Algebraic holography
Encyclopedia
Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the holographic principle
of quantum gravity
within the framework of algebraic quantum field theory, due to Karl-Henning Rehren
. It is sometimes described as an alternative formulation of the AdS/CFT correspondence
of string theory
, but some string theorists reject this statement http://golem.ph.utexas.edu/~distler/blog/archives/000987.html. The theories discussed in algebraic holography do not satisfy the usual holographic principle because their entropy follows a higher-dimensional power law.
(or its universal covering space) is the conformal Minkowski space
(or its universal covering space) with one fewer dimension. Let's work with the universal covering spaces. In AQFT, a QFT in the conformal space is given by a conformally covariant net of C* algebras over the conformal space and the QFT in AdS is given a covariant net of C* algebras over AdS. Any two distinct null geodesic hypersurfaces of codimension 1 which intersect at more than just a point in AdS divides AdS into four distinct regions, two of which are spacelike. Any of the two spacelike regions is called a wedge. It's a geometrical fact that the conformal boundary of a wedge is a double cone in the conformal boundary and that any double cone
in the conformal boundary is associated with a unique wedge. In other words, we have a one-to-one correspondence between double cones in CFT and wedges in AdS. It's easy to check that any CFT defined in terms of algebras over the double cones which satisfy the Haag-Kastler axioms also gives rise to a net over AdS which satisfies these axioms if we assume that the algebra associated with a wedge is the same as the algebra associated with its corresponding double cone and vice versa. This correspondence between AQFTs on both sides is called algebraic holography.
Unlike the usual AdS/CFT correspondence, the Rehren-dual theory on the AdS side does not appear to be a theory of quantum gravity as there is no apparent diffeomorphism covariance on the AdS side. Also, if the algebra associated with any double cone in AdS is nontrivial (i.e. contains more than just the identity), the corresponding CFT does not satisfy primitive causality. From this, we can conclude that the AdS Rehren-dual of any realistic CFT does not have any local degrees of freedom (wedges are noncompact).
Holographic principle
The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
of quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
within the framework of algebraic quantum field theory, due to Karl-Henning Rehren
Karl-Henning Rehren
Karl-Henning Rehren is a German physicist who focuses on algebraic quantum field theory.Rehren studied physics in Heidelberg, Paris and Freiburg. In Freiburg he received his PhD in 1984. Habilitation 1991 in Berlin...
. It is sometimes described as an alternative formulation of the AdS/CFT correspondence
AdS/CFT correspondence
In physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory and gravity defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, but some string theorists reject this statement http://golem.ph.utexas.edu/~distler/blog/archives/000987.html. The theories discussed in algebraic holography do not satisfy the usual holographic principle because their entropy follows a higher-dimensional power law.
Rehren's duality
The conformal boundary of an anti de Sitter spaceAnti de Sitter space
In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
(or its universal covering space) is the conformal Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
(or its universal covering space) with one fewer dimension. Let's work with the universal covering spaces. In AQFT, a QFT in the conformal space is given by a conformally covariant net of C* algebras over the conformal space and the QFT in AdS is given a covariant net of C* algebras over AdS. Any two distinct null geodesic hypersurfaces of codimension 1 which intersect at more than just a point in AdS divides AdS into four distinct regions, two of which are spacelike. Any of the two spacelike regions is called a wedge. It's a geometrical fact that the conformal boundary of a wedge is a double cone in the conformal boundary and that any double cone
Double cone
Double cones , also known as twin cones in some literature, are two cone cells joined together that may also be coupled optically/electrically...
in the conformal boundary is associated with a unique wedge. In other words, we have a one-to-one correspondence between double cones in CFT and wedges in AdS. It's easy to check that any CFT defined in terms of algebras over the double cones which satisfy the Haag-Kastler axioms also gives rise to a net over AdS which satisfies these axioms if we assume that the algebra associated with a wedge is the same as the algebra associated with its corresponding double cone and vice versa. This correspondence between AQFTs on both sides is called algebraic holography.
Unlike the usual AdS/CFT correspondence, the Rehren-dual theory on the AdS side does not appear to be a theory of quantum gravity as there is no apparent diffeomorphism covariance on the AdS side. Also, if the algebra associated with any double cone in AdS is nontrivial (i.e. contains more than just the identity), the corresponding CFT does not satisfy primitive causality. From this, we can conclude that the AdS Rehren-dual of any realistic CFT does not have any local degrees of freedom (wedges are noncompact).
Differences when compared to AdS/CFT
- "In AdS/CFT, the boundary values of bulk fields are sources for operators of the boundary theory. In Rehren Duality, the boundary values of the bulk fields are the operators of the boundary theory.
- "In AdS/CFT, the bulk theory is necessarily a gravitational one. The source for the conserved stress tensor of the boundary theory is the boundary value of the bulk metric tensor. In Rehren Duality, the bulk theory is an 'ordinary' (non-gravitational) QFT."http://golem.ph.utexas.edu/~distler/blog/archives/001702.html#c017264