Tag Archives: theoretical physics

Yang-Mills Exists: What Does That Mean, and What Is It Good For?

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Last week, OpenAI announced they tackled a longstanding problem in mathematical physics: the existence of quantum Yang-Mills theory. Since one of this blog’s first posts was an explainer about what Yang-Mills theory is, I figured I should probably comment.

In 1954, Yang and Mills wrote down a type of theory that would turn out to describe three of the four fundamental force of nature. Electromagnetism uses the simplest version of the theory, while the two nuclear forces are more complicated. Gravity is the only force that isn’t described by Yang-Mills theory.

But while Yang and Mills wrote down a theory that any physicist would be happy with, that’s not enough for mathematicians. Physicists take shortcuts and make assumptions in order to get a theory they can readily use, and for quantum theories those shortcuts and assumptions can paper over a lot of subtleties. Normally, that doesn’t matter much: when physicists want to use Yang-Mills theories to understand the real world, they use well-tested approximations and simulation techniques. But in a controversial-enough corner case, where you might want ironclad mathematical proof to convince skeptics, physicists wouldn’t have that option. The true theory, without any approximation tricks, wasn’t defined in terms of axioms that were proven to make sense, not with a mathematician’s carefulness. In short, despite its successes, Yang-Mills wasn’t proven to exist.

Over the years, people tried to fix this. Not many people, mind you: most physicists didn’t know enough math, most mathematicians didn’t know enough physics. But mathematical physicists had managed to find axioms and proved they worked for other theories, simpler than Yang-Mills. And a few people made progress towards more complicated theories. It’s a problem that has a lot of prestige associated with it: it’s the first step in one of the million-dollar Millennium Prize Problems in mathematics (though to be clear, just the first step: put a pin in that). But that prestige didn’t draw many people to work on it: it was viewed as too difficult and not particularly urgent, not the kind of productive direction people could build careers on.

It definitely wasn’t a type of research I understood. Hearing about it occasionally from the outside, it seemed not only mysterious but messy. It felt like there was a zoo of different assumptions and objections to those assumptions, theorems and loopholes to those theorems. It felt like the kind of research area where someone out there would be convinced that they had already solved the problem, while someone else was equally convinced that that solution was totally unsatisfactory. So even after OpenAI announced they had tackled a different Millennium problem, I didn’t expect Yang-Mills to be next on the list. Not because it was hard: it was clear to me by then that AI could solve hard problems. Rather, because it seemed like not just a mathematical problem, but a social one, of convincing people that some specific kind of approach actually did the job.

So, what happened?

I’m probably wrong about that, though we don’t quite know yet.

AI has made prestigious problems a lot more tempting. If you don’t need to dedicate your career to becoming the right kind of expert, then why not give it a shot?

Not every prestigious problem can be solved, of course. And trying to solve one you can’t solve yet is more likely to just make the AI “cheat”, like how an impossible hacking challenge got an AI to hack the website HuggingFace.

But at this point, some people with the right AI tools and the right expertise had been working on this particular problem. Some of them were Ethan Sussman and Levent Alpöge (the Anthropic employee who was involved in the controversy around OpenAI’s solution of the Navier-Stokes Millennium problem). Others, apparently, were working at OpenAI.

Much like with Navier-Stokes, OpenAI got there first. They haven’t said much about how, besides a rough timeline, which puts their result before Sussman and Alpöge posted their in-progress work online.

Instead of posting their own results online, OpenAI reached out to David Gross. David Gross isn’t known for this kind of mathematical work, but he is a Nobel prizewinner who made an important discovery about Yang-Mills. He’s also one of the founding figures of string theory.

Reaching out to David Gross was pretty clearly a way for OpenAI to try to get approval from someone considered a pillar of the theoretical physics community (at least, by the people who don’t think of him as an arrogant jerk). David Gross in turn put together a team of people to check the result. Of them, Arthur Jaffe is a major figure in making these types of theories mathematically consistent, while Michael Douglas is someone who has been trying to get people more involved in these kinds of prestigious or mathematically deep problems, and has been keeping track of progress on this problem specifically for the last twenty years. The other two, Igor Klebanov and Nikita Nekrasov, are more generally known as smart, mathematically savvy physicists and don’t have a direct connection to the topic that I’m aware of.

The team has decided not to release the result until they’ve checked it and written a decent explanation. This has annoyed a lot of people, including the Advisory Group on Mathematics and Artificial Intelligence, who think the result should be released immediately, so everyone has a fair chance to try to understand and use the result. I can appreciate some of the logic behind waiting: the AI-written papers OpenAI has released before have been pretty poorly written, the kind of thing that would be heavily edited before being published if they were written by a human. And even if the math works, the important question is whether it actually represents Yang-Mills theory, which is something you need physics expertise to answer. Still, it’s hard to argue against more transparency. My goofy recommendation is that they don’t release the paper, but do release the version of the result in the mathematical programming language Lean, and people can work to understand that.

Until they do make something public, or release a decision, we shouldn’t count on the result actually doing what it was supposed to, and proving Yang-Mills exists. It still probably does, these AI systems are increasingly reliable. But there could well be a loophole that wasn’t considered.

What is it good for?

I’ve seen physicists shrug at this result. But I think some of the intuitions there are outdated.

We already know Yang-Mills is fine. Not in a mathematically rigorous way, but through experiments and tests and more or less consistent procedures. Nobody was doubting that Yang-Mills existed in the everyday sense of the word.

Now, mathematical rigor can still help. With a nice functional set of axioms, it can become possible to prove other things. The rest of the Yang-Mills Milennium problem is proving one of those things, namely that the massless gluon particles in Yang-Mills theories group up into massive glueballs. That’s still not really in doubt, it’s been checked in computer simulations pretty extensively. But the idea is, if you can prove that, maybe you can prove other interesting things.

This hasn’t been high priority, because math is slow. Often, by the time mathematicians have carefully nailed down a result, physicists have raced ahead and done something reckless. So while some physicists will pay lip service to the idea of getting mathematicians interested, most are pessimistic about it helping.

But I’ve seen a few people put two and two together here, and notice that math is no longer slow.

AI is good at math, better than it is at most other things. That isn’t because it’s some ultra-logical beep boop sci-fi robot. It’s because the best way to get an AI good at something is to train it on a huge number of examples of that thing, and you can generate a lot more examples automatically than you can by hand. So an AI trained on math can use the Lean programming language during training to check it got the answer right, rather than waiting for human graders. That makes it a lot easier to train AI to be good at math.

So when a part of physics becomes expressible in Lean code, it gets a lot easier for AI to get good at it.

What happens when an AI gets good at Yang-Mills, or similar physics theories?

First, the rest of the Millennium problem. If it can’t do that, it probably can’t do much else.

But after that?

Physics is an experimental science. You can’t prove a theory is correct with math alone, you have to test it.

But many of the most controversial questions in physics aren’t actually questions about experiments. They’re questions about theory.

You can’t use mathematics alone to prove which theory of quantum gravity is correct, you eventually need experiments. But as it happens, every theory of quantum gravity is controversial for mathematical reasons as well. String theorists believe there is one unique theory that unites the different types of string theory, others think they’re full of hot air. Loop quantum gravity researchers think their theory can describe a world like ours, others think the theory forces space and time into a crumpled up wad of nonsense. Asymptotic safety researchers assemble evidence that gravity has a special property that keeps it safe from the problems that make quantum gravity seem hard, others think the evidence doesn’t go far enough.

If any of those theories had axioms, maybe someone could write a proof to settle one of these controversies. If AI gets good enough at this, it might be able to.

Now, all of this still might not matter very much. Right now, AI contributions to math and physics seem to be a mix of public relations and advertising, benchmarks (where AI companies test how powerful their setup is by seeing how many math problems it can solve) and individual employees playing around with the system in their free time. That mix makes it hard to predict what will be tackled next. If making a system able to resolve big longstanding physics questions is an expensive detour and doesn’t look like it will generate good PR, then it might not make sense for AI companies to pursue it, and if the technology stays expensive it may be out of reach of users. But if it’s cheap, or for some reason they think it’s a high priority, then it might happen.

As always these days, who knows?

Showing Me Your Amateur Breakthrough Is Evidence That It’s Wrong

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Ok, people who read my post on the Anthropic science blog: we need to talk.

It’s understandable. You saw me write about AI being used to tackle a serious, publishable physics problem. Naturally, some of you want to share that you accomplished something even more impressive with AI, something the scientific journals and arXiv moderators won’t acknowledge. Maybe you’re hoping I’ll write about it, maybe you want me to check your math. Maybe you just think it’s neat.

But here’s the thing: most of the time, I’m not the right person to send those ideas to. And the fact that you didn’t figure that out is a bad sign.

Outside of the movies, scientists are specialists. Not just “physics” or “particle physics” or “theoretical particle physics”, but so specific that they often focus on particular methods, or are geared around particular goals. For any idea you propose, if it’s good, there will be a small list of people who will be excited to see it and able to evaluate it. They’ll be people who are trying to accomplish the same thing, or people who used the methods you’re using for other things. Those people aren’t hard to find, if your idea is in a field you’re actually interested in: they’re the people who write papers on the subject. If you find the topic interesting, you’ll have read their papers, and know who to reach out to.

So even back when I was active as a physicist, there was only a small list of ideas where I’d be the right person to talk to. Now, though, as I mentioned in the Anthropic post, I’m not a scientist anymore. I haven’t been following the field in enough detail to actually verify anything in detail, even on topics I used to research. Part of the punchline of the post was that I don’t have a clear idea of what’s possible in the field right now. So even if you’ve figured out something in scattering amplitudes, you should reach out to someone who’s still researching the subject, a few of whom were even mentioned in the post!

Now, it might be you’re not interested in someone checking your work, and instead you’re writing to me in my capacity as a science journalist, because you feel like you did something newsworthy. But as a science journalist, I’m not supposed to evaluate your work by myself. I’m supposed to reach out to an independent expert, and if I can’t find any who will speak positively about your result, then the editors won’t publish it. So if you think you did something big enough to merit news coverage, the first thing to do is once again, reach out to a scientist who actually has the expertise to evaluate your work, and get them excited about it. Then you two can contact me together!

If you don’t know who to contact, that’s a sign you don’t read papers in the field. It makes it more likely you managed to convince yourself of something nonsensical, because you don’t have the expertise to evaluate it. It’s a pretty clear sign that I shouldn’t trust you or your AI methods about the result.

It’s also just plain bad for you. If you produce a paper without ever wanting to read any, it suggests your priorities are in a dangerous place: that you’re trying to do something that sounds cool and important over doing something you actually care about.

I’m coming from personal experience here.

As an academic, I spent years telling myself that not enjoying reading papers or going to talks was just part of the job. Then it really cost me something, and I realized the job wasn’t worth my superficial desire to be a physicist. So I left.

Then, I spent a year trying to get a job in industry, and was told the same thing. They wanted to hire people who cared about the job. The finance guys wanted to hire people who read books about finance. The data scientists wanted to hire people who did projects in data science. They wanted, if not passion, then at least grace. Someone who could take to the job easily, and naturally.

Now, what I do for a living is something I was doing once a week, for no more incentive than a vague sense of personal obligation, for over a decade. And I’m doing a lot better.

You want to do something impressive with AI? You think you’re a whiz with Claude Code, that your custom workflow is immune to AI hallucinations?

Then use it for something else. Pick something you actually enjoy reading about, not just the pop news, but the real technical literature. Something you don’t need the AI to explain to you. Something that matters to you, in a way that actually counts.

At the very least, it’ll work out better than writing to me.

It Got to My Field

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For the folks who found this blog from Anthropic’s site, welcome!

For everyone else, I’d better give some context.

Last month, I had a blog post titled “It Only Counts When AI Gets to My Field”. The title was a joke, the content less so. I said that if some of the longstanding problems of my old research field got solved by AI, then I’d sit up and take notice.

The post turned out to be a bit of a self-fulfilling prophecy, after some folks at Anthropic read it and decided to tackle one of the outstanding problems I mentioned. They then invited me to write a post about it on their science blog.

The post is here. I recommend reading it, then coming back. The rest of this post will be a little Q&A.

Q: At the end of that post, it says Anthropic paid you for your time. Can we trust what you wrote?

A: When I agreed to write the post, I made it clear that I was going to give my own opinion, not write an ad for Anthropic. They could propose light edits, but that’s it. And I avoided signing anything with them, not even an NDA, so I could freely tell you if they pushed my boundaries.

They didn’t push my boundaries. They asked me to clarify a few things, and to give more detail on the science. I didn’t change the message or the takeaways.

The reason I asked them to pay me is that, as a freelance writer, I don’t have a salary to fall back on. Time I spend on a project like that post is time I’m not spending on journalistic projects, so if I worked on it for free I’d essentially be using my vacation time for it. The rate I’m charging them is roughly in the middle of what I’d have gotten paid if I used that time on journalism: a bit more than I would have made working for the lower-paying outlets, a bit less than I would have made with the higher-paying ones.

I’m also not expecting this to be the start of a longer-term business relationship or anything like that. So overall, I don’t think I’m incentivized to lie on their behalf. You can trust me.

Q: How confident are you that they did what they said they did?

A: I didn’t get the feeling they were lying. But I didn’t start out skeptical.

I haven’t seen the logs from the LLM, or anything like that. That’s the kind of thing I would dig into if I were more suspicious of their story. If there’s a reason to be suspicious, I’ll ask.

But so far, I don’t feel that I have much reason to be suspicious. They were able to look up details on the fly when I asked them, they didn’t seem to have a polished message they were trying to push past me. And more importantly, as I’ll mention in a bit, I don’t think anything they described is all that outlandish. It lines up, largely, with the capabilities I’d expect their Claude Science platform to have.

It’s also relevant that they didn’t use an internal model for this. It means that scientists will likely be trying out similar problems soon, so if it does turn out they exaggerated something, people are going to figure out quite quickly.

Q: So how big is this? We just saw AI solve a Millennium problem, after all.

A: This was a lot easier than a Millennium problem. But it was also a lot cheaper.

The problem they solved is one with a clear recipe, honed and explained over multiple papers. It’s something I expected to be hard to do without access to a lot of computer power, so I thought it would need to be approached with a novel technique, in order to avoid using that computer power.

In the end, it didn’t need that. As I mention in the post, another group got most of the result at around the same time, with a much smaller amount of AI assistance.

It’s not an easy recipe, to be clear. I think other people will be surprised that a reasonably affordable program like Claude Science can do this without a lot of guidance. I’m not surprised, mostly, because I’ve been paying enough attention to what people have been doing with these things, and carrying out this kind of recipe consistently is actually something the good science AIs can do right now. If you haven’t been following as closely, that’s going to be a lot more surprising.

So maybe the best way to answer this is: instead of paying $15 million to solve one of the most famous problems in mathematics, they paid $1,000 or so to make a significant next step in an ongoing research program in theoretical physics. This doesn’t tell you much about whether AI can achieve field-defining breakthroughs, but it tells you a lot about the kinds of things a theoretical physicist with $1000 spare budget can do right now.

(As an aside, it feels crazy to me that 90% or more of the compute cost was from the LLM, not the calculation itself. On the one hand, it feels nuts to essentially use ten times more computer power to do this than would have been needed if a human had done the coding. On the other hand, I’ve applied for grants that budgeted more than that per year for travel costs, and spending this kind of money on getting a result seems a lot more useful than spending it on airfare.)

Q: Isn’t it reckless to publicly post a problem for AI like that?

A: I posted my challenge before OpenAI announced their Navier-Stokes result. At that point, there had been a few awkward surprises, but for the most part AI companies had a pattern of talking to an academic first before trying to solve a problem. It’s what they did back in March.

I expected that was what they’d do this time, and got more than a little blindsided when they just solved the problem on their own and reached out to Lance afterwards. I’m lucky that Lance doesn’t seem mad at me about it, it would have been quite understandable if he was.

I did tell them that if they wanted to tackle any of the other problems in that post, they should reach out to one of the scientists involved first, before attempting it. In addition to being polite, it’s a way to make sure that they understand the problem correctly and have a plan to verify the result. They happened to pick a challenge that was particularly easy to verify, but it didn’t have to go that way.

Q: Any thoughts about AI’s future impact on science?

A: If there’s a problem you can’t solve, but you think someone with expertise in another field or better programming skills could, then it’s probably solvable with AI.

A lot of problems don’t fall into that category. I’ve seen people talking online about AI solving quantum gravity, but solving quantum gravity is a question of which bullets you’re willing to bite, not a question of technical skill. That doesn’t mean AI will never be able to address it, but if so it will come from some sort of superpersuader aspect, not merely scientific capabilities.

And of course, some fields do require experiments. People are increasingly building AI-powered labs. I’ll leave it to people from actual experimental fields to think about the potential there.

Q: Can you say something explicit about the bigger risks from AI, now?

A: As I mention in the post, I didn’t learn that much about the bigger questions from this. I’m still not an expert.

But there’s one thing I think is worth emphasizing:

This technology is clearly getting more effective over time. I don’t think they could have done this six months ago. If you’re trying to predict what will happen next, you shouldn’t just assume this is the most powerful it will get, or the cheapest. If you think there’s a limit, you need to argue for it.

Edit: One more thing I should mention, now that I’ve confirmed she’s ok with it: the idea for my challenge came from some discussions at Lancefest, a conference in honor of Lance back in June, and particularly from Anastasia Volovich, whose talk declared the nine-loop calculation “a benchmark – or a ‘challenge’ – against which any “AI takeover” should be measured”.

Amplitudes 2026, Part II

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This is a continuation of my conference coverage from last week. The same warnings apply: this is much more technical than my usual posts, readers beware!

Last week I covered the talks from Monday through Wednesday, so I’ll jump right in here with Thursday morning, where the first speaker was Francesco Riva, who after a brief ad for his new board game Tutti Quantum gave a review of positivity constraints, first-principles restrictions on quantum field theories based on their behavior at high energies. After covering some of the research program’s successes like arguments against Galileons and massive gravity, he talked about how new methods allow one to take into account the possibility of loops of massless particles, essentially by invoking a formal version of the idea that real experiments have finite size. He was followed by Grant Remmen, who talked about his work deriving string theory-like amplitudes from increasingly minimal assumptions. While one can always quibble with the assumptions they impose, I do find it encouraging that they are now managing to do this game with both gravity and gauge theory, and with five-particle amplitudes, not just four.

Paul Heslop then covered applications of bootstrap techniques to non-supersymmetric theories, where analytic superspace still finds a way to be useful. Tomasz Taylor covered progress in calculating Yang-Mills amplitudes in de Sitter space. Andrea Puhm and Nima Arkani-Hamed don’t have slides online yet: Puhm’s title suggests her talk was part of the celestial holography field, while Nima’s was likely similar to his talk at Lancefest, where he talked about calculations of amplitudes in the limit of a very large number of loops (represented in the field with a capital L) or very large numbers of particles (represented in the field with a lower-case n). Given the different context, I’m guessing he left out the self-effacing jokes where he was “little n” and Lance was “big L”, though I’m hoping he at least mentioned his students had been checking their results against what they called the “lanswer”.

The evening ended with a gong show, which for the non-initiated is a series of short student talks with a strict time limit (hence the gong). I’m not quite so intrepid as to read all of the slides for these, commenters who attended are welcome to highlight special examples.

Friday began with a talk by Agnese Bissi, who reviewed the connection between holographic correlators and amplitudes in AdS space. I liked her emphasis on this as a lab to find nice new representations of amplitudes, and her summary at the end of the current frontiers. Axel Kleinschmidt followed with a talk on one-loop string amplitudes, where he and his collaborators have gotten gradually more proficient at manipulating the rich structure of elliptic functions that make an appearance. Piotr Tourkine talked about his work using the S-matrix bootstrap to find scattering amplitudes in higher dimensions, a context where accounting for thresholds presented a new challenge. This is a problem people are approaching with genuine supercomputers, he quoted one calculation at 100,000 CPU hours. Lauren Williams is one of a small community of mathematicians who have been intrigued by the amplituhedron, her talk was a walk through a series of conjectures, some made by physicists, some by mathematicians, most with counterexamples found in the last few years.

Finally, Zvi Bern closed the conference with a talk on the frontier of amplitudes-based gravitational wave calculations, referring to it, probably to David Kosower’s annoyance, as 5PM. Some parts of this frontier have been calculated, but a few have proved hard going, bottlenecked by immensely challenging integrals, which go beyond the capabilities of publicly available codes. The new integrals have a variety of strange functions, including the elliptics and Calabi-Yaus I spent time on in my own career, as well as Heun integrals, which I imagine I will have to learn more about from the slides of Elliptics & beyond ’26, as I don’t remember people talking about them when I was in the field. New integration strategies have led to improvements in the public codes and seem to be making good progress, but Zvi highlighted that ideally they want to not need supercomputers at all, as they’ll need to go higher in loops to see effects from, for example, the deformability of neutron stars.

Amplitudes 2027 will be in Munich, with the summer school in Mainz under Stefan Weinzierl’s capable hands. I’m looking forward to seeing what the state of the art looks like then!

Amplitudes 2026

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This week was Amplitudes, my old subfield’s big yearly conference. This year, it’s at Queen Mary University of London.

I’m too busy to attend these days, now that nobody is paying me a salary to do that sort of thing. But it’s still a good chance to keep up with the field, which helps me find stories. And I know I have a few readers who are interested. So I read the slides when I can, and fill you all in.

As of writing this, I’ve read through slides from the first few days of talks. I’ll likely post on the rest next week. As usual when I’m conference-blogging, this post will be quite a bit more technical than my average post, so readers beware: I’ll be mentioning a lot of amplitude-ish ideas without much explanation. I’m happy to explain in the comments if you’re curious, though!

Before I launch into talking about the content, I should mention something I’ve heard about the venue. Apparently this year registration filled up surprisingly quickly. The rumor is that the folks at Queen Mary weren’t able to find a venue with enough space to host the full community, leading to a smaller conference than usual. As the Amplitudes subfield grows, I suspect this will be more and more of a challenge. I remember back when I was organizing in 2021, rooms that could hold enough people were shockingly expensive to book. I wouldn’t be surprised if this becomes more of an issue going forward.

David Kosower opened the conference with a review of the state of the art in amplitudes for gravitational waves. I enjoyed an early slide showing how LIGO has increased in precision over the last ten years, which really helped illustrate the value that good theoretical predictions can bring as the experiment gets better. I also appreciated his attempt to get people to stop saying “post-Minkowskian” and start using a more normal term like Relativistic Perturbation Theory, though based on the other talks it doesn’t look like it’s catching on. After covering the overall state of the art (currently around four loops) he talked about his own work looking for ways to more directly get waveforms for orbiting black holes out of an amplitudes-style calculation, a theme his collaborator Donal O’Connell covered in more detail later that day. (The trick, apparently, is background fields!) Other gravitational wave-related talks came from Gustav Jakobsen, who covered four-loop results using the worldline method, Graham Brown, who explained how to calculate the Magnusian Jakobsen mentioned (it’s the log of the amplitude, basically), Canxin Shi, who talked about how a concept called Stratonovich-Weyl quantization helps explain structures that keep showing up in classical observables, Giulia Isabella, who covered a way to get six-loop contributions to gravitational waves via a wave equation, Lara Bohnenblust, who showed how to get strong-field information from a shockwave limit, and Gang Chen, whose slides were brief enough that I didn’t really get a clear idea of what he was up to. Rounding out the day were two talks that didn’t involve gravitational waves: Zihan Zhou, reporting on work with Nima on a water wave polytope called the hydrotope that they found a general formula for with help from Claude (the AI, not Duhr, to steal a recurring joke from Lancefest), and Michael Ruf, who probably snuck in due to the fact that he has done a lot of work with gravitational waves, but was reporting on a QCD calculation using a tool called Scorpio.

Tuesday had more of a QCD theme. Thomas Gehrmann gave the day’s opening review talk, where he pointed out that to get 1% precision for predictions for the LHC, we’ll likely need three-loop calculations. He pointed out the important role IR real emission calculations and improvements in parton distribution functions need to play, and pointed out that collider physics isn’t just the LHC: between reanalyses of older electron-positron collider data and the upcoming electron-ion collider and FCC, there are even more contexts where high-precision QCD will matter. Simone Zoia talked about one aspect of the current state of the art, five-particle processes involving massive particles, where the functions get strange and elliptical and one has to think hard about the methods one uses (for example, do you want canonical differential equations, or almost-canonical? Bulirsch-Stoer or AMFlow for numerics?) He also included an excellent Laplace quote, “Nature Laughs at the Difficulties of Integration”. Yang Zhang talked about a different calculational frontier, covering progress in the planar limit and with no masses, but for two-loop six-particle and three-loop five-particle scattering. The talk included some nice branding, like an “epsilon collaboration” proposing an algorithm to find epsilon forms for any Feynman integral and a program called Effortless to find symbol letters. Yang Zhang is also a skilled amateur photographer, and he mentions taking the conference photo for Amplitudes five times before. Personally, I’m surprised it’s only been five times! Dmitry Chicherin presented results bootstrapping QCD amplitudes. This was a dream of mine back in my hexagon function days, and while they aren’t quite at the point of being useful (my understanding is they’re still only getting the leading transcendentality, and none of the amplitudes they’re finding are new) it’s still pretty cool that this is even possible now. Bo Feng proposed what he claims is a general algorithm to find generating functions for IBP reduction. It’s not clear to me whether his setup bypasses the computational difficulties in existing methods (mostly involving solving large systems of equations), or whether it shifts the issue elsewhere, though. Pierre Vanhove’s talk also involved QCD, though at an effective field theory mediated step removed, with chiral perturbation theory, a theory of low-energy QCD that he used to shore up the lower energy ranges of lattice calculations for contributions to the muon anomalous magnetic moment (a natural calculation to involve him due to the presence of novel elliptic integrals). Overall, the QCD talks impressed me with the wide range of software packages mentioned, some of which already existed when I was in the field but many of which are new. I do wonder if this is just a result of many research paths maturing at the same time, or if people are finding it easier to write packages with tools like Claude Code.

The remaining three Tuesday talks were more mathematical or theoretical in focus. Henrik Johansson talked about black hole Compton scattering in N=8 supergravity, where it’s possible to use the wave equation to extract all-loop results. Cristian Vergu reported on his progress with Landau analysis. It’s been really fun watching this grow from a small reading group at NBI to what appears to be at this point quite a deep understanding, including a picture of how to understand amplitude singularities with quite a lot of breadth and detail, and the persistent hope that this could allow one to manipulate singular quantities without needing dimensional regularization. Michael Borinsky gave an update on tropicalization, where he showcased a theorem he proved demonstrating a way to compute certain classes of amplitudes in polynomial time in the loop order, even when the number of Feynman diagrams increases factorially. The examples he talked through were impressive, but I’m still a bit skeptical this could work for the Standard Model. I’d want to talk to him about it to figure it out, anyway!

Wednesday was a short day, as is tradition, to give people some time for tourism in the afternoon. Alexander Zhiboedov began the day with a talk on energy correlators in N=4 super Yang-Mills at finite coupling, where he is now able to do a bootstrap with two-sided bounds to hone in on the actual quantity even outside of the planar limit. Arthur Lipstein and Daniel Baumann both talked about cosmological correlators, the former with amplitudeologists’ favorite toy model of the conformally coupled scalar and the latter with Yang-Mills and gravity in de Sitter space. Dave Dunbar closed the day with a historical talk, walking through the milestone amplitudes papers before the Amplitudes conference existed. It’s the kind of talk he could have given at Lancefest the week before if others hadn’t already covered the material!

I’ll cover the rest of the conference (Thursday and Friday) in next week’s post, so see you then!

At Lancefest

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It’s been a while since I’ve said this: I’m at a conference this week!

Specifically, I’m at Lancefest, which is not just any old conference, but a birthday conference for Lance Dixon. When a renowned academic turns 60 or so, their students and collaborators hold a birthday party-flavored conference for them. The conferences are usually a mix of academic talks and reminiscences, with the occasional roast thrown in.

I went to my advisor’s birthday conference four years ago. Lance wasn’t my advisor, but in many ways he might as well have been. When my advisor took a sabbatical in the middle of my PhD, he sent me to work with Lance. It was my first real experience doing research in a team, not just puzzling away by myself with occasional feedback. And I was hooked: I spent the rest of my academic career in Lance’s field. We collaborated time after time, and even when I started to branch out he remained a frequent presence.

In part, that’s because Lance’s field was really Lance’s field. Amplitudeology has grown a lot since I started out, with several hundred people going to the field’s big yearly conference and subfields like Elliptics having their own yearly conferences. In such a world, it’s tough for anyone to feel like a truly central figure. But Lance tends to. He’s been able to keep up with that growing world, to keep finding important problems and keep understanding others’ ideas. While some have specialized, or stepped back, Lance seems to somehow manage to be a father figure for the whole field all at once. It’s a capability his advisor Jeff Harvey might have predicted, he mentioned in his talk that Lance’s interests were always broad. Over the years the young students I met who joined when the amplitudes field was already large saw Lance as a kind of mysterious titan, and were occasionally awed that I had worked with him. “What was that like?”

Well, it was like working with Lance. Lance isn’t a manager at heart, like some senior academics end up. He wants to understand everything he works on, and will happily dive in, Maple subscription in hand, and try to figure things out for himself. He wants you to keep up with him, understanding on your own terms, to keep him honest, to provide an independent check. But that often wasn’t possible, because the man is just so damn fast. He’d be miles ahead of me, with his Maple and his laptop, while I was churning overnight Mathematica runs on a twenty-machine cluster.

(Was the difference due to our taste in software? Partly. I did learn to use Maple later, it genuinely is faster at some things. But Lance is faster at almost everything.)

And he cared so damn much sometimes. About getting things right, like a scientist should. About making things nice, too: finding a pretty basis of functions, a better notation for the paper, something that might jostle out the next big insight. Working with him, you could feel like that one paper was the most important thing in the universe.

Others at the event have had similar stories. Fernando Febres Cordero remembers noticing a potential issue, emailing Lance about it, and in a few minutes hearing back with a potential explanation.

Lance is someone who became a leader without really being a politician. He doesn’t have the legions of students in tenured positions that some do. I trimmed that count by one, and it wasn’t huge to begin with. But for someone who isn’t “everywhere” in that sense, he manages to be “everywhere” all the same.

So Lance, happy birthday! You’re really the only person who could have had a birthday conference quite like this, a cross-section of the field, all with something kind to say. Thanks for putting up with any embarrassment associated with having this much attention for three days, and I wish you many Maple-fueled mysteries to come.

An AI Opinions Chart

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You ever read something and suddenly a whole classification scheme lights up in your head?

A thread on X from “stringking42069” showed me a combination of opinions I hadn’t seen before. stringking42069 is a pro-string theory commentator with a macho gym bro memer gimmick. He’s openly contemptuous of many physicists who describe themselves as string theorists, arguing that only a smaller number really deserve the name.

To be clear, none of that is the new combination. Long-time readers of this blog will remember a frequent commenter with a very similar attitude, if much less tendency to use the word “bro”.

The new thing, from my perspective, is how he thinks about AI. As he explains in that thread, he sees AI as great at certain kinds of physics calculations, ones where the methods and goals are mostly known and the challenge is working out the math. He doesn’t expect it to be able to contribute real creativity or judgement, the messy decision-making that physicists use to decide what is worth building in the first place.

Others with that perspective tend to argue that this will be a boon for scientists, who AI will free up to do creative work, multiplying their output. The difference is, stringking42069 thinks a lot of scientists are not doing creative work in the first place, including most of the people making extensive use of AI. So if anything he’s happy to see them go, and only pissed that they’re sucking up resources and attention on the way out, and discouraging students who could be joining the parts of the field that do real creative work.

It made me realize that there are two axes to thinking about AI in physics.

On the one hand, there’s where you think AI capabilities are. Is AI going to lead to “a nation of geniuses in a data center”, an AI-powered super-(cyber-)Ed Witten for everything and everyone? Is AI great at routine work and coding, but will never be able to do anything really creative or novel? Or is AI total hype, almost always a waste of time?

On the other hand, there’s another axis: misanthropy about science. For some of the people arguing about AI online, most scientists are good people trying their best to do worthwhile things. For others, most scientists are complacent and cliquish, wasting time and money on ideas that are going nowhere and forcing the real geniuses out of the field.

Put those together, and you get the table below:

Thinks academia is mostly fineMisanthrope
AI geniuses are comingThe practice of science will change. We’ll play at science like chess, and have fun trying to read and understand amazing AI insights.Soon all scientists will be out of a job when the public notices AI can do it all better. Then the real breakthroughs will come.
AI can do routine workAI frees scientists to focus on what we do best: creativity. We should think carefully about how to train junior scientists now, though.AI is comparable to bad scientists who only do derivative work. If they leave, we real paradigm-changers could inherit the field.
AI is complete hypeMost scientists don’t use AI. AI is worrying because it misleads students and the public, who should listen to real scientists.Scientists are shilling for AI companies, as you should expect for people who waste the public’s money on reputation games.

This classification is missing a lot, of course. One important question is not just what AI can do in principle, but what it can do cost-effectively, and whether anyone is actually willing to pay for it. A point where I agree with stringking42069 is that companies get a lot of good PR out of building AI physicists right now, and that PR benefit won’t be relevant forever. I’m also leaving out the more general questions of AI’s effect on society, for example people who think AI geniuses will lead to the end of the world as we know it.

But I suspect if you look at this table, you can already start matching the scientists you see on social media. I’ve seen examples of all of these in the wild (though the bottom-left is somewhat rare, as far as I can tell). Where do you fall?

Breakthrough Prize 2026

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Because of last week’s “bonus info” post, I’m only now getting around to commenting on this year’s Breakthrough Prizes in Fundamental Physics. While I don’t comment on them every year, I know enough about several of this year’s winners that I figured a post would be helpful.

For those who haven’t heard of it, the Breakthrough Prizes are a bit like the Nobel, if it was created by a 21st century rich person instead of a 19th century one. They give out more money, and instead of an organization like the Swedish Academy of Sciences they pick winners via a committee of past winners. They’re more flexible in structure than the Nobel, with extra prizes for early-career researchers and a tendency to reward accomplishments that are either entirely theoretical or solid experimental work that doesn’t show a new discovery, both of which are things the Nobel Prize is structured to avoid. They’ve also shown willingness to reward large collaborations, rather than following the Nobel’s informal rule to only give the award to three people at a time.

This last was on display this year in their main award in physics this year, for the muon g-2 collaborations. The award is going to collaborations of scientists and engineers at three different particle colliders, for work done over a span of over fifty years to measure the magnetic properties of the muon. These measurements have shown a tantalizing discrepancy with predictions that inspired many to conjecture new physics. However, in the last few years it’s looked more and more like the discrepancy was due to an imprecise prediction, and better methods seem to be converging to the experimental value. At this point, smart money is that there is no disagreement with the Standard Model here, but as always in science there’s a chance some mystery remains.

The Breakthrough Prize also offered a special, out-of-schedule prize to David Gross. Already a Nobel laureate, Gross had a crucial role in our understanding of the force of quantum chromodynamics that binds protons and neutrons together. He was also a major founding figure in string theory, and since the Breakthrough Prize is more comfortable recognizing theoretical contributions they get to mention this as well. Gross is also known in the community for his personality, which tends to fill up any room he’s in. I can only imagine the conversations that led to Breakthrough’s decision to add a special prize for him this year.

Breakthrough is also adding a new recurring prize, the Vera Rubin New Frontiers Prize, honoring women who make important contributions to physics within two years of their PhD. The prize is a bit smaller than the exiting early-career New Horizons in Physics Prizes, presumably because it goes to even younger researchers. This year’s winner is from my old field, scattering amplitudes. Carolina Figueiredo is part of the latest evolution of the research program behind the amplituhedron. The new framework of “surfaceology” seems like a promising geometry-flavored way to understand particle physics calculations in more realistic theories, and unlike its predecessors may have some practical value eventually as well. Congrats Carolina!

Finally, the New Horizons in Physics Prizes are for impressive early-career researchers. I don’t know much about the first recipient, Benjamin Safdi, who works on searches for axions and axion-like particles, today’s most trendy dark matter candidate. I know a bit more about the work done by Clay Córdova, Thomas Dumitrescu, Shu-Heng Shao, and Yifan Wang, having met several of them in my physics career. They work on what are called generalized symmetries, concepts which go beyond the usual idea of how symmetry is supposed to work by involving more complicated tensors. I saw these crop up a fair bit in talks, but they were distant enough from my area that I never had a particularly clear grasp of what people were doing with them. I know even less about the work of the last three, Dillon Brout, J. Colin Hill, Mathew Madhavacheril, Maria Vincenzi, Daniel Scolnic, and W. L. Kimmy Wu, on cosmological measurements, but I was friends with Mathew in grad school and am impressed that he’s now working on cosmology given how little cosmology research there was at Stony Brook at the time.

A Window on Absolutely Everything

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It’s often said that in quantum physics, everything that can happen will happen.

One way this comes up is in something called a path integral, used to calculate the probabilities of quantum events. If you want to find what happens to a particle traveling from point A to point B, you have to add up a contribution for every path, no matter how windy, that goes between A and B. These contributions mostly cancel out, and matter less the further they are from a straight line, so the straight-line path is, for the most part, a good description of what happens. But in principle, all of the other paths matter too.

The same thing happens in quantum field theory, in more elaborate form. Instead of a path from one place to another, the paths are from one configuration of quantum fields to another, via all the different ways fields can in principle interact. We are almost never able to take account of all these possibilities mathematically, so we have to approximate, organizing the interactions into more and more complicated pictures called Feynman diagrams, each with a smaller and smaller effect.

In principle, these diagrams need to contain every single combination of interactions that might result in the end-state we’re interested in. These combinations can have a Rube Goldberg flavor, with one field activating another, which activates another, only to all cancel out in the end. Because of this, any field that exists, any particle no matter how rare, can matter, if only a little.

And from that, physicists can learn something.

Because absolutely everything matters, physicists get to reason about absolutely everything that exists.

The best example involves something called an anomaly. These aren’t the anomalies of experimental physics, unexpected results that have a tendency to go away with better measurements. Instead of something unexpected, a theorist’s anomaly is something impossible.

Anomalies are combinations of particles that, if they were to show up together in a sum of Feynman diagrams, would break the rules that the theory was made with in the first place. If they show up, they’re a sign of an inconsistent theory, one that doesn’t obey its own rules and thus doesn’t make sense.

In order to have a theory without anomalies, different calculations involving different particles need to cancel. For example, it might be that the charge of different particles has to add up to zero. This means that if you’ve only discovered a few particles, and their charges don’t add up to zero, then you know you’re missing one. There is an extra particle there, which you haven’t observed, that together makes charge add up to zero.

This logic actually works! It was used to predict the top quark. Before the top quark was discovered, the list of quarks, electrons, and neutrinos had electric charges that didn’t add up to zero. One particle was missing, with the same charge as the up quark and charm quark. It was found in 1995, after being proposed almost 20 years earlier.

What AI Physicists Are Missing and What They Aren’t

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I’ve seen a couple more thoughtful takes on use of LLMs for physics lately. This blog post by Minas Karamis is particularly nice.

He points out something that I’ve said a version of: an AI that must be supervised like a student isn’t very useful, because the main point of student projects isn’t the paper at the end: it’s training the student. If students don’t struggle through all the mistakes of a project, they won’t get the expertise to one day do greater things.

Someone might object that not all suffering is educational. In the 1700’s, Leonhard Euler calculated digit after digit of transcendental numbers by hand. Nobody asks students to do that anymore, and they still seem to turn out alright. Why would using an LLM for science be worse than using a computer for numerical calculations?

In a word: different skills. Programming numerics teaches you some of the same skills as calculating the numbers by hand: skills at being specific about what you mean, aware of the consequences of the details and their implications. Prompting an AI still requires those skills, to check whether the AI’s output is correct. But it’s much worse at teaching them: unlike programming or calculating, when prompting AI, the consequences of your actions aren’t predictable.

For some, though, there is another objection. Sure, using AI reliably might require those skills now. But when it gets better, surely being careful will stop mattering. Surely the AI will end up doing science on its own, and all that training will be as useful as if we trained the students to play football.

I’m skeptical, but not as strongly as some. I think we’re still living in a time when it makes sense to hire scientists, and train people to think, and invest in your retirement.

I don’t think I have any knock-down arguments for that, though. Just some suggestive ones.

One I’ve talked about before is that a lot of the most important parts of thinking aren’t written down. An AI physicist is going to have a hard time replicating the kinds of methods and approaches that people use behind the scenes, but rarely describe or spell out. It will be easier to suss this out over time, as more data accumulates of people working with LLMs and correcting them. But ultimately there isn’t going to be a lot of documentation of this kind of thing.

Another limitation is memory. A mature scientist can draw from experiences across their entire career. For an LLM, any problem it’s solved in the past is by default lost in each new session. People build structures around this, taking notes and reminding the AI when it “wakes up”, or making documents the AI can be prompted to check. But nothing in this vein so far seems to get nearly as wide-scope or powerful as human memory. A scientist career is still the best way we have to build durable, functional expertise.

Finally, there is a question of costs, and efficiency. Here I’m not an expert, and I get the impression the actual experts disagree. I don’t know whether we should expect scaling to hit a wall, but I wouldn’t be that surprised if it did.

There are other common reasons for skepticism that seem more dubious to me. I don’t think AI is inherently worse at creativity just because they’re trained on existing work, though some of the skills we associate with creativity aren’t very well-documented, and thus are hard to train for. I don’t think AI’s randomness or unreliability is a deal-breaker, because human intuition is also random and unreliable: we solve that with tools, and that’s something AI can in principle do as well. I don’t think humans are “more agentic” or something, except in the sense that most AIs are made by companies who need to make them behave in a customer-friendly way. But an agent is just a game-theoretic construct, a way to figure out can win or lose in situations with defined stakes, and anything you can train or engineer to try to win can be modeled by that construct.

Coming from a place of uncertainty, my main appeal to you is to not get hung up on the bad reasons, either yourself, or from the people you’re arguing with. Focus on the best arguments, and see where they take you.