Tag Archives: theoretical physics

What You Don’t Know, You Can Parametrize

In physics, what you don’t know can absolutely hurt you. If you ignore that planets have their own gravity, or that metals conduct electricity, you’re going to calculate a lot of nonsense. At the same time, as physicists we can’t possibly know everything. Our experiments are never perfect, our math never includes all the details, and even our famous Standard Model is almost certainly not the whole story. Luckily, we have another option: instead of ignoring what we don’t know, we can parametrize it, and estimate its effect.

Estimating the unknown is something we physicists have done since Newton. You might think Newton’s big discovery was the inverse-square law for gravity, but others at the time, like Robert Hooke, had also been thinking along those lines. Newton’s big discovery was that gravity was universal: that you need to know the effect of gravity, not just from the sun, but from all the other planets as well. The trouble was, Newton didn’t know how to calculate the motion of all of the planets at once (in hindsight, we know he couldn’t have). Instead, he estimated, using what he knew to guess how big the effect of what he didn’t would be. It was the accuracy of those guesses, not just the inverse square law by itself, that convinced the world that Newton was right.

If you’ve studied electricity and magnetism, you get to the point where you can do simple calculations with a few charges in your sleep. The world doesn’t have just a few charges, though: it has many charges, protons and electrons in every atom of every object. If you had to keep all of them in your calculations you’d never pass freshman physics, but luckily you can once again parametrize what you don’t know. Often you can hide those charges away, summarizing their effects with just a few numbers. Other times, you can treat materials as boundaries, and summarize everything beyond in terms of what happens on the edge. The equations of the theory let you do this, but this isn’t true for every theory: for the Navier-Stokes equation, which we use to describe fluids, it still isn’t known whether you can do this kind of trick.

Parametrizing what we don’t know isn’t just a trick for college physics, it’s key to the cutting edge as well. Right now we have a picture for how all of particle physics works, called the Standard Model, but we know that picture is incomplete. There are a million different theories you could write to go beyond the Standard Model, with a million different implications. Instead of having to use all those theories, physicists can summarize them all with what we call an effective theory: one that keeps track of the effect of all that new physics on the particles we already know. By summarizing those effects with a few parameters, we can see what they would have to be to be compatible with experimental results, ruling out some possibilities and suggesting others.

In a world where we never know everything, there’s always something that can hurt us. But if we’re careful and estimate what we don’t know, if we write down numbers and parameters and keep our options open, we can keep from getting burned. By focusing on what we do know, we can still manage to understand the world.

Which Things Exist in Quantum Field Theory

If you ever think metaphysics is easy, learn a little quantum field theory.

Someone asked me recently about virtual particles. When talking to the public, physicists sometimes explain the behavior of quantum fields with what they call “virtual particles”. They’ll describe forces coming from virtual particles going back and forth, or a bubbling sea of virtual particles and anti-particles popping out of empty space.

The thing is, this is a metaphor. What’s more, it’s a metaphor for an approximation. As physicists, when we draw diagrams with more and more virtual particles, we’re trying to use something we know how to calculate with (particles) to understand something tougher to handle (interacting quantum fields). Virtual particles, at least as you’re probably picturing them, don’t really exist.

I don’t really blame physicists for talking like that, though. Virtual particles are a metaphor, sure, a way to talk about a particular calculation. But so is basically anything we can say about quantum field theory. In quantum field theory, it’s pretty tough to say which things “really exist”.

I’ll start with an example, neutrino oscillation.

You might have heard that there are three types of neutrinos, corresponding to the three “generations” of the Standard Model: electron-neutrinos, muon-neutrinos, and tau-neutrinos. Each is produced in particular kinds of reactions: electron-neutrinos, for example, get produced by beta-plus decay, when a proton turns into a neutron, an anti-electron, and an electron-neutrino.

Leave these neutrinos alone though, and something strange happens. Detect what you expect to be an electron-neutrino, and it might have changed into a muon-neutrino or a tau-neutrino. The neutrino oscillated.

Why does this happen?

One way to explain it is to say that electron-neutrinos, muon-neutrinos, and tau-neutrinos don’t “really exist”. Instead, what really exists are neutrinos with specific masses. These don’t have catchy names, so let’s just call them neutrino-one, neutrino-two, and neutrino-three. What we think of as electron-neutrinos, muon-neutrinos, and tau-neutrinos are each some mix (a quantum superposition) of these “really existing” neutrinos, specifically the mixes that interact nicely with electrons, muons, and tau leptons respectively. When you let them travel, it’s these neutrinos that do the traveling, and due to quantum effects that I’m not explaining here you end up with a different mix than you started with.

This probably seems like a perfectly reasonable explanation. But it shouldn’t. Because if you take one of these mass-neutrinos, and interact with an electron, or a muon, or a tau, then suddenly it behaves like a mix of the old electron-neutrinos, muon-neutrinos, and tau-neutrinos.

That’s because both explanations are trying to chop the world up in a way that can’t be done consistently. There aren’t electron-neutrinos, muon-neutrinos, and tau-neutrinos, and there aren’t neutrino-ones, neutrino-twos, and neutrino-threes. There’s a mathematical object (a vector space) that can look like either.

Whether you’re comfortable with that depends on whether you think of mathematical objects as “things that exist”. If you aren’t, you’re going to have trouble thinking about the quantum world. Maybe you want to take a step back, and say that at least “fields” should exist. But that still won’t do: we can redefine fields, add them together or even use more complicated functions, and still get the same physics. The kinds of things that exist can’t be like this. Instead you end up invoking another kind of mathematical object, equivalence classes.

If you want to be totally rigorous, you have to go a step further. You end up thinking of physics in a very bare-bones way, as the set of all observations you could perform. Instead of describing the world in terms of “these things” or “those things”, the world is a black box, and all you’re doing is finding patterns in that black box.

Is there a way around this? Maybe. But it requires thought, and serious philosophy. It’s not intuitive, it’s not easy, and it doesn’t lend itself well to 3d animations in documentaries. So in practice, whenever anyone tells you about something in physics, you can be pretty sure it’s a metaphor. Nice describable, non-mathematical things typically don’t exist.

To Elliptics and Beyond!

I’ve been busy running a conference this week, Elliptics and Beyond.

After Amplitudes was held online this year, a few of us at the Niels Bohr Institute were inspired. We thought this would be the perfect time to hold a small online conference, focused on the Calabi-Yaus that have been popping up lately in Feynman diagrams. Then we heard from the organizers of Elliptics 2020. They had been planning to hold a conference in Mainz about elliptic integrals in Feynman diagrams, but had to postpone it due to the pandemic. We decided to team up and hold a joint conference on both topics: the elliptic integrals that are just starting to be understood, and the mysterious integrals that lie beyond. Hence, Elliptics and Beyond.

I almost suggested Buzz Lightyear for the logo but I chickened out

The conference has been fun thus far. There’s been a mix of review material bringing people up to speed on elliptic integrals and exciting new developments. Some are taking methods that have been successful in other areas and generalizing them to elliptic integrals, others have been honing techniques for elliptics to make them “production-ready”. A few are looking ahead even further, to higher-genus amplitudes in string theory and Calabi-Yaus in Feynman diagrams.

We organized the conference along similar lines to Zoomplitudes, but with a few experiments of our own. Like Zoomplitudes, we made a Slack space for the conference, so people could chat physics outside the talks. Ours was less active, though. I suspect that kind of space needs a critical mass of people, and with a smaller conference we may just not have gotten there. Having fewer people did allow us a more relaxed schedule, which in turn meant we could mostly keep things on-time. We had discussion sessions in the morning (European time), with talks in the afternoon, so almost everyone could make the talks at least. We also had a “conference dinner”, which went much better than I would have expected. We put people randomly into Zoom Breakout Rooms of five or six, to emulate the tables of an in-person conference, and folks chatted while eating their (self-brought of course) dinner. People seemed to really enjoy the chance to just chat casually with the other folks at the conference. If you’re organizing an online conference soon, I’d recommend trying it!

Holding a conference online means that a lot of people can attend who otherwise couldn’t. We had over a hundred people register, and while not all of them showed up there were typically fifty or sixty people on the Zoom session. Some of these were specialists in elliptics or Calabi-Yaus who wouldn’t ordinarily make it to a conference like this. Others were people from the rest of the amplitudes field who joined for parts of the conference that caught their eye. But surprisingly many weren’t even amplitudeologists, but students and young researchers in a variety of topics from all over the world. Some seemed curious and eager to learn, others I suspect just needed to say they had been to a conference. Both are responding to a situation where suddenly conference after conference is available online, free to join. It will be interesting to see if, and how, the world adapts.

Zero-Point Energy, Zero-Point Diagrams

Listen to a certain flavor of crackpot, or a certain kind of science fiction, and you’ll hear about zero-point energy. Limitless free energy drawn from quantum space-time itself, zero-point energy probably sounds like bullshit. Often it is. But lurking behind the pseudoscience and the fiction is a real physics concept, albeit one that doesn’t really work like those people imagine.

In quantum mechanics, the zero-point energy is the lowest energy a particular system can have. That number doesn’t actually have to be zero, even for empty space. People sometimes describe this in terms of so-called virtual particles, popping up from nothing in particle-antiparticle pairs only to annihilate each other again, contributing energy in the absence of any “real particles”. There’s a real force, the Casimir effect, that gets attributed to this, a force that pulls two metal plates together even with no charge or extra electromagnetic field. The same bubbling of pairs of virtual particles also gets used to explain the Hawking radiation of black holes.

I’d like to try explaining all of these things in a different way, one that might clear up some common misconceptions. To start, let’s talk about, not zero-point energy, but zero-point diagrams.

Feynman diagrams are a tool we use to study particle physics. We start with a question: if some specific particles come together and interact, what’s the chance that some (perhaps different) particles emerge? We start by drawing lines representing the particles going in and out, then connect them in every way allowed by our theory. Finally we translate the diagrams to numbers, to get an estimate for the probability. In particle physics slang, the number of “points” is the total number of particles: particles in, plus particles out. For example, let’s say we want to know the chance that two electrons go in and two electrons come out. That gives us a “four-point” diagram: two in, plus two out. A zero-point diagram, then, means zero particles in, zero particles out.

A four-point diagram and a zero-point diagram

(Note that this isn’t why zero-point energy is called zero-point energy, as far as I can tell. Zero-point energy is an older term from before Feynman diagrams.)

Remember, each Feynman diagram answers a specific question, about the chance of particles behaving in a certain way. You might wonder, what question does a zero-point diagram answer? The chance that nothing goes to nothing? Why would you want to know that?

To answer, I’d like to bring up some friends of mine, who do something that might sound equally strange: they calculate one-point diagrams, one particle goes to none. This isn’t strange for them because they study theories with defects.

For some reason, they didn’t like my suggestion to use this stamp on their papers

Normally in particle physics, we think about our particles in an empty, featureless space. We don’t have to, though. One thing we can do is introduce features in this space, like walls and mirrors, and try to see what effect they have. We call these features “defects”.

If there’s a defect like that, then it makes sense to calculate a one-point diagram, because your one particle can interact with something that’s not a particle: it can interact with the defect.

A one-point diagram with a wall, or “defect”

You might see where this is going: let’s say you think there’s a force between two walls, that comes from quantum mechanics, and you want to calculate it. You could imagine it involves a diagram like this:

A “zero-point diagram” between two walls

Roughly speaking, this is the kind of thing you could use to calculate the Casimir effect, that mysterious quantum force between metal plates. And indeed, it involves a zero-point diagram.

Here’s the thing, though: metal plates aren’t just “defects”. They’re real physical objects, made of real physical particles. So while you can think of the Casimir effect with a “zero-point diagram” like that, you can also think of it with a normal diagram, more like the four-point diagram I showed you earlier: one that computes, not a force between defects, but a force between the actual electrons and protons that make up the two plates.

A lot of the time when physicists talk about pairs of virtual particles popping up out of the vacuum, they have in mind a picture like this. And often, you can do the same trick, and think about it instead as interactions between physical particles. There’s a story of roughly this kind for Hawking radiation: you can think of a black hole event horizon as “cutting in half” a zero-point diagram, and see pairs of particles going out from the black hole…but you can also do a calculation that looks more like particles interacting with a gravitational field.

This also might help you understand why, contra the crackpots and science fiction writers, zero-point energy isn’t a source of unlimited free energy. Yes, a force like the Casimir effect comes “from the vacuum” in some sense. But really, it’s a force between two particles. And just like the gravitational force between two particles, this doesn’t give you unlimited free power. You have to do the work to move the particles back over and over again, using the same amount of power you gained from the force to begin with. And unlike the forces you’re used to, these are typically very small effects, as usual for something that depends on quantum mechanics. So it’s even less useful than more everyday forces for this.

Why do so many crackpots and authors expect zero-point energy to be a massive source of power? In part, this is due to mistakes physicists made early on.

Sometimes, when calculating a zero-point diagram (or any other diagram), we don’t get a sensible number. Instead, we get infinity. Physicists used to be baffled by this. Later, they understood the situation a bit better, and realized that those infinities were probably just due to our ignorance. We don’t know the ultimate high-energy theory, so it’s possible something happens at high energies to cancel those pesky infinities. Without knowing exactly what happened, physicists would estimate by using a “cutoff” energy where they expected things to change.

That kind of calculation led to an estimate you might have heard of, that the zero-point energy inside single light bulb could boil all the world’s oceans. That estimate gives a pretty impressive mental image…but it’s also wrong.

This kind of estimate led to “the worst theoretical prediction in the history of physics”, that the cosmological constant, the force that speeds up the expansion of the universe, is 120 orders of magnitude higher than its actual value (if it isn’t just zero). If there really were energy enough inside each light bulb to boil the world’s oceans, the expansion of the universe would be quite different than what we observe.

At this point, it’s pretty clear there is something wrong with these kinds of “cutoff” estimates. The only unclear part is whether that’s due to something subtle or something obvious. But either way, this particular estimate is just wrong, and you shouldn’t take it seriously. Zero-point energy exists, but it isn’t the magical untapped free energy you hear about in stories. It’s tiny quantum corrections to the forces between particles.

A Non-Amplitudish Solution to an Amplitudish Problem

There was an interesting paper last week, claiming to solve a long-standing problem in my subfield.

I calculate what are called scattering amplitudes, formulas that tell us the chance that two particles scatter off each other. Formulas like these exist for theories like the strong nuclear force, called Yang-Mills theories, they also exist for the hypothetical graviton particles of gravity. One of the biggest insights in scattering amplitude research in the last few decades is that these two types of formulas are tied together: as we like to say, gravity is Yang-Mills squared.

A huge chunk of my subfield grew out of that insight. For one, it’s why some of us think we have something useful to say about colliding black holes. But while it’s been used in a dozen different ways, an important element was missing: the principle was never actually proven (at least, not in the way it’s been used).

Now, a group in the UK and the Czech Republic claims to have proven it.

I say “claims” not because I’m skeptical, but because without a fair bit more reading I don’t think I can judge this one. That’s because the group, and the approach they use, isn’t “amplitudish”. They aren’t doing what amplitudes researchers would do.

In the amplitudes subfield, we like to write things as much as possible in terms of measurable, “on-shell” particles. This is in contrast to the older approach that writes things instead in terms of more general quantum fields, with formulas called Lagrangians to describe theories. In part, we avoid the older Lagrangian framing to avoid redundancy: there are many different ways to write a Lagrangian for the exact same physics. We have another reason though, which might seem contradictory: we avoid Lagrangians to stay flexible. There are many ways to rewrite scattering amplitudes that make different properties manifest, and some of the strangest ones don’t seem to correspond to any Lagrangian at all.

If you’d asked me before last week, I’d say that “gravity is Yang-Mills squared” was in that category: something you couldn’t make manifest fully with just a Lagrangian, that you’d need some stranger magic to prove. If this paper is right, then that’s wrong: if you’re careful enough you can prove “gravity is Yang-Mills squared” in the old-school, Lagrangian way.

I’m curious how this is going to develop: what amplitudes people will think about it, what will happen as the experts chime in. For now, as mentioned, I’m reserving judgement, except to say “interesting if true”.

Science as Hermeneutics: Closer Than You’d Think

This post is once again inspired by a Ted Chiang short story. This time, it’s “The Evolution of Human Science”, which imagines a world in which super-intelligent “metahumans” have become incomprehensible to the ordinary humans they’ve left behind. Human scientists in that world practice “hermeneutics“: instead of original research, they try to interpret what the metahumans are doing, reverse-engineering their devices and observing their experiments.

Much like a blogger who, out of ideas, cribs them from books.

It’s a thought-provoking view of what science in the distant future could become. But it’s also oddly familiar.

You might think I’m talking about machine learning here. It’s true that in recent years people have started using machine learning in science, with occasionally mysterious results. There are even a few cases of physicists using machine-learning to suggest some property, say of Calabi-Yau manifolds, and then figuring out how to prove it. It’s not hard to imagine a day when scientists are reduced to just interpreting whatever the AIs throw at them…but I don’t think we’re quite there yet.

Instead, I’m thinking about my own work. I’m a particular type of theoretical physicist. I calculate scattering amplitudes, formulas that tell us the probabilities that subatomic particles collide in different ways. We have a way to calculate these, Feynman’s famous diagrams, but they’re inefficient, so researchers like me look for shortcuts.

How do we find those shortcuts? Often, it’s by doing calculations the old, inefficient way. We use older methods, look at the formulas we get, and try to find patterns. Each pattern is a hint at some new principle that can make our calculations easier. Sometimes we can understand the pattern fully, and prove it should hold. Other times, we observe it again and again and tentatively assume it will keep going, and see what happens if it does.

Either way, this isn’t so different from the hermeneutics scientists practice in the story. Feynman diagrams already “know” every pattern we find, like the metahumans in the story who already know every result the human scientists can discover. But that “knowledge” isn’t in a form we can understand or use. We have to learn to interpret it, to read between the lines and find underlying patterns, to end up with something we can hold in our own heads and put into action with our own hands. The truth may be “out there”, but scientists can’t be content with that. We need to get the truth “in here”. We need to interpret it for ourselves.

Unification That Does Something

I’ve got unification on the brain.

Recently, a commenter asked me what physicists mean when they say two forces unify. While typing up a response, I came across this passage, in a science fiction short story by Ted Chiang.

Physics admits of a lovely unification, not just at the level of fundamental forces, but when considering its extent and implications. Classifications like ‘optics’ or ‘thermodynamics’ are just straitjackets, preventing physicists from seeing countless intersections.

This passage sounds nice enough, but I feel like there’s a misunderstanding behind it. When physicists seek after unification, we’re talking about something quite specific. It’s not merely a matter of two topics intersecting, or describing them with the same math. We already plumb intersections between fields, including optics and thermodynamics. When we hope to find a unified theory, we do so because it does something. A real unified theory doesn’t just aid our calculations, it gives us new ways to alter the world.

To show you what I mean, let me start with something physicists already know: electroweak unification.

There’s a nice series of posts on the old Quantum Diaries blog that explains electroweak unification in detail. I’ll be a bit vaguer here.

You might have heard of four fundamental forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force. You might have also heard that two of these forces are unified: the electromagnetic force and the weak nuclear force form something called the electroweak force.

What does it mean that these forces are unified? How does it work?

Zoom in far enough, and you don’t see the electromagnetic force and the weak force anymore. Instead you see two different forces, I’ll call them “W” and “B”. You’ll also see the Higgs field. And crucially, you’ll see the “W” and “B” forces interact with the Higgs.

The Higgs field is special because it has what’s called a “vacuum” value. Even in otherwise empty space, there’s some amount of “Higgsness” in the background, like the color of a piece of construction paper. This background Higgs-ness is in some sense an accident, just one stable way the universe happens to sit. In particular, it picks out an arbitrary kind of direction: parts of the “W” and “B” forces happen to interact with it, and parts don’t.

Now let’s zoom back out. We could, if we wanted, keep our eyes on the “W” and “B” forces. But that gets increasingly silly. As we zoom out we won’t be able to see the Higgs field anymore. Instead, we’ll just see different parts of the “W” and “B” behaving in drastically different ways, depending on whether or not they interact with the Higgs. It will make more sense to talk about mixes of the “W” and “B” fields, to distinguish the parts that are “lined up” with the background Higgs and the parts that aren’t. It’s like using “aft” and “starboard” on a boat. You could use “north” and “south”, but that would get confusing pretty fast.

My cabin is on the west side of the ship…unless we’re sailing east….

What are those “mixes” of the “W” and “B” forces? Why, they’re the weak nuclear force and the electromagnetic force!

This, broadly speaking, is the kind of unification physicists look for. It doesn’t have to be a “mix” of two different forces: most of the models physicists imagine start with a single force. But the basic ideas are the same: that if you “zoom in” enough you see a simpler model, but that model is interacting with something that “by accident” picks a particular direction, so that as we zoom out different parts of the model behave in different ways. In that way, you could get from a single force to all the different forces we observe.

That “by accident” is important here, because that accident can be changed. That’s why I said earlier that real unification lets us alter the world.

To be clear, we can’t change the background Higgs field with current technology. The biggest collider we have can just make a tiny, temporary fluctuation (that’s what the Higgs boson is). But one implication of electroweak unification is that, with enough technology, we could. Because those two forces are unified, and because that unification is physical, with a physical cause, it’s possible to alter that cause, to change the mix and change the balance. This is why this kind of unification is such a big deal, why it’s not the sort of thing you can just chalk up to “interpretation” and ignore: when two forces are unified in this way, it lets us do new things.

Mathematical unification is valuable. It’s great when we can look at different things and describe them in the same language, or use ideas from one to understand the other. But it’s not the same thing as physical unification. When two forces really unify, it’s an undeniable physical fact about the world. When two forces unify, it does something.

Formal Theory and Simulated Experiment

There are two kinds of theoretical physicists. Some, called phenomenologists, make predictions about the real world. Others, the so-called “formal theorists”, don’t. They work with the same kinds of theories as the phenomenologists, quantum field theories of the sort that have been so successful in understanding the subatomic world. But the specific theories they use are different: usually, toy models that aren’t intended to describe reality.

Most people get this is valuable. It’s useful to study toy models, because they help us tackle the real world. But they stumble on another point. Sure, they say, you can study toy models…but then you should call yourself a mathematician, not a physicist.

I’m a “formal theorist”. And I’m very much not a mathematician, I’m definitely a physicist. Let me explain why, with an analogy.

As an undergrad, I spent some time working in a particle physics lab. The lab had developed a new particle detector chip, designed for a future experiment: the International Linear Collider. It was my job to test this chip.

Naturally, I couldn’t test the chip by flinging particles at it. For one, the collider it was designed for hadn’t been built yet! Instead, I had to use simulated input: send in electrical signals that mimicked the expected particles, and see what happens. In effect, I was using a kind of toy model, as a way to understand better how the chip worked.

I hope you agree that this kind of work counts as physics. It isn’t “just engineering” to feed simulated input into a chip. Not when the whole point of that chip is to go into a physics experiment. This kind of work is a large chunk of what an experimental physicist does.

As a formal theorist, my work with toy models is an important part of what a theoretical physicist does. I test out the “devices” of theoretical physics, the quantum-field-theoretic machinery that we use to investigate the world. Without that kind of careful testing on toy models, we’d have fewer tools to work with when we want to understand reality.

Ok, but you might object: an experimental physicist does eventually build the real experiment. They don’t just spend their career on simulated input. If someone only works on formal theory, shouldn’t that at least make them a mathematician, not a physicist?

Here’s the thing, though: after those summers in that lab, I didn’t end up as an experimental physicist. After working on that chip, I didn’t go on to perfect it for the International Linear Collider. But it would be rather bizarre if that, retroactively, made my work in that time “engineering” and not “physics”.

Oh, I should also mention that the International Linear Collider might not ever be built. So, there’s that.

Formal theory is part of physics because it cares directly about the goals of physics: understanding the real world. It is just one step towards that goal, it doesn’t address the real world alone. But neither do the people testing out chips for future colliders. Formal theory isn’t always useful, similarly, planned experiments don’t always get built. That doesn’t mean it’s not physics.

Zoomplitudes Retrospective

During Zoomplitudes (my field’s big yearly conference, this year on Zoom) I didn’t have time to write a long blog post. I said a bit about the format, but didn’t get a chance to talk about the science. I figured this week I’d go back and give a few more of my impressions. As always, conference posts are a bit more technical than my usual posts, so regulars be warned!

The conference opened with a talk by Gavin Salam, there as an ambassador for LHC physics. Salam pointed out that, while a decent proportion of speakers at Amplitudes mention the LHC in their papers, that fraction has fallen over the years. (Another speaker jokingly wondered which of those mentions were just in the paper’s introduction.) He argued that there is still useful work for us, LHC measurements that will require serious amplitudes calculations to understand. He also brought up what seems like the most credible argument for a new, higher-energy collider: that there are important properties of the Higgs, in particular its interactions, that we still have not observed.

The next few talks hopefully warmed Salam’s heart, as they featured calculations for real-world particle physics. Nathaniel Craig and Yael Shadmi in particular covered the link between amplitudes and Standard Model Effective Field Theory (SMEFT), a method to systematically characterize corrections beyond the Standard Model. Shadmi’s talk struck me because the kind of work she described (building the SMEFT “amplitudes-style”, directly from observable information rather than more complicated proxies) is something I’d seen people speculate about for a while, but which hadn’t been done until quite recently. Now, several groups have managed it, and look like they’ve gotten essentially “all the way there”, rather than just partial results that only manage to replicate part of the SMEFT. Overall it’s much faster progress than I would have expected.

After Shadmi’s talk was a brace of talks on N=4 super Yang-Mills, featuring cosmic Galois theory and an impressively groan-worthy “origin story” joke. The final talk of the day, by Hofie Hannesdottir, covered work with some of my colleagues at the NBI. Due to coronavirus I hadn’t gotten to hear about this in person, so it was good to hear a talk on it, a blend of old methods and new priorities to better understand some old discoveries.

The next day focused on a topic that has grown in importance in our community, calculations for gravitational wave telescopes like LIGO. Several speakers focused on new methods for collisions of spinning objects, where a few different approaches are making good progress (Radu Roiban’s proposal to use higher-spin field theory was particularly interesting) but things still aren’t quite “production-ready”. The older, post-Newtonian method is still very much production-ready, as evidenced by Michele Levi’s talk that covered, among other topics, our recent collaboration. Julio Parra-Martinez discussed some interesting behavior shared by both supersymmetric and non-supersymmetric gravity theories. Thibault Damour had previously expressed doubts about use of amplitudes methods to answer this kind of question, and part of Parra-Martinez’s aim was to confirm the calculation with methods Damour would consider more reliable. Damour (who was actually in the audience, which I suspect would not have happened at an in-person conference) had already recanted some related doubts, but it’s not clear to me whether that extended to the results Parra-Martinez discussed (or whether Damour has stated the problem with his old analysis).

There were a few talks that day that didn’t relate to gravitational waves, though this might have been an accident, since both speakers also work on that topic. Zvi Bern’s talk linked to the previous day’s SMEFT discussion, with a calculation using amplitudes methods of direct relevance to SMEFT researchers. Clifford Cheung’s talk proposed a rather strange/fun idea, conformal symmetry in negative dimensions!

Wednesday was “amplituhedron day”, with a variety of talks on positive geometries and cluster algebras. Featured in several talks was “tropicalization“, a mathematical procedure that can simplify complicated geometries while still preserving essential features. Here, it was used to trim down infinite “alphabets” conjectured for some calculations into a finite set, and in doing so understand the origin of “square root letters”. The day ended with a talk by Nima Arkani-Hamed, who despite offering to bet that he could finish his talk within the half-hour slot took almost twice that. The organizers seemed to have planned for this, since there was one fewer talk that day, and as such the day ended at roughly the usual time regardless.

We also took probably the most unique conference photo I will ever appear in.

For lack of a better name, I’ll call Thursday’s theme “celestial”. The day included talks by cosmologists (including approaches using amplitudes-ish methods from Daniel Baumann and Charlotte Sleight, and a curiously un-amplitudes-related talk from Daniel Green), talks on “celestial amplitudes” (amplitudes viewed from the surface of an infinitely distant sphere), and various talks with some link to string theory. I’m including in that last category intersection theory, which has really become its own thing. This included a talk by Simon Caron-Huot about using intersection theory more directly in understanding Feynman integrals, and a talk by Sebastian Mizera using intersection theory to investigate how gravity is Yang-Mills squared. Both gave me a much better idea of the speakers’ goals. In Mizera’s case he’s aiming for something very ambitious. He wants to use intersection theory to figure out when and how one can “double-copy” theories, and might figure out why the procedure “got stuck” at five loops. The day ended with a talk by Pedro Vieira, who gave an extremely lucid and well-presented “blackboard-style” talk on bootstrapping amplitudes.

Friday was a grab-bag of topics. Samuel Abreu discussed an interesting calculation using the numerical unitarity method. It was notable in part because renormalization played a bigger role than it does in most amplitudes work, and in part because they now have a cool logo for their group’s software, Caravel. Claude Duhr and Ruth Britto gave a two-part talk on their work on a Feynman integral coaction. I’d had doubts about the diagrammatic coaction they had worked on in the past because it felt a bit ad-hoc. Now, they’re using intersection theory, and have a clean story that seems to tie everything together. Andrew McLeod talked about our work on a Feynman diagram Calabi-Yau “bestiary”, while Cristian Vergu had a more rigorous understanding of our “traintrack” integrals.

There are two key elements of a conference that are tricky to do on Zoom. You can’t do a conference dinner, so you can’t do the traditional joke-filled conference dinner speech. The end of the conference is also tricky: traditionally, this is when everyone applauds the organizers and the secretaries are given flowers. As chair for the last session, Lance Dixon stepped up to fill both gaps, with a closing speech that was both a touching tribute to the hard work of organizing the conference and a hilarious pile of in-jokes, including a participation award to Arkani-Hamed for his (unprecedented, as far as I’m aware) perfect attendance.

The Sum of Our Efforts

I got a new paper out last week, with Andrew McLeod, Henrik Munch, and Georgios Papathanasiou.

A while back, some collaborators and I found an interesting set of Feynman diagrams that we called “Omega”. These Omega diagrams were fun because they let us avoid one of the biggest limitations of particle physics: that we usually have to compute approximations, diagram by diagram, rather than finding an exact answer. For these Omegas, we figured out how to add all the infinite set of Omega diagrams up together, with no approximation.

One implication of this was that, in principle, we now knew the answer for each individual Omega diagram, far past what had been computed before. However, writing down these answers was easier said than done. After some wrangling, we got the answer for each diagram in terms of an infinite sum. But despite tinkering with it for a while, even our resident infinite sum expert Georgios Papathanasiou couldn’t quite sum them up.

Naturally, this made me think the sums would make a great Master’s project.

When Henrik Munch showed up looking for a project, Andrew McLeod and I gave him several options, but he settled on the infinite sums. Impressively, he ended up solving the problem in two different ways!

First, he found an old paper none of us had seen before, that gave a general method for solving that kind of infinite sum. When he realized that method was really annoying to program, he took the principle behind it, called telescoping, and came up with his own, simpler method, for our particular case.

Picture an old-timey folding telescope. It might be long when fully extended, but when you fold it up each piece fits inside the previous one, resulting in a much smaller object. Telescoping a sum has the same spirit. If each pair of terms in a sum “fit together” (if their difference is simple), you can rearrange them so that most of the difficulty “cancels out” and you’re left with a much simpler sum.

Henrik’s telescoping idea worked even better than expected. We found that we could do, not just the Omega sums, but other sums in particle physics as well. Infinite sums are a very well-studied field, so it was interesting to find something genuinely new.

The rest of us worked to generalize the result, to check the examples and to put it in context. But the core of the work was Henrik’s. I’m really proud of what he accomplished. If you’re looking for a PhD student, he’s on the market!