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Further information: string theory
String theory
Superstring theory
Theory
String theory
Superstrings
Bosonic string theory
M-theory (simplified)

Type I string · Type II string
Heterotic string
String field theory
Holographic principle
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Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings See also String theory Superstring theory is an attempt to explain all of the particles and Fundamental forces of nature in one theory by modelling Bosonic string theory is the original version of String theory, developed in the late 1960s In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified In non-technical terms M-theory presents an idea about the basic substance of the universe In Theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions In Theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. In Physics, a heterotic string is a peculiar mixture (or hybrid of the bosonic string and the Superstring (the adjective heterotic comes from In Theoretical physics, string field theory is a proposal to define String theory in such a way that the Background independence is respected The holographic principle is a physical property of Quantum gravity theories proposed by Gerard 't Hooft and Leonard Susskind, which resolves the In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that It is considered one of the most promising candidate theories of quantum gravity. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature 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. Bosonic string theory is the original version of String theory, developed in the late 1960s String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that

Contents

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. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter Superclusters are large groupings of smaller galaxy groups and clusters, and are among the largest structures of the Cosmos. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms

The development of a quantum field theory of a force invariably results in infinite (and therefore useless) probabilities. In quantum field theory (QFT the forces between particles are mediated by other particles 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. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection In Physics, the electromagnetic force is the force that the Electromagnetic field exerts on electrically charged particles In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Gravitation is a natural Phenomenon by which objects with Mass attract one another The development of a quantum theory of gravity must therefore come about by different means than those used for the other forces. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature

Basic idea

The basic idea is that the fundamental constituents of reality are strings of the Planck length (about 10−33 m) which vibrate at resonant frequencies. Planck units are Units of measurement named after the German physicist Max Planck, who first proposed them in 1899 In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's 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 (1044 newtons). Planck force is the derived unit of Force resulting from the definition of the base Planck units for time length and mass The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical 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. In Physics, the graviton is a hypothetical Elementary particle, a Boson to be exact that mediates the force of Gravity in the framework 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 Physical cosmology, the Big Crunch is one possible scenario for the Ultimate fate of the universe, in which the Metric expansion of space eventually 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. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS The conflict between observation and theory is resolved by making the unobserved dimensions compactified. In Physics, compactification means changing a theory with respect to one of its Space-time 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 Calabi-Yau shapes can account for the additional dimensions required by superstring theory. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial 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 10−33 m.

Superstring theory is not the first theory to propose extra spatial dimensions, the Kaluza-Klein theory did already. In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and 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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Unsolved problems in physics: Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything," or just a blind alley?

Number of superstring theories

Theoretical physicists were troubled by the existence of five separate string theories. This is a list of some of the major unsolved problems in Physics. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena 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 second superstring revolution was the intense wave of breakthroughs in String theory that took place approximately between 1994 and 1997. In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified

String Theories
Type Spacetime dimensions
 Details
Bosonic 26 Only bosons, no fermions means only forces, no matter, with both open and closed strings; major flaw: a particle with imaginary mass, called the tachyon
I 10 Supersymmetry between forces and matter, with both open and closed strings, no tachyon, group symmetry is SO(32)
IIA 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions spin both ways (nonchiral)
IIB 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions only spin one way (chiral)
HO 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is SO(32)
HE 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is E8×E8

The five consistent superstring theories are:

Chiral gauge theories can be inconsistent due to anomalies. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Quantum physics an anomaly or quantum anomaly is the failure of a Symmetry of a theory's classical action to be a symmetry of any regularization This happens when certain one-loop Feynman diagrams cause a quantum mechanical breakdown of the gauge symmetry. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The anomalies were canceled out via the Green-Schwarz mechanism. The Green-Schwarz mechanism (sometimes called the Green-Schwarz anomaly cancellation mechanism) is the main discovery that started the First superstring revolution

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). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The two are very rarely used together, and the most common case in which they are combined is in the study of black holes. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e 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

The five superstring interactions
The five superstring interactions

There are five ways open and closed strings can interact. An interaction in superstring theory is a topology changing event. Since superstring theory has to be a local theory to obey causality the topology change must only occur at a single point. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence If O represents a closed string and C an open string, 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 tachyons, with momentum ki which connect to a string world surface at the surface points zi. A tachyon (from the Greek, takhyónion, from, takhýs, ie swift fast is any hypothetical particle that travels at Faster-than-light It is given by the following functional integral which integrates (sums) over all possible embeddings of this 2D surface in 26 dimensions. You may also be looking for Functional integration (neurobiology or Functional integration (sociology.

A_N = \int{D\mu \int{D[X] 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) }}

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

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 the various points zi. 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):

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)) }

Which is a beta function. In Mathematics, the beta function, also called the Euler integral of the first kind is a Special function defined by It was this beta function which was apparently 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 θ. 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 theta functions and elliptic functions such as the Dedekind eta function. In Mathematics, theta functions are Special functions of Several complex variables. The Dedekind eta function, named after Richard Dedekind, is a function defined on the Upper half-plane of Complex numbers whose imaginary part is positive 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 occurring as a result of a Kaluza-Klein compactification of 11D M-Theory which contains membranes. In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and Because compactification of a geometric theory produces extra vector fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

\partial_z \rightarrow \partial_z +iA_z(z,\overline{z})

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:

Superstring Model Invariant
Heterotic \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}
IIA \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}-i\overline{\theta_{R}}\Gamma^\mu\partial_z\theta_{R}
IIB \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}

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 Gaussian the functional integrals are very difficult to solve and so this has confounded the top theoretical physicists. Carl Friedrich Gauss (1777 &ndash 1855 is the Eponym of all of the topics listed below Edward Witten has popularised the concept of a theory in 11 dimensions M-Theory involving membranes interpolating from the known symmetries of superstring theory. Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified 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 noncommutative 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 Standard Model. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles 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:

\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} } }

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

Compactification

Investigating theories of higher dimensions often involves looking at the 10 dimensional superstring theory and interpreting some of the more obscure results in terms of compactified dimensions. For example D-branes are seen as compactified membranes from 11D M-Theory. In String theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions after which they are named Theories of higher dimensions such as 12D F-theory and beyond will produce other effects such as gauge terms higher than U(1). The components of the extra vector fields (A) in the D-brane actions can be thought of as extra coordinates (X) in disguise. However, the known symmetries including supersymmetry currently restrict the spinors to have 32-components which limits the number of dimensions to 11 (or 12 if you include two time dimensions. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the ) Some commentators (e. g. John Baez et al) have speculated that the exceptional lie groups E6, E7 and E8 having maximum orthogonal subgroups O(10), O(12) and O(16) may be related to theories in 10, 12 and 16 dimensions; 10 dimensions corresponding to string theory and the 12 and 16 dimensional theories being yet undiscovered but would be theories bases on 3-branes and 7-branes respectively. John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings However this is a minority view within the string community. Since E7 is some sense F4 quaternified and E8 is F4 octonified, then the 12 and 16 dimensional theories, if they did exist, may involve the noncommutative geometry based on the quaternions and octonions respectively. Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real From the above discussion it can be seen that physicists have many ideas for to extend superstring theory beyond the current 10 dimensional theory but so far none have been successful.

Kac-Moody algebras

Since strings can have an infinite number of modes, the symmetry used to describe string theory is based on infinite dimensional Lie algebras. Some Kac-Moody algebras that have been considered as symmetries for M-Theory have been E10 and E11 and their supersymmetric extensions. In Mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional that can be defined by generators and relations through a Generalized Cartan In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified

See also

References


For the relation of the AdS/CFT correspondence to the general context of string theory see String theory. Grand Unification, grand unified theory, or GUT refers to any of several very similar unified field theories or models in Physics that This is a list of String theory topics. See also List of mathematical topics in quantum theory. In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In Theoretical physics, string field theory is a proposal to define String theory in such a way that the Background independence is respected
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