Author Archives: 4gravitons

A Tale of Two Donuts

I’ve got a new paper up this week, with Hjalte Frellesvig, Cristian Vergu, and Matthias Volk, about the elliptic integrals that show up in Feynman diagrams.

You can think of elliptic integrals as integrals over a torus, a curve shaped like the outer crust of a donut.

Do you prefer your integrals glazed, or with powdered sugar?

Integrals like these are showing up more and more in our field, the subject of bigger and bigger conferences. By now, we think we have a pretty good idea of how to handle them, but there are still some outstanding mysteries to solve.

One such mystery came up in a paper in 2017, by Luise Adams and Stefan Weinzierl. They were working with one of the favorite examples of this community, the so-called sunrise diagram (sunrise being a good time to eat donuts). And they noticed something surprising: if they looked at the sunrise diagram in different ways, it was described by different donuts.

What do I mean, different donuts?

The integrals we know best in this field aren’t integrals on a torus, but rather integrals on a sphere. In some sense, all spheres are the same: you can make them bigger or smaller, but they don’t have different shapes, they’re all “sphere-shaped”. In contrast, integrals on a torus are trickier, because toruses can have different shapes. Think about different donuts: some might have a thin ring, others a thicker one, even if the overall donut is the same size. You can’t just scale up one donut and get the other.

This donut even has a marked point

My colleague, Cristian Vergu, was annoyed by this. He’s the kind of person who trusts mathematics like an old friend, one who would never lead him astray. He thought that there must be one answer, one correct donut, one natural way to represent the sunrise diagram mathematically. I was skeptical, I don’t trust mathematics nearly as much as Cristian does. To sort it out, we brought in Hjalte Frellesvig and Matthias Volk, and started trying to write the sunrise diagram every way we possibly could. (Along the way, we threw in another “donut diagram”, the double-box, just to see what would happen.)

Rather than getting a zoo of different donuts, we got a surprise: we kept seeing the same two. And in the end, we stumbled upon the answer Cristian was hoping for: one of these two is, in a meaningful sense, the “correct donut”.

What was wrong with the other donut? It turns out when the original two donuts were found, one of them involved a move that is a bit risky mathematically, namely, combining square roots.

For readers who don’t know what I mean, or why this is risky, let me give a simple example. Everyone else can skip to after the torus gif.

Suppose I am solving a problem, and I find a product of two square roots:

\sqrt{x}\sqrt{x}

I could try combining them under the same square root sign, like so:

\sqrt{x^2}

That works, if x is positive. But now suppose x=-1. Plug in negative one to the first expression, and you get,

\sqrt{-1}\sqrt{-1}=i\times i=-1

while in the second,

\sqrt{(-1)^2}=\sqrt{1}=1

Torus transforming, please stand by

In this case, it wasn’t as obvious that combining roots would change the donut. It might have been perfectly safe. It took some work to show that indeed, this was the root of the problem. If the roots are instead combined more carefully, then one of the donuts goes away, leaving only the one, true donut.

I’m interested in seeing where this goes, how many different donuts we have to understand and how they might be related. But I’ve also been writing about donuts for the last hour or so, so I’m getting hungry. See you next week!

This Week, at Scattering-Amplitudes.com

I did a guest post this week, on an outreach site for the Max Planck Institute for Physics. The new Director of their Quantum Field Theory Department, Johannes Henn, has been behind a lot of major developments in scattering amplitudes. He was one of the first to notice just how symmetric N=4 super Yang-Mills is, as well as the first to build the “hexagon functions” that would become my stock-in-trade. He’s also done what we all strive to do, and applied what he learned to the real world, coming up with an approach to differential equations that has become the gold standard for many different amplitudes calculations.

Now in his new position, he has a swanky new outreach site, reached at the conveniently memorable scattering-amplitudes.com and managed by outreach-ologist Sorana Scholtes. They started a fun series recently called “Talking Terms” as a kind of glossary, explaining words that physicists use over and over again. My guest post for them is part of that series. It hearkens all the way back to one of my first posts, defining what “theory” means to a theoretical physicist. It covers something new as well, a phrase I don’t think I’ve ever explained on this blog: “working in a theory”. You can check it out on their site!

Physical Intuition From Physics Experience

One of the most mysterious powers physicists claim is physical intuition. Let the mathematicians have their rigorous proofs and careful calculations. We just need to ask ourselves, “Does this make sense physically?”

It’s tempting to chalk this up to bluster, or physicist arrogance. Sometimes, though, a physicist manages to figure out something that stumps the mathematicians. Edward Witten’s work on knot theory is a classic example, where he used ideas from physics, not rigorous proof, to win one of mathematics’ highest honors.

So what is physical intuition? And what is its relationship to proof?

Let me walk you through an example. I recently saw a talk by someone in my field who might be a master of physical intuition. He was trying to learn about what we call Effective Field Theories, theories that are “effectively” true at some energy but don’t include the details of higher-energy particles. He calculated that there are limits to the effect these higher-energy particles can have, just based on simple cause and effect. To explain the calculation to us, he gave a physical example, of coupled oscillators.

Oscillators are familiar problems for first-year physics students. Objects that go back and forth, like springs and pendulums, tend to obey similar equations. Link two of them together (couple them), and the equations get more complicated, work for a second-year student instead of a first-year one. Such a student will notice that coupled oscillators “repel” each other: their frequencies get father apart than they would be if they weren’t coupled.

Our seminar speaker wanted us to revisit those second-year-student days, in order to understand how different particles behave in Effective Field Theory. Just as the frequencies of the oscillators repel each other, the energies of particles repel each other: the unknown high-energy particles could only push the energies of the lighter particles we can detect lower, not higher.

This is an example of physical intuition. Examine it, and you can learn a few things about how physical intuition works.

First, physical intuition comes from experience. Using physical intuition wasn’t just a matter of imagining the particles and trying to see what “makes sense”. Instead, it required thinking about similar problems from our experience as physicists: problems that don’t just seem similar on the surface, but are mathematically similar.

Second, physical intuition doesn’t replace calculation. Our speaker had done the math, he hadn’t just made a physical argument. Instead, physical intuition serves two roles: to inspire, and to help remember. Physical intuition can inspire new solutions, suggesting ideas that you go on to check with calculation. In addition to that, it can help your mind sort out what you already know. Without the physical story, we might not have remembered that the low-energy particles have their energies pushed down. With the story though, we had a similar problem to compare, and it made the whole thing more memorable. Human minds aren’t good at holding a giant pile of facts. What they are good at is holding narratives. “Physical intuition” ties what we know into a narrative, building on past problems to understand new ones.

Finally, physical intuition can be risky. If the problem is too different then the intuition can lead you astray. The mathematics of coupled oscillators and Effective Field Theories was similar enough for this argument to work, but if it turned out to be different in an important way then the intuition would have backfired, making it harder to find the answer and harder to keep track once it was found.

Physical intuition may seem mysterious. But deep down, it’s just physicists using our experience, comparing similar problems to help keep track of what we need to know. I’m sure chemists, biologists, and mathematicians all have similar stories to tell.

Physics Acculturation

We all agree physics is awesome, right?

Me, I chose physics as a career, so I’d better like it. And you, right now you’re reading a physics blog for fun, so you probably like physics too.

Ok, so we agree, physics is awesome. But it isn’t always awesome.

Read a blog like this, or the news, and you’ll hear about the more awesome parts of physics: the black holes and big bangs, quantum mysteries and elegant mathematics. As freshman physics majors learn every year, most of physics isn’t like that. It’s careful calculation and repetitive coding, incremental improvements to a piece of a piece of a piece of something that might eventually answer a Big Question. Even if intellectually you can see the line from what you’re doing to the big flashy stuff, emotionally the two won’t feel connected, and you might struggle to feel motivated.

Physics solves this through acculturation. Physicists don’t just work on their own, they’re part of a shared worldwide culture of physicists. They spend time with other physicists, and not just working time but social time: they eat lunch together, drink coffee together, travel to conferences together. Spending that time together gives physics more emotional weight: as humans, we care a bit about Big Questions, but we care a lot more about our community.

This isn’t unique to physics, of course, or even to academics. Programmers who have lunch together, philanthropists who pat each other on the back for their donations, these people are trying to harness the same forces. By building a culture around something, you can get people more motivated to do it.

There’s a risk here, of course, that the culture takes over, and we lose track of the real reasons to do science. It’s easy to care about something because your friends care about it because their friends care about it, looping around until it loses contact with reality. In science we try to keep ourselves grounded, to respect those who puncture our bubbles with a good argument or a clever experiment. But we don’t always succeed.

The pandemic has made acculturation more difficult. As a scientist working from home, that extra bit of social motivation is much harder to get. It’s perhaps even harder for new students, who haven’t had the chance to hang out and make friends with other researchers. People’s behavior, what they research and how and when, has changed, and I suspect changing social ties are a big part of it.

In the long run, I don’t think we can do without the culture of physics. We can’t be lone geniuses motivated only by our curiosity, that’s just not how people work. We have to meld the two, mix the social with the intellectual…and hope that when we do, we keep the engines of discovery moving.

What Tells Your Story

I watched Hamilton on Disney+ recently. With GIFs and songs from the show all over social media for the last few years, there weren’t many surprises. One thing that nonetheless struck me was the focus on historical evidence. The musical Hamilton is based on Ron Chernow’s biography of Alexander Hamilton, and it preserves a surprising amount of the historian’s care for how we know what we know, hidden within the show’s other themes. From the refrain of “who tells your story”, to the importance of Eliza burning her letters with Hamilton (not just the emotional gesture but the “gap in the narrative” it created for historians), to the song “The Room Where It Happens” (which looked from GIFsets like it was about Burr’s desire for power, but is mostly about how much of history is hidden in conversations we can only partly reconstruct), the show keeps the puzzle of reasoning from incomplete evidence front-and-center.

Any time we try to reason about the past, we are faced with these kinds of questions. They don’t just apply to history, but to the so-called historical sciences as well, sciences that study the past. Instead of asking “who” told the story, such scientists must keep in mind “what” is telling the story. For example, paleontologists reason from fossils, and thus are limited by what does and doesn’t get preserved. As a result after a century of studying dinosaurs, only in the last twenty years did it become clear they had feathers.

Astronomy, too, is a historical science. Whenever astronomers look out at distant stars, they are looking at the past. And just like historians and paleontologists, they are limited by what evidence happened to be preserved, and what part of that evidence they can access.

These limitations lead to mysteries, and often controversies. Before LIGO, astronomers had an idea of what the typical mass of a black hole was. After LIGO, a new slate of black holes has been observed, with much higher mass. It’s still unclear why.

Try to reason about the whole universe, and you end up asking similar questions. When we see the movement of “standard candle” stars, is that because the universe’s expansion is accelerating, or are the stars moving as a group?

Push far enough back and the evidence doesn’t just lead to controversy, but to hard limits on what we can know. No matter how good our telescopes are, we won’t see light older than the cosmic microwave background: before that background was emitted the universe was filled with plasma, which would have absorbed any earlier light, erasing anything we could learn from it. Gravitational waves may one day let us probe earlier, and make discoveries as surprising as feathered dinosaurs. But there is yet a stronger limit to how far back we can go, beyond which any evidence has been so diluted that it is indistinguishable from random noise. We can never quite see into “the room where it happened”.

It’s gratifying to see questions of historical evidence in a Broadway musical, in the same way it was gratifying to hear fractals mentioned in a Disney movie. It’s important to think about who, and what, is telling the stories we learn. Spreading that lesson helps all of us reason better.

A Physicist New Year

Happy New Year to all!

Physicists celebrate the new year by trying to sneak one last paper in before the year is over. Looking at Facebook last night I saw three different friends preview the papers they just submitted. The site where these papers appear, arXiv, had seventy new papers this morning, just in the category of theoretical high-energy physics. Of those, nine of them were in my, or a closely related subfield.

I’d love to tell you all about these papers (some exciting! some long-awaited!), but I’m still tired from last night and haven’t read them yet. So I’ll just close by wishing you all, once again, a happy new year.

Newtonmas in Uncertain Times

Three hundred and eighty-two years ago today (depending on which calendars you use), Isaac Newton was born. For a scientist, that’s a pretty good reason to celebrate.

Reason’s Greetings Everyone!

Last month, our local nest of science historians at the Niels Bohr Archive hosted a Zoom talk by Jed Z. Buchwald, a Newton scholar at Caltech. Buchwald had a story to tell about experimental uncertainty, one where Newton had an important role.

If you’ve ever had a lab course in school, you know experiments never quite go like they’re supposed to. Set a room of twenty students to find Newton’s constant, and you’ll get forty different answers. Whether you’re reading a ruler or clicking a stopwatch, you can never measure anything with perfect accuracy. Each time you measure, you introduce a little random error.

Textbooks worth of statistical know-how has cropped up over the centuries to compensate for this error and get closer to the truth. The simplest trick though, is just to average over multiple experiments. It’s so obvious a choice, taking a thousand little errors and smoothing them out, that you might think people have been averaging in this way through history.

They haven’t though. As far as Buchwald had found, the first person to average experiments in this way was Isaac Newton.

What did people do before Newton?

Well, what might you do, if you didn’t have a concept of random error? You can still see that each time you measure you get a different result. But you would blame yourself: if you were more careful with the ruler, quicker with the stopwatch, you’d get it right. So you practice, you do the experiment many times, just as you would if you were averaging. But instead of averaging, you just take one result, the one you feel you did carefully enough to count.

Before Newton, this was almost always what scientists did. If you were an astronomer mapping the stars, the positions you published would be the last of a long line of measurements, not an average of the rest. Some other tricks existed. Tycho Brahe for example folded numbers together pair by pair, averaging the first two and then averaging that average with the next one, getting a final result weighted to the later measurements. But, according to Buchwald, Newton was the first to just add everything together.

Even Newton didn’t yet know why this worked. It would take later research, theorems of statistics, to establish the full justification. It seems Newton and his later contemporaries had a vague physics analogy in mind, finding a sort of “center of mass” of different experiments. This doesn’t make much sense – but it worked, well enough for physics as we know it to begin.

So this Newtonmas, let’s thank the scientists of the past. Working piece by piece, concept by concept, they gave use the tools to navigate our uncertain times.

Inevitably Arbitrary

Physics is universal…or at least, it aspires to be. Drop an apple anywhere on Earth, at any point in history, and it will accelerate at roughly the same rate. When we call something a law of physics, we expect it to hold everywhere in the universe. It shouldn’t depend on anything arbitrary.

Sometimes, though, something arbitrary manages to sneak in. Even if the laws of physics are universal, the questions we want to answer are not: they depend on our situation, on what we want to know.

The simplest example is when we have to use units. The mass of an electron is the same here as it is on Alpha Centauri, the same now as it was when the first galaxies formed. But what is that mass? We could write it as 9.1093837015×10−31 kilograms, if we wanted to, but kilograms aren’t exactly universal. Their modern definition is at least based on physical constants, but with some pretty arbitrary numbers. It defines the Planck constant as 6.62607015×10−34 Joule-seconds. Chase that number back, and you’ll find references to the Earth’s circumference and the time it takes to turn round on its axis. The mass of the electron may be the same on Alpha Centauri, but they’d never write it as 9.1093837015×10−31 kilograms.

Units aren’t the only time physics includes something arbitrary. Sometimes, like with units, we make a choice of how we measure or calculate something. We choose coordinates for a plot, a reference frame for relativity, a zero for potential energy, a gauge for gauge theories and regularization and subtraction schemes for quantum field theory. Sometimes, the choice we make is instead what we measure. To do thermodynamics we must choose what we mean by a state, to call two substances water even if their atoms are in different places. Some argue a perspective like this is the best way to think about quantum mechanics. In a different context, I’d argue it’s why we say coupling constants vary with energy.

So what do we do, when something arbitrary sneaks in? We have a few options. I’ll illustrate each with the mass of the electron:

  • Make an arbitrary choice, and stick with it: There’s nothing wrong with measuring an electron in kilograms, if you’re consistent about it. You could even use ounces. You just have to make sure that everyone else you compare with is using the same units, or be careful to convert.
  • Make a “natural” choice: Why not set the speed of light and Planck’s constant to one? They come up a lot in particle physics, and all they do is convert between length and time, or time and energy. That way you can use the same units for all of them, and use something convenient, like electron-Volts. They even have electron in the name! Of course they also have “Volt” in the name, and Volts are as arbitrary as any other metric unit. A “natural” choice might make your life easier, but you should always remember it’s still arbitrary.
  • Make an efficient choice: This isn’t always the same as the “natural” choice. The units you choose have an effect on how difficult your calculation is. Sometimes, the best choice for the mass of an electron is “one electron-mass”, because it lets you calculate something else more easily. This is easier to illustrate with other choices: for example, if you have to pick a reference frame for a collision, picking one in which one of the objects is at rest, or where they move symmetrically, might make your job easier.
  • Stick to questions that aren’t arbitrary: No matter what units we use, the electron’s mass will be arbitrary. Its ratios to other masses won’t be though. No matter where we measure, dimensionless ratios like the mass of the muon divided by the mass of the electron, or the mass of the electron divided by the value of the Higgs field, will be the same. If we can make sure to ask only this kind of question, we can avoid arbitrariness. Note that we can think of even a mass in “kilograms” as this kind of question: what’s the ratio of the mass of the electron to “this arbitrary thing we’ve chosen”? In practice though, you want to compare things in the same theory, without the historical baggage of metric.

This problem may seem silly, and if we just cared about units it might be. But at the cutting-edge of physics there are still areas where the arbitrary shows up. Our choices of how to handle it, or how to avoid it, can be crucial to further progress.

QCD Meets Gravity 2020, Retrospective

I was at a Zoomference last week, called QCD Meets Gravity, about the many ways gravity can be thought of as the “square” of other fundamental forces. I didn’t have time to write much about the actual content of the conference, so I figured I’d say a bit more this week.

A big theme of this conference, as in the past few years, was gravitational waves. From LIGO’s first announcement of a successful detection, amplitudeologists have been developing new methods to make predictions for gravitational waves more efficient. It’s a field I’ve dabbled in a bit myself. Last year’s QCD Meets Gravity left me impressed by how much progress had been made, with amplitudeologists already solidly part of the conversation and able to produce competitive results. This year felt like another milestone, in that the amplitudeologists weren’t just catching up with other gravitational wave researchers on the same kinds of problems. Instead, they found new questions that amplitudes are especially well-suited to answer. These included combining two pieces of these calculations (“potential” and “radiation”) that the older community typically has to calculate separately, using an old quantum field theory trick, finding the gravitational wave directly from amplitudes, and finding a few nice calculations that can be used to “generate” the rest.

A large chunk of the talks focused on different “squaring” tricks (or as we actually call them, double-copies). There were double-copies for cosmology and conformal field theory, for the celestial sphere, and even some version of M theory. There were new perspectives on the double-copy, new building blocks and algebraic structures that lie behind it. There were talks on the so-called classical double-copy for space-times, where there have been some strange discoveries (an extra dimension made an appearance) but also a more rigorous picture of where the whole thing comes from, using twistor space. There were not one, but two talks linking the double-copy to the Navier-Stokes equation describing fluids, from two different groups. (I’m really curious whether these perspectives are actually useful for practical calculations about fluids, or just fun to think about.) Finally, while there wasn’t a talk scheduled on this paper, the authors were roped in by popular demand to talk about their work. They claim to have made progress on a longstanding puzzle, how to show that double-copy works at the level of the Lagrangian, and the community was eager to dig into the details.

From there, a grab-bag of talks covered other advancements. There were talks from string theorists and ambitwistor string theorists, from Effective Field Theorists working on gravity and the Standard Model, from calculations in N=4 super Yang-Mills, QCD, and scalar theories. Simon Caron-Huot delved into how causality constrains the theories we can write down, showing an interesting case where the common assumption that all parameters are close to one is actually justified. Nima Arkani-Hamed began his talk by saying he’d surprise us, which he certainly did (and not by keeping on time). It’s tricky to explain why his talk was exciting. Comparing to his earlier discovery of the Amplituhedron, which worked for a toy model, this is a toy calculation in a toy model. While the Amplituhedron wasn’t based on Feynman diagrams, this can’t even be compared with Feynman diagrams. Instead of expanding in a small coupling constant, this expands in a parameter that by all rights should be equal to one. And instead of positivity conditions, there are negativity conditions. All I can say is that with all of that in mind, it looks like real progress on an important and difficult problem from a totally unanticipated direction. In a speech summing up the conference, Zvi Bern mentioned a few exciting words from Nima’s talk: “nonplanar”, “integrated”, “nonperturbative”. I’d add “differential equations” and “infinite sums of ladder diagrams”. Nima and collaborators are trying to figure out what happens when you sum up all of the Feynman diagrams in a theory. I’ve made progress in the past for diagrams with one “direction”, a ladder that grows as you add more loops, but I didn’t know how to add “another direction” to the ladder. In very rough terms, Nima and collaborators figured out how to add that direction.

I’ve probably left things out here, it was a packed conference! It’s been really fun seeing what the community has cooked up, and I can’t wait to see what happens next.

QCD Meets Gravity 2020

I’m at another Zoom conference this week, QCD Meets Gravity. This year it’s hosted by Northwestern.

The view of the campus from wonder.me

QCD Meets Gravity is a conference series focused on the often-surprising links between quantum chromodynamics on the one hand and gravity on the other. By thinking of gravity as the “square” of forces like the strong nuclear force, researchers have unlocked new calculation techniques and deep insights.

Last year’s conference was very focused on one particular topic, trying to predict the gravitational waves observed by LIGO and VIRGO. That’s still a core topic of the conference, but it feels like there is a bit more diversity in topics this year. We’ve seen a variety of talks on different “squares”: new theories that square to other theories, and new calculations that benefit from “squaring” (even surprising applications to the Navier-Stokes equation!) There are talks on subjects from String Theory to Effective Field Theory, and even a talk on a very different way that “QCD meets gravity”, in collisions of neutron stars.

With still a few more talks to go, expect me to say a bit more next week, probably discussing a few in more detail. (Several people presented exciting work in progress!) Until then, I should get back to watching!