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Juan Maldacena
3-dimensional anti-de Sitter space
Juan Maldacena’s 1997 paper: The AdS/CFT Correspondence
Our story of holography and quantum gravity brings us to an Argentinian theoretical physicist, who will propose the most reliable and concrete version of the holographic principle in 1997. This conjecture will be known as the AdS/CFT correspondence. It is a proposed duality between two kinds of theories in physics:
On the first side of the correspondence (the bulk):
M.C. Escher's "Circle Limit IV: Heaven and Hell" drawing that illustrates the geometry of anti-de Sitter space. The curvature of space changes, as one moves outside the circle along its radius. Thus angels and demons, which all would have the same size in our Euclidean space, become smaller and smaller in the anti-de Sitter space.
Anti-de Sitter space: On the bulk of the correspondence, will be string theory, formulated in an anti-de Sitter geometry. Anti-de Sitter space is a particular vacuum solution to the field equations of Albert Einstein, with a negative curvature or cosmological constant. In the geometry of anti-de Sitter space, the metric, which is the notion of distance between points, is different than it is in Euclidean space. Anti-de Sitter space can be described or understood as a tessellation (or a tiling) of triangles and squares in the shape of a disk. Anti-de Sitter space is curved, in such a fashion, or, distance between points are described in such a way, that:
all of the triangles and squares are the same size.
any point on the interior of the disk is infinitely far from any point on the boundary.
If one were to stack these disks directly on top of the other, the resulting geometry would resemble a cylinder. This would represent a hypothetical universe with 1 dimension of time and 2 dimensions of space. This is three-dimensional anti-de Sitter space! However, the AdS/CFT Correspondence, can be formulated in any arbitrary number of dimensions.
The first and simplest non-Euclidean geometry to be discovered was the hyperbolic space. This discovery was made by Nikolai Lobachevsky. Anti-de Sitter space has a geometry, similar to hyperbolic space. There is constant negative curvature. This was the first curved classical geometry to be understood. It's quantum geometry has also been understood to an extent. Hyperbolic space is when there is an expansion along the spatial direction, not the time direction. All directions are space-like. Anti-de Sitter space is the same, except, one of these dimensions is time-like. It is a space with constant negative curvature. The quantum version of hyperbolic space is related to conformal field theories on the boundary, as we will see.
In anti-de Sitter space, one of the coordinates is turned into time. We have spatial sections which look like Minkowski space. As you move along the radial direction, they are also expanding.
"On the left is a positively curved surface area, and on the right a surface area with negative curvature. Our four-dimensional space-time can be positively and negatively curved in a similar way. "
This is why anti-de Sitter space is sometimes regarded as being shaped like a horse saddle or a pringles chip.
De Sitter space (left) is an expanding circle (sphere in the appropriate number of dimensions). Anti-de Sitter space (right) is a saddle point that folds in on itself in time."
On the other side of the correspondence (the boundary):
Conformal field theory: The CFT, as it is known, is a kind of quantum field theory that describes elementary particles. The CFT lives on the boundary of the anti-de Sitter space, in one lower number of dimensions.
Local points on the conformal boundary, resemble Minkowski space, hence, there is no curvature, in the sense that, angles are preserved and the theory itself is well-behaved and symmetric. Points on this boundary will obey special relativity (the spacetime of Minkowski with no curvature).
There is also a property of the CFT known as scale invariance. This is when their is no fixed scale of distance or time. This is to give us a better picture of some phenomenon. If we rescale all of the coordinates, we will see the same physics. Most of everyday physics is not scale invariant. The physics is not the same at our scale and the atomic scale. Things are quite different. Scale invariant theories, usually have another symmetry known as conformal invariance. These are transformations that preserve angels. They change the local lengths, however, preserve angles between different lines. The duality does hold, however, for some non-scale invariant theories. A CFT, is a quantum field theory that preserves angles under conformal transformations. It is possible for a quantum field theory to be scale invariant, however, not conformally invariant. The situations, however, are rare. That being said, the terms are often used interchangeably in quantum field theory.
The idea behind the AdS/CFT correspondence is that the boundary of AdS, is the spacetime for the CFT. Both of the theories, despite living in a different number of dimensions, have, encoded on them, completely equivalent information. The CFT acts as a hologram, capturing equivalent information to bulk AdS.
In the case of AdS, the wavefunction of the universe can be viewed in AdS in terms of a system on the boundary that describes the gravitational physics in the interior. Quantum field theory, can be equal to a theory with dynamical spacetime. This could be string theory or some other gravitational dual theory.
In the AdS/CFT correspondence, a conformal field theory (a symmetric kind of quantum field theory that describes subatomic particles), lives on the boundary to a theory of anti-de Sitter space (a theory of gravitation in string theory or M-theory), in 1 lower dimension. However, the theories are exactly equivalent. This is our most reliable realization of the holographic principle, since, the relationship between the two theories, is analogous to the relationship between a 3-dimensional hologram and the 2-dimensional film that produced it.
Maldacena, pondering quantum gravity by a pond...
The conformal field theory acts as a holographic description of the higher dimensional theory of gravitation. They are equivalent, despite living in different numbers of dimensions.
The idea is that the wavefunction of the universe in AdS can be computed in terms of a field theory on the boundary to that AdS space. All physics in the AdS can be reduced to a quantum field theory that lives on the boundary to that space. The idea is that these two theories should be equivalent. A 4-dimensional QFT is equivalent, on the boundary to a 5-dimensional anti-de Sitter space.
Before I get into the applications of the AdS/CFT correspondence (there are many), I would like to look at 3 of the most famous realizations of the correspondence itself and give a brief description:
Type IIB string theory with the compactification: AdS5 x S^5 with N=4 supersymmetric Yang-Mills theory
The boundary here, is 4 dimensional. However, the spacetime of the theory of gravity is 5 dimensional. The S^5 indicates that there are 5 compact dimensions. Of course, spacetime is 4 dimensional in our world, thus, this model of gravity is not entirely realistic. However, since the boundary theory is similar to QCD, the theory of the strong force, it is able to answer questions in nuclear physics, as I will get into in a moment. The idea that QCD-related theories can give rise to strings, goes back to an idea by Gerard 't Hooft in 1974. A gluon, has a color and anticolor. These are 3 different kinds of charges that gluons can have. T' Hooft considered that the number of different colors could be infinity. He argued that, after this condition is met, some calculations in quantum field theory resemble some calculations in string theory.
M-theory with the compactification: AdS7 x S^4 with 6D (2,0) superconformal field theory
Here, the boundary of the theory is 6 dimensional and the theory of gravitation is 7 dimensional. Also, this boundary theory is not entirely understood, however, it’s existence is predicted. Nevertheless, this theory is still an object of interest is physics, as well as in mathematics, such as in knot theory.
M-theory with the compactification: AdS4 x S^7 with ABJM superconformal field theory
In this version of the AdS/CFT correspondence, space time is 4-dimensional and the boundary theory is 3 dimensional. This model of the correspondence is a bit more realistic than the others, since the spacetime is 4-dimensional.
It is now time to discuss the applications of the AdS/CFT correspondence. The AdS/CFT correspondence can provide answers to questions in a number of different fields, in both quantum gravity and in quantum field theory:
The AdS/CFT correspondence applications in quantum gravity:
Black hole information paradox:
The black hole information paradox is a fundamental disagreement between spacetime and quantum mechanics. Stephen Hawking proposed that a black hole, isn’t so black. Hawking put forth that a black hole is actually emitting a dim radiation at the event horizon and that the black hole, was evaporating away physical information. This is going to be a problem for quantum mechanics. In quantum mechanics, physical information cannot be destroyed or permanently lost, however, it can evolve, by means of the Schrodinger equation. This is the black hole information paradox.
How does the AdS/CFT correspondence help to answer these questions? Well, AdS/CFT, can, to an extent, provide an answer. The correspondence shows how a black hole, can evolve, in time, without violating quantum mechanics. A black hole, can be formulated in anti-de Sitter space, and can be seen as equivalent to a collection of particles on a dual thermal quantum field theory. The AdS/CFT correspondence, shows how a black hole can evolve in a manner consistent with quantum mechanics, while preserving unitarity. Stephen Hawking, himself, admitted defeat in 2005, thanks to AdS/CFT and admitted a way that information can be preserved.
A black hole in AdS corresponds to thermal configurations on the boundary theory. The entropy of the black hole can be computed by computing the area of the horizon. It can be done. This problem is difficult in field theory, however, is much more simple in gravity. A gas of particles on the boundary, can describe a black hole. That is the duality and no information is lost at black holes.
Non-perturbative formulation of string theory: Another application of the AdS/CFT correspondence to quantum gravity that I believe is worth mentioning is its ability to provide theorists with a non-perturbative formulation of string theory. What is perturbation theory? Perturbation theory is an approach at making calculations in physics. This is when some phenomenon are described as some small deviations or oscillations of some stable state. It could also be the interactions among those oscillations. Perturbative quantum field theory is used for physical problems that involve the interactions between particles.
This is the use of Feynman diagrams, which were introduced by Richard Feynman in 1948, and they describe the behavior of subatomic particles. They are pictorial representations of how these particles interact. This works very well for making predictions. However, it only works when the coupling constant, which determines the strength of the interaction is small enough. That being said, if one were to go about constructing a perturbative formulation of string theory, one would replace the world line traced out by a point particle in a Feynman diagram, with a worldsheet traced out by a 1-dimensional string in string theory. The AdS/CFT correspondence is a strong-weak duality. This means that it is useful for translating strongly interacting calculations in quantum field theory into weakly interacting calculations in string theory. The calculation in quantum field theory, becomes an easier calculation in string theory.
Thus, the AdS/CFT correspondence, could be useful for this purpose, of translating strongly interacted quantum field theories, that can be understood perturbatively, into weakly interacting calculations in string theory, which can also be understood perturbatively! However, since string theory does not yet have a full non-perturbative formulation, there is all the more motivation for constructing the AdS/CFT correspondence.
Juan Maldacena
However, not any CFT can give rise to a theory of quantum gravity. It can, if the CFT is strongly coupled. The gravity waves, would have to be weakly interacting and behave like the Einstein theory of gravity form. There are some string theories that are weakly coupled, however, it is not equivalent to Einstein gravity. We need these two conditions. What does the CFT theory, to have a weakly coupled description of gravity need? Weak coupling means that it is hard to change the metric. We need a large number of fields to have a weakly coupled description of gravity. That does not ensure that we have the Einstein theory of gravity. It will not necessarily be equal. We need a little more.
What is the other condition that must be met? We don't fully understand yet how to get the local theory. We know that we do need some extra conditions. The mass of the bulk, higher spin particles should be very large. We have the spin-2 particle. This is the only massless, higher spin (spin greater than 2) elementary particle that we see. There also must be strong interactions in the boundary theory, to prevent these higher spin particles and to keep space almost flat. The expectation is that this will yield the Einstein theory of gravity with some extra particles. We do not have a complete understanding of all of the conditions.
Why does strong coupling in a CFT make the calculations easier in quantum gravity? In other words: why does a problem in a strongly coupled quantum field theory become an easy problem in a weakly interacting theory of quantum gravity? If particles are interacting strongly enough, already established physics can be used.
The AdS/CFT correspondence applications in quantum field theory:
Dam Thanh Son
Nuclear physics: The AdS/CFT correspondence has been used to understand an exotic state of matter called the quark-gluon plasma. The quark-gluon plasma will arise at very high temperatures. We are talking, about, 2 trillion kelvins. The last time we saw 2 trillion kelvins was at 10^-11 seconds after the Big Bang. Indeed, these are extreme conditions. It can arise very briefly in particle collider experiments. What happens is: the nuclei, are so hot, they literally melt, and the constituent quarks deconfine. Quantum chromodynamics, our current theory of the strong force, quarks and gluons, is insufficient at understanding some of the properties of the quark-gluon plasma.
Dam Thanh Son, in 2005, is going to propose that the AdS/CFT correspondence, can provide some insight into understanding some properties of the quark-gluon plasma. What he showed was that a particular realization of the AdS/CFT correspondence with 5 dimensional spacetime, could have a dual understanding of the quark-gluon plasma. He proceeded to calculate a ratio, for the “shear viscosity” and the “volume density of entropy” for the quark-gluon plasma. These properties were put into proportion to a constant that was proven in 2008 at the RHIC. He also proposed a phenomenon known as the “jet quenching parameter” which was a relationship between the quark’s energy and it’s distance traveled through the plasma (only a few femtometers) before it is stopped.
Subir Sachdev
Condensed matter physics: Condensed matter physicists are concerned with exotic states of matter, such as:
Superconductors: Systems, that, when cooled to a sufficient temperature, have 0 electric resistance.
Superfluids: A fluid with 0 viscosity, that can flow without loss of any kinetic energy. These atoms are electrically neutral, and flow without any friction.
Indeed, quantum field theory can be used to understand much about these exotic states of matter. However, some properties and phenomenon remain out of reach using the standard techniques of quantum field theory. Subir Sachdev, has used the AdS/CFT correspondence to attempt to understand these states of matter, perhaps, within the language of string theory. There has been some success: the discovery of the transition of a superfluid to an insulator can be understood using the theoretical methods of string theory. The way this is done experimentally involves several steps:
Trillions of cold atoms (in an initial state of superfluidity) are placed in a lattice of lasers criss crossing each other, in kind of a tic-tac-toe arrangement.
The intensity of the lasers are increased.
The atoms begin to lose their mobility.
That’s it: the atoms have transitioned to an insulating state.
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