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Quantum Geometry and a new notion of distance.

  1. Jul 11, 2009 #1
    I've been reading through Brian Greene's "The Elegant Universe". It was going great until the chapter on Quantum Geometry. Here he says that a wound string over a circular garden hose universe of radius R would measure the radius to be 1/R. The measurement of radius to be R is only from the perspective of the unwound 'light' strings. He says asking why the heavy strings do not measure the same thing is not a meaningful question.

    I'm not sure I understand this. I get that reducing a particle's energy would increase the uncertainty in the position of whatever it is probing, hence our resolution of whatever it is that we are probing is also reduced. But this doesn't affect the actual spatial dimensions of whatever is being probed right? Why are we coming to the conclusion that there is no 'actual spatial dimension'? Is that the conclusion or am I misinterpreting this?

    If anyone has read The Elegant Universe and remembers the chapter, please give me your thoughts. I have a pdf version if you need to refresh you memory.

    I've excerpted the part of the explanation relevant to the argument: the circumference of the garden-hose universe is different depending on whether the string is wound or unwound:
    Unwound strings can move around freely and probe the full circumference of the circle, a length proportional to R. By the uncertainty principle, their energies are proportional to 1/R (recall from Chapter 6 the inverse relation between the energy of a probe and the distances to which it is sensitive). On the other hand, we have seen that wound strings have minimum energy proportional to R; as probes of distances the uncertainty principle tells us that they are therefore sensitive to the reciprocal of this value, 1/R. The mathematical embodiment of this idea shows that if each is used to measure the radius of a circular dimension of space, unwound string probes will measure R while wound strings will measure 1/R, where, as before, we are measuring distances in multiples of the Planck length.​

    The note attached to this descriptions reads:
    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 pointparticle 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.​
     
  2. jcsd
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