On the Classical and the Quantum Mechanics in a Magnetic Field

This paper studies the relationship between the classical mechanics in a magnetic field and its quantized system. The phase space for classical motion in a magnetic field is derived through the Marsden-Weinstein reduction starting with a principal U(1)-bundle. The quantized system associated to the classical system is defined as the Bochner-Laplacian on a line bundle associated to the principal bundle. In this context we obtain a generalization of Helton's theorem which gives a characterization of the periodicity of the classical trajectories by the spectrum of the associated quantum Hamiltonian (the Bochner-Laplacian). 1991 Mathematics Subject Classification. Primary 58G25: Secondary 58F05

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。