by Brian Greene
5. From this description, you might conclude that there are infinitely many locations that the electron could be found: to properly fill out the gradually varying quantum wave you would need an infinite number of spiked shapes, each associated with a possible position of the electron. How does this relate to Chapter 2 in which we discussed there being finitely many distinct configurations for particles? To avoid constant qualifications that would be of minimal relevance to the major points I am explaining in this chapter, I have not emphasized the fact, encountered in Chapter 2, that to pinpoint the electron’s location with ever-greater accuracy your device would need to exert ever-greater energy. As physically realistic situations have access to finite energy, resolution is thus imperfect. For the spiked quantum waves, this means that in any finite energy context, the spikes have nonzero width. In turn, this implies that in any bounded domain (such as a cosmic horizon) there are finitely many measurably distinct electron locations. Moreover, the thinner the spikes are (the more refined the resolution of the particle’s position) the wider are the quantum waves describing the particle’s energy, illustrating the trade-off necessitated by the uncertainty principle.
6. For the philosophically inclined reader, I’ll note that the two-tiered story for scientific explanation which I’ve outlined has been the subject of philosophical discussion and debate. For related ideas and discussions see Frederick Suppe, The Semantic Conception of Theories and Scientific Realism (Chicago: University of Illinois Press, 1989); James Ladyman, Don Ross, David Spurrett, and John Collier, Every Thing Must Go (Oxford: Oxford University Press, 2007).
7. Physicists often speak loosely of there being infinitely many universes associated with the Many Worlds approach to quantum mechanics. Certainly, there are infinitely many possible probability wave shapes. Even at a single location in space you can continuously vary the value of a probability wave, and so there are infinitely many different values it can have. However, probability waves are not the physical attribute of a system to which we have direct access. Instead, probability waves contain information about the possible distinct outcomes in a given situation, and these need not have infinite variety. Specifically, the mathematically inclined reader will note that a quantum wave (a wavefunction) lies in a Hilbert space. If that Hilbert space is finite-dimensional, then there are finitely many distinct possible outcomes for measurements on the physical system described by that wavefunction (that is, any Hermitian operator has finitely many distinct eigenvalues). This would entail finitely many worlds for a finite number of observations or measurements. It is believed that the Hilbert space associated with physics taking place within any finite volume of space, and limited to having a finite amount of energy, is necessarily finite dimensional (a point we will take up more generally in Chapter 9), which suggests that the number of worlds would similarly be finite.
8. See Peter Byrne, The Many Worlds of Hugh Everett III (New York: Oxford University Press, 2010), p. 177.
9. Over the years, a number of researchers including Neill Graham; Bryce DeWitt; James Hartle; Edward Farhi, Jeffrey Goldstone, and Sam Gutmann; David Deutsch; Sidney Coleman; David Albert; and others, including me, have independently come upon a striking mathematical fact that seems central to understanding the nature of probability in quantum mechanics. For the mathematically inclined reader, here’s what it says: Let be the wavefunction for a quantum mechanical system, a vector that’s an element of the Hilbert space H. The wavefunction for n-identical copies of the system is thus . Let A be any Hermitian operator with eigenvalues αk, and eigenfunctions. Let Fk(A) be the “frequency” operator that counts the number of times appears in a given state lying in . The mathematical result is that lim. That is, as the number of identical copies of the system grows without bound, the wavefunction of the composite system approaches an eigenfunction of the frequency operator, with eigenvalue . This is a remarkable result. Being an eigenfunction of the frequency operator means that, in the stated limit, the fractional number of times an observer measuring A will find αk is —which looks like the most straightforward derivation of the famous Born rule for quantum mechanical probability. From the Many Worlds perspective, it suggests that those worlds in which the fractional number of times that αk is observed fails to agree with the Born rule have zero Hilbert space norm in the limit of arbitrarily large n. In this sense, it seems as though quantum mechanical probability has a direct interpretation in the Many Worlds approach. All observers in the Many Worlds will see results with frequencies that match those of standard quantum mechanics, except for a set of observers whose Hilbert space norm becomes vanishingly small as n goes to infinity. As promising as this seems, on reflection it is less convincing. In what sense can we say that an observer with a small Hilbert space norm, or a norm that goes to zero as n goes to infinity, is unimportant or doesn’t exist? We want to say that such observers are anomalous or “unlikely,” but how do we draw a link between a vector’s Hilbert space norm and these characterizations? An example makes the issue manifest. In a two-dimensional Hilbert space, say with states spin-up , and spin-down , consider a state . This state yields the probability for measuring spin-up of about .98 and for measuring spin-down to be about .02. If we consider n copies of this spin system, , then as n goes to infinity, the vast majority of terms in the expansion of this vector have roughly equal numbers of spin-up and spin-down states. So from the standpoint of observers (copies of the experimenter) the vast majority would see spin-ups and spin-downs in a ratio that does not agree with the quantum mechanical predictions. Only the very few terms in the expansion of that have 98 percent spin-ups and 2 percent spin-downs are consistent with the quantum mechanical expectation; the result above tells us that these states are the only ones with nonzero Hilbert space norm as n goes to infinity. In some sense, then, the vast majority of terms in the expansion of (the vast majority of copies of the experimenter) need to be considered as “non existent.” The challenge lies in understanding what, if anything, that means.
I also independently found the mathematical result described above, while preparing lectures for a course on quantum mechanics I was teaching. It was a notable thrill to have the probabilistic interpretation of quantum mechanics seemingly fall out directly from the mathematical formalism—I would imagine the list of physicists (on this page) who found this result before me had the same experience. I’m surprised at how little known the result is among mainstream physics. For instance, I don’t know of any standard quantum physics textbook that includes it. My take on the result is that it is best thought of as (1) a strong mathematical motivation for the Born probability interpretation of the wavefunction—had Born not “guessed” this interpretation, the math would have led someone there eventually; (2) a consistency check on the probability interpretation—had this mathematical result not held, it would have challenged the internal sensibility of the probability interpretation of the wavefunction.
10. I’ve been using the phrase “Zaxtarian-type reasoning” to denote a framework in which probability enters through the ignorance of each inhabitant of the Many Worlds as to which particular world he or she inhabits. Lev Vaidman has suggested taking more of the particulars of the Zaxtarian scenario to heart. He argues that probability enters the Many Worlds approach in the temporal window between an experimenter completing a measurement and reading the result. But, skeptics counter, this is too late in the game: it’s incumbent on quantum mechanics, and science more generally, to make predictions about what will happen in an experiment, not what did happen. What’s more, it seems precarious for the bedrock of quantum probability to rely on what seems to be an avoidable time delay: if a scientist gains immediate access to the result of his or her experiment, quantum probability seems in danger of being squeezed out of the picture. (For a detailed discussion see David Albert, “Probability in the Everett Picture” in Many Worlds: Everett, Quantum Theory, and Reality, eds. Simon Saunders, Jonathan Barrett, Adrian Kent, and David Wallace (Oxford: Oxford University
Press, 2010) and “Uncertainty and Probability for Branching Selves,” Peter Lewis, philsciarchive.pitt.edu/archive/00002636.) A final issue of relevance to Vaidman’s suggestion and also to this type of ignorance probability is this: when I flip a fair coin in the familiar context of a single universe, the reason I say there’s a 50 percent chance the coin will land heads is that while I’ll experience only one outcome, there are two outcomes that I could have experienced. But let me now close my eyes and imagine I’ve just measured the position of the somber electron. I know that my detector display says either Strawberry Fields or Grant’s Tomb, but I don’t know which. You then confront me. “Brian,” you say, “what’s the probability that your screen says Grant’s Tomb?” To answer, I think back on the coin toss, and just as I’m about to follow the same reasoning, I hesitate. “Hmmm,” I think. “Are there really two outcomes that I could have experienced? The only detail that differentiates me from the other Brian is the reading on my screen. To imagine that my screen could have returned a different reading is to imagine that I’m not me. It’s to imagine I’m the other Brian.” So even though I don’t know what my screen says, I—this guy talking in my head right now—couldn’t have experienced any other outcome; that suggests that my ignorance doesn’t lend itself to probabilistic thinking.
11. Scientists are meant to be objective in their judgments. But I feel comfortable admitting that because of its mathematical economy and far-reaching implications for reality, I’d like the Many Worlds approach to be right. At the same time, I maintain a healthy skepticism, fueled by the difficulties of integrating probability into the framework, so I’m fully open to alternative lines of attack. Two of these provide good bookends for the discussion in the text. One tries to develop the incomplete Copenhagen approach into a full theory; the other can be viewed as Many Worlds without the many worlds.
The first direction, spearheaded by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber, tries to make sense of the Copenhagen scheme by changing Schrödinger’s math so that it does allow probability waves to collapse. This is easier said than done. The modified math should barely affect the probability waves for small things like individual particles or atoms, since we don’t want to change the theory’s successful descriptions in this domain. But the modifications must kick in with a vengeance when a large object like a piece of laboratory equipment comes into play, causing the commingled probability wave to collapse. Ghirardi, Rimini, and Weber developed math that does just that. The upshot is that with their modified equations, measuring does indeed make a probability wave collapse; it sets in motion the evolution pictured in Figure 8.6.
The second approach, initially developed by Prince Louis de Broglie back in the 1920s, and then more fully decades later by David Bohm, starts from a mathematical premise that resonates with Everett. Schrödinger’s equation should always, in every circumstance, govern the evolution of quantum waves. So, in the de Broglie–Bohm theory, probability waves evolve just as they do in the Many Worlds approach. The de Broglie–Bohm theory goes on, however, to propose the very idea I emphasized earlier as being wrongheaded: in the de Broglie–Bohm approach, all but one of the many worlds encapsulated in a probability wave are merely possible worlds; only one world is singled out as real.
To accomplish this, the approach jettisons the traditional quantum haiku of wave or particle (an electron is a wave until it’s measured, whereupon it reverts to being a particle) and instead advocates a picture that embraces waves and particles. Contrary to the standard quantum view, de Broglie and Bohm envision particles as tiny, localized entities that travel along definite trajectories, yielding an ordinary, unambiguous reality, much as in the classical tradition. The only “real” world is the one in which the particles inhabit their unique, definite positions. Quantum waves then play a very different role. Rather than embodying a multitude of realities, a quantum wave acts to guide the motion of particles. The quantum wave pushes particles toward locations where the wave is large, making it likely that particles will be found at such locations, and away from locations where the wave is small, making it unlikely that particles will be found at those. To account for the process, de Broglie and Bohm needed an additional equation describing the effect of a quantum wave on a particle, so in their approach, Schrödinger’s equation, while not superseded, shares the stage with another mathematical player. (The mathematically inclined reader can see these equations below.)
For many years, the word on the street was that the de Broglie–Bohm approach was not worth considering, laden as it was with unnecessary baggage—not only a second equation but also, since it involves both particles and waves, a doubly long list of ingredients. More recently, there has been a growing recognition that these criticisms need context. As the Ghirardi-Rimini-Weber work makes explicit, even a sensible version of the standard-bearer Copenhagen approach requires a second equation. Additionally, the inclusion of both waves and particles yields an enormous benefit: it restores the notion of objects moving from here to there along definite trajectories, a return to a basic and familiar feature of reality that the Copenhagenists may have persuaded everyone to relinquish a little too quickly. More technical criticisms are that the approach is nonlocal (the new equation shows that influences exerted at one location appear to instantaneously affect distant locations) and that it is difficult to reconcile the approach with special relativity. The potency of the former criticism is diminished by the recognition that even the Copenhagen approach has non-local features that, moreover, have been confirmed experimentally. The latter point regarding relativity, though, is certainly an important one that has yet to be fully resolved.
Part of the resistance to the de Broglie–Bohm theory arose because the theory’s mathematical formalism has not always been presented in its most straightforward form. Here, for the mathematically inclined reader, is the most direct derivation of the theory.
Begin with Schrödinger’s equation for the wavefunction of a particle: , where the probability density for the particle to be at position x, p(x), is given by the standard equation . Then, imagine assigning a definite trajectory to the particle, with velocity at x given by a function v(x). What physical condition should this velocity function satisfy? Certainly, it should ensure conservation of probability: if the particle is moving with velocity v(x) from one region into another, the probability density should adjust accordingly: . It is now straightforward to solve for v(x) and find , where m is the particle’s mass.
Together with Schrödinger’s equation, this latter equation defines the de Broglie–Bohm theory. Note that this latter equation is nonlinear, but this has no bearing on Schrödinger’s equation, which retains its full linearity. The proper interpretation, then, is that this approach to filling in the gaps left by the Copenhagen approach adds a new equation, which depends nonlinearly on the wavefunction. All of the power and beauty of the underlying wave equation, that of Schrödinger, is fully preserved.
I might also add that the generalization to many particles is immediate: on the right-hand side of the new equation, we substitute the wavefunction of the multiparticle system: ψ(x1, x2, x3, … xn), and in calculating the velocity of the kth particle, we take the derivative with respect to the k-th coordinate (working, for ease, in a one-dimensional space; for higher dimensions, we suitably increase the number of coordinates). This generalized equation manifests the nonlocality of this approach: the velocity of the kth particle depends, instantaneously, on the positions of all other particles (as the particles’ coordinate locations are the arguments of the wavefunction).
12. Here is a concrete in-principle experiment for distinguishing the Copenhagen and Many Worlds approaches. An electron, like all other elementary particles, has a property known as spin. Somewhat as a top can spin about an axis, an electron can too, with one significant difference being that the rate of this spin—regardless of the direction of the axis—is always the same. It is an intrinsic property of the electron, like its mass or its electrical
charge. The only variable is whether the spin is clockwise or counterclockwise about a given axis. If it is counterclockwise, we say the electron’s spin about that axis is up; if it is clockwise, we say the electron’s spin is down. Because of quantum mechanical uncertainty, if the electron’s spin about a given axis is definite—say, with 100 percent certainty its spin is up about the z-axis—then its spin about the x- or y-axis is uncertain: about the x-axis the spin would be 50 percent up and 50 percent down; and similarly for the y-axis.
Imagine, then, starting with an electron whose spin about the z-axis is 100 percent up and then measuring its spin about the x-axis. According to the Copenhagen approach, if you find spin-down, that means the probability wave for the electron’s spin has collapsed: the spin-up possibility has been erased from reality, leaving the sole spike at spin-down. In the Many Worlds approach, by contrast, both the spin-up and spin-down outcomes occur, so, in particular, the spin-up possibility survives fully intact.
To adjudicate between these two pictures, imagine the following. After you measure the electron’s spin about the x-axis, have someone fully reverse the physical evolution. (The fundamental equations of physics, including that of Schrödinger, are time-reversal invariant, which means, in particular, that, at least in principle, any evolution can be undone. See The Fabric of the Cosmos for an in-depth discussion of this point.) Such reversal would be applied to everything: the electron, the equipment, and anything else that’s part of the experiment. Now, if the Many Worlds approach is correct, a subsequent measurement of the electron’s spin about the z-axis should yield, with 100 percent certainty, the value with which we began: spin-up. However, if the Copenhagen approach is correct (by which I mean a mathematically coherent version of it, such as the Ghirardi-Rimini-Weber formulation), we would find a different answer. Copenhagen says that upon measurement of the electron’s spin about the x-axis, in which we found spin-down, the spin-up possibility was annihilated. It was wiped off reality’s ledger. And so, upon reversing the measurement we don’t get back to our starting point because we’ve permanently lost part of the probability wave. Upon subsequent measurement of the electron’s spin about the z-axis, then, there is not 100 percent certainty that we will get the same answer we started with. Instead, it turns out that there’s a 50 percent chance that we will and a 50 percent chance that we won’t. If you were to undertake this experiment repeatedly, and if the Copenhagen approach is correct, on average, half the time you would not recover the same answer you initially did for the electron’s spin about the z-axis. The challenge, of course, is in carrying out the full reversal of a physical evolution. But, in principle, this is an experiment that would provide insight into which of the two theories is correct.