General relativity describes the force of gravity and is usually applied to large objects. Quantum mechanics is most relevant in describing the smallest structures in the universe such as atoms, electrons and quarks. Any calculation which simultaneously uses General Relativity and Quantum Mechanics yields nonsensical answers. In order to solve this problem, the basic idea is to replace point-like elementary particles with fundamental strings, which are line-like objects of very small length, based on the (deceptively simple) premise that at Planckian scales, where the quantum effects of gravity are strong, particles are actually one-dimensional extended objects.
All the usual particles emerge as excitations of the string and the interactions are simply given by the geometric splitting and joining of these strings. These strings have certain vibrational modes which can be characterized by various quantum numbers such as mass, spin, etc. The basic idea is that each mode carries a set of quantum numbers that correspond to a distinct type of fundamental particle. Among the particles arising as vibrations of the string, we find some which are very similar to electrons, muons, neutrinos and quarks -- the known matter particles. There are others similar to photons, W and Z bosons and gluons -- the known force carriers. And there is one particle similar to the graviton, the elusive fourth force carrier.
Kaluza-Klein theory or compactification.
But what also follows from the theory of strings is that spacetime has more than four dimensions. This is required to keep the theory both consistent and finite. As we observe a 4-dimensional spacetime, somehow we need to find where the extra dimensions are if superstrings are to describe our universe. The extra dimensions at every point in our familiar spacetime are "curled up" into structures on the scale of the [Planck length]?.
In Kaluza's original work it was shown that if we start with a theory of general relativity in five spacetime dimensions and then curl up one of the dimensions into a circle, we end up with a 4-dimensional theory of general relativity plus electromagnetism! The reason why this works is that electromagnetism is a U(1) [gauge theory]? (U(1) is just the group of rotations around a circle). If we assume that the electron has a degree of freedom corresponding to a point moving freely on a circle in spacetime, we find that the theory must contain the photon and that the electron obeys the equations of motion of electromagnetism (Maxwells equations). The Kaluza-Klein mechanism simply gives a geometrical explanation for this circle: it comes from an actual fifth dimension that has been curled up. In this simple example we see that even though the compact dimensions maybe too small to detect directly, they still can have profound physical implications.
The mathematics carried out on the initial string theory found that relativistic, quantum strings fixed the number of space-time dimensions at 26. Also, the initial theory predicted the existence of a particle with an imaginary mass called the tachyon. This was problematic.
Superstring Theory
The next step in string theory development, then, was to eliminate the tachyon. Supersymmetry was seen as the way to solve this problem. It is a theory which unifies bosons and fermions. Every known particle would be paired with a "superpartner" of the opposite type (boson vs. fermions). The major goal of the newest high energy accelerators is to discover these superpartners and to find evidence for supersymmetry. It is an essential ingredient in realistic string theory models, hence the "super" in "superstring".
The consequences are that the tachyon disappears as a predicted particle and the number of space-time dimensions determined by the theory drops from 26 to 10. The new developments had led to five different superstring theories depending on a number of variables within superstring theory itself (properties such as chirality and heteroticity, a vibrating string can either be open or closed, and different mathematical symmetries that the theory can have). Five equally mathematically valid theories were possible.
Table of super string theories:
| Type | Spacetime Dimensions |
Details |
|---|---|---|
| I | 10 | both open and closed strings, group symmetry is SO(32) |
| IIA | 10 | closed strings only, massless fermions spin both ways (nonchiral) |
| IIB | 10 | closed strings only, massless fermions only spin one way (chiral) |
| HO | 10 | closed strings only, heterotic, meaning right moving and left moving strings differ, group symmetry is SO(32) |
| HE | 10 | closed strings only, heterotic, meaning right moving and left moving strings differ, group symmetry is E8 x E8 |
The mathematical apparatus of string theory is very involved and is based on ideas from [conformal field theory]?, [infinite-dimensional algebras]? and Einstein's General Relativity. Although a complete picture of string theory is not yet available, there are indications that string theory is at this moment a promising candidate theory for a unified description of the fundamental particles and forces in nature including gravity.
In the past years, however, strings have been subsumed by M-theory.
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