2003/06/30 by Wellington da Cruz
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.NT
published as Published in "FOCUS ON MATHEMATICAL PHYSICS RESEARCH", ed. by C. V. Benton (Nova Science Publishers, Inc) (2004), pp.177-192.) · Misprints corrected. Latex, 15 pages
arxiv created 2004/06/18 · arxiv updated 2009/11/30
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.