These notes are based on a course entitled "Symplectic Geometry and Geometric Quantization" taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.

This series is jointly published between the AMS and the Center for Pure and Applied Mathematics at the University of California at Berkeley (UCB CPAM).

Readership

Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Table of Contents

• Introduction: The harmonic oscillator
• The WKB method
• Symplectic manifolds
• Quantization in cotangent bundles
• The symplectic category
• Fourier integral operators
• Geometric quantization
• Algebraic quantization
• Densities (Appendix A)
• The method of stationary phase (Appendix B)
• Čech cohomology (Appendix C)
• Principal \({\mathbb{T}} _{\hbar}\) bundles (Appendix D)
• References