Introduction to Quantum Mechanics and the Quantum-Classical transition

Abstract

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.

Survey paper based on the lectures delivered at the XV International Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11-16, 2006. To appear in Publ. de la RSME

References in corpus (11)

• Geometric Quantum Mechanics
• Alternative commutation relations, star products and tomography
• Quantum Bi-Hamiltonian Systems
• Banach Lie-Poisson spaces and reduction
• Star products, duality and double Lie algebras
• Between classical and quantum
• Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics
• Duality symmetry for star products
• Alternative Structures and Bihamiltonian Systems
• A tomographic setting for quasi-distribution functions
• Alternative linear structures associated with regular Lagrangians. Weyl quantization and the Von Neumann uniqueness theorem