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String theory
Fundamental objects
String Cosmic string Brane D-brane
Perturbative theory
Bosonic Superstring (Type I, Type II, Heterotic)
Non-perturbative results
S-duality T-duality U-duality M-theory F-theory AdS/CFT correspondence
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Phenomenology Cosmology Landscape
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Geometric Langlands correspondence Mirror symmetry Monstrous moonshine Vertex algebra K-theory
Related concepts Theory of everything Conformal field theory Quantum gravity Supersymmetry Supergravity Twistor string theory N = 4 supersymmetric Yang–Mills theory Kaluza–Klein theory Multiverse Holographic principle
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History Glossary
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Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantum gravity. Superstring theory is a shorthand for supersymmetric string theory because unlike bosonic string theory, it is the version of string theory that incorporates fermions and supersymmetry.

Background

The deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, galaxies, super clusters), with quantum mechanics, which describes the other three fundamental forces acting on the atomic scale.

The development of a quantum field theory of a force invariably results in infinite (and therefore useless) probabilities. Physicists have developed mathematical techniques (renormalization) to eliminate these infinities which work for three of the four fundamental forces – electromagnetic, strong nuclear and weak nuclear forces - but not for gravity. The development of a quantum theory of gravity must therefore come about by different means than those used for the other forces.

Basic idea

The basic idea is that the fundamental constituents of reality are strings of the Planck length (about 10 m) which vibrate at resonant frequencies. Every string in theory has a unique resonance, or harmonic. Different harmonics determine different fundamental forces. The tension in a string is on the order of the Planck force (10 newtons). The graviton (the proposed messenger particle of the gravitational force), for example, is predicted by the theory to be a string with wave amplitude zero. Another key insight provided by the theory is that no measurable differences can be detected between strings that wrap around dimensions smaller than themselves and those that move along larger dimensions (i.e., effects in a dimension of size R equal those whose size is 1/R). Singularities are avoided because the observed consequences of "Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of a string, at which point it would actually begin expanding.

Extra dimensions

See also: Why does consistency require 10 dimensions?

Our physical space is observed to have only three large dimensions — and taken together with time as the fourth dimension — a physical theory must take this into account. However, nothing prevents a theory from including more than 4 dimensions, per se. In the case of string theory, consistency requires spacetime to have 10, 11 or 26 dimensions. The conflict between observation and theory is resolved by making the unobserved dimensions compactified.

Our minds have difficulty visualizing higher dimensions because we can only move in three spatial dimensions. One way of dealing with this limitation is not to try to visualize higher dimensions at all, but just to think of them as extra numbers in the equations that describe the way the world works. This opens the question of whether these 'extra numbers' can be investigated directly in any experiment (which must show different results in 1, 2, or 2+1 dimensions to a human scientist). This, in turn, raises the question of whether models that rely on such abstract modeling (and potentially impossibly huge experimental apparatus) can be considered 'scientific.' Six-dimensional Calabi-Yau shapes can account for the additional dimensions required by superstring theory.The theory states that every point in space(or whatever we considered as point) is in fact a very small 'sphere'(better say manifold) with a diameter of 10 m

Superstring theory is not the first theory to propose extra spatial dimensions, the Kaluza-Klein theory did already. Modern string theory relies on the mathematics of folds, knots, and topology, which was largely developed after Kaluza and Klein, and has made physical theories relying on extra dimensions much more credible.

Number of superstring theories

Theoretical physicists were troubled by the existence of five separate string theories. This has been solved by the second superstring revolution in the 1990s during which the five string theories were discovered to be different limits of a single underlying theory: M-theory.

The five consistent superstring theories are:

• The type I string has one supersymmetry in the ten-dimensional sense (16 supercharges). This theory is special in the sense that it is based on unoriented open and closed strings, while the rest are based on oriented closed strings.
• The type II string theories have two supersymmetries in the ten-dimensional sense (32 supercharges). There are actually two kinds of type II strings called type IIA and type IIB. They differ mainly in the fact that the IIA theory is non-chiral (parity conserving) while the IIB theory is chiral (parity violating).
• The heterotic string theories are based on a peculiar hybrid of a type I superstring and a bosonic string. There are two kinds of heterotic strings differing in their ten-dimensional gauge groups: the heterotic E₈×E₈ string and the heterotic SO(32) string. (The name heterotic SO(32) is slightly inaccurate since among the SO(32) Lie groups, string theory singles out a quotient Spin(32)/Z₂ that is not equivalent to SO(32).)

Chiral gauge theories can be inconsistent due to anomalies. This happens when certain one-loop Feynman diagrams cause a quantum mechanical breakdown of the gauge symmetry. Having anomalies cancel puts a severe constraint on possible superstring theories.

Integrating general relativity and quantum mechanics

General relativity typically deals with situations involving large mass objects in fairly large regions of spacetime whereas quantum mechanics is generally reserved for scenarios at the atomic scale (small spacetime regions). The two are very rarely used together, and the most common case in which they are combined is in the study of black holes. Having "peak density", or the maximum amount of matter possible in a space, and very small area, the two must be used in synchrony in order to predict conditions in such places; yet, when used together, the equations fall apart, spitting out impossible answers, such as imaginary distances and less than one dimension.

The major problem with their congruence is that, at sub-Planck (an extremely small unit of length) lengths, general relativity predicts a smooth, flowing surface, while quantum mechanics predicts a random, warped surface, neither of which are anywhere near compatible. Superstring theory resolves this issue, replacing the classical idea of point particles with loops. These loops have an average diameter of the Planck length, with extremely small variances, which completely ignores the quantum mechanical predictions of sub-Planck length dimensional warping, there being no matter that is of sub-Planck length.

The Mathematics

The single most important equation in (first quantisized bosonic) string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are tachyons, with momentum ${\displaystyle k_{i}}$ which connect to a string world surface at the surface points ${\displaystyle z_{i}}$. It is given by the following functional integral which integrates (sums) over all possible embeddings of this 2D surface in 10 dimensions.

${\displaystyle A_{n}=\int {D\mu \int {Dexp\left(\partial _{z}X_{\mu }(z,{\overline {z}})\partial _{\overline {z}}X^{\mu }(z,{\overline {z}})d^{2}z+i\sum _{i=1}^{N}{k_{i\mu }X^{\mu }(z_{i},{\overline {z}}_{i})}\right)}}}$

The functional integral can be done because it is a Gaussian to become:

${\displaystyle A_{n}=\int {D\mu \prod _{0<i<j<N+1}{|z_{i}-z_{j}|^{2\alpha k_{i}.k_{j}}}}}$

This is integrated over a complex region. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. When this is taken into account it can be used to calculate the 4-point scattering amplitude:

${\displaystyle A_{4}={\frac {\Gamma (-(1+{\frac {1}{2}}(k_{1}+k_{2})^{2}))\Gamma (-(1+{\frac {1}{2}}(k_{2}+k_{3})^{2}))}{\Gamma (-(1+{\frac {1}{2}}((k_{1}+k_{2})^{2}+(k_{2}+k_{3})^{2})))}}}$

Which is a beta function. It was this beta function which was apparantly found before full string theory was invented.

See also

• AdS/CFT
• Grand unification theory
• List of string theory topics
• M-theory
• Quantum gravity
• String theory
• Large Hadron Collider

References

http://www.nuclecu.unam.mx/~alberto/physics/string.html http://www.superstringtheory.com/ http://www.superstringtheory.com/basics/basic4.html http://www.pbs.org/wgbh/nova/elegant/ http://www.pbs.org/wgbh/nova/elegant/scale.html http://www.pbs.org/wgbh/nova/elegant/resonance.html http://www.sukidog.com/jpierre/strings/ http://www.superstringtheory.com/blackh/blackh4.html

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• String theory
• Supersymmetry