Tag Archives: quantum field theory

The Rippling Pond Universe

[Background: Someone told me they couldn’t imagine popularizing Quantum Field Theory in the same flashy way people popularize String Theory. Naturally I took this as a challenge. Please don’t take any statements about what “really exists” here too seriously, this isn’t intended as metaphysics, just metaphor.]


You probably learned about atoms in school.

Your teacher would have explained that these aren’t the same atoms the ancient Greeks imagined. Democritus thought of atoms as indivisible, unchanging spheres, the fundamental constituents of matter. We know, though, that atoms aren’t indivisible. They’re clouds of electrons, buzzing in their orbits around a nucleus of protons and neutrons. Chemists can divide the electrons from the rest, nuclear physicists can break the nucleus. The atom is not indivisible.

And perhaps your teacher remarked on how amazing it is, that the nucleus is such a tiny part of the atom, that the atom, and thus all solid matter, is mostly empty space.


You might have learned that protons and neutrons, too, are not indivisible. That each proton, and each neutron, is composed of three particles called quarks, particles which can be briefly freed by powerful particle colliders.

And you might have wondered, then, even if you didn’t think to ask: are quarks atoms? The real atoms, the Greek atoms, solid indestructible balls of fundamental matter?


They aren’t, by the way.


You might have gotten an inkling of this, learning about beta decay. In beta decay, a neutron transforms, becoming a proton, an electron, and a neutrino. Look for an electron inside a neutron, and you won’t find one. Even if you look at the quarks, you see the same transformation: a down quark becomes an up quark, plus an electron, plus a neutrino. If quarks were atoms, indivisible and unchanging, this couldn’t happen. There’s nowhere for the electron to hide.


In fact, there are no atoms, not the way the Greeks imagined. Just ripples.

Water Drop

Picture the universe as a pond. This isn’t a still pond: something has disturbed it, setting ripples and whirlpools in motion. These ripples and whirlpools skim along the surface of the pond, eddying together and scattering apart.

Our universe is not a simple pond, and so these are not simple ripples. They shine and shimmer, each with their own bright hue, colors beyond our ordinary experience that mix in unfamiliar ways. The different-colored ripples interact, merge and split, and the pond glows with their light.

Stand back far enough, and you notice patterns. See that red ripple, that stays together and keeps its shape, that meets other ripples and interacts in predictable ways. You might imagine the red ripple is an atom, truly indivisible…until it splits, transforms, into ripples of new colors. The quark has changed, down to up, an electron and a neutrino rippling away.

All of our world is encoded in the colors of these ripples, each kind of charge its own kind of hue. With a wink (like your teacher’s, telling you of empty atoms), I can tell you that distance itself is just a kind of ripple, one that links other ripples together. The pond’s very nature as a place is defined by the ripples on it.


This is Quantum Field Theory, the universe of ripples. Democritus said that in truth there are only atoms and the void, but he was wrong. There are no atoms. There is only the void. It ripples and shimmers, and each of us lives as a collection of whirlpools, skimming the surface, seeming concrete and real and vital…until the ripples dissolve, and a new pattern comes.


At the GGI Lectures on the Theory of Fundamental Interactions

I’m at the Galileo Galilei Institute for Theoretical Physics in Florence at their winter school, the GGI Lectures on the Theory of Fundamental Interactions. Next week I’ll be helping Lance Dixon teach Amplitudeology, this week, I’m catching the tail end of Ira Rothstein’s lectures.


The Galileo Galilei Institute, at the end of a long, winding road filled with small, speedy cars and motorcycles, in classic Italian fashion

Rothstein has been heavily involved in doing gravitational wave calculations using tools from quantum field theory, something that has recently captured a lot of interest from amplitudes people. Specifically, he uses Effective Field Theory, theories that are “effectively” true at some scale but hide away higher-energy physics. In the case of gravitational waves, these theories are a powerful way to calculate the waves that LIGO and VIRGO can observe without using the full machinery of general relativity.

After seeing Rothstein’s lectures, I’m reminded of something he pointed out at the QCD Meets Gravity conference in December. He emphasized then that even if amplitudes people get very good at drawing diagrams for classical general relativity, that won’t be the whole story: there’s a series of corrections needed to “match” between the quantities LIGO is able to see and the ones we’re able to calculate. Different methods incorporate these corrections in different ways, and the most intuitive approach for us amplitudes folks may still end up cumbersome once all the corrections are included. In typical amplitudes fashion, this just makes me wonder if there’s a shortcut: some way to compute, not just a piece that gets plugged in to an Effective Field Theory story, but the waves LIGO sees in one fell swoop (or at least, the part where gravity is weak enough that our methods are still useful). That’s probably a bit naive of me, though.

4gravitons Meets QCD Meets Gravity

I’m at UCLA this week, for the workshop QCD Meets Gravity. I haven’t worked on QCD or gravity yet, so I’m mostly here as an interested observer, and as an excuse to enjoy Los Angeles in December.


I think there’s a song about this…

QCD Meets Gravity is a conference centered around the various ways that “gravity is Yang-Mills squared”. There are a number of tricks that let you “square” calculations in Yang-Mills theories (a type of theory that includes QCD) to get calculations in gravity, and this conference showcased most of them.

At Amplitudes this summer, I was disappointed there were so few surprises. QCD Meets Gravity was different, with several talks on new or preliminary results, including one by Julio Parra-Martinez where the paper went up in the last few minutes of the talk! Yu-tin Huang talked about his (still-unpublished) work with Nima Arkani-Hamed on “UV/IR Polytopes”. The story there is a bit like the conformal bootstrap, with constraints (in this case based on positivity) marking off a space of “allowed” theories. String theory, interestingly, is quite close to the boundary of what is allowed. Enrico Herrmann is working on a way to figure out which gravity integrands are going to diverge without actually integrating them, while Simon Caron-Huot, in his characteristic out-of-the-box style, is wondering whether supersymmetric black holes precess. We also heard a bit more about a few recent papers. Oliver Schlotterer’s talk cleared up one thing: apparently the GEF functions he defines in his paper on one-loop “Z theory” are pronounced “Jeff”. I kept waiting for him to announce “Jeff theory”, but unfortunately no such luck. Sebastian Mizera’s talk was a very clear explanation of intersection theory, the subject of his recent paper. As it turns out, intersection theory is the study of mathematical objects like the Beta function (which shows up extensively in string theory), taking them apart in a way very reminiscent of the “squaring” story of Yang-Mills and gravity.

The heart of the workshop this year was gravitational waves. Since LIGO started running, amplitudes researchers (including, briefly, me) have been looking for ways to get involved. This conference’s goal was to bring together amplitudes people and the gravitational wave community, to get a clearer idea of what we can contribute. Between talks and discussions, I feel like we all understand the problem better. Some things that the amplitudes community thought were required, like breaking the symmetries of special relativity, turn out to be accidents of how the gravitational wave community calculates things: approximations that made things easier for them, but make things harder for us. There are areas in which we can make progress quite soon, even areas in which amplitudes people have already made progress. The detectors for which the new predictions matter might still be in the future (LIGO can measure two or three “loops”, LISA will see up to four), but they will eventually be measured. Amplitudes and gravitational wave physics could turn out to be a very fruitful partnership.


Interesting Work at the IAS

I’m visiting the Institute for Advanced Study this week, on the outskirts of Princeton’s impressively Gothic campus.


A typical Princeton reading room

The IAS was designed as a place for researchers to work with minimal distraction, and we’re taking full advantage of it. (Though I wouldn’t mind a few more basic distractions…dinner closer than thirty minutes away for example.)

The amplitudes community seems to be busily working as well, with several interesting papers going up on the arXiv this week, four with some connection to the IAS.

Carlos Mafra and Oliver Schlotterer’s paper about one-loop string amplitudes mentions visiting the IAS in the acknowledgements. Mafra and Schlotterer have found a “double-copy” structure in the one-loop open string. Loosely, “double-copy” refers to situations in which one theory can be described as two theories “multiplied together”, like how “gravity is Yang-Mills squared”. Normally, open strings would be the “Yang-Mills” in that equation, with their “squares”, closed strings, giving gravity. Here though, open strings themselves are described as a “product” of two different pieces, a Yang-Mills part and one that takes care of the “stringiness”. You may remember me talking about something like this and calling it “Z theory”. That was at “tree level”, for the simplest string diagrams. This paper updates the technology to one-loop, where the part taking care of the “stringiness” has a more sophisticated mathematical structure. It’s pretty nontrivial for this kind of structure to survive at one loop, and it suggests something deeper is going on.

Yvonne Geyer (IAS) and Ricardo Monteiro (non-IAS) work on the ambitwistor string, a string theory-like setup for calculating particle physics amplitudes. Their paper shows how this setup can be used for one-loop amplitudes in a wide range of theories, in particular theories without supersymmetry. This makes some patterns that were observed before quite a bit clearer, and leads to a fairly concise way of writing the amplitudes.

Nima-watchers will be excited about a paper by Nima Arkani-Hamed and his student Yuntao Bai (IAS) and Song He and his student Gongwang Yan (non-IAS). This paper is one that has been promised for quite some time, Nima talked about it at Amplitudes last summer. Nima is famous for the amplituhedron, an abstract geometrical object that encodes amplitudes in one specific theory, N=4 super Yang-Mills. Song He is known for the Cachazo-He-Yuan (or CHY) string, a string-theory like picture of particle scattering in a very general class of theories that is closely related to the ambitwistor string. Collaborating, they’ve managed to link the two pictures together, and in doing so take the first step to generalizing the amplituhedron to other theories. In order to do this they had to think about the amplituhedron not in terms of some abstract space, but in terms of the actual momenta of the particles they’re colliding. This is important because the amplituhedron’s abstract space is very specific to N=4 super Yang-Mills, with supersymmetry in some sense built in, while momenta can be written down for any particles. Once they had mastered this trick, they could encode other things in this space of momenta: colors of quarks, for example. Using this, they’ve managed to find amplituhedron-like structure in the CHY string, and in a few particular theories. They still can’t do everything the amplituhedron can, in particular the amplituhedron can go to any number of loops while the structures they’re finding are tree-level. But the core trick they’re using looks very powerful. I’ve been hearing hints about the trick from Nima for so long that I had forgotten they hadn’t published it yet, now that they have I’m excited to see what the amplitudes community manages to do with it.

Finally, last night a paper by Igor Prlina, Marcus Spradlin, James Stankowicz, Stefan Stanojevic, and Anastasia Volovich went up while three of the authors were visiting the IAS. The paper deals with Landau equations, a method to classify and predict the singularities of amplitudes. By combining this method with the amplituhedron they’ve already made substantial progress, and this paper serves as a fairly thorough proof of principle, using the method to comprehensively catalog the singularities of one-loop amplitudes. In this case I’ve been assured that they have papers at higher loops in the works, so it will be interesting to see how powerful this method ends up being.

Amplitudes Papers I Haven’t Had Time to Read

Interesting amplitudes papers seem to come in groups. Several interesting papers went up this week, and I’ve been too busy to read any of them!

Well, that’s not quite true, I did manage to read this paper, by James Drummond, Jack Foster, and Omer Gurdogan. At six pages long, it wasn’t hard to fit in, and the result could be quite useful. The way my collaborators and I calculate amplitudes involves building up a mathematical object called a symbol, described in terms of a string of “letters”. What James and collaborators have found is a restriction on which “letters” can appear next to each other, based on the properties of a mathematical object called a cluster algebra. Oddly, the restriction seems to have the same effect as a more physics-based condition we’d been using earlier. This suggests that the abstract mathematical restriction and the physics-based restriction are somehow connected, but we don’t yet understand how. It also could be useful for letting us calculate amplitudes with more particles: previously we thought the number of “letters” we’d have to consider there was going to be infinite, but with James’s restriction we’d only need to consider a finite number.

I didn’t get a chance to read David Dunbar, John Godwin, Guy Jehu, and Warren Perkins’s paper. They’re computing amplitudes in QCD (which unlike N=4 super Yang-Mills actually describes the real world!) and doing so for fairly complicated arrangements of particles. They claim to get remarkably simple expressions: since that sort of claim was what jump-started our investigations into N=4, I should probably read this if only to see if there’s something there in the real world amenable to our technique.

I also haven’t read Rutger Boels and Hui Lui’s paper yet. From the abstract, I’m still not clear which parts of what they’re describing is new, or how much it improves on existing methods. It will probably take a more thorough reading to find out.

I really ought to read Burkhard Eden, Yunfeng Jiang, Dennis le Plat, and Alessandro Sfondrini’s paper. They’re working on a method referred to as the Hexagon Operator Product Expansion, or HOPE. It’s related to an older method, the Pentagon Operator Product Expansion (POPE), but applicable to trickier cases. I’ve been keeping an eye on the HOPE in part because my collaborators have found the POPE very useful, and the HOPE might enable something similar. It will be interesting to find out how Eden et al.’s paper modifies the HOPE story.

Finally, I’ll probably find the time to read my former colleague Sebastian Mizera’s paper. He’s found a connection between the string-theory-like CHY picture of scattering amplitudes and some unusual mathematical structures. I’m not sure what to make of it until I get a better idea of what those structures are.

When It Rains It Amplitudes

The last few weeks have seen a rain of amplitudes papers on arXiv, including quite a few interesting ones.


As well as a fair amount of actual rain in Copenhagen

Over the last year Nima Arkani-Hamed has been talking up four or five really interesting results, and not actually publishing any of them. This has understandably frustrated pretty much everybody. In the last week he published two of them, Cosmological Polytopes and the Wavefunction of the Universe with Paolo Benincasa and Alexander Postnikov and Scattering Amplitudes For All Masses and Spins with Tzu-Chen Huang and Yu-tin Huang. So while I’ll have to wait on the others (I’m particularly looking forward to seeing what he’s been working on with Ellis Yuan) this can at least tide me over.

Cosmological Polytopes and the Wavefunction of the Universe is Nima & co.’s attempt to get a geometrical picture for cosmological correlators, analogous to the Ampituhedron. Cosmological correlators ask questions about the overall behavior of the visible universe: how likely is one clump of matter to be some distance from another? What sorts of patterns might we see in the Cosmic Microwave Background? This is the sort of thing that can be used for “cosmological collider physics”, an idea I mention briefly here.

Paolo Benincasa was visiting Perimeter near the end of my time there, so I got a few chances to chat with him about this. One thing he mentioned, but that didn’t register fully at the time, was Postnikov’s involvement. I had expected that even if Nima and Paolo found something interesting that it wouldn’t lead to particularly deep mathematics. Unlike the N=4 super Yang-Mills theory that generates the Amplituhedron, the theories involved in these cosmological correlators aren’t particularly unique, they’re just a particular class of models cosmologists use that happen to work well with Nima’s methods. Given that, it’s really surprising that they found something mathematically interesting enough to interest Postnikov, a mathematician who was involved in the early days of the Amplituhedron’s predecessor, the Positive Grassmannian. If there’s something that mathematically worthwhile in such a seemingly arbitrary theory then perhaps some of the beauty of the Amplithedron are much more general than I had thought.

Scattering Amplitudes For All Masses and Spins is on some level a byproduct of Nima and Yu-tin’s investigations of whether string theory is unique. Still, it’s a useful byproduct. Many of the tricks we use in scattering amplitudes are at their best for theories with massless particles. Once the particles have masses our notation gets a lot messier, and we often have to rely on older methods. What Nima, Yu-tin, and Tzu-Chen have done here is to build a notation similar to what we use for massless particle, but for massive ones.

The advantage of doing this isn’t just clean-looking papers: using this notation makes it a lot easier to see what kinds of theories make sense. There are a variety of old theorems that restrict what sorts of theories you can write down: photons can’t interact directly with each other, there can only be one “gravitational force”, particles with spins greater than two shouldn’t be massless, etc. The original theorems were often fairly involved, but for massless particles there were usually nice ways to prove them in modern amplitudes notation. Yu-tin in particular has a lot of experience finding these kinds of proofs. What the new notation does is make these nice simple proofs possible for massive particles as well. For example, you can try to use the new notation to write down an interaction between a massive particle with spin greater than two and gravity, and what you find is that any expression you write breaks down: it works fine at low energies, but once you’re looking at particles with energies much higher than their mass you start predicting probabilities greater than one. This suggests that particles with higher spins shouldn’t be “fundamental”, they should be explained in terms of other particles at higher energies. The only way around this turns out to be an infinite series of particles to cancel problems from the previous ones, the sort of structure that higher vibrations have in string theory. I often don’t appreciate papers that others claim are a pleasure to read, but this one really was a pleasure to read: there’s something viscerally satisfying about seeing so many important constraints manifest so cleanly.

I’ve talked before about the difference between planar and non-planar theories. Planar theories end up being simpler, and in the case of N=4 super Yang-Mills this results in powerful symmetries that let us do much more complicated calculations. Non-planar theories are more complicated, but necessary for understanding gravity. Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector, a new paper by Zvi Bern, Michael Enciso, Harald Ita, and Mao Zeng, works on bridging the gap between these two worlds.

Most of the paper is concerned with using some of the symmetries of N=4 super Yang-Mills in other, more realistic (but still planar) theories. The idea is that even if those symmetries don’t hold one can still use techniques that respect those symmetries, and those techniques can often be a lot cleaner than techniques that don’t. This is probably the most practically useful part of the paper, but the part I was most curious about is in the last few sections, where they discuss non-planar theories. For a while now I’ve been interested in ways to treat a non-planar theory as if it were planar, to try to leverage the powerful symmetries we have in planar N=4 super Yang-Mills elsewhere. Their trick is surprisingly simple: they just cut the diagram open! Oddly enough, they really do end up with similar symmetries using this method. I still need to read this in more detail to understand its limitations, since deep down it feels like something this simple couldn’t possibly work. Still, if anything like the symmetries of planar N=4 holds in the non-planar case there’s a lot we could do with it.

There are a bunch of other interesting recent papers that I haven’t had time to read. Some look like they might relate to weird properties of N=4 super Yang-Mills, others say interesting things about the interconnected web of theories tied together by their behavior when a particle becomes “soft”. Another presents a method for dealing with elliptic functions, one of the main obstructions to applying my hexagon function technique to more situations. And of course I shouldn’t fail to mention a paper by my colleague Carlos Cardona, applying amplitudes techniques to AdS/CFT. Overall, a lot of interesting stuff in a short span of time. I should probably get back to reading it!

Topic Conferences, Place Conferences

I spent this week at Current Themes in High Energy Physics and Cosmology, a conference at the Niels Bohr Institute.

Most conferences focus on a particular topic. Usually the broader the topic, the bigger the conference. A workshop on flux tubes is smaller than Amplitudes, which is smaller than Strings, which is smaller than the March Meeting of the American Physical Society.

“Current Themes in High Energy Physics and Cosmology” sounds like a very broad topic, but it was a small conference. The reason why is that it wasn’t a “topic conference”, it was a “place conference”.

Most conferences focus on a topic, but some are built around a place. These conferences are hosted by a particular institute year after year. Sometimes each year has a loose theme (for example, the Simons Summer Workshop this year focused on theories without supersymmetry) but sometimes no attempt is made to tie the talks together (“current themes”).

Instead of a theme, the people who go to these conferences are united by their connections to the institute. Some of them have collaborators there, or worked there in the past. Others have been coming for many years. Some just happened to be in the area.

While they may seem eclectic, “place” conferences have a valuable role: they help to keep our interests broad. In physics, there’s a natural tendency to specialize. Left alone, we end up reading papers and going to talks only when they’re directly relevant for what we’re working on. By doing this we lose track of the wider field, losing access to the insights that come from different perspectives and methods.

“Place” conferences, like seminars, help pull things in the other direction. When you’re hearing talks from “everyone connected to the Simons Center” or “everyone connected to the Niels Bohr Institute”, you’re exposed to a much broader range of topics than a conference for just your sub-field. You get a broad overview of what’s going on in the field, but unlike a big conference like Strings there are few enough people that you can actually talk to everyone.

Physicists’ attachment to places is counter-intuitive. We’re studying mathematical truths and laws of nature, surely it shouldn’t matter where we work. In practice, though, we’re still human. Out of the vast span of physics we still pick our interests based on the people around us. That’s why places, why institutes with a wide range of excellent people, are so important: they put our social instincts to work studying the universe.