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Tables of Contents for Maslov Classes, Metaplectic Representation and Lagrangian Quantization
Chapter/Section Title
Page #
Page Count
Introduction
9
5
Notaions
14
1
Chapter 1. Introduction to symplectic geometry
15
32
1.1 The standard symplectic space (Z,w)
15
3
1.2 The symplectic group Sp
18
6
1.3 The Lagrangian Grassmannian Lag of (Z,w)
24
4
1.4 Symplectic manifolds
28
6
1.5 Darboux' theorem and its variants
34
1
1.6 State and phase space
35
3
1.7 Hamiltonian Mechanics in (Z,w)
38
4
1.8 Hamiltonian Mechanics on Symplectic manifolds
42
3
Comments and references
45
2
Chapter 2. Maslov classes
47
34
2.1 The signature of a triple of Lagrangian planes
47
6
2.2 The cocycle property of the signature
53
1
2.3 The fundamental groups of Sp and Lag
54
4
2.4 The index u on Lag
58
6
2.5 Maslov Classes on Lag and Sp
64
3
2.6 Maslov Classes and q-symplectic geometry
70
6
2.7 Oriented and q-oriented Lagrangian manifolds
76
3
Comments and references
79
2
Chapter 3. The metaplectic representation of Sp2
81
38
3.1 The Fourier transform
81
2
3.2 Quadratic Fourier transforms
83
7
3.3 The metaplectic group Mp
90
11
3.4 Definition of the Maslov index on Mp
101
7
3.5 The relation between m and the Maslov index on Sp2
108
6
3.6 Reconstruction of the ALM index modulo 4
114
3
Comments and references
117
2
Chapter 4. Lagrangian quantization
119
28
4.1 The intrinsic Hilbert space
119
5
4.2 Metaplectic half-forms
124
4
4.3 The catalogue bundle
6
4.4 Lagrangian quantization
134
11
Comments and references
145
2
Chapter 5. Quantum Mechanics
147
33
5.1 A non-example of physical quantization
147
9
5.2 Classical interpretation of the wave function
156
7
5.3 The action of Hamiltonian flows on MHFs
163
4
5.4 Quasi-classical approximation
167
2
5.5 Metaplectic covariance
169
9
Comments and references
178
2
Bibliography
180
5
Subject index
185