Tag Archives: mathematics

Energy Is That Which Is Conserved

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In school, kids learn about different types of energy. They learn about solar energy and wind energy, nuclear energy and chemical energy, electrical energy and mechanical energy, and potential energy and kinetic energy. They learn that energy is conserved, that it can never be created or destroyed, but only change form. They learn that energy makes things happen, that you can use energy to do work, that energy is different from matter.

Some, between good teaching and good students, manage to impose order on the jumble of concepts and terms. Others end up envisioning the whole story a bit like Pokemon, with different types of some shared “stuff”.

Energy isn’t “stuff”, though. So what is it? What relates all these different types of things?

Energy is something which is conserved.

The mathematician Emmy Noether showed that, when the laws of physics are symmetrical, they come with a conserved quantity. For example, because the laws of the physics are the same from place to place, momentum is conserved. Similarly, because the laws of physics are the same from one time to another, Noether’s theorem states that there must be some quantity related to time, some number we can calculate, that is conserved, even as other things change. We call that number energy.

If energy is that simple, why are there all those types?

Energy is a number we can calculate. It’s a number we can calculate for different things. If you have a detailed description of how something in physics works, you can use that description to calculate that thing’s energy. In school, you memorize formulas like \frac{1}{2}m v^2 and m g h. These are all formulas that, with a bit more knowledge, you could calculate. They are the things that, for a something that meets the conditions, are conserved. They are things that, according to Noether’s theorem, stay the same.

Because of this, you shouldn’t think of energy as a substance, or a fuel. Energy is something we can do: we physicists, and we students of physics. We can take a physical system, and see what about it ought to be conserved. Energy is an action, a calculation, a conceptual tool that can be used to make predictions.

Most things are, in the end.

Some Dumb AI Ideas

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Sometimes, when I write a post about AI, I’ve been sitting on an idea for a long time. I’ve talked to experts, I’ve tried to understand the math, I’ve honed my points and cleared away clutter.

This is not one of those times. The ideas in this post almost certainly have something deeply wrong with them. But hopefully they’re interesting food for thought.

My first dumb idea: instruction tuning was a mistake.

I’m drawing the seeds of this one from a tumblr post by nostalgebraist, someone known for making a popular bot trained on his tumblr posts in the early days before GPT became ChatGPT.

AIs like ChatGPT are based on Large Language Models, insanely complicated mathematical formulas that predict, given part of a text, what the rest of that text is likely to look like. In the early days, this was largely how they were used. Loosely described nostalgebraist’s bot, called nostalgebraist-autoresponder, began with a list of tumblr posts and asks and determines what additional posts would best fit in.

If you think about it, though, ChatGPT doesn’t really work like that. ChatGPT has conversations: you send it messages, it sends you responses. The text it creates is a dialogue, with you supplying half the input. But most texts aren’t dialogues, and ChatGPT draws on a lot of non-dialogue texts to make its dialogue-like responses.

The reason it does this is something called instruction tuning. ChatGPT has been intentionally biased, not to give the most likely completion to a task in general, but to give completions that fit this dialogue genre. What I didn’t know until I read nostalgebraist’s post was that this genre was defined artificially: AI researchers made up fake dialogues with AI, cheesy sci-fi conversations imagining how an AI might respond to instructions from a user, and then biased the Large Language Model so that rather than giving the most likely text in general, it gives a text that is more likely to look like these cheesy sci-fi conversations. It’s why ChatGPT sounds kind of like a fictional robot: not because sci-fi writers accurately predicted what AI would sound like, but because AI was created based on sci-fi texts.

For nostalgebraist, this leads into an interesting reflection of how a sci-fi AI should behave, how being warped around a made-up genre without history or depth creates characters which act according to simple narratives and express surprising anxiety.

For myself, though, I can’t help but wonder if the goal of dialogue itself is the problem. Dialogue is clearly important commercially: people use ChatGPT because they can chat with it. But Large Language Models aren’t inherently chatbots: they produce plausible texts, of any sort you could imagine. People seem to want a machine that can, for example, answer scientific questions as part of a conversation. But most competent answers to scientific questions aren’t conversations, they’re papers. If people stuck with the “raw” model, producing excerpts of nonexistent papers rather than imitating a dialogue with a non-existent expert, wouldn’t you expect the answers to be more accurate, with the model no longer biased by an irrelevant goal? Is the need to make a sell-able chatbot making these AIs worse at everything else people are trying to use them for?

I’m imagining a world where, instead of a chatbot, OpenAI built an “alternate universe simulator”. You give it some context, some texts or parts of texts from a universe you made up, and it completes them in a plausible way. By imagining different universes, you can use it to answer different questions. Such a gimmick would get fewer customers, and fewer investors, it would probably do worse. But I have to wonder if the actual technology might have been more useful.

My second idea is dumber, to the point where I mostly know why it doesn’t work. But thinking about it might help clarify how things work for people unused to AI.

I saw someone point out that, unlike something like Wikipedia, AI doesn’t give you context. You shouldn’t trust Wikipedia, or a source you find on Google, blindly. If you want to, you can look through the edit history on Wikipedia, or figure out who wrote a page you found on Google and how. If ChatGPT tells you something, by default you don’t know where that knowledge came from. You can tell it to search, and then you’ll get links, but that’s because it’s using Google or the like behind the scenes anyway. You don’t know where the model is getting its ideas.

Why couldn’t we get that context, though?

Every text produced by a Large Language Model is causally dependent on its training data. Different data, different model, different text. That doesn’t mean that each text draws from one source, or just a few sources: ChatGPT isn’t copying the training data, at least not so literally.

But it does mean that, if ChatGPT says something is true, you should in principle be able to ask which data was most important in making it say that. If you leave a piece of data out of the training, and get similar answers, you can infer that the response you got doesn’t have much to do with that piece of data. But if you leave out a text in training, and now ChatGPT gives totally different responses to the same question…then there’s a pretty meaningful sense that it got the information from that source.

If this were the type of non-AI statistical model people use in physics, this would be straightforward. Researchers do this all the time: take one experiment out of the data, see how their analysis changes, and thereby figure out which experiments are most important to check. One can even sometimes calculate, given a model, where you should look.

Unfortunately, you can’t do this with ChatGPT. The model is just too big. You can’t calculate anything explicitly about it, the giant mathematical formulas behind it are so complicated that the most you can do is get probabilities out case by case, you can’t “unwind” them and see where the numbers come from. And you can’t just take out sources one by one, and train the model again: not when training takes months of expensive computer time.

So unlike with the previous idea, I understand even on a technical level why you can’t do this. But it helped me to be able to think about what I would like to do, if it were possible. Maybe it helps you too!

Amplitudes 2025 This Week

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Summer is conference season for academics, and this week held my old sub-field’s big yearly conference, called Amplitudes. This year, it was in Seoul at Seoul National University, the first time the conference has been in Asia.

(I wasn’t there, I don’t go to these anymore. But I’ve been skimming slides in my free time, to give you folks the updates you crave. Be forewarned that conference posts like these get technical fast, I’ll be back to my usual accessible self next week.)

There isn’t a huge amplitudes community in Korea, but it’s bigger than it was back when I got started in the field. Of the organizers, Kanghoon Lee of the Asia Pacific Center for Theoretical Physics and Sangmin Lee of Seoul National University have what I think of as “core amplitudes interests”, like recursion relations and the double-copy. The other Korean organizers are from adjacent areas, work that overlaps with amplitudes but doesn’t show up at the conference each year. There was also a sizeable group of organizers from Taiwan, where there has been a significant amplitudes presence for some time now. I do wonder if Korea was chosen as a compromise between a conference hosted in Taiwan or in mainland China, where there is also quite a substantial amplitudes community.

One thing that impresses me every year is how big, and how sophisticated, the gravitational-wave community in amplitudes has grown. Federico Buccioni’s talk began with a plot that illustrates this well (though that wasn’t his goal):

At the conference Amplitudes, dedicated to the topic of scattering amplitudes, there were almost as many talks with the phrase “black hole” in the title as there were with “scattering” or “amplitudes”! This is for a topic that did not even exist in the subfield when I got my PhD eleven years ago.

With that said, gravitational wave astronomy wasn’t quite as dominant at the conference as Buccioni’s bar chart suggests. There were a few talks each day on the topic: I counted seven in total, excluding any short talks on the subject in the gong show. Spinning black holes were a significant focus, central to Jung-Wook Kim’s, Andres Luna’s and Mao Zeng’s talks (the latter two showing some interesting links between the amplitudes story and classic ideas in classical mechanics) and relevant in several others, with Riccardo Gonzo, Miguel Correia, Ira Rothstein, and Enrico Herrmann’s talks showing not just a wide range of approaches, but an increasing depth of research in this area.

Herrmann’s talk in particular dealt with detector event shapes, a framework that lets physicists think more directly about what a specific particle detector or observer can see. He applied the idea not just to gravitational waves but to quantum gravity and collider physics as well. The latter is historically where this idea has been applied the most thoroughly, as highlighted in Hua Xing Zhu’s talk, where he used them to pick out particular phenomena of interest in QCD.

QCD is, of course, always of interest in the amplitudes field. Buccioni’s talk dealt with the theory’s behavior at high-energies, with a nice example of the “maximal transcendentality principle” where some quantities in QCD are identical to quantities in N=4 super Yang-Mills in the “most transcendental” pieces (loosely, those with the highest powers of pi). Andrea Guerreri’s talk also dealt with high-energy behavior in QCD, trying to address an experimental puzzle where QCD results appeared to violate a fundamental bound all sensible theories were expected to obey. By using S-matrix bootstrap techniques, they clarify the nature of the bound, finding that QCD still obeys it once correctly understood, and conjecture a weird theory that should be possible to frame right on the edge of the bound. The S-matrix bootstrap was also used by Alexandre Homrich, who talked about getting the framework to work for multi-particle scattering.

Heribertus Bayu Hartanto is another recent addition to Korea’s amplitudes community. He talked about a concrete calculation, two-loop five-particle scattering including top quarks, a tricky case that includes elliptic curves.

When amplitudes lead to integrals involving elliptic curves, many standard methods fail. Jake Bourjaily’s talk raised a question he has brought up again and again: what does it mean to do an integral for a new type of function? One possible answer is that it depends on what kind of numerics you can do, and since more general numerical methods can be cumbersome one often needs to understand the new type of function in more detail. In light of that, Stephen Jones’ talk was interesting in taking a common problem often cited with generic approaches (that they have trouble with the complex numbers introduced by Minkowski space) and finding a more natural way in a particular generic approach (sector decomposition) to take them into account. Giulio Salvatori talked about a much less conventional numerical method, linked to the latest trend in Nima-ology, surfaceology. One of the big selling points of the surface integral framework promoted by people like Salvatori and Nima Arkani-Hamed is that it’s supposed to give a clear integral to do for each scattering amplitude, one which should be amenable to a numerical treatment recently developed by Michael Borinsky. Salvatori can currently apply the method only to a toy model (up to ten loops!), but he has some ideas for how to generalize it, which will require handling divergences and numerators.

Other approaches to the “problem of integration” included Anna-Laura Sattelberger’s talk that presented a method to find differential equations for the kind of integrals that show up in amplitudes using the mathematical software Macaulay2, including presenting a package. Matthias Wilhelm talked about the work I did with him, using machine learning to find better methods for solving integrals with integration-by-parts, an area where two other groups have now also published. Pierpaolo Mastrolia talked about integration-by-parts’ up-and-coming contender, intersection theory, a method which appears to be delving into more mathematical tools in an effort to catch up with its competitor.

Sometimes, one is more specifically interested in the singularities of integrals than their numerics more generally. Felix Tellander talked about a geometric method to pin these down which largely went over my head, but he did have a very nice short description of the approach: “Describe the singularities of the integrand. Find a map representing integration. Map the singularities of the integrand onto the singularities of the integral.”

While QCD and gravity are the applications of choice, amplitudes methods germinate in N=4 super Yang-Mills. Ruth Britto’s talk opened the conference with an overview of progress along those lines before going into her own recent work with one-loop integrals and interesting implications of ideas from cluster algebras. Cluster algebras made appearances in several other talks, including Anastasia Volovich’s talk which discussed how ideas from that corner called flag cluster algebras may give insights into QCD amplitudes, though some symbol letters still seem to be hard to track down. Matteo Parisi covered another idea, cluster promotion maps, which he thinks may help pin down algebraic symbol letters.

The link between cluster algebras and symbol letters is an ongoing mystery where the field is seeing progress. Another symbol letter mystery is antipodal duality, where flipping an amplitude like a palindrome somehow gives another valid amplitude. Lance Dixon has made progress in understanding where this duality comes from, finding a toy model where it can be understood and proved.

Others pushed the boundaries of methods specific to N=4 super Yang-Mills, looking for novel structures. Song He’s talk pushes an older approach by Bourjaily and collaborators up to twelve loops, finding new patterns and connections to other theories and observables. Qinglin Yang bootstraps Wilson loops with a Lagrangian insertion, adding a side to the polygon used in previous efforts and finding that, much like when you add particles to amplitudes in a bootstrap, the method gets stricter and more powerful. Jaroslav Trnka talked about work he has been doing with “negative geometries”, an odd method descended from the amplituhedron that looks at amplitudes from a totally different perspective, probing a bit of their non-perturbative data. He’s finding more parts of that setup that can be accessed and re-summed, finding interestingly that multiple-zeta-values show up in quantities where we know they ultimately cancel out. Livia Ferro also talked about a descendant of the amplituhedron, this time for cosmology, getting differential equations for cosmological observables in a particular theory from a combinatorial approach.

Outside of everybody’s favorite theories, some speakers talked about more general approaches to understanding the differences between theories. Andreas Helset covered work on the geometry of the space of quantum fields in a theory, applying the method to a general framework for characterizing deviations from the standard model called the SMEFT. Jasper Roosmale Nepveu also talked about a general space of theories, thinking about how positivity (a trait linked to fundamental constraints like causality and unitarity) gets tangled up with loop effects, and the implications this has for renormalization.

Soft theorems, universal behavior of amplitudes when a particle has low energy, continue to be a trendy topic, with Silvia Nagy showing how the story continues to higher orders and Sangmin Choi investigating loop effects. Callum Jones talks about one of the more powerful results from the soft limit, Weinberg’s theorem showing the uniqueness of gravity. Weinberg’s proof was set up in Minkowski space, but we may ultimately live in curved, de Sitter space. Jones showed how the ideas Weinberg explored generalize in de Sitter, using some tools from the soft-theorem-inspired field of dS/CFT. Julio Parra-Martinez, meanwhile, tied soft theorems to another trendy topic, higher symmetries, a more general notion of the usual types of symmetries that physicists have explored in the past. Lucia Cordova reported work that was not particularly connected to soft theorems but was connected to these higher symmetries, showing how they interact with crossing symmetry and the S-matrix bootstrap.

Finally, a surprisingly large number of talks linked to Kevin Costello and Natalie Paquette’s work with self-dual gauge theories, where they found exact solutions from a fairly mathy angle. Paquette gave an update on her work on the topic, while Alfredo Guevara talked about applications to black holes, comparing the power of expanding around a self-dual gauge theory to that of working with supersymmetry. Atul Sharma looked at scattering in self-dual backgrounds in work that merges older twistor space ideas with the new approach, while Roland Bittelson talked about calculating around an instanton background.

Also, I had another piece up this week at FirstPrinciples, based on an interview with the (outgoing) president of the Sloan Foundation. I won’t have a “bonus info” post for this one, as most of what I learned went into the piece. But if you don’t know what the Sloan Foundation does, take a look! I hadn’t known they funded Jupyter notebooks and Hidden Figures, or that they introduced Kahneman and Tversky.

Bonus info for Reversible Computing and Megastructures

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After some delay, a bonus info post!

At FirstPrinciples.org, I had a piece covering work by engineering professor Colin McInnes on stability of Dyson spheres and ringworlds. This was a fun one to cover, mostly because of how it straddles the borderline between science fiction and practical physics and engineering. McInnes’s claim to fame is work on solar sails, which seem like a paradigmatic example of that kind of thing: a common sci-fi theme that’s surprisingly viable. His work on stability was interesting to me because it’s the kind of work that a century and a half ago would have been paradigmatic physics. Now, though, very few physicists work on orbital mechanics, and a lot of the core questions have passed on to engineering. It’s fascinating to see how these classic old problems can still have undiscovered solutions, and how the people best equipped to find them now are tinkerers practicing their tools instead of cutting-edge mathematicians.

At Quanta Magazine, I had a piece about reversible computing. Readers may remember I had another piece on that topic at the end of March, a profile on the startup Vaire Computing at FirstPrinciples.org. That piece talked about FirstPrinciples, but didn’t say much about reversible computing. I figured I’d combine the “bonus info” for both posts here.

Neither piece went into much detail about the engineering involved, as it didn’t really make sense in either venue. One thing that amused me a bit is that the core technology that drove Vaire into action is something that actually should be very familiar to a physics or engineering student: a resonator. Theirs is obviously quite a bit more sophisticated than the base model, but at its heart it’s doing the same thing: storing charge and controlling frequency. It turns out that those are both essential to making reversible computers work: you need to store charge so it isn’t lost to ground when you empty a transistor, and you need to control the frequency so you can have waves with gentle transitions instead of the more sharp corners of the waves used in normal computers, thus wasting less heat in rapid changes of voltage. Vaire recently announced they’re getting 50% charge recovery from their test chips, and they’re working on raising that number.

Originally, the Quanta piece was focused more on reversible programming than energy use, as the energy angle seemed a bit more physics-focused than their computer science desk usually goes. The emphasis ended up changing as I worked on the draft, but it meant that an interesting parallel story got lost on the cutting-room floor. There’s a community of people who study reversible computing not from the engineering side, but from the computer science side, studying reversible logic and reversible programming languages. It’s a pursuit that goes back to the 1980’s, where at Caltech around when Feynman was teaching his course on the physics of computing a group of students were figuring out how to set up a reversible programming language. Called Janus, they sent their creation to Landauer, and the letter ended up with Michael Frank after Landauer died. There’s a lovely quote from it regarding their motivation: “We did it out of curiosity over whether such an odd animal as this was possible, and because we were interested in knowing where we put information when we programmed. Janus forced us to pay attention to where our bits went since none could be thrown away.”

Being forced to pay attention to information, in turn, is what has animated the computer science side of the reversible computing community. There are applications to debugging, where you can run code backwards when it gets stuck, to encryption and compression, where you want to be able to recover the information you hid away, and to security, where you want to keep track of information to make sure a hacker can’t figure out things they shouldn’t. Also, for a lot of these people, it’s just a fun puzzle. Early on my attention was caught by a paper by Hannah Earley describing a programming language called Alethe, a word you might recognize from the Greek word for truth, which literally means something like “not-forgetting”.

(Compression is particularly relevant for the “garbage data” you need to output in a reversible computation. If you want to add two numbers reversibly, naively you need to keep both input numbers and their output, but you can be more clever than that and just keep one of the inputs since you can subtract to find the other. There are a lot of substantially more clever tricks in this vein people have figured out over the years.)

I didn’t say anything about the other engineering approaches to reversible computing, that try to do something outside of traditional computer chips. There’s DNA computing, which tries to compute with a bunch of DNA in solution. There’s the old concept of ballistic reversible computing, where you imagine a computer that runs like a bunch of colliding billiard balls, conserving energy. Coordinating such a computer can be a nightmare, and early theoretical ideas were shown to be disrupted by something as tiny as a few stray photons from a distant star. But people like Frank figured out ways around the coordination problem, and groups have experimented with superconductors as places to toss those billiard balls around. The early billiard-inspired designs also had a big impact on quantum computing, where you need reversible gates and the only irreversible operation is the measurement. The name “Toffoli” comes up a lot in quantum computing discussions, I hadn’t known before this that Toffoli gates were originally for reversible computing in general, not specifically quantum computing.

Finally, I only gestured at the sci-fi angle. For reversible computing’s die-hards, it isn’t just a way to make efficient computers now. It’s the ultimate future of the technology, the kind of energy-efficiency civilization will need when we’re covering stars with shells of “computronium” full of busy joyous artificial minds.

And now that I think about it, they should chat with McInnes. He can tell them the kinds of stars they should build around.

Some FAQ for Microsoft’s Majorana 1 Chip

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Recently, Microsoft announced a fancy new quantum computing chip called Majorana 1. I’ve noticed quite a bit of confusion about what they actually announced, and while there’s a great FAQ page about it on the quantum computing blog Shtetl Optimized, the post there aims at a higher level, assuming you already know the basics. You can think of this post as a complement to that one, that tries to cover some basic things Shtetl Optimized took for granted.

Q: In the announcement, Microsoft said:

“It leverages the world’s first topoconductor, a breakthrough type of material which can observe and control Majorana particles to produce more reliable and scalable qubits, which are the building blocks for quantum computers.”

That sounds wild! Are they really using particles in a computer?

A: All computers use particles. Electrons are particles!

Q: You know what I mean!

A: You’re asking if these are “particle physics” particles, like the weird types they try to observe at the LHC?

No, they’re not.

Particle physicists use a mathematical framework called quantum field theory, where particles are ripples in things called quantum fields that describe properties of the universe. But they aren’t the only people to use that framework. Instead of studying properties of the universe you can study properties of materials, weird alloys and layers of metal and crystal that do weird and useful things. The properties of these materials can be approximately described with the same math, with quantum fields. Just as the properties of the universe ripple to produce particles, these properties of materials ripple to produce what are called quasiparticles. Ultimately, these quasiparticles come down to movements of ordinary matter, usually electrons in the original material. They’re just described with a kind of math that makes them look like their own particles.

Q: So, what are these Majorana particles supposed to be?

A: In quantum field theory, most particles come with an antimatter partner. Electrons, for example, have partners called positrons, with a positive electric charge instead of a negative one. These antimatter partners have to exist due to the math of quantum field theory, but there is a way out: some particles are their own antimatter partner, letting one particle cover both roles. This happens for some “particle physics particles”, but all the examples we’ve found are a type of particle called a “boson”, particles related to forces. In 1937, the physicist Ettore Majorana figured out the math you would need to make a particle like this that was a fermion instead, the other main type of particle that includes electrons and protons. So far, we haven’t found one of these Majorana fermions in nature, though some people think the elusive neutrino particles could be an example. Others, though, have tried instead to find a material described by Majorana’s theory. This should in principle be easier, you can build a lot of different materials after all. But it’s proven quite hard for people to do. Back in 2018, Microsoft claimed they’d managed this, but had to retract the claim. This time, they seem more confident, though the scientific community is still not convinced.

Q: And what’s this topoconductor they’re talking about?

A: Topoconductor is short for topological superconductor. Superconductors are materials that conduct electricity much better than ordinary metals.

Q: And, topological means? Something about donuts, right?

A: If you’ve heard anything about topology, you’ve heard that it’s a type of mathematics where donuts are equivalent to coffee cups. You might have seen an animation of a coffee cup being squished and mushed around until the ring of the handle becomes the ring of a donut.

This isn’t actually the important part of topology. The important part is that, in topology, a ball is not equivalent to a donut.

Topology is the study of which things can change smoothly into one another. If you want to change a donut into a ball, you have to slice through the donut’s ring or break the surface inside. You can’t smoothly change one to another. Topologists study shapes of different kinds of things, figuring out which ones can be changed into each other smoothly and which can’t.

Q: What does any of that have to do with quantum computers?

A: The shapes topologists study aren’t always as simple as donuts and coffee cups. They can also study the shape of quantum fields, figuring out which types of quantum fields can change smoothly into each other and which can’t.

The idea of topological quantum computation is to use those rules about what can change into each other to encode information. You can imagine a ball encoding zero, and a donut encoding one. A coffee cup would then also encode one, because it can change smoothly into a donut, while a box would encode zero because you can squash the corners to make it a ball. This helps, because it means that you don’t screw up your information by making smooth changes. If you accidentally drop your box that encodes zero and squish a corner, it will still encode zero.

This matters in quantum computing because it is very easy to screw up quantum information. Quantum computers are very delicate, and making them work reliably has been immensely challenging, requiring people to build much bigger quantum computers so they can do each calculation with many redundant backups. The hope is that topological superconductors would make this easier, by encoding information in a way that is hard to accidentally change.

Q: Cool. So does that mean Microsoft has the best quantum computer now?

A: The machine Microsoft just announced has only a single qubit, the quantum equivalent of just a single bit of computer memory. At this point, it can’t do any calculations. It can just be read, giving one or zero. The hope is that the power of the new method will let Microsoft catch up with companies that have computers with hundred of qubits, and help them arrive faster at the millions of qubits that will be needed to do anything useful.

Q: Ah, ok. But it sounds like they accomplished some crazy Majorana stuff at least, right?

A: Umm…

Read the Shtetl-Optimized FAQ if you want more details. The short answer is that this is still controversial. So far, the evidence they’ve made public isn’t enough to show that they found these Majorana quasiparticles, or that they made a topological superconductor. They say they have more recent evidence that they haven’t published yet. We’ll see.

Bonus Material for “How Hans Bethe Stumbled Upon Perfect Quantum Theories”

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I had an article last week in Quanta Magazine. It’s a piece about something called the Bethe ansatz, a method in mathematical physics that was discovered by Hans Bethe in the 1930’s, but which only really started being understood and appreciated around the 1960’s. Since then it’s become a key tool, used in theoretical investigations in areas from condensed matter to quantum gravity. In this post, I thought I’d say a bit about the story behind the piece and give some bonus material that didn’t fit.

When I first decided to do the piece I reached out to Jules Lamers. We were briefly office-mates when I worked in France, where he was giving a short course on the Bethe ansatz and the methods that sprung from it. It turned out he had also been thinking about writing a piece on the subject, and we considered co-writing for a bit, but that didn’t work for Quanta. He helped me a huge amount with understanding the history of the subject and tracking down the right sources. If you’re a physicist who wants to learn about these things, I recommend his lecture notes. And if you’re a non-physicist who wants to know more, I hope he gets a chance to write a longer popular-audience piece on the topic!

If you clicked through to Jules’s lecture notes, you’d see the word “Bethe ansatz” doesn’t appear in the title. Instead, you’d see the phrase “quantum integrability”. In classical physics, an “integrable” system is one where you can calculate what will happen by doing an integral, essentially letting you “solve” any problem completely. Systems you can describe with the Bethe ansatz are solvable in a more complicated quantum sense, so they get called “quantum integrable”. There’s a whole research field that studies these quantum integrable systems.

My piece ended up rushing through the history of the field. After talking about Bethe’s original discovery, I jumped ahead to ice. The Bethe ansatz was first used to think about ice in the 1960’s, but the developments I mentioned leading up to it, where experimenters noticed extra variability and theorists explained it with the positions of hydrogen atoms, happened earlier, in the 1930’s. (Thanks to the commenter who pointed out that this was confusing!) Baxter gets a starring role in this section and had an important role in tying things together, but other people (Lieb and Sutherland) were involved earlier, showing that the Bethe ansatz indeed could be used with thin sheets of ice. This era had a bunch of other big names that I didn’t have space to talk about: C. N. Yang makes an appearance, and while Faddeev comes up later, I didn’t mention that he had a starring role in the 1970’s in understanding the connection to classical integrability and proposing a mathematical structure to understand what links all these different integrable theories together.

I vaguely gestured at black holes and quantum gravity, but didn’t have space for more than that. The connection there is to a topic you might have heard of before if you’ve read about string theory, called AdS/CFT, a connection between two kinds of world that are secretly the same: a toy model of gravity called Anti-de Sitter space (AdS) and a theory without gravity that looks the same at any scale (called a Conformal Field Theory, or CFT). It turns out that in the most prominent example of this, the theory without gravity is integrable! In fact, it’s a theory I spent a lot of time working with back in my research days, called N=4 super Yang-Mills. This theory is kind of like QCD, and in some sense it has integrability for similar reasons to those that Feynman hoped for and Korchemsky and Faddeev found. But it actually goes much farther, outside of the high-energy approximation where Korchemsky and Faddeev’s result works, and in principle seems to include everything you might want to know about the theory. Nowadays, people are using it to investigate the toy model of quantum gravity, hoping to get insights about quantum gravity in general.

One thing I didn’t get a chance to mention at all is the connection to quantum computing. People are trying to build a quantum computer with carefully-cooled atoms. It’s important to test whether the quantum computer functions well enough, or if the quantum states aren’t as perfect as they need to be. One way people have been testing this is with the Bethe ansatz: because it lets you calculate the behavior of special systems perfectly, you can set up your quantum computer to model a Bethe ansatz, and then check how close to the prediction your results are. You know that the theoretical result is complete, so any failure has to be due to an imperfection in your experiment.

I gave a quick teaser to a very active field, one that has fascinated a lot of prominent physicists and been applied in a wide variety of areas. I hope I’ve inspired you to learn more!

Integration by Parts, Evolved

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I posted what may be my last academic paper today, about a project I’ve been working on with Matthias Wilhelm for most of the last year. The paper is now online here. For me, the project has been a chance to broaden my horizons, learn new skills, and start to step out of my academic comfort zone. For Matthias, I hope it was grant money well spent.

I wanted to work on something related to machine learning, for the usual trendy employability reasons. Matthias was already working with machine learning, but was interested in pursuing a different question.

When is machine learning worthwhile? Machine learning methods are heuristics, unreliable methods that sometimes work well. You don’t use a heuristic if you have a reliable method that runs fast enough. But if all you have are heuristics to begin with, then machine learning can give you a better heuristic.

Matthias noticed a heuristic embedded deep in how we do particle physics, and guessed that we could do better. In particle physics, we use pictures called Feynman diagrams to predict the probabilities for different outcomes of collisions, comparing those predictions to observation to look for evidence of new physics. Each Feynman diagram corresponds to an integral, and for each calculation there are hundreds, thousands, or even millions of those integrals to do.

Luckily, physicists don’t actually have to do all those integrals. It turns out that most of them are related, by a slightly more advanced version of that calculus class mainstay, integration by parts. Using integration by parts you can solve a list of equations, finding out how to write your integrals in terms of a much smaller list.

How big a list of equations do you need, and which ones? Twenty-five years ago, Stefano Laporta proposed a “golden rule” to choose, based on his own experience, and people have been using it (more or less, with their own tweaks) since then.

Laporta’s rule is a heuristic, with no proof that it is the best option, or even that it will always work. So we probably shouldn’t have been surprised when someone came up with a better heuristic. Watching talks at a December 2023 conference, Matthias saw a presentation by Johann Usovitsch on a curious new rule. The rule was surprisingly simple, just one extra condition on top of Laporta’s. But it was enough to reduce the number of equations by a factor of twenty.

That’s great progress, but it’s also a bit frustrating. Over almost twenty-five years, no-one had guessed this one simple change?

Maybe, thought Matthias and I, we need to get better at guessing.

We started out thinking we’d try reinforcement learning, a technique where a machine is trained by playing a game again and again, changing its strategy when that strategy brings it a reward. We thought we could have the machine learn to cut away extra equations, getting rewarded if it could cut more while still getting the right answer. We didn’t end up pursuing this very far before realizing another strategy would be a better fit.

What is a rule, but a program? Laporta’s golden rule and Johann’s new rule could both be expressed as simple programs. So we decided to use a method that could guess programs.

One method stood out for sheer trendiness and audacity: FunSearch. FunSearch is a type of algorithm called a genetic algorithm, which tries to mimic evolution. It makes a population of different programs, “breeds” them with each other to create new programs, and periodically selects out the ones that perform best. That’s not the trendy or audacious part, though, people have been doing that sort of genetic programming for a long time.

The trendy, audacious part is that FunSearch generates these programs with a Large Language Model, or LLM (the type of technology behind ChatGPT). Using an LLM trained to complete code, FunSearch presents the model with two programs labeled v0 and v1 and asks it to complete v2. In general, program v2 will have some traits from v0 and v1, but also a lot of variation due to the unpredictable output of LLMs. The inventors of FunSearch used this to contribute the variation needed for evolution, using it to evolve programs to find better solutions to math problems.

We decided to try FunSearch on our problem, modifying it a bit to fit the case. We asked it to find a shorter list of equations, giving a better score for a shorter list but a penalty if the list wasn’t able to solve the problem fully.

Some tinkering and headaches later, it worked! After a few days and thousands of program guesses, FunSearch was able to find a program that reproduced the new rule Johann had presented. A few hours more, and it even found a rule that was slightly better!

But then we started wondering: do we actually need days of GPU time to do this?

An expert on heuristics we knew had insisted, at the beginning, that we try something simpler. The approach we tried then didn’t work. But after running into some people using genetic programming at a conference last year, we decided to try again, using a Python package they used in their work. This time, it worked like a charm, taking hours rather than days to find good rules.

This was all pretty cool, a great opportunity for me to cut my teeth on Python programming and its various attendant skills. And it’s been inspiring, with Matthias drawing together more people interested in seeing just how much these kinds of heuristic methods can do there. I should be clear though, that so far I don’t think our result is useful. We did better than the state of the art on an example, but only slightly, and in a way that I’d guess doesn’t generalize. And we needed quite a bit of overhead to do it. Ultimately, while I suspect there’s something useful to find in this direction, it’s going to require more collaboration, both with people using the existing methods who know better what the bottlenecks are, and with experts in these, and other, kinds of heuristics.

So I’m curious to see what the future holds. And for the moment, happy that I got to try this out!

How Small Scales Can Matter for Large Scales

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For a certain type of physicist, nothing matters more than finding the ultimate laws of nature for its tiniest building-blocks, the rules that govern quantum gravity and tell us where the other laws of physics come from. But because they know very little about those laws at this point, they can predict almost nothing about observations on the larger distance scales we can actually measure.

“Almost nothing” isn’t nothing, though. Theoretical physicists don’t know nature’s ultimate laws. But some things about them can be reasonably guessed. The ultimate laws should include a theory of quantum gravity. They should explain at least some of what we see in particle physics now, explaining why different particles have different masses in terms of a simpler theory. And they should “make sense”, respecting cause and effect, the laws of probability, and Einstein’s overall picture of space and time.

All of these are assumptions, of course. Further assumptions are needed to derive any testable consequences from them. But a few communities in theoretical physics are willing to take the plunge, and see what consequences their assumptions have.

First, there’s the Swampland. String theorists posit that the world has extra dimensions, which can be curled up in a variety of ways to hide from view, with different observable consequences depending on how the dimensions are curled up. This list of different observable consequences is referred to as the Landscape of possibilities. Based on that, some string theorists coined the term “Swampland” to represent an area outside the Landscape, containing observations that are incompatible with quantum gravity altogether, and tried to figure out what those observations would be.

In principle, the Swampland includes the work of all the other communities on this list, since a theory of quantum gravity ought to be consistent with other principles as well. In practice, people who use the term focus on consequences of gravity in particular. The earliest such ideas argued from thought experiments with black holes, finding results that seemed to demand that gravity be the weakest force for at least one type of particle. Later researchers would more frequently use string theory as an example, looking at what kinds of constructions people had been able to make in the Landscape to guess what might lie outside of it. They’ve used this to argue that dark energy might be temporary, and to try to figure out what traits new particles might have.

Second, I should mention naturalness. When talking about naturalness, people often use the analogy of a pen balanced on its tip. While possible in principle, it must have been set up almost perfectly, since any small imbalance would cause it to topple, and that perfection demands an explanation. Similarly, in particle physics, things like the mass of the Higgs boson and the strength of dark energy seem to be carefully balanced, so that a small change in how they were set up would lead to a much heavier Higgs boson or much stronger dark energy. The need for an explanation for the Higgs’ careful balance is why many physicists expected the Large Hadron Collider to discover additional new particles.

As I’ve argued before, this kind of argument rests on assumptions about the fundamental laws of physics. It assumes that the fundamental laws explain the mass of the Higgs, not merely by giving it an arbitrary number but by showing how that number comes from a non-arbitrary physical process. It also assumes that we understand well how physical processes like that work, and what kinds of numbers they can give. That’s why I think of naturalness as a type of argument, much like the Swampland, that uses the smallest scales to constrain larger ones.

Third is a host of constraints that usually go together: causality, unitarity, and positivity. Causality comes from cause and effect in a relativistic universe. Because two distant events can appear to happen in different orders depending on how fast you’re going, any way to send signals faster than light is also a way to send signals back in time, causing all of the paradoxes familiar from science fiction. Unitarity comes from quantum mechanics. If quantum calculations are supposed to give the probability of things happening, those probabilities should make sense as probabilities: for example, they should never go above one.

You might guess that almost any theory would satisfy these constraints. But if you extend a theory to the smallest scales, some theories that otherwise seem sensible end up failing this test. Actually linking things up takes other conjectures about the mathematical form theories can have, conjectures that seem more solid than the ones underlying Swampland and naturalness constraints but that still can’t be conclusively proven. If you trust the conjectures, you can derive restrictions, often called positivity constraints when they demand that some set of observations is positive. There has been a renaissance in this kind of research over the last few years, including arguments that certain speculative theories of gravity can’t actually work.

Why Quantum Gravity Is Controversial

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Merging quantum mechanics and gravity is a famously hard physics problem. Explaining why merging quantum mechanics and gravity is hard is, in turn, a very hard science communication problem. The more popular descriptions tend to lead to misunderstandings, and I’ve posted many times over the years to chip away at those misunderstandings.

Merging quantum mechanics and gravity is hard…but despite that, there are proposed solutions. String Theory is supposed to be a theory of quantum gravity. Loop Quantum Gravity is supposed to be a theory of quantum gravity. Asymptotic Safety is supposed to be a theory of quantum gravity.

One of the great virtues of science and math is that we are, eventually, supposed to agree. Philosophers and theologians might argue to the end of time, but in math we can write down a proof, and in science we can do an experiment. If we don’t yet have the proof or the experiment, then we should reserve judgement. Either way, there’s no reason to get into an unproductive argument.

Despite that, string theorists and loop quantum gravity theorists and asymptotic safety theorists, famously, like to argue! There have been bitter, vicious, public arguments about the merits of these different theories, and decades of research doesn’t seem to have resolved them. To an outside observer, this makes quantum gravity seem much more like philosophy or theology than like science or math.

Why is there still controversy in quantum gravity? We can’t do quantum gravity experiments, sure, but if that were the problem physicists could just write down the possibilities and leave it at that. Why argue?

Some of the arguments are for silly aesthetic reasons, or motivated by academic politics. Some are arguments about which approaches are likely to succeed in future, which as always is something we can’t actually reliably judge. But the more justified arguments, the strongest and most durable ones, are about a technical challenge. They’re about something called non-perturbative physics.

Most of the time, when physicists use a theory, they’re working with an approximation. Instead of the full theory, they’re making an assumption that makes the theory easier to use. For example, if you assume that the velocity of an object is small, you can use Newtonian physics instead of special relativity. Often, physicists can systematically relax these assumptions, including more and more of the behavior of the full theory and getting a better and better approximation to the truth. This process is called perturbation theory.

Other times, this doesn’t work well. The full theory has some trait that isn’t captured by the approximations, something that hides away from these systematic tools. The theory has some important aspect that is non-perturbative.

Every proposed quantum gravity theory uses approximations like this. The theory’s proponents try to avoid these approximations when they can, but often they have to approximate and hope they don’t miss too much. The opponents, in turn, argue that the theory’s proponents are missing something important, some non-perturbative fact that would doom the theory altogether.

Asymptotic Safety is built on top of an approximation, one different from what other quantum gravity theorists typically use. To its proponents, work using their approximation suggests that gravity works without any special modifications, that the theory of quantum gravity is easier to find than it seems. Its opponents aren’t convinced, and think that the approximation is missing something important which shows that gravity needs to be modified.

In Loop Quantum Gravity, the critics think their approximation misses space-time itself. Proponents of Loop Quantum Gravity have been unable to prove that their theory, if you take all the non-perturbative corrections into account, doesn’t just roll up all of space and time into a tiny spiky ball. They expect that their theory should allow for a smooth space-time like we experience, but the critics aren’t convinced, and without being able to calculate the non-perturbative physics neither side can convince the other.

String Theory was founded and originally motivated by perturbative approximations. Later, String Theorists figured out how to calculate some things non-perturbatively, often using other simplifications like supersymmetry. But core questions, like whether or not the theory allows a positive cosmological constant, seem to depend on non-perturbative calculations that the theory gives no instructions for how to do. Some critics don’t think there is a consistent non-perturbative theory at all, that the approximations String Theorists use don’t actually approximate to anything. Even within String Theory, there are worries that the theory might try to resist approximation in odd ways, becoming more complicated whenever a parameter is small enough that you could use it to approximate something.

All of this would be less of a problem with real-world evidence. Many fields of science are happy to use approximations that aren’t completely rigorous, as long as those approximations have a good track record in the real world. In general though, we don’t expect evidence relevant to quantum gravity any time soon. Maybe we’ll get lucky, and studies of cosmology will reveal something, or an experiment on Earth will have a particularly strange result. But nature has no obligation to help us out.

Without evidence, though, we can still make mathematical progress. You could imagine someone proving that the various perturbative approaches to String Theory become inconsistent when stitched together into a full non-perturbative theory. Alternatively, you could imagine someone proving that a theory like String Theory is unique, that no other theory can do some key thing that it does. Either of these seems unlikely to come any time soon, and most researchers in these fields aren’t pursuing questions like that. But the fact the debate could be resolved means that it isn’t just about philosophy or theology. There’s a real scientific, mathematical controversy, one rooted in our inability to understand these theories beyond the perturbative methods their proponents use. And while I don’t expect it to be resolved any time soon, one can always hold out hope for a surprise.

Amplitudes 2024, Continued

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I’ve now had time to look over the rest of the slides from the Amplitudes 2024 conference, so I can say something about Thursday and Friday’s talks.

Thursday was gravity-focused. Zvi Bern’s review talk was actually a review, a tour of the state of the art in using amplitudes techniques to make predictions for gravitational wave physics. Bern emphasized that future experiments will require much more precision: two more orders of magnitude, which in our lingo amounts to two more “loops”. The current state of the art is three loops, but they’ve been hacking away at four, doing things piece by piece in a way that cleverly also yields publications (for example, they can do just the integrals needed for supergravity, which are simpler). Four loops here is the first time that the Feynman diagrams involve Calabi-Yau manifolds, so they will likely need techniques from some of the folks I talked about last week. Once they have four loops, they’ll want to go to five, since that is the level of precision you need to learn something about the material in neutron stars. The talk covered a variety of other developments, some of which were talked about later on Thursday and some of which were only mentioned here.

Of that day’s other speakers, Stefano De Angelis, Lucile Cangemi, Mikhail Ivanov, and Alessandra Buonanno also focused on gravitational waves. De Angelis talked about the subtleties that show up when you try to calculate gravitational waveforms directly with amplitudes methods, showcasing various improvements to the pipeline there. Cangemi talked about a recurring question with its own list of subtleties, namely how the Kerr metric for spinning black holes emerges from the math of amplitudes of spinning particles. Gravitational waves were the focus of only the second half of Ivanov’s talk, where he talked about how amplitudes methods can clear up some of the subtler effects people try to take into account. The first half was about another gravitational application, that of using amplitudes methods to compute the correlations of galaxy structures in the sky, a field where it looks like a lot of progress can be made. Finally, Buonanno gave the kind of talk she’s given a few times at these conferences, a talk that puts these methods in context, explaining how amplitudes results are packaged with other types of calculations into the Effective-One-Body framework which then is more directly used at LIGO. This year’s talk went into more detail about what the predictions are actually used for, which I appreciated. I hadn’t realized that there have been a handful of black hole collisions discovered by other groups from LIGO’s data, a win for open science! Her slides had a nice diagram explaining what data from the gravitational wave is used to infer what black hole properties, quite a bit more organized than the statistical template-matching I was imagining. She explained the logic behind Bern’s statement that gravitational wave telescopes will need two more orders of magnitude, pointing out that that kind of precision is necessary to be sure that something that might appear to be a deviation from Einstein’s theory of gravity is not actually a subtle effect of known physics. Her method typically is adjusted to fit numerical simulations, but she shows that even without that adjustment they now fit the numerics quite well, thanks in part to contributions from amplitudes calculations.

Of the other talks that day, David Kosower’s was the only one that didn’t explicitly involve gravity. Instead, his talk focused on a more general question, namely how to find a well-defined basis of integrals for Feynman diagrams, which turns out to involve some rather subtle mathematics and geometry. This is a topic that my former boss Jake Bourjaily worked on in a different context for some time, and I’m curious whether there is any connection between the two approaches. Oliver Schlotterer gave the day’s second review talk, once again of the “actually a review” kind, covering a variety of recent developments in string theory amplitudes. These include some new pictures of how string theory amplitudes that correspond to Yang-Mills theories “square” to amplitudes involving gravity at higher loops and progress towards going past two loops, the current state of the art for most string amplitude calculations. (For the experts: this does not involve taking the final integral over the moduli space, which is still a big unsolved problem.) He also talked about progress by Sebastian Mizera and collaborators in understanding how the integrals that show up in string theory make sense in the complex plane. This is a problem that people had mostly managed to avoid dealing with because of certain simplifications in the calculations people typically did (no moduli space integration, expansion in the string length), but taking things seriously means confronting it, and Mizera and collaborators found a novel solution to the problem that has already passed a lot of checks. Finally, Tobias Hansen’s talk also related to string theory, specifically in anti-de-Sitter space, where the duality between string theory and N=4 super Yang-Mills lets him and his collaborators do Yang-Mills calculations and see markedly stringy-looking behavior.

Friday began with Kevin Costello, whose not-really-a-review talk dealt with his work with Natalie Paquette showing that one can use an exactly-solvable system to learn something about QCD. This only works for certain rather specific combinations of particles: for example, in order to have three colors of quarks, they need to do the calculation for nine flavors. Still, they managed to do a calculation with this method that had not previously been done with more traditional means, and to me it’s impressive that anything like this works for a theory without supersymmetry. Mina Himwich and Diksha Jain both had talks related to a topic of current interest, “celestial” conformal field theory, a picture that tries to apply ideas from holography in which a theory on the boundary of a space fully describes the interior, to the “boundary” of flat space, infinitely far away. Himwich talked about a symmetry observed in that research program, and how that symmetry can be seen using more normal methods, which also lead to some suggestions of how the idea might be generalized. Jain likewise covered a different approach, one in which one sets artificial boundaries in flat space and sees what happens when those boundaries move.

Yifei He described progress in the modern S-matrix bootstrap approach. Previously, this approach had gotten quite general constraints on amplitudes. She tries to do something more specific, and predict the S-matrix for scattering of pions in the real world. By imposing compatibility with knowledge from low energies and high energies, she was able to find a much more restricted space of consistent S-matrices, and these turn out to actually match pretty well to experimental results. Mathieu Giroux addresses an important question for a variety of parts of amplitudes research, how to predict the singularities of Feynman diagrams. He explored a recursive approach to solving Landau’s equations for these singularities, one which seems impressively powerful, in one case being able to find a solution that in text form is approximately the length of Harry Potter. Finally, Juan Maldacena closed the conference by talking about some progress he’s made towards an old idea, that of defining M theory in terms of a theory involving actual matrices. This is a very challenging thing to do, but he is at least able to tackle the simplest possible case, involving correlations between three observations. This had a known answer, so his work serves mostly as a confirmation that the original idea makes sense at at least this level.