Table of Contents

Preface to the First Edition xv

Preface to the Second Edition xxi

Foreword Basil Hiley xxiii

1 From Kepler to Schrödinger… and Beyond 1

1.1 Classical Mechanics 2

1.1.1 Newton's Laws and Mach's Principle 3

1.1.2 Mass, Force, and Momentum 5

1.2 Symplectic Mechanics 6

1.2.1 Hamilton's Equations 6

1.2.2 Gauge Transformations 8

1.2.3 Hamiltonian Fields and Flows 8

1.2.4 The "Symplectization of Science" 9

1.3 Action and Hamilton-Jacobi's Theory 12

1.3.1 Action 12

1.3.2 Hamilton-Jacobi's Equation 13

1.4 Quantum Mechanics 14

1.4.1 Matter Waves 15

1.4.2 "If There Is a Wave, There Must Be a Wave Equation!" 17

1.4.3 Schrodinger's Quantization Rule and Geometric Quantization 18

1.5 The Statistical Interpretation of ψ 20

1.5.1 Heisenberg's Inequalities 20

1.6 Quantum Mechanics in Phase Space 23

1.6.1 Schrödinger's "firefly" Argument 23

1.6.2 The Symplectic Camel 24

1.7 Feynman's "Path Integral" 26

1.7.1 The "Sum Over All Paths" 26

1.7.2 The Metaplectic Group 27

1.8 Bohmian Mechanics 28

1.8.1 Quantum Motion: The Bell-DGZ Theory 29

1.8.2 Bohm's Theory 30

1.9 Interpretations 32

1.9.1 Epistemology or Ontology? 32

1.9.2 The Copenhagen Interpretation 33

1.9.3 The Bohmian Interpretation 35

1.9.4 The Platonic Point of View 36

2 Newtonian Mechanics 37

2.1 Maxwell's Principle and the Lagrange Form 37

2.1.1 The Hamilton Vector Field 38

2.1.2 Force Fields 38

2.1.3 Statement of Maxwell's Principle 40

2.1.4 Magnetic Monopoles and the Dirac String 42

2.1.5 The Lagrange Form 44

2.1.6 AT-Particle Systems 47

2.2 Hamilton's Equations 50

2.2.1 The Poincaré-Cartan Form and Hamilton's Equations 51

2.2.2 Hamiltonians for N-Particle Systems 54

2.2.3 The Transformation Law for Hamilton Vector Fields 56

2.2.4 The Suspended Hamiltonian Vector Field 57

2.3 Galilean Covariance 59

2.3.1 Inertial Frames 59

2.3.2 The Galilean Group Gal(3) 60

2.3.3 Galilean Covariance of Hamilton's Equations 62

2.4 Constants of the Motion and Integrable Systems 66

2.4.1 The Poisson Bracket 66

2.4.2 Constants of the Motion and Liouville's Equation 67

2.4.3 Constants of the Motion in Involution 69

2.5 Liouville's Equation and Statistical Mechanics 72

2.5.1 Liouville's Condition 73

2.5.2 Marginal Probabilities 74

2.5.3 Distributional Densities: An Example 76

3 The Symplectic Group 79

3.1 Symplectic Matrices and Sp(n) 79

3.2 Symplectic Invariance of Hamiltonian Flows 82

3.2.1 Notations and Terminology 83

3.2.2 Proof of the Symplectic Invariance of Hamiltonian Flows 84

3.2.3 Another Proof of the Symplectic Invariance of Flows∗ 85

3.3 The Properties of Sp(n) 86

3.3.1 The Subgroups U(n) and O(n) of Sp(n) 87

3.3.2 The Lie Algebra sp(n) 88

3.3.3 Sp(n) as a Lie Group 89

3.4 Quadratic Hamiltonians 91

3.4.1 The Linear Symmetric Triatomic Molecule 92

3.4.2 Electron in a Uniform Magnetic Field 93

3.4.3 Small Oscillations Near Equilibrium 95

3.5 The Inhomogeneous Symplectic Group 98

3.5.1 Galilean Transformations and ISp(n) 99

3.6 An Illuminating Analogy 100

3.6.1 The Optical Hamiltonian 100

3.6.2 Paraxial Optics 102

3.7 Gromov's Non-Squeezing Theorem 105

3.7.1 Liouville's Theorem Revisited 107

3.7.2 Gromov's Theorem 109

3.7.3 The Uncertainty Principle in Classical Mechanics 112

3.8 Symplectic Capacity and Periodic Orbits 114

3.8.1 The Capacity of an Ellipsoid 117

3.8.2 Symplectic Area and Volume 119

3.9 Capacity and Periodic Orbits 120

3.9.1 Periodic Hamiltonian Orbits 121

3.9.2 Action of Periodic Orbits and Capacity 123

3.10 Phase Space Quantization and Quantum Blobs 124

3.10.1 Stationary States of Schrödinger's Equation 125

3.10.2 Quantum Blobs and the Minimum Capacity Principle 127

3.10.3 Quantization of the N-Dimensional Harmonic Oscillator 128

3.10.4 Quantum Blobs and the Uncertainty Principle 133

4 Action and Phase 137

4.1 Introduction 137

4.2 The Fundamental Property of the Poincaré-Cartan Form 138

4.2.1 Helmholtz's Theorem: The Case n = 1 139

4.2.2 Helmholtz's Theorem: The General Case 140

4.3 Free Symplectomorphisms and Generating Functions 142

4.3.1 Generating Functions 144

4.3.2 Optical Analogy: The Eikonal 147

4.4 Generating Functious and Action 148

4.4.1 The Generating Function Determined by H 148

4.4.2 Action vs. Generating Function 152

4.4.3 Gauge Transformations and Generating Functions 153

4.4.4 Solving Hamilton's Equations with W 154

4.4.5 The Cauchy Problem for Hamilton-Jacobi's Equation 156

4.5 Short-Time Approximations to the Action 158

4.5.1 The Case of a Scalar Potential 159

4.5.2 One Particle in a Gauge (A, U) 162

4.5.3 Many Particle Systems in a Gauge (A, U) 165

4.6 Lagrangian Manifolds 167

4.6.1 Definitions and Basic Properties 167

4.6.2 Lagrangian Manifolds in Mechanics 170

4.7 The Phase of a Lagrangian Manifold 172

4.7.1 The Phase of an Exact Lagrangian Manifold 172

4.7.2 The Universal Covering of a Manifold∗ 175

4.7.3 The Phase: General Case 176

4.7.4 Phase and Hamiltonian Motion 177

4.8 Keller-Maslov Quantization 179

4.8.1 The Maslov Index for Loops 179

4.8.2 Quantization of Lagrangian Manifolds 184

4.8.3 Illustration: The Plane Rotator 186

5 Semi-Classical Mechanics 191

5.1 Bohmian Motion and Half-Densities 192

5.1.1 Wave-Forms on Exact Lagrangian Manifolds 192

5.1.2 Semi-Classical Mechanics 194

5.1.3 Wave-Forms: Introductory Example 196

5.2 The Leray Index and the Signature Function∗ 198

5.2.1 Cohomological Notations 198

5.2.2 The Leray Index: n = 1 199

5.2.3 The Leray Index: General Case 201

5.2.4 Properties of the Leray Index 207

5.2.5 More on the Signature Function 210

5.2.6 The Reduced Leray Index 212

5.3 De Rhan Forms 213

5.3.1 Volumes and their Absolute Values 214

5.3.2 Construction of de Rham Forms on Manifolds 216

5.3.3 De Rham Forms on Lagrangian Manifolds 220

5.4 Wave-Forms on a Lagrangian Manifold 225

5.4.1 Definition of Wave Forms 225

5.4.2 The Classical Motion of Wave-Forms 228

5.4.3 The Shadow of a Wave-Form 230

6 Metaplectic Group, Maslov Index, and Quantization 235

6.1 Introduction 235

6.1.1 Could Schrodinger have Done it Rigorously? 235

6.1.2 Schrödinger's Idea 236

6.1.3 Sp(n)'s "Big Brother" Mp(n) 237

6.2 Free Symplertic Matrices and their Generating Functions 239

6.2.1 Free Symplectic Matrices 240

0.2.2 The Case of Affine Symplectomorphisms 242

6.2.3 The Generators of Sp(n) 243

6.3 The Metaplectic Group Mp(n) 246

6.3.1 Quadratic Fourier Transforms 246

6.3.2 The Operators M_L,m and VP 248

6.3.3 The Metaplectic Group and Wave Optics 251

6.4 The Projections II and II^ε 253

6.4.1 Construction of the Projection II 254

6.4.2 The Covering Groups Mp^ε(n) 257

6.5 The Maslov Index on Mp(n) 259

6.5.1 Maslov Index:. a "Simple" Example 260

6.5.2 Definition of the Maslov Index on Mp(n) 262

6.6 The Cohomological Meaning of the Maslov Index∗ 264

6.6.1 Group Cocycles on Sp(n) 265

6.6.2 The Fundamental Property of m(.) 267

6.7 The Inhomogeneous Metaplectic Group 270

6.7.1 The Heisenberg Group 271

6.7.2 The Group IMp 273

6.7.3 The Heisenberg-Weyl Operators 275

6.8 The Groups Symp(n) and Ham(n)∗ 277

6.8.1 A Topological Property of Symp(n) 277

6.8.2 The Group Ham(n) of Hamiltonian Symplectomorphisms 278

6.9 Canonical Quantization 281

6.9.1 The Ordering Problem for Monomials 284

6.9.2 The Case of General Observables 287

6.9.3 The Groenewold-van Hove Theorem 292

7 Schrödinger's Equation and the Metatron 295

7.1 Schrödinger's Equation for the Free Particle 296

7.1.1 The Free Particle's Phase 296

7.1.2 The Free Particle Propagator 297

7.1.3 An Explicit Expression for G 299

7.1.4 The Metaplectic Representation of the Free Flow 301

7.1.5 More Quadratic Hamiltonians 302

7.2 Van Vleck's Determinant 305

7.2.1 Trajectory Densities 305

7.2.2 The Continuity Equation for van Vleek's Density 308

7.2.3 The Continuity Equation 310

7.3 The Short-Time Propagator 312

7.3.1 Properties of the Short-Time Propagator 313

7.4 The Case of Quadratic Hamiltonians 316

7.4.1 Exact Green Function 316

7.4.2 Exact Solutions of Schrödinger's Equation 317

7.5 Solving Schrodinger's Equation: General Case 319

7.5.1 The Short-Time Propagator and Causality 319

7.5.2 Statement of the Main Theorem 321

7.5.3 The Formula of Stationary Phase 322

7.5.4 Two Lemmas - and the Proof 324

7.6 Metatrons and the Implicate Order 329

7.6.1 Unfolding and Implicate Order 330

7.6.2 Prediction and Retrodiction 331

7.6.3 The Lie-Trotter Formula for Flows 333

7.6.4 The "Unfolded" Metatron 336

7.6.5 Quantum Zeno Effect and the Mott Paradox 340

7.6.6 The Generalized Metaplectic Representation 341

7.7 Phase Space and Schrödinger's Equation 343

7.7.1 Mixed Representations in Quantum Mechanics 347

7.7.2 Complementarity and the Implicate Order 351

A Symplectic Linear Algebra 355

B The Lie-Trotter Formula for Flows 359

B.1 The Lie-Trotter Formula for Vector Fields 360

C The Heisenberg Groups 363

D The Bundle of s-Densities 367

E The Lagrangian Grassmannian 371

E.1 Lag(n) as a Set of Matrices 371

E.1.1 The Universal Coverings U∞ (n,C) and W∞ (n,C) 374

Bibliography 377

Index 391