Lectures on the Geometry of Quantization Sean Bates Alan Weinstein Department of Mathematics Department of Mathematics Columbia University University of California New York, NY 10027 USA Berkeley, CA 94720 USA
[email protected] [email protected] 1 Contents 1 Introduction: The Harmonic Oscillator 5 2 The WKB Method 8 2.1 Some Hamilton-Jacobi preliminaries . 8 2.2 The WKB approximation . 11 3 Symplectic Manifolds 17 3.1 Symplectic structures . 17 3.2 Cotangent bundles . 28 3.3 Mechanics on manifolds . 32 4 Quantization in Cotangent Bundles 36 4.1 Prequantization . 36 4.2 The Maslov correction . 41 4.3 Phase functions and lagrangian submanifolds . 46 4.4 WKB quantization . 56 5 The Symplectic Category 64 5.1 Symplectic reduction . 64 5.2 The symplectic category . 76 5.3 Symplectic manifolds and mechanics . 79 6 Fourier Integral Operators 83 6.1 Compositions of semi-classical states . 83 6.2 WKB quantization and compositions . 87 7 Geometric Quantization 93 7.1 Prequantization . 93 7.2 Polarizations and the metaplectic correction . 98 7.3 Quantization of semi-classical states . 109 8 Algebraic Quantization 111 8.1 Poisson algebras and Poisson manifolds . 111 8.2 Deformation quantization . 112 8.3 Symplectic groupoids . 114 A Densities 119 B The method of stationary phase 121 C Cechˇ cohomology 124 D Principal T~ bundles 125 2 Preface These notes are based on a course entitled “Symplectic geometry and geometric quantization” taught by Alan Weinstein at the University of California, Berkeley, in the fall semester of 1992 and again at the Centre Emile Borel (Institut Henri Poincar´e)in the spring semester of 1994.