Holographic principle

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In the theories of quantum gravity, the holographic principle is the hypothesis that for every description of the dynamics of a space-time region there is an equivalent description that is only localized on the edge of this region . This has u. a. As a result, the maximum possible entropy of a spatial area does not depend on the volume, but only on its surface. This is the case with the Bekenstein-Hawking entropy of black holes .

The holographic principle expresses that, taking into account gravity, the " information content ", i. H. the number of possible arrangements of particles and fields can not be a purely local quantity, because then it would be proportional to the volume.

The term holographic is based on the analogy to the hologram , which saves a three-dimensional image on a two-dimensional photo plate. The holographic principle was developed by Gerardus' t Hooft and Leonard Susskind , among others .

Coding at the event horizon

An important argument for the holographic principle is the entropy of black holes . The area measure of the event horizon , the interface of the black hole formed by the Schwarzschild radius, is a direct measure of the entropy or the information content of the enclosed volume of space and thus of the masses contained therein. A black hole always represents the maximum possible concentration of matter in a spatial area and thus also the upper limit of possible entropy or information in the spatial volume it occupies ( Bekenstein limit ).

The holographic principle postulates that any information that exceeds the surface area of ​​the event horizon of a black hole is completely encoded on the boundary surface spanned by the Schwarzschild radius, similar to a two-dimensional hologram that contains three-dimensional image information.

Since the Schwarzschild radius of a black hole is only directly proportional to its mass, the codable volume increases with the square of the surface. In order to encode four times the volume, all that is required is a doubling of the interface, or to put it another way, the information density of a spatial area decreases with increasing volume (just as its average mass density decreases with the size of a black hole). Or more briefly: information equals surface.

Suspected AdS / CFT correspondence

A particularly well elaborated special case is a 1997 resulting correspondence between presumption Anti-de Sitter space AdS (Engl. Anti-de Sitter space ) and conformal field theory CFT (Engl. Conformal Field Theory ). The anti-de-sitter space represents a possible solution to Albert Einstein's field equations with a negative cosmological constant ( negative to describe the attractive effect of gravity). Conformal field theories have a particularly high degree of symmetry.

In mathematics, correspondence is understood to be a sharp duality relation when describing physical phenomena using two different theories . Such dual theories are u. A. interesting for the following reasons:

  • What is a weak effect in one theory can produce strong effects in the other .
  • What is difficult to solve in one theory can be an easy problem in the other .

Originally, duality, and the relationship to the holographic principle, was formulated by Juan Maldacena between two concrete theories:

The first theory was an essentially five- dimensional Type IIB string theory (more precisely: a product of a five- dimensional anti-de-sitter space and a 5- sphere corresponding to the compactified dimensions). The dual theory equivalent to this was a special conformal field theory, the N = 4 supersymmetric Yang-Mills theory (SYM) , defined on the four- dimensional edge of AdS space. This situation corresponds exactly to the holographic principle.

There are now generalizations of this specific situation, for example in the algebraic quantum field theory of Rudolf Haag and Alfred Kastler , and it is assumed that the conjecture is confirmed in greater generality, although there is no mathematical proof of it . However, there are a large number of hints that arise in borderline cases of correspondence in which both sides (both string theory and conformal field theory) can be calculated.

An AdS / CFT correspondence between gauge field theories with higher spin (containing suggestions for any high even-numbered spin), which, according to Mikhail Vasiliev, was assumed in four dimensions and O (N) vector models in three dimensions, was demonstrated by Simone Giombi and Xi Yin ( For this they received the New Horizons in Physics Prize in 2017 ) and thus confirmed a conjecture by Igor Klebanov and Alexander Markowitsch Polyakow .

Applications

The equivalence u. A. when calculating the viscosity of a quark-gluon plasma , an extremely dense and hot state of matter that presumably prevailed a few fractions of a second after the big bang and can be generated in particle accelerators . The viscosity in the equivalent higher-dimensional space , which is very difficult to calculate in terms of quantum chromodynamics, corresponds to an absorption of gravitational waves through a black hole (more precisely defined higher-dimensional black holes, black branes), which can be calculated more easily using string theory . Dam Thanh Son and colleagues say for the ratio of shear viscosity and entropy density for the quark-gluon plasma

ahead (with as Boltzmann constant and as reduced Planck's quantum of action ). They also hypothesized that this is a lower limit for many quantum field theories at finite temperature. Experiments at the RHIC showed good agreement with the limit value (the quark-gluon plasma behaves like an ideal liquid, which corresponds to the lower limit value).

In solid-state physics, the correspondence should make it possible to determine the universal properties of high-temperature superconductors (based on the work of Subir Sachdev and others). There are also applications in hydrodynamics (dual description of the Navier-Stokes equations in the scaling limit case by Einstein gravity, Shiraz Minwalla ).

Literature and web links

Original work

Reviews and books:

Popular science and introductory presentations

Video lecture

Textbooks

  • B. Zwiebach: A First Course in String Theory , Cambridge University Press, Chapter 23: Strong interactions and AdS / CFT online (PDF; 34 MB)

Individual evidence

  1. ^ Leonard Susskind: The war over the black hole 1st edition, ISBN 978-3-518-42205-2 , p. 182.
  2. ↑ K.-H. Rehren: Algebraic Holography , May 1999, arxiv : hep-th / 9905179 .
  3. ↑ G. Policastro, DT Son, AO Starinets, From AdS / CFT correspondence to hydrodynamics, JHEP 0209: 043, 2002, Arxiv
  4. ↑ P. Kovtun, DT Son, AO Starinets, Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics, Phys. Rev. Lett., Vol. 94, 2005, p. 111601, Arxiv
  5. ↑ Son's homepage , popular scientific articles on his work
  6. ↑ Johanna Erdmenger, Max Planck Institute for Physics , quoted in: Ulf von Rauchhaupt : Absonderliche Fäden und Kugelige Kühe , Frankfurter Allgemeine Sonntagszeitung, July 29, 2012, p. 54.