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Quantum algorithmic integrability: the metaphor of classical polygonal billiards
1International Center for the Study of Dynamical Systems, Universita della Insubria, via Lucini 3, 22100 Como, Italy; and Istituto Nazionale di Fisica della Materia, Unita di Milano; and Istituto Nazionale di Fisica Nucleare, sezione di Milan.
Summary
Algorithmic complexity in polygonal billiards mirrors quantum system quantization. As sides increase, billiard complexity scales similarly to quantum systems, offering insights into integrable and chaotic dynamics.
Area of Science:
- Mathematical Physics
- Dynamical Systems Theory
- Quantum Mechanics
Background:
- Polygonal billiards offer a discrete model for studying complex dynamics.
- The transition from polygonal to curved billiards relates to the semiclassical limit in quantum mechanics.
Purpose of the Study:
- To investigate the algorithmic complexity of motion in polygonal billiards.
- To establish an equivalence between billiard complexity and quantum system quantization.
- To analyze scaling relations with respect to the number of sides in billiard systems.
Main Methods:
- Analysis of symbolic trajectories in polygonal billiards.
- Comparison of scaling relations with those in quantum systems.
- Examination of polygonal approximations of the circle (integrable) and stadium (chaotic).
Main Results:
- The average complexity of symbolic trajectories in polygonal billiards exhibits scaling relations analogous to quantum systems.
- This complexity scales with the number of sides, mirroring the variation of a semiclassical parameter.
Conclusions:
- Polygonal billiards serve as a valuable model for understanding quantum quantization.
- The study provides paradigms for the quantization of both integrable and chaotic systems through billiard approximations.