I did a guest post this week, on an outreach site for the Max Planck Institute for Physics. The new Director of their Quantum Field Theory Department, Johannes Henn, has been behind a lot of major developments in scattering amplitudes. He was one of the first to notice just how symmetric N=4 super Yang-Mills is, as well as the first to build the “hexagon functions” that would become my stock-in-trade. He’s also done what we all strive to do, and applied what he learned to the real world, coming up with an approach to differential equations that has become the gold standard for many different amplitudes calculations.
Now in his new position, he has a swanky new outreach site, reached at the conveniently memorable scattering-amplitudes.com and managed by outreach-ologist Sorana Scholtes. They started a fun series recently called “Talking Terms” as a kind of glossary, explaining words that physicists use over and over again. My guest post for them is part of that series. It hearkens all the way back to one of my first posts, defining what “theory” means to a theoretical physicist. It covers something new as well, a phrase I don’t think I’ve ever explained on this blog: “working in a theory”. You can check it out on their site!
One of the most mysterious powers physicists claim is physical intuition. Let the mathematicians have their rigorous proofs and careful calculations. We just need to ask ourselves, “Does this make sense physically?”
It’s tempting to chalk this up to bluster, or physicist arrogance. Sometimes, though, a physicist manages to figure out something that stumps the mathematicians. Edward Witten’s work on knot theory is a classic example, where he used ideas from physics, not rigorous proof, to win one of mathematics’ highest honors.
So what is physical intuition? And what is its relationship to proof?
Oscillators are familiar problems for first-year physics students. Objects that go back and forth, like springs and pendulums, tend to obey similar equations. Link two of them together (couple them), and the equations get more complicated, work for a second-year student instead of a first-year one. Such a student will notice that coupled oscillators “repel” each other: their frequencies get father apart than they would be if they weren’t coupled.
Our seminar speaker wanted us to revisit those second-year-student days, in order to understand how different particles behave in Effective Field Theory. Just as the frequencies of the oscillators repel each other, the energies of particles repel each other: the unknown high-energy particles could only push the energies of the lighter particles we can detect lower, not higher.
This is an example of physical intuition. Examine it, and you can learn a few things about how physical intuition works.
First, physical intuition comes from experience. Using physical intuition wasn’t just a matter of imagining the particles and trying to see what “makes sense”. Instead, it required thinking about similar problems from our experience as physicists: problems that don’t just seem similar on the surface, but are mathematically similar.
Second, physical intuition doesn’t replace calculation. Our speaker had done the math, he hadn’t just made a physical argument. Instead, physical intuition serves two roles: to inspire, and to help remember. Physical intuition can inspire new solutions, suggesting ideas that you go on to check with calculation. In addition to that, it can help your mind sort out what you already know. Without the physical story, we might not have remembered that the low-energy particles have their energies pushed down. With the story though, we had a similar problem to compare, and it made the whole thing more memorable. Human minds aren’t good at holding a giant pile of facts. What they are good at is holding narratives. “Physical intuition” ties what we know into a narrative, building on past problems to understand new ones.
Finally, physical intuition can be risky. If the problem is too different then the intuition can lead you astray. The mathematics of coupled oscillators and Effective Field Theories was similar enough for this argument to work, but if it turned out to be different in an important way then the intuition would have backfired, making it harder to find the answer and harder to keep track once it was found.
Physical intuition may seem mysterious. But deep down, it’s just physicists using our experience, comparing similar problems to help keep track of what we need to know. I’m sure chemists, biologists, and mathematicians all have similar stories to tell.
Physicists celebrate the new year by trying to sneak one last paper in before the year is over. Looking at Facebook last night I saw three different friends preview the papers they just submitted. The site where these papers appear, arXiv, had seventy new papers this morning, just in the category of theoretical high-energy physics. Of those, nine of them wereinmy, oracloselyrelatedsub–field.
I’d love to tell you all about these papers (some exciting! some long-awaited!), but I’m still tired from last night and haven’t read them yet. So I’ll just close by wishing you all, once again, a happy new year.
A big theme of this conference, as in the past few years, was gravitational waves. From LIGO’s first announcement of a successful detection, amplitudeologists have been developing new methods to make predictions for gravitational waves more efficient. It’s a field I’ve dabbledin a bit myself. Last year’s QCD Meets Gravity left me impressed by how much progress had been made, with amplitudeologists already solidly part of the conversation and able to produce competitive results. This year felt like another milestone, in that the amplitudeologists weren’t just catching up with other gravitational wave researchers on the same kinds of problems. Instead, they found new questions that amplitudes are especially well-suited to answer. These included combining two pieces of these calculations (“potential” and “radiation”) that the older community typically has to calculate separately, using an old quantum field theory trick, finding the gravitational wave directly from amplitudes, and finding a few nice calculations that can be used to “generate” the rest.
A large chunk of the talks focused on different “squaring” tricks (or as we actually call them, double-copies). There were double-copies for cosmology and conformal field theory, for the celestial sphere, and even some version of M theory. There were new perspectives on the double-copy, new building blocks and algebraic structures that lie behind it. There were talks on the so-called classical double-copy for space-times, where there have been some strange discoveries (an extra dimension made an appearance) but also a more rigorous picture of where the whole thing comes from, using twistor space. There were not one, but two talks linking the double-copy to the Navier-Stokes equation describing fluids, from two different groups. (I’m really curious whether these perspectives are actually useful for practical calculations about fluids, or just fun to think about.) Finally, while there wasn’t a talk scheduled on this paper, the authors were roped in by popular demand to talk about their work. They claim to have made progress on a longstanding puzzle, how to show that double-copy works at the level of the Lagrangian, and the community was eager to dig into the details.
From there, a grab-bag of talks covered other advancements. There were talks from string theorists and ambitwistor string theorists, from Effective Field Theorists working on gravity and the Standard Model, from calculations in N=4 super Yang-Mills, QCD, and scalar theories. Simon Caron-Huot delved into how causality constrains the theories we can write down, showing an interesting case where the common assumption that all parameters are close to one is actually justified. Nima Arkani-Hamed began his talk by saying he’d surprise us, which he certainly did (and not by keeping on time). It’s tricky to explain why his talk was exciting. Comparing to his earlier discovery of the Amplituhedron, which worked for a toy model, this is a toy calculation in a toy model. While the Amplituhedron wasn’t based on Feynman diagrams, this can’t even be compared with Feynman diagrams. Instead of expanding in a small coupling constant, this expands in a parameter that by all rights should be equal to one. And instead of positivity conditions, there are negativity conditions. All I can say is that with all of that in mind, it looks like real progress on an important and difficult problem from a totally unanticipated direction. In a speech summing up the conference, Zvi Bern mentioned a few exciting words from Nima’s talk: “nonplanar”, “integrated”, “nonperturbative”. I’d add “differential equations” and “infinite sums of ladder diagrams”. Nima and collaborators are trying to figure out what happens when you sum up all of the Feynman diagrams in a theory. I’ve made progress in the past for diagrams with one “direction”, a ladder that grows as you add more loops, but I didn’t know how to add “another direction” to the ladder. In very rough terms, Nima and collaborators figured out how to add that direction.
I’ve probably left things out here, it was a packed conference! It’s been really fun seeing what the community has cooked up, and I can’t wait to see what happens next.
Last year’s conference was very focused on one particular topic, trying to predict the gravitational waves observed by LIGO and VIRGO. That’s still a core topic of the conference, but it feels like there is a bit more diversity in topics this year. We’ve seen a variety of talks on different “squares”: new theories that square to other theories, and new calculations that benefit from “squaring” (even surprising applications to the Navier-Stokes equation!) There are talks on subjects from String Theory to Effective Field Theory, and even a talk on a very different way that “QCD meets gravity”, in collisions of neutron stars.
With still a few more talks to go, expect me to say a bit more next week, probably discussing a few in more detail. (Several people presented exciting work in progress!) Until then, I should get back to watching!
In physics, what you don’t know can absolutely hurt you. If you ignore that planets have their own gravity, or that metals conduct electricity, you’re going to calculate a lot of nonsense. At the same time, as physicists we can’t possibly know everything. Our experiments are never perfect, our math never includes all the details, and even our famous Standard Model is almost certainly not the whole story. Luckily, we have another option: instead of ignoring what we don’t know, we can parametrize it, and estimate its effect.
Estimating the unknown is something we physicists have done since Newton. You might think Newton’s big discovery was the inverse-square law for gravity, but others at the time, like Robert Hooke, had also been thinking along those lines. Newton’s big discovery was that gravity was universal: that you need to know the effect of gravity, not just from the sun, but from all the other planets as well. The trouble was, Newton didn’t know how to calculate the motion of all of the planets at once (in hindsight, we know he couldn’t have). Instead, he estimated, using what he knew to guess how big the effect of what he didn’t would be. It was the accuracy of those guesses, not just the inverse square law by itself, that convinced the world that Newton was right.
If you’ve studied electricity and magnetism, you get to the point where you can do simple calculations with a few charges in your sleep. The world doesn’t have just a few charges, though: it has many charges, protons and electrons in every atom of every object. If you had to keep all of them in your calculations you’d never pass freshman physics, but luckily you can once again parametrize what you don’t know. Often you can hide those charges away, summarizing their effects with just a fewnumbers. Other times, you can treat materials as boundaries, and summarize everything beyond in terms of what happens on the edge. The equations of the theory let you do this, but this isn’t true for every theory: for the Navier-Stokes equation, which we use to describe fluids, it still isn’t known whether you can do this kind of trick.
Parametrizing what we don’t know isn’t just a trick for college physics, it’s key to the cutting edge as well. Right now we have a picture for how all of particle physics works, called the Standard Model, but we know that picture is incomplete. There are a million different theories you could write to go beyond the Standard Model, with a million different implications. Instead of having to use all those theories, physicists can summarize them all with what we call an effective theory: one that keeps track of the effect of all that new physics on the particles we already know. By summarizing those effects with a few parameters, we can see what they would have to be to be compatible with experimental results, ruling out some possibilities and suggesting others.
In a world where we never know everything, there’s always something that can hurt us. But if we’re careful and estimate what we don’t know, if we write down numbers and parameters and keep our options open, we can keep from getting burned. By focusing on what we do know, we can still manage to understand the world.
The thing is, this is a metaphor. What’s more, it’s a metaphor for an approximation. As physicists, when we draw diagrams with more and more virtual particles, we’re trying to use something we know how to calculate with (particles) to understand something tougher to handle (interacting quantum fields). Virtual particles, at least as you’re probably picturing them, don’t really exist.
I don’t really blame physicists for talking like that, though. Virtual particles are a metaphor, sure, a way to talk about a particular calculation. But so is basically anything we can say about quantum field theory. In quantum field theory, it’s pretty tough to say which things “really exist”.
You might have heard that there are three types of neutrinos, corresponding to the three “generations” of the Standard Model: electron-neutrinos, muon-neutrinos, and tau-neutrinos. Each is produced in particular kinds of reactions: electron-neutrinos, for example, get produced by beta-plus decay, when a proton turns into a neutron, an anti-electron, and an electron-neutrino.
Leave these neutrinos alone though, and something strange happens. Detect what you expect to be an electron-neutrino, and it might have changed into a muon-neutrino or a tau-neutrino. The neutrino oscillated.
Why does this happen?
One way to explain it is to say that electron-neutrinos, muon-neutrinos, and tau-neutrinos don’t “really exist”. Instead, what really exists are neutrinos with specific masses. These don’t have catchy names, so let’s just call them neutrino-one, neutrino-two, and neutrino-three. What we think of as electron-neutrinos, muon-neutrinos, and tau-neutrinos are each some mix (a quantum superposition) of these “really existing” neutrinos, specifically the mixes that interact nicely with electrons, muons, and tau leptons respectively. When you let them travel, it’s these neutrinos that do the traveling, and due to quantum effects that I’m not explaining here you end up with a different mix than you started with.
This probably seems like a perfectly reasonable explanation. But it shouldn’t. Because if you take one of these mass-neutrinos, and interact with an electron, or a muon, or a tau, then suddenly it behaves like a mix of the old electron-neutrinos, muon-neutrinos, and tau-neutrinos.
That’s because both explanations are trying to chop the world up in a way that can’t be done consistently. There aren’t electron-neutrinos, muon-neutrinos, and tau-neutrinos, and there aren’t neutrino-ones, neutrino-twos, and neutrino-threes. There’s a mathematical object (a vector space) that can look like either.
Whether you’re comfortable with that depends on whether you think of mathematical objects as “things that exist”. If you aren’t, you’re going to have trouble thinking about the quantum world. Maybe you want to take a step back, and say that at least “fields” should exist. But that still won’t do: we can redefine fields, add them together or even use more complicated functions, and still get the same physics. The kinds of things that exist can’t be like this. Instead you end up invoking another kind of mathematical object, equivalence classes.
If you want to be totally rigorous, you have to go a step further. You end up thinking of physics in a very bare-bones way, as the set of all observations you could perform. Instead of describing the world in terms of “these things” or “those things”, the world is a black box, and all you’re doing is finding patterns in that black box.
Is there a way around this? Maybe. But it requires thought, and serious philosophy. It’s not intuitive, it’s not easy, and it doesn’t lend itself well to 3d animations in documentaries. So in practice, whenever anyone tells you about something in physics, you can be pretty sure it’s a metaphor. Nice describable, non-mathematical things typically don’t exist.
After Amplitudes was held online this year, a few of us at the Niels Bohr Institute were inspired. We thought this would be the perfect time to hold a small online conference, focused on the Calabi-Yaus that have been poppinguplately in Feynman diagrams. Then we heard from the organizers of Elliptics 2020. They had been planning to hold a conference in Mainz about elliptic integrals in Feynman diagrams, but had to postpone it due to the pandemic. We decided to team up and hold a joint conference on both topics: the elliptic integrals that are just starting to be understood, and the mysterious integrals that lie beyond. Hence, Elliptics and Beyond.
The conference has been fun thus far. There’s been a mix of review material bringing people up to speed on elliptic integrals and exciting new developments. Some are taking methods that have been successful in other areas and generalizing them to elliptic integrals, others have been honing techniques for elliptics to make them “production-ready”. A few are looking ahead even further, to higher-genus amplitudes in string theory and Calabi-Yaus in Feynman diagrams.
We organized the conference along similar lines to Zoomplitudes, but with a few experiments of our own. Like Zoomplitudes, we made a Slack space for the conference, so people could chat physics outside the talks. Ours was less active, though. I suspect that kind of space needs a critical mass of people, and with a smaller conference we may just not have gotten there. Having fewer people did allow us a more relaxed schedule, which in turn meant we could mostly keep things on-time. We had discussion sessions in the morning (European time), with talks in the afternoon, so almost everyone could make the talks at least. We also had a “conference dinner”, which went much better than I would have expected. We put people randomly into Zoom Breakout Rooms of five or six, to emulate the tables of an in-person conference, and folks chatted while eating their (self-brought of course) dinner. People seemed to really enjoy the chance to just chat casually with the other folks at the conference. If you’re organizing an online conference soon, I’d recommend trying it!
Holding a conference online means that a lot of people can attend who otherwise couldn’t. We had over a hundred people register, and while not all of them showed up there were typically fifty or sixty people on the Zoom session. Some of these were specialists in elliptics or Calabi-Yaus who wouldn’t ordinarily make it to a conference like this. Others were people from the rest of the amplitudes field who joined for parts of the conference that caught their eye. But surprisingly many weren’t even amplitudeologists, but students and young researchers in a variety of topics from all over the world. Some seemed curious and eager to learn, others I suspect just needed to say they had been to a conference. Both are responding to a situation where suddenly conference after conference is available online, free to join. It will be interesting to see if, and how, the world adapts.
Listen to a certain flavor of crackpot, or a certain kind of science fiction, and you’ll hear about zero-point energy. Limitless free energy drawn from quantum space-time itself, zero-point energy probably sounds like bullshit. Often it is. But lurking behind the pseudoscience and the fiction is a real physics concept, albeit one that doesn’t really work like those people imagine.
In quantum mechanics, the zero-point energy is the lowest energy a particular system can have. That number doesn’t actually have to be zero, even for empty space. People sometimes describe this in terms of so-called virtual particles, popping up from nothing in particle-antiparticle pairs only to annihilate each other again, contributing energy in the absence of any “real particles”. There’s a real force, the Casimir effect, that gets attributed to this, a force that pulls two metal plates together even with no charge or extra electromagnetic field. The same bubbling of pairs of virtual particles also gets used to explain the Hawking radiation of black holes.
I’d like to try explaining all of these things in a different way, one that might clear up some common misconceptions. To start, let’s talk about, not zero-point energy, but zero-point diagrams.
Feynman diagrams are a tool we use to study particle physics. We start with a question: if some specific particles come together and interact, what’s the chance that some (perhaps different) particles emerge? We start by drawing lines representing the particles going in and out, then connect them in every way allowed by our theory. Finally we translate the diagrams to numbers, to get an estimate for the probability. In particle physics slang, the number of “points” is the total number of particles: particles in, plus particles out. For example, let’s say we want to know the chance that two electrons go in and two electrons come out. That gives us a “four-point” diagram: two in, plus two out. A zero-point diagram, then, means zero particles in, zero particles out.
(Note that this isn’t why zero-point energy is called zero-point energy, as far as I can tell. Zero-point energy is an older term from before Feynman diagrams.)
Remember, each Feynman diagram answers a specific question, about the chance of particles behaving in a certain way. You might wonder, what question does a zero-point diagram answer? The chance that nothing goes to nothing? Why would you want to know that?
To answer, I’d like to bring up some friends of mine, who do something that might sound equally strange: they calculate one-point diagrams, one particle goes to none. This isn’t strange for them because they study theories with defects.
Normally in particle physics, we think about our particles in an empty, featureless space. We don’t have to, though. One thing we can do is introduce features in this space, like walls and mirrors, and try to see what effect they have. We call these features “defects”.
If there’s a defect like that, then it makes sense to calculate a one-point diagram, because your one particle can interact with something that’s not a particle: it can interact with the defect.
You might see where this is going: let’s say you think there’s a force between two walls, that comes from quantum mechanics, and you want to calculate it. You could imagine it involves a diagram like this:
Roughly speaking, this is the kind of thing you could use to calculate the Casimir effect, that mysterious quantum force between metal plates. And indeed, it involves a zero-point diagram.
Here’s the thing, though: metal plates aren’t just “defects”. They’re real physical objects, made of real physical particles. So while you can think of the Casimir effect with a “zero-point diagram” like that, you can also think of it with a normal diagram, more like the four-point diagram I showed you earlier: one that computes, not a force between defects, but a force between the actual electrons and protons that make up the two plates.
A lot of the time when physicists talk about pairs of virtual particles popping up out of the vacuum, they have in mind a picture like this. And often, you can do the same trick, and think about it instead as interactions between physical particles. There’s a story of roughly this kind for Hawking radiation: you can think of a black hole event horizon as “cutting in half” a zero-point diagram, and see pairs of particles going out from the black hole…but you can also do a calculation that looks more like particles interacting with a gravitational field.
This also might help you understand why, contra the crackpots and science fiction writers, zero-point energy isn’t a source of unlimited free energy. Yes, a force like the Casimir effect comes “from the vacuum” in some sense. But really, it’s a force between two particles. And just like the gravitational force between two particles, this doesn’t give you unlimited free power. You have to do the work to move the particles back over and over again, using the same amount of power you gained from the force to begin with. And unlike the forces you’re used to, these are typically very small effects, as usual for something that depends on quantum mechanics. So it’s even less useful than more everyday forces for this.
Why do so many crackpots and authors expect zero-point energy to be a massive source of power? In part, this is due to mistakes physicists made early on.
Sometimes, when calculating a zero-point diagram (or any other diagram), we don’t get a sensible number. Instead, we get infinity. Physicists used to be baffled by this. Later, they understood the situation a bit better, and realized that those infinities were probably just due to our ignorance. We don’t know the ultimate high-energy theory, so it’s possible something happens at high energies to cancel those pesky infinities. Without knowing exactly what happened, physicists would estimate by using a “cutoff” energy where they expected things to change.
That kind of calculation led to an estimate you might have heard of, that the zero-point energy inside single light bulb could boil all the world’s oceans. That estimate gives a pretty impressive mental image…but it’s also wrong.
This kind of estimate led to “the worst theoretical prediction in the history of physics”, that the cosmological constant, the force that speeds up the expansion of the universe, is 120 orders of magnitude higher than its actual value (if it isn’t just zero). If there really were energy enough inside each light bulb to boil the world’s oceans, the expansion of the universe would be quite different than what we observe.
At this point, it’s pretty clear there is something wrong with these kinds of “cutoff” estimates. The only unclear part is whether that’s due to something subtle or something obvious. But either way, this particular estimate is just wrong, and you shouldn’t take it seriously. Zero-point energy exists, but it isn’t the magical untapped free energy you hear about in stories. It’s tiny quantum corrections to the forces between particles.
I calculate what are called scattering amplitudes, formulas that tell us the chance that two particles scatter off each other. Formulas like these exist for theories like the strong nuclear force, called Yang-Mills theories, they also exist for the hypothetical graviton particles of gravity. One of the biggest insights in scattering amplitude research in the last few decades is that these two types of formulas are tied together: as we like to say, gravity is Yang-Mills squared.
A huge chunk of my subfield grew out of that insight. For one, it’s why some of us think we have something useful to say about colliding black holes. But while it’s been used in a dozen different ways, an important element was missing: the principle was never actually proven (at least, not in the way it’s been used).
Now, a group in the UK and the Czech Republic claims to have proven it.
I say “claims” not because I’m skeptical, but because without a fair bit more reading I don’t think I can judge this one. That’s because the group, and the approach they use, isn’t “amplitudish”. They aren’t doing what amplitudes researchers would do.
In the amplitudes subfield, we like to write things as much as possible in terms of measurable, “on-shell” particles. This is in contrast to the older approach that writes things instead in terms of more general quantum fields, with formulas called Lagrangians to describe theories. In part, we avoid the older Lagrangian framing to avoid redundancy: there are many different ways to write a Lagrangian for the exact same physics. We have another reason though, which might seem contradictory: we avoid Lagrangians to stay flexible. There are many ways to rewrite scattering amplitudes that make different properties manifest, and some of the strangest ones don’t seem to correspond to any Lagrangian at all.
If you’d asked me before last week, I’d say that “gravity is Yang-Mills squared” was in that category: something you couldn’t make manifest fully with just a Lagrangian, that you’d need some stranger magic to prove. If this paper is right, then that’s wrong: if you’re careful enough you can prove “gravity is Yang-Mills squared” in the old-school, Lagrangian way.
I’m curious how this is going to develop: what amplitudes people will think about it, what will happen as the experts chime in. For now, as mentioned, I’m reserving judgement, except to say “interesting if true”.