re a minority. They’re called String Phenomenologists. By itself, “phenomenologist” refers to particle physicists who try to propose theories that can be tested in the real world. “String phenomenology” is actually a bit misleading, since most string phenomenologists aren’t actually in the business of creating new testable theories. Rather, they try to reproduce some of the more common proposals of phenomenologists, like the MSSM, from within the framework of string theory. While string theory can reproduce many possible descriptions of the world (10^500 by some estimates), that doesn’t mean it covers every possible theory; making sure it can cover realistic options is an important, ongoing technical challenge. Beyond that, a minority within a minority of string phenomenologists actually try to make testable predictions, though often these are controversial.

None of these people need non-empirical confirmation. For the majority of string theorists, string theory doesn’t need to be “confirmed” at all. And for the minority who work on string phenomenology, empirical confirmation is still the order of the day, either directly from experiment or indirectly from the particle phenomenologists struggling to describe it.

What about inflation?

Cosmic inflation was proposed to solve an empirical problem, the surprising uniformity of the observed universe. Look through a few papers in the field, and you’ll notice that most are dedicated to finding empirical confirmation: they’re proposing observable effects on the cosmic microwave background, or on the distribution of large-scale structures in the universe. Cosmologists who study inflation aren’t claiming to be certain, and they aren’t rejecting experiment: overall, they don’t actually want non-empirical confirmation.

To be honest, though, I’m being a little unfair to Dawid here. The reason that string theory and inflation are in the name of his workshop aren’t because he thinks they independently use non-empirical confirmation. Rather, it’s because, if you view both as confirmed (and make a few other assumptions), then you’ve got a multiverse.

In this case, it’s again important to compare what people are doing in their actual work to what they’re saying in public. While a lot of people have made public claims about the existence of a multiverse, very few of them actually work on it. In fact, the two sets of people seem to be almost entirely disjoint.

People who make public statements about the multiverse tend to be older prominent physicists, often ones who’ve worked on supersymmetry as a solution to the naturalness problem. For them, the multiverse is essentially an excuse. Naturalness predicted new particles, we didn’t find new particles, so we need an excuse to have an “unnatural” universe, and for many people the multiverse is that excuse. As I’ve argued before, though, this excuse doesn’t have much of an impact on research. These people aren’t discouraged from coming up with new ideas because they believe in the multiverse, rather, they’re talking about the multiverse because they’re currently out of new ideas. Nima Arkani-Hamed is a pretty clear case of someone who has supported the multiverse in pieces like Particle Fever, but who also gets thoroughly excited about new ideas to rescue naturalness.

By contrast, there are many fewer people who actually work on the multiverse itself, and they’re usually less prominent. For the most part, they actually seem concerned with empirical confirmation, trying to hone tricks like anthropic reasoning to the point where they can actually make predictions about future experiments. It’s unclear whether this tiny group of people are on the right track…but what they’re doing definitely doesn’t seem like something that merits non-empirical confirmation, at least at this point.

It’s a shame that Dawid chose the focus he did for his workshop. Non-empirical theory confirmation is an interesting idea (albeit one almost certainly known to philosophy long before Dawid), and there are plenty of places in physics where it could use some examination. We seem to have come to our current interpretation of renormalization non-empirically, and while string theory itself doesn’t rely on non-empirical conformation many of its arguments with loop quantum gravity seem to rely on non-empirical considerations, in particular arguments about what is actually required for a proper theory of quantum gravity. But string theory, inflation, and the multiverse aren’t the examples he’s looking for.

Map Your Dead Ends

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I’m at Brown this week, where I’ve been chatting with Mark Spradlin and Anastasia Volovich, two of the founding figures of my particular branch of amplitudeology. Back in 2010 they figured out how to turn this seventeen-page two-loop amplitude:

into a formula that just takes up two lines:This got everyone very excited, it inspired some of my collaborators to do work that would eventually give rise to the Hexagon Functions, my main research project for the past few years.

Unfortunately, when we tried to push this to higher loops, we didn’t get the sort of nice, clean-looking formulas that the Brown team did. Each “loop” is an additional layer of complexity, a series of approximations that get closer to the exact result. And so far, our answers look more like that first image than the second: hundreds of pages with no clear simplifications in sight.

At the time, people wondered whether some simple formula might be enough. As it turns out, you can write down a formula similar to the one found by Spradlin and Volovich, generalized to a higher number of loops. It’s clean, it’s symmetric, it makes sense…and it’s not the right answer.

That happens in science a lot more often than science fans might expect. When you hear about this sort of thing in the news, it always works: someone suggests a nice, simple answer, and it turns out to be correct, and everyone goes home happy. But for every nice simple guess that works, there are dozens that don’t: promising ideas that just lead to dead ends.

One of the postdocs here at Brown worked on this “wrong” formula, and while chatting with him here he asked a very interesting question: why is it wrong? Sure, we know that it’s wrong, we can check that it’s wrong…but what, specifically, is missing? Is it “part” of the right answer in some sense, with some predictable corrections?

As it turns out, this is a very interesting question! We’ve been looking into it, and the “wrong” answer has some interesting relationships with some of our Hexagon Functions. It may have been a “dead end”, but it still could turn out to be a useful one.

A good physics advisor will tell their students to document their work. This doesn’t just mean taking notes: most theoretical physicists will maintain files, in standard journal article format, with partial results. One reason to do this is that, if things work out, you’ll have some of your paper already written. But if something doesn’t work out, you’ll end up with a pdf on your hard drive carefully explaining an idea that didn’t quite work. Physicists often end up with dozens of these files squirreled away on their computers. Put together, they’re a map: a map of dead ends.

There’s a handy thing about having a map: it lets you retrace your steps. Any one of these paths may lead nowhere, but each one will contain some substantive work. And years later, often enough, you end up needing some of it: some piece of the calculation, some old idea. You follow the map, dig it up…and build it into something new.

Hexagon Functions III: Now with More Symmetry

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I’ve got a new paper up this week.

It’s a continuation of my previous work, understanding collisions involving six particles in my favorite theory, N=4 super Yang-Mills.

This time, we’re pushing up the complexity, going from three “loops” to four. In the past, I could have impressed you with the number of pages the formulas I’m calculating take up (eight hundred pages for the three-loop formula from that first Hexagon Functions paper). Now, though, I don’t have that number: putting my four-loop formula into a pdf-making program just crashes the program. Instead, I’ll have to impress you with file sizes: 2.6 MB for the three-loop formula, 96 MB for the four-loop one.

Calculating such a formula sounds like a pretty big task, and it was, the first time. But things got a lot simpler after a chat I had at Amplitudes.

We calculate these things using an ansatz, a guess for what the final answer should look like. The more vague our guess, the more parameters we need to fix, and the more work we have in general. If we can guess more precisely, we can start with fewer parameters and things are a lot easier.

Often, more precise guesses come from understanding the symmetries of the problem. If we can know that the final answer must be the same after making some change, we can rule out a lot of possibilities.

Sometimes, these symmetries are known features of the answer, things that someone proved had to be correct. Other times, though, they’re just observations, things that have been true in the past and might be true again.

We started out using an observation from three loops. That got us pretty far, but we still had a lot of work to do: 808 parameters, to be fixed by other means. Fixing them took months of work, and throughout we hoped that there was some deeper reason behind the symmetries we observed.

Finally, at Amplitudes, I ran into fellow amplitudeologist Simon Caron-Huot and asked him if he knew the source of our observed symmetry. In just a few days he was able to link it to supersymmetry, giving us justification for our jury rigged trick. However, we figured out that his explanation went further than any of us expected. In the end, rather than 808 parameters we only really needed to consider 34.

Thirty-four options to consider. Thirty-four possible contributions to a ~100 MB file. That might not sound like a big deal, but compared to eight hundred and eight it’s a huge deal. More symmetry means easier calculations, meaning we can go further. At this point going to the next step in complexity, to five loops rather than four, might be well within reach.

Scooped Is a Spectrum

Want to Open up Your Work? Try a Data Mine!

Where Do the Experts Go When They Need an Expert?

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If your game crashes, or Windows keeps spitting out bizarre error messages, you google the problem. Chances are, you find someone on a help forum who had the same problem, and hopefully someone else posted the answer.

(If your preferred strategy is to ask a younger relative, then I’m sorry, but nine times out of ten they’re just doing that.)

What do scientists do, though? We’re at the cutting-edge of knowledge. When we have a problem, who do we turn to?

Typically, Stack Exchange.

The thing is, when we’re really confused about something, most of the time it’s not really a physics problem. We get mystified by the intricacies of Mathematica, or we need some quality trick from numerical methods. And while I haven’t done much with them yet, there are communities dedicated to answering actual physics questions, like Physics Overflow.

The idea I was working on last week? That came from a poster on the Mathematica Stack Exchange, who mentioned a handy little function called Association that I hadn’t heard of before. (It worked, by the way.)

Science is a collaborative process. Sometimes that means actual collaborators, but sometimes we need a little help from folks online, just like everyone else.

Why I Spent Convergence Working