| Acknowledgment |
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vii | |
| 1 Introduction |
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1 | (12) |
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1 | (3) |
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4 | (2) |
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1.3 Geometry of the Space of Models |
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6 | (2) |
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8 | (1) |
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1.5 Organization of the Work: Leitfaden |
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9 | (4) |
| 2 Dynamics |
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13 | (26) |
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2.1 The Phase Space Functor |
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13 | (9) |
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14 | (3) |
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2.1.2 The deformation functor of representations |
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17 | (2) |
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2.1.3 Blow-ups and desingularizations |
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19 | (3) |
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22 | (1) |
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2.2 The Iterated Phase Space Functor Ph* and the Dirac Derivation |
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22 | (9) |
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2.2.1 Formal curves of representations |
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30 | (1) |
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2.3 The Generalized de Rham Complex |
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31 | (8) |
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2.3.1 Excursion into the Jacobian conjecture |
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36 | (3) |
| 3 Non-Commutative Algebraic Geometry |
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39 | (14) |
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3.1 Moduli of Representations |
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40 | (1) |
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3.2 Moduli of Simple Modules |
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41 | (3) |
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3.2.1 Evolution in the moduli of simple modules |
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42 | (2) |
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3.3 Non-Commutative Deformations of Swarms |
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44 | (9) |
| 4 The Dirac Derivation and Dynamical Structures |
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53 | (56) |
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53 | (4) |
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4.2 Gauge Groups and Invariant Theory |
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57 | (7) |
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4.2.1 The global gauge group and invariant theory |
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57 | (3) |
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4.2.2 The local gauge group |
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60 | (4) |
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4.3 The Generic Dynamical Structures Associated to a Metric |
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64 | (5) |
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4.3.1 The commutative case, metrics, and gravitation |
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64 | (3) |
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4.3.2 The Lie algebra of isometries |
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67 | (2) |
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4.4 Metrics, Gravitation, and Energy |
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69 | (13) |
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4.4.1 The case of subspaces, spectral triples |
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79 | (2) |
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4.4.2 Relations to Clifford algebras |
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81 | (1) |
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4.5 Potentials and the Classical Gauge Invariance |
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82 | (6) |
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4.5.1 Infinitesimal structure on Rep(C(σg)) |
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83 | (3) |
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4.5.2 Physics and the Chern-Simons class |
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86 | (2) |
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4.6 A Generalized Yang-Mills Theory |
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88 | (7) |
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4.7 Reuniting GR, YM, and General QFT |
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95 | (10) |
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4.8 Family of Representations versus Family of Metrics |
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105 | (4) |
| 5 Time-Space and Space-Times |
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109 | (24) |
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5.1 The Cylindrical Coordinates, Newton, and Kepler |
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111 | (12) |
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5.2 Thermodynamics, the Heat Equation and Navier-Stokes |
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123 | (10) |
| 6 Entropy |
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133 | (14) |
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6.1 The Classical Commutative Case |
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133 | (1) |
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134 | (3) |
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6.3 Representations of Phinfinity |
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137 | (10) |
| 7 Cosmology, Cosmos, and Cosmological Time |
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147 | (24) |
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7.1 Background, and Some Remarks on Philosophy of Science |
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147 | (3) |
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7.2 Deformations of Associative Algebras |
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150 | (6) |
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7.3 The Universal Gauge Groups and SUSY |
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156 | (9) |
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7.4 The Singular Sub-Scheme of SUSY |
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165 | (6) |
| 8 The Universe as a Versal Base Space |
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171 | (16) |
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172 | (5) |
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8.2 Density of Mass, Inflation, and Cyclical Cosmology |
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177 | (2) |
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8.3 A Conformally Trivial Cosmological Model |
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179 | (3) |
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8.4 Where Are We, the Observers, in This Universe? |
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182 | (3) |
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8.5 The Speed of Photons, and the Red-Shift |
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185 | (2) |
| 9 Worked Out Formulas |
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187 | (16) |
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187 | (2) |
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9.2 Action of g, and a Canonical Basis for Vector Fields |
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189 | (3) |
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9.3 The 8-Fold Way of Gell-Mann: The "Real" Story |
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192 | (7) |
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9.3.1 Charge, and the charge conjugation operator C |
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197 | (2) |
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199 | (4) |
| 10 Summing Up the Model |
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203 | (40) |
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10.1 Metrics, Particles, and the Furniture |
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205 | (6) |
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10.2 Time, Gravitation, and Einstein's Equation |
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211 | (7) |
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10.2.1 Einsteins field equations |
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214 | (4) |
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10.3 Energy, Dirac, and Maxwell |
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218 | (16) |
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219 | (3) |
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222 | (3) |
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10.3.3 Classical Maxwell equations |
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225 | (5) |
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10.3.4 Photons, tenebrous, and electrons |
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230 | (4) |
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10.4 Black Mass and Energy |
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234 | (4) |
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10.5 Ensembles, Bi-Algebras, and Quantum Groups |
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238 | (4) |
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10.6 Black Mass and Gravitational "Waves" |
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242 | (1) |
| 11 Particles, Fields, and Probabilities |
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243 | (10) |
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11.1 Elementary Particles |
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244 | (4) |
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11.2 Time as a Source for Probabilities |
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248 | (2) |
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11.3 Quantum Field Theory, Wightman's Axioms |
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250 | (3) |
| 12 Interactions |
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253 | (22) |
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12.1 Interaction and Non-Commutative Deformations |
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254 | (1) |
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12.2 The Weak and Strong Interactions |
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255 | (11) |
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12.3 Graphs and Sub-Categories Generated by a Family of Modules |
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266 | (4) |
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12.3.1 Interactions and dynamics |
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268 | (2) |
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12.4 Creating New Particles from Old Ones |
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270 | (2) |
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12.5 Entanglement, Consciousness |
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272 | (13) |
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273 | (2) |
| 13 Comparing the Toy Model with the Standard Model |
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275 | (10) |
| 14 End Words |
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285 | (14) |
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14.1 Relations to Non-Commutative Geometry (NCG) |
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287 | (2) |
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14.2 Models for Quantum Gravitation |
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289 | (1) |
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14.3 The General Dynamical Model |
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290 | (2) |
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14.4 Time, Lagrangians, Probabilities, Reality |
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292 | (5) |
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14.4.1 Unsolved problems in physics |
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294 | (1) |
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14.5 Relations to Classical Cosmologies |
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295 | (2) |
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297 | (2) |
| Bibliography |
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299 | (4) |
| Index |
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303 | |
