The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos - Brian Greene (2011)
Chapter 8. The Many Worlds of Quantum Measurement
The Quantum Multiverse
The most reasonable assessment of the parallel universe theories we’ve so far encountered is that the jury is out. An infinite spatial expanse, eternal inflation, braneworlds, cyclical cosmology, string theory’s landscape—these intriguing ideas have emerged from a range of scientific developments. But each remains tentative, as do the multiverse proposals each has spawned. While many physicists are willing to offer their opinions, pro and con, regarding these multiverse schemes, most recognize that future insights—theoretical, experimental, and observational—will determine whether any become part of the scientific canon.
The multiverse we’ll now take up, emerging from quantum mechanics, is viewed very differently. Many physicists have already reached a final verdict on this particular multiverse. The thing is, they haven’t all reached the same verdict. The differences come down to the deep and as yet unresolved problem of navigating from the probabilistic framework of quantum mechanics to the definite reality of common experience.
In 1954, nearly thirty years after the foundations of quantum theory had been set down by luminaries like Niels Bohr, Werner Heisenberg, and Erwin Schrödinger, an unknown graduate student from Princeton University named Hugh Everett III came to a startling realization. His analysis, which focused on a gaping hole that Bohr, the grand master of quantum mechanics, had danced around but failed to fill, revealed that a proper understanding of the theory might require a vast network of parallel universes. Everett’s was one of the earliest mathematically motivated insights suggesting that we might be part of a multiverse.
Everett’s approach, which in time would be called the Many Worlds interpretation of quantum mechanics, has had a checkered history. In January 1956, having worked out the mathematical consequences of his new proposal, Everett submitted a draft of his thesis to John Wheeler, his doctoral adviser. Wheeler, one of twentieth-century physics’ most celebrated thinkers, was thoroughly impressed. But that May, when Wheeler visited Bohr in Copenhagen and discussed Everett’s ideas, the reception was icy. Bohr and his followers had spent decades refining their view of quantum mechanics. To them, the questions Everett raised, and the outlandish ways in which he thought they should be addressed, were of little merit.
Wheeler held Bohr in the highest regard, and so placed particular value on appeasing his elder colleague. In response to the criticisms, Wheeler delayed granting Everett his Ph.D. and compelled him to modify the thesis substantially. Everett was to cut out those parts blatantly critical of Bohr’s methodology and emphasize that his results were meant to clarify and extend the conventional formulation of quantum theory. Everett resisted, but he had already accepted a job in the Defense Department (where he would soon play an important behind-the-scenes role in the Eisenhower and Kennedy administrations’ nuclear-weapons policy) that required a doctorate, so he reluctantly acquiesced. In March of 1957, Everett submitted a substantially trimmed-down version of his original thesis; by April it was accepted by Princeton as fulfilling his remaining requirements, and in July it was published in the Reviews of Modern Physics.1 But with Everett’s approach to quantum theory having already been dismissed by Bohr and his entourage, and with the muting of the grander vision articulated in the original thesis, the paper was ignored.2
Ten years later, the renowned physicist Bryce DeWitt plucked Everett’s work from obscurity. DeWitt, who was inspired by the results of his graduate student Neill Graham that further developed Everett’s mathematics, became a vocal proponent of the Everettian rethinking of quantum theory. Besides publishing a number of technical papers that brought Everett’s insights to a small but influential community of specialists, in 1970 DeWitt wrote a general level summary for Physics Today that reached a much broader scientific audience. And unlike Everett’s 1957 paper, which shied away from talk of other worlds, DeWitt underscored this feature, highlighting it with an unusually candid reflection regarding his “shock” on learning Everett’s conclusion that we are part of an enormous “multiworld.” The article generated a significant response in a physics community that had become more receptive to tampering with orthodox quantum ideology and ignited a debate, still going on, that concerns the nature of reality when, as we believe they do, quantum laws hold sway.
Let me set the stage.
The upheaval in understanding that took place between roughly 1900 and 1930 resulted in a ferocious assault on intuition, common sense, and the well-accepted laws that the new vanguard soon began calling “classical physics”—a term that carries the weight and respect given to a picture of reality that is at once venerable, immediate, satisfying, and predictive. Tell me how things are now, and I’ll use the laws of classical physics to predict how things will be at any moment in the future, or how they were at any moment in the past. Subtleties such as chaos (in the technical sense: slight changes in how things are now can result in huge errors in the predictions) and the complexity of the equations challenge the practicality of this program in all but the simplest situations, but the laws themselves are unwavering in their viselike grip on a definitive past and future.
The quantum revolution required that we give up the classical perspective because new results established that it was demonstrably wrong. For the motion of big objects like the earth and the moon, or of everyday objects like rocks and balls, the classical laws do a fine job of prediction and description. But pass into the microworld of molecules, atoms, and subatomic particles and the classical laws fail. In contradiction of the very heart of classical reasoning, if you run identical experiments on identical particles that have been set up identically, you will generally not get identical results.
Imagine, for example, that you have 100 identical boxes, each containing one electron, set up according to an identical laboratory procedure. After exactly 10 minutes, you and 99 cohorts measure the positions of each of the 100 electrons. Despite what Newton, Maxwell, or even a young Einstein would have anticipated—would likely have been willing to bet their lives on—the 100 measurements won’t yield the same result. In fact, at first blush the results will look random, with some electrons found near their box’s front lower left corner, some near the back upper right, some around the middle, and so on.
The regularities and patterns that make physics a rigorous and predictive discipline become apparent only if you run this same experiment, with 100 boxed electrons, over and over again. Were you to do so, here’s what you’d find. If your first batch of 100 measurements found 27 percent of the electrons near the lower left corner, 48 percent near the upper right corner, and 25 percent near the middle, then the second batch will yield a very similar distribution. So will the third batch, the fourth, and those that follow. The regularity, therefore, isn’t evident in any single measurement; you can’t predict where any given electron will be. Instead, the regularity is found in the statistical distribution of many measurements. The regularity, that is, speaks to the likelihood, or probability, of finding an electron at any particular location.
The breathtaking achievement of quantum mechanics’ founders was to develop a mathematical formalism that dispensed with the absolute predictions intrinsic to classical physics and instead predicted such probabilities. Working from an equation Schrödinger published in 1926 (and an equivalent though somewhat more awkward equation Heisenberg wrote down in 1925), physicists can input the details of how things are now, and then calculate the probability that they will be one way, or another, or another still, at any moment in the future.
But don’t be misled by the simplicity of my little electron example. Quantum mechanics applies not just to electrons but to all types of particles, and it tells us not only about their positions but about also their velocities, their angular momenta, their energies, and how they behave in a wide range of situations, from the barrage of neutrinos now wafting through your body, to the frenzied atomic fusions taking place in the cores of distant stars. Across such a broad sweep, the probabilistic predictions of quantum mechanics match experimental data. Always. In the more than eighty years since these ideas were developed, there has not been a single verifiable experiment or astrophysical observation whose results conflict with quantum mechanical predictions.
For a generation of physicists to have confronted such a radical departure from the intuitions formed out of thousands of years of collective experience, and in response to have recast reality within a wholly new framework based on probabilities, is a virtually unmatched intellectual achievement. Yet one uncomfortable detail has been hovering over quantum mechanics since its inception—a detail that eventually opened a pathway to parallel universes. To understand it, we need to look a little more closely at the quantum formalism.
The Puzzle of Alternatives
In April 1925, during an experiment at Bell Labs undertaken by two American physicists, Clinton Davisson and Lester Germer, a glass tube containing a hot chunk of nickel suddenly exploded. Davisson and Germer had been spending their days firing beams of electrons at specimens of nickel to investigate various aspects of the metal’s atomic properties; the equipment failure was a nuisance, albeit one all too familiar in experimental work. On cleaning up the glass shards, Davisson and Germer noticed that the nickel had been tarnished during the explosion. Not a big deal, of course. All they had to do was heat the sample, vaporize the contaminant, and start again. And so they did. But that choice, to clean the sample instead of opting for a new one, proved fortuitous. When they directed the electron beam at the newly cleaned nickel, the results were completely different from any they or anyone else had ever encountered. By 1927, it was clear that Davisson and Germer had established a vital feature of the rapidly developing quantum theory. And within a decade, their serendipitous discovery would be honored with the Nobel Prize.
Although Davisson and Germer’s demonstration predates talking movies and the Great Depression, it’s still the most widely used method for introducing quantum theory’s essential ideas. Here’s how to think about it. When Davisson and Germer heated the tarnished sample, they caused numerous small nickel crystals to meld into fewer larger ones. In turn, their electron beam no longer reflected off a highly uniform surface of nickel but instead bounced back from a few concentrated locations where the larger nickel crystals were centered. A simplified version of their experiment, the setup of Figure 8.1, in which electrons are fired at a barrier containing two narrow slits, highlights the essential physics. Electrons emanating from one slit or the other are like electrons bouncing back from one nickel crystal or its neighbor. Modeled in this way, Davisson and Germer were carrying out the first version of what’s now called the double-slit experiment.
To grasp Davisson and Germer’s startling result, imagine closing off either the left or the right slit and capturing the electrons that pass through, one by one, on a detector screen. After many such electrons are fired, the detector screens will look like those in Figure 8.2a and Figure 8.2b. A rational, nonquantum trained mind would therefore expect that when both slits are open, the data would be an amalgam of these two results. But the astounding fact is that this is not what happens. Instead, Davisson and Germer found data, much like those illustrated in Figure 8.2c, consisting of light and dark bands indicating a series of positions where electrons do and do not land.
Figure 8.1 The essence of the Davisson and Germer experiment is captured by the “double-slit” setup in which electrons are fired at a barrier that has two narrow slits. In the Davisson and Germer experiment, two streams of electrons are produced when incident electrons bounce off neighboring nickel crystals; in the double-slit experiment, two similar streams are produced by electrons that pass through the neighboring slits.
These results deviate from expectations in a way that’s particularly peculiar. The dark bands are locations where electrons are copiously detected if only the left slit or only the right slit is open (the corresponding regions in Figures 8.2a and 8.2b are bright), but which are apparently unreachable when both slits are available. The presence of the left slit thus changes the possible landing locations of electrons passing through the right slit, and vice versa. Which is thoroughly perplexing. On the scale of a tiny particle like an electron, the distance between the slits is huge. So when the electron passes through one slit, how could the presence or absence of the other have any effect, let alone the dramatic influence evident in the data? It’s as if for many years you happily enter an office building using one door, but when the management finally adds a second door on the building’s other side, you can no longer reach your office.
What are we to make of this? The double-slit experiment leads us inescapably to a conclusion hard to fathom. Regardless of which slit it passes through, each individual electron somehow “knows” about both. There’s something associated with, or connected to, or part of each individual electron that is affected by both slits.
But what could that something be?
Figure 8.2 (a) The data obtained when electrons are fired and only the left slit is open. (b) The data obtained when electrons are fired and only the right slit is open. (c) The data obtained when electrons are fired and both slits are open.
For a clue as to how an electron traveling through one slit “knows” about the other, look more closely at the data in Figure 8.2c. The light-dark-light-dark pattern is as recognizable to a physicist as a mother’s face is to her baby. The pattern says—no, it screams—waves. If you’ve ever dropped two pebbles into a pond and watched as the resulting ripples spread and overlap, you know what I mean. Where the peak of one wave crosses the peak of another, the combined wave height is big; where the trough of one crosses the trough of another, the combined wave depression is deep; and most important of all, where the peak of one crosses the trough of the other, the waves cancel and the water remains level. This is illustrated in Figure 8.3. If you were to insert a detector screen across the top of the figure that recorded the water’s agitation at each location—the larger the agitation, the brighter the reading—the result would be a series of alternating bright and dark regions on the screen. Bright regions would be where the waves reinforce each other, yielding much agitation; dark regions would be where the waves cancel, yielding no agitation. Physicists say the overlapping waves interfere with one another, and call the bright-dark-bright data they produce an interference pattern.
The similarity to Figure 8.2c is unmistakable, so in trying to explain the electron data we’re led to think about waves. Good. That’s a start. But the details are still murky. What kind of waves? Where are they? And what have they to do with particles such as electrons?
The next clue comes from the experimental fact I emphasized at the outset. Reams of data on the motion of particles show that regularities emerge only statistically. The same measurements performed on identically prepared particles will generally reveal them to be in different places; yet many such measurements establish that, on average, the particles have the same probability of being found at any given location. In 1926, the German physicist Max Born joined these two clues together and with them made a leap that nearly three decades later would earn him a Nobel Prize. You’ve got experimental evidence that waves play a role. You’ve got experimental evidence that probabilities play a role. Perhaps, Born suggested, the wave associated with a particle is a probability wave.
Figure 8.3 When two water waves overlap, they “interfere,” creating alternating regions of more or less agitation called an interference pattern.
It was an unprecedented and spectacularly original contribution. The idea is that in analyzing the motion of a particle we shouldn’t think of it as a rock hurtling from here to there. Instead, we should think of it as a wave undulating from here to there. Locations where the wave’s values are large, near its peaks and troughs, are locations where the particle is likely to be found. Locations where the probability wave’s values are small are locations where the particle is unlikely to be found. Locations where the wave’s values vanish are places where the particle won’t be found. As the wave rolls onward, the values evolve, going up in some locations, down in others. And since we’re interpreting the undulating values as undulating probabilities, the wave is justly called a probability wave.
To flesh out the picture, consider how it explains the double-slit data. As an electron travels toward the barrier in Figure 8.2c, quantum mechanics tells us to think of it as an undulating wave, as in Figure 8.4. When the wave encounters the barrier, two wave fragments make it through the slits and undulate onward toward the detector screen. What happens next is key. Much like overlapping water waves, the probability waves emerging from the two slits overlap and interfere, yielding a combined form that looks much like that in Figure 8.3: a pattern of high and low values that, according to quantum mechanics, corresponds with a pattern of high and low probabilities for where the electron will land. When electron after electron is fired, the cumulative landing positions conform to this probability profile. Many electrons land where the probability is high, few where it’s low, and none where the probability vanishes. The result is the bright and dark bands of Figure 8.2c.3
And that’s how quantum theory explains the data. The description makes manifest that each electron does “know” about both slits, since each electron’s probability wave passes through both. It’s the union of these two partial waves that dictates the probabilities for where the electrons land. That’s why the mere presence of a second slit affects the results.
Figure 8.4 When we describe the motion of an electron in terms of an undulating probability wave, the puzzling interference data are explained.
Not So Fast
Although I’ve focused on electrons, similar experiments have established the same probability-wave picture for all of nature’s basic constituents. Photons, neutrinos, muons, quarks—every fundamental particle—all are described by waves of probability. But before we declare victory, three questions immediately present themselves. Two are straightforward. One is a bear. It’s the latter that Everett sought to answer back in the 1950s, and it led him to a quantum version of parallel worlds.
First, if quantum theory is right and the world unfolds probabilistically, why is Newton’s nonprobabilistic framework so good at predicting the motion of things from baseballs to planets to stars? The answer is that probability waves for big things usually (but not always, as we will shortly see) have a very particular shape. They’re extraordinarily narrow, as in Figure 8.5a, meaning there’s a huge probability, just shy of 100 percent, that the object is located where the wave is peaked and a minuscule probability, just a shade above 0 percent, that it’s located anywhere else.4 Moreover, the quantum laws show that the peaks of such narrow waves move along the very same trajectories that emerge from Newton’s equations. And so, while Newton’s laws predict precisely the trajectory of a baseball, quantum theory offers only the most minimal refinement, saying there’s a nearly 100 percent probability that the ball will land where Newton says it should, and a nearly 0 percent probability that it won’t.
Figure 8.5 (a) The probability wave for a macroscopic object is generally narrowly peaked. (b) The probability wave for a microscopic object, say, a single particle, is typically widely spread.
In fact, the words “just shy” and “nearly” don’t do the physics justice. The chance of a macroscopic body deviating from Newton’s predictions is so fantastically tiny that if you’d been keeping tabs on the cosmos for the last few billion years, the odds are overwhelming that you’d have never seen it happen. But according to quantum theory, the smaller an object, the more spread-out its probability wave typically is. For example, a typical electron’s wave might look like that in Figure 8.5b, with substantial probabilities of being at a variety of locations, a totally foreign concept in a Newtonian world. And that’s why it’s the microrealm where the probabilistic nature of reality comes to the fore.
Second, can we see the probability waves on which quantum mechanics relies? Is there any way to directly access the unfamiliar probabilistic haze, such as that illustrated schematically in Figure 8.5b, in which a single particle has a chance of being found in a variety of locations? No. The standard approach to quantum mechanics, developed by Bohr and his group, and called the Copenhagen interpretation in their honor, envisions that whenever you try to see a probability wave, the very act of observation thwarts your attempt. When you look at an electron’s probability wave, where “look” means “measure its position,” the electron responds by snapping to attention and coalescing at one definite location. Correspondingly, the probability wave surges to 100 percent at that spot, while collapsing to 0 percent everywhere else, as in Figure 8.6. Look away, and the needle-thin probability wave rapidly spreads, indicating that once again there’s a reasonable chance of finding the electron at a variety of locations. Look back, and the electron’s wave collapses anew, eliminating the range of possible places the electron might be found in favor of its occupying a single definite spot. In short, every time you attempt to see the probabilistic haze it disappears—it collapses—and is supplanted by familiar reality. The detector screen in Figure 8.2c provides a case in point: it measures the impinging probability wave of an electron and thus immediately causes it to collapse. The detector forces the electron to relinquish the many available options for where it could hit and settle upon a definite landing location, which is then evidenced by a tiny dot on the screen.
Figure 8.6 The Copenhagen approach to quantum mechanics envisions that when measured or observed, a particle’s probability wave instantaneously collapses at all but one location. The range of possible positions for the particle transforms into one definite outcome.
I understand full well if this explanation leaves you shaking your head. There’s no denying that quantum dogma sounds a lot like snake oil. I mean, along comes a theory that proposes a startling new picture of reality founded on waves of probability and then, in the very next breath, announces that the waves can’t be seen. Imagine Lucille claiming she’s a blonde—until someone looks, when she immediately transforms into a redhead. Why would physicists accept an approach that’s not only strange but that seems so downright slippery?
Fortunately, for all its mysterious and hidden features, quantum mechanics is testable. According to the Copenhagenists, the larger a probability wave is at a particular location, the greater the chance that when the wave collapses, its sole remaining spike—and hence the electron itself—will be situated there. That statement yields predictions. Run a given experiment over and over again, count how often you find the particle at various locations, and assess whether the frequencies you observe agree with the probabilities dictated by the probability wave. If the wave is 2.874 times as big here as it is there, do you find the particle here 2.874 times as often as you find it there? Predictions like these have been enormously successful. Wily as the quantum perspective may seem, it’s hard to argue with such phenomenal results.
But not impossible.
Which takes us to the third and most difficult question. The collapsing of probability waves upon measurement, Figure 8.6, is a centerpiece of the Copenhagen approach to quantum theory. The confluence of its successful predictions and Bohr’s forceful proselytizing led most physicists to accept it, but even polite prodding quickly reveals an uncomfortable feature. Schrödinger’s equation, the mathematical engine of quantum mechanics, dictates how the shape of a probability wave evolves in time. Give me an initial wave shape, say, that of Figure 8.5b, and I can use Schrödinger’s equation to draw a picture of what the wave will look like in a minute, or an hour, or at any other moment. But straightforward analysis of the equation reveals that the evolution depicted in Figure 8.6—the instantaneous collapse of a wave at all but one point, like a lone parishioner in a megachurch accidentally standing while everyone else kneels—can’t possibly emerge from Schrödinger’s math. Waves surely can have a needle-thin spiked shape; we’ll make ample use of some spiked waves shortly. But they can’t become spiked in the manner envisioned by the Copenhagen approach. The math simply doesn’t allow it. (We’ll see why in just a moment.)
Bohr advanced a heavyhanded remedy: evolve probability waves according to Schrödinger’s equation whenever you’re not looking or performing any kind of measurement. But when you do look, Bohr continued, you should throw Schrödinger’s equation aside and declare that your observation has caused the wave to collapse.
Now, not only is this prescription ungainly, not only is it arbitrary, not only does it lack a mathematical underpinning, it’s not even clear. For instance, it doesn’t precisely define “looking” or “measuring.” Must a human be involved? Or, as Einstein once asked, will a sidelong glance from a mouse suffice? How about a computer’s probe, or even a nudge from a bacterium or virus? Do these “measurements” cause probability waves to collapse? Bohr announced that he was drawing a line in the sand separating small things, such as atoms and their constituents, to which Schrödinger’s equation would apply, and big things, such as experimenters and their equipment, to which it wouldn’t. But he never said where exactly that line would be. The reality is, he couldn’t. With each passing year, experimenters confirm that Schrödinger’s equation works, without modification, for increasingly large collections of particles, and there’s every reason to believe that it works for collections as hefty as those making up you and me and everything else. Like floodwaters slowly rising from your basement, rushing into your living room, and threatening to engulf your attic, the mathematics of quantum mechanics has steadily spilled beyond the atomic domain and has succeeded on ever-larger scales.
So the way to think about the problem is this. You and I and computers and bacteria and viruses and everything else material are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrödinger’s equation works for electrons and quarks, and all evidence points to its working for things made of these constituents, regardless of the number of particles involved. This means that Schrödinger’s equation should continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer …) coming into contact with another (the particle or particles being measured). But if that’s the case, if Schrödinger’s math refuses to bow down, then Bohr is in trouble. Schrödinger’s equation doesn’t allow waves to collapse. An essential element of the Copenhagen approach would therefore be undermined.
So the third question is this: If the reasoning just recounted is right and probability waves don’t collapse, how do we pass from the range of possible outcomes that exist before a measurement to the single outcome the measurement reveals? Or to put it in more general terms, what happens to a probability wave during a measurement that allows a familiar, definite, unique reality to take hold?
Everett pursued this question in his Princeton doctoral dissertation and came to an unforeseen conclusion.
Linearity and Its Discontents
To understand Everett’s path of discovery, you need to know a little more about Schrödinger’s equation. I’ve emphasized that it doesn’t allow probability waves to suddenly collapse. But why not? And what doesit allow? Let’s get a feel for how Schrödinger’s math guides a probability wave as it evolves through time.
This is fairly straightforward, because Schrödinger’s is one of the simplest kinds of mathematical equations, characterized by a property known as linearity—a mathematical embodiment of the whole being the sum of its parts. To see what this means, imagine that the shape in Figure 8.7a is the probability wave at noon for a given electron (for visual clarity, I will use a probability wave that depends on location in the one dimension represented by the horizontal axis, but the ideas are general). We can use Schrödinger’s equation to follow the evolution of this wave forward in time, yielding its shape at, say, one p.m., schematically illustrated in Figure 8.7b. Now notice the following. You can decompose the initial wave shape in Figure 8.7a into two simpler pieces, as in Figure 8.8a; if you combine the two waves in the figure, adding their values point by point, you recover the original wave shape. The linearity of Schrödinger’s equation means that you can use it on each piece in Figure 8.8a separately, determining what each wave fragment will look like at one p.m., and then combine the results as in Figure 8.8b to recover the full result shown in Figure 8.7b. And there’s nothing sacred about decomposition into two pieces; you can break the initial shape up into any number of pieces, evolve each separately, and combine the results to get the final wave shape.
Figure 8.7 (a) An initial probability wave shape at one moment evolves via Schrödinger’s equation to a different shape (b) at a later time.
This may sound like a mere technical nicety, but linearity is an extraordinarily powerful mathematical feature. It allows for an all-important divide-and-conquer strategy. If an initial wave shape is complicated, you are free to divide it up into simpler pieces and analyze each separately. At the end, you just put the individual results back together. We’ve actually already seen an important application of linearity through our analysis of the double-slit experiment in Figure 8.4. To determine how the electron’s probability wave evolves, we divided the task: we noted how the piece passing through the left slit evolves, we noted how the piece passing through the right slit evolves, and we then added the two waves together. That’s how we found the famous interference pattern. Look at a quantum theorist’s blackboard, and it is this very approach you’ll see underlying a great many of the mathematical manipulations.
Figure 8.8 (a) An initial probability wave shape can be decomposed as the union of two simpler shapes. (b) The evolution of the initial probability wave can be reproduced by evolving the simpler pieces and combining the results.
Figure 8.9 An electron’s probability wave, at a given moment, is spiked at Thirty-fourth Street and Broadway. A measurement of the electron’s position, at that moment, confirms that it is located where its wave is spiked.
But linearity not only makes quantum calculations manageable; it’s also at the heart of the theory’s difficulty in explaining what happens during a measurement. This is best understood by applying linearity to the act of measurement itself.
Imagine you’re an experimentalist with great nostalgia for your childhood in New York, so you’re measuring the positions of electrons that you inject into a miniature tabletop model of the city. You start your experiments with one electron whose probability wave has a particularly simple shape—it’s nice and spiked, as in Figure 8.9, indicating that with essentially 100 percent probability the electron is momentarily sitting at the corner of Thirty-fourth Street and Broadway. (Don’t worry about how the electron got this wave shape; just take it as a given.)* If at that very moment you measure the electron’s position with a well-made piece of equipment, the result should be accurate—the device’s readout should say “Thirty-fourth Street and Broadway.” Indeed, if you do this experiment, that’s just what happens, as in Figure 8.9.
It would be extraordinarily complicated to work out how Schrödinger’s equation entwines the probability wave of the electron with that of the trillion trillion or so atoms that make up the measuring device, coaxing a collection of the latter to arrange themselves in the readout to spell “Thirty-fourth Street and Broadway,” but whoever designed the device has done the heavy lifting for us. It’s been engineered so that its interaction with such an electron causes the readout to display the single definite position where, at that moment, the electron is located. If the device did anything else in this situation, we’d be wise to exchange it for a new one that functions properly. And, of course, Macy’s notwithstanding, there’s nothing special about Thirty-fourth and Broadway; if we do the same experiment with the electron’s probability wave spiked at the Hayden Planetarium near Eighty-first and Central Park West, or at Bill Clinton’s office on 125th near Lenox Avenue, the device’s readout will return these locations.
Let’s now consider a slightly more complicated wave shape, as in Figure 8.10. This probability wave indicates that, at the given moment, there are two places the electron might be found—Strawberry Fields, the John Lennon memorial in Central Park, and Grant’s Tomb in Riverside Park. (The electron’s in one of its dark moods.) If we measure the electron’s position but, in opposition to Bohr and in keeping with the most refined experiments, assume that Schrödinger’s equation continues to apply—to the electron, to the particles in the measuring device, to everything—what will the device’s output read? Linearity is the key to the answer. We know what happens when we measure spiked waves individually. Schrödinger’s equation causes the device’s display to spell out the spike’s location, as in Figure 8.9. Linearity then tells us that to find the answer for two spikes, we combine the results of measuring each spike separately.
Here’s where things get weird. At first blush, the combined results suggest that the display should simultaneously register the locations of both spikes. As in Figure 8.10, the words “Strawberry Fields” and “Grant’s Tomb” should flash simultaneously, one location commingled with the other, like the confused monitor of a computer that’s about to crash. Schrödinger’s equation also dictates how the probability waves of the photons emitted by the measuring device’s display entangle with those of the particles in your rods and cones, and subsequently those rushing through your neurons, creating a mental state reflecting what you see. Assuming unlimited Schrödinger hegemony, linearity applies here too, so not only will the device simultaneously display both locations but also your brain will be caught up in the confusion, thinking that the electron is simultaneously positioned at both.
Figure 8.10 An electron’s probability wave is spiked at two locations. The linearity of Schrödinger’s equation suggests that a measurement of the electron’s position would yield a confusing amalgam of both locations.
For yet more complicated wave shapes, the confusion becomes yet wilder. A shape with four spikes doubles the dizziness. With six, it triples. Notice that if you keep on going, putting wave spikes of various heights at every location in the model Manhattan, their combined shape fills out an ordinary, more gradually varying quantum wave shape, as schematically illustrated in Figure 8.11. Linearity still holds, and this implies that the final device reading, as well as your final brain state and mental impression, are dictated by the union of the results for each spike individually.
Figure 8.11 A general probability wave is the union of many spiked waves, each representing a possible position of the electron.
The device should simultaneously register the location of each and every spike—each and every location in Manhattan—as your mind becomes profoundly puzzled, being unable to settle on a single definite location for the electron.5
But, of course, this seems grossly at odds with experience. No properly functioning device, when taking a measurement, displays conflicting results. No properly functioning person, on performing a measurement, has the mental impression of a dizzying mélange of simultaneous yet distinct outcomes.
You can now see the appeal of Bohr’s prescription. Hold the Dramamine, he’d declare. According to Bohr, we don’t see ambiguous meter readings because they don’t happen. He’d argue that we’ve come to an incorrect conclusion because we’ve overextended the reach of Schrödinger’s equation into the domain of big things: laboratory equipment that takes measurements, and scientists who read the results. Although Schrödinger’s equation and its feature of linearity dictate that we should combine the results from distinct possible outcomes—nothing collapses—Bohr tells us that this is wrong because the act of measurement tosses Schrödinger’s math out the window. Instead, he’d pronounce, the measurement causes all but one of the spikes in Figure 8.10 or Figure 8.11 to collapse to zero; the probability that a particular spike will be the sole survivor is proportional to the spike’s height. That unique remaining spike determines the device’s unique reading, as well as your mind’s recognition of a unique result. Dizziness done.
But for Everett, and later DeWitt, the cost of Bohr’s approach was too high. Schrödinger’s equation is meant to describe particles. All particles. Why would it somehow not apply to particular configurations of particles—those constituting the equipment that takes measurements, and those in the experimenters who monitor the equipment? This just didn’t make sense. Everett therefore suggested that we not dispense with Schrödinger so quickly. Instead, he advocated that we analyze where Schrödinger’s equation takes us from a decidedly different perspective.
The challenge we’ve encountered is that it’s bewildering to think of a measuring device or a mind as simultaneously experiencing distinct realities. We can have conflicting opinions on this or that issue, mixed emotions regarding this or that person, but when it comes to the facts that constitute reality, everything we know attests to there being an unambiguous objective description. Everything we know attests that one device and one measurement will yield one reading; one reading and one mind will yield one mental impression.
Everett’s idea was that Schrödinger’s math, the core of quantum mechanics, is compatible with such basic experiences. The source of the supposed ambiguity in device readings and mental impressions is the manner in which we’ve executed that math—the manner, that is, in which we’ve combined the results of the measurements illustrated in Figure 8.10 and Figure 8.11. Let’s think it through.
When you measure a single spiked wave, such as that in Figure 8.9, the device registers the spike’s location. If it’s spiked at Strawberry Fields, that’s what the device reads; if you look at the result, your brain registers that location and you become aware of it. If it’s spiked at Grant’s Tomb, that’s what the device registers; if you look, your brain registers that location and you become aware of it. When you measure the double spiked wave in Figure 8.10, Schrödinger’s math tells you to combine the two results you just found. But, says Everett, be careful and precise when you combine them. The combined result, he argued, does not yield a meter and a mind each simultaneously registering two locations. That’s sloppy thinking.
Instead, proceeding slowly and literally, we find that the combined result is a device and a mind registering Strawberry Fields, and a device and a mind registering Grant’s Tomb. And what does that mean? I’ll use broad strokes in painting the general picture, which I’ll refine shortly. To accommodate Everett’s suggested outcome, the device and you and everything else must split upon measurement, yielding two devices, two yous, and two everything elses—the only difference between the two being that one device and one you registers Strawberry Fields, while the other device and the other you registers Grant’s Tomb. As in Figure 8.12, this implies that we now have two parallel realities, two parallel worlds. To the you occupying each, the measurement and your mental impression of the result are sharp and unique and thus feel like life as usual. The peculiarity, of course, is that there are two of you who feel this way.
To keep the discussion accessible, I’ve focused on the position measurement of a single particle, and one that has a particularly simple probability wave. But Everett’s proposal applies generally. If you measured the position of a particle whose probability wave has any number of spikes, say, five, the result, according to Everett, would be five parallel realities differing only by the location registered on each reality’s device, and within the mind of each reality’s you. If one of these yous then measured the position of another particle whose wave had seven spikes, that you and that world would split again, into seven more, one for each possible outcome. And if you measured a wave like that of Figure 8.11, which can be partitioned into a great many tightly packed spikes, the result would be a great many parallel realities in which each possible particle location would be recorded on a device and read by a copy of you. In Everett’s approach, everything that is possible, quantum-mechanically speaking (that is, all those outcomes to which quantum mechanics assigns a nonzero probability), is realized in its own separate world. These are the “many worlds” of the Many Worlds approach to quantum mechanics.
Figure 8.12 In Everett’s approach, the measurement of a particle whose probability wave has two spikes yields both outcomes. In one world, the particle is found at the first location; in another world, it is found at the second.
If we apply the terminology we’ve been using in earlier chapters, these many worlds would properly be described as many universes, composing a multiverse, the sixth we’ve encountered. I’ll call it the Quantum Multiverse.
A Tale of Two Tales
In describing how quantum mechanics may generate many realities, I used the word “split.” Everett used it. So did DeWitt. Nevertheless, in this context it’s a loaded verb with the potential to grossly mislead, and I’d intended not to invoke it. But I gave in to temptation. In my defense, it’s sometimes more effective to use a sledgehammer to break down a barrier separating us from an unfamiliar proposal about the workings of reality, and to subsequently repair the damage, than it is to delicately carve a pristine window that directly reveals the new vista. I’ve been using that sledgehammer; in this and the next section I’ll undertake the necessary repairs. Some of the ideas are a touch more difficult than those we’ve so far encountered, and the explanatory chains are a bit longer as well, but I encourage you to stay with me. I’ve found that all too often, people who learn about, or are even somewhat familiar with, the Many Worlds idea have the impression that it emerged from speculation of the most extravagant sort. But nothing could be further from the truth. As I will explain, the Many Worlds approach is, in some ways, the most conservative framework for defining quantum physics, and it’s important to understand why.
The essential point is that physicists must always tell two kinds of stories. One is the mathematical story of how the universe evolves according to a given theory. The other, also essential, is the physical story, which translates the abstract mathematics into experiential language. This second story describes how the mathematical evolution will appear to observers like you and me, and more generally, what the theory’s mathematical symbols tell us about the nature of reality.6 In the time of Newton, the two stories were essentially identical, as I suggested with my remarks in Chapter 7 about Newtonian “architecture” being immediate and palpable. Every mathematical symbol in Newton’s equations has a direct and transparent physical correlate. The symbol x? Oh, that’s the ball’s position. The symbol v? The ball’s velocity. By the time we get to quantum mechanics, however, translation between the mathematical symbols and what we can see in the world around us becomes far more subtle. In turn, the language used and the concepts deemed relevant to each of the two stories become so different that you need both to acquire a full understanding. But it’s important to keep straight which story is which: to understand fully which ideas and descriptions are invoked as part of the theory’s fundamental mathematical structure and which are used to build a bridge to human experience.
Let’s tell the two stories for the Many Worlds approach to quantum mechanics. Here’s the first.
The mathematics of Many Worlds, unlike that of Copenhagen, is pure, simple, and constant. Schrödinger’s equation determines how probability waves evolve over time, and it is never set aside; it is always in effect. Schrödinger’s math guides the shape of probability waves, causing them to shift, morph, and undulate over time. Whether it’s addressing the probability wave for a particle, or for a collection of particles, or for the various assemblages of particles that constitute you and your measuring equipment, Schrödinger’s equation takes the particles’ initial probability wave shape as input and then, like the graphics program driving an elaborate screen saver, provides the wave’s shape at any future time as output. And that, according to this approach, is how the universe evolves. Period. End of story. Or, more precisely, end of first story.
Notice that in telling the first story I did not need the word “split” nor the terms “many worlds,” “parallel universes,” or “Quantum Multiverse.” The Many Worlds approach does not hypothesize these features. They play no role in the theory’s fundamental mathematical structure. Rather, as we will now see, these ideas are called upon in the theory’s second story, when, following Everett and others who’ve since extended his pioneering work, we investigate what the mathematics tells us about our observations and measurements.
Let’s start simply—or, as simply as we can. Consider measuring an electron that has a spiked probability wave, as in Figure 8.9. (Again, don’t worry about how it got this wave shape; just take it as a given.) As noted earlier, to tell the first story of even this measurement process in detail is beyond what we can do. We’d need to use Schrödinger’s math to figure out how the probability wave describing the positions of the huge number of particles that constitute you and your measuring device joins with the probability wave of the electron, and how their union evolves forward in time. My undergraduate students, many of whom are quite able, often struggle to solve Schrödinger’s equation for even a single particle. Between you and the device, there are something like 1027 particles. Working out Schrödinger’s math for that many constituents is virtually impossible. Even so, we understand qualitatively what the math entails. When we measure the electron’s position, we cause a mass particle migration. Some 1024 or so particles in the device’s display, like performers in a crisply choreographed halftime show, race to the appropriate spot so that they collectively spell out “Thirty-fourth Street and Broadway,” while a similar number in my eyes and brain do whatever’s required for me to develop a firm mental grasp of the result. Schrödinger’s math—however impenetrable explicit analysis of it might be when faced with so many particles—describes such a particle shift.
To visualize this transformation at the level of a probability wave is also far beyond reach. In Figure 8.9 and others in that sequence, I used two axes, the north-south and east-west street grid of our model Manhattan, to denote the possible positions of a single particle. The probability wave’s value at each location was denoted by the wave’s height. This already simplifies things because I’ve left out the third axis, the particle’s vertical position (whether it’s on the second floor of Macy’s, or the fifth). Including the vertical would have been awkward, because if I’d used it to denote position, I’d have no axis left for recording the size of the wave. Such are the limitations of a brain and a visual system that evolution has firmly rooted in three spatial dimensions. To properly visualize the probability wave for roughly 1027 particles, I’d need to include three axes for each, allowing me to account mathematically for every possible position each particle could occupy.* Adding even a single vertical axis to Figure 8.9 would have made it difficult to visualize; to contemplate adding a billion billion billion more is, well, silly.
But a mental image of the key ideas is important; so, however imperfect the result, let’s give it a try. In sketching the probability wave for the particles making up you and your device, I’ll abide by the two-axis flat-page limit but will use an unconventional interpretation of what the axes mean. Roughly speaking, I’ll think of each axis as comprising an enormous bundle of axes, tightly grouped together, which will symbolically delineate the possible positions of a similarly enormous number of particles. A wave drawn using these bundled axes will therefore lay out the probabilities for the positions of a huge group of particles. To emphasize the distinction between the many-particle and single-particle situations, I’ll use a glowing outline for the many-particle probability wave, as in Figure 8.13.
Figure 8.13 A schematic depiction of the combined probability wave for all the particles making up you and your measuring device.
The many-particle and single-particle illustrations have some features in common. Just as the spiked wave shape in Figure 8.6 indicates probabilities that are sharply skewed (being almost 100 percent at the spike’s location and almost 0 percent everywhere else), so the peaked wave in Figure 8.13 also denotes sharply skewed probabilities. But you need to exercise care, because understanding based on the single-particle illustrations can take you only so far. For example, based on Figure 8.6 it is natural to think that Figure 8.13 represents particles that are all clustered around the same location. Yet, that’s not right. The peaked shape in Figure 8.13 symbolizes that each of the particles making up you and each of the particles making up the device starts out in the ordinary, familiar state of having a position that is nearly 100 percent definite. But they are not all positioned at the same location. The particles constituting your hand, shoulder, and brain are, with near certainty, clustered within the location of your hand, shoulder, and brain; the particles constituting the measuring device are, with near certainty, clustered within the location of the device. The peaked wave shape in Figure 8.13 denotes that each of these particles has only the most remote chance of being found anywhere else.
If you now perform the measurement illustrated in Figure 8.14, the many-particle probability wave (for the particles inside you and the device), by virtue of the interaction with the electron, evolves (as illustrated schematically in Figure 8.14a). All the particles involved still have nearly definite positions (within you; within the device), which is why the wave in Figure 8.14a maintains a spiked shape. But a mass particle rearrangement occurs that results in the words “Strawberry Fields” forming in the device’s readout and also in your brain (as in Figure 8.14b). Figure 8.14a represents the mathematical transformation dictated by Schrödinger’s equation, the first kind of story. Figure 8.14b illustrates the physical description of such mathematical evolution, the second kind of story. Similarly, if we perform the experiment in Figure 8.15, an analogous wave shift takes place (Figure 8.15a). This shift corresponds to a mass particle rearrangement that spells out “Grant’s Tomb” in the display and generates within you the associated mental impression (Figure 8.15b).
Now use linearity to put the two together. If you measure the position of an electron whose probability wave is spiked at two locations, the probability wave for you and your device commingles with that of the electron, resulting in the evolution shown in Figure 8.16a—the combined evolutions depicted in Figure 8.14a and Figure 8.15a. So far, this is nothing but an illustrated and annotated version of the first type of quantum story. We start with a probability wave of a given shape, Schrödinger’s equation evolves it forward in time, and we end up with a probability wave of a new shape. But the details we’ve overlaid now let us tell this mathematical story in more qualitative, type-two story language.
Physically, each spike in Figure 8.16a represents a configuration of an enormous number of particles that results in a device having a particular reading and your mind acquiring that information. In the left spike, the reading is Strawberry Fields; in the right, it’s Grant’s Tomb. Besides that difference, nothing distinguishes one spike from the other. I emphasize this because it’s essential to realize that neither is somehow more real than the other. Nothing but the device’s particular reading, and your reading of that reading, distinguishes the two multiparticle wave spikes.
Which means that our type-two story, as illustrated in Figure 8.16b, involves two realities.
In fact, the focus on the device and your mind is merely another simplification. I could also have included the particles that make up the laboratory and everything therein, as well as those of the earth, the sun, and so on, and the whole discussion would have been the same, essentially verbatim. The only difference would have been that the glowing probability wave in Figure 8.16a would now have information about all those other particles, too. But because the measurement we’re discussing has essentially no impact on them, they’d just come along for the ride. It’s useful to include those particles, though, because our second story can now be augmented to comprise not only a copy of you examining a device that’s undertaken a measurement, but also copies of the surrounding laboratory, the rest of the earth in orbit around the sun, and so on. This means that each spike, in story-two language, corresponds to what we’d traditionally call a bona fide universe. In one such universe, you see “Strawberry Fields” on the display’s reading; in the other, “Grant’s Tomb.”
Figure 8.14 (a) A schematic illustration of the evolution, dictated by Schrödinger’s equation, of the combined probability wave for all the particles making up you and the measuring device, when you measure the position of an electron. The electron’s own probability wave is spiked at Strawberry Fields.
Figure 8.14 (b) The corresponding physical, or experiential, story.
Figure 8.15 (a) The same type of mathematical evolution as in Figure 8.14a, but with the electron’s probability wave spiked at Grant’s Tomb.
Figure 8.15 (b) The corresponding physical, or experiential, story.
Figure 8.16 (a) A schematic illustration of the evolution of the combined probability wave of all the particles making up you and your device, when measuring the position of an electron whose probability wave is spiked at two locations.
Figure 8.16 (b) The corresponding physical, or experiential, story.
If the electron’s original probability wave had, say, four spikes, or five, or a hundred, or any number, the same would follow: the wave evolution would result in four, or five, or a hundred, or any number of universes. In the most general case, as in Figure 8.11, a spread-out wave is composed of spikes at every location, and so the wave evolution would yield a vast collection of universes, one for each possible position.7
As advertised, though, the only thing that happens in any of these scenarios is that a probability wave enters Schrödinger’s equation, his math goes to work, and out comes a wave with a modified shape. There’s no “cloning machine.” There’s no “splitting machine.” This is why I said earlier that such words can give a misleading impression. There’s nothing but a probability-wave-evolution “machine” driven by the lean mathematical law of quantum mechanics. When the resulting waves have a particular shape, as in Figure 8.16a, we retell the mathematical story in type-two language, and conclude that in each spike there’s a sentient being, situated within a normal-looking universe, certain he sees one and only one definite result for the given experiment, as in Figure 8.16b. If I could somehow interview all these sentient beings, I’d find each to be an exact replica of the others. Their only point of departure would be that each would attest to a different definite result.
And so, whereas Bohr and the Copenhagen gang would argue that only one of these universes would exist (because the act of measurement, which they claim lies outside of Schrödinger’s purview, would collapse away all the others), and whereas a first-pass attempt to go beyond Bohr and extend Schrödinger’s math to all particles, including those constituting equipment and brains, yielded dizzying confusion (because a given machine or mind seemed to internalize all possible outcomes simultaneously), Everett found that a more careful reading of Schrödinger’s math leads somewhere else: to a plentiful reality populated by an ever-growing collection of universes.
Prior to the publication of Everett’s 1957 paper, a preliminary version was circulated to a number of physicists around the world. Under Wheeler’s guidance, the paper’s language had been abbreviated so aggressively that many who read it were unsure as to whether Everett was arguing that all the universes in the mathematics were real. Everett became aware of this confusion and decided to clarify it. In a “note added in proof” that he seems to have slipped in just before publication, and apparently without Wheeler’s notice, Everett sharply articulated his stance on the reality of the different outcomes: “From the viewpoint of the theory, all … are ‘actual,’ none any more ‘real’ than the rest.”8
When Is an Alternative a Universe?
Besides the loaded words “splitting” and “cloning,” we’ve freely invoked two other grand terms in our type-two stories—“world” and, interchangeably in this context, “universe.” Are there guidelines for determining when this usage is appropriate? When we consider a probability wave for a single electron that has two (or more) spikes, we don’t speak of two (or more) worlds. Instead, we speak of one world—ours—containing an electron whose position is ambiguous. Yet, in Everett’s approach, when we measure or observe that electron, we speak in terms of multiple worlds. What is it that distinguishes the unmeasured and the measured particle, yielding descriptions that sound so radically different?
One quick answer is that for a single isolated electron, we don’t tell a type-two story because without a measurement or an observation there’s no link to human experience that’s in need of articulation. The type-one story of a probability wave evolving via Schrödinger’s math is all that’s needed. And without a type-two story, there’s no opportunity to invoke multiple realities. Although this explanation is adequate, it proves worthwhile to delve a little deeper, revealing a special feature of quantum waves that comes into play when many particles are involved.
To grasp the essential idea, it’s easiest to look back at the double-slit experiment of Figures 8.2 and 8.4. Recall that an electron’s probability wave encounters the barrier, and two wave fragments make it through the slits and travel onward to the detector screen. Inspired by our Many Worlds discussion, you might be tempted to think of the two racing waves as representing separate realities. In one, an electron whisks through the left slit; in the other, an electron whisks through the right slit. But you promptly realize that the intermingling of these supposedly “distinct realities” profoundly affects the experiment’s outcome; the intermingling is why an interference pattern is produced. So it doesn’t make much sense, nor does it yield any particular insight, to consider the two wave trajectories as existing in separate universes.
If we change the experiment, however, by placing a meter behind each slit that records whether or not an electron passes through it, the situation is radically different. Because macroscopic equipment is now involved, the two distinct trajectories of an electron generate differences in a huge number of particles—the huge number of particles in the meters’ displays that register “electron passed through left slit” or “electron passed through right slit.” And because of this, the respective probability waves for each possibility become so disparate that it’s virtually impossible for them to have any subsequent influence on each other. Much as in Figure 8.16a, the differences between the billions and billions of particles in the meters cause the waves for the two outcomes to shift away from each other, leaving negligible overlap. With no overlap, the waves don’t engage in any of the hallmark interference phenomena of quantum physics. Indeed, with the meters in place, the electrons no longer yield the striped pattern of Figure 8.2c; instead, they generate a simple, non-interfering amalgam of the results in Figure 8.2a and Figure 8.2b. Physicists say that the probability waves have decohered (something you can read about in more detail, for example, in Chapter 7 of The Fabric of the Cosmos).
The point, then, is that once decoherence sets in, the waves for each outcome evolve independently—there’s no intermingling between the distinct possible outcomes—and each can thus be called a world or a universe of its own. For the case at hand, in one such universe the electron goes through the left slit, and the meter displays left; in another universe the electron goes through the right slit, and the meter records right.
In this sense, and only in this sense, there’s resonance with Bohr. According to the Many Worlds approach, big things made of many particles do differ from small things made from one particle or a mere handful. Big things don’t stand outside the basic mathematical law of quantum mechanics, as Bohr thought, but they do allow probability waves to acquire enough variations that their capacity to interfere with one another becomes negligible. And once two or more waves can’t affect one another, they become mutually invisible; each “thinks” the others have disappeared. So, whereas Bohr argued away by fiat all but one outcome in a measurement, the Many Worlds approach, combined with decoherence, ensures that within each universe it appears as though the other outcomes have vanished. Within each universe, that is, it’s as if the probability wave has collapsed. But, compared with the Copenhagen approach, the “as if” provides for a very different picture of the expanse of reality. In the Many Worlds view, all outcomes, not just one, are realized.
Uncertainty at the Cutting Edge
This might seem like a good place to end the chapter. We’ve seen how the bare-bones mathematical structure of quantum mechanics leads us by the nose to a new conception of parallel universes. Yet you’ll note that the chapter still has a fair way to go. In those pages I’ll explain why the Many Worlds approach to quantum physics remains controversial; we will see that the resistance goes well beyond the queasiness some feel about the conceptual leap into such an unfamiliar perspective on reality. But in case you’ve reached saturation and feel compelled to skip ahead to the next chapter, here is a short summary.
In day-to-day life, probability enters our thinking when we face a range of possible outcomes, but for one reason or another we’re unable to figure out which will actually happen. Sometimes we have enough information to determine which outcomes are more or less likely to occur, and probability is the tool that makes such insights quantitative. Our confidence in a probabilistic approach grows when we find that the outcomes deemed likely happen often and those deemed unlikely happen rarely. The challenge facing the Many Worlds approach is that it needs to make sense of probability—quantum mechanics’ probabilistic predictions—in a wholly different context, one that envisions all possible outcomes happening. The dilemma is simple to state: How can we speak of some outcomes being likely and others being unlikely when all take place?
In the remaining sections, I’ll explain the issue more fully and discuss attempts to address it. Be warned: we are now deep into cutting-edge research, so opinions vary widely on where we currently stand.
A Probable Problem
A frequent criticism of the Many Worlds approach is that it’s just too baroque to be true. The history of physics teaches us that successful theories are simple and elegant; they explain data with a minimum of assumptions and provide an understanding that’s precise and economical. A theory that introduces an ever-growing cornucopia of universes falls way short of this ideal.
Proponents of the Many Worlds approach argue, credibly, that in assessing the complexity of a scientific proposal, you shouldn’t focus on its implications. What matters is the fundamental features of the proposal itself. The Many Worlds approach assumes that a single equation—Schrödinger’s—governs all probability waves all the time, so for simplicity of formulation and economy of assumptions, it’s hard to beat. The Copenhagen approach is surely no simpler. It, too, invokes Schrödinger’s equation, but it also includes a vague, ill-defined prescription for when Schrödinger’s equation should be turned off, and then an even less detailed prescription regarding the process of wave collapse that is meant to take its place. That the Many Worlds approach leads to an exceptionally rich picture of reality is no more a black mark against it than the rich diversity of life on earth is a black mark against Darwinian natural selection. Mechanisms that are fundamentally simple can give rise to complicated consequences.
Nevertheless, while this establishes that Occam’s razor isn’t sharp enough to pare away the Many Worlds approach, the proposal’s surfeit of universes does yield a potential problem. Earlier I said that in applying a theory, physicists need to tell two kinds of stories—the story describing how the world evolves mathematically and the story that links the math to our experiences. But there’s actually a third story, related to these two, that the physicist must also tell. It’s the story of how we’ve come to have confidence in a given theory. For quantum mechanics, the third story generally goes like this: our confidence in quantum mechanics comes from its phenomenal success in explaining data. If a quantum expert uses the theory to calculate that in repeating a given experiment we expect one outcome to happen, say, 9.62 times more often than another, that’s what experimenters invariably see. Turning this around, had results not agreed with the quantum predictions, experimenters would have concluded that quantum mechanics wasn’t right. Actually, being careful scientists, they would have been more cautious. They would have called it doubtful that quantum mechanics was right but would have noted that their results didn’t rule out the theory definitively. Even a fair coin tossed 1,000 times can have surprising runs that defy the odds. But the larger the deviation, the more one suspects the coin is not fair; the larger the experimental deviations from those predicted by quantum mechanics, the more strongly the experimenters would have suspected that quantum theory was mistaken.
That confidence in quantum mechanics could have been undermined by data is essential; with any proposed scientific theory that has been suitably developed and understood, we should be able to say, at least in principle, that if upon doing such and such an experiment we don’t find such and such results, our belief in the theory should diminish. And the more that observations deviate from predictions, the greater the loss of credibility should be.
The potential problem with the Many Worlds approach, and the reason it remains controversial, is that it may undercut this means for assessing the credibility of quantum mechanics. Here’s why. When I flip a coin, I know there’s a 50 percent chance that it will land heads and a 50 percent chance that it will land tails. But that conclusion rests on the usual assumption that a coin toss yields a unique result. If a coin toss yields heads in one world and tails in another, and moreover, if there’s a copy of me in each world who witnesses the outcome, what sense can we make of the usual odds? There’ll be someone who looks just like me, has all my memories, and emphatically claims to be me who sees heads, and another being, equally convinced that he’s me, who sees tails. Since both outcomes happen—there’s a Brian Greene who sees heads and a Brian Greene who sees tails—the familiar probability of there being an equal chance that Brian Greene will see either heads or tails seems nowhere to be found.
The same concern applies to an electron whose probability wave is hovering near Strawberry Fields and Grant’s Tomb, as in Figure 8.16b. Traditional quantum reasoning says that you, the experimenter, have a 50 percent chance of finding the electron at either location. But in the Many Worlds approach, both outcomes happen. There’s a you who will find the electron at Strawberry Fields and another you who will find the electron at Grant’s Tomb. So, how can we make sense of the traditional probabilistic predictions, which in this case say that with equal odds you’ll see one result or the other?
The natural inclination of many people when they first encounter this issue is to think that among the various yous in the Many Worlds approach, there’s one who’s somehow more real than the others. Even though each you in each world looks identical and has the same memories, the common thought is that only one of these beings is really you. And, this line of thought continues, it’s that you, who sees one and only one outcome, to whom the probabilistic predictions apply. I appreciate this response. Years ago, when I first learned about these ideas, I had it too. But the reasoning runs completely counter to the Many Worlds approach. Many Worlds practices minimalist architecture. Probability waves simply evolve by Schrödinger’s equation. That’s it. To imagine that one of the copies of you is the “real” you is to slip in through the back door something closely akin to Copenhagen. Wave collapse in the Copenhagen approach is a brutish means for making one and only one of the possible outcomes real. If in the Many Worlds approach you imagine that one and only one of the yous is really you, you’re doing the same thing, just a little more quietly. Such a move would erase the very reason for introducing the Many Worlds scheme. Many Worlds emerged from Everett’s attempt to address the failings of Copenhagen, and his strategy was to invoke nothing beyond the battle-tested Schrödinger equation.
This realization shines an uncomfortable light on the Many Worlds approach. We have confidence in quantum mechanics because experiments confirm its probabilistic predictions. Yet, in the Many Worlds approach, it’s hard to see how probability even plays a role. How, then, can we tell the third kind of story, the one that should provide the basis of our confidence in the Many Worlds scheme? That’s the quandary.
On reflection, it’s not surprising that we’ve bumped into this wall. There’s nothing at all chancy in the Many Worlds approach. Waves simply evolve from one shape to another in a manner described fully and deterministically by Schrödinger’s equation. No dice are thrown; no roulette wheels are spun. By contrast, in the Copenhagen approach, probability enters through the hazily defined measurement-induced wave collapse (again, the larger the wave’s value at a given location, the larger the probability that the collapse will put the particle there). That’s the point in the Copenhagen approach where “dice throwing” makes an appearance. But since the Many Worlds approach abandons collapse, it abandons the traditional entry point for probability.
So, is there a place for probability in the Many Worlds approach?
Probability and Many Worlds
Everett surely thought there was. The bulk of his 1956 draft dissertation, as well as the truncated 1957 version, was devoted to explaining how to incorporate probability in the Many Worlds approach. But a half century later, the debate still rages. Among those physicists and philosophers who spend their professional lives puzzling over the issue, there is a wide range of opinions on how, and whether, Many Worlds and probability come together. Some have argued that the problem is insoluble, and so the Many Worlds approach should be discarded. Others have argued that probability, or at least something that masquerades as probability, can indeed be incorporated.
Everett’s original proposal provides a good example of the difficult points that arise. In everyday settings, we invoke probability because we generally have incomplete knowledge. If, when a coin is tossed, we know enough details (the coin’s precise dimensions and weight, precisely how the coin was thrown, and so on), we’d be able to predict the outcome. But since we generally don’t have that information, we resort to probability. Similar reasoning applies to the weather, the lottery, and every other familiar example where probability plays a role: we deem the outcomes chancy only because our knowledge of each situation is limited. Everett argued that probabilities find their way into the Many Worlds approach because an analogous ignorance, from a thoroughly different source, necessarily creeps in. Inhabitants of the Many Worlds only have access to their own single world; they do not experience the others. Everett argued that with such a limited perspective comes an infusion of probability.
To get a feel for how, leave quantum mechanics for a moment and consider an imperfect but helpful analogy. Imagine that aliens from the planet Zaxtar have succeeded in building a cloning machine that can make identical copies of you, me, or anyone. Were you to step into the cloning machine, and were two of you then to step out, both would be absolutely convinced that they were the real you, and both would be right. The Zaxtarians delight in subjecting less intelligent life-forms to existential dilemmas, so they swoop down to earth and make you the following offer. Tonight, when you go to sleep, you’ll be carefully wheeled into the cloning machine; five minutes later two of you will be wheeled out. When one of you awakes, life will be normal—except that you will have been granted any wish of your choosing. When the other you awakes, life won’t be normal; you will be escorted to a torture chamber back on Zaxtar, never to leave. And no, your lucky clone is not allowed to wish for your release. Do you accept the offer?
For most people, the answer is no. Since each of the clones really, truly is you, in accepting the offer you’d be guaranteeing that there will be a you who awakens to a lifetime of torment. Sure, there will also be a you who awakens to your usual life, augmented by the unlimited power of an arbitrary wish, but for the you on Zaxtar there’ll be nothing but torture. The price is too high.
Anticipating your reluctance, the Zaxtarians up the ante. Same deal, but now they’ll make a million and one copies of you. A million will wake up on a million identical-looking earths, with the power to fulfill any wish; one will get the Zaxtarian torture. Do you accept? At this point, you begin to waver. “Heck,” you think, “the odds seem pretty good that I won’t end up on Zaxtar but instead will wake up right here at home, wish in hand.”
This last intuition is particularly relevant to the Many Worlds approach. If odds entered your thinking because you imagine that only one of the million and one clones is the “real” you, then you’ve not taken in the scenario fully. Each copy is you. There’s a 100 percent certainty that one of you will wake up to an unbearable future. If this was indeed what led you to think in terms of odds, you need to let it go. However, probability may have entered your thinking in a more refined way. Imagine that you just agreed to the Zaxtarian offer and are now contemplating what it will be like to wake up tomorrow morning. Curled up under a warm duvet, just regaining consciousness but not yet having opened your eyes, you’ll remember the Zaxtarian deal. At first it will seem like an unusually vivid nightmare, but as your heart starts to pound you’ll recognize that it is real—that a million and one copies of you are in the process of waking up, with one of you destined for Zaxtar and the others about to be granted extraordinary power. “What are the odds,” you’ll ask yourself nervously, “that when I open my eyes I’ll be shipping out to Zaxtar?”
Before the cloning there was no sensible way to speak of whether it was or wasn’t likely that you’d be Zaxtar bound—it is absolutely certain that there will be such a you, so how could it be unlikely? But after the cloning, the situation seems different. Each clone experiences itself as the real you; indeed, each is the real you. But each copy is also a separate and distinct individual who can inquire about his or her own future. Each of the million and one copies can ask for the probability that they will go to Zaxtar. And since each knows that only one of the million and one will wake up to that outcome, each reckons that the odds of being that unlucky individual are low. Upon waking, a million will find their cheery expectation confirmed, and only one will not. So although there’s nothing uncertain, nothing chancy, nothing probabilistic in the Zaxtarian scenario—again, no dice are rolled and no roulette wheels spun—probability nevertheless seems to enter. It does so through the subjective ignorance experienced by each individual clone regarding which outcome he or she will witness.
This suggests a tack for injecting probabilities into the Many Worlds approach. Before you undertake a given experiment, you are much like your precloned self. You contemplate all outcomes allowed by quantum mechanics and know that there’s a 100 percent certainty that a copy of you will see each. Nothing at all chancy has made an appearance. You then undertake the experiment. At that point, as with the Zaxtarian scenario, a notion of probability presents itself. Each copy of you is an independent sentient being capable of wondering about which world he or she happens to inhabit—the likelihood, that is, that when the experiment’s results are revealed, he or she will see this or that particular outcome. Probability enters through each inhabitant’s subjective experience.
Everett’s approach, which he described as “objectively deterministic” with probability “reappearing at the subjective level,” resonated with this strategy. And he was thrilled by the direction. As he noted in the 1956 draft of his dissertation, the framework offered to bridge the position of Einstein (who famously believed that a fundamental theory of physics should not involve probability) and the position of Bohr (who was perfectly happy with a fundamental theory that did). According to Everett, the Many Worlds approach accommodated both positions, the difference between them merely being one of perspective. Einstein’s perspective is the mathematical one in which the grand probability wave of all particles relentlessly evolves by the Schrödinger equation, with chance playing absolutely no role.* I like to picture Einstein soaring high above the many worlds of Many Worlds, watching as Schrödinger’s equation fully dictates how the entire panorama unfolds, and happily concluding that even though quantum mechanics is correct, God doesn’t play dice. Bohr’s perspective is that of an inhabitant in one of the worlds, also happy, using probabilities to explain, with stupendous precision, those observations to which his limited perspective gives him access.
It’s a captivating vision—Einstein and Bohr agreeing on quantum mechanics. But there are pesky details that for more than half a century have convinced many that it’s still too early to sign on. Those who have studied Everett’s thesis generally agree that while his intent was clear—a deterministic theory that to its inhabitants nevertheless appears probabilistic—he didn’t convincingly spell out how to achieve it. For example, much in the spirit of material covered in Chapter 7, Everett sought to determine what a “typical” inhabitant of the many worlds would observe in any given experiment. But (unlike our focus in Chapter 7) in the Many Worlds approach, the inhabitants we need to contend with are all the same person; if you’re the experimenter, they are all you, and collectively they will see a range of different outcomes. So who is the “typical” you?
Inspired by the Zaxtarian scenario, a natural suggestion is to count the number of yous who will see a given result; the outcome seen by the greatest number of yous would then qualify as typical. Or, more quantitatively, define the probability of a result to be proportional to the number of yous who see it. For simple examples, this works: in Figure 8.16, there’s one of you who sees each outcome, and so you peg the odds at 50:50 for seeing one result or the other. That’s good; the usual quantum mechanical prediction is also 50:50, because the probability wave heights at the two locations are equal.
Figure 8.17 The combined probability wave for you and your device encounters a probability wave that has multiple spikes of different magnitudes.
However, consider a more general situation, such as that in Figure 8.17, in which the probability wave heights are unequal. If the wave is a hundred times larger at Strawberry Fields than at Grant’s Tomb, then quantum mechanics predicts that you are a hundred times more likely to find the electron at Strawberry Fields. But in the Many Worlds approach, your measurement still generates one you who sees Strawberry Fields and another you who sees Grant’s Tomb; the odds based on counting the number of yous is thus still 50:50—the wrong result. The origin of the mismatch is clear. The number of yous who see one result or another is determined by the number of spikes in the probability wave. But the quantum mechanical probabilities are determined by something else—not by the number of spikes but by their relative heights. And it’s these predictions, the quantum mechanical predictions, which have been convincingly confirmed by experiments.
Everett developed a mathematical argument that was meant to address this mismatch; many others have since pushed it further.9 In broad strokes, the idea is that in calculating the odds of seeing one or another outcome, we should place ever-less weight on universes whose wave heights are ever smaller, as depicted symbolically in Figure 8.18. But this is perplexing. And controversial. Is the universe in which you find the electron at Strawberry Fields somehow a hundred times as genuine, or a hundred times as likely, or a hundred times as relevant as the one in which you find it at Grant’s Tomb? These suggestions would surely create tension with the belief that every world is just as real as every other.
After more than fifty years, during which distinguished scientists have revisited, revised, and extended Everett’s arguments, many agree that the puzzles persist. Yet it remains seductive to imagine that the mathematically simple, totally bare-bones, profoundly revolutionary Many Worlds approach yields the probabilistic predictions that form the foundation of belief in quantum theory. This has inspired many other ideas, beyond the Zaxtarian-type reasoning, for joining probability and Many Worlds.10
A prominent proprosal comes from a leading group of researchers at Oxford, including, among others, David Deutsch, Simon Saunders, David Wallace, and Hilary Greaves. They’ve developed a sophisticated line of attack that focuses on a seemingly boorish question. If you’re a gambler, and you believe in the Many Worlds approach, what’s the optimum strategy for placing bets on quantum mechanical experiments? Their answer, which they argue for mathematically, is that you’d bet just as Neils Bohr would. When speaking of maximizing your return, these authors have in mind something that would have sent Bohr into a tizzy—they’re considering an average over the many inhabitants of the multiverse who claim to be you. But even so, their conclusion is that the numbers that Bohr and everyone since have been calculating and calling probabilities are the very numbers that should guide how you wager. That is, even though quantum theory is fully deterministic, you should treat the numbers as if they were probabilities.
Some are convinced that this completes Everett’s program. Some are not.
The lack of consensus on the crucial question of how to treat probability in the Many Worlds approach is not all that unexpected. The analyses are highly technical and also deal with a topic—probability—that is notoriously tricky even outside its application to quantum theory. When you roll a die, we all agree that you have a 1 in 6 chance of getting a 3, and so we’d predict that over the course of, say, 1,200 rolls the number 3 will turn up about 200 times. But since it’s possible, in fact likely, that the number of 3s will deviate from 200, what does the prediction mean? We want to say that it’s highly probable that ⅙th of the outcomes will be 3s, but if we do that, then we’ve defined the probability of getting a 3 by invoking the concept of probability. We’ve gone circular. That’s just a small taste of how the issues, beyond their intrinsic mathematical complexity, are conceptually slippery. Throw into the mix the added Many Worlds intricacy of “you” no longer referring to a single person, and it’s no wonder researchers find ample points of contention. I have little doubt that full clarity will one day emerge, but not yet, and perhaps not for some time.
Figure 8.18 (a) A schematic illustration of the evolution, dictated by Schrödinger’s equation, of the combined probability wave for all the particles making up you and the measuring device, when you measure the position of an electron. The electron’s own probability wave is spiked at two locations, but with unequal wave heights.
Figure 8.18 (b) Some proposals suggest that in the Many Worlds approach, unequal wave heights imply that some worlds are less genuine, or less relevant, than others. There is controversy over what, if anything, this means.
Predictions and Understanding
For all these controversies, quantum mechanics itself remains as successful as any theory in the history of ideas. The reason, as we’ve seen, is that for the kinds of experiments we can do in the laboratory, and for many of the observations we can make of astrophysical processes, we have a “quantum algorithm” that produces testable predictions. Use Schrödinger’s equation to calculate the evolution of the relevant probability waves and use the results—the various wave heights—to predict the probability that you’ll find one outcome or another. As far as predictions are concerned, why this algorithm works—whether the wave collapses upon measurement, whether all possibilities are realized in their own universes, whether some other process is at work—is secondary.
Some physicists argue that even calling the issue secondary accords it more status than it deserves. In their view, physics is only about making predictions, and as long as different approaches don’t affect those predictions, why should we care which is ultimately correct? I offer three thoughts.
First, beyond making predictions, physical theories need to be mathematically coherent. The Copenhagen approach is a valiant effort, but it fails to meet this standard: at the critical moment of observation, it retreats into mathematical silence. That’s a substantial gap. The Many Worlds approach attempts to fill it.11
Second, in some situations, the predictions of the Many Worlds approach would differ from those of the Copenhagen approach. In Copenhagen, the process of collapse would revise Figure 8.16a to have a single spike. So if you could cause the two waves depicted in the figure—representing macroscopically distinct situations—to interfere, generating a pattern similar to that in Figure 8.2c, it would establish that Copenhagen’s hypothesized wave collapse didn’t happen. Because of decoherence, as discussed earlier, it is an extraordinarily formidable task to do this, but, at least theoretically speaking, the Copenhagen and Many Worlds approaches yield different predictions.12 It is an important point of principle. The Copanhagen and Many Worlds approaches are often referred to as different “interpretations” of quantum mechanics. This is an abuse of language. If two approaches can yield different predictions, you can’t call them mere interpretations. Well, you can. And people do. But the terminology is off the mark.
Third, physics is not just about making predictions. If one day we were to find a black box that always and accurately predicted the outcome of our particle physics experiments and our astronomical observations, the existence of the box would not bring inquiry in these fields to a close. There’s a difference between making predictions and understanding them. The beauty of physics, its raison d’être, is that it offers insights into why things in the universe behave the way they do. The ability to predict behavior is a big part of physics’ power, but the heart of physics would be lost if it didn’t give us a deep understanding of the hidden reality underlying what we observe. And should the Many Worlds approach be right, what a spectacular reality our unwavering commitment to understanding predictions will have uncovered.
I don’t expect theoretical or experimental consensus to come in my lifetime concerning which version of reality—a single universe, a multiverse, something else entirely—quantum mechanics embodies. But I have little doubt that future generations will look back upon our work in the twentieth and twenty-first centuries as having nobly laid the basis for whatever picture finally emerges.
*For simplicity, we won’t consider the electron’s position in the vertical direction—we focus solely on its position on a map of Manhattan. Also, let me re-emphasize that while this section will make clear that Schrödinger’s equation doesn’t allow waves to undergo an instantaneous collapse as in Figure 8.6, waves can be carefully prepared by the experimenter in a spiked shape (or, more precisely, very close to a spiked shape).
*For a mathematical depiction, see note 4.
*This non-chancy perspective would argue strongly for abandoning the colloquial terminology that I’ve used, “probability wave,” in favor of the technical name, “wavefunction.”