Fractal statistics, fractal index and fractons

The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge $c[\nu]$ and the particle-hole duality $\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value $\nu$ of the statistical parameter. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. We also establish a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.Comment: Latex, 7 pages, references update, To appear Proceedings Workshop on Geometrical Aspects Of Quantum Fields, 17 to 22 April 2000, State University of Londrina (Londrina, Parana, Brazil

A quantum-geometrical description of fracton statistics

We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$$$<$$$$h$$$$<$$$$2$, a fractal distribution function associated with a fractal von Neumann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational values of spin. This formulation can be considered in the context of the fractional quantum Hall effect-FQHE and number theory.Comment: Typos corrected, latex, 8 pages, Talk given at the 2nd International Londrina Winter School: Mathematical Methods in Physics, August, 26-30 (2002), Universidade Estadual de Londrina, Paran\'a, Brazil. Version to be published in Int. J. Mod. Phys. {\bf A}, (2003