Modern Canonical Quantum General Relativity

Cover......Page 1
Abstract......Page 2
Series contents......Page 3
Title......Page 6
Date-line......Page 7
Secondary abstract......Page 8
Contents......Page 10
Foreword, by Chris Isham......Page 18
Preface......Page 20
Notation and conventions......Page 24
Why quantum gravity in the twenty-first century?......Page 28
The role of background independence ......Page 35
Approaches to quantum gravity ......Page 38
Motivation for canonical quantum general relativity ......Page 50
Outline of the book ......Page 52
I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE CANONICAL QUANTISATION PROGRAMME ......Page 64
1.1 The ADM action ......Page 66
1.2 Legendre transform and Dirac analysis of constraints ......Page 73
1.3 Geometrical interpretation of the gauge transformations ......Page 77
1.4 Relation between the four-dimensional diffeomorphism group and the transformations generated by the constraints ......Page 83
1.5.1 Boundary conditions ......Page 87
1.5.2 Symmetries and gauge transformations ......Page 92
2 The problem of time, locality and the interpretation of quantum mechanics ......Page 101
2.1 The classical problem of time: Dirac observables ......Page 102
2.2 Partial and complete observables for general constrained systems ......Page 108
2.2.1 Partial and weak complete observables ......Page 109
2.2.2 Poisson algebra of Dirac observables ......Page 112
2.2.3 Evolving constants ......Page 116
2.2.4 Reduced phase space quantisation of the algebra of Dirac observables and unitary implementation of the multi-fingered time evolution ......Page 117
2.3 Recovery of locality in General Relativity ......Page 120
2.4.1 Physical inner product ......Page 122
2.4.2 Interpretation of quantum mechanics ......Page 125
3 The programme of canonical quantisation ......Page 134
3.1 The programme ......Page 135
4.1 Historical overview ......Page 145
4.2.1 Extension of the ADM phase space ......Page 150
4.2.2 Canonical transformation on the extended phase space ......Page 153
II FOUNDATIONS OF MODERN CANONICAL QUANTUM GENERAL RELATIVITY ......Page 166
5.1 Outline and historical overview ......Page 168
6.1 Motivation for the choice of $\\mathfrac{B}$ ......Page 184
6.2.1 Semianalytic paths and holonomies ......Page 189
6.2.2 A natural topology on the space of generalised connections ......Page 195
6.2.3 Gauge invariance: distributional gauge transformations ......Page 202
6.2.4 The $C^*$ algebraic viewpoint and cylindrical functions ......Page 210
6.3 Definition of $\\mathfrac{B}$: (2) surfaces, electric fields, fluxes and vector fields ......Page 218
6.4 Definition of $\\mathfrac{B}$: (3) regularisation of the holonomy-flux Poisson algebra ......Page 221
6.5 Definition of $\\mathfrac{B}$: (4) Lie algebra of cylindrical functions and flux vector fields ......Page 229
7.1 Definition of $\\mathfrac{U}$ ......Page 233
7.2 (Generalised) bundle automorphisms of $\\mathfrac{U}$ ......Page 236
8.1 General considerations ......Page 239
8.2 Uniqueness proof: (1) existence ......Page 246
8.2.1 Regular Borel measures on the projective limit: the uniform measure ......Page 247
8.2.2 Functional calculus on a projective limit ......Page 253
8.2.3 + Density and support properties of $\\mathcal{A}$, $\\mathcal{A}/\\mathcal{G}$ with respect to $\\overline\\mathcal{A}$, $\\overline\\mathcal{A}/\\mathcal{G}$ ......Page 260
8.2.4 Spin-network functions and loop representation ......Page 264
8.2.5 Gauge and diffeomorphism invariance of $\\mu_0$ ......Page 269
8.2.6 + Ergodicity of $\\mu_0$ with respect to spatial diffeomorphisms ......Page 272
8.2.7 Essential self-adjointness of electric flux momentum operators ......Page 273
8.3 Uniqueness proof: (2) uniqueness ......Page 274
8.4 Uniqueness proof: (3) irreducibility ......Page 279
9.1.1 Derivation of the Gau\\ss constraint operator ......Page 291
9.1.2 Complete solution of the Gau\\ss constraint ......Page 293
9.2.1 Derivation of the spatial diffeomorphism constraint operator ......Page 296
9.2.2 General solution of the spatial diffeomorphism constraint ......Page 298
10.1 Outline of the construction ......Page 306
10.2 Heuristic explanation for UV fmiteness due to background independence ......Page 309
10.3 Derivation of the Hamiltonian constraint operator ......Page 313
10.4.1 Concrete implementation ......Page 318
10.4.2 Operator limits ......Page 323
10.4.3 Commutator algebra ......Page 327
10.4.4 The quantum Dirac algebra ......Page 336
10.5 The kernel of the Wheeler-DeWitt constraint operator ......Page 338
10.6.1 Motivation for the Master Constraint Programme in General Relativity ......Page 344
10.6.2 Definition of the Master Constraint ......Page 347
10.6.3 Physical inner product and Dirac observables ......Page 353
10.6.4 Extended Master Constraint ......Page 356
10.6.5 Algebraic Quantum Gravity (AQG) ......Page 358
10.7.1 The Wick transform ......Page 361
10.7.2 Testing the new regularisation technique by models of quantum gravity ......Page 367
10.7.3 Quantum Poincare algebra ......Page 368
10.7.4 Vasiliev invariants and discrete quantum gravity ......Page 371
11 Step V: semiclassical analysis ......Page 372
11.1 + Weaves ......Page 376
11.2 Coherent states ......Page 380
11.2.1 Semiclassical states and coherent states ......Page 381
11.2.2 Construction principle: the complexifier method ......Page 383
11.2.3 Complexifier coherent states for diffeomorphism-invariant theories of connections ......Page 389
11.2.4 Concrete example of complexifier ......Page 394
11.2.5 Semiclassical limit of loop quantum gravity: graph-changing operators, shadows and diffeomorphism-invariant coherent states ......Page 403
11.2.6 + The infinite tensor product extension ......Page 412
11.3 Graviton and photon Fock states from $L_2(\\overline\\mathcal{A},d\\mu_0)$ ......Page 417
III PHYSICAL APPLICATIONS ......Page 424
12 Extension to standard matter ......Page 426
12.1 The classical standard model coupled to gravity ......Page 427
12.1.1 Fermionic and Einstein contribution ......Page 428
12.1.2 Yang-Mills and Higgs contribution ......Page 432
12.2.1 Fermionic sector ......Page 433
12.2.2 Higgs sector ......Page 438
12.2.3 Gauge and diffeomorphism-invariant subspace ......Page 444
12.3 Quantisation of matter Hamiltonian constraints ......Page 445
12.3.1 Quantisation of Einstein-Yang-Mills theory ......Page 446
12.3.2 Fermionic sector ......Page 449
12.3.3 Higgs sector ......Page 452
12.3.4 A general quantisation scheme ......Page 456
13 Kinematical geometrical operators ......Page 458
13.1 Derivation of the area operator ......Page 459
13.2 Properties of the area operator ......Page 461
13.3 Derivation of the volume operator ......Page 465
13.4.1 Cylindrical consistency ......Page 474
13.4.3 Discreteness and anomaly-freeness ......Page 475
13.4.4 Matrix elements ......Page 476
13.5 Uniqueness of the volume operator, consistency with the flux operator and pseudo-two-forms ......Page 480
13.6 Spatially diffeomorphism-invariant volume operator ......Page 482
14.1 Heuristic motivation from the canonical framework ......Page 485
14.2 Spin foam models from BF theory ......Page 489
14.3.1 Plebanski action and simplicity constraints ......Page 493
14.3.2 Discretisation theory ......Page 499
14.3.3 Discretisation and quantisation of BF theory ......Page 503
14.3.4 Imposing the simplicity constraints ......Page 509
14.3.5 Summary of the status of the Barrett-Crane model ......Page 521
14.4 Triangulation dependence and group field theory ......Page 522
14.5 Discussion ......Page 529
15 Quantum black hole physics ......Page 538
15.1.1 Null geodesic congruences ......Page 541
15.1.2 Event horizons, trapped surfaces and apparent horizons ......Page 544
15.1.3 Trapping, dynamical, non-expanding and (weakly) isolated horizons ......Page 546
15.1.4 Spherically symmetric isolated horizons ......Page 553
15.1.5 Boundary symplectic structure for SSIHs ......Page 562
15.2 Quantisation of the surface degrees of freedom ......Page 567
15.2.1 Quantum U(l) Chern-Simons theory with punctures ......Page 568
15.3 Implementing the quantum boundary condition ......Page 573
15.4 Implementation of the quantum constraints ......Page 575
15.4.1 Remaining U(l) gauge transformations ......Page 576
15.5 Entropy counting ......Page 577
15.6 Discussion ......Page 584
16.1 Quantum gauge fixing ......Page 589
16.2 Loop Quantum Cosmology ......Page 590
17 Loop Quantum Gravity phenomenology ......Page 599
IV MATHEMATICAL TOOLS AND THEIR CONNECTION TO PHYSICS ......Page 602
18.1 Generalities ......Page 604
18.2 Specific results ......Page 608
19.1.1 Manifolds ......Page 612
19.1.2 Passive and active diffeomorphisms ......Page 614
19.1.3 Differential calculus ......Page 617
19.2 Riemannian geometry ......Page 633
19.3.1 Symplectic geometry ......Page 641
19.3.2 Symplectic reduction ......Page 643
19.3.3 Symplectic group actions ......Page 648
19.4 Complex, Hermitian and Kahler manifolds ......Page 650
20.1 Semianalytic structures on $\\mathds{R}^n$ ......Page 654
20.2 Semianalytic manifolds and submanifolds ......Page 658
21.1 General fibre bundles and principal fibre bundles ......Page 661
21.2 Connections on principal fibre bundles ......Page 663
22.1 The groupoid of equivariant maps ......Page 671
22.2 Holonomies and transition functions ......Page 674
23.1 Prequantisation ......Page 679
23.2 Polarisation ......Page 689
23.3 Quantisation ......Page 695
24.1 The Dirac algorithm ......Page 698
24.2 First- and second-class constraints and the Dirac bracket ......Page 701
25.1 Generalities and the Riesz-Markov theorem ......Page 707
25.2 Measure theory and ergodicity ......Page 714
26.1 Metric spaces and normed spaces ......Page 716
26.2 Hubert spaces ......Page 718
26.3 Banach spaces ......Page 720
26.5 Locally convex spaces ......Page 721
26.6 Bounded operators ......Page 722
26.7 Unbounded operators ......Page 724
26.8 Quadratic forms ......Page 726
27.1 Banach algebras and their spectra ......Page 728
27.2 The Gel\'fand transform and the Gel\'fand isomorphism ......Page 736
28.1 Definition and properties ......Page 740
28.2 Analogy with loop quantum gravity ......Page 742
29.1 Operator *-algebras, representations and GNS construction ......Page 746
29.2 Spectral theorem, spectral measures, projection valued measures, functional calculus ......Page 750
30.1 RAQ ......Page 756
30.2 Master Constraint Programme (MCP) and DID ......Page 762
31.1 Representations and Haar measures ......Page 773
31.2 The Peter and Weyl theorem ......Page 779
32.1 Basics of the representation theory of SU(2) ......Page 782
32.2 Spin-network functions and recoupling theory ......Page 784
32.3 Action of holonomy operators on spin-network functions ......Page 789
32.4 Examples of coherent state calculations ......Page 792
33.1 Infinite-dimensional (symplectic) manifolds ......Page 797
References ......Page 802
Index ......Page 836