String theory: Encyclopedia II - String theory - Basic properties

String theory - Basic properties

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories created by adding supersymmetry. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'.

While understanding the details of string and superstring theories requires considerable mathematical sophistication, some qualitative properties of quantum strings can be understood in a fairly intuitive fashion. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg's uncertainty principle. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". Consequently, the minimum size of a string must be related to the string tension.

String theory - Dualities

Before the 1990s, string theorists believed there were five distinct superstring theories: type I, types IIA and type IIB, and the two heterotic string theories (SO(32) and E₈×E₈). The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. But now it is known that this naive picture was wrong, and that the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory, of which there is only one. These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation.

These dualities link quantities that were also thought to be separate. Large and small distance scales, strong and weak coupling strengths – these quantities have always marked very distinct limits of behavior of a physical system, in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.

Suppose we're in ten spacetime dimensions, which means we have nine space and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized momentum around the circle, and this will contribute to the total energy of the particle. But a string is very different, because in addition to traveling around the circle, the string can wrap around the circle. The number of times the string winds around the circle is called the winding number, and that is also quantized. Now the weird thing about string theory is that these momentum modes and the winding modes can be interchanged, as long as we also interchange the radius R of the circle with the quantity , where L_st is the string length. If R is very much smaller than the string length, then the quantity is going to be very large. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.

This type of duality is called T-duality. T-duality relates type IIA superstring theory to type IIB superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle, then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories.

On the other hand, every force has a coupling constant. For electromagnetism, the coupling constant is proportional to the square of the electric charge. When physicists study the quantum behavior of electromagnetism, they can't solve the whole theory exactly, so they break it down to little pieces, and each little piece that they can solve has a different power of the coupling constant in front of it. At normal energies in electromagnetism, the coupling constant is small, and so the first few little pieces make a good approximation to the real answer. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the oscillation modes of the string, called the dilaton. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of expanding in a series, but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

Superstring theories related by S-duality are: type I superstring theory with heterotic SO(32) superstring theory, and type IIB theory with itself.

String theory - Extra dimensions

One intriguing feature of string theory is that it predicts the number of dimensions which the universe should possess. Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions "by hand". The first person to add a fifth dimension to Einstein's four was the German mathematician Theodor Kaluza in 1919. The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926.

Instead, string theory allows one to compute the number of spacetime dimensions from first principles. Technically, this happens because Lorentz invariance can only be satisfied in a certain number of dimensions. This is roughly like saying that if an observer measures the distance between two points, then rotates by some angle and measures again, the observed distance only stays the same if the universe has a particular number of dimensions.

The only problem is that when the calculation is done, the universe's dimensionality is not four as one may expect (three axes of space and one of time), but twenty-six. More precisely, bosonic string theories are 26-dimensional, while superstring and M-theories turn out to involve 10 or 11 dimensions. In bosonic string theories, the 26 dimensions come from the Polyakov equation

(see technical details in the preprint "Quantum Geometry of Bosonic Strings - Revisited").

However, these models appear to contradict observed phenomena. Physicists usually solve this problem in one of two different ways. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable in our phenomenal experience. The 6-dimensional model's resolution is achieved with Calabi-Yau spaces. In 7 dimensions, they are termed G₂ manifolds. Essentially these extra dimensions are compactified by causing them to loop back upon themselves.

A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. If, however, one approaches the hose, one discovers that it contains a second dimension, its circumference. This "extra dimension" is only visible within a relatively close range to the hose, just as the extra dimensions of the Calabi-Yau space are only visible at extremely small distances, and thus are not easily detected.

(Of course, everyday garden hoses exist in three spatial dimensions, but for the purpose of the analogy, its thickness is neglected and only motion on the surface of the hose is considered. A point on the hose's surface can be specified by two numbers, a distance along the hose and a distance along the circumference, just as points on the Earth's surface can be uniquely specified by latitude and longitude. In either case, the object has two spatial dimensions. Like the Earth, garden hoses have an interior, a region that requires an extra dimension; however, unlike the Earth, a Calabi-Yau space has no interior.)

Another possibility is that we are stuck in a 3+1 dimensional subspace of the full universe, where the "3+1" reminds us that time is a different kind of dimension than space. Because it involves mathematical objects called D-branes, this is known as a braneworld theory.

In either case, gravity acting in the hidden dimensions produces other non-gravitational forces such as electromagnetism. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility.

Adapted from the Wikipedia article "Basic properties", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki