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Covariant Loop Quantum Gravity

Book Description

Quantum gravity is among the most fascinating problems in physics. It modifies our understanding of time, space and matter. The recent development of the loop approach has allowed us to explore domains ranging from black hole thermodynamics to the early Universe. This book provides readers with a simple introduction to loop quantum gravity, centred on its covariant approach. It focuses on the physical and conceptual aspects of the problem and includes the background material needed to enter this lively domain of research, making it ideal for researchers and graduate students. Topics covered include quanta of space; classical and quantum physics without time; tetrad formalism; Holst action; lattice QCD; Regge calculus; ADM and Ashtekar variables; Ponzano-Regge and Turaev-Viro amplitudes; kinematics and dynamics of 4D Lorentzian quantum gravity; spectrum of area and volume; coherent states; classical limit; matter couplings; graviton propagator; spinfoam cosmology and black hole thermodynamics.

Table of Contents

  1. Cover
  2. Half-title page
  3. Title
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Part I FOUNDATIONS
    1. 1 Spacetime as a quantum object
      1. 1.1 The problem
      2. 1.2 The end of space and time
      3. 1.3 Geometry quantized
      4. 1.3.1 Quanta of area and volume
      5. 1.4 Physical consequences of the existence of the Planck scale
      6. 1.4.1 Discreteness: scaling is finite
      7. 1.4.2 Fuzziness: disappearance of classical space and time
      8. 1.5 Graphs, loops, and quantum Faraday lines
      9. 1.6 The landscape
      10. 1.7 Complements
      11. 1.7.1 SU(2) representations and spinors
      12. 1.7.2 Pauli matrices
      13. 1.7.3 Eigenvalues of the volume
    2. 2 Physics without time
      1. 2.1 Hamilton function
      2. 2.1.1 Boundary terms
      3. 2.2 Transition amplitude
      4. 2.2.1 Transition amplitude as an integral over paths
      5. 2.2.2 General properties of the transition amplitude
      6. 2.3 General covariant form of mechanics
      7. 2.3.1 Hamilton function of a general covariant system
      8. 2.3.2 Partial observables
      9. 2.3.3 Classical physics without time
      10. 2.4 Quantum physics without time
      11. 2.4.1 Observability in quantum gravity
      12. 2.4.2 Boundary formalism
      13. 2.4.3 Relational quanta, relational space
      14. 2.5 Complements
      15. 2.5.1 Example of a timeless system
      16. 2.5.2 Symplectic structure and Hamilton function
    3. 3 Gravity
      1. 3.1 Einstein’s formulation
      2. 3.2 Tetrads and fermions
      3. 3.2.1 An important sign
      4. 3.2.2 First-order formulation
      5. 3.3 Holst action and Barbero–Immirzi coupling constant
      6. 3.3.1 Linear simplicity constraint
      7. 3.3.2 Boundary term
      8. 3.4 Hamiltonian general relativity
      9. 3.4.1 ADM variables
      10. 3.4.2 What does this mean? Dynamics
      11. 3.4.3 Ashtekar connection and triads
      12. 3.5 Euclidean general relativity in three spacetime dimensions
      13. 3.6 Complements
      14. 3.6.1 Working with general covariant field theory
      15. 3.6.2 Problems
    4. 4 Classical discretization
      1. 4.1 Lattice QCD
      2. 4.1.1 Hamiltonian lattice theory
      3. 4.2 Discretization of covariant systems
      4. 4.3 Regge calculus
      5. 4.4 Discretization of general relativity on a two-complex
      6. 4.5 Complements
      7. 4.5.1 Holonomy
      8. 4.5.2 Problems
  9. Part II THREE-DIMENSIONAL THEORY
    1. 5 Three-dimensional euclidean theory
      1. 5.1 Quantization strategy
      2. 5.2 Quantum kinematics: Hilbert space
      3. 5.2.1 Length quantization
      4. 5.2.2 Spin networks
      5. 5.3 Quantum dynamics: transition amplitudes
      6. 5.3.1 Properties of the amplitude
      7. 5.3.2 Ponzano–Regge model
      8. 5.4 Complements
      9. 5.4.1 Elementary harmonic analysis
      10. 5.4.2 Alternative form of the transition amplitude
      11. 5.4.3 Poisson brackets
      12. 5.4.4 Perimeter of the universe
    2. 6 Bubbles and the cosmological constant
      1. 6.1 Vertex amplitude as gauge-invariant identity
      2. 6.2 Bubbles and spikes
      3. 6.3 Turaev–Viro amplitude
      4. 6.3.1 Cosmological constant
      5. 6.4 Complements
      6. 6.4.1 A few notes on SU(2)q
      7. 6.4.2 Problem
  10. Part III THE REAL WORLD
    1. 7 The real world: four-dimensional lorentzian theory
      1. 7.1 Classical discretization
      2. 7.2 Quantum states of gravity
      3. 7.2.1 Yγ map
      4. 7.2.2 Spin networks in the physical theory
      5. 7.2.3 Quanta of space
      6. 7.3 Transition amplitudes
      7. 7.3.1 Continuum limit
      8. 7.3.2 Relation with QED and QCD
      9. 7.4 Full theory
      10. 7.4.1 Face amplitude, wedge amplitude, and the kernel P
      11. 7.4.2 Cosmological constant and IR finiteness
      12. 7.4.3 Variants
      13. 7.5 Complements
      14. 7.5.1 Summary of the theory
      15. 7.5.2 Computing with spin networks
      16. 7.5.3 Spectrum of the volume
      17. 7.5.4 Unitary representation of the Lorentz group and the Yγ map
    2. 8 Classical limit
      1. 8.1 Intrinsic coherent states
      2. 8.1.1 Tetrahedron geometry and SU(2) coherent states
      3. 8.1.2 Livine–Speziale coherent intertwiners
      4. 8.1.3 Thin and thick wedges and time-oriented tetrahedra
      5. 8.2 Spinors and their magic
      6. 8.2.1 Spinors, vectors, and bivectors
      7. 8.2.2 Coherent states and spinors
      8. 8.2.3 Representations of SU(2) and SL(2,C) on functions of spinors and Yγ map
      9. 8.3 Classical limit of the vertex amplitude
      10. 8.3.1 Transition amplitude in terms of coherent states
      11. 8.3.2 Classical limit versus continuum limit
      12. 8.4 Extrinsic coherent states
    3. 9 Matter
      1. 9.1 Fermions
      2. 9.2 Yang–Mills fields
  11. Part IV PHYSICAL APPLICATIONS
    1. 10 Black holes
      1. 10.1 Bekenstein–Hawking entropy
      2. 10.2 Local thermodynamics and Frodden–Ghosh–Perez energy
      3. 10.3 Kinematical derivation of the entropy
      4. 10.4 Dynamical derivation of the entropy
      5. 10.4.1 Entanglement entropy and area fluctuations
      6. 10.5 Complements
      7. 10.5.1 Accelerated observers in Minkowski and Schwarzschild metrics
      8. 10.5.2 Tolman law and thermal time
      9. 10.5.3 Algebraic quantum theory
      10. 10.5.4 KMS and thermometers
      11. 10.5.5 General covariant statistical mechanics and quantum gravity
    2. 11 Cosmology
      1. 11.1 Classical cosmology
      2. 11.2 Canonical loop quantum cosmology
      3. 11.3 Spinfoam cosmology
      4. 11.3.1 Homogeneous and isotropic geometry
      5. 11.3.2 Vertex expansion
      6. 11.3.3 Large-spin expansion
      7. 11.4 Maximal acceleration
      8. 11.5 Physical predictions?
    3. 12 Scattering
      1. 12.1 n-Point functions in general covariant theories
      2. 12.2 Graviton propagator
    4. 13 Final remarks
      1. 13.1 Brief historical note
      2. 13.2 What is missing
  12. References
  13. Index