Editing String theory (section)

== AdS/CFT correspondence ==
{{main|AdS/CFT correspondence}}

One approach to formulating string theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. This is a theoretical result that implies that string theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.<ref name="Klebanov and Maldacena 2009"/> The AdS/CFT correspondence was first proposed by [[Juan Maldacena]] in late 1997.<ref name=Maldacena1998/> Important aspects of the correspondence were elaborated in articles by [[Steven Gubser]], [[Igor Klebanov]], and [[Alexander Markovich Polyakov]],<ref name=Gubser/> and by Edward Witten.<ref name=Witten1998/> By 2010, Maldacena's article had over 7000 citations, becoming the most highly cited article in the field of [[high energy physics]].{{efn|{{cite web |url=http://www.slac.stanford.edu/spires/topcites/2010/eprints/to_hep-th_annual.shtml |title=Top Cited Articles during 2010 in hep-th |author=<!--Staff writer(s); no by-line.--> |access-date=25 July 2013}}}}

=== Overview of the correspondence ===

[[Image:Uniform tiling 433-t0 (formatted).svg|thumb|left|alt=A disk tiled by triangles and quadrilaterals which become smaller and smaller near the boundary circle.|A [[tessellation]] of the [[hyperbolic plane]] [[tritetragonal tiling|by triangles and squares]]]]

In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain [[vacuum solution]] of [[Einstein's equation]] called [[anti-de Sitter space]].<ref name="Klebanov and Maldacena 2009"/> In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the [[metric tensor|metric]]) is different from the notion of distance in ordinary [[Euclidean geometry]]. It is closely related to [[hyperbolic space]], which can be viewed as a [[Poincaré disk model|disk]] as illustrated on the left.<ref name="Maldacena 2005, p. 60">[[#Maldacena2005|Maldacena 2005]], p. 60</ref> This image shows a [[tessellation]] of a disk by triangles and squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.<ref name="Maldacena 2005, p. 61">[[#Maldacena2005|Maldacena 2005]], p. 61</ref>

One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensional anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/> It looks like a solid [[cylinder (geometry)|cylinder]] in which any [[cross section (geometry)|cross section]] is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is [[curvature|curved]] in such a way that any point in the interior is actually infinitely far from this boundary surface.<ref name="Maldacena 2005, p. 61"/>

[[File:AdS3.svg|thumb|right|alt=A cylinder formed by stacking copies of the disk illustrated in the previous figure.|upright=1.6|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincaré disk model|hyperbolic disks]], each one representing the state of the universe at a given time. The resulting [[spacetime]] looks like a solid [[cylinder (geometry)|cylinder]].]]

This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/>

An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like [[Minkowski space]], the model of spacetime used in non-gravitational physics.<ref>[[#Zwiebach|Zwiebach]], p. 552</ref> One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. The claim is that this quantum field theory is equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.<ref>[[#Maldacena2005|Maldacena 2005]], pp. 61–62</ref>

=== Applications to quantum gravity ===

The discovery of the AdS/CFT correspondence was a major advance in physicists' understanding of string theory and quantum gravity. One reason for this is that the correspondence provides a formulation of string theory in terms of quantum field theory, which is well understood by comparison. Another reason is that it provides a general framework in which physicists can study and attempt to resolve the paradoxes of black holes.<ref name="de Haro et al. 2013, p.2"/>

In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black but emit a dim radiation due to quantum effects near the [[event horizon]].<ref name=Hawking1975/> At first, Hawking's result posed a problem for theorists because it suggested that black holes destroy information. More precisely, Hawking's calculation seemed to conflict with one of the basic [[postulates of quantum mechanics]], which states that physical systems evolve in time according to the [[Schrödinger equation]]. This property is usually referred to as [[Unitarity (physics)|unitarity]] of time evolution. The apparent contradiction between Hawking's calculation and the unitarity postulate of quantum mechanics came to be known as the [[black hole information paradox]].<ref name=Susskind2008/>

The AdS/CFT correspondence resolves the black hole information paradox, at least to some extent, because it shows how a black hole can evolve in a manner consistent with quantum mechanics in some contexts. Indeed, one can consider black holes in the context of the AdS/CFT correspondence, and any such black hole corresponds to a configuration of particles on the boundary of anti-de Sitter space.<ref>[[#Zwiebach|Zwiebach]], p. 554</ref> These particles obey the usual rules of quantum mechanics and in particular evolve in a unitary fashion, so the black hole must also evolve in a unitary fashion, respecting the principles of quantum mechanics.<ref>[[#Maldacena2005|Maldacena 2005]], p. 63</ref> In 2005, Hawking announced that the paradox had been settled in favor of information conservation by the AdS/CFT correspondence, and he suggested a concrete mechanism by which black holes might preserve information.<ref name=Hawking2005/>

=== Applications to nuclear physics ===
{{main|AdS/QCD correspondence}}

[[File:Meissner effect p1390048.jpg|thumb|left|alt=A magnet levitating over a superconducting material.|A [[magnet]] [[Meissner effect|levitating]] above a [[high-temperature superconductor]]. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.<ref name="Merali 2011"/>]]

In addition to its applications to theoretical problems in quantum gravity, the AdS/CFT correspondence has been applied to a variety of problems in quantum field theory. One physical system that has been studied using the AdS/CFT correspondence is the [[quark–gluon plasma]], an exotic [[state of matter]] produced in [[particle accelerator]]s. This state of matter arises for brief instants when heavy [[ions]] such as [[gold]] or [[lead]] nuclei are collided at high energies. Such collisions cause the [[quarks]] that make up atomic nuclei to [[deconfinement|deconfine]] at temperatures of approximately two [[1,000,000,000,000|trillion]] [[kelvin]], conditions similar to those present at around {{math|10<sup>−11</sup>}} seconds after the [[Big Bang]].<ref>[[#Zwiebach|Zwiebach]], p. 559</ref>

The physics of the quark–gluon plasma is governed by a theory called [[quantum chromodynamics]], but this theory is mathematically intractable in problems involving the quark–gluon plasma.{{efn|More precisely, one cannot apply the methods of perturbative quantum field theory.}} In an article appearing in 2005, [[Đàm Thanh Sơn]] and his collaborators showed that the AdS/CFT correspondence could be used to understand some aspects of the quark-gluon plasma by describing it in the language of string theory.<ref name="Kovtun, Son, and Starinets 2001"/> By applying the AdS/CFT correspondence, Sơn and his collaborators were able to describe the quark-gluon plasma in terms of black holes in five-dimensional spacetime. The calculation showed that the ratio of two quantities associated with the quark-gluon plasma, the [[shear viscosity]] and volume density of entropy, should be approximately equal to a certain universal [[constant (mathematics)|constant]]. In 2008, the predicted value of this ratio for the quark-gluon plasma was confirmed at the [[Relativistic Heavy Ion Collider]] at [[Brookhaven National Laboratory]].<ref name="Merali 2011"/><ref name=Luzum/>

=== Applications to condensed matter physics ===
{{main|AdS/CMT correspondence}}

The AdS/CFT correspondence has also been used to study aspects of condensed matter physics. Over the decades, [[experimental physics|experimental]] condensed matter physicists have discovered a number of exotic states of matter, including [[superconductors]] and [[superfluids]]. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including [[Subir Sachdev]] hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.<ref name="Merali 2011"/>

So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an [[insulator (electricity)|insulator]]. A superfluid is a system of [[electrically neutral]] [[atoms]] that flows without any [[friction]]. Such systems are often produced in the laboratory using [[liquid helium]], but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing [[lasers]]. These atoms initially behave as a superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the [[temperature]] and on the [[Planck constant]], the fundamental parameter of quantum mechanics, which does not enter into the description of the other [[phase (matter)|phases]]. This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole.<ref name=Sachdev/>