2002/07/24 by Maurice A. de Gosson, Maurice de Gosson · 22 citations
Mathematics · Physics and Astronomy · #Classical mechanics #Computer science #Constant (computer programming) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical physics #Mathematics #Phase space #Philosophy #Physics #Planck constant #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Semiclassical physics #Simple (philosophy) #Symplectic geometry #math.DS #math.SG
paper · pdf · doi:10.1088/0305-4470/35/32/305
published in Journal of Physics A Mathematical and General 35(32), 6825-6851 (Institute of Physics) · no figures. to appear in J. Phys. Math A. (2002)
arxiv created 2002/07/24 · openalex publication_date 2002/07/30 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We propose a theory of semiclassical mechanics in phase space based on the notion of quantized symplectic area. The definition of symplectic area makes use of a deep topological property of symplectic mappings, known as the 'principle of the symplectic camel' which places stringent conditions on the global geometry of Hamiltonian mechanics. Following this principle, symplectic mappings—and hence Hamiltonian flows—are much more rigid than Liouville's theorem suggests. The dynamical objects of our semiclassical theory are 'waveforms', whose definition requires the notion of square root of de Rham forms. The arguments of these square roots are calculated by using the properties of a generalized Maslov index. The motion of waveforms is determined by Hamiltonian mechanics, and the local expressions of these moving waveforms on configuration space are the usual approximate solutions of WKB-Maslov theory.