by Brian Greene
15. These four authors were the first to show fully that by judicious choices of Calabi-Yau shapes, and the fluxes threading their holes, they could realize string models with small, positive cosmological constants, like those found by observations. Together with Juan Maldacena and Liam McAllister, this group subsequently wrote a highly influential paper on how to combine inflationary cosmology with string theory.
16. More precisely, this mountainous terrain would inhabit a roughly 500-dimensional space, whose independent directions—axes—would correspond to different field fluxes. Figure 6.4 is a rough pictorial depiction but gives a feel for the relationships between the various forms for the extra dimensions. Additionally, when speaking of the string landscape, physicists generally envision that the mountainous terrain encompasses, in addition to the possible flux values, all the possible sizes and shapes (the different topologies and geometries) of the extra dimensions. The valleys in the string landscape are locations (specific forms for the extra dimensions and the fluxes they carry) where a bubble universe naturally settles, much as a ball would settle in such a spot in a real mountain terrain. When described mathematically, valleys are (local) minima of the potential energy associated with the extra dimensions. Classically, once a bubble universe acquired an extra dimensional form corresponding to a valley that feature would never change. Quantum mechanically, however, we will see that tunneling events can result in the form of the extra dimensions changing.
17. Quantum tunneling to a higher peak is possible but substantially less likely according to quantum calculations.
Chapter 7: Science and the Multiverse
1. The duration of the bubble’s expansion prior to collision determines the impact, and attendant disruption, of the ensuing crash. Such collisions also raise an interesting point to do with time, harking back to the example with Trixie and Norton in Chapter 3. When two bubbles collide, their outer edges—where the inflaton field’s energy is high—come into contact. From the perspective of someone within either one of the colliding bubbles, high inflaton energy value corresponds to early moments in time, near that bubble’s big bang. And so, bubble collisions happen at the inception of each universe, which is why the ripples created can affect another early universe process, the formation of the microwave background radiation.
2. We will take up quantum mechanics more systematically in Chapter 8. As we will see there, the statement I’ve made, “slither outside the arena of everyday reality” can be interpreted on a number of levels. What I have in mind here is the conceptually simplest: the equation of quantum mechanics assumes that probability waves generally don’t inhabit the spatial dimensions of common experience. Instead, the waves reside in a different environment that takes account not only of the everyday spatial dimensions but also of the number of particles being described. It is called configuration space and is explained for the mathematically inclined reader in note 4 of Chapter 8.
3. If the accelerated expansion of space that we’ve observed is not permanent, then at some time in the future the expansion of space will slow down. The slowing would allow light from objects that are now beyond our cosmic horizon to reach us; our cosmic horizon would grow. It would then be yet more peculiar to suggest that realms now beyond our horizon are not real since in the future we would have access to those very realms. (You may recall that toward the end of Chapter 2, I noted that the cosmic horizons illustrated in Figure 2.1 will grow larger as time passes. That’s true in a universe in which the pace of spatial expansion is not quickening. However, if the expansion is accelerating, there is distance beyond that we can never see, regardless of how long we wait. In an accelerating universe, the cosmic horizons can’t grow larger than a size determined mathematically by the rate of acceleration.)
4. Here is a concrete example of a feature that can be common to all universes in a particular multiverse. In Chapter 2, we noted that current data point strongly toward the curvature of space being zero. Yet, for reasons that are mathematically technical, calculations establish that all bubble universes in the Inflationary Multiverse have negative curvature. Roughly speaking, the spatial shapes swept out by equal inflaton values—shapes determined by connecting equal numbers in Figure 3.8b—are more like potato chips than like flat tabletops. Even so, the Inflationary Multiverse remains compatible with observation, because as any shape expands its curvature drops; the curvature of a marble is obvious, while that of the earth’s surface escaped notice for millennia. If our bubble universe has undergone sufficient expansion, its curvature could be negative yet so exceedingly small that today’s measurements can’t distinguish it from zero. That gives rise to a potential test. Should more precise observations in the future determine that the curvature of space is very small but positive that would provide evidence against our being part of an Inflationary Multiverse as argued by B. Freivogel, M. Kleban, M. Rodríguez Martínez, and L. Susskind, (see “Observational Consequences of a Landscape,” Journal of High Energy Physics 0603, 039 [2006]), measurement of positive curvature of 1 part in 105 would make a strong case against the kind of quantum tunneling transitions (Chapter 6) envisioned to populate the string landscape.
5. The many cosmologists and string theorists who have advanced this subject include Alan Guth, Andrei Linde, Alexander Vilenkin, Jaume Garriga, Don Page, Sergei Winitzki, Richard Easther, Eugene Lim, Matthew Martin, Michael Douglas, Frederik Denef, Raphael Bousso, Ben Freivogel, I-Sheng Yang, Delia Schwartz-Perlov, among many others.
6. An important caveat is that while the impact of modest changes to a few constants can reliably be deduced, more significant changes to a larger number of constants make the task far more difficult. It is at least possible that such significant changes to a variety of nature’s constants cancel out one another’s effects, or work together in novel ways, and are thus compatible with life as we know it.
7. A little more precisely, if the cosmological constant is negative, but sufficiently tiny, the collapse time would be long enough to allow galaxy formation. For ease, I am glossing over this subtlety.
8. Another point worthy of note is that the calculations I’ve described were undertaken without making a specific choice for the multiverse. Instead, Weinberg and his collaborators proceeded by positing a multiverse in which features could vary and calculated the abundance of galaxies in each of their constituent universes. The more galaxies a universe had, the more weight Weinberg and collaborators gave to its properties in their calculation of the average features a typical observer would encounter. But because they didn’t commit to an underlying multiverse theory, the calculations necessarily failed to account for the probability that a universe with this or that property would actually be found in the multiverse (the probabilities, that is, that we discussed in the previous section). Universes with cosmological constants and primordial fluctuations in certain ranges might be ripe for galaxy formation, but if such universes are rarely created in a given multiverse, it would nevertheless be highly unlikely for us to find ourselves in one of them.
To make the calculations manageable, Weinberg and collaborators argued that since the range of cosmological constant values they were considering was so narrow (between 0 and about 10–120), the intrinsic probabilities that such universes would exist in a given multiverse were not likely to vary wildly, much as the probabilities that you’ll encounter a 59.99997-pound dog or one weighing 59.99999 pounds also don’t differ substantially. They thus assumed that every value for the cosmological constant in the small range consistent with the formation of galaxies is as intrinsically probable as any other. With our rudimentary understanding of multiverse formation, this might seem like a reasonable first pass. But subsequent work has questioned the validity of this assumption, emphasizing that a full calculation needs to go further: committing to a definite multiverse proposal and determining the actual distribution of universes with various properties. A self-contained anthropic calculation that relies on a bare minimum of assumptions is the only way to judge
whether this approach will ultimately bear explanatory fruit.
9. The very meaning of “typical” is also burdened, as it depends on how it’s defined and measured. If we use numbers of kids and cars as our delimeter, we arrive at one kind of “typical” American family. If we use different scales such as interest in physics, love of opera, or immersion in politics, the characterization of a “typical” family will change. And what’s true for the “typical” American family is likely true for “typical” observers in the multiverse: consideration of features beyond just population size would yield a different notion of who is “typical.” In turn, this would affect the predictions for how likely it is that we will see this or that property in our universe. For an anthropic calculation to be truly convincing, it would have to address this issue. Alternatively, as indicated in the text, the distributions would need to be so sharply peaked that there would be minimal variation from one life-supporting universe to another.
10. The mathematical study of sets with an infinite number of members is rich and well developed. The mathematically inclined reader may be familiar with the fact that research going back to the nineteenth century established there are different “sizes” or, more commonly, “levels” of infinity. That is, one infinite quantity can be larger than another. The level of infinity that gives the size of the set containing all the whole numbers is called N0. This infinity was shown by Georg Cantor to be smaller than that giving the number of members contained in the set of real numbers. Roughly speaking, if you try to match up whole numbers and real numbers, you necessarily exhaust the former before the latter. And if you consider the set of all subsets of real numbers, the level of infinity grows larger still.
Now, in all of the examples we discuss in the main text, the relevant infinity is N0. since we are dealing with infinite collections of discrete, or “countable,” objects—various collections, that is, of whole numbers. In the mathematical sense, then, all of the examples have the same size; their total membership is described by the same level of infinity. However, for physics, as we will shortly see, a conclusion of this sort would not be particularly useful. The goal instead is to find a physically motivated scheme for comparing infinite collections of universes that would yield a more refined hierarchy, one that reflects the relative abundance across the multiverse of one set of physical features compared with another. A typical physics approach to a challenge of this sort is to first make comparisons between finite subsets of the relevant infinite collections (since in the finite case, all of the puzzling issues evaporate), and then allow the subsets to include ever more members, ultimately embracing the full infinite collections. The hurdle is finding a physically justifiable way of picking out the finite subsets for comparison, and then also establishing that comparisons remain sensible as the subsets grow larger.
11. Inflation is credited with other successes too, including the solution to the magnetic monopole problem. In attempts to meld the three nongravitational forces into a unified theoretical structure (known as grand unification) researchers found that the resulting mathematics implied that just after the big bang a great many magnetic monopoles would have been formed. These particles would be, in effect, the north pole of a bar magnet without the usual pairing with a south pole (or vice versa). But no such particles have ever been found. Inflationary cosmology explains the absence of monopoles by noting that the brief but stupendous expansion of space just after the big bang would have diluted their presence in our universe to nearly zero.
12. Currently, there are differing views on how great a challenge this presents. Some view the measure problem as a knotty technical issue that once solved will provide inflationary cosmology with an important additional detail. Others (for example, Paul Steinhardt) have expressed the belief that solving the measure problem will require stepping so far outside the mathematical formulation of inflationary cosmology that the resulting framework will need to be interpreted as a completely new cosmological theory. My view, one held by a small but growing number of researchers, is that the measure problem is tapping into a deep problem at the very root of physics, one that may require a substantial overhaul of foundational ideas.
Chapter 8: The Many Worlds of Quantum Measurement
1. Both Everett’s original 1956 thesis and the shortened 1957 version can be found in The Many-Worlds Interpretation of Quantum Mechanics, edited by Bryce S. DeWitt and Neill Graham (Princeton: Princeton University Press, 1973).
2. On January 27, 1998, I had a conversation with John Wheeler to discuss aspects of quantum mechanics and general relativity that I would be writing about in The Elegant Universe. Before getting into the science proper, Wheeler noted how important it was, especially for young theoreticians, to find the right language for expressing their results. At the time, I took this as nothing more than sagely advice, perhaps inspired by his speaking with me, a “young theoretician” who’d expressed interest in using ordinary language to describe mathematical insights. On reading the illuminating history laid out in The Many Worlds of Hugh Everett III by Peter Byrne (New York: Oxford University Press, 2010), I was struck by Wheeler’s emphasis of the same theme some forty years earlier in his dealings with Everett, but in a context whose stakes were far higher. In response to Everett’s first draft of his thesis, Wheeler told Everett that he needed to “get the bugs out of the words, not the formalism” and warned him of “the difficulty of expressing in everyday words the goings-on in a mathematical scheme that is about as far removed as it could be from the everyday description; the contradictions and misunderstandings that will arise; the very very heavy burden and responsibility you have to state everything in such a way that these misunderstandings can’t arise.” Byrne makes a compelling case that Wheeler was walking a delicate line between his admiration for Everett’s work and his respect for the quantum mechanical framework that Bohr and many other renowned physicists had labored to build. On the one hand, he didn’t want Everett’s insights to be summarily dismissed by the old guard because the presentation was deemed overreaching, or because of hot-button words (like universes “splitting”) that could appear fanciful. On the other hand, Wheeler didn’t want the established community of physicists to conclude that he was abandoning the demonstrably successful quantum formalism by spearheading an unjustified assault. The compromise Wheeler was imposing on Everett and his dissertation was to keep the mathematics he’d developed but frame its meaning and utility in a softer, more conciliatory tone. At the same time, Wheeler strongly encouraged Everett to visit Bohr and make his case in person, at a blackboard. In 1959 Everett did just that, but what Everett thought would be a two-week showdown amounted to a few unproductive conversations. No minds changed; no positions altered.
3. Let me clarify one imprecision. Schrödinger’s equation shows that the values attained by a quantum wave (or, in the language of the field, the wavefunction) can be positive or negative; more generally, the values can be complex numbers. Such values cannot be interpreted directly as probabilities—what would a negative or complex probability mean? Instead, probabilities are associated with the squared magnitude of the quantum wave at a given location. Mathematically, this means that to determine the probability that a particle will be found at a given location, we take the product of wave’s value at that point and its complex conjugate. This clarification also addresses an important related issue. Cancellations between overlapping waves are vital to creating an interference pattern. But if the waves themselves were properly described as probability waves, such cancellation couldn’t happen because probabilities are positive numbers. As we now see, however, quantum waves do not only have positive values; this allows cancellations to take place between positive and negative numbers, as well as, more generally, between complex numbers. Because we will only need qualitative features of such waves, for ease of discussion in the main text I will not distinguish between a quantum wave and the associated probability wave (derived from its squared magnitude).
4. For
the mathematically inclined reader, note that the quantum wave (wavefunction) for a single particle with large mass would conform to the description I’ve given in the text. However, very massive objects are generally composed of many particles, not one. In such a situation, the quantum mechanical description is more involved. In particular, you might have thought that all of the particles could be described by a quantum wave defined on the same coordinate grid we employ for a single particle—using the same three spatial axes. But that’s not right. The probability wave takes as input the possible position of each particle and produces the probability that the particles occupy those positions. Consequently, the probability wave lives in a space with three axes for each particle—that is, in total three times as many axes as there are particles (or ten times as many, if you embrace string theory’s extra spatial dimensions). This means that the wavefunction for a composite system made of n fundamental particles is a complex-valued function whose domain is not ordinary three-dimensional space but rather 3n-dimensional space; if the number of spatial dimensions is not 3 but rather m, the number 3 in these expressions would be replaced by m. This space is called configuration space. That is, in the general setting, the wavefunction would be a map . When we speak of such a wavefunction as being sharply peaked, we mean that this map would have support in a small mn-dimensional ball within its domain. Note, in particular, that wavefunctions don’t generally reside in the spatial dimensions of common experience. It is only in the idealized case of the wavefunction for a completely isolated single particle that its configuration space coincides with the familiar spatial environment. Note as well that when I say that the quantum laws show that the sharply peaked wavefunction for a massive object traces the same trajectory that Newton’s equations imply for the object itself, you can think of the wavefunction describing the object’s center of mass motion.