The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos

by Brian Greene

  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 does it 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 theor
ist’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.

  Many Worlds

  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.