Shinsei Ryu and
Tadashi Takayanagi published 2006 a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
within
holography
Holography is a technique that allows a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interfe ...
that posits a quantitative relationship between the
entanglement entropy of a
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
and the geometry of an associated
anti-de Sitter spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
.
The formula characterizes "holographic screens" in the bulk; that is, it specifies which regions of the bulk geometry are "responsible to particular information in the dual
CFT".
[
] The authors were awarded the 2015
Breakthrough Prize in Fundamental Physics
The Breakthrough Prize in Fundamental Physics is one of the Breakthrough Prizes, awarded by the Breakthrough Prize Board. Initially named Fundamental Physics Prize, it was founded in July 2012 by Russia-born Israeli entrepreneur, venture capit ...
for "fundamental ideas about
entropy
Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
in quantum field theory and quantum gravity", and awarded the 2024
Dirac Medal of the ICTP for "their insights on quantum entropy in quantum gravity and quantum field theories". The formula was generalized to a
covariant form in 2007.
Motivation
The
thermodynamics of black holes suggests certain relationships between the entropy of black holes and their geometry. Specifically, the Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area:
:
The Bekenstein–Hawking entropy
is a measure of the information lost to external observers due to the presence of the horizon. The horizon of the black hole acts as a "screen" distinguishing one region of the spacetime (in this case the exterior of the black hole) that is not affected by another region (in this case the interior). The Bekenstein–Hawking area law states that the area of this surface is proportional to the entropy of the information lost behind it.
The Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system; however, there is another type of entropy that is important in quantum information theory, namely the
entanglement (or von Neumann) entropy. This form of entropy provides a measure of how far from a pure state a given quantum state is, or, equivalently, how entangled it is. The entanglement entropy is a useful concept in many areas, such as in condensed matter physics and quantum many-body systems. Given its use, and its suggestive similarity to the Bekenstein–Hawking entropy, it is desirable to have a holographic description of entanglement entropy in terms of gravity.
Holographic preliminaries
The holographic principle states that gravitational theories in a given dimension are dual to a
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
in one lower dimension. The
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) that are used ...
is one example of such duality. Here, the field theory is defined on a fixed background and is equivalent to a quantum gravitational theory whose different states each correspond to a possible spacetime geometry. The conformal field theory is often viewed as living on the boundary of the higher dimensional space whose gravitational theory it defines. The result of such a duality is a dictionary between the two equivalent descriptions. For example, in a CFT defined on
dimensional
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole.
Important for the present discussion is that the thermal state of a CFT defined on the
dimensional sphere corresponds to the
dimensional Schwarzschild black hole in AdS space.
The Bekenstein–Hawking area law, while claiming that the area of the black hole horizon is proportional to the black hole's entropy, fails to provide a sufficient microscopic description of how this entropy arises. The holographic principle provides such a description by relating the black hole system to a quantum system which does admit such a microscopic description. In this case, the CFT has discrete eigenstates and the thermal state is the canonical ensemble of these states.
[ The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law. This turns out to be a special case of the Ryu–Takayanagi conjecture.
]
Conjecture
Consider a spatial slice of an AdS space time on whose boundary we define the dual CFT. The Ryu–Takayanagi formula states:
where is the entanglement entropy of the CFT in some spatial sub-region with its complement , and is the Ryu–Takayanagi surface in the bulk. This surface must satisfy three properties:
# has the same boundary as .
# is homologous to A.
# extremizes the area. If there are multiple extremal surfaces, is the one with the least area.
Because of property (3), this surface is typically called the ''minimal surface'' when the context is clear. Furthermore, property (1) ensures that the formula preserves certain features of entanglement entropy, such as and . The conjecture provides an explicit geometric interpretation of the entanglement entropy of the boundary CFT, namely as the area of a surface in the bulk.
Example
In their original paper, Ryu and Takayanagi show this result explicitly for an example in where an expression for the entanglement entropy is already known. For an space of radius , the dual CFT has a central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
given by
Furthermore, has the metric
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
in (essentially a stack of hyperbolic disks). Since this metric diverges at , is restricted to . This act of imposing a maximum is analogous to the corresponding CFT having a UV cutoff. If is the length of the CFT system, in this case the circumference of the cylinder calculated with the appropriate metric, and is the lattice spacing, we have
.
In this case, the boundary CFT lives at coordinates . Consider a fixed slice and take the subregion A of the boundary to be