Fractals, coherent states and self-similarity induced noncommutative geometry

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.

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References in corpus (4)

• The mathematical basis for deterministic quantum mechanics
• Golden Quantum Oscillator and Binet-Fibonacci Calculus
• Cortical phase transitions, non-equilibrium thermodynamics and the time-dependent Ginzburg-Landau equation
• Dissipation and quantization for composite systems

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