New paths towards quantum gravity [electronic resource] / Bernhelm Booss-Bavnbek, Giampiero Esposito, Matthias Lesch (eds.).
- Published
- Berlin ; New York : Springer, [2010]
- Copyright Date
- ©2010
- Physical Description
- xix, 359 pages : illustrations ; 24 cm.
- Additional Creators
- Booss, Bernhelm, 1941-, Esposito, Giampiero, and Lesch, Matthias, 1961-
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- Contents
- Machine generated contents note: pt. I Three Physics Visions -- 1.Notes on "Quantum Gravity" and Noncommutative Geometry / J.M. Gracia-Bondia -- 1.1.Introduction -- 1.2.Gravity and Experiment: Expect the Unexpected -- 1.2.1.Noncommutative Geometry I -- 1.2.2.Whereto Diffeomorphism Invariance? -- 1.3.Gravity from Gauge Invariance in Field Theory -- 1.3.1.Preliminary Remarks -- 1.3.2.Exempli Gratiae -- 1.3.3.The Free Lagrangian -- 1.3.4.A Canonical Setting -- 1.3.5.What to Expect -- 1.3.6.Causal Gauge Invariance by Brute Force -- 1.3.7.CGI at All Orders: Going for It -- 1.3.8.Details on Quantization and Graviton Helicities -- 1.3.9.Final Remarks -- 1.3.10.Other Ways -- 1.4.The Unimodular Theories -- 1.5.The Noncommutative Connection -- 1.5.1.Prolegomena -- 1.5.2.Ironies of History -- 1.5.3.Spectral Triples -- 1.5.4.On the Reconstruction Theorem -- 1.5.5.The Noncommutative Torus -- 1.5.6.The Noncompact Case -- 1.5.7.Nc Toric Manifolds (Compact and Noncompact) -- 1.5.8.Closing Points -- 1.5.9.Some Interfaces with Quantum Gravity -- 1.6.More on the "Cosmological Constant Problem" and the Astroparticle Interface -- References -- 2.Quantum Gravity as Sum over Spacetimes / R. Loll -- 2.1.Introduction -- 2.2.CDT -- 2.3.Numerical Results -- 2.3.1.The Emergent de Sitter Background -- 2.3.2.Fluctuations Around de Sitter Space -- 2.3.3.The Size of the Universe and the Flow of G -- 2.4.Two-Dimensional Euclidean Quantum Gravity -- 2.4.1.Continuum Formulation -- 2.4.2.The Lattice Regularization -- 2.4.3.Counting Graphs -- 2.4.4.The Continuum Limit -- 2.5.Two-Dimensional Lorentzian Quantum Gravity -- 2.6.Matrix Model Representation -- 2.6.1.The Loop Equations -- 2.6.2.Summation over All Genera in the CDT Matrix Model -- 2.7.Discussion and Perspectives -- References -- 3.Lectures on Quantization of Gauge Systems / N. Reshetikhin -- 3.1.Introduction -- 3.2.Local Lagrangian Classical Field Theory -- 3.2.1.Space-Time Categories -- 3.2.2.Local Lagrangian Classical Field Theory -- 3.2.3.Classical Mechanics -- 3.2.4.First-Order Classical Mechanics -- 3.2.5.Scalar Fields -- 3.2.6.Pure Euclidean d-Dimensional Yang-Mills -- 3.2.7.Yang-Mills Field Theory with Matter -- 3.2.8.Three-Dimensional Chern-Simons Theory -- 3.3.Hamiltonian Local Classical Field Theory -- 3.3.1.The Framework -- 3.3.2.Hamiltonian Formulation of Local Lagrangian Field Theory -- 3.4.Quantum Field Theory Framework -- 3.4.1.General Framework of Quantum Field Theory -- 3.4.2.Constructions of Quantum Field Theory -- 3.5.Feynman Diagrams -- 3.5.1.Formal Asymptotic of Oscillatory Integrals -- 3.5.2.Integrals Over Grassmann Algebras -- 3.5.3.Formal Asymptotics of Oscillatory Integrals Over Super-manifolds -- 3.5.4.Charged Fermions -- 3.6.Finite-Dimensional Faddeev-Popov Quantization and the BRST Differential -- 3.6.1.Faddeev-Popov Trick -- 3.6.2.Feynman Diagrams with Ghost Fermions -- 3.6.3.Gauge Independence -- 3.6.4.Feynman Diagrams for Linear Constraints -- 3.6.5.The BRST Differential -- 3.7.Semiclassical Quantization of a Scalar Bose Field -- 3.7.1.Formal Semiclassical Quantum Mechanics -- 3.7.2.d > 1 and Ultraviolet Divergencies -- 3.8.The Yang-Mills Theory -- 3.8.1.The Gauge Fixing -- 3.8.2.The Faddeev-Popov Action and Feynman Diagrams -- 3.8.3.The Renormalization -- 3.9.The Chern-Simons Theory -- 3.9.1.The Gauge Fixing -- 3.9.2.The Faddeev-Popov Action in the Chern-Simons Theory -- 3.9.3.Vacuum Feynman Diagrams and Invariants of 3-Manifolds -- 3.9.4.Wilson Loops and Invariants of Knots -- 3.9.5.Comparison with Combinatorial Invariants -- References -- pt. II Novel Mathematical Tools -- 4.Mathematical Tools for Calculation of the Effective Action in Quantum Gravity / I.G. Avramidi -- 4.1.Introduction -- 4.2.Effective Action in Quantum Field Theory and Quantum Gravity -- 4.2.1.Non-gauge Field Theories -- 4.2.2.Gauge Field Theories -- 4.2.3.Quantum General Relativity -- 4.3.Heat Kernel Method -- 4.3.1.Laplace-Type Operators -- 4.3.2.Spectral Functions -- 4.3.3.Heat Kernel -- 4.3.4.Asymptotic Expansion of the Heat Kernel -- 4.3.5.Zeta-Function and Determinant -- 4.4.Green Function -- 4.5.Heat Kernel Coefficients -- 4.5.1.Non-recursive Solution of the Recursion Relations -- 4.5.2.Covariant Taylor Basis -- 4.5.3.Matrix Algorithm -- 4.5.4.Diagramatic Technique -- 4.5.5.General Structure of Heat Kernel Coefficients -- 4.6.High-Energy Approximation -- 4.7.Low-Energy Approximation -- 4.7.1.Algebraic Approach -- 4.7.2.Covariantly Constant Background in Flat Space -- 4.7.3.Homogeneous Bundles over Symmetric Spaces -- 4.8.Low-Energy Effective Action in Quantum General Relativity -- References -- 5.Lectures on Cohomology, T-Duality, and Generalized Geometry / R Bouwknegt -- 5.1.Cohomology and Differential Characters -- 5.1.1.A Brief Review of de Rham and Cech Cohomology -- 5.1.2.Electromagnetism -- 5.1.3.The Cech - de Rham Complex -- 5.1.4.Differential Cohomologies -- 5.2.T-Duality -- 5.2.1.Introduction to T-Duality -- 5.2.2.The Buscher Rules -- 5.2.3.Gysin Sequences and Dimensional Reduction -- 5.2.4.T-Duality = Takai Duality -- 5.2.5.T-Duality as a Duality of Loop Group Bundles -- 5.3.Generalized Geometry -- 5.3.1.Cartan Relations -- 5.3.2.Lie Algebroids -- 5.3.3.Generalized Geometry -- 5.3.4.Courant Algebroids -- 5.3.5.Generalized Complex Geometry -- 5.3.6.T-Duality in Generalized Geometry -- References -- 6.Stochastic Geometry and Quantum Gravity: Some Rigorous Results / H. Zessin -- 6.1.An Axiomatic Introduction into Regge's Model -- 6.2.The Zero-Infinity Law of Stochastic Geometry and the Cluster Process -- 6.2.1.Basic Concepts -- 6.2.2.Cluster Properties and the Zero-Infinity Law of Stochastic Geometry -- 6.2.3.Cluster Properties and the Zero-Infinity Law for Marked Configurations -- 6.2.4.The Cluster Process -- 6.2.5.The Poisson Process Pρ -- 6.3.Poisson-Delaunay Surfaces with Intrinsic Random Metric -- 6.3.1.The Delaunay Cluster Property -- 6.3.2.Poisson-Delaunay Surfaces -- 6.3.3.Scholion: The Voronoi Cluster Property -- 6.4.Ergodic Behaviour of PD-Surfaces -- 6.4.1.Palm Measures and Palm Distributions -- 6.4.2.An Ergodic Theorem for Intrinsic Metric Quantities of Stationary Random Surfaces -- 6.4.3.Curvature Properties of the Poisson-Delaunay Surface -- 6.5.The Two-Dimensional Regge Model of Pure Quantum Gravity -- 6.6.Comments and Final Dreams -- References -- pt. III Afterthoughts -- 7.Steps Towards Quantum Gravity and the Practice of Science: Will the Merger of Mathematics and Physics Work? / B. Booß-Bavnbek -- 7.1.Regarding the Need and the Chances of Unification -- 7.2.The Place of Physics in John Dee's Groundplat of Sciences and Artes, Mathematicall of 1570 -- 7.3.Delimitation Between Mathematics and Physics -- 7.4.Variety of Modelling Purposes -- 7.4.1.Production of Data, Model-Based Measurements -- 7.4.2.Simulation -- 7.4.3.Prediction -- 7.4.4.Control -- 7.4.5.Explain Phenomena -- 7.4.6.Theory Development -- 7.5."The Trouble with Physics" -- 7.6.Theory-Model-Experiment -- 7.6.1.First Principles -- 7.6.2.Towards a Taxonomy of Models -- 7.6.3.The Scientific Status of Quantum Gravity as Compared to Medicine and Economics -- 7.7.General Trends of Mathematization and Modelling -- 7.7.1.Deep Divide -- 7.7.2.Charles Sanders Peirce's Semiotic View -- References.
- Summary
- Aside from the obvious statement that it should be a theory capable of unifying general relativity and quantum field theory, not much is known about the true nature of quantum gravity. --
Because it is an exciting area of research, there are many new ideas about quantum gravity, but they often diverge to such an incredible degree that it seems impossible to decide which of the many possible directions the ongoing developments should be pursued. --
The division of this text into two overlapping parts reflects the duality between the physical vision and the mathematical construction of quantum gravity research. The former is represented by tutorial reviews on non-commutative geometry, on space-time discretization and renormalization, and on gauge field path integrals. The latter contains lectures on cohomology, on stochastic geometry and on mathematical tools for the effective action in quantum gravity. --
This book will benefit everyone working in or entering the field of quantum gravity research. --Book Jacket. - Subject(s)
- Genre(s)
- ISBN
- 9783642118968 (pbk. : alk. paper)
3642118968 (pbk. : alk. paper)
9783642118975 (e-ISBN)
3642118976 (e-ISBN) - Bibliography Note
- Includes bibliographical references and index.
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