Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime / Что-то глубоко скрытое: квантовые миры и появление пространства-времени (by Sean Carroll, 2019) - аудиокнига на английском

PROLOGUE Don’t Be Afraid You don’t need a PhD in theoretical physics to be afraid of quantum mechanics. But it doesn’t hurt. That might seem strange. Quantum mechanics is our best theory of the microscopic world. It describes how atoms and particles interact through the forces of nature, and makes incredibly precise experimental predictions. To be sure, quantum mechanics has something of a reputation for being difficult, mysterious, just this side of magic. But professional physicists, of all people, should be relatively comfortable with a theory like that. They are constantly doing intricate calculations involving quantum phenomena, and building giant machines dedicated to testing the resulting predictions. Surely we’re not suggesting that physicists have been faking it all this time? They haven’t been faking, but they haven’t exactly been honest with themselves either. On the one hand, quantum mechanics is the heart and soul of modern physics. Astrophysicists, particle physicists, atomic physicists, laser physicists—everyone uses quantum mechanics all the time, and they’re very good at it. It’s not just a matter of esoteric research. Quantum mechanics is ubiquitous in modern technology. Semiconductors, transistors, microchips, lasers, and computer memory all rely on quantum mechanics to function. For that matter, quantum mechanics is necessary to make sense of the most basic features of the world around us. Essentially all of chemistry is a matter of applied quantum mechanics. To understand how the sun shines, or why tables are solid, you need quantum mechanics. Imagine closing your eyes. Hopefully things look pretty dark. You might think that makes sense, because no light is coming in. But that’s not quite right; infrared light, with a slightly longer wavelength than visible light, is being emitted all the time by any warm object, and that includes your own body. If our eyes were as sensitive to infrared light as they are to visible light, we would be blinded even when our lids were closed, from all the light emitted by our eyeballs themselves. But the rods and cones that act as light receptors in our eyes are cleverly sensitive to visible light, not infrared. How do they manage that? Ultimately, the answer comes down to quantum mechanics. Quantum mechanics isn’t magic. It is the deepest, most comprehensive view of reality we have. As far as we currently know, quantum mechanics isn’t just an approximation of the truth; it is the truth. That’s subject to change in the face of unexpected experimental results, but we’ve seen no hints of any such surprises thus far. The development of quantum mechanics in the early years of the twentieth century, involving names like Planck, Einstein, Bohr, Heisenberg, Schr?dinger, and Dirac, left us by 1927 with a mature understanding that is surely one of the greatest intellectual accomplishments in human history. We have every reason to be proud. On the other hand, in the memorable words of Richard Feynman, “I think I can safely say that nobody understands quantum mechanics.” We use quantum mechanics to design new technologies and predict the outcomes of experiments. But honest physicists admit that we don’t truly understand quantum mechanics. We have a recipe that we can safely apply in certain prescribed situations, and which returns mind-bogglingly precise predictions that have been triumphantly vindicated by the data. But if you want to dig deeper and ask what is really going on, we simply don’t know. Physicists tend to treat quantum mechanics like a mindless robot they rely on to perform certain tasks, not as a beloved friend they care about on a personal level. This attitude among the professionals seeps into how quantum mechanics gets explained to the wider world. What we would like to do is to present a fully formed picture of Nature, but we can’t quite do that, since physicists don’t agree about what quantum mechanics actually says. Instead, popular treatments tend to emphasize that quantum mechanics is mysterious, baffling, impossible to understand. That message goes against the basic principles that science stands for, which include the idea that the world is fundamentally intelligible. We have something of a mental block when it comes to quantum mechanics, and we need a bit of quantum therapy to help get past it. When we teach quantum mechanics to students, they are taught a list of rules. Some of the rules are of a familiar type: there’s a mathematical description of quantum systems, plus an explanation of how such systems evolve over time. But then there are a bunch of extra rules that have no analogue in any other theory of physics. These extra rules tell us what happens when we observe a quantum system, and that behavior is completely different from how the system behaves when we’re not observing it. What in the world is going on with that? There are basically two options. One is that the story we’ve been telling our students is woefully incomplete, and in order for quantum mechanics to qualify as a sensible theory we need to understand what a “measurement” or “observation” is, and why it seems so different from what the system does otherwise. The other option is that quantum mechanics represents a violent break from the way we have always thought about physics before, shifting from a view where the world exists objectively and independently of how we perceive it, to one where the act of observation is somehow fundamental to the nature of reality. In either case, the textbooks should by all rights spend time exploring these options, and admit that even though quantum mechanics is extremely successful, we can’t claim to be finished developing it just yet. They don’t. For the most part, they pass over this issue in silence, preferring to stay in the physicist’s comfort zone of writing down equations and challenging students to solve them. That’s embarrassing. And it gets worse. You might think, given this situation, that the quest to understand quantum mechanics would be the single biggest goal in all of physics. Millions of dollars of grant money would flow to researchers in quantum foundations, the brightest minds would flock to the problem, and the most important insights would be rewarded with prizes and prestige. Universities would compete to hire the leading figures in the area, dangling superstar salaries to lure them away from rival institutions. Sadly, no. Not only is the quest to make sense of quantum mechanics not considered a high-status specialty within modern physics; in many quarters it’s considered barely respectable at all, if not actively disparaged. Most physics departments have nobody working on the problem, and those who choose to do so are looked upon with suspicion. (Recently while writing a grant proposal, I was advised to concentrate on describing my work in gravitation and cosmology, which is considered legitimate, and remain silent about my work on the foundations of quantum mechanics, as that would make me appear less serious.) There have been important steps forward over the last ninety years, but they have typically been made by headstrong individuals who thought the problems were important despite what all of their colleagues told them, or by young students who didn’t know any better and later left the field entirely. In one of Aesop’s fables, a fox sees a juicy bunch of grapes and leaps to reach it, but can’t quite jump high enough. In frustration he declares that the grapes were probably sour, and he never really wanted them anyway. The fox represents “physicists,” and the grapes are “understanding quantum mechanics.” Many researchers have decided that understanding how nature really works was never really important; all that matters is the ability to make particular predictions. Scientists are trained to value tangible results, whether they are exciting experimental findings or quantitative theoretical models. The idea of working to understand a theory we already have, even if that effort might not lead to any specific new technologies or predictions, can be a tough sell. The underlying tension was illustrated in the TV show The Wire, where a group of hardworking detectives labored for months to meticulously gather evidence that would build a case against a powerful drug ring. Their bosses, meanwhile, had no patience for such incremental frivolity. They just wanted to see drugs on the table for their next press conference, and encouraged the police to bang heads and make splashy arrests. Funding agencies and hiring committees are like those bosses. In a world where all the incentives push us toward concrete, quantifiable outcomes, less pressing big-picture concerns can be pushed aside as we race toward the next immediate goal. This book has three main messages. The first is that quantum mechanics should be understandable—even if we’re not there yet—and achieving such understanding should be a high-priority goal of modern science. Quantum mechanics is unique among physical theories in drawing an apparent distinction between what we see and what really is. That poses a special challenge to the minds of scientists (and everyone else), who are used to thinking about what we see as unproblematically “real,” and working to explain things accordingly. But this challenge isn’t insuperable, and if we free our minds from certain old-fashioned and intuitive ways of thinking, we find that quantum mechanics isn’t hopelessly mystical or inexplicable. It’s just physics. The second message is that we have made real progress toward understanding. I will focus on the approach I feel is clearly the most promising route, the Everett or Many-Worlds formulation of quantum mechanics. Many-Worlds has been enthusiastically embraced by many physicists, but it has a sketchy reputation among people who are put off by a proliferation of other realities containing copies of themselves. If you are one of those people, I want to at least convince you that Many-Worlds is the purest way of making sense of quantum mechanics—it’s where we end up if we just follow the path of least resistance in taking quantum phenomena seriously. In particular, the multiple worlds are predictions of the formalism that is already in place, not something added in by hand. But Many-Worlds isn’t the only respectable approach, and we will mention some of its main competitors. (I will endeavor to be fair, if not necessarily balanced.) The important thing is that the various approaches are all well-constructed scientific theories, with potentially different experimental ramifications, not just woolly-headed “interpretations” to be debated over cognac and cigars after we’re finished doing real work. The third message is that all this matters, and not just for the integrity of science. The success to date of the existing adequate-but-not-perfectly-coherent framework of quantum mechanics shouldn’t blind us to the fact that there are circumstances under which such an approach simply isn’t up to the task. In particular, when we turn to understanding the nature of spacetime itself, and the origin and ultimate fate of the entire universe, the foundations of quantum mechanics are absolutely crucial. I’ll introduce some new, exciting, and admittedly tentative proposals that draw provocative connections between quantum entanglement and how spacetime bends and curves—the phenomenon you and I know as “gravity.” For many years now, the search for a complete and compelling quantum theory of gravity has been recognized as an important scientific goal (prestige, prizes, stealing away faculty, and all that). It may be that the secret is not to start with gravity and “quantize” it, but to dig deeply into quantum mechanics itself, and find that gravity was lurking there all along. We don’t know for sure. That’s the excitement and anxiety of cutting-edge research. But the time has come to take the fundamental nature of reality seriously, and that means confronting quantum mechanics head-on. 1 What’s Going On Looking at the Quantum World Albert Einstein, who had a way with words as well as with equations, was the one who stuck quantum mechanics with the label it has been unable to shake ever since: spukhaft, usually translated from German to English as “spooky.” If nothing else, that’s the impression we get from most public discussions of quantum mechanics. We’re told that it’s a part of physics that is unavoidably mystifying, weird, bizarre, unknowable, strange, baffling. Spooky. Inscrutability can be alluring. Like a mysterious, sexy stranger, quantum mechanics tempts us into projecting all sorts of qualities and capacities onto it, whether they are there or not. A brief search for books with “quantum” in the title reveals the following list of purported applications: Quantum Success Quantum Leadership Quantum Consciousness Quantum Touch Quantum Yoga Quantum Eating Quantum Psychology Quantum Mind Quantum Glory Quantum Forgiveness Quantum Theology Quantum Happiness Quantum Poetry Quantum Teaching Quantum Faith Quantum Love For a branch of physics that is often described as only being relevant to microscopic processes involving subatomic particles, that’s a pretty impressive r?sum?. To be fair, quantum mechanics—or “quantum physics,” or “quantum theory,” the labels are all interchangeable—is not only relevant to microscopic processes. It describes the whole world, from you and me to stars and galaxies, from the centers of black holes to the beginning of the universe. But it is only when we look at the world in extreme close-up that the apparent weirdness of quantum phenomena becomes unavoidable. One of the themes in this book is that quantum mechanics doesn’t deserve the connotation of spookiness, in the sense of some ineffable mystery that it is beyond the human mind to comprehend. Quantum mechanics is amazing; it is novel, profound, mind-stretching, and a very different view of reality from what we’re used to. Science is like that sometimes. But if the subject seems difficult or puzzling, the scientific response is to solve the puzzle, not to pretend it’s not there. There’s every reason to think we can do that for quantum mechanics just like any other physical theory. Many presentations of quantum mechanics follow a typical pattern. First, they point to some counterintuitive quantum phenomenon. Next, they express bafflement that the world can possibly be that way, and despair of it making sense. Finally (if you’re lucky), they attempt some sort of explanation. Our theme is prizing clarity over mystery, so I don’t want to adopt that strategy. I want to present quantum mechanics in a way that will make it maximally understandable right from the start. It will still seem strange, but that’s the nature of the beast. What it won’t seem, hopefully, is inexplicable or unintelligible. We will make no effort to follow historical order. In this chapter we’ll look at the basic experimental facts that force quantum mechanics upon us, and in the next we’ll quickly sketch the Many-Worlds approach to making sense of those observations. Only in the chapter after that will we offer a semi-historical account of the discoveries that led people to contemplate such a dramatically new kind of physics in the first place. Then we’ll hammer home exactly how dramatic some of the implications of quantum mechanics really are. With all that in place, over the rest of the book we can set about the fun task of seeing where all this leads, demystifying the most striking features of quantum reality. Physics is one of the most basic sciences, indeed one of the most basic human endeavors. We look around the world, we see it is full of stuff. What is that stuff, and how does it behave? These are questions that have been asked ever since people started asking questions. In ancient Greece, physics was thought of as the general study of change and motion, of both living and nonliving matter. Aristotle spoke a vocabulary of tendencies, purposes, and causes. How an entity moves and changes can be explained by reference to its inner nature and to external powers acting upon it. Typical objects, for example, might by nature be at rest; in order for them to move, it is necessary that something be causing that motion. All of this changed thanks to a clever chap named Isaac Newton. In 1687 he published Principia Mathematica, the most important work in the history of physics. It was there that he laid the groundwork for what we now call “classical” or simply “Newtonian” mechanics. Newton blew away any dusty talk of natures and purposes, revealing what lay underneath: a crisp, rigorous mathematical formalism with which teachers continue to torment students to this very day. Whatever memory you may have of high-school or college homework assignments dealing with pendulums and inclined planes, the basic ideas of classical mechanics are pretty simple. Consider an object such as a rock. Ignore everything about the rock that a geologist might consider interesting, such as its color and composition. Put aside the possibility that the basic structure of the rock might change, for example, if you smashed it to pieces with a hammer. Reduce your mental image of the rock down to its most abstract form: the rock is an object, and that object has a location in space, and that location changes with time. Classical mechanics tells us precisely how the position of the rock changes with time. We’re very used to that by now, so it’s worth reflecting on how impressive this is. Newton doesn’t hand us some vague platitudes about the general tendency of rocks to move more or less in this or that fashion. He gives us exact, unbreakable rules for how everything in the universe moves in response to everything else—rules that can be used to catch baseballs or land rovers on Mars. Here’s how it works. At any one moment, the rock will have a position and also a velocity, a rate at which it’s moving. According to Newton, if no forces act on the rock, it will continue to move in a straight line at constant velocity, for all time. (Already this is a major departure from Aristotle, who would have told you that objects need to be constantly pushed if they are to be kept in motion.) If a force does act on the rock, it will cause acceleration—some change in the velocity of the rock, which might make it go faster, or slower, or merely alter its direction—in direct proportion to how much force is applied. That’s basically it. To figure out the entire trajectory of the rock, you need to tell me its position, its velocity, and what forces are acting on it. Newton’s equations tell you the rest. Forces might include the force of gravity, or the force of your hand if you pick up the rock and throw it, or the force from the ground when the rock comes to land. The idea works just as well for billiard balls or rocket ships or planets. The project of physics, within this classical paradigm, consists essentially of figuring out what makes up the stuff of the universe (rocks and so forth) and what forces act on them. Classical physics provides a straightforward picture of the world, but a number of crucial moves were made along the way to setting it up. Notice that we had to be very specific about what information we required to figure out what would happen to the rock: its position, its velocity, and the forces acting on it. We can think of those forces as being part of the outside world, and the important information about the rock itself as consisting of just its position and velocity. The acceleration of the rock at any moment in time, by contrast, is not something we need to specify; that’s exactly what Newton’s laws allow us to calculate from the position and the velocity. Together, the position and velocity make up the state of any object in classical mechanics. If we have a system with multiple moving parts, the classical state of that entire system is just a list of the states of each of the individual parts. The air in a normal-sized room will have perhaps 1027 molecules of different types, and the state of that air would be a list of the position and velocity of every one of them. (Strictly speaking, physicists like to use the momentum of each particle, rather than its velocity, but as far as Newtonian mechanics is concerned the momentum is simply the particle’s mass times its velocity.) The set of all possible states that a system could have is known as the phase space of the system. The French mathematician Pierre-Simon Laplace pointed out a profound implication of the classical mechanics way of thinking. In principle, a vast intellect could know the state of literally every object in the universe, from which it could deduce everything that would happen in the future, as well as everything that had happened in the past. Laplace’s demon is a thought experiment, not a realistic project for an ambitious computer scientist, but the implications of the thought experiment are profound. Newtonian mechanics describes a deterministic, clockwork universe. The machinery of classical physics is so beautiful and compelling that it seems almost inescapable once you grasp it. Many great minds who came after Newton were convinced that the basic superstructure of physics had been solved, and future progress lay in figuring out exactly what realization of classical physics (which particles, which forces) was the right one to describe the universe as a whole. Even relativity, which was world-transforming in its own way, is a variety of classical mechanics rather than a replacement for it. Then along came quantum mechanics, and everything changed. Alongside Newton’s formulation of classical mechanics, the invention of quantum mechanics represents the other great revolution in the history of physics. Unlike anything that had come before, quantum theory didn’t propose a particular physical model within the basic classical framework; it discarded that framework entirely, replacing it with something profoundly different. The fundamental new element of quantum mechanics, the thing that makes it unequivocally distinct from its classical predecessor, centers on the question of what it means to measure something about a quantum system. What exactly a measurement is, and what happens when we measure something, and what this all tells us about what’s really happening behind the scenes: together, these questions constitute what’s called the measurement problem of quantum mechanics. There is absolutely no consensus within physics or philosophy on how to solve the measurement problem, although there are a number of promising ideas. Attempts to address the measurement problem have led to the emergence of a field known as the interpretation of quantum mechanics, although the label isn’t very accurate. “Interpretations” are things that we might apply to a work of literature or art, where people might have different ways of thinking about the same basic object. What’s going on in quantum mechanics is something else: a competition between truly distinct scientific theories, incompatible ways of making sense of the physical world. For this reason, modern workers in this field prefer to call it “foundations of quantum mechanics.” The subject of quantum foundations is part of science, not literary criticism. Nobody ever felt the need to talk about “interpretations of classical mechanics”—classical mechanics is perfectly transparent. There is a mathematical formalism that speaks of positions and velocities and trajectories, and oh, look: there is a rock whose actual motion in the world obeys the predictions of that formalism. There is, in particular, no such thing as a measurement problem in classical mechanics. The state of the system is given by its position and its velocity, and if we want to measure those quantities, we simply do so. Of course, we can measure the system sloppily or crudely, thereby obtaining imprecise results or altering the system itself. But we don’t have to; just by being careful, we can precisely measure everything there is to know about the system without altering it in any noticeable way. Classical mechanics offers a clear and unambiguous relationship between what we see and what the theory describes. Quantum mechanics, for all its successes, offers no such thing. The enigma at the heart of quantum reality can be summed up in a simple motto: what we see when we look at the world seems to be fundamentally different from what actually is. Think about electrons, the elementary particles orbiting atomic nuclei, whose interactions are responsible for all of chemistry and hence almost everything interesting around you right now. As we did with the rock, we can ignore some of the electron’s specific properties, like its spin and the fact that it has an electric field. (Really we could just stick with the rock as our example—rocks are quantum systems just as much as electrons are—but switching to a subatomic particle helps us remember that the features distinguishing quantum mechanics only become evident when we consider very tiny objects indeed.) Unlike in classical mechanics, where the state of a system is described by its position and velocity, the nature of a quantum system is something a bit less concrete. Consider an electron in its natural habitat, orbiting the nucleus of an atom. You might think, from the word “orbit” as well as from the numerous cartoon depictions of atoms you have doubtless been exposed to over the years, that the orbit of an electron is more or less like the orbit of a planet in the solar system. The electron (so you might think) has a location, and a velocity, and as time passes it zips around the central nucleus in a circle or maybe an ellipse. Quantum mechanics suggests something different. We can measure values of the location or velocity (though not at the same time), and if we are sufficiently careful and talented experimenters we will obtain some answer. But what we’re seeing through such a measurement is not the actual, complete, unvarnished state of the electron. Indeed, the particular measurement outcome we will obtain cannot be predicted with perfect confidence, in a profound departure from the ideas of classical mechanics. The best we can do is to predict the probability of seeing the electron in any particular location or with any particular velocity. The classical notion of the state of a particle, “its location and its velocity,” is therefore replaced in quantum mechanics by something utterly alien to our everyday experience: a cloud of probability. For an electron in an atom, this cloud is more dense toward the center and thins out as we get farther away. Where the cloud is thickest, the probability of seeing the electron is highest; where it is diluted almost to imperceptibility, the probability of seeing the electron is vanishingly small. This cloud is often called a wave function, because it can oscillate like a wave, as the most probable measurement outcome changes over time. We usually denote a wave function by ?, the Greek letter Psi. For every possible measurement outcome, such as the position of the particle, the wave function assigns a specific number, called the amplitude associated with that outcome. The amplitude that a particle is at some position x0, for example, would be written ?(x0). The probability of getting that outcome when we perform a measurement is given by the amplitude squared. Probability of a particular outcome = |Amplitude for that outcome|2 This simple relation is called the Born rule, after physicist Max Born.* Part of our task will be to figure out where in the world such a rule came from. We’re most definitely not saying that there is an electron with some position and velocity, and we just don’t know what those are, so the wave function encapsulates our ignorance about those quantities. In this chapter we’re not saying anything at all about what “is,” only what we observe. In chapters to come, I will pound the table and insist that the wave function is the sum total of reality, and ideas such as the position or the velocity of the electron are merely things we can measure. But not everyone sees things that way, and for the moment we are choosing to don a mask of impartiality. Let’s place the rules of classical and quantum mechanics side by side to compare them. The state of a classical system is given by the position and velocity of each of its moving parts. To follow its evolution, we imagine something like the following procedure: Rules of Classical Mechanics 1. Set up the system by fixing a specific position and velocity for each part. 2. Evolve the system using Newton’s laws of motion. That’s it. The devil is in the details, of course. Some classical systems can have a lot of moving pieces. In contrast, the rules of standard textbook quantum mechanics come in two parts. In the first part, we have a structure that exactly parallels that of the classical case. Quantum systems are described by wave functions rather than by positions and velocities. Just as Newton’s laws of motion govern the evolution of the state of a system in classical mechanics, there is an equation that governs how wave functions evolve, called Schr?dinger’s equation. We can express Schr?dinger’s equation in words as: “The rate of change of a wave function is proportional to the energy of the quantum system.” Slightly more specifically, a wave function can represent a number of different possible energies, and the Schr?dinger equation says that high-energy parts of the wave function evolve rapidly, while low-energy parts evolve very slowly. Which makes sense, when we think about it. What matters for our purposes is simply that there is such an equation, one that predicts how wave functions evolve smoothly through time. That evolution is as predictable and inevitable as the way objects move according to Newton’s laws in classical mechanics. Nothing weird is happening yet. The beginning of the quantum recipe reads something like this: Rules of Quantum Mechanics (Part One) 1. Set up the system by fixing a specific wave function ?. 2. Evolve the system using Schr?dinger’s equation. So far, so good—these parts of quantum mechanics exactly parallel their classical predecessors. But whereas the rules of classical mechanics stop there, the rules of quantum mechanics keep going. All the extra rules deal with measurement. When you perform a measurement, such as the position or spin of a particle, quantum mechanics says there are only certain possible results you will ever get. You can’t predict which of the results it will be, but you can calculate the probability for each allowed outcome. And after your measurement is done, the wave function collapses to a completely different function, with all of the new probability concentrated on whatever result you just got. So if you measure a quantum system, in general the best you can do is predict probabilities for various outcomes, but if you were to immediately measure the same quantity again, you will always get the same answer—the wave function has collapsed onto that outcome. Let’s write this out in gory detail. Rules of Quantum Mechanics (Part Two) 3. There are certain observable quantities we can choose to measure, such as position, and when we do measure them, we obtain definite results. 4. The probability of getting any one particular result can be calculated from the wave function. The wave function associates an amplitude with every possible measurement outcome; the probability for any outcome is the square of that amplitude. 5. Upon measurement, the wave function collapses. However spread out it may have been pre-measurement, afterward it is concentrated on the result we obtained. In a modern university curriculum, when physics students are first exposed to quantum mechanics, they are taught some version of these five rules. The ideology associated with this presentation—treat measurements as fundamental, wave functions collapse when they are observed, don’t ask questions about what’s going on behind the scenes—is sometimes called the Copenhagen interpretation of quantum mechanics. But people, including the physicists from Copenhagen who purportedly invented this interpretation, disagree on precisely what that label should be taken to describe. We can just refer to it as “standard textbook quantum mechanics.” The idea that these rules represent how reality actually works is, needless to say, outrageous. What precisely do you mean by a “measurement”? How quickly does it happen? What exactly constitutes a measuring apparatus? Does it need to be human, or have some amount of consciousness, or perhaps the ability to encode information? Or maybe it just has to be macroscopic, and if so how macroscopic does it have to be? When exactly does the measurement occur, and how quickly? How in the world does the wave function collapse so dramatically? If the wave function were very spread out, does the collapse happen faster than the speed of light? And what happens to all the possibilities that were seemingly allowed by the wave function but which we didn’t observe? Were they never really there? Do they just vanish into nothingness? To put things most pointedly: Why do quantum systems evolve smoothly and deterministically according to the Schr?dinger equation as long as we aren’t looking at them, but then dramatically collapse when we do look? How do they know, and why do they care? (Don’t worry, we’re going to answer all these questions.) Science, most people think, seeks to understand the natural world. We observe things happening, and science hopes to provide an explanation for what is going on. In its current textbook formulation, quantum mechanics has failed in this ambition. We don’t know what’s really going on, or at least the community of professional physicists cannot agree on what it is. What we have instead is a recipe that we enshrine in textbooks and teach to our students. Isaac Newton could tell you, starting with the position and velocity of a rock that you have thrown into the air in the Earth’s gravitational field, just what the subsequent trajectory of that rock was going to be. Analogously, starting with a quantum system prepared in some particular way, the rules of quantum mechanics can tell you how the wave function will change over time, and what the probability of various possible measurement outcomes will be should you choose to observe it. The fact that the quantum recipe provides us with probabilities rather that certainties might be annoying, but we could learn to live with it. What bugs us, or should, is our lack of understanding about what is actually happening. Imagine that some devious genius figured out all the laws of physics, but rather than revealing them to the rest of the world, they programmed a computer to answer questions concerning specific physics problems, and put an interface to the program on a web page. Anyone who was interested could just surf over to that site, type in a well-posed physics question, and get the correct answer. Such a program would obviously be of great use to scientists and engineers. But having access to the site wouldn’t qualify as understanding the laws of physics. We would have an oracle that was in the business of providing answers to specific questions, but we ourselves would be completely lacking in any intuitive idea of the underlying rules of the game. The rest of the world’s scientists, presented with such an oracle, wouldn’t be moved to declare victory; they would continue with their work of figuring out what the laws of nature actually were. Quantum mechanics, in the form in which it is currently presented in physics textbooks, represents an oracle, not a true understanding. We can set up specific problems and answer them, but we can’t honestly explain what’s happening behind the scenes. What we do have are a number of good ideas about what that could be, and it’s past time that the physics community started taking these ideas seriously. * There’s a slight technicality, which we’ll mention here and then pretty much forget аbout: the amplitude for any given outcome is actually a complex number, not a real number. Real numbers are the ones that appear on the number line, any number between minus infinity and plus infinity. Anytime you take the square of a real number, you get another real number that is greater than or equal to zero, so as far as real numbers are concerned there’s no such thing as the square root of a negative number. Mathematicians long ago realized that square roots of negative numbers would be really useful things to have, so they defined the “imaginary unit” i as the square root of -1. An imaginary number is just a real number, called “the imaginary part,” times i. Then a complex number is just a combination of a real number and an imaginary one. The little bars in the notation |Amplitude|2 in the Born rule mean that we actually add the squares of the real and the imaginary parts. All that is just for the sticklers out there; henceforth we’ll be happy to say “the probability is the amplitude squared” and be done with it. 2 The Courageous Formulation Austere Quantum Mechanics The attitude inculcated into young students by modern quantum mechanics textbooks has been compactly summarized by physicist N. David Mermin as “Shut up and calculate!” Mermin himself wasn’t advocating such a position, but others have. Every decent physicist spends a good deal of time calculating things, whatever their attitude toward quantum foundations might be. So really the admonition could be shortened to simply “Shut up!”* It wasn’t always thus. Quantum mechanics took decades to piece together, but rounded into its modern form around 1927. In that year, at the Fifth International Solvay Conference in Belgium, the world’s leading physicists came together to discuss the status and meaning of quantum theory. By that time the experimental evidence was clear, and physicists were at long last in possession of a quantitative formulation of the rules of quantum mechanics. It was time to roll up some sleeves and figure out what this crazy new worldview actually amounted to. The discussions at this conference help set the stage, but our goal here isn’t to get the history right. We want to understand the physics. So we’ll sketch out a logical path by which we will be led to a full-blown scientific theory of quantum mechanics. No vague mysticism, no seemingly ad hoc rules. Just a simple set of assumptions leading to some remarkable conclusions. With this picture in mind, many things that might otherwise have seemed ominously mysterious will suddenly start to make perfect sense. The Solvay Conference has gone down in history as the beginning of a famous series of debates between Albert Einstein and Niels Bohr over how to think about quantum mechanics. Bohr, a Danish physicist based in Copenhagen who is rightfully regarded as the godfather of quantum theory, advocated an approach similar to the textbook recipe we discussed in the last chapter: use quantum mechanics to calculate the probabilities for measurement outcomes, but don’t ask of it anything more than that. Do not, in particular, worry too much about what is really happening behind the scenes. Supported by his younger colleagues Werner Heisenberg and Wolfgang Pauli, Bohr insisted that quantum mechanics was a perfectly fine theory as it was. Einstein would have none of it. He was firmly convinced that the duty of physics was precisely to ask what was going on behind the scenes, and that the state of quantum mechanics in 1927 fell far short of providing a satisfactory account of nature. With his own sympathizers, such as Erwin Schr?dinger and Louis de Broglie, Einstein advocated looking more deeply, and attempting to extend and generalize quantum mechanics into a satisfactory physical theory. Participants in the 1927 Solvay Conference. Among the more well-known were: 1. Max Planck, 2. Marie Curie, 3. Paul Dirac, 4. Erwin Schr?dinger, 5. Albert Einstein, 6. Louis de Broglie, 7. Wolfgang Pauli, 8. Max Born, 9. Werner Heisenberg, and 10. Niels Bohr. (Courtesy of Wikipedia) Einstein and his compatriots had reason to be cautiously optimistic that such a new-and-improved theory was out there to be found. Just a few decades before, in the later years of the nineteenth century, physicists had developed the theory of statistical mechanics, which described the motion of large numbers of atoms and molecules. A key step in that development—which all took place under the rubric of classical mechanics, before quantum theory came on the scene—was the idea that we can talk profitably about the behavior of a large collection of particles even if we don’t know precisely the position and velocity of each one of them. All we need to know is a probability distribution describing the likelihood that the particles might be behaving in various ways. In statistical mechanics, in other words, we think that there actually is some particular classical state of all the particles, but we don’t know it, all we have is a distribution of probabilities. Happily, such a distribution is all we need to do a great deal of useful physics, since it fixes properties such as the temperature and pressure of the system. But the distribution isn’t a complete description of the system; it’s simply a reflection of what we know (or don’t) about it. To tag this distinction with philosophical buzzwords, in statistical mechanics the probability distribution is an epistemic notion—describing the state of our knowledge—rather than an ontological one—describing some objective feature of reality. Epistemology is the study of knowledge; ontology is the study of what is real. It was natural, in 1927, to suspect that quantum mechanics should be thought of along similar lines. After all, by that time we had figured out that what we use wave functions for is to calculate the probability of any particular measurement outcome. Surely it makes sense to imagine that nature itself knows precisely what the outcome is going to be, but the formalism of quantum theory simply doesn’t completely capture that knowledge, and thus needs to be improved. The wave function, in this view, isn’t the whole story; there are additional “hidden variables” that fix what the actual measurement outcomes are going to be, even if we don’t know (and perhaps can’t ever determine ahead of the measurement) what their values are. Maybe. But in subsequent years a number of results have been obtained, most notably by the physicist John Bell in the 1960s, implying that the most simple and straightforward attempts along these lines are doomed to failure. People tried—de Broglie actually put forward a specific theory, which was rediscovered and extended by David Bohm in the 1950s, and Einstein and Schr?dinger both batted around ideas. But Bell’s theorem implies that any such theory requires “action at a distance”—a measurement at one location can instantly affect the state of the universe arbitrarily far away. This seems to be in violation of the spirit if not the letter of the theory of relativity, which says that objects and influences cannot propagate faster than the speed of light. The hidden-variable approach is still being actively pursued, but all known attempts along these lines are ungainly and hard to reconcile with modern theories such as the Standard Model of particle physics, not to mention speculative ideas about quantum gravity, as we’ll discuss later. Perhaps this is why Einstein, the pioneer of relativity, never found a satisfactory theory of his own. In the popular imagination, Einstein lost the Bohr-Einstein debates. We are told that Einstein, a creative revolutionary in his youth, had grown old and conservative, and was unable to accept or even understand the dramatic implications of the new quantum theory. (At the time of the Solvay Conference Einstein was forty-eight years old.) Physics subsequently went on without him, as the great man retreated to pursue idiosyncratic attempts at finding a unified field theory. Nothing could be further from the truth. While Einstein failed to put forward a complete and compelling generalization of quantum mechanics, his insistence that physics needs to do better than shut up and calculate was directly on point. It is wildly off base to think that he failed to understand quantum theory. Einstein understood it as well as anyone, and continued to make fundamental contributions to the subject, including demonstrating the importance of quantum entanglement, which plays a central role in our current best picture of how the universe really works. What he failed to do was to convince his fellow physicists of the inadequacy of the Copenhagen approach, and the importance of trying harder to understand the foundations of quantum theory. If we want to follow Einstein’s ambition of a complete, unambiguous, realistic theory of the natural world, but we are discouraged by the difficulties of tacking new hidden variables onto quantum mechanics, is there any remaining strategy left? One approach is to forget about new variables, throw away all the problematic ideas about the measurement process, strip quantum mechanics down to its absolute essentials, and ask what happens. What’s the leanest, meanest version of quantum theory we can invent, and still hope to explain the experimental results? Every version of quantum mechanics (and there are plenty) employs a wave function or something equivalent, and posits that the wave function obeys Schr?dinger’s equation, at least most of the time. These are going to have to be ingredients in just about any theory we can take seriously. Let’s see if we can be stubbornly minimalist, and get away with adding little or nothing else to the formalism. This minimalist approach has two aspects. First, we take the wave function seriously as a direct representation of reality, not just a bookkeeping device to help us organize our knowledge. We treat it as ontological, not epistemic. That’s the most austere strategy we can imagine adopting, since anything else would posit additional structure over and above the wave function. But it’s also a dramatic step, since wave functions are very different from what we observe when we look at the world. We don’t see wave functions; we see measurement outcomes, like the position of a particle. But the theory seems to demand that wave functions play a central role, so let’s see how far we can get by imagining that reality is exactly described by a quantum wave function. Second, if the wave function usually evolves smoothly in accordance with the Schr?dinger equation, let’s suppose that’s what it always does. In other words, let’s erase all of those extra rules about measurement in the quantum recipe entirely, and bring things back to the stark simplicity of the classical paradigm: there is a wave function, and it evolves according to a deterministic rule, and that’s all there is to say. We might call this proposal “austere quantum mechanics,” or AQM for short. It stands in contrast with textbook quantum mechanics, where we appeal to collapsing wave functions and try to avoid talking about the fundamental nature of reality altogether. A bold strategy. But there’s an immediate problem with it: it sure seems like wave functions collapse. When we make measurements of a quantum system with a spread-out wave function, we get a specific answer. Even if we think an electron wave function is a diffuse cloud centered on the nucleus, when we actually look at it we don’t see such a cloud, we see a point-like particle at some particular location. And if we look immediately again, we see the electron in basically the same location. There’s a good reason why the pioneers of quantum mechanics invented the idea of wave functions collapsing—because that’s what they appear to do. But maybe that’s too quick. Let’s turn the question around. Rather than starting with what we see and trying to invent a theory to explain it, let’s start with austere quantum mechanics (wave functions evolving smoothly, that’s it), and ask what people in a world described by that theory would actually experience. Think about what this could mean. In the last chapter, we were careful to talk about the wave function as a kind of mathematical black box from which predictions for measurement outcomes could be extracted: for any particular outcome, the wave function assigns an amplitude, and the probability of getting that outcome is the amplitude squared. Max Born, who proposed the Born rule, was one of the attendees at Solvay in 1927. Now we’re saying something deeper and more direct. The wave function isn’t a bookkeeping device; it’s an exact representation of the quantum system, just as a set of positions and velocities would be a representation of a classical system. The world is a wave function, nothing more nor less. We can use the phrase “quantum state” as a synonym for “wave function,” in direct parallel with calling a set of positions and velocities a “classical state.” This is a dramatic claim about the nature of reality. In ordinary conversation, even among grizzled veterans of quantum physics, people are always talking about concepts like “the position of the electron.” But this wave-function-is-everything view implies that such talk is wrongheaded in an important way. There is no such thing as “the position of the electron.” There is only the electron’s wave function. Quantum mechanics implies a profound distinction between “what we can observe” and “what there really is.” Our observations aren’t revealing pre-existing facts of which we were previously ignorant; at best, they reveal a tiny slice of a much bigger, fundamentally elusive reality. Consider an idea you will often hear: “Atoms are mostly empty space.” Utterly wrong, according to the AQM way of thinking. It comes from a stubborn insistence on thinking of an electron as a tiny classical dot zipping around inside of the wave function, rather than the electron actually being the wave function. In AQM, there’s nothing zipping around; there is only the quantum state. Atoms aren’t mostly empty space; they are described by wave functions that stretch throughout the extent of the atom. The way to break out of our classical intuition is to truly abandon the idea that the electron has some particular location. An electron is in a superposition of every possible position we could see it in, and it doesn’t snap into any one specific location until we actually observe it to be there. “Superposition” is the word physicists use to emphasize that the electron exists in a combination of all positions, with a particular amplitude for each one. Quantum reality is a wave function; classical positions and velocities are merely what we are able to observe when we probe that wave function. So the reality of a quantum system, according to austere quantum mechanics, is described by a wave function or quantum state, which can be thought of as a superposition of every possible outcome of some observation we might want to make. How do we get from there to the annoying reality that wave functions appear to collapse when we make such measurements? Start by examining the statement “we measure the position of the electron” a little more carefully. What does this measurement process actually involve? Presumably some lab equipment and a bit of experimental dexterity, but we don’t need to worry about specifics. All we need to know is that there is some measuring apparatus (a camera or whatever) that somehow interacts with the electron, and then lets us read off where the electron was seen. In the textbook quantum recipe, that’s as much insight as we would ever get. Some of the people who pioneered this approach, including Niels Bohr and Werner Heisenberg, would go a little bit further, making explicit the idea that the measuring apparatus should be thought of as a classical object, even if the electron it was observing was quantum-mechanical. This line of division between the parts of the world that should be treated using quantum versus classical descriptions is sometimes called the Heisenberg cut. Rather than accepting that quantum mechanics is fundamental and classical mechanics is just a good approximation to it in appropriate circumstances, textbook quantum mechanics puts the classical world at center stage, as the right way to talk about people and cameras and other macroscopic things that interact with microscopic quantum systems. This doesn’t smell right. One’s first guess should be that the quantum/ classical divide is a matter of our personal convenience, not a fundamental aspect of nature. If atoms obey the rules of quantum mechanics and cameras are made of atoms, presumably cameras obey the rules of quantum mechanics too. For that matter, you and I presumably obey the rules of quantum mechanics. The fact that we are big, lumbering, macroscopic objects might make classical physics a good approximation to what we are, but our first guess should be that it’s really quantum from top to bottom. If that’s true, it’s not just the electron that has a wave function. The camera should have a wave function of its own. So should the experimenter. Everything is quantum. That simple shift of perspective suggests a new angle on the measurement problem. The AQM attitude is that we shouldn’t treat the measurement process as anything mystical or even in need of its own set of rules; the camera and the electron simply interact with each other according to the laws of physics, just like a rock and the earth do. A quantum state describes systems as superpositions of different measurement outcomes. The electron will, in general, start out in a superposition of various locations—all the places we could see it were we to look. The camera starts out in some wave function that might look complicated, but amounts to saying “This is a camera, and it hasn’t yet looked at the electron.” But then it does look at the electron, which is a physical interaction governed by the Schr?dinger equation. And after that interaction, we might expect that the camera itself is now in a superposition of all the possible measurement outcomes it might have observed: the camera saw the electron in this location, or the camera saw the electron in that location, and so on. If that were the whole story, AQM would be an untenable mess. Electrons in superpositions, cameras in superpositions, nothing much resembling the robust approximately classical world of our experience. Fortunately we can appeal to another startling feature of quantum mechanics: given two different objects (like an electron and a camera), they are not described by separate, individual wave functions. There is only one wave function, which describes the entire system we care about, all the way up to the “wave function of the universe” if we’re talking about the whole shebang. In the case under consideration, there is a wave function describing the combined electron camera system. So what we really have is a superposition of all possible combinations of where the electron might have been located, and where the camera actually observed it to be. Although such a superposition in principle includes every possibility, most of the possible outcomes are assigned zero weight in the quantum state. The cloud of probability vanishes into nothingness for most possible combinations of electron location and camera image. In particular, there is no probability that the electron was in one location but the camera saw it somewhere else (as long as you have a relatively functional camera). This is the quantum phenomenon known as entanglement. There is a single wave function for the combined electron camera system, consisting of a superposition of various possibilities of the form “the electron was at this location, and the camera observed it at the same location.” Rather than the electron and the camera doing their own thing, there is a connection between the two systems. Now let’s take every appearance of “camera” in the above discussion and replace it with “you.” Rather than taking a picture with a mechanical apparatus, we (fancifully) imagine that you have really good eyesight and can see where electrons are just by looking at them. Otherwise, nothing changes. According to the Schr?dinger equation, an initially unentangled situation—the electron is in a superposition of various possible locations, and you haven’t looked at the electron yet—evolves smoothly into an entangled one—a superposition of each location the electron could have been observed, and you having seen the electron in just that location. That’s what the rules of quantum mechanics would say, if we hadn’t tacked on all of those extra annoying bits about the measurement process. Maybe all of those extra rules were just a waste of time. In AQM, the story we just told, about you and the electron entangling and evolving into a superposition, is the complete story. There isn’t anything special about measurement; it’s just something that happens when two systems interact in an appropriate way. And afterward, you and the system you interacted with are in a superposition, in each part of which you have seen the electron in a slightly different location. The problem is, this story still doesn’t match onto what you actually experience when you observe a quantum system. You never feel like you have evolved into a superposition of different possible measurement outcomes; you simply think you’ve seen some specific outcome, which can be predicted with a definite probability. That’s why all of those extra measurement rules were added in the first place. Otherwise you seemingly have a very pretty and elegant formalism (quantum states, smooth evolution) that just doesn’t match up to reality. Time to get a little philosophical. What exactly do we mean by “you” in the above paragraph? Constructing a scientific theory isn’t simply a matter of writing down some equations; we also need to indicate how those equations map onto the world. When it comes to you and me, we tend to think that the process of matching ourselves onto some part of a scientific formalism is pretty straightforward. Certainly in the story told above, where an observer measures the position of an electron, it definitely seems as if that observer evolves into an entangled superposition of the different possible measurement outcomes. But there’s an alternative possibility. Before the measurement happened, there was one electron and one observer (or camera, if you prefer—it doesn’t matter how we think about the thing that interacts with the electron as long as it’s a big, macroscopic object). After they interact, however, rather than thinking of that one observer having evolved into a superposition of possible states, we could think of them as having evolved into multiple possible observers. The right way to describe things after the measurement, in this view, is not as one person with multiple ideas about where the electron was seen, but as multiple worlds, each of which contains a single person with a very definite idea about where the electron was seen. Here’s the big reveal: what we’ve described as austere quantum mechanics is more commonly known as the Everett, or Many-Worlds, formulation of quantum mechanics, first put forward by Hugh Everett in 1957. The Everett view arises from a fundamental annoyance with all of the special rules about measurements that are presented as part of the standard textbook quantum recipe, and suggests instead that there is just a single kind of quantum evolution. The price we pay for this vastly increased elegance of theoretical formalism is that the theory describes many copies of what we think of as “the universe,” each slightly different, but each truly real in some sense. Whether the benefit is worth the cost is an issue about which people disagree. (It is.) In stumbling upon the Many-Worlds formulation, at no point did we take ordinary quantum mechanics and tack on a bunch of universes. The potential for such universes was always there—the universe has a wave function, which can very naturally describe superpositions of many different ways things could be, including superpositions of the whole universe. All we did is to point out that this potential is naturally actualized in the course of ordinary quantum evolution. Once you admit that an electron can be in a superposition of different locations, it follows that a person can be in a superposition of having seen the electron in different locations, and indeed that reality as a whole can be in a superposition, and it becomes natural to treat every term in that superposition as a separate “world.” We didn’t add anything to quantum mechanics, we just faced up to what was there all along. We might reasonably call Everett’s approach the “courageous” formulation of quantum mechanics. It embodies the philosophy that we should take seriously the simplest version of underlying reality that accounts for what we see, even if that reality differs wildly from our everyday experience. Do we have the courage to accept it? This brief introduction to Many-Worlds leaves many questions unanswered. When exactly does the wave function split into many worlds? What separates the worlds from one another? How many worlds are there? Are the other worlds really “real”? How would we ever know, if we can’t observe them? (Or can we?) How does this explain the probability that we’ll end up in one world rather than another one? All of these questions have good answers—or at least plausible ones—and much of the book to come will be devoted to answering them. But we should also admit that the whole picture might be wrong, and something very different is required. Every version of quantum mechanics features two things: (1) a wave function, and (2) the Schr?dinger equation, which governs how wave functions evolve in time. The entirety of the Everett formulation is simply the insistence that there is nothing else, that these ingredients suffice to provide a complete, empirically adequate account of the world. (“Empirically adequate” is a fancy way that philosophers like to say “it fits the data.”) Any other approach to quantum mechanics consists of adding something to that bare-bones formalism, or somehow modifying what is there. The most immediately startling implication of pure Everettian quantum mechanics is the existence of many worlds, so it makes sense to call it Many-Worlds. But the essence of the theory is that reality is described by a smoothly evolving wave function and nothing else. There are extra challenges associated with this philosophy, especially when it comes to matching the extraordinary simplicity of the formalism to the rich diversity of the world we observe. But there are corresponding advantages of clarity and insight. As we’ll see when we ultimately turn to quantum field theory and quantum gravity, taking wave functions as primary in their own right, free of any baggage inherited from our classical experience, is extraordinarily helpful when tackling the deep problems of modern physics. Given the necessity of these two ingredients (wave functions and the Schr?dinger equation), there are a few alternatives to Many-Worlds we might also consider. One is to imagine adding new physical entities over and above the wave function. This approach leads to hidden-variable models, which were in the back of the minds of people like Einstein from the start. These days the most popular such approach is called the de Broglie–Bohm theory, or simply Bohmian mechanics. Alternatively, we could leave the wave function by itself but imagine changing the Schr?dinger equation, for example, to introduce real, random collapses. Finally, we might imagine that the wave function isn’t a physical thing at all, but simply a way of characterizing what we know about reality. Such approaches are broadly known as epistemic models, and a currently popular version is QBism, or quantum Bayesianism. All of these options—and there are many more not listed here—represent truly distinct physical theories, not simply “interpretations” of the same underlying idea. The existence of multiple incompatible theories that all lead (at least thus far) to the observable predictions of quantum mechanics creates a conundrum for anyone who wants to talk about what quantum theory really means. While the quantum recipe is agreed upon by working scientists and philosophers, the underlying reality—what any particular phenomenon actually means—is not. I am defending one particular view of that reality, the Many-Worlds version of quantum mechanics, and for most of this book I will simply be explaining things in Many-Worlds terms. This shouldn’t be taken to imply that the Everettian view is unquestionably right. I hope to explain what the theory says, and why it’s reasonable to assign a high credence to it being the best view of reality we have; what you personally end up believing is up to you. * If you look on the Internet, you will find numerous attributions of “Shut up and calculate!” to Richard Feynman, a physicist who was an all-time great at doing difficult calculations. But he never said any such thing, nor would he have found the sentiment congenial; Feynman thought carefully about quantum mechanics, and nobody ever accused him of shutting up. It’s common for quotations to be reattributed to plausible speakers who are more famous than the actual source of the quote. Sociologist Robert Merton has dubbed this the Matthew Effect, after a line from the Gospel of Matthew: “For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken away even that which he hath.” 3 Why Would Anybody Think This? How Quantum Mechanics Came to Be “Sometimes I’ve believed as many as six impossible things before breakfast,” notes the White Queen to Alice in Through the Looking Glass. That can seem like a useful skill as one comes to grips with quantum mechanics in general, and Many-Worlds in particular. Fortunately, the impossible-seeming things we’re asked to believe aren’t whimsical inventions or logic-busting Zen koans; they are features of the world that we are nudged toward accepting because actual experiments have dragged us, kicking and screaming, in that direction. We don’t choose quantum mechanics; we only choose to face up to it. Physics aspires to figure out what kinds of stuff the world is made of, how that stuff naturally changes over time, and how various bits of stuff interact with one another. In my own environment, I can immediately see many different kinds of stuff: papers and books and a desk and a computer and a cup of coffee and a wastebasket and two cats (one of whom is extremely interested in what’s inside the wastebasket), not to mention less solid things like air and light and sound. By the end of the nineteenth century, scientists had managed to distill every single one of these things down to two fundamental kinds of substances: particles and fields. Particles are point-like objects at a definite location in space, while fields (like the gravitational field) are spread throughout space, taking on a particular value at every point. When a field is oscillating across space and time, we call that a “wave.” So people will often contrast particles with waves, but what they really mean is particles and fields. Quantum mechanics ultimately unified particles and fields into a single entity, the wave function. The impetus to do so came from two directions: first, physicists discovered that things they thought were waves, like the electric and magnetic fields, had particle-like properties. Then they realized that things they thought were particles, like electrons, manifested field-like properties. The reconciliation of these puzzles is that the world is fundamentally field-like (it’s a quantum wave function), but when we look at it by performing a careful measurement, it looks particle-like. It took a while to get there. Particles seem to be pretty straightforward things: objects located at particular points in space. The idea goes back to ancient Greece, where a small group of philosophers proposed that matter was made up of point-like “atoms,” for the Greek word for “indivisible.” In the words of Democritus, the original atomist, “Sweet is by convention, bitter by convention, hot by convention, cold by convention, color by convention; in truth there are only atoms and the void.” At the time there wasn’t that much actual evidence in favor of the proposal, so it was largely abandoned until the beginning of the 1800s, when experimenters had begun to study chemical reactions in a quantitative way. A crucial role was played by tin oxide, a compound made of tin and oxygen, which was discovered to come in two different forms. The English scientist John Dalton noted that for a fixed amount of tin, the amount of oxygen in one form of tin oxide was exactly twice the amount in the other. We could explain this, Dalton argued in 1803, if both elements came in the form of discrete particles, for which he borrowed the word “atom” from the Greeks. All we have to do is to imagine that one form of tin oxide was made of single tin atoms combined with single oxygen atoms, while the other form consisted of single tin atoms combined with two oxygen atoms. Every kind of chemical element, Dalton suggested, was associated with a unique kind of atom, and the tendency of the atoms to combine in different ways was responsible for all of chemistry. A simple summary, but one with world-altering implications. Dalton jumped the gun a little bit with his nomenclature. For the Greeks, the whole point of atoms was that they were indivisible, the fundamental building blocks out of which everything else is made. But Dalton’s atoms are not at all indivisible—they consist of a compact nucleus surrounded by orbiting electrons. It took over a hundred years to realize that, however. First the English physicist J. J. Thomson discovered electrons in 1897. These seemed to be an utterly new kind of particle, electrically charged and only 1/1800th the mass of hydrogen, the lightest atom. In 1909 Thomson’s former student Ernest Rutherford, a New Zealand physicist who had moved to the UK for his advanced studies, showed most of the mass of the atom was concentrated in a central nucleus, while the atom’s overall size was set by the orbits of much lighter electrons traveling around that nucleus. The standard cartoon picture of an atom, with electrons circling the nucleus much like planets orbit the sun in our solar system, represents this Rutherford model of atomic structure. (Rutherford didn’t know about quantum mechanics, so this cartoon deviates from reality in significant ways, as we shall see.) Further work, initiated by Rutherford and followed up by others, revealed that nuclei themselves aren’t elementary, but consist of positively charged protons and uncharged neutrons. The electric charges of electrons and protons are equal in magnitude but opposite in sign, so an atom with an equal number of each (and however many neutrons you like) will be electrically neutral. It wasn’t until the 1960s and ’70s that physicists established that protons and neutrons are also made of smaller particles, called quarks, held together by new force-carrying particles called gluons. Chemically speaking, electrons are where it’s at. Nuclei give atoms their heft, but outside of rare radioactive decays or fission/fusion reactions, they basically go along for the ride. The orbiting electrons, on the other hand, are light and jumpy, and their tendency to move around is what makes our lives interesting. Two or more atoms can share electrons, leading to chemical bonds. Under the right conditions, electrons can change their minds about which atoms they want to be associated with, which gives us chemical reactions. Electrons can even escape their atomic captivity altogether in order to move freely through a substance, a phenomenon we call “electricity.” And when you shake an electron, it sets up a vibration in the electric and magnetic fields around it, leading to light and other forms of electromagnetic radiation. To emphasize the idea of being truly point-like, rather than a small object but with some definite nonzero size, we sometimes distinguish between “elementary” particles, which define literal points in space, and “composite” particles that are really made of even smaller constituents. As far as anyone can tell, electrons are truly elementary particles. You can see why discussions of quantum mechanics are constantly referring to electrons when they reach for examples—they’re the easiest fundamental particle to make and manipulate, and play a central role in the behavior of the matter of which we and our surroundings are made. In bad news for Democritus and his friends, nineteenth-century physics didn’t explain the world in terms of particles alone. It suggested, instead, that two fundamental kinds of stuff were required: both particles and fields. Fields can be thought of as the opposite of particles, at least in the context of classical mechanics. The defining feature of a particle is that it’s located at one point in space, and nowhere else. The defining feature of a field is that it is located everywhere. A field is something that has a value at literally every point in space. Particles need to interact with each other somehow, and they do so through the influence of fields. Think of the magnetic field. It’s a vector field—at every point in space it looks like a little arrow, with a magnitude (the field can be strong, or weak, or even exactly zero) and also a direction (it points along some particular axis). We can measure the direction in which the magnetic field points just by pulling out a magnetic compass and observing what direction the needle points in. (It will point roughly north, if you are located at most places on Earth and not standing too close to another magnet.) The important thing is that the magnetic field exists invisibly everywhere throughout space, even when we’re not observing it. That’s what fields do. There is also the electric field, which is also a vector with a magnitude and a direction at every point in space. Just as we can detect a magnetic field with a compass, we can detect the electric field by placing an electron at rest and seeing if it accelerates. The faster the acceleration, the stronger the electric field.* One of the triumphs of nineteenth-century physics was when James Clerk Maxwell unified electricity and magnetism, showing that both of these fields could be thought of as different manifestations of a single underlying “electromagnetic” field. The other field that was well known in the nineteenth century is the gravitational field. Gravity, Isaac Newton taught us, stretches over astronomical distances. Planets in the solar system feel a gravitational pull toward the sun, proportional to the sun’s mass and inversely proportional to the square of the distance between them. In 1783 Pierre-Simon Laplace showed that we can think of Newtonian gravity as arising from a “gravitational potential field” that has a value at every point in space, just as the electric and magnetic fields do. By the end of the 1800s, physicists could see the outlines of a complete theory of the world coming into focus. Matter was made of atoms, which were made of smaller particles, interacting via various forces carried by fields, all operating under the umbrella of classical mechanics. What the World Is Made Of (Nineteenth-Century Edition) • Particles (point-like, making up matter). • Fields (pervading space, giving rise to forces). New particles and forces would be discovered over the course of the twentieth century, but in the year 1899 it wouldn’t have been crazy to think that the basic picture was under control. The quantum revolution lurked just around the corner, largely unsuspected. If you’ve read anything about quantum mechanics before, you’ve probably heard the question “Is an electron a particle, or a wave?” The answer is: “It’s a wave, but when we look at (that is, measure) that wave, it looks like a particle.” That’s the fundamental novelty of quantum mechanics. There is only one kind of thing, the quantum wave function, but when observed under the right circumstances it appears particle-like to us. What the World Is Made Of (Twentieth Century and Beyond) • A quantum wave function. It took a number of conceptual breakthroughs to go from the nineteenth-century picture of the world (classical particles and classical fields) to the twentieth-century synthesis (a single quantum wave function). The story of how particles and fields are different aspects of the same underlying thing is one of the underappreciated successes of the quest for unification in physics. To get there, early twentieth-century physicists needed to appreciate two things: fields (like electromagnetism) can behave in particle-like ways, and particles (like electrons) can behave in wave-like ways. The particle-like behavior of fields was appreciated first. Any particle with an electrical charge, such as an electron, creates an electric field everywhere around it, fading in magnitude as you get farther away from the charge. If we shake an electron, oscillating it up and down, the field oscillates along with it, in ripples that gradually spread out from its location. This is electromagnetic radiation, or “light” for short. Every time we heat up a material to sufficient temperature, electrons in its atoms start to shake, and the material begins to glow. This is known as black-body radiation, and every object with a uniform temperature gives off a form of blackbody radiation. Red light corresponds to slowly oscillating, low-frequency waves, while blue light is rapidly oscillating, high-frequency waves. Given what physicists knew about atoms and electrons at the turn of the century, they could calculate how much radiation a blackbody should emit at every different frequency, the so-called blackbody spectrum. Their calculations worked well for low frequencies, but became less and less accurate as they went to higher frequencies, ultimately predicting an infinite amount of radiation coming from every material body. This was later dubbed the “ultraviolet catastrophe,” referring to the invisible frequencies even higher than blue or violet light. Finally in 1900, German physicist Max Planck was able to derive a formula that fit the data exactly. The important trick was to propose a radical idea: that every time light was emitted, it came in the form of a particular amount—a “quantum”—of energy, which was related to the frequency of the light. The faster the electromagnetic field oscillates, the more energy each emission will have. In the process, Planck was forced to posit the existence of a new fundamental parameter of nature, now known as Planck’s constant and denoted by the letter h. The amount of energy contained in a quantum of light is proportional to its frequency, and Planck’s constant is the constant of proportionality: the energy is the frequency times h. Very often it’s more convenient to use a modified version ?, pronounced “h-bar,” which is just Planck’s original constant h divided by 2?. The appearance of Planck’s constant in an expression is a signal that quantum mechanics is at work. Planck’s discovery of his constant suggested a new way of thinking about physical units, such as energy, mass, length, or time. Energy is measured in units such as ergs or joules or kilowatt-hours, while frequency is measured in units of 1/time, since frequency tells us how many times something happens in a given amount of time. To make energy proportional to frequency, Planck’s constant therefore has units of energy times time. Planck himself realized that his new quantity could be combined with the other fundamental constants—G, Newton’s constant of gravity, and c, the speed of light—to form universally defined measures of length, time, and so forth. The Planck length is about 10-33 centimeters, while the Planck time is about 10-43 seconds. The Planck length is a very short distance indeed, but presumably it has physical relevance, as a scale at which quantum mechanics (h), gravity (G), and relativity (c) all simultaneously matter. Amusingly, Planck’s mind immediately went to the possibility of communicating with alien civilizations. If we someday start chatting with extraterrestrial beings using interstellar radio signals, they won’t know what we mean if we were to say human beings are “about two meters tall.” But since they will presumably know at least as much about physics as we do, they should be aware of Planck units. This suggestion hasn’t yet been put to practical use, but Planck’s constant has had an immense impact elsewhere. The idea that light is emitted in discrete quanta of energy related to its frequency is puzzling, when you think about it. From what we intuitively know about light, it might make sense if someone suggested that the amount of energy it carried depended on how bright it was, but not on what color it was. But the assumption led Planck to derive the right formula, so something about the idea seemed to be working. It was left to Albert Einstein, in his singular way, to brush away conventional wisdom and take a dramatic leap into a new way of thinking. In 1905, Einstein suggested that light was emitted only at certain energies because it literally consisted of discrete packets, not a smooth wave. Light was particles, in other words—“photons,” as they are known today. This idea, that light comes in discrete, particle-like quanta of energy, was the true birth of quantum mechanics, and was the discovery for which Einstein was awarded the Nobel Prize in 1921. (He deserved to win at least one more Nobel for the theory of relativity, but never did.) Einstein was no dummy, and he knew that this was a big deal; as he told his friend Conrad Habicht, his light quantum proposal was “very revolutionary.” Note the subtle difference between Planck’s suggestion and Einstein’s. Planck says that light of a fixed frequency is emitted in certain energy amounts, while Einstein says that’s because light literally is discrete particles. It’s the difference between saying that a certain coffee machine makes exactly one cup at a time, and saying that coffee only exists in the form of one-cup-size amounts. That might make sense when we’re talking about matter particles like electrons and protons, but just a few decades earlier Maxwell had triumphantly explained that light was a wave, not a particle. Einstein’s proposal was threatening to undo that triumph. Planck himself was reluctant to accept this wild new idea, but it did explain the data. In a wild new idea’s search for acceptance, that’s a powerful advantage to have. Meanwhile another problem was lurking over on the particle side of the ledger, where Rutherford’s model explained atoms in terms of electrons orbiting nuclei. Remember that if you shake an electron, it emits light. By “shake” we just mean accelerate in some way. An electron that does anything other than move in a straight line at a constant velocity should emit light. From the picture of the Rutherford atom, with electrons orbiting around the nucleus, it certainly looks like those electrons are not moving in straight lines. They’re moving in circles or ellipses. In a classical world, that unambiguously means that the electrons are being accelerated, and equally unambiguously that they should be giving off light. Every single atom in your body, and in the environment around you, should be glowing, if classical mechanics was right. That means the electrons should be losing energy as they emit radiation, which in turn implies that they should spiral downward into the central nucleus. Classically, electron orbits should not be stable. Perhaps all of your atoms are giving off light, but it’s just too faint to see. After all, identical logic applies to the planets in the solar system. They should be giving off gravitational waves—an accelerating mass should cause ripples in the gravitational field, just like an accelerating charge causes ripples in the electromagnetic field. And indeed they are. If there was any doubt that this happens, it was swept away in 2016, when researchers at the LIGO and Virgo gravitational-wave observatories announced the first direct detection of gravitational waves, created when black holes over a billion light years away spiraled into each other. But the planets in the solar system are much smaller, and move more slowly, than those black holes, which were each over thirty times the mass of the sun. As a result, the emitted gravitational waves from our planetary neighbors are very weak indeed. The power emitted in gravitational waves by the orbiting Earth amounts to about 200 watts—equivalent to the output of a few lightbulbs, and completely insignificant compared to other influences such as solar radiation and tidal forces. If we pretend that the emission of gravitational waves were the only thing affecting the Earth’s orbit, it would take over 1023 years for it to crash into the sun. So perhaps the same thing is true for atoms: maybe electron orbits aren’t really stable, but they’re stable enough. This is a quantitative question, and it’s not hard to plug in the numbers and see what falls out. The answer is catastrophic, because electrons should move much faster than planets and electromagnetism is a much stronger force than gravity. The amount of time it would take an electron to crash into the nucleus of its atom works out to about ten picoseconds. That’s one-hundred-billionth of a second. If ordinary matter made of atoms only lasted for that long, someone would have noticed by now. This bothered a lot of people, most notably Niels Bohr, who had briefly worked under Rutherford in 1912. In 1913, Bohr published a series of three papers, later known simply as “the trilogy,” in which he put forth another of those audacious, out-of-the-blue ideas that characterized the early years of quantum theory. What if, he asked, electrons can’t spiral down into atomic nuclei because electrons simply aren’t allowed to be in any orbit they want, but instead have to stick to certain very specific orbits? There would be a minimum-energy orbit, another one with somewhat higher energy, and so on. But electrons weren’t allowed to go any closer to the nucleus than the lowest orbit, and they weren’t allowed to be in between the orbits. The allowed orbits were quantized. Bohr’s proposal wasn’t quite as outlandish as it might seem at first. Physicists had studied how light interacted with different elements in their gaseous form—hydrogen, nitrogen, oxygen, and so forth. They found that if you shined light through a cold gas, some of it would be absorbed; likewise, if you passed electrical current through a tube of gas, the gas would start glowing (the principle behind fluorescent lights still used today). But they only emitted and absorbed certain very specific frequencies of light, letting other colors pass right through. Hydrogen, the simplest element with just a single proton and a single electron, in particular had a very regular pattern of emission and absorption frequencies. For a classical Rutherford atom, that would make no sense at all. But in Bohr’s model, where only certain electron orbits were allowed, there was an immediate explanation. Even though electrons couldn’t linger in between the allowed orbits, they could jump from one to another. An electron could fall from a higher-energy orbit to a lower-energy one by emitting light with just the right energy to compensate, or it could leap upward in energy by absorbing an appropriate amount of energy from ambient light. Because the orbits themselves were quantized, we should only see specific energies of light interacting with the electrons. Together with Planck’s idea that the frequency of light is related to its energy, this explained why physicists saw only certain frequencies being emitted or absorbed. By comparing his predictions to the observed emission of light by hydrogen, Bohr was able to not simply posit that only some electron orbits were allowed, but calculate which ones they were. Any orbiting particle has a quantity called the angular momentum, which is easy to calculate—it’s just the mass of the particle, times its velocity, times its distance from the center of the orbit. Bohr proposed that an allowed electron orbit was one whose angular momentum was a multiple of a particular fundamental constant. And when he compared the energy that electrons should emit when jumping between orbits to what was actually seen in light emitted from hydrogen gas, he could figure out what that constant needed to be in order to fit the data. The answer was Planck’s constant, h. Or more specifically, the modified h-bar version, ? = h/2?. That’s the kind of thing that makes you think you’re on the right track. Bohr was trying to account for the behavior of electrons in atoms, and he posited an ad hoc rule according to which they could only move along certain quantized orbits, and in order to fit the data his rule ended up requiring a new constant of nature, and that new constant was the same as the new constant that Planck was forced to invent when he was trying to account for the behavior of photons. All of this might seem ramshackle and a bit sketchy, but taken together it appeared as if something profound was happening in the realm of atoms and particles, something that didn’t fit comfortably with the sacred rules of classical mechanics. The ideas of this period are now sometimes described under the rubric of “the old quantum theory,” as opposed to “the new quantum theory” of Heisenberg and Schr?dinger that came along in the late 1920s. As provocative and provisionally successful as the old quantum theory was, nobody was really happy with it. Planck and Einstein’s idea of light quanta helped make sense of a number of experimental results, but was hard to reconcile with the enormous success of Maxwell’s theory of light as electromagnetic waves. Bohr’s idea of quantized electron orbits helped make sense of the light emitted and absorbed by hydrogen, but seemed to be pulled out of a hat, and didn’t really work for elements other than hydrogen. Even before the “old quantum theory” was given that name, it seemed clear that these were just hints at something much deeper going on. One of the least satisfying features of Bohr’s model was the suggestion that electrons could “jump” from one orbit to another. If a low-energy electron absorbed light with a certain amount of energy, it makes sense that it would have to jump up to another orbit with just the right amount of additional energy. But when an electron in a high-energy orbit emitted light to jump down, it seemed to have a choice about exactly how far down to go, which lower orbit to end up in. What made that choice? Rutherford himself worried about this in a letter to Bohr: There appears to me one grave difficulty in your hypothesis, which I have no doubt you fully realize, namely, how does an electron decide what frequency it is going to vibrate at when it passes from one stationary state to the other? It seems to me that you would have to assume that the electron knows beforehand where it is going to stop. This business about electrons “deciding” where to go foreshadowed a much more drastic break with the paradigm of classical physics than physicists in 1913 were prepared to contemplate. In Newtonian mechanics one could imagine a Laplace demon that could predict, at least in principle, the entire future history of the world from its present state. At this point in the development of quantum mechanics, nobody was really confronting the prospect that this picture would have to be completely discarded. It took more than ten years for a more complete framework, the “new quantum theory,” to finally come on the scene. In fact, two competing ideas were proposed at the time, matrix mechanics and wave mechanics, before they were ultimately shown to be mathematically equivalent versions of the same thing, which can now simply be called quantum mechanics. Matrix mechanics was formulated initially by Werner Heisenberg, who had worked with Niels Bohr in Copenhagen. These two men, along with their collaborator Wolfgang Pauli, are responsible for the Copenhagen interpretation of quantum mechanics, though who exactly believed what is a topic of ongoing historical and philosophical debate. Heisenberg’s approach in 1926, reflecting the boldness of a younger generation coming on the scene, was to put aside questions of what was really happening in a quantum system, and to focus exclusively on explaining what was observed by experimenters. Bohr had posited quantized electron orbits without explaining why some orbits were allowed and others were not. Heisenberg dispensed with orbits entirely. Forget about what the electron is doing; ask only what you can observe about it. In classical mechanics, an electron would be characterized by position and momentum. Heisenberg kept those words, but instead of thinking of them as quantities that exist whether we are looking at them or not, he thought of them as possible outcomes of measurements. For Heisenberg, the unpredictable jumps that had bothered Rutherford and others became a central part of the best way of talking about the quantum world. Heisenberg was only twenty-four years old when he first formulated matrix mechanics. He was clearly a prodigy, but far from an established figure in the field, and wouldn’t obtain a permanent academic position until a year later. In a letter to Max Born, another of his mentors, Heisenberg fretted that he “had written a crazy paper and did not dare to send it in for publication.” But in a collaboration with Born and the even younger physicist Pascual Jordan, they were able to put matrix mechanics on a clear and mathematically sound footing. It would have been natural for Heisenberg, Born, and Jordan to share the Nobel Prize for the development of matrix mechanics, and indeed Einstein nominated them for the award. But it was Heisenberg alone who was honored by the Nobel committee in 1932. It has been speculated that Jordan’s inclusion would have been problematic, as he became known for aggressive right-wing political rhetoric, ultimately becoming a member of the Nazi Party and joining a Sturmabteilung (Storm trooper) unit. At the same time, however, he was considered unreliable by his fellow Nazis, due to his support for Einstein and other Jewish scientists. In the end, Jordan never won the prize. Born was also left off the prize for matrix mechanics, but that omission was made up for when he was awarded a separate Nobel in 1954 for his formulation of the probability rule. That was the last time a Nobel Prize has been awarded for work in the foundations of quantum mechanics. After the onset of World War II, Heisenberg led a German government program to develop nuclear weapons. What Heisenberg actually thought about the Nazis, and whether he truly tried as hard as possible to push the weapons program forward, are matters of some historical dispute. It seems that, like a number of other Germans, Heisenberg was not fond of the Nazi Party, but preferred a German victory in the conflict to the prospect of being run over by the Soviets. There is no evidence that he actively worked to sabotage the nuclear bomb program, but it is clear that his team made very little progress. In part that must be attributed to the fact that so many brilliant Jewish physicists had fled Germany as the Nazis took power. As impressive as matrix mechanics was, it suffered from a severe marketing flaw: the mathematical formalism was highly abstract and difficult to understand. Einstein’s reaction to the theory was typical: “A veritable sorcerer’s calculation. This is sufficiently ingenious and protected by its great complexity, to be immune to any proof of its falsity.” (This from the guy who had proposed describing spacetime in terms of non-Euclidean geometry.) Wave mechanics, developed immediately thereafter by Erwin Schr?dinger, was a version of quantum theory that used concepts with which physicists were already very familiar, which greatly helped accelerate acceptance of the new paradigm. Physicists had studied waves for a long time, and with Maxwell’s formulation of electromagnetism as a theory of fields, they had become adept at thinking about them. The earliest intimations of quantum mechanics, from Planck and Einstein, had been away from waves and toward particles. But Bohr’s atom suggested that even particles weren’t what they seemed to be. In 1924, the young French physicist Louis de Broglie was thinking about Einstein’s light quanta. At this point the relationship between photons and classical electromagnetic waves was still murky. An obvious thing to contemplate was that light consisted of both a particle and a wave: particle-like photons could be carried along by the well-known electromagnetic waves. And if that’s true, there’s no reason we couldn’t imagine the same thing going on with electrons—maybe there is something wave-like that carries along the electron particles. That’s exactly what de Broglie suggested in his 1924 doctoral thesis, proposing a relationship between the momentum and wavelength of these “matter waves” that was analogous to Planck’s formula for light, with larger momenta corresponding to shorter wavelengths. Like many suggestions at the time, de Broglie’s hypothesis may have seemed a little ad hoc, but its implications were far-reaching. In particular, it was natural to ask what the implications of matter waves might be for electrons orbiting around a nucleus. A remarkable answer suggested itself: for the wave to settle down into a stationary configuration, its wavelength had to be an exact multiple of the circumference of a corresponding orbit. Bohr’s quantized orbits could be derived rather than simply postulated, simply by associating waves with the electron particles surrounding the nucleus. Consider a string with its ends held fixed, such as on a guitar or violin. Even though any one point can move up or down as it likes, the overall behavior of the string is constrained by being tied down at either end. As a result, the string only vibrates at certain special wavelengths, or combinations thereof; that’s why the strings on musical instruments emit clear notes rather than an indistinct noise. These special vibrations are called the modes of the string. The essentially “quantum” nature of the subatomic world, in this picture, comes about not because reality is actually subdivided into distinct chunks but because there are natural vibrational modes for the waves out of which physical systems are made. The word “quantum,” referring to some definite amount of stuff, can give the impression that quantum mechanics describes a world that is fundamentally discrete and pixelated, like when you zoom in closely on a computer monitor or TV screen. It’s actually the opposite; quantum mechanics describes the world as a smooth wave function. But in the right circumstances, where individual parts of the wave function are tied down in a certain way, the wave takes the form of a combination of distinct vibrational modes. When we observe such a system, we see those discrete possibilities. That’s true for orbits of electrons, and it will also explain why quantum fields look like sets of individual particles. In quantum mechanics, the world is fundamentally wavy; its apparent quantum discreteness comes from the particular way those waves are able to vibrate. De Broglie’s ideas were intriguing, but they fell short of providing a comprehensive theory. That was left to Erwin Schr?dinger, who in 1926 put forth a dynamical understanding of wave functions, including the equation they obey, later named after him. Revolutions in physics are generally a young person’s game, and quantum mechanics was no different, but Schr?dinger bucked the trend. Among the leaders of the discussions at Solvay in 1927, Einstein at forty-eight years old, Bohr at forty-two, and Born at forty-four were the grand old men. Heisenberg was twenty-five, Pauli twenty-seven, and Dirac twenty-five. Schr?dinger, at the ripe old age of thirty-eight, was looked upon as someone suspiciously long in the tooth to appear on the scene with radical new ideas like this. Note the shift here from de Broglie’s “matter waves” to Schr?dinger’s “wave function.” Though Schr?dinger was heavily influenced by de Broglie’s work, his concept went quite a bit further, and deserves a distinct name. Most obviously, the value of a matter wave at any one point was some real number, while the amplitudes described by wave functions are complex numbers—the sum of a real number and an imaginary one. More important, the original idea was that each kind of particle would be associated with a matter wave. That’s not how Schr?dinger’s wave function works; you have just one function that depends on all the particles in the universe. It’s that simple shift that leads to the world-altering phenomenon of quantum entanglement. What made Schr?dinger’s ideas an instant hit was the equation he proposed, which governs how wave functions change with time. To a physicist, a good equation makes all the difference. It elevates a pretty-sounding idea (“particles have wave-like properties”) to a rigorous, unforgiving framework. Unforgiving might sound like a bad quality in a person, but it’s just what you want in a scientific theory. It’s the feature that lets you make precise predictions. When we say that quantum textbooks spend a lot of time having students solve equations, it’s mostly the Schr?dinger equation we have in mind. Schr?dinger’s equation is what a quantum version of Laplace’s demon would be solving as it predicted the future of the universe. And while the original form in which Schr?dinger wrote down his equation was meant for systems of individual particles, it’s actually a very general idea that applies equally well to spins, fields, superstrings, or any other system you might want to describe using quantum mechanics. Unlike matrix mechanics, which was expressed in terms of mathematical concepts most physicists at the time had never been exposed to, Schr?dinger’s wave equation was not all that different in form from Maxwell’s electromagnetic equations that adorn T-shirts worn by physics students to this day. You could visualize a wave function, or at least you might convince yourself that you could. The community wasn’t sure what to make of Heisenberg, but they were ready for Schr?dinger. The Copenhagen crew—especially the youngsters, Heisenberg and Pauli—didn’t react graciously to the competing ideas from an undistinguished old man in Z?rich. But before too long they were thinking in terms of wave functions, just like everyone else. Schr?dinger’s equation involves unfamiliar symbols, but its basic message is not hard to understand. De Broglie had suggested that the momentum of a wave goes up as its wavelength goes down. Schr?dinger proposed a similar thing, but for energy and time: the rate at which the wave function is changing is proportional to how much energy it has. Here is the celebrated equation in its most general form: We don’t need the details here, but it’s nice to see the real way that physicists think of an equation like this. There’s some maths involved, but ultimately it’s just a translation into symbols of the idea we wrote down in words. ? (the Greek letter Psi) is the wave function. The left-hand side is the rate at which the wave function is changing over time. On the right-hand side we have a proportionality constant involving Planck’s constant ?, the fundamental unit of quantum mechanics, and i, the square root of minus one. The wave function ? is acted on by something called the Hamiltonian, or H. Think of the Hamiltonian as an inquisitor who asks the following question: “How much energy do you have?” The concept was invented in 1833 by Irish mathematician William Rowan Hamilton, as a way to reformulate the laws of motion of a classical system, long before it gained a central role in quantum mechanics. When physicists start modeling different physical systems, the first thing they try to do is work out a mathematical expression for the Hamiltonian of that system. The standard way of figuring out the Hamiltonian of something like a collection of particles is to start with the energies of the particles themselves, and then add in additional contributions describing how the particles interact with each other. Maybe they bump off each other like billiard balls, or perhaps they exert a mutual gravitational interaction. Each such possibility suggests a particular kind of Hamiltonian. And if you know the Hamiltonian, you know everything; it’s a compact way of capturing all the dynamics of a physical system. If a quantum wave function describes a system with some definite value of the energy, the Hamiltonian simply equals that value, and the Schr?dinger equation implies that the system just keeps doing the same thing, maintaining a fixed energy. More often, since wave functions are superpositions of different possibilities, the system will be a combination of multiple energies. In that case the Hamiltonian captures a bit of all of them. The bottom line is that the right-hand side of Schr?dinger’s equation is a way of characterizing how much energy is carried by each of the contributions to a wave function in a quantum superposition; high-energy components evolve quickly, low-energy ones evolve more slowly. What really matters is that there is some specific deterministic equation. Once you have that, the world is your playground. Wave mechanics made a huge splash, and before too long Schr?dinger, English physicist Paul Dirac, and others demonstrated that it was essentially equivalent to matrix mechanics, leaving us with a unified theory of the quantum world. Still, all was not peaches and cream. Physicists were left with the question that we are still struggling with today: What is the wave function, really? What physical thing does it represent, if any? In de Broglie’s view, his matter waves served to guide particles around, not to replace them entirely. (He later developed this idea into pilot-wave theory, which remains a viable approach to quantum foundations today, although it is not popular among working physicists.) Schr?dinger, by contrast, wanted to do away with fundamental particles entirely. His original hope was that his equation would describe localized packets of vibrations, confined to a relatively small region of space, so that each packet would appear particle-like to a macroscopic observer. The wave function could be thought of as representing the density of mass in space. Alas, Schr?dinger’s aspirations were undone by his own equation. If we start with a wave function describing a single particle approximately localized in some empty region of space, the Schr?dinger equation is clear about what happens next: it quickly spreads out all over the place. Left to their own devices, Schr?dinger’s wave functions don’t look particle-like at all.* It was left to Max Born, one of Heisenberg’s collaborators on matrix mechanics, to provide the final missing piece: we should think about the wave function as a way of calculating the probability of seeing a particle in any given position when we look for it. In particular, we should take both the real and imaginary parts of the complex-valued amplitude, square them both individually, and add the two numbers together. The result is the probability of observing the corresponding outcome. (The suggestion that it’s the amplitude squared, rather than the amplitude itself, appears in a footnote added at the last minute to Born’s 1926 paper.) And after we observe it, the wave function collapses to be localized at the place where we saw the particle. You know who didn’t like the probability interpretation of the Schr?dinger equation? Schr?dinger himself. His goal, like Einstein’s, was to provide a definite mechanistic underpinning for quantum phenomena, not just to create a tool that could be used to calculate probabilities. “I don’t like it, and I’m sorry I ever had anything to do with it,” he later groused. The point of the famous Schr?dinger’s Cat thought experiment, in which the wave function of a cat evolves (via the Schr?dinger equation) into a superposition of “alive” and “dead,” was not to make people say, “Wow, quantum mechanics is really mysterious.” It was to make people say, “Wow, this can’t possibly be correct.” But to the best of our current knowledge, it is. A lot of intellectual action was packed into the first three decades of the twentieth century. Over the course of the 1800s, physicists had put together a promising picture of the nature of matter and forces. Matter was made of particles, and forces were carried by fields, all under the umbrella of classical mechanics. But confrontation with experimental data forced them to think beyond this paradigm. In order to explain radiation from hot objects, Planck suggested that light was emitted in discrete amounts of energy, and Einstein pushed this further by suggesting that light actually came in the form of particle-like quanta. Meanwhile, the fact that atoms are stable and the observation of how light was emitted from gases inspired Bohr to suggest that electrons could only move along certain allowed orbits, with occasional jumps between them. Heisenberg, Born, and Jordan elaborated this story of probabilistic jumps into a full theory, matrix mechanics. From another angle, de Broglie pointed out that if we think of matter particles such as electrons as actually being waves, we can derive Bohr’s quantized orbits rather than postulating them. Schr?dinger developed this suggestion into a full-blown quantum theory of its own, and it was ultimately demonstrated that wave mechanics and matrix mechanics were equivalent ways of saying the same thing. Despite initial hopes that wave mechanics could explain away the apparent need for probabilities as a fundamental part of the theory, Born showed that the right way to think about Schr?dinger’s wave function was as something that you square to get the probability of a measurement outcome. Whew. That’s quite a journey, taken in a remarkably short period of time, from Planck’s observation in 1900 to the Solvay Conference in 1927, when the new quantum mechanics was fleshed out once and for all. It’s to the enormous credit of the physicists of the early twentieth century that they were willing to face up to the demands of the experimental data, and in doing so to completely upend the fantastically successful Newtonian view of the classical world. They were less successful, however, at coming to grips with the implications of what they had wrought. * Annoyingly, the electron accelerates in precisely the opposite direction that the electric field points, because by human convention we’ve decided to call the charge on the electron “negative” and that on a proton “positive.” For that we can blame Benjamin Franklin in the eighteenth century. He didn’t know about electrons and protons, but he did figure out there was a unified concept called “electric charge.” When he went to arbitrarily label which substances were positively charged and which were negatively charged, he had to choose something, and the label he picked for positive charge corresponds to what we would now call “having fewer electrons than it should.” So be it. * I’ve emphasized that there is only one wave function, the wave function of the universe, but the alert reader will notice that I often talk about “the wave function of a particle.” This latter construction is perfectly okay if—and only if—the particle is unentangled from the rest of the universe. Happily, that is often the case, but in general we have to keep our wits about us. 4 What Cannot Be Known, Because It Does Not Exist Uncertainty and Complementarity A police officer pulls over Werner Heisenberg for speeding. “Do you know how fast you were going?” asks the cop. “No,” Heisenberg replies, “but I know exactly where I am!” I think we can all agree that physics jokes are the funniest jokes there are. They are less good at accurately conveying physics. This particular chestnut rests on familiarity with the famous Heisenberg uncertainty principle, often explained as saying that we cannot simultaneously know both the position and the velocity of any object. But the reality is deeper than that. It’s not that we can’t know position and momentum, it’s that they don’t even exist at the same time. Only under extremely special circumstances can an object be said to have a location—when its wave function is entirely concentrated on one point in space, and zero everywhere else—and similarly for velocity. And when one of the two is precisely defined, the other could be literally anything, were we to measure it. More often, the wave function includes a spread of possibilities for both quantities, so neither has a definite value. Back in the 1920s, all this was less clear. It was still natural to think that the probabilistic nature of quantum mechanics simply indicated that it was an incomplete theory, and that there was a more deterministic, classical-sounding picture waiting to be developed. Wave functions, in other words, might be a way of characterizing our ignorance of what was really going on, rather than being the total truth about what is going on, as we’re advocating here. One of the first things people did when learning about the uncertainty principle was to try to find loopholes in it. They failed, but in doing so we learned a lot about how quantum reality is fundamentally different from the classical world we had been used to. The absence of definite quantities at the heart of reality that map more or less straightforwardly onto what we can eventually observe is one of the deep features of quantum mechanics that can be hard to accept upon first encounter. There are quantities that are not merely unknown but do not even exist, even though we can seemingly measure them. Quantum mechanics forces us to confront this yawning chasm between what we see and what really is. In this chapter we’ll see how that gap manifests itself in the uncertainty principle, and in the next chapter we’ll see it again more forcefully in the phenomenon of entanglement. The uncertainty principle owes its existence to the fact that the relationship between position and momentum (mass times velocity) is fundamentally different in quantum mechanics from what it was in classical mechanics. Classically, we can imagine measuring the momentum of a particle by tracking its position over time, and seeing how fast it moves. But if all we have access to is a single moment, position and momentum are completely independent from each other. If I tell you that a particle has a certain position at one instant, and I tell you nothing else, you have no idea what its speed is, and vice versa. Physicists refer to the different numbers we use to specify something as that system’s “degrees of freedom.” In Newtonian mechanics, to tell me the complete state of a bunch of particles, you have to tell me the position and momentum of every one of them, so the degrees of freedom are the positions and the momenta. Acceleration is not a degree of freedom, since it can be calculated once we know the forces acting on the system. The essence of a degree of freedom is that it doesn’t depend on anything else. When we switch to quantum mechanics and start thinking about Schr?dinger’s wave functions, things become a little different. To make a wave function for a single particle, think of every location where the particle could possibly be found, were we to observe it. Then to each location assign an amplitude, a complex number with the property that the square of each number is the probability of finding the particle there. There is a constraint that the squares of all these numbers add up to precisely one, since the total probability that the particle is found somewhere must equal one. (Sometimes we speak of probabilities in terms of percentages, which are numerically 100 times the actual probability; a 20 percent chance is the same as a 0.2 probability.) Notice we didn’t mention “velocity” or “momentum” there. That’s because we don’t have to separately specify the momentum in quantum mechanics, as we did in classical mechanics. The probability of measuring any particular velocity is completely determined by the wave function for all the possible positions. Velocity is not a separate degree of freedom, independent of position. The basic reason why is that the wave function is, you know, a wave. Unlike for a classical particle, we don’t have a single position and a single momentum, we have a function of all possible positions, and that function typically oscillates up and down. The rate of those oscillations determines what we’re likely to see if we were to measure the velocity or momentum. Consider a simple sine wave, oscillating up and down in a regular pattern throughout space. Plug such a wave function into the Schr?dinger equation and ask how it will evolve. We find that a sine wave has a definite momentum, with shorter wavelengths corresponding to faster velocity. But a sine wave has no definite position; on the contrary, it’s spread out everywhere. And a more typical shape, which is neither localized at one point nor spread out in a perfect sine wave of fixed wavelength, won’t correspond to either a definite position or a definite momentum, but some mixture of each. We see the basic dilemma. If we try to localize a wave function in space, its momentum becomes more and more spread out, and if we try to limit it to one fixed wavelength (and therefore momentum) it becomes more spread out in position. That’s the uncertainty principle. It’s not that we can’t know both quantities at the same time; it’s just a fact about how wave functions work that if position is concentrated near some location, momentum is completely undetermined, and vice versa. The old-fashioned classical properties called position and momentum aren’t quantities with actual values, they’re possible measurement outcomes. People sometimes refer to the uncertainty principle in everyday contexts, outside of the equation-filled language of physics texts. So it’s important to emphasize what the principle does not say. It’s not an assertion that “everything is uncertain.” Either position or momentum could be certain in an appropriate quantum state; they just can’t be certain at the same time. And the uncertainty principle doesn’t say we necessarily disturb a system when we measure it. If a particle has a definite momentum, we can go ahead and measure that without changing it at all. The point is that there are no states for which both position and momentum are simultaneously definite. The uncertainty principle is a statement about the nature of quantum states and their relationship to observable quantities, not a statement about the physical act of measurement. Finally, the principle is not a statement about limitations on our knowledge of the system. We can know the quantum state exactly, and that’s all there is to know about it; we still can’t predict the results of all possible future observations with perfect certainty. The idea that “there’s something we don’t know,” given a certain wave function, is an outdated relic of our intuitive insistence that what we observe is what really exists. Quantum mechanics teaches us otherwise. You’ll sometimes hear the idea, provoked by the uncertainty principle, that quantum mechanics violates logic itself. That’s silly. Logic deduces theorems from axioms, and the resulting theorems are simply true. The axioms may or may not apply to any given physical situation. Pythagoras’s theorem—the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides—is correct as a formal deduction from the axioms of Euclidean geometry, even though those axioms do not hold if we’re talking about curved surfaces rather than a flat tabletop. The idea that quantum mechanics violates logic lives in the same neighborhood of the idea that atoms are mostly empty space (a bad neighborhood). Both notions stem from a deep conviction that, despite everything we’ve learned, particles are really points with some position and momentum, rather than being wave functions that are spread out. Consider a particle in a box, where we’ve drawn a line dividing the box into left and right sides. It has some wave function that is spread throughout the box. Let proposition P be “the particle is on the left side of the box,” and proposition Q be “the particle is on the right side of the box.” We might be tempted to say that both of these propositions are false, since the wave function stretches over both sides of the box. But the proposition “P or Q” has to be true, since the particle is in the box. In classical logic, we can’t have both P and Q be false but “P or Q” be true. So something fishy is going on. What’s fishy is neither logic nor quantum mechanics but our casual disregard for the nature of quantum states when assigning truth values to the statements P and Q. These statements are neither true nor false; they’re just ill defined. There is no such thing as “the side of the box the particle is on.” If the wave function were concentrated entirely on one side of the box and exactly vanished on the other, we could get away with assigning truth values to P and Q; but in that case one would be true and the other would be false, and classical logic would be fine. Despite the fact that classical logic is perfectly valid whenever it is properly applied, quantum mechanics has inspired more general approaches known as quantum logic, pioneered by John von Neumann and his collaborator Garrett Birkhoff. By starting with slightly different logical axioms from the standard ones, we can derive a set of rules obeyed by the probabilities implied by the Born rule in quantum mechanics. Quantum logic in this sense is both interesting and useful, but its existence does not invalidate the correctness of ordinary logic in appropriate circumstances. Niels Bohr, in an attempt to capture what makes quantum theory so unique, proposed the concept of complementarity. The idea is that there can be more than two ways of looking at a quantum system, each of them equally valid, but with the property that you can’t employ them simultaneously. We can describe the wave function of a particle in terms of either position or momentum, but not both at the same time. Similarly, we can think of electrons as exhibiting either particle-like or wave-like properties, just not at the same time. Nowhere is this feature made more evident than in the famous double-slit experiment. This experiment wasn’t actually performed until the 1970s, long after it was proposed. It wasn’t one of those surprising experimental results that theorists had to invent a new way of thinking in order to understand, but rather a thought experiment (suggested in its original form by Einstein during his debates with Bohr, and later popularized by Richard Feynman in his lectures to Caltech undergraduates) meant to show the dramatic implications of quantum theory. The idea of the experiment is to home in on the distinction between particles and waves. We start with a source of classical particles (maybe a pellet gun that tends to spray in somewhat unpredictable directions), shoot them through a single thin slit, then detect them at a screen on the other side of the slit. Mostly the particles will pass right through, with perhaps very slight deviations if they bump up against the sides of the slit. So what we see at the detector is a pattern of individual points where we detect the particles, arranged in more or less a slit-like pattern. We could also do the same thing with waves, for example, by placing the slit in a tub of water and creating waves that pass through it. When the waves pass through, they spread out in a semicircular pattern before eventually reaching the screen. Of course, we don’t observe particle-like points when the water wave hits the screen, but let’s imagine we have a special screen that lights up with a brightness that depends on the amplitude the waves reach at any particular point. They will be brightest at the point of the screen that is closest to the slit, and gradually fade as we get farther away. Now let’s do the same thing, but with two slits in the way rather than just one. The particle case isn’t that much different; as long as our source of particles is sufficiently random that particles pass through both slits, what we’ll see on the other side is two lines of points, one for each slit (or one thick line, if the slits themselves are sufficiently close together). But the wave case is altered in an interesting way. Waves can oscillate downward as well as upward, and two waves oscillating in opposite directions will cancel each other out—a phenomenon known as interference. So the waves pass through both slits at once, emanating outward in semicircles, but then set up an interference pattern on the other side. As a result, if we observe the amplitude of the resultant wave at the final screen, we don’t simply see two bright lines; rather, there will be a bright line in the middle (closest to both slits), with alternating dark/bright regions that gradually fade to either side. So far, that’s the classical world we know and love, where particles and waves are different things and everyone can easily distinguish between them. Now let’s replace our pellet gun or wave machine with a source of electrons, in all their quantum-mechanical glory. There are several twists on this setup, each with provocative consequences. First consider just a single slit. In this case the electrons behave just as if they were classical particles. They pass through the slit, then are detected by the screen on the other side, each electron leaving a single particle-like mark. If we let numerous electrons through, their marks are scattered around a central line in the image of the slit that they passed through. Nothing funny yet. Now let’s introduce two slits. (The slits have to be very close together for this to work, which is one reason it took so long for the experiment to actually be carried out.) Once again, electrons pass through the slits and leave individual marks on the screen on the other side. However, their marks do not clump into two lines, as the classical pellets did. Rather, they form a series of lines: a high-density one in the middle, surrounded by parallel lines with gradually fewer marks, each separated by dark regions with almost no marks at all. In other words, electrons going through two slits leave what is unmistakably an interference pattern, just like waves do, even though they hit the screen with individual marks just like particles. This phenomenon has launched a thousand unhelpful discussions about whether electrons are “really” particles or waves, or are sometimes particle-like and other times wave-like. One way or another, it’s indisputable that something went through both slits as the electrons traveled to the screen. At this point this is no surprise to us. The electrons passing through the slits are described by a wave function, which just like our classical wave will go through both slits and oscillate up and down, and therefore it makes sense that we see interference patterns. Then when they hit the screen they are being observed, and it’s at that point they appear to us as particles. Let’s introduce one additional wrinkle. Imagine that we set up little detectors at each slit, so we can tell whether an electron goes through it. That will settle this crazy idea that an electron can travel through two slits once and for all. You should be able to figure out what we see. The detectors don’t measure half of an electron going through each of the two slits; they measure a full electron going through one, and nothing through the other, every time. That’s because the detector acts as a measuring device, and when we measure electrons we see particles. But that’s not the only consequence of looking at the electron as it passes through the slits. At the screen, on the other side of the slits, the interference pattern disappears, and we are back to seeing two bands of marks made by the detected electrons, one for each slit. With the detectors doing their job, the wave function collapses as the electron goes through the slits, so we don’t see interference from a wave passing through both slits at once. When we’re looking at them, electrons behave like particles. The double-slit experiment makes it difficult to cling to the belief that the electron is just a single classical point, and the wave function simply represents our ignorance about where that point is. Ignorance doesn’t cause interference patterns. There is something real about the wave function. Wave functions may be real, but they’re undeniably abstract, and once we start considering more than one particle at a time they become hard to visualize. As we move forward with increasingly subtle examples of quantum phenomena in action, it will be very helpful to have a simple, readily graspable example we can refer to over and over. The spin of a particle—a degree of freedom in addition to its position or momentum—is just what we’re looking for. We have to think a bit about what spin means within quantum mechanics, but once we do, it will make our lives much easier. The notion of spin itself isn’t hard to grasp: it’s just rotation around an axis, as the Earth does every day or a pirouetting ballet dancer does on their tiptoes. But just like the energies of an electron orbiting an atomic nucleus, in quantum mechanics there are only certain discrete results we can obtain when we measure a particle’s spin. For an electron, for example, there are two possible measurement outcomes for spin. First pick an axis with respect to which we measure the spin. We always find that the electron is spinning either clockwise or counterclockwise when we look along that axis, and always at the same rate. These are conventionally referred to as “spin-up” and “spin-down.” Think of the “right-hand rule”: if you wrap the fingers of your right hand in the direction of rotation, your thumb will be pointing along the appropriate up/down axis. A spinning electron is a tiny magnet, with north and south magnetic poles, much like the Earth; the spin axis points toward the north pole. One way of measuring the spin of a particular electron is to shoot it through a magnetic field, which will deflect the electron by a bit depending on how its spin is oriented. (As a technicality, the magnetic field has to be focused in the right way—spread out on one side, pinched tightly on the other—for this to work.) If I told you that the electron had a certain total spin, you might make the following prediction for such an experiment: the electron would be deflected up if its spin axis were aligned with the external field, deflected down if its spin were aligned in the opposite direction, and deflected at some intermediate angle if its spin were somewhere in between. But that’s not what we see. This experiment was first performed in 1922, by German physicists Otto Stern (an assistant to Max Born) and Walter Gerlach, before the idea of spin had been explicitly spelled out. What they saw was remarkable. Electrons are indeed deflected by passing through the magnetic field, but they either go up, or they go down; nothing in between. If we rotate the magnetic field, the electrons are still deflected in the direction of the field they pass through, either along or against it, but no intermediate values. The measured spin, like the energy of an electron orbiting an atomic nucleus, appears to be quantized. That seems surprising. Even if we’ve acclimated ourselves to the idea that the energy of an electron orbiting a nucleus only comes in certain quantized values, at least that energy seems like an objective property of the electron. But this thing we call the “spin” of the electron seems to give us different answers depending on how we measure it. No matter what particular direction we measure the spin along, there are only two possible outcomes we can obtain. To make sure we haven’t lost our minds, let’s be clever and run the electron through two magnets in a row. Remember that the rules of textbook quantum mechanics tell us that if we get a certain measurement outcome, then measure the same system immediately again, we will always get the same answer. And indeed that’s what happens; if an electron is deflected upward by one magnet (and is therefore spin-up), it will always be deflected upward by a following magnet oriented in the same way. What if we rotate one of the magnets by 90 degrees? So we’re splitting an initial beam of electrons into spin-up and spin-down as measured by a vertically oriented magnet, then taking the spin-up electrons and passing them through a horizontally oriented magnet. What happens then? Do they hold their breath and refuse to pass through, because they are vertically oriented spin-up electrons and we’re forcing them to be measured along a horizontal axis? No. Instead, the second magnet splits the spin-up electrons into two beams. Half of them are deflected to the right (along the direction of the second magnet) and half of them are deflected to the left. Madness. Our classical intuition makes us think that there is something called “the axis around which the electron is spinning,” and it makes sense (maybe) that the spin around that axis is quantized. But the experiments show that the axis around which the spin is quantized isn’t predetermined by the particle itself; you can choose any axis you like by rotating your magnet appropriately, and the spin will be quantized with respect to that axis. What we’re bumping up against is another manifestation of the uncertainty principle. The lesson we learned was that “position” and “momentum” aren’t properties that an electron has; they are just things we can measure about it. In particular, no particle can have a definite value of both simultaneously. Once we specify the exact wave function for position, the probability of observing any particular momentum is entirely fixed, and vice versa. The same is true for “vertical spin” and “horizontal spin.”* These are not separate properties an electron can have; they are just different quantities we can measure. If we express the quantum state in terms of the vertical spin, the probability of observing left or right horizontal spin is entirely fixed. The measurement outcomes we can get are determined by the underlying quantum state, which can be expressed in different but equivalent ways. The uncertainty principle expresses the fact that there are different incompatible measurements we can make on any particular quantum state. Systems with two possible measurement outcomes are so common and useful in quantum mechanics that they are given a cute name: qubits. The idea is that a classical “bit” has just two possible values, say, 0 and 1. A qubit (quantum bit) is a system that has two possible measurement outcomes, say, spin-up and spin-down along some specified axis. The state of a generic qubit is a superposition of both possibilities, each weighted by a complex number, the amplitude for each alternative. Quantum computers manipulate qubits in the same way that ordinary computers manipulate classical bits. We can write the wave function of a qubit as The symbols a and b are complex numbers, representing the amplitudes for spin-up and spin-down, respectively. The pieces of the wave function representing the different possible measurement outcomes, in this case spin-up/-down, are the “components.” In this state, the probability of observing the particle to be spin-up would be |a|2, and the probability for spin-down would be |b|2. If, for example, a and b were both equal to the square root of 1/2, the probability of observing spin-up or spin-down would be 1/2. Qubits can help us understand a crucial feature of wave functions: they are like the hypotenuse of a right triangle, for which the shorter sides are the amplitudes for each possible measurement outcome. In other words, the wave function is like a vector—an arrow with a length and a direction. The vector we’re talking about doesn’t point in a direction in real physical space, like “up” or “north.” Rather, it points in a space defined by all possible measurement outcomes. For a single spin qubit, that’s either spin-up or spin-down (once we choose some axis along which to measure). When we say “the qubit is in a superposition of spin-up and spin-down,” what we really mean is “the vector representing the quantum state has some component in the spin-up direction, and another component in the spin-down direction.” It’s natural to think of spin-up and spin-down as pointing in opposite directions. I mean, just look at the arrows. But as quantum states, they are perpendicular to each other: a qubit that is completely spin-up has no component of spin-down, and vice versa. Even the wave function for the position of a particle is a vector, though we normally visualize it as a smooth function throughout space. The trick is to think of every point in space as defining a different component, and the wave function is a superposition of all of them. There are an infinite number of such vectors, so the space of all possible quantum states, called Hilbert space, is infinite-dimensional for the position of a single particle. That’s why qubits are so much easier to think about. Two dimensions are easier to visualize than infinite dimensions. When there are only two components in our quantum state, as opposed to infinitely many, it can be hard to think of the state as a “wave function.” It’s not very wavy, and it doesn’t look like a smooth function of space. The right way to think about it is actually the other way around. The quantum state is not a function of ordinary space, it’s a function of the abstract “space of measurement outcomes,” which for a qubit only includes two possibilities. When the thing we observe is the location of a single particle, the quantum state assigns an amplitude to every possible location, which looks just like a wave in ordinary space. That’s the unusual case, however; the wave function is something more abstract, and when more than one particle is involved, it becomes hard to visualize. But we’re stuck with the “wave function” terminology. Qubits are great because at least the wave function has only two components. This may seem like an unnecessary mathematical detour, but there are immediate payoffs to thinking about wave functions as vectors. One is explaining the Born rule, which says that the probability for any particular measurement outcome is given by its amplitude squared. We’ll dive into details later, but it’s easy to see why the idea makes sense. As a vector, the wave function has a length. You might expect that the length could shrink or grow over time, but it doesn’t; according to Schr?dinger’s equation, the wave function just changes its “direction” while maintaining a constant length. And we can compute that length using Pythagoras’s theorem from high-school geometry. The numerical value of the length of the vector is irrelevant; we can just pick it to be a convenient number, knowing that it will remain constant. Let’s pick it to be one: every wave function is a vector of length one. The vector itself is just like the hypotenuse of a right triangle, with the components forming the shorter sides. So from Pythagoras’s theorem, we have a simple relationship: the squares of the amplitudes add up to unity, |a|2 |b|2 = 1. That’s the simple geometric fact underlying the Born rule for quantum probabilities. Amplitudes themselves don’t add up to one, but their squares do. That is exactly like an important feature of probability: the sum of probabilities for different outcomes needs to equal one. (Something has to happen, and the total probability of all exclusive somethings adds up to unity.) Another rule is that probabilities need to be non-negative numbers. Once again, amplitudes squared fit the bill: amplitudes can be negative (or complex), but their squares are non-negative real numbers. So even before thinking too hard, we can tell that “amplitudes squared” have the right properties to be the probabilities of outcomes—they are a set of non-negative numbers that always add up to one, because that’s the length of the wave function. This is at the heart of the whole matter: the Born rule is essentially Pythagoras’s theorem, applied to the amplitudes of different branches. That’s why it’s the amplitudes squared, not the amplitudes themselves or the square root of the amplitudes or anything crazy like that. The vector picture also explains the uncertainty principle in an elegant way. Remember that spin-up electrons split fifty-fifty into right-and left-spinning electrons when they passed through a subsequent horizontal magnet. That suggests that an electron in a spin-up state is equivalent to a superposition of spin-right and spin-left electron states, and likewise for spin-down. So the idea of being spin-left or spin-right isn’t independent from being spin-up or spin-down; any one possibility can be thought of as a superposition of the others. We say that spin-up and spin-down together form a basis for the state of a qubit—any quantum state can be written as a superposition of those two possibilities. But spin-left and spin-right form another basis, distinct but equally good. Writing it one way completely fixes the other way. Think of this in vector terms. If we draw a two-dimensional plane with spin-up as the horizontal axis and spin-down as the vertical axis, from the above relations we see that spin-right and spin-left point at 45 degrees with respect to them. Given any wave function, we could express it in the up/down basis, but we could equally well express it in the right/left basis. One set of axes is rotated with respect to the other, but they are both perfectly legitimate ways of expressing any vector we like. Now we can see where the uncertainty principle comes from. For a single spin, the uncertainty principle says that the state can’t have a definite value for the spin along the original axes (up/down) and the rotated axes (right/left) at the same time. This is clear from the picture: if the state is purely spin-up, it’s automatically some combination of spin-left and spin-right, and vice versa. Just as there are no quantum states that are simultaneously localized in position and momentum, there are no states that are simultaneously localized in both vertical spin and horizontal spin. The uncertainty principle reflects the relationship between what really exists (quantum states) and what we can measure (one observable at a time). * And for the third perpendicular direction, which we might call “forward spin,” though we didn’t measure that. 5 Entangled Up in Blue Wave Functions of Many Parts Popular discussions of the Einstein-Bohr debates often give the impression that Einstein couldn’t quite handle the uncertainty principle, and spent his time trying to invent clever ways to circumvent it. But what really bugged him about quantum mechanics was its apparent nonlocality—what happens at one point in space can seemingly have immediate consequences for experiments done very far away. It took him a while to codify his concerns into a well-formulated objection, and in doing so he helped illuminate one of the most profound features of the quantum world: the phenomenon of entanglement. Entanglement arises because there is only one wave function for the entire universe, not separate wave functions for each piece of it. How do we know that? Why can’t we just have a wave function for every particle or field? Consider an experiment in which we shoot two electrons at each other, moving with equal and opposite velocities. Because both have a negative electric charge, they will repel each other. Classically, if we were given the initial positions and velocities of the electrons, we could calculate precisely the directions into which each of them would scatter. Quantum-mechanically, all we can do is calculate the probability that they will each be observed on various paths after they interact with each other. The wave function of each particle spreads out in a roughly spherical pattern, until we ultimately observe it and pin down a definite direction in which it was moving. When we actually do this experiment, and observe the electrons after they have scattered, we notice something important. Since the electrons initially had equal and opposite velocities, the total momentum was zero. And momentum is conserved, so the post-interaction momentum should also be zero. This means that while the electrons might emerge moving in various different directions, whatever direction one of them moves in, the other moves in precisely the opposite. That’s funny, when you think about it. The first electron has a probability of scattering at various angles, and so does the second one. But if they each had a separate wave function, those two probabilities would be completely unrelated. We could imagine just observing one of the electrons, and measuring the direction in which it’s moving. The other one would be undisturbed. How could it know that it’s supposed to be moving in the opposite direction when we actually do measure it? We’ve already given away the answer. The two electrons don’t have separate wave functions; their behavior is described by the single wave function of the universe. In this case we can ignore the rest of the universe, and just focus in on these two electrons. But we can’t ignore one of the electrons and focus in on the other; the predictions we make for observations of either one can be dramatically affected by the outcome of observations of the other. The electrons are entangled. A wave function is an assignment of a complex number, the amplitude, to each possible observational outcome, and the square of the amplitude equals the probability that we would observe that outcome were we to make that measurement. When we’re talking about more than one particle, that means we assign an amplitude to every possible outcome of observing all the particles at once. If what we’re observing is positions, for example, the wave function of the universe can be thought of as assigning an amplitude to every possible combination of positions for all the particles in the universe. You might wonder whether it’s possible to visualize something like that. We can do it for the simple case of a single particle that we imagine only moves along one dimension, say, an electron confined to a thin copper wire: we draw a line representing the position of the particle, and plot a function representing the amplitude for each position. (Generally we cheat even in this simple context by just plotting a real number rather than a complex number, but so be it.) For two particles confined to the same one-dimensional motion, we could draw a two-dimensional plane representing the positions of each of the two particles, and then do a three-dimensional contour plot for the wave function. Note that this isn’t one particle in two-dimensional space; it’s two particles, each on a one-dimensional space, so the wave function is defined on the two-dimensional plane describing both positions. Because of the finite speed of light and a finite time since the Big Bang, we can see only a finite region of the cosmos, which we label “the observable universe.” There are approximately 1088 particles in the observable universe, mostly photons and neutrinos. That is a number much greater than two. And each particle is located in three-dimensional space, not just a one-dimensional line. How in the world are we supposed to visualize a wave function that assigns an amplitude to every possible configuration of 1088 particles distributed through three-dimensional space? We’re not. Sorry. The human imagination wasn’t designed to visualize the enormously big mathematical spaces that are routinely used in quantum mechanics. For just one or two particles, we can muddle through; more than that, and we have to describe things in words and equations. Fortunately, the Schr?dinger equation is straightforward and definite in what it says about how the wave function behaves. Once we understand what’s going on for two particles, the generalization to 1088 particles is just maths. The fact that wave functions are so big can make thinking about them a little unwieldy. Happily we can cast almost everything interesting to say about entanglement into the much simpler context of just a few qubits. Borrowing from a whimsical tradition in the literature on cryptography, quantum physicists like to consider two people named Alice and Bob who share qubits with each other. So let’s imagine two electrons, A belonging to Alice and B belonging to Bob. The spins of those two electrons constitute a two-qubit system, and are described by a corresponding wave function. The wave function assigns an amplitude to each configuration of the system as a whole, with respect to something we might observe about it, such as its spin in the vertical direction. So there are four possible measurement outcomes: both spins are up, both spins are down, A is up and B is down, and A is down and B is up. The state of the system is some superposition of these four possibilities, which are the basis states. Within each set of parentheses, the first spin is Alice’s, and the second is Bob’s. Just because we have two qubits, it doesn’t mean they are necessarily entangled. Consider a state that is simply one of the basis states, say, the one where both qubits are spin-up. If Alice measures her qubit along the vertical axis, she will always obtain spin-up, and likewise for Bob. If Alice measures her spin along the horizontal axis, she has a fifty-fifty chance of getting spin-right or spin-left, and again likewise for Bob. But in each case, we don’t learn anything about what Bob will see by learning what Alice saw. That’s why we can often casually speak of “the wave function of a particle,” even though we know better—when different parts of the system are unentangled with each other, it’s just as if they have their own wave functions. Instead, let’s consider an equal superposition of two basis states, one with both spins up, and the other with both spins down: If Alice measures her spin along the vertical axis, she has a fifty-fifty chance of getting spin-up or spin-down, and likewise for Bob. The difference now is that if we learn Alice’s outcome before Bob does his measurement, we know what Bob will see with 100 percent confidence—he’s going to see the same thing that Alice did. In the language of textbook quantum mechanics, Alice’s measurement collapses the wave function onto one of the two basis states, leaving Bob with a deterministic outcome. (In Many-Worlds language, Alice’s measurement branches the wave function, creating two different Bobs, each of whom will get a certain outcome.) That’s entanglement in action. In the aftermath of the 1927 Solvay Conference, Einstein remained convinced that quantum mechanics, especially as interpreted by the Copenhagen school, did a very good job at making predictions for experimental outcomes, but fell well short as a complete theory of the physical world. His concerns were finally written up for publication in 1935 with his collaborators Boris Podolsky and Nathan Rosen, in a paper that is universally known as simply EPR. Einstein later said that the primary ideas had been his, Rosen had done the calculations, and Podolsky had done much of the writing. EPR considered the position and momentum of two particles moving in opposite directions, but it’s easier for us to talk about qubits. Consider two spins that are in the entangled state written above. (It’s very easy to create such a state in the lab.) Alice stays home with her qubit, but Bob takes his and embarks on a long journey—say, he jumps in a rocket ship and flies to Alpha Centauri, four light-years away. The entanglement between two particles doesn’t fade away as they are moved apart; as long as neither Alice nor Bob measures the spins of their qubits, the overall quantum state will remain the same. Once Bob arrives safely at Alpha Centauri, Alice finally does measure the spin of her particle, along an agreed-upon vertical axis. Before that measurement, we were completely unsure what such an observation would reveal for her spin, and likewise for Bob’s. Let’s suppose that Alice observes spin-up. Then, by the rules of quantum mechanics, we immediately know that Bob will also observe spin-up, whenever he gets around to doing a measurement. That’s weird. Thirty years earlier, Einstein had established the rules of the special theory of relativity, which says among other things that signals cannot travel faster than the speed of light. And yet here we’re saying that according to quantum mechanics, a measurement that Alice does here and now has an immediate effect on Bob’s qubit, even though it’s four light-years away. How does Bob’s qubit know that Alice’s has been measured, and what the outcome was? This is the “spooky action at a distance” that Einstein so memorably fretted about. It’s not necessarily as bad as it seems. The first thing you might wonder about, upon being informed that quantum mechanics apparently sends influences faster than the speed of light, is whether or not we could take advantage of this phenomenon to communicate instantly across large distances. Can we build a quantum-entanglement phone, for which the speed of light is not a limitation at all? No, we can’t. This is pretty clear in our simple example: if Alice measures spin-up, she instantly knows that Bob will also measure spin-up when he gets around to it. But Bob doesn’t know that. In order for him to know what the spin of his particle is, Alice has to send him her measurement result by conventional means—which are limited by the speed of light. You might think there’s a loophole: What if Alice doesn’t just measure her qubit and find out a random answer, but rather forces her answer to be spin-up? Then Bob would also get spin-up. That would seem like information had been transmitted instantaneously. The problem is that there’s no straightforward way to start with a quantum system that is in a superposition and measure it in such a way that we can force a particular answer. If Alice simply measures her spin, she’ll get up or down with equal probabilities, no ifs, ands, or buts. What Alice can do is to manipulate her spin before she measures it, forcing it to be 100 percent spin-up rather than in a superposition. For example, she can shoot a photon at her electron, with just the right properties that the photon leaves the electron alone if the electron was spin-up, and flips the electron to spin-up if it was spin-down. Now Alice’s original electron will definitely be measured to be spin-up. But that electron is also no longer entangled with Bob’s electron. Rather, the entanglement has been transferred to the photon, which is in a superposition of “left Alice’s electron alone” and “bumped into Alice’s electron.” Bob’s electron is completely unaffected, and he’s going to get spin-up or spin-down with fifty-fifty probability, so no information has been transmitted. This is a general feature of quantum entanglement: the no-signaling theorem, according to which an entangled pair of particles cannot actually be used to transmit information between two parties faster than light. So quantum mechanics seems to be exploiting a subtle loophole, violating the spirit of relativity (nothing travels faster than the speed of light) while obeying the letter of the law (actual physical particles, and whatever useful information they might convey, cannot travel faster than the speed of light). The so-called EPR paradox (which isn’t a paradox at all, just a feature of quantum mechanics) goes beyond simple worries about spooky action at a distance. Einstein aimed to show not only that quantum mechanics was spooky but that it couldn’t possibly be a complete theory—that there had to be some underlying comprehensive model for which quantum mechanics was simply a useful approximation. EPR believed in the principle of locality—the physical quantities describing nature are defined at specific points in spacetime, not spread out all over the place, and they interact directly only with other quantities nearby, not at a distance. Said another way, given the speed-of-light restriction of special relativity, locality would seem to imply that nothing we can do to a particle at one location can instantaneously affect measurements we might perform on another particle very far away. On the face of it, the fact that two widely separated particles can be entangled seems to imply that locality is violated in quantum mechanics. But EPR wanted to be a little more thorough, and establish that there wasn’t some clever work-around that would make everything seem local. They suggested the following principle: if we have a physical system in a specified state, and there is a measurement we can do on that system such that we know with 100 percent certainty what the outcome will be, we associate an element of reality with that measurement outcome. In classical mechanics, the position and the momentum of each particle qualify as elements of reality. In quantum mechanics, if we have a qubit in a pure spin-up state, there is an element of reality corresponding to the spin in the vertical direction, but there need not be an element of reality corresponding to the horizontal spin, as we don’t know what we will get when we measure that. A “complete” theory, in the EPR formulation, is one in which every element of reality has a direct counterpart in the theory itself, and they argued that quantum mechanics couldn’t be complete by this criterion. Let’s take Alice and Bob and their entangled qubits, and imagine that Alice has just measured the vertical spin of her particle, finding that it points upward. We now know that Bob will also measure spin-up, even if Bob doesn’t know it himself. So by EPR’s lights, there is an element of reality attached to Bob’s particle, saying that the spin is up. It’s not that this element of reality came into existence when Alice did her measurement, as Bob’s particle is very far away, and locality says that the element of reality must be located where the particle is; it must have been there all along. But now imagine that Alice didn’t do the vertical-spin measurement at all, but instead measured the spin of her particle along the horizontal axis. Let’s say she measures spin-right for the particle. The entangled quantum state we started with ensures us that Bob will get the same result that Alice did, no matter what direction she chooses to measure her spin in. So we know that Bob would also measure spin-right, and by EPR’s lights there is—and was all along—an element of reality that says “spin-right for Bob’s qubit if it’s measured along the horizontal axis.” There’s no way for either Alice’s particle or Bob’s to know ahead of time which measurement Alice was going to make. Hence, Bob’s qubit must come equipped with elements of reality guaranteeing that its spin would be up if measured vertically, and right if measured horizontally. That’s exactly what the uncertainty principle says cannot happen. If the vertical spin is exactly determined, the horizontal spin is completely unknown, and vice versa, at least according to the conventional rules of quantum mechanics. There is nothing in the quantum formalism that can determine both a vertical spin and a horizontal spin at the same time. Therefore, EPR triumphantly conclude, there must be something missing—quantum mechanics cannot be a complete description of physical reality. The EPR paper caused a stir that reached far beyond the community of professional physicists. The New York Times, having been tipped off by Podolsky, published a front-page story about the ideas. This outraged Einstein, who penned a stern letter that the Times published, in which he decried advance discussion of scientific results in the “secular press.” It’s been said that he never spoke to Podolsky again. The response from professional scientists was also rapid. Niels Bohr wrote a quick reply to the EPR paper, which many physicists claimed resolved all the puzzles. What is less clear is precisely how Bohr’s paper was supposed to have achieved that; as brilliant and creative as he was as a thinker, Bohr was never an especially clear communicator, as he himself admitted. His paper was full of sentences like “in this stage there arises the essential problem of an influence on the precise conditions which define the possible types of prediction which regard the subsequent behavior of the system.” Roughly, his argument was that we shouldn’t go about attributing elements of reality to systems without taking into account how they are going to be observed. What is real, Bohr seems to suggest, depends not only on what we measure, but on how we choose to measure it. Einstein and his collaborators laid out what they took to be reasonable criteria for a physical theory—locality, and associating elements of reality to deterministically predictable quantities—and showed that quantum mechanics was incompatible with them. But they didn’t conclude that quantum mechanics was wrong, just that it was incomplete. The hope remained alive that we would someday find a better theory that both was local and respected reality. That hope was definitively squashed by John Stewart Bell, a physicist from Northern Ireland who worked at the CERN laboratory in Geneva, Switzerland. He became interested in the foundations of quantum mechanics in the 1960s, at a point in physics history when it was considered thoroughly disreputable to spend time thinking about such things. Today Bell’s theorem on entanglement is considered one of the most important results in physics. The theorem asks us to once again consider Alice and Bob and their entangled qubits with aligned spins. (Such quantum states are now known as Bell states, although it was David Bohm who first conceptualized the EPR puzzle in these terms.) Imagine that Alice measures the vertical spin of her particle, and obtains the result that it is spin-up. We now know that if Bob measures the vertical spin of his particle, he will also obtain spin-up. Furthermore, by the ordinary rules of quantum mechanics we know that if Bob chooses to measure the horizontal spin instead, he will get spin-right and spin-left with fifty-fifty probability. We can say that if Bob measures the vertical spin, the correlation between his result and Alice’s will be 100 percent (we know exactly what he’ll get), whereas if he measures horizontal spin, there will be 0 percent correlation (we have no idea what he will get). So what if Bob, growing bored all by himself in a spaceship orbiting Alpha Centauri, decides to measure the spin of his particle along some axis in between the horizontal and vertical? (For convenience imagine that Alice and Bob actually share a large number of entangled Bell pairs, so they can keep doing these measurements over and over, and we only care about what happens when Alice observes spin-up.) Then Bob will usually, but not always, observe the spin to be pointed along whatever direction is more closely aligned with the vertical “up.” In fact, we can do the maths: if Bob’s axis is at 45 degrees, exactly halfway between vertical and horizontal, there will be a 71 percent correlation between his results and Alice’s. (That’s one over the square root of two, if you’re wondering where the number comes from.) What Bell showed, under certain superficially reasonable assumptions, is that this quantum-mechanical prediction is impossible to reproduce in any local theory. In fact, he proved a strict inequality: the best you can possibly do without some kind of spooky action at a distance would be to achieve a 50 percent correlation between Alice and Bob if their measurements were rotated by 45 degrees. The quantum prediction of 71 percent correlation violates Bell’s inequality. There is a distinct, undeniable difference between the dream of simple underlying local dynamics, and the real-world predictions of quantum mechanics. I presume you are currently thinking to yourself, “Hey, what do you mean that Bell made superficially reasonable assumptions? Spell them out. I’ll decide for myself what I find reasonable and what I don’t.” Fair enough. There are two assumptions behind Bell’s theorem in particular that one might want to doubt. One is contained in the simple idea that Bob “decides” to measure the spin of his qubit along a certain axis. An element of human choice, or free will, seems to have crept into our theorem about quantum mechanics. That’s hardly unique, of course; scientists are always assuming that they can choose to measure whatever they want. But really we think that’s just a convenient way of talking, and even those scientists are composed of particles and forces that themselves obey the laws of physics. So we can imagine invoking superdeterminism—the idea that the true laws of physics are utterly deterministic (no randomness anywhere), and furthermore that the initial conditions of the universe were laid down at the Big Bang in just precisely such a way that certain “choices” are never going to be made. It’s conceivable that one could invent a perfectly local superdeterministic theory that would mimic the predictions of quantum entanglement, simply because the universe was prearranged to make it appear that way. This seems unpalatable to most physicists; if you can delicately arrange your theory to do that, it can basically be arranged to do anything you want, and at that point why are we even doing physics? But some smart people are pursuing the idea. The other potentially doubtable assumption seems uncontroversial at first glance: that measurements have definite outcomes. When you observe the spin of a particle, you get an actual result, either spin-up or spin-down along whatever axis you are measuring it with respect to. Seems reasonable, doesn’t it? But wait. We actually know about a theory where measurements don’t have definite outcomes—austere, Everettian quantum mechanics. There, it’s simply not true that we get either up or down when we measure an electron’s spin; in one branch of the wave function we get up, in the other we get down. The universe as a whole doesn’t have any single outcome for that measurement; it has multiple ones. That doesn’t mean that Bell’s theorem is wrong in Many-Worlds; mathematical theorems are unambiguously right, given their assumptions. It just means that the theorem doesn’t apply. Bell’s result does not imply that we have to include spooky action at a distance in Everettian quantum mechanics, as it does for boring old single-world theories. The correlations don’t come about because of any kind of influence being transmitted faster than light, but because of branching of the wave function into different worlds, in which correlated things happen. For a researcher in the foundations of quantum mechanics, the relevance of Bell’s theorem to your work depends on exactly what it is you’re trying to do. If you have devoted yourself to the task of inventing a new version of quantum mechanics from scratch, in which measurements do have definite outcomes, Bell’s inequality is the most important guidepost you have to keep in mind. If, on the other hand, you’re happy with Many-Worlds and are trying to puzzle out how to map the theory onto our observed experience, Bell’s result is an automatic consequence of the underlying equations, not an additional constraint you need to worry about moving forward. One of the fantastic things about Bell’s theorem is that it turns the supposed spookiness of quantum entanglement into a straightforwardly experimental question—does nature exhibit intrinsically non-local correlations between faraway particles, or not? You’ll be happy to hear that experiments have been done, and the predictions of quantum mechanics have been spectacularly verified every time. There is a tradition in popular media of writing articles with breathless headlines like “Quantum Reality Is Even More Bizarre Than Previously Believed!” But when you look into the results they are actually reporting, it’s another experiment that confirms exactly what a competent quantum mechanic would have predicted all along using the theory that had been established by 1927, or at least by 1935. We understand quantum mechanics enormously better now than we did back then, but the theory itself hasn’t changed. Which isn’t to say that the experiments aren’t important or impressive; they are. The problem with testing Bell’s predictions, for example, is that you are trying to make sure that the extra correlations predicted by quantum mechanics couldn’t have arisen due to some sneaky pre-existing classical correlation. How do we know whether some hidden event in the past secretly affected how we chose to measure our spin, or what the measurement outcome was, or both? Physicists have gone to great lengths to eliminate these possibilities, and a cottage industry has arisen in doing “loophole-free Bell tests.” One recent result wanted to eliminate the possibility that an unknown process in the laboratory worked to influence the choice of how to measure the spin. So instead of letting a lab assistant choose the measurement, or even using a random-number generator sitting on a nearby table, the experiment made that choice based on the polarization of photons emitted from stars many light-years away. If there were some nefarious conspiracy to make the world look quantum-mechanical, it had to have been set up hundreds of years ago, when the light left those stars. It’s possible, but doesn’t seem likely. It seems that quantum mechanics is right again. So far, quantum mechanics has always been right. 6 Splitting the Universe Decoherence and Parallel Worlds The 1935 Einstein-Podolsky-Rosen (EPR) paper on quantum entanglement, and Niels Bohr’s response to it, were the last major public salvos in the Bohr-Einstein debates over the foundations of quantum mechanics. Bohr and Einstein had corresponded about quantum theory soon after Bohr proposed his model of quantized electron orbits in 1913, and their dispute came to a head at the 1927 Solvay Conference. In the popular retelling, Einstein would raise some objection to the rapidly coalescing Copenhagen consensus during conversations at the workshop with Bohr, who would spend the evening fretting about it, and then at breakfast Bohr would triumphantly present his rejoinder to the chastened Einstein. We are told that Einstein simply couldn’t come to grips with the fact of the uncertainty principle and the notion that God plays dice with the universe. That’s not what happened. Einstein’s primary concerns were not with randomness but with realism and locality. His determination to salvage these principles culminated in the EPR paper and their argument that quantum mechanics must be incomplete. But by that time the public-relations battle had been lost, and the Copenhagen approach to quantum mechanics had been adopted by physicists worldwide, who then set about applying quantum mechanics to technical problems in atomic and nuclear physics, as well as the emerging fields of particle physics and quantum field theory. The implications of the EPR paper itself were largely ignored by the community. Wrestling with the confusions at the heart of quantum theory, rather than working on more tangible physics problems, began to be thought of as a somewhat eccentric endeavor. Something that could occupy the time of formerly productive physicists once they reached a certain age and were ready to abandon real work. In 1933, Einstein left Germany and took a position at the new Institute for Advanced Study in Princeton, New Jersey, where he would remain until his death in 1955. His technical work after 1935 focused largely on classical general relativity and his search for a unified theory of gravitation and electromagnetism, but he never stopped thinking about quantum mechanics. Bohr would occasionally visit Princeton, where he and Einstein would carry on their dialogue. John Archibald Wheeler joined the physics faculty at Princeton University, down the road from the Institute and Einstein, as an assistant professor in 1934. In later years Wheeler would become known as one of the world’s experts in general relativity, popularizing the terms “black hole” and “wormhole,” but in his early career he concentrated on quantum problems. He had briefly studied under Bohr in Copenhagen, and in 1939 he and Bohr published a pioneering paper on nuclear fission. Wheeler had great admiration for Einstein, but he venerated Bohr; as he would later put it, “Nothing has done more to convince me that there once existed friends of mankind with the human wisdom of Confucius and Buddha, Jesus and Pericles, Erasmus and Lincoln, than walks and talks under the beech trees of Klampenborg Forest with Niels Bohr.” Wheeler made an impact on physics in a number of ways, one of which was in the mentoring of talented graduate students, including future Nobel laureates such as Richard Feynman and Kip Thorne. One of those students was Hugh Everett III, who would introduce a dramatically new approach to thinking about the foundations of quantum mechanics. We’ve already sketched his basic idea—the wave function represents reality, it evolves smoothly, and that evolution leads to multiple distinct worlds when a quantum measurement takes place—but now we have the tools to do it right. Everett’s proposal, which eventually became his 1957 PhD thesis at Princeton, can be thought of as the purest incarnation of one of Wheeler’s favorite principles—that theoretical physics should be “radically conservative.” The idea is that a successful physical theory is one that has been tested against experimental data, but only in regimes that experimenters are actually able to reach. One should be conservative, in the sense that we should start with the theories and principles that are already established as successful, rather than arbitrarily introducing new approaches whenever new phenomena are encountered. But one should also be radical, in the sense that the predictions and implications of our theories should be taken seriously in regimes well outside where they have been tested. The phrases “we should start” and “should be taken seriously” are crucial here; of course new theories are warranted when old ones are shown to blatantly contradict the data, and just because a prediction is taken seriously doesn’t mean it shouldn’t be revised in light of new information. But Wheeler’s philosophy was that we should start prudently, with aspects of nature we believe we understand, and then act boldly, extrapolating our best ideas to the ends of the universe. Part of Everett’s inspiration was the search for a theory of quantum gravity, which Wheeler had recently become interested in. The rest of physics—matter, electromagnetism, the nuclear forces—seems to fit comfortably within the framework of quantum mechanics. But gravity was (and remains) a stubborn exception. In 1915, Einstein proposed the general theory of relativity, according to which spacetime itself is a dynamical entity whose bends and warps are what you and I perceive as the force of gravity. But general relativity is a thoroughly classical theory, with analogues of position and momentum for the curvature of space-time, and no limits on how we might measure them. Taking that theory and “quantizing” it, constructing a theory of wave functions of space-time rather than particular classical spacetimes has proven difficult. Hugh Everett III (Courtesy of the Hugh Everett III Archive at the University of California, Irvine, and Mark Everett) The difficulties of quantum gravity are both technical—calculations tend to blow up and give infinitely big answers—and also conceptual. Even in quantum mechanics, while you might not be able to say precisely where a certain particle is, the notion of “a point in space” is perfectly well defined. We can specify a location and ask what is the probability of finding the particle nearby. But if reality doesn’t consist of stuff distributed through space, but rather is a quantum wave function describing superpositions of different possible spacetimes, how do we even ask “where” a certain particle is observed? The puzzles become worse when we turn to the measurement problem. By the 1950s the Copenhagen school was established doctrine, and physicists had made their peace with the idea of wave functions collapsing when a measurement occurred. They were even willing to go along with treating the measurement process as a fundamental part of our best description of nature. Or, at least, not to fret too much about it. But what happens when the quantum system under consideration is the entire universe? Crucial to the Copenhagen approach is the distinction between the quantum system being measured and the classical observer doing the measuring. If the system is the universe as a whole, we are all inside it; there’s no external observer to whom we can appeal. Years later, Stephen Hawking and others would study quantum cosmology to discuss how a self-contained universe could have an earliest moment in time, presumably identified with the Big Bang. While Wheeler and others thought about the technical challenges of quantum gravity, Everett became fascinated by these conceptual problems, especially how to handle measurement. The seeds of the Many-Worlds formulation can be traced to a late-night discussion in 1954 with fellow young physicists Charles Misner (also a student of Wheeler’s) and Aage Petersen (an assistant of Bohr’s, visiting from Copenhagen). All parties agree that copious amounts of sherry were consumed on the occasion. Clearly, Everett reasoned, if we’re going to talk about the universe in quantum terms, we can’t carve out a separate classical realm. Every part of the universe will have to be treated according to the rules of quantum mechanics, including the observers within it. There will only be a single quantum state, described by what Everett called the “universal wave function” (and we’ve been calling “the wave function of the universe”). If everything is quantum, and the universe is described by a single wave function, how is measurement supposed to occur? It must be, Everett reasoned, when one part of the universe interacts with another part of the universe in some appropriate way. That is something that’s going to happen automatically, he noticed, simply due to the evolution of the universal wave function according to the Schr?dinger equation. We don’t need to invoke any special rules for measurement at all; things bump into each other all the time. It’s for this reason that Everett titled his eventual paper on the subject “‘Relative State’ Formulation of Quantum Mechanics.” As a measurement apparatus interacts with a quantum system, the two become entangled with each other. There are no wave-function collapses or classical realms. The apparatus itself evolves into a superposition, entangled with the state of the thing being observed. The apparently definite measurement outcome (“the electron is spin-up”) is only relative to a particular state of the apparatus (“I measured the electron to be spin-up”). The other possible measurement outcomes still exist and are perfectly real, just as separate worlds. All we have to do is to courageously face up to what quantum mechanics has been trying to tell us all along. Let’s be a little more explicit about what happens when a measurement is made, according to Everett’s theory. Imagine that we have a spinning electron, which could be observed to be in states of either spin-up or spin-down with respect to some chosen axis. Before measurement, the electron will typically be in some superposition of up and down. We also have a measuring apparatus, which is a quantum system in its own right. Imagine that it can be in superpositions of three different possibilities: it can have measured the spin to be up, it can have measured the spin to be down, or it might not yet have measured the spin at all, which we call the “ready” state. The fact that the measurement apparatus does its job tells us how the quantum state of the combined spin apparatus system evolves according to the Schr?dinger equation. Namely, if we start with the apparatus in its ready state and the spin in a purely spin-up state, we are guaranteed that the apparatus evolves to a pure measured-up state, like so: The initial state on the left can be read as “the spin is in the up state, and the apparatus is in its ready state,” while the one on the right, where the pointer indicates the up arrow, is “the spin is in the up state, and the apparatus has measured it to be up.” Likewise, the ability to successfully measure a pure-down spin implies that the apparatus must evolve from “ready” to “measured down”: What we want, of course, is to understand what happens when the initial spin is not in a pure up or down state, but in some superposition of both. The good news is that we already know everything we need. The rules of quantum mechanics are clear: if you know how the system evolves starting from two different states, the evolution of a superposition of both those states will just be a superposition of the two evolutions. In other words, starting from a spin in some superposition and the measurement device in its ready state, we have: The final state now is an entangled superposition: the spin is up and it was measured to be up, plus the spin is down and it was measured to be down. At this point it’s not strictly correct to say “the spin is in a superposition” or “the apparatus is in a superposition.” Entanglement prevents us from talking about the wave function of the spin, or that of the apparatus, individually, because what we will observe about one can depend on what we observe about the other. The only thing we can say is “the spin apparatus system is in a superposition.” This final state is the clear, unambiguous, definitive final wave function for the combined spin apparatus system, if all we do is evolve it according to the Schr?dinger equation. This is the secret to Everettian quantum mechanics. The Schr?dinger equation says that an accurate measuring apparatus will evolve into a macroscopic superposition, which we will ultimately interpret as branching into separate worlds. We didn’t put the worlds in; they were always there, and the Schr?dinger equation inevitably brings them to life. The problem is that we never seem to come across superpositions involving big macroscopic objects in our experience of the world. The traditional remedy has been to monkey with the fundamental rules of quantum mechanics in one way or another. Some approaches say that the Schr?dinger equation isn’t always applicable, others say that there are additional variables over and above the wave function. The Copenhagen approach is to disallow the treatment of the measurement apparatus as a quantum system in the first place, and treat wave function collapse as a separate way the quantum state can evolve. One way or another, all of these approaches invoke contortions in order to not accept superpositions like the one written above as the true and complete description of nature. As Everett would later put it, “The Copenhagen Interpretation is hopelessly incomplete because of its a priori reliance on classical physics . . . as well as a philosophic monstrosity with a ‘reality’ concept for the macroscopic world and denial of the same for the microcosm.” Everett’s prescription was simple: stop contorting yourself. Accept the reality of what the Schr?dinger equation predicts. Both parts of the final wave function are actually there. They simply describe separate, never-to-interact-again worlds. Everett didn’t introduce anything new into quantum mechanics; he removed some extraneous clunky pieces from the formalism. Every non-Everettian version of quantum mechanics is, as physicist Ted Bunn has put it, a “disappearing worlds” theory. If the multiple worlds bother you, you have to fiddle with either the nature of quantum states or their ordinary evolution in order to get rid of them. Is it worth it? There’s a looming question here. We’re familiar with how wave functions represent superpositions of different possible measurement outcomes. The wave function of an electron can put it in a superposition of various possible locations, as well as in a superposition of spin-up and spin-down. But we were never tempted to say that each part of the superposition was a separate “world.” Indeed, it would have been incoherent to do so. An electron that is in a pure spin-up state with respect to the vertical axis is in a superposition of spin-up and spin-down with respect to the horizontal axis. So does that describe one world, or two? Everett suggested that it is logically consistent to think of superpositions involving macroscopic objects as describing separate worlds. But at the time he was writing, physicists hadn’t yet developed the technical tools necessary to turn this into a complete picture. That understanding only came later, with the appreciation of a phenomenon known as decoherence. Introduced in 1970 by the German physicist Hans Dieter Zeh, the idea of decoherence has become a central part of how physicists think about quantum dynamics. To the modern Everettian, decoherence is absolutely crucial to making sense of quantum mechanics. It explains once and for all why wave functions seem to collapse when you measure quantum systems—and indeed what a “measurement” really is. We know there is only one wave function, the wave function of the universe. But when we’re talking about individual microscopic particles, they can settle into quantum states where they are unentangled from the rest of the world. In that case, we can sensibly talk about “the wave function of this particular electron” and so forth, keeping in mind that it’s really just a useful shortcut we can employ when systems are unentangled with anything else. With macroscopic objects, things aren’t that simple. Consider our spin-measuring apparatus, and let’s imagine we put it in a superposition of having measured spin-up and spin-down. The dial of the apparatus includes a pointer that is pointing either to Up or to Down. An apparatus like that doesn’t stay separate from the rest of the world. Even if it looks like it’s just sitting there, in reality the air molecules in the room are constantly bumping into it, photons of light are bouncing off of it, and so on. Call all that other stuff—the entire rest of the universe—the environment. In ordinary situations, there’s no way to stop a macroscopic object from interacting with its environment, even if very gently. Such interactions will cause the apparatus to become entangled with the environment, for example, because a photon would reflect off the dial if the pointer is in one position, but be absorbed by it if the pointer is pointing somewhere else. So the wave function we wrote down above, where an apparatus became entangled with a qubit, wasn’t quite the whole story. Putting the environment states in curly braces, we should have written It doesn’t really matter what the environment states actually are, so we’ve portrayed them as different backgrounds labeled {E0}, {E1}, and {E2}. We don’t (and generally can’t) keep track of exactly what’s going on in the environment—it’s too complicated. It’s not going to just be a single photon that interacts differently with different parts of the apparatus’s wave function, it will be a huge number of them. Nobody can be expected to keep track of every photon or particle in a room. That simple process—macroscopic objects become entangled with the environment, which we cannot keep track of—is decoherence, and it comes with universe-altering consequences. Decoherence causes the wave function to split, or branch, into multiple worlds. Any observer branches into multiple copies along with the rest of the universe. After branching, each copy of the original observer finds themselves in a world with some particular measurement outcome. To them, the wave function seems to have collapsed. We know better; the collapse is only apparent, due to decoherence splitting the wave function. We don’t know how often branching happens, or even whether that’s a sensible question to ask. It depends on whether there are a finite or infinite number of degrees of freedom in the universe, which is currently an unanswered question in fundamental physics. But we do know that there’s a lot of branching going on; it happens every time a quantum system in a superposition becomes entangled with the environment. In a typical human body, about 5,000 atoms undergo radioactive decay every second. If every decay branches the wave function in two, that’s 25000 new branches every second. It’s a lot. What makes a “world,” anyway? We just wrote down a single quantum state describing a spin, an apparatus, and an environment. What makes us say that it describes two worlds, rather than just one? One thing you would like to have in a world is that different parts of it can, at least in principle, affect each other. Consider the following “ghost world” scenario (not meant as a true description of reality, just a colorful analogy): when living beings die, they all become ghosts. These ghosts can see and talk to one another, but they cannot see or talk to us, nor can we see or talk to them. They live on a separate Ghost Earth, where they can build ghost houses and go to their ghost jobs. But neither they nor their surroundings can interact with us and the stuff around us in any way. In this case it makes sense to say that the ghosts inhabit a truly separate ghost world, for the fundamental reason that what happens in the ghost world has absolutely no bearing on what happens in our world. Now apply this criterion to quantum mechanics. We’re not interested in whether the spin and its measuring apparatus can influence each other—they obviously can. What we care about is whether one component of, say, the apparatus wave function (for example, the piece where the dial is pointing to Up) can possibly influence another piece (for example, where it’s pointing to Down). We’ve previously come across a situation just like this, where the wave function influences itself—in the phenomenon of interference from the double-slit experiment. When we passed electrons through two slits without measuring which one they went through, we saw interference bands on the final screen, and attributed them to the cancellation between the contribution to the total probability from each of the two slits. Crucially, we implicitly assumed that the electron didn’t interact and become entangled with anything along its journey; it didn’t decohere. When instead we did detect which slit the electron went through, the interference bands went away. At the time we attributed this to the fact that a measurement had been performed, collapsing the electron’s wave function at one slit or another. Everett gives us a much more compelling story to tell. What actually happened was that the electron became entangled with the detector as it moved through the slits, and then the detector quickly became entangled with the environment. The process is precisely analogous to what happened to our spin above, except that we’re measuring whether the electron went through the left slit L or the right slit R: No mysterious collapsing; the whole wave function is still there, evolving cheerfully according to the Schr?dinger equation, leaving us in a superposition of two entangled pieces. But note what happens as the electron continues on toward the screen. As before, the state of the electron at any given point on the screen will receive a contribution from what passed through slit L, and another contribution from what passed through slit R. But now those contributions won’t interfere with each other. In order to get interference, we need to be adding up two equal and opposite quantities: 1 (-1) = 0. But there is no point on the screen where we will find equal and opposite contributions to the electron’s wave function from the L and R slits, because passing through those slits entangled the electron with different states of the rest of the world. When we say equal and opposite, we mean precisely equal and opposite, not “equal and opposite except for that thing we’re entangled with.” Being entangled with different states of the detector and environment—being decohered, in other words—means that the two parts of the electron’s wave function can no longer interfere with each other. And that means they can’t interact at all. And that means they are, for all intents and purposes, part of separate worlds.* From the point of view of things entangled with one branch of the wave function, the other branches might as well be populated by ghosts. The Many-Worlds formulation of quantum mechanics removes once and for all any mystery about the measurement process and collapse of the wave function. We don’t need special rules about making an observation: all that happens is that the wave function keeps chugging along in accordance with the Schr?dinger equation. And there’s nothing special about what constitutes “a measurement” or “an observer”—a measurement is any interaction that causes a quantum system to become entangled with the environment, creating decoherence and a branching into separate worlds, and an observer is any system that brings such an interaction about. Consciousness, in particular, has nothing to do with it. The “observer” could be an earthworm, a microscope, or a rock. There’s not even anything special about macroscopic systems, other than the fact that they can’t help but interact and become entangled with the environment. The price we pay for such powerful and simple unification of quantum dynamics is a large number of separate worlds. Everett himself wasn’t familiar with decoherence, so his picture wasn’t quite as robust and complete as the one we’ve painted. But his way of rethinking the measurement problem and offering a unified picture of quantum dynamics was compelling from the start. Even in theoretical physics, people do sometimes get lucky, hitting upon an important idea more because they were in the right place at the right time than because they were particularly brilliant. That’s not the case with Hugh Everett; those who knew him testify uniformly to his incredible intellectual gifts, and it’s clear from his writings that he had a thorough understanding of the implications of his ideas. Were he still alive, he would be perfectly at home in modern discussions of the foundations of quantum mechanics. What was hard was getting others to appreciate those ideas, and that included his advisor. Wheeler was personally very supportive of Everett, but he was also devoted to his own mentor, Bohr, and was convinced of the basic soundness of the Copenhagen approach. He simultaneously wanted Everett’s ideas to get a wide hearing, and to ensure that they weren’t interpreted as a direct assault on Bohr’s way of thinking about quantum mechanics. Yet Everett’s theory was a direct assault on Bohr’s picture. Everett himself knew it, and enjoyed illustrating the nature of this assault in vivid language. In an early draft of his thesis, Everett used the analogy of an amoeba dividing to illustrate the branching of the wave function: “One can imagine an intelligent amoeba with a good memory. As time progresses the amoeba is constantly splitting, each time the resulting amoebas having the same memories as the parent. Our amoeba hence does not have a life line, but a life tree.” Wheeler was put off by the blatantness of this (quite accurate) metaphor, scribbling in the margin of the manuscript, “Split? Better words needed.” Advisor and student were constantly tussling over the best way to express the new theory, with Wheeler advocating caution and prudence while Everett favored bold clarity. In 1956, as Everett was working on finishing his dissertation, Wheeler visited Copenhagen and presented the new scenario to Bohr and his colleagues, including Aage Petersen. He attempted to present it anyway; by this time the wave-functions-collapse-and-don’t-ask-embarrassing-questions-about-exactly-how school of quantum theory had hardened into conventional wisdom, and those who accepted it weren’t interested in revisiting the foundations when there was so much interesting applied work to be done. Letters from Wheeler, Everett, and Petersen flew back and forth across the Atlantic, continuing when Wheeler returned to Princeton and helped Everett craft the final form of his dissertation. The agony of this process is reflected in the evolution of the paper itself: Everett’s first draft was titled “Quantum Mechanics by the Method of the Universal Wave Function,” and a revised version was called “Wave Mechanics Without Probability.” This document, later dubbed the “long version” of the thesis, wasn’t published until 1973. A “short version” was finally submitted for Everett’s PhD as “On the Foundations of Quantum Mechanics,” and eventually published in 1957 as “‘Relative State’ Formulation of Quantum Mechanics.” It omitted many of the juicier sections Everett had originally composed, including examinations of the foundations of probability and information theory and an overview of the quantum measurement problem, focusing instead on applications to quantum cosmology. (No amoebas appear in the published paper, but Everett did manage to insert the word “splitting” in a footnote added in proof while Wheeler wasn’t looking.) Furthermore, Wheeler wrote an “assessment” article that was published alongside Everett’s, which suggested that the new theory was radical and important, while at the same time attempting to paper over its manifest differences with the Copenhagen approach. The arguments continued, without much headway being made. It is worth quoting from a letter Everett wrote to Petersen, in which his frustration comes through: Lest the discussion of my paper die completely, let me add some fuel to the fire with . . . criticisms of the ‘Copenhagen interpretation.’ . . . I do not think you can dismiss my viewpoint as simply a misunderstanding of Bohr’s position. . . . I believe that basing quantum mechanics upon classical physics was a necessary provisional step, but that the time has come . . . to treat [quantum mechanics] in its own right as a fundamental theory without any dependence on classical physics, and to derive classical physics from it. . . . Let me mention a few more irritating features of the Copenhagen Interpretation. You talk of the massiveness of macro systems allowing one to neglect further quantum effects (in discussions of breaking the measuring chain), but never give any justification for this flatly asserted dogma. [And] there is nowhere to be found any consistent explanation for this ‘irreversibility’ of the measuring process. It is again certainly not implied by wave mechanics, nor classical mechanics either. Another independent postulate? But Everett decided not to continue the academic fight. Before finishing his PhD, he accepted a job at the Weapons Systems Evaluation Group for the US Department of Defense, where he studied the effects of nuclear weapons. He would go on to do research on strategy, game theory, and optimization, and played a role in starting several new companies. It’s unclear the extent to which Everett’s conscious decision to not apply for professorial positions was motivated by criticism of his upstart new theory, or simply by impatience with academia in general. He did, however, maintain an interest in quantum mechanics, even if he never published on it again. After Everett defended his PhD and was already working for the Pentagon, Wheeler persuaded him to visit Copenhagen for himself and talk to Bohr and others. The visit didn’t go well; afterward Everett judged that it had been “doomed from the beginning.” Bryce DeWitt, an American physicist who had edited the journal where Everett’s thesis appeared, wrote a letter to him complaining that the real world obviously didn’t “branch,” since we never experience such things. Everett replied with a reference to Copernicus’s similarly daring idea that the Earth moves around the sun, rather than vice versa: “I can’t resist asking: Do you feel the motion of the earth?” DeWitt had to admit that was a pretty good response. After mulling the matter over for a while, by 1970 DeWitt had become an enthusiastic Everettian. He put a great deal of effort into pushing the theory, which had languished in obscurity, toward greater public recognition. His strategies included an influential 1970 article in Physics Today, followed by a 1973 essay collection that included at last the long version of Everett’s dissertation, as well as a number of commentaries. The collection was called simply The Many-Worlds Interpretation of Quantum Mechanics, a vivid name that has stuck ever since. In 1976, John Wheeler retired from Princeton and took up a position at the University of Texas, where DeWitt was also on the faculty. Together they organized a workshop in 1977 on the Many-Worlds theory, and Wheeler coaxed Everett into taking time off from his defense work in order to attend. The conference was a success, and Everett made a significant impression on the assembled physicists in the audience. One of them was the young researcher David Deutsch, who would go on to become a major proponent of Many-Worlds, as well as an early pioneer of quantum computing. Wheeler went so far as to propose a new research institute in Santa Barbara, where Everett could return to full-time work on quantum mechanics, but ultimately nothing came of it. Everett died in 1982, age fifty-one, of a sudden heart attack. He had not lived a healthy lifestyle, overindulging in eating, smoking, and drinking. His son, Mark Everett (who would go on to form the band Eels), has said that he was originally upset with his father for not taking better care of himself. He later changed his mind: “I realize that there is a certain value in my father’s way of life. He ate, smoked and drank as he pleased, and one day he just suddenly and quickly died. Given some of the other choices I’d witnessed, it turns out that enjoying yourself and then dying quickly is not such a hard way to go.” * The set of all branches of the wave function is different from what cosmologists often call “the multiverse.” The cosmological multiverse is really just a collection of regions of space, generally far away from one another, where local conditions look very different. 7 Order and Randomness Where Probability Comes From One sunny day in Cambridge, England, Elizabeth Anscombe ran into her teacher, Ludwig Wittgenstein. “Why do people say,” Wittgenstein opened in his inimitable fashion, “that it was natural to think that the sun went round the earth, rather than that the earth turned on its axis?” Anscombe gave the obvious answer, that it just looks like the sun goes around the Earth. “Well,” Wittgenstein replied, “what would it have looked like if the Earth had turned on its axis?” This anecdote—recounted by Anscombe herself, and which Tom Stoppard retold in his play Jumpers—is a favorite among Everettians. Physicist Sidney Coleman used to relate it in lectures, and philosopher of physics David Wallace used it to open his book The Emergent Multiverse. It even bears a family resemblance to Hugh Everett’s remark to Bryce DeWitt. It’s easy to see why the observation is so relevant. Any reasonable person, when first told about the Many-Worlds picture, has an immediate, visceral objection: it just doesn’t feel like I personally split into multiple people whenever a quantum measurement is performed. And it certainly doesn’t look like there are all sorts of other universes existing parallel to the one I find myself in. Well, the Everettian replies, channeling Wittgenstein: What would it feel and look like if Many-Worlds were true? The hope is that people living in an Everettian universe would experience just what people actually do experience: a physical world that seems to obey the rules of textbook quantum mechanics to a high degree of accuracy, and in many situations is well approximated by classical mechanics. But the conceptual distance between “a smoothly evolving wave function” and the experimental data it is meant to explain is quite large. It’s not obvious that the answer we can give to Wittgenstein’s question is the one we want. Everett’s theory might be austere in its formulation, but there’s still a good amount of work to be done to fully flesh out its implications. In this chapter we’ll confront a major puzzle for Many-Worlds: the origin and nature of probability. The Schr?dinger equation is perfectly deterministic. Why do probabilities enter at all, and why do they obey the Born rule: probabilities equal amplitudes—the complex numbers the wave function associates with each possible outcome—squared? Does it even make sense to speak of the probability of ending up on some particular branch if there will be a future version of myself on every branch? In the textbook or Copenhagen versions of quantum mechanics, there’s no need to “derive” the Born rule for probabilities. We just plop it down there as one of the postulates of the theory. Why couldn’t we do the same thing in Many-Worlds? The answer is that even though the rule would sound the same in both cases—“probabilities are given by the wave function squared”—their meanings are very different. The textbook version of the Born rule really is a statement about how often things happen, or how often they will happen in the future. Many-Worlds has no room for such an extra postulate; we know exactly what will happen, just from the basic rule that the wave function always obeys the Schr?dinger equation. Probability in Many-Worlds is necessarily a statement about what we should believe and how we should act, not about how often things happen. And “what we should believe” isn’t something that really has a place in the postulates of a physical theory; it should be implied by them. Moreover, as we will see, there is neither any room for an extra postulate, nor any need for one. Given the basic structure of quantum mechanics, the Born rule is natural and automatic. Since we tend to see Born rule–like behavior in nature, this should give us confidence that we’re on the right track. A framework in which an important result can be derived from more fundamental postulates should, all else being equal, be preferred to one where it needs to be separately assumed. If we successfully address this question, we will have made significant headway toward showing the world we would expect to see if Many-Worlds were true is the world we actually do see. That is, a world that is closely approximated by classical physics, except for quantum measurement events, during which the probability of obtaining any particular outcome is given by the Born rule. The issue of probabilities is often phrased as trying to derive why probabilities are given by amplitudes squared. But that’s not really the hard part. Squaring amplitudes in order to get probabilities is a very natural thing to do; there weren’t any worries that it might have been the wave function to the fifth power or anything like that. We learned that back in Chapter Five, when we used qubits to explain that the wave function can be thought of as a vector. That vector is like the hypotenuse of a right triangle, and the individual amplitudes are like the shorter sides of that triangle. The length of the vectors equals one, and by Pythagoras’s theorem that’s the sum of the squares of all the amplitudes. So “amplitudes squared” naturally look like probabilities: they’re positive numbers that add up to one. The deeper issue is why there is anything unpredictable about Everettian quantum mechanics at all, and if so, why there is any specific rule for attaching probabilities. In Many-Worlds, if you know the wave function at one moment in time, you can figure out precisely what it’s going to be at any other time, just by solving the Schr?dinger equation. There’s nothing chancy about it. So how in the world is such a picture supposed to recover the reality of our observations, where the decay of a nucleus or the measurement of a spin seems irreducibly random? Consider our favorite example of measuring the spin of an electron. Let’s say we start the electron in an equal superposition of spin-up and spin-down with respect to the vertical axis, and send it through a Stern-Gerlach magnet. Textbook quantum mechanics says that we have a 50 percent chance of the wave function collapsing to spin-up, and a 50 percent chance of it collapsing to spin-down. Many-Worlds, on the other hand, says there is a 100 percent chance of the wave function of the universe evolving from one world into two. True, in one of those worlds the experimenter will have seen spin-up and in the other they will have seen spin-down. But both worlds are indisputably there. If the question we’re asking is “What is the chance I will end up being the experimenter on the spin-up branch of the wave function?,” there doesn’t seem to be any answer. You will not be one or other experimenters; your current single self will evolve, with certainty, into both of them. How are we supposed to talk about probabilities in such a situation? It’s a good question. To answer it, we have get a bit philosophical, and think about what “probability” really means. You will not be surprised to learn that there are competing schools of thought on the issue of probability. Consider tossing a fair coin. “Fair” means that the coin will come up heads 50 percent of the time and tails 50 percent of the time. At least in the long run; nobody is surprised when you toss a coin twice and it comes up tails both times. This “in the long run” caveat suggests a strategy for what we might mean by probability. For just a few coin tosses, we wouldn’t be surprised at almost any outcome. But as we do more and more, we expect the total proportion of heads to come closer to 50 percent. So perhaps we can define the probability of getting heads as the fraction of times we actually would get heads, if the coin were tossed an infinite number of times. This notion of what we mean by probability is sometimes called frequentism, as it defines probability as the relative frequency of an occurrence in a very large number of trials. It matches pretty well with our intuitive notions of how probability functions when we toss coins, roll dice, or play cards. To a frequentist, probability is an objective notion, since it only depends on features of the coin (or whatever other system we’re talking about), not on us or our state of knowledge. Frequentism fits comfortably with the textbook picture of quantum mechanics and the Born rule. Maybe you don’t actually send an infinite number of electrons through a magnetic field to measure their spins, but you could send a very large number. (The Stern-Gerlach experiment is a favorite one to reproduce in undergraduate lab courses for physics majors, so over the years quite a number of spins have been measured this way.) We can gather enough statistics to convince ourselves that the probability in quantum mechanics really is just the wave function squared. Many-Worlds is a different story. Say we put an electron into an equal superposition of spin-up and spin-down, measure its spin, then repeat a large number of times. At every measurement, the wave function branches into a world with a spin-up result and one with a spin-down. Imagine that we record our results, labeling spin-up as “0” and spin-down as “1.” After fifty measurements, there will be a world where the record looks like 10101011111011001011001010100011101100011101000001. That seems random enough, and to obey the proper statistics: there are twenty-four 0’s, and twenty-six 1’s. Not exactly fifty-fifty, but as close as we should expect. But there will also be a world where every measurement returned spin-up, so that the record was just a list of fifty 0’s. And a world where all the spins were observed to be down, so the record was a list of fifty 1’s. And every other possible string of 0’s and 1’s. If Everett is right, there is a 100 percent probability that each possibility is realized in some particular world. In fact, I’ll make a confession: there really are such worlds. The random-looking string above wasn’t something I made up to look random, nor was it created by a classical random-number generator. It was actually created by a quantum random-number generator: a gizmo that makes quantum measurements and uses them to generate random sequences of 0’s and 1’s. According to Many-Worlds, when I generated that random number, the universe split into 250 copies (that’s 1,125,899,906,842,624, or approximately 1 quadrillion), each of which carries a slightly different number. If all of the copies of me in all of those different worlds stuck with the plan of including the obtained number into the text of this book, that means there are over a quadrillion different textual variations of Something Deeply Hidden out there in the wave function of the universe. For the most part the variations will be minor, just rearranging some 0’s and 1’s. But some of those poor versions of me were the unlucky ones who got all 0’s or all 1’s. What are they thinking right now? Probably they thought the random-number generator was broken. They certainly didn’t write precisely the text I am typing at this moment. Whatever I or the other copies of me might think about this situation, it’s quite different from the frequentist paradigm for probabilities. It doesn’t make too much sense to talk about the frequency in the limit of an infinite number of trials when every trial returns every result, just somewhere else in the wave function. We need to turn to another way of thinking about what probability is supposed to mean. Fortunately, an alternative approach to probability exists, and long pre-dates quantum mechanics. That’s the notion of epistemic probability, having to do with what we know rather than some hypothetical infinite number of trials. Consider the question “What is the probability that the Philadelphia 76ers will win the 2020 NBA Championship?” (I put a high value on that personally, but fans of other teams may disagree.) This isn’t the kind of event we can imagine repeating an infinite number of times; if nothing else, the basketball players would grow older, which would affect their play. The 2020 NBA Finals will happen only once, and there is a definite answer to who will win, even if we don’t know what it is. But professional oddsmakers have no qualms about assigning a probability to such situations. Nor do we, in our everyday lives; we are constantly judging the likelihood of different one-shot events, from getting a job we applied for to being hungry by seven p.m. For that matter we can talk about the probability of past events, even though there is a definite thing that happened, simply because we don’t know what that thing was—“I don’t remember what time I left work last Thursday, but it was probably between five p.m. and six p.m., since that’s usually when I head home.” What we’re doing in these cases is assigning “credences”—degrees of belief—to the various propositions under consideration. Like any probability, credences must range between 0 percent and 100 percent, and your total set of credences for the possible outcomes of a specified event should add up to 100 percent. Your credence in something can change as you gather new information; you might have a degree of belief that a word is spelled a certain way, but then you go look it up and find out the right answer. Statisticians have formalized this procedure under the label of Bayesian inference, after Rev. Thomas Bayes, an eighteenth-century Presbyterian minister and amateur mathematician. Bayes derived an equation showing how we should update our credences when we obtain new information, and you can find his formula on posters and T-shirts in statistics departments the world over. So there’s a perfectly good notion of “probability” that applies even when something is only going to happen once, not an infinite number of times. It’s a subjective notion, rather than an objective one; different people, in different states of knowledge, might assign different credences to the same outcomes for some event. That’s okay, as long as everyone agrees to follow the rules about updating their credences when they learn something new. In fact, if you believe in eternalism—the future is just as real as the past; we just haven’t gotten there yet—then frequentism is subsumed into Bayesianism. If you flip a random coin, the statement “The probability of the coin coming up heads is 50 percent” can be interpreted as “Given what I know about this coin and other coins, the best thing I can say about the immediate future of the coin is that it is equally likely to be heads or tails, even though there is some definite thing it will be.” It’s still not obvious that basing probability on our knowledge rather than on frequencies is really a step forward. Many-Worlds is a deterministic theory, and if we know the wave function at one time and the Schr?dinger equation, we can figure out everything that’s going to happen. In what sense is there anything that we don’t know, to which we can assign a credence given by the Born rule? There’s an answer that is tempting but wrong: that we don’t know “which world we will end up in.” This is wrong because it implicitly relies on a notion of personal identity that simply isn’t applicable in a quantum universe. What we’re up against here is what philosophers call our “folk” understanding of the world around us, and the very different view that is suggested by modern science. The scientific view should ultimately account for our everyday experiences. But we have no right to expect that the concepts and categories that have arisen over the course of pre-scientific history should maintain their validity as part of our most comprehensive picture of the physical world. A good scientific theory should be compatible with our experience, but it might speak an entirely different language. The ideas we readily deploy in our day-to-day lives emerge as useful approximations of certain aspects of a more complete story. A chair isn’t an object that partakes of a Platonic essence of chairness; it’s a collection of atoms arranged in a certain configuration that makes it sensible for us to include it in the category “chair.” We have no trouble recognizing that the boundaries of this category are somewhat fuzzy—does a sofa count? What about a barstool? If we take something that is indubitably a chair, and remove atoms from it one by one, it gradually becomes less and less chairlike, but there’s no hard-and-fast threshold that it crosses to jump suddenly from chair to non-chair. And that’s okay. We have no trouble accepting this looseness in our everyday speech. When it comes to the notion of “self,” however, we’re a little more protective. In our everyday experience, there’s nothing very fuzzy about our self. We grow and learn, our body ages, and we interact with the world in a variety of ways. But at any one moment I have no trouble identifying a specific person that is undeniably “myself.” Quantum mechanics suggests that we’re going to have to modify this story somewhat. When a spin is measured, the wave function branches via decoherence, a single world splits into two, and there are now two people where I used to be just one. It makes no sense to ask which one is “really me.” Likewise, before the branching happens, it makes no sense to wonder which branch “I” will end up in. Both of them have every right to think of themselves as “me.” In a classical universe, identifying a single individual as a person aging through time is generally unproblematic. At any moment a person is a certain arrangement of atoms, but it’s not the individual atoms that matter; to a large extent our atoms are replaced over time. What matters is the pattern that we form, and the continuity of that pattern, especially in the memories of the person under consideration. The new feature of quantum mechanics is the duplication of that pattern when the wave function branches. That’s no reason to panic. We just have to adjust our notion of personal identity through time to account for a situation that we never had reason to contemplate over the millennia of pre-scientific human evolution. As stubborn as our identity is, the concept of a single person extending from birth to death was always just a useful approximation. The person you are right now is not exactly the same as the person you were a year ago, or even a second ago. Your atoms are in slightly different locations, and some of your atoms might have been exchanged for new ones. (If you’re eating while reading, you might have more atoms now than you had a moment ago.) If we wanted to be more precise than usual, rather than talking about “you,” we should talk about “you at 5:00 p.m.,” “you at 5:01 p.m.,” and so on. The idea of a unified “you” is useful not because all of these different collections of atoms at different moments of time are literally the same, but because they are related to one another in an obvious way. They describe a real pattern. You at one moment descend from you at an earlier moment, through the evolution of the individual atoms within you and the possible addition or subtraction of a few of them. Philosophers have thought this through, of course; Derek Parfit, in particular, suggested that identity through time is a matter of one instance in your life “standing in Relation R” to another instance, where Relation R says that your future self shares psychological continuity with your past self. The situation in Many-Worlds quantum mechanics is exactly the same way, except that now more than one person can descend from a single previous person. (Parfit would have had no problem with that, and in fact investigated analogous situations featuring duplicator machines.) Rather than talking about “you at 5:01 p.m.,” we need to talk about “the person at 5:01 p.m. who descended from you at 5:00 p.m. and who ended up on the spin-up branch of the wave function,” and likewise for the person on the spin-down branch. Every one of those people has a reasonable claim to being “you.” None of them is wrong. Each of them is a separate person, all of whom trace their beginnings back to the same person. In Many-Worlds, the life-span of a person should be thought of as a branching tree, with multiple individuals at any one time, rather than as a single trajectory—much like a splitting amoeba. And nothing about this discussion really hinges on what we’re talking about being a person rather than a rock. The world duplicates, and everything within the world goes along with it. We’re now set up to confront this issue of probabilities in Many-Worlds. It might have seemed natural to think the proper question is “Which branch will I end up on?” But that’s not how we should be thinking about it. Think instead about the moment immediately after decoherence has occurred and the world has branched. Decoherence is an extraordinarily rapid process, generally taking a tiny fraction of a second to happen. From a human perspective, the wave function branches essentially instantaneously (although that’s just an approximation). So the branching happens first, and we only find out about it slightly later, for example, by looking to see whether the electron went up or down when it passed through the magnetic field. For a brief while, then, there are two copies of you, and those two copies are precisely identical. Each of them lives on a distinct branch of the wave function, but neither of them knows which one it is on. You can see where this is going. There is nothing unknown about the wave function of the universe—it contains two branches, and we know the amplitude associated with each of them. But there is something that the actual people on these branches don’t know: which branch they’re on. This state of affairs, first emphasized in the quantum context by physicist Lev Vaidman, is called self-locating uncertainty—you know everything there is to know about the universe, except where you are within it. That ignorance gives us an opening to talk about probabilities. In that moment after branching, both copies of you are subject to self-locating uncertainty, since they don’t know which branch they’re on. What they can do is assign a credence to being on one branch or the other. What should that credence be? There are two plausible ways to go. One is that we can use the structure of quantum mechanics itself to pick out a preferred set of credences that rational observers should assign to being on various branches. If you’re willing to accept that, the credences you’ll end up assigning are exactly those you would get from the Born rule. The fact that the probability of a quantum measurement outcome is given by the wave function squared is just what we would expect if that probability arose from credences assigned in conditions of self-locating uncertainty. (And if you’re willing to accept that and don’t want to be bothered with the details, you’re welcome to skip the rest of this chapter.) But there’s another school of thought, which basically denies that it makes sense to assign any definite credences at all. I can come up with all sorts of wacky rules for calculating probabilities for being on one branch of the wave function or another. Maybe I assign higher probability to being on a branch where I’m happier, or where spins are always pointing up. Philosopher David Albert has (just to highlight the arbitrariness, not because he thinks it’s reasonable) suggested a “fatness measure,” where the probability is proportional to the number of atoms in your body. There’s no reasonable justification for doing so, but who’s to stop me? The only “rational” thing to do, according to this attitude, is to admit that there’s no right way to assign credences, and therefore refuse to do so. That is a position one is allowed to take, but I don’t think it’s the best one. If Many-Worlds is correct, we are going to find ourselves in situations of self-locating uncertainty whether we like it or not. And if our goal is to come up with the best scientific understanding of the world, that understanding will necessarily involve an assignment of credences in these situations. After all, part of science is predicting what will be observed, even if only probabilistically. If there were an arbitrary collection of ways to assign credences, and each of them seemed just as reasonable as the other, we would be stuck. But if the structure of the theory points unmistakably to one particular way to assign such credences, and that way is in agreement with our experimental data, we should adopt it, congratulate ourselves on a job well done, and move on to other problems. Let’s say we buy into the idea that there could be a clearly best way to assign credences when we don’t know which branch of the wave function we’re on. Before, we mentioned that, at heart, the Born rule is just Pythagoras’s theorem in action. Now we can be a little more careful and explain why that’s the rational way to think about credences in the presence of self-locating uncertainty. This is an important question, because if we didn’t already know about the Born rule, we might think that amplitudes are completely irrelevant to probabilities. When you go from one branch to two, for example, why not just assign equal probability to each, since they’re two separate universes? It’s easy to show that this idea, known as branch counting, can’t possibly work. But there’s a more restricted version, which says that we should assign equal probabilities to branches when they have the same amplitude. And that, wonderfully, turns out to be all we need to show that when branches have different amplitudes, we should use the Born rule. Let’s first dispatch the wrong idea of branch counting before turning to the strategy that actually works. Consider a single electron whose vertical spin has been measured by an apparatus, so that decoherence and branching has occurred. Strictly speaking, we should keep track of the states of the apparatus, observer, and environment, but they just go along for the ride, so we won’t write them explicitly. Let’s imagine that the amplitudes for spin-up and spin-down aren’t equal, but rather we have an unbalanced state ?, with unequal amplitudes for the two directions. Those numbers outside the different branches are the corresponding amplitudes. Since the Born rule says the probability equals the amplitude squared, in this example we should have a 1/3 probability of seeing spin-up and a 2/3 probability of seeing spin-down. Imagine that we didn’t know about the Born rule, and were tempted to assign probabilities by simple branch counting. Think about the point of view of the observers on the two branches. From their perspective, those amplitudes are just invisible numbers multiplying their branch in the wave function of the universe. Why should they have anything to do with probabilities? Both observers are equally real, and they don’t even know which branch they’re on until they look. Wouldn’t it be more rational, or at least more democratic, to assign them equal credences? The obvious problem with that is that we’re allowed to keep on measuring things. Imagine that we agreed ahead of time that if we measured spin-up, we would stop there, but if we measured spin-down, an automatic mechanism would quickly measure another spin. This second spin is in a state of spin-right, which we know can be written as a superposition of spin-up and spin-down. Once we’ve measured it (only on the branch where the first spin was down), we have three branches: one where the first spin was up, one where we got down and then up, and one where we got down twice in a row. The rule of “assign equal probability to each branch” would tell us to assign a probability of 1/3 to each of these possibilities. That’s silly. If we followed that rule, the probability of the original spin-up branch would suddenly change when we did a measurement on the spin-down branch, going from 1/2 to 1/3. The probability of observing spin-up in our initial experiment shouldn’t depend on whether someone on an entirely separate branch decides to do another experiment later on. So if we’re going to assign credences in a sensible way, we’ll have to be a little more sophisticated than simple branch counting. Instead of simplistically saying “Assign equal probability to each branch,” let’s try something more limited in scope: “Assign equal probability to branches when they have equal amplitudes.” For example, a single spin in a spin-right state can be written as an equal superposition of spin-up and spin-down. This new rule says we should give 50 percent credence to being on either the spin-up or spin-down branches, were we to observe the spin along the vertical axis. That seems reasonable, as there is a symmetry between the two choices; really, any reasonable rule should assign them equal probability.* One nice thing about this more modest proposal is that no inconsistency arises with repeated measurements. Doing an extra measurement on one branch but not the other would leave us with branches that have unequal amplitudes again, so the rule doesn’t seem to say anything at all. But in fact it’s way better than that. If we start with this simple equal-amplitudes-imply-equal-probabilities rule, and ask whether that is a special case of a more general rule that never leads to inconsistencies, we end up with a unique answer. And that answer is the Born rule: probability equals amplitude squared. We can see this by returning to our unbalanced case, with one amplitude equal to the square root of 1/3 and the other equal to the square root of 2/3. This time we’ll explicitly include a second horizontal spin-right qubit from the start. At first, this second qubit just goes along for the ride. Insisting on equal probability for equal amplitudes doesn’t tell us anything yet, since the amplitudes are not equal. But we can play the same game we did before, measuring the second spin along the vertical axis if the first spin is down. The wave function evolves into three components, and we can figure out what their amplitudes are by looking back at the decomposition of a spin-right state into vertical spins above. Multiplying the square root of 2/3 by the square root of 1/2 gives the square root of 1/3, so we get three branches, all with equal amplitudes. Since the amplitudes are equal, we can now safely assign them equal probabilities. Since there are three of them, that’s 1/3 each. And if we don’t want the probability of one branch to suddenly change when something happens on another branch, that means we should have assigned probability 1/3 to the spin-up branch even before we did the second measurement. But 1/3 is just the square of the amplitude of that branch—exactly as the Born rule would predict. There are a couple of lingering worries here. You may object that we considered an especially simple example, where one probability was exactly twice the other one. But the same strategy works whenever we can subdivide our states into the right number of terms so that all of the amplitudes are equal in magnitude. That works whenever the amplitudes squared are all rational numbers (one integer divided by another one), and the answer is the same: probability equals amplitude squared. There are plenty of irrational numbers out there, but as a physicist if you’re able to prove that something works for all rational numbers, you hand the problem to a mathematician, mumble something about “continuity,” and declare that your work here is done. We can see Pythagoras’s theorem at work. It’s the reason why a branch that is bigger than another branch by the square root of two can split into two branches of equal size to the other one. That’s why the hard part isn’t deriving the actual formula, it’s providing a solid grounding for what probability means in a deterministic theory. Here we’ve explored one possible answer: it comes from the credences we have for being on different branches of the wave function immediately after the wave function branches. You might worry, “But I want to know what the probability of getting a result will be even before I do the measurement, not just afterward. Before the branching, there’s no uncertainty about anything—you’ve already told me it’s not right to wonder which branch I’m going to end up on. So how do I talk about probabilities before the measurement is made?” Never fear. You’re right, imaginary interlocutor, it makes no sense to worry about which branch you’ll end up on. Rather, we know with certainty that there will be two descendants of your present state, and each of them will be on a different branch. They will be identical, and they’ll be uncertain as to which branch they’re on, and they should assign credences given by the Born rule. But that means that all of your descendants will be in exactly the same epistemic position, assigning Born-rule probabilities. So it makes sense that you go ahead and assign those probabilities right now. We’ve been forced to shift the meaning of what probability is from a simple frequentist model to a more robust epistemic picture, but how we calculate things and how we act on the basis of those calculations goes through exactly as before. That’s why physicists have been able to do interesting work while avoiding these subtle questions all this time. Intuitively, this analysis suggests that the amplitudes in a quantum wave function lead to different branches having a different “weight,” which is proportional to the amplitude squared. I wouldn’t want to take that mental image too literally, but it provides a concrete picture that helps us make sense of probabilities, as well as of other issues like energy conservation that we’ll talk about later. Weight of a branch = |Amplitude of that branch|2 When there are two branches with unequal amplitudes, we say that there are only two worlds, but they don’t have equal weight; the one with higher amplitude counts for more. The weights of all the branches of any particular wave function always add up to one. And when one branch splits into two, we don’t simply “make more universe” by duplicating the existing one; the total weight of the two new worlds is equal to that of the single world we started with, and the overall weight stays the same. Worlds get thinner as branching proceeds. This isn’t the only way to derive the Born rule in the Many-Worlds theory. A strategy that is even more popular in the foundations-of-physics community appeals to decision theory—the rules by which a rational agent makes choices in an uncertain world. This approach was pioneered in 1999 by David Deutsch (one of the physicists who had been impressed by Hugh Everett at the Texas meeting in 1977), and later made more rigorous by David Wallace. Decision theory posits that rational agents attach different amounts of value, or “utility,” to different things that might happen, and then prefer to maximize the expected amount of utility—the average of all the possible outcomes, weighted by their probabilities. Given two outcomes A and B, an agent that assigns exactly twice the utility to B as to A should be indifferent between A happening with certainty and B happening with 50 percent probability. There are a bunch of reasonable-sounding axioms that any good assignment of utilities should obey; for example, if an agent prefers A to B and also prefers B to C, they should definitely prefer A to C. Anyone who goes through life violating the axioms of decision theory is deemed to be irrational, and that’s that. To use this framework in the context of Many-Worlds, we ask how a rational agent should behave, knowing that the wave function of the universe was about to branch and knowing what the amplitudes of the different branches were going to be. For example, an electron in an equal superposition of spin-up and spin-down is going to travel through a Stern-Gerlach magnet and have its spin be measured. Someone offers to pay you $2 if the result is spin-up, but only if you promise to pay them $1 if the result is spin-down. Should you take the offer? If we trust the Born rule, the answer is obviously yes, since our expected payoff is 0.5($2) 0.5(-$1) = $0.50. But we’re trying to derive the Born rule here; how are you supposed to find an answer knowing that one of your future selves will be $2 richer but another one will be $1 poorer? (Let’s assume you’re sufficiently well-off that gaining or losing a dollar is something you care about, but not life-changing.) The manipulations are trickier here than in the previous case where we were explaining probabilities as credences in a situation of self-locating uncertainty, so we won’t go through them explicitly, but the basic idea is the same. First we consider a case where the amplitudes on two different branches are equal, and we show that it’s rational to calculate your expected value as the simple average of the two different utilities. Then suppose we have an unbalanced state like ? above, and I ask you to give me $1 if the spin is measured to be up and promise to give you $1 if the spin is down. By a bit of mathematical prestidigitation, we can show that your expected utility in this situation is exactly the same as if there were three possible outcomes with equal amplitudes, such that you give me $1 for one outcome and I give you $1 for the other two. In that case, the expected value is the average of the three different outcomes. At the end of the day, a rational agent in an Everettian universe acts precisely as if they live in a nondeterministic universe where probabilities are given by the Born rule. Acting otherwise would be irrational, if we accept the various plausible-seeming axioms about what it means to be rational in this context. One could stubbornly maintain that it’s not good enough to show that people should act “as if” something is true; it needs to actually be true. That’s missing the point a little bit. Many-Worlds quantum mechanics presents us with a dramatically different view of reality from an ordinary one-world view with truly random events. It’s unsurprising that some of our most natural-seeming notions are going to have to change along with it. If we lived in the world of textbook quantum mechanics, where wave-function collapse was truly random and obeyed the Born rule, it would be rational to calculate our expected utility in a certain way. Deutsch and Wallace have shown that if we live in a deterministic Many-Worlds universe, it is rational to calculate our expected utility in exactly the same way. From this perspective, that’s what it means to talk about probability: the probabilities of different events actually occurring are equivalent to the weighting we give those events when we calculate our expected utility. We should act exactly as if the probabilities we’re calculating apply to a single chancy universe; but they are still real probabilities, even though the universe is a little richer than that.