The Principles of Newtonian & Quantum Mechanics

CONTENTS......Page 18
FOREWORD BY BASIL HILEY......Page 8
PREFACE......Page 12
1 FROM KEPLER TO SCHRODINGER ... AND BEYOND......Page 25
1.1 Classical Mechanics......Page 26
1.2 Symplectic Mechanics......Page 30
1.3 Action and Hamilton-Jacobi\'s Theory......Page 35
1.4 Quantum Mechanics......Page 37
1.5 The Statistical Interpretation of w......Page 43
1.6 Quantum Mechanics in Phase Space......Page 46
1.7 Feynman\'s \"Path Integral\"......Page 49
1.8 Bohmian Mechanics......Page 51
1.9 Interpretations......Page 55
2.1 Maxwell\'s Principle and the Lagrange Form......Page 61
2.2 Hamilton\'s Equations......Page 73
2.3 Galilean Covariance......Page 82
2.4 Constants of the Motion and Integrable Systems......Page 89
2.5 Liouville\'s Equation and Statistical Mechanics......Page 94
3.1 Symplectic Matrices and Sp(n)......Page 101
3.2 Symplectic Invariance of Hamiitonian Flows......Page 104
3.3 The Properties of Sp(n)......Page 107
3.4 Quadratic Hamiltonians......Page 112
3.5 The Inhomogeneous Symplectic Group......Page 116
3.6 An Illuminating Analogy......Page 118
3.7 Gromov\'s Non-Squeezing Theorem......Page 123
3.8 Symplectic Capacity and Periodic Orbits......Page 132
3.9 Capacity and Periodic Orbits......Page 137
3.10 Cell Quantization of Phase Space......Page 142
4.1 Introduction......Page 151
4.2 The Fundamental Property of the Poincare-Cartan Form......Page 152
4.3 Free Symplectomorphisms and Generating Functions......Page 156
4.4 Generating Functions and Action......Page 161
4.5 Short-Time Approximations to the Action......Page 171
4.6 Lagrangian Manifolds......Page 180
4.7 The Phase of a Lagrangian Manifold......Page 185
4.8 Keller-Maslov Quantization......Page 192
5.1 Bohmian Motion and Half-Densities......Page 203
5.2 The Leray Index and the Signature Function*......Page 210
5.3 De Rham Forms......Page 225
5.4 Wave-Forms on a Lagrangian Manifold......Page 236
6.1 Introduction......Page 245
6.2 Free Symplectic Matrices and their Generating Functions......Page 249
6.3 The Metaplectic Group Mp(n)......Page 255
6.4 The Projections II and IIE......Page 261
6.5 The Maslov Index on Mp(n)......Page 266
6.6 The Cohomological Meaning of the Maslov Index*......Page 271
6.7 The Inhomogeneous Metaplectic Group......Page 277
6.8 The Metaplectic Group and Wave Optics......Page 282
6.9 The Groups Symp(n) and Ham(n)*......Page 284
7.1 Schrodinger\'s Equation for the Free Particle......Page 291
7.2 Van Vleck\'s Determinant......Page 301
7.3 The Continuity Equation for Van Vleck\'s Density......Page 304
7.4 The Short-Time Propagator......Page 308
7.5 The Case of Quadratic Hamiltonians......Page 312
7.6 Solving Schrodinger\'s Equation: General Case......Page 314
7.7 Metatrons and the Implicate Order......Page 324
7.8 Phase Space and Schrodinger\'s Equation......Page 337
A Symplectic Linear Algebra......Page 347
B The Lie-Trotter Formula for Flows......Page 351
C The Heisenberg Groups......Page 355
D The Bundle of S-Densities......Page 359
E The Lagrangian Grassmannian......Page 363
BIBLIOGRAPHY......Page 367
INDEX......Page 377