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THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 6: Notes
1. The expert reader will recognize that this chapter focuses solely on perturbative string theory; nonperturbative aspects are discussed in Chapters 12 and 13. Return to Text
2. Interview with John Schwarz, December 23, 1997. Return to Text
3. Similar suggestions were made independently by Tamiaki Yoneya and by Korkut Bardakci and Martin Halpern. The Swedish physicist Lars Brink also contributed significantly to the early development of string theory. Return to Text
4. Interview with John Schwarz, December 23, 1997. Return to Text
5. Interview with Michael Green, December 20, 1997. Return to Text
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. Return to Text
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. (Epsilon=One: Vector oscillations are referred to as "swing" oscillations. Swing oscillations define the ellipsoid envelopes of a Pulsoid.) 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, (Epsilon=One: Mass is a result of vector oscillations (swing) becoming harmonious with other harmonic oscillations, which sets up a pairs of paired resonances (Resoloids). Said resonance is symbolically described by the inscribed circles within Brunardot triangles.) 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. Return to Text
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. Return to Text
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. (Epsilon=One: Such a dissociation is nonsense.) Return to Text
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 (Epsilon=One: Sum-over-paths framwork is a reflection of the serendipitous occurrence of the uniting of various oscillations of Seminal Motion from chaos that manifests as the symbolic Pulsoid. When the swing oscillation (hypotenuse) is greater than the vibrational oscillation (base) by the value of the Elliptical Constant (EC), there is . . . mass.) through the work of Stanley Mandelstam 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.) Return to Text
2. Interview with John Schwarz, December 23, 1997. Return to Text
3. Similar suggestions were made independently by Tamiaki Yoneya and by Korkut Bardakci and Martin Halpern. The Swedish physicist Lars Brink also contributed significantly to the early development of string theory. Return to Text
4. Interview with John Schwarz, December 23, 1997. Return to Text
5. Interview with Michael Green, December 20, 1997. Return to Text
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. Return to Text
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. (Epsilon=One: Vector oscillations are referred to as "swing" oscillations. Swing oscillations define the ellipsoid envelopes of a Pulsoid.) 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, (Epsilon=One: Mass is a result of vector oscillations (swing) becoming harmonious with other harmonic oscillations, which sets up a pairs of paired resonances (Resoloids). Said resonance is symbolically described by the inscribed circles within Brunardot triangles.) 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. Return to Text
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. Return to Text
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. (Epsilon=One: Such a dissociation is nonsense.) Return to Text
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 (Epsilon=One: Sum-over-paths framwork is a reflection of the serendipitous occurrence of the uniting of various oscillations of Seminal Motion from chaos that manifests as the symbolic Pulsoid. When the swing oscillation (hypotenuse) is greater than the vibrational oscillation (base) by the value of the Elliptical Constant (EC), there is . . . mass.) through the work of Stanley Mandelstam 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.) Return to Text