Symplectic Non-Squeezing Theorems, Quantization of Integrable Systems, and Quantum Uncertainty

Abstract

The ground energy level of an oscillator cannot be zero because of Heisenberg's uncertainty principle. We use methods from symplectic topology (Gromov's non-squeezing theorem, and the existence of symplectic capacities) to analyze and extend this heuristic observation to Liouville-integrable systems, and to propose a topological quantization scheme for such systems, thus extending previous results of ours.