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Notes: Chapter 13
1. For the mathematically inclined reader: I am here referring to conformal symmetry — symmetry under arbitrary angle-preserving transformations on the volume in space-time swept out by the proposed fundamental constituent. Strings sweep out two-spacetime-dimensional surfaces, and the equations of string theory are invariant under the two-dimensional conformal group, which is an infinite dimensional symmetry group. By contrast, in other numbers of space dimensions, associated with objects that are not themselves one-dimensional, the conformal group is finite-dimensional. Return to Text
2. Many physicists contributed significantly to these developments, both by laying the groundwork and through follow-up discoveries: 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 Haava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard deWit, among many others. Return to Text
3. In fact, as explained in Chapter 12 of The Elegant Universe, there is an even tighter connection between the overlooked tenth spatial dimension and p-branes. As you increase the size of the tenth spatial dimension in, say, the type IIA formulation, one-dimensional strings stretch into two-dimensional inner-tube-like membranes. If you assume the tenth dimension is very small, as had always been implicitly done prior to these discoveries, the inner tubes look and behave like strings. As is the case for strings, the question of whether these newly found branes are indivisible or, instead, are made of yet finer constituents, remains unanswered. Researchers are open to the possibility that the ingredients so far identified in string/M-theory will not bring to a close the search for the elementary constituents of the universe. However, it's also possible that they will. Since much of what follows is insensitive to this issue, we'll adopt the simplest perspective and imagine that all the ingredients — strings and branes of various dimensions — are fundamental. And what of the earlier reasoning, which suggested that fundamental higher dimensions objects could not be incorporated into a physically sensible framework? Well, that reason ing was itself rooted in another quantum mechanical approximation scheme—one that is standard and fully battle tested but that, like any approximation, has limitations. Althougl researchers have yet to figure out all the subtleties associated with incorporating higher dimensional objects into a quantum theory, these ingredients fit so perfectly and consis tently within all five string formulations that almost everyone believes that the feared violations of basic and sacred physical principles are absent. Return to Text
4. In fact, we could be living on an even higher-dimensional brane (a four-brane, a five-brane .) three of whose dimensions fill ordinary space, and whose other dimensions fill some of the smaller, extra dimensions the theory requires. Return to Text
5. The mathematically inclined reader should note that for many years string theo¬rists have known that closed strings respect something called `T-duality (as explained fur¬ther in Chapter 16, and in Chapter 10 of The Elegant Universe). Basically, T-duality is the statement that if an extra dimension should be in the shape of a circle, string theory is completely insensitive to whether the circle's radius is R or 1/R. The reason is that strings can move around the circle ("momentum modes") and/or wrap around the circle ("wind¬ing modes") and, under the replacement of R with 1/R, physicists have realized that the roles of these two modes simply interchange, keeping the overall physical properties of the theory unchanged. Essential to this reasoning is that the strings are closed loops, since if they are open there is no topologically stable notion of their winding around a circular dimension. So, at first blush, it seems that open and closed strings behave completely dif¬ferently under T-duality. With closer inspection, and by making use of the Dirichlet boundary conditions for open strings (the "D" in D-branes), Polchinski, Dai, Leigh, as well as Hotava, Green, and other researchers resolved this puzzle. Return to Text
6. Proposals that have tried to circumvent the introduction of dark matter or dark energy have suggested that even the accepted behavior of gravity on large scales may dif¬fer from what Newton or Einstein would have thought, and in that way attempt to account for gravitational effects incompatible with solely the material we can see. As yet, these proposals are highly speculative and have little support, either experimental or the¬oretical. Return to Text
7. The physicists who introduced this idea are S. Giddings and S. Thomas, and S. Dimopoulus and G. Landsberg. Return to Text
8. Notice that the contraction phase of such a bouncing universe is not the same as the expansion phase run in reverse. Physical processes such as eggs splattering and candles melting would happen in the usual "forward" time direction during the expansion phases and would continue to do so during the subsequent contraction phase. That's why entropy would increase during both phases. Return to Text
9. The expert reader will note that the cyclic model can be phrased in the language of four-dimensional effective field theory on one of the three-branes, and in this form it shares many features with more familiar scalar-field-driven inflationary models. When I say "radically new mechanism," I am referring to the conceptual description in terms of colliding branes, which in and of itself is a striking new way of thinking about cosmology. Return to Text
10. Don't get confused on dimension counting. The two three-branes, together with the space interval between them, have four dimensions. Time brings it to five. That leaves six more for the Calabi-Yau space. Return to Text
11. An important exception, mentioned at the end of this chapter and discussed in further detail in Chapter 14, has to do with inhomogeneities in the gravitational field, so-called primordial gravitational waves. Inflationary cosmology and the cyclic model differ in this regard, one way in which there is a chance that they may be distinguished experi¬mentally. Return to Text
12. Quantum mechanics ensures that there is always a nonzero probability that a chance fluctuation will disrupt the cyclic process (e.g., one brane twists relative to the other), causing the model to grind to a halt. Even if the probability is minuscule, sooner or later it will surely come to pass, and hence the cycles cannot continue indefinitely. Return to Text
2. Many physicists contributed significantly to these developments, both by laying the groundwork and through follow-up discoveries: 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 Haava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard deWit, among many others. Return to Text
3. In fact, as explained in Chapter 12 of The Elegant Universe, there is an even tighter connection between the overlooked tenth spatial dimension and p-branes. As you increase the size of the tenth spatial dimension in, say, the type IIA formulation, one-dimensional strings stretch into two-dimensional inner-tube-like membranes. If you assume the tenth dimension is very small, as had always been implicitly done prior to these discoveries, the inner tubes look and behave like strings. As is the case for strings, the question of whether these newly found branes are indivisible or, instead, are made of yet finer constituents, remains unanswered. Researchers are open to the possibility that the ingredients so far identified in string/M-theory will not bring to a close the search for the elementary constituents of the universe. However, it's also possible that they will. Since much of what follows is insensitive to this issue, we'll adopt the simplest perspective and imagine that all the ingredients — strings and branes of various dimensions — are fundamental. And what of the earlier reasoning, which suggested that fundamental higher dimensions objects could not be incorporated into a physically sensible framework? Well, that reason ing was itself rooted in another quantum mechanical approximation scheme—one that is standard and fully battle tested but that, like any approximation, has limitations. Althougl researchers have yet to figure out all the subtleties associated with incorporating higher dimensional objects into a quantum theory, these ingredients fit so perfectly and consis tently within all five string formulations that almost everyone believes that the feared violations of basic and sacred physical principles are absent. Return to Text
4. In fact, we could be living on an even higher-dimensional brane (a four-brane, a five-brane .) three of whose dimensions fill ordinary space, and whose other dimensions fill some of the smaller, extra dimensions the theory requires. Return to Text
5. The mathematically inclined reader should note that for many years string theo¬rists have known that closed strings respect something called `T-duality (as explained fur¬ther in Chapter 16, and in Chapter 10 of The Elegant Universe). Basically, T-duality is the statement that if an extra dimension should be in the shape of a circle, string theory is completely insensitive to whether the circle's radius is R or 1/R. The reason is that strings can move around the circle ("momentum modes") and/or wrap around the circle ("wind¬ing modes") and, under the replacement of R with 1/R, physicists have realized that the roles of these two modes simply interchange, keeping the overall physical properties of the theory unchanged. Essential to this reasoning is that the strings are closed loops, since if they are open there is no topologically stable notion of their winding around a circular dimension. So, at first blush, it seems that open and closed strings behave completely dif¬ferently under T-duality. With closer inspection, and by making use of the Dirichlet boundary conditions for open strings (the "D" in D-branes), Polchinski, Dai, Leigh, as well as Hotava, Green, and other researchers resolved this puzzle. Return to Text
6. Proposals that have tried to circumvent the introduction of dark matter or dark energy have suggested that even the accepted behavior of gravity on large scales may dif¬fer from what Newton or Einstein would have thought, and in that way attempt to account for gravitational effects incompatible with solely the material we can see. As yet, these proposals are highly speculative and have little support, either experimental or the¬oretical. Return to Text
7. The physicists who introduced this idea are S. Giddings and S. Thomas, and S. Dimopoulus and G. Landsberg. Return to Text
8. Notice that the contraction phase of such a bouncing universe is not the same as the expansion phase run in reverse. Physical processes such as eggs splattering and candles melting would happen in the usual "forward" time direction during the expansion phases and would continue to do so during the subsequent contraction phase. That's why entropy would increase during both phases. Return to Text
9. The expert reader will note that the cyclic model can be phrased in the language of four-dimensional effective field theory on one of the three-branes, and in this form it shares many features with more familiar scalar-field-driven inflationary models. When I say "radically new mechanism," I am referring to the conceptual description in terms of colliding branes, which in and of itself is a striking new way of thinking about cosmology. Return to Text
10. Don't get confused on dimension counting. The two three-branes, together with the space interval between them, have four dimensions. Time brings it to five. That leaves six more for the Calabi-Yau space. Return to Text
11. An important exception, mentioned at the end of this chapter and discussed in further detail in Chapter 14, has to do with inhomogeneities in the gravitational field, so-called primordial gravitational waves. Inflationary cosmology and the cyclic model differ in this regard, one way in which there is a chance that they may be distinguished experi¬mentally. Return to Text
12. Quantum mechanics ensures that there is always a nonzero probability that a chance fluctuation will disrupt the cyclic process (e.g., one brane twists relative to the other), causing the model to grind to a halt. Even if the probability is minuscule, sooner or later it will surely come to pass, and hence the cycles cannot continue indefinitely. Return to Text