Symplectic Geometry And Quantum Mechanics

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ISBN: 9783764375744 | 3764375744
Cover: Hardcover
Copyright: 7/21/2006

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This book is devoted to a rather complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Preface

xiii

Introduction

xiii

Organization

xiv

Prerequisites

Bibliography

Acknowledgements

About the Author

xvi

Notation

xvii

Number sets

xvii

Classical matrix groups

xvii

Vector calculus

xviii

Function spaces and multi-index notation

xix

Combinatorial notation

Part I: Symplectic Geometry

Symplectic Spaces and Lagrangian Planes

Symplectic Vector Spaces

(8)

Generalities

(4)

Symplectic bases

(2)

Differential interpretation of σ

(2)

Skew-Orthogonality

(4)

Isotropic and Lagrangian subspaces

(1)

The symplectic Gram-Schmidt theorem

(3)

The Lagrangian Grassmannian

(4)

Lagrangian planes

(3)

The action of Sp(n) on Lag(n)

(1)

The Signature of a Triple of Lagrangian Planes

(8)

First properties

(3)

The cocycle property of τ

(1)

Topological properties of τ

(3)

The Symplectic Group

The Standard Symplectic Group

(11)

Symplectic matrices

(4)

The unitary group U(n)

(3)

The symplectic algebra

(2)

Factorization Results in Sp(n)

(12)

Polar and Cartan decomposition in Sp(n)

(4)

The ``pre-Iwasawa'' factorization

(3)

Free symplectic matrices

(5)

Hamiltonian Mechanics

(16)

Hamiltonian flows

(4)

The variational equation

(3)

The group Ham(n)

(3)

Hamiltonian periodic orbits

(5)

Multi-Oriented Symplectic Geometry

Souriau Mapping and Maslov Index

(8)

The Souriau mapping

(4)

Definition of the Maslov index

(2)

Properties of the Maslov index

(1)

The Maslov index on Sp(n)

(1)

The Arnol'd-Leray--Maslov Index

(10)

The problem

(4)

The Maslov bundle

(1)

Explicit construction of the ALM index

(4)

q-Symplectic Geometry

(11)

The identification Lag∞(n) = Lag(n) x Z

(2)

The universal covering Sp∞(n)

(4)

The action of Sp∞q(n) on Lag∞2q(n)

(4)

Intersection Indices in Lag(n) and Sp(n)

Lagrangian Paths

(5)

The strata of Lag(n)

(1)

The Lagrangian intersection index

(2)

Explicit construction of a Lagrangian intersection index

(2)

Symplectic Intersection Indices

100

(4)

The strata of Sp(n)

100

(1)

Construction of a symplectic intersection index

101

(1)

Example: spectral flows

102

(2)

The Conley--Zehnder Index

104

(19)

Definition of the Conley--Zehnder index

104

(2)

The symplectic Cayley transform

106

(2)

Definition and properties of ν(S∞)

108

(4)

Relation between ν and μlp

112

(11)

Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation

Lagrangian Manifolds and Quantization

Lagrangian Manifolds and Phase

123

(7)

Definition and examples

124

(1)

The phase of a Lagrangian manifold

125

(4)

The local expression of a phase

129

(1)

Hamiltonian Motions and Phase

130

(9)

The Poincare--Cartan Invariant

130

(3)

Hamilton--Jacobi theory

133

(3)

The Hamiltonian phase

136

(3)

Integrable Systems and Lagrangian Tori

139

(6)

Poisson brackets

139

(2)

Angle-action variables

141

(2)

Lagrangian tori

143

(2)

Quantization of Lagrangian Manifolds

145

(7)

The Keller--Maslov quantization conditions

145

(2)

The case of q-oriented Lagrangian manifolds

147

(2)

Waveforms on a Lagrangian Manifold

149

(3)

Heisenberg--Weyl and Grossmann--Royer Operators

152

(8)

Definition of the Heisenberg--Weyl operators

152

(2)

First properties of the operators T(z)

154

(2)

The Grossmann--Royer operators

156

(4)

Heisenberg Group and Weyl Operators

Heisenberg Group and Schrodinger Representation

160

(6)

The Heisenberg algebra and group

160

(3)

The Schrodinger representation of Hn

163

(3)

Weyl Operators

166

(8)

Basic definitions and properties

167

(3)

Relation with ordinary pseudo-differential calculus

170

(4)

Continuity and Composition

174

(11)

Continuity properties of Weyl operators

174

(5)

Composition of Weyl operators

179

(4)

Quantization versus dequantization

183

(2)

The Wigner--Moyal Transform

185

(11)

Definition and first properties

186

(3)

Wigner transform and probability

189

(3)

On the range of the Wigner transform

192

(4)

The Metaplectic Group

Definition and Properties of Mp(n)

196

(12)

Quadratic Fourier transforms

196

(3)

The projection πMp : Mp(n) → Sp(n)

199

(5)

Metaplectic covariance of Weyl calculus

204

(4)

The Metaplectic Algebra

208

(6)

Quadratic Hamiltonians

208

(1)

The Schrodinger equation

209

(3)

The action of Mp(n) on Gaussians: dynamical approach

212

(2)

Maslov Indices on Mp(n)

214

(8)

The Maslov index μ(S)

215

(5)

The Maslov indices μl(S)

220

(2)

The Weyl Symbol of a Metaplectic Operator

222

(16)

The operators Rν(S)

223

(4)

Relation with the Conley--Zehnder index

227

(11)

Part III: Quantum Mechanics in Phase Space

The Uncertainty Principle

States and Observables

238

(1)

Classical mechanics

238

(1)

Quantum mechanics

239

(1)

The Quantum Mechanical Covariance Matrix

239

(5)

Covariance matrices

240

(1)

The uncertainty principle

240

(4)

Symplectic Spectrum and Williamson's Theorem

244

(9)

Williamson normal form

244

(2)

The symplectic spectrum

246

(2)

The notion of symplectic capacity

248

(4)

Admissible covariance matrices

252

(1)

Wigner Ellipsoids

253

(9)

Phase space ellipsoids

253

(2)

Wigner ellipsoids and quantum blobs

255

(3)

Wigner ellipsoids of subsystems

258

(3)

Uncertainty and symplectic capacity

261

(1)

Gaussian States

262

(10)

The Wigner transform of a Gaussian

263

(2)

Gaussians and quantum blobs

265

(1)

Averaging over quantum blobs

266

(6)

The Density Operator

Trace-Class and Hilbert--Schmidt Operators

272

(10)

Trace-class operators

272

(7)

Hilbert-Schmidt operators

279

(3)

Integral Operators

282

(9)

Operators with L2 kernels

282

(3)

Integral trace-class operators

285

(3)

Integral Hilbert-Schmidt operators

288

(3)

The Density Operator of a Quantum State

291

(13)

Pure and mixed quantum states

291

(5)

Time-evolution of the density operator

296

(2)

Gaussian mixed states

298

(6)

A Phase Space Weyl Calculus

Introduction and Discussion

304

(6)

Discussion of Schrodinger's argument

304

(3)

The Heisenberg group revisited

307

(2)

The Stone-von Neumann theorem

309

(1)

The Wigner Wave-Packet Transform

310

(7)

Definition of UΦ

310

(4)

The range of UΦ

314

(3)

Phase-Space Weyl Operators

317

(7)

Useful intertwining formulae

317

(2)

Properties of phase-space Weyl operators

319

(2)

Metaplectic covariance

321

(3)

Schrodinger Equation in Phase Space

324

(7)

Derivation of the equation (10.39)

324

(1)

The case of quadratic Hamiltonians

325

(2)

Probabilistic interpretation

327

(4)

Conclusion

331

(2)

Classical Lie Groups

General Properties

333

(2)

The Baker--Campbell--Hausdorff Formula

335

(1)

One-parameter Subgroups of GL(m, R)

335

(7)

Covering Spaces and Groups

Pseudo-Differential Operators

The Classes Smρδ, Lmρδ

342

(1)

Composition and Adjoint

342

(3)

Basics of Probability Theory

Elementary Concepts

345

(2)

Gaussian Densities

347

(2)

Solutions to Selected Exercises

349

(6)

Bibliography

355

(10)

Index

365

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Symplectic Geometry And Quantum Mechanics

Summary

Table of Contents

Supplemental Materials