Modern Canonical Quantum General Relativity

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 7
Copyright......Page 8
Contents......Page 11
Foreword......Page 19
Preface......Page 21
Notation and conventions......Page 25
Why quantum gravity in the twenty-first century?......Page 29
The role of background independence......Page 36
1. Perturbative approach: string theory......Page 39
– Vacuum degeneracy......Page 40
– Phenomenology match......Page 41
– The landscape......Page 42
– AdS/CFT and cosmology......Page 43
2a. Canonical Quantum General Relativity......Page 44
2b. Continuum functional integral approach......Page 46
2c. Lattice quantum gravity......Page 47
2d. Covariant canonical approaches......Page 48
2e. Non-orthodox approaches......Page 50
Motivation for canonical quantum general relativity......Page 51
(A) Classical formulation......Page 53
(B) Connection formulation......Page 55
(C) General canonical quantisation......Page 56
(D) Application to General Relativity......Page 57
(E) Applications......Page 58
(F) Mathematical tools......Page 62
Part I Classical foundations, interpretation and the canonical quantisation programme......Page 65
1.1 The ADM action......Page 67
1.2 Legendre transform and Dirac analysis of constraints......Page 74
1.3 Geometrical interpretation of the gauge transformations......Page 78
1.4 Relation between the four-dimensional diffeomorphism group and the transformations generated by the constraints......Page 84
1.5.1 Boundary conditions......Page 88
1.5.2 Symmetries and gauge transformations......Page 93
2 The problem of time, locality and the interpretation of quantum mechanics......Page 102
2.1 The classical problem of time: Dirac observables......Page 103
1. Case of a true Hamiltonian......Page 104
2. Case: single constraint......Page 105
4. Case: several, mutually non-Poisson-commuting constraints......Page 107
2.2 Partial and complete observables for general constrained systems......Page 109
2.2.1 Partial and weak complete observables......Page 110
2.2.2 Poisson algebra of Dirac observables......Page 113
2.2.3 Evolving constants......Page 117
2.2.4 Reduced phase space quantisation of the algebra of Dirac observables and unitary implementation of the multi-.ngered time evolution......Page 118
2.3 Recovery of locality in General Relativity......Page 121
2.4.1 Physical inner product......Page 123
2.4.2 Interpretation of quantum mechanics......Page 126
3 The programme of canonical quantisation......Page 135
I. Classical Poisson -subalgebra Beta......Page 136
II. Quantum -algebra…......Page 137
III. Representations of…......Page 139
IV. Solving the quantum constraints, physical inner product and Dirac observables......Page 140
V. Quantum anomalies and classical limit......Page 143
4.1 Historical overview......Page 146
4.2.1 Extension of the ADM phase space......Page 151
4.2.2 Canonical transformation on the extended phase space......Page 154
Affine transformation......Page 155
Part II Foundations of modern canonical Quantum General Relativity......Page 167
5.1 Outline and historical overview......Page 169
1. Formal solutions to the Hamiltonian constraint in the connection representation......Page 171
5. Model systems......Page 172
(ii) 1993–94: measure theory, projective techniques......Page 173
(iv) 1994–2001: relation with constructive quantum (gauge) .eld theory......Page 174
(v) 1995: Hilbert space, adjointness relations and canonical commutation relations......Page 175
(vii) 1995–99: kinematical geometrical operators......Page 176
(ix) 1996–98: Hamiltonian constraint and matter coupling......Page 177
(x) 1997–2006: path integral formulation: spin foam models......Page 178
(xii) 1999–2001: categories and groupoids, hyphs and gauge orbit structure of…......Page 179
(xiv) 2000–2006: semiclassical states......Page 180
(xv) 2000–2006: loop quantum cosmology......Page 181
(xvii) 2002–2006: algebraic methods and representation theory......Page 182
(xviii) 2003–2006: physical inner product and Dirac observables: Master Constraint programme......Page 183
6.1 Motivation for the choice of P......Page 185
6.2.1 Semianalytic paths and holonomies......Page 190
6.2.2 A natural topology on the space of generalised connections......Page 196
6.2.3 Gauge invariance: distributional gauge transformations......Page 203
6.2.4 The C. algebraic viewpoint and cylindrical functions......Page 211
6.3 Definition of P: (2) surfaces, electric fields, fluxes and vector fields......Page 219
6.4 Definition of P: (3) regularisation of the holonomy–flux Poisson algebra......Page 222
6.5 Definition of P: (4) Lie algebra of cylindrical functions and flux vector fields......Page 230
7.1 Definition of A......Page 234
7.2 (Generalised) bundle automorphisms of A......Page 237
8.1 General considerations......Page 240
8.2 Uniqueness proof: (1) existence......Page 247
8.2.1 Regular Borel measures on the projective limit: the uniform measure......Page 248
8.2.2 Functional calculus on a projective limit......Page 254
8.2.3 + Density and support properties of…......Page 261
8.2.4 Spin-network functions and loop representation......Page 265
8.2.5 Gauge and diffeomorphism invariance of Muo......Page 270
8.2.6 + Ergodicity of MU with respect to spatial diffeomorphisms......Page 273
8.2.7 Essential self-adjointness of electric flux momentum operators......Page 274
8.3 Uniqueness proof: (2) uniqueness......Page 275
Step I......Page 276
Step II......Page 278
Step IV......Page 279
8.4 Uniqueness proof: (3) irreducibility......Page 280
Non-Abelian factor......Page 282
9.1.1 Derivation of the GauBeta constraint operator......Page 292
9.1.2 Complete solution of the GauBeta constraint......Page 294
9.2.1 Derivation of the spatial di.eomorphism constraint operator......Page 297
9.2.2 General solution of the spatial di.eomorphism constraint......Page 299
10.1 Outline of the construction......Page 307
10.2 Heuristic explanation for UV finiteness due to background independence......Page 310
10.3 Derivation of the Hamiltonian constraint operator......Page 314
10.4.1 Concrete implementation......Page 319
10.4.2 Operator limits......Page 324
10.4.3 Commutator algebra......Page 328
10.4.4 The quantum Dirac algebra......Page 337
10.5 The kernel of the Wheeler–DeWitt constraint operator......Page 339
3. Absence of true Lie algebra......Page 345
10.6.2 Definition of the Master Constraint......Page 348
First term......Page 353
10.6.3 Physical inner product and Dirac observables......Page 354
10.6.4 Extended Master Constraint......Page 357
10.6.5 Algebraic Quantum Gravity (AQG)......Page 359
10.7.1.1 The general scheme......Page 362
10.7.1.2 Wick transform for quantum gravity......Page 366
10.7.2 Testing the new regularisation technique by models of quantum gravity......Page 368
10.7.3 Quantum Poincar´e algebra......Page 369
10.7.4 Vasiliev invariants and discrete quantum gravity......Page 372
11 Step V: semiclassical analysis......Page 373
11.1 + Weaves......Page 377
11.2 Coherent states......Page 381
11.2.1 Semiclassical states and coherent states......Page 382
11.2.2 Construction principle: the complexi.er method......Page 384
11.2.3 Complexi.er coherent states for diffeomorphism-invariant theories of connections......Page 390
11.2.4 Concrete example of complexifier......Page 395
I. Non-graph-changing approach......Page 404
3. Continuum limit......Page 405
4. Staircase problem......Page 406
2. Kinematical approach......Page 407
(C) Shadows......Page 408
11.2.6 + The in.nite tensor product extension......Page 413
11.3 Graviton and photon Fock states from…......Page 418
Part III Physical applications......Page 425
12 Extension to standard matter......Page 427
12.1 The classical standard model coupled to gravity......Page 428
12.1.1 Fermionic and Einstein contribution......Page 429
12.1.2 Yang–Mills and Higgs contribution......Page 433
12.2.1 Fermionic sector......Page 434
12.2.2.1 Diffeomorphism-invariant Fock representation......Page 439
12.2.2.2 Point holonomies......Page 443
12.2.3 Gauge and diffeomorphism-invariant subspace......Page 445
12.3 Quantisation of matter Hamiltonian constraints......Page 446
12.3.1 Quantisation of Einstein–Yang–Mills theory......Page 447
12.3.2 Fermionic sector......Page 450
12.3.3 Higgs sector......Page 453
12.3.4 A general quantisation scheme......Page 457
13 Kinematical geometrical operators......Page 459
13.1 Derivation of the area operator......Page 460
13.2 Properties of the area operator......Page 462
13.3 Derivation of the volume operator......Page 466
13.4.1 Cylindrical consistency......Page 475
13.4.3 Discreteness and anomaly-freeness......Page 476
13.4.4 Matrix elements......Page 477
13.5 Uniqueness of the volume operator, consistency with the .ux operator and pseudo-two-forms......Page 481
13.6 Spatially di.eomorphism-invariant volume operator......Page 483
14.1 Heuristic motivation from the canonical framework......Page 486
14.2 Spin foam models from BF theory......Page 490
14.3.1 Plebanski action and simplicity constraints......Page 494
Step II......Page 496
Step III......Page 497
14.3.2 Discretisation theory......Page 500
14.3.3 Discretisation and quantisation of BF theory......Page 504
14.3.4 Imposing the simplicity constraints......Page 510
14.3.5 Summary of the status of the Barrett–Crane model......Page 522
14.4 Triangulation dependence and group field theory......Page 523
(i) Spin foams and canonical theory......Page 530
(ii) Spin foam models from the Master Constraint......Page 532
(iii) Semiclassical analysis......Page 533
(iv) Sum over triangulations......Page 534
(v) McDowell–Mansouri action......Page 535
(vi) Other aspects of spin foam models......Page 536
(vii) Graviton propagator......Page 537
15 Quantum black hole physics......Page 539
15.1.1 Null geodesic congruences......Page 542
15.1.2 Event horizons, trapped surfaces and apparent horizons......Page 545
15.1.3 Trapping, dynamical, non-expanding and (weakly) isolated horizons......Page 547
15.1.4 Spherically symmetric isolated horizons......Page 554
15.1.5 Boundary symplectic structure for SSIHs......Page 563
15.2 Quantisation of the surface degrees of freedom......Page 568
15.2.1 Quantum U(1) Chern–Simons theory with punctures......Page 569
15.3 Implementing the quantum boundary condition......Page 574
15.4 Implementation of the quantum constraints......Page 576
15.4.1 Remaining U(1) gauge transformations......Page 577
15.5 Entropy counting......Page 578
15.6 Discussion......Page 585
(ii) Emission spectrum......Page 586
(iii) Quasinormal modes......Page 587
(iv) Open problems......Page 588
(v) First principle calculation......Page 589
16.1 Quantum gauge fixing......Page 590
16.2 Loop Quantum Cosmology......Page 591
17 Loop Quantum Gravity phenomenology......Page 600
Part IV Mathematical tools and their connection to physics......Page 603
18.1 Generalities......Page 605
18.2 Specific results......Page 609
19.1.1 Manifolds......Page 613
19.1.2 Passive and active di.eomorphisms......Page 615
19.1.3 Differential calculus......Page 618
19.2 Riemannian geometry......Page 634
19.3.1 Symplectic geometry......Page 642
19.3.2 Symplectic reduction......Page 644
19.3.3 Symplectic group actions......Page 649
19.4 Complex, Hermitian and Kähler manifolds......Page 651
20.1 Semianalytic structures on Rn......Page 655
20.2 Semianalytic manifolds and submanifolds......Page 659
21.1 General fibre bundles and principal fibre bundles......Page 662
21.2 Connections on principal fibre bundles......Page 664
22.1 The groupoid of equivariant maps......Page 672
22.2 Holonomies and transition functions......Page 675
23.1 Prequantisation......Page 680
23.2 Polarisation......Page 690
23.3 Quantisation......Page 696
24.1 The Dirac algorithm......Page 699
24.2 First- and second-class constraints and the Dirac bracket......Page 702
25.1 Generalities and the Riesz–Markov theorem......Page 708
25.2 Measure theory and ergodicity......Page 715
26.1 Metric spaces and normed spaces......Page 717
26.2 Hilbert spaces......Page 719
26.3 Banach spaces......Page 721
26.5 Locally convex spaces......Page 722
26.6 Bounded operators......Page 723
26.7 Unbounded operators......Page 725
26.8 Quadratic forms......Page 727
27.1 Banach algebras and their spectra......Page 729
27.2 The Gel’fand transform and the Gel’fand isomorphism......Page 737
28.1 Definition and properties......Page 741
28.2 Analogy with loop quantum gravity......Page 743
II. Representations......Page 747
IV. Automorphisms......Page 748
29.2 Spectral theorem, spectral measures, projection valued measures, functional calculus......Page 751
30.1 RAQ......Page 757
30.2 Master Constraint Programme (MCP) and DID......Page 763
2. Structure functions......Page 772
4. Mixed spectrum......Page 773
31.1 Representations and Haar measures......Page 774
31.2 The Peter and Weyl theorem......Page 780
32.1 Basics of the representation theory of SU(2)......Page 783
32.2 Spin-network functions and recoupling theory......Page 785
32.3 Action of holonomy operators on spin-network functions......Page 790
32.4 Examples of coherent state calculations......Page 793
33.1 Infinite-dimensional (symplectic) manifolds......Page 798
References......Page 803
Index......Page 837