One Hundred Years of Gauge Theory: Past, Present and Future Perspectives

Preface
Contents
Contributors
History and Philosophy of Gauge Theory: Weyl’s Raum-Zeit-Materie and Its Legacy
Hermann Weyl's Space-Time  Geometry and the Origin of Gauge Theory 100 Years Ago
	1 Introduction
	2 Weyl's Attempt to Unify Gravitation and Electromagnetism
	3 Weyl's Generalization of Riemannian Geometry
	4 Electromagnetism and Gravitation
		4.1 Einstein's Objection and Reactions of Other Physicists
	5 Weyl's 1929 Classic: ``Electron and Gravitation''
		5.1 Tetrad Formalism
		5.2 The New Form of the Gauge-Principle
	6 Gauge Invariance and QED
	7 On Pauli's Invention of Non-Abelian Kaluza-Klein Theory in 1953
		7.1 The Pauli Letters to Pais
	References
Gauging the Spacetime Metric—Looking Back and Forth a Century Later
	1 Introduction
	2 Weyl's Scale Gauge Geometry 1918–1950
		2.1 Purely Infinitesimal Geometry, 1918–1923
		2.2 Withdrawal of Scale Gauge by Weyl After 1927/29
	3 A New Start for Weyl Geometric Gravity in the 1970s
		3.1 New Interests in Local Scale and Conformal Transformations
		3.2 Weyl Geometric Gravity with a Scalar Field and Dynamical Scale Connection: Omote, Utiyama, Dirac
		3.3 Foundations of General Relativity: EPS
		3.4 Geometrizing Quantum Mechanical Configuration Spaces
	4 Interlude
		4.1 Basic Concepts, Notations, and Integrable Weyl Geometry
		4.2 General Features of Weyl Geometric Gravity
		4.3 Weyl Geometric Gravity in IWG
	5 Interest Today
		5.1 Philosophical Reflections on Gravity
		5.2 Recent Work on the EPS Argument
		5.3 How Does the Standard Model Relate to Gravity?
		5.4 Open Questions in Cosmology and Dark Matter
	6 An Extremely Short Glance Back and Forth
	7 Appendix: An Outline of the Noether Theorems
	References
On Empirical Equivalence and Duality
	1 Introduction
	2 Empirical Equivalence
	3 Duality
	4 Empirical Equivalence of Dual Theories
		4.1 Van Fraassen on Empirical Equivalence
		4.2 Applying the Syntactic Notion of Empirical Equivalence
	5 Conclusion
	References
Gauge Is More Than Mathematical Redundancy
	References
Homotopic Identities and the Limits of the Interpretation of Gauge Symmetries as Descriptive Redundancy
	1 Leibniz's Principle of Identity of Indiscernibles
	2 On the Epistemic Interpretations of Equality Statements
	3 On Equivalence Relations as Contextual Equalities
	4 From Equivalence Relations to Groupoids
	5 Revisiting the Identity Principle
	6 On Truncations and Resolutions
	7 Toward a Homotopic Interpretation of Gauge Theories
	8 On Transformations and Invariants
	References
Weyl’s Raum-Zeit-Materie and Its Philosophical Underpinning
Reichenbach, Weyl, Philosophy  and Gauge
	1 Introduction: Philosophy and Early 20th Century Physics
	2 Reichenbach's Struggle with Kant
	3 Definitions and Conventions
	4 The Rejection of Kantianism as Emancipation from Intuition
	5 Weyl's Phenomenology of Space and Time
	6 Weyl's Gauge Theory
	7 Non-Weylian Routes to Infinitesimal Geometry  and Gauge Freedom
	8 Conclusion: Physics and Intuition
	References
Hermann Weyl, the Gauge Principle, and Symbolic Construction from the “Purely Infinitesimal”
	1 Introduction
	2 Hermann Weyl and the Philosophical Origins of the Gauge Principle
		2.1 Idealism in the Infinitesimal
		2.2 Transcendental Phenomenological Idealism and “Symbolic Construction”
		2.3 Transcendental-Phenomenological Origins of Gauge Invariance
		2.4 From the “Raumproblem” to Lie Groups and Lie Algebras
	3 The Gauge Principle
		3.1 Lie Algebras in Field Theory: Purely Infinitesimal Operations
		3.2 The Gauge Principle Generalized: The SM
	4 Towards an Elucidation of the Gauge Principle
	References
Weyl’s Raum-Zeit-Materie and the Philosophy of Science
	1 Introduction
	2 Philosophy, Mathematics and Physics
	3 Space-Time-Matter and the Foundations of All Happening
		3.1 Raum-Zeit-Materie and Philosophy of Time
		3.2 Raum-Zeit-Materie and the History of Philosophy
	4 The Foundations of Mathematics, Space and Philosophy
	5 Weyl’s Philosophy of Science?
	6 Closing Remarks
	References
Theoretical and Mathematical Physics of Gauge Theory
Space, Time, Matter in Quantum Gravity
	1 From Classical to Quantum Gravity
	2 The Configuration Space of General Relativity
	3 Quantum Geometrodynamics
	4 The Role of Matter
	References
Conservation of Energy-Momentum  of Matter as the Basis for the Gauge  Theory of Gravitation
	1 Yang-Mills Theory, Gauge Theory
	2 Newton-Einstein Gravity
	3 Translational Gauge Theory (TG)
	4 Poincaré Gauge Gravity (PG)
		4.1 Colella-Overhauser-Werner (COW) Experiment Heralds a New Era in Gravitational Physics: The Kibble Laboratory
		4.2 Poincaré Gauge Gravity Kinematics
		4.3 Matter Lagrangian: Currents and Conservation Laws
		4.4 Gravitational Lagrangian: Noether Identities
		4.5 The General Field Equations of Poincaré Gauge Gravity
		4.6 Einstein-Cartan Theory
		4.7 Quadratic Poincaré Gauge Gravity Models
		4.8 Tonti-Diagram of Quadratic Poincaré Gauge Gravity
	5 Discussion and Outlook
	References
Higgs Field in Cosmology
	1 Introduction
	2 Inflation
		2.1 Background Dynamics
		2.2 Perturbations and Inflationary Observables
	3 Standard Model of Particle Physics
	4 Higgs Inflation
		4.1 Minimal Higgs Inflation
		4.2 Non-minimal Higgs Inflation
		4.3 Quantum Corrections and the Renormalization Group
		4.4 Instability of the Electroweak Vacuum
		4.5 Validity of the Effective Field Theory
		4.6 Quantum Cosmology and Initial Conditions for Higgs Inflation
	5 Quantum Field Parametrization Dependence in Cosmology
		5.1 Starobinsky Inflation
		5.2 Classical and Quantum Equivalence in Cosmology
		5.3 Quantum Equivalence and Renormalization
		5.4 Geometry of Field Space and Field Covariant Formalism
	6 Conclusion
	References
The Gauge Theoretical Underpinnings  of General Relativity
	1 Introduction
	2 Field Theoretic Approach
	3 Chrono- and Geo-Metric Approaches
	4 Gauge Theoretic Approach
		4.1 Linear Algebra
		4.2 Connection and Curvature
		4.3 Three Formulations of General Relativity
	5 Concluding Remarks
	References
Past and Future of Gauge Theory
	1 The Early Days
	2 Quantum Electrodynamics, QED
	3 The Weak Interaction
	4 The Brout–Englert–Higgs Mechanism
	5 The Folklore of ``the Origin of Mass''
	6 Unitarity
	7 BRST
	8 Permanent Quark Confinement
	9 Glimpses of the Future
	References