# Algebraic holography

Algebraic holography

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 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.Fact|date=June 2008

Rehren's duality

The conformal boundary of an anti de Sitter space (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).

*"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]

References

*arxiv|archive=hep-th|id=9905179 Karl-Henning Rehren, "Algebraic Holography"For a classical counterpart to Rehren duality see
*arXiv|arcive=hep-th|id=0708.1283 B.S. Kay and P.Larkin, "Pre-Holography"

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• AdS/CFT correspondence — For the relation of the AdS/CFT correspondence to the general context of string theory, see String theory. The conformal field theory, which may be a gauge theory lies on the conformal boundary of anti deSitter space with quantum gravity. In… …   Wikipedia

• Karl-Henning Rehren — (born 1956 in Celle) 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 (advisor Klaus Pohlmeyer) in 1984. Habilitation 1991 in Berlin.… …   Wikipedia

• Karl-Henning Rehren — (* 1956 in Celle) ist ein deutscher Physiker auf dem Gebiet der Quantenfeldtheorie. Karl Henning Rehren (rechts) mit Roberto Longo (links), Klaus Fredenhagen (Mitte) Inhaltsverzeichnis …   Deutsch Wikipedia

• Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

• List of Russian people — The Millennium of Russia monument in Veliky Novgorod, featuring the statues and reliefs of the most celebrated people in the first 1000 years of Russian history …   Wikipedia

• Yuri Manin — am ICM 2006 in Madrid, mit Ksenia Semenova Yuri Manin (russisch Юрий Иванович Манин / Juri Iwanowitsch Manin; * 16. Februar 1937 in Simferopol, heute …   Deutsch Wikipedia

• Список награждённых Национальной медалью науки США — Джошуа Ледерберг (справа) получает Национальную медаль науки из рук Президента США Джорджа Буша старшего Список …   Википедия

• Juri Iwanowitsch Manin — Yuri Manin am ICM 2006 in Madrid, mit Ksenia Semenova Yuri Manin (russisch Юрий Иванович Манин/ Juri Iwanowitsch Manin; * 16. Februar 1937 in Simferopol, heute Ukraine) …   Deutsch Wikipedia

• Juri Manin — Yuri Manin am ICM 2006 in Madrid, mit Ksenia Semenova Yuri Manin (russisch Юрий Иванович Манин/ Juri Iwanowitsch Manin; * 16. Februar 1937 in Simferopol, heute Ukraine) …   Deutsch Wikipedia

• Yurij Manin — Yuri Manin am ICM 2006 in Madrid, mit Ksenia Semenova Yuri Manin (russisch Юрий Иванович Манин/ Juri Iwanowitsch Manin; * 16. Februar 1937 in Simferopol, heute Ukraine …   Deutsch Wikipedia