Editing String theory (section)

== Fundamentals ==
[[Image:Open and closed strings.svg|right|thumb|alt=A wavy open segment and closed loop of string.|The fundamental objects of string theory are open and closed [[string (physics)|strings]].]]

=== Overview ===
In the 20th century, two theoretical frameworks emerged for formulating the laws of physics. The first is [[Albert Einstein]]'s [[general theory of relativity]], a theory that explains the force of [[gravity]] and the structure of [[spacetime]] at the macro-level. The other is [[quantum mechanics]], a completely different formulation, which uses known [[probability]] principles to describe physical phenomena at the micro-level. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the [[universe]], from [[elementary particle]]s to [[atom]]s to the evolution of stars and the universe as a whole.<ref name="Becker, Becker 2007, p. 1">[[#Becker|Becker, Becker and Schwarz]], p. 1</ref>

In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of [[quantum gravity]].<ref name="Becker, Becker 2007, p. 1"/> The general theory of relativity is formulated within the framework of [[classical physics]], whereas the other [[fundamental interaction|fundamental forces]] are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.<ref>[[#Zwiebach|Zwiebach]], p. 6</ref>

String theory is a [[mathematical theory|theoretical framework]] that attempts to address these questions.  

The starting point for string theory is the idea that the [[point particle|point-like particles]] of [[particle physics]] can also be modeled as one-dimensional objects called [[string (physics)|strings]]. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can [[vibration|vibrate]] in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle consistent with non-string models of elementary particles, with its [[mass]], [[charge (physics)|charge]], and other properties determined by the vibrational state of the string. String theory's application as a form of quantum gravity proposes a vibrational state responsible for the [[graviton]], a yet unproven quantum particle that is theorized to carry gravitational force.<ref name="Becker, Becker 2007, pp. 2">[[#Becker|Becker, Becker and Schwarz]], pp. 2–3</ref>

One of the main developments of the past several decades in string theory was the discovery of certain 'dualities', mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as [[M-theory]].<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 9–12</ref>

Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated in the discovery of the [[anti-de Sitter/conformal field theory correspondence]] or AdS/CFT.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 14–15</ref> This is a theoretical result that relates string theory to other physical theories which are better understood theoretically. The AdS/CFT correspondence has implications for the study of black holes and quantum gravity, and it has been applied to other subjects, including [[nuclear physics|nuclear]]<ref name="Klebanov and Maldacena 2009"/> and [[condensed matter physics]].<ref name="Merali 2011"/><ref name=Sachdev/>

Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it will eventually be developed to the point where it fully describes our universe, making it a [[theory of everything]]. One of the goals of current research in string theory is to find a solution of the theory that reproduces the observed spectrum of elementary particles, with a small [[cosmological constant]], containing [[dark matter]] and a plausible mechanism for [[cosmic inflation]]. While there has been progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of details.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 3, 15–16</ref>

One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. The scattering of strings is most straightforwardly defined using the techniques of [[perturbation theory (quantum mechanics)|perturbation theory]], but it is not known in general how to define string theory [[Non-perturbative|nonperturbatively]].<ref>[[#Becker|Becker, Becker and Schwarz]], p. 8</ref> It is also not clear whether there is any principle by which string theory selects its [[vacuum state]], the physical state that determines the properties of our universe.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 13–14</ref> These problems have led some in the community to criticize these approaches to the unification of physics and question the value of continued research on these problems.<ref name="Woit 2006">[[#Woit|Woit]]</ref>

=== Strings ===
{{main|String (physics)}}

[[Image:World lines and world sheet.svg|left|thumb|upright=1.2|Interaction in the quantum world: [[worldline]]s of point-like [[particles]] or a [[worldsheet]] swept up by closed [[string (physics)|strings]] in string theory]]

The application of quantum mechanics to physical objects such as the [[electromagnetic field]], which are extended in space and time, is known as [[quantum field theory]]. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields.<ref name="Zee 2010"/>

In quantum field theory, one typically computes the probabilities of various physical events using the techniques of [[Perturbation theory (quantum mechanics)|perturbation theory]]. Developed by [[Richard Feynman]] and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called [[Feynman diagram]]s to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions.<ref name="Zee 2010"/>

The starting point for string theory is the idea that the point-like particles of quantum field theory can also be modeled as one-dimensional objects called strings.<ref>[[#Becker|Becker, Becker and Schwarz]], p. 2</ref> The interaction of strings is most straightforwardly defined by generalizing the perturbation theory used in ordinary quantum field theory. At the level of Feynman diagrams, this means replacing the one-dimensional diagram representing the path of a point particle by a two-dimensional (2D) surface representing the motion of a string.<ref name="Becker, Becker 2007, p. 6">[[#Becker|Becker, Becker and Schwarz]], p. 6</ref> Unlike in quantum field theory, string theory does not have a full non-perturbative definition, so many of the theoretical questions that physicists would like to answer remain out of reach.<ref>[[#Zwiebach|Zwiebach]], p. 12</ref>

In theories of particle physics based on string theory, the characteristic length scale of strings is assumed to be on the order of the [[Planck length]], or {{math|10<sup>−35</sup>}} meters, the scale at which the effects of quantum gravity are believed to become significant.<ref name="Becker, Becker 2007, p. 6"/> On much larger length scales, such as the scales visible in physics laboratories, such objects would be indistinguishable from zero-dimensional point particles, and the vibrational state of the string would determine the type of particle. One of the vibrational states of a string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force.<ref name="Becker, Becker 2007, pp. 2"/>

The original version of string theory was [[bosonic string theory]], but this version described only [[bosons]], a class of particles that transmit forces between the matter particles, or [[fermions]]. Bosonic string theory was eventually superseded by theories called [[superstring theory|superstring theories]]. These theories describe both bosons and fermions, and they incorporate a theoretical idea called [[supersymmetry]]. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa.<ref>[[#Becker|Becker, Becker and Schwarz]], p. 4</ref>

There are several versions of superstring theory: [[type I string|type I]], [[type IIA string|type IIA]], [[type IIB string|type IIB]], and two flavors of [[heterotic string]] theory ({{math|[[special orthogonal group|''SO''(32)]]}} and {{math|[[E8 (mathematics)|''E''<sub>8</sub>×''E''<sub>8</sub>]]}}). The different theories allow different types of strings, and the particles that arise at low energies exhibit different [[symmetry (physics)|symmetries]]. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA, IIB and heterotic include only closed strings.<ref>[[#Zwiebach|Zwiebach]], p. 324</ref>

=== Extra dimensions <span class="anchor" id="Number of dimensions"></span> ===

[[File:Compactification example.svg|right|thumb|alt=A tubular surface and corresponding one-dimensional curve.|An example of [[compactification (physics)|compactification]]: At large distances, a two dimensional surface with one circular dimension looks one-dimensional.]]

In everyday life, there are three familiar dimensions (3D) of space: height, width and length. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional (4D) [[spacetime]]. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.<ref>[[#Wald|Wald]], p. 4</ref>

In spite of the fact that the Universe is well described by 4D spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily.{{efn|For example, in the context of the [[AdS/CFT correspondence]], theorists often formulate and study theories of gravity in unphysical numbers of spacetime dimensions.}} There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics.<ref name="Zee 2010"/> Finally, there exist scenarios in which there could actually be more than 4D of spacetime which have nonetheless managed to escape detection.<ref>[[#Zwiebach|Zwiebach]], p. 9</ref>

String theories require [[extra dimensions]] of spacetime for their mathematical consistency. In bosonic string theory, spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in [[M-theory]] it is 11-dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.<ref>[[#Zwiebach|Zwiebach]], p. 8</ref>

[[Image:Calabi yau.jpg|left|thumb|alt=Visualization of a complex mathematical surface with many convolutions and self intersections.|A cross section of a quintic [[Calabi–Yau manifold]] ]]

[[Compactification (physics)|Compactification]] is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.<ref name="Yau and Nadis 2010, Ch. 6">[[#Yau|Yau and Nadis]], Ch. 6</ref> In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.

Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a [[Calabi–Yau manifold]].<ref name="Yau and Nadis 2010, Ch. 6"/> A Calabi–Yau manifold is a special [[topological space|space]] which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians [[Eugenio Calabi]] and [[Shing-Tung Yau]].<ref>[[#Yau|Yau and Nadis]], p. ix</ref>

Another approach to reducing the number of dimensions is the so-called [[brane cosmology|brane-world]] scenario. In this approach, physicists assume that the observable universe is a four-dimensional subspace of a higher dimensional space. In such models, the force-carrying bosons of particle physics arise from open strings with endpoints attached to the four-dimensional subspace, while gravity arises from closed strings propagating through the larger ambient space. This idea plays an important role in attempts to develop models of real-world physics based on string theory, and it provides a natural explanation for the weakness of gravity compared to the other fundamental forces.<ref name=Randall/>

=== Dualities ===
[[File:Dualities in String Theory.svg|right|thumb|alt=A diagram indicating the relationships between M-theory and the five superstring theories.|upright=2|A diagram of string theory dualities. Blue edges indicate [[S-duality]]. Red edges indicate [[T-duality]].]]

{{main|S-duality|T-duality}}
A notable fact about string theory is that the different versions of the theory all turn out to be related in highly nontrivial ways. One of the relationships that can exist between different string theories is called [[S-duality]]. This is a relationship that says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. Type I string theory turns out to be equivalent by S-duality to the {{math|''SO''(32)}} heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.<ref name=Becker>[[#Becker|Becker, Becker and Schwarz]]</ref>

Another relationship between different string theories is [[T-duality]]. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius {{math|''R''}} is equivalent to a string propagating around a circle of radius {{math|1/''R''}} in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has [[momentum]] as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the [[winding number]]. If a string has momentum {{math|''p''}} and winding number {{math|''n''}} in one description, it will have momentum {{math|''n''}} and winding number {{math|''p''}} in the dual description. For example, type IIA string theory is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.<ref name=Becker/>

In general, the term ''duality'' refers to a situation where two seemingly different [[physical system]]s turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, [[Montonen–Olive duality]] is an example of an S-duality relationship between quantum field theories. The AdS/CFT correspondence is an example of a duality that relates string theory to a quantum field theory. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be ''dual'' to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.<ref>[[#Zwiebach|Zwiebach]], p. 376</ref>

=== Branes ===
{{main|Brane}}
[[File:D3-brane et D2-brane.PNG|thumb|right|alt=A pair of surfaces joined by wavy line segments.|Open strings attached to a pair of [[D-brane]]s]]

In string theory and other related theories, a [[brane]] is a physical object that generalizes the notion of a point particle to higher dimensions. For instance, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension ''p'', these are called ''p''-branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.<ref name="Moore 2005, p. 214"/>

Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A ''p''-brane sweeps out a (''p''+1)-dimensional volume in spacetime called its ''worldvolume''. Physicists often study [[field (physics)|fields]] analogous to the electromagnetic field which live on the worldvolume of a brane.<ref name="Moore 2005, p. 214"/>

In string theory, [[D-brane]]s are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the [[Dirichlet boundary condition]]. The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory.<ref name="Moore 2005, p. 214"/>

Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain [[category (mathematics)|categories]], such as the [[derived category]] of [[coherent sheaf|coherent sheaves]] on a [[complex algebraic variety]], or the [[Fukaya category]] of a [[symplectic manifold]].<ref name="Aspinwall et al. 2009"/> The connection between the physical notion of a brane and the mathematical notion of a category has led to important mathematical insights in the fields of [[algebraic geometry|algebraic]] and [[symplectic geometry]]<ref name="Kontsevich 1995"/> and [[representation theory]].<ref name=Kapustin/>