Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the

holographic principle The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

within the framework of

algebraic quantum field theory Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in te ...

, due to

Karl-Henning Rehren Karl-Henning Rehren (born 1956 in Celle) is a German physicist who focuses on algebraic quantum field theory. Biography Rehren studied physics in Heidelberg, Paris and Freiburg. In Freiburg he received his PhD (advisor Klaus Pohlmeyer) in 198 ...

. It is sometimes described as an alternative formulation of the

AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

, but some string theorists reject this statemen

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 In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an ext ...

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 Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in te ...

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 mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...

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

References

* For a classical counterpart to Rehren duality see * {{cite journal , last1=Kay , first1=Bernard S. , last2=Larkin , first2=Peter , title=Pre-holography , journal=Physical Review D , volume=77 , issue=12 , date=18 June 2008 , issn=1550-7998 , doi=10.1103/physrevd.77.121501 , page=121501(R), arxiv=0708.1283, bibcode=2008PhRvD..77l1501K , s2cid=263787154 Axiomatic quantum field theory Conformal field theory Quantum gravity