Relationships between quantum and classical mechanics using the representation theory of the Heisenberg group

Introduction......Page 6
Classical Mechanics......Page 9
The Birth and Development of Quantum Mechanics......Page 10
Quantisation......Page 13
The Heisenberg Group and its Representations......Page 15
The Heisenberg Group and its Lie Algebra......Page 16
The Method of Orbits Applied to the Heisenberg group......Page 17
Induced Representations of the Heisenberg Group......Page 18
Relationships Between F2(Oh) and Other Hilbert Spaces......Page 21
The Stone-von Neumann Theorem......Page 25
Square Integrable Covariant Coherent States in F2(Oh)......Page 26
Observables in p-Mechanics......Page 28
p-Mechanical Brackets and the Time Evolution of Observables......Page 31
States and the Pictures of p-Mechanics......Page 32
States......Page 33
Time Evolution of States......Page 38
Eigenvalues and Eigenfunctions......Page 42
Coherent States and Creation/Annihilation Operators......Page 43
The Interaction Picture......Page 49
Relationships Between L2( Rn ) and Hh2......Page 50
The Rigged Hilbert Spaces Associated with Hh2 and F2(Oh)......Page 52
The Harmonic Oscillator......Page 54
The p-Mechanical Forced Oscillator: The Solution and Relation to Classical Mechanics......Page 56
A Periodic Force and Resonance......Page 60
The Interaction Picture of the Forced Oscillator......Page 61
Canonical Transformations in Classical Mechanics, Quantum Mechanics and p-Mechanics......Page 65
Canonical Transformations in Classical Mechanics......Page 66
Canonical Transformations in p-Mechanics......Page 67
The Metaplectic Representation for F2(Oh)......Page 70
Coupled Oscillators: An Application of p-Mechanical Linear Canonical Transformations......Page 76
Non-Linear Canonical Transformations......Page 79
Equations for Non-Linear Transformations Involving Hh2 States......Page 80
A Non-Linear Example......Page 86
The Kepler/Coulomb Problem......Page 95
The p-Mechanisation of the Kepler/Coulomb Problem......Page 96
The p-Dynamic Equation for the Kepler/Coulomb Problem......Page 98
The Kepler/Coulomb Problem in L2 ( R3 ) and F2(Oh)......Page 99
Spherical Polar Coordinates in p-Mechanics......Page 101
Transforming the Position Space......Page 107
The Klauder Coherent States for the Hydrogen atom......Page 112
A Hilbert Space for the Kepler/Coulomb Problem......Page 113
Generalisations......Page 116
Summary......Page 118
Possible Extensions......Page 119
Some Useful Formulae and Results......Page 121
Vector Fields and Differential Forms on  R2n......Page 122
Lie Groups and their Representations......Page 123
Induced Representations......Page 126
Distributions......Page 128