The Elegant Universe - Brian Greene [216]
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 |x> on a circle of radius R can be defined by |x>=veixp|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 |> defined by making use of the winding string states: |>= we|> where |> is a winding eigenstate with = wR. From these definitions we immediately see that x is periodic with period 2R while is periodic with period 2/R, showing that x is a position coordinate on a circle of radius R while is the position coordinate on a circle of radius 1/R. Even more explicitly, we can now imagine taking the two wavepackets |x> and |>, 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 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 h2,1 and h1,1 Hodge numbers, the number