# What Do Theorists Do at Work?

Picture a scientist at work. You’re probably picturing an experiment, test tubes and beakers bubbling away. But not all scientists do experiments. Theoretical physicists work on the mathematical side of the field, making predictions and trying to understand how to make them better. So what does it look like when a theoretical physicist is working?

The first thing you might imagine is that we just sit and think. While that happens sometimes, we don’t actually do that very often. It’s better, and easier, to think by doing something.

Sometimes, this means working with pen and paper. This should be at least a little familiar to anyone who has done math homework. We’ll do short calculations and draw quick diagrams to test ideas, and do a more detailed, organized, “show your work” calculation if we’re trying to figure out something more complicated. Sometimes very short calculations are done on a blackboard instead, it can help us visualize what we’re doing.

Sometimes, we use computers instead. There are computer algebra packages, like Mathematica, Maple, or Sage, that let us do roughly what we would do on pen and paper, but with the speed and efficiency of a computer. Others program in more normal programming languages: C++, Python, even Fortran, making programs that can calculate whatever they are interested in.

Sometimes we read. With most of our field’s papers available for free on arXiv.org, we spend time reading up on what our colleagues have done, trying to understand their work and use it to improve ours.

Sometimes we talk. A paper can only communicate so much, and sometimes it’s better to just walk down the hall and ask a question. Conversations are also a good way to quickly rule out bad ideas, and narrow down to the promising ones. Some people find it easier to think clearly about something if they talk to a colleague about it, even (sometimes especially) if the colleague isn’t understanding much.

And sometimes, of course, we do all the other stuff. We write up our papers, making the diagrams nice and the formulas clean. We teach students. We go to meetings. We write grant applications.

It’s been said that a theoretical physicist can work anywhere. That’s kind of true. Some places are more comfortable, and everyone has different preferences: a busy office, a quiet room, a cafe. But with pen and paper, a computer, and people to talk to, we can do quite a lot.

I build tools, mathematical tools to be specific, and I want those tools to be useful. I want them to be used to study the real world. But when I build those tools, most of the time, I don’t test them on the real world. I use toy models, simpler cases, theories that don’t describe reality and weren’t intended to.

I do this, in part, because it lets me stay one step ahead. I can do more with those toy models, answer more complicated questions with greater precision, than I can for the real world. I can do more ambitious calculations, and still get an answer. And by doing those calculations, I can start to anticipate problems that will crop up for the real world too. Even if we can’t do a calculation yet for the real world, if it requires too much precision or too many particles, we can still study it in a toy model. Then when we’re ready to do those calculations in the real world, we know better what to expect. The toy model will have shown us some of the key challenges, and how to tackle them.

There’s a risk, working with simpler toy models. The risk is that their simplicity misleads you. When you solve a problem in a toy model, could you solve it only because the toy model is easy? Or would a similar solution work in the real world? What features of the toy model did you need, and which are extra?

The only way around this risk is to be careful. You have to keep track of how your toy model differs from the real world. You must keep in mind difficulties that come up on the road to reality: the twists and turns and potholes that real-world theories will give you. You can’t plan around all of them, that’s why you’re working with a toy model in the first place. But for a few key, important ones, you should keep your eye on the horizon. You should keep in mind that, eventually, the simplifications of the toy model will go away. And you should have ideas, perhaps not full plans but at least ideas, for how to handle some of those difficulties. If you put the work in, you stand a good chance of building something that’s useful, not just for toy models, but for explaining the real world.

# Why You Might Want to Bootstrap

A few weeks back, Quanta Magazine had an article about attempts to “bootstrap” the laws of physics, starting from simple physical principles and pulling out a full theory “by its own bootstraps”. This kind of work is a cornerstone of my field, a shared philosophy that motivates a lot of what we do. Building on deep older results, people in my field have found that just a few simple principles are enough to pick out specific physical theories.

There are limits to this. These principles pick out broad traits of theories: gravity versus the strong force versus the Higgs boson. As far as we know they don’t separate more closely related forces, like the strong nuclear force and the weak nuclear force. (Originally, the Quanta article accidentally made it sound like we know why there are four fundamental forces: we don’t, and the article’s phrasing was corrected.) More generally, a bootstrap method isn’t going to tell you which principles are the right ones. For any set of principles, you can always ask “why?”

With that in mind, why would you want to bootstrap?

First, it can make your life simpler. Those simple physical principles may be clear at the end, but they aren’t always obvious at the start of a calculation. If you don’t make good use of them, you might find you’re calculating many things that violate those principles, things that in the end all add up to zero. Bootstrapping can let you skip that part of the calculation, and sometimes go straight to the answer.

Second, it can suggest possibilities you hadn’t considered. Sometimes, your simple physical principles don’t select a unique theory. Some of the options will be theories you’ve heard of, but some might be theories that never would have come up, or even theories that are entirely new. Trying to understand the new theories, to see whether they make sense and are useful, can lead to discovering new principles as well.

Finally, even if you don’t know which principles are the right ones, some principles are better than others. If there is an ultimate theory that describes the real world, it can’t be logically inconsistent. That’s a start, but it’s quite a weak requirement. There are principles that aren’t required by logic itself, but that still seem important in making the world “make sense”. Often, we appreciate these principles only after we’ve seen them at work in the real world. The best example I can think of is relativity: while Newtonian mechanics is logically consistent, it requires a preferred reference frame, a fixed notion for which things are moving and which things are still. This seemed reasonable for a long time, but now that we understand relativity the idea of a preferred reference frame seems like it should have been obviously wrong. It introduces something arbitrary into the laws of the universe, a “why is it that way?” question that doesn’t have an answer. That doesn’t mean it’s logically inconsistent, or impossible, but it does make it suspect in a way other ideas aren’t. Part of the hope of these kinds of bootstrap methods is that they uncover principles like that, principles that aren’t mandatory but that are still in some sense “obvious”. Hopefully, enough principles like that really do specify the laws of physics. And if they don’t, we’ll at least have learned how to calculate better.

# Calculating the Hard Way, for Science!

I had a new paper out last week, with Jacob Bourjaily and Matthias Volk. We’re calculating the probability that particles bounce off each other in our favorite toy model, N=4 super Yang-Mills. And this time, we’re doing it the hard way.

The “easy way” we didn’t take is one I have a lot of experience with. Almost as long as I’ve been writing this blog, I’ve been calculating these particle probabilities by “guesswork”: starting with a plausible answer, then honing it down until I can be confident it’s right. This might sound reckless, but it works remarkably well, letting us calculate things we could never have hoped for with other methods. The catch is that “guessing” is much easier when we know what we’re looking for: in particular, it works much better in toy models than in the real world.

Over the last few years, though, I’ve been using a much more “normal” method, one that so far has a better track record in the real world. This method, too, works better than you would expect, and we’ve managed some quite complicated calculations.

So we have an “easy way”, and a “hard way”. Which one is better? Is the hard way actually harder?

To test that, you need to do the same calculation both ways, and see which is easier. You want it to be a fair test: if “guessing” only works in the toy model, then you should do the “hard” version in the toy model as well. And you don’t want to give “guessing” any unfair advantages. In particular, the “guess” method works best when we know a lot about the result we’re looking for: what it’s made of, what symmetries it has. In order to do a fair test, we must use that knowledge to its fullest to improve the “hard way” as well.

We picked an example in the middle: not too easy, and not too hard, a calculation that was done a few years back “the easy way” but not yet done “the hard way”. We plugged in all the modern tricks we could, trying to use as much of what we knew as possible. We trained a grad student: Matthias Volk, who did the lion’s share of the calculation and learned a lot in the process. We worked through the calculation, and did it properly the hard way.

Which method won?

In the end, the hard way was indeed harder…but not by that much! Most of the calculation went quite smoothly, with only a few difficulties at the end. Just five years ago, when the calculation was done “the easy way”, I doubt anyone would have expected the hard way to be viable. But with modern tricks it wasn’t actually that hard.

This is encouraging. It tells us that the “hard way” has potential, that it’s almost good enough to compete at this kind of calculation. It tells us that the “easy way” is still quite powerful. And it reminds us that the more we know, and the more we apply our knowledge, the more we can do.

# QCD Meets Gravity 2019

I’m at UCLA this week for QCD Meets Gravity, a conference about the surprising ways that gravity is “QCD squared”.

When I attended this conference two years ago, the community was branching out into a new direction: using tools from particle physics to understand the gravitational waves observed at LIGO.

At this year’s conference, gravitational waves have grown from a promising new direction to a large fraction of the talks. While there were still the usual talks about quantum field theory and string theory (everything from bootstrap methods to a surprising application of double field theory), gravitational waves have clearly become a major focus of this community.

This was highlighted before the first talk, when Zvi Bern brought up a recent paper by Thibault Damour. Bern and collaborators had recently used particle physics methods to push beyond the state of the art in gravitational wave calculations. Damour, an expert in the older methods, claims that Bern et al’s result is wrong, and in doing so also questions an earlier result by Amati, Ciafaloni, and Veneziano. More than that, Damour argued that the whole approach of using these kinds of particle physics tools for gravitational waves is misguided.

There was a lot of good-natured ribbing of Damour in the rest of the conference, as well as some serious attempts to confront his points. Damour’s argument so far is somewhat indirect, so there is hope that a more direct calculation (which Damour is currently pursuing) will resolve the matter. In the meantime, Julio Parra-Martinez described a reproduction of the older Amati/Ciafaloni/Veneziano result with more Damour-approved techniques, as well as additional indirect arguments that Bern et al got things right.

Before the QCD Meets Gravity community worked on gravitational waves, other groups had already built a strong track record in the area. One encouraging thing about this conference was how much the two communities are talking to each other. Several speakers came from the older community, and there were a lot of references in both groups’ talks to the other group’s work. This, more than even the content of the talks, felt like the strongest sign that something productive is happening here.

Many talks began by trying to motivate these gravitational calculations, usually to address the mysteries of astrophysics. Two talks were more direct, with Ramy Brustein and Pierre Vanhove speculating about new fundamental physics that could be uncovered by these calculations. I’m not the kind of physicist who does this kind of speculation, and I confess both talks struck me as rather strange. Vanhove in particular explicitly rejects the popular criterion of “naturalness”, making me wonder if his work is the kind of thing critics of naturalness have in mind.

I have a new paper out today, with Jacob Bourjaily, Andrew McLeod, Matthias Wilhelm, Cristian Vergu and Matthias Volk.

There’s a story I’ve told before on this blog, about a kind of “alphabet” for particle physics predictions. When we try to make a prediction in particle physics, we need to do complicated integrals. Sometimes, these integrals simplify dramatically, in unexpected ways. It turns out we can understand these simplifications by writing the integrals in a sort of “alphabet”, breaking complicated mathematical “periods” into familiar logarithms. If we want to simplify an integral, we can use relations between logarithms like these:

$\log(a b)=\log(a)+\log(b),\quad \log(a^n)=n\log(a)$

to factor our “alphabet” into pieces as simple as possible.

The simpler the alphabet, the more progress you can make. And in the nice toy model theory we’re working with, the alphabets so far have been simple in one key way. Expressed in the right variables, they’re rational. For example, they contain no square roots.

Would that keep going? Would we keep finding rational alphabets? Or might the alphabets, instead, have square roots?

After some searching, we found a clean test case. There was a calculation we could do with just two Feynman diagrams. All we had to do was subtract one from the other. If they still had square roots in their alphabet, we’d have proven that the nice, rational alphabets eventually had to stop.

So we calculated these diagrams, doing the complicated integrals. And we found they did indeed have square roots in their alphabet, in fact many more than expected. They even had square roots of square roots!

You’d think that would be the end of the story. But square roots are trickier than you’d expect.

Remember that to simplify these integrals, we break them up into an alphabet, and factor the alphabet. What happens when we try to do that with an alphabet that has square roots?

Suppose we have letters in our alphabet with $\sqrt{-5}$. Suppose another letter is the number 9. You might want to factor it like this:

$9=3\times 3$

Simple, right? But what if instead you did this:

$9=(2+ \sqrt{-5} )\times(2- \sqrt{-5} )$

Once you allow $\sqrt{-5}$ in the game, you can factor 9 in two different ways. The central assumption, that you can always just factor your alphabet, breaks down. In mathematical terms, you no longer have a unique factorization domain.

Instead, we had to get a lot more mathematically sophisticated, factoring into something called prime ideals. We got that working and started crunching through the square roots in our alphabet. Things simplified beautifully: we started with a result that was ten million terms long, and reduced it to just five thousand. And at the end of the day, after subtracting one integral from the other…

We found no square roots!

After all of our simplifications, all the letters we found were rational. Our nice test case turned out much, much simpler than we expected.

It’s been a long road on this calculation, with a lot of false starts. We were kind of hoping to be the first to find square root letters in these alphabets; instead it looks like another group will beat us to the punch. But we developed a lot of interesting tricks along the way, and we thought it would be good to publish our “null result”. As always in our field, sometimes surprising simplifications are just around the corner.

# Knowing When to Hold/Fold ‘Em in Science

The things one learns from Wikipedia. For example, today I learned that the country song “The Gambler” was selected for preservation by the US Library of Congress as being “culturally, historically, or artistically significant.”

You’ve got to know when to hold ’em, know when to fold ’em,

Know when to walk away, know when to run.

Knowing when to “hold ’em” or “fold ’em” is important in life in general, but it’s particularly important in science.

As scientists, we’re often trying to do something no-one else has done before. That’s exciting, but it’s risky too: sometimes whatever we’re trying simply doesn’t work. In those situations, it’s important to recognize when we aren’t making progress, and change tactics. The trick is, we can’t give up too early either: science is genuinely hard, and sometimes when we feel stuck we’re actually close to the finish line. Knowing which is which, when to “hold” and when to “fold”, is an essential skill, and a hard one to learn.

Sometimes, we can figure this out mathematically. Computational complexity theory classifies calculations by how difficult they are, including how long they take. If you can estimate how much time you should take to do a calculation, you can decide whether you’ll finish it in a reasonable amount of time. If you just want a rough guess, you can do a simpler version of the calculation, and see how long that takes, then estimate how much longer the full one will. If you figure out you’re doomed, then it’s time to switch to a more efficient algorithm, or a different question entirely.

Sometimes, we don’t just have to consider time, but money as well. If you’re doing an experiment, you have to estimate how much the equipment will cost, and how much it will cost to run it. Experimenters get pretty good at estimating these things, but they still screw up sometimes and run over budget. Occasionally this is fine: LIGO didn’t detect anything in its first eight-year run, but they upgraded the machines and tried again, and won a Nobel prize. Other times it’s a disaster, and money keeps being funneled into a project that never works. Telling the difference is crucial, and it’s something we as a community are still not so good at.

Sometimes we just have to follow our instincts. This is dangerous, because we have a bias (the “sunk cost fallacy”) to stick with something if we’ve already spent a lot of time or money on it. To counteract that, it’s good to cultivate a bias in the opposite direction, which you might call “scientific impatience”. Getting frustrated with slow progress may not seem productive, but it keeps you motivated to search for a better way. Experienced scientists get used to how long certain types of project take. Too little progress, and they look for another option. This can fail, killing a project that was going to succeed, but it can also prevent over-investment in a failing idea. Only a mix of instincts keeps the field moving.

In the end, science is a gamble. Like the song, we have to know when to hold ’em and fold ’em, when to walk away, and when to run an idea as far as it will go. Sometimes it works, and sometimes it doesn’t. That’s science.