The Hidden Reality Part 13

1. Cla.s.sical Mechanics: Cla.s.sical Mechanics:[image] . Electromagnetism: . Electromagnetism: d*F=*J;dF=0 d*F=*J;dF=0. Quantum mechanics:[image] . General relativity: . General relativity:[image] . .

2. I am referring here to the I am referring here to the fine structure constant, e fine structure constant, e2/hc, whose numerical value (at typical energies for electromagnetic processes) is about 1/137, which is roughly .0073.

3. Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that's dialed small, and vice versa; the Type IIB at large coupling morphs into Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that's dialed small, and vice versa; the Type IIB at large coupling morphs into itself itself, the Type IIB theory but with small coupling. The cases of the Heterotic-E and Type IIA theories are a little more subtle (see The Elegant Universe The Elegant Universe, Chapter 12, for details), but the overall picture is that all five theories partic.i.p.ate in a web of interrelations.

4. For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Cla.s.sically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of s.p.a.cetime dimensions through which the string moves-the dimension, that is, of s.p.a.cetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum a.n.a.lysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the s.p.a.cetime metric) can be systematically "rotated" away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Cla.s.sically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of s.p.a.cetime dimensions through which the string moves-the dimension, that is, of s.p.a.cetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum a.n.a.lysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the s.p.a.cetime metric) can be systematically "rotated" away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, Superstring Theory Superstring Theory, vol. 1 (Cambridge: Cambridge University Press, 1988).

5. As with many major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hoava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt. As with many major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hoava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt.

6. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble's boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally-the multiverses discussed so far all emerge from a.n.a.lyses that focus fundamentally on processes occurring throughout s.p.a.ce. In the multiverse we will now discuss, time is central from the outset. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble's boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally-the multiverses discussed so far all emerge from a.n.a.lyses that focus fundamentally on processes occurring throughout s.p.a.ce. In the multiverse we will now discuss, time is central from the outset.

7. Alexander Friedmann, Alexander Friedmann, The World as s.p.a.ce and Time The World as s.p.a.ce and Time, 1923, published in Russian, as referenced by H. Kragh, in "Continual Fascination: The Oscillating Universe in Modem Cosmology," Science in Context Science in Context 22, no. 4 (2009): 587612. 22, no. 4 (2009): 587612.

8. As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in Chapter 6 Chapter 6). In the last phase of each cycle, the presence of dark energy in the braneworlds ensures agreement with today's observations of accelerated expansion; this accelerated expansion, in turn, dilutes the entropy density, setting the stage for the next cosmological cycle.

9. Large flux values also tend to destabilize a given Calabi-Yau shape for the extra dimensions. That is, the fluxes tend to push the Calabi-Yau shape to grow large, quickly running into conflict with the criterion that extra dimensions not be visible. Large flux values also tend to destabilize a given Calabi-Yau shape for the extra dimensions. That is, the fluxes tend to push the Calabi-Yau shape to grow large, quickly running into conflict with the criterion that extra dimensions not be visible.

Chapter 6: New Thinking About an Old Constant.

1. George Gamow, George Gamow, My World Line My World Line (New York: Viking Adult, 1970); J. C. p.e.c.k.e.r, Letter to the Editor, (New York: Viking Adult, 1970); J. C. p.e.c.k.e.r, Letter to the Editor, Physics Today Physics Today, May 1990, p. 117.

2. Albert Einstein, Albert Einstein, The Meaning of Relativity The Meaning of Relativity (Princeton: Princeton University Press, 2004), p. 127. Note that Einstein used the term "cosmologic member" for what we now call the "cosmological constant"; for clarity, I have made this subst.i.tution in the text. (Princeton: Princeton University Press, 2004), p. 127. Note that Einstein used the term "cosmologic member" for what we now call the "cosmological constant"; for clarity, I have made this subst.i.tution in the text.

3. The Collected Papers of Albert Einstein The Collected Papers of Albert Einstein, edited by Robert Schulmann et al. (Princeton: Princeton University Press, 1998), p. 316.

4. Of course, some things Of course, some things do do change. As pointed out in the change. As pointed out in the notes notes to Chapter 3, galaxies generally have small velocities beyond the spatial swelling. Over the course of cosmological timescales, such additional motion can alter position relations.h.i.+ps; such motion can also result in a variety of interesting astrophysical events such as galaxy collisions and mergers. For the purpose of explaining cosmic distances, however, these complications can be safely ignored. to Chapter 3, galaxies generally have small velocities beyond the spatial swelling. Over the course of cosmological timescales, such additional motion can alter position relations.h.i.+ps; such motion can also result in a variety of interesting astrophysical events such as galaxy collisions and mergers. For the purpose of explaining cosmic distances, however, these complications can be safely ignored.

5. There is one complication that does not affect the essential idea I've explained but which does come into play when undertaking the scientific a.n.a.lyses described. As photons travel to us from a given supernova, their number density gets diluted in the manner I've described. However, there is another diminishment to which they are subject. In the next section, I'll describe how the stretching of s.p.a.ce causes the wavelength of photons to stretch too, and, correspondingly, their energy to decrease-an effect, as we will see, called There is one complication that does not affect the essential idea I've explained but which does come into play when undertaking the scientific a.n.a.lyses described. As photons travel to us from a given supernova, their number density gets diluted in the manner I've described. However, there is another diminishment to which they are subject. In the next section, I'll describe how the stretching of s.p.a.ce causes the wavelength of photons to stretch too, and, correspondingly, their energy to decrease-an effect, as we will see, called reds.h.i.+ft reds.h.i.+ft. As explained there, astronomers use reds.h.i.+ft data to learn about the size of the universe when the photons were emitted-an important step toward determining how the expansion of s.p.a.ce has varied through time. But the stretching of photons-the diminishment of their energy-has another effect: It accentuates the dimming of a distant source. And so, to properly determine the distance of a supernova by comparing its apparent and intrinsic brightness, astronomers must take account not just of the dilution of photon number density (as I've described in the text), but also the additional diminishment of energy coming from reds.h.i.+ft. (More precisely still, this additional dilution factor must be applied twice; the second red s.h.i.+ft factor accounts for the rate at which photons arrive being similarly stretched by the cosmic expansion.) 6. Properly interpreted, the second proposed answer for the meaning of the distance being measured may also be construed as correct. In the example of earth's expanding surface, New York, Austin, and Los Angeles all rush away from one another, yet each continues to occupy the same location on earth it always has. The cities separate because the surface swells, not because someone digs them up, puts them on a flatbed, and transports them to a new site. Similarly, because galaxies separate due to the cosmic swelling, they too occupy the same location in s.p.a.ce they always have. You can think of them as being st.i.tched to the spatial fabric. When the fabric stretches, the galaxies move apart, yet each remains tethered to the very same point it has always occupied. And so, even though the second and third answers appear different-the former focusing on the distance between us and the location a distant galaxy had eons ago, when the supernova emitted the light we now see; the latter focusing on the distance now between us and that galaxy's current location-they're not. The distant galaxy is now, and has been for billions of years, positioned at one and the same spatial location. Only if it moved Properly interpreted, the second proposed answer for the meaning of the distance being measured may also be construed as correct. In the example of earth's expanding surface, New York, Austin, and Los Angeles all rush away from one another, yet each continues to occupy the same location on earth it always has. The cities separate because the surface swells, not because someone digs them up, puts them on a flatbed, and transports them to a new site. Similarly, because galaxies separate due to the cosmic swelling, they too occupy the same location in s.p.a.ce they always have. You can think of them as being st.i.tched to the spatial fabric. When the fabric stretches, the galaxies move apart, yet each remains tethered to the very same point it has always occupied. And so, even though the second and third answers appear different-the former focusing on the distance between us and the location a distant galaxy had eons ago, when the supernova emitted the light we now see; the latter focusing on the distance now between us and that galaxy's current location-they're not. The distant galaxy is now, and has been for billions of years, positioned at one and the same spatial location. Only if it moved through through s.p.a.ce rather than solely ride the wave of swelling s.p.a.ce would its location change. In this sense, the second and third answers are actually the same. s.p.a.ce rather than solely ride the wave of swelling s.p.a.ce would its location change. In this sense, the second and third answers are actually the same.

7. For the mathematically inclined reader, here is how you do the calculation of the distance-now, at time For the mathematically inclined reader, here is how you do the calculation of the distance-now, at time t tnow-that light has traveled since being emitted at time t temitted. We will work in the context of an example in which the spatial part of s.p.a.cetime is flat, and so the metric can be written as ds ds2=c2dt2 a a2(t)dx2, where a(t) a(t) is the scale factor of the universe at time is the scale factor of the universe at time t t, and c c is the speed of light. The coordinates we are using are called is the speed of light. The coordinates we are using are called co-moving co-moving. In the language developed in this chapter, such coordinates can be thought of as labeling points on the static map; the scale factor supplies the information contained in the map's legend.

The special characteristic of the trajectory followed by light is that ds ds2=0 (equivalent to the speed of light always being c) along the path, which implies that (equivalent to the speed of light always being c) along the path, which implies that[image] , or, over a finite time interval such as that between , or, over a finite time interval such as that between[image] . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real s.p.a.ce, we must rescale the formula by today's scale factor; therefore, the total distance the light traveled equals . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real s.p.a.ce, we must rescale the formula by today's scale factor; therefore, the total distance the light traveled equals[image] . If s.p.a.ce were not stretching, the total travel distance would be . If s.p.a.ce were not stretching, the total travel distance would be[image][image] , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light's trajectory is multiplied by the factor , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light's trajectory is multiplied by the factor[image] , which is the amount by which that segment has stretched, since the moment the light traversed it, until today. , which is the amount by which that segment has stretched, since the moment the light traversed it, until today.

8. More precisely, about 7.12 10 More precisely, about 7.12 1030 grams per cubic centimeter. grams per cubic centimeter.

9. The conversion is 7.12 10 The conversion is 7.12 1030 grams/cubic centimeter = (7.12 10 grams/cubic centimeter = (7.12 1030 grams/cubic centimeter) (4.6 10 grams/cubic centimeter) (4.6 104 Planck ma.s.s/gram) (1.62 10 Planck ma.s.s/gram) (1.62 1033 centimeter/Planck length) centimeter/Planck length)3 = 1.38 10 = 1.38 10123 Planck ma.s.s/cubic Planck volume. Planck ma.s.s/cubic Planck volume.

10. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field's potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what's required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field's potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what's required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed.

11. The mathematically inclined reader should note that each such jitter contributes an energy that's inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy. The mathematically inclined reader should note that each such jitter contributes an energy that's inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy.

12. For the mathematically inclined reader, the cancellation occurs because supersymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations. For the mathematically inclined reader, the cancellation occurs because supersymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations.

13. While the a.s.sertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, While the a.s.sertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, The Anthropic Cosmological Principle The Anthropic Cosmological Principle (New York: Oxford University Press, 1986); John Barrow, (New York: Oxford University Press, 1986); John Barrow, The Constants of Nature The Constants of Nature (New York: Pantheon Books, 2003); Paul Davies, (New York: Pantheon Books, 2003); Paul Davies, The Cosmic Jackpot The Cosmic Jackpot (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, Has Science Found G.o.d? Has Science Found G.o.d? (Amherst, N.Y.: Prometheus Books, 2003); and references therein. (Amherst, N.Y.: Prometheus Books, 2003); and references therein.

14. Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say, the Quilted Multiverse, whose infinite spatial expanse contains infinitely many universes. But you need to be careful. Even with infinitely many universes, the list of different cosmological constants represented might not be long. If, for example, the underlying laws don't allow for many different cosmological constant values, then regardless of the number of universes, only the small collection of possible cosmological constants would be realized. So, the question we're asking is whether (a) there are candidate laws of physics that give rise to a multiverse, (b) the multiverse so generated contains far more than 10 Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say, the Quilted Multiverse, whose infinite spatial expanse contains infinitely many universes. But you need to be careful. Even with infinitely many universes, the list of different cosmological constants represented might not be long. If, for example, the underlying laws don't allow for many different cosmological constant values, then regardless of the number of universes, only the small collection of possible cosmological constants would be realized. So, the question we're asking is whether (a) there are candidate laws of physics that give rise to a multiverse, (b) the multiverse so generated contains far more than 10124 different universes, and (c) the laws ensure that the cosmological constant's value varies from universe to universe. different universes, and (c) the laws ensure that the cosmological constant's value varies from universe to universe.

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. 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 s.p.a.ce, whose independent directions-axes-would correspond to different field fluxes. More precisely, this mountainous terrain would inhabit a roughly 500-dimensional s.p.a.ce, whose independent directions-axes-would correspond to different field fluxes. Figure 6.4 Figure 6.4 is a rough pictorial depiction but gives a feel for the relations.h.i.+ps between the various forms for the extra dimensions. Additionally, when speaking of the string landscape, physicists generally envision that the mountainous terrain encompa.s.ses, 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 a.s.sociated with the extra dimensions. Cla.s.sically, 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. is a rough pictorial depiction but gives a feel for the relations.h.i.+ps between the various forms for the extra dimensions. Additionally, when speaking of the string landscape, physicists generally envision that the mountainous terrain encompa.s.ses, 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 a.s.sociated with the extra dimensions. Cla.s.sically, 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. 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 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 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 We will take up quantum mechanics more systematically in Chapter 8 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 a.s.sumes 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 number of particles being described. It is called of particles being described. It is called configuration s.p.a.ce configuration s.p.a.ce and is explained for the mathematically inclined reader in and is explained for the mathematically inclined reader in note 4 note 4 of Chapter 8. of Chapter 8.

3. If the accelerated expansion of s.p.a.ce that we've observed is not permanent, then at some time in the future the expansion of s.p.a.ce 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 If the accelerated expansion of s.p.a.ce that we've observed is not permanent, then at some time in the future the expansion of s.p.a.ce 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 Chapter 2, I noted that the cosmic horizons ill.u.s.trated in Figure 2.1 Figure 2.1 will grow larger as time pa.s.ses. 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.) will grow larger as time pa.s.ses. 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 Here is a concrete example of a feature that can be common to all universes in a particular multiverse. In Chapter 2 Chapter 2, we noted that current data point strongly toward the curvature of s.p.a.ce 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 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 s.p.a.ce is very small but positive positive that would provide evidence against our being part of an Inflationary Multiverse as argued by B. Freivogel, M. Kleban, M. Rodriguez Martinez, and L. Susskind, (see "Observational Consequences of a Landscape," that would provide evidence against our being part of an Inflationary Multiverse as argued by B. Freivogel, M. Kleban, M. Rodriguez Martinez, and L. Susskind, (see "Observational Consequences of a Landscape," Journal of High Energy Physics Journal of High Energy Physics 0603, 039 [2006]), measurement of positive curvature of 1 part in 10 0603, 039 [2006]), measurement of positive curvature of 1 part in 105 would make a strong case against the kind of quantum tunneling transitions ( 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. 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. 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. 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 const.i.tuent 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. 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 const.i.tuent 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 10120), 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 a.s.sumed 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 pa.s.s. But subsequent work has questioned the validity of this a.s.sumption, 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 a.s.sumptions 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. 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 quant.i.ty can be larger than another. The level of infinity that gives the size of the set containing all the whole numbers is called N 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 quant.i.ty 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. 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 members.h.i.+p 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 establis.h.i.+ng that comparisons remain sensible as the subsets grow larger. 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 members.h.i.+p 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 establis.h.i.+ng that comparisons remain sensible as the subsets grow larger.

11. Inflation is credited with other successes too, including the solution to the Inflation is credited with other successes too, including the solution to the magnetic monopole problem magnetic monopole problem. In attempts to meld the three nongravitational forces into a unified theoretical structure (known as grand unification 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 s.p.a.ce 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. 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 Both Everett's original 1956 thesis and the shortened 1957 version can be found in The Many-Worlds Interpretation of Quantum Mechanics 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 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 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 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-b.u.t.ton 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 a.s.sault. 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. 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-b.u.t.ton 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 a.s.sault. 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. Schrodinger'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 a.s.sociated with the Let me clarify one imprecision. Schrodinger'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 a.s.sociated with the squared magnitude 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 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 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 a.s.sociated probability wave (derived from its squared magnitude).

4. For the mathematically inclined reader, note that the quantum wave ( For the mathematically inclined reader, note that the quantum wave (wavefunction) for a single particle with large ma.s.s would conform to the description I've given in the text. However, very ma.s.sive 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 possible position of each particle and produces the probability that the particles occupy those positions. Consequently, the probability wave lives in a s.p.a.ce 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 of each particle and produces the probability that the particles occupy those positions. Consequently, the probability wave lives in a s.p.a.ce 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 n fundamental particles is a complex-valued function whose domain is not ordinary three-dimensional s.p.a.ce but rather 3 fundamental particles is a complex-valued function whose domain is not ordinary three-dimensional s.p.a.ce but rather 3n-dimensional s.p.a.ce; if the number of spatial dimensions is not 3 but rather m m, the number 3 in these expressions would be replaced by m m. This s.p.a.ce is called configuration s.p.a.ce configuration s.p.a.ce. That is, in the general setting, the wavefunction would be a map[image] . 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 s.p.a.ce 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 ma.s.sive 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 ma.s.s motion. . 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 s.p.a.ce 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 ma.s.sive 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 ma.s.s motion.

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 a.s.sociated with a possible position of the electron. How does this relate to 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 a.s.sociated with a possible position of the electron. How does this relate to Chapter 2 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 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 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, ill.u.s.trating 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, 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 The Semantic Conception of Theories and Scientific Realism (Chicago: University of Illinois Press, 1989); James Ladyman, Don Ross, David Spurrett, and John Collier, (Chicago: University of Illinois Press, 1989); James Ladyman, Don Ross, David Spurrett, and John Collier, Every Thing Must Go Every Thing Must Go (Oxford: Oxford University Press, 2007). (Oxford: Oxford University Press, 2007).

7. Physicists often speak loosely of there being infinitely many universes a.s.sociated with the Many Worlds approach to quantum mechanics. Certainly, there are infinitely many possible probability wave shapes. Even at a single location in s.p.a.ce 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 s.p.a.ce. If that Hilbert s.p.a.ce 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 s.p.a.ce a.s.sociated with physics taking place within any finite volume of s.p.a.ce, and limited to having a finite amount of energy, is necessarily finite dimensional (a point we will take up more generally in Physicists often speak loosely of there being infinitely many universes a.s.sociated with the Many Worlds approach to quantum mechanics. Certainly, there are infinitely many possible probability wave shapes. Even at a single location in s.p.a.ce 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 s.p.a.ce. If that Hilbert s.p.a.ce 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 s.p.a.ce a.s.sociated with physics taking place within any finite volume of s.p.a.ce, and limited to having a finite amount of energy, is necessarily finite dimensional (a point we will take up more generally in Chapter 9 Chapter 9), which suggests that the number of worlds would similarly be finite.

8. See Peter Byrne, See Peter Byrne, The Many Worlds of Hugh Everett III The Many Worlds of Hugh Everett III (New York: Oxford University Press, 2010), p. 177. (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 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[image] be the wavefunction for a quantum mechanical system, a vector that's an element of the Hilbert s.p.a.ce H. The wavefunction for n-identical copies of the system is thus be the wavefunction for a quantum mechanical system, a vector that's an element of the Hilbert s.p.a.ce H. The wavefunction for n-identical copies of the system is thus[image] . Let . Let A A be any Hermitian operator with eigenvalues be any Hermitian operator with eigenvalues k, and eigenfunctions. Let Fk(A) be the "frequency" operator that counts the number of times[image] appears in a given state lying in appears in a given state lying in[image] . The mathematical result is that lim . The mathematical result is that lim[image]. 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[image] . 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 . 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 A will find will find k is is[image] -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 -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 k is observed fails to agree with the Born rule have zero Hilbert s.p.a.ce 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 s.p.a.ce norm becomes vanis.h.i.+ngly small as is observed fails to agree with the Born rule have zero Hilbert s.p.a.ce 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 s.p.a.ce norm becomes vanis.h.i.+ngly small as n 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 s.p.a.ce norm, or a norm that goes to zero as goes to infinity. As promising as this seems, on reflection it is less convin