Tag Archives: quantum field theory

Amplitudes 2019 Retrospective

I’m back from Amplitudes 2019, and since I have more time I figured I’d write down a few more impressions.

Amplitudes runs all the way from practical LHC calculations to almost pure mathematics, and this conference had plenty of both as well as everything in between. On the more practical side a standard “pipeline” has developed: get a large number of integrals from generalized unitarity, reduce them to a more manageable number with integration-by-parts, and then compute them with differential equations. Vladimir Smirnov and Johannes Henn presented the state of the art in this pipeline, challenging QCD calculations that required powerful methods. Others aimed to replace various parts of the pipeline. Integration-by-parts could be avoided in the numerical unitarity approach discussed by Ben Page, or alternatively with the intersection theory techniques showcased by Pierpaolo Mastrolia. More radical departures included Stefan Weinzierl’s refinement of loop-tree duality, and Jacob Bourjaily’s advocacy of prescriptive unitarity. Robert Schabinger even brought up direct integration, though I mostly viewed his talk as an independent confirmation of the usefulness of Erik Panzer’s thesis. It also showcased an interesting integral that had previously been represented by Lorenzo Tancredi and collaborators as elliptic, but turned out to be writable in terms of more familiar functions. It’s going to be interesting to see whether other such integrals arise, and whether they can be spotted in advance.

On the other end of the scale, Francis Brown was the only speaker deep enough in the culture of mathematics to insist on doing a blackboard talk. Since the conference hall didn’t actually have a blackboard, this was accomplished by projecting video of a piece of paper that he wrote on as the talk progressed. Despite the awkward setup, the talk was impressively clear, though there were enough questions that he ran out of time at the end and had to “cheat” by just projecting his notes instead. He presented a few theorems about the sort of integrals that show up in string theory. Federico Zerbini and Eduardo Casali’s talks covered similar topics, with the latter also involving intersection theory. Intersection theory also appeared in a poster from grad student Andrzej Pokraka, which overall is a pretty impressively broad showing for a part of mathematics that Sebastian Mizera first introduced to the amplitudes community less than two years ago.

Nima Arkani-Hamed’s talk on Wednesday fell somewhere in between. A series of airline mishaps brought him there only a few hours before his talk, and his own busy schedule sent him back to the airport right after the last question. The talk itself covered several topics, tied together a bit better than usual by a nice account in the beginning of what might motivate a “polytope picture” of quantum field theory. One particularly interesting aspect was a suggestion of a space, smaller than the amplituhedron, that might more accuractly the describe the “alphabet” that appears in N=4 super Yang-Mills amplitudes. If his proposal works, it may be that the infinite alphabet we were worried about for eight-particle amplitudes is actually finite. Ömer Gürdoğan’s talk mentioned this, and drew out some implications. Overall, I’m still unclear as to what this story says about whether the alphabet contains square roots, but that’s a topic for another day. My talk was right after Nima’s, and while he went over-time as always I compensated by accidentally going under-time. Overall, I think folks had fun regardless.

Though I don’t know how many people recognized this guy

Amplitudes 2019

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It’s that time of year again, and I’m at Amplitudes, my field’s big yearly conference. This year we’re in Dublin, hosted by Trinity.

Which also hosts the Book of Kells, and the occasional conference reception just down the hall from the Book of Kells

Increasingly, the organizers of Amplitudes have been setting aside a few slots for talks from people in other fields. This year the “closest” such speaker was Kirill Melnikov, who pointed out some of the hurdles that make it difficult to have useful calculations to compare to the LHC. Many of these hurdles aren’t things that amplitudes-people have traditionally worked on, but are still things that might benefit from our particular expertise. Another such speaker, Maxwell Hansen, is from a field called Lattice QCD. While amplitudeologists typically compute with approximations, order by order in more and more complicated diagrams, Lattice QCD instead simulates particle physics on supercomputers, chopping up their calculations on a grid. This allows them to study much stronger forces, including the messy interactions of quarks inside protons, but they have a harder time with the situations we’re best at, where two particles collide from far away. Apparently, though, they are making progress on that kind of calculation, with some clever tricks to connect it to calculations they know how to do. While I was a bit worried that this would let them fire all the amplitudeologists and replace us with supercomputers, they’re not quite there yet, nonetheless they are doing better than I would have expected. Other speakers from other fields included Leron Borsten, who has been applying the amplitudes concept of the “double copy” to M theory and Andrew Tolley, who uses the kind of “positivity” properties that amplitudeologists find interesting to restrict the kinds of theories used in cosmology.

The biggest set of “non-traditional-amplitudes” talks focused on using amplitudes techniques to calculate the behavior not of particles but of black holes, to predict the gravitational wave patterns detected by LIGO. This year featured a record six talks on the topic, a sixth of the conference. Last year I commented that the research ideas from amplitudeologists on gravitational waves had gotten more robust, with clearer proposals for how to move forward. This year things have developed even further, with several initial results. Even more encouragingly, while there are several groups doing different things they appear to be genuinely listening to each other: there were plenty of references in the talks both to other amplitudes groups and to work by more traditional gravitational physicists. There’s definitely still plenty of lingering confusion that needs to be cleared up, but it looks like the community is robust enough to work through it.

I’m still busy with the conference, but I’ll say more when I’m back next week. Stay tuned for square roots, clusters, and Nima’s travel schedule. And if you’re a regular reader, please fill out last week’s poll if you haven’t already!

Hexagon Functions VI: The Power Cosmic

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I have a new paper out this week. It’s the long-awaited companion to a paper I blogged about a few months back, itself the latest step in a program that has made up a major chunk of my research.

The title is a bit of a mouthful, but I’ll walk you through it:

The Cosmic Galois Group and Extended Steinmann Relations for Planar N = 4 SYM Amplitudes

I calculate scattering amplitudes (roughly, probabilities that elementary particles bounce off each other) in a (not realistic, and not meant to be) theory called planar N=4 super-Yang-Mills (SYM for short). I can’t summarize everything we’ve been doing here, but if you read the blog posts I linked above and some of the Handy Handbooks linked at the top of the page you’ll hopefully get a clearer picture.

We started using the Steinmann Relations a few years ago. Discovered in the 60’s, the Steinmann relations restrict the kind of equations we can use to describe particle physics. Essentially, they mean that particles can’t travel two ways at once. In this paper, we extend the Steinmann relations beyond Steinmann’s original idea. We don’t yet know if we can prove this extension works, but it seems to be true for the amplitudes we’re calculating. While we’ve presented this in talks before, this is the first time we’ve published it, and it’s one of the big results of this paper.

The other, more exotic-sounding result, has to do with something called the Cosmic Galois Group.

Évariste Galois, the famously duel-prone mathematician, figured out relations between algebraic numbers (that is, numbers you can get out of algebraic equations) in terms of a mathematical structure called a group. Today, mathematicians are interested not just in algebraic numbers, but in relations between transcendental numbers as well, specifically a kind of transcendental number called a period. These numbers show up a lot in physics, so mathematicians have been thinking about a Galois group for transcendental numbers that show up in physics, a so-called Cosmic Galois Group.

(Cosmic here doesn’t mean it has to do with cosmology. As far as I can tell, mathematicians just thought it sounded cool and physics-y. They also started out with rather ambitious ideas about it, if you want a laugh check out the last few paragraphs of this talk by Cartier.)

For us, Cosmic Galois Theory lets us study the unusual numbers that show up in our calculations. Doing this, we’ve noticed that certain numbers simply don’t show up. For example, the Riemann zeta function shows up often in our results, evaluated at many different numbers…but never evaluated at the number three. Nor does any number related to that one through the Cosmic Galois Group show up. It’s as if the theory only likes some numbers, and not others.

This weird behavior has been observed before. Mathematicians can prove it happens for some simple theories, but it even applies to the theories that describe the real world, for example to calculations of the way an electron’s path is bent by a magnetic field. Each theory seems to have its own preferred family of numbers.

For us, this has been enormously useful. We calculate our amplitudes by guesswork, starting with the right “alphabet” and then filling in different combinations, as if we’re trying all possible answers to a word jumble. Cosmic Galois Theory and Extended Steinmann have enabled us to narrow down our guess dramatically, making it much easier and faster to get to the right answer.

More generally though, we hope to contribute to mathematicians’ investigations of Cosmic Galois Theory. Our examples are more complicated than the simple theories where they currently prove things, and contain more data than the more limited results from electrons. Hopefully together we can figure out why certain numbers show up and others don’t, and find interesting mathematical principles behind the theories that govern fundamental physics.

For now, I’ll leave you with a preview of a talk I’m giving in a couple weeks’ time:

Two Loops, Five Particles

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There’s a very long-term view of the amplitudes field that gets a lot of press. We’re supposed to be eliminating space and time, or rebuilding quantum field theory from scratch. We build castles in the clouds, seven-loop calculations and all-loop geometrical quantum jewels.

There’s a shorter-term problem, though, that gets much less press, despite arguably being a bigger part of the field right now. In amplitudes, we take theories and turn them into predictions, order by order and loop by loop. And when we want to compare those predictions to the real world, in most cases the best we can do is two loops and five particles.

Five particles here counts the particles coming in and going out: if two gluons collide and become three gluons, we count that as five particles, two in plus three out. Loops, meanwhile, measure the complexity of the calculation, the number of closed paths you can draw in a Feynman diagram. If you use more loops, you expect more precision: you’re approximating nature step by step.

As a field we’re pretty good at one-loop calculations, enough to do them for pretty much any number of particles. As we try for more loops though, things rapidly get harder. Already for two loops, in many cases, we start struggling. We can do better if we dial down the number of particles: there are three-particle and two-particle calculations that get up to three, four, or even five loops. For more particles though, we can’t do as much. Thus the current state of the art, the field’s short term goal: two loops, five particles.

When you hear people like me talk about crazier calculations, we’ve usually got a trick up our sleeve. Often we’re looking at a much simpler theory, one that doesn’t describe the real world. For example, I like working with a planar theory, with lots of supersymmetry. Remove even one of those simplifications, and suddenly our life becomes a lot harder. Instead of seven loops and six particles, we get genuinely excited about, well, two loops five particles.

Luckily, two loops five particles is also about as good as the experiments can measure. As the Large Hadron Collider gathers more data, it measures physics to higher and higher precision. Currently for five-particle processes, its precision is just starting to be comparable with two-loop calculations. The result has been a flurry of activity, applying everything from powerful numerical techniques to algebraic geometry to the problem, getting results that genuinely apply to the real world.

“Two loops, five particles” isn’t as cool of a slogan as “space-time is doomed”. It doesn’t get much, or any media attention. But, steadily and quietly, it’s become one of the hottest topics in the amplitudes field.

Things I’d Like to Know More About

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This is an accountability post, of sorts.

As a kid, I wanted to know everything. Eventually, I realized this was a little unrealistic. Doomed to know some things and not others, I picked physics as a kind of triage. Other fields I could learn as an outsider: not well enough to compete with the experts, but enough to at least appreciate what they were doing. After watching a few string theory documentaries, I realized this wasn’t the case for physics: if I was going to ever understand what those string theorists were up to, I would have to go to grad school in string theory.

Over time, this goal lost focus. I’ve become a very specialized creature, an “amplitudeologist”. I didn’t have time or energy for my old questions. In an irony that will surprise no-one, a career as a physicist doesn’t leave much time for curiosity about physics.

One of the great things about this blog is how you guys remind me of those old questions, bringing me out of my overspecialized comfort zone. In that spirit, in this post I’m going to list a few things in physics that I really want to understand better. The idea is to make a public commitment: within a year, I want to understand one of these topics at least well enough to write a decent blog post on it.

Wilsonian Quantum Field Theory:

When you first learn quantum field theory as a physicist, you learn how unsightly infinite results get covered up via an ad-hoc-looking process called renormalization. Eventually you learn a more modern perspective, that these infinite results show up because we’re ignorant of the complete theory at high energies. You learn that you can think of theories at a particular scale, and characterize them by what happens when you “zoom” in and out, in an approach codified by the physicist Kenneth Wilson.

While I understand the basics of Wilson’s approach, the courses I took in grad school skipped the deeper implications. This includes the idea of theories that are defined at all energies, “flowing” from an otherwise scale-invariant theory perturbed with extra pieces. Other physicists are much more comfortable thinking in these terms, and the topic is important for quite a few deep questions, including what it means to properly define a theory and where laws of nature “live”. If I’m going to have an informed opinion on any of those topics, I’ll need to go back and learn the Wilsonian approach properly.

Wormholes:

If you’re a fan of science fiction, you probably know that wormholes are the most realistic option for faster-than-light travel, something that is at least allowed by the equations of general relativity. “Most realistic” isn’t the same as “realistic”, though. Opening a wormhole and keeping it stable requires some kind of “exotic matter”, and that matter needs to violate a set of restrictions, called “energy conditions”, that normal matter obeys. Some of these energy conditions are just conjectures, some we even know how to violate, while others are proven to hold for certain types of theories. Some energy conditions don’t rule out wormholes, but instead restrict their usefulness: you can have non-traversable wormholes (basically, two inescapable black holes that happen to meet in the middle), or traversable wormholes where the distance through the wormhole is always longer than the distance outside.

I’ve seen a few talks on this topic, but I’m still confused about the big picture: which conditions have been proven, what assumptions were needed, and what do they all imply? I haven’t found a publicly-accessible account that covers everything. I owe it to myself as a kid, not to mention everyone who’s a kid now, to get a satisfactory answer.

Quantum Foundations:

Quantum Foundations is a field that many physicists think is a waste of time. It deals with the questions that troubled Einstein and Bohr, questions about what quantum mechanics really means, or why the rules of quantum mechanics are the way they are. These tend to be quite philosophical questions, where it’s hard to tell if people are making progress or just arguing in circles.

I’m more optimistic about philosophy than most physicists, at least when it’s pursued with enough analytic rigor. I’d like to at least understand the leading arguments for different interpretations, what the constraints on interpretations are and the main loopholes. That way, if I end up concluding the field is a waste of time at least I’d be making an informed decision.

Nonperturbative Methods for Conformal Theories in Natal

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I’m at a conference this week, on Nonperturbative Methods for Conformal Theories, in Natal on the northern coast of Brazil.

Where even the physics institutes have their own little rainforests.

“Nonperturbative” means that most of the people at this conference don’t use the loop-by-loop approximation of Feynman diagrams. Instead, they try to calculate things that don’t require approximations, finding formulas that work even for theories where the forces involved are very strong. In practice this works best in what are called “conformal” theories, roughly speaking these are theories that look the same whichever “scale” you use. Sometimes these theories are “integrable”, theories that can be “solved” exactly with no approximation. Sometimes these theories can be “bootstrapped”, starting with a guess and seeing how various principles of physics constrain it, mapping out a kind of “space of allowed theories”. Both approaches, integrability and bootstrap, are present at this conference.

This isn’t quite my community, but there’s a fair bit of overlap. We care about many of the same theories, like N=4 super Yang-Mills. We care about tricks to do integrals better, or to constrain mathematical guesses better, and we can trade these kinds of tricks and give each other advice. And while my work is typically “perturbative”, I did have one nonperturbative result to talk about, one which turns out to be more closely related to the methods these folks use than I had appreciated.

Hexagon Functions V: Seventh Heaven

Amplitudes in String and Field Theory at NBI

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There’s a conference at the Niels Bohr Institute this week, on Amplitudes in String and Field Theory. Like the conference a few weeks back, this one was funded by the Simons Foundation, as part of Michael Green’s visit here.

The first day featured a two-part talk by Michael Green and Congkao Wen. They are looking at the corrections that string theory adds on top of theories of supergravity. These corrections are difficult to calculate directly from string theory, but one can figure out a lot about them from the kinds of symmetry and duality properties they need to have, using the mathematics of modular forms. While Michael’s talk introduced the topic with a discussion of older work, Congkao talked about their recent progress looking at this from an amplitudes perspective.

Francesca Ferrari’s talk on Tuesday also related to modular forms, while Oliver Schlotterer and Pierre Vanhove talked about a different corner of mathematics, single-valued polylogarithms. These single-valued polylogarithms are of interest to string theorists because they seem to connect two parts of string theory: the open strings that describe Yang-Mills forces and the closed strings that describe gravity. In particular, it looks like you can take a calculation in open string theory and just replace numbers and polylogarithms with their “single-valued counterparts” to get the same calculation in closed string theory. Interestingly, there is more than one way that mathematicians can define “single-valued counterparts”, but only one such definition, the one due to Francis Brown, seems to make this trick work. When I asked Pierre about this he quipped it was because “Francis Brown has good taste…either that, or String Theory has good taste.”

Wednesday saw several talks exploring interesting features of string theory. Nathan Berkovitz discussed his new paper, which makes a certain context of AdS/CFT (a duality between string theory in certain curved spaces and field theory on the boundary of those spaces) manifest particularly nicely. By writing string theory in five-dimensional AdS space in the right way, he can show that if the AdS space is small it will generate the same Feynman diagrams that one would use to do calculations in N=4 super Yang-Mills. In the afternoon, Sameer Murthy showed how localization techniques can be used in gravity theories, including to calculate the entropy of black holes in string theory, while Yvonne Geyer talked about how to combine the string theory-like CHY method for calculating amplitudes with supersymmetry, especially in higher dimensions where the relevant mathematics gets tricky.

Thursday ended up focused on field theory. Carlos Mafra was originally going to speak but he wasn’t feeling well, so instead I gave a talk about the “tardigrade” integrals I’ve been looking at. Zvi Bern talked about his work applying amplitudes techniques to make predictions for LIGO. This subject has advanced a lot in the last few years, and now Zvi and collaborators have finally done a calculation beyond what others had been able to do with older methods. They still have a way to go before they beat the traditional methods overall, but they’re off to a great start. Lance Dixon talked about two-loop five-particle non-planar amplitudes in N=4 super Yang-Mills and N=8 supergravity. These are quite a bit trickier than the planar amplitudes I’ve worked on with him in the past, in particular it’s not yet possible to do this just by guessing the answer without considering Feynman diagrams.

Today was the last day of the conference, and the emphasis was on number theory. David Broadhurst described some interesting contributions from physics to mathematics, in particular emphasizing information that the Weierstrass formulation of elliptic curves omits. Eric D’Hoker discussed how the concept of transcendentality, previously used in field theory, could be applied to string theory. A few of his speculations seemed a bit farfetched (in particular, his setup needs to treat certain rational numbers as if they were transcendental), but after his talk I’m a bit more optimistic that there could be something useful there.

Hadronic Strings and Large-N Field Theory at NBI

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One of string theory’s early pioneers, Michael Green, is currently visiting the Niels Bohr Institute as part of a program by the Simons Foundation. The program includes a series of conferences. This week we are having the first such conference, on Hadronic Strings and Large-N Field Theory.

The bulk of the conference focused on new progress on an old subject, using string theory to model the behavior of quarks and gluons. There were a variety of approaches on offer, some focused on particular approximations and others attempting to construct broader, “phenomenological” models.

The other talks came from a variety of subjects, loosely tied together by the topic of “large N field theories”. “N” here is the number of colors: while the real world has three “colors” of quarks, you can imagine a world with more. This leads to simpler calculations, and often to connections with string theory. Some talks deal with attempts to “solve” certain large-N theories exactly. Others ranged farther afield, even to discussions of colliding black holes.

Changing the Question

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I’ve recently been reading Why Does the World Exist?, a book by the journalist Jim Holt. In it he interviews a range of physicists and philosophers, asking each the question in the title. As the book goes on, he concludes that physicists can’t possibly give him the answer he’s looking for: even if physicists explain the entire universe from simple physical laws, they still would need to explain why those laws exist. A bit disappointed, he turns back to the philosophers.

Something about Holt’s account rubs me the wrong way. Yes, it’s true that physics can’t answer this kind of philosophical problem, at least not in a logically rigorous way. But I think we do have a chance of answering the question nonetheless…by eclipsing it with a better question.

How would that work? Let’s consider a historical example.

Does the Earth go around the Sun, or does the Sun go around the Earth? We learn in school that this is a solved question: Copernicus was right, the Earth goes around the Sun.

The details are a bit more subtle, though. The Sun and the Earth both attract each other: while it is a good approximation to treat the Sun as fixed, in reality it and the Earth both move in elliptical orbits around the same focus (which is close to, but not exactly, the center of the Sun). Furthermore, this is all dependent on your choice of reference frame: if you wish you can choose coordinates in which the Earth stays still while the Sun moves.

So what stops a modern-day Tycho Brahe from arguing that the Sun and the stars and everything else orbit around the Earth?

The reason we aren’t still debating the Copernican versus the Tychonic system isn’t that we proved Copernicus right. Instead, we replaced the old question with a better one. We don’t actually care which object is the center of the universe. What we care about is whether we can make predictions, and what mathematical laws we need to do so. Newton’s law of universal gravitation lets us calculate the motion of the solar system. It’s easier to teach it by talking about the Earth going around the Sun, so we talk about it that way. The “philosophical” question, about the “center of the universe”, has been explained away by the more interesting practical question.

My suspicion is that other philosophical questions will be solved in this way. Maybe physicists can’t solve the ultimate philosophical question, of why the laws of physics are one way and not another. But if we can predict unexpected laws and match observations of the early universe, then we’re most of the way to making the question irrelevant. Similarly, perhaps neuroscientists will never truly solve the mystery of consciousness, at least the way philosophers frame it today. Nevertheless, if they can describe brains well enough to understand why we act like we’re conscious, if they have something in their explanation that looks sufficiently “consciousness-like”, then it won’t matter if they meet the philosophical requirements, people simply won’t care. The question will have been eaten by a more interesting question.

This can happen in physics by itself, without reference to philosophy. Indeed, it may happen again soon. In the New Yorker this week, Natalie Wolchover has an article in which she talks to Nima Arkani-Hamed about the search for better principles to describe the universe. In it, Nima talks about looking for a deep mathematical question that the laws of physics answer. Peter Woit has expressed confusion that Nima can both believe this and pursue various complicated, far-fetched, and at times frankly ugly ideas for new physics.

I think the way to reconcile these two perspectives is to know that Nima takes naturalness seriously. The naturalness argument in physics states that physics as we currently see it is “unnatural”, in particular, that we can’t get it cleanly from the kinds of physical theories we understand. If you accept the argument as stated, then you get driven down a rabbit hole of increasingly strange solutions: versions of supersymmetry that cleverly hide from all experiments, hundreds of copies of the Standard Model, or even a multiverse.

Taking naturalness seriously doesn’t just mean accepting the argument as stated though. It can also mean believing the argument is wrong, but wrong in an interesting way.

One interesting way naturalness could be wrong would be if our reductionist picture of the world, where the ultimate laws live on the smallest scales, breaks down. I’ve heard vague hints from physicists over the years that this might be the case, usually based on the way that gravity seems to mix small and large scales. (Wolchover’s article also hints at this.) In that case, you’d want to find not just a new physical theory, but a new question to ask, something that could eclipse the old question with something more interesting and powerful.

Nima’s search for better questions seems to drive most of his research now. But I don’t think he’s 100% certain that the old questions are wrong, so you can still occasionally see him talking about multiverses and the like.

Ultimately, we can’t predict when a new question will take over. It’s a mix of the social and the empirical, of new predictions and observations but also of which ideas are compelling and beautiful enough to get people to dismiss the old question as irrelevant. It feels like we’re due for another change…but we might not be, and even if we are it might be a long time coming.

4 gravitons

Stories about physics from someone who's been there