Reduction Procedures in Classical and Quantum Mechanics

Abstract

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.

AMS-LaTeX, 35 pages. Expanded version of the Invited review talk delivered by G. Marmo at XXIst International Workshop On Differential Geometric Methods In Theoretical Mechanics, Madrid (Spain), 2006. To appear in Int. J. Geom. Methods in Modern Physics

References in corpus (7)

• Global Theory of Quantum Boundary Conditions and Topology Change
• Geometric Hamilton-Jacobi Theory
• Geometrization of Quantum Mechanics
• Generalized Reduction Procedure: Symplectic and Poisson Formalism
• Ray space Riccati evolution and geometric phases for N-level quantum systems
• Space-time orientations and Maxwell's equations
• Dynamical Aspects of Lie--Poisson Structures

Cited by in corpus (8)

• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Closed forms and multi-moment maps
• Symmetry reduction and superintegrable Hamiltonian systems
• A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities
• A geometric approach to a generalized virial theorem
• An extension of the Marsden-Ratiu reduction for Poisson manifolds
• Tangent bundle geometry from dynamics: application to the Kepler problem
• Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture