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Differential Sheaves and Connections 1st Edition by Anastasios Mallios, Elias Zafiris ISBN 9789814719469, 9814719463

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Differential Sheaves and Connections 1st Edition Anastasios Mallios Digital Instant Download

Author(s): Anastasios Mallios, Elias Zafiris
ISBN(s): 9789814719469, 9814719463
Edition: 1
File Details: PDF, 3.18 MB
Year: 2015
Language: english
SKU: EB-7279946 Category: Tags: ,

Differential Sheaves and Connections 1st Edition by Anastasios Mallios,  Elias Zafiris – Ebook PDF Instant Download/Delivery: 9789814719469, 9814719463
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Product details: 

ISBN 10:  9814719463

ISBN 13: 9789814719469

Author: Anastasios Mallios,  Elias Zafiris 

Kapodistrian University of Athens, Greece) “The book is manifestly dense, idiosyncratic, and ambitious in its aims. It is an interesting approach indeed to a very deep set of questions at the intersection of physics and mathematics, employing novel and even avant-garde techniques and perspectives.” Mathematical Reviews Clippings 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.
 

Table of contents: 

0. Prolegomena

0.1 Exordium

0.2 Basic Working Notions

0.3 Observables and States

0.3.1 Sheaf-Theoretic Observable Localization.

0.3.2 Vector Sheaves of States and Local Gauge Invariance

0.3.3 Exponential Short Exact Sequence

0.4 Connections and Differential Analysis

0.4.1 Kähler-de Rham Paradigm

0.4.2 Kähler’s Algebraic Extension Method

0.4.3 Connections and the Sheaf-Theoretic de Rham Complex

0.4.4 Local Forms of Connection and Curvature on Vector Sheaves of States.

0.4.5 Gauge Equivalence Classes of Differential Line Sheaves

Quantization Condition via Cohomology 0.4.6

Integrable Differential Line Sheaves 0.4.7

0.4.8 Quantum Unitary Rays

0.4.9 Gauge Equivalence of Quantum Unitary Rays

0.4.10 Spectral Beams and Polarization Symmetry

0.4.11 Affine Structure of Spectral Beams.

0.4.12 Monodromy Group and Integrable Phase Factors.

0.4.13 Aharonov-Bohm Effect

0.4.14 Holonomy of Spectral Beams

0.5 The Functorial Imperative

0.5.1 Representable Functors and Natural Transformations

0.5.2 Adjoint Functors: Universals and Equivalence

0.5.3 Probes and Adjoints to Realization Functors

0.5.4 Hom-Tensor Adjunction

0.6 Grothendieck Topos Interpretation of the Hom-Tensor Adjunction..

0.7 The Grothendieck Topology of Epimorphic Families

0.8 Unit and Counit of the Hom-Tensor Adjunction

1. General Theory

1.0 General Introduction

1.1 Basic Assumptions of ADG (: Abstract Differential Geometry)

1.2 Basic Framework

1.2.1 Adjoint Functors

.

1.2.2 Natural Adjunction

1.3 Bohr’s Correspondence

1.4 Functorial, Topos-Theoretic Mechanism of ADG

1.5 Kähler Construction

1.6 Elementary Particles in the Jargon of ADG

1.7 Relational Aspect of Space, Again.

1.8 Dynamical Dressing, Extension: Kähler Construction (Contn’d)

1.9 Adjunction, Least Action Principle

1.9.1 Symmetry

1.9.2 More Thoughts on a Unified Field Theory.

1.10 Transformation Law of Potentials, in Terms of ADG

1.10.1 Lagrangian Perspective via “Abstract Geometric Algebra” (AGA)

1.10.2 More on the Fundamental “Adjunction”

1.11 Characteristics of a Physical Law

1.12 Complementary Remarks.

1.13 Epilogue

2. Applications: Fundamental Adjunctions

2.1 On Utiyama’s Theme/Principle Through “A-invariance”

2.1.1 Introduction

2.1.2 Utiyama’s Theorem

2.1.3 Utiyama’s Theorem (Contn’d: Technical Details).

2.1.4 Dynamical Analogue of the Fundamental Hom-Tensor Adjunction

2.2 “Affine Geometry” and “Quantum”

2.2.1 Introduction

2.2.2 ADG vis-à-vis the “Infinitely Small” (: “Infinitesimal”).

Flow, and the “Quantum” 2.2.3

2.2.4 Final Remarks

2.3 Chasing Feynman

2.3.0 Prelude

2.3.1 Field Interactions

A Non-Spatial Perspective. Whence, ADG 2.3.2

Relational Calculus 2.3.3

2.3.4 “Feynman’s Calculus”, in Terms of ADG

2.3.5 The Exponential

2.3.6 Schrödinger-Hamilton Adjunction

2.3.7 “Everything is Light”

2.4 Stone-von Neumann Adjunction

2.4.1 Introduction

2.4.2 Physical Jargon

2.4.3 Stone-von Neumann Theorem in Action

2.4.4 De Broglie-Einstein-Feynman Adjunction

2.4.5

2.4.6 Conclusions

2.5 Quantized Einstein’s Equation

2.5.1 Introduction

“Invariance”

2.5.2 Einstein’s Fundamental Equation in Vacuo

2.5.3 Einstein’s Equation: The “Standard Model”

2.6 The Essence of ADG

2.6.1 ADG Viewed, as an “Identity”

2.6.2 Final Remarks

2.7 Peroration

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Tags: Anastasios Mallios, Elias Zafiris, Differential Sheaves