by Brian Greene
8. Currently, in addition to string theory, two other approaches for merging general relativity and quantum mechanics are being pursued vigorously. One approach is led by Roger Penrose of Oxford University and is known as twistor theory. The other approach—inspired in part by Penrose's work—is led by Abhay Ashtekar of Pennsylvania State University and is known as the new variables method. Although these other approaches will not be discussed further in this book, there is growing speculation that they may have a deep connection to string theory and that possibly, together with string theory, all three approaches are honing in on the same solution for merging general relativity and quantum mechanics.
Chapter 6
1. The expert reader will recognize that this chapter focuses solely on perturbative string theory; nonperturbative aspects are discussed in Chapters 12 and 13.
2. Interview with John Schwarz, December 23, 1997.
3. Similar suggestions were made independently by Tarmaki Yoneya and by Korkut Bardakci and Martin Halpern. The Swedish physicist Lars Brink also contributed significantly to the early development of string theory.
4. Interview with John Schwarz, December 23, 1997,
5. Interview with Michael Green, December 20, 1997.
6. The standard model does suggest a mechanism by which particles acquire mass—the Higgs mechanism, named after the Scottish physicist Peter Higgs. But from the point of view of explaining the particle masses, this merely shifts the burden to explaining properties of a hypothetical "mass-giving particle"—the so-called Higgs boson. Experimental searches for this particle are underway, but once again, if it is found and its properties measured, these will be input data for the standard model, for which the theory offers no explanation.
7. For the mathematically inclined reader, we note that the association between string vibrational patterns and force charges can be described more precisely as follows. When the motion of a string is quantized, its possible vibrational states are represented by vectors in a Hilbert space, much as for any quantum-mechanical system. These vectors can be labeled by their eigenvalues under a set of commuting hermitian operators. Among these operators are the Hamiltonian, whose eigenvalues give the energy and hence the mass of the vibrational state, as well as operators generating various gauge symmetries that the theory respects. The eigenvalues of these latter operators give the force charges carried by the associated vibrational string state.
8. Based upon insights gleaned from the second superstring revolution (discussed in Chapter 12), Witten and, most notably, Joe Lykken of the Fermi National Accelerator Laboratory have identified a subtle, yet possible, loophole in this conclusion. Lykken, exploiting this realization, has suggested that it might be possible for strings to be under far less tension, and therefore be substantially larger in size, than originally thought. So large, in fact, that they might be observable by the next generation of particle accelerators. If this long-shot possibility turns out to be the case, there is the exciting prospect that many of the remarkable implications of string theory discussed in this and the following chapters will be verifiable experimentally within the next decade. But even in the more "conventional" scenario espoused by string theorists, in which strings are typically on the order of 10-33 centimeters in length, there are indirect ways to search for them experimentally, as we will discuss in Chapter 9.
9. The expert reader will recognize that the photon produced in a collision between an electron and a positron is a virtual photon and therefore must shortly relinquish its energy by dissociating into a particle-antiparticle pair.
10. Of course, a camera works by collecting photons that bounce off the object of interest and recording them on a piece of photographic film. Our use of a camera in this example is symbolic, since we are not imagining bouncing photons off of the colliding strings. Rather, we simply want to record in Figure 6.7(c) the whole history of the interaction. Having said that, we should point out one further subtle point that the discussion in the text glosses over. We learned in Chapter 4 that we can formulate quantum mechanics using Feynman's sum-over-paths method, in which we analyze the motion of objects by combining contributions from all possible trajectories that lead from some chosen starting point to some chosen destination (with each trajectory contributing with a statistical weight determined by Feynman). In Figures 6.6 and 6.7 we show one of the infinite number of possible trajectories followed by point particles (Figure 6.6) or by strings (Figure 6.7) taking them from their initial positions to their final destinations. The discussion in this section, however, applies equally well to any of the other possible trajectories and therefore applies to the whole quantum-mechanical process itself. (Feynman's formulation of point-particle quantum mechanics in the sum-over-paths framework was generalized to string theory through the work of Stanley Mandelstarn of the University of California at Berkeley and by the Russian physicist Alexander Polyakov, who is now on the faculty of the physics department of Princeton University.)
Chapter 7
1. Albert Einstein, as quoted in R. Clark, Einstein: The Life and Times (New York: Avon Books, 1984), p. 287.
2. More precisely, spin-½ means that the angular momentum of the electron from its spin is h/2.
3. The discovery and development of supersymmetry has a complicated history In addition to those cited in the text, essential early contributions were made by R. Haag, M. Sohnius, J. T. Lopuszanski, Y. A. Gol'fand, E. P. Lichtman, J. L. Gervais, B. Sakita, V. P. Akulov, D. V. Volkov, and V. A. Soroka, among many others. Some of their work is documented in Rosanne Di Stefano, Notes on the Conceptual Development of Supersymmetry, Institute for Theoretical Physics, State University of New York at Stony Brook, preprint ITP-SB-8878.
4. For the mathematically inclined reader we note that this extension involves augmenting the familiar Cartesian coordinates of spacetime with new quantum coordinates, say u and v, that are anticommuting: u × v = -v × u. Supersymmetry can then be thought of as translations in this quantum-mechanically augmented form of spacetime.
5. For the reader interested in more details of this technical issue we note the following. In note 6 of Chapter 6 we mentioned that the standard model invokes a "mass-giving particle"—the Higgs boson—to endow the particles of Tables 1.1 and 1.2 with their observed masses. For this procedure to work, the Higgs particle itself cannot be too heavy; studies show that its mass should certainly be no greater than about 1,000 times the mass of a proton. But it turns out that quantum fluctuations tend to contribute substantially to the mass of the Higgs particle, potentially driving its mass all the way to the Planck scale. Theorists have found, however, that this outcome, which would uncover a major defect in the standard model, can be avoided if certain parameters in the standard model (most notably, the so-called bare mass of the Higgs particle) are finely tuned to better than 1 part in 1015 to cancel the effects of these quantum fluctuations on the Higgs particle's mass.
6. One subtle point to note about Figure 7.1 is that the strength of the weak force is shown to be between that of the strong and electromagnetic forces, whereas we have previously said that it is weaker than both. The reason for this lies in Table 1.2, in which we see that the messenger particles of the weak force are quite massive, whereas those of the strong and electromagnetic forces are massless. Intrinsically, the strength of the weak force (as measured by its coupling constant—an idea we will come upon in Chapter 12) is as shown in Figure 7.1, but its massive messenger particles are sluggish conveyers of its influence and diminish its effects. In Chapter 14 we will see how the gravitational force fits into Figure 7.1.
7. Edward Witten, lecture at the Heinz Pagels Memorial Lecture Series, Aspen, Colorado, 1997.
8. For an in-depth discussion of these and related ideas, see Steven Weinberg, Dreams of a Final Theory.
Chapter 8
1. This is a simple idea, but since the imprecision of common language can sometimes lead to confusion, two clarifying remarks are in order. First, we are assuming that the ant is constr
ained to live on the surface of the garden hose. If, on the contrary, the ant could burrow into the interior of the hose—if it could penetrate into the rubber material of the hose—we would need three numbers to specify its position, since we would need to also specify how deeply it had burrowed. But if the ant lives only on the hose's surface, its location can be specified with just two numbers. This leads to our second point. Even with the ant living on the hose's surface, we could, if we so chose, specify its location with three numbers: the ordinary left-right, back-forth, and up-down positions in our familiar three-dimensional space. But once we know that the ant lives on the surface of the hose, the two numbers referred to in the text give the minimal data that uniquely specify the ant's position. This is what we mean by saying that the surface of the hose is two-dimensional.
2. Surprisingly, the physicists Savas Dimopoulos, Nima Arkani-Hamed, and Gia Dvali, building on earlier insights of Ignatios Antomadis and Joseph Lykken, have pointed out that even if an extra curled-up dimension were as large as a millimeter in size, it is possible that it would not yet have been detected experimentally. The reason is that particle accelerators probe the microworld by utilizing the strong, weak, and electromagnetic forces. The gravitational force, being incredibly feeble at technologically accessible energies, is generally ignored. But Dimopoulos and his collaborators note that if the extra curled-up dimension has an impact predominantly on the gravitational force (something, it turns out, that is quite plausible in string theory), all extant experiments could well have overlooked it. New, highly sensitive gravitational experiments will look for such "large" curled-up dimensions in the near future. A positive result would be one of the greatest discoveries of all time.
3. Edwin Abbott, Flatland (Princeton: Princeton University Press, 1991).
4. A. Einstein in letter to T. Kaiuza as quoted in Abraham Pais, "Subtle is the Lord": The Science and the Life of Albert Einstein (Oxford: Oxford University Press, 1982), p. 330.
5. A. Einstein in letter to T. Kaluza as quoted in D. Freedman and P. van Nieuwenhuizen, "The Hidden Dimensions of Spacetime," Scientific American 252 (1985), 62.
6. Ibid.
7. Physicists found that the most difficult feature of the standard model to incorporate through a higher-dimensional formulation is something known as chirality. So as not to overburden the discussion we have not covered this concept in the main text, but for readers who are interested we do so briefly here. Imagine that someone shows you a film of some particular scientific experiment and confronts you with the unusual challenge of determining whether the film shot the experiment directly or whether it shot the experiment by looking at its reflection in a mirror. As the cinematographer was quite expert, there are no telltale signs of a mirror being involved. Is this a challenge you can meet? In the mid-1950s, the theoretical insights of T. D. Lee and C. N. Yang, and the experimental results of C. S. Wu and collaborators, showed that you can meet the challenge, so long as an appropriate experiment had been filmed. Namely, their work established that the laws of the universe are not perfectly mirror symmetric in the sense that the mirror-reflected version of certain processes—those directly dependent on the weak force—cannot happen in our world, even though the original process can. And so, as you watch the film if you see one of these forbidden processes occur, you will know that you are watching a mirror-reflected image of the experiment, as opposed to the experiment itself. Since mirrors interchange left and right, the work of Lee, Yang, and Wu established that the universe is not perfectly left-right symmetric—in the language of the field, the universe is chiral. It is this feature of the standard model (the weak force, in particular) that physicists found nearly impossible to incorporate into a higher-dimensional supergravity framework. To avoid confusion, we note that in Chapter 10 we will discuss a concept in string theory known as "mirror symmetry," but the use of the word "mirror" in that context is completely different from its use here.
8. For the mathematically inclined reader, we note that a Calabi-Yau manifold is a complex Kähler manifold with vanishing first Chern class. In 1957 Calabi conjectured that every such manifold admits a Ricci-flat metric, and in 1977 Yau proved this to be true.
9. This illustration is courtesy of Andrew Hanson of Indiana University, and was made using the Mathematica 3-D graphing package.
10. For the mathematically inclined reader we note that this particular Calabi-Yau space is a real three-dimensional slice through the quintic hypersurface in complex projective four-space.
Chapter 9
1. Edward Witten, "Reflections on the Fate of Spacetime" Physics Today, April 1996, p. 24.
2. Interview with Edward Witten, May 11, 1998.
3. Sheldon Glashow and Paul Ginsparg, "Desperately Seeking Superstrings?" Physics Today, May 1986, p. 7.
4. Sheldon Glashow, in The Superworld I, ed. A. Zichichi (New York: Plenum, 1990), p. 250.
5. Sheldon Glashow, Interactions (New York: Warner Books, 1988), p. 335.
6. Richard Feynman, in Superstrings: A Theory of Everything? ed. Paul Davies and Julian Brown (Cambridge, Eng: Cambridge University Press, 1988).
7. Howard Georgi, in The New Physics, ed. Paul Davies (Cambridge: Cambridge University Press 1989), p. 446.
8. Interview with Edward Witten, March 4, 1998.
9. Interview with Cumrun Vafa, January 12, 1998.
10. Murray Gell-Mann, as quoted in Robert P. Crease and Charles C. Mann, The Second Creation (New Brunswick, N.J.: Rutgers University Press), 1996, p. 414.
11. Interview with Sheldon Glashow, December 28, 1997.
12. Interview with Sheldon Glashow, December 28, 1997.
13. Interview with Howard Georgi, December 28, 1997. During the interview, Georgi also noted that the experimental refutation of the prediction of proton decay that emerged from his and Glashow's first proposed grand unified theory (see Chapter 7) played a significant part in his reluctance to embrace superstring theory. He noted poignantly that his grand unified theory invoked a vastly higher energy realm than any theory previously considered, and when its prediction was proved wrong—when it resulted in his "being slapped down by nature"—his attitude toward studying extremely high energy physics abruptly changed. When I asked him whether experimental confirmation of his grand unified theory might have inspired him to lead the charge to the Planck scale, he responded, "Yes, it likely would have."
14. David Gross, "Superstrings and Unification," in Proceedings of the XXIV International Conference on High Energy Physics, ed. R. Kotthaus and J. Kühn (Berlin: Springer-Verlag, 1988), p. 329.
15. Having said this, it's worth bearing in mind the long-shot possibility, pointed out in endnote 8 of Chapter 6, that strings just might be significantly longer than originally thought and therefore might be subject to direct experimental observation by accelerators within a few decades.
16. For the mathematically inclined reader we note that the more precise mathematical statement is that the number of families is half the absolute value of the Euler number of the Calabi-Yau space. The Euler number itself is the alternating sum of the dimensions of the manifold's homology groups—the latter being what we loosely refer to as multidimensional holes. So, three families emerge from Calabi-Yau spaces whose Euler number is ±6.
17. Interview with John Schwarz, December 23, 1997.
18. For the mathematically inclined reader we note that we are referring to Calabi-Yau manifolds with a finite, nontrivial fundamental group, the order of which, in certain cases, determines the fractional charge denominators.
19. Interview with Edward Witten, March 4, 1998.
20. For the expert we note that some of these processes violate lepton number conservation as well as charge-parity-time (CPT) reversal symmetry.
Chapter 10
1. For completeness, we note that although much of what we have covered to this point in the book applies equally well to open strings (a string with loose ends) or closed-string loops (the strings on which we have focused), the topic
discussed here is one in which the two kinds of strings would appear to have different properties. After all, an open string will not get entangled by looping around a circular dimension. Nevertheless, through work that ultimately has played a pivotal part in the second superstring revolution, in 1989 Joe Polchinski from the University of California at Santa Barbara and two of his students, Jian-Hui Dai and Robert Leigh, showed how open strings fit perfectly into the conclusions we find in this chapter.
2. In case you are wondering why the possible uniform vibrational energies are whole number multiples of 1/R, you need only think back to the discussion of quantum mechanics—the warehouse in particular—from Chapter 4. There we learned that quantum mechanics implies that energy, like money, comes in discrete lumps: whole number multiples of various energy denominations. In the case of uniform vibrational string motion in the Garden-hose universe, this energy denomination is precisely 1/R, as we demonstrated in the text using the uncertainty principle. Thus the uniform vibrational energies are whole number multiples of 1/R.
3. Mathematically, the identity between the string energies in a universe with a circular dimension whose radius is either R or 1/R arises from the fact that the energies are of the form v/R + wR, where v is the vibration number and w is the winding number. This equation is invariant under the simultaneous interchange of v and w as well as R and 1/R—i.e., under the interchange of vibration and winding numbers and inversion of the radius. In our discussion we are working in Planck units, but we can work in more conventional units by rewriting the energy formula in terms of √α'—so-called string scale—whose value is about the Planck length, 10-13 centimeter. We can then express string energies as v/R + wR/α', which is invariant under interchange of v and w as well as R and α'/R, where the latter two are now expressed in terms of conventional units of distance.
4. You may be wondering how it's possible for a string that stretches all the way around a circular dimension of radius R to nevertheless measure the radius to be 1/R. Although a thoroughly justifiable concern, its resolution actually lies in the imprecise phrasing of the question itself. You see, when we say that the string is wrapped around a circle of radius R, we are by necessity invoking a definition of distance (so that the phrase "radius R" has meaning). But this definition of distance is the one relevant for the unwound string modes—that is, the vibration modes. From the point of view of this definition of distance—and only this definition—the winding string configurations appear to stretch around the circular part of space. However, from the second definition of distance, the one that caters to the wound-string configurations, they are every bit as localized in space as are the vibration modes from the viewpoint of the first definition of distance, and the radius they "see" is 1/R, as discussed in the text.