A quantum-geometrical description of the statistical laws of nature

TL;DR

This paper introduces a fractal quantum-geometrical framework for understanding the statistical laws of nature, linking fractons, Hausdorff dimensions, and the fractional quantum Hall effect, and predicts new phenomena based on topological and number-theoretic principles.

Contribution

It develops a novel fractal distribution function and entropy for fractons, connecting quantum paths, topological quantum numbers, and the fractional quantum Hall effect in a unified geometrical approach.

Findings

Fracton distribution functions are associated with Hausdorff dimensions.

Quantization of Hall resistance occurs in pairs of dual topological quantum numbers.

The approach predicts the fractional quantum Hall effect from fractal and topological considerations.

Abstract

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, where we discovered that the quantization of the Hall resistance occurs in pairs of dual topological quantum numbers, the filling factors. In this way, these quantum numbers get their topological character from the Hausdorff dimension associated with the fractal quantum path of such particles termed fractons. On the other hand, the universality classes of the quantum Hall transitions can be classified in terms of $h$. Another consequence of our approach, which is supported by symmetry principles, is the prediction of the FQHE. The connection between Physics and Number Theory appears naturally in this context.