Geometric Quantization in Action

O. Survey of Results.- Some Elementary Quantum Systems.- Examples of Group Representations in Physics.- Asymptotics in Statistical Mechanics.- More Spectral Geometry.- Statistical Mechanics and Representation Theory.- Transformation Groups in Physics.- Fiber Bundles.- Orbit Spaces in Lie Algebras.- Scattering Theory and Statistical Mechanics.- Quantum Field Theory.- 1. Representation Theory.- Basic Ideas of Representation Theory.- Induced Representations.- Schur and Peter-Weyl Theorems.- Lie Groups and Parallelization.- Spectral Theory and Representation Theory.- 2. Euclidean Group.- The Euclidean Group and Semidirect Products.- Fock Space, An Introduction.- 3. Geometry of Symplectic Manifolds.- Elementary Review of Lagrangian and Hamiltonian Mechanics: Notation.- Connections on Principal Bundles.- Riemannian Connections.- Geometry of Symplectic Manifolds.- Classical Mechanics and Symmetry Groups.- Homogeneous Symplectic Manifolds.- 4. Geometry of Contact Manifolds.- Contact Manifolds.- Almost Contact Metric Manifolds.- Dynamical Systems and Contact Manifolds.- Topology of Regular Contact Manifolds.- Infinitesimal Contact Transformations.- Homogeneous Contact Manifolds.- Contact Structures in the Sense of Spencer.- Homogeneous Complex Contact Manifolds.- 5. The Dirac Problem.- Derivations of Lie Algebras.- Geometric Quantization: An introduction.- The Dirac Problem.- Kostant and Souriau Approach.- 6. Geometry of Polarizations.- Polarizations.- Riemanrr-Roch for Polarizations.- Lie Algebra Polarizations.- Spin Structures, Metaplectic Structures and Square Root Bundles.- 7. Geometry of Orbits.- Orbit Theory.- Complete Integrability.- Morse Theory of Orbit Spaces.- 8. Fock Space.- Fock Space and Cohomology.- Nilpotent Lie Groups.- 9. Borel-Weil Theory.- Representation Theory for Compact Semisimple Lie Groups.- Borel-Wei! Theory.- Cocompact Nilradical Groups.- 10. Geometry of C-Spaces and R-Spaces.- The Geometry of C-Manifolds.- Kirillov Character Formula.- Geometry of R-Spaces.- Schubert Cell Decompositions.- 11. Geometric Quantization.- Geometric Quantization of Complex Manifolds.- Harmonic Oscillator.- The Kepler Problem - Hydrogen Atom.- Maslov Quantization.- 12. Principal Series Representations.- Representation Theory for Noncompact Semisimple Lie Groups. Part I: Principal Series Representations.- Applications to the Toda Lattice.- 13. Geometry of De Sitter Spaces.- De Sitter Spaces.- 14. Discrete Series Representations.- Representations of Noncompact Semisimple Lie Groups. Part II: Discrete Series.- 15. Representations and Automorphic Forms.- Geometric Quantization and Automorphic Forms.- Bounded Symmetric Domains and Holomorphic Discrete Series.- 16. Thermodynamics of Homogeneous Spaces.- Density Matrices and Partition Functions.- Epstein Zeta Functions.- Asymptotes of the Density Matrix.- Zeta Functions on Compact Lie Groups.- Ising Models.- 17. Quantum Statistical Mechanics.- Quantum Statistical Mechanics on Compact Symmetric Spaces.- Zeta Functions on Compact Lie Groups.- 18. Selberg Trace Theory.- The Selberg Trace Formula.- The Partition Function and the Length Spectra.- Noncompact Spaces with Finite Volume.- 19. Quantum Field Theory.- Applications to Quantum Field Theory.- Static Space Times and Periodization.- Examples of Zeta Functions in Quantum Field Theory.- 20. Coherent States and Automorphic Forms.- 20.1. Coherent States and Automorphic Forms.- References and Historical Comments.