Mathematical Topics Between Classical and Quantum Mechanics

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.