Fractals, coherent states and self-similarity induced noncommutative geometry

TL;DR

This paper explores the mathematical connection between fractals, coherent states, and noncommutative geometry, revealing how microscopic quantum processes can give rise to macroscopic fractal structures.

Contribution

It introduces a novel framework linking fractal self-similarity with $q$-deformed algebra and coherent states, providing new insights into the geometric origin of fractals.

Findings

Fractals relate to noncommutative geometry and dissipation.

Self-similarity arises from coherent boson condensation.

Fractal properties emerge from microscopic quantum deformations.

Abstract

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the $q$-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.