Mathematical Topics Between Classical and Quantum Mechanics

This monograph draws on two traditions: the algebraic formulation of quantum mechanics as well as 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, which leads on to a discussion of the theory of quantization and the classical limit 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. Accessible to mathematicians with some prior knowledge of classical and quantum mechanics, and to mathematical physicists and theoretical physicists with some background in functional analysis.

Introductory Overview.- I. Observables and Pure States.- Observables.- Pure States.- From Pure States to Observables.- II. Quantization and the Classical Limit.- Foundations.- Quantization on Flat Space.- Quantization on Riemannian Manifolds.- III. Groups, Bundles, and Groupoids.- Lie Groups and Lie Algebras.- Internal Symmetries and External Gauge Fields.- Lie Groupoids and Lie Algebroids.- IV. Reduction and Induction.- Reduction.- Induction.- Applications in Relativistic Quantum Theory.- I Observables and Pure States.- 1 The Structure of Algebras of Observables.- 1.1 Jordan-Lie Algebras and C*-Algebras.- 1.2 Spectrum and Commutative C*-Algebras.- 1.3 Positivity, Order, and Morphisms.- 1.4 States.- 1.5 Representations and the GNS-Construction.- 1.6 Examples of C*-Algebras and State Spaces.- 1.7 Von Neumann Algebras.- 2 The Structure of Pure State Spaces.- 2.1 Pure States and Compact Convex Sets.- 2.2 Pure States and Irreducible Representations.- 2.3 Poisson Manifolds.- 2.4 The Symplectic Decomposition of a Poisson Manifold.- 2.5 (Projective) Hilbert Spaces as Symplectic Manifolds..- 2.6 Representations of Poisson Algebras.- 2.7 Transition Probability Spaces.- 2.8 Pure State Spaces as Transition Probability Spaces.- 3 From Pure States to Observables.- 3.1 Poisson Spaces with a Transition Probability.- 3.2 Identification of the Algebra of Observables.- 3.3 Spectral Theorem and Jordan Product.- 3.4 Unitarity and Leibniz Rule.- 3.5 Orthomodular Lattices.- 3.6 Lattices Associated with States and Observables.- 3.7 The Two-Sphere Property in a Pure State Space.- 3.8 The Poisson Structure on the Pure State Space.- 3.9 Axioms for the Pure State Space of a C*-Algebra.- II Quantization and the Classical Limit.- 1 Foundations.- 1.1 Strict Quantization of Observables.- 1.2 Continuous Fields of C*-Algebras.- 1.3 Coherent States and Berezin Quantization.- 1.4 Complete Positivity.- 1.5 Coherent States and Reproducing Kernels.- 2 Quantization on Flat Space.- 2.1 The Heisenberg Group and its Representations.- 2.2 The Metaplectic Representation.- 2.3 Berezin Quantization on Flat Space.- 2.4 Properties of Berezin Quantization on Flat Space.- 2.5 Weyl Quantization on Flat Space.- 2.6 Strict Quantization and Continuous Fields on Flat Space.- 2.7 The Classical Limit of the Dynamics.- 3 Quantization on Riemannian Manifolds.- 3.1 Some Affine Geometry.- 3.2 Some Riemannian Geometry.- 3.3 Hamiltonian Riemannian Geometry.- 3.4 Weyl Quantization on Riemannian Manifolds.- 3.5 Proof of Strictness.- 3.6 Commutation Relations on Riemannian Manifolds.- 3.7 The Quantum Hamiltonian and its Classical Limit.- III Groups, Bundles, and Groupoids.- 1 Lie Groups and Lie Algebras.- 1.1 Lie Algebra Actions and the Momentum Map.- 1.2 Hamiltonian Group Actions.- 1.3 Multipliers and Central Extensions.- 1.4 The (Twisted) Lie-Poisson Structure.- 1.5 Projective Representations.- 1.6 The Twisted Enveloping Algebra.- 1.7 Group C*-Algebras.- 1.8 A Generalized Peter-Weyl Theorem.- 1.9 The Group C* Algebra as a Strict Quantization.- 1.10 Representation Theory of Compact Lie Groups.- 1.11 Berezin Quantization of Coadjoint Orbits.- 2 Internal Symmetries and External Gauge Fields.- 2.1 Bundles.- 2.2 Connections.- 2.3 Cotangent Bundle Reduction.- 2.4 Bundle Automorphisms and the Gauge Group.- 2.5 Construction of Classical Observables.- 2.6 The Classical Wong Equations.- 2.7 The H-Connection.- 2.8 The Quantum Algebra of Observables.- 2.9 Induced Group Representations.- 2.10 The Quantum Wong Hamiltonian.- 2.11 From the Quantum to the Classical Wong Equations.- 2.12 The Dirac Monopole.- 3 Lie Groupoids and Lie Algebroids.- 3.1 Groupoids.- 3.2 Half-Densities on Lie Groupoids.- 3.3 The Convolution Algebra of a Lie Groupoid.- 3.4 Action *-Algebras.- 3.5 Representations of Groupoids.- 3.6 The C*-Algebra of a Lie Groupoid.- 3.7 Examples of Lie Groupoid C*-Algebras.- 3.8 Lie Algebroids.- 3.9 The Poisson Algebra of a Lie Algebroid.- 3.10 A Generalized Exponential Map.- 3.11 The Groupoid C*-Algebra as a Strict Quantization.- 3.12 The Normal Groupoid of a Lie Groupoid.- IV Reduction and Induction.- 1 Reduction.- 1.1 Basics of Constraints and Reduction.- 1.2 Special Symplectic Reduction.- 1.3 Classical Dual Pairs.- 1.4 The Classical Imprimitivity Theorem.- 1.5 Marsden-Weinstein Reduction.- 1.6 Kazhdan-Kostant-Sternberg Reduction.- 1.7 Proof of the Classical Transitive Imprimitivity Theorem.- 1.8 Reduction in Stages.- 1.9 Coadjoint Orbits of Nilpotent Groups.- 1.10 Coadjoint Orbits of Semidirect Products.- 1.11 Singular Marsden-Weinstein Reduction.- 2 Induction.- 2.1 Hilbert C*-Modules.- 2.2 Rieffel Induction.- 2.3 The C*-Algebra of a Hilbert C*-Module.- 2.4 The Quantum Imprimitivity Theorem.- 2.5 Quantum Marsden-Weinstein Reduction.- 2.6 Induction in Stages.- 2.7 The Imprimitivity Theorem for Gauge Groupoids.- 2.8 Covariant Quantization.- 2.9 The Quantization of Constrained Systems.- 2.10 Quantization of Singular Reduction.- 3 Applications in Relativistic Quantum Theory.- 3.1 Coadjoint Orbits of the Poincaré Group.- 3.2 Orbits from Covariant Reduction.- 3.3 Representations of the Poincaré Group.- 3.4 The Origin of Gauge Invariance.- 3.5 Quantum Field Theory of Photons.- 3.6 Classical Yang-Mills Theory on a Circle.- 3.7 Quantum Yang-Mills Theory on a Circle.- 3.8 Induction in Quantum Yang-Mills Theory on a Circle.- 3.9 Vacuum Angles in Constrained Quantization.- Notes.- I.- II.- III.- IV.- References.