The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  Since the 1920s, physicists have known that particles spin—they execute rotational motion akin to a soccer ball's spinning around as it heads toward the goal. Quantum particle spin, however, differs from this classical image in a number of essential ways, and foremost for us are the following two points. First, particles—for example, electrons and photons— can spin only clockwise or counterclockwise at one never-changing rate about any particular axis; a particle's spin axis can change directions but its rate of spin cannot slow down or speed up. Second, quantum uncertainty applied to spin shows that just as you can't simultaneously determine the position and the velocity of a particle, so also you can't simultaneously determine the spin of a particle about more than one axis. For example, if a soccer ball is spinning about a northeast-pointing axis, its spin is shared between a northward- and an eastward-pointing axis—and by a suitable measurement, you could determine the fraction of spin about each. But if you measure an electron's spin about any randomly chosen axis, you never find a fractional amount of spin. Ever. It's as if the measurement itself forces the electron to gather together all its spinning motion and direct it to be either clockwise or counterclockwise about the axis you happened to have focused on. Moreover, because of your measurement's influence on the electron's spin, you lose the ability to determine how it was spinning about a horizontal axis, about a back-and-forth axis, or about any other axis, prior to your measurement. These features of quantum mechanical spin are hard to picture fully, and the difficulty highlights the limits of classical images in revealing the true nature of the quantum world. Nevertheless, the mathematics of quantum theory, and decades of experiment, assure us that these characteristics of quantum spin are beyond doubt.

  The reason for introducing spin here is not to delve into the intricacies of particle physics. Rather, the example of particle spin will, in just a moment, provide a simple laboratory for extracting wonderfully unexpected answers to the reality question. That is, does a particle simultaneously have a definite amount of spin about each and every axis, although we can never know it for more than one axis at a time because of quantum uncertainty? Or does the uncertainty principle tell us something else? Does it tell us, contrary to any classical notion of reality, that a particle simply does not and cannot possess such features simultaneously? Does it tell us that a particle resides in a state of quantum limbo, having no definite spin about any given axis, until someone or something measures it, causing it to snap to attention and attain—with a probability determined by quantum theory—one particular spin value or another (clockwise or counterclockwise) about the selected axis? By studying this question, essentially the same one we asked in the case of particle positions and velocities, we can use spin to probe the nature of quantum reality (and to extract answers that greatly transcend the specific example of spin). Let's see this.

  As explicitly shown by the physicist David Bohm, 11 the reasoning of Einstein, Podolsky, and Rosen can easily be extended to the question of whether particles have definite spins about any and all chosen axes. Here's how it goes. Set up two detectors capable of measuring the spin of an incoming electron, one on the left side of the laboratory and the other on the right side. Arrange for two electrons to emanate back-to-back from a source midway between the two detectors, such that their spins—rather than their positions and velocities as in our earlier example—are correlated. The details of how this is done are not important; what is important is that it can be done and, in fact, can be done easily. The correlation can be arranged so that if the left and right detectors are set to measure the spins along axes pointing in the same direction, they will get the same result: if the detectors are set to measure the spin of their respective incoming electrons about a vertical axis and the left detector finds that the spin is clockwise, so will the right detector; if the detectors are set to measure spin along an axis 60 degrees clockwise from the vertical and the left detector measures a counterclockwise spin, so will the right detector; and so on. Again, in quantum mechanics the best we can do is predict the probability that the detectors will find clockwise or counterclockwise spin, but we can predict with 100 percent certainty that whatever one detector finds the other will find, too. 7

  Bohm's refinement of the EPR argument is now, for all intents and purposes, the same as it was in the original version that focused on position and velocity. The correlation between the particles' spins allows us to measure indirectly the spin of the left-moving particle about some axis by measuring that of its right-moving companion about that axis. Since this measurement is done far on the right side of the laboratory, it can't possibly influence the left-moving particle in any way. Hence, the latter must all along have had the spin value just determined; all we did was measure it, albeit indirectly. Moreover, since we could have chosen to perform this measurement about any axis, the same conclusion must hold for any axis: the left-moving electron must have a definite spin about each and every axis, even though we can explicitly determine it only about one axis at a time. Of course, the roles of left and right can be reversed, leading to the conclusion that each particle has a definite spin about any axis. 12

  At this stage, seeing no obvious difference from the position/velocity example, you might take Pauli's lead and be tempted to respond that there is no point in thinking about such issues. If you can't actually measure the spin about different axes, what is the point in wondering whether the particle nevertheless has a definite spin—clockwise versus counterclockwise—about each? Quantum mechanics, and physics more generally, is obliged only to account for features of the world that can be measured. And neither Bohm, Einstein, Podolsky, nor Rosen would have argued that the measurements can be done. Instead, they argued that the particles possess features forbidden by the uncertainty principle even though we can never explicitly know their particular values. Such features have come to be known as hidden features, or, more commonly, hidden variables.

  Here is where John Bell changed everything. He discovered that even if you can't actually determine the spin of a particle about more than one axis, still, if in fact it has a definite spin about all axes, then there are testable, observable consequences of that spin.

  Reality Testing

  To grasp the gist of Bell's insight, let's return to Mulder and Scully and imagine that they've each received another package, also containing titanium boxes, but with an important new feature. Instead of having one door, each titanium box has three: one on top, one on the side, and one on the front. 13 The accompanying letter informs them that the sphere inside each box now randomly chooses between flashing red and flashing blue when any one of the box's three doors is opened. If a different door (top versus side versus front) on a given box were opened, the color randomly selected by the sphere might be different, but once one door is opened and the sphere has flashed, there is no way to determine what would have happened had another door been chosen. (In the physics application, this feature captures quantum uncertainty: once you measure one feature you can't determine anything about the others.) Finally, the letter tells them that there is again a mysterious connection, a strange entanglement, between the two sets of titanium boxes: Even though all the spheres randomly choose what color to flash when one of their box's three doors is opened, if both Mulder and Scully happen to open the same door on a box with the same number, the letter predicts that they will see the same color flash. If Mulder opens the top door on his box 1 and sees blue, then the letter predicts that Scully will also see blue if she opens the top door on her box 1; if Mulder opens the side door on his box 2 and sees red, then the letter predicts that Scully will also see red if she opens the side door on her box 2, and so forth. Indeed, when Scully and Mulder open the first few dozen boxes—agreeing by phone which door to open on each—they verify the letter's predictions.

  Although Mulder and Scully are being presented with a somewhat more complicated situation than previously, at first blush it seems that the same reasoning Scully used earlier applies equally well her
e.

  "Mulder," says Scully, "this is as silly as yesterday's package. Once again, there is no mystery. The sphere inside each box must simply be programmed. Don't you see?"

  "But now there are three doors," cautions Mulder, "so the sphere can't possibly 'know' which door we'll choose to open, right?"

  "It doesn't need to," explains Scully. "That's part of the programming. Look, here's an example. Grab hold of the next unopened box, box 37, and I'll do the same. Now, imagine, for argument's sake, that the sphere in my box 37 is programmed, say, to flash red if the top door is opened, to flash blue if the side door is opened, and to flash red if the front door is opened. I'll call this program red, blue, red. Clearly, then, if whoever is sending us this stuff has input this same program into your box 37, and if we both open the same door, we will see the same color flash. This explains the 'mysterious connection': if the boxes in our respective collections with the same number have been programmed with the same instructions, then we will see the same color if we open the same door. There is no mystery!"

  But Mulder does not believe that the spheres are programmed. He believes the letter. He believes that the spheres are randomly choosing between red and blue when one of their box's doors is opened and hence he believes, fervently, that his and Scully's boxes do have some mysterious long-range connection.

  Who is right? Since there is no way to examine the spheres before or during the supposed random selection of color (remember, any such tampering will cause the sphere instantly to choose randomly between red or blue, confounding any attempt to investigate how it really works), it seems impossible to prove definitively whether Scully or Mulder is right.

  Yet, remarkably, after a little thought, Mulder realizes that there is an experiment that will settle the question completely. Mulder's reasoning is straightforward, but it does require a touch more explicit mathematical reasoning than most things we cover. It's definitely worth trying to follow the details—there aren't that many—but don't worry if some of it slips by; we'll shortly summarize the key conclusion.

  Mulder realizes that he and Scully have so far only considered what happens if they each open the same door on a box with a given number. And, as he excitedly tells Scully after calling her back, there is much to be learned if they do not always choose the same door and, instead, randomly and independently choose which door to open on each of their boxes.

  "Mulder, please. Just let me enjoy my vacation. What can we possibly learn by doing that?"

  "Well, Scully, we can determine whether your explanation is right or wrong."

  "Okay, I've got to hear this."

  "It's simple," Mulder continues. "If you're right, then here's what I realized: if you and I separately and randomly choose which door to open on a given box and record the color we see flash, then, after doing this for many boxes we must find that we saw the same color flash more than 50 percent of the time. But if that isn't the case, if we find that we don't agree on the color for more than 50 percent of the boxes, then you can't be right."

  "Really, how is that?" Scully is getting a bit more interested.

  "Well," Mulder continues, "here's an example. Assume you're right, and each sphere operates according to a program. Just to be concrete, imagine the program for the sphere in a particular box happens to be blue, blue, red. Now since we both choose from among three doors, there are a total of nine possible door combinations that we might select to open for this box. For example, I might choose the top door on my box while you might choose the side door on your box; or I might choose the front door and you might choose the top door; and so on."

  "Yes, of course," Scully jumps in. "If we call the top door 1, the side door 2, and the front door 3, then the nine possible door combinations are just (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)."

  "Yes, that's right," Mulder continues. "Now here is the point: Of these nine possibilities notice that five door combinations—(1,1), (2,2), (3,3), (1,2), (2,1)—will result in us seeing the spheres in our boxes flash the same color. The first three door combinations are the ones in which we happen to choose the same door, and as we know, that always results in our seeing the same color. The other two door combinations, (1,2) and (2,1), result in the same color because the program dictates that the spheres will flash the same color—blue—if either door 1 or door 2 is opened. Now, since 5 is more than half of 9, this means that for more than half—more than 50 percent—of the possible combination of doors that we might select to open, the spheres will flash the same color."

  "But wait," Scully protests. "That's just one example of a particular program: blue, blue, red. In my explanation, I proposed that differently numbered boxes can and generally will have different programs."

  "Actually, that doesn't matter. The conclusion holds for all of the possible programs. You see, my reasoning with the blue, blue, red program only relied on the fact that two of the colors in the program are the same, and so an identical conclusion follows for any program: red, red, blue, or red, blue, red, and so on. Any program has to have at least two colors the same; the only programs that are really different are those in which all three colors are the same —red, red, red and blue, blue, blue. But for boxes with either of these programs, we'll get the same color to flash regardless of which doors we happen to open, and so the overall fraction on which we should agree will only increase. So, if your explanation is right and the boxes operate according to programs—even with programs that vary from one numbered box to another—we must agree on the color we see more than 50 percent of the time."

  That's the argument. The hard part is now over. The bottom line is that there is a test to determine whether Scully is correct and each sphere operates according to a program that determines definitively which color to flash depending on which door is opened. If she and Mulder independently and randomly choose which of the three doors on each of their boxes to open, and then compare the colors they see—box by numbered box—they must find agreement for more than 50 percent of the boxes.

  When cast in the language of physics, as it will be in the next section, Mulder's realization is nothing but John Bell's breakthrough.

  Counting Angels with Angles

  The translation of this result into physics is straightforward. Imagine we have two detectors, one on the left side of the laboratory and another on the right side, that measure the spin of an incoming particle like an electron, as in the experiment discussed in the section before last. The detectors require you to choose the axis (vertical, horizontal, back-forth, or one of the innumerable axes that lie in between) along which the spin is to be measured; for simplicity's sake, imagine that we have bargain-basement detectors that offer only three choices for the axes. In any given run of the experiment, you will find that the incoming electron is either spinning clockwise or counterclockwise about the axis you selected.

  According to Einstein, Podolsky, and Rosen, each incoming electron provides the detector it enters with what amounts to a program: Even though it's hidden, even though you can't measure it, EPR claimed that each electron has a definite amount of spin—either clockwise or counterclockwise—about each and every axis. Hence, when an electron enters a detector, the electron definitively determines whether you will measure its spin to be clockwise or counterclockwise about whichever axis you happen to choose. For example, an electron that is spinning clockwise about each of the three axes provides the program clockwise, clockwise, clockwise; an electron that is spinning clockwise about the first two axes and counterclockwise about the third provides the program clockwise, clockwise, counterclockwise, and so forth. In order to explain the correlation between the left-moving and right-moving electrons, Einstein, Podolsky, and Rosen simply claim that such electrons have identical spins and thus provide the detectors they enter with identical programs. Thus, if the same axes are chosen for the left and right detectors, the spin detectors will find identical results.

  Notice that these spin detectors exactly reproduce ever
ything encountered by Scully and Mulder, though with simple substitutions: instead of choosing a door on a titanium box, we are choosing an axis; instead of seeing a red or blue flash, we record a clockwise or counterclockwise spin. So, just as opening the same doors on a pair of identically numbered titanium boxes results in the same color flashing, choosing the same axes on the two detectors results in the same spin direction being measured. Also, just as opening one particular door on a titanium box prevents us from ever knowing what color would have flashed had we chosen another door, measuring the electron spin about one particular axis prevents us, via quantum uncertainty, from ever knowing which spin direction we would have found had we chosen a different axis.

  All of the foregoing means that Mulder's analysis of how to learn who's right applies in exactly the same way to this situation as it does to the case of the alien spheres. If EPR are correct and each electron actually has a definite spin value about all three axes—if each electron provides a "program" that definitively determines the result of any of the three possible spin measurements—then we can make the following prediction. Scrutiny of data gathered from many runs of the experiment—runs in which the axis for each detector is randomly and independently selected—will show that more than half the time, the two electron spins agree, being both clockwise or both counterclockwise. If the electron spins do not agree more than half the time, then Einstein, Podolsky, and Rosen are wrong.

  This is Bell's discovery. It shows that even though you can't actually measure the spin of an electron about more than one axis—even though you can't explicitly "read" the program it is purported to supply to the detector it enters—this does not mean that trying to learn whether it nonetheless has a definite amount of spin about more than one axis is tantamount to counting angels on the head of a pin. Far from it. Bell found that there is a bona fide, testable consequence associated with a particle having definite spin values. By using axes at three angles, Bell provided a way to count Pauli's angels.