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Section 4: Strings and Extra Dimensions

A depiction of one of the six-dimensional spaces that seem promising for string compactification.

Figure 9: A depiction of one of the six-dimensional spaces that seem promising for string compactification.

Source: © Wikimedia Commons, CC Attribution-Share Alike 2.5 Generic license. Author: Lunch, 23 September 2007. More info

We have already mentioned that string theories that correspond to quantum gravity together with the three other known fundamental forces seem to require 10 spacetime dimensions. While this may come as a bit of a shock—after all, we certainly seem to live in four spacetime dimensions—it does not immediately contradict the ability of string theory to describe our universe. The reason is that what we call a "physical theory" is a set of equations that is dictated by the fundamental fields and their interactions. Most physical theories have a unique basic set of fields and interactions, but the equations may have many different solutions. For instance, Einstein's theory of general relativity has many nonphysical solutions in addition to the cosmological solutions that look like our own universe. We know that there are solutions of string theory in which the 10 dimensions take the form of four macroscopic spacetime dimensions and six dimensions curled up in such a way as to be almost invisible. The hope is that one of these is relevant to physics in our world.

To begin to understand the physical consequences of tiny, curled-up extra dimensions, let us consider the simplest relevant example. The simplest possibility is to consider strings propagating in nine-dimensional flat spacetime, with the 10th dimension curled up on a circle of size R. This is clearly not a realistic theory of quantum gravity, but it offers us a tantalizing glimpse into one of the great theoretical questions about gravity: How will a consistent theory of quantum gravity alter our notions of spacetime geometry at short distances? In string theory, the concept of curling up, or compactification, on a circle, is already startlingly different from what it would be in point particle theory.

String theorists generally believe that extra dimensions are compactified, or curled up.

Figure 10: String theorists generally believe that extra dimensions are compactified, or curled up.

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To compare string theory with normal particle theories, we will compute the simplest physical observable in each kind of theory, when it is compactified on a circle from ten to nine dimensions. This simplest observable is just the masses of elementary particles in the lower-dimensional space. It will turn out that a single type of particle (or string) in 10 dimensions gives rise to a whole infinite tower of particles in nine dimensions. But the infinite towers in the string and particle cases have an important difference that highlights the way that strings "see" a different geometry than point particles.

Particles in a curled-up dimension

Let us start by explaining how an infinite tower of nine-dimensional (9D) particles arises in the 10-dimensional (10D) particle theory. To a 9D observer, the velocity and momentum of a given particle in the hidden tenth dimension, which is too small to observe, are invisible. But the motion is real, and a particle moving in the tenth dimension has a nonzero energy. Since the particle is not moving around in the visible dimensions, one cannot attribute its energy to energy of motion, so the 9D observer attributes this energy to the particle's mass. Therefore, for a given particle species in the fundamental 10D theory, each type of motion it is allowed to perform along the extra circle gives rise to a new elementary particle from the 9D perspective.

If a particle is constrained to move on a circle, its wave must resemble the left rather than the right drawing.

Figure 11: If a particle is constrained to move on a circle, its wave must resemble the left rather than the right drawing.

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To understand precisely what elementary particles the 9D observer sees, we need to understand how the 10D particle is allowed to move on the circle. It turns out that this is quite simple. In quantum mechanics, as we will see in Units 5 and 6, the mathematical description of a particle is a "probability wave" that gives the likelihood of the particle being found at any position in space. The particle's energy is related to the frequency of the wave: a higher frequency wave corresponds to a particle with higher energy. When the particle motion is confined to a circle, as it is for our particle moving in the compactified tenth dimension, the particle's probability wave needs to oscillate some definite number of times (0, 1, 2 ...) as one goes around the circle and comes back to the same point. Each possible number of oscillations on the circle corresponds to a distinct value of energy that the 10D particle can have, and each distinct value of energy will look like a new particle with a different mass to the 9D observer. The masses of these particles are related to the size of the circle, and the number of wave oscillations around the circle:

So, as promised, the hidden velocity in the tenth dimension gives rise to a whole tower of particles in nine dimensions.

Strings in a curled-up dimension

Now, let us consider a string theory compactified on the same circle as above. For all intents and purposes, if the string itself is not oscillating, it is just like the 10D particle we discussed above. The 9D experimentalist will see the single string give rise to an infinite tower of 9D particles with distinct masses. But that’s not the end of the story. We can also wind the string around the circular tenth dimension. To visualize this, imagine winding a rubber band around the thin part of a doorknob, which is also a circle. If the string has a tension Tstring = 1/, (the conventional notation for the string tension), then winding the string once, twice, three times ... around a circle of size R, costs an energy:

This is because the tension is defined as the mass per unit length of the string; and if we wind the string n times around the circle, it has a length which is n times the circumference of the circle. Just as a 9D experimentalist cannot see momentum in the 10th dimension, she also cannot see this string's winding number. Instead, she sees each of the winding states above as new elementary particles in the 9D world, with discrete masses that depend on the size of the compactified dimension and the string tension.

Geometry at short distances

One of the problems of quantum gravity raised in Section 2 is that we expect geometry at short distances to be different somehow. After working out what particles our 9D observer would expect to see, we are finally in a position to understand how geometry at short distances is different in a string theory.

The string tension, 1/, is related to the length of the string, , via = 2. Strings are expected to be tiny, with ~ 10-32 centimeter, so the string tension is very high. If the circle is of moderate to macroscopic size, the winding mode particles are incredibly massive since their mass is proportional to the size of the circle multiplied by the string tension. In this case, the 9D elementary particle masses in the string theory look much like that in the point particle theory on a circle of the same size, because such incredibly massive particles are difficult to see in experiments.

The consequences of strings winding around a larger extra dimension are the same as strings moving around a smaller extra dimension.

Figure 12: The consequences of strings winding around a larger extra dimension are the same as strings moving around a smaller extra dimension.

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However, let us now imagine shrinking R until it approaches the scale of string theory or quantum gravity, and becomes less than . Then, the pictures one sees in point particle theory, and in string theory, are completely different. When R is smaller than , the modes m1, m2 ... are becoming lighter and lighter. And at very small radii, they are low-energy excitations that one would see in experiments as light 9D particles.

In the string theory with a small, compactified dimension, then, there are two ways that a string can give rise to a tower of 9D particles: motion around the circle, as in the particle theory, and winding around the circle, which is unique to the string theory. We learn something very interesting about geometry in string theory when we compare the masses of particles in these two towers.

For example, in the "motion" tower, m1 = 1/R; and in the "winding" tower, m1 = R/. If we had a circle of size /R instead of size R, we'd get exactly the same particles, with the roles of the momentum-carrying strings and the wound strings interchanged. Up to this interchange, strings on a very large space are identical (in terms of these light particles, at least) to strings on a very small space. This large/small equivalence extends beyond the simple considerations we have described here. Indeed, the full string theory on a circle of radius R is completely equivalent to the full string theory on a circle of radius /R. This is a very simple illustration of what is sometimes called "quantum geometry" in string theory; string theories see geometric spaces of small size in a very different way than particle theories do. This is clearly an exciting realization, because many of the mysteries of quantum gravity involve spacetime at short distances and high energies.