Geometry and Quantum Physics: Proceedings of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999 (Lecture Notes in Physics (543))

In modern mathematical physics, classical together with quantum, geometrical and functional analytic methods are used simultaneously. Non-commutative geometry in particular is becoming a useful tool in quantum field theories. This book, aimed at advanced students and researchers, provides an introduction to these ideas. Researchers will benefit particularly from the extensive survey articles on models relating to quantum gravity, string theory, and non-commutative geometry, as well as Connes' approach to the standard model. Chapter 1 1 Introduction 2 Localization Formulas 2.1 Stationary Phase Method 2.2 Equivariant Cohomology 2.3 Proof of Localization Formula 2.4 Duistermaat-Heckman Formula 3 Weil Model of Equivariant Cohomology 3.1 Weil Algebra and Weil Di erential 3.2 Weil Model of Equivariant Cohomology 4 Group-Valued Equivariant Localization 4.1 Non-commutative Weil Algebra 4.2 Group-Valued Moment Maps 4.3 Group-Valued Localization References Chapter 2 1 Introduction 2 BF Theory: Classical Field Equations 3 Classical Phase Space 4 Canonical Quantization 5 Observables 6 Canonical Quantization via Triangulations 7 Dynamics 8 Spin Foams 9 q-Deformation and the Cosmological Constant 10 4-Dimensional Quantum Gravity Appendix: Piecewise Linear Cell Complexes Notes 2 BF Theory: Classical Field Equations 3 Classical Phase Space 4 Canonical Quantization 5 Observables 6 Canonical Quantization via Triangulations 7 Dynamics 8 Spin Foams 9 q-Deformation and the Cosmological Constant 10 4-Dimensional Quantum Gravity Appendix Chapter 3 1 Introduction 2 D-Branes and T-Duality 3 R.R Charged D-Branes 4 Microscopic Dp-Brane String Amplitudes and Metrics 6 Near Horizon Geometry 7 Supersymmetry 8 T-Duality and Near Horizon Geometries Acknowledgements References Chapter 4 Foreword 1 Spectral Triples as Generalized Dirac Operators. Sketch of the Quantum Yang-Mills Algorithm. 1.1 The Dirac Operator of a Riemannian Spin Manifold 1.2 Spectral Triples 1.3 Formal Forms 1.4 Quantum DeRham Complex 1.5 Quantum Volume Form 1.6 Quantum Connections and Curvature 1.7 The Quantum Yang-Mills Algorithm (for a Given Even d-Dimensional Spectral Triple (AI; H; D)) 1.8 The Electrodynamics Case 1.9 The Two-Point Algebra CI CI (Embryonal Higgs) 1.10 De nition-Lemma 1.11 Remark 2 The Electroweak Inner Spectral Triple 2.1 Basic -Algebra. Gauge Group 2.2 The Spectral Triple (Aew; Hf ; Df ) 2.3 The Aew-Bimodule DqA 1 of Dq-Quantum One-Forms 2.4 The Aew-Bimodule DqA 2 of Dq-Quantum Two-Forms 2.5 Connections. Curvature. Yang-Mills Action 2.6 Remarks 2.7 Lemma 2.8 Proposition 2.9 Conclusion 3 Sketch of the (Quantum Yang-Mills) Connes-Lott Model. 3.1 Sketch of the Connes-Lott Model 3.2 De nition (Metric Dual Pairs) 3.3 Conceptual Flaws 3.4 The Higgs Mass 4 Real Spectral Triples. 4.1 Defnition 4.2 Remark 4.3 Proposition (The Classical Case) 4.4 Defnitions 4.5 Lemma 4.6 Remark 4.7 Proposition 4.8 Lemma 4.9 Proposition (Tensor Product of Real Spectral Triple 4.10 De nitions 4.11 Proposition 4.12 Remark-Definition 4.13 Remark 5 The Inner S0-Real Spectral Triple of the Standard Model 5.1 Reminder. The Inner Real Metric Dual Pair 5.2 Definition (The Inner-Space S0-Real Spectral Triple) 5.3 Proposition 5.4 Proposition 5.5 Matrix-Form of D (A); J D (A)J; ad(G) and D 5.6 Matrix-Form of Quantum One-Forms 6 The S0-Real Spectral Triple of the Full Standard Model 6.1 De nitions 6.2 Reminder (Tensor Product Structure of Quantum Forms) 6.3 Matrix Form 6.4 Matrix-Form 6.5 Matrices 6.6 Matrix Form of the Covariant Dirac Operator 6.7 Conversion into Classical Objects 6.8 Canonical Decomposition 7 The Spectral Action and Its Heat-Kernel Asymptotic Expansion 7.1 De nition 7.2 Asymptotic Expansion of the Spectral Action 7.3 Remark 7.4 Spectral Action Computation Program 7.5 Computation of Fiber-Traces 7.6 Gathering the Pieces 7.7 Remark 8 Tree-Approximation Results 9 Fermionic Action 9.1 Labeling the Basis 9.2 Euclidean Fermionic Action: 9.3 The Ensuing Minkowskian Fermionic Action 10 Does the Inner Spectral Triple of the Full Standard Model Proceed from a Quantum Group? 10.1 The Hopf Algebra H1 and Its Regular Representation 10.2 The Case of H1: Complexi cation Versus Regular Representation A Heat-Kernel Expansion A.1 General Expansion Result A.2 Properties of the ej(x; P) A.3 Weyl's Theorem A.4 Generalized Laplacians B Generalized Laplacians B.1 Definitions B.2 Remarks B.3 Proposition-De nition B.4 Lemma B.5 Proposition (Local Description of [B.3]) C Cli ord Modules. C.1 C.2 C.3 D Weyl Tensor D.1 Definition D.2 Lemma Bibliography Chapter 5 1 Introduction and Motivation 2 Ultraviolet Regularization 3 Finite-Dimensional Algebras 4 Di erential Calculi 5 Yang-Mills Connections 6 Metrics and Linear Connections 7 In nite-Dimensional Models 8 Gravity References Chapter 6 1 Review 1.1 Introduction 1.2 The Laughlin Argument 1.3 Thouless, Kohomoto, Nightingale, and den Nijs 1.4 J. Avron, R. Seiler, Q. Niu, D. J. Thouless 1.5 J. Bellissard, H. Schulz Baldes, A. Connes 1.6 J. Fr ̈ohlich, Q. Niu, X. G. Wen, A. Zee 2 Adiabatics 2.1 The Adiabatic Setup 2.2 Kato's Equation 2.3 The Adiabatic Theorem 2.4 Adiabatic Curvature and Applications 3 Chern Number Approach 3.1 The QHE for Interacting Fermion Systems 3.2 Fluctuations and Quillen's Formula 3.3 Quantum Viscosity 4 Index Approach, Bulk, and Edge 4.1 The Algebra of Two Projectors 4.2 First Order Calculus on A := R (1 4.3 The Index of a Pair of Projections 4.4 Index Approach to the QHE 4.5 Edge vs. Bulk References Chapter 7 Introduction 1 q-Deformed Heisenberg Algebra in One Dimension 1.1 A Calculus Based on an Algebra 1.2 Field Equations in a Purely Algebraic Context 1.3 Gauge Theories in a Purely Algebraic Context 1.4 q-Fourier Transformations 1.5 Representations 1.6 The De nite Integral and the Hilbert Space L 2 q 1.7 Variational Principle 1.8 The Hilbert Space L 1.9 Gauge Theories on the Factor Spaces References 2 q-Deformed Heisenberg Algebra in n Dimensions 2.1 SLq(2), Quantum Groups and the R-Matrix 2.2 Quantum Planes 2.3 Quantum Derivatives 2.4 Conjugation 2.5 q-Deformed Heisenberg Algebra 2.6 The q-Deformed Lie Algebra slq(2) 2.7 q-Deformed Euclidean Space in Three Dimensions References Chapter 8 References Chapter 9 References Chapter 10 References Chapter 11 References Chapter 12 Chapter 13 Chapter 14 References Chapter 15 References Chapter 16 Chapter 17 References Chapter 18 References Chapter 19 References Chapter 20 References Chapter 21 References Chapter 22 References Chapter 23 References Chapter 24 References Chapter 25 References Chapter 26 Chapter 27 References Chapter 28 References Chapter 29 References Chapter 30 1 Basic Results 1.1 A Strong Coupling Limit 1.2 High Curvature Limit 2 Discussion References Annotation In modern mathematical physics, classical together with quantum, geometrical and functional analytic methods are used simultaneously. Non-commutative geometry in particular is becoming a useful tool in quantum field theories. This book, aimed at advanced students and researchers, provides an introduction to these ideas. Researchers will benefit particularly from the extensive survey articles on models relating to quantum gravity, string theory, and non-commutative geometry, as well as Connes' approach to the standard model