Login | Register

A model of the holographic principle: Randomness and additional dimension

Title:

A model of the holographic principle: Randomness and additional dimension

Boyarsky, Abraham, GÓRA, PAWEŁ and Proppe, Harald (2010) A model of the holographic principle: Randomness and additional dimension. Physics Letters A, 374 (3). pp. 435-438. ISSN 03759601

[img]
Preview
Text (application/pdf)
gora2010.pdf - Accepted Version
128kB

Official URL: http://dx.doi.org/10.1016/j.physleta.2009.11.005

Abstract

In recent years an idea has emerged that a system in a 3-dimensional space can be described from an information point of view by a system on its 2-dimensional boundary. This mysterious correspondence is called the Holographic Principle and has had profound effects in string theory and our perception of space–time. In this note we describe a purely mathematical model of the Holographic Principle using ideas from nonlinear dynamical systems theory. We show that a random map on the surface S2 of a 3-dimensional open ball B has a natural counterpart in B, and the two maps acting in different dimensional spaces have the same entropy. We can reverse this construction if we start with a special 3-dimensional map in B called a skew product. The key idea is to use the randomness, as imbedded in the parameter of the 2-dimensional random map, to define a third dimension. The main result shows that if we start with an arbitrary dynamical system in B with entropy E we can construct a random map on S2 whose entropy is arbitrarily close to E.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Boyarsky, Abraham and GÓRA, PAWEŁ and Proppe, Harald
Journal or Publication:Physics Letters A
Date:2010
Digital Object Identifier (DOI):10.1016/j.physleta.2009.11.005
Keywords:Holographic principle; Random maps; Skew products; Entropy
ID Code:976829
Deposited By: DANIELLE DENNIE
Deposited On:29 Jan 2013 14:21
Last Modified:18 Jan 2018 17:43

References:

[1] L.M. Abramov, V.A. Rohlin Vestnik Leningrad. Univ., 17 (7) (1962), p. 5 (Russian. English summary), MR0140660 (25 #4076)

[2] A. Boyarsky, W. Byers, P. Gauthier Nonlinear Anal., 11 (11) (1987), p. 1317 MR0915528 (89h:58111)

[3] Abraham Boyarsky, Paweł Góra J. Appl. Math. Stoch. Anal., 2 (2004), p. 137 MR2079262 (2006b:37066)

[4] Kifer Yuri Ergodic Theory of Random Transformations Progress in Probability and Statistics, vol. 10Birkhäuser Boston, Inc., Boston, MA (1986) MR0884892 (89c:58069)

[5] Yuri Kifer, Pei-Dong Liu Random Dynamics. Handbook of Dynamical Systems, vol. 1B Elsevier, B.V., Amsterdam (2006) pp. 379–499, MR2186245 (2008a:37002)

[6] Juan Maldacena Int. J. Theoretical Phys., 38 (4) (1999), p. 1113

[7] F.R. Marotto Chaos Solitons Fractals, 25 (1) (2005), p. 25 MR2123626 (2005k:37067)

[8] Martin Bojowold Sci. Amer., 299 (4) (2008), p. 44

[9] T. Morita Osaka J. Math., 22 (1985), p. 481

[10] L. Susskind J. Math. Phys., 36 (1995), p. 63

[11] G. 't Hooft Dimensional reduction in quantum gravity arXiv:gr-qc/9310026v2 (1993)

[12] G. 't Hooft The holographic principle arXiv:hep-th/0003004 (2000)
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Research related to the current document (at the CORE website)
- Research related to the current document (at the CORE website)
Back to top Back to top