Author Archives: 4gravitons

Valentine’s Day Physics Poem 2020

It’s Valentine’s Day, time for my traditional physics poem. I’m trying a new format this year, let me know what you think!

Cherish the Effective

Self-styled wise men waste away, pining for the Ultimate Theory.
I tell you now: spurn their fate.
Scorn the Ultimate.
Cherish the Effective.
 
When you dream of an Ultimate Theory, what do you see?
An answer to your every question?
Every worry,
Every weakness,
Resolved?
A thing of beauty?
A thing of
       your notion
              and only your notion
of beauty?
 
Nature, she has her own worries.
Science, she never stops asking.
Your fairytale ending?
You won’t get it.
And you’ll hurt, and be hurt, in the trying.
 
You need a theory that isn’t an ending.
A theory you
              only
                   start
                          understanding
But can always discover.
No rigid, final truth,
But gentle corrections.
And as you push
                The scale
                           The energy
Your theory always has room for something new.
 
A theory like that, we call Effective.
A theory you can live
                      your
                           life
                                with.
It’s worth more than you think.

You Can’t Anticipate a Breakthrough

As a scientist, you’re surrounded by puzzles. For every test and every answer, ten new questions pop up. You can spend a lifetime on question after question, never getting bored.

But which questions matter? If you want to change the world, if you want to discover something deep, which questions should you focus on? And which should you ignore?

Last year, my collaborators and I completed a long, complicated project. We were calculating the chance fundamental particles bounce off each other in a toy model of nuclear forces, pushing to very high levels of precision. We managed to figure out a lot, but as always, there were many questions left unanswered in the end.

The deepest of these questions came from number theory. We had noticed surprising patterns in the numbers that showed up in our calculation, reminiscent of the fancifully-named Cosmic Galois Theory. Certain kinds of numbers never showed up, while others appeared again and again. In order to see these patterns, though, we needed an unusual fudge factor: an unexplained number multiplying our result. It was clear that there was some principle at work, a part of the physics intimately tied to particular types of numbers.

There were also questions that seemed less deep. In order to compute our result, we compared to predictions from other methods: specific situations where the question becomes simpler and there are other ways of calculating the answer. As we finished writing the paper, we realized we could do more with some of these predictions. There were situations we didn’t use that nonetheless simplified things, and more predictions that it looked like we could make. By the time we saw these, we were quite close to publishing, so most of us didn’t have the patience to follow these new leads. We just wanted to get the paper out.

At the time, I expected the new predictions would lead, at best, to more efficiency. Maybe we could have gotten our result faster, or cleaned it up a bit. They didn’t seem essential, and they didn’t seem deep.

Fast forward to this year, and some of my collaborators (specifically, Lance Dixon and Georgios Papathanasiou, along with Benjamin Basso) have a new paper up: “The Origin of the Six-Gluon Amplitude in Planar N=4 SYM”. The “origin” in their title refers to one of those situations: when the variables in the problem are small, and you’re close to the “origin” of a plot in those variables. But the paper also sheds light on the origin of our calculation’s mysterious “Cosmic Galois” behavior.

It turns out that the origin (of the plot) can be related to another situation, when the paths of two particles in our calculation almost line up. There, the calculation can be done with another method, called the Pentagon Operator Product Expansion, or POPE. By relating the two, Basso, Dixon, and Papathanasiou were able to predict not only how our result should have behaved near the origin, but how more complicated as-yet un-calculated results should behave.

The biggest surprise, though, lurked in the details. Building their predictions from the POPE method, they found their calculation separated into two pieces: one which described the physics of the particles involved, and a “normalization”. This normalization, predicted by the POPE method, involved some rather special numbers…the same as the “fudge factor” we had introduced earlier! Somehow, the POPE’s physics-based setup “knows” about Cosmic Galois Theory!

It seems that, by studying predictions in this specific situation, Basso, Dixon, and Papathanasiou have accomplished something much deeper: a strong hint of where our mysterious number patterns come from. It’s rather humbling to realize that, were I in their place, I never would have found this: I had assumed “the origin” was just a leftover detail, perhaps useful but not deep.

I’m still digesting their result. For now, it’s a reminder that I shouldn’t try to pre-judge questions. If you want to learn something deep, it isn’t enough to sit thinking about it, just focusing on that one problem. You have to follow every lead you have, work on every problem you can, do solid calculation after solid calculation. Sometimes, you’ll just make incremental progress, just fill in the details. But occasionally, you’ll have a breakthrough, something that justifies the whole adventure and opens your door to something strange and new. And I’m starting to think that when it comes to breakthroughs, that’s always been the only way there.

A Scale of “Sure-Thing-Ness” for Experiments

No experiment is a sure thing. No matter what you do, what you test, what you observe, there’s no guarantee that you find something new. Even if you do your experiment correctly and measure what you planned to measure, nature might not tell you anything interesting.

Still, some experiments are more sure than others. Sometimes you’re almost guaranteed to learn something, even if it wasn’t what you hoped, while other times you just end up back where you started.

The first, and surest, type of experiment, is a voyage into the unknown. When nothing is known about your target, no expectations, and no predictions, then as long as you successfully measure anything you’ll have discovered something new. This can happen if the thing you’re measuring was only recently discovered. If you’re the first person who manages to measure the reaction rates of an element, or the habits of an insect, or the atmosphere of a planet, then you’re guaranteed to find out something you didn’t know before.

If you don’t have a total unknown to measure, then you want to test a clear hypothesis. The best of these are the theory killers, experiments which can decisively falsify an idea. History’s most famous experiments take this form, like the measurement of the perihelion of Mercury to test General Relativity or Pasteur’s tests of spontaneous generation. When you have a specific prediction and not much wiggle room, an experiment can teach you quite a lot.

“Not much wiggle room” is key, because these tests can all to easily become theory modifiers instead. If you can tweak your theory enough, then your experiment might not be able to falsify it. Something similar applies when you have a number of closely related theories. Even if you falsify one, you can just switch to another similar idea. In those cases, testing your theory won’t always teach you as much: you have to get lucky and see something that pins your theory down more precisely.

Finally, you can of course be just looking. Some experiments are just keeping an eye out, in the depths of space or the precision of quantum labs, watching for something unexpected. That kind of experiment might never see anything, and never rule anything out, but they can still sometimes be worthwhile.

There’s some fuzziness to these categories, of course. Often when scientists argue about whether an experiment is worth doing they’re arguing about which category to place it in. Would a new collider be a “voyage into the unknown” (new energy scales we’ve never measured before), a theory killer/modifier (supersymmetry! but which one…) or just “just looking”? Is your theory of cosmology specific enough to be “killed”, or merely “modified”? Is your wacky modification of quantum mechanics something that can be tested, or merely “just looked” for?

For any given experiment, it’s worth keeping in mind what you expect, and what would happen if you’re wrong. In science, we can’t do every experiment we want. We have to focus our resources and try to get results. Even if it’s never a sure thing.

Math Is the Art of Stating Things Clearly

Why do we use math?

In physics we describe everything, from the smallest of particles to the largest of galaxies, with the language of mathematics. Why should that one field be able to describe so much? And why don’t we use something else?

The truth is, this is a trick question. Mathematics isn’t a language like English or French, where we can choose whichever translation we want. We use mathematics because it is, almost by definition, the best choice. That is because mathematics is the art of stating things clearly.

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter. The bartender stops them, pours two beers, and says “You guys should know your limits.”

That was an (old) joke about infinite series of numbers. You probably learned in high school that if you add up one plus a half plus a quarter…you eventually get two. To be a bit more precise:

\sum_{i=0}^\infty \frac{1}{2^i} = 1+\frac{1}{2}+\frac{1}{4}+\ldots=2

We say that this infinite sum limits to two.

But what does it actually mean for an infinite sum to limit to a number? What does it mean to sum infinitely many numbers, let alone infinitely many beers ordered by infinitely many mathematicians?

You’re asking these questions because I haven’t yet stated the problem clearly. Those of you who’ve learned a bit more mathematics (maybe in high school, maybe in college) will know another way of stating it.

You know how to sum a finite set of beers. You start with one beer, then one and a half, then one and three-quarters. Sum N beers, and you get

\sum_{i=0}^N \frac{1}{2^i}

What does it mean for the sum to limit to two?

Let’s say you just wanted to get close to two. You want to get \epsilon close, where epsilon is the Greek letter we use for really small numbers.

For every \epsilon>0 you choose, no matter how small, I can pick a (finite!) N and get at least that close. That means that, with higher and higher N, I can get as close to two as a I want.

As it turns out, that’s what it means for a sum to limit to two. It’s saying the same thing, but more clearly, without sneaking in confusing claims about infinity.

These sort of proofs, with \epsilon (and usually another variable, \delta) form what mathematicians view as the foundations of calculus. They’re immortalized in story and song.

And they’re not even the clearest way of stating things! Go down that road, and you find more mathematics: definitions of numbers, foundations of logic, rabbit holes upon rabbit holes, all from the effort to state things clearly.

That’s why I’m not surprised that physicists use mathematics. We have to. We need clarity, if we want to understand the world. And mathematicians, they’re the people who spend their lives trying to state things clearly.

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?

A theoretical physicist, at work in the equation mines

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.

The Road to Reality

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.