Tag Archives: string theory

What Might Lie Beyond, and Why

As the new year approaches, people think about the future. Me, I’m thinking about the future of fundamental physics, about what might lie beyond the Standard Model. Physicists search for many different things, with many different motivations. Some are clear missing pieces, places where the Standard Model fails and we know we’ll need to modify it. Others are based on experience, with no guarantees but an expectation that, whatever we find, it will be surprising. Finally, some are cool possibilities, ideas that would explain something or fill in a missing piece but aren’t strictly necessary.

The Almost-Sure Things

Science isn’t math, so nothing here is really a sure thing. We might yet discover a flaw in important principles like quantum mechanics and special relativity, and it might be that an experimental result we trust turns out to be flawed. But if we chose to trust those principles, and our best experiments, then these are places we know the Standard Model is incomplete:

  • Neutrino Masses: The original Standard Model’s neutrinos were massless. Eventually, physicists discovered this was wrong: neutrinos oscillate, switching between different types in a way they only could if they had different masses. This result is familiar enough that some think of it as already part of the Standard Model, not really beyond. But the masses of neutrinos involve unsolved mysteries: we don’t know what those masses are, but more, there are different ways neutrinos could have mass, and we don’t yet know which is present in nature. Neutrino masses also imply the existence of an undiscovered “sterile” neutrino, a particle that doesn’t interact with the strong, weak, or electromagnetic forces.
  • Dark Matter Phenomena (and possibly Dark Energy Phenomena): Astronomers first suggested dark matter when they observed galaxies moving at speeds inconsistent with the mass of their stars. Now, they have observed evidence for it in a wide variety of situations, evidence which seems decisively incompatible with ordinary gravity and ordinary matter. Some solve this by introducing dark matter, others by modifying gravity, but this is more of a technical difference than it sounds: in order to modify gravity, one must introduce new quantum fields, much the same way one does when introducing dark matter. The only debate is how “matter-like” those fields need to be, but either approach goes beyond the Standard Model.
  • Quantum Gravity: It isn’t as hard to unite quantum mechanics and gravity as you might think. Physicists have known for decades how to write down a naive theory of quantum gravity, one that follows the same steps one might use to derive the quantum theory of electricity and magnetism. The problem is, this theory is incomplete. It works at low energies, but as the energy increases it loses the ability to make predictions, eventually giving nonsensical answers like probabilities greater than one. We have candidate solutions to this problem, like string theory, but we might not know for a long time which solution is right.
  • Landau Poles: Here’s a more obscure one. In particle physics we can zoom in and out in our theories, using similar theories at different scales. What changes are the coupling constants, numbers that determine the strength of the different forces. You can think of this in a loosely reductionist way, with the theories at smaller scales determining the constants for theories at larger scales. This gives workable theories most of the time, but it fails for at least one part of the Standard Model. In electricity and magnetism, the coupling constant increases as you zoom in. Eventually, it becomes infinite, and what’s more, does so at a finite energy scale. It’s still not clear how we should think about this, but luckily we won’t have to very soon: this energy scale is vastly vastly higher than even the scale of quantum gravity.
  • Some Surprises Guarantee Others: The Standard Model is special in a way that gravity isn’t. Even if you dial up the energy, a Standard Model calculation will always “make sense”: you never get probabilities greater than one. This isn’t true for potential deviations from the Standard Model. If the Higgs boson turns out to interact differently than we expect, it wouldn’t just be a violation of the Standard Model on its own: it would guarantee mathematically that, at some higher energy, we’d have to find something new. That was precisely the kind of argument the LHC used to find the Higgs boson: without the Higgs, something new was guaranteed to happen within the energy range of the LHC to prevent impossible probability numbers.

The Argument from (Theoretical) Experience

Everything in this middle category rests on a particular sort of argument. It’s short of a guarantee, but stronger than a dream or a hunch. While the previous category was based on calculations in theories we already know how to write down, this category relies on our guesses about theories we don’t yet know how to write.

Suppose we had a deeper theory, one that could use fewer parameters to explain the many parameters of the Standard Model. For example, it might explain the Higgs mass, letting us predict it rather than just measuring it like we do now. We don’t have a theory like that yet, but what we do have are many toy model theories, theories that don’t describe the real world but do, in this case, have fewer parameters. We can observe how these theories work, and what kinds of discoveries scientists living in worlds described by them would make. By looking at this process, we can get a rough idea of what to expect, which things in our own world would be “explained” in other ways in these theories.

  • The Hierarchy Problem: This is also called the naturalness problem. Suppose we had a theory that explained the mass of the Higgs, one where it wasn’t just a free parameter. We don’t have such a theory for the real Higgs, but we do have many toy models with similar behavior, ones with a boson with its mass determined by something else. In these models, though, the mass of the boson is always close to the energy scale of other new particles, particles which have a role in determining its mass, or at least in postponing that determination. This was the core reason why people expected the LHC to find something besides the Higgs. Without such new particles, the large hierarchy between the mass of the Higgs and the mass of new particles becomes a mystery, one where it gets harder and harder to find a toy model with similar behavior that still predicts something like the Higgs mass.
  • The Strong CP Problem: The weak nuclear force does what must seem like a very weird thing, by violating parity symmetry: the laws that govern it are not the same when you flip the world in a mirror. This is also true when you flip all the charges as well, a combination called CP (charge plus parity). But while it may seem strange that the weak force violates this symmetry, physicists find it stranger that the strong force seems to obey it. Much like in the hierarchy problem, it is very hard to construct a toy model that both predicts a strong force that maintains CP (or almost maintains it) and doesn’t have new particles. The new particle in question, called the axion, is something some people also think may explain dark matter.
  • Matter-Antimatter Asymmetry: We don’t know the theory of quantum gravity. Even if we did, the candidate theories we have struggle to describe conditions close to the Big Bang. But while we can’t prove it, many physicists expect the quantum gravity conditions near the Big Bang to produce roughly equal amounts of matter and antimatter. Instead, matter dominates: we live in a world made almost entirely of matter, with no evidence of large antimatter areas even far out in space. This lingering mystery could be explained if some new physics was biased towards matter instead of antimatter.
  • Various Problems in Cosmology: Many open questions in cosmology fall in this category. The small value of the cosmological constant is mysterious for the same reasons the small value of the Higgs mass is, but at a much larger and harder to fix scale. The early universe surprises many cosmologists by its flatness and uniformity, which has led them to propose new physics. This surprise is not because such flatness and uniformity is mathematically impossible, but because it is not the behavior they would expect out of a theory of quantum gravity.

The Cool Possibilities

Some ideas for physics beyond the standard model aren’t required, either from experience or cold hard mathematics. Instead, they’re cool, and would be convenient. These ideas would explain things that look strange, or make for a simpler deeper theory, but they aren’t the only way to do so.

  • Grand Unified Theories: Not the same as a “theory of everything”, Grand Unified Theories unite the three “particle physics forces”: the strong nuclear force, the weak nuclear force, and electromagnetism. Under such a theory, the different parameters that determine the strengths of those forces could be predicted from one shared parameter, with the forces only seeming different at low energies. These theories often unite the different matter particles too, but they also introduce new particles and new forces. These forces would, among other things, make protons unstable, and so giant experiments have been constructed to try to detect a proton decaying into other particles. So far none has been seen.
  • Low-Energy Supersymmetry: String theory requires supersymmetry, a relationship where matter and force particles share many properties. That supersymmetry has to be “broken”, which means that while the matter and force particles have the same charges, they can have wildly different masses, so that the partner particles are all still undiscovered. Those masses may be extremely high, all the way up at the scale of quantum gravity, but they could also be low enough to test at the LHC. Physicists hoped to detect such particles, as they could have been a good solution to the hierarchy problem. Now that the LHC hasn’t found these supersymmetric particles, it is much harder to solve the problem this way, though some people are still working on it.
  • Large Extra Dimensions: String theory also involves extra dimensions, beyond our usual three space and one time. Those dimensions are by default very small, but some proposals have them substantially bigger, big enough that we could have seen evidence for them at the LHC. These proposals could explain why gravity is so much weaker than the other forces. Much like the previous members of this category though, no evidence for this has yet been found.

I think these categories are helpful, but experts may quibble about some of my choices. I also haven’t mentioned every possible thing that could be found beyond the Standard Model. If you’ve heard of something and want to know which category I’d put it in, let me know in the comments!

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:

If you’ve seen either headline, and haven’t read anything else about it, then please at least read 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.

Machine Learning, Occam’s Razor, and Fundamental Physics

There’s a saying in physics, attributed to the famous genius John von Neumann: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.”

Say you want to model something, like some surprising data from a particle collider. You start with some free parameters: numbers in your model that aren’t decided yet. You then decide those numbers, “fixing” them based on the data you want to model. Your goal is for your model not only to match the data, but to predict something you haven’t yet measured. Then you can go out and check, and see if your model works.

The more free parameters you have in your model, the easier this can go wrong. More free parameters make it easier to fit your data, but that’s because they make it easier to fit any data. Your model ends up not just matching the physics, but matching the mistakes as well: the small errors that crop up in any experiment. A model like that may look like it’s a great fit to the data, but its predictions will almost all be wrong. It wasn’t just fit, it was overfit.

We have statistical tools that tell us when to worry about overfitting, when we should be impressed by a model and when it has too many parameters. We don’t actually use these tools correctly, but they still give us a hint of what we actually want to know, namely, whether our model will make the right predictions. In a sense, these tools form the mathematical basis for Occam’s Razor, the idea that the best explanation is often the simplest one, and Occam’s Razor is a critical part of how we do science.

So, did you know machine learning was just modeling data?

All of the much-hyped recent advances in artificial intelligence, GPT and Stable Diffusion and all those folks, at heart they’re all doing this kind of thing. They start out with a model (with a lot more than five parameters, arranged in complicated layers…), then use data to fix the free parameters. Unlike most of the models physicists use, they can’t perfectly fix these numbers: there are too many of them, so they have to approximate. They then test their model on new data, and hope it still works.

Increasingly, it does, and impressively well, so well that the average person probably doesn’t realize this is what it’s doing. When you ask one of these AIs to make an image for you, what you’re doing is asking what image the model predicts would show up captioned with your text. It’s the same sort of thing as asking an economist what their model predicts the unemployment rate will be when inflation goes up. The machine learning model is just way, way more complicated.

As a physicist, the first time I heard about this, I had von Neumann’s quote in the back of my head. Yes, these machines are dealing with a lot more data, from a much more complicated reality. They literally are trying to fit elephants, even elephants wiggling their trunks. Still, the sheer number of parameters seemed fishy here. And for a little bit things seemed even more fishy, when I learned about double descent.

Suppose you start increasing the number of parameters in your model. Initially, your model gets better and better. Your predictions have less and less error, your error descends. Eventually, though, the error increases again: you have too many parameters so you’re over-fitting, and your model is capturing accidents in your data, not reality.

In machine learning, weirdly, this is often not the end of the story. Sometimes, your prediction error rises, only to fall once more, in a double descent.

For a while, I found this deeply disturbing. The idea that you can fit your data, start overfitting, and then keep overfitting, and somehow end up safe in the end, was terrifying. The way some of the popular accounts described it, like you were just overfitting more and more and that was fine, was baffling, especially when they seemed to predict that you could keep adding parameters, keep fitting tinier and tinier fleas on the elephant’s trunk, and your predictions would never start going wrong. It would be the death of Occam’s Razor as we know it, more complicated explanations beating simpler ones off to infinity.

Luckily, that’s not what happens. And after talking to a bunch of people, I think I finally understand this enough to say something about it here.

The right way to think about double descent is as overfitting prematurely. You do still expect your error to eventually go up: your model won’t be perfect forever, at some point you will really overfit. It might take a long time, though: machine learning people are trying to model very complicated things, like human behavior, with giant piles of data, so very complicated models may often be entirely appropriate. In the meantime, due to a bad choice of model, you can accidentally overfit early. You will eventually overcome this, pushing past with more parameters into a model that works again, but for a little while you might convince yourself, wrongly, that you have nothing more to learn.

(You can even mitigate this by tweaking your setup, potentially avoiding the problem altogether.)

So Occam’s Razor still holds, but with a twist. The best model is simple enough, but no simpler. And if you’re not careful enough, you can convince yourself that a too-simple model is as complicated as you can get.

Image from Astral Codex Ten

I was reminded of all this recently by some articles by Sabine Hossenfelder.

Hossenfelder is a critic of mainstream fundamental physics. The articles were her restating a point she’s made many times before, including in (at least) one of her books. She thinks the people who propose new particles and try to search for them are wasting time, and the experiments motivated by those particles are wasting money. She’s motivated by something like Occam’s Razor, the need to stick to the simplest possible model that fits the evidence. In her view, the simplest models are those in which we don’t detect any more new particles any time soon, so those are the models she thinks we should stick with.

I tend to disagree with Hossenfelder. Here, I was oddly conflicted. In some of her examples, it seemed like she had a legitimate point. Others seemed like she missed the mark entirely.

Talk to most astrophysicists, and they’ll tell you dark matter is settled science. Indeed, there is a huge amount of evidence that something exists out there in the universe that we can’t see. It distorts the way galaxies rotate, lenses light with its gravity, and wiggled the early universe in pretty much the way you’d expect matter to.

What isn’t settled is whether that “something” interacts with anything else. It has to interact with gravity, of course, but everything else is in some sense “optional”. Astroparticle physicists use satellites to search for clues that dark matter has some other interactions: perhaps it is unstable, sometimes releasing tiny signals of light. If it did, it might solve other problems as well.

Hossenfelder thinks this is bunk (in part because she thinks those other problems are bunk). I kind of do too, though perhaps for a more general reason: I don’t think nature owes us an easy explanation. Dark matter isn’t obligated to solve any of our other problems, it just has to be dark matter. That seems in some sense like the simplest explanation, the one demanded by Occam’s Razor.

At the same time, I disagree with her substantially more on collider physics. At the Large Hadron Collider so far, all of the data is reasonably compatible with the Standard Model, our roughly half-century old theory of particle physics. Collider physicists search that data for subtle deviations, one of which might point to a general discrepancy, a hint of something beyond the Standard Model.

While my intuitions say that the simplest dark matter is completely dark, they don’t say that the simplest particle physics is the Standard Model. Back when the Standard Model was proposed, people might have said it was exceptionally simple because it had a property called “renormalizability”, but these days we view that as less important. Physicists like Ken Wilson and Steven Weinberg taught us to view theories as a kind of series of corrections, like a Taylor series in calculus. Each correction encodes new, rarer ways that particles can interact. A renormalizable theory is just the first term in this series. The higher terms might be zero, but they might not. We even know that some terms cannot be zero, because gravity is not renormalizable.

The two cases on the surface don’t seem that different. Dark matter might have zero interactions besides gravity, but it might have other interactions. The Standard Model might have zero corrections, but it might have nonzero corrections. But for some reason, my intuition treats the two differently: I would find it completely reasonable for dark matter to have no extra interactions, but very strange for the Standard Model to have no corrections.

I think part of where my intuition comes from here is my experience with other theories.

One example is a toy model called sine-Gordon theory. In sine-Gordon theory, this Taylor series of corrections is a very familiar Taylor series: the sine function! If you go correction by correction, you’ll see new interactions and more new interactions. But if you actually add them all up, something surprising happens. Sine-Gordon turns out to be a special theory, one with “no particle production”: unlike in normal particle physics, in sine-Gordon particles can neither be created nor destroyed. You would never know this if you did not add up all of the corrections.

String theory itself is another example. In string theory, elementary particles are replaced by strings, but you can think of that stringy behavior as a series of corrections on top of ordinary particles. Once again, you can try adding these things up correction by correction, but once again the “magic” doesn’t happen until the end. Only in the full series does string theory “do its thing”, and fix some of the big problems of quantum gravity.

If the real world really is a theory like this, then I think we have to worry about something like double descent.

Remember, double descent happens when our models can prematurely get worse before getting better. This can happen if the real thing we’re trying to model is very different from the model we’re using, like the example in this explainer that tries to use straight lines to match a curve. If we think a model is simpler because it puts fewer corrections on top of the Standard Model, then we may end up rejecting a reality with infinite corrections, a Taylor series that happens to add up to something quite nice. Occam’s Razor stops helping us if we can’t tell which models are really the simple ones.

The problem here is that every notion of “simple” we can appeal to here is aesthetic, a choice based on what makes the math look nicer. Other sciences don’t have this problem. When a biologist or a chemist wants to look for the simplest model, they look for a model with fewer organisms, fewer reactions…in the end, fewer atoms and molecules, fewer of the building-blocks given to those fields by physics. Fundamental physics can’t do this: we build our theories up from mathematics, and mathematics only demands that we be consistent. We can call theories simpler because we can write them in a simple way (but we could write them in a different way too). Or we can call them simpler because they look more like toy models we’ve worked with before (but those toy models are just a tiny sample of all the theories that are possible). We don’t have a standard of simplicity that is actually reliable.

From the Wikipedia page for dark matter halos

There is one other way out of this pickle. A theory that is easier to write down is under no obligation to be true. But it is more likely to be useful. Even if the real world is ultimately described by some giant pile of mathematical parameters, if a simple theory is good enough for the engineers then it’s a better theory to aim for: a useful theory that makes peoples’ lives better.

I kind of get the feeling Hossenfelder would make this objection. I’ve seen her argue on twitter that scientists should always be able to say what their research is good for, and her Guardian article has this suggestive sentence: “However, we do not know that dark matter is indeed made of particles; and even if it is, to explain astrophysical observations one does not need to know details of the particles’ behaviour.”

Ok yes, to explain astrophysical observations one doesn’t need to know the details of dark matter particles’ behavior. But taking a step back, one doesn’t actually need to explain astrophysical observations at all.

Astrophysics and particle physics are not engineering problems. Nobody out there is trying to steer a spacecraft all the way across a galaxy, navigating the distribution of dark matter, or creating new universes and trying to make sure they go just right. Even if we might do these things some day, it will be so far in the future that our attempts to understand them won’t just be quaint: they will likely be actively damaging, confusing old research in dead languages that the field will be better off ignoring to start from scratch.

Because of that, usefulness is also not a meaningful guide. It cannot tell you which theories are more simple, which to favor with Occam’s Razor.

Hossenfelder’s highest-profile recent work falls afoul of one or the other of her principles. Her work on the foundations of quantum mechanics could genuinely be useful, but there’s no reason aside from claims of philosophical beauty to expect it to be true. Her work on modeling dark matter is at least directly motivated by data, but is guaranteed to not be useful.

I’m not pointing this out to call Hossenfelder a hypocrite, as some sort of ad hominem or tu quoque. I’m pointing this out because I don’t think it’s possible to do fundamental physics today without falling afoul of these principles. If you want to hold out hope that your work is useful, you don’t have a great reason besides a love of pretty math: otherwise, anything useful would have been discovered long ago. If you just try to model existing data as best you can, then you’re making a model for events far away or locked in high-energy particle colliders, a model no-one else besides other physicists will ever use.

I don’t know the way through this. I think if you need to take Occam’s Razor seriously, to build on the same foundations that work in every other scientific field…then you should stop doing fundamental physics. You won’t be able to make it work. If you still need to do it, if you can’t give up the sub-field, then you should justify it on building capabilities, on the kind of “practice” Hossenfelder also dismisses in her Guardian piece.

We don’t have a solid foundation, a reliable notion of what is simple and what isn’t. We have guesses and personal opinions. And until some experiment uncovers some blinding flash of new useful meaningful magic…I don’t think we can do any better than that.

Jumpstarting Elliptic Bootstrapping

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

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

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

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

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

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

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

At Elliptic Integrals in Fundamental Physics in Mainz

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

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

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

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

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

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.

For one, it lets us use remarkably simple signs

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 Most Anthropic of All Possible Worlds

Today, we’d call Leibniz a mathematician, a physicist, and a philosopher. As a mathematician, Leibniz turned calculus into something his contemporaries could actually use. As a physicist, he championed a doomed theory of gravity. In philosophy, he seems to be most remembered for extremely cheaty arguments.

Free will and determinism? Can’t it just be a coincidence?

I don’t blame him for this. Faced with a tricky philosophical problem, it’s enormously tempting to just blaze through with an answer that makes every subtlety irrelevant. It’s a temptation I’ve succumbed to time and time again. Faced with a genie, I would always wish for more wishes. On my high school debate team, I once forced everyone at a tournament to switch sides with some sneaky definitions. It’s all good fun, but people usually end up pretty annoyed with you afterwards.

People were annoyed with Leibniz too, especially with his solution to the problem of evil. If you believe in a benevolent, all-powerful god, as Leibniz did, why is the world full of suffering and misery? Leibniz’s answer was that even an all-powerful god is constrained by logic, so if the world contains evil, it must be logically impossible to make the world any better: indeed, we live in the best of all possible worlds. Voltaire famously made fun of this argument in Candide, dragging a Leibniz-esque Professor Pangloss through some of the most creative miseries the eighteenth century had to offer. It’s possibly the most famous satire of a philosopher, easily beating out Aristophanes’ The Clouds (which is also great).

Physicists can also get accused of cheaty arguments, and probably the most mocked is the idea of a multiverse. While it hasn’t had its own Candide, the multiverse has been criticized by everyone from bloggers to Nobel prizewinners. Leibniz wanted to explain the existence of evil, physicists want to explain “unnaturalness”: the fact that the kinds of theories we use to explain the world can’t seem to explain the mass of the Higgs boson. To explain it, these physicists suggest that there are really many different universes, separated widely in space or built in to the interpretation of quantum mechanics. Each universe has a different Higgs mass, and ours just happens to be the one we can live in. This kind of argument is called “anthropic” reasoning. Rather than the best of all possible worlds, it says we live in the world best-suited to life like ours.

I called Leibniz’s argument “cheaty”, and you might presume I think the same of the multiverse. But “cheaty” doesn’t mean “wrong”. It all depends what you’re trying to do.

Leibniz’s argument and the multiverse both work by dodging a problem. For Leibniz, the problem of evil becomes pointless: any evil might be necessary to secure a greater good. With a multiverse, naturalness becomes pointless: with many different laws of physics in different places, the existence of one like ours needs no explanation.

In both cases, though, the dodge isn’t perfect. To really explain any given evil, Leibniz would have to show why it is secretly necessary in the face of a greater good (and Pangloss spends Candide trying to do exactly that). To explain any given law of physics, the multiverse needs to use anthropic reasoning: it needs to show that that law needs to be the way it is to support human-like life.

This sounds like a strict requirement, but in both cases it’s not actually so useful. Leibniz could (and Pangloss does) come up with an explanation for pretty much anything. The problem is that no-one actually knows which aspects of the universe are essential and which aren’t. Without a reliable way to describe the best of all possible worlds, we can’t actually test whether our world is one.

The same problem holds for anthropic reasoning. We don’t actually know what conditions are required to give rise to people like us. “People like us” is very vague, and dramatically different universes might still contain something that can perceive and observe. While it might seem that there are clear requirements, so far there hasn’t been enough for people to do very much with this type of reasoning.

However, for both Leibniz and most of the physicists who believe anthropic arguments, none of this really matters. That’s because the “best of all possible worlds” and “most anthropic of all possible worlds” aren’t really meant to be predictive theories. They’re meant to say that, once you are convinced of certain things, certain problems don’t matter anymore.

Leibniz, in particular, wasn’t trying to argue for the existence of his god. He began the argument convinced that a particular sort of god existed: one that was all-powerful and benevolent, and set in motion a deterministic universe bound by logic. His argument is meant to show that, if you believe in such a god, then the problem of evil can be ignored: no matter how bad the universe seems, it may still be the best possible world.

Similarly, the physicists convinced of the multiverse aren’t really getting there through naturalness. Rather, they’ve become convinced of a few key claims: that the universe is rapidly expanding, leading to a proliferating multiverse, and that the laws of physics in such a multiverse can vary from place to place, due to the huge landscape of possible laws of physics in string theory. If you already believe those things, then the naturalness problem can be ignored: we live in some randomly chosen part of the landscape hospitable to life, which can be anywhere it needs to be.

So despite their cheaty feel, both arguments are fine…provided you agree with their assumptions. Personally, I don’t agree with Leibniz. For the multiverse, I’m less sure. I’m not confident the universe expands fast enough to create a multiverse, I’m not even confident it’s speeding up its expansion now. I know there’s a lot of controversy about the math behind the string theory landscape, about whether the vast set of possible laws of physics are as consistent as they’re supposed to be…and of course, as anyone must admit, we don’t know whether string theory itself is true! I don’t think it’s impossible that the right argument comes around and convinces me of one or both claims, though. These kinds of arguments, “if assumptions, then conclusion” are the kind of thing that seems useless for a while…until someone convinces you of the conclusion, and they matter once again.

So in the end, despite the similarity, I’m not sure the multiverse deserves its own Candide. I’m not even sure Leibniz deserved Candide. But hopefully by understanding one, you can understand the other just a bit better.

Trapped in the (S) Matrix

I’ve tried to convince you that you are a particle detector. You choose your experiment, what actions you take, and then observe the outcome. If you focus on that view of yourself, data out and data in, you start to wonder if the world outside really has any meaning. Maybe you’re just trapped in the Matrix.

From a physics perspective, you actually are trapped in a sort of a Matrix. We call it the S Matrix.

“S” stands for scattering. The S Matrix is a formula we use, a mathematical tool that tells us what happens when fundamental particles scatter: when they fly towards each other, colliding or bouncing off. For each action we could take, the S Matrix gives the probability of each outcome: for each pair of particles we collide, the chance we detect different particles at the end. You can imagine putting every possible action in a giant vector, and every possible observation in another giant vector. Arrange the probabilities for each action-observation pair in a big square grid, and that’s a matrix.

Actually, I lied a little bit. This is particle physics, and particle physics uses quantum mechanics. Because of that, the entries of the S Matrix aren’t probabilities: they’re complex numbers called probability amplitudes. You have to multiply them by their complex conjugate to get probability out.

Ok, that probably seemed like a lot of detail. Why am I telling you all this?

What happens when you multiply the whole S Matrix by its complex conjugate? (Using matrix multiplication, naturally.) You can still pick your action, but now you’re adding up every possible outcome. You’re asking “suppose I take an action. What’s the chance that anything happens at all?”

The answer to that question is 1. There is a 100% chance that something happens, no matter what you do. That’s just how probability works.

We call this property unitarity, the property of giving “unity”, or one. And while it may seem obvious, it isn’t always so easy. That’s because we don’t actually know the S Matrix formula most of the time. We have to approximate it, a partial formula that only works for some situations. And unitarity can tell us how much we can trust that formula.

Imagine doing an experiment trying to detect neutrinos, like the IceCube Neutrino Observatory. For you to detect the neutrinos, they must scatter off of electrons, kicking them off of their atoms or transforming them into another charged particle. You can then notice what happens as the energy of the neutrinos increases. If you do that, you’ll notice the probability also start to increase: it gets more and more likely that the neutrino can scatter an electron. You might propose a formula for this, one that grows with energy. [EDIT: Example changed after a commenter pointed out an issue with it.]

If you keep increasing the energy, though, you run into a problem. Those probabilities you predict are going to keep increasing. Eventually, you’ll predict a probability greater than one.

That tells you that your theory might have been fine before, but doesn’t work for every situation. There’s something you don’t know about, which will change your formula when the energy gets high. You’ve violated unitarity, and you need to fix your theory.

In this case, the fix is already known. Neutrinos and electrons interact due to another particle, called the W boson. If you include that particle, then you fix the problem: your probabilities stop going up and up, instead, they start slowing down, and stay below one.

For other theories, we don’t yet know the fix. Try to write down an S Matrix for colliding gravitational waves (or really, gravitons), and you meet the same kind of problem, a probability that just keeps growing. Currently, we don’t know how that problem should be solved: string theory is one answer, but may not be the only one.

So even if you’re trapped in an S Matrix, sending data out and data in, you can still use logic. You can still demand that probability makes sense, that your matrix never gives a chance greater than 100%. And you can learn something about physics when you do!

At New Ideas in Cosmology

The Niels Bohr Institute is hosting a conference this week on New Ideas in Cosmology. I’m no cosmologist, but it’s a pretty cool field, so as a local I’ve been sitting in on some of the talks. So far they’ve had a selection of really interesting speakers with quite a variety of interests, including a talk by Roger Penrose with his trademark hand-stippled drawings.

Including this old classic

One thing that has impressed me has been the “interdisciplinary” feel of the conference. By all rights this should be one “discipline”, cosmology. But in practice, each speaker came at the subject from a different direction. They all had a shared core of knowledge, common models of the universe they all compare to. But the knowledge they brought to the subject varied: some had deep knowledge of the mathematics of gravity, others worked with string theory, or particle physics, or numerical simulations. Each talk, aware of the varied audience, was a bit “colloquium-style“, introducing a framework before diving in to the latest research. Each speaker knew enough to talk to the others, but not so much that they couldn’t learn from them. It’s been unexpectedly refreshing, a real interdisciplinary conference done right.