# On Stubbornness and Breaking Down

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

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

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

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

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

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

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

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

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

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

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

# LHC Black Holes for the Terminally Un-Reassured

Could the LHC have killed us all?

No, no it could not.

But…

I’ve had this conversation a few times over the years. Usually, the people I’m talking to are worried about black holes. They’ve heard that the Large Hadron Collider speeds up particles to amazingly high energies before colliding them together. They worry that these colliding particles could form a black hole, which would fall into the center of the Earth and busily gobble up the whole planet.

This pretty clearly hasn’t happened. But also, physicists were pretty confident that it couldn’t happen. That isn’t to say they thought it was impossible to make a black hole with the LHC. Some physicists actually hoped to make a black hole: it would have been evidence for extra dimensions, curled-up dimensions much larger than the tiny ones required by string theory. They figured out the kind of evidence they’d see if the LHC did indeed create a black hole, and we haven’t seen that evidence. But even before running the machine, they were confident that such a black hole wouldn’t gobble up the planet. Why?

The best argument is also the most unsatisfying. The LHC speeds up particles to high energies, but not unprecedentedly high energies. High-energy particles called cosmic rays enter the atmosphere every day, some of which are at energies comparable to the LHC. The LHC just puts the high-energy particles in front of a bunch of sophisticated equipment so we can measure everything about them. If the LHC could destroy the world, cosmic rays would have already done so.

That’s a very solid argument, but it doesn’t really explain why. Also, it may not be true for future colliders: we could build a collider with enough energy that cosmic rays don’t commonly meet it. So I should give another argument.

The next argument is Hawking radiation. In Stephen Hawking’s most famous accomplishment, he argued that because of quantum mechanics black holes are not truly black. Instead, they give off a constant radiation of every type of particle mixed together, shrinking as it does so. The radiation is faintest for large black holes, but gets more and more intense the smaller the black hole is, until the smallest black holes explode into a shower of particles and disappear. This argument means that a black hole small enough that the LHC could produce it would radiate away to nothing in almost an instant: not long enough to leave the machine, let alone fall to the center of the Earth.

This is a good argument, but maybe you aren’t as sure as I am about Hawking radiation. As it turns out, we’ve never measured Hawking radiation, it’s just a theoretical expectation. Remember that the radiation gets fainter the larger the black hole is: for a black hole in space with the mass of a star, the radiation is so tiny it would be almost impossible to detect even right next to the black hole. From here, in our telescopes, we have no chance of seeing it.

So suppose tiny black holes didn’t radiate, and suppose the LHC could indeed produce them. Wouldn’t that have been dangerous?

Here, we can do a calculation. I want you to appreciate how tiny these black holes would be.

From science fiction and cartoons, you might think of a black hole as a kind of vacuum cleaner, sucking up everything nearby. That’s not how black holes work, though. The “sucking” black holes do is due to gravity, no stronger than the gravity of any other object with the same mass at the same distance. The only difference comes when you get close to the event horizon, an invisible sphere close-in around the black hole. Pass that line, and the gravity is strong enough that you will never escape.

We know how to calculate the position of the event horizon of a black hole. It’s the Schwarzchild radius, and we can write it in terms of Newton’s constant G, the mass of the black hole M, and the speed of light c, as follows:

$\frac{2GM}{c^2}$

The Large Hadron Collider’s two beams each have an energy around seven tera-electron-volts, or TeV, so there are 14 TeV of energy in total in each collision. Imagine all of that energy being converted into mass, and that mass forming a black hole. That isn’t how it would actually happen: some of the energy would create other particles, and some would give the black hole a “kick”, some momentum in one direction or another. But we’re going to imagine a “worst-case” scenario, so let’s assume all the energy goes to form the black hole. Electron-volts are a weird physicist unit, but if we divide them by the speed of light squared (as we should if we’re using $E=mc^2$ to create a mass), then Wikipedia tells us that each electron-volt will give us $1.78\times 10^{-36}$ kilograms. “Tera” is the SI prefix for $10^{12}$. Thus our tiny black hole starts with a mass of

$14\times 10^{12}\times 1.78\times 10^{-36} = 2.49\times 10^{-23} \textrm{kg}$

Plugging in Newton’s constant ($6.67\times 10^{-11}$ meters cubed per kilogram per second squared), and the speed of light ($3\times 10^8$ meters per second), and we get a radius of,

$\frac{2\times 6.67\times 10^{-11}\times 14\times 10^{12}\times 1.78\times 10^{-36}}{\left(3\times 10^8\right)^2} = 3.7\times 10^{-50} \textrm{m}$

That, by the way, is amazingly tiny. The size of an atom is about $10^{-10}$ meters. If every atom was a tiny person, and each of that person’s atoms was itself a person, and so on for five levels down, then the atoms of the smallest person would be the same size as this event horizon.

Now, we let this little tiny black hole fall. Let’s imagine it falls directly towards the center of the Earth. The only force affecting it would be gravity (if it had an electrical charge, it would quickly attract a few electrons and become neutral). That means you can think of it as if it were falling through a tiny hole, with no friction, gobbling up anything unfortunate enough to fall within its event horizon.

For our first estimate, we’ll treat the black hole as if it stays the same size through its journey. Imagine the black hole travels through the entire earth, absorbing a cylinder of matter. Using the Earth’s average density of 5515 kilograms per cubic meter, and the Earth’s maximum radius of 6378 kilometers, our cylinder adds a mass of,

$\pi \times \left(3.7\times 10^{-50}\right)^2 \times 2 \times 6378\times 10^3\times 5515 = 3\times 10^{-88} \textrm{kg}$

That’s absurdly tiny. That’s much, much, much tinier than the mass we started out with. Absorbing an entire cylinder through the Earth makes barely any difference.

You might object, though, that the black hole is gaining mass as it goes. So really we ought to use a differential equation. If the black hole travels a distance r, absorbing mass as it goes at average Earth density $\rho$, then we find,

$\frac{dM}{dr}=\pi\rho\left(\frac{2GM(r)}{c^2}\right)^2$

Solving this, we get

$M(r)=\frac{M_0}{1- M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2 r }$

Where $M_0$ is the mass we start out with.

Plug in the distance through the Earth for r, and we find…still about $3\times 10^{-88} \textrm{kg}$! It didn’t change very much, which makes sense, it’s a very very small difference!

But you might still object. A black hole falling through the Earth wouldn’t just go straight through. It would pass through, then fall back in. In fact, it would oscillate, from one side to the other, like a pendulum. This is actually a common problem to give physics students: drop an object through a hole in the Earth, neglect air resistance, and what does it do? It turns out that the time the object takes to travel through the Earth is independent of its mass, and equal to roughly 84.5 minutes.

So let’s ask a question: how long would it take for a black hole, oscillating like this, to double its mass?

We want to solve,

$2=\frac{1}{1- M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2 r }$

so we need the black hole to travel a total distance of

$r=\frac{1}{2M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2} = 5.3\times 10^{71} \textrm{m}$

That’s a huge distance! The Earth’s radius, remember, is 6378 kilometers. So traveling that far would take

$5.3\times 10^{71} \times 84.5/60/24/365 = 8\times 10^{67} \textrm{y}$

Ten to the sixty-seven years. Our universe is only about ten to the ten years old. In another five times ten to the nine years, the Sun will enter its red giant phase, and swallow the Earth. There simply isn’t enough time for this tiny tiny black hole to gobble up the world, before everything is already gobbled up by something else. Even in the most pessimistic way to walk through the calculation, it’s just not dangerous.

I hope that, if you were worried about black holes at the LHC, you’re not worried any more. But more than that, I hope you’ve learned three lessons. First, that even the highest-energy particle physics involves tiny energies compared to day-to-day experience. Second, that gravitational effects are tiny in the context of particle physics. And third, that with Wikipedia access, you too can answer questions like this. If you’re worried, you can make an estimate, and check!

# The Problem of Quantum Gravity Is the Problem of High-Energy (Density) Quantum Gravity

I’ve said something like this before, but here’s another way to say it.

The problem of quantum gravity is one of the most famous problems in physics. You’ve probably heard someone say that quantum mechanics and general relativity are fundamentally incompatible. Most likely, this was narrated over pictures of a foaming, fluctuating grid of space-time. Based on that, you might think that all we have to do to solve this problem is to measure some quantum property of gravity. Maybe we could make a superposition of two different gravitational fields, see what happens, and solve the problem that way.

I mean, we could do that, some people are trying to. But it won’t solve the problem. That’s because the problem of quantum gravity isn’t just the problem of quantum gravity. It’s the problem of high-energy quantum gravity.

Merging quantum mechanics and general relativity is actually pretty easy. General relativity is a big conceptual leap, certainly, a theory in which gravity is really just the shape of space-time. At the same time, though, it’s also a field theory, the same general type of theory as electromagnetism. It’s a weirder field theory than electromagnetism, to be sure, one with deeper implications. But if we want to describe low energies, and weak gravitational fields, then we can treat it just like any other field theory. We know how to write down some pretty reasonable-looking equations, we know how to do some basic calculations with them. This part is just not that scary.

The scary part happens later. The theory we get from these reasonable-looking equations continues to look reasonable for a while. It gives formulas for the probability of things happening: things like gravitational waves bouncing off each other, as they travel through space. The problem comes when those waves have very high energy, and the nice reasonable probability formula now says that the probability is greater than one.

For those of you who haven’t taken a math class in a while, probabilities greater than one don’t make sense. A probability of one is a certainty, something guaranteed to happen. A probability greater than one isn’t more certain than certain, it’s just nonsense.

So we know something needs to change, we know we need a new theory. But we only know we need that theory when the energy is very high: when it’s the Planck energy. Before then, we might still have a different theory, but we might not: it’s not a “problem” yet.

Now, a few of you understand this part, but still have a misunderstanding. The Planck energy seems high for particle physics, but it isn’t high in an absolute sense: it’s about the energy in a tank of gasoline. Does that mean that all we have to do to measure quantum gravity is to make a quantum state out of your car?

Again, no. That’s because the problem of quantum gravity isn’t just the problem of high-energy quantum gravity either.

Energy seems objective, but it’s not. It’s subjective, or more specifically, relative. Due to special relativity, observers moving at different speeds observe different energies. Because of that, high energy alone can’t be the requirement: it isn’t something either general relativity or quantum field theory can “care about” by itself.

Instead, the real thing that matters is something that’s invariant under special relativity. This is hard to define in general terms, but it’s best to think of it as a requirement for not energy, but energy density.

(For the experts: I’m justifying this phrasing in part because of how you can interpret the quantity appearing in energy conditions as the energy density measured by an observer. This still isn’t the correct way to put it, but I can’t think of a better way that would be understandable to a non-technical reader. If you have one, let me know!)

Why do we need quantum gravity to fully understand black holes? Not just because they have a lot of mass, but because they have a lot of mass concentrated in a small area, a high energy density. Ditto for the Big Bang, when the whole universe had a very large energy density. Particle colliders are useful not just because they give particles high energy, but because they give particles high energy and put them close together, creating a situation with very high energy density.

Once you understand this, you can use it to think about whether some experiment or observation will help with the problem of quantum gravity. Does the experiment involve very high energy density, much higher than anything we can do in a particle collider right now? Is that telescope looking at something created in conditions of very high energy density, or just something nearby?

It’s not impossible for an experiment that doesn’t meet these conditions to find something. Whatever the correct quantum gravity theory is, it might be different from our current theories in a more dramatic way, one that’s easier to measure. But the only guarantee, the only situation where we know we need a new theory, is for very high energy density.

# Simulated Wormholes for My Real Friends, Real Wormholes for My Simulated Friends

Maybe you’ve recently seen a headline like this:

Actually, I’m more worried that you saw that headline before it was edited, when it looked like this:

Physicists have not created an actual wormhole. They have simulated a wormhole on a quantum computer.

If you’re willing to read more, then read the rest of this post. There’s a more subtle story going on here, both about physics and about how we communicate it. And for the experts, hold on, because when I say the wormhole was a simulation I’m not making the same argument everyone else is.

[And for the mega-experts, there’s an edit later in the post where I soften that claim a bit.]

The headlines at the top of this post come from an article in Quanta Magazine. Quanta is a web-based magazine covering many fields of science. They’re read by the general public, but they aim for a higher standard than many science journalists, with stricter fact-checking and a goal of covering more challenging and obscure topics. Scientists in turn have tended to be quite happy with them: often, they cover things we feel are important but that the ordinary media isn’t able to cover. (I even wrote something for them recently.)

Last week, Quanta published an article about an experiment with Google’s Sycamore quantum computer. By arranging the quantum bits (qubits) in a particular way, they were able to observe behaviors one would expect out of a wormhole, a kind of tunnel linking different points in space and time. They published it with the second headline above, claiming that physicists had created a wormhole with a quantum computer and explaining how, using a theoretical picture called holography.

This pissed off a lot of physicists. After push-back, Quanta’s twitter account published this statement, and they added the word “Holographic” to the title.

Why were physicists pissed off?

It wasn’t because the Quanta article was wrong, per se. As far as I’m aware, all the technical claims they made are correct. Instead, it was about two things. One was the title, and the implication that physicists “really made a wormhole”. The other was the tone, the excited “breaking news” framing complete with a video comparing the experiment with the discovery of the Higgs boson. I’ll discuss each in turn:

The Title

Did physicists really create a wormhole, or did they simulate one? And why would that be at all confusing?

The story rests on a concept from the study of quantum gravity, called holography. Holography is the idea that in quantum gravity, certain gravitational systems like black holes are fully determined by what happens on a “boundary” of the system, like the event horizon of a black hole. It’s supposed to be a hologram in analogy to 3d images encoded in 2d surfaces, rather than like the hard-light constructions of science fiction.

The best-studied version of holography is something called AdS/CFT duality. AdS/CFT duality is a relationship between two different theories. One of them is a CFT, or “conformal field theory”, a type of particle physics theory with no gravity and no mass. (The first example of the duality used my favorite toy theory, N=4 super Yang-Mills.) The other one is a version of string theory in an AdS, or anti-de Sitter space, a version of space-time curved so that objects shrink as they move outward, approaching a boundary. (In the first example, this space-time had five dimensions curled up in a sphere and the rest in the anti-de Sitter shape.)

These two theories are conjectured to be “dual”. That means that, for anything that happens in one theory, you can give an alternate description using the other theory. We say the two theories “capture the same physics”, even though they appear very different: they have different numbers of dimensions of space, and only one has gravity in it.

Many physicists would claim that if two theories are dual, then they are both “equally real”. Even if one description is more familiar to us, both descriptions are equally valid. Many philosophers are skeptical, but honestly I think the physicists are right about this one. Philosophers try to figure out which things are real or not real, to make a list of real things and explain everything else as made up of those in some way. I think that whole project is misguided, that it’s clarifying how we happen to talk rather than the nature of reality. In my mind, dualities are some of the clearest evidence that this project doesn’t make any sense: two descriptions can look very different, but in a quite meaningful sense be totally indistinguishable.

That’s the sense in which Quanta and Google and the string theorists they’re collaborating with claim that physicists have created a wormhole. They haven’t created a wormhole in our own space-time, one that, were it bigger and more stable, we could travel through. It isn’t progress towards some future where we actually travel the galaxy with wormholes. Rather, they created some quantum system, and that system’s dual description is a wormhole. That’s a crucial point to remember: even if they created a wormhole, it isn’t a wormhole for you.

If that were the end of the story, this post would still be full of warnings, but the title would be a bit different. It was going to be “Dual Wormholes for My Real Friends, Real Wormholes for My Dual Friends”. But there’s a list of caveats. Most of them arguably don’t matter, but the last was what got me to change the word “dual” to “simulated”.

1. The real world is not described by N=4 super Yang-Mills theory. N=4 super Yang-Mills theory was never intended to describe the real world. And while the real world may well be described by string theory, those strings are not curled up around a five-dimensional sphere with the remaining dimensions in anti-de Sitter space. We can’t create either theory in a lab either.
2. The Standard Model probably has a quantum gravity dual too, see this cute post by Matt Strassler. But they still wouldn’t have been able to use that to make a holographic wormhole in a lab.
3. Instead, they used a version of AdS/CFT with fewer dimensions. It relates a weird form of gravity in one space and one time dimension (called JT gravity), to a weird quantum mechanics theory called SYK, with an infinite number of quantum particles or qubits. This duality is a bit more conjectural than the original one, but still reasonably well-established.
4. Quantum computers don’t have an infinite number of qubits, so they had to use a version with a finite number: seven, to be specific. They trimmed the model down so that it would still show the wormhole-dual behavior they wanted. At this point, you might say that they’re definitely just simulating the SYK theory, using a small number of qubits to simulate the infinite number. But I think they could argue that this system, too, has a quantum gravity dual. The dual would have to be even weirder than JT gravity, and even more conjectural, but the signs of wormhole-like behavior they observed (mostly through simulations on an ordinary computer, which is still better at this kind of thing than a quantum computer) could be seen as evidence that this limited theory has its own gravity partner, with its own “real dual” wormhole.
5. But those seven qubits don’t just have the interactions they were programmed to have, the ones with the dual. They are physical objects in the real world, so they interact with all of the forces of the real world. That includes, though very weakly, the force of gravity.

And that’s where I think things break, and you have to call the experiment a simulation. You can argue, if you really want to, that the seven-qubit SYK theory has its own gravity dual, with its own wormhole. There are people who expect duality to be broad enough to include things like that.

But you can’t argue that the seven-qubit SYK theory, plus gravity, has its own gravity dual. Theories that already have gravity are not supposed to have gravity duals. If you pushed hard enough on any of the string theorists on that team, I’m pretty sure they’d admit that.

That is what decisively makes the experiment a simulation. It approximately behaves like a system with a dual wormhole, because you can approximately ignore gravity. But if you’re making some kind of philosophical claim, that you “really made a wormhole”, then “approximately” doesn’t cut it: if you don’t exactly have a system with a dual, then you don’t “really” have a dual wormhole: you’ve just simulated one.

Edit: mitchellporter in the comments points out something I didn’t know: that there are in fact proposals for gravity theories with gravity duals. They are in some sense even more conjectural than the series of caveats above, but at minimum my claim above, that any of the string theorists on the team would agree that the system’s gravity means it can’t have a dual, is probably false.

I think at this point, I’d soften my objection to the following:

Describing the system of qubits in the experiment as a limited version of the SYK theory is in one way or another an approximation. It approximates them as not having any interactions beyond those they programmed, it approximates them as not affected by gravity, and because it’s a quantum mechanical description it even approximates the speed of light as small. Those approximations don’t guarantee that the system doesn’t have a gravity dual. But in order for them to, then our reality, overall, would have to have a gravity dual. There would have to be a dual gravity interpretation of everything, not just the inside of Google’s quantum computer, and it would have to be exact, not just an approximation. Then the approximate SYK would be dual to an approximate wormhole, but that approximate wormhole would be an approximation of some “real” wormhole in the dual space-time.

That’s not impossible, as far as I can tell. But it piles conjecture upon conjecture upon conjecture, to the point that I don’t think anyone has explicitly committed to the whole tower of claims. If you want to believe that this experiment literally created a wormhole, you thus can, but keep in mind the largest asterisk known to mankind.

End edit.

If it weren’t for that caveat, then I would be happy to say that the physicists really created a wormhole. It would annoy some philosophers, but that’s a bonus.

But even if that were true, I wouldn’t say that in the title of the article.

The Title, Again

These days, people get news in two main ways.

Sometimes, people read full news articles. Reading that Quanta article is a good way to understand the background of the experiment, what was done and why people care about it. As I mentioned earlier, I don’t think anything said there was wrong, and they cover essentially all of the caveats you’d care about (except for that last one 😉 ).

Sometimes, though, people just see headlines. They get forwarded on social media, observed at a glance passed between friends. If you’re popular enough, then many more people will see your headline than will actually read the article. For many people, their whole understanding of certain scientific fields is formed by these glancing impressions.

Because of that, if you’re popular and news-y enough, you have to be especially careful with what you put in your headlines, especially when it implies a cool science fiction story. People will almost inevitably see them out of context, and it will impact their view of where science is headed. In this case, the headline may have given many people the impression that we’re actually making progress towards travel via wormholes.

Some of my readers might think this is ridiculous, that no-one would believe something like that. But as a kid, I did. I remember reading popular articles about wormholes, describing how you’d need energy moving in a circle, and other articles about optical physicists finding ways to bend light and make it stand still. Putting two and two together, I assumed these ideas would one day merge, allowing us to travel to distant galaxies faster than light.

If I had seen Quanta’s headline at that age, I would have taken it as confirmation. I would have believed we were well on the way to making wormholes, step by step. Even the New York Times headline, “the Smallest, Crummiest Wormhole You Can Imagine”, wouldn’t have fazed me.

(I’m not sure even the extra word “holographic” would have. People don’t know what “holographic” means in this context, and while some of them would assume it meant “fake”, others would think about the many works of science fiction, like Star Trek, where holograms can interact physically with human beings.)

Quanta has a high-brow audience, many of whom wouldn’t make this mistake. Nevertheless, I think Quanta is popular enough, and respectable enough, that they should have done better here.

At minimum, they could have used the word “simulated”. Even if they go on to argue in the article that the wormhole is real, and not just a simulation, the word in the title does no real harm. It would be a lie, but a beneficial “lie to children”, the basic stock-in-trade of science communication. I think they could have defended it to the string theorists they interviewed on those grounds.

The Tone

Honestly, I don’t think people would have been nearly so pissed off were it not for the tone of the article. There are a lot of physics bloggers who view themselves as serious-minded people, opposed to hype and publicity stunts. They view the research program aimed at simulating quantum gravity on a quantum computer as just an attempt to link a dying and un-rigorous research topic to an over-hyped and over-funded one, pompous storytelling aimed at promoting the careers of people who are already extremely successful.

These people tend to view Quanta favorably, because it covers serious-minded topics in a thorough way. And so many of them likely felt betrayed, seeing this Quanta article as a massive failure of that serious-minded-ness, falling for or even endorsing the hypiest of hype.

To those people, I’d like to politely suggest you get over yourselves.

Quanta’s goal is to cover things accurately, to represent all the facts in a way people can understand. But “how exciting something is” is not a fact.

Excitement is subjective. Just because most of the things Quanta finds exciting you also find exciting, does not mean that Quanta will find the things you find unexciting unexciting. Quanta is not on “your side” in some war against your personal notion of unexciting science, and you should never have expected it to be.

In fact, Quanta tends to find things exciting, in general. They were more excited than I was about the amplituhedron, and I’m an amplitudeologist. Part of what makes them consistently excited about the serious-minded things you appreciate them for is that they listen to scientists and get excited about the things they’re excited about. That is going to include, inevitably, things those scientists are excited about for what you think are dumb groupthinky hype reasons.

I think the way Quanta titled the piece was unfortunate, and probably did real damage. I think the philosophical claim behind the title is wrong, though for subtle and weird enough reasons that I don’t really fault anybody for ignoring them. But I don’t think the tone they took was a failure of journalistic integrity or research or anything like that. It was a matter of taste. It’s not my taste, it’s probably not yours, but we shouldn’t have expected Quanta to share our tastes in absolutely everything. That’s just not how taste works.

# 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 Bohr-100: Current Themes in Theoretical Physics

During the pandemic, some conferences went online. Others went dormant.

Every summer before the pandemic, the Niels Bohr International Academy hosted a conference called Current Themes in High Energy Physics and Cosmology. Current Themes is a small, cozy conference, a gathering of close friends some of whom happen to have Nobel prizes. Holding it online would be almost missing the point.

Instead, we waited. Now, at least in Denmark, the pandemic is quiet enough to hold this kind of gathering. And it’s a special year: the 100th anniversary of Niels Bohr’s Nobel, the 101st of the Niels Bohr Institute. So it seemed like the time for a particularly special Current Themes.

A particularly special Current Themes means some unusually special guests. Our guests are usually pretty special already (Gerard t’Hooft and David Gross are regulars, to just name the Nobelists), but this year we also had Alexander Polyakov. Polyakov’s talk had a magical air to it. In a quiet voice, broken by an impish grin when he surprised us with a joke, Polyakov began to lay out five unsolved problems he considered interesting. In the end, he only had time to present one, related to turbulence: when Gross asked him to name the remaining four, the second included a term most of us didn’t recognize (striction, known in a magnetic context and which he wanted to explore gravitationally), so the discussion hung while he defined that and we never did learn what the other three problems were.

At the big 100th anniversary celebration earlier in the spring, the Institute awarded a few years worth of its Niels Bohr Institute Medal of Honor. One of the recipients, Paul Steinhardt, couldn’t make it then, so he got his medal now. After the obligatory publicity photos were taken, Steinhardt entertained us all with a colloquium about his work on quasicrystals, including the many adventures involved in finding the first example “in the wild”. I can’t do the story justice in a short blog post, but if you won’t have the opportunity to watch him speak about it then I hear his book is good.

An anniversary conference should have some historical elements as well. For this one, these were ably provided by David Broadhurst, who gave an after-dinner speech cataloguing things he liked about Bohr. Some was based on public information, but the real draw were the anecdotes: his own reminiscences, and those of people he knew who knew Bohr well.

The other talks covered interesting ground: from deep approaches to quantum field theory, to new tools to understand black holes, to the implications of causality itself. One out of the ordinary talk was by Sabrina Pasterski, who advocated a new model of physics funding. I liked some elements (endowed organizations to further a subfield) and am more skeptical of others (mostly involving NFTs). Regardless it, and the rest of the conference more broadly, spurred a lot of good debate.

# The Undefinable

If I can teach one lesson to all of you, it’s this: be precise. In physics, we try to state what we mean as precisely as we can. If we can’t state something precisely, that’s a clue: maybe what we’re trying to state doesn’t actually make sense.

Someone recently reached out to me with a question about black holes. He was confused about how they were described, about what would happen when you fall in to one versus what we could see from outside. Part of his confusion boiled down to a question: “is the center really an infinitely small point?”

I remembered a commenter a while back who had something interesting to say about this. Trying to remind myself of the details, I dug up this question on Physics Stack Exchange. user4552 has a detailed, well-referenced answer, with subtleties of General Relativity that go significantly beyond what I learned in grad school.

According to user4552, the reason this question is confusing is that the usual setup of general relativity cannot answer it. In general relativity, singularities like the singularity in the middle of a black hole aren’t treated as points, or collections of points: they’re not part of space-time at all. So you can’t count their dimensions, you can’t see whether they’re “really” infinitely small points, or surfaces, or lines…

This might surprise people (like me) who have experience with simpler equations for these things, like the Schwarzchild metric. The Schwarzchild metric describes space-time around a black hole, and in the usual coordinates it sure looks like the singularity is at a single point where r=0, just like the point where r=0 is a single point in polar coordinates in flat space. The thing is, though, that’s just one sort of coordinates. You can re-write a metric in many different sorts of coordinates, and the singularity in the center of a black hole might look very different in those coordinates. In general relativity, you need to stick to things you can say independent of coordinates.

Ok, you might say, so the usual mathematics can’t answer the question. Can we use more unusual mathematics? If our definition of dimensions doesn’t tell us whether the singularity is a point, maybe we just need a new definition!

According to user4552, people have tried this…and it only sort of works. There are several different ways you could define the dimension of a singularity. They all seem reasonable in one way or another. But they give different answers! Some say they’re points, some say they’re three-dimensional. And crucially, there’s no obvious reason why one definition is “right”. The question we started with, “is the center really an infinitely small point?”, looked like a perfectly reasonable question, but it actually wasn’t: the question wasn’t precise enough.

This is the real problem. The problem isn’t that our question was undefined, after all, we can always add new definitions. The problem was that our question didn’t specify well enough the definitions we needed. That is why the question doesn’t have an answer.

Once you understand the difference, you see these kinds of questions everywhere. If you’re baffled by how mass could have come out of the Big Bang, or how black holes could radiate particles in Hawking radiation, maybe you’ve heard a physicist say that energy isn’t always conserved. Energy conservation is a consequence of symmetry, specifically, symmetry in time. If your space-time itself isn’t symmetric (the expanding universe making the past different from the future, a collapsing star making a black hole), then you shouldn’t expect energy to be conserved.

I sometimes hear people object to this. They ask, is it really true that energy isn’t conserved when space-time isn’t symmetric? Shouldn’t we just say that space-time itself contains energy?

And well yes, you can say that, if you want. It isn’t part of the usual definition, but you can make a new definition, one that gives energy to space-time. In fact, you can make more than one new definition…and like the situation with the singularity, these definitions don’t always agree! Once again, you asked a question you thought was sensible, but it wasn’t precise enough to have a definite answer.

Keep your eye out for these kinds of questions. If scientists seem to avoid answering the question you want, and keep answering a different question instead…it might be their question is the only one with a precise answer. You can define a method to answer your question, sure…but it won’t be the only way. You need to ask precise enough questions to get good answers.

# Duality and Emergence: When Is Spacetime Not Spacetime?

Spacetime is doomed! At least, so say some physicists. They don’t mean this as a warning, like some comic-book universe-destroying disaster, but rather as a research plan. These physicists believe that what we think of as space and time aren’t the full story, but that they emerge from something more fundamental, so that an ultimate theory of nature might not use space or time at all. Other, grumpier physicists are skeptical. Joined by a few philosophers, they think the “spacetime is doomed” crowd are over-excited and exaggerating the implications of their discoveries. At the heart of the argument is the distinction between two related concepts: duality and emergence.

In physics, sometimes we find that two theories are actually dual: despite seeming different, the patterns of observations they predict are the same. Some of the more popular examples are what we call holographic theories. In these situations, a theory of quantum gravity in some space-time is dual to a theory without gravity describing the edges of that space-time, sort of like how a hologram is a 2D image that looks 3D when you move it. For any question you can ask about the gravitational “bulk” space, there is a matching question on the “boundary”. No matter what you observe, neither description will fail.

If theories with gravity can be described by theories without gravity, does that mean gravity doesn’t really exist? If you’re asking that question, you’re asking whether gravity is emergent. An emergent theory is one that isn’t really fundamental, but instead a result of the interaction of more fundamental parts. For example, hydrodynamics, the theory of fluids like water, emerges from more fundamental theories that describe the motion of atoms and molecules.

(For the experts: I, like most physicists, am talking about “weak emergence” here, not “strong emergence”.)

The “spacetime is doomed” crowd think that not just gravity, but space-time itself is emergent. They expect that distances and times aren’t really fundamental, but a result of relationships that will turn out to be more fundamental, like entanglement between different parts of quantum fields. As evidence, they like to bring up dualities where the dual theories have different concepts of gravity, number of dimensions, or space-time. Using those theories, they argue that space and time might “break down”, and not be really fundamental.

The skeptics, though, bring up an important point. If two theories are really dual, then no observation can distinguish them: they make exactly the same predictions. In that case, say the skeptics, what right do you have to call one theory more fundamental than the other? You can say that gravity emerges from a boundary theory without gravity, but you could just as easily say that the boundary theory emerges from the gravity theory. The whole point of duality is that no theory is “more true” than the other: one might be more or less convenient, but both describe the same world. If you want to really argue for emergence, then your “more fundamental” theory needs to do something extra: to predict something that your emergent theory doesn’t predict.

Sometimes this is a fair objection. There are members of the “spacetime is doomed” crowd who are genuinely reckless about this, who’ll tell a journalist about emergence when they really mean duality. But many of these people are more careful, and have thought more deeply about the question. They tend to have some mix of these two perspectives:

First, if two descriptions give the same results, then do the descriptions matter? As physicists, we have a history of treating theories as the same if they make the same predictions. Space-time itself is a result of this policy: in the theory of relativity, two people might disagree on which one of two events happened first or second, but they will agree on the overall distance in space-time between the two. From this perspective, a duality between a bulk theory and a boundary theory isn’t evidence that the bulk theory emerges from the boundary, but it is evidence that both the bulk and boundary theories should be replaced by an “overall theory”, one that treats bulk and boundary as irrelevant descriptions of the same physical reality. This perspective is similar to an old philosophical theory called positivism: that statements are meaningless if they cannot be derived from something measurable. That theory wasn’t very useful for philosophers, which is probably part of why some philosophers are skeptics of “space-time is doomed”. The perspective has been quite useful to physicists, though, so we’re likely to stick with it.

Second, some will say that it’s true that a dual theory is not an emergent theory…but it can be the first step to discover one. In this perspective, dualities are suggestive evidence that a deeper theory is waiting in the wings. The idea would be that one would first discover a duality, then discover situations that break that duality: examples on one side that don’t correspond to anything sensible on the other. Maybe some patterns of quantum entanglement are dual to a picture of space-time, but some are not. (Closer to my sub-field, maybe there’s an object like the amplituhedron that doesn’t respect locality or unitarity.) If you’re lucky, maybe there are situations, or even experiments, that go from one to the other: where the space-time description works until a certain point, then stops working, and only the dual description survives. Some of the models of emergent space-time people study are genuinely of this type, where a dimension emerges in a theory that previously didn’t have one. (For those of you having a hard time imagining this, read my old post about “bubbles of nothing”, then think of one happening in reverse.)

It’s premature to say space-time is doomed, at least as a definite statement. But it is looking like, one way or another, space-time won’t be the right picture for fundamental physics. Maybe that’s because it’s equivalent to another description, redundant embellishment on an essential theoretical core. Maybe instead it breaks down, and a more fundamental theory could describe more situations. We don’t know yet. But physicists are trying to figure it out.

# Classicality Has Consequences

Last week, I mentioned some interesting new results in my corner of physics. I’ve now finally read the two papers and watched the recorded talk, so I can satisfy my frustrated commenters.

Quantum mechanics is a very cool topic and I am much less qualified than you would expect to talk about it. I use quantum field theory, which is based on quantum mechanics, so in some sense I use quantum mechanics every day. However, most of the “cool” implications of quantum mechanics don’t come up in my work. All the debates about whether measurement “collapses the wavefunction” are irrelevant when the particles you measure get absorbed in a particle detector, never to be seen again. And while there are deep questions about how a classical world emerges from quantum probabilities, they don’t matter so much when all you do is calculate those probabilities.

They’ve started to matter, though. That’s because quantum field theorists like me have recently started working on a very different kind of problem: trying to predict the output of gravitational wave telescopes like LIGO. It turns out you can do almost the same kind of calculation we’re used to: pretend two black holes or neutron stars are sub-atomic particles, and see what happens when they collide. This trick has grown into a sub-field in its own right, one I’ve dabbled in a bit myself. And it’s gotten my kind of physicists to pay more attention to the boundary between classical and quantum physics.

The thing is, the waves that LIGO sees really are classical. Any quantum gravity effects there are tiny, undetectably tiny. And while this doesn’t have the implications an expert might expect (we still need loop diagrams), it does mean that we need to take our calculations to a classical limit.

Figuring out how to do this has been surprisingly delicate, and full of unexpected insight. A recent example involves two papers, one by Andrea Cristofoli, Riccardo Gonzo, Nathan Moynihan, Donal O’Connell, Alasdair Ross, Matteo Sergola, and Chris White, and one by Ruth Britto, Riccardo Gonzo, and Guy Jehu. At first I thought these were two groups happening on the same idea, but then I noticed Riccardo Gonzo on both lists, and realized the papers were covering different aspects of a shared story. There is another group who happened upon the same story: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele Veneziano. They haven’t published yet, so I’m basing this on the Gonzo et al papers.

The key question each group asked was, what does it take for gravitational waves to be classical? One way to ask the question is to pick something you can observe, like the strength of the field, and calculate its uncertainty. Classical physics is deterministic: if you know the initial conditions exactly, you know the final conditions exactly. Quantum physics is not. What should happen is that if you calculate a quantum uncertainty and then take the classical limit, that uncertainty should vanish: the observation should become certain.

Another way to ask is to think about the wave as made up of gravitons, particles of gravity. Then you can ask how many gravitons are in the wave, and how they are distributed. It turns out that you expect them to be in a coherent state, like a laser, one with a very specific distribution called a Poisson distribution: a distribution in some sense right at the border between classical and quantum physics.

The results of both types of questions were as expected: the gravitational waves are indeed classical. To make this work, though, the quantum field theory calculation needs to have some surprising properties.

If two black holes collide and emit a gravitational wave, you could depict it like this:

where the straight lines are black holes, and the squiggly line is a graviton. But since gravitational waves are made up of multiple gravitons, you might ask, why not depict it with two gravitons, like this?

It turns out that diagrams like that are a problem: they mean your two gravitons are correlated, which is not allowed in a Poisson distribution. In the uncertainty picture, they also would give you non-zero uncertainty. Somehow, in the classical limit, diagrams like that need to go away.

And at first, it didn’t look like they do. You can try to count how many powers of Planck’s constant show up in each diagram. The authors do that, and it certainly doesn’t look like it goes away:

Luckily, these quantum field theory calculations have a knack for surprising us. Calculate each individual diagram, and things look hopeless. But add them all together, and they miraculously cancel. In the classical limit, everything combines to give a classical result.

You can do this same trick for diagrams with more graviton particles, as many as you like, and each time it ought to keep working. You get an infinite set of relationships between different diagrams, relationships that have to hold to get sensible classical physics. From thinking about how the quantum and classical are related, you’ve learned something about calculations in quantum field theory.

That’s why these papers caught my eye. A chunk of my sub-field is needing to learn more and more about the relationship between quantum and classical physics, and it may have implications for the rest of us too. In the future, I might get a bit more qualified to talk about some of the very cool implications of quantum mechanics.

# Outreach Talk on Math’s Role in Physics

Tonight is “Culture Night” in Copenhagen, the night when the city throws open its doors and lets the public in. Museums and hospitals, government buildings and even the Freemasons, all have public events. The Niels Bohr Institute does too, of course: an evening of physics exhibits and demos, capped off with a public lecture by Denmark’s favorite bow-tie wearing weirder-than-usual string theorist, Holger Bech Nielsen. In between, there are a number of short talks by various folks at the institute, including yours truly.

In my talk, I’m going to try and motivate the audience to care about math. Math is dry of course, and difficult for some, but we physicists need it to do our jobs. If you want to be precise about a claim in physics, you need math simply to say what you want clearly enough.

Since you guys likely don’t overlap with my audience tonight, it should be safe to give a little preview. I’ll be using a few examples, but this one is the most complicated:

I’ll be telling a story I stole from chapter seven of the web serial Almost Nowhere. (That link is to the first chapter, by the way, in case you want to read the series without spoilers. It’s very strange, very unique, and at least in my view quite worth reading.) You follow a warrior carrying a spear around a globe in two different paths. The warrior tries to always point in the same direction, but finds that the two different paths result in different spears when they meet. The story illustrates that such a simple concept as “what direction you are pointing” isn’t actually so simple: if you want to think about directions in curved space (like the surface of the Earth, but also, like curved space-time in general relativity) then you need more sophisticated mathematics (a notion called parallel transport) to make sense of it.

It’s kind of an advanced concept for a public talk. But seeing it show up in Almost Nowhere inspired me to try to get it across. I’ll let you know how it goes!

By the way, if you are interested in learning the kinds of mathematics you need for theoretical physics, and you happen to be a Bachelor’s student planning to pursue a PhD, then consider the Perimeter Scholars International Master’s Program! It’s a one-year intensive at the Perimeter Institute in Waterloo, Ontario, in Canada. In a year it gives you a crash course in theoretical physics, giving you tools that will set you ahead of other beginning PhD students. I’ve witnessed it in action, and it’s really remarkable how much the students learn in a year, and what they go on to do with it. Their early registration deadline is on November 15, just a month away, so if you’re interested you may want to start thinking about it.