Glossary

12.3 AdS/CFT correspondence and holography

8 min read•august 14, 2024

The AdS/CFT correspondence is a groundbreaking concept in theoretical physics. It links a gravitational theory in to a on its boundary. This duality provides a powerful tool for studying and strongly coupled systems.

The correspondence exemplifies the , suggesting that information in a volume of space can be encoded on its boundary. This idea has far-reaching implications for our understanding of black holes, quantum information, and the nature of spacetime itself.

AdS/CFT Correspondence

Basic Concepts and Significance

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  • The AdS/CFT correspondence, also known as the Maldacena duality or , conjectures a relationship between a gravitational theory in Anti-de Sitter (AdS) space and a conformal field theory (CFT) on its boundary
  • States that a theory of gravity in AdS space is equivalent to a CFT on the boundary of that space, providing a holographic description of gravity
  • Specific realization of the holographic principle, suggesting the description of a volume of space can be encoded on a lower-dimensional boundary
  • Powerful tool in theoretical physics, offering insights into quantum gravity, string theory, and strongly coupled quantum field theories
  • Led to new approaches in studying various physical systems (black holes, condensed matter systems, quantum information)

Relationship between Gravitational Theory and Conformal Field Theory

  • Gravitational theory in the bulk of AdS space is typically a string theory or supergravity, while the CFT on the boundary is a gauge theory (N=4 supersymmetric Yang-Mills theory)
  • AdS space is a maximally symmetric spacetime with constant negative curvature, characterized by a cosmological constant Λ<0\Lambda < 0
    • Boundary of AdS space is a flat Minkowski spacetime with one fewer dimension
  • Relates the partition function of the gravitational theory in AdS to the generating functional of correlation functions in the CFT on the boundary
  • AdS/CFT dictionary establishes a one-to-one mapping between operators in the CFT and fields in the gravitational theory, allowing calculation of observables in one theory using the other
  • Relationship between the two theories is a strong-weak coupling duality
    • When the gravitational theory is strongly coupled, the CFT is weakly coupled, and vice versa
    • Allows study of strongly coupled quantum field theories using classical gravity

Gravity vs Conformal Field Theory

Holographic Principle and Gravity

  • Holographic principle states information contained within a region of space can be described by a theory on the boundary of that region, with a density of one bit per Planck area
  • Suggests fundamental degrees of freedom in a gravitational theory are not located in the bulk of spacetime but rather on its boundary, implying gravity and spacetime emerge from a lower-dimensional description
  • AdS/CFT correspondence is a concrete realization of the holographic principle, demonstrating a theory of gravity in the bulk of AdS space is equivalent to a CFT on its boundary
  • Holographic nature of gravity has far-reaching implications for understanding black holes and the information paradox
    • Suggests information about the interior of a black hole is encoded on its event horizon, resolving apparent loss of information when objects fall into a black hole
  • Hints at a deeper connection between quantum mechanics and gravity, potentially providing insights into the nature of quantum gravity and unification of fundamental forces

Implications for Quantum Gravity and Spacetime

  • Holographic principle implies gravity and spacetime are emergent phenomena arising from a more fundamental, lower-dimensional description
  • Suggests a deep connection between quantum mechanics and gravity, as the holographic description of gravity involves a quantum field theory on the boundary
  • Provides a framework to study quantum gravity by relating it to a well-understood quantum field theory (CFT)
    • Allows for the investigation of quantum gravitational effects using tools from quantum field theory and string theory
  • Offers insights into the nature of spacetime at the quantum scale, where the classical notion of spacetime breaks down
    • Suggests spacetime may have a discrete, granular structure at the Planck scale, with the smooth, continuous spacetime of general relativity emerging as an effective description at larger scales
  • Hints at a possible resolution to the incompatibility between quantum mechanics and general relativity, as the holographic principle provides a way to reconcile the two theories within a unified framework

Holographic Principle and Gravity

Black Holes and Information Paradox

  • Holographic principle has significant implications for the understanding of black holes and the information paradox
  • Suggests the information about the interior of a black hole is encoded on its event horizon, rather than being lost forever when an object falls into the black hole
    • Resolves the apparent contradiction between the laws of quantum mechanics (information cannot be lost) and the classical description of black holes (information is lost behind the event horizon)
  • Implies the Bekenstein-Hawking entropy of a black hole, which is proportional to its horizon area, can be interpreted as the amount of information encoded on the horizon
    • Provides a microscopic explanation for the origin of black hole entropy in terms of the degrees of freedom of the holographic boundary theory (CFT)
  • Offers insights into the nature of Hawking radiation and the evaporation of black holes
    • Suggests Hawking radiation may carry the information about the interior of the black hole, allowing for the preservation of information as the black hole evaporates
  • Leads to the development of the "black hole complementarity" principle, which states that the interior and exterior descriptions of a black hole are complementary and not contradictory
    • Resolves the apparent cloning of information (violation of the no-cloning theorem) when an object falls into a black hole, as the interior and exterior descriptions are not simultaneously accessible to an observer

Emergence of Spacetime and Quantum Mechanics

  • Holographic principle suggests that spacetime and gravity are emergent phenomena arising from a more fundamental, lower-dimensional description
  • Implies the smooth, continuous spacetime of general relativity is an effective description that arises from the collective behavior of the degrees of freedom in the holographic boundary theory
    • Analogous to how the smooth, continuous behavior of fluids emerges from the collective motion of individual molecules
  • Provides a framework to understand the relationship between quantum mechanics and gravity
    • Suggests quantum mechanics and gravity are two aspects of a single, unified description, with quantum mechanics governing the boundary theory and gravity emerging in the bulk spacetime
  • Offers insights into the nature of entanglement and its relation to spacetime geometry
    • Entanglement between the degrees of freedom in the boundary theory is related to the connectivity and geometry of the bulk spacetime
    • Leads to the concept of "ER=EPR" (Einstein-Rosen bridges = Einstein-Podolsky-Rosen entanglement), which suggests that entangled particles are connected by a wormhole in the bulk spacetime
  • Opens up new possibilities for understanding the origin of quantum mechanics and its relationship to spacetime
    • Suggests quantum mechanics may be a consequence of the holographic nature of reality, with the probabilistic nature of quantum mechanics arising from the statistical behavior of the degrees of freedom in the boundary theory

Applications of AdS/CFT Correspondence

Condensed Matter Physics

  • AdS/CFT has found applications in condensed matter physics, particularly in the study of strongly correlated systems (high-temperature superconductors, quantum critical points)
    • Allows for the description of certain condensed matter systems using a dual gravitational theory, providing a new perspective on their properties and behavior
  • Examples of applications in condensed matter physics:
    • Holographic description of the quantum Hall effect, which relates the conductivity of a two-dimensional electron gas to the properties of a black hole in the dual gravitational theory
    • Study of non-Fermi liquids, which exhibit behavior that deviates from the standard Fermi liquid theory, using holographic models that capture their unusual properties
    • Investigation of phase transitions and critical phenomena, such as superconductivity and metal-insulator transitions, using the AdS/CFT correspondence to map the problem to a gravitational system
  • Provides a new tool for studying strongly coupled quantum systems, which are difficult to analyze using traditional methods
    • Allows for the calculation of transport coefficients, such as electrical and thermal conductivity, in terms of the properties of the dual gravitational theory
    • Offers insights into the nature of quantum criticality and the behavior of systems near quantum critical points

Heavy-Ion Collisions and Quark-Gluon Plasma

  • In the context of heavy-ion collisions, AdS/CFT has been used to model the quark-gluon plasma (QGP) produced in high-energy collisions of heavy nuclei
    • QGP is a strongly coupled fluid that exhibits properties similar to a perfect fluid with minimal viscosity
    • AdS/CFT provides a framework to calculate transport coefficients and other properties of the QGP
  • Applications of AdS/CFT in heavy-ion collisions:
    • Study of the thermalization process of the QGP, which describes how the system reaches thermal equilibrium after the collision
    • Investigation of jet quenching, which refers to the energy loss of high-energy partons (quarks and gluons) as they traverse the QGP
    • Calculation of the energy loss and momentum broadening of heavy quarks (charm and bottom) in the QGP
  • Allows for the comparison of theoretical predictions with experimental results from heavy-ion colliders, such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC)
    • Helps to constrain the properties of the QGP and improve our understanding of the strong interaction under extreme conditions
  • Provides insights into the transport properties of strongly coupled fluids and the behavior of matter at high temperatures and densities
    • Relevant for understanding the early universe, where the QGP is believed to have existed shortly after the Big Bang

Entanglement Entropy and Quantum Information

  • AdS/CFT has been applied to the study of and quantum information, providing geometric insights into the nature of entanglement and its relation to spacetime geometry
  • Entanglement entropy is a measure of the amount of entanglement between two subsystems of a quantum system
    • In the context of AdS/CFT, entanglement entropy is related to the area of minimal surfaces in the bulk spacetime that separate the two subsystems
    • Provides a holographic interpretation of entanglement entropy in terms of the geometry of the bulk spacetime
  • Allows for the study of entanglement and quantum information in strongly coupled quantum systems
    • Provides a framework to calculate entanglement entropy and other quantum information measures using the dual gravitational theory
    • Offers insights into the nature of and its role in the emergence of spacetime
  • Applications of AdS/CFT in quantum information:
    • Investigation of the properties of quantum error-correcting codes and their relation to the geometry of the bulk spacetime
    • Study of the holographic nature of quantum algorithms and their implementation using the AdS/CFT correspondence
    • Exploration of the connection between quantum entanglement and the structure of spacetime, such as the "ER=EPR" conjecture relating entangled particles to wormholes
  • Provides a new perspective on the interplay between quantum information, entanglement, and gravity
    • Suggests a deep connection between the laws of quantum mechanics and the structure of spacetime
    • Offers potential insights into the nature of quantum gravity and the unification of quantum mechanics and general relativity

Key Terms to Review (16)

Anti-de Sitter Space: Anti-de Sitter space is a spacetime model characterized by a constant negative curvature, which can be thought of as a hyperbolic geometry in higher dimensions. This type of space is crucial in theoretical physics, particularly in the context of the AdS/CFT correspondence, where it provides a framework for understanding the relationship between gravity in a curved spacetime and quantum field theories defined on the boundary of that space.
Black hole thermodynamics: Black hole thermodynamics is the study of the laws governing the behavior and properties of black holes, drawing parallels between thermodynamic laws and black hole physics. This concept suggests that black holes have entropy, temperature, and emit radiation, known as Hawking radiation, which connects gravitational phenomena to quantum mechanics and thermodynamics.
Bulk-boundary correspondence: Bulk-boundary correspondence is a concept in theoretical physics that describes the relationship between the properties of a higher-dimensional bulk space and its lower-dimensional boundary. This principle is especially significant in the context of holography, where the dynamics of a gravitational theory in the bulk can be fully described by a conformal field theory living on the boundary, highlighting a deep connection between gravity and quantum field theories.
Conformal Field Theory: Conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations, which are transformations that preserve angles but not necessarily distances. This property makes CFTs particularly useful in studying critical phenomena and phase transitions, where systems exhibit scale invariance. CFTs also play a crucial role in holographic theories, connecting strongly coupled quantum field theories with gravity in higher-dimensional spaces, and have applications in condensed matter physics, revealing insights into quantum phase transitions and emergent phenomena.
Emergent gravity: Emergent gravity is a theoretical concept suggesting that gravity is not a fundamental force but rather arises from more fundamental underlying phenomena, such as quantum entanglement and thermodynamic principles. This idea challenges traditional views of gravity as a force mediated by spacetime curvature, proposing instead that it emerges from the collective behavior of microscopic degrees of freedom in a system.
Entanglement Entropy: Entanglement entropy is a measure of the quantum correlations between parts of a quantum system, quantified by the von Neumann entropy of the reduced density matrix. It provides insights into how information is distributed in quantum systems, particularly in the context of boundary regions in AdS/CFT correspondence, revealing how the geometry of spacetime can encode quantum information.
Gauge/gravity duality: Gauge/gravity duality is a theoretical framework in which a gauge theory in a certain number of dimensions corresponds to a gravity theory in a higher-dimensional space. This concept is particularly important in the context of the AdS/CFT correspondence, where conformal field theories on the boundary are related to gravity theories in Anti-de Sitter (AdS) space, providing deep insights into quantum gravity and the nature of spacetime.
Holographic duality: Holographic duality is a principle in theoretical physics that suggests a relationship between a higher-dimensional gravitational theory and a lower-dimensional quantum field theory. This idea is a central part of the AdS/CFT correspondence, which posits that certain types of gravity in anti-de Sitter space are equivalent to conformal field theories defined on its boundary. This duality offers deep insights into the nature of quantum gravity and provides a powerful framework for understanding complex quantum systems.
Holographic principle: The holographic principle is a theoretical framework suggesting that all the information contained in a volume of space can be represented as a hologram—a lower-dimensional description on the boundary of that space. This idea is intimately connected to the behavior of black holes and quantum gravity, indicating that our three-dimensional universe might be encoded in two dimensions at its boundaries.
Juan Maldacena: Juan Maldacena is a prominent theoretical physicist best known for his groundbreaking work on the AdS/CFT correspondence, a conjectured relationship between two types of physical theories: Anti-de Sitter (AdS) space and conformal field theory (CFT). His contributions have profoundly influenced the understanding of quantum gravity and string theory, linking gravitational theories in higher dimensions to quantum field theories in lower dimensions. This connection has far-reaching implications in areas such as holography and black hole physics.
Leonard Susskind: Leonard Susskind is a prominent theoretical physicist known for his work in quantum mechanics, string theory, and quantum field theory. He is one of the founding figures of the holographic principle, which connects concepts of gravity and quantum information, and has significantly contributed to understanding spontaneous symmetry breaking and the Higgs mechanism in particle physics.
Maldacena Conjecture: The Maldacena Conjecture, proposed by Juan Maldacena in 1997, is a groundbreaking idea in theoretical physics that suggests a deep relationship between two seemingly different types of physical theories: string theory formulated in Anti-de Sitter (AdS) space and conformal field theory (CFT) defined on the boundary of that space. This conjecture posits that these two theories are equivalent, allowing insights from one to inform the other, which has significant implications for understanding quantum gravity and holography.
N=4 Super Yang-Mills: n=4 Super Yang-Mills is a highly studied quantum field theory that is a special case of Yang-Mills theory, featuring supersymmetry and four supercharges. This theory is significant in theoretical physics, particularly because it plays a crucial role in the AdS/CFT correspondence, linking gauge theories and string theories, and showcasing deep connections between gravity and quantum field theories.
Quantum entanglement: Quantum entanglement is a quantum mechanical phenomenon where two or more particles become interconnected in such a way that the state of one particle cannot be described independently of the state of the other(s), even when the particles are separated by large distances. This non-local connection implies that a measurement performed on one particle will instantaneously affect the state of the other, leading to intriguing implications for the nature of reality and information transfer.
Quantum gravity: Quantum gravity is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It aims to unify general relativity, which describes gravity as a curvature of spacetime, with quantum field theory, which governs the behavior of fundamental particles. This unification is essential for understanding phenomena in extreme environments, such as black holes and the early universe.
Symmetry Breaking: Symmetry breaking occurs when a system that is symmetric under a certain transformation loses that symmetry due to a change in its conditions or configuration. This phenomenon is pivotal in various areas of physics, as it helps explain how certain fundamental forces arise and why certain particles acquire mass while others do not.
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