Differential sheaves and connections : a natural approach to physical geometry / Anastasios Mallios, Elias Zafiris, National and Kapodistrain of Athens, Greece.

Format:

Book

Author/Creator:

Mallios, Anastasios.

Contributor:

Zafiris, Elias, 1970-

Rosengarten Family Fund.

Series:

Series on concrete and applicable mathematics ; v. 18.

Series on concrete and applicable mathematics ; vol. 18

Language:

English

Subjects (All):

Sheaf theory.

Geometry, Differential.

Algebraic topology.

Physical Description:

xv, 285 pages : illustrations ; 26 cm.

Place of Publication:

New Jersey : World Scientific, [2016]

Summary:

"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."

Contents:

Machine generated contents note: 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.Kahler-de Rham Paradigm

0.4.2.Kahler'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

0.4.6.Quantization Condition via Cohomology

0.4.7.Integrable Differential Line Sheaves

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

Note continued: 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.Horn-Tensor Adjunction

0.6.Grothendieck Topos Interpretation of the Horn-Tensor Adjunction

0.7.The Grothendieck Topology of Epimorphic Families

0.8.Unit and Counit of the Horn-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.Kahler Construction

1.6.Elementary Particles in the Jargon of ADG

1.7.Relational Aspect of Space, Again

1.8.Dynamical Dressing, Extension: Kahler Construction (Contn'd)

1.9.Adjunction, Least Action Principle

1.9.1.Symmetry

Note continued: 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 Horn-Tensor Adjunction

2.2."Affine Geometry" and "Quantum"

2.2.1.Introduction

2.2.2.ADG vis-a-vis the "Infinitely Small" (: "Infinitesimal")

2.2.3.Flow, and the "Quantum"

2.2.4.Final Remarks

2.3.Chasing Feynman

2.3.0.Prelude

2.3.1.Field Interactions

2.3.2.A Non-Spatial Perspective. Whence, ADG

2.3.3.Relational Calculus

2.3.4."Feynman's Calculus", in Terms of ADG

Note continued: 2.3.5.The Exponential

2.3.6.Schrodinger

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."Invariance"

2.4.6.Conclusions

2.5.Quantized Einstein's Equation

2.5.1.Introduction

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.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

9789814719469

9814719463

OCLC:

922697782

Publisher Number:

99996450942