Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Motivation and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3 Classical Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Quantum Theory of Large Systems . . . . . . . . . . . . . . . . . . . . . . . 16
2 Geometry of the State Space of Quantum Mechanics . . . . . . . . . . . . 21
2.1 Manifold Structure of PðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Quantum Mechanics as a Classical Hamiltonian Field Theory . . . . 28
3 Classical Mechanical Projections of QM . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Orbits of Lie Group Actions on PðHÞ . . . . . . . . . . . . . . . . . . . . . 35
3.2 Classical Phase Spaces from the Quantal State Space . . . . . . . . . . 38
3.3 Classical Mechanical Projections of Quantal Dynamics . . . . . . . . . 42
4 Examples of Classical Mechanical Projections . . . . . . . . . . . . . . . . . 49
4.1 The Heisenberg Group (CCR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Extension of CCR by a Quadratic Generator . . . . . . . . . . . . . . . . 57
4.3 Notes on Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Macroscopic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Multiple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Generalized Macroscopic Limits . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Dynamics of Quantum Mechanical Macroscopic Systems . . . . . . . . . 113
6.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Spin Systems with Polynomial Local Hamiltonians Q
N . . . . . . . . . 120
6.3 Time Evolution in Generalized Mean-Field Theories . . . . . . . . . . 138
6.4 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 An Example: The B.C.S. Model of Superconductivity . . . . . . . . . 160
ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Motivation and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3 Classical Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Quantum Theory of Large Systems . . . . . . . . . . . . . . . . . . . . . . . 16
2 Geometry of the State Space of Quantum Mechanics . . . . . . . . . . . . 21
2.1 Manifold Structure of PðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Quantum Mechanics as a Classical Hamiltonian Field Theory . . . . 28
3 Classical Mechanical Projections of QM . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Orbits of Lie Group Actions on PðHÞ . . . . . . . . . . . . . . . . . . . . . 35
3.2 Classical Phase Spaces from the Quantal State Space . . . . . . . . . . 38
3.3 Classical Mechanical Projections of Quantal Dynamics . . . . . . . . . 42
4 Examples of Classical Mechanical Projections . . . . . . . . . . . . . . . . . 49
4.1 The Heisenberg Group (CCR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Extension of CCR by a Quadratic Generator . . . . . . . . . . . . . . . . 57
4.3 Notes on Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Macroscopic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Multiple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Generalized Macroscopic Limits . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Dynamics of Quantum Mechanical Macroscopic Systems . . . . . . . . . 113
6.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Spin Systems with Polynomial Local Hamiltonians Q
N . . . . . . . . . 120
6.3 Time Evolution in Generalized Mean-Field Theories . . . . . . . . . . 138
6.4 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 An Example: The B.C.S. Model of Superconductivity . . . . . . . . . 160
ix