Monthly Archives: March 2023

Building the Railroad to Rigor

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As a kid who watched far too much educational television, I dimly remember learning about the USA’s first transcontinental railroad. Somehow, parts of the story stuck with me. Two companies built the railroad from different directions, one from California and the other from the middle of the country, aiming for a mountain in between. Despite the US Civil War happening around this time, the two companies built through, in the end racing to where the final tracks were laid with a golden spike.

I’m a theoretical physicist, so of course I don’t build railroads. Instead, I build new mathematical methods, ways to check our theories of particle physics faster and more efficiently. Still, something of that picture resonates with me.

You might think someone who develops new mathematical methods would be a mathematician, not a physicist. But while there are mathematicians who work on the problems I work on, their goals are a bit different. They care about rigor, about stating only things they can carefully prove. As such, they often need to work with simplified examples, “toy models” well-suited to the kinds of theorems they can build.

Physicists can be a bit messier. We don’t always insist on the same rigor the mathematicians do. This makes our results less reliable, but it makes our “toy models” a fair amount less “toy”. Our goal is to try to tackle questions closer to the actual real world.

What happens when physicists and mathematicians work on the same problem?

If the physicists worked alone, they might build and build, and end up with an answer that isn’t actually true. The mathematicians, keeping rigor in mind, would be safe in the truth of what they built, but might not end up anywhere near the physicists’ real-world goals.

Together, though, physicists and mathematicians can build towards each other. The physicists can keep their eyes on the mathematicians, correcting when they notice something might go wrong and building more and more rigor into their approach. The mathematicians can keep their eyes on the physicists, building more and more complex applications of their rigorous approaches to get closer and closer to the real world. Eventually, like the transcontinental railroad, the two groups meet: the mathematicians prove a rigorous version of the physicists’ approach, or the physicists adopt the mathematicians’ ideas and apply them to their own theories.

A sort of conference photo

In practice, it isn’t just two teams, physicists and mathematicians, building towards each other. Different physicists themselves work with different levels of rigor, aiming to understand different problems in different theories, and the mathematicians do the same. Each of us is building our own track, watching the other tracks build towards us on the horizon. Eventually, we’ll meet, and science will chug along over what we’ve built.

At Geometries and Special Functions for Physics and Mathematics in Bonn

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I’m at a workshop this week. It’s part of a series of “Bethe Forums”, cozy little conferences run by the Bethe Center for Theoretical Physics in Bonn.

You can tell it’s an institute for theoretical physics because they have one of these, but not a “doing room”

The workshop’s title, “Geometries and Special Functions for Physics and Mathematics”, covers a wide range of topics. There are talks on Calabi-Yau manifolds, elliptic (and hyper-elliptic) polylogarithms, and cluster algebras and cluster polylogarithms. Some of the talks are by mathematicians, others by physicists.

In addition to the talks, this conference added a fun innovative element, “my favorite problem sessions”. The idea is that a speaker spends fifteen minutes introducing their “favorite problem”, then the audience spends fifteen minutes discussing it. Some treated these sessions roughly like short talks describing their work, with the open directions at the end framed as their favorite problem. Others aimed broader, trying to describe a general problem and motivate interest in people of other sub-fields.

This was a particularly fun conference for me, because the seemingly distinct topics all connect in one way or another to my own favorite problem. In our “favorite theory” of N=4 super Yang-Mills, we can describe our calculations in terms of an “alphabet” of pieces that let us figure out predictions almost “by guesswork”. These alphabets, at least in the cases we know how to handle, turn out to correspond to mathematical structures called cluster algebras. If we look at interactions of six or seven particles, these cluster algebras are a powerful guide. For eight or nine, they still seem to matter, but are much harder to use.

For ten particles, though, things get stranger. That’s because ten particles is precisely where elliptic curves, and their related elliptic polylogarithms, show up. Things then get yet more strange, and with twelve particles or more we start seeing Calabi-Yau manifolds magically show up in our calculations.

We don’t know what an “alphabet” should look like for these Calabi-Yau manifolds (but I’m working on it). Because of that, we don’t know how these cluster algebras should appear.

In my view, any explanation for the role of cluster algebras in our calculations has to extend to these cases, to elliptic polylogarithms and Calabi-Yau manifolds. Without knowing how to frame an alphabet for these things, we won’t be able to solve the lingering mysteries that fill our field.

Because of that, “my favorite problem” is one of my biggest motivations, the question that drives a large chunk of what I do. It’s what’s made this conference so much fun, and so stimulating: almost every talk had something I wanted to learn.

Talking and Teaching

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Someone recently shared with me an article written by David Mermin in 1992 about physics talks. Some aspects are dated (our slides are no longer sheets of plastic, and I don’t think anyone writing an article like that today would feel the need to put it in the mouth of a fictional professor (which is a shame honestly)), but most of it still holds true. I particularly recognized the self-doubt of being a young physicist sitting in a talk and thinking “I’m supposed to enjoy this?”

Mermin’s basic point is to keep things as light as possible. You want to convey motivation more than content, and background more than your own contributions. Slides should be sparse, both because people won’t be able to see everything but also because people can get frustrated “reading ahead” of what you say.

Mermin’s suggestion that people read from a prepared text was probably good advice for him, but maybe not for others. It can be good if you can write like he does, but I don’t think most people’s writing is that much better than what they say in talks (you can judge this by reading peoples’ papers!) Some are much clearer speaking impromptu. I agree with him that in practice people end up just reading from their slides, which indeed is bad, but reading from a normal physics paper isn’t any better.

I also don’t completely agree with him about the value of speech over text. Yes, putting text on your slides means people can read ahead (unless you hide some of the text, which is easier to do these days than in the days of overhead transparencies). But just saying things means that if someone’s attention lapses for just a moment, they’ll be lost. Unless you repeat yourself a lot (good practice in any case), you should avoid just saying anything you need your audience to remember, and make sure they can read it somewhere if they need it as well.

That said, “if they need it” is doing a lot of work here, and this is where I agree again with Mermin. Fundamentally, you don’t need to convey everything you think you do. (I don’t usually need to convey everything I think I do!) It’s a lesson I’ve been learning this year from pedagogy courses, a message they try to instill in everyone who teaches at the university. If you want to really convey something well, then you just can’t convey that much. You need to focus, pick a few things and try to get them across, and structure the rest of what you say to reinforce those things. When teaching, or when speaking, less is more.

On Stubbornness and Breaking Down

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In physics, we sometimes say that an idea “breaks down”. What do we mean by that?

When a theory “breaks down”, we mean that it stops being accurate. Newton’s theory of gravity is excellent most of the time, but for objects under strong enough gravity or high enough speed its predictions stop matching reality and a new theory (relativity) is needed. This is the sense in which we say that Newtonian gravity breaks down for the orbit of mercury, or breaks down much more severely in the area around a black hole.

When a symmetry is “broken”, we mean that it stops holding true. Most of physics looks the same when you flip it in a mirror, a property called parity symmetry. Take a pile of electric and magnetic fields, currents and wires, and you’ll find their mirror reflection is also a perfectly reasonable pile of electric and magnetic fields, currents and wires. This isn’t true for all of physics, though: the weak nuclear force isn’t the same when you flip it in a mirror. We say that the weak force breaks parity symmetry.

What about when a more general “idea” breaks down? What about space-time?

In order for space-time to break down, there needs to be a good reason to abandon the idea. And depending on how stubborn you are about it, that reason can come at different times.

You might think of space-time as just Einstein’s theory of general relativity. In that case, you could say that space-time breaks down as soon as the world deviates from that theory. In that view, any modification to general relativity, no matter how small, corresponds to space-time breaking down. You can think of this as the “least stubborn” option, the one with barely any stubbornness at all, that will let space-time break down with a tiny nudge.

But if general relativity breaks down, a slightly more stubborn person could insist that space-time is still fine. You can still describe things as located at specific places and times, moving across curved space-time. They just obey extra forces, on top of those built into the space-time.

Such a person would be happy as long as general relativity was a good approximation of what was going on, but they might admit space-time has broken down when general relativity becomes a bad approximation. If there are only small corrections on top of the usual space-time picture, then space-time would be fine, but if those corrections got so big that they overwhelmed the original predictions of general relativity then that’s quite a different situation. In that situation, space-time may have stopped being a useful description, and it may be much better to describe the world in another way.

But we could imagine an even more stubborn person who still insists that space-time is fine. Ultimately, our predictions about the world are mathematical formulas. No matter how complicated they are, we can always subtract a piece off of those formulas corresponding to the predictions of general relativity, and call the rest an extra effect. That may be a totally useless thing to do that doesn’t help you calculate anything, but someone could still do it, and thus insist that space-time still hasn’t broken down.

To convince such a person, space-time would need to break down in a way that made some important concept behind it invalid. There are various ways this could happen, corresponding to different concepts. For example, one unusual proposal is that space-time is non-commutative. If that were true then, in addition to the usual Heisenberg uncertainty principle between position and momentum, there would be an uncertainty principle between different directions in space-time. That would mean that you can’t define the position of something in all directions at once, which many people would agree is an important part of having a space-time!

Ultimately, physics is concerned with practicality. We want our concepts not just to be definable, but to do useful work in helping us understand the world. Our stubbornness should depend on whether a concept, like space-time, is still useful. If it is, we keep it. But if the situation changes, and another concept is more useful, then we can confidently say that space-time has broken down.

Visiting CERN

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So, would you believe I’ve never visited CERN before?

I was at CERN for a few days this week, visiting friends and collaborators and giving an impromptu talk. Surprisingly, this is the first time I’ve been, a bit of an embarrassing admission for someone who’s ostensibly a particle physicist.

Despite that, CERN felt oddly familiar. The maze of industrial buildings and winding roads, the security gates and cards (and work-arounds for when you arrive outside of card-issuing hours, assisted by friendly security guards), the constant construction and remodeling, all of it reminded me of the times I visited SLAC during my PhD. This makes a lot of sense, of course: one accelerator is at least somewhat like another. But besides a visit to Fermilab for a conference several years ago, I haven’t been in many other places like that since then.

(One thing that might have also been true of SLAC and Fermilab but I never noticed: CERN buildings not only have evacuation instructions for the building in case of a fire, but also evacuation instructions for the whole site.)

CERN is a bit less “pretty” than SLAC on average, without the nice grassy area in the middle or the California sun that goes with it. It makes up for it with what seems like more in terms of outreach resources, including a big wooden dome of a mini-museum sponsored by Rolex, and a larger visitor center still under construction.

The outside, including a sculpture depicting the history of science with the Higgs boson discovery on the “cutting edge”
The inside. Bubbles on the ground contain either touchscreens or small objects (detectors, papers, a blackboard with the string theory genus expansion for some reason). Bubbles in the air were too high for me to check.

CERN hosts a variety of theoretical physicists doing various different types of work. I was hosted by the “QCD group”, but the string theorists just down the hall include a few people I know as well. The lounge had a few cardboard signs hidden under the table, leftovers of CERN’s famous yearly Christmas play directed by John Ellis.

It’s been a fun, if brief, visit. I’ll likely get to see a bit more this summer, when they host Amplitudes 2023. Until then, it was fun reconnecting with that “accelerator feel”.