AdS/CFT Correspondence
In theoretical physics, the AdS/CFT (anti-de Sitter space/conformal field theory) correspondence also called the Maldacena conjecture, Maldacena duality or gauge/gravity duality, is a conjectured relationship between two types of physical theories. On one side are the anti-de Sitter (AdS) spaces used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, which include theories similar to the Yang-Mills theories that describe elementary particles.
Duality represents a breakthrough in our understanding of string theory and quantum gravity. This is because it provides a non-perturbative formulation of string theory with certain boundary conditions, and because it is the most successful realization of the holographic principle, an idea in quantum gravity originally proposed by Gerard 't Hooft and pioneered by Leonard Susskind.
In physics, the AdS/CFT correspondence is the equivalence between a string theory or supergravity defined in a certain kind of anti-de Sitter space and a conformal field theory defined at its boundary with dimension smaller by one.
The anti-de Sitter (AdS) space corresponds to a solution to Einstein's equations with a negative cosmological constant, and is a classical theory of gravity; while conformal field theory (CFT: Conformal Field Theory) is a quantum theory. This correspondence between a classical theory of gravity and a quantum one may be the path to quantum gravity.
The AdS/CFT correspondence was originally proposed by Argentine physicist Juan Maldacena in late 1997, and some of its technical properties were soon clarified in an article by Edward Witten and another article by Gubser, Klebanov and Polyakov. By 2015, Maldacena's paper had more than 10,000 citations, making it the most cited paper in the field of particle physics.
Summary of correspondence
The geometry of anti-de Sitter space
In the AdS/CFT correspondence, either string theory or M-theory is considered against an anti-de Sitter background. This means that the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called the anti-de Sitter.
In very elementary terms, anti-de Sitter space is a mathematical model of space-time in which the notion of distance between points (the metric) is different from the notion of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be seen as a disk as illustrated to the right. This image shows a tessellation of a disk by triangles and squares. One can define the distance between points on this disk in such a way that all triangles and squares are the same size and the circular outer boundary is infinitely far from any point on the interior.
Now imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given moment. The resulting geometric object is anti-de Sitter three-dimensional space. It looks like a solid cylinder in which any cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this image. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, the anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.
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This construction describes a hypothetical universe with only two spatial dimensions and one time dimension, but it can be generalized to any number of dimensions. In fact, hyperbolic space can have more than two dimensions and one can "stack" copies of hyperbolic space to obtain higher-dimensional models of anti-de Sitter space.
The idea of AdS/CFT
An important feature of anti-de Sitter space is its boundary (which resembles a cylinder in the case of three-dimensional anti-de Sitter space). One property of this limit is that, locally around any point, it resembles Minkowski space, the model of space-time used in lay physics.
Therefore, it can be considered an auxiliary theory in which "space-time" is given by the limit of the anti-de Sitter space. This observation is the starting point for the AdS/CFT correspondence, which establishes that the limit of anti-de Sitter space can be considered as "space-time" for a conformal field theory. The claim is that this conformal field theory is equivalent to the gravitational theory in bulk anti-de Sitter space in the sense that there is a "dictionary" to translate calculations in one theory into calculations in the other. Every entity in one theory has a counterpart in the other theory. For example, a single particle in the gravitational theory could correspond to some collection of particles in the limit theory. Furthermore, 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 limit theory would also have a 40 percent chance of colliding..
Note that the boundary of the anti-de Sitter space has fewer dimensions than the anti-de Sitter space itself. For example, in the three-dimensional example illustrated above, the boundary is a two-dimensional surface. The AdS/CFT correspondence is often described as a "holographic duality", because this relationship between the two theories is similar to the relationship between a three-dimensional object and its image as a hologram. Although a hologram is two-dimensional, encodes information about the three dimensions of the object it represents. In the same way, theories that are related by the AdS/CFT correspondence are conjectured to be exactly equivalent, despite living in different numbers of dimensions. Conformal field theory is like a hologram that captures information about the higher-dimensional theory of quantum gravity.
Examples of correspondence
Following Maldacena's insight in 1997, theorists have discovered many different realizations of the AdS/CFT mapping. These relate various conformal theories of the field to compactifications of string theory and M-theory in various numbers of dimensions. The theories involved are generally not viable models of the real world, but have certain characteristics, such as their particle content or high degree of symmetry, that make them useful for solving problems in quantum field theory and quantum gravity.
The most famous example of AdS/CFT correspondence indicates that IIB type string theory in the product space AdS5× × S5{displaystyle AdS_{5}times S^{5} is equivalent to the supersimetric theory N = 4 of Yang-Mills in the four-dimensional limit. In this example, the space-time in which gravitational theory lives is effectively five-dimensional (then notation AdS5{displaystyle AdS_{5}}), and there are five additional compact dimensions (coded by style S5{displaystyle S^{5}} factor). In the real world, space-time is four dimensions, at least macroscopically, so this version of correspondence does not provide a realistic model of gravity. Similarly, dual theory is not a viable model of any real world system, as it assumes a large amount of supersymmetry. However, as explained below, this theory of limits shares some characteristics in common with quantum chromodynamics, the fundamental theory of strong force. It describes particles similar to the gluons of quantum chromodynamics along with certain femiions. As a result, it has found applications in nuclear physics, particularly in the study of quark-gluon plasma.
Another realization of correspondence indicates that M theory in AdS7× × S4{displaystyle AdS_{7}times S^{4} is equivalent to the call (2.0) Theory in six dimensions. In this example, the space-time of gravitational theory is indeed seven dimensions. The existence of the theory (2.0) that appears on one side of duality is predicted by the classification of superconform field theories. It is still little understood because it is a quantum mechanical theory without a classic limit. Despite the difficulty inherent in the study of this theory, it is considered to be an interesting object for a variety of reasons, both physical and mathematical.
Another realization of correspondence states that the M theory in AdS4× × S7{displaystyle AdS_{4}times S^{7}} is equivalent to the theory of the ABJM superconformal field in three dimensions. Here gravitational theory has four non-compact dimensions, so this correspondence version provides a somewhat more realistic description of gravity.
History and development
String Theory and Nuclear Physics
The discovery of the AdS/CFT correspondence in late 1997 was the culmination of a long history of efforts to relate string theory to nuclear physics. In fact, string theory was originally developed in late the 1960s and early 1970s as a theory of hadrons, subatomic particles like the proton and neutron held together by the strong nuclear force. The idea was that each of these particles could be seen as a different oscillation mode of a string. By the late 1960s, experimentalists had found that hadrons fell into families called Regge trajectories with energy squared proportional to angular momentum, and theorists showed that this relationship arises naturally from the physics of a rotating relativistic string.
On the other hand, attempts to model hadrons as strings ran into serious problems. One problem is that string theory includes a massless spin-2 particle, while no particle appears in hadron physics. Such a particle would mediate a force with the properties of gravity. In 1974, Joel Scherk and John Schwarz suggested that string theory was therefore not a theory of nuclear physics, as many theorists had thought, but a theory of quantum gravity. At the same time, they noted that hadrons are actually made of quarks, and the sequence theory approach was abandoned in favor of quantum chromodynamics.
In quantum chromodynamics, quarks have a kind of load that comes in three varieties called colors. In a work of 1974, Gerard 't Hooft studied the relationship between string theory and nuclear physics from another point of view when considering theories similar to quantum chromodynamics, where the number of colors is an arbitrary number N{displaystyle N}instead of Three. In this article, 't Hooft considered a certain limit where N{displaystyle N} tends to infinity and argued that in this limit certain calculations in quantum field theory resemble calculations in string theory.
Black holes and holography
In 1975, Stephen Hawking published a calculation that suggested that black holes are not completely black, as they emit faint radiation due to quantum effects near the event horizon. This work expanded upon previous results by Jacob Bekenstein that he had suggested that black holes have a well-defined entropy. At first, Hawking's result seemed to contradict one of the main postulates of quantum mechanics, namely the unitary evolution of time. Intuitively, the Unitarity Postulate says that quantum mechanical systems do not destroy information as they evolve from one state to another. For this reason, the apparent contradiction came to be known as the black hole information paradox.
Later in 1993, Gerard 't Hooft wrote a speculative paper on quantum gravity in which he reviewed Hawking's work on the thermodynamics of black holes, concluding that the total number of degrees of freedom in a The region of spacetime surrounding a black hole is proportional to the surface area of the horizon. This idea was pioneered by Leonard Susskind and is now known as the holographic principle. The holographic principle and its realization in string theory via the AdS/CFT correspondence have helped to elucidate the black hole mysteries suggested by Hawking's work and are believed to provide a resolution of the hole information paradox. black. In 2004, Hawking admitted that black holes do not violate quantum mechanics, and suggested a concrete mechanism by which they could preserve information.
Maldacena's work
In late 1997, Juan Maldacena published a landmark paper that initiated the study of AdS/CFT. According to Alexander Markovich Polyakov, "[Maldacena's] work opened the floodgates". it immediately aroused great interest in the string theory community and was considered in papers by Steven Gubser, Igor Klebanov and Alexander Polyakov, and by Edward Witten. These papers made Maldacena's conjecture more accurate and showed that field theory The conformal that appears in the correspondence lives on the edge of the anti-de Sitter space.
A special case of Maldacena's proposal says that the N=4 super-Yang-Mills theory, a gauge theory similar in some respects to quantum chromodynamics, is equivalent to string theory in anti-de space. Sitter in five dimensions. This result helped to clarify 't Hooft's earlier work on the relationship between string theory and quantum chromodynamics, taking string theory back to its roots as a theory of nuclear physics. Maldacena's results also provided a concrete realization of the holographic principle with important implications for quantum gravity and black hole physics. By 2015, Maldacena's paper had become the most cited paper in high-energy physics with over 10,000 citations. These subsequent papers have provided considerable evidence that the correspondence is correct, although it has not been proven so far. rigorously.
AdS/CFT finds applications
Main articles: AdS/QCD and AdS/CMT
In 1999, after taking a job at Columbia University, nuclear physicist Đàm Thanh Sơn paid a visit to Andrei Starinets, a friend from Sơn's student days who happened to be doing a PhD. in string theory at New York University. Although the two men had no intention of collaborating, Sơn soon realized that the AdS/CFT calculations Starinets was doing could shed light on some aspects of quark-plasma. gluon, an exotic state of matter produced when heavy ions collided at high energies. In collaboration with Starinets and Pavel Kovtun, Sơn was able to use the AdS/CFT correspondence to calculate a key plasma parameter. As Sơn later recalled: "We did the calculation in his head to give us a prediction of the value of the shear viscosity of a plasma... A friend of mine in nuclear physics joked that ours was the first useful article to come out of String Theory".
Physicists today continue to search for applications of the AdS/CFT correspondence in quantum field theory. In addition to the applications to nuclear physics advocated by Đàm Thanh Sơn and his collaborators, condensed matter physicists such as Top Sachdev have used methods from string theory to understand some aspects of condensed matter physics. A notable result in this direction was the description, through the AdS/CFT correspondence, of the transition from a superfluid to an insulator. Another emerging subject is the fluid/gravity correspondence, which uses the AdS/CFT correspondence to translate problems in fluid dynamics into problems in general relativity.
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