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Fractal sets of dual topological quantum numbers | | Wellington da Cruz
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27 Jun 2003 | | Journal: | Published in "FOCUS ON MATHEMATICAL PHYSICS RESEARCH", ed. by C. V. Benton (Nova Science Publishers, Inc) (2004), pp.177-192.) | | Subject: | Mathematical Physics; Number Theory | math-ph math.MP math.NT | | Abstract: | The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval $1 < h < 2$ and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of the quantum paths of particles termed fractons which obey a fractal distribution function associated with a fractal von Neumann entropy. This way, the quantum Hall transitions satisfy some properties related to the Farey sequences of rational numbers and so our theoretical description of the FQHE establishes a connection between physics, fractal geometry and number theory. The FQHE as a convenient physical system for a possible prove of the Riemann hypothesis is suggested. | | Source: | arXiv, math-ph/0306071 | | Services: | Forum | Review | PDF | Favorites | |
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