Differential Sheaves and Connections

A Natural Approach to Physical Geometry

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:

• Prolegomena:

  • Exordium
  • Basic Working Notions
  • Observables and States
  • Connections and Differential Analysis
  • The Functorial Imperative
  • Grothendieck Topos Interpretation of the Hom-Tensor Adjunction
  • The Grothendieck Topology of Epimorphic Families
  • Unit and Counit of the Hom-Tensor Adjunction

• General Theory:

  • General Introduction
  • Basic Assumptions of ADG (Abstract Differential Geometry)
  • Basic Framework
  • Bohr's Correspondence
  • Functorial, Topos-Theoretic Mechanism of ADG
  • Kähler Construction
  • Elementary Particles in the Jargon of ADG
  • Relational Aspect of Space, Again
  • Dynamical Dressing, Extension: Kähler Construction (Contn'd)
  • Adjunction, Least Action Principle
  • Transformation Law of Potentials, in Terms of ADG
  • Characteristics of a Physical Law
  • Complementary Remarks
  • Epilogue

• Applications: Fundamental Adjunctions:

  • On Utiyama's Theme/Principle Through "A-Invariance"
  • "Affine Geometry" and "Quantum"
  • Chasing Feynman
  • Stone–von Neumann Adjunction
  • Quantized Einstein's Equation
  • The Essence of ADG
  • Peroration

Readership: Graduate students in mathematics and/or physics; post-graduate mathematics/physics students interested in the conceptual aspects and the applicability of categorical/algebraic methods in fundamental problems of physics; theoretical/mathematical physicists, pure and applied mathematicians, general system scientists and philosophers of mathematics/physics.
Key Features:

• For the first time, a self-contained conceptual and technical introduction to the theory of differential sheaves is presented along with their applications in modeling physical geometry
• The text takes an insight-oriented approach based on physical principles to elucidate a background-independent, natural and relational approach to physical geometry
• The main emphasis is on the exposition of the new concepts and methods from sheaf theory and homological algebra in relation to the interpretation of gauge-theoretic physical geometric spectrums