From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  In other words, the Boltzmann-brain argument makes an implicit assumption: that we are somehow “typical observers” in the universe, and that therefore we should make predictions by asking what most observers would see.190 That sounds innocuous, even humble. But upon closer inspection, it leads to conclusions that seem stronger than we can really justify.

  Imagine we have two theories of the universe that are identical in every way, except that one predicts that an Earth-like planet orbiting the star Tau Ceti is home to a race of 10 trillion intelligent lizard beings, while the other theory predicts there are no intelligent beings of any kind in the Tau Ceti system. Most of us would say that we don’t currently have enough information to decide between these two theories. But if we are truly typical observers in the universe, the first theory strongly predicts that we are more likely to be lizards on the planet orbiting Tau Ceti, not humans here on Earth, just because there are so many more lizards than humans. But that prediction is not right, so we have apparently ruled out the existence of that many observers without collecting any data at all about what’s actually going on in the Tau Ceti system.

  Assuming we are typical might seem like a self-effacing move on our part, but it actually amounts to an extremely strong claim about what happens throughout the rest of the universe. Not only “we are typical observers,” but “typical observers are like us.” Put that way, it seems like a stronger assumption than we have any right to make. (In the literature this is known as the “Presumptuous Philosopher Problem.”) So perhaps we shouldn’t be comparing the numbers of different kinds of observers in the universe at all; we should only ask whether a given theory predicts that observers like us appear somewhere, and if they do we should think of the theory as being consistent with the data. If that were the right way to think about it, we wouldn’t have any reason to reject the Boltzmann-Lucretius scenario. Even though most observers would be alone in the universe, some would find themselves in regions like ours, so the theory would be judged to be in agreement with our experience.191

  The difficulty with this minimalist approach is that it offers us too little handle on what is likely to happen in the universe, instead of too much. Statistical mechanics relies on the Principle of Indifference—the assumption that all microstates consistent with our current macrostate are equally likely, at least when it comes to predicting the future. That’s essentially an assumption of typicality: Our microstate is likely to be a typical member of our macrostate. If we’re not allowed to assume that, all sorts of statistical reasoning suddenly become inaccessible. We can’t say that an ice cube is likely to melt in a glass of warm water, because in an eternal universe there will occasionally be times when the opposite happens. We seem to have taken our concerns about typicality too far.

  Instead, we should aim for a judicious middle ground. It’s too much to ask that we are typical among all observers in the universe, because that’s making a strong claim about parts of the universe we’ve never observed. But we can at least say that we are typical among observers exactly like us—that is, observers with the basic physiology and the same set of memories that we have, the same coarse-grained experience of the universe.192 That assumption doesn’t allow us to draw any unwarranted conclusions about the possible existence of other kinds of intelligent beings elsewhere in the universe. But it is more than enough to rule out the Boltzmann-Lucretius scenario. If the universe fluctuates around thermal equilibrium for all eternity, not only will most observers appear all by themselves from the surrounding chaos, but the same is true for the subset of observers with precisely the features that you or I have—complete with our purported memories of the past. Those memories will generally be false, and fluctuating into them is very unlikely, but it’s still much more unlikely than fluctuating the entire universe. Even this minimal necessary condition for carrying out statistical reasoning—we should take ourselves to be chosen randomly from the set of observers exactly like us—is more than enough to put the Boltzmann-Lucretius scenario to rest.

  The universe we observe is not a fluctuation—at least, to be more careful, a statistical fluctuation in an eternal universe that spends most of its time in equilibrium. So that’s what the universe is not; what it is, we still have to work out.

  ENDINGS

  On the evening of September 5, 1906, Ludwig Boltzmann took a piece of cord, tied it to a curtain rod in the hotel room where he was vacationing in Italy with his family, and hanged himself. His body was discovered by his daughter Emma when she returned to their room that evening. He was sixty-two years old.

  The reasons for Boltzmann’s suicide remain unclear. Some have suggested that he was despondent over the failure of his ideas concerning atomic theory to gain wider acceptance. But, while many German-speaking scientists of the time remained skeptical about atoms, kinetic theory had become standard throughout much of the world, and Boltzmann’s status as a major scientist was unquestioned in Austria and Germany. Boltzmann had been suffering from health problems and was prone to fits of depression; it was not the first time he had attempted suicide.

  But his depression was intermittent; only months before his death, he had written an engaging and high-spirited account of his previous year’s trip to America to lecture at the University of California at Berkeley, and circulated it among his friends. He referred to California as “Eldorado,” but found American water undrinkable, and would drink only beer and wine. This was problematic, as the Temperance movement was strong in America at the time, and Berkeley in particular was completely dry; a recurring theme in Boltzmann’s account is his attempts to smuggle wine into various forbidden places.193 We will probably never know what mixture of failing health, depression, and scientific controversy contributed to his ultimate act.

  On the question of the existence of atoms and their utility in understanding the properties of macroscopic objects, any lingering doubts that Boltzmann was right were rapidly dissipating when he died. One of Albert Einstein’s papers in his “miraculous year” of 1905 was an explanation of Brownian motion (the seemingly random motion of small particles suspended in air) in terms of collisions with individual atoms; most remaining skepticism on the part of physicists was soon swept away.

  Questions about the nature of entropy and the Second Law remain with us, of course. When it comes to explaining the low entropy of our early universe, we won’t ever be able to say, “Boltzmann was right,” because he suggested a number of different possibilities without ever settling on one in particular. But the terms of the debate were set by him, and we’re still arguing over the questions that puzzled him more than a century ago.

  11

  QUANTUM TIME

  Sweet is by convention, bitter by convention, hot by convention, cold by convention, color by convention; in truth there are but atoms and the void.

  —Democritus194

  Many people who have sat through introductory physics courses in high school or college might disagree with the claim “Newtonian mechanics makes intuitive sense to us.” They may remember the subject as a bewildering merry-go-round of pulleys and vectors and inclined planes, and think that “intuitive sense” is the last thing that Newtonian mechanics should be accused of making.

  But while the process of actually calculating something within the framework of Newtonian mechanics—doing a homework problem, or getting astronauts to the moon—can be ferociously complicated, the underlying concepts are pretty straightforward. The world is made of tangible things that we can observe and recognize: billiard balls, planets, pulleys. These things exert forces, or bump into one another, and their motions change in response to those influences. If Laplace’s Demon knew all of the positions and momenta of every particle in the universe, it could predict the future and the past with perfect fidelity; we know that this is outside of our capabilities, but we can imagine knowing the positions and momenta of a few billiard balls on a frictionless table, and at least in principle we can imagine doing the math. After that it’s just a matt
er of extrapolation and courage to encompass the entire universe.

  Newtonian mechanics is usually referred to as “classical” mechanics by physicists, who want to emphasize that it’s not just a set of particular rules laid down by Newton. Classical mechanics is a way of thinking about the deep structure of the world. Different types of things—baseballs, gas molecules, electromagnetic waves—will follow different specific rules, but those rules will share the same pattern. The essence of that pattern is that everything has some kind of “position,” and some kind of “momentum,” and that information can be used to predict what will happen next.

  This structure is repeated in a variety of contexts: Newton’s own theory of gravitation, Maxwell’s nineteenth-century theory of electricity and magnetism, and Einstein’s general relativity all fit into the classical framework. Classical mechanics isn’t a particular theory; it’s a paradigm, a way of conceptualizing what a physical theory is, and one that has demonstrated an astonishing range of empirical success. After Newton published his 1687 masterwork, Philosophiæ Naturalis Principia Mathematica, it became almost impossible to imagine doing physics any other way. The world is made of things, characterized by positions and momenta, pushed about by certain sets of forces; the job of physics was to classify the kinds of things and figure out what the forces were, and we’d be done.

  But now we know better: Classical mechanics isn’t correct. In the early decades of the twentieth century, physicists trying to understand the behavior of matter on microscopic scales were gradually forced to the conclusion that the rules would have to be overturned and replaced with something else. That something else is quantum mechanics, arguably the greatest triumph of human intelligence and imagination in all of history. Quantum mechanics offers an image of the world that is radically different from that of classical mechanics, one that scientists never would have seriously contemplated if the experimental data had left them with any other choice. Today, quantum mechanics enjoys the status that classical mechanics held at the dawn of the twentieth century: It has passed a variety of empirical tests, and most researchers are convinced that the ultimate laws of physics are quantum mechanical in nature.

  But despite its triumphs, quantum mechanics remains somewhat mysterious. Physicists are completely confident in how they use quantum mechanics—they can build theories, make predictions, and test against experiments, and there is never any ambiguity along the way. Nevertheless, we’re not completely sure we know what quantum mechanics really is. There is a respectable field of intellectual endeavor, occupying the time of a substantial number of talented scientists and philosophers, that goes under the name “interpretations of quantum mechanics.” A century ago, there was no such field as “interpretations of classical mechanics”—classical mechanics is perfectly straightforward to interpret. We’re still not sure what is the best way to think and talk about quantum mechanics.

  This interpretational anxiety stems from the single basic difference between quantum mechanics and classical mechanics, which is both simple and world shattering in its implications:

  According to quantum mechanics, what we can observe about the world is only a tiny subset of what actually exists.

  Attempts at explaining this principle often water it down beyond recognition. “It’s like that friend of yours who has such a nice smile, except when you try to take his picture it always disappears.” Quantum mechanics is a lot more profound than that. In the classical world, it might be difficult to obtain a precise measurement of some quantity; we need to be very careful not to disturb the system we’re looking at. But there is nothing in classical physics that prevents us from being careful. In quantum mechanics, on the other hand, there is an unavoidable obstacle to making complete and nondisruptive observations of a physical system. It simply can’t be done, in general. What exactly happens when you try to observe something, and what actually counts as a “measurement”—those are the locus of the mystery. This is what is helpfully known as the “measurement problem,” much as having an automobile roll off a cliff and smash into pieces on the rocks hundreds of feet below might be known as “car trouble.” Successful physical theories aren’t supposed to have ambiguities like this; the very first thing we ask about them is that they be clearly defined. Quantum mechanics, despite all its undeniable successes, isn’t there yet.

  None of which should be taken to mean that all hell has broken loose, or that the mysteries of quantum mechanics offer an excuse to believe whatever you want. In particular, quantum mechanics doesn’t mean you can change reality just by thinking about it, or that modern physics has rediscovered ancient Buddhist wisdom. 195 There are still rules, and we know how the rules operate in the regimes of interest to our everyday lives. But we’d like to understand how the rules operate in every conceivable situation.

  Most modern physicists deal with the problems of interpreting quantum mechanics through the age-old strategy of “denial.” They know how the rules operate in cases of interest, they can put quantum mechanics to work in specific circumstances and achieve amazing agreement with experiment, and they don’t want to be bothered with pesky questions about what it all means or whether the theory is perfectly well-defined. For our purposes in this book, that is often a pretty good strategy. The problem of the arrow of time was there for Boltzmann and his collaborators before quantum mechanics was ever invented; we can go very far talking about entropy and cosmology without worrying about the details of quantum mechanics.

  At some point, however, we need to face the music. The arrow of time is, after all, a fundamental puzzle, and it’s possible that quantum mechanics will play a crucial role in resolving that puzzle. And there’s something else of more direct interest: That process of measurement, where all of the interpretational tangles of quantum mechanics are to be found, has the remarkable property that it is irreversible . Alone among all of the well-accepted laws of physics, quantum measurement is a process that defines an arrow of time: Once you do it, you can’t undo it. And that’s a mystery.

  It’s very possible that this mysterious irreversibility is of precisely the same character as the mysterious irreversibility in thermodynamics, as codified in the Second Law: It’s a consequence of making approximations and throwing away information, even though the deep underlying processes are all individually reversible. I’ll be advocating that point of view in this chapter. But the subject remains controversial among the experts. The one sure thing is that we have to confront the measurement problem head-on if we’re interested in the arrow of time.

  THE QUANTUM CAT

  Thanks to the thought-experiment stylings of Erwin Schrödinger, it has become traditional in discussions of quantum mechanics to use cats as examples.196 Schrödinger’s cat was proposed to help illustrate the difficulties involved in the measurement problem, but we’re going start with the basic features of the theory before diving into the subtleties. And no animals will be harmed in our thought experiments.

  Imagine your cat, Miss Kitty, has two favorite places in your house: on the sofa and under the dining room table. In the real world, there are an infinite number of positions in space that could specify the location of a physical object such as a cat; likewise, there are an infinite number of momenta, even if your cat tends not to move very fast. We’re going to be simplifying things dramatically, in order to get at the heart of quantum mechanics. So let’s imagine that we can completely specify the state of Miss Kitty—as it would be described in classical mechanics—by saying whether she is on the sofa or under the table. We’re throwing out any information about her speed, or any knowledge of exactly what part of the sofa she’s on, and we’re disregarding any possible positions that are not “sofa” or “table.” From the classical point of view, we are simplifying Miss Kitty down to a two-state system. (Two-state systems actually exist in the real world; for example, the spin of an electron or photon can either point up or point down. The quantum state of a two-state system is described by a “qubit.”)
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br />   Here is the first major difference between quantum mechanics and classical mechanics: In quantum mechanics, there is no such thing as “the location of the cat.” In classical mechanics, it may happen that we don’t know where Miss Kitty is, so we may end up saying things like “I think there’s a 75 percent chance that she’s under the table.” But that’s a statement about our ignorance, not about the world; there really is a fact of the matter about where the cat is, whether we know it or not.

  In quantum mechanics, there is no fact of the matter about where Miss Kitty (or anything else) is located. The space of states in quantum mechanics just doesn’t work that way. Instead, the states are specified by something called a wave function . And the wave function doesn’t say things like “the cat is on the sofa” or “the cat is under the table.” Rather, it says things like “if we were to look, there would be a 75 percent chance that we would find the cat under the table, and a 25 percent chance that we would find the cat on the sofa.”

  This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. If the wave function tells us there is a 75 percent chance of observing the cat under the table and a 25 percent chance of observing her on the sofa, that does not mean there is a 75 percent chance that the cat is under the table and a 25 percent chance that she is on the sofa. There is no such thing as “where the cat is.” Her quantum state is described by a superposition of the two distinct possibilities we would have in classical mechanics. It’s not even that “they are both true at once”; it’s that there is no “true” place where the cat is. The wave function is the best description we have of the reality of the cat.