The Hidden Reality Part 8

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 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 const.i.tuents, to which Schrodinger'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 pa.s.sing year, experimenters confirm that Schrodinger'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 bas.e.m.e.nt, rus.h.i.+ng 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. Schrodinger's equation works for electrons and quarks, and all evidence points to its working for things made of these const.i.tuents, regardless of the number of particles involved. This means that Schrodinger'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 Schrodinger's math refuses to bow down, then Bohr is in trouble. Schrodinger'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 pa.s.s 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 Schrodinger's equation. I've emphasized that it doesn't allow probability waves to suddenly collapse. But why not? And what does does it allow? Let's get a feel for how Schrodinger's math guides a probability wave as it evolves through time. it allow? Let's get a feel for how Schrodinger's math guides a probability wave as it evolves through time.

This is fairly straightforward, because Schrodinger's is one of the simplest kinds of mathematical equations, characterized by a property known as linearity 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 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 Schrodinger's equation to follow the evolution of this wave forward in time, yielding its shape at, say, one p.m., schematically ill.u.s.trated in 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 Schrodinger's equation to follow the evolution of this wave forward in time, yielding its shape at, say, one p.m., schematically ill.u.s.trated in Figure 8.7b Figure 8.7b. Now notice the following. You can decompose the initial wave shape in Figure 8.7a Figure 8.7a into two simpler pieces, as in into two simpler pieces, as in Figure 8.8a 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 Schrodinger's equation means that you can use it on each piece in Figure 8.8a Figure 8.8a separately, determining what each wave fragment will look like at one p.m., and then combine the results as in separately, determining what each wave fragment will look like at one p.m., and then combine the results as in Figure 8.8b Figure 8.8b to recover the full result shown in to recover the full result shown in Figure 8.7b Figure 8.7b. And there's nothing sacred about decomposition into two 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. 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 Schrodinger's equation to a different shape An initial probability wave shape at one moment evolves via Schrodinger's equation to a different shape (b) (b) at a later time 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 a.n.a.lyze 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 a.n.a.lysis of the double-slit experiment in Figure 8.4 Figure 8.4. To determine how the electron's probability wave evolves, we divided the task: we noted how the piece pa.s.sing through the left slit evolves, we noted how the piece pa.s.sing 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 An initial probability wave shape can be decomposed as the union of two simpler shapes. (b) (b) The evolution of the initial probability wave can be reproduced by evolving the simpler pieces and combining the results 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 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 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 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 Figure 8.9.

It would be extraordinarily complicated to work out how Schrodinger'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 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, a.s.sume that Schrodinger'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. Schrodinger's equation causes the device's display to spell out the spike's location, as in Figure 8.9 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 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. Schrodinger'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 rus.h.i.+ng through your neurons, creating a mental state reflecting what you see. a.s.suming unlimited Schrodinger 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 Schrodinger's equation suggests that a measurement of the electron's position would yield a confusing amalgam of both locations An electron's probability wave is spiked at two locations. The linearity of Schrodinger'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 ill.u.s.trated in Figure 8.11 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 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 melange 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 Schrodinger's equation into the domain of big things: laboratory equipment that takes measurements, and scientists who read the results. Although Schrodinger'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 Schrodinger's math out the window. Instead, he'd p.r.o.nounce, the measurement causes all but one of the spikes in Figure 8.10 Figure 8.10 or or Figure 8.11 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. 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. Schrodinger's equation is meant to describe particles. All particles. Why would it somehow not apply to particular configurations of particles-those const.i.tuting 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 Schrodinger so quickly. Instead, he advocated that we a.n.a.lyze where Schrodinger'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 const.i.tute 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 Schrodinger's math, the core of quantum mechanics, is 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 ill.u.s.trated in 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 ill.u.s.trated in Figure 8.10 Figure 8.10 and and Figure 8.11 Figure 8.11. Let's think it through.

When you measure a single spiked wave, such as that in Figure 8.9 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 Figure 8.10, Schrodinger'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 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 Figure 8.11, which can be part.i.tioned 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 possible, quantum-mechanically speaking (that is, all those outcomes to which quantum mechanics a.s.signs a nonzero probability), is realized realized in its own separate world. These are the "many worlds" of the in its own separate world. These are the "many worlds" of the Many Worlds Many Worlds approach to quantum mechanics. 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 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 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 In the time of Newton, the two stories were essentially identical, as I suggested with my remarks in Chapter 7 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 about Newtonian "architecture" being immediate and palpable. Every mathematical symbol in Newton's equations has a direct and transparent physical correlate. The symbol x x? Oh, that's the ball's position. The symbol v 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. Schrodinger's equation determines how probability waves evolve over time, and it is never set aside; it is always always in effect. Schrodinger's math guides the shape of probability waves, causing them to s.h.i.+ft, 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 a.s.semblages of particles that const.i.tute you and your measuring equipment, Schrodinger'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. in effect. Schrodinger's math guides the shape of probability waves, causing them to s.h.i.+ft, 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 a.s.semblages of particles that const.i.tute you and your measuring equipment, Schrodinger'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 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 Schrodinger's math to figure out how the probability wave describing the positions of the huge number of particles that const.i.tute 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 Schrodinger's equation for even a single particle. Between you and the device, there are something like 1027 particles. Working out Schrodinger's math for that many const.i.tuents is virtually impossible. Even so, we understand qualitatively what the math entails. When we measure the electron's position, we cause a ma.s.s particle migration. Some 10 particles. Working out Schrodinger's math for that many const.i.tuents is virtually impossible. Even so, we understand qualitatively what the math entails. When we measure the electron's position, we cause a ma.s.s particle migration. Some 1024 or so particles in the device's display, like performers in a crisply ch.o.r.eographed 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. Schrodinger's math-however impenetrable explicit a.n.a.lysis of it might be when faced with so many particles-describes such a particle s.h.i.+ft. or so particles in the device's display, like performers in a crisply ch.o.r.eographed 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. Schrodinger's math-however impenetrable explicit a.n.a.lysis of it might be when faced with so many particles-describes such a particle s.h.i.+ft.

To visualize this transformation at the level of a probability wave is also far beyond reach. In Figure 8.9 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 10 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. 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 Adding even a single vertical axis to Figure 8.9 Figure 8.9 would have made it difficult to visualize; to contemplate adding a billion billion billion more is, well, silly. 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.

Figure 8.13 A schematic depiction of the combined probability wave for all the particles making up you and your measuring device 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 ill.u.s.trations have some features in common. Just as the spiked wave shape in Figure 8.6 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 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 Figure 8.13 also denotes sharply skewed probabilities. But you need to exercise care, because understanding based on the single-particle ill.u.s.trations can take you only so far. For example, based on also denotes sharply skewed probabilities. But you need to exercise care, because understanding based on the single-particle ill.u.s.trations can take you only so far. For example, based on Figure 8.6 Figure 8.6 it is natural to think that it is natural to think that Figure 8.13 Figure 8.13 represents particles that are all cl.u.s.tered around the same location. Yet, that's not right. The peaked shape in represents particles that are all cl.u.s.tered around the same location. Yet, that's not right. The peaked shape in Figure 8.13 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 const.i.tuting your hand, shoulder, and brain are, with near certainty, cl.u.s.tered within the location of your hand, shoulder, and brain; the particles const.i.tuting the measuring device are, with near certainty, cl.u.s.tered within the location of the device. The peaked wave shape in 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 const.i.tuting your hand, shoulder, and brain are, with near certainty, cl.u.s.tered within the location of your hand, shoulder, and brain; the particles const.i.tuting the measuring device are, with near certainty, cl.u.s.tered within the location of the device. The peaked wave shape in Figure 8.13 Figure 8.13 denotes that each of these particles has only the most remote chance of being found anywhere else. denotes that each of these particles has only the most remote chance of being found anywhere else.

If you now perform the measurement ill.u.s.trated in Figure 8.14 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 ill.u.s.trated schematically in Figure 8.14a 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 Figure 8.14a maintains a spiked shape. But a ma.s.s particle rearrangement occurs that results in the words "Strawberry Fields" forming in the device's readout and also in your brain (as in maintains a spiked shape. But a ma.s.s 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 8.14b). Figure 8.14a Figure 8.14a represents the mathematical transformation dictated by Schrodinger's equation, the first kind of story. represents the mathematical transformation dictated by Schrodinger's equation, the first kind of story. Figure 8.14b Figure 8.14b ill.u.s.trates the physical description of such mathematical evolution, the second kind of story. Similarly, if we perform the experiment in ill.u.s.trates the physical description of such mathematical evolution, the second kind of story. Similarly, if we perform the experiment in Figure 8.15 Figure 8.15, an a.n.a.logous wave s.h.i.+ft takes place (Figure 8.15a). This s.h.i.+ft corresponds to a ma.s.s particle rearrangement that spells out "Grant's Tomb" in the display and generates within you the a.s.sociated 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 Figure 8.16a-the combined evolutions depicted in Figure 8.14a Figure 8.14a and and Figure 8.15a Figure 8.15a. So far, this is nothing but an ill.u.s.trated and annotated version of the first type of quantum story. We start with a probability wave of a given shape, Schrodinger'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 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, 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 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. 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 ill.u.s.trated in Figure 8.16b 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 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." 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 ill.u.s.tration of the evolution, dictated by Schrodinger'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 A schematic ill.u.s.tration of the evolution, dictated by Schrodinger'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 The corresponding physical, or experiential, story.

Figure 8.15 (a) The same type of mathematical evolution as in The same type of mathematical evolution as in Figure 8.14a 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 The corresponding physical, or experiential, story.

Figure 8.16 (a) A schematic ill.u.s.tration 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 A schematic ill.u.s.tration 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 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 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 Schrodinger'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 particular shape, as in shape, as in Figure 8.16a 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 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 Schrodinger's purview, would collapse away all the others), and whereas a first-pa.s.s attempt to go beyond Bohr and extend Schrodinger's math to all particles, including those const.i.tuting 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 Schrodinger'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 Schrodinger'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 Figures 8.2 and and 8.4. 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. 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 pa.s.ses 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 pa.s.sed through left slit" or "electron pa.s.sed 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 Figure 8.16a, the differences between the billions and billions of particles in the meters cause the waves for the two outcomes to s.h.i.+ft 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 Figure 8.2c; instead, they generate a simple, non-interfering amalgam of the results in Figure 8.2a Figure 8.2a and and Figure 8.2b Figure 8.2b. Physicists say that the probability waves have decohered decohered (something you can read about in more detail, for example, in Chapter 7 of (something you can read about in more detail, for example, in Chapter 7 of The Fabric of the Cosmos 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 appears as though the other outcomes have vanished. Within each universe, that is, it's as though the other outcomes have vanished. Within each universe, that is, it's as if 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. 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 quant.i.tative. 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 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? 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 a.s.sumptions 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 a.s.sessing the complexity of a scientific proposal, you shouldn't focus on its implications implications. What matters is the fundamental features of the proposal itself. The Many Worlds approach a.s.sumes that a single equation-Schrodinger's-governs all probability waves all the time, so for simplicity of formulation and economy of a.s.sumptions, it's hard to beat. The Copenhagen approach is surely no simpler. It, too, invokes Schrodinger's equation, but it also includes a vague, ill-defined prescription for when Schrodinger'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 a.s.sessing 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 a.s.sumption 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 or tails seems nowhere to be found. 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 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 or the other? 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 really you. And, this line of thought continues, it's you. And, this line of thought continues, it's that 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 Schrodinger'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 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 Schrodinger'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 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 Schrodinger equation. 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 Schrodinger equation.

This realization s.h.i.+nes 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 b.u.mped 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 Schrodinger'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 a.n.a.logous 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 a.n.a.logy. 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 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. 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.

Antic.i.p.ating 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 I won't end up on Zaxtar but instead will wake up right here at home, wish in hand." 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 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 o