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{{string-theory}}
{{see|string theory}}

'''Superstring theory''' is an attempt to explain all of the ] and ]s of nature in one theory by modelling them as vibrations of tiny ] strings. It is considered one of the most promising candidate theories of ]. Superstring theory is a shorthand for '''supersymmetric string theory''' because unlike ], it is the version of ] that incorporates ] and ].

== Background ==

The deepest problem in ] is harmonizing the theory of ], which describes gravitation and applies to large-scale structures (]s, ], ]s), with ], which describes the other three ] acting on the atomic scale.

The development of a ] of a force invariably results in infinite (and therefore useless) probabilities. Physicists have developed mathematical techniques (]) to eliminate these infinities which work for three of the four fundamental forces – ], ] and ] forces - but not for ]. The development of a ] 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 ] (about 10<sup>&minus;33</sup> m) which vibrate at ] 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 ] (10<sup>44</sup> ]s). The ] (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 "]es" 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 ]?''
Our ] is observed to have only three large ]s &mdash; and taken together with time as the fourth dimension &mdash; 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 ], ] requires ] to have 10, 11 or 26 dimensions. The conflict between observation and theory is resolved by making the unobserved dimensions ].

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 modelling (and potentially impossibly huge experimental apparatus) can be considered 'scientific.' Six-dimensional ] 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<sup>&minus;33</sup> m

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

{{unsolved|physics|Is ], superstring theory, or ], or some other variant on this theme, a step on the road to a "]," or just a blind alley?}}

== Number of superstring theories ==

Theoretical physicists were troubled by the existence of five separate string theories. This has been solved by the ] in the 1990s during which the five string theories were discovered to be different limits of a single underlying theory: ].
{| border="1" cellpadding="1" cellspacing="1" bordercolorlight="#666699" bordercolordark="#666699" bgcolor="#CCFFCC" mm_noconvert="TRUE"
|- bgcolor="#FFFFFF"
! colspan="3" class="dark" | String Theories
|-
! class="dark" | Type
! class="dark" | Spacetime dimensions<br>
! class="dark" | Details
|-
! bgcolor="#FFCCCC" class="dark" | Bosonic
| align="CENTER" class="dark" | 26
| bgcolor="#FFFFCC" class="dark" | Only ]s, no ]s means only forces, no matter, with both open and closed strings; major flaw: a ] with imaginary ], called the ]
|-
! bgcolor="#FFCCCC" class="dark" | I
| align="CENTER" class="dark" | 10
| bgcolor="#FFFFCC" class="dark" | ] between forces and matter, with both open and closed strings, no ], group symmetry is ]
|-
! bgcolor="#FFCCCC" class="dark" | IIA
| align="CENTER" class="dark" | 10
| bgcolor="#FFFFCC" class="dark" | ] between forces and matter, with closed strings only, no ], massless ]s spin both ways (nonchiral)
|-
! bgcolor="#FFCCCC" class="dark" | IIB
| align="CENTER" class="dark" | 10
| bgcolor="#FFFFCC" class="dark" | ] between forces and matter, with closed strings only, no ], massless ]s only spin one way (chiral)
|-
! bgcolor="#FFCCCC" class="dark" | HO
| align="CENTER" class="dark" | 10
| bgcolor="#FFFFCC" class="dark" | ] between forces and matter, with closed strings only, no ], ], meaning right moving and left moving strings differ, group symmetry is ]
|-
! bgcolor="#FFCCCC" class="dark" | HE
| align="CENTER" class="dark" | 10
| bgcolor="#FFFFCC" class="dark" | ] between forces and matter, with closed strings only, no ], ], meaning right moving and left moving strings differ, group symmetry is ]
|}

The five consistent superstring theories are:
* The ] has one supersymmetry in the ten-dimensional sense (16 supercharges). This theory is special in the sense that it is based on unoriented ] and ]s, while the rest are based on oriented closed strings.
* The ] 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-] (parity conserving) while the IIB theory is chiral (parity violating).
* The ] 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 ]s: the heterotic ] string and the heterotic ] string. (The name heterotic SO(32) is slightly inaccurate since among the SO(32) ]s, string theory singles out a quotient Spin(32)/Z<sub>2</sub> that is not equivalent to SO(32).)

Chiral ] can be inconsistent due to ]. This happens when certain one-loop ]s cause a quantum mechanical breakdown of the gauge symmetry. The anomalies were canceled out via the ].

==Integrating general relativity and quantum mechanics==
] typically deals with situations involving large mass objects in fairly large regions of ] whereas ] 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 ]s. 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 Five Superstring Interactions==
]
There are five ways open and closed strings can interact. An interaction in superstring theory is a ] event. Since superstring theory has to be a ] to obey ] the topology change must only occur at a single point. If O represents a closed string and C an open string. (We are using what the strings look like not the name!) Then the five interactions are, symbollically:

'''CCC + OOO + CCCC + CO + CCO'''

All open superstring theories also contain closed superstrings since closed superstrings can be seen from the fifth interaction, they are unavoidable. Although all these interactions are possible, in practice the most used superstring model is the closed heterotic E8xE8 superstring which only has closed strings and so only the second interaction (OOO) is needed.

==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 ], with momentum <math>k_i</math> which connect to a string world surface at the surface points <math>z_i</math>. It is given by the following ] which integrates (sums) over all possible embeddings of this 2D surface in 26 dimensions.

<math> A_N = \int{D\mu \int{D exp \left( -\frac{1}{4\pi\alpha} \int{ \partial_z X_{\mu}(z,\overline{z}) \partial_{\overline{z}} X^{\mu}(z,\overline{z})}dz^2 + i \sum_{i=1}^{N}{k_{i \mu} X^{\mu}(z_i,\overline{z}_i) } \right) }} </math>

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

<math> A_N = \int{D\mu \prod_{0<i<j<N+1}{ |z_i-z_j|^{2\alpha k_i.k_j} } }</math>

This is integrated over the various points <math>z_i</math>. 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. Also you need to make sure you are not integrating multiple times over different paramaterisations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):

<math> A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2) } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } </math>

Which is a ]. It was this beta function which was aparantly found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the grassman coordinates <math>\theta</math>. Since there are various ways this can be done this leads to different string theories.

When integrating over surfaces such as the torus, we end up with equations in terms of ] and elliptic functions such as the ]. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface.

===D-Branes===
D-Branes are membrane-like objects in 10D string theory. They can be thought of as occuring as a result of a ] compactification of 11D M-Theory which contains membranes. Because compactification of a geometric theory produces extra ] the D-branes can be included in the action by adding an extra U(1) vector field to the string action.

E=MC2

In '''type I''' open string theory, the ends of open strings are always attached to D-brane surfaces. A string theory with more gauge fields such as SU(2) gauge fields would then correspond to the compactification of some higher dimensional theory above 11 dimensions which is not thought to be possible to date.

===Why Five Superstring Theories?===
For a 10 dimensional supersymmetric theory we are allowed a 32-component Majorana spinor. This can be decomposed into a pair of 16-component Majorana-Weyl (chiral) spinors. There are then various ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities:
{| class="wikitable" style="text-align:center"; border="5"
! Superstring Model !! Invariant
|-
| Heterotic || <math>\partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}</math>
|-
| IIA || <math>\partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}-i\overline{\theta_{R}}\Gamma^\mu\partial_z\theta_{R}</math>
|-
| IIB || <math>\partial_zX^\mu-i\overline{\theta^1_{L}}\Gamma^\mu\partial_z\theta^1_{L}-i\overline{\theta^2_{L}}\Gamma^\mu\partial_z\theta^2_{L}</math>
|}
The heterotic superstrings come in two types SO(32) and E8xE8 as indicated above and the type I superstrings include open strings.

==Beyond Superstring Theory==
It is commonly believed that the 5 superstring theories are approximated to a theory in higher dimensions possibly involving membranes. Unfortunately because the action for this involves quartic terms and higher so is not ] the functional integrals are very difficult to solve and so this has confounded the top theoretical physicists. ] has popularised the concept of a theory in 11 dimensions ] involving membranes interpolating from the known symmetries of superstring theory. It may turn out that there exist membrane models or other non-membrane models in higher dimensions which may become acceptable when new unknown symmetries of nature are found, such as noncommutable geometry for example. It is thought, however, that 16 is probably the maximum since O(16) is a maximal subgroup of E8 the largest exceptional lie group and also is more than large enough to contain the ].
Quartic integrals of the non-functional kind are easier to solve so there is hope for the future. This is the series solution which is always convergent when a is non-zero and negative:

<math> \int_{-\infty}^{\infty}{\exp({a x^4+b x^3+c x^2+d x+f})dx}
= e^f\sum_{n,m,p=0}^{\infty}{ \frac{ b^{4n}}{(4n)!}\frac{c^{2m}}{(2m)!}\frac{d^{4p}}{(4p)!} \frac{ \Gamma(3n+m+p+\frac14) }{a^{3n+m+p+\frac14} } } </math>

In the case of membranes the series would correspond to sums of various membrane interactions that are not seen in string theory.


== See also ==

* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]

== 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<!----~~~~ Princess Janay added references. :-)-->

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Revision as of 11:59, 18 March 2008

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