Tag Archives: quantum field theory

In Scientific American, With a Piece on Vacuum Decay

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I had a piece in Scientific American last week. It’s paywalled, but if you’re a subscriber there you can see it, or you can buy the print magazine.

(I also had two pieces out in other outlets this week. I’ll be saying more about them…in a couple weeks.)

The Scientific American piece is about an apocalyptic particle physics scenario called vacuum decay. It’s a topic I covered last year in Quanta Magazine, an unlikely event where the Higgs field which gives fundamental particles their mass changes value, suddenly making all other particles much more massive and changing physics as we know it. It’s a change that physicists think would start as a small bubble and spread at (almost) the speed of light, covering the universe.

What I wrote for Quanta was a short news piece covering a small adjustment to the calculation, one that made the chance of vacuum decay slightly more likely. (But still mind-bogglingly small, to be clear.)

Scientific American asked for a longer piece, and that gave me space to dig deeper. I was able to say more about how vacuum decay works, with a few metaphors that I think should make it a lot easier to understand. I also got to learn about some new developments, in particular, an interesting story about how tiny primordial black holes could make vacuum decay dramatically more likely.

One thing that was a bit too complicated to talk about were the puzzles involved in trying to calculate these chances. In the article, I mention a calculation of the chance of vacuum decay by a team including Matthew Schwartz. That calculation wasn’t the first to estimate the chance of vacuum decay, and it’s not the most recent update either. Instead, I picked it because Schwartz’s team approached the question in what struck me as a more reliable way, trying to cut through confusion by asking the most basic question you can in a quantum theory: given that now you observe X, what’s the chance that later you observe Y? Figuring out how to turn vacuum decay into that kind of question correctly is tricky (for example, you need to include the possibility that vacuum decay happens, then reverses, then happens again).

The calculations of black holes speeding things up didn’t work things out in quite as much detail. I like to think I’ve made a small contribution by motivating them to look at Schwartz’s work, which might spawn a more rigorous calculation in future. When I talked to Schwartz, he wasn’t even sure whether the picture of a bubble forming in one place and spreading at light speed is correct: he’d calculated the chance of the initial decay, but hadn’t found a similarly rigorous way to think about the aftermath. So even more than the uncertainty I talk about in the piece, the questions about new physics and probability, there is even some doubt about whether the whole picture really works the way we’ve been imagining it.

That makes for a murky topic! But it’s also a flashy one, a compelling story for science fiction and the public imagination, and yeah, another motivation to get high-precision measurements of the Higgs and top quark from future colliders! (If maybe not quite the way this guy said it.)

Antimatter Isn’t Magic

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You’ve heard of antimatter, right?

For each type of particle, there is a rare kind of evil twin with the opposite charge, called an anti-particle. When an anti-proton meets a proton, they annihilate each other in a giant blast of energy.

I see a lot of questions online about antimatter. One recurring theme is people asking a very general question: how does antimatter work?

If you’ve just heard the pop physics explanation, antimatter probably sounds like magic. What about antimatter lets it destroy normal matter? Does it need to touch? How long does it take? And what about neutral particles like neutrons?

You find surprisingly few good explanations of this online, but I can explain why. Physicists like me don’t expect antimatter to be confusing in this way, because to us, antimatter isn’t doing anything all that special. When a particle and an antiparticle annihilate, they’re doing the same thing that any other pair of particles do when they do…basically anything else.

Instead of matter and antimatter, let’s talk about one of the oldest pieces of evidence for quantum mechanics, the photoelectric effect. Scientists shone light at a metal, and found that if the wavelength of the light was short enough, electrons would spring free, causing an electric current. If the wavelength was too long, the metal wouldn’t emit any electrons, no matter how much light they shone. Einstein won his Nobel prize for the explanation: the light hitting the metal comes in particle-sized pieces, called photons, whose energy is determined by the wavelength of the light. If the individual photons don’t have enough energy to get an electron to leave the metal, then no electron will move, no matter how many photons you use.

What happens to the photons after they hit the metal?

They go away. We say they are absorbed, an electron absorbs a photon and speeds up, increasing its kinetic energy so it can escape.

But we could just as easily say the photon is annihilated, if we wanted to.

In the photoelectric effect, you start with one electron and one photon, they come together, and you end up with one electron and no photon. In proton-antiproton annihilation, you start with a proton and an antiproton, they come together, and you end up with no protons or antiprotons, but instead “energy”…which in practice, usually means two photons.

That’s all that happens, deep down at the root of things. The laws of physics are rules about inputs and outputs. Start with these particles, they come together, you end up with these other particles. Sometimes one of the particles stays the same. Sometimes particles seem to transform, and different kinds of particles show up. Sometimes some of the particles are photons, and you think of them as “just energy”, and easy to absorb. But particles are particles, and nothing is “just energy”. Each thing, absorption, decay, annihilation, each one is just another type of what we call interactions.

What makes annihilation of matter and antimatter seem unique comes down to charges. Interactions have to obey the laws of physics: they conserve energy, they conserve momentum, and they conserve charge.

So why can an antiproton and a proton annihilate to pure photons, while two protons can’t? A proton and an antiproton have opposite charge, a photon has zero charge. You could combine two protons to make something else, but it would have to have the same charge as two protons.

What about neutrons? A neutron has no electric charge, so you might think it wouldn’t need antimatter. But a neutron has another type of charge, called baryon number. In order to annihilate one, you’d need an anti-neutron, which would still have zero electric charge but would have the opposite baryon number. (By the way, physicists have been making anti-neutrons since 1956.)

On the other hand, photons actually have no charge. So do Higgs bosons. So one Higgs boson can become two photons, without annihilating with anything else. Each of these particles can be called its own antiparticle: a photon is also an antiphoton, a Higgs is also an anti-Higgs.

Because particle-antiparticle annihilation follows the same rules as other interactions between particles, it also takes place via the same forces. When a proton and an antiproton annihilate each other, they typically do this via the electromagnetic force. This is why you end up with light, which is an electromagnetic wave. Like everything in the quantum world, this annihilation isn’t certain. Is has a chance to happen, proportional to the strength of the interaction force involved.

What about neutrinos? They also appear to have a kind of charge, called lepton number. That might not really be a conserved charge, and neutrinos might be their own antiparticles, like photons. However, they are much less likely to be annihilated than protons and antiprotons, because they don’t have electric charge, and thus their interaction doesn’t depend on the electromagnetic force, but on the much weaker weak nuclear force. A weaker force means a less likely interaction.

Antimatter might seem like the stuff of science fiction. But it’s not really harder to understand than anything else in particle physics.

(I know, that’s a low bar!)

It’s just interactions. Particles go in, particles go out. If it follows the rules, it can happen, if it doesn’t, it can’t. Antimatter is no different.

Some FAQ for Microsoft’s Majorana 1 Chip

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Recently, Microsoft announced a fancy new quantum computing chip called Majorana 1. I’ve noticed quite a bit of confusion about what they actually announced, and while there’s a great FAQ page about it on the quantum computing blog Shtetl Optimized, the post there aims at a higher level, assuming you already know the basics. You can think of this post as a complement to that one, that tries to cover some basic things Shtetl Optimized took for granted.

Q: In the announcement, Microsoft said:

“It leverages the world’s first topoconductor, a breakthrough type of material which can observe and control Majorana particles to produce more reliable and scalable qubits, which are the building blocks for quantum computers.”

That sounds wild! Are they really using particles in a computer?

A: All computers use particles. Electrons are particles!

Q: You know what I mean!

A: You’re asking if these are “particle physics” particles, like the weird types they try to observe at the LHC?

No, they’re not.

Particle physicists use a mathematical framework called quantum field theory, where particles are ripples in things called quantum fields that describe properties of the universe. But they aren’t the only people to use that framework. Instead of studying properties of the universe you can study properties of materials, weird alloys and layers of metal and crystal that do weird and useful things. The properties of these materials can be approximately described with the same math, with quantum fields. Just as the properties of the universe ripple to produce particles, these properties of materials ripple to produce what are called quasiparticles. Ultimately, these quasiparticles come down to movements of ordinary matter, usually electrons in the original material. They’re just described with a kind of math that makes them look like their own particles.

Q: So, what are these Majorana particles supposed to be?

A: In quantum field theory, most particles come with an antimatter partner. Electrons, for example, have partners called positrons, with a positive electric charge instead of a negative one. These antimatter partners have to exist due to the math of quantum field theory, but there is a way out: some particles are their own antimatter partner, letting one particle cover both roles. This happens for some “particle physics particles”, but all the examples we’ve found are a type of particle called a “boson”, particles related to forces. In 1937, the physicist Ettore Majorana figured out the math you would need to make a particle like this that was a fermion instead, the other main type of particle that includes electrons and protons. So far, we haven’t found one of these Majorana fermions in nature, though some people think the elusive neutrino particles could be an example. Others, though, have tried instead to find a material described by Majorana’s theory. This should in principle be easier, you can build a lot of different materials after all. But it’s proven quite hard for people to do. Back in 2018, Microsoft claimed they’d managed this, but had to retract the claim. This time, they seem more confident, though the scientific community is still not convinced.

Q: And what’s this topoconductor they’re talking about?

A: Topoconductor is short for topological superconductor. Superconductors are materials that conduct electricity much better than ordinary metals.

Q: And, topological means? Something about donuts, right?

A: If you’ve heard anything about topology, you’ve heard that it’s a type of mathematics where donuts are equivalent to coffee cups. You might have seen an animation of a coffee cup being squished and mushed around until the ring of the handle becomes the ring of a donut.

This isn’t actually the important part of topology. The important part is that, in topology, a ball is not equivalent to a donut.

Topology is the study of which things can change smoothly into one another. If you want to change a donut into a ball, you have to slice through the donut’s ring or break the surface inside. You can’t smoothly change one to another. Topologists study shapes of different kinds of things, figuring out which ones can be changed into each other smoothly and which can’t.

Q: What does any of that have to do with quantum computers?

A: The shapes topologists study aren’t always as simple as donuts and coffee cups. They can also study the shape of quantum fields, figuring out which types of quantum fields can change smoothly into each other and which can’t.

The idea of topological quantum computation is to use those rules about what can change into each other to encode information. You can imagine a ball encoding zero, and a donut encoding one. A coffee cup would then also encode one, because it can change smoothly into a donut, while a box would encode zero because you can squash the corners to make it a ball. This helps, because it means that you don’t screw up your information by making smooth changes. If you accidentally drop your box that encodes zero and squish a corner, it will still encode zero.

This matters in quantum computing because it is very easy to screw up quantum information. Quantum computers are very delicate, and making them work reliably has been immensely challenging, requiring people to build much bigger quantum computers so they can do each calculation with many redundant backups. The hope is that topological superconductors would make this easier, by encoding information in a way that is hard to accidentally change.

Q: Cool. So does that mean Microsoft has the best quantum computer now?

A: The machine Microsoft just announced has only a single qubit, the quantum equivalent of just a single bit of computer memory. At this point, it can’t do any calculations. It can just be read, giving one or zero. The hope is that the power of the new method will let Microsoft catch up with companies that have computers with hundred of qubits, and help them arrive faster at the millions of qubits that will be needed to do anything useful.

Q: Ah, ok. But it sounds like they accomplished some crazy Majorana stuff at least, right?

A: Umm…

Read the Shtetl-Optimized FAQ if you want more details. The short answer is that this is still controversial. So far, the evidence they’ve made public isn’t enough to show that they found these Majorana quasiparticles, or that they made a topological superconductor. They say they have more recent evidence that they haven’t published yet. We’ll see.

Bonus Material for “How Hans Bethe Stumbled Upon Perfect Quantum Theories”

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I had an article last week in Quanta Magazine. It’s a piece about something called the Bethe ansatz, a method in mathematical physics that was discovered by Hans Bethe in the 1930’s, but which only really started being understood and appreciated around the 1960’s. Since then it’s become a key tool, used in theoretical investigations in areas from condensed matter to quantum gravity. In this post, I thought I’d say a bit about the story behind the piece and give some bonus material that didn’t fit.

When I first decided to do the piece I reached out to Jules Lamers. We were briefly office-mates when I worked in France, where he was giving a short course on the Bethe ansatz and the methods that sprung from it. It turned out he had also been thinking about writing a piece on the subject, and we considered co-writing for a bit, but that didn’t work for Quanta. He helped me a huge amount with understanding the history of the subject and tracking down the right sources. If you’re a physicist who wants to learn about these things, I recommend his lecture notes. And if you’re a non-physicist who wants to know more, I hope he gets a chance to write a longer popular-audience piece on the topic!

If you clicked through to Jules’s lecture notes, you’d see the word “Bethe ansatz” doesn’t appear in the title. Instead, you’d see the phrase “quantum integrability”. In classical physics, an “integrable” system is one where you can calculate what will happen by doing an integral, essentially letting you “solve” any problem completely. Systems you can describe with the Bethe ansatz are solvable in a more complicated quantum sense, so they get called “quantum integrable”. There’s a whole research field that studies these quantum integrable systems.

My piece ended up rushing through the history of the field. After talking about Bethe’s original discovery, I jumped ahead to ice. The Bethe ansatz was first used to think about ice in the 1960’s, but the developments I mentioned leading up to it, where experimenters noticed extra variability and theorists explained it with the positions of hydrogen atoms, happened earlier, in the 1930’s. (Thanks to the commenter who pointed out that this was confusing!) Baxter gets a starring role in this section and had an important role in tying things together, but other people (Lieb and Sutherland) were involved earlier, showing that the Bethe ansatz indeed could be used with thin sheets of ice. This era had a bunch of other big names that I didn’t have space to talk about: C. N. Yang makes an appearance, and while Faddeev comes up later, I didn’t mention that he had a starring role in the 1970’s in understanding the connection to classical integrability and proposing a mathematical structure to understand what links all these different integrable theories together.

I vaguely gestured at black holes and quantum gravity, but didn’t have space for more than that. The connection there is to a topic you might have heard of before if you’ve read about string theory, called AdS/CFT, a connection between two kinds of world that are secretly the same: a toy model of gravity called Anti-de Sitter space (AdS) and a theory without gravity that looks the same at any scale (called a Conformal Field Theory, or CFT). It turns out that in the most prominent example of this, the theory without gravity is integrable! In fact, it’s a theory I spent a lot of time working with back in my research days, called N=4 super Yang-Mills. This theory is kind of like QCD, and in some sense it has integrability for similar reasons to those that Feynman hoped for and Korchemsky and Faddeev found. But it actually goes much farther, outside of the high-energy approximation where Korchemsky and Faddeev’s result works, and in principle seems to include everything you might want to know about the theory. Nowadays, people are using it to investigate the toy model of quantum gravity, hoping to get insights about quantum gravity in general.

One thing I didn’t get a chance to mention at all is the connection to quantum computing. People are trying to build a quantum computer with carefully-cooled atoms. It’s important to test whether the quantum computer functions well enough, or if the quantum states aren’t as perfect as they need to be. One way people have been testing this is with the Bethe ansatz: because it lets you calculate the behavior of special systems perfectly, you can set up your quantum computer to model a Bethe ansatz, and then check how close to the prediction your results are. You know that the theoretical result is complete, so any failure has to be due to an imperfection in your experiment.

I gave a quick teaser to a very active field, one that has fascinated a lot of prominent physicists and been applied in a wide variety of areas. I hope I’ve inspired you to learn more!

How Small Scales Can Matter for Large Scales

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For a certain type of physicist, nothing matters more than finding the ultimate laws of nature for its tiniest building-blocks, the rules that govern quantum gravity and tell us where the other laws of physics come from. But because they know very little about those laws at this point, they can predict almost nothing about observations on the larger distance scales we can actually measure.

“Almost nothing” isn’t nothing, though. Theoretical physicists don’t know nature’s ultimate laws. But some things about them can be reasonably guessed. The ultimate laws should include a theory of quantum gravity. They should explain at least some of what we see in particle physics now, explaining why different particles have different masses in terms of a simpler theory. And they should “make sense”, respecting cause and effect, the laws of probability, and Einstein’s overall picture of space and time.

All of these are assumptions, of course. Further assumptions are needed to derive any testable consequences from them. But a few communities in theoretical physics are willing to take the plunge, and see what consequences their assumptions have.

First, there’s the Swampland. String theorists posit that the world has extra dimensions, which can be curled up in a variety of ways to hide from view, with different observable consequences depending on how the dimensions are curled up. This list of different observable consequences is referred to as the Landscape of possibilities. Based on that, some string theorists coined the term “Swampland” to represent an area outside the Landscape, containing observations that are incompatible with quantum gravity altogether, and tried to figure out what those observations would be.

In principle, the Swampland includes the work of all the other communities on this list, since a theory of quantum gravity ought to be consistent with other principles as well. In practice, people who use the term focus on consequences of gravity in particular. The earliest such ideas argued from thought experiments with black holes, finding results that seemed to demand that gravity be the weakest force for at least one type of particle. Later researchers would more frequently use string theory as an example, looking at what kinds of constructions people had been able to make in the Landscape to guess what might lie outside of it. They’ve used this to argue that dark energy might be temporary, and to try to figure out what traits new particles might have.

Second, I should mention naturalness. When talking about naturalness, people often use the analogy of a pen balanced on its tip. While possible in principle, it must have been set up almost perfectly, since any small imbalance would cause it to topple, and that perfection demands an explanation. Similarly, in particle physics, things like the mass of the Higgs boson and the strength of dark energy seem to be carefully balanced, so that a small change in how they were set up would lead to a much heavier Higgs boson or much stronger dark energy. The need for an explanation for the Higgs’ careful balance is why many physicists expected the Large Hadron Collider to discover additional new particles.

As I’ve argued before, this kind of argument rests on assumptions about the fundamental laws of physics. It assumes that the fundamental laws explain the mass of the Higgs, not merely by giving it an arbitrary number but by showing how that number comes from a non-arbitrary physical process. It also assumes that we understand well how physical processes like that work, and what kinds of numbers they can give. That’s why I think of naturalness as a type of argument, much like the Swampland, that uses the smallest scales to constrain larger ones.

Third is a host of constraints that usually go together: causality, unitarity, and positivity. Causality comes from cause and effect in a relativistic universe. Because two distant events can appear to happen in different orders depending on how fast you’re going, any way to send signals faster than light is also a way to send signals back in time, causing all of the paradoxes familiar from science fiction. Unitarity comes from quantum mechanics. If quantum calculations are supposed to give the probability of things happening, those probabilities should make sense as probabilities: for example, they should never go above one.

You might guess that almost any theory would satisfy these constraints. But if you extend a theory to the smallest scales, some theories that otherwise seem sensible end up failing this test. Actually linking things up takes other conjectures about the mathematical form theories can have, conjectures that seem more solid than the ones underlying Swampland and naturalness constraints but that still can’t be conclusively proven. If you trust the conjectures, you can derive restrictions, often called positivity constraints when they demand that some set of observations is positive. There has been a renaissance in this kind of research over the last few years, including arguments that certain speculative theories of gravity can’t actually work.

Which String Theorists Are You Complaining About?

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Do string theorists have an unfair advantage? Do they have an easier time getting hired, for example?

In one of the perennial arguments about this on Twitter, Martin Bauer posted a bar chart of faculty hires in the US by sub-field. The chart was compiled by Erich Poppitz from data in the US particle physics rumor mill, a website where people post information about who gets hired where for the US’s quite small number of permanent theoretical particle physics positions at research universities and national labs. The data covers 1994 to 2017, and shows one year, 1999, when there were more string theorists hired than all other topics put together. The years around then also had many string theorists hired, but the proportion starts falling around the mid 2000’s…around when Lee Smolin wrote a book, The Trouble With Physics, arguing that string theorists had strong-armed their way into academic dominance. After that, the percentage of string theorists falls, oscillating between a tenth and a quarter of total hires.

Judging from that, you get the feeling that string theory’s critics are treating a temporary hiring fad as if it was a permanent fact. The late 1990’s were a time of high-profile developments in string theory that excited a lot of people. Later, other hiring fads dominated, often driven by experiments: I remember when the US decided to prioritize neutrino experiments and neutrino theorists had a much easier time getting hired, and there seem to be similar pushes now with gravitational waves, quantum computing, and AI.

Thinking about the situation in this way, though, ignores what many of the critics have in mind. That’s because the “string” column on that bar chart is not necessarily what people think of when they think of string theory.

If you look at the categories on Poppitz’s bar chart, you’ll notice something odd. “String” its itself a category. Another category, “lattice”, refers to lattice QCD, a method to find the dynamics of quarks numerically. The third category, though, is a combination of three things “ph/th/cosm”.

“Cosm” here refers to cosmology, another sub-field. “Ph” and “th” though aren’t really sub-fields. Instead, they’re arXiv categories, sections of the website arXiv.org where physicists post papers before they submit them to journals. The “ph” category is used for phenomenology, the type of theoretical physics where people try to propose models of the real world and make testable predictions. The “th” category is for “formal theory”, papers where theoretical physicists study the kinds of theories they use in more generality and develop new calculation methods, with insights that over time filter into “ph” work.

“String”, on the other hand, is not an arXiv category. When string theorists write papers, they’ll put them into “th” or “ph” or another relevant category (for example “gr-qc”, for general relativity and quantum cosmology). This means that when Poppitz distinguishes “ph/th/cosm” from “string”, he’s being subjective, using his own judgement to decide who counts as a string theorist.

So who counts as a string theorist? The simplest thing to do would be to check if their work uses strings. Failing that, they could use other tools of string theory and its close relatives, like Calabi-Yau manifolds, M-branes, and holography.

That might be what Poppitz was doing, but if he was, he was probably missing a lot of the people critics of string theory complain about. He even misses many people who describe themselves as string theorists. In an old post of mine I go through the talks at Strings, string theory’s big yearly conference, giving them finer-grained categories. The majority don’t use anything uniquely stringy.

Instead, I think critics of string theory have two kinds of things in mind.

First, most of the people who made their reputations on string theory are still in academia, and still widely respected. Some of them still work on string theory topics, but many now work on other things. Because they’re still widely respected, their interests have a substantial influence on the field. When one of them starts looking at connections between theories of two-dimensional materials, you get a whole afternoon of talks at Strings about theories of two-dimensional materials. Working on those topics probably makes it a bit easier to get a job, but also, many of the people working on them are students of these highly respected people, who just because of that have an easier time getting a job. If you’re a critic of string theory who thinks the founders of the field led physics astray, then you probably think they’re still leading physics astray even if they aren’t currently working on string theory.

Second, for many other people in physics, string theorists are their colleagues and friends. They’ll make fun of trends that seem overhyped and under-thought, like research on the black hole information paradox or the swampland, or hopes that a slightly tweaked version of supersymmetry will show up soon at the LHC. But they’ll happily use ideas developed in string theory when they prove handy, using supersymmetric theories to test new calculation techniques, string theory’s extra dimensions to inspire and ground new ideas for dark matter, or the math of strings themselves as interesting shortcuts to particle physics calculations. String theory is available as reference to these people in a way that other quantum gravity proposals aren’t. That’s partly due to familiarity and shared language (I remember a talk at Perimeter where string theorists wanted to learn from practitioners from another area and the discussion got bogged down by how they were using the word “dimension”), but partly due to skepticism of the various alternate approaches. Most people have some idea in their heads of deep problems with various proposals: screwing up relativity, making nonsense out of quantum mechanics, or over-interpreting on limited evidence. The most commonly believed criticisms are usually wrong, with objections long-known to practitioners of the alternate approaches, and so those people tend to think they’re being treated unfairly. But the wrong criticisms are often simplified versions of correct criticisms, passed down by the few people who dig deeply into these topics, criticisms that the alternative approaches don’t have good answers to.

The end result is that while string theory itself isn’t dominant, a sort of “string friendliness” is. Most of the jobs aren’t going to string theorists in the literal sense. But the academic world string theorists created keeps turning. People still respect string theorists and the research directions they find interesting, and people are still happy to collaborate and discuss with string theorists. For research communities people are more skeptical of, it must feel very isolating, like the world is still being run by their opponents. But this isn’t the kind of hegemony that can be solved by a revolution. Thinking that string theory is a failed research program, and people focused on it should have a harder time getting hired, is one thing. Thinking that everyone who respects at least one former string theorist should have a harder time getting hired is a very different goal. And if what you’re complaining about is “string friendliness”, not actual string theorists, then that’s what you’re asking for.

The Nowhere String

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Space and time seem as fundamental as anything can get. Philosophers like Immanuel Kant thought that they were inescapable, that we could not conceive of the world without space and time. But increasingly, physicists suspect that space and time are not as fundamental as they appear. When they try to construct a theory of quantum gravity, physicists find puzzles, paradoxes that suggest that space and time may just be approximations to a more fundamental underlying reality.

One piece of evidence that quantum gravity researchers point to are dualities. These are pairs of theories that seem to describe different situations, including with different numbers of dimensions, but that are secretly indistinguishable, connected by a “dictionary” that lets you interpret any observation in one world in terms of an equivalent observation in the other world. By itself, duality doesn’t mean that space and time aren’t fundamental: as I explained in a blog post a few years ago, it could still be that one “side” of the duality is a true description of space and time, and the other is just a mathematical illusion. To show definitively that space and time are not fundamental, you would want to find a situation where they “break down”, where you can go from a theory that has space and time to a theory that doesn’t. Ideally, you’d want a physical means of going between them: some kind of quantum field that, as it shifts, changes the world between space-time and not space-time.

What I didn’t know when I wrote that post was that physicists already knew about such a situation in 1993.

Back when I was in pre-school, famous string theorist Edward Witten was trying to understand something that others had described as a duality, and realized there was something more going on.

In string theory, particles are described by lengths of vibrating string. In practice, string theorists like to think about what it’s like to live on the string itself, seeing it vibrate. In that world, there are two dimensions, one space dimension back and forth along the string and one time dimension going into the future. To describe the vibrations of the string in that world, string theorists use the same kind of theory that people use to describe physics in our world: a quantum field theory. In string theory, you have a two-dimensional quantum field theory stuck “inside” a theory with more dimensions describing our world. You see that this world exists by seeing the kinds of vibrations your two-dimensional world can have, through a type of quantum field called a scalar field. With ten scalar fields, ten different ways you can push energy into your stringy world, you can infer that the world around you is a space-time with ten dimensions.

String theory has “extra” dimensions beyond the three of space and one of time we’re used to, and these extra dimensions can be curled up in various ways to hide them from view, often using a type of shape called a Calabi-Yau manifold. In the late 80’s and early 90’s, string theorists had found a similarity between the two-dimensional quantum field theories you get folding string theory around some of these Calabi-Yau manifolds and another type of two-dimensional quantum field theory related to theories used to describe superconductors. People called the two types of theories dual, but Witten figured out there was something more going on.

Witten described the two types of theories in the same framework, and showed that they weren’t two equivalent descriptions of the same world. Rather, they were two different ways one theory could behave.

The two behaviors were connected by something physical: the value of a quantum field called a modulus field. This field can be described by a number, and that number can be positive or negative.

When the modulus field is a large positive number, then the theory behaves like string theory twisted around a Calabi-Yau manifold. In particular, the scalar fields have many different values they can take, values that are smoothly related to each other. These values are nothing more or less than the position of the string in space and time. Because the scalars can take many values, the string can sit in many different places, and because the values are smoothly related to each other, the string can smoothly move from one place to another.

When the modulus field is a large negative number, then the theory is very different. What people thought of as the other side of the duality, a theory like the theories used to describe superconductors, is the theory that describes what happens when the modulus field is large and negative. In this theory, the scalars can no longer take many values. Instead, they have one option, one stable solution. That means that instead of there being many different places the string could sit, describing space, there are no different places, and thus no space. The string lives nowhere.

These are two very different situations, one with space and one without. And they’re connected by something physical. You could imagine manipulating the modulus field, using other fields to funnel energy into it, pushing it back and forth from a world with space to a world of nowhere. Much more than the examples I was aware of, this is a super-clear example of a model where space is not fundamental, but where it can be manipulated, existing or not existing based on physical changes.

We don’t know whether a model like this describes the real world. But it’s gratifying to know that it can be written down, that there is a picture, in full mathematical detail, of how this kind of thing works. Hopefully, it makes the idea that space and time are not fundamental sound a bit more reasonable.

Replacing Space-Time With the Space in Your Eyes

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Nima Arkani-Hamed thinks space-time is doomed.

That doesn’t mean he thinks it’s about to be destroyed by a supervillain. Rather, Nima, like many physicists, thinks that space and time are just approximations to a deeper reality. In order to make sense of gravity in a quantum world, seemingly fundamental ideas, like that particles move through particular places at particular times, will probably need to become more flexible.

But while most people who think space-time is doomed research quantum gravity, Nima’s path is different. Nima has been studying scattering amplitudes, formulas used by particle physicists to predict how likely particles are to collide in particular ways. He has been trying to find ways to calculate these scattering amplitudes without referring directly to particles traveling through space and time. In the long run, the hope is that knowing how to do these calculations will help suggest new theories beyond particle physics, theories that can’t be described with space and time at all.

Ten years ago, Nima figured out how to do this in a particular theory, one that doesn’t describe the real world. For that theory he was able to find a new picture of how to calculate scattering amplitudes based on a combinatorical, geometric space with no reference to particles traveling through space-time. He gave this space the catchy name “the amplituhedron“. In the years since, he found a few other “hedra” describing different theories.

Now, he’s got a new approach. The new approach doesn’t have the same kind of catchy name: people sometimes call it surfaceology, or curve integral formalism. Like the amplituhedron, it involves concepts from combinatorics and geometry. It isn’t quite as “pure” as the amplituhedron: it uses a bit more from ordinary particle physics, and while it avoids specific paths in space-time it does care about the shape of those paths. Still, it has one big advantage: unlike the amplituhedron, Nima’s new approach looks like it can work for at least a few of the theories that actually describe the real world.

The amplituhedron was mysterious. Instead of space and time, it described the world in terms of a geometric space whose meaning was unclear. Nima’s new approach also describes the world in terms of a geometric space, but this space’s meaning is a lot more clear.

The space is called “kinematic space”. That probably still sounds mysterious. “Kinematic” in physics refers to motion. In the beginning of a physics class when you study velocity and acceleration before you’ve introduced a single force, you’re studying kinematics. In particle physics, kinematic refers to the motion of the particles you detect. If you see an electron going up and to the right at a tenth the speed of light, those are its kinematics.

Kinematic space, then, is the space of observations. By saying that his approach is based on ideas in kinematic space, what Nima is saying is that it describes colliding particles not based on what they might be doing before they’re detected, but on mathematics that asks questions only about facts about the particles that can be observed.

(For the experts: this isn’t quite true, because he still needs a concept of loop momenta. He’s getting the actual integrands from his approach, rather than the dual definition he got from the amplituhedron. But he does still have to integrate one way or another.)

Quantum mechanics famously has many interpretations. In my experience, Nima’s favorite interpretation is the one known as “shut up and calculate”. Instead of arguing about the nature of an indeterminately philosophical “real world”, Nima thinks quantum physics is a tool to calculate things people can observe in experiments, and that’s the part we should care about.

From a practical perspective, I agree with him. And I think if you have this perspective, then ultimately, kinematic space is where your theories have to live. Kinematic space is nothing more or less than the space of observations, the space defined by where things land in your detectors, or if you’re a human and not a collider, in your eyes. If you want to strip away all the speculation about the nature of reality, this is all that is left over. Any theory, of any reality, will have to be described in this way. So if you think reality might need a totally new weird theory, it makes sense to approach things like Nima does, and start with the one thing that will always remain: observations.

I Ain’t Afraid of No-Ghost Theorems

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In honor of Halloween this week, let me say a bit about the spookiest term in physics: ghosts.

In particle physics, we talk about the universe in terms of quantum fields. There is an electron field for electrons, a gluon field for gluons, a Higgs field for Higgs bosons. The simplest fields, for the simplest particles, can be described in terms of just a single number at each point in space and time, a value describing how strong the field is. More complicated fields require more numbers.

Most of the fundamental forces have what we call vector fields. They’re called this because they are often described with vectors, lists of numbers that identify a direction in space and time. But these vectors actually contain too many numbers.

These extra numbers have to be tidied up in some way in order to describe vector fields in the real world, like the electromagnetic field or the gluon field of the strong nuclear force. There are a number of tricks, but the nicest is usually to add some extra particles called ghosts. Ghosts are designed to cancel out the extra numbers in a vector, leaving the right description for a vector field. They’re set up mathematically such that they can never be observed, they’re just a mathematical trick.

Mathematical tricks aren’t all that spooky (unless you’re scared of mathematics itself, anyway). But in physics, ghosts can take on a spookier role as well.

In order to do their job cancelling those numbers, ghosts need to function as a kind of opposite to a normal particle, a sort of undead particle. Normal particles have kinetic energy: as they go faster and faster, they have more and more energy. Said another way, it takes more and more energy to make them go faster. Ghosts have negative kinetic energy: the faster they go, the less energy they have.

If ghosts are just a mathematical trick, that’s fine, they’ll do their job and cancel out what they’re supposed to. But sometimes, physicists accidentally write down a theory where the ghosts aren’t just a trick cancelling something out, but real particles you could detect, without anything to hide them away.

In a theory where ghosts really exist, the universe stops making sense. The universe defaults to the lowest energy it can reach. If making a ghost particle go faster reduces its energy, then the universe will make ghost particles go faster and faster, and make more and more ghost particles, until everything is jam-packed with super-speedy ghosts unto infinity, never-ending because it’s always possible to reduce the energy by adding more ghosts.

The absence of ghosts, then, is a requirement for a sensible theory. People prove theorems showing that their new ideas don’t create ghosts. And if your theory does start seeing ghosts…well, that’s the spookiest omen of all: an omen that your theory is wrong.

Transforming Particles Are Probably Here to Stay

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It can be tempting to imagine the world in terms of lego-like building-blocks. Atoms stick together protons, neutrons, and electrons, and protons and neutrons are made of stuck-together quarks in turn. And while atoms, despite the name, aren’t indivisible, you might think that if you look small enough you’ll find indivisible, unchanging pieces, the smallest building-blocks of reality.

Part of that is true. We might, at some point, find the smallest pieces, the things everything else is made of. (In a sense, it’s quite likely we’ve already found them!) But those pieces don’t behave like lego blocks. They aren’t indivisible and unchanging.

Instead, particles, even the most fundamental particles, transform! The most familiar example is beta decay, a radioactive process where a neutron turns into a proton, emitting an electron and a neutrino. This process can be explained in terms of more fundamental particles: the neutron is made of three quarks, and one of those quarks transforms from a “down quark” to an “up quark”. But the explanation, as far as we can tell, doesn’t go any deeper. Quarks aren’t unchanging, they transform.

Beta decay! Ignore the W, which is important but not for this post.

There’s a suggestion I keep hearing, both from curious amateurs and from dedicated crackpots: why doesn’t this mean that quarks have parts? If a down quark can turn into an up dark, an electron, and a neutrino, then why doesn’t that mean that a down quark contains an up quark, an electron, and a neutrino?

The simplest reason is that this isn’t the only way a quark transforms. You can also have beta-plus decay, where an up quark transforms into a down quark, emitting a neutrino and the positively charged anti-particle of the electron, called a positron.

Also, ignore the directions of the arrows, that’s weird particle physics notation that doesn’t matter here.

So to make your idea work, you’d somehow need each down quark to contain an up quark plus some other particles, and each up quark to contain a down quark plus some other particles.

Can you figure out some complicated scheme that works like that? Maybe. But there’s a deeper reason why this is the wrong path.

Transforming particles are part of a broader phenomenon, called particle production. Reactions in particle physics can produce new particles that weren’t there before. This wasn’t part of the earliest theories of quantum mechanics that described one electron at a time. But if you want to consider the quantum properties of not just electrons, but the electric field as well, then you need a more complete theory, called a quantum field theory. And in those theories, you can produce new particles. It’s as simple as turning on the lights: from a wiggling electron, you make light, which in a fully quantum theory is made up of photons. Those photons weren’t “part of” the electron to start with, they are produced by its motion.

If you want to avoid transforming particles, to describe everything in terms of lego-like building-blocks, then you want to avoid particle production altogether. Can you do this in a quantum field theory?

Actually, yes! But your theory won’t describe the whole of the real world.

In physics, we have examples of theories that don’t have particle production. These example theories have a property called integrability. They are theories we can “solve”, doing calculations that aren’t possible in ordinary theories, named after the fact that the oldest such theories in classical mechanics were solved using integrals.

Normal particle physics theories have conserved charges. Beta decay conserves electric charge: you start out with a neutral particle, and end up with one particle with positive charge and another with negative charge. It also conserves other things, like “electron-number” (the electron has electron-number one, the neutrino that comes out with it has electron-number minus one), energy, and momentum.

Integrable theories have those charges too, but they have more. In fact, they have an infinite number of conserved charges. As a result, you can show that in these theories it is impossible to produce new particles. It’s as if each particle’s existence is its own kind of conserved charge, one that can never be created or destroyed, so that each collision just rearranges the particles, never makes new ones.

But while we can write down these theories, we know they can’t describe the whole of the real world. In an integrable theory, when you build things up from the fundamental building-blocks, their energy follows a pattern. Compare the energy of a bunch of different combinations, and you find a characteristic kind of statistical behavior called a Poisson distribution.

Look at the distribution of energies of nuclei of atoms, and you’ll find a very different kind of behavior. It’s called a Wigner-Dyson distribution, and it indicates the opposite of integrability: chaos. Chaos is behavior that can’t be “solved” like integrable theories, behavior that has to be approached by simulations and approximations.

So if you really want there to be un-changing building-blocks, if you think that’s really essential? Then you should probably start looking at integrable theories. But I wouldn’t hold my breath if I were you: the real world seems pretty clearly chaotic, not integrable. And probably, that means particle production is here to stay.