Geometry of Time-Spaces: Non-Commutative Algebraic Geometry, Applied to Quantum Theory

Contents......Page 10
Preface......Page 8
1.1 Philosophy......Page 12
1.2 Phase Spaces, and the Dirac Derivation......Page 14
1.3 Non-commutative Algebraic Geometry, andModuli of Simple Modules......Page 15
1.4 Dynamical Structures......Page 16
1.5 Quantum Fields and Dynamics......Page 17
1.7 Planck’s Constants, and Fock Space......Page 20
1.8 General Quantum Fields, Lagrangians and Actions......Page 21
1.10 Connections and the Generic Dynamical Structure......Page 23
1.12 Time-Space and Space-Times......Page 24
1.14 Interaction and Non-commutative Algebraic Geometry......Page 25
1.15 Apology......Page 26
2.1 Phase Spaces......Page 28
2.2 The Dirac Derivation......Page 33
3.1 Non-commutative Deformations......Page 38
3.2 The O-construction......Page 40
3.3 Iterated Extensions......Page 42
3.4 Non-commutative Schemes......Page 43
3.4.1 Localization, Topology and the Scheme Structure on Simp(A)......Page 44
3.4.2 Completions of Simpn(A)......Page 53
3.5 Morphisms, Hilbert Schemes, Fields and Strings......Page 57
4.1 Dynamical Structures......Page 62
4.2 Quantum Fields and Dynamics......Page 63
4.3 Classical Quantum Theory......Page 69
4.4 Planck's Constant(s) and Fock Space......Page 71
4.5 General Quantum Fields, Lagrangians and Actions......Page 75
4.6 Grand Picture: Bosons, Fermions, and Supersymmetry......Page 80
4.7 Connections and the Generic Dynamical Structure......Page 87
4.8 Clocks and Classical Dynamics......Page 113
4.9 Time-space and Space-times......Page 114
4.10 Cosmology, Big Bang and All That......Page 131
5.1 Interactions......Page 136
5.2 Examples and Some Ideas......Page 139
Bibliography......Page 148
Index......Page 152