Mathematical Models in Science

Cover Page
Title Page
Copyright
Dedication
Acknowledgment
Contents
1. Introduction
    1.1 Philosophy
    1.2 Mathematical Models
    1.3 Geometry of the Space of Models
    1.4 Cosmology
    1.5 Organization of the Work: Leitfaden
2. Dynamics
    2.1 The Phase Space Functor
        2.1.1 First properties
        2.1.2 The deformation functor of representations
        2.1.3 Blow-ups and desingularizations
        2.1.4 Chern classes
    2.2 The Iterated Phase Space Functor Ph∗ and the Dirac Derivation
        2.2.1 Formal curves of representations
    2.3 The Generalized de Rham Complex
        2.3.1 Excursion into the Jacobian conjecture
3. Non-Commutative Algebraic Geometry
    3.1 Moduli of Representations
    3.2 Moduli of Simple Modules
        3.2.1 Evolution in the moduli of simple modules
    3.3 Non-Commutative Deformations of Swarms
4. The Dirac Derivation and Dynamical Structures
    4.1 Dynamical Structures
    4.2 Gauge Groups and Invariant Theory
        4.2.1 The global gauge group and invariant theory
        4.2.2 The local gauge group
    4.3 The Generic Dynamical Structures Associated to a Metric
        4.3.1 The commutative case, metrics, and gravitation
        4.3.2 The Lie algebra of isometries
    4.4 Metrics, Gravitation, and Energy
        4.4.1 The case of subspaces, spectral triples
        4.4.2 Relations to Clifford algebras
    4.5 Potentials and the Classical Gauge Invariance
        4.5.1 Infinitesimal structure on Rep(C(σg))
        4.5.2 Physics and the Chern–Simons class
    4.6 A Generalized Yang–Mills Theory
    4.7 Reuniting GR, YM, and General QFT
    4.8 Family of Representations versus Family of Metrics
5. Time–Space and Space–Times
    5.1 The Cylindrical Coordinates, Newton, and Kepler
    5.2 Thermodynamics, the Heat Equation and Navier–Stokes
6. Entropy
    6.1 The Classical Commutative Case
    6.2 The General Case
    6.3 Representations of Ph∞
7. Cosmology, Cosmos, and Cosmological Time
    7.1 Background, and Some Remarks on Philosophy of Science
    7.2 Deformations of Associative Algebras
    7.3 The Universal Gauge Groups and SUSY
    7.4 The Singular Sub-Scheme of SUSY
8. The Universe as a Versal Base Space
    8.1 First Properties
    8.2 Density of Mass, Inflation, and Cyclical Cosmology
    8.3 A Conformally Trivial Cosmological Model
    8.4 Where Are We, the Observers, in This Universe?
    8.5 The Speed of Photons, and the Red-Shift
9. Worked Out Formulas
    9.1 Some Examples
    9.2 Action of g, and a Canonical Basis for Vector Fields
    9.3 The 8-Fold Way of Gell-Mann: The “Real” Story
        9.3.1 Charge, and the charge conjugation operator C
    9.4 Adjoint Actions of g
10. Summing Up the Model
    10.1 Metrics, Particles, and the Furniture
    10.2 Time, Gravitation, and Einstein’s Equation
        10.2.1 Einsteins field equations
    10.3 Energy, Dirac, and Maxwell
        10.3.1 Energy
        10.3.2 Dirac
        10.3.3 Classical Maxwell equations
        10.3.4 Photons, tenebrons, and electrons
    10.4 Black Mass and Energy
    10.5 Ensembles, Bi-Algebras, and Quantum Groups
    10.6 Black Mass and Gravitational “Waves”
11. Particles, Fields, and Probabilities
    11.1 Elementary Particles
    11.2 Time as a Source for Probabilities
    11.3 Quantum Field Theory, Wightman’s Axioms
12. Interactions
    12.1 Interaction and Non-Commutative Deformations
    12.2 The Weak and Strong Interactions
    12.3 Graphs and Sub-Categories Generated by a Family of Modules
        12.3.1 Interactions and dynamics
    12.4 Creating New Particles from Old Ones
    12.5 Entanglement, Consciousness
        12.5.1 Self-reflection
13. Comparing the Toy Model with the Standard Model
14. End Words
    14.1 Relations to Non-Commutative Geometry (NCG)
    14.2 Models for Quantum Gravitation
    14.3 The General Dynamical Model
    14.4 Time, Lagrangians, Probabilities, Reality
        14.4.1 Unsolved problems in physics
    14.5 Relations to Classical Cosmologies
    14.6 So What?
Bibliography
Index