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

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. Read more... 1. Introduction. 1.1. Philosophy. 1.2. Phase spaces, and the Dirac derivation. 1.3. Non-commutative algebraic geometry, and moduli of simple modules. 1.4. Dynamical structures. 1.5. Quantum fields and dynamics. 1.6. Classical quantum theory. 1.7. Planck's constants, and Fock space. 1.8. General quantum fields, Lagrangians and actions. 1.9. Grand picture. Bosons, fermions, and supersymmetry. 1.10. Connections and the generic dynamical structure. 1.11. Clocks and classical dynamics. 1.12. Time-space and space-times. 1.13. Cosmology, big bang and all that. 1.14. Interaction and non-commutative algebraic geometry. 1.15. Apology -- 2. Phase spaces and the Dirac derivation. 2.1. Phase spaces. 2.2. The Dirac derivation -- 3. Non-commutative deformations and the structure of the Moduli space of simple representations. 3.1. Non-commutative deformations. 3.2. The O-construction. 3.3. Iterated extensions. 3.4. Non-commutative schemes. Morphisms, Hilbert schemes, fields and strings -- 4. Geometry of time-spaces and the general dynamical law. 4.1. Dynamical structures. 4.2. Quantum fields and dynamics. 4.3. Classical quantum theory. 4.4. Planck's Constant(s) and Fock space. 4.5. General quantum fields, Lagrangians and actions. 4.6. Grand picture : Bosons, fermions, and supersymmetry. 4.7. Connections and the generic dynamical structure. 4.8. Clocks and classical dynamics. 4.9. Time-space and space-times. 4.10. Cosmology, big bang and all that -- 5. Interaction and non-commutative algebraic geometry. 5.1. Interactions. 5.2. Examples and some ideas