Ryutakayanagi FormulaEdit

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The Ryū–Takayanagi formula The Ryū–Takayanagi formula, named after Shinsei Ryu and Tadashi Takayanagi, is a foundational result in the holographic principle and the AdS/CFT correspondence. It provides a precise geometric representation of quantum entanglement in certain quantum field theories by relating it to areas of minimal surfaces in a higher-dimensional gravitational spacetime. In its standard static form, the entanglement entropy S_A of a spatial region A in a conformal field theory on the boundary is proportional to the area of a minimal surface γ_A in the bulk whose boundary matches the boundary of A: S_A = Area(γ_A) / (4 G_N), where G_N is Newton’s constant in the bulk. This striking relation connects quantum information to geometry and supports the view that spacetime itself can emerge from quantum entanglement.

Historical context and significance Proposed in 2006, the formula emerged from studies of black hole thermodynamics and the holographic principle, which posits that a gravitational theory in a bulk spacetime can be described by a non-gravitational theory on the boundary. The Ryu–Takayanagi construction provided a concrete prescription for computing boundary entanglement entropy using classical bulk geometry, offering a bridge between quantum information theory and gravitational dynamics. It has since become a central tool in the study of holography and the emergence of spacetime.

Foundations and precise statement - Setting: A static or stationary spacetime with negative cosmological constant, often in the context of the AdS/CFT correspondence, and a boundary region A with boundary ∂A. - Bulk geometry: A bulk spacetime that satisfies the classical equations of motion (often Einstein gravity with possible matter fields) in the large-N, strong-coupling limit where the bulk is well-approximated by a classical geometry. - Surface: γ_A is a bulk surface of codimension two that is anchored on ∂A and extremizes area (minimal in the static case). - Entropy: S_A is proportional to the area of γ_A via S_A = Area(γ_A) / (4 G_N). This mirrors the Bekenstein–Hawking entropy formula for black holes, highlighting a deep link between geometry and information. - Key references: The original proposal is commonly cited as the Ryu–Takayanagi formula, with subsequent refinements and generalizations explored in the literature.

Geometric intuition and interpretations - The minimal surface γ_A serves as a geometric proxy for quantum correlations between A and its complement. Regions of strong entanglement in the boundary theory correspond to larger areas of the corresponding bulk surface. - The formula encodes a form of holographic duality: boundary entanglement data is encoded in the bulk geometry, reinforcing the idea that spacetime geometry is emergent from quantum information.

Extensions and refinements - Time dependence: For dynamical or time-dependent settings, the covariant generalization uses extremal surfaces rather than strictly minimal ones. This is known as the Hubeny–Rangamani–Takayanagi (HRT) formula. - See also: Hubeny–Rangamani–Takayanagi formula - Quantum corrections: Realistic theories include quantum effects in the bulk. The leading quantum-corrected version adds the bulk entanglement entropy S_bulk contained within the bulk region bounded by γ_A, leading to a quantum extremal surface (QES) framework. - See also: Quantum extremal surface - Relevant developments: Faulkner, Lewkowycz, Maldacena, and others in the early 2010s established the framework for including bulk quantum corrections to holographic entanglement entropy. - Subregion duality and entanglement wedge: The RT/HRT prescriptions underpin ideas about how bulk information in certain regions is encoded in boundary subregions, leading to the concept of the entanglement wedge and related reconstruction theorems. - See also: Entanglement wedge and Subregion duality

Applications and implications - Gauge/gravity duality as a calculational tool: The formula enables concrete computations of entanglement entropy in strongly coupled quantum field theories by solving a classical geometric problem in the bulk. - Black holes and information: The area-law behavior encoded in the RT formula resonates with black hole entropy results and has informed discussions about information recovery in gravitational systems. - Condensed matter and quantum many-body physics: Holographic duality, guided by the RT prescription, has inspired models of strongly correlated systems and provided intuition for entanglement structure in complex quantum phases. - Foundational questions: The geometric view of entanglement informs broader questions about the emergence of spacetime, quantum gravity, and the role of information in fundamental physics.

Controversies and debates - Regimes of validity: The RT/HRT prescriptions are derived under specific conditions—large-N, strong coupling, and a classical bulk limit. There is ongoing discussion about how broadly these results apply to non-holographic theories or real-world materials. - Extensions to non-AdS spacetimes: Applications to cosmological spacetimes (e.g., de Sitter space) face conceptual and technical challenges, and a fully established, universal prescription in those contexts remains an area of active research. - Quantum corrections and uniqueness: While the leading area term captures the leading entanglement contribution, incorporating bulk quantum effects raises questions about scheme dependence, regulator choices, and the precise definition of S_bulk in interacting theories. - Bulk reconstruction and information flow: The connection between boundary entanglement and bulk geometry raises deep questions about how and when bulk information can be reconstructed from boundary data, with various proposals and partial results leading to ongoing debate. - Interpretational diversity: The relationship between entanglement, geometry, and causality invites multiple interpretational viewpoints about how spacetime and gravity emerge, and some researchers advocate alternative or complementary pictures to understand holography.

See also - Entanglement entropy - Ryu–Takayanagi formula - Hubeny–Rangamani–Takayanagi formula - AdS/CFT correspondence - Quantum extremal surface - Bekenstein–Hawking entropy - Black hole information paradox - Holographic principle