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.
Author(s): Sean Bates, Alan Weinstein.
Year: 1997
Language: English
Pages: 141
1 Introduction: The Harmonic Oscillator......Page 11
2.1 Some Hamilton-Jacobi preliminaries......Page 14
2.2 The WKB approximation......Page 17
3.1 Symplectic structures......Page 23
3.2 Cotangent bundles......Page 34
3.3 Mechanics on manifolds......Page 38
4.1 Prequantization......Page 42
4.2 The Maslov correction......Page 47
4.3 Phase functions and lagrangian submanifolds......Page 52
4.4 WKB quantization......Page 62
5.1 Symplectic reduction......Page 70
5.2 The symplectic category......Page 82
5.3 Symplectic manifolds and mechanics......Page 85
6.1 Compositions of semi-classical states......Page 89
6.2 WKB quantization and compositions......Page 93
7.1 Prequantization......Page 99
7.2 Polarizations and the metaplectic correction......Page 104
7.3 Quantization of semi-classical states......Page 115
8.1 Poisson algebras and Poisson manifolds......Page 117
8.2 Deformation quantization......Page 118
8.3 Symplectic groupoids......Page 120
A Densities......Page 125
B The method of stationary phase......Page 127
C Cech cohomology......Page 130
D Principal Th bundles......Page 131
