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El. knyga: Differential Sheaves And Connections: A Natural Approach To Physical Geometry

(National & Kapodistrian Univ Of Athens, Greece), (National & Kapodistrian Univ Of Athens, Greece)
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Tackling a perennial issue in mathematical physics, Mallios and Zafiris ask whether there exists a natural description of physical geometry, that is one not based on ad hoc conventions involving a "God-given" geometric coordinate substratum of any local or global form, but refers to the pertinent physical relations themselves together with their empirical realization in terms of observed events. Their topics include connections and differential analysis, the functional imperative, Bohr's correspondence, elementary particles in the jargon of abstract differential geometry, affine geometry and quantum, and quantized Einstein's equation. Annotation ©2016 Ringgold, Inc., Portland, OR (protoview.com)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums.
Preface vii
0 Prolegomena
1(88)
0.1 Exordium
1(6)
0.2 Basic Working Notions
7(7)
0.3 Observables and States
14(10)
0.3.1 Sheaf-Theoretic Observable Localization
14(6)
0.3.2 Vector Sheaves of States and Local Gauge Invariance
20(3)
0.3.3 Exponential Short Exact Sequence
23(1)
0.4 Connections and Differential Analysis
24(35)
0.4.1 Kahler-de Rham Paradigm
24(2)
0.4.2 Kahler's Algebraic Extension Method
26(4)
0.4.3 Connections and the Sheaf-Theoretic de Rham Complex
30(3)
0.4.4 Local Forms of Connection and Curvature on Vector Sheaves of States
33(3)
0.4.5 Gauge Equivalence Classes of Differential Line Sheaves
36(2)
0.4.6 Quantization Condition via Cohomology
38(4)
0.4.7 Integrable Differential Line Sheaves
42(1)
0.4.8 Quantum Unitary Rays
43(2)
0.4.9 Gauge Equivalence of Quantum Unitary Rays
45(1)
0.4.10 Spectral Beams and Polarization Symmetry
46(3)
0.4.11 Affine Structure of Spectral Beams
49(3)
0.4.12 Monodromy Group and Integrable Phase Factors
52(2)
0.4.13 Aharonov-Bohm Effect
54(2)
0.4.14 Holonomy of Spectral Beams
56(3)
0.5 The Functorial Imperative
59(16)
0.5.1 Representable Functors and Natural Transformations
59(2)
0.5.2 Adjoint Functors: Universals and Equivalence
61(5)
0.5.3 Probes and Adjoints to Realization Functors
66(5)
0.5.4 Horn-Tensor Adjunction
71(4)
0.6 Grothendieck Topos Interpretation of the Horn-Tensor Adjunction
75(6)
0.7 The Grothendieck Topology of Epimorphic Families
81(2)
0.8 Unit and Counit of the Horn-Tensor Adjunction
83(6)
1 General Theory
89(86)
1.0 General Introduction
89(2)
1.1 Basic Assumptions of ADG (: Abstract Differential Geometry)
91(4)
1.2 Basic Framework
95(9)
1.2.1 Adjoint Functors
98(3)
1.2.2 Natural Adjunction
101(3)
1.3 Bohr's Correspondence
104(10)
1.4 Functorial, Topos-Theoretic Mechanism of ADG
114(5)
1.5 Kahler Construction
119(2)
1.6 Elementary Particles in the Jargon of ADG
121(4)
1.7 Relational Aspect of Space, Again
125(2)
1.8 Dynamical Dressing, Extension: Kahler Construction (Contn'd)
127(9)
1.9 Adjunction, Least Action Principle
136(16)
1.9.1 Symmetry
142(6)
1.9.2 More Thoughts on a Unified Field Theory
148(4)
1.10 Transformation Law of Potentials, in Terms of ADG
152(13)
1.10.1 Lagrangian Perspective via "Abstract Geometric Algebra" (AGA)
157(6)
1.10.2 More on the Fundamental "Adjunction"
163(2)
1.11 Characteristics of a Physical Law
165(4)
1.12 Complementary Remarks
169(2)
1.13 Epilogue
171(4)
2 Applications: Fundamental Adjunctions
175(90)
2.1 On Utiyama's Theme/Principle Through "A-invariance"
175(11)
2.1.1 Introduction
175(1)
2.1.2 Utiyama's Theorem
176(2)
2.1.3 Utiyama's Theorem (Contn'd: Technical Details)
178(6)
2.1.4 Dynamical Analogue of the Fundamental Horn-Tensor Adjunction
184(2)
2.2 "Affine Geometry" and "Quantum"
186(9)
2.2.1 Introduction
186(1)
2.2.2 ADG vis-a-vis the "Infinitely Small" (: "Infinitesimal")
187(2)
2.2.3 Flow, and the "Quantum"
189(5)
2.2.4 Final Remarks
194(1)
2.3 Chasing Feynman
195(34)
2.3.0 Prelude
195(1)
2.3.1 Field Interactions
196(3)
2.3.2 A Non-Spatial Perspective. Whence, ADG
199(2)
2.3.3 Relational Calculus
201(1)
2.3.4 "Feynman's Calculus", in Terms of ADG
202(3)
2.3.5 The Exponential
205(2)
2.3.6 Schrodinger--Hamilton Adjunction
207(9)
2.3.7 "Everything is Light"
216(13)
2.4 Stone--von Neumann Adjunction
229(21)
2.4.1 Introduction
229(1)
2.4.2 Physical Jargon
230(1)
2.4.3 Stone--von Neumann Theorem in Action
231(2)
2.4.4 De Broglie--Einstein--Feynman Adjunction
233(5)
2.4.5 "Invariance"
238(11)
2.4.6 Conclusions
249(1)
2.5 Quantized Einstein's Equation
250(4)
2.5.1 Introduction
250(1)
2.5.2 Einstein's Fundamental Equation in Vacuo
250(3)
2.5.3 Einstein's Equation: The "Standard Model"
253(1)
2.6 The Essence of ADG
254(4)
2.6.1 ADG Viewed, as an "Identity"
254(3)
2.6.2 Final Remarks
257(1)
2.7 Peroration
258(7)
Bibliography 265(14)
Index 279
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