Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory

Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then, is that they can't see the problem. one day, perhaps you will fmd the final question. G. K. Chesterton, The Scandal of Father Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. Van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geo­ metry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical progmmming profit from homotopy theory; Lie algebras are relevant to fIltering; and prediction and electrical engineering can use Stein spaces.

Table of Contents:
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.