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According to the theory of superstrings, the fundamental constituents of the material world are not point-like elementary particles, but tiny one-dimensional strings having a length of about 10^{-33} cm (the Planck length). Like the string of a violin, they can vibrate in many different ways (different modes), which correspond to the different elementary particles observed in nature. It is a quantum theory that incorporates gravity naturally. In its larger framework of M-theory, the strengths of all the four fundamental forces merge together at very small distance (~10^{-33} cm.) as shown in Figure 15-13.

There are two classes of strings, those with ends (open strings), and those without (closed strings -- a loop) as shown in Figure 15-14(a) and (b). The particles associated with the open strings are the spin-1 gauge bosons and fermions. Their movement is restricted on the surface of a membrane by some boundary conditions. Graviton with spin-2 is an example of closed string, which can travel freely in all spatial dimensions. These are the ingredients for the theory of manyfold universe. When a point particle moves through space-time, it follows a geodesic (a path of minimum length) and sweeps out a one-dimensional curve which is referred to as its world-line (Figure 15-14(c)). However, when a string propagates through | ||

## Figure 15-13 Super-unifi- |
## Figure 15-14 Strings [view large image] |
space-time, it sweeps out a two-dimensional surface which, by analogy, is called its world-sheet, and moves along a surface of minimum area. (See |

When supersymmetry is incorporated into the original string theory, it resolves the problem with tachyon (square of mass is negative), accommodates the ferminonic vibrational pattern, and merge general relativity with quantum mechanics. The Theory of Strings becomes the Theory of Superstrings.

The "heterotic" superstring theory is a theory of closed strings. In contrast to open strings with gauge charges at the endpoints, here the gauge charges are "smeared" over the entire heterotic string. Vibrations (waves) can travel around any closed string in two directions, but the unusual feature of the heterotic string is that the waves moving in each direction are completely different. The clockwise moving waves are the waves of the 10-dimensional superstring, whereas the waves moving anticlockwise are those of the original 26-dimensional bosonic string. To obtain a consistent 10 dimensional theory, 16 of the extra dimensions are interpreted as internal degrees of freedom, which are found to be related to local gauge symmetry.

A ten-dimensional space is required in order to eliminate ghosts (negative probability) in the formalism. To specify a point in this 10-dimensional space requires the usual four (x, y, z, t), plus an additional six more coordinates. Suppose one of these extra coordinates is curled into a small circle, | ||

## Figure 15-15 Compactification |
## Figure 15-16 Calabi-Yau Space [view large image] |

Topology becomes an important tool in superstring when it is treated as quantum mechanical object. This branch of mathematics is concerned with smooth, gradual, continuous change of geometric shape. For example, a square can be continuously deformed into a circle by pushing in the corners and rounding the sides. The essential rule is that no new hole can be created in the new form by tearing. Some topological equivalent objects are shown in Figure 15-17. Thus, as shown in Figure 15-18a, several one-loop Feynman | ||

## Figure 15-17 Topology |
## Figure 15-18a Feynman Diagrams, Closed-string [view large image] |
diagrams for point particle can now be represented by just one closed-string diagram of the same topology. |

String theory is invariant under conformal transformations (Figure 15-18b). Such change in topology makes it feasible to evaluate string diagrams. Among other things this makes it possible to compactify the world sheet, closing off the holes corresponding to incoming and outgoing strings. For example, a world sheet with one incoming and one outgoing string (as in (a) of Figure 15-18b) can be conformally mapped to the plane of (a') with the incoming string appearing at the origin and the outgoing string at infinity (not shown) or to the sphere of (a") with the incoming and outgoing strings appearing at the south and north poles. The external string states in (b) of Figure 15-18b with four awkward legs are projected to points as indicated in (b'). | ||

## Figure 15-18b Conformal Changes [view large image] |
## Figure 15-18c String History [view large image] |

Interaction between two strings can be portrayed by a diagram similar to the Feynman diagram for the interaction of two point particles. In place of lines and points in the latter case, the paths of the strings become tubes. The two strings do not meet at a point, they interact by merging (from the incoming) and splitting (to the outgoing). Such a smearing of the interaction avoids the singularity at the point where the two particles meet and thus the theory of string is not plagued by the infinities in point particle | |

## Figure 15-19 Strings Interaction [view large image] |

According to the differences in the number of supersymmetry, gauge groups, kinds of strings or branes, etc. there are five different versions of superstrings theories as shown in Figure 15-20. They are connected by the S-duality, which relates the strong coupling limit of one theory to the weak coupling limit of another theory; and the T-duality, which relates a theory which is compactified on a circle with radius R, to another theory compactified on a circle with radius 1/R. These 10-dimension theories are ultimately linked to the 11-dimension M theory. | |

## Figure 15-20 Superstrings Theories [view large image] |

Currently there is no testable predictions from superstrings. However, it can be shown that at energies below 10^{16} Gev, the heterotic string theory effectively leads to an ordinary grand unified theory. Meanwhile at this moment, superstrings is the only viable theory that can unify the four interactions (See Figure 15-04a), and have the potential to provide explanations for all the fundamental phenomena. It could take the place at the end of the long journey toward the ultimate theory as depicted in Figure 15-05a and 15-05b.

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