Mathematical Topics Between Classical and Quantum Mechanics

This Monograph Draws On Two Traditions: The Algebraic Formulation Of Quantum Mechanics And Quantum Field Theory, And The Geometric Theory Of Classical Mechanics. These Are Combined In A Unified Treatment Of The Theory Of Poisson Algebras Of Observables And Pure State Spaces With A Transition Probability. The Theory Of Quantization And The Classical Limit Is Discussed From This Perspective. A Prototype Of Quantization Comes From The Analogy Between The C*- Algebra Of A Lie Groupoid And The Poisson Algebra Of The Corresponding Lie Algebroid. The Parallel Between Reduction Of Symplectic Manifolds In Classical Mechanics And Induced Representations Of Groups And C*- Algebras In Quantum Mechanics Plays An Equally Important Role. Examples From Physics Include Constrained Quantization, Curved Spaces, Magnetic Monopoles, Gauge Theories, Massless Particles, And $theta$- Vacua. The Book Should Be Accessible To Mathematicians With Some Prior Knowledge Of Classical And Quantum Mechanics, To Mathematical Physicists And To Theoretical Physicists Who Have Some Background In Functional Analysis. Introductory Overview -- Observables And Pure States -- Quantization And The Classical Limit -- Groups, Bundles, And Groupoids -- Reduction And Induction -- Notes -- References -- Index. By N. P. Landsman.