**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 10: Notes**

**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.

*Return to Text***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.*

*Return to Text***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 Va'—the so-called string scale—whose value is about the Planck length, 10^-33 centimeter. We can then express string energies as

*v*/

*R*+

*wR*/

*a*', which is invariant under interchange of

*v*and

*w*as well as

*R*and

*a*'/

*R*, where the latter two are now expressed in terms of conventional units of distance.

*Return to Text***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.

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 Ix> on a circle of radius R can be defined by lx>=Eerlp> where p = v/R and Ip> 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: Vc>= E„el'Ip> where I p> is a winding eigenstate with p = wR. From these definitions we immediately see that xis periodic with period 27tR while "X is periodic with period 27t/R, showing that x is a position coordinate on a circle of radius R while .2 is the position coordinate on a circle of radius 1/R. Even more explicitly, we can now imagine taking the two wavepackets lx> and ji->, 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.

*Return to Text***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

*h*^2,1 and

*h*^1,1, where

*h*^2,1h

*h*^p,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 two-cycles ("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' and Hodge numbers, the number of particle families—and the total number of "holes"—will not change.

*Return to Text***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.

*Return to Text***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.

*Return to Text*