Category Archives: Amplitudes Methods

Jumpstarting Elliptic Bootstrapping

I was at a mini-conference this week, called Jumpstarting Elliptic Bootstrap Methods for Scattering Amplitudes.

I’ve done a lot of work with what we like to call “bootstrap” methods. Instead of doing a particle physics calculation in all its gory detail, we start with a plausible guess and impose requirements based on what we know. Eventually, we have the right answer pulled up “by its own bootstraps”: the only answer the calculation could have, without actually doing the calculation.

This method works very well, but so far it’s only been applied to certain kinds of calculations, involving mathematical functions called polylogarithms. More complicated calculations involve a mathematical object called an elliptic curve, and until very recently it wasn’t clear how to bootstrap them. To get people thinking about it, my colleagues Hjalte Frellesvig and Andrew McLeod asked the Carlsberg Foundation (yes, that Carlsberg) to fund a mini-conference. The idea was to get elliptic people and bootstrap people together (along with Hjalte’s tribe, intersection theory people) to hash things out. “Jumpstart people” are not a thing in physics, so despite the title they were not invited.

Anyone remember these games? Did you know that they still exist, have an educational MMO, and bought neopets?

Having the conference so soon after the yearly Elliptics meeting had some strange consequences. There was only one actual duplicate talk, but the first day of talks all felt like they would have been welcome additions to the earlier conference. Some might be functioning as “overflow”: Elliptics this year focused on discussion and so didn’t have many slots for talks, while this conference despite its discussion-focused goal had a more packed schedule. In other cases, people might have been persuaded by the more relaxed atmosphere and lack of recording or posted slides to give more speculative talks. Oliver Schlotterer’s talk was likely in this category, a discussion of the genus-two functions one step beyond elliptics that I think people at the previous conference would have found very exciting, but which involved work in progress that I could understand him being cautious about presenting.

The other days focused more on the bootstrap side, with progress on some surprising but not-quite-yet elliptic avenues. It was great to hear that Mark Spradlin is making new progress on his Ziggurat story, to hear James Drummond suggest a picture for cluster algebras that could generalize to other theories, and to get some idea of the mysterious ongoing story that animates my colleague Cristian Vergu.

There was one thing the organizers couldn’t have anticipated that ended up throwing the conference into a new light. The goal of the conference was to get people started bootstrapping elliptic functions, but in the meantime people have gotten started on their own. Roger Morales Espasa presented his work on this with several of my other colleagues. They can already reproduce a known result, the ten-particle elliptic double-box, and are well on-track to deriving something genuinely new, the twelve-particle version. It’s exciting, but it definitely makes the rest of us look around and take stock. Hopefully for the better!

Cabinet of Curiosities: The Nested Toy

I had a paper two weeks ago with a Master’s student, Alex Chaparro Pozo. I haven’t gotten a chance to talk about it yet, so I thought I should say a few words this week. It’s another entry in what I’ve been calling my cabinet of curiosities, interesting mathematical “objects” I’m sharing with the world.

I calculate scattering amplitudes, formulas that give the probability that particles scatter off each other in particular ways. While in principle I could do this with any particle physics theory, I have a favorite: a “toy model” called N=4 super Yang-Mills. N=4 super Yang-Mills doesn’t describe reality, but it lets us figure out cool new calculation tricks, and these often end up useful in reality as well.

Many scattering amplitudes in N=4 super Yang-Mills involve a type of mathematical functions called polylogarithms. These functions are especially easy to work with, but they aren’t the whole story. One we start considering more complicated situations (what if two particles collide, and eight particles come out?) we need more complicated functions, called elliptic polylogarithms.

A few years ago, some collaborators and I figured out how to calculate one of these elliptic scattering amplitudes. We didn’t do it as well as we’d like, though: the calculation was “half-done” in a sense. To do the other half, we needed new mathematical tools, tools that came out soon after. Once those tools were out, we started learning how to apply them, trying to “finish” the calculation we started.

The original calculation was pretty complicated. Two particles colliding, eight particles coming out, meant that in total we had to keep track of ten different particles. That gets messy fast. I’m pretty good at dealing with six particles, not ten. Luckily, it turned out there was a way to pretend there were six particles only: by “twisting” up the calculation, we found a toy model within the toy model: a six-particle version of the calculation. Much like the original was in a theory that doesn’t describe the real world, these six particles don’t describe six particles in that theory: they’re a kind of toy calculation within the toy model, doubly un-real.

Not quintuply-unreal though

With this nested toy model, I was confident we could do the calculation. I wasn’t confident I’d have time for it, though. This ended up making it perfect for a Master’s thesis, which is how Alex got into the game.

Alex worked his way through the calculation, programming and transforming, going from one type of mathematical functions to another (at least once because I’d forgotten to tell him the right functions to use, oops!) There were more details and subtleties than expected, but in the end everything worked out.

Then, we were scooped.

Another group figured out how to do the full, ten-particle problem, not just the toy model. That group was just “down the hall”…or would have been “down the hall” if we had been going to the office (this was 2021, after all). I didn’t hear about what they were working on until it was too late to change plans.

Alex left the field (not, as far as I know, because of this). And for a while, because of that especially thorough scooping, I didn’t publish.

What changed my mind, in part, was seeing the field develop in the meantime. It turns out toy models, and even nested toy models, are quite useful. We still have a lot of uncertainty about what to do, how to use the new calculation methods and what they imply. And usually, the best way to get through that kind of uncertainty is with simple, well-behaved toy models.

So I thought, in the end, that this might be useful. Even if it’s a toy version of something that already exists, I expect it to be an educational toy, one we can learn a lot from. So I’ve put it out into the world, as part of this year’s cabinet of curiosities.

At Elliptic Integrals in Fundamental Physics in Mainz

I’m at a conference this week. It’s named Elliptic Integrals in Fundamental Physics, but I think of it as “Elliptics 2022”, the latest in a series of conferences on elliptic integrals in particle physics.

It’s in Mainz, which you can tell from the Gutenberg street art

Elliptics has been growing in recent years, hurtling into prominence as a subfield of amplitudes (which is already a subfield of theoretical physics). This has led to growing lists of participants and a more and more packed schedule.

This year walked all of that back a bit. There were three talks a day: two one-hour talks by senior researchers and one half-hour talk by a junior researcher. The rest, as well as the whole last day, are geared to discussion. It’s an attempt to go back to the subfield’s roots. In the beginning, the Elliptics conferences drew together a small group to sort out a plan for the future, digging through the often-confusing mathematics to try to find a baseline for future progress. The field has advanced since then, but some of our questions are still almost as basic. What relations exist between different calculations? How much do we value fast numerics, versus analytical understanding? What methods do we want to preserve, and which aren’t serving us well? To answer these questions, it helps to get a few people together in one place, not to silently listen to lectures, but to question and discuss and hash things out. I may have heard a smaller range of topics at this year’s Elliptics, but due to the sheer depth we managed to probe on those fewer topics I feel like I’ve learned much more.

Since someone always asks, I should say that the talks were not recorded, but they are posting slides online, so if you’re interested in the topic you can watch there. A few people discussed new developments, some just published and some yet to be published. I discussed the work I talked about last week, and got a lot of good feedback and ideas about how to move forward.

Cabinet of Curiosities: The Coaction

I had two more papers out this week, continuing my cabinet of curiosities. I’ll talk about one of them today, and the other in (probably) two weeks.

This week, I’m talking about a paper I wrote with an excellent Master’s student, Andreas Forum. Andreas came to me looking for a project on the mathematical side. I had a rather nice idea for his project at first, to explain a proof in an old math paper so it could be used by physicists.

Unfortunately, the proof I sent him off to explain didn’t actually exist. Fortunately, by the time we figured this out Andreas had learned quite a bit of math, so he was ready for his next project: a coaction for Calabi-Yau Feynman diagrams.

We chose to focus on one particular diagram, called a sunrise diagram for its resemblance to a sun rising over the sea:

This diagram

Feynman diagrams depict paths traveled by particles. The paths are a metaphor, or organizing tool, for more complicated calculations: computations of the chances fundamental particles behave in different ways. Each diagram encodes a complicated integral. This one shows one particle splitting into many, then those many particles reuniting into one.

Do the integrals in Feynman diagrams, and you get a variety of different mathematical functions. Many of them integrate to functions called polylogarithms, and we’ve gotten really really good at working with them. We can integrate them up, simplify them, and sometimes we can guess them so well we don’t have to do the integrals at all! We can do all of that because we know how to break polylogarithm functions apart, with a mathematical operation called a coaction. The coaction chops polylogarithms up to simpler parts, parts that are easier to work with.

More complicated Feynman diagrams give more complicated functions, though. Some of them give what are called elliptic functions. You can think of these functions as involving a geometrical shape, in this case a torus.

Other functions involve more complicated geometrical shapes, in some cases very complicated. For example, some involve the Calabi-Yau manifolds studied by string theorists. These sunrise diagrams are some of the simplest to involve such complicated geometry.

Other researchers had proposed a coaction for elliptic functions back in 2018. When they derived it, though, they left a recipe for something more general. Follow the instructions in the paper, and you could in principle find a coaction for other diagrams, even the Calabi-Yau ones, if you set it up right.

I had an idea for how to set it up right, and in the grand tradition of supervisors everywhere I got Andreas to do the dirty work of applying it. Despite the delay of our false start and despite the fact that this was probably in retrospect too big a project for a normal Master’s thesis, Andreas made it work!

Our result, though, is a bit weird. The coaction is a powerful tool for polylogarithms because it chops them up finely: keep chopping, and you get down to very simple functions. Our coaction isn’t quite so fine: we don’t chop our functions into as many parts, and the parts are more mysterious, more difficult to handle.

We think these are temporary problems though. The recipe we applied turns out to be a recipe with a lot of choices to make, less like Julia Child and more like one of those books where you mix-and-match recipes. We believe the community can play with the parameters of this recipe, finding new version of the coaction for new uses.

This is one of the shiniest of the curiosities in my cabinet this year, I hope it gets put to good use.

Cabinet of Curiosities: The Cubic

Before I launch into the post: I got interviewed on Theoretically Podcasting, a new YouTube channel focused on beginning grad student-level explanations of topics in theoretical physics. If that sounds interesting to you, check it out!

This Fall is paper season for me. I’m finishing up a number of different projects, on a number of different things. Each one was its own puzzle: a curious object found, polished, and sent off into the world.

Monday I published the first of these curiosities, along with Jake Bourjaily and Cristian Vergu.

I’ve mentioned before that the calculations I do involve a kind of “alphabet“. Break down a formula for the probability that two particles collide, and you find pieces that occur again and again. In the nicest cases, those pieces are rational functions, but they can easily get more complicated. I’ve talked before about a case where square roots enter the game, for example. But if square roots appear, what about something even more complicated? What about cubic roots?

What about 1024th roots?

Occasionally, my co-authors and I would say something like that at the end of a talk and an older professor would scoff: “Cube roots? Impossible!”

You might imagine these professors were just being unreasonable skeptics, the elderly-but-distinguished scientists from that Arthur C. Clarke quote. But while they turned out to be wrong, they weren’t being unreasonable. They were thinking back to theorems from the 60’s, theorems which seemed to argue that these particle physics calculations could only have a few specific kinds of behavior: they could behave like rational functions, like logarithms, or like square roots. Theorems which, as they understood them, would have made our claims impossible.

Eventually, we decided to figure out what the heck was going on here. We grabbed the simplest example we could find (a cube root involving three loops and eleven gluons in N=4 super Yang-Mills…yeah) and buckled down to do the calculation.

When we want to calculate something specific to our field, we can reference textbooks and papers, and draw on our own experience. Much of the calculation was like that. A crucial piece, though, involved something quite a bit less specific: calculating a cubic root. And for things like that, you can tell your teachers we use only the very best: Wikipedia.

Check out the Wikipedia entry for the cubic formula. It’s complicated, in ways the quadratic formula isn’t. It involves complex numbers, for one. But it’s not that crazy.

What those theorems from the 60’s said (and what they actually said, not what people misremembered them as saying), was that you can’t take a single limit of a particle physics calculation, and have it behave like a cubic root. You need to take more limits, not just one, to see it.

It turns out, you can even see this just from the Wikipedia entry. There’s a big cube root sign in the middle there, equal to some variable “C”. Look at what’s inside that cube root. You want that part inside to vanish. That means two things need to cancel: Wikipedia labels them \Delta_1, and \sqrt{\Delta_1^2-4\Delta_0^3}. Do some algebra, and you’ll see that for those to cancel, you need \Delta_0=0.

So you look at the limit, \Delta_0\rightarrow 0. This time you need not just some algebra, but some calculus. I’ll let the students in the audience work it out, but at the end of the day, you should notice how C behaves when \Delta_0 is small. It isn’t like \sqrt[3]{\Delta_0}. It’s like just plain \Delta_0. The cube root goes away.

It can come back, but only if you take another limit: not just \Delta_0\rightarrow 0, but \Delta_1\rightarrow 0 as well. And that’s just fine according to those theorems from the 60’s. So our cubic curiosity isn’t impossible after all.

Our calculation wasn’t quite this simple, of course. We had to close a few loopholes, checking our example in detail using more than just Wikipedia-based methods. We found what we thought was a toy example, that turned out to be even more complicated, involving roots of a degree-six polynomial (one that has no “formula”!).

And in the end, polished and in their display case, we’ve put our examples up for the world to see. Let’s see what people think of them!

Why the Antipode Was Supposed to Be Useless

A few weeks back, Quanta Magazine had an article about a new discovery in my field, called antipodal duality.

Some background: I’m a theoretical physicist, and I work on finding better ways to make predictions in particle physics. Folks in my field make these predictions with formulas called “scattering amplitudes” that encode the probability that particles bounce, or scatter, in particular ways. One trick we’ve found is that these formulas can often be written as “words” in a kind of “alphabet”. If we know the alphabet, we can make our formulas much simpler, or even guess formulas we could never have calculated any other way.

Quanta’s article describes how a few friends of mine (Lance Dixon, Ömer Gürdoğan, Andrew McLeod, and Matthias Wilhelm) noticed a weird pattern in two of these formulas, from two different calculations. If you flip the “words” around, back to front (an operation called the antipode), you go from a formula describing one collision of particles to a formula for totally different particles. Somehow, the two calculations are “dual”: two different-seeming descriptions that secretly mean the same thing.

Quanta quoted me for their article, and I was (pleasantly) baffled. See, the antipode was supposed to be useless. The mathematicians told us it was something the math allows us to do, like you’re allowed to order pineapple on pizza. But just like pineapple on pizza, we couldn’t imagine a situation where we actually wanted to do it.

What Quanta didn’t say was why we thought the antipode was useless. That’s a hard story to tell, one that wouldn’t fit in a piece like that.

It fits here, though. So in the rest of this post, I’d like to explain why flipping around words is such a strange, seemingly useless thing to do. It’s strange because it swaps two things that in physics we thought should be independent: branch cuts and derivatives, or particles and symmetries.

Let’s start with the first things in each pair: branch cuts, and particles.

The first few letters of our “word” tell us something mathematical, and they tell us something physical. Mathematically, they tell us ways that our formula can change suddenly, and discontinuously.

Take a logarithm, the inverse of e^x. You’re probably used to plugging in positive numbers, and getting out something reasonable, that changes in a smooth and regular way: after all, e^x is always positive, right? But in mathematics, you don’t have to just use positive numbers. You can use negative numbers. Even more interestingly, you can use complex numbers. And if you take the logarithm of a complex number, and look at the imaginary part, it looks like this:

Mostly, this complex logarithm still seems to be doing what it’s supposed to, changing in a nice slow way. But there is a weird “cut” in the graph for negative numbers: a sudden jump, from \pi to -\pi. That jump is called a “branch cut”.

As physicists, we usually don’t like our formulas to make sudden changes. A change like this is an infinitely fast jump, and we don’t like infinities much either. But we do have one good use for a formula like this, because sometimes our formulas do change suddenly: when we have enough energy to make a new particle.

Imagine colliding two protons together, like at the LHC. Colliding particles doesn’t just break the protons into pieces: due to Einstein’s famous E=mc^2, it can create new particles as well. But to create a new particle, you need enough energy: mc^2 worth of energy. So as you dial up the energy of your protons, you’ll notice a sudden change: you couldn’t create, say, a Higgs boson, and now you can. Our formulas represent some of those kinds of sudden changes with branch cuts.

So the beginning of our “words” represent branch cuts, and particles. The end represents derivatives and symmetries.

Derivatives come from the land of calculus, a place spooky to those with traumatic math class memories. Derivatives shouldn’t be so spooky though. They’re just ways we measure change. If we have a formula that is smoothly changing as we change some input, we can describe that change with a derivative.

The ending of our “words” tell us what happens when we take a derivative. They tell us which ways our formulas can smoothly change, and what happens when they do.

In doing so, they tell us about something some physicists make sound spooky, called symmetries. Symmetries are changes we can make that don’t really change what’s important. For example, you could imagine lifting up the entire Large Hadron Collider and (carefully!) carrying it across the ocean, from France to the US. We’d expect that, once all the scared scientists return and turn it back on, it would start getting exactly the same results. Physics has “translation symmetry”: you can move, or “translate” an experiment, and the important stuff stays the same.

These symmetries are closely connected to derivatives. If changing something doesn’t change anything important, that should be reflected in our formulas: they shouldn’t change either, so their derivatives should be zero. If instead the symmetry isn’t quite true, if it’s what we call “broken”, then by knowing how it was “broken” we know what the derivative should be.

So branch cuts tell us about particles, derivatives tell us about symmetries. The weird thing about the antipode, the un-physical bizarre thing, is that it swaps them. It makes the particles of one calculation determine the symmetries of another.

(And lest you’ve heard about particles with symmetries, like gluons and SU(3)…this is a different kind of thing. I don’t have enough room to explain why here, but it’s completely unrelated.)

Why the heck does this duality exist?

A commenter on the last post asked me to speculate. I said there that I have no clue, and that’s most of the answer.

If I had to speculate, though, my answer might be disappointing.

Most of the things in physics we call “dualities” have fairly deep physical meanings, linked to twisting spacetime in complicated ways. AdS/CFT isn’t fully explained, but it seems to be related to something called the holographic principle, the idea that gravity ties together the inside of space with the boundary around it. T duality, an older concept in string theory, is explained: a consequence of how strings “see” the world in terms of things to wrap around and things to spin around. In my field, one of our favorite dualities links back to this as well, amplitude-Wilson loop duality linked to fermionic T-duality.

The antipode doesn’t twist spacetime, it twists the mathematics. And it may be it matters only because the mathematics is so constrained that it’s forced to happen.

The trick that Lance Dixon and co. used to discover antipodal duality is the same trick I used with Lance to calculate complicated scattering amplitudes. It relies on taking a general guess of words in the right “alphabet”, and constraining it: using mathematical and physical principles it must obey and throwing out every illegal answer until there’s only one answer left.

Currently, there are some hints that the principles used for the different calculations linked by antipodal duality are “antipodal mirrors” of each other: that different principles have the same implication when the duality “flips” them around. If so, then it could be this duality is in some sense just a coincidence: not a coincidence limited to a few calculations, but a coincidence limited to a few principles. Thought of in this way, it might not tell us a lot about other situations, it might not really be “deep”.

Of course, I could be wrong about this. It could be much more general, could mean much more. But in that context, I really have no clue what to speculate. The antipode is weird: it links things that really should not be physically linked. We’ll have to see what that actually means.

Amplitudes 2022 Retrospective

I’m back from Amplitudes 2022 with more time to write, and (besides the several papers I’m working on) that means writing about the conference! Casual readers be warned, there’s no way around this being a technical post, I don’t have the space to explain everything!

I mostly said all I wanted about the way the conference was set up in last week’s post, but one thing I didn’t say much about was the conference dinner. Most conference dinners are the same aside from the occasional cool location or haggis speech. This one did have a cool location, and a cool performance by a blind pianist, but the thing I really wanted to comment on was the setup. Typically, the conference dinner at Amplitudes is a sit-down affair: people sit at tables in one big room, maybe getting up occasionally to pick up food, and eventually someone gives an after-dinner speech. This time the tables were standing tables, spread across several rooms. This was a bit tiring on a hot day, but it did have the advantage that it naturally mixed people around. Rather than mostly talking to “your table”, you’d wander, ending up at a new table every time you picked up new food or drinks. It was a good way to meet new people, a surprising number of which in my case apparently read this blog. It did make it harder to do an after-dinner speech, so instead Lance gave an after-conference speech, complete with the now-well-established running joke where Greta Thunberg tries to get us to fly less.

(In another semi-running joke, the organizers tried to figure out who had attended the most of the yearly Amplitudes conferences over the years. Weirdly, no-one has attended all twelve.)

In terms of the content, and things that stood out:

Nima is getting close to publishing his newest ‘hedron, the surfacehedron, and correspondingly was able to give a lot more technical detail about it. (For his first and most famous amplituhedron, see here.) He still didn’t have enough time to explain why he has to use category theory to do it, but at least he was concrete enough that it was reasonably clear where the category theory was showing up. (I wasn’t there for his eight-hour lecture at the school the week before, maybe the students who stuck around until 2am learned some category theory there.) Just from listening in on side discussions, I got the impression that some of the ideas here actually may have near-term applications to computing Feynman diagrams: this hasn’t been a feature of previous ‘hedra and it’s an encouraging development.

Alex Edison talked about progress towards this blog’s namesake problem, the question of whether N=8 supergravity diverges at seven loops. Currently they’re working at six loops on the N=4 super Yang-Mills side, not yet in a form it can be “double-copied” to supergravity. The tools they’re using are increasingly sophisticated, including various slick tricks from algebraic geometry. They are looking to the future: if they’re hoping their methods will reach seven loops, the same methods have to make six loops a breeze.

Xi Yin approached a puzzle with methods from String Field Theory, prompting the heretical-for-us title “on-shell bad, off-shell good”. A colleague reminded me of a local tradition for dealing with heretics.

While Nima was talking about a new ‘hedron, other talks focused on the original amplituhedron. Paul Heslop found that the amplituhedron is not literally a positive geometry, despite slogans to the contrary, but what it is is nonetheless an interesting generalization of the concept. Livia Ferro has made more progress on her group’s momentum amplituhedron: previously only valid at tree level, they now have a picture that can accomodate loops. I wasn’t sure this would be possible, there are a lot of things that work at tree level and not for loops, so I’m quite encouraged that this one made the leap successfully.

Sebastian Mizera, Andrew McLeod, and Hofie Hannesdottir all had talks that could be roughly summarized as “deep principles made surprisingly useful”. Each took topics that were explored in the 60’s and translated them into concrete techniques that could be applied to modern problems. There were surprisingly few talks on the completely concrete end, on direct applications to collider physics. I think Simone Zoia’s was the only one to actually feature collider data with error bars, which might explain why I singled him out to ask about those error bars later.

Likewise, Matthias Wilhelm’s talk was the only one on functions beyond polylogarithms, the elliptic functions I’ve also worked on recently. I wonder if the under-representation of some of these topics is due to the existence of independent conferences: in a year when in-person conferences are packed in after being postponed across the pandemic, when there are already dedicated conferences for elliptics and practical collider calculations, maybe people are just a bit too tired to go to Amplitudes as well.

Talks on gravitational waves seem to have stabilized at roughly a day’s worth, which seems reasonable. While the subfield’s capabilities continue to be impressive, it’s also interesting how often new conceptual challenges appear. It seems like every time a challenge to their results or methods is resolved, a new one shows up. I don’t know whether the field will ever get to a stage of “business as usual”, or whether it will be novel qualitative questions “all the way up”.

I haven’t said much about the variety of talks bounding EFTs and investigating their structure, though this continues to be an important topic. And I haven’t mentioned Lance Dixon’s talk on antipodal duality, largely because I’m planning a post on it later: Quanta Magazine had a good article on it, but there are some aspects even Quanta struggled to cover, and I think I might have a good way to do it.

At Amplitudes 2022 in Prague

It’s that time of year again! I’m at the big yearly conference of my subfield, Amplitudes, this year in Prague.

The conference poster included a picture of Prague’s famous clock, which is admittedly cool. But I think this computer-generated anachronism from Matt Schwartz’s machine learning talk is much more fun.

Amplitudes has grown, and keeps growing. The last time we met in person, there were 175 of us. This year, many people are skipping: some avoiding travel due to COVID, others just exhausted from a summer filled with long-postponed conferences. Nonetheless, we have more people here than then: 222 registered participants!

The large number of people means a large number of talks. Almost all were quite short, 25+5 minutes. Some speakers took advantage of the short length to deliver very accessible talks. Others seemed to think of the time limit as an excuse to cut short the introduction and dive right into technical details. We had just a few 40+5 minute talks, each a review from an adjacent field.

It’s been fun seeing people in person again. I think half of my conversations started with “It’s been a long time!” It’s easy for motivation to wane when you don’t have regular contact with the wider field, getting enthusiastic about shared goals and brainstorming big questions.

I’ll probably give a longer retrospective later: the packed schedule means I don’t have much time to write! But I can say that I’ve largely enjoyed this, the organizers were organized and the presenters presented and things felt a bit more like they ought to in the world.

Carving Out the Possible

If you imagine a particle physicist, you probably picture someone spending their whole day dreaming up new particles. They figure out how to test those particles in some big particle collider, and for a lucky few their particle gets discovered and they get a Nobel prize.

Occasionally, a wiseguy asks if we can’t just cut out the middleman. Instead of dreaming up particles to test, why don’t we just write down every possible particle and test for all of them? It would save the Nobel committee a lot of money at least!

It turns out, you can sort of do this, through something called Effective Field Theory. An Effective Field Theory is a type of particle physics theory that isn’t quite true: instead, it’s “effectively” true, meaning true as long as you don’t push it too far. If you test it at low energies and don’t “zoom in” too much then it’s fine. Crank up your collider energy high enough, though, and you expect the theory to “break down”, revealing new particles. An Effective Field Theory lets you “hide” unknown particles inside new interactions between the particles we already know.

To help you picture how this works, imagine that the pink and blue lines here represent familiar particles like electrons and quarks, while the dotted line is a new particle somebody dreamed up. (The picture is called a Feynman diagram, if you don’t know what that is check out this post.)

In an Effective Field Theory, we “zoom out”, until the diagram looks like this:

Now we’ve “hidden” the new particle. Instead, we have a new type of interaction between the particles we already know.

So instead of writing down every possible new particle we can imagine, we only have to write down every possible interaction between the particles we already know.

That’s not as hard as it sounds. In part, that’s because not every interaction actually makes sense. Some of the things you could write down break some important rules. They might screw up cause and effect, letting something happen before its cause instead of after. They might screw up probability, giving you a formula for the chance something happens that gives a number greater than 100%.

Using these rules you can play a kind of game. You start out with a space representing all of the interactions you can imagine. You begin chipping at it, carving away parts that don’t obey the rules, and you see what shape is left over. You end up with plots that look a bit like carving a ham.

People in my subfield are getting good at this kind of game. It isn’t quite our standard fare: usually, we come up with tricks to make calculations with specific theories easier. Instead, many groups are starting to look at these general, effective theories. We’ve made friends with groups in related fields, building new collaborations. There still isn’t one clear best way to do this carving, so each group manages to find a way to chip a little farther. Out of the block of every theory we could imagine, we’re carving out a space of theories that make sense, theories that could conceivably be right. Theories that are worth testing.

Geometry and Geometry

Last week, I gave the opening lectures for a course on scattering amplitudes, the things we compute to find probabilities in particle physics. After the first class, one of the students asked me if two different descriptions of these amplitudes, one called CHY and the other called the amplituhedron, were related. There does happen to be a connection, but it’s a bit subtle and indirect, not the sort of thing the student would have been thinking of. Why then, did he think they might be related? Well, he explained, both descriptions are geometric.

If you’ve been following this blog for a while, you’ve seen me talk about misunderstandings. There are a lot of subtle ways a smart student can misunderstand something, ways that can be hard for a teacher to recognize. The right question, or the right explanation, can reveal what’s going on. Here, I think the problem was that there are multiple meanings of geometry.

One of the descriptions the student asked about, CHY, is related to string theory. It describes scattering particles in terms of the path of a length of string through space and time. That path draws out a surface called a world-sheet, showing all the places the string touches on its journey. And that picture, of a wiggly surface drawn in space and time, looks like what most people think of as geometry: a “shape” in a pretty normal sense, which here describes the physics of scattering particles.

The other description, the amplituhedron, also uses geometric objects to describe scattering particles. But the “geometric objects” here are much more abstract. A few of them are familiar: straight lines, the area between them forming shapes on a plane. Most of them, though are generalizations of this: instead of lines on a plane, they have higher dimensional planes in higher dimensional spaces. These too get described as geometry, even though they aren’t the “everyday” geometry you might be familiar with. Instead, they’re a “natural generalization”, something that, once you know the math, is close enough to that “everyday” geometry that it deserves the same name.

This week, two papers presented a totally different kind of geometric description of particle physics. In those papers, “geometric” has to do with differential geometry, the mathematics behind Einstein’s theory of general relativity. The descriptions are geometric because they use the same kinds of building-blocks of that theory, a metric that bends space and time. Once again, this kind of geometry is a natural generalization of the everyday notion, but now in once again a different way.

All of these notions of geometry do have some things in common, of course. Maybe you could even write down a definition of “geometry” that includes all of them. But they’re different enough that if I tell you that two descriptions are “geometric”, it doesn’t tell you all that much. It definitely doesn’t tell you the two descriptions are related.

It’s a reasonable misunderstanding, though. It comes from a place where, used to “everyday” geometry, you expect two “geometric descriptions” of something to be similar: shapes moving in everyday space, things you can directly compare. Instead, a geometric description can be many sorts of shape, in many sorts of spaces, emphasizing many sorts of properties. “Geometry” is just a really broad term.