Preface ....................................................... vii
0 Prolegomena ................................................ 1
0.1 Exordium ................................................... 1
0.2 Basic Working Notions ...................................... 7
0.3 Observables and States .................................... 14
0.3.1 Sheaf-Theoretic Observable Localization ............ 14
0.3.2 Vector Sheaves of States and Local Gauge
Invariance ......................................... 20
0.3.3 Exponential Short Exact Sequence ................... 23
0.4 Connections and Differential Analysis ..................... 24
0.4.1 Kähler-de Rham Paradigm ............................ 24
0.4.2 Kähler's Algebraic Extension Method ................ 26
0.4.3 Connections and the Sheaf-Theoretic de Rham
Complex ............................................ 30
0.4.4 Local Forms of Connection and Curvature on Vector
Sheaves of States .................................. 33
0.4.5 Gauge Equivalence Classes of Differential Line
Sheaves ............................................ 36
0.4.6 Quantization Condition via Cohomology .............. 38
0.4.7 Integrable Differential Line Sheaves ............... 42
0.4.8 Quantum Unitary Rays ............................... 43
0.4.9 Gauge Equivalence of Quantum Unitary Rays .......... 45
0.4.10 Spectral Beams and Polarization Symmetry ........... 46
0.4.11 Affine Structure of Spectral Beams ................. 49
0.4.12 Monodromy Group and Integrable Phase Factors ....... 52
0.4.13 Aharonov-Bohm Effect ............................... 54
0.4.14 Holonomy of Spectral Beams ......................... 56
0.5 The Functorial Imperative ................................. 59
0.5.1 Representable Functors and Natural
Transformations .................................... 59
0.5.2 Adjoint Functors: Universale and Equivalence ....... 61
0.5.3 Probes and Adjoints to Realization Functors ........ 66
0.5.4 Horn-Tensor Adjunction ............................. 71
0.6 Grothendieck Topos Interpretation of the Horn-Tensor
Adjunction ................................................ 75
0.7 The Grothendieck Topology of Epimorphic Families ..... 81
0.8 Unit and Counit of the Horn-Tensor Adjunction ........ 83
1 General Theory ............................................ 89
1.0 General Introduction ...................................... 89
1.1 Basic Assumptions of ADG (Abstract Differential
Geometry) ................................................. 91
1.2 Basic Framework ........................................... 95
1.2.1 Adjoint Functors ................................... 98
1.2.2 Natural Adjunction ................................ 101
1.3 Bohr's Correspondence .................................... 104
1.4 Functorial, Topos-Theoretic Mechanism of ADG ............. 114
1.5 Kähler Construction ...................................... 119
1.6 Elementary Particles in the Jargon of ADG ................ 121
1.7 Relational Aspect of Space, Again ........................ 125
1.8 Dynamical Dressing, Extension: Kähler Construction
(Contn'd) ................................................ 127
1.9 Adjunction, Least Action Principle ....................... 136
1.9.1 Symmetry .......................................... 142
1.9.2 More Thoughts on a Unified Field Theory ........... 148
1.10 Transformation Law of Potentials, in Terms of ADG ........ 152
1.10.1 Lagrangian Perspective via "Abstract Geometric
Algebra" (AGA) .................................... 157
1.10.2 More on the Fundamental "Adjunction" .............. 163
1.11 Characteristics of a Physical Law ........................ 165
1.12 Complementary Remarks .................................... 169
1.13 Epilogue ................................................. 171
2 Applications: Fundamental Adjunctions .................... 175
2.1 On Utiyama's Theme/Principle Through " -invariance" ...... 175
2.1.1 Introduction ...................................... 175
2.1.2 Utiyama's Theorem ................................. 176
2.1.3 Utiyama's Theorem (Contn'd: Technical Details) .... 178
2.1.4 Dynamical Analogue of the Fundamental Horn-
Tensor Adjunction ................................. 184
2.2 "Affine Geometry" and "Quantum" .......................... 186
2.2.1 Introduction ...................................... 186
2.2.2 ADG vis-à-vis the "Infinitely Small"
(:"Infinitesimal") ................................ 187
2.2.3 Flow, and the "Quantum" ........................... 189
2.2.4 Final Remarks ..................................... 194
2.3 Chasing Feynman .......................................... 195
2.3.0 Prelude ........................................... 195
2.3.1 Field Interactions ................................ 196
2.3.2 A Non-Spatial Perspective. Whence, ADG ............ 199
2.3.3 Relational Calculus ............................... 201
2.3.4 "Feynman's Calculus", in Terms of ADG ............. 202
2.3.5 The Exponential ................................... 205
2.3.6 Schrödinger-Hamilton Adjunction ................... 207
2.3.7 "Everything is Light" ............................. 216
2.4 Stone-von Neumann Adjunction ............................. 229
2.4.1 Introduction ...................................... 229
2.4.2 Physical Jargon ................................... 230
2.4.3 Stone-von Neumann Theorem in Action ............... 231
2.4.4 De Broglie-Einstein -Feynman Adjunction ........... 233
2.4.5 "Invariance" ...................................... 238
2.4.6 Conclusions ....................................... 249
2.5 Quantized Einstein's Equation ............................ 250
2.5.1 Introduction ...................................... 250
2.5.2 Einstein's Fundamental Equation in Vacuo .......... 250
2.5.3 Einstein's Equation: The "Standard Model" ......... 253
2.6 The Essence of ADG ....................................... 254
2.6.1 ADG Viewed, as an "Identity" ...................... 254
2.6.2 Final Remarks ..................................... 257
2.7 Peroration ............................................... 258
Bibliography .................................................. 265
Index ......................................................... 279
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