Multi-dimensional Cantor sets in classical and quantum mechanics
type: Item
instance of: scholarly article
title: Multi-dimensional Cantor sets in classical and quantum mechanics
zbMATH Open document ID: 0758.58012
DOI: 10.1016/0960-0779(92)90010-K
published in: Chaos, Solitons and Fractals
publication date: 1992-09-27T00:00:00Z
review text: The author's summary: ``We give two different descriptions of an abstract \(n\)-dimensional dynamical system. First we use a Sierpinski space setting and subsequently we use a statistical cellular space setting. The results of the analysis elucidate certain universal behaviour which was observed in a wide category of cellular automata. The results further show that in four dimensions the phase space dynamics is Peano-like and resembles an Anosov diffeomorphism of a compact manifold which is dense and quasi- ergodic. The fractal dimension in this case is \(d_ c^{(4)}=3.981\cong 4\) and we conjecture that fully developed turbulence is related to \(d_ c^{(5)}=6.3\). The corresponding Shannon information entropy of the second analysis are \(S_ s^{(4)}=3.68\) and \(S_ s^{(5)}=6.12\). In the case of eight dimensional phase space both descriptions lead to almost identical numerical results. Possible implications of these theoretical results to physical spatio-temporal chaos and the reduction of complexity are discussed. In conclusion, the relevance of Cantor-like space-time for the Copenhagen interpretation of quantum mechanics and the connection to non-standard analysis and Boscovich covariance are touched upon.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 82B99
Mathematics Subject Classification ID: 81S99
Mathematics Subject Classification ID: 76F99
zbMATH DE Number: 64962
zbMATH Keywords: abstract \(n\)-dimensional dynamical system
zbMATH Keywords: Sierpinski space
zbMATH Keywords: statistical cellular space
zbMATH Keywords: turbulence
zbMATH Keywords: information entropy
zbMATH Keywords: Cantor-like space-time
zbMATH Keywords: Copenhagen interpretation
zbMATH Keywords: quantum mechanics
author: Mohamed Saladin El Naschie
reviewed by: Piotr Kosiński
MaRDI profile type: Publication
cites work: Q4023253
cites work: Q3994385
cites work: Q4118531
cites work: Q5732499
cites work: On the Susceptibility of Local Elastic Buckling to Chaos
cites work: Q3822882
cites work: On strange nonchaotic attractors and their dimensions
cites work: Differentiable dynamical systems
cites work: Statistical mechanics of cellular automata
cites work: Q5569065
cites work: Quantum mechanics and the possibility of a Cantorian space-time
cites work: Q4740120
cites work: The canard unchained or how fast/slow dynamical systems bifurcate
cites work: Nonstandard analysis - recent developments
full work available at URL: 0960-0779(92)90010-k
OpenAlex ID: W2142992109
description: scientific article; zbMATH DE number 64962