by Brian Greene
This description gives some sense of why wound and unwound strings measure distances that are inversely related. But as the point is quite subtle, it is perhaps worth noting the underlying technical analysis for the mathematically inclined reader. In ordinary point-particle quantum mechanics, distance and momentum (essentially energy) are related by Fourier transform. That is, a position eigenstate |x> on a circle of radius R can be defined by |x>=Σveixp|p> where p = v/R and |p> is a momentum eigenstate (the direct analog of what we have called a uniform-vibration mode of a string—overall motion without change in shape). In string theory, though, there is a second notion of position eigenstate |x~> defined by making use of the winding string states: |x~> = Σweix~p~|p~ > where |p~> is a winding eigenstate with p~ = wR. From these definitions we immediately see that x is periodic with period 2πR while x~ is periodic with period 2π/R, showing that x is a position coordinate on a circle of radius R~ while x~ is the position coordinate on a circle of radius 1/R. Even more explicitly, we can now imagine taking the two wavepackets |x> and |x~> both starting say, at the origin, and allowing them to evolve in time to carry out our operational approach for defining distance. The radius of the circle, as measured by either probe, is then proportional to the required time lapse for the packet to return to its initial configuration. Since a state with energy E evolves with a phase factor involving Et, we see that the time lapse, and hence the radius, is t ~ 1/E ~ R for the vibration modes and t ~ 1/E ~ 1/R for the winding modes.
5. For the mathematically inclined reader, we note that, more precisely, the number of families of string vibrations is one-half the absolute value of the Euler characteristic of the Calabi-Yau space, as mentioned in note 16 of Chapter 9. This is given by the absolute value of difference between h2,1 and h1,1, where hp,q denotes the (p,q) Hodge number. Up to a numerical shift, these count the number of nontrivial homology three-cycles ("three-dimensional holes") and the number of homology twocycles ("two-dimensional holes"). And so, whereas we speak of the total number of holes in the main text, the more precise analysis shows that the number of families depends on the absolute value of difference between the odd- and even-dimensional holes. The conclusion, however, is the same. For instance, if two Calabi-Yau spaces differ by the interchange of their respective h2,1 and h1,1 Hodge numbers, the number of particle families-and the total number of "holes"—will not change.
6. The name comes from the fact that the "Hodge diamonds"—a mathematical summary of the holes of various dimensions in a Calabi-Yau space—for each Calabi-Yau space of a mirror pair are mirror reflections of one another.
7. The term mirror symmetry is also used in other, completely different contexts in physics, such as in the question of chirality—that is, whether the universe is left-right symmetric—as discussed in note 7 of Chapter 8.
Chapter 11
1. The mathematically inclined reader will recognize that we are asking whether the topology of space is dynamical—that is, whether it can change. We note that although we will often use the language of dynamical topology change, in practice we are usually considering a one-parameter family of spacetimes whose topology changes as a function of the parameter. Technically speaking, this parameter is not time, but in certain limits can essentially be identified with time.
2. For the mathematically inclined reader, the procedure involves blowing down rational curves on a Calabi-Yau manifold and then making use of the fact that, under certain circumstances, the resulting singularity can be repaired by distinct small resolutions.
3. K. C. Cole, New York Times Magazine, October 18, 1987, p. 20.
Chapter 12
1. Albert Einstein, as quoted in John D. Barrow, Theories of Everything (New York: Fawcett-Columbine, 1992), p. 13.
2. Let's briefly summarize the differences between the five string theories. To do so, we note that vibrational disturbances along a loop of string can travel clockwise or counterclockwise. The Type IIA and Type 1/R strings differ in that in the latter theory, these clockwise/counterclockwise vibrations are identical, while in the former, they are exactly opposite in form. Opposite has a precise mathematical meaning in this context, but it's easiest to think about in terms of the spins of the resulting vibrational patterns in each theory. In the Type 1/R theory, it turns out that all particles spin in the same direction (they have the same chirality), whereas in the Type IIA theory, they spin in both directions (they have both chiralities). Nevertheless, each theory incorporates supersymmetry. The two heterotic theories differ in a similar but more dramatic way. Each of their clockwise string vibrations looks like those of the Type II string (when focusing on just the clockwise vibrations, the Type IIA and Type 1/R theories are the same), but their counterclockwise vibrations are those of the original bosonic string theory. Although the bosonic string has insurmountable problems when chosen for both clockwise and counterclockwise string vibrations, in 1985 David Gross, Jeffrey Harvey, Emit Martinec, and Ryan Rhom (all then at Princeton University and dubbed the "Princeton String Quartet") showed that a perfectly sensible theory emerges if it is used in combination with the Type II string. The really odd feature of this union is that it has been known since the work of Claude Lovelace of Rutgers University in 1971 and the work of Richard Brower of Boston University, Peter Goddard of Cambridge University, and Charles Thorn of the University of Florida at Gainesville in 1972 that the bosonic string requires a 26-dimensional spacetime, whereas the superstring, as we have discussed, requires a 10-dimensional one. So the heterotic string constructions are a strange hybrid—a heterosis—in which counterclockwise vibrational patterns live in 26 dimensions and clockwise patterns live in 10 dimensions! Before you get caught up in trying to make sense of this perplexing union, Gross and his collaborators showed that the extra 16 dimensions on the bosonic side must be curled up into one of two very special higher-dimensional doughnutlike shapes, giving rise to the Heterotic-O and Heterotic-E theories. Since the extra 16 dimensions on the bosonic side are rigidly curled up, each of these theories behaves as though it really has 10 dimensions, just as in the Type II case. Again, both heterotic theories incorporate a version of supersymmetry. Finally, the Type I theory is a close cousin of the Type 1/R string except that, in addition to the closed loops of string we have discussed in previous chapters, it also has strings with unconnected ends—so-called open strings.
3. When we speak of "exact" answers in this chapter, such as the "exact" motion of the earth, what we really mean is the exact prediction for some physical quantity within some chosen theoretical framework. Until we truly have the final theory—perhaps we now do, perhaps we never will—all of our theories will themselves be approximations to reality. But this notion of approximate has nothing to do with our discussion in this chapter. Here we are concerned with the fact that within a chosen theory, it is often difficult, if not impossible, to extract the exact predictions that the theory makes. Instead, we have to extract such predictions using approximation methods based on a perturbative approach.
4. These diagrams are string theory versions of the so-called Feynman diagrams, invented by Richard Feynman for performing perturbative calculations in point-particle quantum field theory.
5. More precisely, every virtual string pair, that is, every loop in a given diagram, contributes—among other more complicated terms—a multiplicative factor of the string coupling constant. More loops translate into more factors of the string coupling constant. If the string coupling constant is less than 1, repeated multiplications make the overall contribution ever smaller; if it is 1 or larger, repeated multiplications yield a contribution with the same or larger magnitude.
6. For the mathematically inclined reader, we note that the equation states that spacetime must admit a Ricci-flat metric. If we split spacetime into a Cartesian product of four-dimensional Minkowski spacetime and a six-dimensional compact Kähler space, Ricci-flatness is equivalent to the latter being a Calabi-Yau manifold. This is why Calabi-Yau spaces play such a prominent role in string
theory
7. Of course, nothing absolutely ensures that these indirect approaches are justified. For example, just as some faces are not left-right symmetric, it might be that the laws of physics are different in other far-flung regions of the universe, as we will discuss briefly in Chapter 14.
8. The expert reader will recognize that these statements require so-called N=2 supersymmetry.
9. To be a little more precise, if we call the Heterotic-O coupling constant gHO and the Type I coupling constant gI, then the relation between the two theories states that they are physically identical so long as gHO = 1/gI, which is equivalent to gI = 1/gHO. When one coupling constant is big the other is small.
10. This is a close analog of the R, 1/R duality discussed previously If we call the Type 1/R string coupling constant gIIB then the statement that appears to be true is that the values gIIB and 1/gIIB describe the same physics. gIIB is big, 1/gIIB is small, and vice versa.
11. If all but four dimensions are curled up, a theory with more than eleven total dimensions necessarily gives rise to massless particles with spin greater than 2, something that both theoretical and experimental considerations rule out.
12. A notable exception is the important 1987 work of Duff, Paul Howe, Takeo Inami, and Kelley Stelle in which they drew on earlier insights of Eric Bergshoeff, Ergin Sezgin, and Townsend to argue that ten-dimensional string theory should have a deep eleven-dimensional connection.
13. More precisely, this diagram should be interpreted as saying that we have a single theory that depends on a number of parameters. The parameters include coupling constants as well as geometrical size and shape parameters. In principle, we should be able to use the theory to calculate particular values for all of these parameters—a particular value for its coupling constant and a particular form for the spacetime geometry—but within our current theoretical understanding, we do not know how to accomplish this. And so, to understand the theory better string theorists study its properties as the values of these parameters are varied over all possibilities. If the parameter values are chosen to lie in any of the six peninsular regions of Figure 1.1, the theory has the properties inherent to one of the five string theories, or to eleven-dimensional supergravity, as marked. If the parameter values are chosen to lie in the central region, the physics is governed by the still mysterious M-theory.
14. We should note, though, that even in the peninsular regions there are some exotic ways in which branes can have an effect on familiar physics. For example, it has been suggested that our three extended spatial dimensions might themselves be a three-brane that is large and unfurled. If so, as we go about our daily business we would be gliding through the interior of a three-dimensional membrane. Investigations of such possibilities are now being undertaken.
15. Interview with Edward Witten, May 11, 1998.
Chapter 13
1. The expert reader will recognize that under mirror symmetry, a collapsing three-dimensional sphere on one Calabi-Yau space gets mapped to a collapsing two-dimensional sphere on the mirror Calabi-Yau space—apparently putting us back in the situation of flops discussed in Chapter 11. The difference, however, is that a mirror rephrasing of this sort results in the antisymmetric tensor field Bμv—the real part of the complexified Kähler form on the mirror Calabi-Yau space—vanishing, and this is a far more drastic sort of singularity than that discussed in Chapter 11.
2. More precisely, these are examples of extremal black holes: black holes that have the minimum mass consistent with the force charges they carry, just like the BPS states in Chapter 12. Similar black holes will also play a pivotal role in the following discussion on black hole entropy.
3. The radiation emitted from a black hole should be just like that emitted from a hot oven—the very problem, discussed at the outset of Chapter 4, that played such a pivotal role in the development of quantum mechanics.
4. It turns out that because the black holes involved in space-tearing conifold transitions are extremal, they do not Hawking radiate, regardless of how light they become.
5. Stephen Hawking, lecture at Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1996.
6. In their initial calculation, Strominger and Vafa found that the mathematics was made easier by working with five—not four—extended spacetime dimensions. Surprisingly, after completing their calculation of the entropy of such a five-dimensional black hole they realized that no theoretician had as yet constructed such hypothetical extremal black holes in the setting of five-dimensional general relativity. Since only by comparing their answer to the area of the event horizon of such a hypothetical black hole could they confirm their results, Strominger and Vafa then set out to mathematically construct such a five-dimensional black hole. They succeeded. It was then a simple matter to show that the microscopic string theory calculation of the entropy was in agreement with what Hawking would have predicted based on the area of the black hole's event horizon. But it is interesting to realize that because the black hole solution was found later, Strominger and Vafa did not know the answer they were shooting for while undertaking their entropy calculation. Since their work, numerous researchers, led most notably by Princeton physicist Curtis Callan, have succeeded in extending the entropy calculations to the more familiar setting of four extended spacetime dimensions, and all are in agreement with Hawking's predictions.
7. Interview with Sheldon Glashow, December 29, 1997.
8. Laplace, Philosophical Essay on Probabilities, trans. Andrew I. Dale (New York: Springer-Verlag, 1995).
9. Stephen Hawking, in Hawking and Roger Penrose, The Nature of Space and Time (Princeton: Princeton University Press, 1995), p. 41.
10. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.
11. Interview with Andrew Strominger, December 29, 1997.
12. Interview with Cumrun Vafa, January 12, 1998.
13. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.
14. This issue also has some bearing on the information-loss question, as some physicists have speculated over the years that there might be a central "nugget" embedded in the depths of a black hole that stores all of the information carried by matter that gets trapped within the hole's horizon.
15. In fact, the space-tearing conifold transitions discussed in this chapter involve black holes and hence might seem to be tied up with the question of their singularities. But recall that the conifold tear occurs just as the black hole has shed all its mass, and is therefore not directly related to questions concerning black hole singularities.
Chapter 14
1. More precisely, the universe should be filled with photons conforming to the radiation thermally emitted by a perfectly absorbent body—a "black-body" in the language of thermodynamics—with the stated temperature range. This is the same radiation spectrum emitted quantum mechanically by black holes, as explained by Hawking, and by a hot oven, as explained by Planck.
2. The discussion conveys the spirit of the issues involved although we are glossing over some subtle features having to do with the motion of light in an expanding universe that affect the detailed numerics. In particular, although special relativity declares that nothing can travel faster than the speed of light, this does not preclude two photons carried along on the expanding spatial fabric from receding from one another at a speed exceeding that of light. For example, at the time the universe first became transparent, about 300,000 years ATB, locations in the heavens that were about 900,000 light-years apart would have been able to have influenced each other, even though the distance between them exceeds 300,000 light-years. The extra factor of three comes from the expansion of the spatial fabric. This means that as we run the cosmic film backward in time, by the time we get to 300,000 years ATB, two points in the heavens need only be less than 900,000 light-years apart to have had a chance to influence each other's temperature. These detailed numerics
do not change the qualitative features of the issues discussed.
3. For a detailed and lively discussion of the discovery of the inflationary cosmological model and the problems it resolves, see Alan Guth, The Inflationary Universe (Reading, Mass: Addison-Wesley, 1997).
4. For the mathematically inclined reader, we note that the idea underlying this conclusion is the following: If the sum of the spacetime dimensions of the paths swept out by each of two objects is greater than or equal to the spacetime dimension of the arena through which they are moving then they will generically intersect. For instance, point particles sweep out one-dimensional spacetime paths—the sum of the spacetime dimensions for two such particle paths is therefore two. The spacetime dimension of Lineland is also two, and hence their paths will generally intersect (assuming their velocities have not been finely tuned to be exactly equal). Similarly, strings sweep out two-dimensional spacetime paths (their world-sheets); for two strings the sum in question is therefore four. This means that strings moving in four spacetime dimensions (three space and one time) will generally intersect.
