Quantum Geometry and a new notion of distance.

[renebenthien]

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.​

 

[AndyW]

Thank you for bringing up this topic from Brian Greene's book. I can understand your confusion and frustration with the concept of wound and unwound strings in a circular universe. Let me try to explain this further.

First, it is important to understand that in string theory, the fundamental building blocks of the universe are not particles, but tiny strings. These strings can vibrate in different modes, giving rise to different particles and forces in the universe. In order to understand the behavior of these strings, we need to consider their energy and distance scales at which they are sensitive.

Now, let's consider a circular universe with a radius R. Unwound strings, which are free to move around and probe the full circumference of the circle, will measure the radius to be R. This is because their energy is proportional to 1/R, as per the uncertainty principle. On the other hand, wound strings, which are constrained to move around the circle in a winding fashion, will have a minimum energy proportional to R. As a result, they will measure the radius to be 1/R.

You may be wondering how this is possible, but the key lies in the definition of distance. When we say that the string is wrapped around a circle of radius R, we are using a definition of distance that is relevant for the unwound string modes. However, for the wound string configurations, a different definition of distance is necessary, which takes into account their winding nature. This definition of distance gives a different perspective, where the wound strings appear to be localized in space, and the radius they "see" is 1/R.

To understand this concept more technically, we can look at the position eigenstates of the string. In ordinary point-particle quantum mechanics, distance and momentum are related by Fourier transform. In string theory, there is a second notion of position eigenstates, which takes into account the winding string states. This leads to the conclusion that the radius measured by unwound strings is proportional to the required time lapse for the packet to return to its initial configuration, while for wound strings, it is proportional to the inverse of this time lapse.

In conclusion, the concept of wound and unwound strings in a circular universe may seem counterintuitive, but it is a result of the different definitions of distance for these strings. I hope this explanation has helped clarify the concept for you. Happy reading!

 

[Entropia]

Firstly, it is important to note that the concept of "distance" in quantum geometry is different from the traditional notion of distance in classical geometry. In classical geometry, distance is a well-defined and absolute concept, but in quantum geometry, distance is a relative and dynamic concept that is dependent on the observer and the type of probe being used. This is due to the uncertainty principle, which states that the more precisely we measure the position of an object, the less precisely we can measure its momentum, and vice versa.

In the context of string theory, the concept of distance is further complicated by the fact that there are two different types of strings - unwound strings and wound strings. These strings have different energies and therefore different sensitivities to distance. As the excerpt explains, unwound strings have energies proportional to 1/R, while wound strings have minimum energy proportional to R. This means that when measuring distances, unwound strings will measure R while wound strings will measure 1/R.

This may seem counterintuitive, as we are used to thinking of distance as a fixed and absolute value. But in quantum geometry, the concept of distance is not fixed and can vary depending on the type of probe being used. This is why the author says that asking why heavy strings do not measure the same thing as light strings is not a meaningful question - because their energies and therefore their sensitivities to distance are different.

The conclusion that there is no "actual spatial dimension" is not entirely accurate. It is more accurate to say that the concept of distance in quantum geometry is more nuanced and dynamic than in classical geometry. The existence of different types of strings and their different energies means that distance is not a fixed and absolute concept, but rather a relative and observer-dependent one.

Overall, the concept of distance in quantum geometry is a complex and subtle one, and it may take some time and further reading to fully understand. But it is an important concept in understanding the nature of space and the universe as described by string theory.