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Part Ⅰ Relativistic foundations 1

1 General ideas and heuristic picture 3

1.1 The problem of quantum gravity 3

1.1.1 Unfinished revolution 3

1.1.2 How to search for quantum gravity? 4

1.1.3 The physical meaning of general relativity 9

1.1.4 Background-independent quantum field theory 10

1.2 Loop quantum gravity 13

1.2.1 Why loops? 14

1.2.2 Quantum space:spin networks 17

1.2.3 Dynamics in background-independent QFT 22

1.2.4 Quantum spacetime:spinfoam 26

1.3 Conceptual issues 28

1.3.1 Physics without time 29

2 General Relativity 33

2.1 Formalism 33

2.1.1 Gravitational field 33

2.1.2 "Matter" 37

2.1.3 Gauge invariance 40

2.1.4 Physical geometry 42

2.1.5 Holonomy and metric 44

2.2 The conceptual path to the theory 48

2.2.1 Einstein's first problem:a field theory for the newtonian interaction 48

2.2.2 Einstein's second problem:relativity of motion 52

2.2.3 The key idea 56

2.2.4 Active and passive diffeomorphisms 62

2.2.5 General covariance 65

2.3 Interpretation 71

2.3.1 Observables,predictions and coordinates 71

2.3.2 The disappearance of spacetime 73

2.4 Complements 75

2.4.1 Mach principles 75

2.4.2 Relationalism versus substantivalism 76

2.4.3 Has general covariance any physical content?Kretschmann's objection 78

2.4.4 Meanings of time 82

2.4.5 Nonrelativistic coordinates 87

2.4.6 Physical coordinates and GPS observables 88

3 Mechanics 98

3.1 Nonrelativistic mechanics:mechanics is about time evolution 98

3.2 Relativistic mechanics 105

3.2.1 Structure of relativistic systems:partial observables,relativistic states 105

3.2.2 Hamiltonian mechanics 108

3.2.3 Nonrelativistic systems as a special case 114

3.2.4 Discussion:mechanics is about relations between observables 118

3.2.5 Space of boundary data G and Hamilton function S 120

3.2.6 Evolution parameters 126

3.2.7 Complex variables and reality conditions 128

3.3 Field theory 129

3.3.1 Partial observables in field theory 130

3.3.2 Relativistic hamiltonian mechanics 131

3.3.3 The space of boundary data G and the Hamilton function S 134

3.3.4 Hamilton-Jacobi 137

3.4 Thermal time hypothesis 140

4 Hamiltonian general relativity 145

4.1 Einstein-Hamilton-Jacobi 145

4.1.1 3d fields."The length of the electric field is the area" 147

4.1.2 Hamilton function of GR and its physical meaning 151

4.2 Euclidean GR and real connection 153

4.2.1 Euclidean GR 153

4.2.2 Lorentzian GR with a real connection 155

4.2.3 Barbero connection and Immirzi parameter 156

4.3 Hamiltonian GR 157

4.3.1 Version 1:real SO(3,1)connection 157

4.3.2 Version 2:complex SO(3)connection 157

4.3.3 Configuration space and hamiltonian 158

4.3.4 Derivation of the Hamilton-Jacobi formalism 159

4.3.5 Reality conditions 162

5 Quantum mechanics 164

5.1 Nonrelativistic QM 164

5.1.1 Propagator and spacetime states 166

5.1.2 Kinematical state space K and"projector"P 169

5.1.3 Partial observables and probabilities 172

5.1.4 Boundary state space K and covariant vacuum ｜0〉 174

5.1.5 Evolving constants of motion 176

5.2 Relativistic QM 177

5.2.1 General structure 177

5.2.2 Quantization and classical limit 179

5.2.3 Examples:pendulum and timeless double pendulum 180

5.3 Quantum field theory 184

5.3.1 Functional representation 186

5.3.2 Field propagator between parallel boundary surfaces 190

5.3.3 Arbitrary boundary surfaces 193

5.3.4 What is a particle? 195

5.3.5 Boundary state space K and covariant vacuum ｜0〉 197

5.3.6 Lattice scalar product,intertwiners and spin network states 198

5.4 Quantum gravity 200

5.4.1 Transition amplitudes in quantum gravity 200

5.4.2 Much ado about nothing:the vacuum 202

5.5 Complements 204

5.5.1 Thermal time hypothesis and Tomita flow 204

5.5.2 The"choice"of the physical scalar product 206

5.5.3 Reality conditions and scalar product 208

5.6 Relational interpretation of quantum theory 209

5.6.1 The observer observed 210

5.6.2 Facts are interactions 215

5.6.3 Information 218

5.6.4 Spacetime relationalism versus quantum relationalism 220

Part Ⅱ Loop quantum gravity 223

6 Quantum space 225

6.1 Structure of quantum gravity 225

6.2 The kinematical state space K 226

6.2.1 Structures in K 230

6.2.2 Invariances of the scalar product 231

6.2.3 Gauge-invariant and diffeomorphism-invariant states 233

6.3 Internal gauge invariance.The space K0 234

6.3.1 Spin network states 234

6.3.2 Details about spin networks 236

6.4 Diffeomorphism invariance.The space Kdiff 238

6.4.1 Knots and s-knot states 240

6.4.2 The Hilbert space Kdiff is separable 241

6.5 Operators 242

6.5.1 The connection A 242

6.5.2 The conjugate momentum E 243

6.6 Operators on K0 246

6.6.1 The operator A(S) 246

6.6.2 Quanta of area 249

6.6.3 n-hand operators and recoupling theory 250

6.6.4 Degenerate sector 253

6.6.5 Quanta of volume 259

6.7 Quantum geometry 262

6.7.1 The texture of space:weaves 268

7 Dynamics and matter 276

7.1 Hamiltonian operator 277

7.1.1 Finiteness 280

7.1.2 Matrix elements 282

7.1.3 Variants 284

7.2 Matter:kinematics 286

7.2.1 Yang-Mills 286

7.2.2 Fermions 287

7.2.3 Scalars 288

7.2.4 The quantum states of space and matter 289

7.3 Matter:dynamics and finiteness 289

7.4 Loop quantum gravity 291

7.4.1 Variants 293

8 Applications 296

8.1 Loop quantum cosmology 296

8.1.1 Inflation 301

8.2 Black-hole thermodynamics 301

8.2.1 The statistical ensemble 303

8.2.2 Derivation of the Bekenstein-Hawking entropy 308

8.2.3 Ringing modes frequencies 311

8.2.4 The Bekenstein-Mukhanov effect 312

8.3 Observable effects 315

9 Quantum spacetime:spinfoams 320

9.1 From loops to spinfoams 321

9.2 Spinfoam formalism 327

9.2.1 Boundaries 328

9.3 Models 329

9.3.1 3d quantum gravity 331

9.3.2 BF theory 340

9.3.3 The spinfoam/GFT duality 343

9.3.4 BC models 348

9.3.5 Group field theory 356

9.3.6 Lorentzian models 359

9.4 Physics from spinfoams 361

9.4.1 Particles'scattering and Minkowski vacuum 363

10 Conclusion 366

10.1 The physical picture of loop gravity 366

10.1.1 GR and QM 366

10.1.2 Observables and predictions 367

10.1.3 Space,time and unitarity 368

10.1.4 Quantum gravity and other open problems 370

10.2 What has been achieved and what is missing? 371

Part Ⅲ Appendices 375

Appendix A Groups and recoupling theory 377

A1 SU(2):spinors,intertwiners,n-j symbols 377

A2 Recoupling theory 383

A2.1 Penrose binor calculus 383

A2.2 KL recoupling theory 385

A2.3 Normalizations 388

A3 SO(n)and simple representations 390

Appendix B History 393

B1 Three main directions 393

B2 Five periods 396

B2.1 The Prehistory:1930-1957 398

B2.2 The Classical Age:1958-1969 400

B2.3 The Middle Ages:1970-1983 403

B2.4 The Renaissance:1984-1994 407

B2.5 Nowadays:1995- 410

B3 The divide 412

Appendix C On method and truth 415

C1 The cumulative aspects of scientific knowledge 415

C2 On realism 420

C3 On truth 422

References 424

Index 450