Geometry of time-spaces : non-commutative algebraic geometry, applied to quantum theory

Title:

Personal Author:

Laudal, Olav Arnfinn

Physical Description:

vii, 143 pages : illustrations ; 24 cm.

ISBN:

9789814343343

Subject Term:

Geometry, Algebraic.

Noncommutative differential geometry

Quantum theory

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010280464	QC20.7.A37 L38 2011	Open Access Book	Book	Searching... Unknown

This is a monograph about non-commutative algebraic geometry, and its application to physics. The main mathematical inputs are the non-commutative deformation theory, moduli theory of representations of associative algebras, a new non-commutative theory of phase spaces, and its canonical Dirac derivation. The book starts with a new definition of time, relative to which the set of mathematical velocities form a compact set, implying special and general relativity. With this model in mind, a general Quantum Theory is developed and shown to fit with the classical theory. In particular the "toy"-model used as example, contains, as part of the structure, the classical gauge groups u(1), su(2) and su(3), and therefore also the theory of spin and quarks, etc.

Preface	p. vii
1 Introduction	p. 1
1.1 Philosophy	p. 1
1.2 Phase Spaces, and the Dirac Derivation	p. 3
1.3 Non-commutative Algebraic Geometry, and Moduli of Simple Modules	p. 4
1.4 Dynamical Structures	p. 5
1.5 Quantum Fields and Dynamics	p. 6
1.6 Classical Quantum Theory	p. 9
1.7 Planck's Constants, and Fock Space	p. 9
1.8 General Quantum Fields, Lagrangians and Actions	p. 10
1.9 Grand Picture. Bosons, Fermions, and Supersymmetry	p. 12
1.10 Connections and the Generic Dynamical Structure	p. 12
1.11 Clocks and Classical Dynamics	p. 13
1.12 Time-Space and Space-Times	p. 13
1.13 Cosmology, Big Bang and All That	p. 14
1.14 Interaction and Non-commutative Algebraic Geometry	p. 14
1.15 Apology	p. 15
2 Phase Spaces and the Dirac Derivation	p. 17
2.1 Phase Spaces	p. 17
2.2 The Dirac Derivation	p. 22
3 Non-commutative Deformations and the Structure of the Moduli Space of Simple Representations	p. 27
3.1 Non-commutative Deformations	p. 27
3.2 The O-construction	p. 29
3.3 Iterated Extensions	p. 31
3.4 Non-commutative Schemes	p. 32
3.4.1 Localization, Topology and the Scheme Structure on Simp(A)	p. 33
3.4.2 Completions of Simp n (A)	p. 42
3.5 Morphisms, Hilbert Schemes, Fields and Strings	p. 46
4 Geometry of Time-spaces and the General Dynamical Law	p. 51
4.1 Dynamical Structures	p. 51
4.2 Quantum Fields and Dynamics	p. 52
4.3 Classical Quantum Theory	p. 58
4.4 Planck's Constant(s) and Fock Space	p. 60
4.5 General Quantum Fields, Lagrangians and Actions	p. 64
4.6 Grand Picture: Bosons, Fermions, and Supersymmetry	p. 69
4.7 Connections and the Generic Dynamical Structure	p. 76
4.8 Clocks and Classical Dynamics	p. 102
4.9 Time-space and Space-times	p. 103
4.10 Cosmology, Big Bang and All That	p. 120
5 Interaction and Non-commutative Algebraic Geometry	p. 125
5.1 Interactions	p. 125
5.2 Examples and Some Ideas	p. 128
Bibliography	p. 137
Index	p. 141

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