Vll l

Contents

Chapter 3. Hamiltonian Vector Fields and the Poisson Bracket 71

§3.1. Preliminaries 71

§3.2. Hamiltonian systems 74

§3.3. Poisson brackets 79

§3.4. Contact manifolds 85

Chapter 4. The Moment Map 93

§4.1. Definitions 93

§4.2. Constructions and examples 97

§4.3. Reduction of phase spaces by the consideration of symmetry 104

Chapter 5. Quantization 111

§5.1. Homogeneous quadratic polynomials and 5^ H I

§5.2. Polynomials of degree 1 and the Heisenberg group 114

§5.3. Polynomials of degree 2 and the Jacobi group 120

§5.4. The Groenewold-van Hove theorem 124

§5.5. Towards the general case 128

Appendix A. Differentiable Manifolds and Vector Bundles 135

§A.l. Differentiable manifolds and their tangent spaces 135

§A.2. Vector bundles and their sections 144

§A.3. The tangent and the cotangent bundles 146

§A.4. Tensors and differential forms 150

§A.5. Connections 158

Appendix B. Lie Groups and Lie Algebras 163

§B.l. Lie algebras and vector fields 163

§B.2. Lie groups and invariant vector fields 165

§B.3. One-parameter subgroups and the exponent map 167

Appendix C. A Little Cohomology Theory 171

§C.l. Cohomology of groups 171

§C2. Cohomology of Lie algebras 173

§C3. Cohomology of manifolds 174

Appendix D. Representations of Groups 177

§D.l. Linear representations 177

§D.2. Continuous and unitary representations 179

§D.3. On the construction of representations 180