MODERN CANONICAL QUANTUM GENERAL RELATIVITY Modern physics rests on two fundamental building blocks: general relativity and quantum theory General relativity is a geometric interpretation of gravity, while quantum theory governs the microscopic behaviour of matter According to Einstein’s equations, geometry is curved when and where matter is localized Therefore, in general relativity, geometry is a dynamical quantity that cannot be prescribed a priori but is in interaction with matter The equations of nature are background independent in this sense; there is no space-time geometry on which matter propagates without backreaction of matter on geometry Since matter is described by quantum theory, which in turn couples to geometry, we need a quantum theory of gravity The absence of a viable quantum gravity theory to date is due to the fact that quantum (field) theory as currently formulated assumes that a background geometry is available, thus being inconsistent with the principles of general relativity In order to construct quantum gravity, one must reformulate quantum theory in a background-independent way Modern Canonical Quantum General Relativity is about one such candidate for a background-independent quantum gravity theory: loop quantum gravity This book provides a complete treatise of the canonical quantization of general relativity The focus is on detailing the conceptual and mathematical framework, describing the physical applications, and summarizing the status of this programme in its most popular incarnation: loop quantum gravity Mathematical concepts and their relevance to physics are provided within this book, so it is suitable for graduate students and researchers with a basic knowledge of quantum field theory and general relativity T h o m a s T h i e m a n n is Staff Scientist at the Max Planck Institut fă ur Gravitationsphysik (Albert Einstein Institut), Potsdam, Germany He is also a long-term researcher at the Perimeter Institute for Theoretical Physics and Associate Professor at the University of Waterloo, Canada Thomas Thiemann obtained his Ph.D in theoretical physics from the Rheinisch-Westfă alisch Technische Hochschule, Aachen, Germany He held two-year postdoctoral positions at The Pennsylvania State University and Harvard University As of 2005 he holds a guest professor position at Beijing Normal University, China CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General editors: P V Landshoff, D R Nelson, S Weinberg S J Aarseth Gravitational N-Body Simulations J Ambjørn, B Durhuus and T Jonsson Quantum Geometry: A Statistical Field Theory Approach A M Anile Relativistic Fluids and Magneto-Fluids: With Applications in Astrophysics and Plasma Physics J A de Azc´ arrage and J M Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O Babelon, D Bernard and M Talon Introduction to Classical Integrable Systems† F Bastianelli and P van Nieuwenhuizen Path Integrals and Anomalies in Curved Space V Belinkski and E Verdaguer Gravitational Solitons J Bernstein Kinetic Theory in the Expanding Universe G F Bertsch and R A Broglia Oscillations in Finite Quantum Systems N D Birrell and P C W Davies Quantum Fields in Curved space† M Burgess Classical Covariant Fields S Carlip Quantum Gravity in + Dimensions† P Cartier and C DeWitt-Morette Functional Integration: Action and Symmetries J C Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion† M Creutz Quarks, Gluons and Lattices† P D D’Eath Supersymmetric Quantum Cosmology F de Felice and C J S Clarke Relativity on Curved Manifolds† B S DeWitt Supermanifolds, 2nd edition† P G O Freund Introduction to Supersymmetry† J Fuchs Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory† J Fuchs and C Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y Fujii and K Maeda The Scalar–Tensor Theory of Gravitation A S Galperin, E A Ivanov, V I Orievetsky and E S Sokatchev Harmonic Superspace R Gambini and J Pullin Loops, Knots, Gauge Theories and Quantum Gravity† T Gannon Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics M Gă ockeler and T Schă ucker Dierential Geometry, Gauge Theories and Gravity† C G´ omez, M Ruiz-Altaba and G Sierra Quantum Groups in Two-dimensional Physics M B Green, J H Schwarz and E Witten Superstring Theory, Volume 1: Introduction† M B Green, J H Schwarz and E Witten Superstring Theory, Volume 2: Loop Amplitudes, Anomalies and Phenomenology† V N Gribov The Theory of Complex Angular Momenta: Gribov Lectures an Theoretical Physics S W Hawking and G F R Ellis The Large-Scale Structure of Space-Time† F Iachello and A Arima The Interacting Boson Model F Iachello and P van Isacker The Interacting Boson–Fermion Model C Itzykson and J.-M Drouffe Statistical Field Theory, Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C Itzykson and J.-M Drouffe Statistical Field Theory, Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems† C Johnson D-Branes† J I Kapusta and C Gale Finite-Temperature Field Theory, 2nd edition V E Korepin, A G Izergin and N M Boguliubov The Quantum Inverse Scattering Method and Correlation Functions M Le Bellac Thermal Field Theory† Y Makeenko Methods of Contemporary Gauge Theory N Manton and P Sutcliffe Topological Solitons N H March Liquid Metals: Concepts and Theory I M Montvay and G Mă unster Quantum Fields on a Lattice† L O’Raifeartaigh Group Structure of Gauge Theories† T Ort´in Gravity and Strings A Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† R Penrose and W Rindler Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields† R Penrose and W Rindler Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry† S Pokorski Gauge Field Theories, 2nd edition J Polchinski String Theory, Volume 1: An Introduction to the Bosonic String J Polchinski String Theory, Volume 2: Superstring Theory and Beyond V N Popov Functional Integrals and Collective Excitations† R J Rivers Path Integral Methods in Quantum Field Theory† R G Roberts The Structure of the Proton: Deep Inelastic Scattering† C Rovelli Quantum Gravity W C Saslaw Gravitational Physics of Stellar and Galactic Systems† H Stephani, D Kramer, M A H MacCallum, C Hoenselaers and E Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition J M Stewart Advanced General Relativity† T Thiemann Modern Canonical Quantum General Relativity A Vilenkin and E P S Shellard Cosmic Strings and Other Topological Defects† R S Ward and R O Wells Jr Twistor Geometry and Field Theory† J R Wilson and G J Mathews Relativistic Numerical Hydrodynamics † Issued as a paperback Modern Canonical Quantum General Relativity THOMAS THIEMANN Max Planck Institut fă ur Gravitationsphysik, Germany h G c CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521842631 © T Thiemann 2007 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2007 eBook (NetLibrary) ISBN-13 978-0-511-36743-4 ISBN-10 0-511-36743-0 eBook (NetLibrary) ISBN-13 ISBN-10 hardback 978-0-521-84263-1 hardback 0-521-84263-8 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Figure Copyright: Max Planck Institute for Gravitational Physics (Albert Einstein Institute), MildeMarketing Science Communication, Exozet To see the animation, please visit the URL http://www.einstein-online.info/de/vertiefung/Spinnetzwerke/ index.html Quantum spin dynamics This is a still from an animation which illustrates the dynamical evolution of quantum geometry in Loop Quantum Gravity (LQG), which is a particular incarnation of canonical Quantum General Relativity The faces of the tetrahedra are elementary excitations (atoms) of geometry Each face is coloured, where red and violet respectively means that the face carries low or high area respectively The colours or areas are quantised in units of the Planck area 2P ≈ 10−66 cm2 Thus the faces not have area as they appear to have in the figure, rather one would have to shrink red and stretch violet faces accordingly in order to obtain the correct picture The faces are dual to a four-valent graph, that is, each face is punctured by an edge which connects the centres of the tetrahedra with a common face These edges are ‘charged’ with half-integral spin-quantum numbers and these numbers are proportional to the quantum area of the faces The collection of spins and edges defines a spin-network state The spin quantum numbers are created and annihilated at each Planck time step of τP ≈ 10−43 s in a specific way as dictated by the quantum Einstein equations Hence the name Quantum Spin Dynamics (QSD) in analogy to Quantum Chromodynamics (QCD) Spin zero corresponds to no edge or face at all, hence whole tetrahedra are created and annihilated all the time Therefore, the free space not occupied by tetrahedra does not correspond to empty (matter-free) space but rather to space without geometry, it has zero volume and therefore is a hole in the quantum spacetime The tetrahedra are not embedded in space, they are the space Matter can only exist where geometry is excited, that is, on the edges (bosons) and vertices (fermions) of the graph Thus geometry is completely discrete and chaotic at the Planck scale, only on large scales does it appear smooth In this book, this fascinating physics is explained in mathematical detail Contents Foreword, by Chris Isham Preface Notation and conventions page xvii xix xxiii Introduction: Defining quantum gravity Why quantum gravity in the twenty-first century? The role of background independence Approaches to quantum gravity Motivation for canonical quantum general relativity Outline of the book 1 11 23 25 I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE CANONICAL QUANTISATION PROGRAMME 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 Classical Hamiltonian formulation of General Relativity The ADM action Legendre transform and Dirac analysis of constraints Geometrical interpretation of the gauge transformations Relation between the four-dimensional diffeomorphism group and the transformations generated by the constraints Boundary conditions, gauge transformations and symmetries 1.5.1 Boundary conditions 1.5.2 Symmetries and gauge transformations The problem of time, locality and the interpretation of quantum mechanics The classical problem of time: Dirac observables Partial and complete observables for general constrained systems 2.2.1 Partial and weak complete observables 2.2.2 Poisson algebra of Dirac observables 2.2.3 Evolving constants 2.2.4 Reduced phase space quantisation of the algebra of Dirac observables and unitary implementation of the multi-fingered time evolution Recovery of locality in General Relativity 39 39 46 50 56 60 60 65 74 75 81 82 85 89 90 93 References 805 [808] S Hod Bohr’s correspondence principle and the area spectrum of quantum black holes Phys Rev Lett 81 (1998) 4293 [gr-qc12002] [809] L Motl An analytical computation of asymptotic Schwarzschild quasinormal frequencies Adv Theor Math Phys (2003) 1135 [gr-qc/0212096] [810] L Motl and A Neitzke Asymptotic black hole quasinormal frequencies Adv Theor Math Phys (2003) 307–30 [hep-th/0301173] [811] O Dreyer Quasinormal modes, the area spectrum and black hole entropy Phys Rev Lett 90 (2003) 081301 [gr-qc/0211076] [812] A Ashtekar, C Beetle and J Lewandowski Mechanics of rotating isolated horizons Phys Rev D64 (2001) 044016 [gr-qc/0103026] [813] A Ashtekar, J Engle, T Pawlowski and C Van Den Broeck Multipole moments of isolated horizons Class Quant Grav 21 (2004) 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A Ashtekar, T Pawlowski and P Singh Quantum nature of the big bang: an analytical and numerical investigation I Phys Rev D73 (2006) 124038 [gr-qc/0604013] [870] A Ashtekar, T Pawlowski and P Singh Quantum nature of the big bang: improved dynamics [gr-qc/0607039] [871] V F Mukhanov, H A Feldman and R H Brandenberger Theory of cosmological perturbations; Part Classical perturbations; Part Quantum Theory of Perturbations; Part Extensions Phys Rept 215 (1992) 203–333 [872] F Markopoulou Planck scale models of the universe [gr-qc/0210086] [873] S D Biller et al Limits to quantum gravity effects from observations of TeV flares in active galaxies Phys Rev Lett 83 (1999) 2108–11 [gr-qc/9810044] [874] J Kowalski-Glikman Introduction to doubly special relativity Lect Notes Phys 669 (2005) 131–59 [hep-th/0405273] [875] J Kowalski-Glikman and S Nowak Doubly special relativity theories as different bases of kappa Poincar´e algebra Phys Lett B539 (2002) 126–32 [hep-th/0203040] [876] J Lukierski and 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[893] J Sniatycki Constraints and quantisation In Non-linear Partial Differential Operators and Quantisation Procedures, S Anderson and H.-D Doebner (eds) (Lecture Notes in Mathematics 1037, Springer-Verlag, Berlin, 1983) [894] M Blau On the geometric quantisation of constrained systems Class Quant Grav (1988) 1033–44 [895] A Hanson, T Regge and C Teitelboim Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Roma, 1978) [896] E Hewitt and K A Ross Abstract Harmonic Analysis I (Springer-Verlag, Berlin, 1987) [897] M S Birman and M Z Solomjak Spectral Theory of Self-adjoint Operators in Hilbert Space (D Reidel, Dordrecht, 1987) [898] H Boerner Representations of Groups (North-Holland, Amsterdam, 1970) [899] E Hebey Sobolev Spaces on Riemannian Manifolds (Lecture Notes in Mathematics 1635, Springer-Verlag, Berlin, 1996) [900] S Lang Differential Manifolds (Addison-Wesley, Reading (MA), 1972) Index abstract index notation 593 acceleration of geodesic congruence 514 adjoint action 637 of an operator, see operator representation 637 ADM energy 65 quantum 341 ADM formulation 31 ADM momentum 63 AdS/CFT correspondence 15 affine parametrisation 514 algebra Abelian 186, 701 of almost periodic functions 713 Banach 186, 701 of cylindrical functions 153 C ∗ - 186, 701 Grassmann 594 holonomy–flux 205 normed 701 spectrum of 186, 702 sub- 701 unital 186, 701 von Neumann 387, 719 algebraically special spacetime 524 algebraic quantum field theory almost periodic function 713 amenable group 718 annihilation operator 355 annihilator subspace 619 anomaly 115 antisymmetrisation 593 area gap 437 operator 432 of black hole 550 Arnowitt–Deser–Misner (ADM) action 46, 51 Ashtekar connection, see new connection variables, see new variables –Isham space, see distributional connection –Lewandowski measure, see uniform measure asymptotically flat 61 Minkoswki 61 atlas 585 locally finite 585 automorphism of ∗ -algebra 160, 720 of principal bundle 209 axiom of choice 649, 689 background independence Banach–Steinhaus theorem 694, 704 Barrett–Crane model 466 barycentre of simplex 472 barycentric refinement 475 subdivision 475 Bekenstein–Hawking entropy 511 Bergmann–Komar group 56 Bessel’s inequality 691 BF theory 462 Bianchi identity 131, 641 bi-vector, bi-co-vector 471 black body spectrum 558 black hole 511 region 517 thermodynamics zeroth law 526 entropy 511, 550 quasinormal modes 559 Bogol’ubov transformation 214 Bohr compactification 184, 213, 566, 713 Bolzano–Weierstrass theorem 394, 579 Borel measure, see measure sum, 506 Boulatov–Ooguri matrix model 496 boundary data of fundamental atom 487 boundary operator 467 bounded linear functional (BLT) theorem 691 810 Calabi–Yau space 12, 626 canonical commutation relations 110, 110, 206 group 621 quantisation 108 transformation, see symplectomorphism Cantor aleph 321, 386 Cartan structure equation 641 category 166 Cauchy sequence 689 causal set Ansatz 22 caustic 517 ˇ Cech cohomology 655 cell complex, see simplicial complex central extension, see Lie algebra chain 473, 604 boundary 604 cycle 604 character on Banach algebra 702 on compact group 747, 752 chart 585 Chern class 626 Chern–Simons action 314 theory 541 level 543 Christoffel symbols 43, 609 C ∞ -vector 219 classical limit 345 Clebsch–Gordan coefficient 759 theorem 759 closed graph theorem 694 closure of densely defined operator 700, 697 of metric space 690 of quadratic form 700 coarse graining 103 coboundary operator 467 cochain 467 Codacci equation 44 co-final subset 230 cohomology group 604 compactification 12 complete metric space, see space complex line bundle 544, 653 manifold 623 null tetrad 516 structure 358, 624 almost 624 Index complexification 663 complexifier 357 configuration coordinate 48, 615 space 48, 615 conformal field theory (CFT) 15 invariance 610 isometry 611 congruence of curves 514 connection affine 607 Levi–Civita 609 new 128 potential 638 one-form 637 spin 126, 614 congruence null 514 geodesic 514 expansion 515 shear 515 twist 515 constraint first-class 674 Gauß 128, 264 Hamiltonian 48, 279 Euclidean 287 primary 672 secondary 673 second-class 674 spatial diffeomorphism 48, 269 contraction semigroup 505 coordinates 585 cosmic censorship 518 cosmological constant problem covariance general, see background independence of Gaußian measure, see Gaußian measure covariant derivative 640 differential 607, 642 phase space approach 22 Crane–Yetter model 496 crossing symmetry 461 curvature 608 field strength of 641 scalar 610 two-form 640 curve 163 beginning point 164 composition 164 Index final point 164 integral 597 range 164 smooth 597 cutoff state 348 cycle 497 cylindrical function 153 projection 184 family of measures 220 Darboux coordinates 358 theorem 615 dark energy 1, matter d-Bein 123 decoherence 100 functional 103 deficiency index 699 density 283, 612 deparametrisation 79 de Rham cohomology 605 isomorphism 656 map 474 theorem 605 derivation, see vector field diffeomorphism 586 active 587 analytic 586 group 585 passive 587 semianalytic, see semianalytic diffeomorphism diffeomorphism constraint classical ADM form 48 new variable form 133 quantum 269 differential form coboundary 604 cocycle 604 integration of 600 one- 591 n- 593 structure 585 Dirac algebra 50 bracket 675 observable, see observable quantisation 677 811 direct integral decomposition 735 method 114, 735 of Hilbert spaces 114, 737, 738 directed set 141 system of Hilbert spaces 230 of operators 230 Dirichlet–Voronoi construction 351 discretisation theory 472 distribution complex 666 horizontal 637 of tangent spaces 616 characteristic 619 integrable 616 integral manifold of 616 reducible 619 tempered 693 vertical 637 distributional connection 171 modulo distributional gauge transformation 176 connections modulo gauge transformations 179 gauge transformation 175 divergence ultraviolet avoidance 6, 11, 282 of vector field 228 domain questions 111 doubly special relativity 574 dual simplicial complex 473 dual space algebraic 731 topological 691, 731 duality transformations 13 dynamical triangulation 19 edge 166, 633 amplitude 481 germ 207 independence 168 Ehrenfest property 354 Einstein equation classical quantum –Hilbert Lagrangian 39 Elliot–Biedenharn identity 452 embedding 163, 586 regular 586 electric field 405 812 energy condition dominant 517 strong 517 weak 517 entropy, see black hole entropy enveloping algebra 110 Epstein and Glaser renormalisation equivariant 584, 640, 644 ergodic mean 81, 115, 736 group action 245, 687 Euler characteristic 534 Euler–Poincar´e theorem 605 Everett interpretation 104 evolution equation 49, 50, 60 evolving constant 78 expansion 514 expectation value property 354 exponential map 608 exterior derivation 594 product 593 face 472 amplitude 481 in dual of 4D simplicial complex 477 of flux 192, 633 of simplicial complex 472 factor of von Neumann algebra 388 factor ordering ambiguity 112 singularity 112 Fell’s theorem 722 Feynman diagram 497 Feynman–Kac formula 505 fermion coupling 406, 422 fibre bundle base space 634 complex line 544 local trivialisation 634 of linear frames 613 of orthonormal frames 614 principal 635 projection 634 section 635 cross- 635 structure group 634 total space 634 transition function 634 trivial 635 typical fibre 634 vector 636 fibre metric 544, 654 field 701 finiteness (UV), see divergence Index fluctuation property 354 flux classical 159 operator 219 vector fields 202 Fock space foliation, see integral manifold of distribution leaf of 616 reducible 616 folium, see state four-simplex 477 fractal 380 Fr´ echet space 694, 730, 773 free tensor algebra 110 Friedmann–Robertson–Walker (FRW) model 564 Friedrich extension 114, 324, 700 Frobenius’ theorem 42, 517, 617 Fubini’s theorem 224, 261, 682 function continuous 577 cylindrical 160, 153 smooth 590 functional calculus 728 functor 166 fundamental atom 486 fundamental form first 42 second 42 γ-ray burst 1, 572 gauge fixing 678 gauge transformation 60, 639 generator of 674 Gauß constraint classical 128, 133 quantum 264 Gauß equation 43 Gauß–Bonnet theorem 534 Gel’fand isomorphism 186, 711 –Naimark–Segal construction, see GNS construction theorem 705 topology 708 transform 709 triple 731 generalised eigenvector 112, 732 geodesic completeness 608 geodesic equation 514, 608 affine parametrisation 514 GLAST detector globally hyperbolic 40, 517 GNS construction 111, 720 graph 168 Index of densely defined operator 700 of bounded operator 694 gravitational electric field, see new electric field gravitons 390 Gribov copies 678 Groenwald–van Hove theorem 662 group action 582, 621 effective 621 ergodic 687 free 621 measure preserving 687 transitive 621 group averaging 113, 733 group field theory (GFT) 495 groupoid 166 group theoretical quantisation 621 Haag’s theorem 178 Haag–Kastler axioms 722 Haar measure, see measure Hahn–Banach theorem 693, 721 Hamiltonian constraint classical ADM form 48 new variable form 133 quantum 286 Hamiltonian equations of motion 50 Hamilton–Jacobi equation 618 Hartle–Hawking wave function, see path integral Hausdorff 170, 578 Hawking effect 512 Heine–Borel theorem 579 HellingerTă oplitz theorem 694 Hermitian manifold 625 structure 625 Higgs field 411, 425 Hilbert–Schmidt operator 697, 747 theorem 696 history bracket formulation 20 Hodge operator 466, 474 holographic principle 15 holonomy 158, 168 –flux algebra 205 operator 219 point 415 homeomorphism 577 homology group 604 hoop 167 independence 172 tame 172 Hopf algebra 508, 574 horizon apparent 517 dynamical 519 event 517 isolated 520 weakly 520 spherically symmetric 526 Killing 526 non-expanding 433 non-rotating 520 trapping 519 horizontal lift 639 hypersurface 585 deformation algebra 50 ideal 701 left 701 maximal 703 right 701 immersion 163, 586 Immirzi parameter 127 inductive limit of Hilbert spaces 230 of operators 230 inflation 563 inflaton 563 initial singularity 563 inner product 691 interior product 594 intertwiner 237, 752, 757 involution 701 isometric monomorphism 230, 375 isometry of Hilbert space 695 of spacetime metric 611 Jacobi identity 590 Jones polynomial 144 Kă ahler form 625 manifold 625 metric 626 polarisation 358, 665 potential 626 Kaluza–Klein modes 12 theory 12 Killing field 611 kinematical Hilbert space 96 representation 111 state 96 813 814 knot theory 144 Kodama state 314 Lagrangian Einstein–Hilbert, see Einstein–Hilbert Lagrangian singular 671 type of subspace 665 Kă ahler 665 non-negative 665 positive 665 real 665 landscape 15 lapse function 41 lattice gauge theory 329 quantum gravity 19 Lebesgue decomposition theorem 743 integral 681 Legendre transform 671 Leibniz rule, see vector field length operator 431, 453 Leray cover 656 Levi–Civita connection 609 totally skew symbol 611 Lie algebra 590 central extension of 622 coboundary 622 cochain 622 cocycle 622 cohomology 622 obstruction cocycle 622 bracket 590 derivative 597 Liouville form 615 measure 354, 361 local quantum physics 4, 722 loop representation 237 LQG string 215 Lusin’s theorem 685 magnetic field 405, 419 manifold 585 complex analytic, see holomorphic, see complex manifold differentiable C k sub- 585 dimension of 585 embedded sub- 585 holomorphic 585, 623 Kă ahler, see Kă ahler manifold orientable 585 Index paracompact 585 Poisson, see symplectic manifold real analytic 585 smooth 585 symplectic, see symplectic manifold with boundary 585 mapping class group 272 master constraint 80, 317, 735 algebra 319 extended 329 programme 80 equation 80 matrix model 496 m-(co)-bein 614 McDowell–Mansouri action 507 measurable function 221, 680 set 680 space 680 measure absolutely continuous 683 Borel 220, 684 class 738 disjoint 738 complex 680 consistent family of 220 equivalent 738 faithful 222, 685 Gaußian 392, 413 covariance of 392, 413 generating functional of 392 white noise covariance 413 Haar 223, 748 left 748 right 748 mutually singular 683 positive 680 definite 681 probability 220 pushforward of 220 regular 220, 684 σ-finite 683 space 680 completion of 683 spectral 728 support of 236, 683 uniform 223 metaplectic correction 669 metric space 577, 689 tensor 609 microlocal analysis minimal uncertainty relation 355 mixing 688 Index modular theory 388 module 701 momentum coordinate 48, 615 momentum map 622 momentum operator 229 M-theory 13 multiplicity 738 multisymplectic Ansatz 20 neighbourhood 577 Nelson’s analytic vector theorem 337, 339 net 173, 578 convergence 578 of local algebras 722 subnet 578 universal 578 Newlander and Nirenberg theorem 625 Newman–Penrose coefficient 524, 532 new connection 128 electric field 124 variables 123 Nijenhuis tensor 625 non-commutative geometry 22 non-observable 78 norm on normed space 690 normal bundle 633 nuclear operator 747 topology 297, 418, 730 null congruence 514 infinity 517 normal 517 surface 517 tetrad 516 observable complete 78 Dirac strong 674 weak 674 partial 78 operator adjoint of 695, 697 algebra 719 bounded 695 closable 697 closure of 697 compact 696 singular values of 696 domain of 697 graph of 697 Hilbert–Schmidt 697, 747 815 nuclear = trace class 697, 747 positive 688 resolvent 697 set 697 self-adjoint 695, 697 essentially 697 spectrum of 697 symmetric 697 topology strong 695 uniform 695 weak 695 weak∗ 695 trace class 697, 747 unbounded 697 unitary 692 orbit 582 orthonormal basis 692 Osterwalder–Schrader reconstruction 147 overcompleteness 355 overlap function 355, 766 pairing 669 parallel transport 607, 643 equation 639 partial isometry 695 final subspace 695 initial subspace 695 kernel 695 range 695 partial order 141 partial trace 100 partition 582 function 464 of unity 587 path 164 path integral Euclidean 19 Hartle–Hawking proposal 19 non-Gaußian fixed point 19 peakedness property 355 pentagon diagram 482 perturbative quantum gravity Peter and Weyl theorem 239, 753 Petrov type 524 Pfaff system 617 phase space 46, 614 phenomenology match of string theory 12 physical Hamiltonian 90 inner product 96 state 96 Plancherel formula 258 theorem 148 816 PLANCK satellite Plebanski action 466 constraints 466 Pohlmeyer string 23 Poincar´ e algebra 72 Poincar´ e’s lemma 605 Poisson resummation formula 383, 766 polar decomposition 696 polarisation 334, 543, 662 admissible 666 complex 666 Kă ahler, see Kă ahler polarisation polarised 666 section 668 Ponzano–Regge model 496 positive linear functional 221, 684 pre-Hilbert space 691 prequantum bundle 662 Hilbert space 662 operator 544, 662 problem of time 74 projection-valued measure 728 projective family 141 limit 141 propagator 497 pseudo-tensor 612 quadratic form 700 quantisation canonical 107 Dirac, see canonical gauge fixed 562 geometric 652 group theoretical 621 map 110 reduced phase space 90 refined algebraic 729 quantum constraint equation 96 quantum group 549, 574 quantum spin dynamics (QSD) 286 quasinormal modes 559 quasiperiodic functions 565 quotient map 582 Racah formula 451 Radon–Nikodym derivative 683 theorem 683 rapid decrease 693 reality conditions 110, 135, 206, 334 recoupling scheme 451, 758, 759 refined algebraic quantisation 264, 729 Index Regge calculus 19 relational Ansatz 74 representation of ∗ -algebra 111, 719 cyclic 719 equivalent 719 faithful 719 irreducible 719 non-degenerate 719 of group 746 character of 746 completely reducible 746 conjugacy class of 746 contragredient 746 dimension of 746 dual 746 equivalent 746 faithful 746 induced 752 invariant subspace of 746 irreducible 746 reducible 746 tensor product 746 unitary 746 of holonomy–flux algebra 219 resolvent 703 set 703 Ricci tensor 609 Riemannian space 609 Riemann tensor 610 Riemann Theta function 546 Riesz lemma 697, 692 Riesz–Markov theorem 222, 684 Riesz representation theorem, see Riesz–Markov theorem Riesz–Schauder theorem 696 rigged Hilbert space, see Gel’fand triple rigging map 96, 114, 732 ring 701 root of unity 549 Rovelli–Smolin Wilson loop functions, see spin-network spin-networks, see spin-network scale factor 564 Schur’s lemma 752 Schwarz inequality 692 Schwinger function 497 Segal–Bargmann representation 358 transform 142, 146 self-adjointness 697 basic criterion of 697 essential 697 basic criterion of 697 Index self-dual connection 134 spinor 527 tensor Euclidean signature 467 Lorentzian signature 528 semianalytic atlas 631 bundle automorphism 210 diffeomorphism 210, 269, 630 function 628 manifold 631 map 631 partition 627 analytic 631 structure 163, 631 submanifold 631 semiclassical limit, see classical limit seminorm 694 semi-semianalytic partition 630 separable Hilbert space 692 topological space 578 separating the points 581, 694 set Borel 684 inner regular 684 outer regular 684 closed 577 of measure zero 681 open 577 σ-compact 683 σ-finite 683 thick 683 shadow 153, 349, 380 shear 514 shift vector field 41 σ-algebra 170, 680 Borel 680 signature Euclidean 609 Lorentzian 609 simple intertwiner 494 representation 490 simplex 472 simplicial complex 472 simplicity constraint 463, 482 singularity 518 big bang 564 curvature 567 initial 518 naked 518 resolution 564 6j-symbol 451 817 skeletonisation 505 Smolin–Rovelli Wilson loop functions, see spin-network spin-networks, see spin-network Sobolev topology 770, 772 soldering form 123, 527 space Banach 690, 693 Hilbert 691 locally convex 694 metric 689 complete 689 normed 690 reflexive 691 topological 577 spatial diffeomorphism constraint, see diffeomorphism constraint spectral measure 723 projection 728 radius 703 theorem 726 spectrum of normal operator 723 continuous 697 discrete 697 essential 697 pure point 697 of normed, unital algebra element 703 of unital Banach algebra 702 spin foam model 458 spin-network 241 function 237, 755 spinor calculus 527 standard model 399 state on ∗ -algebra 720 as positive linear functional 720 coherent 354 faithful 722 folium of 390, 721 invariant 720 mixed 720 normalised 720 polarised 544, 668 pure 720 regular 215 semiclassical 354 vector 720 state sum model 458 Stokes’ theorem 602 ˇ Stone–Cech compactification 713 Stone–von Neumann theorem 213 Stone–Weierstrass theorem 581 string field theory 14 818 string theory 11 subgroupoid 166 tame 142 submanifold co-isotropic 619 isotropic 619 Lagrangian 358, 619 symplectic 619 subsimplex 472 supergravity 8, 16 superstring theory, see string theory supersymmetry 10 support of function 587, 684 of measure, see thick set surface gravity 514 symmetrisation 593 symmetry 65, 341 symplectic group action 621 isometry, see symplectomorphism manifold 614 presymplectic 619 potential 614 reduction 616 structure 614 submanifold 619 symplectomorphism 616 tangent space 592 tempered distribution 693 tensor bundle 592 density, see density field 591 contravariant 591 covariant 591 invariant 612 pull-back 595 push-forward 595 transformation law 595 tetrad, see m-(co)-bein θ-moduli 149, 272, 326, 347 time coordinate 74 parameter 74 physical 74 slice axiom unphysical 74 topological inclusion 577 isomorphism, see homeomorphism quantum field theory 458 space 577 compact 578 Index disconnected 578 first countable 578 Hausdorff 578 locally compact 683 locally convex, see Fr´ echet space normal 578 regular 578 second countable 578 separable 578 topology 577 base for 577 change 40, 384 coarser 577 finer 577 induced 577 quotient 582 relative 577 stronger 577 strong operator 297 subset 581 Tychonov 579 uniform 297 URST 298 weaker 577 weak operator 297 weak ∗ operator 297, 708 topos theory 22 Torre–Varadarajan obstruction 216 torsion 608 trace class operator 721, 747 trapped region 517 total 517 surface 517 inner marginally 517 marginally 517 outer marginally 517 triad 123, see m-(co)-bein triangulation 288, 465, 472 dual 473 independence 495 Turarev–Ooguri–Crane–Yetter model 465 Turarev–Viro model 496 twist 514 twistor theory 22 Tychonov topology 170, 173, 579 ultraviolet finiteness 282 uniqueness theorem for LQG 214 existence 219 irreducibility 252 uniqueness 247 Stone–von Neumann 213 Index Unruh effect 512 Urysohn’s lemma 581 vacuum degeneracy in string theory 12 Varadarajan map 394 vector field 590 contraction with respect to 592 flow of 597 Hamiltonian 616 vertex 168 amplitude 481 of Feynman diagram 497 volume operator 290, 438 consistency with flux operator 453 von Neumann mean ergodic theorem 687 self-adjointness criterion 339 wave function of the universe, see Hartle–Hawking proposal weak continuity 213 weave 349 819 wedge of fundamental atom 486 wedge product, see exterior product Weierstrass theorem, see Stone–Weierstrass theorem Weil integrality criterion 543, 658, 659 Weingarten map 525 Weyl element 110, 206 Weyl group 766 Weyl tensor 524, 610 Wheeler–DeWitt equation 17, 311 white noise covariance 413 Whitney map 474 Wick transform 287, 334 Wightman axioms WMAP satellite Yang–Mills field 419 Young tableaux 755 zeroth law of black hole thermodynamics 526 Zorn’s lemma 649, 689 .. .MODERN CANONICAL QUANTUM GENERAL RELATIVITY Modern physics rests on two fundamental building blocks: general relativity and quantum theory General relativity is a geometric... the principles of general relativity In order to construct quantum gravity, one must reformulate quantum theory in a background-independent way Modern Canonical Quantum General Relativity is about... Issued as a paperback Modern Canonical Quantum General Relativity THOMAS THIEMANN Max Planck Institut fă ur Gravitationsphysik, Germany h G c CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne,